updating of local non-linear stiffness- and damping

M. Böswald: Updating of Local Non-Linear Stiffness and Damping Parameters in Large Order Finite Element Models by Using Vibration Test Data, Dissertation (PhD Thesis), University of Kassel, August 2005 Published as: DLR Forschungsbericht DLR-FB--2005-25 (ISSN 1434-8454, ISNR DLR-FB--2005-25), Deutsches Zentrum für Luft- und Raumfahrt e.V.

UPDATING OF LOCAL NON-LINEAR STIFFNESS- AND DAMPING PARAMETERS IN LARGE ORDER FINITE ELEMENT MODELS BY USING VIBRATION TEST DATA IM FACHBEREICH BAUINGENIEURWESEN DER

UNIVERSITÄT KASSEL ANGENOMMENE

DISSERTATION ZUR ERLANGUNG DES AKADEMISCHEN GRADES EINES DOKTOR-INGENIEURS (DR.-ING.)

VON

DIPL.-ING. MARC BÖSWALD AUS MOSHEIM

KASSEL, AUGUST 2005

Vorwort Die vorliegende Dissertation entstand während meiner Tätigkeit als wissenschaftlicher Mitarbeiter am Fachgebiet Leichtbau der Universität Kassel unter der Leitung von Herrn Prof. Dr.-Ing. Michael Link. An dieser Stelle möchte ich Herrn Prof. Dr.-Ing. Michael Link dafür danken, dass er mir die Möglichkeit gegeben hat, meine Forschungstätigkeit unter hervorragenden Arbeitsbedingungen und in einer sehr angenehmen Atmosphäre durchführen zu können. Seine umfangreiche Erfahrung auf dem Gebiet der Strukturmechanik und seine wichtigen Anregungen und Impulse waren für das Gelingen dieser Arbeit von entscheidender Bedeutung. Dieser Dank schließt sein bereitwilliges Engagement bei der Akquisition des Drittmittelforschungsprojektes CERES ein, welches mir den Weg zur Promotion ermöglicht hat. Bedanken möchte ich mich auch bei Herrn Prof. Dr. John Mottershead von der University of Liverpool für die bereitwillige Übernahme des Zweitgutachtens. Herrn Prof. Dr.-Ing. Horst Irretier von der Universität Kassel möchte ich für die Übernahme des Drittgutachten danken und dafür, dass er durch seine sehr anschaulich gestalteten Vorlesungen mein Interesse für die Strukturmechanik geweckt hat. Des Weiteren möchte ich mich bei Herrn Prof. Dr.-Ing. Anton Matzenmiller und Herrn Prof. Dr.-Ing. Friedel Hartmann für die Mitarbeit in der Prüfungskommission bedanken. Mein Dank gilt insbesondere auch den vielen Mitarbeitern des Fachgebiets Leichtbau, die mich während meiner 5-jährigen Tätigkeit begleitet haben. An erster Stelle möchte ich Herrn Dr.-Ing. Matthias Weiland danken. In vielen fachlichen und nichtfachlichen Gesprächen konnte er einen Teil seines umfangreichen Wissens auf dem Gebiet der experimentellen Strukturmechanik an mich vermitteln. Des Weiteren möchte ich mich bei Herrn Dr.-Ing. Stefan Dominico (geb. Meyer) bedanken, der durch seine hervorragende Forschungsarbeit das Fundament für meine eigene Arbeit gelegt hat. Mein Dank gilt außerdem den Herren Dipl.-Ing. Edgar Görl, Dipl.-Ing. Yves Govers und Dipl.-Ing. Horst Ulrich. Sie waren jederzeit ansprechbar wenn fachliche oder nichtfachliche Probleme die ansonsten ideale Arbeitsatmosphäre am Fachgebiet Leichtbau zu trüben versuchten. Bedanken möchte ich mich auch bei ehemaligen Mitarbeitern des Fachgebiets Leichtbau. Namentlich seinen hier Herr Dr.-Ing. Carsten Schedlinski und Herr Dr.-Ing. Dennis Göge erwähnt. Der größte Dank richtet sich an meine Eltern. Ihre großartige Unterstützung vom Anfang meines Studiums bis zum Ende meiner Promotion war beispiellos. Ihnen möchte ich daher diese Arbeit widmen. Kassel, im August 2005

Marc Böswald

Kurzfassung

i

Kurzfassung Die Reduzierung von Entwicklungszeiten komplexer Maschinen und Geräte1 erfordert in der Regel eine Unterteilung in Substrukturen. Dadurch wird erreicht, dass einzelne Substrukturen zeitlich parallel von verschiedenen Personen oder sogar von verschiedenen Firmen konstruiert und gefertigt werden können. Während eine Zerlegung in Substrukturen klare Vorteile bei Planung und Entwicklung von Systemen bringen kann, so kann sie sich durchaus nachteilig auf deren Betriebseigenschaften auswirken. Das gilt insbesondere für Leichtbaustrukturen unter dynamischen Belastungen, wie sie zum Beispiel in der Luft- und Raumfahrt Anwendung finden. Bei strukturdynamischen Experimenten zeigen komplexe zusammengesetzte Maschinen und Geräte häufig Abweichungen vom erwarteten bzw. erwünschten linearen Strukturverhalten. Dieses nichtlineare Verhalten führt zu Problemen bei der Identifikation der dynamischen Eigenschaften im Rahmen von Modaltests und somit zu einer Verschlechterung der Qualität der Testdaten. Nichtlineares Verhalten ist nicht nur bei der modalen Identifikation unerwünscht, sondern häufig auch bei strukturdynamischen Simulationen mit Hilfe von Finite Elemente (FE) Modellen, da der Berechnungsaufwand durch vorhandene Nichtlinearitäten stark erhöht werden kann. Häufig nimmt die Genauigkeit der Simulationsergebnisse bei nichtlinearen Strukturen ab, besonders dann, wenn die Nichtlinearitäten nicht angemessen in den FE Modellen abgebildet wurden. Aber selbst bei einer angemessenen nichtlinearen Modellierung können die Simulationsergebnisse erheblich von realem gemessenem Strukturverhalten abweichen. Ursache hierfür ist dann die relativ große Unsicherheit die den Parametern nichtlinearer Elemente anhaftet. Diese Dissertation widmet sich der Problematik der Identifikation von Parametern nichtlinearer Elemente in großen FE Modellen, um dadurch eine höhere Genauigkeit von Simulationen des dynamischen Verhaltens nichtlinearer Strukturen zu erzielen. Hierbei werden lokale Nichtlinearitäten betrachtet, wie sie häufig an Verbindungsstellen zusammengesetzter Strukturen auftreten. Solche lokalen Nichtlinearitäten können meist in geeigneter Weise durch 2-Freiheitsgrad-Elemente zur nichtlinearen Kopplung von ansonsten linearen Substrukturen modelliert werden. Die vorliegende Dissertation gliedert sich in 13 Kapitel und einen Anhang. Nach der Einleitung in Kapitel 1 wird in Kapitel 2 ein Überblick über den Stand der Forschung gegeben. Ein umfassender Überblick wird über die für diese Arbeit relevanten Themen gegeben. Dazu zählen unter anderem die automatisierte Modellanpassung, die Identifikation nichtlinearer Parameter, die analytische Bestimmung nichtlinearer Frequenzgänge und die Anwendung von Modellreduktionsverfahren. Ausgehend vom Stand der Forschung erscheint es sinnvoll und Erfolg versprechend, ein Verfahren zu entwickeln, welches die Differenzen zwischen gemessenen und berechneten nichtlinearen Frequenzgängen benutzt, um daraus die Parameter nichtlinearer Elemente des FE Modells zu identifizieren. Gemäß dieser Zielsetzung beschäftigen sich die anschließenden 1 Maschinen und Geräte werden in dieser Arbeit allgemein als Systeme oder Strukturen bezeichnet

ii

Kurzfassung

Kapitel verstärkt mit den Methoden zur Berechnung nichtlinearer Schwingungsantworten im Frequenzbereich, sowie mit den Methoden der computerunterstützten Modellanpassung. In Kapitel 3 wird die Modellierung von nichtlinearen Verbindungsstellen mit Hilfe von FE Modellen behandelt. Insbesondere wird die praxisrelevante Modellierung von Flanschverbindungen diskutiert, für die ein vereinfachter Modellierungsansatz aus einer Lastpfadanalyse hergeleitet wird. Dieser vereinfachte Modellierungsansatz kann dabei sowohl das zugrunde liegende lineare Verhalten abbilden, als auch das nichtlineare Verhalten, welches häufig erst bei größeren Schwingungsamplituden zu beobachten ist. Die Theorie zur Berechnung nichtlinearer Frequenzgänge mit Hilfe von FE Modellen wird ausführlich in Kapitel 4 behandelt. Dabei wird das Verfahren der Harmonischen Balance vorgestellt, welches als Näherungsverfahren zur Berechnung von Schwingungsantworten im eingeschwungenen Zustand bei harmonischer Anregung verwendet werden kann. Zwei Varianten dieses Verfahrens werden diskutiert. Zum einen die Single-Harmonic Balance Method, mit der die nichtlineare Schwingungsantwort der Grundharmonischen berechnet werden kann, und zum anderen die Multi-Harmonic Balance Method zur näherungsweisen Bestimmung der vollständigen (periodischen) nichtlinearen Schwingungsantwort. Als Ergänzung wird eine Erweiterung der Strukturmodifikationsmethode vorgestellt, welche die Berechnung nichtlinearer Frequenzgänge ausgehend von den bekannten Frequenzgängen des zugrunde liegenden linearen Systems ermöglicht. Im Kapitel 5 werden anschließend die Eigenschaften nichtlinearer Frequenzgänge diskutiert und auf Besonderheiten bei der Verwendung von Schrittsinus oder Gleitsinus als Anregungssignal in nichtlinearen Schwingungsmessungen hingewiesen. Die Anwendung von Modellreduktionsverfahren zur Berechnung dynamischer Antworten großer FE Modelle wird in Kapitel 6 behandelt. Die am häufigsten verwendeten Verfahren werden im Hinblick auf ihre Eignung zur Reduktion von großen FE Modellen mit lokalen Nichtlinearitäten diskutiert. Die Craig-Bampton Methode stellte sich hierbei als besonders geeignet heraus und es wird auf die Möglichkeit hingewiesen, eine Modellreduktion in mehreren, aufeinander folgenden Schritten durchzuführen. In Kapitel 7 werden Methoden zur Modellierung von Dämpfungen in FE Modellen aufgezeigt. Der Dämpfung kommt bei der Antwortberechnung im Frequenzbereich eine besondere Bedeutung zu, da diese einen unmittelbaren Einfluss auf die Größe der Schwingungsantworten im eingeschwungenen Zustand besitzt. Der klassische Ansatz der proportionalen Dämpfung und die Verwendung von gemessenen modalen Dämpfungsmaßen werden behandelt. Die Synthese einer Dämpfungsmatrix aus gemessenen modalen Dämpfungsmaßen wird empfohlen. Aspekte der Messung nichtlinearer Frequenzgänge werden in Kapitel 8 diskutiert. Sowohl die Möglichkeit zur Linearisierung nichtlinearen Strukturverhaltens durch gezielte Einstellung der aufgebrachten Erregerkräfte, als auch die gezielte Messung verzerrter nichtlinearer Frequenzgänge wird behandelt. Im Anschluss werden Verfahren zur Detektierung und Charakterisierung nichtlinearen Strukturverhaltens aus gemessenen Schwingungssignalen werden vorgestellt. Nach einer kurzen Behandlung von Qualitätsaspekten für Testdaten und FE Modelle in Kapitel 9 widmet sich Kapitel 10 der Theorie zur Identifikation nichtlinearer Parameter. Die parametrische (iterative) Identifikation wird dabei aufgrund ihrer physikalischen Interpretierbarkeit favorisiert. Der klassische Ansatz der Substrukturmatrizen wird vorgestellt und auf nichtlineare Elemente erweitert. Eine Diskussion der Berechnung von Sensitivitäten nichtlinearer Frequenzgänge

Kurzfassung

iii

bzgl. der Parameter nichtlinearer Elemente schließt sich an. Dabei werden Ansätze zur Sensitivitätsanalyse auf der Basis von Finite-Differenzen-Quotienten und auf der Basis von Strukturmodifikationsmethoden vorgestellt. Eine Variante der Tikhonov-Regularisierung wird eingeführt, um die mögliche schlechte Kondition der Gleichungssysteme bei der computerunterstützten Modellanpassung zu verbessern. Kapitel 11 zeigt drei Anwendungen der entwickelten Verfahren zur computerunterstützten Identifikation nichtlinearer Parameter in großen FE Modellen. In der ersten Anwendung wird zunächst anhand eines akademischen Beispiels die Leistungsfähigkeit und Genauigkeit der entwickelten Verfahren geprüft. Des Weiteren werden wichtige Hinweise für die Anwendung auf große Systeme abgeleitet. Die Anwendbarkeit der nichtlinearen Parameteridentifikationsverfahren auf größere FE Modelle wird in den beiden folgenden Anwendungen demonstriert. Dazu werden in der zweiten Anwendung Testdaten herangezogen, welche im Rahmen von Rütteltischversuchen an einer realen Laborstruktur aufgezeichnet wurden. Ein FE Modell mit ca. 9000 Freiheitsgraden wird benutzt um die nichtlinearen Steifigkeits- und Dämpfungseigenschaften einer Flanschverbindung der Laborstruktur aus gemessenen Übertragungsfunktionen heraus zu identifizieren. Die Modellierung der Flanschverbindung wird dabei gemäß den Ergebnissen von Kapitel 3 vorgenommen. Die letzte Anwendung ist die Identifikation der nichtlinearen Steifigkeits- und Dämpfungseigenschaften einer Flanschverbindung eines Flugtriebwerks. Dabei kommt ein FE Modell mit ca. 90000 Freiheitsgraden zum Einsatz, welches eine geeignete nichtlineare Modellierung aufweist. Diese Anwendung ist im Prinzip der zweiten Anwendung ähnlich, stellt aber eine praxisrelevante Anwendung dar, die im Rahmen eines Europäischen Forschungsprojektes durchgeführt wurde. Ein Vergleich der gemessenen nichtlinearen Frequenzgänge nachdem die nichtlinearen Parameter der Flanschverbindung identifiziert wurden zeigt eine gute Übereinstimmung. Aus der guten Übereinstimmung der gemessenen und berechneten nichtlinearen Frequenzgänge nach der Parameteridentifikation kann gefolgert werden, dass sich das in dieser Dissertation entwickelte Verfahren zur Identifikation von nichtlinearen Steifigkeits- und Dämpfungsparametern aus gemessenen Schwingungstestdaten zur Verbesserung der Vorhersagefähigkeit von nichtlinearen FE Modellen eignet.

Table of Contents Chapter 1: 1.1 1.2

Motivation and Objectives................................................................................. 2 Contents and Organization of the Text ............................................................ 4

Chapter 2: 2.1 2.2 2.3 2.4

Introduction .............................................................................................. 1

Literature Review..................................................................................... 5

Computational Model Updating ....................................................................... 5 Identification of Non-Linear Parameters......................................................... 7 Non-Linear Frequency Response Analysis ...................................................... 9 Model Reduction Techniques .......................................................................... 12

Chapter 3:

Non-Linear Joint Modeling Approach ................................................... 14

3.1 Bolted Joints .................................................................................................... 15 3.1.1 Concentrically Clamped Single Bolted Joint.......................................... 15 3.1.2 Eccentrically Loaded Bolted Joints and Flanged Joints........................ 20 3.2 Deformation Behavior of Bolted Flange Joints.............................................. 22 Chapter 4:

Non-Linear Response Analysis .............................................................. 28

4.1 Review of Procedures for Linear Response Analysis..................................... 28 4.2 From Linear to Non-Linear............................................................................. 29 4.3 Numerical Integration Schemes for Time Domain Response Analysis........ 30 4.4 Different Procedures for Frequency Domain Response Analysis ................. 32 4.4.1 Single-Harmonic Balance Method........................................................... 32 4.4.2 Multi-Harmonic Balance Method............................................................ 38 4.4.3 Non-Linear Response Analysis Based on Structural Modification Theory ....................................................................................................... 42 Chapter 5: 5.1

Accuracy Assessment by Comparison with Time Domain Results............... 54

Chapter 6: 6.1 6.2 6.3 6.4 6.5

Properties of the Non-Linear Fundamental Harmonic Response......... 50

Model Reduction Techniques for Non-Linear Response Analysis ........ 57

Mode Superposition ......................................................................................... 57 GUYAN Reduction ............................................................................................. 59 Exact Dynamic Condensation ......................................................................... 61 Craig-Bampton Reduction............................................................................... 62 Successive Steps of Model Reduction ............................................................. 66

Chapter 7:

Approaches for the Generation of Damping Matrices ........................... 67

7.1 Damping of Linear Systems............................................................................ 67 7.1.1 Proportional Damping / RAYLEIGH Damping.......................................... 67 7.1.2 Expansion of Modal Viscous Damping Ratios ........................................ 68 7.1.3 Generation of a Non-Proportional Damping Matrix .............................. 70 7.2 Non-Linear Damping Supplement ................................................................. 70

Chapter 8:

Non-Linear Vibration Tests and Test Data Processing.........................72

8.1 8.2

Example: Structure with Coulomb-Friction Non-linearity ...........................75 Detection and Characterization of Non-Linearities from Measured Response Data...................................................................................................................78 8.2.1 FRF Distortions and Describing Functions ............................................78 8.2.2 Hilbert Transformation ............................................................................89 8.2.3 Distortions of Frequency Isochrones in a Nyquist Plot..........................91 8.2.4 Inverse FRF Plot.......................................................................................91 8.2.5 Restoring Force Surface and Force-State Mapping................................92 8.2.6 Final Remark on Detection and Characterization Methods ..................93

Chapter 9:

Quality Assessment for FE Model and Test Data .................................94

9.1 Finite Element Model Quality Assessment....................................................94 9.1.1 Mathematical Checks...............................................................................94 9.1.2 Engineering Checks..................................................................................94 9.1.3 Comparison of Model Predictions with Experimental Results ..............94 9.1.4 Different Sources of Model Errors ...........................................................95 9.2 Test Data Quality Assessment........................................................................96 Chapter 10: Theory of Non-Linear Parameter Updating...........................................97 10.1 Parameterization of Finite Element Models ..................................................98 10.2 Determination of Parameter Changes............................................................99 10.3 Residuals for Linear Parameter Updating ...................................................101 10.4 A Residual for Linear and Non-Linear Parameter Updating .....................102 10.5 Calculation of Sensitivities............................................................................103 10.5.1 Eigenfrequency and Mode Shape Sensitivity .......................................103 10.5.2 Linear Frequency Response Function Sensitivity................................103 10.5.3 Non-Linear Frequency Response Function Sensitivity by Finite Differences ..............................................................................................103 10.5.4 Non-Linear Frequency Response Function Sensitivity Based on Structural Modification Theory .............................................................104 10.6 Regularization ................................................................................................107 Chapter 11: Applications...........................................................................................109 11.1

Updating of a Non-linear Cantilever Beam by using Simulated Test Data .........................................................................................................................112 11.1.1 Non-Linear Updating using Direct Frequency Response Analysis .....115 11.1.2 Non-Linear Updating using Modal Frequency Response Analysis.....116 11.1.3 Preliminary Conclusions ........................................................................121 11.2 CTS Joint Identification ................................................................................123 11.2.1 Interpretation of Updating Results .......................................................134 11.3 Identification of Non-Linear Joint Parameters of an Aero-Engine Casing Joint ................................................................................................................138 11.3.1 Updating of the Underlying Linear Model............................................139 11.3.2 Non-Linear Updating of Joint Parameters ...........................................144 11.3.3 Interpretation of Updating Results .......................................................155

Chapter 12: Summary............................................................................................... 160 12.1 12.2

Conclusions .................................................................................................... 162 Outlook ........................................................................................................... 165

Chapter 13: References ............................................................................................. 167 Chapter 14: Appendix ............................................................................................... 177 14.1 14.2 14.3

MSC.Nastran Input Deck of the Cantilever Beam Model........................... 177 Modes of the Cantilever Beam Model........................................................... 179 Modes of the CTS Shell Model ...................................................................... 180

1. Introduction

Chapter 1:

1

Introduction

Finite element models are extensively used in product development cycles of engineering structures. Among other things, they are used to optimize the structural behavior already in early stages of development. The criteria for optimization can be manifold, e.g. the safe design with regard to mechanical strength and durability, noise- and vibration reduction, comfort improvement, etc. The objective to gradually replace expensive tests on prototypes with numerical simulations and, hence, to achieve cost-effective structural design within shorter time scales is closely linked to the increasing demand for reliable finite element (FE) simulations. The reliability of FE simulation results is largely affected by the accuracy of the models used and is thus determined by the assumptions made during model generation. Modern FE software packages provide means for the generation of highly accurate models for continuous structures, e.g. by directly supporting the use of CAD (computer aided design) models for meshing with 3D volume elements. Despite of sophisticated modeling tools, the generation of predictive models for complex assembled structures cannot be automated completely so that such models often appear to be inaccurate. Sources of possible inaccuracies of complex assembled models can of course be found in inaccurate substructure models, but more importantly, in insufficient modeling of structural interfaces (joints). Computational model updating (CMU) procedures were developed to improve the accuracy of FE models by adjusting uncertain model parameters based on the minimization of discrepancies between model predictions and experimental observations. Model parameters in this respect can be material parameters like Young’s modulus and density, but also properties of structural elements like thicknesses of plates and shells or cross sectional properties of beam elements. Even the geometrical shape of FE models can be updated by CMU, e.g. by adjusting the position of nodes. Adequate parameterization of the FE model geometry is a prerequisite for geometry updating. When acquiring vibration test data for CMU, it can often be observed that assembled structures reveal different dynamic behaviors at different excitation force levels (and hence different vibration amplitude levels). Such structures violate the principles of superposition and reciprocity and are called non-linear structures if the change of the response with increasing excitation force level can be expressed by a non-linear function. For example, Figure 1.1 shows a frequency response function (FRF) which was measured at different levels of constant excitation force amplitudes. It can be observed that the first and the fifth resonance peak show non-linear behavior, whereas the other resonance peaks are unaffected. This indicates non-linear behavior of the first and the fifth mode. When non-linear behavior is observed in a vibration test, it is common practice to validate the corresponding FE model in a “best linear fit” manner by using the test data acquired at a level of excitation which is representative of the operational load level of the structure. Modeling of non-linearities is disregarded in most cases.

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1. Introduction

Figure 1.1: Non-linear acceleration FRF for different levels of constant excitation force

The increasing demand for predictive models which are valid in a wide range of excitation force levels led to the inclusion of non-linearity in FE models. However, due to the computer resource and storage requirements of iterative procedures, nonlinear analysis was considered prohibitive for large order assembled models in the past. This aspect gradually changed with the increasing computer power over the last decades. Nonetheless, the inclusion of non-linearity in FE models involves two basic problems. The first one is related to modeling of non-linear effects. Even though it is possible in many practical cases to assess the type of non-linearity inherent in a structure, it is nonetheless difficult to derive meaningful properties for non-linear elements to represent the non-linearity observed in a test. A prominent example is the determination of stiffness and damping properties of a simple bolted joint. This can already be a difficult task for linear analysis but becomes even more difficult for nonlinear analysis. The second problem is related to the type of analysis to be performed. While different analysis methods are available for linear systems (e.g. transient response analysis, frequency response analysis, eigenvalue analysis), most commercial FE software packages only provide transient response analysis for non-linear systems. This poses some problems on the validation of non-linear models, because transient responses (time history responses) are less well suited for model validation than modal data or frequency domain response data.

1.1

Motivation and Objectives

Basically, this thesis focuses on improvements for the validation of non-linear models. The term “non-linear model” shall be restricted to models with local nonlinearities like those introduced at structural joints. Material non-linearities which can occur in a distributed way anywhere inside a structure, or geometric nonlinearities due to large displacements shall not be considered here. Even though the non-linearities considered are local, the overall structural behavior is non-linear and thus requires adequate analysis procedures.

1. Introduction

3

The fundamental idea for non-linear model validation followed in this thesis is to make use of the linear model validation expertise and experience of engineers also for non-linear model validation. This requires that the model validation procedures which will be developed in the following shall use as much as possible of the linear model validation procedures also for non-linear model validation. However, modifications of the well established linear model validation procedures might be necessary to meet the specific requirements for non-linear data. In contrast to transient time domain data, non-linear frequency response functions represent some kind of condensed information which is suited for the characterization of non-linear behavior. It is therefore reasonable to assume that non-linear FRFs can also be used for the identification of non-linear joint properties in a CMU procedure, because the theoretical basis for linear model updating using frequency response data is well developed. This is illustrated in Figure 1.2, where the shaded areas indicate the differences between an analytical and an experimental fundamental harmonic non-linear FRF. Such non-linear FRFs can be obtained experimentally by performing constant level step-sine measurements, and analytically by applying the Harmonic Balance Method to a non-linear FE model. The non-linear FRF differences can be used as a frequency response residual in CMU for the identification of the non-linear element properties which are assumed to be the cause of the FRF deviations.

Figure 1.2: Differences between analytical and experimental non-linear FRFs

It must be checked, however, if there is enough information contained in the distortion characteristic of non-linear FRFs to properly identify joint non-linearities in FE models, especially when the FE models are large. This is a fundamental question which will be investigated in this thesis. From the analytical side, all the FE model preparation steps necessary for non-linear frequency response analysis and non-linear model updating will be discussed one at a time. From the experimental side, test design for non-linear vibration tests will be reviewed to provide a database for non-linear updating. Finally, the non-linear updating method will be applied to a large order aero-engine finite element model with the objective to identify the non-linear stiffness and damping properties of a bolted flange casing joint which revealed some kind of nonlinear behavior in constant excitation force step-sine measurements.

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1.2

1. Introduction

Contents and Organization of the Text

In computational model updating, analytical and experimental data are brought together with the objective to improve the prediction capability of finite element models. This is illustrated in Figure 1.3, where the output of analysis and test serve as an input to model updating. This thesis is focused on non-linear model updating so that special issues concerning non-linear FE models and non-linear tests must be discussed. The organization of the text can be observed from the corresponding chapter numbers for each item in Figure 1.3. Non-linear Analysis: • modeling of non-linearities (chapter 3) - bolted joints - bolted flange joints • non-linear frequency response analysis (chapter 4+5) - harmonic balance method - non-linear structural modification • model reduction techniques (chapter 6) - mode superposition - Craig-Bampton reduction • generation of damping matrices (chapter 7)

Non-linear Test: • FRF measurements with level control (chapter 8) - constant force levels - constant response levels • detection/characterization of non-linearities (chapter 8) - FRF distortions - Hilbert transformation - frequency isochrones - inverse FRF plot - restoring force surface

Model Updating: • quality assessment of FE model and test data (chapter 9) • sensitivity based parameter updating (chapter 10) • updating residuals (chapter 10) for - linear model updating - non-linear model updating • sensitivity analysis (chapter 10) based on - finite differences - structural modification Figure 1.3: Different aspects of non-linear model updating and their location in the text

The items mentioned in the above figure are supplemented with a preceding literature review given in chapter 2 and subsequent applications of non-linear updating presented in chapter 11. The applications include an analytical example and practical examples. The analytical example is used to demonstrate the capabilities and limitations of non-linear computational model updating, whereas the practical applications shall demonstrate the applicability of the methods developed to large order systems. The practical applications include the identification non-linear stiffness and damping properties of an aero-engine casing joint from measured non-linear FRFs. This work was partly conducted within the European research project CERES together with the European aero-engine manufacturers. CERES stands for Cost-Effective Rotordynamics Engineering Solutions and was funded by the European Community under the ‘Competitive and Sustainable Growth’ program (framework 5, 1998-2002). A summary of the thesis and conclusions are given in chapter 12. A list of specific literature and references is provided in chapter 13.

2. Literature Review

Chapter 2:

5

Literature Review

The methods developed in this thesis shall contribute to the validation of non-linear models as an extension to the well established linear model validation techniques. Computational model updating procedures can be considered as an integral part of the linear model validation process which is schematically sketched in Figure 2.1. The basic idea followed here is to make use of the linear model validation expertise and experience of engineers also for non-linear model validation. This requires that the model validation procedures developed in the following shall use as much as possible of the linear model validation procedures and to introduce modifications to meet specific requirements for non-linear response data. In the following section, an outline of basic literature is discussed which gives an overview about the different issues involved in non-linear model validation using frequency response residuals.

Figure 2.1: Model validation flow chart

2.1

Computational Model Updating

Direct and iterative methods can be distinguished in model updating. Direct methods can adjust the complete analytical system matrices of a finite element model in a single updating step so that afterwards the model is capable to exactly reproduce the experimental data to which it was fitted. The major drawback of the direct methods is the loss of physical significance of the updated system matrices. Changes of the system matrices cannot be related to changes of particular model areas which are assumed to be erroneous. This can lead to poor model prediction capability, i.e. the

6


updated model may not be able to predict the response due to other loading conditions or due to structural modifications, or the updated model may not be suited to serve as a substructure in an overall assembled model. The term computational model updating (CMU) denotes model updating methods which iteratively adjust initially defined uncertain model parameters in order to minimize an objective function which is a measure for the deviation of the model predictions from experimental observation. In many practical cases the objective function represents the weighted square sum of the so-called residuals. These residuals are the differences between experimental and analytical response data, e.g. differences between static deflections in case of static model updating (e.g. [Lallement 1992b]), or eigenfrequency differences in case of dynamic model updating (e.g. [Collins 1974]), etc. The advantage of the iterative updating methods over the direct updating methods can be found in the parameterization of the FE model. Parameterization allows for the direct relation of changes in the mass-, stiffness-, and damping matrix to changes of uncertain model parameters so that the updated system matrices retain their physical meaning. The nature of the objective function to be minimized requires linearization and thus iterative minimization. Even though this brings along all the problems possibly involved in iterative procedures (e.g. convergence problems and evaluation of analytical response data in each iteration step) the advantages of CMU methods overbalance their drawbacks so that they became one of the core applications in the whole model validation process as it is sketched in Figure 2.1. The earliest publications in the field of computational model updating are certainly those of Collins et al. and Natke et al., [Collins 1974], [Natke 1974]. They used Fox’s approach for first order sensitivities of eigenvalues and mode shapes to improve uncertain mass- and stiffness parameters of dynamic systems, [Fox 1968]. Since then, much effort was spent on improving the CMU procedures, or respectively, to find other fields of application. Among other things, different residuals were proposed for different purposes of model updating. Prominent examples for updating dynamic models are the eigenvalue and mode shape residuals introduced by Collins et al., [Collins 1974], and the frequency response residual introduced by Natke, [Natke 1977]. The advantage of the frequency response residual over the eigenvalue and mode shape residuals is its ability to update uncertain damping parameters. In addition, experimental modal analysis (EMA) is not necessary to extract eigenfrequencies and mode shapes from measured FRFs. Instead, FRFs can be used directly. Another type of residual is the so-called equation error originally introduced by Cottin, Felgenhauer, and Natke [Cottin 1984], [Cottin 1986]. The equation error residual seeks to balance the eigenvalue equation using measured eigenvalues and eigenvectors together with analytical mass- and stiffness matrices. It suffers from the incompleteness of test data, i.e. information of less measurement degrees of freedom (DoFs) than finite element DoFs is available. Lallement and Piranda used the equation error as a tool for FE model error localization, [Lallement 1990]. Link and Santiago further developed the error localization based on the equation error and used it as a tool for model updating, [Link 1991]. Apart from the derivation of different updating residuals the problem of illconditioning of the updating equations was investigated. Mottershead and Foster proposed methods based on singular value decomposition and regularization to improve the least-squares parameter estimation in case of incomplete or noisy test data, [Mottershead 1991].


7

From an overview of publications in the field of computational model updating, it can be concluded that modal data and frequency domain response data are the preferred sources of data for updating dynamic models. They represent some kind of condensed information about the behavior of a system and is therefore better suited for updating than time domain response data. Today, computational model updating are well established techniques to improve finite element models. However, only a few textbooks were published on that subject. [Natke 1992] and [Friswell 1995] are certainly the most popular ones. Link contributed significantly to establish the application of CMU in engineering practice. [Link 1999a] and [Link 1999b] are generic publications which summarize the basic CMU theory as well as application aspects. Many other publications of Link et al. are referenced there which focus on the application of CMU.

2.2

Identification of Non-Linear Parameters

When compared to the identification of linear parameters, it can be said that nonlinear parameter identification mainly suffers from the problems involved in calculating non-linear analytical data. In fact, frequency domain response data and modal data (i.e. the preferred data for linear model updating) cannot be calculated directly for non-linear systems. Consequently, the first systematic attempts in the field of non-linear parameter identification utilized experimental non-linear time history responses. One of the fundamental publications on the identification of parameters of non-linear systems was published by Masri et al., [Masri 1979]. They developed a method which fully relies on experimental time history response data with the restriction to nonlinear single DoF oscillators. The non-linear restoring force is calculated from measured acceleration response and excitation force. The nature of the non-linear restoring force can be observed from the so-called restoring force surface. This is a three-dimensional surface plot of the non-linear restoring force over displacement and velocity response from which non-linear parameters can be identified by curvefitting. Displacement and velocity response must either be measured or must be calculated numerically by integrating the measured acceleration response. It should be noted that the restoring force surface method was developed independently by Crawley et al. and was introduced as force-state mapping, [Crawley 1986]. Pitfalls and drawbacks of the restoring force surface method (or respectively the force-state mapping) can be traced back to the restriction to single DoF systems, which is rarely fulfilled by practical structures. But even if the single DoF assumption is violated, one can generate a characteristic non-linear restoring force surface which can provide valuable information about the type of non-linearity simply by visual inspection of the 3D surface plot (provided that the system does not have multiple non-linearities of different types). The identification of non-linear parameters is not possible with the original restoring force surface method in case of multi DoF systems. Enhancements of the original restoring force surface method were published by Wright and co-workers, [Wright 2001], [Platten 2002]. They investigated the nonlinear response of a plate-like structure, whose non-linearity was caused by distributed geometric non-linear effects. Force appropriation was used to enforce single DoF behavior in order to overcome the limitation of the original method. Nonlinear modal properties were identified for the plate-like structure from resonant decay measurements.

8


A similar approach and also a further enhancement of the force-state mapping was developed by Göge, e.g. [Göge 2003] and [Göge 2004]. Göge identified non-linear modal stiffness and non-linear modal damping properties from curve-fitting of modal restoring forces. These modal restoring forces were derived from modal acceleration responses and modal excitation forces acquired during phase resonance testing of aero-space structures. Force appropriation was also used in this case to enforce the structure to exclusively vibrate in a single mode. In contrast to the plate-like structure investigated by Wright, the type of structures investigated by Göge leads to the assumption that the non-linearity was caused by concentrated local effects at structural joints. However, Göge’s method only provides the identification of global non-linear behavior on a modal basis. Local non-linear effects somehow contribute to the identified modal non-linear behavior but cannot be separated, nor be isolated or identified. Additional applications of force-state mapping with enhancement to multi-degree of freedom systems can also be found in [Dimitriadis 1998] and [Dimitriadis 2001]. A significant contribution to the identification of friction type non-linearities from measured time histories was provided by Gaul and co-workers. They designed a test setup which allowed the derivation of the hysteresis loop of the non-linear restoring force (shear force transmission) of a single bolted joint from measured excitation force and acceleration response, [Gaul 1987], [Bohlen 1987]. Lenz and Gaul used the same test setup to study the influence of microslip and macroslip on the dynamic behavior of bolted joints, [Lenz 1997], [Gaul 1997]. A detailed finite element model of the friction contact area was utilized to provide understanding of the different slipstick mechanisms which may occur multiple times in a single period of vibration. They also investigated different friction models including microslip behavior with their friction parameters fitted to the experimentally acquired hysteresis loops for shear force transmission of bolted joints. Good agreement with experimental transient responses was found after assembling these “updated” friction joints in a complex finite element model. Further application of friction damping at bolted joints can be found in [Gaul 2000]. The investigation of non-linear systems based on non-linear frequency domain response data was initiated by Tomlinson, e.g. [Tomlinson 1984] and [Tomlinson 1986]. He revealed that certain non-linearities cause characteristic distortions to non-linear frequency response functions obtained with constant excitation force level testing (or respectively, analytical non-linear FRFs obtained with the Harmonic Balance Method which can be considered as the analytical equivalent to constant excitation force level step-sine testing). In addition, Simon and Tomlinson were the first ones to use the Hilbert transform of FRFs as an indicator for non-linear behavior, [Simon 1984]. Even though the Hilbert transform essentially indicates non-causality, it is widely accepted that the Hilbert transform is suitable to indicate non-linearity as well. Based on Tomlinson’s non-linear frequency domain methods, Weiland introduced a method which uses the Harmonic Balance Method to identify non-linear distortions of experimental FRFs caused by polynomial stiffness- and damping non-linearities, [Weiland 1995]. The objective of his work was to filter out the identified non-linear distortions such that pure linear FRFs are available for subsequent experimental modal analysis of the underlying linear system. Meyer introduced a computational model updating approach for non-linear parameter identification in order to overcome the problems of the methods based on non-linear time history responses, [Meyer 2003]. His basic idea was to calculate frequency responses of non-linear finite element models by using the Harmonic


9

Balance Method. These non-linear frequency responses were then used together with experimental non-linear frequency responses as a residual for updating the parameters of local non-linear elements in otherwise linear finite element models. This approach seemed quite promising, because it combines the well established theory of computational model updating using frequency response residuals with the non-linear frequency domain response methods based on the Harmonic Balance Method. The numerical effort involved in the calculation of analytical non-linear frequency responses caused the method to be inefficient for complex large order models so that only relatively small finite element models of laboratory structures could be investigated by Meyer. The work presented in this thesis is based on Meyer’s work and seeks to overcome the limitations of his method with the aim to make it applicable to large order systems typically encountered in engineering practice.

2.3

Non-Linear Frequency Response Analysis

Efficient non-linear frequency response analysis is the key to efficient non-linear computational model updating when considering the method developed by Meyer. Different approaches are available for non-linear frequency response analysis and will be reviewed in the following. The classical Harmonic Balance Method was introduced by Krylov and Bogoljubov as a tool for the approximate calculation of steady-state responses of non-linear systems, [Krylov 1947], [Bogoljubov 1965]. It is based on the assumption that a nonlinear system produces an almost harmonic response when excited harmonically. This method will be referred to as the Single-Harmonic Balance Method (SHBM) in the following, because it only accounts for the fundamental harmonic response. The SHBM allows for the calculation of equivalent stiffness- and damping parameters of non-linear elements by equivalent linearization over one period of harmonic vibration using a given magnitude of displacement response. These equivalent nonlinear stiffness- and damping parameters are generally response magnitude dependent but no longer time dependent. They provide the basis for non-linear frequency response analysis which must be performed iteratively, since the response magnitude which is used to determine the equivalent non-linear parameters is generally unknown at the beginning of the analysis. Due to the response magnitude dependence of the equivalent non-linear stiffness- and damping parameters, the analytical non-linear frequency responses are also response magnitude dependent and, thus, also depend on the magnitude of excitation. Apart from its application in structural dynamics, the SHBM also found application in other engineering disciplines. For example, it is known as the Describing Function Approach in control engineering and it is used there for the equivalent linearization of non-linear input-output relations of non-linear subsystems. Gelb and Vander Velde, [Gelb 1968], published an excellent textbook on the Describing Function Approach with an extensive list of analytical formulations for equivalent non-linear stiffness- and damping parameters for general types of non-linearities typically encountered in engineering practice. Magnus and Popp and Worden and Tomlinson published textbooks on non-linearity in structural dynamics, [Magnus 1997], [Worden 2001]. The latter one specifically summarizes frequency domain methods for the detection and identification of non-linearities in dynamic systems. Tanrikulu, Kuran, and Özgüven, [Tanrikulu 1993], developed a procedure for nonlinear frequency response analysis of multi-degree of freedom systems based on the

10


SHBM. In their method, the non-linear restoring forces of non-linear elements assembled in a finite element model are expressed by a matrix product of the socalled quasi-linear matrix, which contains response magnitude dependent equivalent stiffness- and damping parameters, and the vector of complex displacement response magnitudes. This procedure was also applied by Budak and Özgüven in [Budak 1993] and Cömert and Özgüven in [Cömert 1995] where they analyzed the non-linear response of systems with polynomial type non-linearities or respectively the non-linear response of linear substructures coupled by non-linear elements. Sanliturk and Ewins, [Sanliturk 1996], used the SHBM to analyze the forced response of a two dimensional planar friction contact model with the aim to derive an improved contact model for inter-shroud contact conditions of bladed disks used in aero-engines. They used a so-called Masing model for the friction representation. The Masing friction model allows to investigate macroslip effects (3-parameter elasto-slip model), but also microslip effects by increasing the number of discrete friction elements assembled in the Masing model (up to 10 discrete friction elements arranged in parallel were studied by Sanliturk and Ewins to approximate microslip effects). Gaul published an analytical expression for the equivalent non-linear stiffness- and damping parameters of the n-parameter Masing model (elasto-slip model) in [Gaul 2000]. Mickens investigated the accuracy of the SHBM from a mathematical point of view, [Mickens 1984]. He stated that the SHBM produces excellent results for approximate solutions of non-linear differential equations which cannot be solved in closed form. This statement was formulated for non-linearities having odd restoring force functions (odd non-linearities). Mickens found out that in case of odd non-linearities, it is often not necessary to calculate the response of higher harmonics to obtain accurate response predictions. He also found out that the response accuracy degrades in case of non-odd non-linearities. Even though the fundamental harmonic response can be calculated accurately using the SHBM, the total response of a nonlinear system with non-odd non-linearities contains other response components which are disregarded by the SHBM. Other frequency domain response methods may be better suited for accurate analysis of the total non-linear response of systems with non-odd non-linearities. The Multi-Harmonic Balance Method (MHBM) was introduced by Pierre, Ferri, and Dowell as a frequency domain response analysis tool for the calculation of the total steady-state response of non-linear systems, [Pierre 1985]. They investigated friction damped systems with one and respectively two DoFs by calculating the steady-state response, once with a time marching numerical integration scheme, and once with the MHBM using different numbers of harmonics. The method was found to be very accurate when compared with existing time domain techniques. From a comparison of the frequency response curves obtained with MHBM and SHBM they observed only little response differences in the resonance region, whereas the response differences were more pronounced in the regime of moderate amplitudes where slipstick motion is likely to occur. In this case, the total non-linear response of the system cannot be described accurately by harmonic motion which is the reason for the discrepancies between SHBM and MHBM solutions. Pierre, Ferri, and Dowell also mentioned the tradeoff between the level of accuracy and the computational cost and stated that three harmonics were usually found to be adequate for good accuracy at reasonable cost. A further application of the MHBM to friction damped systems is given in [Ferri 1988]. Chen, Yang, and Menq extended the one-dimensional friction model to include planar friction damped motion with possible separation of contacting friction


11

partners and calculated the MHBM response of a 3-DoF oscillator with the planar friction model included, [Chen 2000b]. Petrov also used a planar friction model with variable normal load and possible separation of friction partners to simulate the blade-to-disk and inter-shroud contact conditions of aero-engine bladed disks. He applied the method to a finite element model of a bladed disk and analyzed the multi-harmonic non-linear response by using the MHBM. An interesting study of non-linear frequency domain response methods was performed by Kuran and Özgüven, [Kuran 1996]. They compared the responses obtained with SHBM and MHBM of non-linear systems with dry friction, cubic stiffness, and piecewise-linear stiffness. They applied mode superposition using the modes of the underlying linear system to reduce the number of effective DoFs and concluded that the MHBM response in the vicinity of the resonance of an isolated mode can be represented accurately using a single mode (i.e. a single generalized coordinate), whereas the SHBM response turned out to be only reasonably accurate when using mode superposition, but less accurate than the MHBM response. A basic problem of SHBM and MHBM is their limited applicability to large order systems. This problem can either be addressed by applying appropriate model reduction techniques or by considering the non-linear elements as response amplitude dependent structural modifications to an otherwise linear system. In this case, structural modification theory can be applied for response analysis by using relatively small FRF matrices instead of large order system matrices. One of the first papers in the field of structural modification theory (SMT) was published by Duncan, [Duncan 1941]. Klosterman can be considered as one of the pioneers in the field of SMT and introduced it as a frequency response analysis tool which utilizes experimental FRFs of a linear system to analyze the response of the same system after structural modifications have been applied, [Klosterman 1971]. Possible modifications include the attachment of masses, the attachment of spring/damper elements, and the introduction of additional boundary conditions. Theory and applications of the SMT can also be found in [Crowley 1984] and [Özgüven 1990]. Mottershead, Mares, and James applied the SMT to separate close modes of axisymmetric structures to finally improve the conditions for modal correlation. This was achieved by sequential application of fictitious modifications, [Mottershead 2002]. Lallement and Cogan presented a method to enlarge the knowledge space for FE model updating, [Lallement 1992a], [Lallement 1999]. They proposed to perform fictitious tests (pseudo-tests) based on SMT by using test data of an initial test of a structure. The advantage of SMT based pseudo-tests is that no additional unknowns such as support stiffness are introduced as it is usually done in real tests. The basic idea of their method is to provide additional test data of a structure e.g. with other boundary conditions or due to structural modifications, which can be for computational model updating of structural parameters. It was stated by Brandon, [Brandon 1990], that SMT is not limited to linear systems and that the method would in principle be applicable to non-linear systems as well. Indeed, first attempts of non-linear structural modifications were performed by Malhalingam to modify a linear structure by friction damper inserts, [Malhalingam 1975]. He derived a constant equivalent viscous damper element for the non-linear friction damper and actually performed a linear structural modification. Ferreira and Ewins developed a receptance coupling approach based on SMT which can be used to couple linear substructures by non-linear joint elements in order to calculate the non-linear fundamental harmonic response of the coupled

12


system, [Ferreira 1996]. The FRF matrices of the linear substructures are needed as an input together with the describing function (equivalent non-linear parameters) of the non-linear coupling elements. The method developed by Ferreira and Ewins is a promising technique, because model reduction and mode superposition are not necessary for the method to be efficient. It directly operates with a relatively small FRF matrix defined at only a few DoFs (e.g. DoFs at which the response shall be calculated, exciter DoFs, and the non-linear coupling DoFs at the joints). Due to their capabilities for fundamental harmonic frequency response analysis, SMT based methods were also applied in the field of joint identification. Maia, Silva, and Silva used an SMT based substructure coupling approach to identify the transfer behavior of a joint structure which couples two substructures, [Maia 2000]. Liu and Ewins developed a joint identification method based on SMT for the identification of linear joint stiffness properties between linear substructures, [Liu 2000a], [Liu 2000b]. The method performed well in numerical simulations but less well when the FRF data of the substructures was contaminated with Gaussian noise. Link developed a computational model updating approach for the identification of linear joint stiffness and damping parameters, [Link 2004]. He used the differences between measured frequency responses and analytical frequency responses as a residual for updating. The advantage of the method is that the analytical frequency responses were calculated by using SMT so that during updating response analysis of the updated FE model is not necessary. Instead, frequency response analysis is performed by using SMT with the analytical FRF matrix obtained from the initial FE model. Stiffness and damping changes at the joint are considered as structural modifications. Link also derived an improved formulation for frequency response sensitivity based on structural modification theory. In a recent publication, Özer, Özgüven, and Royston presented an approach based on SMT for the identification of structural non-linearities, [Özer 2005]. This approach can be used to identify the describing function of a non-linear unit rank modification from a comparison of the underlying linear FRF (e.g. analytical FRF) and the corresponding non-linear FRF (e.g. non-linear experimental FRF). Özer, Özgüven, and Royston identified the describing function of a non-linear spring of a single DoF oscillator by using simulated test data. Formulations for general types of structural modifications were not given there.

2.4

Model Reduction Techniques

It was mentioned before that non-linear frequency response analysis often suffers from large computational effort. Model reduction techniques are commonly applied in engineering practice to condense linear models in order to increase the numerical efficiency of analysis, or respectively, to make analysis of complex large order models possible at all. The first model reduction method was developed by Guyan and is also known as the static condensation, [Guyan 1965]. The system under consideration is partitioned into independent master DoFs and dependent slave DoFs. The motion of the slave DoFs is expressed in terms of the motion of the master DoFs. The transformation matrix utilized for model reduction is derived by neglecting internal dynamic forces at the slave DoFs. Consequently, statically condensed models are only exact for static analysis but gradually degrade with increasing frequency in case of dynamic analysis.


13

Craig and Bampton extended the static condensation approach used by Guyan to approximate the disregarded internal dynamic forces by adding modal information of the condensed model parts, [Craig 1968]. They derived a transformation matrix which combines physical and modal information for model reduction. A review of the most commonly applied model reduction techniques can be found in [Craig 1987]. O’Callahan performed a comparative study of different model reduction techniques, [O’Callahan 1990], among those, Guyan reduction, Craig-Bampton reduction, and SEREP (system equivalent reduction-expansion process), a model reduction technique which was developed by himself. Numerous papers have been published in the field of model reduction techniques. Quite frequently they were aimed at improving the accuracy by proper selection of master DoFs, e.g. [Bouhaddi 1992]. But also the application of model reduction techniques was investigated in case of local non-linear systems, e.g. [Deloo 1991], [Balmes 1997].

14

Chapter 3:

3. Non-Linear Joint Modeling Approach

Non-Linear Joint Modeling Approach

Software for CAD and FE analysis are commonly used in product development. Data exchange interfaces are available between popular CAD and FE software packages to make product development more efficient. For example, modern FE software supports CAD geometry import by means of standard interface files (e.g. IGES files). FE models can thus be generated directly from 3D geometry models without using paper drawings. When assuming that in the near future the number of DoFs is no longer a limitation for FE model generation, it would be possible to generate highly accurate FE models by importing CAD geometry into FE software and subsequently performing adaptive meshing using 3D elements which can represent a generic state of three dimensional stress. Proceeding this way can help to minimize modeling uncertainties like geometric shape discrepancies and idealization errors introduced by the use of inappropriate element types. In many practical cases, engineering structures are assembled of several components or substructures. Frequently, the FE modeling work is shared among different people who individually generate FE models of the components. It can nonetheless happen that the overall structural model, which was assembled from accurate component models, turns out to be inaccurate, even though the component models used were validated individually. The reason for the poor accuracy of the overall assembled model can then be found in insufficient joint modeling. It might be relatively easy in the near future to generate accurate models for continuous structures (such as for the components of an overall assembled model), but it will remain difficult to derive meaningful finite element representations for the joints which connect accurate component models. The types of joints between substructures can be manifold and different categories can be distinguished, e.g. by the mechanisms of force transmission. Another criterion could be to distinguish between detachable and non-detachable connections. Welded joints and bonded joints, for example, belong to the group of non-detachable connections. Forces are transmitted by additional material which is fully combined with the joined parts. Form-locking joints (positive joints) belong to the group of detachable connections where force transmission is achieved by the geometric shape of the joined parts (e.g. pinned joints, taper-pinned joints, and pivots). Another group of detachable connections are joints with force transmission by friction. Riveted and bolted joints as well as interference fits and cone fits for shaft-hub connections are members of that group. The number of possible joints is large and many more examples can be found which would fit into the above mentioned categories. This is not the objective of this thesis. Instead, emphasis shall be given to bolted joints. They are the most commonly used type of joint in mechanical engineering due to their high degree of versatility which results form the large variety of different designs and standards defined for bolts.


3.1

15

Bolted Joints

Failure of highly loaded bolted joints in mechanical engineering structures can have severe consequences, not only in terms of maintenance costs. Therefore, the analysis of forces and displacements of bolts and clamped parts is essential for their safe design. The engineering guideline [VDI 2230] represents the standard for the systematic calculation of high duty bolted joints in mechanical engineering and will be used as a baseline for the derivation of a meaningful FE modeling approach for bolted joints. Even though bolted joints in most cases consist of multiple bolts arranged in a certain pattern, their analysis is based on single bolted joints. Consequently, the force resultant of an operational load applied to a bolted joint must be analyzed and distributed in a meaningful way to the single bolts which are members of the joint under consideration. Systematic bolt calculation is then performed for the bolt with the highest loading. Single bolted joints are considered next because of their importance for generic bolted joint analysis.

3.1.1

Concentrically Clamped Single Bolted Joint

The calculation of single bolted joints is based on the elastic behavior in the immediate surroundings of the bolt. This region has considerable effect on the bolt deformation during assembly and in service and consequently also on the loading of the bolt. The principal mechanical behavior of bolted joints will be discussed in the following on a concentrically clamped single bolted joint. Concentrically, in this respect, means that operational loads are introduced in a symmetric way so that no bending moments are being produced. It is further assumed that the single bolted joint is located far away from obstacles which might disturb the extension of the elastic deformation zone in the immediate surrounding of the bolt. The transmission of axial and transverse operational loads will be discussed separately, because they are based on different physical effects.

3.1.1.1 Transmission of Axial Loads A single bolted joint shall be considered with the operational load FA introduced in a symmetric way directly under the bolt head and under the nut, see Figure 3.1. During assembly of the joint, a pre-load FM is being produced in the bolt which causes an elastic extension uSM of the bolt and an elastic compression uPM of the clamped parts. The forces and displacements of the bolted joint can be illustrated in a simplified manner by means of a bolted joint loads diagram shown on the left hand side of Figure 3.2. The bolt pre-load FM equals the clamp force FK in the absence of operational loads FA . In the analysis of single bolted joints it is assumed that the clamp force FK produces an elastic deformation of the clamped parts in a narrow region around the bolt hole. This pre-stressed deformation zone can be idealized by a double-cone and whose size can be assessed in a simplified way from the geometry of the clamped parts and the bolt according to [VDI 2230].

16


1 2

FA

1 2

FA

pre-stressed deformation cone

interface 1 2

FA

1 2

FA

Figure 3.1: Axial load transmission of a single bolted joint

after joint assembly, without operational load

force

force

The axial operational load FA is transmitted proportionally by the clamped region of the interface and the bolt. The load portion applied to the bolt (in addition to the preload FM ) is denoted by the additional bolt load FSA , whereas the complementary load portion FPA relieves the clamped parts. The division of the (tensile) operational load FA into the load portions FSA and FPA can be observed in the right hand side diagram of Figure 3.2. The factor of proportion of the load division depends on the elastic properties of the joint partners, i.e. the bolt stiffness k S and the stiffness of the deformation cone k P . after joint assembly, with operational load

FSA FA

FM = FK

kP kS uSM

FPA FKR

uPM

uSA

displ.

displ.

Figure 3.2: Bolted joint loads diagram without and with tensile operational load FA>0

The additional bolt load FSA produces an additional elastic extension of the bolt uSA which equals the elastic relieve of the clamped parts. The load portion FPA reduces the clamp force between the joined parts so that only a residual clamp force FKR = FK − FPA remains. The surface pressure at the clamped interface is reduced which is necessary for the bolted joint to fulfill its function. The relation between forces and displacements of the single bolted joint is linear until the (tensile) operational load FA exceeds the critical limit force:

FA+,cr = nF

kS + kP FM , kP

(3.1)

with nF being a factor which takes into account the effects of load introduction ( nF ≈ 1 in case of a concentrically clamped bolted joint according to Figure 3.1). After passing this transition point (indicated by the critical bolt extension ucr ), the residual clamp force FKR between the joint parts vanishes and the operational load


17

force

FA is no longer split up proportionally into the load portions FSA and FPA . Instead, the clamp force reduction FPA remains constant after it has reached its maximum, whereas the additional bolt load FSA further increases. This can be observed in the bolted joint loads diagram for large tensile loads shown in Figure 3.3.

FSA

FA ucr uSA

FPA

displ.

Figure 3.3: Bolted joint loads diagram for large tensile operational load FA>0

force

Such high operational loads are usually not taken into account, because bolted joints are designed to avoid gap opening at the clamped interface at any time. Bolted joints may not only be loaded by tensile loads ( FA > 0 ), but also by compressive loads ( FA < 0 ). Figure 3.4 shows the force and displacement conditions of a single bolted joint in case of compressive operational loads.

FPA

FA

FSA

FKR FSR uSA

displ.

Figure 3.4: Bolted joint loads diagram for compressive operational load FA ucr

(3.3)

F kS FA+,cr

kS + kP

ucr

u

Figure 3.5: Force-displacement diagram of single bolted joint for axial load transmission idealized by a pre-loaded bilinear curve

3.1.1.2 Transmission of Transverse Loads In case of operational loads FQ which act perpendicular to the bolt axis (see Figure 3.6) the bolted joint has to avoid relative motion between the clamped parts which would cause undesired shear loads to the bolts. The residual clamp force FKR generates a surface pressure at the clamped interface which is utilized for the transmission of transverse loads FQ by static friction. The surface pressure is assumed to be limited to the deformation cone whose size depends on the geometry of the bolted joint and can be derived according to [VDI 2230]. pre-stressed deformation cone

FQ

FQ

Figure 3.6: Shear force transmission of a single bolted joint

The cross sectional area of the deformation cone at the interface determines the magnitude of the surface pressure. The deformation cone of the single bolted joint of


19

Figure 3.6 is sketched in Figure 3.7 where the ring-shaped clamped interface area can be observed. bolt head contact area

bolt hole assumed deformation cone

clamped interface area bolt nut contact area

Figure 3.7: Deformation cone of a single bolted joint

In case of uniformly distributed surface pressure at the clamped interface, transverse loads can be transmitted by static friction until the friction force limit is reached so that undesired relative motion of the clamped parts can occur (sliding friction). In case of harmonic transverse loads (whose magnitudes exceed the friction force limit), a hysteresis loop is being produced in the force-displacement diagram with periodically alternating states of static friction and sliding friction, see Figure 3.8. The area enclosed by the hysteresis loop is a measure for the energy dissipated by friction and thus is a measure for the amount of damping produced by the joint. microslip effect

F

k SS

sliding friction

k SP + k SS

static friction

u static friction sliding friction

Figure 3.8: Hysteresis loop in case of harmonic shear force transmission with alternating states of static friction and sliding friction

The stiffness of the bolted joint in case of static friction is determined by: • the shear stiffness of the clamped parts, • the interaction of asperities of the contacting surfaces, • the shear stiffness of the bolt. The total shear stiffness of the clamped parts is denoted by k SP and combines the first two items of the above list. The stiffness for sliding friction is only governed by the shear stiffness k SS of the bolt. The transition from static friction to sliding friction depends on the static coefficient of friction which is a function of the contact surface conditions like: • surface roughness, • lubrication conditions, • material combination of the clamped parts.

20


The distribution of the surface pressure at the clamped interface is generally uneven. Nonetheless, it can be assumed that the surface pressure is distributed symmetrically around the bolt hole with the maximum pressure underneath the bolt head. Furthermore, it can be assumed that the surface pressure decreases with increasing distance from the bolt center line. The distribution of the surface pressure becomes more uneven: • the larger the extent of the interface compared to the height of the clamped parts (finite bending stiffness of clamped parts), • the closer the distance between load introduction point and interface, • in case of bending moments caused by non-symmetric axial loads. Uneven surface pressure distribution can cause different local friction states at the clamped interface, i.e. one part of the clamped interface is in a state of static friction, while other parts of the interface have already exceeded their local friction force limit due to low local surface pressure and are thus in a state of sliding friction. The simultaneous existence of two different friction states at the clamped interface is called microslip effect and produces damping while gross slip motion (macroslip) between the clamped parts does not necessarily occur, [Lenz 1997]. Microslip produces a relatively smooth transition from static friction to sliding friction and is indicated by a “fillet radius” in the corresponding corners of the friction hysteresis loop in the force-displacement diagram, see Figure 3.8. Many factors which may also influence the force-displacement conditions of single bolted joints have not been discussed here. For example, operational loads are usually not introduced directly underneath the bolt head and directly underneath the nut. Forces are rather introduced at other positions so that the stress resultants in the vicinity of the bolt joint must be considered. These are considered to be introduced somewhere inside the clamped parts. Consequently, the shear stiffness of the bolted joint can change which has not been considered here. The influence of load introduction and other influence factors are discussed in the engineering guideline [VDI 2230] and shall not be reviewed here.

3.1.2

Eccentrically Loaded Bolted Joints and Flanged Joints

The concentrically clamped single bolted joint is suited to discuss the principal mechanical behavior of bolted joints but is seldom encountered in engineering structures due to the simplified loading assumptions. Eccentrically loaded bolted joints like the one shown in Figure 3.9 are a more general type of bolted joints. The analysis of such joints is discussed in the engineering guideline [VDI 2230] and an outline will be given here to point out the influence of geometry and loading conditions on the force-displacement diagram.


21

FA surface pressure distribution

pre-stressed deformation cone

FA Figure 3.9: Axial load transmission of eccentrically loaded bolted joint

Eccentrically applied axial loads result in an uneven surface pressure distribution at the clamped interface, see Figure 3.9. One-sided gap opening of the interface can occur if the axial load is high enough so that the surface pressure vanishes at the respective boundary of the interface. In this case, the force-displacement relation can no longer be derived from the bolted joint loads diagram, because the bending flexibility of the clamped parts and the eccentricity of the applied loads become dominant factors which can no longer be disregarded. Similar conditions can be observed for bolted flange joints. The deformation cone of the bolted flange joint shown in Figure 3.10 does not reach the base of the flange, i.e. no surface pressure is available there for the transmission of axial loads. Consequently, a gap opening can occur even for very small tensile loads. This effect can either be reduced by increasing the flange thickness (and thereby increasing the size of the deformation cone), or by reducing the bolt eccentricity. Such a bolted flange joint is shown in Figure 3.11 where it can be seen that the extension of the surface pressure reaches the flange base. Other factors which also affect the forcedisplacement relation are e.g. the bending stiffness of the flanges and the plates, and in case of excessively high tensile loads also the bending stiffness of the bolt. flange

base of the flange

FA

plate

FA

Figure 3.10: Flexible bolted flange joint without surface pressure at the flange base

22


flange

plate

FA

FA

Figure 3.11: Stiff bolted flange joint with surface pressure at the flange base

3.2

Deformation Behavior of Bolted Flange Joints

It can be concluded from the preceding text that the derivation of a meaningful stiffness representation of eccentrically loaded bolted joints is a difficult task and is influenced by many factors. In the following section, bolted flange joints shall be investigated in more detail. A simplified but meaningful FE representation for bolted flange joints shall be derived which takes into account the fundamental linear and non-linear behavior and which is suitable for the inclusion in large order dynamic models for efficient non-linear dynamic analysis. Engineering guidelines and standards such as [VDI 2230] and [DIN 2505] are a useful basis for the analysis of bolted flange joints in terms of mechanical strength, however, they are less useful when it comes to the determination of joint stiffness and damping properties to be used in finite element modeling. Nonetheless, it is believed that the fundamental linear and non-linear stiffness characteristic of the single bolted joint (i.e. pre-loaded bilinear stiffness for axial load transmission and friction characteristic for transverse load transmission) is retained for bolted flange joints as well. However, their stiffness and damping properties generally depend on the properties of the single bolted joint and on the geometry of the flanges. Stiffness and damping properties of bolted flange joints can therefore not be specified in a closed form. The basic idea which is followed here is to derive a simplified joint model for dynamic analysis of large order models from load path analysis of a refined model of a joint subsection. Load path analysis can be performed in terms of static analysis by using appropriate boundary conditions and loading conditions. The deformation behavior and the load path conditions can be analyzed in detail when a 3D model of a joint subsection is used, for example, when a single bolted joint is considered detached from the rest of the structure. The results of such a local joint analysis can be used to derive a simplified joint model suitable for the inclusion in large order FE models while the underlying linear stiffness and the fundamental non-linear stiffness and damping properties are sufficiently represented. The derivation of a modeling approach for bolted flange joints will be discussed in the following on the Cylindrical Test Structure (CTS) shown in Figure 3.12. The CTS was designed and manufactured in the course of the European research project CERES. It was used as a vehicle for validating non-linear test methods and nonlinear joint modeling approaches. The CTS is assembled of two nominal identical components. Essentially, each component is a thin-walled cylindrical shell (shell thickness 2mm, diameter ca. 250mm, length ca. 250mm, weight ca. 3.65kg)


23

manufactured of medium quality steel (ST-52). The CTS components are connected by a bolted flange joint with variable flange height in circumferential direction, see Figure 3.12. This type of joint is called scalloped flange joint and is representative of typical aero-engine casing joints. 250mm

flange, 16 bolts, M6, thickness 5mm

250mm

250mm

scalloped flange, 12 bolts, M4, thickness 2mm flange, 16 bolts, M6, thickness 5mm

Figure 3.12: CTS assembly and details of bolted scalloped flange joint

A simplified but physically meaningful FE representation shall be derived for the bolted scalloped flange joint with the objective to perform efficient non-linear analysis using the CTS assembly model shown in Figure 3.13. In addition, the simplified joint model should represent both, the underlying linear and the fundamental non-linear joint stiffness and damping properties. component 2

flanges represented by circumferential beam rings

FE representation of the joint?

component 1

Figure 3.13: FE model of the CTS for dynamic analysis

According to the strategy for the derivation of simplified joint models mentioned above, a detailed 3D FE model was generated based on the geometry of a subsection of the scalloped flange joint. Boundary conditions for cyclic symmetry were applied to simulate realistic boundary conditions. A distributed (static) tensile load was applied to the joint subsection to simulate the joint subsection loading conditions in case of global bending of the CTS, see Figure 3.14.

24


distributed load

boundary conditions for cyclic symmetry distributed load

boundary conditions for cyclic symmetry

single bolted joint 30°

axis of symmetry of the complete structure

Figure 3.14: Geometry model of a segment of the bolted scalloped flange joint of the CTS with boundary conditions and loading conditions

The contact conditions of the flanges were modeled using contact elements at the top and at the base of the flanges. The surface pressure distribution in the immediate surrounding of the bolt was modeled by pre-loaded contact elements distributed over the area spanned by the assumed deformation cone (according to [VDI 2230]). Figure 3.15 shows some contact modeling details. contact elements at flange top

pre-loaded contact elements

bolt assumed deformation cone

contact elements at flange base

Figure 3.15: Modeling details of contact conditions at bolted flange joint

The pre-loads for the discrete contact elements at the clamped interface were calculated from integrating the assumed surface pressure distribution over a representative area in the vicinity of each contact element. The distribution of the discrete pre-loaded contact elements in the vicinity of the bolt can be observed in Figure 3.16 together with the assumed surface pressure distribution as a function of the radial distance from the bolt center line. It can be seen that the surface pressure


25

pre-loaded contact element

surface pressure

was assumed to be constant underneath the bolt head and linearly decreasing with increasing distance from the bolt center line. The surface pressure vanishes at the outer boundary of the deformation cone. The magnitude of the bolt pre-load, which determines the magnitude of the surface pressure, was calculated in accordance with [VDI 2230] and is a function of the bolt torque and the friction conditions between the bolt and the clamped parts. bolt radius

assumed surface pressure distribution

distance from bolt center line

bolt head radius

surface pressure area represented by a single preloaded contact element Figure 3.16: Clamped interface area of bolted joint represented by pre-loaded contact elements

The deformation behavior of the CTS joint due to the distributed tensile load can be observed from Figure 3.17. Gap opening occurs at the flange base while contact is retained in the region between the bolt and the top of the flange. Furthermore, it can be seen that the load is passed from one component through the bolted joint into the other component. Bending of the cylindrical casing shells is activated by the bolt eccentricity and it can be observed that there is also significant bending deformation in the flanges in the region between the bolt and the flange base. These effects should be taken into account when deriving a simplified joint model suitable for the inclusion in the dynamic model of the CTS shown in Figure 3.13.

casing shell flange

Figure 3.17: Deformation of 3D FE model of flange segment due to distributed tensile load

It has to be mentioned that shear load transmission was not investigated with the 3D joint model. Such transverse loads are transmitted by friction at the clamped

26


interface. Friction effects are also included in the contact modeling used in the 3D model, since the contact elements act as friction elements in transverse directions. However, these effects would not be activated in the 3D model used, because the bolt was modeled in a way that it perfectly fits the bolt hole without clearance. This prevents the occurrence of macroslip. The FE model of the CTS shown in Figure 3.13 uses a shell element representation for the cylindrical casings. Different simplified joint models can be derived with respect to different modeling approaches of the flanges. For example, Figure 3.18 shows a sketch of a simplified joint model for the important practical case of a beam element representation of the flanges (i.e. flanges are represented by circumferential rings of beam elements). In principle, there is no gap between the components so that the nodes of the shell elements of the adjacent components are coincident. For visualization purposes, however, the simplified joint model is sketched in an exploded view. The beam elements which represent the flanges are eccentrically connected to the nodes of the shell elements which represent the cylindrical casing shells. The load path conditions are modeled by rigid links which are connected by a hinge located at the level of the bolt center line. This allows for a gap opening at the flange base in case of tensile loading, while the bending stiffness of the casing shells is activated by the eccentricity of the hinge. Non-linear joint stiffness and damping effects can be introduced between the nodes at the flange base by a combination of appropriate linear and non-linear 2-DoF spring/damper elements. component 1

offset beam representing flange

component 2

hinge

additional node at bolt level

rigid link

rigid link

casing shells

combination of linear and non-linear spring/damper elements

Figure 3.18: Simplified joint model for beam element representation of flanges

Figure 3.19 shows a simplified joint model for the important practical case of a shell element representation of the flanges. The adjacent components are rigidly connected at the top of the flange and at the bolt center line level. A gap opening can occur at the base of the flange in case of tensile loading due to the bending stiffness of the shell elements representing the flanges. This simplified joint model can be supplemented with a combination of appropriate linear and non-linear 2-DoF spring/damper elements to account for the non-linear joint behavior.


component 1

27

component 2 rigid link

shell elements representing flange

casing shells

rigid link

combination of linear and non-linear spring/damper elements

Figure 3.19: Simplified joint model for shell element representation of flanges

The use of simplified joint models yields an overall assembled model whose substructures are still linear but connected in a non-linear way by only a few local non-linear 2-DoF elements introduced at the joints. Even though the effort for the generation of the detailed 3D model of the CTS joint subsection was quite high (especially modeling of the surface pressure at the interface by pre-loaded contact elements), it was still not possible to derive meaningful parameters from 3D model simulation results for the non-linear 2-DoF spring/damper elements to be used in the simplified joint models. Consequently, the properties (parameters) of the non-linear spring/damper elements must be adjusted based on experimental observations in order to obtain a predictive model. In a first step it is recommended to introduce only a single linear spring at the flange base to represent the underlying linear behavior. An almost rigid joint can then be simulated by assigning an arbitrarily high stiffness value to the underlying linear joint spring. In case of the CTS model for dynamic analysis shown in Figure 3.13, the connection of two components followed the modeling approach shown in Figure 3.18. Such simplified joint models were introduced at 12 equidistant positions on the circumference between the cylindrical casings according to the bolt positions of the scalloped flange joint. In principle, the strategy for simplified joint modeling discussed here on the CTS example is applicable to other types of joints as well. The basic steps to be followed are summarized below: • generation of a detailed model of a joint subsection, • investigation of the deformation behavior and the load path for a typical loading condition, • derivation of a simplified joint model which is suitable for the inclusion in a large order dynamic model while taking into account the specific load path conditions analyzed with the detailed model of a joint subsection, • supplementing the simplified joint model by appropriate linear and non-linear spring/damper elements between those DoFs which have significant relative displacement in case of typical loading conditions.

28

4. Non-Linear Response Analysis

Chapter 4: 4.1

Non-Linear Response Analysis

Review of Procedures for Linear Response Analysis

The motion of a linear system of finite elements is governed by a second order nonhomogeneous ordinary differential equation with constant coefficients: (4.1) [ M ]{u(t )} + [C ]{u(t )} + [ K ]{u(t )} = { f (t )} . In this equation, { f (t )} is a vector of time dependent excitation forces, {u (t )} , {u (t )} , and {u (t )} are the displacement-, velocity-, and acceleration vector, and [ K ] , [C ] , and [ M ] are the stiffness-, viscous damping-, and mass matrix. According to standard differential equation theory, the general solution of equation (4.1) is given by the superposition of the complementary solution {uc (t )} and the particular solution u p (t ) :

{

}

{u(t )} = {uc (t )} + {u p (t )} .

(4.2)

The complementary solution is a unique solution for the homogeneous equation of motion and must be adapted to the initial conditions of the system. The particular solution (also known as steady-state response) does not depend on initial conditions and persists after the complementary solution has decayed away. The characteristic of the particular solution depends on the type of the dynamic excitation function { f (t )} . It can be analyzed directly by considering the system in a state where the complementary solution has already decayed away. In the important case of harmonic excitation, it is reasonable to assume that the motion of the system will also be harmonic but amplified (or attenuated) and phase shifted with respect to the loading function. It is convenient to use complex notation for harmonic excitation, while only the real part is considered (the operator ℜ ( ) for the real part is often dropped for convenience). The following complex approach is used for the harmonic excitation and for the harmonic steady-state response:

{ f (t )} = ℜ ({ fˆ }e jΩt ) = { fˆre }cos Ωt − { fîm }sin Ωt ,

(4.3)

{u (t )} = ℜ ({uˆ}e ) = {uˆ }cos Ωt − {uˆ }sin Ωt .

(4.4)

jΩt

p

re

im

Here, Ω represents the angular frequency of harmonic excitation, j = −1 is the imaginary unit, fˆ = fˆre + j fîm and {uˆ} = {uˆre } + j {uîm } are vectors of complex excitation force amplitudes and complex steady-state displacement response amplitudes. Real and imaginary parts of the complex amplitudes represent the phase shift between excitation and response as indicated in equations (4.3) and (4.4). The steady-state velocity- and acceleration response are the derivatives of the steady-state displacement response:

{} { } { }

{u (t )} = {u (t )} = ℜ ( jΩ {uˆ}e ) = −Ω {uˆ }sin Ωt − Ω {uˆ }cos Ωt , {u (t )} = {u (t )} = ℜ ( −Ω {uˆ}e ) = −Ω {uˆ }cos Ωt + Ω {uˆ }sin Ωt . p

j Ωt

d dt

p

d2 dt 2

p

re

2

p

jΩt

2

im

2

re

im

(4.5) (4.6)


29

Inserting equations (4.3) to (4.6) into equation (4.1) yields the frequency domain representation of the equation of motion which is no longer a differential equation but a complex algebraic equation:

( −Ω [ M ] + jΩ [C ] + [ K ]) {uˆ} = { fˆ } . 2

(4.7)

[ Z ( Ω )]

The steady-state response amplitudes can be calculated by inverting the frequency dependent coefficient matrix [ Z (Ω)] which is called dynamic stiffness matrix or impedance matrix:

{uˆ} = [ Z (Ω)]

−1

{ fˆ }.

(4.8)

This is the direct approach for frequency response analysis, because the dynamic stiffness matrix must be inverted directly at each frequency of interest to calculate the response. Once the complex displacement response vector {uˆ} was calculated from equation (4.8), the time history of the particular displacement response u p (t ) can be obtained from equation (4.4). Time histories of velocity and acceleration response can be obtained from equations (4.5) and (4.6).

{

4.2

}

From Linear to Non-Linear

The motion of a non-linear system of finite elements is governed by a second order non-homogeneous ordinary differential equation with the coefficients being dependent on the yet unknown response:

[ M ]{u(t )} + ⎡⎣C ({u}, t )⎤⎦ {u(t )} + ⎡⎣ K ({u})⎤⎦ {u(t )} = { f (t )} .

(4.9)

Non-linear systems violate the principles of superposition and reciprocity. This means that the general response cannot be expressed by the superposition of the complementary solution and the particular solution as it was stated for linear systems in equation (4.2). Even though the superposition principle is no longer valid, the complementary non-linear solution and the particular non-linear solution still exist and can be analyzed independently. However, the general non-linear transient response cannot be obtained by the superposition of these solutions since they are no longer independent. The solution of equation (4.9) can involve considerable numerical effort. In case of non-linear dynamic analysis, most commercial finite element codes only provide numerical integration schemes for time domain response analysis for the calculation of the general non-linear response which contains both, the complementary and the particular solution. Procedures for non-linear frequency domain response analysis aim at calculating the particular solution only, i.e. the steady-state response of the non-linear system due to harmonic excitation. These approximate methods rely on assumptions which have to be checked (a priori or a posteriori) to justify their application. Consequently, non-linear frequency domain response analysis procedures are less widespread and cannot be considered as a standard in non-linear dynamic analysis. In the remainder of this chapter, different procedures for non-linear time domain and frequency domain response analysis will be reviewed and their inherent assumptions will be discussed.

30

4.3


Numerical Integration Schemes for Time Domain Response Analysis

Numerical integration schemes are implemented in most commercial finite element codes. They are commonly applied to compute the response of structures due to almost arbitrary transient excitation in a step-by-step integration of the non-linear equation of motion. Prominent examples for such analyses are crashworthiness, airbag deployment, driving dynamics simulations in the car industry, or containment test simulations for fan blade-off and bird strike in the aero-engine industry. Numerical integration schemes utilize the dynamic equilibrium among internal forces (element forces) and externally applied loads which must be satisfied at discrete times ti only. These discrete times are spaced by the time interval Δt , i.e. ti = i ⋅ Δt with i = 0,1,… , n . Different numerical integration schemes use different assumptions for the variation of displacements, velocities, and accelerations within each interval Δt . The form of these assumptions determines accuracy, stability, and cost of the solution procedure, [Bathe 1996]. Numerical integration schemes may be subdivided into explicit and implicit schemes. Explicit integration schemes calculate the response at time t + Δt based on the equilibrium conditions at time t (which is assumed to be fulfilled). Implicit integration schemes formulate the equilibrium conditions at time t + Δt to calculate the response at the same point in time. The most commonly applied explicit integration scheme is the Central Difference Method. It uses a central difference expression for acceleration and velocity in terms of displacement. The Central Difference Method is very efficient when using a diagonal mass matrix and when velocity dependent damping forces are negligible. The major drawback of the Central Difference Method is stability. It is only conditionally stable which requires that the time interval Δt must be sufficiently small for the method to be stable and accurate. The critical limit Δtcr for the time step size Δt depends on the highest natural frequency f n of the system:

Δt ≤ Δtcr = π 1f n .

(4.10)

Since f n can be very high if the system contains stiff elements, the critical time step size Δtcr can be excessively small which decreases the numerical efficiency. The most commonly applied implicit integration scheme is the Newmark Method. It assumes a constant-average acceleration within each time interval Δt . The constant acceleration is calculated from the weighted average of the accelerations at two subsequent times t and t + Δt . Two weighting factors are available to adjust the accuracy and stability of the method. It can be shown that the Newmark Method passes over into the trapezoidal rule of integration for a particular combination of the two weighting factors, [Bathe 1996]. The Newmark Method (like other implicit integration schemes as well) is unconditionally stable, i.e. the solution is stable regardless of the time step size Δt . This means that the integration scheme is not bounded to the highest natural frequency of the system. However, the response can only be calculated accurately up to a certain frequency limit which depends on the time step size Δt . The response at frequencies above this frequency limit will suffer from time period elongations and/or amplitude attenuations. The time step size for implicit integration schemes thus has to be selected based on the spectral content of the response to be calculated accurately, and of course based on the spectral content of the excitation. A rule of thumb for estimating the time step size is:

Δt ≈ 80 f1max ,

(4.11)


31

which ensures that the spectral content of the response is analyzed accurately up to four times the maximum frequency f max when using the Newmark Method. This rule is based on the assumption that a single period of vibration must be subdivided into 20 intervals to obtain accurate results, [Bathe 1996]. Stability, in this respect, shall not be mistaken as accuracy! If the excitation force has spectral contents above f max it may well happen that the solution converges towards wrong values. This can be traced back to the erroneous analysis of the spectral response portions above f max which may appear attenuated and phase shifted. In case of non-linear systems, the internal inertia-, damping-, and elastic restoring forces depend in a non-linear way on the displacement and velocity response. It is thus necessary to iterate in the solution of the equation of motion (4.9) at each time ti to enforce dynamic equilibrium. The classical Newton-Raphson scheme and its modifications are frequently utilized for this purpose, [Bathe 1996], [Nastran 1992]. The convergence criteria for the termination of the equilibrium iteration at each time ti largely affect efficiency and accuracy of the analysis. In addition, non-linear integration schemes often use adaptive time stepping, i.e. the time interval Δt is adjusted based on the performance of the equilibrium iterations. The systems analyzed by numerical integration schemes can be highly non-linear. For example, non-linear material characteristics, large deformations, and variable contact conditions with progressing analysis are often considered in non-linear transient response analysis. However, a smaller time step size than stated in equation (4.11) might be necessary to achieve convergence in a non-linear response analysis. Despite these advantages, the calculation of the non-linear steady-state response for the important case of harmonic excitation can be expensive in terms of computation time. The numerical integration schemes calculate the general non-linear response which contains both, the decaying complementary solution and the steady-state response. The decaying time of the complementary solution can be considerable, especially in case of lightly damped structures. This can significantly degrade the efficiency of the numerical integration schemes for the computation of steady-state responses of non-linear systems, especially when taking into account that a small time step size must be used and that several equilibrium iterations might be necessary at each time step. Another aspect to be mentioned here is the amount of data usually produced by the non-linear integration schemes. A huge amount of data has to be processed after response analysis, from which characteristic information about the influence of nonlinearities on the dynamic response must be extracted. Therefore, raw non-linear time domain responses are less well suited for model validation from an updating point of view.

32

4.4


Different Procedures for Frequency Domain Response Analysis

Frequency response analysis procedures are aimed at directly calculating the steadystate response in case of harmonic excitation. For linear systems, this was achieved by inserting the assumptions for harmonic excitation and harmonic response according to equations (4.3) and (4.4) into the equation of motion (4.1). This yields the complex algebraic frequency domain equation of motion (4.7) which can be solved for the unknown complex displacement response magnitudes. Even though the procedure is straightforward for linear systems, the derivation of the non-linear frequency domain equation of motion can generally not be achieved unless certain assumptions are made for the behavior of the system. The basic problem encountered in the development of non-linear frequency domain methods is the violation of the principle of superposition which causes that the assumption for the response to be of the same form as the excitation is no longer valid. This is indicated in equation (4.12), where it can be seen that harmonic excitation produces a non-harmonic steady-state response when propagated through a non-linear system:

{ f (t )} = ℜ ({ fˆ }e jΩt )

Non − Linear ⎯⎯⎯⎯→ System

{u (t )} ≠ ℜ ({uˆ}e ) . j Ωt

p

(4.12)

The reason for the response to have a different form than the excitation can be traced back to the system matrices being non-linearly dependent on the response. Consequently, the non-linear differential equation of motion for harmonic excitation cannot be solved for the steady-state response in the same way as it was done for linear systems. Some non-linear systems produce an irregularly oscillating response in case of harmonic excitation which may in addition be highly sensitive to the initial conditions of the system. Such systems are called chaotic systems according to [Magnus 1997] and shall not be considered here. Instead, emphasis shall be given to non-linear systems which produce a periodic response in case of harmonic excitation. Such a periodic response can be approximated by a discrete number of harmonics in the sense of a Fourier series decomposition. The non-linear system shall be called weakly non-linear, if the fundamental harmonic (i.e. in the frequency of excitation) dominates the whole periodic response. The procedures for frequency domain response analysis discussed in the following rely on weakly non-linear behavior as defined in this paragraph.

4.4.1

Single-Harmonic Balance Method

A rather simple approach for the steady-state forced response analysis of non-linear systems is the Single-Harmonic Balance Method (SHBM) initially introduced by Krylov and Bogoljubov, [Krylov 1947]. The SHBM shall be discussed on a single degree of freedom (DoF) oscillator before applying it to the general case of multidegree of freedom oscillators. Non-linear single DoF systems can be described by a second order differential equation with the linear and non-linear terms separated:

mu(t ) + cu(t ) + ku(t ) + f R (u, u, t ) = f (t ) . linear

non − linear

(4.13)


33

In this equation, m , c , and k represent the mass, viscous damping, and stiffness of the system. u(t ) , u(t ) , and u(t ) are the displacement-, velocity-, and acceleration response. f R (u, u, t ) is a general non-linear forcing function which comprises the nonlinear elastic restoring force (conservative) and non-linear damping force (nonconservative) of the system, for example, f R (u, u, t ) = knl u 3 (t ) in case of a cubic hardening spring with the non-linear spring constant knl . For the sake of simplicity the general non-linear function f R (u, u, t ) shall be called non-linear restoring force, even though it may also comprise non-conservative damping forces. In the development of a frequency domain response analysis procedure the frequency domain non-linear equation of motion is sought, which allows for the direct analysis of the non-linear steady-state response. The basic assumption of the SHBM is that harmonic excitation with the angular frequency Ω approximately produces a harmonic response in the same frequency:

(

f (t ) = ℜ fˆ e jΩt

)

Non − Linear ⎯⎯⎯⎯→ u p (t ) ≈ ℜ ( uˆ e jΩt ) . System

(4.14)

The complex quantities fˆ = fˆre + j fîm and uˆ = uˆre + j uîm are the excitation force amplitude and the steady-state displacement response amplitude. The following relations exist among displacement, velocity, and acceleration:

u p (t ) = ℜ ( uˆ e jΩt ) ,

(4.15)

u p (t ) = ℜ ( jΩuˆ e jΩt ) ,

(4.16)

u p (t ) = ℜ ( −Ω 2uˆ e jΩt ) .

(4.17)

Using the harmonic response assumption for the evaluation of the non-linear restoring force f R (u, u, t ) yields a periodic function of time which can be expressed by a complex Fourier series:

(

)

∞ ⎛ ∞ ⎞ f R (u, u, t ) = ℜ ⎜ ∑ fˆRn e jnΩt ⎟ = fˆR 0 + ∑ ℜ fˆRn e jnΩt . n =1 ⎝ n =0 ⎠

(4.18)

The evaluation of the periodic non-linear restoring force function is sketched in Figure 4.1. The given harmonic response u p (t ) is projected onto the non-linear forcedisplacement diagram f R (u ) of a non-linear spring to produce a periodic function of time f R (u, t ) for the non-linear restoring force.

34


fR(u)

fR(u,t)

force-displacement diagram u

t

periodic non-linear restoring force

up(t)

imposed harmonic response t Figure 4.1: Periodic restoring force function in case of harmonic response

The complex force amplitudes fˆR 0 = fˆR 0,re + j 0 and fˆRn = fˆRn ,re + j fˆRn ,im in equation (4.18) can be calculated according to the following equations, where the substitution ϕ = Ωt is used for the phase angle to eliminate the time dependency so that the integration can be performed over one period of vibration 0 ≤ ϕ ≤ 2π : 2π

1 fˆR 0 = 2π 1 fˆRn =

∫

(4.19)

f R (u, u, ϕ ) e − jnϕ dϕ .

(4.20)

0

2π

∫

π

f R (u , u, ϕ ) dϕ ,

0

When using real notation, equation (4.18) can be written as: ∞

(

)

f R (u, u, t ) = fˆR 0 + ∑ fˆRn ,re cos nΩt − fˆRn ,im sin nΩt , n =1

(4.21)

where the force amplitudes can be calculated according to the well known equations for the Fourier coefficients with the substitution ϕ = Ωt for the phase angle: 2π

1 fˆR 0 = 2π 1 fˆRn ,re =

2π

π

∫

1 fˆRn ,im = −

π

∫

f R (u , u, ϕ ) dϕ ,

(4.22)

f R (u, u, ϕ ) cos nϕ dϕ ,

(4.23)

0

0

2π

∫

f R (u, u, ϕ ) sin nϕ dϕ .

(4.24)

0

The non-linear frequency domain equation of motion can be derived by considering only the fundamental term (i.e. n = 1 ) of the Fourier series of the non-linear restoring force which corresponds to the angular frequency of excitation:

(

)

f R (u, u, t ) ≈ ℜ fˆR1 e jΩt .

(4.25)


35

When neglecting all other terms of the Fourier series of f R (u, u, t ) the restoring force can be expressed by yet unknown equivalent stiffness and equivalent viscous damping parameters:

f R (u, u, t ) ≈ kequ(t ) + cequ(t ) .

(4.26)

Inserting the assumptions for harmonic response into equation (4.26) yields:

(

)

f R (u, u, t ) ≈ ℜ ( keq + jΩceq ) uˆ e jΩt .

(4.27)

The equivalent parameters keq and ceq can be determined from a comparison of the coefficients of equations (4.25) and (4.27). Note that both, the restoring force amplitude fˆR1 and the displacement amplitude uˆ are complex quantities (the operator ℑ ( ) denotes the imaginary part of a complex quantity):

⎛ fˆ ⎞ fˆ uˆ + fˆR1,imuîm , keq (uˆ ) = ℜ ⎜ R1 ⎟ = R1,re re2 2 uˆre + uîm ⎝ uˆ ⎠ ceq (uˆ ) =

1 ⎛ fˆR1 ⎞ 1 ⎛ fˆR1,imuˆre − fˆR1,reuîm ⎞ ℑ⎜ ⎟⎟ . ⎟= ⎜ 2 uˆre2 + uîm Ω ⎝ uˆ ⎠ Ω ⎝⎜ ⎠

(4.28)

(4.29)

It can be seen from equations (4.28) and (4.29) that the equivalent stiffness and viscous damping parameters are functions of the complex displacement amplitude uˆ which is unknown at the beginning of the analysis. The calculation of the equivalent parameters keq (uˆ ) and ceq (uˆ ) thus requires an estimate for the response amplitude (e.g. linear response amplitude) to start a subsequent equilibrium iteration. Better estimates for the equivalent stiffness and damping parameters are obtained which can be used in turn to calculate better estimates of the true response amplitude. Finally, the iteration converges towards the true response amplitude and thus towards the true equivalent non-linear parameters (provided that the initial response estimate is not too far away from the true non-linear response). When introducing the response magnitude dependent equivalent non-linear parameters into the non-linear equation of motion the non-linear frequency domain equation of motion is obtained (complex algebraic equation):

( −Ω m + jΩ ( c + c 2

eq

)

(uˆ ) ) + k + keq (uˆ ) uˆ = fˆ .

(4.30)

Even though the SHBM is frequently referred to as a linearization method, it can be seen from equation (4.30) that the frequency domain equation of motion is still nonlinear and has to be solved iteratively at each frequency of interest. Nonetheless, the application of the SHBM allows for the elimination of time dependency at the cost of response magnitude dependence. Standard Newton-Raphson iteration can be applied for equilibrium iteration in nonlinear frequency response analysis. The quantities which must be handled during the iteration are complex in this case. It was stated above that the linear response can be used as a first estimate for the non-linear response to start the iteration. However, when approaching resonance, the non-linear response can deviate significantly from the linear response which may cause the iteration scheme to diverge. It is therefore recommended to start the response analysis at a frequency point far away from a resonance where the linear and the non-linear response are expected to almost coincide. It is quite easy to obtain a converged non-linear response far away from a resonance and in most cases only two iterations are required. When approaching resonance, the initial estimate for the non-linear

36


response can be extrapolated from the non-linear responses obtained at the last converged frequency points. Cubic spline extrapolation was found to be efficient for this purpose. Even though the calculation of equivalent non-linear parameters for non-linear elements according to the Harmonic Balance method seems to be quite complicated, it should be noted that the equivalent non-linear parameters can often be derived in a closed form by solving equations (4.20), (4.28), and (4.29) analytically. This can be done with reasonable effort for many different types of non-linearities encountered in engineering practice for which the non-linear force displacement diagram is known. In addition, the SHBM is also known as the Describing Function Method in control engineering and is applied there for equivalent linearization of non-linear electrical subsystems. For example, [Gelb 1968] is an excellent reference for the Describing Function Method and contains a comprehensive list of closed form analytical solutions for the equivalent non-linear parameters for the most common types of non-linearities. The SHBM for single DoF systems presented here uses complex notation for the harmonic response and loading functions. An approach based on real notation was published in [Böswald 2004a], [Böswald 2004b], and [Böswald 2005a] but shall not be repeated here. The extension of the SHBM to multi-DoF systems is relatively simple when a restriction to systems with local non-linearities is applied, i.e. structures assembled of linear substructures which are coupled in a non-linear way by appropriate 2-DoF elements. In this case, the non-linear equation of motion is:

(−Ω [ M ] + jΩ ([C ] + ⎡⎣C 2

eq

({uˆ})⎤⎦ ) + [ K ] + ⎡⎣ K ({uˆ})⎤⎦ ){uˆ} = { fˆ } . eq

(4.31)

In this equation, the response dependent equivalent viscous damping matrix ⎡Ceq ({uˆ}) ⎤ and the response dependent equivalent stiffness matrix ⎡ K eq ({uˆ})⎤ can be ⎣ ⎦ ⎣ ⎦ calculated from the summation of all non-linear equivalent element matrices transformed to global degrees of freedom by so-called coincidence matrices: E

⎡Ceq ({uˆ})⎤ = ∑ [Te ] ⎡⎣Ceq ,e ( Δuê ) ⎤⎦ [Te ] , ⎣ ⎦ T

(4.32)

e =1 E

⎡ K eq ({uˆ})⎤ = ∑ [Te ] ⎡⎣ K eq ,e ( Δuê ) ⎤⎦ [Te ] . ⎣ ⎦ T

(4.33)

e =1

The coincidence matrices [Te ] contain the coincidence information among the two local element DoFs {ue } and the global DoFs {u} of the assembled system:

{ue } = [Te ]{u} .

(4.34)

The non-linear element matrices are simple 2-DoF element matrices with a nonlinear factor depending on the relative response between the two element DoFs. In case of a non-linear element defined between the DoFs ui and u j , the equivalent non-linear element matrices can be written as:

⎡ 1 −1⎤ ⎡⎣Ceq ( Δuîj ) ⎤⎦ = ceq ( Δuîj ) ⎢ ⎥, ⎣ −1 1 ⎦

(4.35)

⎡ 1 −1⎤ ⎡⎣ K eq ( Δuîj ) ⎤⎦ = keq ( Δuîj ) ⎢ ⎥, ⎣ −1 1 ⎦

(4.36)


37

with the complex relative response magnitude:

Δuîj = uˆ j − uî .

(4.37)

The equivalent non-linear stiffness and damping parameters keq ( Δuîj ) and ceq ( Δuîj ) do not purely depend on the magnitude of the relative displacement Δuîj . They also depend on the physical non-linear parameters which determine the non-linear restoring force function f R (u, u, t ) . The equivalent parameters can be calculated according to equations (4.20), (4.28), and (4.29). If non-linear elements with more than two DoFs are considered it is no longer possible to have one common response dependent factor for each element matrix. For example, in case of a 4-DoF beam element, 6 different relative responses exist and in case of a 4-noded plane stress membrane element with 8 DoFs, altogether 28 different relative responses exist. For such non-linear elements it would be best to formulate a non-linear material law so that an equivalent non-linear Young’s modulus can be derived which would depend on the magnitude of the von Mises strain. This approach shall not be further investigated here. The application of the SHBM shall briefly be discussed on a cantilever beam model with a non-linear joint as it is sketched in Figure 4.2. This non-linear beam model will be used later on as an analytical example for non-linear updating. A detailed discussion of the properties of the cantilever beam model is therefore postponed until then. The underlying linear MSC.Nastran model can be found in the Appendix. linear grounded rotational spring

non-linear rotational spring and damper

F

hinge hinged support

component 1

component 2

tip mass

Figure 4.2: Cantilever beam with non-linear joint

The non-linear joint between the two beam components consists of a cubic “softening” spring (i.e. decreasing stiffness with increasing response amplitudes) and a quadratic “hardening” damper (i.e. increasing damper constant with increasing response amplitudes). The following non-linear frequency response functions (FRFs) were obtained at the driving point for different levels of constant excitation force amplitudes, see Figure 4.3. The frequency range was situated around the resonance of the first bending mode.

38


Figure 4.3: Response dependent FRFs of the cantilever beam with the non-linear joint

The non-linear FRFs shown in Figure 4.3 were obtained by iteratively solving the non-linear frequency domain equation of motion (4.31) for different levels of constant excitation force amplitude. This was done by inverting the non-linear dynamic stiffness matrix (impedance matrix) at each frequency point of interest (direct approach for non-linear frequency response analysis). It can be observed that the distortion of the FRFs increases with increasing excitation force levels (and hence increasing response levels). The softening stiffness characteristic is indicated by the shift of the resonance peak towards lower frequencies. The hardening damping characteristic can be observed in the resonance peak attenuation with increasing excitation force levels. A detailed discussion of the properties of non-linear fundamental harmonic FRFs shall be postponed until the next chapter. The FRFs shown in Figure 4.3 shall only illustrate the principal appearance of response dependent fundamental harmonic FRFs calculated by using the SHBM.

4.4.2

Multi-Harmonic Balance Method

The Multi-Harmonic Balance Method (MHBM) was introduced by Pierre, Ferri, and Dowell, [Pierre 1985]. They analyzed the steady-state time domain response of a friction damped system due to harmonic excitation by using a frequency domain method for the calculation of the magnitudes and phase angles of multiple response harmonics. The theory of the MHBM presented here is based on the formulation published by Kuran and Özgüven [Kuran 1996]. It will be discussed on a non-linear single degree of freedom oscillator according to equation (4.13) and subsequently on multi degree of freedom systems. In contrast to the SHBM, the MHBM assumes that a non-linear system produces a periodic response when excited harmonically (no restriction to the fundamental harmonic response):

(

f (t ) = ℜ fˆ e jΩt

)

⎛ ∞ Non − Linear jnΩt ⎞ ( ) ⎯⎯⎯⎯→ ≈ ℜ u t ⎜ ∑ uˆn e ⎟ . System ⎝ n =0 ⎠

(4.38)


39

The complex quantities fˆ = fˆre + j fîm and uˆn = uˆn ,re + j uˆn ,im are the excitation force amplitude and the displacement response amplitudes of the different harmonics. Since the response is periodic, the following relations exist among displacement, velocity, and acceleration (all expressed by complex Fourier series):

⎛ ∞ ⎞ u(t ) = ℜ ⎜ ∑ uˆn e jnΩt ⎟ , ⎝ n =0 ⎠

(4.39)

⎛ ∞ ⎞ u(t ) = ℜ ⎜ ∑ jnΩuˆn e jnΩt ⎟ , ⎝ n =0 ⎠

(4.40)

⎛ ∞ ⎞ u(t ) = ℜ ⎜ ∑ −( nΩ) 2 uˆn e jnΩt ⎟ . ⎝ n =0 ⎠

(4.41)

It is reasonable to assume that the non-linear restoring force is also a periodic function and can be expressed by a complex Fourier series:

(

)

∞ ⎛ ∞ ⎞ f R (u, u, t ) = ℜ ⎜ ∑ fˆRn e jnΩt ⎟ = fˆR 0 + ∑ ℜ fˆRn e jnΩt , (4.42) n =1 ⎝ n =0 ⎠ with the complex force amplitudes fˆRn = fˆRn ,re + j fˆRn ,im calculated according to equations (4.19) and (4.20). It was already stated for the SHBM that the calculation of the non-linear restoring force amplitudes fˆRn requires an estimate for the unknown response amplitudes uˆn . In case of periodic response and MHBM it is usually sufficient to make only an estimate for the fundamental harmonic response amplitude uˆ1 which then serves as a starting point for the subsequent response iteration and which finally converges towards the true multi-harmonic response amplitudes (provided that the initial response estimate is not too far away from the true response). It is not feasible in practice to consider an infinite number of terms in the Fourier decomposition of the response and of the non-linear restoring force. It is rather recommended to concentrate on only a few harmonics ( n = 1,… , N ), which should already yield a reasonably accurate approximation of the true non-linear steadystate response (Pierre, Ferri and Dowell found out that three harmonics are often sufficient, [Pierre 1985]). The MHBM yields N equations for the calculation of N harmonic response amplitudes:

fˆ n = 0 : kuˆ0 + fˆR 0 = 0 → uˆ0 = − R 0 k n = 1: n = 2,

,N :

( −Ω2m + jΩc + k ) uˆ1 + fˆR1 = fˆ ( −(nΩ) m + jnΩc + k ) uˆ 2

n

→ uˆ1 =

(4.43)

fˆ − fˆR1 −Ω 2m + jΩc + k

+ fˆRn = 0 → uˆn = −

(4.44)

fˆRn (4.45) −( nΩ) 2 m + jnΩc + k

It is obvious from equations (4.43) to (4.45) that the efficiency of the procedure degrades with increasing number of harmonics considered for analysis. Thus, there is a tradeoff between accuracy improvement and numerical efficiency which was already mentioned by Pierre, Ferri, and Dowell. The extension of the MHBM to MDoF system is straightforward. In essence, equations (4.43) to (4.45) have to be formulated for MDoF systems:

40


n = 0:

−1

{uˆ1} = ( −Ω2 [ M ] +

n = 1: n = 2,

{uˆ0 } = − [ K ]

,N :

{ fˆ }

jΩ [C ] + [ K ])

{uˆn } = − ( −(nΩ)2 [ M ] +

(4.46)

R0

−1

({ fˆ } − { fˆ }) R1

jnΩ [C ] + [ K ])

−1

{ }

{ fˆ } Rn

(4.47) (4.48)

The magnitudes fˆRn of the harmonics of the non-linear restoring force function are calculated according to equations (4.19) and (4.20) with the extension to vector quantities. Note that the substitution ϕ = Ωt for the phase angle is used here as well to eliminate time dependency for integration purposes:

{ }

1 fˆR 0 = 2π

{ }

1 fˆRn =

π

2π

∫ { f ({u},{u},ϕ )}dϕ , R

(4.49)

0

2π

∫ { f ({u},{u},ϕ )}e

− jnϕ

R

dϕ .

(4.50)

0

When the same restriction to systems with local non-linear 2-DoF elements already used in the development of the SHBM is also applied here for the MHBM, then the total non-linear restoring force vector f R ({u},{u}, t ) can be expressed by a summation which comprises the contributions of all non-linear elements e = 1,… , E transformed to global DoFs by using the coincidence matrices [Te ] :

{

}

{ f ({u},{u}, t )} = ∑[T ] { f ({u },{u }, t )}. E

R

e =1

T

e

R ,e

e

e

(4.51)

The coincidence matrices [Te ] were already derived in equation (4.34) for the SHBM. It can be observed from equations (4.46) to (4.48) that each harmonic considered for analysis requires the solution of a unique system of equations. This makes the application of the MHBM prohibitive for large order systems, especially when a relatively large number of harmonics is used to improve the accuracy. Therefore, the theoretical basis for the MHBM was presented here only for the sake of completeness, but the MHBM is not considered for efficient use in non-linear updating. The application of the MHBM shall briefly be presented on a bilinear single DoF oscillator with the following equation of motion:

mu(t ) + cu(t ) + k (u )u (t ) = f (t ) .

(4.52)

fR (u)

The mass-, stiffness-, and damping properties are as listed below: • m = 1 kg , • c = 0.04 Ns m , i.e. modal viscous damping ratio of 2% of the underlying linear •

•

system, ⎧ 5 N , ∀u ≤ 0 , i.e. compression regime stiffness is 5 times higher than k (u ) = ⎨ 3 m 1 N ⎩ 3 m , ∀u > 0 the tension regime stiffness N the equivalent stiffness according to the SHBM is keq = 12 ( 53 + 13 ) N m = 1 m ≠ keq (uˆ ) .


41

The following response was obtained by numerical integration of the non-linear equation of motion with the initial conditions being zero. Harmonic sine excitation in the resonance of the underlying linear system was used with unit excitation force amplitude.

Figure 4.4: Transient response of bilinear single degree of freedom oscillator (Ω=1s-1)

It can be observed from Figure 4.4 that the system is vibrating with an offset around the zero-displacement position. Such an offset usually occurs when the non-linear restoring force function deviates from an odd function, i.e. f R (u ) ≠ − f R ( −u ) , and can be interpreted as a shift of the position of static equilibrium due to the constant restoring force fˆR 0 generated by the non-linearity. fˆR 0 is the constant term of the Fourier series of the non-linear restoring force f R (u, u, t ) and can be considered as a static pre-load to the system. Since fˆR 0 is generally response dependent, the static pre-load of the system will increase when approaching a resonance and will decrease again when the resonance was passed. When using the MHBM, the steady-state response can be obtained directly and is compared with the transient response in Figure 4.5.

Figure 4.5: Comparison of non-linear time-domain responses (Ω=1s-1)

When comparing the steady-state response of the last vibration periods in more detail (see Figure 4.6), it can be observed that both, magnitude and phase are accurately represented by the MHBM solution. Five harmonics plus the constant term were used for response analysis. The corresponding magnitudes can be observed in the bar chart of Figure 4.6. Note that this bar chart comprises the displacement magnitudes. The acceleration magnitudes of the higher harmonics are

42


usually more significant and can be obtained from the displacement magnitudes by 2 multiplication with ( nΩ ) .

Figure 4.6: Comparison of steady-state response and magnitudes of different harmonics

4.4.3

Non-Linear Response Modification Theory

Analysis

Based

on

Structural

Structural Modification Theory (SMT) is a frequency response analysis tool which utilizes the FRFs of a linear system to analyze the response of the same system after structural modifications have been applied, [Klosterman 1971]. Possible modifications include the attachment of masses, springs, and dampers, or the introduction of additional boundary conditions. SMT is discussed in detail in [Brandon 1990] together with the theory of response sensitivity which will be discussed later on in the chapter dedicated to updating theory. It is stated (but not proved) in [Brandon 1990] that the response of non-linear systems with linear modification can also be analyzed using SMT. The more general case of structural modification of a linear system with a non-linear modification cannot be analyzed using standard SMT. An iterative SMT approach will be developed in the following, which is based on the SMT formulations given in [Liu 2000a], [Liu 2000b], and [Link 2004]. This approach shall be called Non-Linear Structural Modification Theory (NLSMT) and is suited for frequency response analysis of linear or non-linear systems with linear or non-linear modifications. It should be noted that the NLSMT approach to be presented is an extension of the classical SMT and is therefore only suited for the calculation of fundamental harmonic non-linear responses.

4.4.3.1 Review of Linear Structural Modification Theory (SMT) According to equation (4.8), the fundamental harmonic steady-state response {uˆ} of a structure can be obtained by direct inversion of the dynamic stiffness matrix [ Z (Ω)] (also known as impedance matrix):

{uˆ} = ( −Ω 2 [ M ] +

jΩ [C ] + [ K ])

−1

{ fˆ } = [Z (Ω)] { fˆ } = [ H (Ω)]{ fˆ } . −1

(4.53)

The complex frequency dependent matrix [ H (Ω)] is the inverse dynamic stiffness matrix and contains the frequency response functions in terms of displacement (cross-receptances and point-receptances). This matrix is called FRF matrix or receptance matrix. The frequency dependence of the FRF matrix and the impedance matrix will be dropped for convenience in the following equations, i.e. [ H ] = [ H (Ω) ] and [ Z ] = [ Z (Ω)] .


43

According to Figure 4.7, the modification of an initial structure is assumed to affect only a subset of all DoFs of the structure. Partitioning of equation (4.53) into DoFs with modifications (subscript m ) and without modifications (subscript u ) yields:

⎧⎪{uû } ⎫⎪ ⎡ [ H uu ] ⎨ uˆ ⎬ = ⎢ H ⎩⎪{ m }⎭⎪ ⎣⎢[ mu ]

[ H um ] ⎤ ⎪⎧{ f u }⎪⎫ . [ H mm ]⎥⎦⎥ ⎨⎪{ fˆm }⎬⎪ ˆ

⎩

(4.54)

⎭

unmodified DoFs, index u

structural modification

initial structure

modification DoFs, index m

Figure 4.7: Degree of freedom sets of the initial structure and the structural modification

Equation (4.54) is formulated for the initial structure without modification but can also be specified for the modified structure (superscript m ) after a structural modification has been applied:

{ }⎪⎫ ⎬. { }⎪⎭

m m ⎧{uûm }⎫ ⎡ ⎡ H uum ⎤ ⎡ H um ⎤⎦ ⎤ ⎧ fû ⎦ ⎣ ⎪ ⎪ ⎢⎣ ⎪ ⎥⎨ ⎨ m ⎬= m m ˆm ⎢ ˆ ⎡ ⎤ ⎡ ⎤ ⎪⎩{um }⎪⎭ ⎣ ⎣ H mu ⎦ ⎣ H mm ⎦ ⎥⎦ ⎪ f m ⎩

(4.55)

SMT aims at calculating the FRF matrix ⎡⎣ H m ⎤⎦ of the modified structure by using the FRF matrix [ H ] of the initial structure together with the dynamic stiffness matrix [ ΔZ ] of the structural modification. Note that such a structural modification matrix has non-zero entries only at the modification DoFs:

⎡[0] [0] ⎤ [ ΔZ ] = −Ω2 [ΔM ] + jΩ [ΔC ] + [ΔK ] = ⎢ 0 ΔZ ⎥ . ⎣⎢[ ] [ mm ]⎦⎥

(4.56)

The conditions of compatibility and equilibrium between the initial structure and the structural modification must be fulfilled for the calculation of the modified FRF matrix. The condition of equilibrium requires that the structural modification itself is in a state of dynamic equilibrium (upper case letters apply to the structural modification):

{Fˆ } = [ΔZ ]{Uˆ } . m

mm

m

(4.57)

44


The equilibrium of forces of the modified structure requires the summation of the forces of the initial structure and the forces of the structural modification at the modification DoFs:

{ fˆ } = { fˆ } + {Fˆ } . m m

m

(4.58)

m

The conditions of compatibility between the structural modification and the modified structure requires that the motion of the modification DoFs of the initial structure and the structural modification must be equal and identical to the motion of the modification DoFs of the modified structure:

{uˆm } = {Uˆ m } = {uˆmm } .

(4.59)

Equations (4.58) and (4.59) can be interpreted as a rigid attachment of a structural modification to the initial structure at the modification DoFs to build the modified structure. Inserting equations (4.58) and (4.59) into equation (4.57) yields:

{ fˆ } = { fˆ } − [ΔZ m

m m

mm

]{uˆm } .

(4.60)

Substituting {uˆm } in equation (4.60) by the second equation of (4.54) leads to:

{ fˆ } = { fˆ } − [ΔZ ] ([ H ]{ fˆ } + [ H ]{ fˆ }) , m

m m

mm

mu

u

mm

(4.61)

m

from which equal force terms can be collected:

{ fˆ } = ([ I ] + [ΔZ m

mm

][ H mm ])

−1

({ fˆ } − [ΔZ m m

mm

][ H mu ]{ fû }) .

(4.62)

According to the conditions of equilibrium, the forces and displacements at the unmodified DoFs of the initial and the modified structure keep no change before and after attaching the modification:

{ fˆ } = { fˆ } ,

(4.63)

{uû } = {uûm } .

(4.64)

m u

u

Introducing equation (4.63) into (4.62) and subsequently inserting equation (4.62) into (4.55) and using equation (4.64) yields the following equation system:

⎧{uûm }⎫ ⎡ [ H ] ⎪ ⎪ uu ⎨ m ⎬=⎢ ⎪⎩{uˆm }⎪⎭ ⎢⎣[ H mu ]

{ fu } [ H um ] ⎤ ⎪ ⎪ . (4.65) ⎥⎨ −1 [ H mm ]⎥⎦ ⎪([ I ] + [ΔZ mm ][ H mm ]) ({ fˆmm } − [ ΔZ mm ][ H mu ]{ fûm }) ⎪⎬ ⎧

⎫

ˆm

⎩

⎭

After some algebraic transformations, it can be written as:

⎧{uûm }⎫ ⎡ [ H ] − [ H ][ I ]−1 [ ΔZ ][ H ] uu um mm mu Δ ⎪ ⎪ ⎨ m ⎬=⎢ −1 ⎪⎩{uˆm }⎪⎭ ⎢⎣[ H mu ] − [ H mm ][ I Δ ] [ ΔZ mm ][ H mu ]

[ H um ][ I Δ ] ⎤ ⎪⎧{ f u }⎪⎫ ⎥⎨ ⎬, [ H mm ][ I Δ ]−1 ⎥⎦ ⎪{ fˆmm }⎪ ˆm

−1

⎩

(4.66)

⎭

⎡H m ⎤ ⎣ ⎦

with the abbreviation:

[ I Δ ] = [ I ] + [ ΔZ mm ][ H mm ] .

(4.67)


45

Equation (4.66) is formally identical with equation (4.55) so that the partitions of the FRF matrix of the modified system can be identified as:

⎡⎣ H uum ⎤⎦ = [ H uu ] − [ H um ] ([ I ] + [ ΔZ mm ][ H mm ])

−1

[ΔZ mm ][ H mu ] ,

m ⎡⎣ H um ⎤⎦ = [ H um ] ([ I ] + [ ΔZ mm ][ H mm ]) , −1

m ⎡⎣ H mu ⎤⎦ = [ H mu ] − [ H mm ] ([ I ] + [ ΔZ mm ][ H mm ])

−1

[ΔZ mm ][ H mu ] ,

m ⎡⎣ H mm ⎤⎦ = [ H mm ] ([ I ] + [ ΔZ mm ][ H mm ]) . −1

(4.68) (4.69) (4.70) (4.71)

It should be noted that the FRF matrix of the modified system given in equations (4.66) to (4.71) is symmetric, even though this property cannot be observed immediately. Equations (4.68) to (4.71) are directly applicable for frequency response analysis of linear systems with linear structural modifications. An important feature is that these equations are exact, regardless of the magnitude of the modification. This is one of the major advantages of SMT based methods over the classical sensitivity based modification approaches, which are accurate only in case of small modifications, [Brandon 1990]. It can be concluded from equations (4.68) to (4.71) that the FRF matrix of the initial system must be known a priori for the calculation of the modified FRF matrix. The initial FRF matrix must be available at all DoFs to which modifications shall be applied and at all DoFs which are considered for response analysis. This can involve quite an effort if the FRF matrix has to be measured experimentally. But even when SMT is applied to an analytical system, the calculation of the FRF matrix can become tedious if the number of modification DoFs is large. Furthermore, SMT requires the inversion of the complex matrix [ I Δ ] ∈ M ×M , where M is the number of modification DoFs. It is known from experience that the inversion of an experimental FRF matrix can involve problems like ill-conditioning and error-proneness, especially at resonances and anti-resonances. The inversion of an analytical FRF matrix does not have these limitations, unless the analytical FRF matrix shall be inverted exactly at a resonance frequency where rank deficiencies might occur. Nonetheless, SMT is efficient when the number of modification DoFs is small and consequently when the order of the matrix [ I Δ ] to be inverted is small, too. The FRF matrix [ H ] represents the inverse of the dynamic stiffness matrix [ Z ] . However, FRF matrices are often only available at relatively small sets of DoFs of large systems. In this case, the FRF matrix represents the inverse of a dynamic stiffness matrix which has been exactly condensed to this subset of DoFs. This is a very important feature, because this actually means that model reduction is not necessary for efficient frequency response analysis of large order systems when using SMT based response analysis procedures.

4.4.3.2 Non-Linear Structural Modification Theory (NLSMT) In case of non-linear structural modification (NLSMT), the initial system is considered to be the underlying linear system and the modification matrix [ ΔZ mm ] is considered as a structural modification (stiffness and damping) caused by the attachment of non-linear elements. This results in a response dependent structural modification matrix as indicated below:

46

4. Non-Linear Response Analysis Non − Linear System

Linear System

⇒ ⎡⎣ ΔZ mm ({u}, {u}, t )⎤⎦ .

[ ΔZ mm ]

(4.72)

SMT can be used to calculate the fundamental harmonic frequency response of the modified system. NLSMT is an extension of the classical SMT to non-linear systems and can also only be used to calculate the fundamental harmonic non-linear frequency response. Furthermore, the application of the Harmonic Balance method is an integral part of NLSMT. It is utilized to calculate equivalent non-linear stiffness and damping parameters for general types of non-linear structural modifications. Consequently, the equivalent non-linear structural modification matrix depends in a non-linear way on the response magnitude and contains the equivalent non-linear stiffness and damping matrix of the non-linear elements introduced at the modifications DoFs:

(

)

(

)

Harmonic ⎡ K eq , mm {uˆ m } ⎤ + jΩ ⎡Ceq , mm {uˆ m } ⎤ . ⎡ ⎤ Single Balance Method ⎣ ΔZ mm ({u}, {u}, t )⎦ ⎯⎯⎯⎯⎯→ ⎣ ⎦ ⎣ ⎦

(4.73)

({ })

⎡ ΔZ uˆ m ⎤⎥ ⎣⎢ mm ⎦

{ }

The response of the modified system uˆ m is unknown at the beginning of the analysis so that an estimate for the response is required to start the subsequent equilibrium iteration. This means that NLSMT is an iterative SMT approach for non-linear frequency response analysis. The NLSMT procedure can be summarized in four steps (superscript on the left hand side indicates the iteration step): 1. calculation of the linear response using the linear FRF matrix:

{}

⎡⎣ 0 H ⎤⎦ fˆ = { 0 uˆ}

(4.74)

2. evaluation of the response dependent equivalent non-linear structural modification matrix according to SHBM:

⎡⎣ 0 ΔZ mm ⎤⎦ = ⎡ ΔZ mm ⎣

({ uˆ})⎤⎦ 0

(4.75)

3. modification of the underlying linear system according to equations (4.68) to (4.71) by using the equivalent non-linear structural modification matrix:

(

)

⎡⎣ 1H m ⎤⎦ = ⎡ H m ⎡⎣ 0 ΔZ mm ⎤⎦ ⎤ ⎣ ⎦

(4.76)

4. loop of Newton-Raphson iterations ( i = 0,1, 2,… ) until the following out-ofbalance response vector i ε vanishes:

{ } { ε } = ⎡⎣ i

i +1

{}

H m ⎤⎦ fˆ − { i uˆ} → min.

(4.77)

A practical convergence criterion for the termination of the iteration loop is to check T i T 0 i 0 the ratio of the scalar products ε ε and ε ε :

{ }{ } { }{ } { ε } { ε } ≤ 10 . { ε} { ε} i

T

i

0

T

0

−9

(4.78)

Other criteria for the termination of equilibrium iterations can be found in [Bathe 1996]. It is worth noting that during the Newton-Raphson iteration an additional response is calculated in each iteration step so that a new equivalent nonlinear structural modification matrix has to be evaluated in each iteration step. It is


47

also worth noting that always the same underlying linear FRF matrix [ H ] is used for structural modification according to equations (4.68) to (4.71), but each time a different equivalent non-linear structural modification matrix ⎡⎣ i ΔZ mm ⎤⎦ is used. An interesting feature of SMT is that equations (4.68) to (4.71) are exact, even when the initial system is non-linear. Approximations are only introduced in the evaluation of the equivalent non-linear structural modification matrix ⎡⎣ i ΔZ mm ⎤⎦ according to SHBM, where the unknown response has to be estimated in the first step and is then iterated in the following steps. Numerical efficiency of the NLSMT can only be retained if the number of modification DoFs is kept small so that a relatively small complex matrix [ I Δ ] has to be inverted.

4.4.3.3 NLSMT for Unit Rank Modifications Important cases of structural modifications are so-called unit rank modifications. They shall be discussed next, because the resulting equations can be developed in closed form. Unit rank modifications include the attachment of a (linear or nonlinear) spring/damper element which shall be introduced between the two DoFs i and j of the initial system. The structural modification matrix for this type of modification can be expressed by the following dyadic product:

[ ΔZ ] = Δz {eij }{eij }

T

.

(4.79)

{ }

The location of the structural modification is described by the vector eij constructed from the difference between the i -th and the j -th unit vector:

⎧⎪ = − = e e e { } { ij } i { j } ⎨ ⎪⎩

i

j

+1

−1

which is

T

⎫⎪ ⎬ . ⎭⎪

(4.80)

The FRF matrix of the modified system can now be obtained by the following equation: T Δz ⎡⎣ H m ⎤⎦ = [ H ] − [ H ] eij }{eij } [ H ] . { 1 + Δz ( H ii + H jj − 2 H ij )

(4.81)

An analytical expression for a single element H klm of the modified FRF matrix (i.e. response at DoF k when the excitation was applied at DoF l ) can be derived by T exploiting the sparsity of the dyadic product eij eij :

H klm = H kl −

Δz 1 + Δz ( H ii + H jj

{ }{ } ( H − H )( H − 2H ) ki

kj

il

− H jl ) .

(4.82)

ij

When using this equation, it is straightforward to derive an expression for the l-th column of the modified FRF matrix:

Δz ( H il − H jl )

{H } = {H } − 1 + Δz m l

l

(H

ii

+ H jj − 2 H ij )

(

)

⋅ {H i } − {H j } .

(4.83)

Equations (4.81), (4.82), and (4.83) represent the SMT for linear systems in case of linear unit rank modifications. In case of non-linear modifications, the response dependent equivalent non-linear structural modification can be derived according to the SHBM:

48


⎡⎣ ΔZ ( Δuîj ) ⎤⎦ = ( keq ( Δuîj ) + jΩceq ( Δuîj ) ){eij }{eij } , T

(4.84)

Δz ( Δuîj )

where Δuîj is the relative displacement response amplitude between DoFs i and j according to equation (4.37). The response dependence of the structural modification matrix requires an iterative analysis of the FRFs of the modified system according to equations (4.74) to (4.77) for the general case of NLSMT. This iterative procedure shall not be repeated here. As an example for the calculation of non-linear FRFs using NLSMT the cantilever beam model shall be considered which has already been used as an example for the SHBM, see Figure 4.2. In accordance with the SHBM example, non-linear FRFs shall be analyzed for different levels of constant excitation force amplitude. The FRF matrix of the underlying linear beam model is used as the initial FRF matrix to be modified. The cubic softening spring and the quadratic hardening damper are considered as response dependent structural modifications attached to the initial system at the joint DoFs, where the two beam components are connected. In principle, the same results were obtained by the application of the NLSMT as were already shown in Figure 4.3 for the SHBM. Thus, non-linear FRFs obtained with the NLSMT will not be shown here again. Instead, a comparison plot between the FRFs obtained with SHBM and NLSMT is shown in Figure 4.8. The non-linear FRF obtained with SHBM was calculated by using the direct approach for non-linear frequency response analysis, i.e. direct inversion of the non-linear dynamic stiffness matrix according to equation (4.8). No visual difference can be observed between the SHBM and the NLSMT FRFs. From the convergence chart it can be seen that up to 150 iterations were required to achieve a converged solution using NLSMT. Such a poor convergence behavior was observed quite often when using NLSMT and must be considered as one of the drawbacks of this method.

Figure 4.8: Comparison plot of non-linear FRFs at 50N excitation force level


49

Even though the convergence criterion stated in equation (4.78) is relatively tight, the computation of the FRFs by NLSMT was quite efficient. For example, the computation time for a single non-linear FRF took only 6.72 seconds on a Windows XP computer with an 1800MHz Intel Pentium IV CPU and 512 MB RAM under MATLAB 6.5.1. When considering the accuracy of NLSMT, it can be concluded that almost identical results can be obtained with the NLSMT as with the SHBM in conjunction with the direct approach for frequency response analysis. This is indicated in the error plot shown in Figure 4.9, where it can be observed that the absolute error of the NLSMT is smaller than 0.0005%. This is not a surprise, because SMT was derived based on equation (4.53) which is the equation for the direct frequency response analysis.

Figure 4.9: Magnitude error between NLSMT and SHBM of the non-linear FRF obtained at 50N excitation force level

50

5. Properties of the Non-Linear Fundamental Harmonic Response

Chapter 5: Properties of the Non-Linear Fundamental Harmonic Response After the discussion of the theory of non-linear frequency response analysis some of the basic properties of fundamental harmonic non-linear frequency responses shall briefly be discussed on a non-linear single-DoF oscillator with the following equation of motion:

mu(t ) + cu(t ) + knl (u )u(t ) = f (t ) .

(5.1)

The mass-, stiffness-, and damping parameters of the single DoF oscillator are summarized below: • m = 1 kg , • c = 0.04 Ns m , i.e. underlying linear modal viscous damping ratio ξ = 2% ,

⎧k1ugap + k2 ( u − ugap ) , ∀u > ugap ⎪⎪ N • knl (u ) = ⎨ k1u, ∀ u ≤ ugap , with k1 = 1 N m , k 2 = 2 m , u gap = 10 m . ⎪ ⎪⎩ − k1ugap + k2 ( u + ugap ) , ∀u < −ugap The piecewise linear force-displacement diagram (restoring force function) of the non-linear spring is shown in Figure 5.1 (hardening stiffness).

Figure 5.1: Non-Linear restoring force function of hardening piecewise linear spring

The response amplitude dependent equivalent non-linear stiffness of the piecewise linear spring according to SHBM can be expressed by the following equation:

⎧ k1 , ∀ uˆ ≤ ugap ⎪⎪ 2 ⎛ uˆ 2 − ugap keq (uˆ ) = ⎨ k2 − k1 ⎛⎜ −1 ⎛ u gap ⎞ ⎜ ⎪ k2 − ⎜ π ⎜ 2sin ⎜ uˆ ⎟ + 2ugap uˆ 2 ⎝ ⎠ ⎝ ⎝ ⎩⎪

⎞⎞ . ⎟ ⎟ , ∀ uˆ > ugap ⎟⎟ ⎠⎠

(5.2)

Figure 5.2 shows the equivalent stiffness plotted over the displacement amplitude. This type of curve is frequently referred to as describing function. It can be seen that


51

for small amplitudes the equivalent stiffness is equal to the stiffness k1 , whereas for large amplitudes the equivalent stiffness converges towards the stiffness k 2 .

Figure 5.2: Describing function of a piecewise linear spring with hardening stiffness characteristic

The fundamental harmonic frequency response of a non-linear single DoF system can be calculated in a non-iterative way according to [Magnus 1997], pp. 226-230. The non-linear frequency response of the piecewise linear oscillator is shown in Figure 5.3 where an excitation force amplitude of 3N was used in this case. The hardening stiffness character of the piecewise linear spring causes a shift of the resonance peak towards higher frequencies. A resonance peak distortion can only be observed for response amplitudes which exceed the limit ugap where the stiffness transition occurs. The linear and the non-linear response coincide for response amplitudes below ugap .

Figure 5.3: Linear and non-linear fundamental harmonic frequency responses

The shift of the equivalent resonance frequency is also plotted in Figure 5.3. It was calculated from the equivalent non-linear stiffness and the constant mass by the well known equation:

52


ωeq (uˆ ) =

keq (uˆ ) m

.

(5.3)

It can be observed that the equivalent resonance frequency equals the underlying linear eigenfrequency, until the response amplitude exceeds the limit ugap . After passing this limit, the equivalent resonance frequency increases in the same way as the equivalent stiffness keq (uˆ ) . A stable and an unstable response branch are both indicated in the non-linear frequency response plot shown in Figure 5.3. The unstable response occurs in the frequency range with multiple possible response solutions between Ω = 1.1s −1 and Ω = 1.3s −1 . For example, three different response solutions are available when the non-linear single DoF oscillator is excited harmonically at a frequency of Ω = 1.2s −1 . The first one is located on the lower branch which is equal to the linear response in this case, the second one is located on the upper stable response branch, and the third one is located between the first and the second one but is unstable. Unstable, in this respect, means that the system would either vibrate with the amplitude of the upper or the lower response branch, but never with the amplitude of the unstable response branch. Consequently, the unstable response branch cannot be observed in a sine excitation test and is therefore less important in practice. The response algorithm for SHBM which is used for non-linear frequency response analysis in this thesis does not account for the unstable solution branch. Thus, only the upper and the lower response branches are calculated. A so-called jump phenomenon occurs whenever a response branch comes to an end so that the response curve has to be continued on another response branch. This can be observed in Figure 5.4, where the linear response and the stable nonlinear response branches are plotted. The question, which one of the stable response branches is the right one, depends on the “initial conditions”. For example, consider a structure which is excited at a frequency which falls into the frequency interval with multiple solutions. Let us assume that the structure vibrates in steady-state with the response amplitude of the upper branch. When the excitation frequency is slightly changed while the excitation force amplitude is kept constant (as it is done, for example, in a constant level step-sine test), then the upper branch is followed until a frequency is reached where the upper branch ends (point A in Figure 5.3). Consequently, the vibration amplitude “jumps” from the upper branch to the lower branch (from A to B). On the other hand, when the structure vibrates with the lower branch amplitude, then the lower branch is followed as the excitation frequency is changed. A jump occurs at a frequency where the lower branch comes to an end (point C) and the vibration amplitude jumps from the lower branch to the upper branch (from C to D) to continue the vibration.


53

A unstable region

D C

B

Figure 5.4: Linear response and upper and lower branch of non-linear frequency response calculated with SHBM

In vibration testing of non-linear systems with hardening stiffness characteristic, it can be observed that when using sweep-sine excitation the upper branch is followed when sweeping in upward direction and the lower branch is followed when sweeping in downward direction. For non-linear structures with softening stiffness character the situation is vice versa. When using step-sine testing one has to consider how the frequency change is achieved by the measurement software. For example, when the frequency is incremented by sweeping from one discrete frequency to the next, then the same rules apply as for the sweep excitation discussed above. In some cases of step-sine testing the frequency is incremented by stopping the signal source, incrementing the frequency, and starting the signal source again. In this case, it might be impossible to measure the upper branch of the stable response, because the structural vibration is given time to decay when the signal source is stopped. When the signal source is started again, the initial conditions might be closer to the lower branch response and, consequently, this branch is followed. Another possible pitfall in non-linear vibration testing is the wrong interpretation of the response curves when the lower response branch has been measured. It can be observed from Figure 5.4 that the response peak magnitude of the lower response branch (where the jump from C to D occurs) is much smaller than the corresponding linear response peak magnitude. When only the lower response branch was measured in a step-sine test, this can be misunderstood as a dramatic increase of damping, which is obviously not the case (remember that the non-linear oscillator considered here has no damping non-linearity).

54


5.1

Accuracy Assessment by Comparison with Time Domain Results

The SHBM is most frequently used for the calculation of steady-state fundamental harmonic response amplitudes of non-linear systems. In principle, this can also be achieved by using numerical integration schemes with harmonic excitation provided that the analysis time is long enough so that the homogeneous solution is given time to decay away. In the following section, the accuracy of the SHBM response shall be investigated by comparison with those obtained with the Newmark integration scheme. The system used for this comparative study is the planar cantilever beam which has already been used in chapter 4.4.1. This time, however, a different non-linear joint characteristic is used. The model consists of two nominal identical beam components and a mass attached to the free beam tip, see Figure 5.5. Each beam component is modeled by four discrete beam elements according to the Timoshenko beam theory. The beam components are connected by a hinge, i.e. equal lateral displacements but different rotational displacements at the joint. The underlying linear MSC.Nastran model can be found in the Appendix. linear grounded rotational spring

piecewise linear rotational spring

F=50N

M

1 3

kϕ kϕ

component 1

Δϕ

component 2 hinge

tip mass

hinged support Figure 5.5: Cantilever beam model with non-linear joint spring

A rotational softening non-linear spring was introduced at the joint whose restoring force function is piecewise linear. The underlying linear stiffness of the piecewise linear spring is about three times higher than the stiffness for large displacements. The total damping of the underlying linear system is represented by proportional damping (MSC.Nastran analysis parameters ALPHA1 and ALPHA2) which equals 0.3% modal viscous damping at the eigenfrequency of the first mode at 114.2 Hz. A harmonic excitation force with 50N amplitude was introduced at the tip mass and the frequency range around the resonance of the first bending mode was analyzed in a non-linear frequency response analysis using SHBM. The response curves are shown in Figure 5.6 and the different response branches can be observed which indicate the frequency region with multiple stable response solutions.


55

Figure 5.6: Underlying linear and non-linear frequency response obtained with SHBM

Non-linear transient response analyses were performed on the cantilever beam model for correlation purposes. Steady-state transient responses were calculated for 50N harmonic excitation by using different excitation frequencies one at a time, among those, 112 Hz, 113.2 Hz, 114 Hz and 115.5 Hz. The fundamental harmonic response amplitudes were extracted from curve-fitting the non-linear steady-state transient responses. A comparison of the response amplitudes obtained with SHBM and with Newmark integration (time step size 10-4 s) is shown in Figure 5.7. The consistency of the damping models used in time- and frequency domain was checked by comparing the steady-state response amplitudes obtained with linear transient- and linear frequency response analysis. The circles at the above mentioned frequencies in Figure 5.7 indicate a good correlation between time domain and frequency domain results and is a prove for consistent damping models. The stars in Figure 5.7 indicate the steady-state fundamental harmonic response amplitudes obtained with the Newmark integration. Even in the non-linear case a good correlation was achieved so that the SHBM can be considered as an efficient and reasonably accurate tool for fundamental harmonic non-linear response analysis.

Figure 5.7: Response magnitude comparison of SHBM response and non-linear transient response

56


It should be mentioned that is was not possible to achieve a converged steady-state transient response at 112 Hz using MSC.Nastran (note that the “star” at 112 Hz is missing in Figure 5.7). One possible reason for the failing non-linear transient analysis could be that the solution oscillates between the two stable solutions available at this frequency. This oscillation may interfere with the convergence criteria checked for the termination of the response iteration so that the oscillation is interpreted as a non-converged solution after the maximum number of iterations has been performed. The non-linear transient analysis may be stabilized by applying artificial numerical damping, see [Nastran 1992]. However, this would corrupt the steady-state solution which is very sensitive to damping, especially in case of lightly damped structures. Hint for MSC.Nastran users: When using Sol 129 for non-linear transient response analysis it is worth checking the settings for the analysis parameters BETA and NDAMP. The former one is used to set the weighting factors for the Newmark integration scheme and should be set to 0.25. The latter one controls the so-called numerical damping which stabilizes the solution in case of undamped systems. This numerical damping is added to other sources of damping otherwise defined for the system. Thus, the parameter NDAMP should be set to zero when the analysis is aimed at the calculation of the steady-state response of lightly damped systems. Refer to [Nastran 1992] for details of these analysis parameters.

6. Model Reduction Techniques for Non-Linear Response Analysis

57

Chapter 6: Model Reduction Techniques for Non-Linear Response Analysis Model reduction techniques have been developed since the late sixties of the last century when computer power was the limiting factor for detailed FE analysis. At that time, much effort had to be spent on building reasonably accurate FE models due to the restrictions on the maximum allowable number of DoFs. Reduction techniques were found to be an attractive way for solving large order FE systems whose analysis might otherwise be impossible due to limitations posed on computer resources. With increasing computer power the application of model reduction techniques was doomed to become inefficient and condensation of large order models was no longer considered necessary. The power of recent computers almost poses no limitation on the number of DoFs of FE models. Consequently, today’s FE models appear to be much more refined than they would have been some years ago. The trend to even more refined models together with the increasing demand for the inclusion of non-linear effects in FE simulations again led to the request for model reduction techniques for efficient analysis. Different model reduction techniques will be discussed in the following, and in particular, their applicability to non-linear frequency domain response analysis will be investigated.

6.1

Mode Superposition

Mode superposition uses the transformation of the equation of motion from physical coordinates to generalized modal coordinates. The so-called mode shape matrix [Φ ] is utilized for this purpose. Its columns are the real normal modes {φn } which can be obtained from eigenvalue analysis of the undamped system: N

{u} = [Φ ]{q} = ∑{φn } qn .

(6.1)

n =1

Introducing the modal transformation of equation (6.1) into the frequency domain equation of motion (4.7) and subsequently left hand side multiplication with the transposed modal matrix yields:

[Φ ]T ( −Ω2 [ M ] + jΩ [C ] + [ K ]) [Φ ]{qˆ} = [Φ ]T { fˆ } ,

(6.2)

( −Ω [ μ ] + jΩ [Δ ] + [γ ]) {qˆ} = {rˆ} .

(6.3)

or respectively: 2

{}

T Vector {rˆ} = [Φ ] fˆ is the generalized modal excitation force vector, whereas the T T diagonal matrices [ μ ] = [Φ ] [ M ][Φ ] and [γ ] = [Φ ] [ K ][Φ ] are the so-called the modal mass- and modal stiffness matrices. The modal damping matrix [ Δ ] = [Φ ]T [C ][Φ ] is generally non-diagonal, unless the physical damping matrix is proportional to the mass and to the stiffness matrix, i.e. [C ] = α [ M ] + β [ K ] . The offdiagonal terms of [ Δ ] are generally small in case of lightly damped systems. It is convenient to assume that the modal damping matrix is also diagonal without

58


loosing accuracy. This approach is frequently applied in engineering practice, because it leads to uncoupled modal equations of motion which can be solved efficiently. Equations (6.2) and (6.3) represent an exact transformation if the mode shape matrix [Φ ] is complete. In this case, the mode shape matrix [Φ ] is a square matrix and comprises all real normal modes of the system, i.e. [Φ ] ∈ N ×N , with N being the number of DoFs. When [Φ ] contains only an incomplete subset of modes ( n = 1,… , R , with R < N ), then the modal transformation of equation (6.1) is truncated and the mode shape matrix becomes a rectangular matrix, i.e. [Φ ] ∈ N ×R which can be applied for order reduction of the equation system. The resulting truncated modal model of equation (6.3) is an equivalent model whose validity is limited to the frequency range spanned by the eigenfrequencies of the modes comprised in [Φ ] . Significant order reduction can be achieved by the mode superposition method for frequency response analysis, in particular when the frequency band to be analyzed is narrow. In this case, only those modes must be contained in the mode shape matrix, whose eigenfrequencies are located within the frequency band of analysis. Nonetheless, it is recommended to supplement the mode shape matrix with additional out of band modes, whose eigenfrequencies are located slightly outside the frequency limits (residual effects caused by upper and lower modes). After mode superposition, the response is calculated in modal coordinates {qˆ} by inverting the relatively small diagonal modal dynamic stiffness matrix according to equation (6.3). For this reason, mode superposition in frequency response analysis is known as the modal approach (in contrast to the direct approach according to equation (4.7), which involves the direct inversion of a large order physical dynamic stiffness matrix). The response {uˆ} in physical coordinates can be obtained from the superposition of the modal responses {qˆ} according to equation (6.1). The major drawback of the mode superposition method is that mode shapes are not defined for non-linear systems. Nonetheless, weakly non-linear systems can be reduced efficiently by mode superposition using the mode shapes of the so-called underlying linear system. The following modal transformation can be established for non-linear systems:

[Φ ]

T

(−Ω [ M ] + jΩ ([C ] + ⎡⎣C ({uˆ})⎤⎦ ) + [ K ] + ⎡⎣ K ({uˆ})⎤⎦) [Φ ]{qˆ} = [Φ ] { fˆ }, (−Ω [μ ] + jΩ ([Δ] + ⎡⎣Δ ({uˆ})⎤⎦ ) + [γ ] + ⎡⎣γ ({uˆ})⎤⎦){qˆ} = {rˆ} . T

2

eq

eq

2

eq

eq

(6.4) (6.5)

The equivalent non-linear stiffness- and damping matrices were already introduced in equations (4.32) and (4.33) and were derived using the SHBM. The modal equivalent non-linear stiffness- and damping matrices ⎡⎣γ eq ({uˆ}) ⎤⎦ and ⎡⎣ Δ eq ({uˆ})⎤⎦ are generally non-diagonal and their off-diagonal terms can often not be disregarded as it was proposed for the linear modal damping matrix. Instead, the off-diagonal terms of these matrices couple the modal DoFs of the system so that even when a single mode is excited, multiple modes may contribute significantly to the modal response of the system. For example, consider a non-linear system excited harmonically at the eigenfrequency of the n-th undamped mode of the underlying linear system. Let us assume that a perfectly appropriated excitation force vector is used to suppress the responses of other modes. The response of the non-linear system may nonetheless contain contributions of other modes, even though precautions were undertaken to suppress them. The reason for these unexpected modal response contributions can be found in the coupling of other modes with the n-th mode by off-diagonal elements of


59

the non-linear modal matrices. Consequently, the non-linear system is vibrating in an operating deflection shape which is a superposition of the n-th mode and contributions of other modes which are coupled. The resulting operating deflection shape can be considered as a mode shape of the non-linear system but is generally response magnitude dependent. For example, a change of the excitation force level will cause a change in the response, and consequently, a change of the equivalent non-linear stiffness- and damping matrices. This will affect the operating deflection shape and thus the non-linear mode shape. Due to the response dependence of the non-linear mode shapes it can be concluded that it is more convenient to use the mode shapes of the undamped underlying linear system for mode superposition at the cost of a coupled modal equation system whose coupling terms are response dependent. Systems condensed by mode superposition are defined in modal space and a transformation back to physical coordinates according to equation (6.1) is required to evaluate the equivalent non-linear matrices. When the number of non-linear elements is small, the equivalent non-linear modal stiffness and damping matrices can efficiently be calculated by performing modal transformation for each non-linear element individually. The assembly of equivalent non-linear system matrices is not necessary in this case, because the modal equivalent non-linear matrices can be assembled from the summation of modal non-linear equivalent element matrices: E

E

⎡γ eq ({uˆ}) ⎤ = ∑ [Φ e ] ⎡⎣ K eq ,e ( Δuê )⎤⎦ [Φ e ] = ∑ ⎡⎣γ eq ,e ( Δuê )⎤⎦ , ⎣ ⎦ T

e =1 E

E

⎡ ⎤ ⎣ Δ eq ({uˆ})⎦ = ∑ [Φ e ] ⎡⎣Ceq ,e ( Δuê )⎤⎦ [Φ e ] = ∑ ⎡⎣ Δ eq ,e ( Δuê )⎤⎦ . e =1

(6.6)

e =1

T

(6.7)

e =1

When considering the important case of weakly non-linear systems assembled of linear substructures coupled by non-linear 2-DoF elements, the equivalent nonlinear element stiffness and damping matrices ⎡⎣ K eq ,e ( Δuê )⎤⎦ ∈ 2×2 and ⎡⎣Ceq ,e ( Δuê )⎤⎦ ∈ 2×2 are response dependent element matrices. Δuê is the complex relative displacement amplitude between the DoFs of the non-linear element and matrix made up of the two rows of the truncated mode [Φ e ] ∈ 2×R is a modeN ×shape R shape matrix [Φ ] ∈ which correspond to the non-linear element DoFs. The major drawback of the mode superposition method for non-linear systems can be found in the presence of modal coupling. Even though modal truncation leads to efficient model reduction for linear systems, it may well be the case for non-linear systems that higher modes are coupled to the modes of the active frequency band, which were not contained in the truncated series of equation (6.1). Modes which are not represented by the truncated mode shape matrix cannot contribute to the total physical response of the system. Thus, the selection of modes to be contained in [Φ ] directly affects the accuracy of the response analysis and is therefore an essential step for the mode superposition method to be successful in case of non-linear systems.

6.2

GUYAN Reduction

The Guyan reduction [Guyan 1965] is possibly the oldest model reduction technique and is also known as static condensation. It uses the representation of the complete motion {u} of a system in terms of the motion of a few independent DoFs (master DoFs) comprised in the subset {uu } :

60


{u} = [TG ]{uu } .

(6.8)

Neglecting dynamic effects and partitioning of the system into dependent (slave) DoFs {us } , ( s = 1,… , S ), and master DoFs {uu } , ( u = 1,… ,U with S U ), leads to:

⎡[ K uu ] ⎢ ⎢⎣[ K su ]

[ Kus ]⎤ ⎪⎧{uu }⎪⎫ ⎪⎧{ f u }⎪⎫ = . [ K ss ]⎥⎥⎦ ⎩⎨⎪{us }⎭⎬⎪ ⎩⎨⎪ {0} ⎭⎬⎪

(6.9)

It is assumed in equation (6.9) that no external loads are applied to the slave DoFs. The following transformation matrix [TG ] ∈ N ×U ( N = S + U ) can be derived from equation (6.9):

⎤ [I ] ⎥. −1 ⎢⎣ − [ K ss ] [ K su ]⎥⎦ ⎡

[TG ] = ⎢

(6.10)

Each column {ψ i } of the matrix product − [ K ss ] [ K su ] ∈ S ×U can be interpreted as the motion of the slave DoFs when imposing a unit displacement at the master DoF i , while all other master DoFs u ≠ i are kept fixed: −1

− [ K ss ]

−1

[ K su ] = [ Ψ ] = ⎡⎣{ψ 1}

{ψ U }⎤⎦ .

(6.11)

The vectors {ψ i } are frequently called constraint modes and the matrix [ Ψ ] represents a basis of non-orthogonal Ritz vectors. The Guyan reduction can be interpreted as the relocation of stiffness-, mass-, and damping properties from the dependent slave DoFs (to be omitted) to the independent master DoFs. Efficient model reduction can be achieved if the number of slave DoFs S largely exceeds the number of master DoFs U , i.e. S > U :

[TG ]T ( −Ω2 [ M ] + jΩ [C ] + [ K ]) [TG ]{uû } = [TG ]T { fˆ } ,

( −Ω

2

(6.12)

{ }

)

G G G ˆ ⎣⎡ M ⎦⎤ + jΩ ⎣⎡C ⎦⎤ + ⎣⎡ K ⎦⎤ {uû } = f u .

(6.13)

Due to the relocation of stiffness-, mass-, and damping properties, the reduced system matrices are no longer sparse and loose their small bandwidths. In case of non-linear systems, the original system matrices are response dependent and their evaluation can be performed efficiently when all non-linear elements are fully located among master DoFs:

[TG ]

T

(−Ω [ M ] + jΩ ([C ] + ⎡⎣C

(−Ω ⎡⎣ M 2

G

(

({uˆ })⎤⎦ ) + [ K ] + ⎡⎣ K ({uˆ })⎤⎦ ) [T ]{uˆ } = [T ]

T

2

eq

u

eq

)

u

G

u

)

G

{ }

{ fˆ } ,(6.14)

⎤⎦ + jΩ ⎡⎣C G ⎤⎦ + ⎡⎣CeqG ({uû })⎤⎦ + ⎡⎣ K G ⎤⎦ + ⎡⎣ K eqG ({uû }) ⎤⎦ {uû } = fû .

(6.15)

The master DoFs of a statically condensed system retain their physical significance, and consequently, the results obtained with the condensed system can be interpreted directly as physical displacements. The expansion of the master DoF displacements to the full system according to equation (6.8) is only necessary if the motion at the slave DoFs is explicitly requested. The major drawback of the Guyan reduction is its lack of accuracy in case of dynamic analysis. The condensed model can only yield accurate results for quasi-static analysis but tends to be too stiff for dynamic analysis. Consequently, the accuracy of the results will degrade with increasing frequency of excitation. A proper selection of master DoFs can improve the accuracy of the condensed model in the lower


61

frequency range. A meaningful selection of the master DoFs can be performed e.g. by using all DoFs with high mass concentrations as master DoFs. This, however, might significantly increase the number of master DoFs and is contradictory to the objective of model reduction. Other approaches for the selection of master DoFs have been developed, e.g. [Bouhaddi 1992], but shall not be discussed here.

6.3

Exact Dynamic Condensation

The Exact Dynamic Condensation is an extension of the Guyan reduction and seeks to overcome the accuracy limitations of static condensation. As with the Guyan reduction, the system is partitioned into master and slave DoFs. The motion of the system shall be expressed in terms of the motion of the master DoFs:

{u} = [TD ]{uu } .

(6.16)

The transformation matrix [TD ] can be derived from the second equation of the partitioned equation of motion (6.17) with dynamic terms included:

⎛ ⎡[ M uu ] ⎜ −Ω 2 ⎢ ⎜ ⎣⎢[ M su ] ⎝

⎡[Cuu ] [Cus ]⎤ ⎡[ K uu ] [ K us ]⎤ ⎞ ⎧⎪{uû }⎫⎪ ⎧⎪{ f u }⎫⎪ [ M us ]⎤ ⎥ + jΩ ⎢ ⎥+⎢ ⎥ ⎟⎟ ⎨ ˆ ⎬ = ⎨ ⎬, [ M ss ]⎦⎥ ⎣⎢[Csu ] [Css ]⎦⎥ ⎣⎢[ K su ] [ K ss ]⎦⎥ ⎠ ⎪⎩{us }⎭⎪ ⎩⎪ {0} ⎭⎪

(6.17)

⎡ ⎤ [I ] ⎥ . (6.18) ⎡⎣TD ( Ω )⎤⎦ = ⎢ − 1 ⎢ − ( −Ω 2 [ M ss ] + jΩ [Css ] + [ K ss ]) ( −Ω 2 [ M su ] + jΩ [Csu ] + [ K su ]) ⎥ ⎣ ⎦ The assumption that no external forces are applied to the slave DoFs is used here as well. It can be seen from equation (6.18) that the transformation matrix is rectangular, complex, and frequency dependent, i.e. [TD (Ω)] ∈ N ×U . Consequently, the condensed system matrices will be also frequency dependent and complex:

⎡⎣TD ( Ω )⎤⎦

T

( −Ω

( −Ω [ M ] + jΩ [C ] + [ K ]) ⎡⎣T ( Ω )⎤⎦ {uˆ } = ⎡⎣T ( Ω )⎤⎦ { fˆ } , T

2

D

2

u

(6.19)

D

)

{ }

⎡⎣ M D ( Ω )⎤⎦ + jΩ ⎡⎣C D ( Ω )⎤⎦ + ⎡⎣ K D ( Ω )⎤⎦ {uû } = fû .

(6.20)

Reduction of non-linear systems can be performed in the same way as with the Guyan reduction (notation for frequency dependence is dropped for convenience in equations (6.21) and (6.22)):

[TD ]

(−Ω [ M ] + jΩ ([C ] + ⎡⎣C (−Ω ⎣⎡ M 2

D

(

({uˆ })⎤⎦ ) + [ K ] + ⎡⎣ K ({uˆ })⎤⎦ ) [T ]{uˆ } = [T ]

T

2

eq

u

eq

)

u

D

u

)

D

{ }

{ fˆ } ,(6.21)

D D D D ˆ ⎦⎤ + jΩ ⎣⎡C ⎦⎤ + ⎡⎣Ceq ({uû }) ⎤⎦ + ⎣⎡ K ⎦⎤ + ⎡⎣ K eq ({uû }) ⎤⎦ {uû } = f u .

(6.22)

The major drawback of the exact dynamic condensation is the frequency dependence of the transformation matrix [TD ] . While model reduction using mode superposition or Guyan reduction only has to be performed once, the exact dynamic condensation requires model reduction at each frequency of interest. Modifications of the exact dynamic condensation have been proposed, where the frequency band for response analysis is divided into discrete frequency intervals with a constant transformation matrix to be used within each interval. These modifications of the exact dynamic condensation have been proved to produce less accurate results, because of

62


discontinuities of the response curves which occur at the interval limits, [Meyer 2003]. Due to the relocation of stiffness, mass, and damping properties, the condensed system matrices are no longer sparse and loose their small bandwidths. Furthermore, the condensed system matrices are complex so that model reduction requires complex arithmetic which even more degrades the efficiency of this method compared to other model reduction methods. Despite the drawbacks of the exact dynamic condensation, it is nonetheless exact and significant order reduction can be achieved. Theoretically, a linear model can be condensed down to a single degree of freedom without loosing accuracy! A non-linear model can theoretically be reduced to the set of non-linear DoFs.

6.4

Craig-Bampton Reduction

Component Mode Synthesis represents a group of techniques originally developed for coupling of linear substructures. They found application in the field of model reduction as well and utilize a combination of physical and modal information for this purpose. An overview about Component Mode Synthesis methods can be found in [Craig 1987]. The Craig-Bampton reduction is probably the most commonly applied method of that group and shall be discussed next, [Craig 1968]. The idea of the Craig-Bampton reduction is to improve the transformation method introduced by Guyan to take into account dynamic effects at the dependent DoFs. In contrast to the exact dynamic condensation, the Craig-Bampton reduction utilizes modal information to approximate dynamic effects:

{us } = [ Ψ ]{uu } + [Φ CB ]{qCB } . physical term

(6.23)

modal term

The matrix [ Ψ ] was already introduced by Guyan and contains the non-orthogonal constraint modes according to equation (6.11). {qCB } is a vector of generalized modal coordinates. The modal matrix [Φ CB ] contains orthogonal mode shapes of the undamped system fixed at the master DoFs {uu } . These modes are frequently called fixed interface modes. It is obvious from equation (6.23) that eigenvalue analysis of the undamped system clamped at the master DoFs is necessary for model reduction. The Craig-Bampton transformation matrix can be derived when partitioning the system into master DoFs (index u , u = 1,… ,U ) and slave DoFs (index s , s = 1,… , S ):

⎧{u }⎫ ⎡ [ I ] [0] ⎤ ⎧ {u } ⎫ {u} = ⎪⎨ uu ⎪⎬ = ⎢ Ψ Φ ⎥ ⎪⎨ q u ⎪⎬ , ⎪⎩{ s }⎪⎭ ⎢⎣[ ] [ CB ]⎥⎦ ⎪⎩{ CB }⎪⎭

(6.24)

{u} = [TCB ]{uCB } .

(6.25)

or respectively:

Significant order reduction can be achieved if the number of additional modal DoFs comprised in {qCB } is small compared to the number of slave DoFs comprised in {us } . The reduction of the system matrices is performed by introducing equation (6.25) into the equation of motion and subsequently left hand side multiplication with the transposed transformation matrix:

[TCB ]T ( −Ω2 [ M ] + jΩ [C ] + [ K ]) [TCB ]{uˆCB } = [TCB ]T { fˆ } ,

(6.26)


( −Ω

2

)

63

{ }

⎡⎣ M CB ⎤⎦ + jΩ ⎡⎣C CB ⎤⎦ + ⎡⎣ K CB ⎤⎦ {uˆCB } = fˆCB .

(6.27)

When assuming that no external loads are applied to the slave DoFs, then the reduced force vector has only non-zero elements at the master DoFs:

⎧{ f }⎫ { fCB } = ⎪⎨ 0u ⎪⎬ . ⎩⎪ { } ⎭⎪

(6.28)

The mixture of non-orthogonal (physical) constraint modes and orthogonal (modal) fixed interface modes in the transformation matrix [TCB ] causes the condensed system matrices to have physical and modal partitions. In addition, they have coupling partitions which couple the modal DoFs with the physical master DoFs. The principal structure of a condensed system matrix after Craig-Bampton reduction is sketched in Figure 6.1. physical partition

coupling between physical and modal DoFs

coupling between physical and modal DoFs diagonal modal partition

Figure 6.1: Sketch of a condensed system matrix after Craig-Bampton reduction

The physical partitions of the system matrices are equal to the statically condensed system matrices obtained by Guyan reduction. These partitions are generally nondiagonal and non-sparse. The modal partitions are diagonal matrices due to the orthogonality of the fixed interface modes (the modal partition of the condensed damping matrix can be non-diagonal in case of non-proportional damping). The coupling partitions of the condensed system matrices activate the modal term in equation (6.23) in case of motion at the physical master DoFs. It should be noted that the coupling partitions of the reduced stiffness matrix are zero, i.e. coupling between physical and modal DoFs is only achieved by mass- and damping terms. The Craig-Bampton reduction (like other component mode synthesis methods as well) offers an interesting approach for the analysis of complex assembled structures. The basic idea of the Component Mode Synthesis methods is to disassemble a structure into its components (frequently referred to as superelements) and to perform model reduction for each component individually. Finally, the overall condensed model is assembled from the condensed component models. Even though the reduction of the component models is performed individually, the choice of the master DoFs is constrained by the need to reassemble the condensed component models to form the overall condensed model. Consequently, the interface DoFs which are used for the connection of component models must be used as master DoFs. Furthermore, DoFs with large mass concentration should also be used as master DoFs to improve the quality of the reduction. The procedure of CraigBampton model reduction is schematically shown in Figure 6.2.

64


assembled structure

superelement 2 (component 2)

superelement 1 (component 1)

common interface DoFs master DoFs SE1 master DoFs SE2

Reduction of superelement 1

Reduction of superelement 2

Assembly of reduced superelements

Analysis of reduced system

Data recovery for superelement 1 (optional)

Data recovery for superelement 2 (optional)

Figure 6.2: Flowchart of Craig-Bampton reduction and subsequent analysis with two superelements

The principal structure of a Craig-Bampton reduced system matrix assembled from the contributions of two superelements (SEs) is sketched in Figure 6.3. It can be observed that the two individually reduced superelements are coupled by interface DoFs which are shared among the two superelements. The modal partitions and the coupling partitions of the reduced system matrices remain uncoupled.


65

physical partition SE 1

coupling partition SE 1

interface DoFs of SE1 and SE2


physical partition SE 2 modal partition SE 1 modal partition SE 2 Figure 6.3: Sketch of Craig-Bampton reduced system matrix assembled of two superelements

The advantage of disassembling the structure into superelements is that the reduction process with the inherent eigenvalue analysis to calculate the fixed interface modes can be performed on relatively small subsystems (i.e. on component models). Eigenvalue analysis of the full assembled structure is not necessary. When considering non-linear systems assembled of linear substructures which are coupled in a non-linear way, it can be concluded that the Craig-Bampton reduction can be performed in a similar way as for linear systems. Non-linear elements are introduced at the component interfaces to represent non-linear effects, e.g. nonlinear joints represented by appropriate 2-DoF elements. It is thus required that adjacent component models do not share the same interface DoFs but rather use coincident interface DoFs which can be connected in a non-linear way. Coincident interface DoFs, which are not shared among adjacent superelements, means that there is no overlap of the physical partitions of the superelement matrices in the assembled reduced system matrices. Coupling of the superelements is rather achieved by introducing linear and/or non-linear joint elements at the coincident interface DoFs after the reduced matrices have been assembled, see Figure 6.4. The linear or non-linear joint elements do neither contribute to the modal partitions, nor to the coupling partitions of the reduced system matrices, because they are fully located among master DoFs which retain their physical significance even after reduction and after assemblage of the reduced system. (Note the differences between Figure 6.3 and Figure 6.4.

66


physical partition SE 1


non-linear joint elements


physical partition SE 2 modal partition SE 1 modal partition SE 2 Figure 6.4: Sketch of Craig-Bampton reduced system matrix assembled of two superelements joined by non-linear elements

The advantage of the Craig-Bampton method over the exact dynamic condensation is that the transformation matrix is neither frequency dependent, nor complex. However, eigenvalue analysis is required for each individual superelement to establish the transformation matrices. Another big advantage is that the CraigBampton reduction only has to be performed once, whereas the exact dynamic condensation has to be performed at each frequency point of interest. Craig-Bampton reduced models have similar drawbacks in terms of accuracy like modal models obtained from mode superposition. Since a part of the Craig-Bampton reduced model is represented by truncated modal information, it is clear that such models are only accurate up to a certain frequency limit which is sufficiently represented by the truncated modal term in equation (6.23). Since Craig-Bampton reduction is performed for each component individually, it is obvious that the quality of the reduction is different among the components depending on the amount of additional modal information and on the quality of the master DoF selection. It should be noted, however, that a single poorly reduced component model can corrupt the quality of the complete assembled Craig-Bampton reduced model. Therefore, attention should be given to the meaningful selection of modal DoFs for each component to be condensed.

6.5

Successive Steps of Model Reduction

Model reduction methods aim at order reduction of system matrices of complex assembled models. In principle, a model which has been reduced once can be reduced a second time by applying a different (or even the same) model reduction technique. For example, a statically condensed model may be further reduced by applying mode superposition. In some circumstances, however, it might be difficult to justify a second step of model reduction. For example, the application of Guyan reduction to a modal model obtained from mode superposition is hard to interpret from a physical point of view. It is possible, however, to perform multiple steps of model reduction in a subsequent manner. A physically meaningful order of application must be determined as the case arises but cannot be specified in general due to the large number of possible combinations of these methods.

7. Approaches for the Generation of Damping Matrices

67

Chapter 7: Approaches for the Generation of Damping Matrices When considering transient response analysis of linear systems, it can be stated that damping plays a less important role, because internal inertia forces and elastic restoring forces are much more pronounced than internal damping forces in short duration events such as crashworthiness analysis and other impact simulations. However, if periodic excitation is considered in time domain response analysis, then damping becomes sensitive and affects the magnitude of the steady-state response. Damping is also a sensitive parameter in frequency response analysis and can largely affect the steady-state response amplitudes. For linear systems, damping can conveniently be expressed in terms of modal damping ratios. Such damping ratios can either be obtained from experiments (modal survey testing, half-power bandwidth method, logarithmic decrement), or can be estimated based on experience for different categories of structures. Consequently, there is no need to explicitly express damping of linear systems in terms of a physical damping matrix. For nonlinear systems, the modal approach might not always be favorable for frequency response analysis. The generation of a meaningful physical damping matrix is therefore important for the accuracy of the calculated response. Different approaches for the definition of damping in linear and non-linear systems will be discussed in the following. The non-linear systems focused in this thesis are assumed to have an underlying linear system. The generation of damping matrices of linear systems will therefore be discussed prior to the generation of the non-linear damping supplement.

7.1

Damping of Linear Systems

7.1.1

Proportional Damping / RAYLEIGH Damping

A rather simple approach for the generation of a damping matrix is the proportional damping approach, also known as Rayleigh damping (after Lord Rayleigh who first proposed it). The damping matrix is considered to be proportional to the mass- and stiffness matrix:

[C ] = α [ M ] + β [ K ] .

(7.1)

In general, modal damping Δ i of the i –th mode of vibration can be expressed in terms of the modal mass μi , the angular eigenfrequency ωi , and the modal viscous damping ratio ξi :

Δ i = 2 μiωiξi .

(7.2)

The modal damping approach assumes that this general expression for modal damping Δ i can be approximated by a term which is proportional to the modal mass μi and to the modal stiffness γ i :

Δ i = 2 μiωiξi ≅ αμi + βγ i = μi (α + βωi2 ) .

(7.3)

68


When using the Rayleigh quotient as a relation between modal mass, modal stiffness, and eigenfrequency, it can be shown that the mass proportional damping term is inversely proportional to the frequency while the stiffness proportional damping term produces a damping ratio which is directly in proportion to the frequency, [Clough 1993].

ξ combined stiffness proportional

ξn ξm

mass proportional

ωm

ωn

Ω

Figure 7.1: Relationship between damping ratio and frequency in case of Rayleigh damping

The factors α and β of equations (7.1) or respectively (7.3) can be adjusted in such a way that specific damping ratios ξ m and ξ n are obtained for certain modes whose eigenfrequencies are ωm and ωn :

⎧α ⎫ 2ωmωn ⎨ ⎬= 2 2 ⎩ β ⎭ ωn − ωm

−ωm ⎤ ⎧ξ m ⎫ ⎡ ωn ⎢ ⎥⎨ ⎬. ⎣ −1/ ωn 1/ ωm ⎦ ⎩ ξ n ⎭

(7.4)

It is recommended that ωm is generally taken as the fundamental eigenfrequency of the system and that ωn is set among the eigenfrequencies of higher modes which contribute significantly to the dynamic response. By proceeding this way, it is ensured that the damping ratios ξ m and ξ n are obtained for modes m and n . Damping ratios ξ in the frequency interval ωm < Ω < ωn can be determined from the following equation (derived from equation (7.3)):

1⎛α ⎞ ξ = ⎜ + β Ω⎟. 2⎝Ω ⎠

(7.5)

It can be observed from Figure 7.1 that responses of higher modes are effectively eliminated by high damping ratios, because stiffness proportional damping becomes the dominant source of damping in the higher frequency range and increases linearly with increasing frequency. Extensions of the classical Rayleigh damping can be found in [Clough 1993]. The approaches presented there are less popular and can even yield negative damping ratios for high frequency modes. Extensions of the Rayleigh damping shall therefore not be discussed here.

7.1.2

Expansion of Modal Viscous Damping Ratios

This approach for the generation of a physical damping matrix is associated with a given set of modal damping ratios which are known a priori, either from experiments or from experience. When assuming a diagonal modal damping matrix [ Δ ] , damping can be defined on a modal basis by the following equation which utilizes the relation between modal masses μi , angular eigenfrequencies ωi , and modal viscous damping ratios ξi already stated in equation (7.2):


⎡ [ Δ ] = ⎢⎢ ⎢⎣ 0

2 μiωiξi

69

0⎤ ⎥. ⎥ ⎥⎦

(7.6)

According to this equation, modal damping can be defined for all modes whose eigenfrequencies are located within the frequency range to be analyzed in a forced harmonic response analysis. It is not necessary to explicitly define modal damping for all modes of the system. When taking into account that the modal damping matrix [ Δ ] can be obtained from modal transformation of the physical damping matrix [C ] , it can be concluded that the physical damping matrix can be retrieved by inverse modal transformation of the modal damping matrix:

[ Δ ] = [Φ ] [C ][Φ ] → [C ] = [Φ ] [Δ ][Φ ] . The mode shape matrix [Φ ] is usually rectangular and it can be proven that: −1 −1 T [Φ ] = [ μ ] [Φ ] [ M ] . −T

T

−1

(7.7)

(7.8)

Inserting this relation into equation (7.7) yields an expression for the physical damping matrix:

[C ] = [ M ][Φ ][ μ ] [Δ ][ μ ][Φ ] [ M ] = ⎡⎣ M ⎤⎦ [d ] ⎡⎣ M ⎤⎦ −1

⎣⎡ M ⎦⎤

T

[d ]

⎡⎣ M ⎤⎦

T

,

(7.9)

T

with:

⎡ ⎢ [d ] = ⎢⎢ ⎢ ⎣0

2ξiωi

μi

0⎤ ⎥ ⎥. ⎥ ⎥ ⎦

(7.10)

This damping matrix can be considered as a superposition of modal damping contributions. Processing the matrix products in equation (7.9) yields:

⎡ R ≤ N ⎛ 2ξ nωn

[C ] = [ M ] ⎢ ∑ ⎜

⎣ n =1 ⎝ μn

{φn }{φn }

T

⎞⎤ ⎟⎥ [ M ] . ⎠⎦

(7.11)

It can be seen from this equation that the modal contributions to the physical damping matrix are directly in proportion to the modal damping ratios which can be defined for each mode individually. Consequently, only those modes will be damped which are explicitly included in the summation of equation (7.11). Other modes are undamped! The damping matrix obtained from equation (7.9), or respectively (7.11), is a full matrix, i.e. each DoF of the system is coupled to all other DoFs by internal damping forces. A full damping matrix is inefficient not only in terms of computation time, but also in terms of storage requirements in numerical analysis. Stiffness- and mass matrices are usually sparse and special procedures are available for efficient numerical operations among sparse matrices. The full damping matrix can have an excessively high storage requirement, especially in case of large order systems so that the generation of a damping matrix according to equation (7.9) or respectively (7.11) becomes prohibitive.

70


A practical approach for the generation of large order damping matrices is achieved by retaining only significant entries in the matrix ⎡⎣ M ⎤⎦ = [ M ][Φ ] while small entries are intentionally set to zero. This is accomplished by searching in a column by column manner for elements in the matrix ⎡⎣ M ⎤⎦ , whose magnitudes are larger than a certain fraction of the magnitude of the largest element in the current column. For example, only those entries are retained in a column of ⎡⎣ M ⎤⎦ which have a magnitude larger than 0.0001 times the largest magnitude found in that column. By proceeding this way, a sparse viscous damping matrix can be obtained in accordance with equation (7.9). This matrix is efficient in terms of storage requirements and in terms of numerical computation. It should be noted that damping matrices obtained from the expansion of modal damping ratios have rank deficiencies. The rank of the physical damping matrix is determined by the number of independent modal damping contributions which is typically less than the order of the physical damping matrix.

7.1.3

Generation of a Non-Proportional Damping Matrix

Complex structures are often assembled of several components which may even consist of different materials. In some cases, the different materials of the components can provide energy-loss mechanisms where the distribution of internal damping forces will not be proportional to the distribution of internal inertia forces and elastic restoring forces. Such structures are called non-proportionally damped, [Clough 1993]. Each component of a non-proportionally damped structure can be considered individually and a proportional damping matrix [Cc ] can be generated for each component c = 1,… , C using unique proportion factors α c and β c : C

C

c =1

c =1

[C ] = ∑ [C c ] = ∑ (α c [ M c ] + β c [ K c ] ) .

(7.12)

The proportion factors α c and β c represent the damping behavior of the individual component c. The assembly of the proportional component damping matrices yields a non-proportional overall assembled damping matrix. Modal transformation of such a non-proportional damping matrix results in a non-diagonal modal damping matrix and, hence, produces damping coupling of the modal equations of motion in case of mode superposition.

7.2

Non-Linear Damping Supplement

It was mentioned before that the non-linear systems considered in this thesis are assumed to have underlying linear systems for which damping can be defined according to one of the methods discussed above. Non-linear damping can therefore be considered as a supplement to the damping of the underlying linear system which is being produced by non-linear elements having non-conservative restoring force functions. Consequently, the total damping matrix of a non-linear system can be constructed from the superposition of the underlying linear damping matrix and the response dependent non-linear damping matrix:

⎡Ctotal ({u}, t ) ⎤ = [C ] + ⎡Cnl ({u}, t )⎤ . ⎣ ⎦ ⎣ ⎦

(7.13)


71

When considering forced harmonic response analysis of non-linear systems, it was shown in chapter 4 that a response magnitude dependent equivalent non-linear damping matrix can be generated according to the Harmonic Balance method. This is achieved by the summation of equivalent non-linear element damping matrices transformed to global degrees of freedom by using coincidence matrices (see equation (4.32)): E

Harmonic Balance ⎡Cnl ({u}, t )⎤ ⎯⎯⎯⎯⎯⎯ → ⎡⎣Ceq ({uˆ})⎤⎦ = ∑ [Te ] ⎡⎣Ceq ,e ( Δuê ) ⎤⎦ [Te ] . Method ⎣ ⎦ T

(7.14)

e =1

Damping of the underlying linear system is frequently expressed in terms of modal damping ratios so that a physical damping matrix is not explicitly required. Frequency response analysis can then be performed in the modal domain by solving the (coupled) modal equations of motion according to equation (6.3), instead of using the direct frequency response approach according to equation (4.7) which is based on the inversion of the dynamic stiffness matrix. Modal frequency response analysis can efficiently be performed for non-linear systems as well, see equation (6.5). In this case, the modal equivalent non-linear damping matrix is assembled from the summation of the modal equivalent non-linear element matrices (see equations (4.32), (4.35), and (6.7) for details): E

⎡ Δ eq ({uˆ})⎤ = ∑ [Φ e ] ⎡⎣Ceq ,e ( Δuê )⎤⎦ [Φ e ] . ⎣ ⎦ T

(7.15)

e =1

The total modal damping matrix of the non-linear system can be calculated from the superposition of the underlying linear modal damping matrix and the modal equivalent non-linear damping matrix:

⎡ Δ total ({uˆ})⎤ = [ Δ ] + ⎡ Δ eq ({uˆ}) ⎤ . ⎣ ⎦ ⎣ ⎦

(7.16)

It can be seen from equation (7.15) that modal frequency response analysis of nonlinear systems does not only provide efficient order reduction, it does also provide an effective way for the generation of a non-linear damping matrix.

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8. Non-Linear Vibration Tests and Test Data Processing

Chapter 8: Non-Linear Vibration Tests and Test Data Processing Real structures are seldom completely linear. This can be proved by the violation of the principles of superposition and reciprocity. Nonetheless, many of them closely approximate linear behavior so that most of the theory developed for modal analysis relies on linear behavior. However, there are complex structures which do behave in a non-linear way, especially in the vicinity of resonances where the response amplitudes are large. This can give rise to concerns and problems when they are being tested. Signs of non-linear behavior include: • natural frequencies varying with position and strength of excitation, • distorted FRF plots especially near resonances, • presence of sub- and/or superharmonics in the response, • unstable or unrepeatable data. Probably the most obvious way of checking for the existence of non-linearity is to check for higher harmonics in the response signals in case of harmonic excitation. This is an elemental prove which is fulfilled by non-linear systems but which can also be the result of other circumstances, such as vibration exciter misalignment or improper sensor installation, [Worden 2001]. Another effective check for the existence of non-linearity is to repeat a certain FRF measurement a number of times using different levels of excitation (and hence response), or using different types of excitation signals. If the resulting response curves differ from one such measurement to the next, especially around the resonances, then there is a strong possibility of non-linearity. This is a check which will work with most types of excitation signals, [Ewins 2000]. If non-linear behavior was detected by a simple non-linearity check, a strategy is needed for how to proceed with testing in order to provide meaningful data for model validation. In general, FRFs of non-linear structures can be measured by applying appropriate filters to filter out higher harmonic responses. In this case, the nonlinear fundamental harmonic FRF can be measured which can appear distorted if no special precautions were taken during the measurements. These distortions are dependent not only on the characteristic of the non-linearity, but also on the type of the excitation signal used. It is widely accepted that sine excitation produces the most clearly distorted FRFs while other excitation signals are less effective in detecting the presence of non-linearities. The reason for this can be found in the spectral distribution of the excitation energy. When using broadband excitation signals such as impact excitation or random excitation, the excitation energy is spread over a number of spectral lines such that the excitation energy associated with a single spectral line is relatively low. When using sine excitation, the excitation energy is concentrated on a single spectral line. Relatively large vibration amplitudes are being produced when the system is given time to develop steady-state vibration which then sufficiently activates structural non-linearities. One of the best ways of dealing with weak non-linearities in practical structures is to exercise a degree of amplitude control on the vibration levels developed during the measurements. For example, Figure 8.1 shows the FRFs of a single degree of freedom oscillator with a cubic stiffness non-linearity obtained from constant


73

excitation force amplitude measurements. The system under consideration is governed by the following equation of motion:

mu(t ) + cu(t ) + ku(t ) + knl u 3 (t ) = f (t ) ,

(8.1)

fR (u )

with the following mass-, damping-, and stiffness properties: • m = 1, • c = 0.04 , • k = 1, • knl = 7.5 ⋅ 10−5 . Figure 8.2 shows the FRFs of the same non-linear system obtained from constant response amplitude measurements.

Figure 8.1: FRFs obtained from different constant excitation force level measurements

Figure 8.2: FRFs obtained from different constant response level measurements

74


It can be seen that the non-linear constant force level FRFs are distorted in the vicinity of the resonance, whereas the constant response level FRFs appear to be the FRFs of slightly different linear systems. Indeed, the equivalent non-linear stiffness of the cubic spring depends on the amplitude of the displacement response, so that the behavior of the non-linear system can be linearized by keeping the displacement response amplitude constant. A linear FRF can be measured at each level of constant response amplitude and it would be possible to apply experimental modal analysis to each linear FRF to extract modal data which can be used subsequently for response level dependent model validation. In contrast to the constant response level measurements, distorted FRFs are obtained when the excitation force amplitudes are kept constant. Distorted means that the constant force level FRFs deviate from the linear ones due to the influence of non-linearities and it is frequently observed that the distortion increases with increasing force levels. These distorted FRFs are well suited to characterize the nonlinearity of a structure and they can easily be correlated with analytical non-linear FRFs obtained with the SHBM. This offers an attractive way for non-linear model updating, because the differences between experimental and analytical constant force level FRFs can be used as a frequency response residual in a CMU procedure for the identification of non-linear FE model parameters. This approach will be followed in this thesis and applications will be presented. Despite the advantages of constant force level and constant response level testing, it should be noted that these methods require the adjustment of the excitation force level at each frequency of measurement, until the control parameter (excitation force amplitude or response amplitude) is steady at the desired level. When the structure is non-linear, this can be more difficult to achieve than expected. Depending on the type of non-linearity (stiffness or damping) different response quantities must be utilized as a control parameter. For example, stiffness non-linearities require constant displacement amplitudes for linearization, while damping non-linearities require constant velocity amplitudes. If the structure under test is expected to have both types of non-linearity (stiffness and damping), then it is more or less impossible to linearize the structure in order to measure “closest to linear” FRFs. Linearization by constant response level becomes even more complicated in the general case of multi-degree of freedom systems. In this case, only the response at a single measurement point can be kept constant (e.g. at the driving point) and the structure can only be linearized when the frequency band for FRF measurement is sufficiently small so that the structure exclusively vibrates in a single mode. The response levels at all other measurement points would be directly in proportion to the single response level which should be kept constant throughout the frequency band of interest. The factors of proportion of the response levels can be obtained from the mode shape which was mainly excited. An advantage of constant response level testing over constant force level testing is that the structure can be prevented from overtesting. The response which will develop at a resonance is known in advance when the response levels are controlled. This is not the case in constant force level testing and it might happen, that response amplitudes will be produced in this case which might cause damage to the structure.


8.1

Example: linearity

Structure

with

75

Coulomb-Friction

Non-

A practical application for constant level measurements is presented next on the 6-mass oscillator shown in Figure 8.3. The 6-mass oscillator is nominally a symmetric structure which consists of two vertical cantilever beams, each having three masses attached. The coupling between the two cantilever beams is achieved by a soft arch spring whose stiffness is orders of magnitude smaller than the stiffness of the vertical cantilever beams. Repeated roots must be expected due to the symmetry of the structure, i.e. the modes of the structure appear as pairs of modes with close eigenfrequencies. The symmetry of the 6-mass oscillator is disturbed by a grounded friction damper attached to DoF 3 (see Figure 8.3). The friction damper is essentially an aluminum sheet connected to mass number 3 by means of a hinge. This aluminum sheet is allowed to slide on a fixed surface and thereby producing friction. The grounded friction damper results in a non-proportional damping matrix and hence results in significant damping coupling of the closely spaced modes. According to the SHBM, the equivalent non-linear damper constant ceq of the friction damper depends on the displacement response amplitude uˆ , or respectively, on the velocity response amplitude vˆ at DoF number 3:

ceq (uˆ ) =

4h π Ω uˆ

vˆ =Ωuˆ ←⎯⎯ → ceq ( vˆ ) =

4h . π vˆ

(8.2)

Ω is the angular frequency of excitation and h is the sliding friction force of the friction contact which depends on the contact force and on the coefficient of friction. arch spring aluminum sheet friction contact hinge cantilever beam

6

3

shaker 2

cantilever beam

4

1 force sensor

5

fixed column

Figure 8.3: 6-Mass oscillator with coulomb friction damper

The excitation force was introduced at DoF 1 by using an electrodynamic vibration exciter (shaker). Sine excitation measurements were performed in the vicinity of the resonances of modes 3 and 4 (see Figure 8.4). The acceleration response was measured at all 6 DoFs and, in addition, a laser vibrometer was utilized to measure

76


the velocity response at DoF 3 where the grounded friction damper is attached. Higher harmonic responses were filtered out so that the non-linear FRF of the fundamental harmonic could be measured. A possibly existing constant response portion may well exist (in addition to higher harmonic responses) when the structure is excited at a resonance frequency, however, such a constant (static) response can only be observed in the displacement response which was not measured. It should also be mentioned that constant response portions cannot be measured using piezoelectric sensors.

Figure 8.4: Mode 3 (42.4 Hz) and mode 4 (43.4 Hz) of the 6-mass oscillator

Constant force level measurements were performed which clearly indicate the nonlinear “softening” damping effect by the variation of the resonance peak magnitude, see Figure 8.5. Softening damping, in this respect, means that the level of damping decreases with increasing force levels and hence with increasing response levels. A weak stiffness non-linearity can also be observed from the frequency shift of the resonance peak in Figure 8.5. This stiffness non-linearity can be considered as a combination of the following non-linear effects: • clearance type non-linearity of the hinge used for connecting the aluminum sheet with mass number 3 • non-linear effects caused by temporal lift-off of the sliding aluminum sheet at the friction contact (separation of contact partners), especially when approaching resonance


77

Figure 8.5: FRFs obtained at DoF 2 from constant force level measurements

Constant velocity level measurements were carried out which aimed at linearization of the behavior of the structure by keeping the equivalent non-linear damping parameter ceq (uˆ ) of the grounded friction damper at a constant level throughout the measurements. The FRFs obtained from these measurements are shown in Figure 8.6. Each response curve can be interpreted as “closest to linear” behavior. The decreasing damping with increasing response levels already observed in the constant force level measurements can be confirmed by the constant velocity level measurements. The extraction of modal data from the constant force level FRFs can be quite difficult since the FRFs are not consistent with the linear behavior assumed by modal analysis procedures. The linearized constant response level FRFs are better suited for this purpose, however, at the cost of different sets of modal data for each level of constant response. It can be observed from the response curves of both measurements that mode 4 (43.2 Hz) is poorly excited by the single point excitation introduced at DoF 1. This can be confirmed by the experimental mode shapes shown in Figure 8.4, where it can be seen that the deformation at the excitation DoF 1 is small for mode 4, but not for mode 3. Thus, it can be expected that mode 3 can be excited much better than mode 4.

78


Figure 8.6: FRFs obtained at DoF 2 from constant velocity level measurements

8.2

Detection and Characterization of Non-Linearities from Measured Response Data

Detection and characterization of non-linear behavior should already be performed at the stage of FRF measurement. Procedures and methods for the characterization of non-linearities are therefore presented in the following. These methods can help to develop a strategy for FRF measurements of non-linear systems, or respectively, can support non-linear finite element modeling when it turned out that the influence of the non-linearities on the dynamic behavior is strong so that it should not be disregarded in the FE model.

8.2.1

FRF Distortions and Describing Functions

A simple approach for the detection and characterization of non-linearities is the overlaid plot of FRFs measured at different levels of constant excitation force. Such a plot was already presented for the 6-mass oscillator in Figure 8.5, where the increasing FRF distortion with increasing force level is a characteristic form from which the type of non-linearity can roughly be categorized (softening or hardening stiffness, increasing or decreasing damping). The different FRF distortion characteristics shall be discussed in the following on a single degree of freedom oscillator with the following harmonically balanced nonlinear equation of motion:

( −Ω m + jΩc 2

eq

(uˆ ) + keq (uˆ ) ) uˆ = fˆ .

The mass-, stiffness-, and damping properties of this non-linear oscillator are: • constant unit mass, i.e. m = 1 , • underlying linear stiffness equal to 1, i.e. keq (uˆ ≈ 0) = 1 ,

(8.3)


•

79

underlying linear damping expressed by 2% modal damping ratio, i.e. ceq (uˆ ≈ 0) = 0.04 .

With these properties, the angular eigenfrequency of the underlying linear system is ω = 1 so that more or less “normalized” distorted FRF can be obtained depending on the characteristic of the equivalent stiffness and damping parameters. The following types of non-linearities shall be investigated in more detail: a) pre-loaded bilinear spring b) clearance type non-linearity (piecewise linear spring) c) cubic spring (polynomial type stiffness non-linearity) d) quadratic damper (polynomial type damping non-linearity) e) elasto-slip friction non-linearity (3 parameter Masing model) f) combined stiffness and damping non-linearities a) Pre-loaded bilinear spring The pre-loaded bilinear spring considered here shall have a force-displacement diagram as shown in Figure 8.7. The force-displacement relation is governed by three parameters: • underlying linear stiffness k1 = 1 (compression regime stiffness) •

location of stiffness transition point uc = 20 ,

•

tension regime stiffness k2 = 2 .

According to the SHBM, the equivalent non-linear stiffness of the pre-loaded bilinear spring can be expressed by the following equation, see e.g. [Gelb 1968]:

⎧ k1 , ∀ uˆ ≤ uc ⎪ keq (uˆ ) = ⎨ 4uc k2 − k1 ⎛ ⎛ ⎞⎞ ⎪ k1 + 2π ⎜ π − 2α + sin ⎜⎝ 2α − uˆ cos α ⎟⎠ ⎟ , ∀ uˆ > uc ⎝ ⎠ ⎩

(8.4)

with:

sin α =

uc . uˆ

(8.5)

1 2

k2 uc

uc

( k1 + k2 )

k1

k1

Figure 8.7: Force-displacement diagram and describing function of the pre-loaded bilinear spring

It can be seen from the describing function plot (equivalent stiffness plotted over displacement amplitude) that the equivalent non-linear stiffness keq equals the underlying linear stiffness k1 until the displacement amplitude uˆ exceeds the stiffness transition point uc . For displacement amplitudes larger than the stiffness transition point the equivalent non-linear stiffness changes significantly and finally

80


converges towards the average stiffness 12 ( k1 + k2 ) for very large displacement amplitudes. When assembled with the single-DoF oscillator in equation (8.3), the amplitude dependent equivalent non-linear spring causes characteristic FRF distortions. The analytical non-linear FRFs of the single-DoF oscillator simulated by using different constant excitation force levels are shown in Figure 8.8. It can be seen that for low excitation force levels (and hence low vibration amplitudes) the non-linear FRF does not deviate much from the linear one. In fact, the system remains linear as long as the vibration amplitudes are smaller than the stiffness transition point. The FRF distortions increase with increasing vibration amplitudes. The resonance frequency shift decreases at large vibration levels which can be explained by the displacement amplitude dependent behavior of the equivalent non-linear stiffness:

ωeq (uˆ ) =

keq (uˆ ) m

.

(8.6)

Figure 8.8: FRF distortion caused by pre-loaded bilinear spring (upper stable response branch only)

It should be noted that the equivalent non-linear stiffness of the bilinear spring without pre-load (i.e. uc = 0 ) is independent on the response and equals the average stiffness 12 ( k1 + k2 ) . Even though the force-displacement curve of the pure bilinear spring is definitely non-linear, it does not cause distortions to the fundamental harmonic FRF. Thus, the bilinear spring without pre-load cannot be analyzed with the Single-Harmonic Balance Method, because the main non-linear effects are contained in the higher harmonics. The Multi-Harmonic Balance Method would generally be better suited for frequency response analysis.


81

b) Clearance type non-linearity (backlash or piecewise linear spring) The clearance type non-linearity considered here shall have a force-displacement diagram as shown in Figure 8.9. The force-displacement relation is governed by the following parameters: • underlying linear stiffness k1 = 1 (opened gap regime stiffness) •

half width of the gap uc = 20 ,

•

closed gap regime stiffness k2 = 3 .

According to the SHBM, the equivalent non-linear stiffness of the clearance type non-linearity can be expressed by the following equation, see e.g. [Gelb 1968]:

⎧ k1 , ∀ uˆ ≤ uc ⎪⎪ ⎛k −k ⎛ keq (uˆ ) = ⎨ uˆ 2 − uc2 −1 ⎛ uc ⎞ 2 1 ⎜ ⎪ k2 − ⎜ π ⎜⎜ 2sin ⎜⎝ uˆ ⎟⎠ + 2uc uˆ 2 ⎝ ⎝ ⎩⎪

⎞⎞ ⎟ ⎟ , ∀ uˆ > uc ⎟⎟ ⎠⎠

(8.7)

It can be seen from the describing function plot (equivalent stiffness plotted over displacement amplitude) that the equivalent non-linear stiffness equals the opened gap stiffness k1 until the displacement amplitude uˆ exceeds the half gap width uc . For displacement amplitudes larger than uc , the equivalent non-linear stiffness changes significantly and converges towards the closed gap stiffness k 2 for very large displacement amplitudes.

k2 k1

k2

−uc

k1

uc k2

uc

Figure 8.9: Force-displacement diagram and describing function of clearance type non-linearity

Figure 8.10 shows the analytical non-linear FRFs simulated with the single-DoF oscillator and different constant excitation force levels.

82


Figure 8.10: FRF distortion caused by clearance type non-linearity (upper stable response branch only)

c) Cubic stiffness non-linearity Polynomial type non-linearities should only be used as a supplement to the underlying linear system. They can be used to describe deviations from the linear system when the source of non-linearity of a structure is not fully understood. It is convenient to approximate non-linear restoring forces by polynomial functions. It should be kept in mind, however, that there is often lack of physical reasoning to justify the use of polynomial type non-linearities. A prominent member of the group of polynomial stiffness non-linearities is the cubic spring whose non-linear forcedisplacement relation is governed by three parameters: • underlying linear stiffness klin = 1 •

non-linear stiffness factor knl = 7.5 ⋅ 10−5

order of non-linearity p , i.e. the exponent which is p = 3 in case of the cubic spring According to the SHBM, the equivalent non-linear stiffness of the cubic spring is, see e.g. [Gelb 1968]: •

keq (uˆ ) = klin + 43 knl uˆ 2 .

(8.8)

It can be seen from this equation that the equivalent stiffness increases quadratically with the displacement amplitude. This can also be observed in the describing function plot in Figure 8.11. It is a common phenomenon of all polynomial type non-linearities that the describing function is also a polynomial function which is one order less than the original restoring force function, i.e. a cubic spring has a quadratic describing function, a quadratic spring has a linear describing function, etc.


klin

83

klin

Figure 8.11: Force-displacement diagram and describing function of cubic hardening spring

The FRF distortions caused by the hardening cubic spring (i.e. knl > 0 ) can be observed in Figure 8.12. The resonance peak shift increases with increasing vibration amplitudes and in contrast to the other stiffness non-linearities discussed so far, the equivalent non-linear stiffness does not converge towards a steady value but rather diverges towards infinity for very large vibration amplitudes. This divergence behavior can lead to numerical instability when computing non-linear responses at large amplitudes. This effect becomes even worse when using softening cubic springs (i.e. knl < 0 ). In this case, the equivalent non-linear stiffness decreases quadratically so that even a negative equivalent stiffness can be obtained when a certain threshold amplitude is exceeded. From that point of view, models with polynomial type non-linearities cannot really be considered predictive. They can be used as equivalent models to approximate the true non-linear behavior of a structure within a limited range of vibration amplitudes. The prediction of the non-linear response for very large vibration amplitudes far away from the amplitude levels which were used for stiffness adjustment can be inaccurate.

Figure 8.12: FRF distortion caused by cubic hardening spring (upper stable response branch only)

84


d) Quadratic damper It was stated above that polynomial type non-linearities should only be used as a supplement to the underlying linear system and, thus, should only be used to describe the deviation from the linear system. One representative of the polynomial type of damping non-linearities is the quadratic damper with the following restoring force function:

f R (u, t ) = clin u(t ) + cnl u(t ) u(t ) .

(8.9)

The non-linear restoring force function is governed by three parameters: • underlying linear damping clin = 0.04 •

non-linear damping factor cnl = 1.5 ⋅ 10−3

order of non-linearity p , i.e. the exponent which is p = 2 in case of the quadratic damper According to the SHBM, the equivalent non-linear damper constant of the quadratic damper can be expressed by the following equation, see e.g. [Gelb 1968]: •

vˆ =Ωuˆ ceq (uˆ ) = clin + 38π cnl Ωuˆ ←⎯⎯ → ceq ( vˆ ) = clin + 38π cnl vˆ .

(8.10)

It can be seen from this equation that the equivalent damping increases linearly with the displacement amplitude uˆ and also with the angular frequency of excitation Ω , i.e. the equivalent damping increases linearly with the velocity amplitude vˆ = Ωuˆ . This can also be observed in the describing function plot in Figure 8.13. The describing function is linear while the original restoring force is a quadratic function of velocity.

clin clin

Figure 8.13: Force-displacement diagram and describing function of quadratic “hardening” damper

The FRF distortions caused by the “hardening” quadratic damper (i.e. cnl > 0 ) can be observed in Figure 8.14 where the influence of increasing damping is indicated by the reduction of the resonance peak magnitude. Hardening, in this respect, means that the level of damping increases with increasing response levels.


85

Figure 8.14: FRF distortion caused by quadratic “hardening” damper (upper stable response branch only)

e) Elasto-slip friction non-linearity The elasto-slip non-linearity (3-parameter Masing model) is a special type of friction non-linearity. It combines both, non-linear stiffness and non-linear damping effects.

k0 f1 , u1

f 2 , u2 fN k1

μ

Figure 8.15: Equivalent mechanical system for the elasto-slip non-linearity

In case of harmonic response, a hysteresis loop is obtained for the non-linear restoring force function (provided that the response amplitude exceeds a certain threshold so that sliding friction occurs). The area enclosed by the hysteresis is a measure for the energy dissipated during one cycle of vibration and thus is a measure for the amount of damping. The force-displacement curve of the elasto-slip non-linearity is governed by three parameters: • sliding friction stiffness k0 = 0.6 , •

additional stiffness in case of static friction k1 = 0.4 ,

•

friction force limit h = μ f N = 8 .

The friction hysteresis (force-displacement diagram) can be observed from Figure 8.16. Furthermore, it can be seen that the force-velocity curve deviates from the elliptical form which would be obtained in case of a linear Kelvin-Voigt model (linear spring and linear viscous damper arranged in parallel).

86


2h

k0 + k1

2uc k0

Figure 8.16: Force-displacement diagram and force-velocity curve of elasto-slip non-linearity

According to the SHBM, the equivalent non-linear stiffness and equivalent nonlinear damping parameters can be expressed by the following equations, see e.g. [Gaul 2000]:

∀ uˆ ≤ uc ⎧ k0 + k1 , ⎪ keq (uˆ ) = ⎨ , k1 1 sin 2α ˆ k u u α + − ∀ > ( ) ( ) 0 c 2 ⎪⎩ π

(8.11)

∀ uˆ ≤ uc ⎧0, ⎪ ˆ ceq (u ) = ⎨ k1 , ⎪⎩ 2πΩ (1 − cos ( 2α ) ) , ∀ uˆ > uc

(8.12)

uc =

h , k1

cos α = 1 − 2

h . uˆ k1

(8.13) (8.14)

The describing functions of the 3-parameter elasto-slip model are shown in Figure 8.17. It can be seen that the equivalent stiffness is constant at k0 + k1 until the displacement amplitude reaches the threshold uc where the transition from static friction to sliding friction occurs. The equivalent stiffness decreases if the displacement amplitude is further increased and finally the equivalent stiffness converges towards the sliding friction stiffness k0 for very large displacement amplitudes. The equivalent non-linear damping is zero until the displacement amplitude exceeds the threshold where the friction force limit is reached. Damping increases significantly when the displacement amplitude is further increased. The maximum damping is reached when the displacement amplitude is twice as high as the threshold amplitude. The maximum equivalent damping constant is ceq,max = k1 / π Ω at uˆ = 2uc but decreases again for larger amplitudes and finally seems to converge hyperbolically towards zero.


k0 + k1

87

ceq ,max

uc k0 uc

2uc

Figure 8.17: Equivalent stiffness and equivalent damping of elasto-slip non-linearity

The FRF distortions caused by the elasto-slip element are shown in Figure 8.18. The effect of increasing and subsequently decreasing damping can be observed together with a continuous shift of the resonance peak towards lower frequencies caused by the continuously decreasing equivalent stiffness.

Figure 8.18: FRF distortion caused by 3-parameter elasto-slip non-linearity (upper stable response branch only)

The elasto-slip model as shown in Figure 8.15 only distinguishes between static friction and sliding friction. The transition from static friction to sliding friction does not occur gradually but rather abruptly, i.e. microslip effects are not included in the elasto-slip model. Microslip effects can be taken into account either by using more sophisticated elasto-slip models with more than three parameters, or by using other formulations for the description of the friction force, such as the Valanis friction model discussed in [Lenz 1997] and [Gaul 2000]. The Valanis friction model is originated in material science to describe plasticity effects such as yielding. However, the relations between stresses and strains in yielding of material can also be used to

88


describe the relations between forces and displacements of a friction element. The non-linear restoring force of the Valanis friction element can be expressed by the following differential equation:

f R ( u, u , f R ) =

(

E0 u 1 + 1+κ

λ

E0

λ

E0

sgn ( u )( Et u − f R )

sgn ( u )( Et u − f R )

),

(8.15)

with the following similarities among the parameters of the Valanis model and the elasto-slip model:

E0 = k0 + k1 ,

(8.16)

Et = k 0 ,

(8.17)

h=

(

E0

λ 1−κ

Et E0

)

,

(8.18)

and an additional parameter to control microslip>

0 ≤ κ < 1.

(8.19)

When considering the use of the Valanis model for non-linear frequency response analysis, it can be concluded that the major drawback of the Valanis friction model is that the non-linear restoring force is not represented by an algebraic equation. Instead, the non-linear restoring force has to be calculated from solving the statedependent first order differential equation (8.15). This poses some problems to the calculation of the equivalent stiffness and damping values. Nonetheless, the influence of microslip on the describing function was investigated in a numerical study, where the governing parameters of the Valanis friction element were adjusted in such a way that the resulting hysteresis curve is fitted to the hysteresis of the 3-parameter elasto-slip element discussed above. The hysteresis curves of the Valanis friction element are shown in Figure 8.19 for different values of the microslip control parameter κ . Almost no microslip is present when κ = 0.99 , but microslip increases for smaller values of κ .

microslip

microslip

Figure 8.19: Force-displacement curves of Valanis friction element including microslip effects

The calculation of equivalent stiffness and damping values of the Valanis friction element was achieved by numerically integrating the differential equation (8.15) for


89

different amplitudes of enforced harmonic motion ranging from 1 to 200 in this case. The time history of the non-linear restoring force obtained from numerical integration was used subsequently to calculate the equivalent non-linear stiffness and damping values according to equations (4.20), (4.28), and (4.29) of the harmonic balance method. The results of the numerical study of the Valanis friction model are shown in Figure 8.20, from which the influence of microslip on the equivalent stiffness and the damping can be observed.

Figure 8.20: Equivalent non-linear stiffness and damping of the Valanis friction element

Different describing functions were obtained for different values of κ . The describing functions of the elasto-slip element are obtained as κ approaches 1. It can be observed that microslip is only effective in the small amplitude regime ( uˆ ≤ 2uc ) and that the Valanis friction element converges towards the elasto-slip element for large amplitudes, regardless of the microslip control parameter κ . Due to the specific problems introduced by the differential equation representation of the Valanis model, it cannot be recommended to be used in a non-linear frequency response analysis, because the calculation of the equivalent non-linear parameters involves numerical integration of the governing differential equation and would make the procedure numerically inefficient. f) Combined stiffness and damping non-linearities For linear systems, the influence of stiffness and damping can be separated and investigated individually. A change of stiffness causes a resonance peak shift, whereas a change of damping causes attenuation of the peak magnitude. For nonlinear systems, the influence of stiffness and damping non-linearities cannot be separated. For example, an increase of damping due to damping non-linearities causes a change of the displacement response amplitude. The reduced displacement response amplitude in turn affects the stiffness non-linearities which would be less activated due to the lower displacement amplitudes. This is kind of a pitfall which must be taken into account when identifying non-linearities. Essentially, the coupling between stiffness and damping non-linearities requires the simultaneous identification of these quantities instead of performing separate steps for the identification of stiffness and damping parameters as it is often done in linear model updating.

8.2.2

Hilbert Transformation

A popular approach for the detection of non-linear behavior is the so-called Hilbert transformation. In contrast to the Fourier transformation it does not provide a transformation from time domain to frequency domain. It is rather used to calculate an estimate of the real part FRF from the corresponding imaginary part FRF and

90


vice versa. These calculated estimates of real and imaginary part FRF are called the Hilbert transform and can readily be compared with the original FRF. Differences between the original FRF and its Hilbert transform indicate non-linear behavior, [Simon 1984]. It is frequently stated that the nature of the difference between FRF and Hilbert transform can also be used as an indicator for the type of the nonlinearity, however, it is known from practice that this is a very optimistic statement an cannot be fully agreed with. A recommended reference for the Hilbert transformation is [Worden 2001], where theory and application aspects are discussed in more detail. Regarding to the application of the Hilbert transformation, it has to be said that the computation of the Hilbert transform is not a trivial task. It requires the solution of a principal value integral which extents from minus infinity to plus infinity in the frequency domain. Since FRFs can only be measured in a limited frequency band, the calculation of the Hilbert transform suffers from truncation errors which can partly be balanced by correction terms to a certain extent. Even though, it might still be the case that differences between the original FRF and its Hilbert transform can appear, which are the results of truncation errors instead of non-linearity, [Worden 2001]. It should be noted here, however, that the Hilbert transformation can essentially only be used to detect non-causality of structures, instead of non-linearity. Nonetheless, it is widely accepted that non-causality detected by the Hilbert transform is also an indicator for non-linearity. An example for the Hilbert transform is presented in Figure 8.21 where a non-linear FRF obtained from constant force level measurements is compared with its Hilbert transform. The original FRF has only a slight degree of non-linearity which cannot be detected at first sight (the excitation force level was relatively low so that nonlinear effects are not very pronounced). However, when comparing the FRF with its Hilbert transform the differences between the two curves indicate the presence of non-linearity. The effect of the non-linear distortion of the Hilbert transform can also be observed in the Nyquist plot (imaginary part response plotted over real part response), which is in some cases a more powerful tool for the detection of nonlinearity than the Bode plots (response plotted vs. frequency).

Figure 8.21: Bode plots and Nyquist plot of FRF and Hilbert transform of a non-linear system


8.2.3

91

Distortions of Frequency Isochrones in a Nyquist Plot

The Nyquist plot itself can be used as a tool for the detection of non-linearities. Linear responses appear as circular curves in the Nyquist plot, whereas nonlinearities cause distortions from the circular form. When the frequency responses (absolute response, not FRF) obtained at different levels of excitation are plotted together in a single Nyquist plot, then the deviation of the frequency isochrones (lines which connect points of equal frequency among the different response curves) from straight lines indicate the presence of non-linearity, [Tomlinson 1986]. This can be observed in Figure 8.22 where the Nyquist plots with frequency isochrones of a linear and a non-linear system are shown.

Figure 8.22: Nyquist plots with frequency isochrones of linear and non-linear system

8.2.4

Inverse FRF Plot

The inverse FRF method seeks to separate the effects of non-linear stiffness and nonlinear damping by using the inverse of an FRF. It is essentially a single degree of freedom technique but can in principle also be applied to systems with well separated modes. It is expected for linear systems that the real part of the inverse FRF is linear with frequency squared (when mass is assumed to be constant). The imaginary part of the inverse FRF is expected to be directly proportional to the frequency in case of viscous damping, or respectively, to be constant in case of structural damping (hysteretic damping). If the stiffness is amplitude dependent, then non-linearity is indicated by the real part of the inverse FRF plot deviating from the straight line characteristic. On the other hand, if the damping is amplitude dependent, then non-linearity is indicated by the imaginary part of the inverse FRF plot deviating from the straight line characteristic. An example is given in Figure 8.23 where it can be seen that the real part of the inverse FRF deviates from a straight line when plotted over the frequency squared. The imaginary part of the inverse FRF is a straight line when plotted versus frequency. Consequently, the system has a stiffness non-linearity only.

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Figure 8.23: Inverse FRF plot of system with stiffness non-linearity only

8.2.5

Restoring Force Surface and Force-State Mapping

Time domain response data can also be used for the detection and characterization of non-linear behavior. A very popular approach is the restoring force surface method according to [Masri 1979], or respectively, the force-state mapping according to [Crawley 1986]. Extensions of these methods were recently published in [Dimitriadis 2001], [Wright 2001], and [Göge 2003]. This group of methods is based on the non-linear equation of motion of a single degree of freedom system according to the following equation:

mu(t ) + f R (u, u, t ) = f (t ) .

(8.20)

The identification of the non-linear restoring force

f R (u, u, t ) requires the measurement of the excitation force f (t ) and the acceleration response u(t ) . If the mass of the system is known the non-linear restoring force can be obtained: f R (u, u, t ) = f (t ) − mu(t ) .

(8.21)

If the displacement response u (t ) and the velocity response u (t ) were also measured, the restoring force can be plotted as a 3 dimensional surface over displacement and velocity. This is shown, for example, in Figure 8.24 where the restoring force of a single DoF oscillator is shown with a cubic hardening stiffness according to the following equation:

mu(t ) + cu(t ) + ku(t ) + knl u 3 (t ) = f (t ) .

(8.22)

f R ( u , u ,t )

Even though the measurement of the acceleration response and the excitation force can easily be done by using piezo-electric sensors, it can be quite difficult to measure the velocity- and the displacement responses. Modern laser vibrometers provide means for such measurements so that the method can efficiently be applied. If velocity and displacement response cannot be measured directly, they have to be calculated by numerically integrating the measured acceleration response.


93

Figure 8.24: Restoring force surface (force-state map) of a system with cubic non-linear stiffness

The three-dimensional restoring force surface produced by this method can be used not only to characterize the type of stiffness- and/or damping non-linearity. By curve fitting of the non-linear restoring force surface it is possible to derive a meaningful model for the non-linearity. The parameters of this model can be identified in a subsequent step, e.g. by minimizing the differences between analytical and experimental non-linear FRFs as it is proposed in this thesis.

8.2.6

Final Remark on Detection and Characterization Methods

Even though there are many more methods available for the detection and characterization of non-linear behavior, only a few can be mentioned here. Emphasis was given to methods which use frequency domain response data, because this type of data will be used later on for the identification of non-linear parameters of a finite element model. Other methods for the detection and characterization of nonlinearities can be found in specific literature, e.g. [Worden 2001], [Ewins 2000]. [Vanhoenacker 2002] provides a comprehensive summary of the recent techniques developed for the detection and characterization of non-linearities.

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9. Quality Assessment for FE Model and Test Data

Chapter 9: Quality Assessment for FE Model and Test Data 9.1

Finite Element Model Quality Assessment

Once an FE model was built it should undergo a number of quality checks to assess the accuracy of the results to be obtained using this model. These quality checks should involve basic mathematical checks, engineering checks, and the comparison of model predictions with experimental results.

9.1.1

Mathematical Checks

Basic mathematical checks for the assessment of the FE model quality are in most cases described in the FE code specific software manuals, e.g. [Nastran 1994]. Only a brief outline of some basic checks will be given here. A simple mathematical check is to prove the ability of the model to undergo rigid body motion in case of statically underdetermined boundary conditions. The number of zero frequency modes (rigid body modes) should not exceed 6 in case of a totally unconstrained spatial model. Generally less than 6 rigid body modes are obtained in case of statically underdetermined constraints. The presence of more than 6 rigid body modes either indicates the existence of unconstrained mechanisms in the model, or insufficient connectivity between substructures of an assembled model. The ratio of the eigenfrequencies of the first elastic mode and the highest rigid body mode is frequently used as a mathematical check and should be >104 for simple models and >103 for large order assembled models of complex structures. Good numerical conditioning can be proved by the ratio of the largest and the smallest main diagonal factor of the stiffness matrix being smaller than 105. In addition, highly distorted elements should generally be avoided. Acceptable limits for element distortions like aspect ratio, skewness, taper, and warping are element type specific and have to be checked with the corresponding FE code software manuals, e.g. [Nastran 1994].

9.1.2

Engineering Checks

A simple but nonetheless effective engineering check is the comparison of the mass of the model with the real weighted mass of the structure. It would generally be desired to compare other inertia properties as well, like the position of the center of gravity (CoG) or mass moments of inertia. However, these inertia properties are sometimes difficult to measure. Different methods for the measurement of inertia properties are summarized in [Schedlinski 1997] and [Schedlinski 2001].

9.1.3

Comparison of Model Predictions with Experimental Results

A detailed comparison of model predictions with experimental results is part of the FE model validation process. For example, experimental modal data is frequently used for correlation purposes and the FE model quality can be assessed in terms of eigenfrequency deviations and MAC values (Modal Assurance Criterion,


95

[Allemang 1982]). Model validation shall not be discussed in detailed in this chapter, but rather the aspect of the comparison of some fundamental model predictions with experimental results for the rough assessment of the model quality. For example, the comparison of the first few experimental eigenfrequencies with the analytical eigenfrequencies can readily be performed and gives a quick overview about the FE model quality. When a model of a new design is first completed, it may well be the case that experimental data does not yet exist. It is usually still possible to assess the FE model quality in some more limited way, such as to reference to hand calculations. For instance, the first few eigenfrequencies of a simple aero-engine casing may be checked by using formulae for natural frequencies of a plain cylinder.

9.1.4

Different Sources of Model Errors

According to [Link 1999a], FE model errors mainly arise from three different sources. These are idealization errors, discretisation errors, and erroneous assumptions for model parameters. Idealization errors arise from the assumptions made to characterize the mechanical behavior of the structure. This involves the choice of element types and element formulations for the representation of certain parts of the structure to be modeled. For example, can spokes and vanes of aero-engine casings appropriately be modeled by beam elements, or are other element types better suited? Discretization errors are introduced by the FE inherent numerical methods for the discretization of continuous structures. This involves typical meshing problems, for example, if a certain mesh of finite elements with their inherent shape functions is appropriate to represent the true deformation of a structure under a given loading condition, or if either mesh refinement (h-convergence) or shape function enhancement (p-convergence) is necessary to improve the discretization. Erroneous assumptions for model parameters are introduced whenever the properties of discrete elements (stiffness, mass, and damping) can only be estimated roughly. This leads to a low level of confidence with these element properties. Prominent examples for this type of model error are the determination of stiffness and damping properties of a simple bolted joint and modeling of a shell-like area of varying thickness by using a mesh of shell elements with constant overall thickness. Idealization and discretization errors cannot be detected and corrected automatically. Engineering judgment is required for the detection of such model errors within initial model quality checks. A set of test problems for the detection of discretization errors is proposed in [MacNeal 1985]. The detection of idealization errors is largely dependent on the experience of the personnel responsible for the FE model generation. Erroneous model parameters can be corrected automatically by using computational model updating. A number of model parameters can be corrected simultaneously which cannot be done manually within reasonable timescales. However, computational model updating will only yield meaningful results for the model parameters if the initial model structure is physically meaningful. This means that idealization and discretization errors must be eliminated prior to computational model updating (as far as possible) and the predictions of the initial model should not deviate too much from experimental results. The requirements posed on FE models for computational model updating are summarized in [Link 1999a]. Furthermore, the quality requirements for updated models are discussed there with respect to their final utilization.

96

9.2


Test Data Quality Assessment

Eigenfrequencies and mode shapes are commonly used for FE model validation. These modal quantities are usually extracted from measured FRFs. Thus, test data quality affects computational model updating and is therefore discussed here. Test data quality comprises the quality of the complete test setup, the measured FRFs, and the quality of the modal analysis carried out on the measured FRFs. In many cases, the person who is responsible for FE model validation is supplied with experimental modal data, but is not supplied with information about the test data quality. This information is necessary, however, to justify the use or omission of some of the data for model updating. According to [DTA 1995], a quality control checklist for FRF measurements encloses items like: • measurement- and test uncertainties, • calibration of measurement chain, • comparison of FRFs, • coherence functions, • over- and under ranging of measured signals during A/D conversion, • excitation spectrum criteria, • presence of unmeasured excitation, • aliasing and signal filtering, • noise reduction, • point FRF criteria, • reciprocity checks, • effects of non-linearity, • exercise of the structure, • transverse sensitivity of transducers, • repeatability of measurements (stability of data). In addition to these items, the quality of the modal analysis must be considered as well. This would involve the following items: • robustness of extracted modes with respect to different modal extraction methods, • comparison of synthesized and experimental FRFs, • check for excessive numerical values of modal damping and/or modal mass, • orthogonality check of experimental modes. A discussion of these items is certainly beyond the scope of this thesis but must be mentioned here, because it is known from experience that when it comes to updating, the attention of the person in charge of the work is focused on the selection of updating parameters, the performance of the updating code, and subsequently the interpretation of the updating results. Test data is seldom considered uncertain at this stage of model validation (because it was actually measured!). [Brüel 1982] provides an overview about practical issues of modal testing. For example, sensor and cable mounting are discussed there and the influence of environmental conditions on the measurements. [DTA 1995] gives a broad overview about most items involved in vibration measurements and indicates possible pitfalls. [Maia 1997] and [Ewins 2000] provide more scientific insight into vibration measurements together with the theory behind modal analysis procedures.

10. Theory of Non-Linear Parameter Updating

97

Chapter 10: Theory of Non-Linear Parameter Updating Finite element modeling aims at generating predictive mathematical models to be used e.g. for numerical simulations of the dynamic behavior of structures. A basic requirement for meaningful numerical simulations is the ability of the models to predict the mechanical behavior of real structures with reasonable accuracy. In practice, FE models are never exact and can only approximate the true behavior of structures. This applies in particular to models of complex assembled structures, where the representation of the interface conditions between substructures is often a major source of model error. Model updating is a tool for improving the prediction capability of FE models. This is achieved by adjusting uncertain model parameters in such a way that the differences between experimental data and model response predictions (analytical data) are minimized. Parameters, in this respect, are the FE model properties. For example, global parameters like Young’s modulus and material density, but also local parameters such as cross sectional areas of beams and rods, thicknesses of shells and plates, or stiffness and damping coefficients of discrete spring/damper elements are prominent examples for model parameters. Model updating can be performed manually when only a few uncertain parameters must be adjusted. Engineering judgment can help in this case to adjust the parameters one at a time. With increasing complexity of a model, however, the number of uncertain model parameters increases inevitably so that manual model updating can no longer be performed within reasonable timescales. Procedures for automated parameter adjustment were developed in the last decades. These procedures are known as computational model updating (CMU) and are aimed at minimizing the differences between experimental and analytical data by simultaneously adjusting a number of uncertain model parameters which would be impossible by using engineering judgment alone, [Natke 1992], [Friswell 1995]. Since the CMU procedures involve automated correlation of experimental and analytical data, they can effectively be applied as a tool within model validation, see Figure 2.1. The important steps of CMU are discussed in the following. Among those are the parameterization of the FE model, the calculation of parameter changes, the choice of appropriate updating residuals, the calculation of parameter sensitivities, and tools for regularization.

98


10.1 Parameterization of Finite Element Models Computational model updating is based on the parameterization of the FE model system matrices according to the so-called substructure matrix approach, [Natke 1992], [Friswell 1995], [Link 1999b]. In order to generalize the updating theory to include the non-linear case some extensions have to be introduced. A general formulation for the parameterization of an FE model is achieved by a Taylor series expansion of the system matrices at a certain linearization point (index 0): I

[M ] = [M 0 ] + ∑ i =1

∂ [M ] Δmi , ∂mi 0

(10.1)

∂ [K ] Δk j , j =1 ∂k j

(10.2)

∂ [C ] Δck . k =1 ∂ck 0

(10.3)

J

[ K ] = [ K0 ] + ∑

0

K

[C ] = [C0 ] + ∑ [ M 0 ] , [ K0 ] , [C0 ]

are the mass-, stiffness-, and damping matrix at the linearization point, Δmi = mi − mi 0 , Δk j = k j − k j 0 , Δck = ck − ck 0 denote the changes of the uncertain mass-, stiffness-, and damping parameters mi , k j , and ck from their initial values mi 0 , k j 0 , ck 0 at the linearization point. It is convenient to convert the general formulation of FE model parameterization into the following form: I

[ M ] = [ M 0 ] + ∑ mi 0 i =1 J

[ K ] = [ K0 ] + ∑ k j 0 j =1 K

(10.4)

∂ [K ] βj , ∂k j

(10.5)

∂ [C ] γk , ∂ck 0

(10.6)

0

[C ] = [C0 ] + ∑ ck 0 k =1

∂ [M ] αi , ∂mi 0

where dimensionless design parameters:

αi = Δmi mi 0 ,

(10.7)

β j = Δk j k j 0 ,

(10.8)

γ k = Δck ck 0 ,

(10.9)

indicate the rate of change of the uncertain mass-, stiffness-, and damping parameters at the linearization point. Type and location of the assumed FE model errors are described by the so-called substructure matrices:

[ M i ] = mi 0

∂ [M ] , ∂mi 0

(10.10)


99

∂ [K ] ⎡⎣ K j ⎤⎦ = k j 0 , ∂k j

(10.11)

∂ [C ] . ∂ck 0

(10.12)

0

[Ck ] = ck 0

These substructure matrices may either affect a single element, a group of elements (e.g. all elements which refer to the same material property), or even a complete component model of an overall assembled model. The dimensionless design parameters αi , β j , γ k describe the magnitudes of the assumed FE model errors related to the substructure matrices [ M i ] , ⎡⎣ K j ⎤⎦ , [Ck ] . For simplification, the dimensionless design parameters are comprised in the parameter vector { p} :

{ p} = {α1

αI

β1

{ p} = { p1

γK} , T

βJ γ1

(10.13)

pnp } , np = I + J + K . T

(10.14)

10.2 Determination of Parameter Changes The parameterization of the FE model according to equations (10.4) to (10.6) allows for local updating of uncertain model parameters. Using these equations together with appropriate residuals containing test/analysis differences, the following objective function J can be established which shall be minimized by the updating process:

{

} [W ]{r ({ p})} → min . {r ({ p})} is a function of the

J ({ p}) = r ({ p})

T

(10.15)

The so-called residual vector unknown design parameters and [W ] is a diagonal weighting matrix. The residual vector r ({ p}) contains the difference between the test data vector u t and the analytical data vector u a ({ p}) (superscript t denotes “test data”, whereas superscript a denotes “analytical data”):

{

{ }

}

{r ({ p})} = {u } − {u ({ p})} . t

a

{

}

(10.16)

It can be seen from the quadratic form of equation (10.15) that the objective function represents the weighted square sum of the test/analysis differences. The analytical data vector u a ({ p}) usually depends in a non-linear way on the design parameters. Thus, the minimization problem stated in equation (10.15) is also non-linear and must be solved iteratively. This can be achieved by using the classical sensitivity approach [Fox 1968], where the analytical data vector is linearized at the linearization point 0 by a Taylor series expansion truncated after the linear term:

{

}

{u ({ p})} ≈ {u } + [G ]{Δp} . a

{ }

a 0

0

(10.17)

In this equation, u0a is the analytical data vector at the linearization point. The socalled sensitivity matrix [G0 ] contains the first order derivatives of the analytical data with respect to the design parameters:

100


[G0 ] = {Δp}

{

}

∂ u a ({ p}) ∂ { p}

.

(10.18)

0

are the design parameter changes with respect to the initial values at the

linearization point:

{Δp} = { p} − { p0 } .

(10.19)

Introducing equation (10.17) into equation (10.16) yields a linearized residual vector which is a linear function of the parameter changes {Δp} :

{r ({Δp})} = {u } − {u } − [G ]{Δp} = {r } − [G ]{Δp} . t

a 0

0

0

0

(10.20)

Introducing the linearized residual vector into the objective function (10.15) and furthermore introducing a so-called regularization term to account for possibly illconditioned sensitivity matrices yields the following quadratic objective function:

{

} [W ]{r ({Δp})} + {Δp}

J ({Δp}) = r ({Δp})

T

T

⎡⎣W p ⎤⎦ {Δp} → min .

(10.21)

regularization term

residual term

The diagonal weighting matrix ⎡⎣W p ⎤⎦ is the so-called regularization matrix which can be used to constrain excessive parameter variation in case of ill-conditioned updating equations. A discussion of regularization is postponed until later on. If the design parameters are not bounded, the minimization of the objective function (10.21)

J ({Δp}) → min

⇒

∂J ({Δp}) ∂ {Δp}

= {0}

(10.22)

leads to the following equation for the calculation of design parameter changes:

{Δp} = ([G0 ] [W ][G0 ] + ⎡⎣Wp ⎤⎦ ) [G0 ] [W ]{r0 } . T

−1

T

(10.23)

The parameter changes calculated from equation (10.23) are the rates of change of the dimensionless design parameters at the current linearization point. With these design parameter changes, better estimates of the system matrices can be calculated according to equations (10.4) to (10.6), which better approximate the experimental data. The fit of the system matrices to the experimental data will not be exact due to the Taylor series approximation introduced for the analytical data. However, the new system matrices represent a new model which can be used in turn as a new initial system for a subsequent step of model updating. Consequently, computational model updating is an iterative procedure which keeps on iterating until the objective function of equation (10.21) drops below a certain threshold value, or respectively, until the test/analysis differences converge towards acceptable levels with respect to the intended usage of the FE model. It should be noted that the Taylor series approximation for the analytical data is only accurate within a small range of parameter changes around the linearization point. Therefore, the parameter changes calculated in each iteration step (i.e. at each linearization point) should be restricted in some way. A maximum change of about 20% per iteration was found to be practical. Larger parameter changes can be allowed for less sensitive parameters whose sensitivities are not expected to change significantly.


101

10.3 Residuals for Linear Parameter Updating The residuals used in computational model updating can be manifold and many of them have actually been proposed. Most frequently, eigenfrequency and mode shape residuals are utilized for linear model updating, where the residual vector comprises eigenfrequency and mode shape differences obtained at the linearization point 0:

⎧ ω1t − ω1a ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ωnt − ωna ⎪ {r0 } = ⎪⎨ φ t − φ a ⎪⎬ . ⎪{ 1 } { 1 }⎪ ⎪ ⎪ ⎪ t ⎪ ⎪⎩{φn } − {φna }⎭⎪ 0

(10.24)

Here, ωit are the test eigenfrequencies and ωia are the analytical eigenfrequencies obtained at the linearization point. φit are the test mode shape vectors and φia are the analytical mode shape vectors obtained at linearization point. The sensitivity matrix for the eigenfrequency and mode shape residual according to equation (10.24) can be expressed in the following way:

{ }

⎡ ∂ω1a ⎢ ∂p 1 ⎢ ⎢ ⎢ a ⎢ ∂ωn ⎢ ∂p 1 [G0 ] = ⎢⎢ ∂ φ a { 1} ⎢ ⎢ ∂p1 ⎢ ⎢ ⎢ ∂ {φ a } n ⎢ ⎣⎢ ∂p1

{ }

∂ω1a ⎤ ∂pnp ⎥ ⎥ ⎥ ⎥ ∂ωna ⎥ ∂pnp ⎥ ⎥ a ⎥ . ∂ {φ1 } ⎥ ∂pnp ⎥ ⎥ ⎥ a ⎥ ∂ {φn } ⎥ ∂pnp ⎦⎥

(10.25)

0

The upper partition of the sensitivity matrix contains the partial derivatives of the analytical eigenfrequencies with respect to the design parameters, whereas the lower partition contains the partial derivatives of the analytical eigenvectors with respect to the design parameters.

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10.4 A Residual for Linear and Non-Linear Parameter Updating In case of models with linear and non-linear parameters to be updated, the theory of computational model updating can be applied in the same way. However, the type of residual must be adapted to the specific application. Eigenfrequencies and mode shapes are not defined for non-linear systems so that it is reasonable to use nonlinear response data for model updating. The updating residual proposed here contains the differences between experimental and analytical non-linear frequency response functions (FRFs) obtained at different constant excitation force levels. FRFs are typically available at discrete frequencies only. Hence, the residual vector in case of non-linear updating comprises with the non-linear FRF differences at the discrete frequencies Ωi ( i = 1,… , n ):

⎧ H tjk ( Ω1 ) − H ajk ( Ω1 ) ⎫ ⎪ {r0 } = ⎪⎨ ⎬ . t a ⎪ H (Ω ) − H (Ω )⎪ jk n ⎭ ⎩ jk n

(10.26)

0

H tjk ( Ωi ) is the experimental non-linear FRF at the frequency point Ωi measured at the DoF j while the excitation force was applied at DoF k . H ajk ( Ωi ) denotes the corresponding analytical non-linear FRF obtained at the linearization point, i.e. with the initial set of parameters. The sensitivity matrix for the frequency response residual contains the partial derivatives of the analytical FRFs with respect to the design parameters:

⎡ ∂H ajk ( Ω1 ) ⎢ ⎢ ∂p1 [G0 ] = ⎢⎢ ⎢ ∂H ajk ( Ω n ) ⎢ ∂p ⎢⎣ 1

∂H ajk ( Ω1 ) ⎤ ⎥ ∂pnp ⎥ ⎥ . ⎥ ∂H ajk ( Ω n ) ⎥ ∂pnp ⎥⎥⎦

(10.27)

0

The residual vector for the frequency response residual is stated for a single FRF H jk (Ω) in equation (10.26). In case of non-linear updating, the FRFs of different levels of constant excitation force can be stacked in the residual vector (10.26) and consequently must be stacked in the sensitivity matrix (10.27).


103

10.5 Calculation of Sensitivities 10.5.1 Eigenfrequency and Mode Shape Sensitivity It was already stated above that eigenfrequencies and mode shapes are commonly used for updating of linear models. An efficient way for calculating eigenfrequency and mode shape sensitivities was derived by Fox, [Fox 1968]. Nelson derived a simplified method for the calculation of eigenvector sensitivities, [Nelson 1976]. These methods for sensitivity analysis are mentioned here for the sake of completeness but shall not be discussed further, because they can be considered as a standard in computational model updating.

10.5.2 Linear Frequency Response Function Sensitivity The calculation of linear FRF sensitivities is subject of discussion in textbooks on model updating. For example, an analytical expression for frequency response sensitivities can be found in [Natke 1992]. Brandon derived an expression for FRF sensitivities based on structural modification theory, [Brandon 1990]. This sensitivity formulation is in particular efficient when the number of DoFs affected by stiffness and damping parameter changes is relatively small compared to the overall number of DoFs.

10.5.3 Non-Linear Frequency Response Function Sensitivity by Finite Differences A closed form analytical solution for the calculation of non-linear FRF sensitivities does not yet exist, at least not to the knowledge of the author. In general, sensitivities (linear and non-linear) can be approximated by finite differences. The sensitivity of the analytical FRF H kla ( Ω ) with respect to the parameter p j can be approximated by the following finite difference expression:

∂H kla ({ p}, Ω ) ∂p j

=

H kla

({ p} + δ p {e }, Ω ) − H ({ p}, Ω ) . j

δp

a kl

(10.28)

In this approximation, δ p is a small parameter perturbation which is applied to parameter p j by using the j-th unit vector e j . It can be seen from equation (10.28) that two different non-linear FRFs are required to compute the non-linear FRF sensitivity: One FRF which was obtained by using the initial parameters { p} , and another one obtained with a small perturbation δ p applied to the j-th parameter, i.e. { p} + δ p e j . When considering that the non-linear FRF obtained with the initial parameters is known in advance (e.g. the result of the previous iteration step of updating), it can be concluded that the effort for computing the whole sensitivity matrix is directly proportional to the number of updating parameters. A non-linear frequency response analysis is required for each parameter to be identified. The finite difference approximation of the true sensitivity requires that the imposed parameter perturbation δ p is sufficiently small. It is known from experience that 0.01% is usually adequate. It must be noted, however, that the parameter perturbation should not be too small. Erroneous sensitivities can be the result of numerical round-off errors which may arise in case of relatively insensitive parameters. For example, the FRF sensitivities can be corrupted when the difference

{ }

{ }

104


between the non-linear FRFs required for finite difference approximation cannot be observed properly due to limited numerical accuracy in case of an excessively small parameter perturbation. Even though the accuracy of the sensitivities calculated by finite differences depends on the magnitude of the parameter perturbation, it has to be noted that the finite difference approach provides a reliable means for calculating sensitivities for almost every type of parameter, including geometry parameters like the position of nodes of the FE model.

10.5.4 Non-Linear Frequency Response Function Sensitivity Based on Structural Modification Theory The structural modification approach for the calculation of FRF sensitivities presented in [Brandon 1990] offers a new way for the calculation of non-linear FRF sensitivities as well. In this approach, the non-linear elements are considered as structural modifications which are applied to only few so-called modification DoFs of the initial system. The FRF sensitivity of linear systems will be discussed first and the extension to FRF sensitivity of non-linear systems will be discussed afterwards. The FRF matrix [ H ] is the inverse of the dynamic stiffness matrix [ Z ] (impedance matrix). This applies not only to the original system, but also to the modified system:

[H ] = [Z ]

−1

→ ⎡⎣ H m ⎤⎦ = ([ Z ] + [ ΔZ ]) .

The structural modification matrix modifications and is generally sparse:

−1

[ ΔZ ]

(10.29)

shall contain stiffness and damping

[ ΔZ ] = [ΔK ] + jΩ [ΔC ] .

(10.30)

According to [Brandon 1990] and [Natke 1992] is the FRF sensitivity the partial derivative of the FRF matrix with respect to a certain parameter pm and can be expressed by the following equation:

∂[H ] ∂ [ ΔZ ] = −[H ] [H ] . ∂pm ∂pm

(10.31)

If the parameter pm is a dimensionless design parameter β j , e.g. related to a stiffness modification according to [ ΔK ] = β j ⎡⎣ K j ⎤⎦ , then the partial derivative of the modification matrix in equation (10.31) becomes:

(

)

∂ [ ΔZ ] ∂ ([ ΔK ] + jΩ [ ΔC ]) ∂ β j ⎣⎡ K j ⎦⎤ = = = ⎡⎣ K j ⎤⎦ . ∂β j ∂β j ∂β j

(10.32)

On the other hand, if the parameter pm is a dimensionless design parameter γ k related to a damping modification according to [ ΔC ] = γ k [Ck ] , then the partial derivative in equation (10.31) becomes:

∂ ( γ k [Ck ] ) ∂ [ ΔZ ] ∂ ([ ΔK ] + jΩ [ ΔC ]) = = jΩ = jΩ [Ck ] . ∂γ k ∂γ k ∂γ k

(10.33)

When considering the important case of a unit rank modification, i.e. when the structural modification only affects the two DoFs i and j of the initial system, then the structural modification matrix [ ΔZ ] can be expressed by the dyadic product of equation (4.79):


105

[ ΔZ ] = Δz {eij }{eij }

T

,

(10.34)

where Δz describes the magnitude and the type of the modification:

Δz = Δk + jΩ Δc .

(10.35)

{ }

The vector eij , already derived in equation (4.80), defines the location of the structural modification:

⎧ {eij } = {ei } − {e j } = ⎪⎨ ⎩⎪

T

i

j

+1

−1

⎫⎪ ⎬ . ⎭⎪

(10.36)

When using the equations for structural modification, the sensitivity of the FRF matrix with respect to a parameter pm can be expressed by the relatively simple equation: T ∂ ( Δz ) ∂ [H ] . = − [ H ]{eij }{eij } [ H ] ∂pm ∂pm

(10.37)

{ }{ }

T

When exploiting the sparsity of the dyadic product eij eij , the sensitivity of a single FRF with respect to the parameter pm can be written as:

∂ ( Δz ) ∂H kl , = − ( H ki − H kj )( H il − H jl ) ∂pm ∂pm

(10.38)

where H kl is the FRF of the initial system observed at DoF k while the excitation force was introduced at DoF l . If the parameter pm is a dimensionless design parameter β j , e.g. related to a stiffness modification of the j-th 2-DoF-spring-element according to Δk = β j k j , then the last term of equations (10.37) and (10.38) becomes:

∂ ( Δz ) ∂ ( Δk + jΩ Δc ) ∂ ( β j k j ) = = = kj. ∂β j ∂β j ∂β j

(10.39)

On the other hand, if the parameter pm is a dimensionless design parameter γ k related to a damping modification of the k-th 2-DoF-damper-element according to Δc = γ k ck , then the partial derivative in equations (10.37) and (10.38) becomes:

∂ ( Δz ) ∂ ( Δk + jΩ Δc ) ∂ (γ k ck ) = = jΩ = jΩ ck . ∂γ k ∂γ k ∂γ k

(10.40)

Even though these equations for FRF sensitivity analysis based on structural modification were stated for linear systems, they can be applied to non-linear systems as well. In this case, the matrix [ H ] represents the initial non-linear FRF matrix obtained at a given level of excitation and is known a priori, e.g. as the result of the last iteration step of non-linear updating. The structural modification matrix [ ΔZ ] becomes response dependent in this case and may contain contributions of linear and non-linear spring and damper elements. As an example, the non-linear FRF sensitivity shall be calculated with respect to a dimensionless design parameter β j which shall be related to the physical non-linear spring stiffness knl of a 2-DoF cubic spring element. The non-linear element shall be located between the DoFs i and j of the original system and shall have a cubic restoring force function:

106


f R (u, t ) = knl ( u j (t ) − ui (t ) ) . 3

(10.41)

When applying the Harmonic Balance method to such a cubic non-linear spring an equivalent response dependent structural modification can be obtained (see also chapter 8.2.1c and equation 8.8):

⎡⎣ ΔK ({uˆ})⎤⎦ = β j ⎡⎣ K eq , j ({uˆ})⎤⎦ =

3 4

( β k ) Δuˆ {e }{e } j nl

2 ij

T

ij

ij

,

(10.42)

keq ( Δuîj )

where Δuîj is the magnitude of the complex relative displacement response between the two DoFs i and j where the non-linear structural modification is located (see equation (4.37)):

Δuîj = uˆ j − uî .

(10.43)

The corresponding FRF sensitivity with respect to the dimensionless design parameter β j (which is related to the physical non-linear parameter knl , not to the equivalent non-linear parameter keq ) can be written as follows:

∂ ⎡ ΔK ({uˆ})⎤⎦ ∂k ( Δuˆ ) T ∂[H ] = −[H ] ⎣ [ H ] = − [ H ]{eij }{eij } [ H ] eq ijj . ∂β j ∂β j ∂β j

(10.44)

In case of the simple cubic spring a closed form solution can be obtained for the partial derivative of the equivalent non-linear parameter keq with respect to the dimensionless non-linear design parameter β j :

∂keq ( Δuîj ) ∂β j

= 43 knl Δuîj2 .

(10.45)

It can be seen from equation (10.44) that the non-linear FRF sensitivity is response dependent. This means that the sensitivity of a single non-linear FRF will change with the response level and consequently with the excitationT force level. When exploiting the sparsity of the dyadic product eij eij in equation (10.44), the sensitivity of a single non-linear FRF with respect to non-linear design parameter β j can be written as:

{ }{ }

∂keq ( Δuîj ) ∂H kl = − ( H ki − H kj )( H il − H jl ) . ∂β j ∂β j

(10.46)

Here, H kl is the FRF of the initial system observed at DoF k while the excitation force was introduced at DoF l . The sensitivity of a single column of the FRF matrix can be derived from equation (10.46). For example, the sensitivity of the l-th column of the FRF matrix can be calculated from the difference between the i-th and the j-th column of the initial FRF matrix:

∂keq ( Δuîj ) ∂ {H l } . = − {H i } − {H j } ( H il − H jl ) ∂β j ∂β j

(

)

(10.47)

The partial derivative of the equivalent non-linear stiffness parameter k eq (or respectively of the equivalent non-linear damping parameter ceq ) with respect to a non-linear design parameter is a scalar quantity. A closed form solution can be obtained for simple non-linear elements such as polynomial type non-linearities.


107

However, this partial derivative can also be computed by finite differences. This can be efficient for non-linearities where the equivalent non-linear parameters depend in a complicated way on the non-linear design parameters related to the physical nonlinear parameters that govern the non-linear restoring force function. Even though equations (10.44), (10.46), and (10.47) for FRF sensitivity derived from structural modification theory look rather complicated, it must be stated that they offer an attractive way for response sensitivity analysis required during non-linear updating. If the FRF matrix is known a priori than these equations offer an efficient way for the calculation of non-linear FRF sensitivities in case of unit rank modifications, because in contrast to the finite difference approach for sensitivity analysis, additional non-linear frequency response analyses with perturbed parameters are no longer required.

10.6 Regularization Regularization was introduced in equation (10.21) as a modification to the original objective function with the aim to constrain excessive parameter variation in case of ill-conditioned updating equations. The effect of regularization can be observed from equation (10.23) when regularization is disregarded:

{Δp} = ([G0 ] [W ][G0 ]) [G0 ] [W ]{r0 } . T

−1

T

(10.48)

This is a weighted least-squares equation for estimating the parameter changes {Δp} from the residual {r0 } by using the sensitivity matrix [G0 ] . It is known from experience that the parameter estimation can yield unsatisfactory results when the T matrix product [G0 ] [W ][G0 ] is ill-conditioned. Ill-conditioning can be a result of almost linear dependent rows or columns in the sensitivity matrix and can lead to rank deficiencies of the sensitivity matrix. Illconditioning can also be introduced by updating of very sensitive and almost insensitive parameters at the same time. Mottershead and Foster proposed tools based on singular value decomposition and regularization to improve the parameter estimation, [Mottershead 1991]. The idea of regularization is to improve the condition of the matrix product T [G0 ] [W ][G0 ] prior to inversion. This is achieved by adding Ta diagonal matrix ⎡⎣Wp ⎤⎦ which has full rank to the rank deficient matrix product [G0 ] [W ][G0 ] :

{Δp} = ([G0 ] [W ][G0 ] + ⎡⎣Wp ⎤⎦ ) [G0 ] [W ]{r0 } . T

−1

T

(10.49)

This approach for improving the condition of a matrix prior to inversion is known as Tikhonov regularization, [Tikhonov 1977]. It can be observed from equation (10.49) that the regularization matrix affects the results of the parameter estimation. In fact, the problem with regularization is to select a meaningful regularization matrix which is suitable to improve the condition of (10.49) while the parameter changes do not depart significantly from the solution that would be obtained without regularization and without the problem of ill-conditioning. For example, if the regularization matrix was selected carelessly so that the diagonal elements of ⎡⎣W p ⎤⎦ are too large, the influence of regularization is too strong and no parameter changes will occur. This is inconsistent with the objective of model updating. Link developed an approach for the calculation of the regularization matrix based on Tikhonov regularization, [Link 2000a]:

108


α

( (

))

⎡⎣Wp ⎤⎦ = w p 1 diag [G0 ] [W ][G0 ] α 2

T

−1

.

(10.50)

The mathematical operator diag (…) means that only the main diagonal of the matrix product in the brackets is retained whereas the off-diagonal terms are disregarded. The factor α1 is the average of the main diagonal factors of the matrix T product [G0 ] [W ][G0 ] . The factor α 2 is−1the average of the main diagonal factors of T the diagonal matrix diag [G0 ] [W ][G0 ] . The advantage of this regularization method is that the definition of the complete regularization matrix can be reduced to the definition of a single multiplication factor w p . This approach will be used throughout this thesis whenever regularization is required. It is known from experience that 0 ≤ w p ≤ 0.2 usually produces reasonable results while the updating performance is not corrupted.

(

)

11. Applications

109

Chapter 11: Applications The following examples shall demonstrate the field of application for non-linear updating. The software for non-linear frequency response analysis using SHBM and the software for non-linear updating based on frequency response residuals which were applied to the examples presented in this chapter were developed within the European research project CERES (Cost-Effective Rotordynamics Engineering Solutions funded by the European Commission under the Competitive and Sustainable Growth Programme, Framework V, Key Action 4, 1998-2002). The MATLAB based non-linear frequency response analysis software HBResp (Harmonic Balance Response analysis, see [HBResp 2003]) uses an interface to the commercial finite element software MSC.Nastran so that finite element modeling of the underlying linear system can be performed within the MSC.Nastran environment. The export of the underlying linear system is achieved by DMAPs (Direct Matrix Abstraction Programming) which must be included in the standard solution sequence for eigenvalue analysis. These DMAPs enforce the export of the underlying linear mass- and stiffness matrix to binary files, after the following steps have been performed: 1. assembly of system matrices, 2. processing of multi-point constraints (rigid elements), 3. introduction of boundary conditions (single point constraints), 4. application of model reduction techniques (Guyan reduction or Craig-Bampton reduction). In addition, the DMAPs enforce the export of eigenfrequencies and mode shapes of the undamped underlying linear system to binary files. These binary files can be imported into HBResp, where non-linear modeling is performed. Non-linear analysis control parameters can also be defined in HBResp and involves the specification of the: • effective frequency range, • excitation force magnitudes vs. frequency of excitation, • (multiple) excitation DoFs, • (multiple) response DoFs, • analysis approach (direct or modal), • type and amount of damping. Subsequently, non-linear frequency response analysis is performed within MATLAB by using the Harmonic Balance method. A typical appearance of the graphical user interface (GUI) of the non-linear frequency response analysis software HBResp is shown in Figure 11.1.

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11. Applications

HBResp Main Window

HBResp Parameter Definition Window MATLAB Command Window Figure 11.1: Typical GUI appearance of HBResp used for non-linear frequency response analysis

The MATLAB based non-linear updating software Update_NL (see [Update_NL 2004]) was developed within the European research project CERES. Update_NL does not aim at updating the equivalent non-linear stiffness and damping parameters, but rather the physical non-linear parameters which govern the non-linear restoring force functions and from which the equivalent non-linear parameters must be derived together with the response amplitudes. In principle, this would mean that the non-linear physical parameters identified with Update_NL can also be used for transient time domain response analyzes, i.e. the non-linear parameters are identified in the frequency domain to improve the accuracy of time domain response analysis. Update_NL provides efficient means for the import of experimental non-linear FRFs from universal files (type 58). Analytical data must be provided in HBResp database format. The element properties (linear and non-linear) of the HBResp model can be parameterized for updating within Update_NL. Database management is also provided by Update_NL. This is required for residual generation, especially when using response data of: • multiple load levels, • multiple measurement positions, • multiple resonances which show non-linear behavior, • multiple boundary conditions, for updating a unique set of linear and non-linear parameters. Update_NL runs HBResp automatically whenever non-linear frequency response analysis is required (e.g. for sensitivity analysis by using finite difference, or for calculating the non-linear response with a new set of parameters at the end of an updating step). Updating control parameters like: • absolute limits for the variation of updating parameters, • limits for the variation of updating parameters per iteration step • type of the updating residual (acceleration, velocity, or displacement, absolute response or FRF), • regularization, • coincidence information between experimental and analytical DoFs,

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111

• response plot options and sensitivity plot options, are defined within Update_NL. A typical GUI appearance of the MATLAB based software Update_NL can be observed in Figure 11.2.

Figure 11.2: Typical GUI appearance of the non-linear updating software Update_NL

Table 11-1 gives an overview about the applications which will be discussed in the following and about the different procedures applied during test, analysis, and model updating for each application. Table 11-1: Summary of non-linear updating applications Cantilever Beam

General Description

Non-Linear Vibration Test Data

Model Reduction Response Analysis

Model Updating

CTS

RB199

comment

analytical example

laboratory structure

aero-engine subassembly

model type

beam model

shell model

shell model

number of DoFs

18

ca. 9500

ca. 90000

214 beam elements

2360 beam elements

1440 shell elements

15930 shell elements

number of elements

8 beam elements

test procedure

--

step-sine base accel.

step-sine

level control

--

constant base accel.

constant exciation force exp. FRFs

response data

simulated responses

exp. transmissibilities

number of meas. DoFs

18

9

7

applied method

--

Craig-Bampton

Craig-Bampton

DoFs after reduction

18

686

5336

applied method

direct + modal SHBM

direct + modal SHBM

modal SHBM

number of modes used

study of multiple sets

5

10

number of parameters

3 and 2

3

3

type of residual

displ. response magn.

transmissibility magn.

displ. FRF magnitude

number of meas. DoFs used in residual

4

1

4

number of force levels

1

3

2

underlying linear system

variable and constant

constant

constant

regularization

no

yes, wp=0.05

yes, wp=0.05

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11.1 Updating of a Non-linear Cantilever Beam by using Simulated Test Data In this section, non-linear updating is discussed on an academic example using simulated test data. The system under consideration is a planar cantilever beam which is assembled of two beam components, see Figure 11.3. Each of the beam components is assembled of four beam elements according to Timoshenko beam theory. The two components are connected by a hinge, i.e. the translational DoFs at the joint are rigidly connected, whereas the rotational DoFs at the joint are connected by a combination of linear and non-linear 2-DoF elements. In the example discussed here, the joint is modeled by a linear spring in conjunction with a nonlinear softening cubic spring and a non-linear hardening quadratic damper. The support of the cantilever beam is modeled in a flexible way. The translational support motion is clamped, whereas the rotational support motion is constrained by a grounded linear rotational spring. Damping of the underlying linear system is represented by the approach according to equation (7.11). Modal viscous damping ratios of 0.54% were defined for the first two modes. Other modes are undamped. linear grounded rotational spring

hinged support

non-linear rotational spring and damper

component 1

component 2

F

tip mass

Figure 11.3: Sketch of non-linear cantilever beam

The undamped underlying linear MSC.Nastran model can be found in the Appendix. It already includes the linear support spring and the underlying linear joint spring. hinge: equal translational displacements, unequal rotational displacements

4

3

2

1

Figure 11.4: Degrees of freedom of the planar cantilever beam FE model

The planar cantilever beam model has 18 DoFs as indicated in Figure 11.4. Test data was simulated by perturbing three (linear and non-linear) parameters of the initial model. The non-linear updating software Update_NL was applied to simultaneously identify the imposed parameter changes from the differences between the simulated non-linear frequency responses obtained with the perturbed model and those obtained with the initial model.

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The parameters which were perturbed for test data simulation are the: • stiffness of the linear support spring k sup , •

non-linear stiffness of the softening cubic joint spring knl ,

•

damper constant of the hardening quadratic joint damper cnl .

These are physical non-linear parameters (not amplitude dependent equivalent nonlinear parameters). Table 11-2 summarizes the initial and perturbed parameter values (note the physical units of the non-linear parameters). Table 11-2: Initial parameter values, perturbed parameter values, and imposed parameter changes Parameter

Initial Value

k sup

1 ⋅ 1010

knl

−1.4 ⋅ 1015

cnl

3.75 ⋅ 105

Perturbed Value

1.25 ⋅ 1010

Nmm rad Nmm rad 3

N mm s rad 2

2

Nmm rad

−1.12 ⋅ 1015

4.875 ⋅ 105

+25%

Nmm rad 3

N mm s rad 2

Change

2

-20% +30%

Table 11-3 summarizes the correlation results of the underlying linear modal data of initial and perturbed model. It can be observed that the 25% change of the support stiffness has an influence on some of the lower eigenfrequencies but also on some higher frequencies. However, it seems that the modes are not strongly affected by the perturbation of the linear support spring. This is indicated by the relatively high MAC values. Pictures of the modes can be found in the Appendix. Table 11-3: Correlation table of initial model and perturbed model Mode Initial Model Number Freq. [Hz] 1 114,19 2 649,23 3 1930,83 4 4275,03 5 6171,04 6 9136,91 7 11062,30 8 13875,92 9 14924,44 10 104160,35 11 133767,50 12 195805,24 13 202148,33 14 205382,66 15 209315,13 16 218093,77 17 237936,10 18 353762,30

Perturbed Model Frequency MAC Freq. [Hz] Deviation [%] Value [%] 118,28 3,58 99,98 656,33 1,09 99,99 1948,52 0,92 100,00 4276,33 0,03 100,00 6171,24 0,00 100,00 9137,16 0,00 100,00 11064,40 0,02 100,00 13875,97 0,00 100,00 14924,77 0,00 100,00 106545,88 2,29 99,49 134028,53 0,20 99,88 196646,05 0,43 99,54 202415,98 0,13 99,63 205591,36 0,10 98,20 209412,02 0,05 99,61 218322,61 0,10 99,57 243497,48 2,34 99,42 353762,34 0,00 100,00

The updating residual used in this application contains the differences between the non-linear acceleration frequency response magnitudes (responses, not FRFs!) of DoFs 1, 2, 3, and 4, see Figure 11.4. These DoFs are well distributed over the model and are therefore suited to monitor the bending deformation of the structure. Test data was simulated by using the direct frequency response approach, i.e. the inversion of the non-linear dynamic stiffness matrix at each frequency of interest according to equation (4.31). This approach does not suffer from truncation errors

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11. Applications

and can be considered as the most accurate approach for non-linear frequency response analysis. An 80 N constant harmonic excitation force amplitude was introduced at the tip mass of the cantilever beam and the non-linear acceleration frequency response was analyzed in the vicinity of the resonance of the first bending mode (301 frequency points equally spaced in the frequency range from 100 Hz to 130 Hz). The underlying linear and the non-linear drive point acceleration responses of the initial and the perturbed model are shown in Figure 11.5.

Figure 11.5: Acceleration response for 80N excitation force level of the initial model and the perturbed model

It can be seen that the resonance peak of the perturbed model is located at higher frequencies due to the increased support stiffness. The effect of the hardening quadratic damper can clearly be observed from the peak magnitude attenuation. The effect of the softening cubic stiffness can be observed in the peak deformation towards lower frequencies. During non-linear updating, the parameters of the initial model are changed in such a way that the differences of the acceleration response magnitudes between the initial and the perturbed model are minimized. Updating was performed with constrained parameter changes per iteration step (max. 10%) but with unconstrained total parameter variation. Regularization has not been applied. The computation was performed under MATLAB version 6.5 on a Windows XP computer with an Intel Pentium IV 1800 MHz CPU and 512 MB RAM.

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11.1.1 Non-Linear Updating using Direct Frequency Response Analysis In this section, updating was performed by using the direct approach for non-linear frequency response analysis according to equation (4.31) for the calculation of analytical non-linear frequency responses. This means that the response dependent non-linear elements were assembled with the physical system matrices after harmonic balancing. Subsequently, the non-linear frequency response was calculated by direct inversion of the non-linear dynamic stiffness matrix at each frequency of interest. The non-linear updating track plots are shown in Figure 11.6. It can be seen that the imposed parameter changes were exactly identified after 10 iteration steps and that the parameters fully converged.

Figure 11.6: Evolution of updating parameters and objective function

The non-linear joint stiffness parameter knl was changed in the wrong direction within the first three iteration steps. A possible explanation for this adverse effect is the dominating sensitivity of the underlying linear stiffness parameter k sup compared to the sensitivities of the non-linear joint stiffness and damping parameters. Once the underlying linear stiffness parameter converged to its final value, the other parameters were changed in the right direction so that finally the imposed parameter changes were identified and the objective function was minimized to zero. The computation time for a single non-linear response analysis was 9.94 seconds (301 frequency points in the frequency range from 100 to 130 Hz). As expected, a response comparison after non-linear updating revealed no difference between the responses of the perturbed model and the updated model.

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11. Applications

11.1.2 Non-Linear Updating using Modal Frequency Response Analysis In this section, updating was performed by using the modal approach for non-linear frequency response analysis according to equation (6.5) for the calculation of analytical non-linear frequency responses. This means that mode superposition is applied to the system matrices so that non-linear frequency response analysis is performed in generalized modal coordinates. Significant order reduction can be achieved when only a few modes are considered for mode superposition. In this case, the resulting modal model is an equivalent model, which is only valid in a limited frequency range. Furthermore, coupling of higher modes is disregarded when only a few lower modes are used for mode superposition so that the modal frequency response will be less accurate than the response obtained with the direct approach. A study was performed which aimed at investigating the application of the less accurate modal frequency response approach for non-linear updating. The updating track plots shown in Figure 11.7 were obtained by using modal frequency response analysis when using only the first two modes for mode superposition. Even though the characteristic of the parameter evolution is quite similar to the one obtained with the exact direct frequency response analysis, it was not possible to identify the imposed parameter changes. This is also indicated by the objective function which could not be reduced to zero after 9 iteration steps.

Figure 11.7: Evolution of parameters and objective function when using modes 1 and 2 for modal frequency response analysis

The poor updating results can be improved by the inclusion of additional modes for mode superposition. Finally, when using all modes it is possible to identify the exact parameter values. Indeed, exactly the same parameter evolution was obtained when using all modes as it was obtained by using the direct approach. This indicates that there is a tradeoff between identification accuracy and the number of modes used for mode superposition, and hence, the numerical effort involved in the computation of the non-linear frequency response. Table 11-4 summarizes the results of a study where updating was performed with different sets of modes used for modal frequency response analysis. It can be seen that the accuracy of the parameter estimation increases with increasing number of modes. However, the numerical effort increases as well, so that the 6-DoF modal model is numerically less efficient than the full physical model with 18-DoFs when using the direct approach. It has to be noted, however, that the order of the cantilever beam model is too small for efficient modal response analysis, i.e. the savings achieved by order reduction due to mode superposition are overbalanced by

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117

the effort involved in modal transformation and back-transformation for evaluating the non-linear element matrices. It is therefore not justified to conclude from this study that the modal approach for non-linear frequency response analysis is less efficient than the direct approach. The last row of Table 11-4 summarizes the updating results obtained with modal frequency response analysis in conjunction with a “properly selected” set of modes. These results emphasize the tradeoff between numerical efficiency and accuracy of the parameter estimation when using mode superposition for non-linear frequency response analysis in non-linear updating. The selection of modes will be discussed in the last application of this chapter. Table 11-4: Summary of parameter identification using different models for non-linear modal frequency response analysis Modes used 1+2 1 to 6 1 to 18 1,2,3,4,5,7,9,11

k sup =+21.98 k sup =+22.38 k sup =+25.00 k sup =+22.46

Parameter Changes k nl =-3.98 k nl =-10.42 k nl =-20.00 k nl =-14.42

c nl =+50.65 c nl =+42.60 c nl =+30.00 c nl =+37.40

Objective Computation Function [%] Time [s] 0.8 6.95 0.14 13.89 0 38.86 0.09 19.51

When looking at the relatively poor results of non-linear updating using modal frequency response analysis, a question may arise concerning the reason for these inaccuracies. It can be seen from Table 11-3 that the underlying linear system is affected by the imposed parameter change of the support stiffness. However, it can also be seen that the modes are only slightly affected by the stiffness change. From that point of view, errors in the mode shapes are not expected to be a major source of error. The major source of error is supposed to be modal truncation. Due to the nonlinear spring and damper introduced at the joint, non-diagonal modal stiffness and damping matrices are being produced which couple the higher modes with the lower ones. When only using a few of the fundamental modes for mode superposition, the coupling with higher modes would be disregarded so that they cannot contribute to the total non-linear response. When looking at the comparison of the initial model acceleration responses obtained with the direct approach and with the modal approach by using the first two modes (see Figure 11.8) it can be seen that these responses correlate quite well so that the assumption that inaccurate modal response analysis is the major source of error in non-linear updating is obviously wrong.

Figure 11.8: Comparison of real and imaginary part acceleration response at the drive point obtained with the initial model

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11. Applications

Indeed, the comparison of the magnitude responses shown in Figure 11.9 reveals that the error of the modal response is smaller than 3% (the direct response is always considered as the reference).

Figure 11.9: Comparison of magnitude acceleration response at the drive point and corresponding magnitude response error obtained with the initial model

However, as updating progresses the parameters are changed and consequently the underlying linear system changes as well. When comparing the response of the perturbed model (obtained with direct frequency response analysis) with the response obtained with the initial model by using modal frequency response analysis with the exact parameter changes imposed, one would expect that these response curves would be close to each other. However, it can be observed from Figure 11.10 and Figure 11.11 that this is not the case. Indeed, the corresponding response curves deviate significantly from each other. The error in the response magnitudes increases up to 30% but this error can also be interpreted to be the result of a slight resonance peak shift between the two curves. The deviation between the response curves can be reduced by the inclusion of additional modes for mode superposition and ultimately, the modal response curve passes over into the direct response curve when all modes are included in mode superposition.

Figure 11.10: Comparison of real and imaginary part acceleration response at the drive point obtained with the perturbed model

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119

Figure 11.11: Comparison of magnitude acceleration response at the drive point and corresponding magnitude response error obtained with the perturbed model

The reason for the response deviation shown in Figure 11.10 and Figure 11.11 can be found in the mode superposition being performed with the “wrong modes”. The modes of the underlying linear system change due to the imposed change of 25% in the linear support stiffness. This change of the mode shapes is disregarded by the modal frequency response algorithm implemented in HBResp, i.e. the mode shapes of the initial underlying linear system are used for mode superposition throughout the complete response analysis. Consequently, the initial underlying linear mode shapes do no longer decouple the underlying linear system when parameter changes are imposed which affect the underlying linear system. Thereby, the modal truncation error inherent in non-linear modal frequency response analysis is increased significantly. The situation can be improved by applying a two-step strategy for updating. In the first step, the underlying linear system is updated independently. Well established techniques for computational model updating can be used for this purpose, e.g. using modal data obtained from low level modal survey testing. In the second step, nonlinear parameters are updated while the already updated underlying linear system is kept constant. The application of this two-step strategy reduces the modal truncation error so that more accurate parameter estimates can be obtained from non-linear updating by using the modal approach for non-linear frequency response analysis. The accuracy improvement for modal frequency response analysis achieved with the underlying linear system updated in a previous step can be observed in Figure 11.12. There, the response obtained with direct frequency response analysis of the perturbed model is compared with the response obtained with modal frequency response analysis of the (exactly) updated underlying linear initial model with the exact parameter changes imposed for the non-linear parameters. Even though only the first two modes were used for mode superposition, it can be seen from Figure 11.13 that the magnitude response error could be reduced to 1.5%. This indicates that updating of the underlying linear system prior to non-linear updating can improve the response accuracy of modal non-linear frequency response analysis. The response accuracy improvement is quite impressive, especially when considering that the error was 20 times larger without updating the underlying linear model.

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Figure 11.12: Real part and imaginary part acceleration response comparison of perturbed model after updating the underlying linear system

Figure 11.13: Magnitude acceleration response comparison and corresponding magnitude response error of perturbed model after updating the underlying linear system

With the accuracy improvement in modal response analysis, an improvement can also be expected for non-linear updating. The following updating results were obtained after the underlying linear system of the initial model had been updated exactly. Consequently, only the non-linear parameters knl and cnl must be identified by the non-linear updating software Update_NL. It can be seen from the updating track plots shown in Figure 11.14, that even when the underlying linear system was updated exactly, the true parameter values cannot be identified by using modal frequency response analysis. The plots shown in Figure 11.14 were obtained by using the first two modes for mode superposition. The results of non-linear updating are nonetheless much better than those shown in Table 11-4. It is worth noting that the non-linear parameters were changed in the right direction already from the first iteration step. The updating results can be further improved by using additional modes for mode superposition. The results of a study are summarized in Table 11-5, where non-linear updating was performed after the underlying linear system had been updated. Different sets of modes were used for non-linear modal frequency response analysis and it can be seen that the accuracy of the parameter estimation increases with increasing number of modes.

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Figure 11.14: Evolution of parameters and objective function when using modes 1 and 2 for mode superposition and exactly updated underlying linear system

Table 11-5: Summary of non-linear parameter identification using different sets of modes for nonlinear modal frequency response analysis and exactly updated underlying linear system Modes used 1+2 1 to 6 1 to 18 1,2,3,4,5,7,9,11

Parameter Objective Computation Changes Function [%] Time [s] k nl =-18.02 c nl =+35.23 2.26 6.01 k nl =-18.42 c nl =+34.13 1.72 12.16 k nl =-20.00 c nl =+30.00 0 32.421 k nl =-19.00 c nl =+32.62 1.12 15.72

11.1.3 Preliminary Conclusions From the investigations performed on the analytical cantilever beam model it can be concluded that linear and non-linear parameters should be updated separately when modal frequency response analysis is used for non-linear updating. This condition can be relaxed, either when all modes of the system are used for mode superposition, or when the modes of the underlying linear system are re-calculated in each iteration step when the underlying linear system is changed. Both options are considered prohibitive for large order systems and are therefore not discussed further. It was shown that the error of non-linear updating by using modal frequency response analysis could be reduced significantly by updating the underlying linear system prior to updating the non-linear parameters. But even then it was not possible to identify the true non-linear parameters. The reason for this can be found in the non-linear frequency response being calculated inaccurately when a truncated modal model is used. Nonetheless, reasonably accurate updating results can be obtained when a properly selected set of modes is used for mode superposition. The major drawback is, however, that the modes used for mode superposition can generally be selected only from a subset of all modes (especially in case of large order systems). The selection of modes can thus not be done on the complete set of modes and modal truncation errors cannot be avoided. The fact that even slight differences between modal response and direct response can yield such poor updating results is closely linked to the low sensitivities of the nonlinear parameters. They are only activated sufficiently in the resonance region but are generally less sensitive at the outer limits of the frequency interval considered for updating. Due to the low parameter sensitivities, the slight response deviations between modal and direct response are compensated by relatively large parameter

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errors. Since the non-linear parameters are response dependent, their sensitivities are response dependent as well. Consequently, it can be expected that the parameter estimation using modal frequency response analysis would yield better results when the FRFs of a higher excitation force level would have been used for updating. On the other hand, the parameter estimation would yield even worse results when FRFs of a very low excitation force level would have been used. From a theoretical point of view it is possible to identify linear and non-linear parameters in a single updating run. This was proved in chapter 11.1.1, where the direct frequency response analysis has been used, but also in chapter 11.1.2, where modal frequency response analysis has been used with the full set of modes. However, due to the dominant sensitivities of the parameters affecting the underlying linear system, it may well happen that the non-linear parameters are changed in the wrong direction until the underlying linear parameters approximately reach their final values. This is an undesired effect in computational model updating and can be avoided by updating linear parameters separately. Separate updating of the underlying linear system also improves the accuracy of the updating results when using modal frequency response analysis for the calculation of analytical non-linear frequency responses. It is therefore recommended to perform updating in two steps. In the first step, the underlying linear system is updated, e.g. by using eigenfrequency and mode shape residuals. In the second step, non-linear parameters are updated using modal frequency response analysis, while the already updated underlying linear system is kept constant to reduce modal truncation errors.

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11.2 CTS Joint Identification The cylindrical test structure (CTS), shown in Figure 11.15, was already introduced in chapter 3, where the simplified joint modeling approach for the bolted scalloped flange was developed. The CTS was designed and manufactured within the European research project CERES. It was used there as a vehicle to approve nonlinear vibration test procedures and to validate non-linear joint modeling approaches. Furthermore, the CTS was used to check the performance of the nonlinear frequency response analysis software HBResp and of the non-linear parameter updating software Update_NL for their final application, i.e. the identification of non-linear joint parameters of aero-engine-like structures. The CTS shall be used here as an example for the application of non-linear updating to an FE model of moderate order. In particular, the advantages of non-linear updating in conjunction with modal frequency response analysis shall be emphasized here. tuning mass (ca. 22kg) component 2 (ca. 3.65kg)

accelerometers bolted scalloped flange joint

component 1 (ca. 3.65kg) adapter base acceleration slip table Figure 11.15: CTS with tuning mass and accelerometers mounted on a slip table

A finite element model was generated for the CTS which uses a shell element representation for the cylindrical casing shells and a beam element representation for the flanges. The simplified joint modeling was already developed in chapter 3 and is sketched in Figure 11.16. It can be observed that non-linear joint effects are represented by non-linear 2-DoF elements introduced at the base of the flange. This type of simplified joint model was introduced at 12 equidistant positions on the circumference of the scalloped flange joint according to 12 bolt positions. All 12 nonlinear spring/damper elements of the simplified joint models have the same properties (since the single bolted joints are nominally identical). However, due to

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the different relative deformations which occur at the different bolt positions (e.g. in case of bending), the actual stiffness and damping properties of the single non-linear spring/damper elements will be different depending on the relative displacement at the corresponding position. Details of the scalloped flange joint can be found in Figure 3.12. Details of the simplified joint model can be found in Figure 3.18.

hinged joint at bolt center line level

combination of linear and non-linear spring/damper elements Figure 11.16: Simplified joint modeling approach for bolted scalloped flange (acc. to Figure 3.18)

The finite element model of the CTS assembly shown in Figure 11.16 consists of 1440 shell elements, 214 beam elements (representing flanges and weld seams), 12 lumped masses (representing masses of bolts and nuts), and 60 rigid beam elements (used for load path modeling at the joint). The overall number of DoFs is ca. 9500. According to the model updating strategy published in [Böswald 2005b] and [Schedlinski 1998], updating was performed in subsequent steps. In the first step, the single components were updated using experimental modal data obtained from modal survey test with hammer excitation and free/free boundary conditions. Uncertain model parameters were updated, among those, the cross sectional properties of the beam elements representing the scalloped flanges. The test setup for modal testing of the CTS components can be observed in Figure 11.17. In the second step of model updating, experimental modal data of the CTS assembly was used to update the underlying linear joint stiffness. Free boundary conditions turned out to be inadequate for this purpose, because the modes observed in the free boundary condition assembly test (see Figure 11.17) were exclusively ovalizing shell modes to which the joint stiffness was not sufficiently sensitive. The joint stiffness was much more sensitive to the bending modes so that the test setup for linear CTS assembly testing was modified by mounting the CTS onto a foundation (large seismic mass) and subsequently attaching a heavy tuning mass to its upper unconstrained flange. The test setup for the constrained boundary condition test on the CTS assembly can be observed in Figure 11.17. The eigenfrequencies of the bending modes of the CTS assembly in constrained boundary conditions were used to manually adjust the underlying linear joint stiffness.

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setup for free/free component test

125

setup for free/free assembly test

setup for assembly test with fixed boundary

Figure 11.17: Different test setups for modal survey testing of CTS

In the third step of computational model updating, non-linear elements were introduced at the joints and their non-linear stiffness and damping parameters were updated from measured non-linear frequency response data, while the already updated underlying linear system was kept constant. This means that updating was performed in accordance with the conclusions gained from the cantilever beam example presented in chapter 11.1. The selection of the type of non-linear elements introduced at the joints was based on the non-linear effects observed in the measurements. For example, the distortion characteristic of the non-linear experimental FRFs obtained from constant excitation force testing can be used to characterize the type of non-linearity. In case of the CTS, non-linear frequency responses were measured from base excitation tests conducted on a slip table. The setup of these non-linear tests can be observed in Figure 11.15 and it can be seen that the boundary conditions of this setup are similar to those used for modal testing of the CTS assembly in constrained boundary conditions. Sinusoidal acceleration was imposed in a step-sine manner at the constrained base. The acceleration response was measured at several positions on the structure together with the (controlled) enforced base acceleration. Transmissibility functions were obtained (instead of FRFs) by dividing the measured acceleration responses by the measured base acceleration. The transmissibility measurements were repeated for different levels of constant base acceleration amplitudes. The frequency range investigated in the non-linear step-sine tests was situated around the resonance of the first bending mode, i.e. around 114.5 Hz. This bending mode was expected to reveal non-linear behavior so that the non-linear vibration tests were specifically designed to excite this mode (pictures of the modes can be found in the Appendix). Figure 11.18 shows the experimental non-linear transmissibilities measured at the tuning mass in excitation direction (see Figure 11.15). It can be observed that the resonance peak slightly shifts towards lower frequencies as the base acceleration level increases. In addition, the damping level increases which is indicated by the attenuation of the response peak magnitude. This type of non-linearity can be characterized as a “softening” stiffness in conjunction with a “hardening” damping non-linearity. The non-linearity of the bending mode of the CTS is also indicated by the distortions of the frequency isochrones in the Nyquist plot of the frequency responses (not transmissibilities).

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Figure 11.18: Non-linear transmissibilities of CTS obtained from base excitation tests

With regard to the discussion on bolted flange joints in chapter 3, it is quite reasonable to assume a pre-loaded bilinear stiffness characteristic to represent the softening stiffness behavior of the CTS joint. The non-linear restoring force function of a pre-loaded bilinear spring is governed by three parameters (see Figure 11.19). These are the compression regime stiffness k1 , the tension regime stiffness k 2 , and the position of the stiffness transition point uc . The compression regime stiffness equals the underlying linear stiffness (when assuming that the stiffness transition occurs in the tension regime, i.e. at a positive relative displacement). Consequently, the stiffness parameter k1 was already identified by updating the underlying linear system using linear modal data and k1 was therefore kept constant in non-linear updating. The tension regime stiffness k2 and the position of the stiffness transition point uc are non-linear stiffness parameters which must be identified from experimental response data. The source of the damping non-linearity is not fully understood so that an equivalent non-linear damping model must be used. A quadratic hardening damper was chosen here, whose damper constant cnl increases linearly with increasing displacement amplitude. This type of non-linear damper is suited to describe vibration level dependent resonance peak attenuations similar to those observed from experimental transmissibilities. The non-linear damper constant cnl must be identified from experimental response data. Polynomial type non-linear damping is only a supplement to the underlying linear damping which was specified in terms of modal damping with the corresponding damping ratios taken from the modal survey tests performed on the CTS assembly. Figure 11.19 shows the non-linear restoring force functions of the pre-loaded bilinear spring and the hardening quadratic damper.

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fR

k1

fR

k2

uc

u

⎧ k1u, u ≤ uc fR = ⎨ ⎩ k2 ( u − uc ) + k1uc , u > uc

u

f R = clin u + cnl u u

Figure 11.19: Non-linear restoring force of pre-loaded bilinear spring and hardening quadratic damper

It should be noted that whenever equivalent non-linear element types are used, the prediction capability of the non-linear model is restricted to the vicinity of the displacement amplitudes which where used for updating the non-linear parameters. The prediction of non-linear responses for vibration levels far away from the experimental vibration levels can be inaccurate when the polynomial non-linear restoring force function deviates significantly from the true non-linear restoring force function. Consequently, FE models with polynomial type non-linearities must be considered as equivalent non-linear FE models, whose range of validity is limited to a certain range of displacement amplitudes. In order to assure that the distortions of the non-linear transmissibilities shown in Figure 11.18 were really caused by the local joint non-linearity (instead of other nonlinear effects such as yielding of material), the non-linear step-sine tests were performed in a specific way. It is obvious that the test campaign started with the lowest base acceleration level and that the base acceleration level was gradually increased from measurement to measurement. However, each time a measurement of a high base acceleration level was finished, a control measurement was performed by again using the lowest base acceleration level in order to check for possible differences with the original measurement. These control measurement were used as a check for yielding of material which would cause a reduction of the bolt pre-load and, thus, would cause a change of the bolted joint non-linearity. Apart from the plasticity control measurements, non-linear transmissibilities were measured at each base acceleration level by stepping from the lowest towards the highest frequency, and vice versa. The comparison of the two measurements can help to detect the existence of multiple stable response solutions which are usually accompanied by the so-called “jump phenomenon” already discussed in chapter 5. A jump phenomenon was not detected during the CTS tests. Nonetheless, it was observed form the plasticity control measurements that after finishing the 4.0 m/s2 base acceleration level measurements, it was no longer possible to reproduce the transmissibility of the 0.5 m/s2 base acceleration level. Instead, the resonance peak appeared at a slightly lower frequency than it was before and it was assumed that plastic deformation of the scalloped flanges were the reason for this effect. From that point of view, the level of confidence is relatively low with the 4.0 m/s2 transmissibilities, and indeed, a visual inspection of the transmissibilities reveals that the frequency shift between the 3.5 m/s2 level and the 4.0 m/s2 level is much more pronounced than the frequency shifts between all other levels. This may be an indicator for a change of the dominating source of non-linearity from bilinear

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stiffness at the joint to material yielding in the flanges. Consequently, the non-linear base acceleration tests had to be stopped at the 4.0 m/s2 level to prevent the CTS from overtesting, or respectively, to avoid damaging of the test equipment. Another point to consider is the type of excitation used in the non-linear vibration tests. The CTS was excited by enforced motion introduced at the clamped base. This actually means that discrete points are not available to apply external forces to the FE model to calculate analytical non-linear frequency responses. Instead, the enforced motion at the base DoFs (index b ) of the FE model is known, while the response at the unconstrained DoFs (index a ) is unknown. Thus, the frequency domain equation of motion must be partitioned into DoFs with unknown response {ua } and DoFs with known (enforced) response {ub } :

⎛ ⎡[ M aa ] ⎜ −Ω 2 ⎢ ⎜ ⎢⎣[ M ba ] ⎝

⎡[Caa ] [Cab ]⎤ ⎡[ K aa ] [ K ab ]⎤ ⎞ ⎧⎪{uâ }⎫⎪ ⎧⎪{0}⎫⎪ [ M ab ]⎤ ⎥ + jΩ ⎢ ⎥+⎢ ⎥⎟⎨ ⎬ = ⎨ ⎬. [ M bb ]⎥⎦ ⎢⎣[Cba ] [Cbb ]⎥⎦ ⎢⎣[ K ba ] [ K bb ]⎥⎦ ⎟⎠ ⎩⎪{uˆb }⎭⎪ ⎩⎪{0}⎭⎪

(11.1)

It is indicated in Figure 11.20, that the response {uâ } at the unconstrained DoFs due to the enforced base motion {uˆb } can be calculated from the superposition of the socalled quasi-static response uâstatic , which would be obtained in case of an infinitely stiff structure, and a relative dynamic response uârel caused by the so-called effective excitation force vector fêff which acts on the unconstrained DoFs:

{

}

{ }

{ }

{uâ } = {uâstatic } + {uârel } ,

(11.2)

( −Ω [ M ] + jΩ [C ] + [ K ]){uˆ } = { fˆ } . 2

aa

quasi-static response

aa

aa

rel a

(11.3)

eff

relative dynamic response

effective excitation forces

mzz fy

enforced base acceleration x

x

fixed support

ub y

y Figure 11.20: Base acceleration and effective excitation forces

In case of lightly damped structures like the CTS, internal inertia forces are much more pronounced than internal damping forces. It is convenient to disregard the latter ones without loosing accuracy. When the coupling between constrained and unconstrained DoFs by internal inertia forces is also neglected (i.e. [ M ab ] ≈ [0] ), the following expression can be derived for the effective excitation force vector [Link 2002]:

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{ fˆ } = −Ω [ M

][ K aa ] [ K ab ]{uˆb } . −1

2

eff

aa

(11.4)

−[TG ]

It can be seen from this equation that the effective excitation force vector depends on the mass distribution at the unconstrained DoFs represented by [ M aa ] , on the enforced harmonic base acceleration {ub } = −Ω 2 {uˆb } , and on the so-called geometry matrix [TG ] which relates the motion at the unconstrained DoFs {ua } to the base motion {ub } . Enforced motion is not implemented in the non-linear frequency response analysis software HBResp. Consequently, the relative dynamic response was calculated by using an approximation of the effective excitation force vector f eff . When exploiting the highly uneven mass distribution of the CTS (ca. 3.6 kg distributed mass of each CTS component and ca. 22 kg rigid tuning mass represented by a single lumped mass with mass moments of inertia) it was possible to approximate the effective excitation force vector by a single translational force and a single moment of force, both introduced at the center of gravity of the tuning mass. All other forces were orders of magnitude smaller and could be disregarded. The following relationship could be established for the CTS shell model between the enforced base acceleration ub , the translational excitation force f y , and the moment of force mzz (see Figure 11.20):

{ }

f y = −22.8356

Ns 2 m

ub ,

mzz = +9.97 ⋅ 10−3 Ns 2 ub .

(11.5) (11.6)

The quasi-static response can readily be calculated by using the geometry matrix [TG ] and was added to the relative dynamic response in a subsequent step. Transmissibilities were obtained by dividing the total acceleration response by the enforced base acceleration. It should be noted that in contrast to FRFs, the acceleration transmissibilities are dimensionless and are equal to the displacement transmissibilities and equal to the velocity transmissibilities. In order to increase the numerical efficiency for non-linear frequency response analysis, the CTS shell model was condensed by applying the Craig-Bampton reduction according to chapter 6.4. Each component of the CTS was considered as a superelement. The interface nodes at the joint (where the non-linear elements were introduced) must be used as master DoFs in this case. In addition, the node at the center of gravity of the tuning mass was used as a master node due to the large mass concentration. Furthermore, one ring of nodes in the middle of each component were also used as master nodes to improve the reduction process. The total number of DoFs after Craig-Bampton reduction was 686 (i.e. 666 physical master DoFs plus 10 additional modal DoFs per superelement to account for dynamic effects at the omitted model parts). A Craig-Bampton reduced model usually yields slightly different results for dynamic analysis than the original full model. This should be avoided as far as possible when the reduced model is used for non-linear frequency response analysis. Indeed, it can be observed from Figure 11.18 that the frequency shift caused by the joint non-linearity is only about 2% of the resonance frequency and it can be expected that non-linear updating would fail when the frequency shift caused by model reduction would be about the same order. In case of the CTS, it was checked that the eigenfrequency deviations between the full model and the CraigBampton reduced model were less than 0.01% for the first 20 modes in the frequency range up to 1500Hz. Pictures of the first 20 modes can be found in the Appendix. A

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comparison of the results of the full model and the Craig-Bampton reduced model is skipped for convenience. Reasonable initial values for the non-linear parameters are required for non-linear frequency response analysis. These were found in a “trial-and-error” manner and are listed in Table 11-6. Table 11-6: Initial values for non-linear joint parameters Parameter

Description

Initial Value

k1

tension regime stiffness

N 11750 mm

k2

compression regime stiffness

N 8163 mm

uc

stiffness transition point

0.035mm

cnl

non-linear quadratic damper constant

Ns 0.04 mm 2

2

Since the order of the Craig-Bampton reduced model is moderate, the direct frequency response analysis approach can be used for non-linear updating. A comparison of the non-linear frequency responses obtained with the direct approach in conjunction with the Craig-Bampton reduced model and with the modal approach in conjunction with the full model is shown in Figure 11.21. Only the first three modes were used for mode superposition, but even though it can be observed that the modal response error is below 1.5%. The computation time for the modal response was 16.03 seconds compared to 822 seconds for the direct response. This is a significant reduction of computation time compared to the direct response. When using the first 5 modes for mode superposition, the modal response error can be further reduced to only 0.5%. However, the improved accuracy is obtained at the cost of increased computation time. In case of the first 5 modes, the computation time of a single non-linear response took 29.67 seconds, which is still acceptable. Further reduction of the computation time can be achieved when modal frequency response analysis is performed in conjunction with the Craig-Bampton reduced model. In this case, the computation of the modal response when using the first three modes for mode superposition is 10.53 seconds and 17.03 seconds when using five modes.

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Figure 11.21: Comparison of different modal frequency responses with the direct frequency response for a base acceleration level of 2.0m/s2

Based on these investigations, it is expected that non-linear updating in conjunction with modal frequency response analysis would yield about the same results as with the direct approach, provided that a properly selected set of modes is used for mode superposition. In contrast to the cantilever beam model discussed above (which only had bending modes), the CTS shell model has mostly ovalizing shell modes. These modes almost contribute nothing to the response in the vicinity of the resonance of the first bending mode. The ovalizing shell modes can therefore be disregarded for modal frequency response analysis. In addition to the bending modes and the ovalizing shell modes, there is also an axial extension mode and a torsion mode. These modes should also be considered for non-linear modal response analysis. A procedure for the selection of the most influential modes is presented later in the aero-engine application but shall not be discussed here. The following analytical non-linear transmissibilities were obtained at the tuning mass by using the roughly adjusted initial parameters listed in Table 11-6. According to the experimental transmissibilities, different levels of enforced base acceleration ranging from 0.5 m/s2 to 4.0 m/s2 were used. It can be observed from Figure 11.22 that the fundamental non-linear characteristic (attenuation of peak magnitude and resonance frequency shift) can already be obtained with the roughly adjusted initial parameters. The amount of resonance peak shift and peak attenuation, however, are not consistent with the experimental observations. These discrepancies between experimental and analytical non-linear transmissibilities must be reduced by updating of the non-linear parameters.

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Figure 11.22: Comparison of initial analytical and experimental transmissibilities at the tuning mass

The following non-linear parameters were selected for updating: • tension regime stiffness k 2 of the pre-loaded bilinear joint spring, •

position of the stiffness transition uc of the pre-loaded bilinear spring,

•

non-linear damper constant cnl of the hardening quadratic damper.

The already updated underlying linear joint stiffness was kept constant during nonlinear updating. The updating residual contained the difference between the experimental and the analytical magnitude transmissibility obtained at the tuning mass. Three different levels of base acceleration were used, i.e. a low level transmissibility at 0.5 m/s2, a medium level transmissibility at 2.0 m/s2, and a high level transmissibility at 3.5 m/s2. The active frequency range was situated between 107 Hz to 122 Hz, with a frequency spacing of 0.15 Hz (i.e. 101 frequency points). A weak regularization ( w p = 0.05 ) according to equation (10.50) was used to constrain the parameter variation per iteration. Damping of the underlying linear system was represented by a damping ratio of 0.54% for the first two bending modes and 0.2% for the other modes. At first, non-linear updating was performed by using direct frequency response analysis in conjunction with the Craig-Bampton reduced model. The evolution of the updating parameters and the evolution of the objective function can be observed in Figure 11.23. It can be seen that the parameters converged after 5 iteration steps and that the normalized error (normalized objective function) was reduced to 51.5 %.

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Figure 11.23: Evolution of updating parameters and objective function (Craig-Bampton reduced model, direct frequency response analysis)

Almost identical results were obtained when non-linear updating is performed with modal frequency response analysis in conjunction with the full model. In this case, the five most influential modes were used for modal frequency response analysis. These are the first pair of bending modes (modes 1+2), the axial extensional mode (mode 3), and the second pair of bending modes (modes 11+12). The same characteristic parameter evolution was obtained as with the direct approach but with slightly different parameter values and slightly different values of the objective function. Identical results were obtained when using modal frequency response analysis in conjunction with either the full model or with the Craig-Bampton reduced model. This in an interesting feature which can be exploited for large order systems. Table 11-7 summarizes the updating results obtained with the direct approach and the Craig-Bampton reduced model, whereas Table 11-8 summarizes the results obtained with the modal approach when using only the five most influential modes (results of full model and Craig-Bampton reduced model are identical in case of modal response analysis). Both tables summarize the parameter values before and after updating together with the parameter changes of the non-linear joint element parameters. Table 11-7: Summary of parameter values and parameter changes when using direct response (Craig-Bampton reduced model) Parameter

Description

Initial Value

Final Value

Change

k1


N 11750 mm

N 11750 mm

--

k2


N 8163 mm

N 10081.3 mm

+23.5%

uc


0.035mm

8.51 ⋅ 10−3 mm

−75.7%

cnl


Ns 0.04 mm 2

1.58 ⋅ 10−2

−60.6%

2

Ns 2 mm 2

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Table 11-8: Summary of parameter values and parameter changes when using modal response (full model and Craig-Bampton reduced model yield identical results) Parameter

Description

Initial Value

Final Value

Change

k1


N 11750 mm

N 11750 mm

--

k2


N 8163 mm

N 10081.3 mm

+23.5%

uc


0.035mm

8.44 ⋅ 10−3 mm

−75.9%

1.54 ⋅ 10−2

−61.4%

cnl


2

Ns 0.04 mm 2

2

Ns mm 2

It can be seen from Table 11-7 and Table 11-8 that the final parameter values are very similar, regardless of the approach used for frequency response analysis. This also applies to the objective function. It was reduced to 51.5% in case of the direct approach (see Figure 11.23) and to 51.1% when using the modal approach. Even though the results are quite similar, there are significant differences in the computation time. The total computation time of the updating run (9 iteration steps) in case of direct frequency response analysis with the Craig-Bampton reduced model was 40010 seconds, whereas the computation time when using the modal response with the full model was only 4280 seconds (i.e. 10.7% of the computation time when using the direct approach). The modal approach in conjunction with the CraigBampton reduced model was even faster. The computation time was in this case only 2957 seconds (i.e. 7.4% of the computation time when using direct approach). This clearly emphasizes the efficiency of non-linear updating in conjunction with modal frequency response analysis in case of large order systems and also emphasizes the savings which can be achieved by the application of subsequent steps of model reduction.

11.2.1 Interpretation of Updating Results When looking at the results of non-linear updating, it can be seen that in all three updating runs the tension regime stiffness k2 was increased so that its final value is close to the compression regime stiffness k1 (ca. 86% of the compression regime stiffness). In addition, the position of the stiffness transition point of the pre-loaded bilinear spring was shifted towards zero, i.e. towards the origin of the forcedisplacement diagram (see Figure 11.19). From the interpretation of the updating results it can thus be concluded that during updating the pre-loaded bilinear spring was almost converted into a linear spring. The reduction of the quadratic damper constant cnl by ca. 61% indicates that damping was overestimated in the initial model. The comparison of the transmissibility magnitudes of the three different base acceleration levels used for updating can be observed in Figure 11.24, Figure 11.25, and Figure 11.26. The upper diagrams show the response comparison (at the tuning mass) before updating, whereas the lower diagrams show the response comparison after updating. All levels were used in the updating residual and it can be observed that in all three cases the analytical transmissibilities better fit the experimental ones after updating. From that point of view, updating can be considered successful. However, a closer inspection of the experimental transmissibilities reveals that they look quite rough. A discussion on the quality of the test data shall be postponed until the next paragraphs.

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Figure 11.24: Comparison of the magnitude transmissibility of the 0.5m/s2 base acceleration level



The comparison of the transmissibilities of all base acceleration levels shall be used as a predictivity check and is shown in Figure 11.27. It can be seen that the fundamental trend of the resonance peak shift and the peak magnitude attenuation is better represented by the non-linear analytical model after updating. The characteristic of the resonance peak distortion with increasing base acceleration level is however not satisfactory and indicates that the updated model is not really predictive.

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Figure 11.27: Comparison of analytical and experimental transmissibilities at the tuning mass after updating

It was mentioned earlier that the level of confidence is not very high with the 4.0 m/s2 transmissibility. When looking at the magnitude response comparisons for the three different levels shown in Figure 11.24 to Figure 11.26, it can be observed that the experimental magnitude transmissibilities look somewhat strange. A possible reason can be found in the performance of the feedback loop controller which was utilized to maintain the base acceleration at constant amplitude while stepping through the resonance. When looking at the base acceleration amplitudes of the different base acceleration levels shown in Figure 11.28, it can be observed that drop-outs occur near the resonance which might influence the distortion characteristics of the non-linear transmissibilities.

Figure 11.28: Amplitudes of different base acceleration levels

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When looking at the time domain acceleration responses it can be concluded that non-linear behavior was sufficiently activated. This can be proved, for example, by the presence of higher harmonics in the relative acceleration response at the scalloped flange joint which was calculated from the difference between the acceleration responses measured with two accelerometers placed inside the CTS (see Figure 11.29). The first accelerometer was placed directly above and the second one directly below the scalloped flange joint (see Figure 11.30). It can be seen that the relative acceleration response is pretty much sinusoidal at the 0.5 m/s2 base acceleration level but becomes more and more distorted as the base acceleration level is increased. Even though the presence of higher harmonics usually indicates nonlinearity, it seems that this type of non-linearity is less effective in terms of resonance frequency shift. Indeed, if the joint non-linearity was a pure bilinear spring (i.e. uc = 0 ) it would cause no peak shift at all. In this case, the SHBM would not be suited to identify the joint non-linearity, because the information content of the fundamental harmonic of a pure bilinear spring would not sufficient to identify the non-linear parameters.

Figure 11.29: Relative acceleration at bolted scalloped flange joint

Figure 11.30: Accelerometers placed inside the CTS directly above and directly below the bolted flange joint to monitor the gap opening at the joint

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11.3 Identification of Non-Linear Joint Parameters of an Aero-Engine Casing Joint In this section, non-linear model updating shall be applied to an aero-engine FE model in order to identify the non-linear stiffness and damping properties of an aeroengine casing joint from measured non-linear FRFs. The structure considered here is a subassembly of the RB199 aero-engine and consists of the following components (non-rotating parts only): • fan casing 1st stage (FC1), • fan casing 2nd stage (FC2), • front bearing housing (FBH), • intermediate casing (IMC), • combustion chamber outer casing (CCOC), • turbine casing (TC), • rear bearing support structure (RBSS, not visible, mounted inside the TC). A cut through the complete engine is shown in Figure 11.31, whereas the subassembly investigated here is shown in Figure 11.32. The total mass of the engine subassembly is about 119 kg.

Figure 11.31: RB199 aero-engine (courtesy of MTU)

FC1

FC2

FBH

IMC

CCOC

TC

Figure 11.32: Subassembly of RB199 aero-engine supported by bungee cords with shaker attached

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The engine components are connected by bolted flange joints, except for the TCRBSS interface. The RBSS is mounted inside the TC and is connected to it by 10 radial spokes which are bolted to the TC casing shells. Vibration testing (linear and non-linear) was performed by Imperial College staff at the MTU vibration test laboratories in Munich. The test campaign was performed within the European research project CERES. The author gratefully acknowledges the work of Dr. Suresh Perinpanayagam during the test campaign, who worked as a PhD student of Prof. D.J. Ewins at the Imperial College of Science, Technology, and Medicine in London.

11.3.1 Updating of the Underlying Linear Model Updating of the underlying linear system was found to be a prerequisite for successful non-linear updating when using the modal approach for the calculation of analytical non-linear frequency responses. Updating of the underlying linear system can be achieved by computational model updating using experimental modal data. Since the assembled aero-engine is a quite complex structure, it was decided to disassemble it and to perform modal survey testing on the individual components. Standard modal survey tests in free/free configuration with hammer excitation were conducted on each component. Subsequently, computational model updating was performed for the individual components. This allowed for updating of parameters which cannot be updated by using modal data of the complete engine assembly. The testing and updating work was shared among the different partners involved in the European research project CERES. After the component models had been validated, a modal survey test was performed on the RB199 subassembly shown in Figure 11.32 to provide a database for updating of the linear joint stiffness between the component models. The spatial distribution of the measurement DoFs of the modal survey test can be observed in Figure 11.33. Eight rings of measurement points were used in a roving accelerometer test. Each ring comprises eight measurement points equally spaced in circumferential direction, so that 64 measurement points were used in total. At each point of the test mesh, the response was measured using a tri-axial accelerometer. The excitation force was introduced at the thrust trunnion mount located at the top dead center of the IMC (reference DoF in Figure 11.33). The excitation force was applied with a hammer in radial direction. reference DoF

reference DoF

Figure 11.33: Spatial distribution of measurement DoFs

A summary of the modal data extracted from the 192 FRFs is listed in Table 11-9. Twenty modes were identified in the frequency range up to 500 Hz. It is obvious from the relatively low modal viscous damping ratios that the structure is lightly damped. Modes 3 and 5 represent local bending modes of the vanes of FC1 and FC2. Vanes

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are radial spokes which are used to direct the airflow through the engine. Even though the resonances of these modes could be observed in the measured FRFs, the corresponding modes could not be identified by the chosen set of measurement DoFs due to insufficient spatial resolution of the test mesh. Thus, the level of confidence of these modes is low. The same applies to experimental modes 18, 19, and 20. A static expansion of these modes to the FE model DoFs yields unreasonable deformations and led to the conclusion that they are not reliable and should not be used for model updating. The AutoMAC matrix of the experimental modes (EMA data) is shown in Figure 11.34 (experimental modes paired with themselves). The off-diagonal elements of the AutoMAC matrix support the decision for the exclusion of these modes. Table 11-9: Summary of experimental modal data of RB199 subassembly (ND = nodal diameter) Mode Frequency [Hz] Damping [%] 1 181,25 0,15 2 184,31 0,17 3 190,15 0,02 4 211,63 0,03 5 217,90 0,02 6 232,62 0,12 7 233,27 0,11 8 269,48 0,02 9 286,16 0,20 10 292,06 0,17 11 324,88 0,03 12 342,96 0,07 13 343,93 0,06 14 346,75 0,37 15 350,73 0,35 16 424,53 0,23 17 427,50 0,18 18 453,36 0,22 19 458,58 0,05 20 481,36 0,05

Description 2ND Shell Mode of FCs, Antiphase 2ND Shell Mode of FCs, Antiphase Vane Bending FC1 Vane Bending FBH Vane Bending FC2 3ND Shell Mode of FCs, in Phase 3ND Shell Mode of FCs, in Phase Rotation of RBSS vs. TC 2ND global Shell Mode, in Phase 2ND global Shell Mode, in Phase Rotation of inner vs. outer Core 4ND Shell Mode of FCs, in Phase 4ND Shell Mode of FCs, in Phase Global Bending, X-Z Plane Global Bending, X-Y Plane 2ND Shell Mode of TC, in Phase 2ND Shell Mode of TC, in Phase 3ND Shell Mode of TC, in Phase 3ND Shell Mode of TC, in Phase --

Figure 11.34: AutoMAC matrix of experimental modes

It was mentioned before, that the FE model was assembled from updated component models. A summary of the assembled model is listed below which emphasizes its complexity:

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• • • • • •

141

15641 nodes, 15930 shell elements, 2360 beam elements, 240 spring elements, 7 lumped masses, 1531 rigid body elements (geometric constraints).

The bolted flange joints of the assembled FE model were modeled in a flexible way according to the simplified joint modeling approach shown in Figure 3.18. However, due to rigid elements introduced at the joints, the combination of linear and nonlinear springs and dampers could not be introduced between the nodes at the flange base (as sketched in Figure 3.18). Instead, the linear and non-linear spring/damper elements had to be introduced between the rotational DoFs at the hinge located at the bolt center line level. This is sketched in Figure 11.35. The rotational springs and dampers at the hinge can be converted into equivalent translational springs and dampers at the flange base by using the distance between the flange base and the hinge (eccentricity of 8.375 mm) in conjunction with the geometric constraints among these DoFs introduced by the rigid elements. combination of linear and non-linear rotational spring/damper elements offset beam representing flange hinged rigid link

additional node at bolt level rigid link eccentricity

casing shells component 1

component 2

Figure 11.35: Non-linear joint elements introduced as rotational elements at the bolt center line level

At this stage of underlying linear model updating, only linear elements were introduced at the joints in order to update the underlying linear joint stiffness. The stiffnesses of the underlying linear joint springs were set to a more or less arbitrarily chosen numerical value of 108 Nmm rad in order to simulate almost rigid joint behavior. A summary of the analytical modal data (FEA data) obtained with the FE model and the almost rigid joint stiffnesses can be found in Table 11-10.

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Table 11-10: Summary of analytical modal data of RB199 subassembly with almost rigid joints Mode Frequency [Hz] 7 163,57 8 176,01 9 177,91 10 180,04 11 203,55 12 229,89 13 230,00 14 272,69 15 280,91 16 284,43 17 331,68 18 337,66 19 338,34 20 349,86 21 351,43 22 391,60 23 396,36 24 440,63 25 445,48 26 451,12

Description Vane Bending FC1 2ND Shell Mode of FCs, Antiphase 2ND Shell Mode of FCs, Antiphase Vane Bending FC2 Vane Bending FBH 3ND Shell Mode of FCs, in Phase 3ND Shell Mode of FCs, in Phase Rotation of RBSS vs. TC 2ND global Shell Mode, in Phase 2ND global Shell Mode, in Phase Rotation of inner vs. outer Core 4ND Shell Mode of FCs, in Phase 4ND Shell Mode of FCs, in Phase Global Bending, X-Z Plane Global Bending, X-Y Plane 3ND Shell Mode of TC, Antiphase 3ND Shell Mode of TC, Antiphase -2ND Shell Mode of TC, in Phase 2ND Shell Mode of TC, in Phase

The AutoMAC matrix of the analytical modes (FEA data) is shown in Figure 11.36. The MAC matrix was calculated by using only those DoFs of the FE model, which correspond to the measurement DoFs used in modal survey testing, see Figure 11.33. The off-diagonal elements of the FEA AutoMAC matrix indicate that the spatial distribution of the measurement DoFs is not sufficient to exactly distinguish between the modes.

Figure 11.36: FEA AutoMAC matrix using measurement DoFs only (instead of all FE DoFs)

Table 11-11 shows the correlation table of experimental modal data and analytical modal data obtained with almost rigid joints. The unreliable experimental modes (EMA data) were intentionally removed from the correlation table.

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Table 11-11: Correlation table of EMA and FEA data before updating of joint stiffness Absolute Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14

EMA Mode Number 1 2 4 6 7 8 9 10 11 13 14 15 16 17

FEA Mode Number 8 9 11 12 13 14 15 16 17 19 20 21 25 26

EMA Freq. [Hz] 181,25 184,31 211,63 232,62 233,27 269,48 286,16 292,06 324,88 343,93 346,75 350,73 424,53 427,49

FEA Freq. [Hz] 176,01 177,91 203,55 229,89 230,00 272,69 280,91 284,43 331,68 338,34 349,86 351,43 445,48 451,12

Freq. Dev. [%] -2,89 -3,47 -3,82 -1,18 -1,40 1,19 -1,83 -2,61 2,09 -1,62 0,90 0,20 4,93 5,53

MAC [%] 72,39 98,26 84,06 72,34 58,14 84,82 70,54 96,78 91,88 76,73 91,05 55,03 71,62 64,93

Modal Damping [%] 0,15 0,17 0,03 0,12 0,11 0,02 0,20 0,17 0,03 0,06 0,37 0,35 0,23 0,18

Figure 11.37 shows the MAC matrix between experimental (EMA) and analytical (FEA) modes and gives an overview about the cross pairing of the modes. Concerning the relatively poor mode pairing of some of the modes, it must be stated that the FE model predicts some local modes which could not be measured by the chosen test mesh due to the insufficient spatial resolution. This applies, for example, to FE modes 7 and 10 and EMA modes 4 and 5 which cannot be correlated properly. This was already indicated by the off-diagonal elements of the FEA AutoMAC matrix shown in Figure 11.36 and the off-diagonal elements of the EMA AutoMAC matrix shown in Figure 11.34. Nonetheless, 14 modes out of 20 could be paired and the average absolute frequency deviation of these 14 modes is 2.4% while the average MAC of the paired modes is 78%. This correlation result is not too bad so that it can be concluded that the initial FE model (which was assembled from the updated component models) is already quite accurate in the frequency range up to 500 Hz. Updating of the underlying linear joint stiffness is therefore not considered necessary at this stage.

Figure 11.37: MAC matrix of EMA and FEA modes before updating of joint stiffness

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11.3.2 Non-Linear Updating of Joint Parameters The RB199 subassembly shown in Figure 11.32 was subjected to non-linear step-sine tests with constant excitation force levels to provide a database for the identification of the non-linear joint properties. The test setup was already shown in Figure 11.32 with the structure supported by bungee cords to simulate free boundary conditions. Harmonic excitation was introduced by a shaker which was attached to a relatively stiff position of the IMC by means of a flexible drive rod (see Figure 11.38). This test setup aimed at a proper excitation of the bending modes of the structure which were expected to exhibit some degree of non-linear behavior.

Figure 11.38: Shaker attachment to RB199 structure

A broad frequency range was investigated in preliminary tests with respect to nonlinear behavior. These tests confirmed that mostly the bending modes showed nonlinear behavior, whereas other modes, such as the typical ovalizing shell modes, almost behaved linearly. Actually, only one global bending mode in the X-Z plane (mode number 14 in Table 11-9) showed non-linear behavior in the frequency range up to 400 Hz. The corresponding experimental bending mode is shown in Figure 11.39. For better visualization, this mode was statically expanded to the FE model DoFs.

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IMC-CCOC interface

Figure 11.39: Experimental bending mode (ca 346 Hz) statically expanded to the FE DoFs

The resonance frequency of this non-linear bending mode is close to the eigenfrequency of the 4-nodal-diameter (ND) ovalizing shell mode of the fan casings (mode number 13 in Table 11-9, see Figure 11.40). Even worse, it was observed that the resonance frequency of the non-linear bending mode shifted into the resonance region of the ovalizing shell mode when excited at a high force level. That meant that the distortion characteristic of the non-linear FRFs cannot be observed properly due to the interference of the two resonances. A 1 kg tuning mass was therefore attached to the top dead center of the front flange of FC1 to separate the resonance of the ovalizing shell mode from that of the bending mode (the frequency shift of the ovalizing shell mode due to tuning mass attachment was about -30Hz). The statically expanded experimental ovalizing shell mode is shown in Figure 11.40 together with the tuning mass attachment.

front flange FC1 Figure 11.40: Tuning mass attachment for the separation of experimental modes 13 and 14

After the tuning mass attachment, non-linear step sine tests were performed in a narrow frequency band around the resonance of the global X-Z plane bending mode at different constant excitation force levels. The following levels of constant excitation force amplitudes were used: 1N, 10N, 20N, 30N, 40N, 50N, 60N, and 70N. The frequency range and the frequency spacing were adjusted for each level of excitation based on the measured FRFs of the preceding force level. By proceeding this way it was ensured that the resonance peak was properly located at the center of the frequency range investigated in the step-sine tests. Figure 11.41 shows the experimental non-linear acceleration FRFs measured at the driving point. The linear FRF, which is also shown there, was measured using a

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relatively low excitation force without level control. It can be seen that the resonance frequency shifts towards lower frequencies with increasing excitation force level. In addition, the magnitude of the resonance peak increases with increasing force levels. Such a non-linear behavior can be characterized as softening stiffness non-linearity in conjunction with decreasing damping non-linearity.

Figure 11.41: Experimental driving point acceleration FRF for different levels of excitation

The characteristic FRF distortion shown in Figure 11.41 can be observed not only at the driving point, but at all other measurement points as well. Altogether 7 measurement points were used in the non-linear step-sine tests. Their distribution over the structure can be observed in Figure 11.42. These measurement DoFs are suited to monitor the bending deformation in the X-Z plane. In addition, measurement DoFs 5 and 7 can be used to check for the coupling with the bending mode in the X-Y plane. MP 7 Node 160104

MP 5 Node 310301 MP 4 Node 561413

MP 2 Node 160113 MP 1 (DP) Node 260933

MP 3 Node 310113

MP 6 Node 311213

Figure 11.42: FE model with distribution of measurement point used in non-linear tests

The variation of resonance frequency and damping ratio with increasing excitation force level can be observed from the linearity plots (also known as modal characterization functions) shown in Figure 11.43. In this case, the resonance frequency was assumed to be the peak frequency of the drive point FRF and the damping ratio was estimated from the half-power bandwidth method. It can be seen

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that the resonance frequency of the lowest excitation force level is even lower than the eigenfrequency of the bending mode identified from modal survey tests on the underlying linear system. This frequency difference was caused by the tuning mass attachment and possibly also by the shaker attachment. It was thus decided to include the tuning mass in the FE model as a lumped mass, whose mass moments of inertia were derived from engineering formulae.

Figure 11.43: Variation of resonance frequency and damping ratio with excitation force level

Due to the limited amount of experimental data available for non-linear updating (only the response in the vicinity of the resonance of a single bending mode), the number of non-linear joint parameters which can be updated is restricted. From a closer inspection of the modal strain energies of the underlying linear joint springs, it was concluded that only the IMC-CCOC interface was sufficiently activated, while the other joints were activated much less (see Figure 11.44). According to [Fox 1968], the modal strain energy is directly proportional to the eigenvalue sensitivity. The decision to focus on updating of the IMC-CCOC joint parameters only thus corresponds to a selection of the most sensitive updating parameters with respect to eigenvalue sensitivity. It would not make sense to try to update the other interface parameters from the vibration test data available, because their sensitivities are much less so that they would suffer unreasonable parameter changes.

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Figure 11.44: Normalized element strain energies of joint springs at the different interfaces

A comparison of the linear experimental and analytical FRFs revealed that there were still minor discrepancies between experimental and analytical FRFs. These discrepancies were reduced prior to non-linear updating by manual adjustment of the underlying linear IMC-CCOC joint stiffness. It was already stated above that all joint stiffnesses were initially set to 108 Nmm as a first estimate. After manual rad updating, the underlying linear IMC-CCOC joint stiffness was reduced to 1.58 ⋅ 107 Nmm rad . This value is not too far away from the initial estimate, because the initial model already correlated quite well with the test data. After the underlying linear joint stiffness had been adjusted, a relatively good agreement of the experimental and analytical linear FRFs was achieved so that an adjustment of the underlying linear modal damping ratios (which were taken from the modal survey test data) was not considered necessary. Table 11-12 summarizes the eigenfrequencies of the RB199 subassembly model after attaching the tuning mass and after adjusting the underlying linear IMC-CCOC joint stiffness. It can be seen that the tuning mass separates the ovalizing shell modes which would otherwise appear as repeated roots. In principle, the underlying linear joint parameters can be updated together with the non-linear joint parameters. However, it is known from experience (and it was proven in the cantilever beam example) that the underlying linear joint parameters are often much more sensitive than the non-linear parameters. This can lead to illconditioning of the updating equations which should be avoided. Another aspect for the separation of linear and non-linear updating involves the accuracy of non-linear modal frequency response analysis and has also been discussed in the cantilever beam example in chapter 11.1.

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Table 11-12: Summary of analytical modal data of RB199 subassembly with tuning mass attached and after adjustment of underlying linear stiffness of IMC-CCOC interface Mode Frequency [Hz] 7 160,37 8 163,56 9 174,86 10 180,02 11 195,67 12 203,50 13 223,14 14 225,40 15 272,28 16 274,82 17 282,59 18 313,06 19 327,86 20 331,80 21 342,86 22 343,98 23 390,78 24 395,57 25 440,62 26 443,73 27 449,38 28 455,32 29 477,32

Description 2ND Shell Mode of FCs, Antiphase Vane Bending FC1 2ND Shell Mode of FCs, Antiphase Vane Bending FC2 -Vane Bending FBH 3ND Shell Mode of FCs, in Phase 3ND Shell Mode of FCs, in Phase Rotation of RBSS vs. TC 2ND global Shell Mode, in Phase 2ND global Shell Mode, in Phase 4ND Shell Mode of FCs, in Phase 4ND Shell Mode of FCs, in Phase Rotation of inner vs. outer Core Global Bending, X-Z Plane Global Bending, X-Y Plane 3ND Shell Mode of TC, Antiphase 3ND Shell Mode of TC, Antiphase -2ND Shell Mode of TC, in Phase 2ND Shell Mode of TC, in Phase 5ND Shell Mode of FCs, in Phase 5ND Shell Mode of FCs, in Phase

The order of the FE model is too large for efficient non-linear frequency response analysis. Thus, Craig-Bampton reduction was applied, where each single component of the assembled FE model was used as a superelement. According to the discussion in chapter 6.4, the nodes at the component model interfaces must be used as master DoFs. This is a requirement for the application of the linear Craig-Bampton model reduction to a model with local non-linearities. In addition, nodes with high mass concentrations and nodes which correspond to the measurement positions were also used as master DoFs. High mass concentrations were checked by the ratio of the main diagonal factors of the mass and stiffness matrix, i.e. αi = M ii / Kii , i = 1,… , nDoF . DoFs with high numerical values for α i were used as master DoFs to improve the quality of the model reduction process. Altogether 4788 master DoFs were selected out of approximately 90000 DoFs of the full model. The Craig-Bampton reduction approximates dynamic effects at the omitted model parts by additional modal degrees of freedom. The eigenfrequencies of the fixed interface modes of the individual superelements were used as a criterion for the selection of additional modal DoFs. In this case, the number of additional modal DoFs for each superelement is equal to the number of fixed interface modes below 3 kHz. The following list summarizes the distribution of additional modal DoFs over the superelements: • FC1: 100 modal DoFs, • FC2: 48 modal DoFs, • FBH: 80 modal DoFs, • IMC: 140 modal DoFs, • CCOC: 80 modal DoFs, • TC: 50 modal DoFs, • RBSS: 50 modal DoFs, • Total: 548 modal DoFs.

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The Craig-Bampton reduced model had altogether 5336 DoFs (4788 physical master DoFs plus 548 additional modal DoFs), i.e. the order of the reduced model is only 6% of the original model order. Despite of the significant order reduction, the CraigBampton reduced model was still accurate in the frequency range up to 500 Hz. In fact, the largest frequency deviation between the full model and the reduced model was less than 1.5% and most importantly, the eigenfrequency deviation of the bending mode under consideration was only 0.04%. The MAC values between the modes of the full model and the reduced model showed an optimal correlation of 100% so that the correlation of the modal results of the full and the reduced model is skipped here. After the underlying linear system had been updated and the Craig-Bampton reduction had been applied, non-linear elements were introduced at the IMC-CCOC interface. According to the discussion on bolted flange joints in chapter 3 and according to the distortion characteristic of the experimental FRFs shown in Figure 11.41, it was a reasonable choice to use pre-loaded bilinear springs at the IMC-CCOC joint to represent the softening stiffness non-linearity observed in the tests. A softening quadratic damper was chosen to represent the softening damping characteristic of the experimental FRFs. The quadratic damper was used as a supplement to the underlying linear damping which was represented by modal damping ratios taken from the modal survey tests. Such a combination of distributed underlying linear modal damping together with local non-linear (physical) dampers can be used to represent the decreasing damping level with increasing excitation force levels. However, it should be noted that this can only be considered as an equivalent damping model, because the real mechanisms of energy dissipation at the IMC-CCOC joint are not fully understood and can therefore only be approximated by the polynomial damping model. 48 non-linear spring and damper elements were introduced at the IMC-CCOC joint in accordance with the simplified joint model shown in Figure 11.35. The non-linear elements share all the same response dependent properties but may have different actual stiffness and damping properties depending on the local relative response level. The restoring force diagrams of the non-linear springs and dampers are shown in Figure 11.45. The compression regime stiffness k1 of the pre-loaded bilinear spring is the underlying linear joint stiffness and was already updated manually. The underlying linear damper constant clin does not need to be defined explicitly. It is rather represented by distributed modal damping and needs not to be updated.

fR

k1

fR

k2

uc

u

⎧ k1u, u ≤ uc fR = ⎨ ⎩ k2 ( u − uc ) + k1uc , u > uc

u

fR = clinu + cnl u u, cnl < 0

Figure 11.45: Force-displacement diagrams of non-linear joint elements

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Three non-linear parameters remain for updating, these are the: • tension regime stiffness k2 , •

position of the stiffness transition uc ,

•

non-linear damper constant cnl .

After type and location of the non-linearity were defined, meaningful initial values must be determined for the three remaining non-linear parameters k2 , uc , and cnl . “Trial-and-Error” is always a possibility to find such initial values. However, in order to approximate the distortion characteristic of the experimental FRFs with the nonlinear FE model, a large number of successive non-linear frequency response analyses may be necessary to find appropriate initial parameter values. A more scientific approach for the rough adjustment of the initial non-linear parameters was published in [Böswald 2004b]. This approach is based on the comparison of linearity plots (modal characterization functions) obtained from test data and from the nonlinear FE model with an estimated set of initial non-linear parameters. The advantage of this method is that the linearity plots can be produced quite rapidly without the need to calculate non-linear FRFs in an iterative procedure. The basic assumptions of the procedure for the rough adjustment of the initial nonlinear parameters are that the non-linear modal equations of motion are uncoupled and that the non-linear response magnitude does not deviate significantly from the linear one. In addition, it is assumed that there is no coupling between non-linear stiffness and non-linear damping. It is obvious that these assumptions are violated by non-linear systems, but nonetheless, slightly erroneous analytical linearity plots such as the one shown in Figure 11.46 can be produced when relying on these assumptions. The analytical linearity plots can be compared with the experimental ones and the initial non-linear parameters can be adjusted manually until the experimental and analytical linearity plots roughly correlate. There is no need to achieve a perfect match of the analytical and experimental linearity plots, because the analytical ones are anyway slightly erroneous. This procedure leads to meaningful initial values for the non-linear parameters within a few minutes. “Fine tuning” of the non-linear parameters can then be performed by non-linear updating using frequency response residuals. Figure 11.46 shows the comparison of experimental and analytical linearity plots. The analytical linearity plots were produced with the initial set of non-linear parameters and the experimental linearity plots were extracted from the measured non-linear FRFs. Even though there are still discrepancies between the two curves, they nonetheless indicate that the initial non-linear parameters are already suited to approximate the principal non-linear behavior, i.e. the change of resonance frequency and damping level with increasing excitation force amplitudes.

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Figure 11.46: Comparison of experimental and analytical linearity plots

The initial values for the non-linear joint parameters are listed in Table 11-13. Table 11-13: Initial values for non-linear joint parameters Parameter

Description

Initial Value

k1


1.58 ⋅ 107

k2


5 ⋅ 106

uc


cnl

non-linear softening damper constant

Nmm rad

Nmm rad

5 ⋅ 10−6 rad −3 ⋅ 103

Nmm s 2 rad 2

In order to effectively perform non-linear updating, modal frequency response analysis is the recommended type of non-linear analysis. A proper selection of the modes for mode superposition is a prerequisite for accurate modal response analysis. For example, the linear frequency response can be calculated by using the following modal superposition (when assuming real normal modes obtained from the FE model):

{φr }{φr }

T

n

{uˆ} = ∑ r =1

μr (ω − Ω 2 r

2

fˆ } . { + j 2ξ ω Ω ) r

(11.7)

r

The selection of effective modes for non-linear modal frequency response analysis used here is based on a criterion which uses a combination of three different factors. These factors are: T 1. α = {φ } fˆ , i.e. modal excitation 1

r

{}

2. α 2 = ωr2 − Ω c2

−1

, i.e. distance of eigenfrequency ωr to the center of the active

frequency range Ω c

1 E ∑ φe1,r − φe2,r , i.e. average absolute modal relative deformation at the E e=1 DoFs of the non-linear elements

3. α 3 =

Here, E is the number of non-linear elements (index e ), and φe1,r , φe 2,r are the elements of the r-th mode shape which correspond to the non-linear element DoFs e1 and e2 the of element e . The combination of the first two factors can be used for the

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selection of the most influential modes for linear modal frequency response analysis. The third factor takes into account how strong a mode may be affected by non-linear 2-DoF elements. The selection of the most influential modes of the RB199 subassembly for modal frequency response analysis was performed by using the influence factors α1 , α 2 , α 3 and is shown in the bar chart in Figure 11.47. The influence factors were normalized to visualize their relative importance for accurate modal frequency response analysis. The normalized linear factor shown in the left hand side bar chart is the product of the first two influence factors, i.e. α1 ⋅ α 2 (normalized to unit maximum value). The normalized total factor shown in the right hand side bar chart is the normalized product of all three factors of the above list, i.e. α1 ⋅ α 2 ⋅ α 3 . It can be observed from these bar charts that the bending mode in the X-Z plane (mode 21) is the most important mode for linear and for non-linear frequency response analysis. The influence of all other modes is one order of magnitude less (note the logarithmic scale). In principle, the X-Y plane bending mode (mode 22) would have the same influence, however, this mode is less well excited (small factor α1 ) and is thus less important.

Figure 11.47: Factors for mode selection for modal frequency response analysis

The 10 most influential modes were retained for modal non-linear frequency response analysis. Consequently, a 10x10 coupled system of modal equations must be solved in the non-linear frequency response analysis. This means that the number of DoFs was reduced from initially ca. 90000 to 5336 by Craig-Bampton reduction, and subsequently from 5336 to only 10 generalized modal DoFs by mode superposition. Figure 11.48 shows the analytical non-linear driving point FRFs calculated by modal frequency response analysis using the 10 most influential modes and the initial nonlinear parameters derived from the comparison of linearity plots. A visual comparison of the analytical non-linear FRFs shown in Figure 11.48 with the experimental ones shown in Figure 11.41 reveals that the FE model is already able to roughly predict the resonance peak shift and peak magnitude amplification with increasing excitation force levels. However, the initial non-linear parameters still need to be updated to improve the prediction capability of the FE model.

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Figure 11.48: Analytical acceleration FRF at driving point obtained with initial parameters

The differences between the experimental and the analytical non-linear FRFs were assembled in the residual vector for computational model updating. The displacement FRF magnitudes at the measurement DoFs 1 to 4 were used because these measurement DoFs are particularly suited to monitor the X-Z bending deformation. The FRFs of the 20N and the 70N excitation force level were used, i.e. a moderate level FRF and a high level FRF. Altogether 8 FRFs were used in the residual (4 measurement DoFs times 2 levels). The non-linear FRFs of other load levels will be used later on to check the prediction capability of the updated model. The evolution of the updating parameters within 10 iteration steps can be observed in Figure 11.49. It can be seen that all three parameters were increased. The tension regime stiffness k 2 increased by +14.1%, the location of the stiffness transition point

uc increased by +53.7%, and the damper constant of the softening quadratic damper increased by +38.3%. The overall differences between analytical and experimental FRFs were reduced which is indicated by the drop of the objective function down to 51%. The parameter changes per iteration were constrained to ±20% but no limits were imposed on the absolute parameter variation. A weak regularization was applied to smoothen the parameter evolution. The regularization matrix was calculated according to equation (10.50) by using w p = 0.05 .

Figure 11.49: Evolution of updating parameters and objective function

The following table summarizes the updating results obtained after 10 iteration steps of computational model updating.

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Table 11-14: Summary of parameter values and parameter changes Parameter

Description

Initial Value

k1


1.58 ⋅ 107

k2


5 ⋅ 106

uc


5 ⋅ 10−6 rad

cnl

non-linear softening damper constant

−3 ⋅ 103

Nmm rad

rad 2

1.58 ⋅ 107 5.705 ⋅ 106

Nmm rad

Nmm s

Final Value

2

Change

Nmm rad

--

+14.1%

Nmm rad

7.685 ⋅ 10−6 rad

+53.7%

2

+38.3%

−4.15 ⋅ 103

Nmm s rad 2

11.3.3 Interpretation of Updating Results A predictivity check was performed to verify the non-linear updating results. This was done by comparing the experimental non-linear FRFs with the corresponding analytical non-linear FRFs calculated with the identified set of non-linear parameters, especially for the excitation force levels which have not been used for updating. Figure 11.50 to Figure 11.53 show the comparison plots of the analytical (left) and experimental (right) non-linear acceleration FRFs obtained at measurement points 1 to 4. The non-linear FRF distortion can be observed at all measurement points and a relatively good agreement can be found between experimental and analytical data. The responses at other measurement DoFs were too small for meaningful correlation with analytical responses.

Figure 11.50: FRF comparison at measurement point 1 (drive point)

156

11. Applications

Figure 11.51: FRF comparison at measurement point 2


11. Applications

157


A closer inspection of the FRFs was used to verify the local results after non-linear updating. Therefore, the drive point acceleration FRF is plotted for three different levels of excitation force (20N, 40N and 70N). The FRF of the linear model, the initial non-linear model and the updated non-linear model are compared with the experimental non-linear FRF. It can be seen from the FRF comparison of the lowest excitation force level shown in Figure 11.54 that the different models (linear, initial non-linear, updated non-linear) do not deviate significantly from each other and that they show a relatively good agreement with the test data.

Figure 11.54: Magnitude of drive point FRF, 20N excitation force level

It can be observed from the medium force level FRF plot shown in Figure 11.55 that the linear model already deviates significantly from the FRFs obtained with the nonlinear models and also from the non-linear experimental FRF. No significant difference can be observed between the initial and the updated non-linear model at this level of constant excitation force.

158

11. Applications


Finally, it can be observed in Figure 11.56 that for a high excitation force level the linear FRF is far away from the measured FRF due to the strong activation of the non-linearity. Furthermore, better results can be obtained with the updated nonlinear model than with the initial non-linear model.


From the comparison of the experimental and the analytical non-linear FRFs after non-linear updating it can be concluded that non-linear updating of the IMC-CCOC joint stiffness and damping properties was successful. The fact, however, that the compression regime stiffness of the pre-loaded bilinear spring is only 2.8 times higher after non-linear updating than the tension regime stiffness is suspicious from a physical point of view and does not comply with the real stiffness conditions of bolted flange joints (see Table 11-14). This may possibly be a result of the excitation force levels being not sufficiently high to significantly activate the non-linear joint stiffness by reducing the compressive pre-stress between the IMC and CCOC. It would be interesting to see what updating results would be obtained when using the direct approach for frequency response analysis, because it was found out in the cantilever beam example that modal non-linear frequency response is not optimal for non-linear updating. Indeed, attempts were made to perform non-linear updating using the direct approach in conjunction with the Craig-Bampton reduced model. However, the non-linear frequency response analysis was aborted after a few iteration steps so that non-linear responses obtained with the direct approach are not available. In the CTS example it was shown that the non-linear updating results obtained with modal frequency response analysis are almost identical to those obtained with the direct frequency response approach when a properly selected set of

11. Applications

159

modes is used, or respectively, when modes are disregarded which do not contribute to the response in the vicinity of a certain resonance. This approach was followed in this aero-engine application as well. From that point of view it is assumed that the direct approach would have produced almost the same results as the modal approach. Another point to consider is the load introduction. The excitation force was measured by a force sensor which was mounted to a stiff position of the IMC (see Figure 11.38). It may be possible that non-linear effects were activated at sensor/structure interface before any joint non-linearity was activated at the casing interfaces. Unfortunately, these non-linear effects were not investigated within the test campaign so that no reliable statement can be made to support or to discard this.

160

12. Summary and Conclusions

Chapter 12: Summary The theoretical basis of a computational model updating procedure for the identification of local non-linear stiffness and damping parameters in large order finite element models was developed in this thesis. This updating procedure uses the differences between experimental and analytical fundamental harmonic non-linear steady-state responses as an updating residual to identify the physical non-linear parameters of non-linear elements which govern the restoring force functions. The application of the model updating procedure is restricted to weakly non-linear systems. These are systems which produce a periodic response when excited harmonically. Furthermore, the fundamental harmonic dominates the total nonlinear steady-state response such that the Harmonic Balance method is a suitable approach for non-linear response analysis. The non-linearities are assumed to be concentrated local non-linearities which can be modeled by a set of non-linear 2-DoF elements to connect linear substructures of an overall assembled finite element model. On the experimental side, two approaches for the measurement of non-linear fundamental harmonic frequency response functions (FRFs) were discussed. One approach uses sine excitation in conjunction with constant response amplitudes, whereas the other approach also uses sine excitation, but with a constant excitation force amplitude. In the constant response level measurements, the non-linear behavior of a structure is linearized so that “almost linear” FRFs can be acquired. These FRFs are suited for linear modal analysis and thus for linear model validation which must be performed for each level of constant response. FRFs obtained from constant excitation force level measurements usually deviate from the linear ones and appear distorted, especially in the vicinity of resonances where the response is typically large. The distortion characteristics of the constant force level FRFs are suited to detect and to characterize structural non-linearities. It was therefore reasonable to use constant force level FRFs as an input to non-linear computational model updating for the identification of the non-linearities which caused the FRF distortions. Besides the two non-linear FRF measurement approaches, tools for the detection and characterization of non-linearities from measured response data were discussed. Methods which utilize frequency domain response data and methods which rely on time domain response data were summarized. Once non-linear behavior was detected by the application of these tools, they can help to prepare strategies for non-linear FRF measurements, or respectively, can support the selection of appropriate non-linear element types to represent the detected nonlinearity in a finite element model. On the analytical side, different methods for non-linear frequency response analysis were presented. The classical Harmonic Balance method was discussed, which can be used for the calculation of the fundamental harmonic non-linear steady-state response due to harmonic excitation. In addition, a non-linear frequency response analysis tool based on Structural Modification Theory was developed in this thesis. It uses the FRF matrix of the underlying linear system as a basis and treats non-linear elements as response dependent structural modifications applied to the underlying linear system. This allows for the iterative calculation of the non-linear fundamental harmonic response based on the well known equations for structural modification applied to the underlying linear FRF matrix. The properties of fundamental


161

harmonic non-linear responses were discussed to provide better understanding of experimental and analytical non-linear response curves. The presence of frequency intervals with multiple stable response solutions was discussed, together with the socalled jump phenomenon, which occurs whenever a stable branch of the response curve comes to an end so that the response curve must be continued on another stable response branch. The theory of the Multi-Harmonic Balance method was also presented for the sake of completeness. This is a tool for the calculation of the general periodic steady-state response of non-linear systems in case of harmonic excitation. The application of the Multi-Harmonic Balance method was considered prohibitive for large order systems due to the numerical effort involved in the solution of the equation systems generated. One of the requirements posed on the non-linear model updating procedure developed in this thesis is the applicability to large order systems. Different well established model reduction techniques were therefore reviewed. Emphasis was given to their applicability to systems with local non-linearities. The Craig-Bampton reduction and the mode superposition method were investigated in more detail. Non-linear updating was applied to three different examples, not only to demonstrate the field of application for non-linear updating, but also to assess the limitations of non-linear frequency response analysis and non-linear updating. Table 12-1: Summary of non-linear updating applications

General Description

Non-Linear Vibration Test Data

Model Reduction Response Analysis

Model Updating

Cantilever Beam

CTS

RB199

comment

analytical example

laboratory structure

aero-engine subassembly

model type

beam model

shell model

shell model

number of DoFs

18

ca. 9500

ca. 90000

214 beam elements

2360 beam elements

number of elements

8 beam elements

test procedure

--

1440 shell elements

15930 shell elements

step-sine base accel.

step-sine

level control

--

constant base accel.

constant exciation force

response data

simulated responses

exp. transmissibilities

exp. FRFs

number of meas. DoFs

18

9

7

applied method

--

Craig-Bampton

Craig-Bampton

DoFs after reduction

18

686

5336

applied method

direct + modal SHBM

direct + modal SHBM

modal SHBM

number of modes used

study of multiple sets

5

10

number of parameters

3 and 2

3

3

type of residual

displ. response magn.

transmissibility magn.

displ. FRF magnitude

number of meas. DoFs used in residual

4

1

4

number of force levels

1

3

2

underlying linear system

variable and constant

constant

constant

regularization

no

yes, wp=0.05

yes, wp=0.05

The first application of non-linear updating was a small order analytical cantilever beam model which was updated in a non-linear way by using simulated test data. It was shown that linear and non-linear parameters can be updated simultaneously when the direct approach for non-linear frequency response analysis is used. When the modal approach shall be used, a two-step strategy for non-linear updating was recommended. The underlying linear system is updated in the first step and the nonlinear parameters are updated in the second step while the already updated underlying linear system is kept unchanged. By proceeding this way, it was shown

162


that truncation errors inherent in non-linear modal frequency response analysis were reduced so that the non-linear updating results could also be improved. The second application of non-linear updating was the identification of the nonlinear stiffness and damping properties of the bolted flange joint of a cylindrical test structure (CTS). A medium order shell model of the CTS was updated by using experimental non-linear transmissibilities. These were obtained from base excitation tests performed on a slip table with different levels of constant base acceleration amplitudes. A simplified model for bolted flange joints was proposed, which had been derived from a load path analysis of a 3D model of a typical joint segment. CraigBampton reduction and mode superposition were applied to effectively reduce the order of the system. Large savings in terms of computation time were achieved for non-linear frequency response analysis and also for non-linear updating by the application of model reduction techniques. In particular, the modal approach for nonlinear frequency response analysis was shown to be efficient for updating of large order systems. Almost identical updating results were obtained with the direct and the modal approach for non-linear response analysis in case of the CTS example. The applicability of non-linear model updating to large order systems was demonstrated by the identification of non-linear stiffness and damping properties of a bolted flange casing joint of an aero-engine from measured non-linear frequency response functions. A bolted flange casing joint of the aero-engine was modeled in the same simplified way as it was done in the CTS example. The 90000 degree of freedom system was reduced by Craig-Bampton reduction in a first step, and subsequently, by mode superposition in a second step. Non-linear updating was successfully performed by using modal frequency response analysis in conjunction with the Craig-Bampton reduced model.

12.1 Conclusions It was demonstrated in the cantilever beam example in chapter 11.1 that non-linear parameters can be identified from non-linear fundamental harmonic frequency responses. Furthermore, it was shown that linear and non-linear parameters can be identified simultaneously in a single updating run, provided that the direct approach is used for the calculation of the analytical non-linear frequency responses. The modal approach was shown to be an attractive (i.e. numerically efficient) way for non-linear frequency response analysis. It was demonstrated that modal truncation errors only yield little difference in the calculated response when compared to the response obtained with the direct approach. However, these almost negligible response errors can lead to significant differences in non-linear updating when the parameters are insensitive. The truncation errors of the modal frequency response can be reduced by including additional modes for mode superposition. Reasonably accurate non-linear updating results can be obtained with the modal approach for frequency response analysis when updating is performed in two successive steps, i.e. updating of the underlying linear system is performed prior to updating of the non-linear parameters. This two step strategy was found to reduce the modal truncation errors in non-linear modal frequency response analysis and led to more accurate response predictions and thus to more accurate updating results. Indeed, it was shown in the CTS application in chapter 11.2 that non-linear updating with modal frequency response analysis yields almost identical results as with the direct approach.


163

When considering the numerical efficiency, it can be concluded that the modal approach is the first choice for non-linear frequency response analysis of large order systems. For example, non-linear updating of the CTS example took about 40010 seconds when the direct approach was used in conjunction with the Craig-Bampton reduced model (reduction from initially ca. 9500 to 686 DoFs). The same updating run took only 4280 seconds when the modal approach was used (reduction from initially ca. 9500 DoFs to only 5 modal DoFs by mode superposition). The numerical efficiency could be even more improved when updating was performed by applying mode superposition to the Craig-Bampton reduced model (reduction from initially ca. 9500 to 686 DoFs by Craig-Bampton and subsequently from 686 DoFs to only 5 modal DoFs by mode superposition). In this case, an identical updating run with the same set of parameters and the same number of iteration steps took only 2957 seconds. Figure 12.1 summarizes the normalized computation times of the different CTS updating runs performed with different reduced order models. This clearly emphasizes the efficiency of the modal approach, especially when taking into account that the updating results of the different updating runs only deviate about 1% from each other. The efficiency of non-linear model updating in case of large order systems was achieved at the cost of an increased effort for FE model preparation. For example, the initial FE model must be reduced by Craig-Bampton reduction which involves the meaningful selection of master DoFs and additional modal DoFs for the model reduction to be accurate. In addition, the most influential modes must be selected to achieve accurate modal response analysis with relatively low numerical effort. This altogether results in a significant amount of time which must be spent to prepare large order models for non-linear updating.

Figure 12.1: Comparison of computation time of CTS updating when using different order reduced models and different response analysis approaches

From the updating results of the analytical cantilever beam example it can be concluded that the exact parameters could be identified by using the fundamental harmonic response. This can be considered as a prerequisite for the application to real systems, because the cantilever beam model is an analytical example where type and location of the non-linearity were exactly known a priori. The updating results of the CTS example are somewhat unsatisfactory. Even though the overall trend of resonance frequency shift and resonance peak attenuation with increasing excitation levels could be predicted with the updated non-linear model, it must be stated that the characteristic of the resonance frequency shift with increasing excitation level and the characteristic of the resonance peak attenuation, is not fully consistent with the experimental observations. This can be observed in Figure 11.27. The final values of the updating parameters are also hard to interpret

164


from a physical point of view. The fact that the pre-loaded bilinear spring was almost converted into a linear spring in the course of updating is not consistent with the expected stiffness values of bolted flange joints. It must be stated, however, that the CTS joint belongs to the group of flexible bolted joints without surface pressure at the flange base (according to Figure 3.10) and that the non-linearity of such flexible bolted flange joints can be described by a bilinear stiffness characteristic which has almost no pre-load. The equivalent non-linear stiffness of such a bilinear spring is almost independent of the response level and it was stated before that the Harmonic Balance method is not well suited for fundamental harmonic response analysis of such systems, because it can no longer be assumed that the fundamental harmonic will dominate the response of such a system. When looking at the updating results of the aero-engine application it must be stated that the characteristic of the resonance peak shift and of the resonance peak amplification could be predicted quite well by the updated non-linear FE model. Also local results like the responses of excitation force levels which were not used in the updating residual could be predicted quite accurately by the updated non-linear FE model. From that point of view it can be concluded that non-linear updating was very successful in case of the aero-engine application. However, it is hard to interpret by physical reasoning that the compression regime stiffness of the bolted flange joint is only about 3 times higher than the tension regime stiffness (even though the updated model was able to accurately predict the FRF distortions). This result is somewhat unsatisfactory, because the updating results are inconsistent with the expectations for the stiffness values in tension and compression regime, in particular when looking at the bolted joint loads diagrams published in the engineering guideline [VDI 2230]. A possible reason for this effect can be found in the excitation force levels which were probably too small to sufficiently activate the joint nonlinearity. Indeed, it can be observed from the bolted joint loads diagrams that tension and compression regime stiffness are identical, unless the operational load exceeds the threshold stated in equation (3.1). From that point of view, it would generally be desired to excite the structure at even higher excitation force levels while overtesting must be avoided at any time (risk of damage). But then another problem may arise which is concerned with load introduction. A harmonic excitation force is usually introduced by means of a shaker and a flexible drive rod which is bolted in some way to the structure. Non-linear effects must be expected at the shaker/structure interface in case of large excitation forces. It is not clear how these possibly non-linear effects can affect the FRF measurements. Base excitation tests, like those performed on the CTS, do not suffer from load introduction problems. Instead, the excitation force is effectively distributed over the structure. However, base excitation tests have other disadvantages, like additional unknown boundary conditions (e.g. the slip table is floating on a thin oil film with unknown stiffness) and noise contamination of the measured signals due to the significant electromagnetic radiation caused by the huge shakers and amplifiers which typically drive the slip table. From a theoretical point of view, it would be possible to use the non-linear parameters identified from non-linear FRFs for other types of non-linear response analysis as well. For example, the physical non-linear parameters of the pre-loaded bilinear joint stiffness of the aero-engine casing joint were identified from non-linear FRFs. Even though these non-linear FRFs were measured in a special-purpose test using sinusoidal excitation, it would nonetheless be possible to use the identified non-linear parameters for subsequent non-linear transient response analysis, such as simulation of bird strike or fan blade-off events in case of the aero-engine


165

example. It should be noted, however, that polynomial type non-linearities were used, which can only be considered as an equivalent non-linear damping model to approximate the true non-linear softening damping non-linearity observed in the experimental FRFs. This equivalent non-linear damping model had to be used, however, since the true mechanisms of non-linear damping were not known. It is of course possible to introduce polynomial non-linear dampers in an FE model for nonlinear transient response analysis, but numerical instabilities may arise when response amplitudes are being produced in the transient analysis, which are far away from the response levels used for non-linear updating. In this case, the polynomial approximation of non-linear damping might deviate significantly from the true non-linear damping and can cause inaccurate response predictions or may even cause the analysis to fail.

12.2 Outlook It was stated above that the Harmonic Balance method for non-linear response analysis is not suited for systems with pure bilinear stiffness. In this case, the fundamental harmonic is no longer sufficient to describe the total non-linear response and is consequently no longer sufficient to properly identify the non-linear parameters from measured non-linear FRFs. The use of higher harmonic FRFs can help to overcome this limitation. From the experimental side, this would require that the test equipment used for FRF measurements must be modified to allow for the measurement of higher harmonic FRFs. These higher harmonic FRFs could then be used as additional information for the identification of non-linear parameters. From the analytical side, the Multi-Harmonic Balance method would be appropriate to calculate the non-linear FRFs. This method was considered prohibitive for updating of large order systems due to the immense computational effort. However, if it turns out that a few additional higher harmonic FRFs would be sufficient to improve the identification of non-linear parameters (e.g. of bilinear systems), then it would be worth to further investigate this approach. The use of the Multi-Harmonic Balance method for non-linear updating (using residuals of multiple harmonics) in conjunction with appropriate model reduction techniques is certainly a promising approach to be investigated. It was unsatisfactory that the true of non-linear damping mechanisms of the applications presented in this thesis could only be approximated by polynomial type non-linearities. The use of polynomial type non-linearities restricts the prediction capability of non-linear models to a narrow range of vibration amplitude levels which are not too far away from those measured in the non-linear tests. The derivation of non-linear damping models based on physical reasoning would certainly help to overcome this limitation. For example, the non-linearity of bolted joints in case of shear force transmission was discussed in chapter 3 and it was found that the corresponding non-linear restoring force can be represented by a friction type nonlinearity with microslip effects. The investigation of the physical effects which are really the cause of stiffness and damping non-linearity can be achieved by so-called micro-modeling approaches, where the interfaces which are expected to cause nonlinearity are modeled in a highly detailed way based on statistical information about the measured surface roughness. These micro-modeling approaches are expected to introduce many additional parameters which may influence in some way the nonlinear stiffness and non-linear damping parameters (some of these may even not be

166


structural parameters), but it is doubtful whether these parameters can be identified from measured response data. The non-linear frequency response analysis method presented in this thesis relies on structures with local non-linearities which can be represented by 2-DoF elements. The extension of the non-linear response analysis software to include other types of non-linear elements would certainly offer new fields of application. The approach of a Young’s modulus which depends on the amplitude of the “von Mises” equivalent strain would be an interesting approach for the calculation of response amplitude dependent beam-, shell-, or solid elements. Further development of the non-linear Structural Modification Theory is certainly an interesting field and should be investigated. The major advantage of this method is that it operates directly with the FRF matrix which is the inverse of the exactly condensed dynamic stiffness matrix, i.e. application of model reduction techniques would no longer be required. Unfortunately, the idea of using such an approach for non-linear response analysis came almost too late during my time as a research assistant of Prof. Michael Link. It was therefore only possible to develop the theory and software with limited functionality. It was not possible to develop reliable software which could be introduced as an additional feature into the already existing harmonic balance response software HBResp. Another point to consider is the variability of test data. It was stated in chapter 4 and chapter 8 that in some cases of non-linear systems it is difficult to obtain stable or repeatable response data. One example for that group of non-linear systems are friction type non-linearities where the friction surface conditions may change in the course of a non-linear vibration test campaign, e.g. due to wear of the friction surfaces. Therefore, it should be kept in mind that in some cases it would not make sense to identify a unique set of non-linear parameters for such structures, because this unique parameter set can only represent one possible parameter combination in the whole parameter space. It would be better to identify confidence intervals for those parameters which are expected to change during testing, instead of identifying crisp numbers. Two different approaches were developed in recent years to handle the problem of uncertain model parameters. One is a probabilistic approach which can be used to calculate the response of FE models with uncertain parameters defined by statistical information. Another approach is known as the Fuzzy Finite Element Method and uses interval arithmetic to propagate parameter intervals through an FE model which are linked to confidence levels. Finally, it has to be mentioned that the methods developed in this thesis do not claim to be complete and readily applicable to any type of problem concerned with of nonlinear parameter identification and non-linear response analysis. This thesis rather comprises a practical approach to non-linear parameter identification, where different well established techniques are combined in an ably way and where extensions to existing methods have been developed to make them suitable to the weakly non-linear systems.

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14. Appendix

177

Chapter 14: Appendix 14.1 MSC.Nastran Input Deck of the Cantilever Beam Model $ -----------------------------------------------------------------------------$ $ This is a Nastran Input Deck of the Underlying Linear System of the Cantilever $ Beam Model $ $ -----------------------------------------------------------------------------$ $ Create OP2-File for Modal Data (Created by Fortran Unit 12) -----------------ASSIGN OUTPUT2 = 'CTS_Beam.op2', UNIT = 12 $ $ Create OP4-File for System Matrices (Created by Fortran Unit 31) ------------ASSIGN OUTPUT4 = 'CTS_Beam.op4', UNIT = 31 $ -----------------------------------------------------------------------------$ Perform Eigenvalue Analysis SOL 103 $ $ Include HBResp DMAP to Export System Matrices to OP4 File -------------------INCLUDE 'd:\ukl\hbresp\DMAP\MEXPORT.V2001' $ -----------------------------------------------------------------------------$ CEND TITLE = Cantilever Beam Model of Cylindrical Test Structure CTS ECHO = NONE SUBCASE 1 METHOD = 1 SPC = 2 VECTOR(SORT1,REAL)=ALL $ BEGIN BULK $ $ General Analysis Parameters -------------------------------------------------PARAM,POST,-1 PARAM,COUPMASS,1 PARAM,GRDPNT,0 PARAM,NOCOMPS,-1 PARAM,PRTMAXIM,YES $ $ Parameters for Eigenvalue Analysis ------------------------------------------EIGRL 1 6 0 MASS $ -----------------------------------------------------------------------------$ $ Nodes of the Entire Model ---------------------------------------------------GRID 1 0. 0. 0. GRID 2 0. 0. 62.125 GRID 3 0. 0. 124.25 GRID 4 0. 0. 186.375 GRID 5 0. 0. 248.5 GRID 6 0. 0. 248.5 GRID 7 0. 0. 310.625 GRID 8 0. 0. 372.75 GRID 9 0. 0. 434.875 GRID 10 0. 0. 497. GRID 11 0. 0. 512. $ -----------------------------------------------------------------------------$ $ Material Record: Steel ------------------------------------------------------MAT1 1 204750. .3 8.05-9

178

14. Appendix

$ -----------------------------------------------------------------------------$ $ Beam Elements Representing the Cylindrical Components -----------------------PBAR 2 1 1577.07 1.242+7 1.242+7 2.484+7 + .5 .5 0. CBAR 1 2 1 2 1. 0. 0. CBAR 2 2 2 3 1. 0. 0. CBAR 3 2 3 4 1. 0. 0. CBAR 4 2 4 5 1. 0. 0. CBAR 5 2 6 7 1. 0. 0. CBAR 6 2 7 8 1. 0. 0. CBAR 7 2 8 9 1. 0. 0. CBAR 8 2 9 10 1. 0. 0. $ -----------------------------------------------------------------------------$ $ Rotational Spring at Fixed Beam End to Simulate Support Stiffness -----------PELAS 1 1.+10 CELAS1 11 1 1 5 $ -----------------------------------------------------------------------------$ $ Rotational Spring to Connect the Rotational DOFs at the Joint ---------------PELAS 3 2.825+9 CELAS1 10 3 5 5 6 5 $ -----------------------------------------------------------------------------$ $ Tuning Mass Attached to the Free Beam Tip -----------------------------------CONM2 9 10 0 .02295 357.171 357.171 355.450 $ -----------------------------------------------------------------------------$ $ Masses of the Different Flanges ---------------------------------------------CONM2 14 10 0 4.55E-4 CONM2 15 5 0 7.7E-5 CONM2 16 6 0 7.7E-5 $ -----------------------------------------------------------------------------$ $ Rigid Body Elements ---------------------------------------------------------$ Connect CoG of Tuning Mass and Free Beam Tip Rigidly RBAR 12 10 11 123456 15 $ Connect Translational DoFs at the Joint Rigidly RBAR 13 6 5 123456 1 $ -----------------------------------------------------------------------------$ $ Boundary Conditions ---------------------------------------------------------SPCADD 2 1 3 $ Planar Beam Problem: Constrain all DOF which would cause out of plane motion SPC1 1 2346 1 THRU 11 $ Constrain translational DOF at Support SPC1 3 1 1 $ -----------------------------------------------------------------------------$ ENDDATA

14. Appendix

179

14.2 Modes of the Cantilever Beam Model

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Mode 11

Mode 12

Mode 13

Mode 14

Mode 15

Mode 16

Mode 17

Mode 18

180

14. Appendix

14.3 Modes of the CTS Shell Model

Mode 1, 114.7 Hz (bending mode)

Mode 2, 114.7 Hz (bending mode)

Mode 3, 370 Hz (extension mode)

Mode 4, 489 Hz (torsion mode)

Mode 5, 593 Hz (3-ND shell mode)






Mode 11, 918 Hz (bending mode)

Mode 12, 918 Hz (bending mode)

14. Appendix

181



Mode 15, 1108 Hz (5-ND shell mode, anti-phase)




