reduced model preserves the exact eigen-structure of the full model. For model reduction of nonlinear systems, a "linear-based reduction," which utilizes the ...
LINEAR AND NONLINEAR MODEL REDUCTION IN STRUCTURAL DYNAMICS WITH APPLICATION TO MODEL UPDATING by WINSTON RHEE, B.S.M.E., M.S.M.E. A DISSERTATION IN MECHANICAL ENGINEERING Submitted t o the Graduate Faculty of Texas Tech University in P a r t i a l Fulfillment of the Requirements for t h e Degree of DOCTOR OF PHILOSOPHY
Accepted
August, 2000
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor, Dr Thomas D. Burton, for giving his patience and guidance throughout this research. I would also like to thank my Ph.D. committee for their guidance: Dr. Alan A. Barhorst, Dr. Stephen Ekwaro-Osire. Dr. Atila Ertas, and Dr. H. Scott Norville. I would like to extend my appreciation to my mentors, Dr. Francois Hemez and Scott Doebling of Engineering Sciences and Applications-Engineering Analysis (ESAEA) at Los Alamos National Laboratories. Their support, encouragement, and guidance to complete this degree are greatly acknowledged. My deepest appreciation is to my fiancee, Hyun-Ae. I am most thankful for her love and emotional support to complete this degree. Lastly, I am grateful for the financial support that I received from both the Department of Mechanical Engineering and from Los Alamos National Laboratories.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
li
ABSTRACT
vii
LIST OF TABLES
ix
LIST OF FIGURES
xi
CHAPTER I.
II.
INTRODUCTION
1
1.1
Nonlinear Structural Dynamics
1
1.2
Model Reduction
2
1.3
Model Updating and Damage Detection in Structural Systems
3
1.4
Objectives of Dissertation
4
BACKGROUND AND PREVIOUS RESEARCH
7
2.1
Model Reduction for Linear System
7
2.1.1
Direct Model Reduction
7
2.1.2
Reduction Based on Generalized Modal Analysis
10
2.1.3
Summary of Model Reduction for Linear System
14
2.2
2.3
Model Reduction for Nonlinear System
15
2.2.1
20
Summary of Model Reduction for Nonlinear System
Model Updating and Damage Detection
20
2.3.1
Frequency Domain Model Updating
21
2.3.2
Time Domain Model Updating
25
2.3.3
Summary of Model Updating and Damage Detection
28
111
2.4 III.
Current State-Of-The-Art
29
RFTZ VECTOR BASED LINEAR MODEL REDUCTION AND MODEL UPDATING 3.1
Ritz Vector Based Model Reduction 3.1.1
32 32
A Three DOF Spring-Mass System with Non-proportional Damping
33
3.1.2
Free Vibration Response and Reaction Forces
38
3.1.3
A Three DOF Spring-Mass System with Damage
42
3.2
Mode Shape Expansion Using Damage Ritz Vector (DRV)
45
3.3
Damage Ritz Vector Based Model Updating
49
3.3.1
Model Updating Using Expanded Mode Shapes
49
3.3.2
Numerical Results
51
3.4
3.5
3.6
Model Updating Based on Reduced Model
52
3.4.1
General Procedure of Model Updating Via Reduced Model
53
3.4.2
Numerical Results
55
Finite Element Beam
57
3.5.1
DRV Based Mode Shape Expansion
58
3.5.2
Model Updating and Damage Detection
63
3.5.3
Model Updating Based on Reduced Model
64
Experimental Validation
68
3.6.1
Introduction
68
3.6.2
Experimental Results
70
3.6.3
Model Updating Based on Mode Shape Expansion
73
i\
3.6.4
Model Updating and Damage Detection Using Reduced Model
3.7 IV.
NONLmEAR MODEL REDUCTION 4.1
76 79
Model Reduction Methodology
79
4.1.1
Linear-Based Reduction
82
4.1.2
NNM-Based Model Reduction
83
4.2
Features of the Two DOF to One DOF Reduction
83
4.3
Results for Systems with Analytic Nonlinearity
86
4.3.1
A Two DOF System with a Cubically Nonlinear Spring
87
4.3.2
A Variation of the System of Example 4.3.1
93
4.3.3
Internal Resonance Effect in A Two DOF System
99
4.3.4
Example: A 4 DOF System Reduced to One DOF
102
4.3.5
Example: A 4 DOF System Reduced to 2 DOF
105
4.4
4.5 V.
Chapter Summary
76
Results for Systems with Non-Analytic Nonlinearity
108
4.4.1
Deadbend Nonlinearity
108
4.4.2
Discontinuous Nonlinearity
119
Chapter Summary
127
MODEL UPDATING OF NONLINEAR SYSTEM
129
5.1
Model Updating Procedure
129
5.2
Numerical Results on a 4 DOF Spring-Mass System
132
5.2.1
1 ^4
Main Motivation of This Work
5.3
VI.
5.2.2
Results for the Linear System
134
5.2.3
Results for the Nonhnear System
136
5.2.4
Influence of the Initial Condition
139
5.2.5
Sensitivity to Measurement Noise
140
5.2.6
Coupled Influence of Nonlinearity and IC
141
Chapter Summary
142
CONCLUSIONS AND FUTURE W O R K
144
6.1
Conclusions
144
6.2
Future Work
147
BIBILIOGRAPHY
148
\i
ABSTRACT
The development of state-of-art methodologies in the areas of model reduction and model updating/damage detection for linear and nonlinear systems is addressed in this dissertation. Direct model reduction (MR) seeks a transformation between master states, which are to be retained in the reduced model and slave states, which are to be eliminated. Direct model reduction for a linear system seeks an exact transformation, such that the reduced model preserves the exact eigen-structure of the full model. For model reduction of nonlinear systems, a "linear-based reduction," which utilizes the exact-for-the linear case master/slave transformation, is developed here. This is compared to a reduced model-based on calculation of the nonlinear normal modes of the nonlinear system ("NNM-based reduction"). Numerical studies illustrate that the linear-based reduction is much simpler to implement than the NNM-based reduction and provides competitive results in terms of accuracy. The linear-based model reduction captures the essential nature of non-analytic nonlinearities such as deadband and discontinuous saturation, whereas these would be extremely difficult to characterize using the NNM-based reduction. Ritz vector based reduction for linear systems is presented as an efficient model reduction technique to approximate eigen-properties of large structures under the influence of structural changes due to localized non-proportional damping or damage. This method enables the reduced model to capture the effect of structural changes, so that the changes in the lowest mode eigen-properitics are approximated accurately Numerical
vii
studies illustrate that the reduced model obtained using this method has an abilit\ to predict the complete system response with exceptional accuracy. The application of the Ritz vector is extended to characterize damage in a finite element beam model. Mode shape expansion based on Ritz vectors is formulated for damaged structural systems. This is shown to be a simple and accurate method to do mode shape expansion. The objective of the model updating/damage detection is to improve or "'update" the structural dynamics model by comparing experimental modal properties to those from the model. The problem is that the number of measured coordinates is generally much fewer than the number of degrees of freedom (DOF) in the model. Two methods are presented to reconcile the differences: (1) "expanded mode shape*' method, in which the measured mode shapes are expanded to the full model dimension, and (2) model reduction, in which the large model is reduced to the measurement dimension. Singular Value Decomposition (SVD) is utilized in the iterative optimization process used to obtain the updated model. Numerical results illustrate that the model updating/damage detection based on the expanded model shape or on the model reduction is feasible and is effective in terms of accuracy for the example systems given in this dissertation.
Mil
LIST OF TABLES
3.1
DRV based model updating results. Mode shape expansion is performed using the three measured coordinates 2, 5,and 7. The damage is simulated by the 20% spring stiffness reduction at spring 6.
52
Model updating results via reduced model. The damage is simulated by 20% stiffness reduction at spring 7.
56
DRV based model updating results for mode 2. Mode shape expansion is performed using four measured coordinates 13. 14, 15. and 16. The damage simulated with 20% beam stiffness reduction in the 6'*^ element.
63
3.4
Mass properties of the undamaged system.
71
3.5
Identified frequencies from experiment.
72
3.6
Calculated frequencies from the mathematical model.
72
3.7
Mass and stiffness properties of the initial updated undamaged system.
73
3.8
Frequency results.
73
3.9
Updated results of the 8-DOF testbed for mode 2. The 2, 5, and 7'^ entries are corresponding to sensor locations. The damage is simulated with 14% spring stiffness reduction at spring 5.
75
3.2
3.3
4.1
Table 4.1. Calculations of coefficients in (4.41) and (4.42).
100
5.1
Updating results for the linear system. (No random noise is added to the data.)
135
Updating results for the nonlinear system. (No random noise is added to the data.)
137
Sensitivity to Gaussian random noise for the linear system. (Two master DOF are assumed. The IC is generated from the first two mode shapes.)
140
Sensitivity to Gaussian random noise for the nonlinear system.(Three master DOF are assumed. The IC is generated from the first mode shape. The nonlinear spring stiffness is assumed to be known.)
141
5.2 5.3
5.4
IX
5.5
Coupling the influence of the nonlinearity and initial condition amplitudes. (Three master DOF are assumed. The IC is generated from the first mode shape. The nonlinear spring stiffness is assumed to be known.)
142
LIST OF FIGURES
1.1
A Dissertation Outline Flowchart.
3.1
System with non-proportional damping (c =2.5). Unit values are assigned to all the mass and spring elements.
33
A five DOF system with light (c =1) non-proportional damping. Unit values are assigned to all the mass and spring elements.
38
Free vibration response of displacement at xs. Light damping (C = 1) is considered.
40
Free vibration response of displacement at X5. Heavy damping (C = 3) is considered.
40
Reaction force exerted at the absolute wall. A damper (C =1) is placed between mass 2 and 3.
41
Reaction force exerted at the absolute wall. A damper (C=3) is placed between mass 2 and 3.
42
3.2 3.3 3.4 3.5 3.6 3.7
5
An Eight DOF spring-mass system. Unit values are assigned to all the mass and spring elements with consistent units.
46
3.8
A Cantilever beam with damage in 6 element.
58
3.9
An 8 DOF spring-mass system.
69
3.10
Experimental apparatus.
70
4.1
A two degree of freedom (DOF) oscillator system with a cubically nonlinear spring ( e).
87
Comparison of reduced model simulations: NNM-based reduced model, equation (4.23): "dashed line"; Linear-based reduced model, equation (4.21): "dotted line"; Original two DOF model, equations (4.20): "Solid line". Amplitude a =1. Initial conditions for equations (4.20): X2(0) = 1. X2(0) = 0, xi(0)= 1.3848, x,(0) = 0.
91
Frequency-amplitude dependence of periodic motion of the system of Figure 4.1 obtained by numerical integration: "Linear-Based" is from equation (4.21); NNM-based is from equation (4.23); Exact is from equation (4.20).
92
4.2
4.3
Xl
4.4
First modal manifold projection onto xi, X2 plane: linear: X2 = xj; Cubic: equation (4.22): Quintic: equation (4.26); Implied cubic: equation (4.27); Exact: equation (4.20).
92
First modal manifold projection onto X2, x, plane: Solid line based on exact IC's ((X2(0) = 1 . x, (0) = 0, x,(0) = 1.3848, x, (0) = 0): Dashed line based on IC's initiated from the vicinity of exact IGs (X2(0) = 1 . x. (0) = 0,x,(0)=1.30, x,(0) = 0). "
93
4.6
Spring-mass system with a cubically nonlinear spring
94
4.7
Comparison of reduced model simulations: NNM-based reduced model equation (4.32): dashed line ; Linear-based reduced model, equation (4.34) ; dotted line ; Full model, equations (4.34): solid line Amplitude a = 1. Initial conditions for equations (4.29): X2(0) = 1, x, (0) = 0, xi(0) = 0.8223, x,(0) = 0.
97
Frequency-amplitude dependence of periodic motion of the system shown in Figure 4.6. "Linear-Based" from equation (4.34); "NNMbased" from equation (4.32); "Exact" from equation (4.29).
98
First modal manifold projection onto xi, X2 plane for the system of example 4.3.2. Cubic: terms up to cubic power terms in equation (4.30); Quintic: equation (4.30); Implied cubic: equation (4.35); Exact: equation (4.29)
98
Comparison of reduced model simulations for the example of section 4.3.2 with k = 3.5: NNM-based reduced model, equation (4.42): dashed line ; Linear-based reduced model, equation (4.40): dotted line ; Full two DOF model, equations (4.39): solid line. Amplitude a =1. Initial conditions for equations (4.39): X2(0)= 1, x, (0) = 0, Xi(0) = 1.032, x,(0) = 0.
101
Comparison of reduced model simulations for the example of section 4.3.2 with k = 4.5: NNM-based reduced model, equation (4.42): dashed line ; Linear-based reduced model, equation (4.40): dotted line ; Full two DOF model, equations(4.39): solid line. Amplitude a =1. Initial conditions for equations (4.39): X2(0)= 1, x, (0) = 0, X|(()) = 0.92, .x,(0) = 0.
101
A four degree of freedom system w ith a cubic nonlinear spring (e).
103
4.5
4.8
4.9
4.10
4.11
4.12
XII
4.13
Comparison of reduced model simulations for 4 DOF model of Figure 4.12. NNM-based reduced model, equation (4.46): dashed line; Linearbased reduced model, equation (4.45): dotted line ; Full 4 DOF model, equations (4.43): solid line. Amplitude a = 0.5. Initial conditions for 4 DOF model: X4(0) = 0.5, X3(0) = 0.4999, X2(0) = 0.372. xi(0) = 0.1854. and all initial velocities zero.
104
Comparison of reduced model simulations for 4 DOF model of Figure 4.12. NNM-based reduced model, equation (4.51): dashed line ; Linearbased reduced model, equation (4.50): dotted line ; Full 4 DOF model, equations (4.43): solid line. Amplitude a = 0.5. Initial conditions for the 4 DOF model: X4(0) = 0.5, X3(0) = 0.4999, X2(0) = 0.372, x,(0) = 0.1854; for the 2 DOF NNM-based reduced model: X4(0) = 0.5, X3(0) = 0.5178; for the 2 DOF Linear-based reduced model: X4(0) = 0.5, X3(0) = 0.457. and all initial velocities zero.
107
A two degree of freedom oscillator system with the nonlinear force (N). All masses and linear stiffnesses are unity, in appropriate units.
109
4.16
A Deadband Nonlinear Force (N).
110
4.17
A Phase Plot of Deadband Nonlineariy Model. Initial conditions for equations (4.52) and (4.53): Solid line: X2(0) = 1.2, x-, (0) = 0, x,(0) = 0.73, X, (0) = 0; Dotted line: x.(0) = 4, X2 (0) = 0. x,(0) = 3.34, x, (0) =
4.14
4.15
0.
11^
4.18
Collection of Exact Manifolds in 3D (xi ,X2,x2).
112
4.19
Projectionof manifolds in 3D (xi,X2,X2) onto X1-X2 plane.
113
4.20
Frequency-amplitude dependence of periodic motion of the system shown in Figure 4.15 obtained by numerical integration: linear-based is from equation (4.55); exact is from equation (4.52) and (4.53). Reduction is done using the first mode.
113
Frequency-amplitude dependence of periodic motion of the system shown in Figure 4.15 obtained by numerical integration: linear-based is from equation (4.57); exact is from equation (4.52) and (4.53). The reduction is done using the second mode.
114
4.21
XIII
4.22
Comparison of reduced model simulations: Linear-based reduced model, equation (4.55). Original two DOF model (Exact), equations (4.52) and (4.53): Amplitude a = 3. Initial conditions for equations (4.52) and (4.53): X2(0) = 3, x, (0) = 0, Xi(0) = 2.35, x, (0) = 0. Reduction is done using the first mode.
114
Comparison of reduced model simulations: Linear-based reduced model, equation (4.57). Original two DOF model (Exact), equations (4.52) and (4.53): Amplitude a = 3. Initial conditions for equations (4.52) and (4.53): X2(0) = 3, X2 (0) = 0, Xi(0) = -3.67, x, (0) = 0. Reduction is done using the second mode.
115
First modal manifold projection onto xi, X2 plane: linear: X2 = 0.618 Xi; implied linear-based equation (4.58); exact: determined as described in the text.
117
Second modal manifold projection onto xi. X2 plane: linear: X2 = -1.618 xi; implied linear-based in equation (4.59) ; exact: determined as described in the text.
118
4.26
A Discontinuous Nonlinear Force.
120
4.27
A Phase Plot of Deadband Nonlineariy Model. Initial conditions for equations (4.52) and (4.61): Solid line: X2(0) = 0.6, X2 (0) = 0, xi(0) = 1.033, X, (0) = 0; Dotted line: X2(0) = 3, x^ (0) = 0, x,(0) = 2.29, x, (0) =
4.23
4.24
4.25
0.
121
4.28
Collection of Exact Manifolds in 3D (xi ,X2,X2).
122
4.29
Projection of exact manifolds in 3D (xi,X2, x^) onto X1-X2 plane.
4.30
Frequency-amplitude dependence of periodic motion of the system of Figure 4.15 obtained by numerical integration: linear-based is from equation (4.55); exact is from equation (4.52) and (4.60). Reduction is done using the first mode.
123
Frequency-amplitude dependence of periodic motion of the system of Figure 4.15 obtained by numerical integration: linear-ba.sed is from equation (4.57); exact is from equation (4.52) and (4.60). Reduction is done using the second mode.
123
4.31
Xl\
|22
4.32
4.33
4.34
4.35
Comparison of reduced model simulations: linear-based reduced model in equation (4.55). Original two DOF model (exact) in equations (4.52) and (4.60): Amplitude a = 1, Initial conditions for equations (4.52): X2(0) = 1, X2 (0) = 0, xi(0) =1.17, X, (0) = 0. Reduction is done using the first mode.
124
Comparison of reduced model simulations: linear-based reduced model in equation (4.57). Original two DOF model (Exact) in equations (4.52) and (4.60): Amplitude a = 1. Initial conditions for equations (4.52): X2(0) = 1, X2 (0) = 0, x,(0) = -0.8812, X, (0) = 0. Reduction is done using the second mode.
124
First modal manifold projection onto Xi, X2 plane: linear: X2 = 0.618 X|; implied linear-based in equations (4.58) and (4.60); exact: determined as described in text.
126
Second modal manifold projection onto xi, X2 plane: linear: X2 = -1.618 Xj; implied linear-based in equations (4.59) and (4.60); exact: determined as described in text.
127
5.1
Model updating procedure.
131
5.2
Illustration of the 4-DOF, mass/spring system with cubic nonlinearity (K4 = k( 1 +Ex;)). Test-Analysis Correlation (TAC) for the linear system with the MR/RMS cost function.
136
Test-Analysis Correlation for the nonlinear system with the MR/RMS cost function. Refer to the case c) of Table 5.2.
137
Test-Analysis Correlation for the nonlinear system with the MR/SVD cost function. Refer to the case f) of Table 5.2.
138
5.3
5.4
5.5
x\
133
CHAPTER I INTRODUCTION
This chapter presents the structural dynamics problems of interest and the motivations of this dissertation work. First, the basic elements of model reduction, model updating/damage detection, and the applications for linear and nonlinear structural systems are discussed. Lastly, objectives of this research are presented.
1.1 Nonlinear Structural Dynamics The proposed research involves the vibration of multi-degree of freedom nonlinear structural system models of the form [M](x} + [C](x} + [K]{x} + {N(x,x)} = {F(t)}
(1.1)
where [M], [C], and [K] are the (n x n) symmetric mass, damping, and stiffness matrices, and {x} and {x} are the n-vectors of coordinates and velocities. The {N} in (1.1) is an nvector of nonlinear functions of the state, and {F(t)} is an n-vector of external forces. The study of predicting the dynamic response of structural systems (1.1) has been widely investigated over several decades, mostly for linear systems ({N} =0). Typical practical problems are systems (1.1) in which there exist nonlinearities due to large displacements/rotations or material behavior. It is intended to investigate a simplified version of equation (1.1), with no damping and no forcing, [MHx} + [K]{x} + {N(x)} = {0}
(1.2)
where {N} is an n-vector of nonlinear functions that depend on the coordinates only. The objective of this research is to develop two methodologies that will improve the capability to model real world structures that may contain nonlinearities: (1) computationally simple model reduction for the nonlinear system (1.2), and (2) model updating and damage detection for linear and nonlinear structural systems. These areas of research are described briefly in the following two sections.
1.2 Model Reduction Model reduction is a computational cost-saving tool that enables an analyst to extract dynamic quantities of interest from a reduced dynamic model. The reduction consists of condensing out some degrees of freedom (DOF) from the full-size, n-DOF finite element (FE) model, so that a reduced model having m DOF is obtained (with m < n). The reduced model is intended to retain maximum information from the full model, such as the m lowest eigen-values and eigen-vectors of the full system. In model reduction for linear systems, the reduced system matrices are obtained using a linear coordinate transformation matrix developed from the full model mass and stiffness matrices. Extensive work in this area has been reported. When a system is subjected to localized structural change such as damage or localized damping, retaining the effects of this localized change in the reduced model may be difficult. One of the results presented in this work is that a "Ritz vector" based model reduction has the abilit\ to capture the effects of localized changes in the reduced model.
Several methods for nonlinear model reduction have been proposed over the last decade. One of the objectives of this research is to compare two of these reduction methods for nonlinear systems: (1) a "nonlinear normal mode" (NNM) based reduction proposed by Shaw and Pierre (1993), and (2) a "linear-based" reduction for nonlinear systems proposed by Burton (1997) and Burton and Rhee (2000). "Linear-based" reduction means that a linear coordinate transformation is used to reduce the nonlinear model. "NNM-based" reduction means that a nonlinear coordinate transformation, based on the calculation of the modal manifolds of the nonlinear systems, is used to reduce the nonlinear model. The NNM-based reduction is more expensive computationally than the linear-based reduced model, although the NNM-based method should be more accurate. The linear-based model reduction has been proposed as an efficient alternative for reducing nonlinear structural models. Burton and Rhee (2000) have shown that the linear based reduction is computationally simple and effective and in some cases actually produces a more accurate reduced model than the NNM-based reduction. In this thesis the objective of nonlinear model reduction is to determine whether the linear-based model reduction is competitive with the NNM-based reduction in terms of accuracy for several simple vibratory systems.
1.3 Model Updating and Damage Detection in Structural Systems Model updating is the process of refining a stmctural dynamics model by comparing simulated and experimental responses: some system parameters are changed or updated in order to bring the simulated responses into closer agreement with the
experimental responses. Model updating is also employed for damage identification in structural systems which may undergo changes in their mass and stiffness properties, because the location and magnitude of structural property change may be associated with damage. The majority of test-analysis correlation techniques have been developed for experimental analysis in the frequency domain, and the application is generally restricted to linear systems. One of the objectives of this work is to extend the concept of testanalysis correlation to nonlinear systems. Nonlinear FE updating requires the analysis to take place in the time domain, because linear modal quantities are no longer relevant for the analysis of nonlinear systems.
1.4 Objectives of Dissertation The objectives of this work are summarized as follows: a. Develop the Ritz Vector based linear model reduction and study the efficienc\ of this reduction method. b. Apply frequency domain model updating and damage detection techniques based on 1) expanded mode shapes using the "Damage Ritz Vectors" (DRV) and 2) model reduction. c. Develop and assess model reduction methods for nonlinear systems. d. Apply model reduction (MR) techniques to linear and nonlinear models for updating and damage detection in the time domain.
4
Figure 1.1 shows the dissertation outline flowchart that features model reduction and model updating. In Chapter II, we present the technical background related to model reduction, Ritz reduction, nonlinear model reduction, damage detection, and time domain model updating. Chapter III presents the utility of Ritz vectors for the study of isolated effects such as damage or alteration in structural systems when the number of measurements is less than the number of full DOF. First, the ability of the Ritz based model reduction to retain eigen-structure in reduced models is presented. The Ritz \ ector based mode shape expansion, model updating and damage detection techniques based on model reduction are presented. The Ritz vector based model updating and damage
IVIodel Reduction (Chapter I, II)
4^
;
Nonlinear IVIodel Reducti o n (Chapter i\o
Linear IVIodel Reduction
i Direct IVIodel Reduction
i
Damage Ritz Vectors
i
Linear IVIodel LTpdating (Chapter III, AO
~s» Basic Nonlinear B ehavior (Ch apter IV) Nonlinear IVIodel Updating (Chapter \^
Figure 1.1 A Dissertation outline flowchart.
detection methods are applied to experimental data for validation. Chapter IV presents the model reduction methods for nonlinear systems. Numerical results are presented to show the efficiency of two model reduction methods: the linear-based reduction and the NNM-based reduction. Chapter V presents model reduction (MR) based updating for linear and nonlinear structural dynamics systems. In this Chapter, two objectixe functions are defined to update parameters of the model: (1) the Root Mean Square (RMS) cost function, and (2) the Singular Value Decomposition (SVD) cost function. Both rely on model reduction techniques to condense the equation (1.2) of nonlinear structural dynamics. The efficiency of these two methods will be studied through examples. Lastly, in Chapter VI, conclusions and future work are outlined.
CHAPTER n BACKGROUND AND PREVIOUS RESEARCH
The objective of this chapter is to provide technical background and relevant issues for the main work of this dissertation, presented in Chapters III through V. Two main areas are discussed for linear and nonlinear structural systems: model reduction and model updating/damage detection. The relation of this work to previous work is presented, along with the current understanding of the state-of-the-art.
2.1 Model Reduction for Linear Systems This section presents two model reduction methods: direct model reduction and standard modal analysis. Detailed descriptions of each technique are presented in the following sections 2.1.1 and 2.1.2.
2.1.1 Direct Model Reduction The equation of motion for an undamped, unforced linear structural system may be written as [M]{x} + [KHx}-{0}
(2.1
where {x} is the n - dimensional displacement vector, and [M] and [K] are the n x n symmetric mass and stiffness matrices. "Direct model reduction" involves the partitioning of the coordinate vector {xj into two subsets: a set {xj of so called "slave coordinates," which are eventually to be eliminated, and a set {x^} of so called "master
coordinates," in terms of which the reduced model is to be expressed. The partitioned form of (2.1) is defined in terms of the submatrices of [M] and [K] as ^ss .
ms
msm ^"mm
>+
'^ss
ms
•mj
sm
/ ' 4 J \0]
mm_
l^mj
>
0
(2.2)
where jxs} is the s x 1 vector of slave coordinates, and {Xm} is the m x 1 vector of master coordinates. The integer s denotes the number of slave DOF. and m denotes the number of master DOF. The direct model reduction assumes the existence of an appropriate linear transformation which expresses the slave coordinates as functions of the masters: {Xs} = [T]{Xn,}
(2.3)
where [T] is the (s x m) transformation matrix. Guyan (1965) defined a transformation matrix by ignoring all inertia effects on the transformation matrix [T] in (2.3), so that the transformation matrix [Td defined by Guyan is based on static terms only. [T^l = [-k,J-'[k,, J
(2.4)
Guyan's transformation is based on the assumption that, for the dynamics of the lower modes of the system, the relation between the master DOF and slave DOF can be well approximated by the static relation (2.4). This method has been used extensively by man> structural engineers, and it is employed as an industry standard in commercial finite element codes due to its computational simplicity. The disadvantage of using the Guyan method is, however, that the eigen-solutions obtained from the reduced model can be quite inaccurate. The static assumption, which ignores the inertial terms of the complete
system in the coordinate transformation matrix, has been shown to yield inaccurate eigenvalue approximations to actual solutions. Over the years a number of researchers have proposed improvements or alterations to the Guyan reduction, including Irons (1965), Downs (1980), Flax (1975), Johnson et al. (1980), Kidder (1973), Kim (1985), Kuhar and Stable (1974), Leung (1978), Miller (1980), Paz (1983, 1984), and Suarez and Singh (1992). In these work, discussed in more detail by Young (1994), the transformation connecting the master and secondary coordinates has not been improved upon very much, and Guyan reduction is still the industry standard for doing direct model reduction. Recently Young (1994) and Burton and Young (1994) have presented an exact (eigen-structure preserving) version of the Guyan reduction in which both inertia and static properties are used to determine the master/secondary coordinate transformation [Tl (dropping matrix notation), T=-[k,,-(m33T + m3^)(m^J-hm^,J-'k^3]"' •[ksm -(n^ssT + m , ^ ) ( m ^ J + m„,„,)"'k,^,^ 1 Note that in the special case for which a lumped mass model is used, so that [M] is diagonal, one may then easily multiply equation (2.5) by [M] ^ so that the mass matrix becomes the identity matrix. In this special case the preceding results reduce to the following (dropping matrix notation): T = -(k3,-Tk„3)-^(k,,-Tk„,J.
(2.6)
Once the solution to Equation (2.5) or (2.6) for [T] has been found iteratively, the reduced model is obtained by forming the "reduction matrix" [R], defined as
[Tl [R]=
(2.7)
where [R] is (n x m) and [Imj is the (m x m) identity matrix. The reduction matrix [R] expresses the complete coordinate vector {x} in terms of the master coordinates {x,n} as {x} = [R]{x,}.
(2.8)
Substitution of equation (2.8) into the linear model defined by equation (2.1), followed by pre-multiplication by [Rl^, produces the desired reduced model which contains the m master degrees of freedom, [m]*{x,K[k]*{x,}={0}
(2.9)
where the reduced mass and stiffness matrices are defined by [m]* = [Rf [M][Rj
^^^
[k]* = [Rf[K][Rl. Note that the reduction method described here is used for the model updating/damage detection via reduced model in Chapter III. Also this "exact-for-thelinear-problem" reduction is what we employ in Chapter IV to obtain approximate reductions for nonlinear systems.
2.1.2 Reduction Based on Generalized Modal Analysis As an alternative model reduction approach, the classical linear modal analysis (Clough and Penzien (1993) and Meirovich (1970)), has been extensively used. Modal analysis means that the free vibration mode shapes are used to construct the linear
10
transformation connecting the n physical coordinates {x} and a set of m modal coordinates {y}, as {x}=[(t)]{y}
(2.11)
where {x} is the n x 1 displacement vector (coordinates), [({)] is the "reduced modal matrix," an n x m matrix in which the columns are the m lowest frequency free vibration mode shapes, and {y} is the m-vector of modal coordinates. When equation (2.11) is substituted into (2.1), the resulting reduced model will have the form defined in equation (2.9). The benefit of using this method is that the responses of structural systems can be easily obtained because the m columns of the matrix [(j)! are the undamped free vibration mode shapes, and the resulting reduced mass and stiffness matrices in (2.10) are diagonalized. (If linear damping is present, the reduced damping matrix [c 1 = [R] [C] [R] is also diagonal as long as the damping is proportional.) For any Rayleigh-Ritz type method, in which [(j)] in (2.11) consists of any set of m linearly independent n-vectors (see, Clough and Penzien (1993)), the utility of (2.11) is dictated by the nature of the system damping in the structural model. We use Rayleigh damping (Clough and Penzien (1993)) to define a proportional damping, for which the damping matrix [C] can be written as a linear combination of [M] and [K]: [C] = a [ M U b [ K l
(2.12)
where a and b are real constants. Systems with proportional damping maintain selfsimilar displacement patterns at all times, having fixed relations between all displacements and velocities preserved at all times in a given mode. However, for systems with non-proportional damping (that is. the reduced damping matrix [c ] will not
be diagonal, [c*] ij 9^ 0 for [c*l = [(|)]^ [C] [([)].), the classical normal modes are destroyed due to the physical effect of the non-proportional damping: the displacement pattern is no longer self-similar at all times, hi this non-proportional case, modal analysis methods derived assuming proportional damping may be inaccurate if they are applied to systems with non-proportional damping. In past work related to approximating eigensolutions to systems with nonproportional damping, Hurty and Rubinstein (1964) used complex modes based on a generalized modal analysis. Warburton and Soni (1977) attempted to define equivalent diagonal damping matrices and establish a criterion for when non-proportional damping can be approximated by proportional damping. Thomson et al. (1974) showed more complex procedures, though similar in spirit to the idea of diagonalization. The drawback of these methods is that when the damping in the system is increased, the error produced by the diagonalization is significant, due to the increase of the non-diagonal terms in the generalized damping matrix. Clough and Mojtahedi (1976) compared various methods for simulating the responses of systems with non- proportional damping, and concluded that direct integration of reduced coupled equations resulting using undamped mode shape coordinates is the most efficient method to use. Pacheco and Fujino (1989) proposed a numerical perturbation technique to approximate moderately or weakly defined non-proportional damping. Most stmctural systems with this type of weakly defined damping can be approximated by the standard modal analysis w ith a relati\ cl\ small error produced. In situations where strong, isolated damping mechanisms are used to increase passive damping, however, the reduced model obtained using (2.11) produces
12
significant errors in the approximation of eigen-solutions. In a situation in which there exist strong and localized effects such as damage or non-proportional damping in the system, the basis formed by the combination of free vibration mode shapes and Ritz vectors can be used effectively. The Ritz vectors are defined so that they physical!} characterize the change in the system due to the damage, and when the model reduction is performed on the complete system, the mode shape/Ritz vector combination provides an effective way to preserve the eigensolutions in the reduced model. This process is described in the paper of Chu and Milman (1992), who used Ritz vectors to do modal analysis for a large-scale space structure with a few isolated and relatively powerful dampers. When "Ritz vectors" are incorporated as a way to account for the physical effects of the non-proportional damping, the reduced models are shown to yield an excellent eigenvalue approximation to the actual solutions. The damping matrix due to a localized damper can be expressed as [C]=c{b}{b}'^
(2.13)
where c is a scalar proportional to the change in damping. The vector (b) identifies the location (i.e.. the affected coordinates) of the damper or damage. The Ritz vector is then defined as the static displacement pattern resulting from application of a static load {b}, {r}=[K]-'{b}.
(2.14)
This Ritz vector {r} is then used to augment the linear mode shapes to obtain a reduction of the form, {x}=[(l),rl{y}.
(2.15)
This type of Ritz reduction has been employed by Chu and Milman (1992) in the optimization of damping introduced by a few passive elements placed in a space structure to eliminate unwanted vibration. In Chapter III, we will study how accurately the Ritz reduction method can predict the eigen-solutions of systems with non-proportional damping.
2.1.3 Summary of Model Reduction for Linear System For direct model reduction, an exact method has been presented by Burton and Young (1994). This method preserves the full system eigen-structure (m modes) in the reduced model. The exact transformation matrix [Tl given by the implicit equation (2.5) is (dropping matrix notation): T = -[k,,-(m,J-hm,^)(m^J-hm^^)-'k^J
-1
•[ksm -(n^ssT + m , , J ( m ^ J + m^,J"'k,„^ 1 The reduced model will have the form (equation (2.9)) [m]*{x^}+[k]*{x,}={0} where the reduced mass and stiffness matrices are defined by equation (2.10). [m]* = [Rr[M][R] [k]* = [Rf[K][R]. The reduction matrix [R] is [R] = [[T] [ I J ^ . The reduction method described here is used for the model updating/damage detection via the reduced model in Chapter III. Also, this "exact-for-the-linear-problem" reduction is used as the basis to obtain approximate reduction for nonlinear systems in Chapter IV.
14
For modal reduction, Ritz vectors are introduced as a way to account for physical changes due to localized effect such as non-proportional damping or damage. The basis (equation (2.15)) formed by the combination of free vibration mode shapes {(j)} and Ritz vector {r}. {x}=[(l),r]{y} provides an effective way to preserve the eigensolutions in the reduced model. In Chapter III, the efficiency of reduction method based on Ritz vector is addressed. Also, this Ritz vector is used for model reduction and mode shape expansion for damaged systems. Further, the expanded mode shapes are used in model updating/damage detection for linear systems with the damage.
2.2 Model Reduction for Nonlinear System To introduce the issues of interest and the objectives, we consider model reduction in structural dynamics for situations in which static nonlinear effects need to be taken into account. The dynamic model is assumed to have the form (1.2). The nonlinearity jN} in (1.2) is assumed to be static and conservative, so that the total energy is conserved. Equation (1.2) is to describe the motions in the vicinity of the equilibrium {x} = {0}. For this system (1.2), it is assumed the associated linear problem ({Nj = {0}) admits n distinct nonzero natural frequencies coi, .... (On with associated mode shapes ^\. (bp. Each mode shape defines a flat two dimensional surface in the 2n dimensional phase space; the n independent modal motions li\e in these Oat spaces. We assume that
the natural frequencies and mode shapes have been determined for those modes in the frequency range of interest. As in the case of linear model reduction described in section 2.1.1, the objectixe of the model reduction considered here is to obtain an accurate dynamic model in terms of a reduced set of the original physical coordinates. We term this reduced set the "master" coordinates, designated by the m-vector {Xm}. The remaining s = (n-m) coordinates, designated by the vector (Xs}, are termed "secondary" coordinates and are to be eliminated from the model. In terms of the master and secondary coordinates, equation (1.2) is partitioned as mss ms
k
m,^ mm
+ m
f
k ss
k
sm
k ms
\
^n^{x^.xj >+ {
L-^mJ
= {0}.
(2.16)
"mC^s'^m)
mm
The matrices a__,a,^,a^_,a_-^ (where a = mork) are of dimension (s x s), (s x m), (m xs), and (m x m), respectively. Equation (2.16) is written in the following form, with matrix-vector notation dropped: mss ^s+ m^,^ x,„-h k,3X,+ k^^x^-h n^{x^.xj
=0
m ^ s ^ s + ^mn. ^ m + k^s Xs+ ^mm ^m+ n , J X , , X , J = 0
(2.17) (2.IS)
The reduced model which is sought has the form [mf {Xn.}+[tf{x,)+{n*(x,„.x,)} = {0}
(2.19
where [ml* and [k]* are the (m x m) reduced mass and stiffness matrices. Generally in this reduced model (2.19), the nonlinear terms which result will depend on both the master coordinates and the master velocities. Here a nonlinear reduction is sought for which the linear kernel ({n* = 0}) will reproduce exactly the m lowest frequency modes
16
of linear motion and for which the important nonlinear effects are represented with sufficient accuracy. Any model reduction method requires one to define a coordinate (or state) transformation in which the secondary coordinates (or states) are expressed as functions of the master coordinates (states), allowing the secondaries to be eliminated from the model. The primary objective here is to compare two methods of nonlinear model reduction which differ in the manner in which the master/secondary state transformation is defined: (1) a "linear-based" reduction, in which the exact coordinate transformation for the linear case is used to describe the leading nonlinear terms, and (2) a nonlinear normal mode or "NNM-based" reduction, in which the manifolds defining the nonlinear normal modes associated with the m lowest frequency linear modes are calculated. This procedure results in a nonlinear state transformation, in which the secondary states are expressed as nonlinear functions of the master states. Both of these reductions result in the same linear kernel in the reduced model, but the nonlinear terms which result differ for the two models. One would expect the NNM-based nonlinear reduction to be more accurate but more computationally involved than the linear-based reduction. There are two main areas of research which are relevant to the nonlinear structural dynamics model reduction problem: (1) the associated linear model reduction problem and (2) nonlinear normal modes (NNM's) in discrete systems. Discussion of some of the relevant work in linear model reduction has been given in section 2.1. Here we discuss existing results in the study of nonlinear normal modes (NNM\s) in freely oscillating conservative nonlinear systems. The concept of the .\NM was
17
originally presented by Rosenberg (1966). He studied "similar nonlinear normal modes," for which the displacement pattern maintains a self-similar-at-all-times character, exacth as in the case of a linear normal mode. During the past ten years there has been a lot of interest in NNM's, motivated by the work of Vakakis (1990). Vakakis and Caughe\ (1991), and Shaw and Pierre (1991, 1992). A number of more recent investigations have been reported, including those of Shaw and Pierre (1994), Aubrecht and Vakakis (1996). Nayfeh and Nayfeh (1994, 1995), Nayfeh (1995), Nayfeh et al. (1996). and Burton and Hamdan (1996). Shaw and Pierre (1991, 1993) defined the NNM as a two dimensional invariant manifold in the 2n-dimensional phase space, with this manifold tangent to the associated linear mode eigenplane at the origin of the phase space. They point out that the nonlinear modal motions confined to these manifolds need not exhibit the self-similar-atall-times behavior of the classical normal mode. Much of the work to date on NNM's has been directed toward determination of the leading term approximations to the state transformation equations which describe the modal manifolds associated with individual NNM's, rather than with collapsing the dynamics onto the subspace inhabited by a set of two or more NNM's (see, Shaw and Pierre (1991, 1993) and Burton and Young (1994), for example). These NNM calculation methods lead to sets of coupled linear algebraic equations which are solved for the coefficients in the polynomial expansions which define the individual NNM's. Na\ teh and co-workers have used a complex algebra approach to calculate the modal manifolds (see Nayfeh (1994), Nayfeh and Nayfeh (1994), and Nayfeh et al. (1996)). The comput-
tational advantage of their procedure is that the linear algebraic equations defining the needed polynomial expansions are uncoupled, rendering the solution quite direct. Phenomenologically, works of special interest here are those of Aubrecht and Vakakis (1996) and Nayfeh et al. (1996). Aubrecht and Vakakis (1996) studied NNM's in a multi-span beam with and without mode localization, for which some modes have appreciable motions only in spatially localized regions. Aubrecht and Vakakis (1996) note that mode localization can lead to complicated structure of the NNM's. In the w ork presented here, we assume that mode localization is not an issue. The influence of mode localization on nonlinear model reduction is a subject for useful future work, especialh since mode localization in real structures is not uncommon. Also of relevance to our work is the study by Nayfeh et al. (1996) on the effect of internal resonance on the determination of NNM's. Nayfeh et al. (1996) used the complex algebra/normal form approach to calculate the four dimensional manifolds which contain two nonlinear modes which are in a one to one or a three to one internal resonance. They note that the attempt to calculate individual NNM's which are in internal resonance with other modes will fail: the algebraic equations defining the leading polynomial terms become singular. Thus, if two (or more) modes are in internal resonance, one must calculate the four (or higher) dimensional manifold containing all of the interacting modes. This internal resonance phenomenon is of special interest to us, because we have found that the accuracy of model reduction based on calculation of NNM's is quite sensitive to the "nearness" of an internal resonance condition.
19
Chapter IV presents details of the model reduction methods for nonlinear systems. Numerical results are presented to show the efficiency of the two model reduction methods of interest here: linear-based reduction and NNM-based reduction.
2.2.1 Summary of Model Reduction for Nonlinear System The reduced model for the nonlinear system (1.2) has the form (equation (2.19)) [mf{x,J-F[kf{x^}-F{n*(x„,,x,J} = {0} where [m] and [k] are the (m x m) reduced mass and stiffness matrices. Generally in this reduced model (2.19), the nonlinear terms which result will depend on both the master coordinates and the master velocities. Two methods have been presented for obtaining the reduced model in (2.19): (1) a "linear-based" reduction, in which the exact coordinate transformation for the linear case is used to describe the leading nonlinear terms, and (2) a nonlinear normal mode or "NNM-based" reduction, in which the manifolds defining the nonlinear normal modes associated with the m lowest frequency linear modes are calculated. In Chapter IV, numerical results are presented to show the effectiveness of these methods.
2.3 Model Updating and Damage Detection There exist numerous methods of health monitoring of structures, pertaining to identifying the structural change or alteration in the system. According to the Los Alamos National Laboratory report (Doebling et al. (1996)), damage identification/detection techniques have been studied. These include techniques tor delecting changes in
20
frequency, mode shapes, curvature and strain mode shape, and dynamically measured flexibility. In this section we focus on model updating and damage detection methods that have been investigated over the past few decades. Model updating is the process of improving a FE model by comparing the numerical simulation to experimental data. Model updating refines the FE models through two procedures: (1) locates the errors (damage), and (2) corrects them (update the change in system). The majority of test-analysis correlation techniques have been developed to handle experimental analysis in the frequency domain, and these methods are generally restricted to linear systems. The objective of this work is to extend the concept of testanalysis correlation to nonlinear systems. Nonlinear FE updating requires the analysis to take place in the time domain, since linear modal quantities are not relevant for the analysis of nonlinear systems.
2.3.1 Frequency Domain Model Updating Model updating for linear systems involves a formulation of continuous test analysis and correlation between test data and analytical data in the frequency domain. The correlation matrix is defined in terms of dynamic quantities such as frequencies, mode shapes, and modal dampings. An optimization solution is obtained by minimizing a correlation metric with respect to design variables. Then, the system matrices are updated using the optimum design. The procedure is repeated until test and analysis data are in good agreement.
21
Much of the work in the area of frequency domain based model updating for linear systems has already been done (Doebling et al., 1996). A discussion of some of the relevant work in the model updating for linear systems is given here: (1) optimum matrix update method, (2) sensitivity methods. As an improvement of the state-of-art technoIog\ in model updating, we then introduce (3) damage Ritz vector based model updating and damage detection. Baruch (1978, 1984), Kabe (1985), and Smith and Beattie (1991) used optimal matrix update methods which use closed-form equations to compute the updated model matrices at the global level. The drawback of this method is that many constraints are added in the objective function in order to maintain matrix properties such as sparsity. positivity and symmetry at the global level. Therefore, solving the differential equations is more computationally involved. Hemez (1993) and Hemez and Farhat (1993) studied the Sensitivity-Based Matrix Update (SBMU) which is based on the solution of a firstorder Taylor series that minimize a dynamic error residual. Using the SBMU method, Hemez (1993) formulated the sensitivities at the element level. The advantage of this method is that it is computationally more efficient than forming the sensitivities at the global matrix level. Among all of the model updating and damage detection methods recently introduced, the "damage Ritz vector" or DRV has been formulated by Burton et al. (1998) as a way to character the error [AKl in the system stiffness matrix. This method, which is based on minimization of errors in the modal residual that is obtained from the difference between simulated and experimental data, identifies local changes in the stiffness matrix that are assumed to occur due to damage of a localized nature.
-•?
Mathematically the objective is to determine the change [AKl in the stiffness matrix due to the damage. Burton et al. (1998) noted that the damage Ritz vectors essentially characterize or define the effect of damage on the modal displacement patterns. These Ritz vectors, combined with undamaged mode shapes, form an effective basis to do model reduction or mode shape expansion. The use of damage Ritz vectors in model updating is discussed in detail in the next paragraph. Model updating using damage Ritz vectors employs an iterative scheme to calculate [AK], which may exist due to structural change or damage. The algorithm utilizes the static Ritz vector to characterize the damage. The change in stiffness [AK] is computed directly from the Ritz vectors, in such a way that matrix sparsity is preser\ ed, assuming that the location of damage is known approximately ahead of time. Several methods (see, Pandey and Biswas, 1994; Farrar and Jauregui, 1996; and Cornwell et al., 1997) have been proposed for locating damage, and we assume that such methods ha\e been applied prior to the application of the method discussed here. The change in stiffness due to damage can be written as [AK] = a{b}{b}'
(2.20)
where a is a scalar proportional to the change in stiffness. The vector (b) identifies the location of the damage (Burton et al., 1998). The "damage Ritz vector" (DRV) is then defined as {r} = [Kr'{b}.
(2.21)
The damage Ritz vector jr} in (2.21) is combined with the linear undamaged mode shapes to form a basis to express the expanded measured shapes of the altered (damaged)
2^
system. These expanded mode shapes are used in an iterative scheme for estimating [AK] and a new damage scale coefficient a. The advantage of the Ritz vector {r} is that it is physically related to the effect of localized damage, because jr} defines the static displacement pattern when a static loading vector {b} is applied. To date the damage Ritz vector methodology has been applied to simple spring mass type systems and to a beam finite element model. In review of development of the Ritz vectors, the relevance to the proposed research can be found from several sources. A Ritz vector is the displacement vector resulting from the application of a static loading pattern applied to the structure. Ritz vectors, also called "Lanczos coordinates" (see, e.g., Nour-Omid and Clough, 1984), have been used to provide many advantages such as computational efficiency and more accurate prediction of system response over the natural mode. Clough and Penzien (1993) provided a general procedure to formulate Lanczos vectors (Ritz vectors). In their procedure, the first Ritz vector is the static deformation o\' a structure due to a load applied to the structure. The subsequent vectors account for the inertial effects of the loading and are generated by matrix iteration and orthogonalization. A similar procedure is used by Cao and Zimmerman (1997, 1997) and Cao et al. (1998) to extract Ritz vectors from dynamic or ambient testing data. They found that, in general, load dependent vectors are more sensitive to damage than the corresponding natural modal vectors, and concluded that the Ritz vector set provides a better basis than the undamaged mode shapes (natural mode shapes). In a similar spirit, the Ritz vector defined in equation (2.21) is calculated based on the static deflection of the structural system due to the loading pattern {b} defined in
24
(2.20). The Ritz vector in (2.21) differs from that of Cao and Zimmerman (1997. 1997) in that the equation (2.21) is formulated so that the Ritz vector is directly related to the damage in the structure, whereas the load dependent Ritz vectors are calculated from the loading distribution applied experimentally to the stmcture. Therefore, they are unrelated to the damage. The original formulation of damage Ritz vectors is found in the paper of Chu and Milman (1992), who used Ritz vectors to characterize the effect of isolated, localized damping elements in a large space truss. Utilizing the idea of Chu and Milman (1992), Burton et al. (1998) used damage Ritz vectors (DRV) effectively in mode shape expansion and model updating/damage detection. In Chapter III, the focus of this thesis research is limited to extending the Ritz vector calculation/updating methodology to the finite element beam model, which admits both translation and rotation in the response of the system. For such systems, the formulation of equation (2.20) and (2.21) involves multiple Ritz vectors, and the specification of these Ritz vectors and their use to do model updating will be the focus of this research. In addition, experimental data have been generated for an 8 DOF spring mass system for a baseline (undamaged) and a locally "damaged" system, in which the "damage" is represented by an altered spring constant. The data from this experiment will be used as a testbed for the updating method.
2.3.2 Model Updating for Nonlinear Systems Most of the available updating techniques utilize analysis in the frequency domain. The application of model updating in the frequency domain is generally
^s
restricted to linear systems. One objective of this research is to extend updating to nonlinear systems, which require the analysis to take place in the time domain. In this research we will explore an updating procedure utilizing a combination of linear-based model reduction and singular value decomposition (SVD). Hemez (1997) investigated the problem of updating nonlinear finite element models in which explicit time integration is implemented and he proposed a formulation of the updating problem in the time domain. In addition, this work explores the state-of-the-art practices in nonlinear model updating and we rely on this study to improve on the existing technology. Hasselman et al. (1998) investigated the method of Principal Component Decomposition (PCD) analysis for a nonlinear system based on the Singular Value Decomposition (SVD) of a collection of response time-histories. The data to be analyzed are collected in a matrix [X] ^x,(t,)
•••
x,(tj (2.22
[X] = x,„(t,)
•••
x,(tj
where the rows of [X] correspond to different measurement locations (xi columns correspond to the measurement times (ti
x„J. and
tn). The SVD of matrix X in
equation (2.22) results is expressed as [X] = [O] [D] [n (t)]
(2.23)
where the [D] is the diagonal n x n matrix of singular values. The size of [0]([r|]) are n x n (m X n) and the columns (rows) of [0]([r|]) are called the left (right) principal singular vectors of [X] and they are orthogonal vectors: [(D^][(D] = [Ti]^[n] = [Il
26
(124)
where [I] denotes is the identity matrix. The SVD has been extensively used in the electrical engineering field to filter out noise in measured data. The general construction of SVD is given by Sontag (1990). For linear systems, the representation (2.23) is similar to the representation using modal analysis, [X] = [(t)] [a] [y(t)]
(2.25)
where (j) is the modal matrix, a is the diagonal matrix of modal amplitudes, and y(t) is the generalized modal time response matrix. The PCD is analogous to the modal response time histories of linear structural analysis (see, e.g.. Eraser, 1989). In the updating approach of Hasselman et al. (1998) the cost function to be minimized is defined in terms of the entries of the SVD matrices in equation (2.23):
J=I I i
[AT,j]'+I[ADijJ- + Z Z [ A v ^ J i
j
(2.26)
J k
where the matrices [A ^ ] ([Av]) represent the difference between the experimental and analytical matrices [0]([ri]) in (2.23), and [AD] is the normalized differences between experimental and analytical singular values. The main features of the SVD updating method are the following: (1) the cost function J in equation (2.26) is formulated in terms of the SVD parameters in (2.23), (2) the SVD method uses time series data, so that it is applicable to both linear and nonlinear systems, (3) the SVD method is intrinsically useful for noise reduction, and (4) the SVD reduces the dimension of the system analyzed, because fully measured data sets are not needed to do the SVD.
27
The primary objective of this thesis research is to develop the SVD time series model updating based on a reduced model (MR) obtained via the nonlinear model reduction described in section 2.2. Note that computing the response of the full model may be computer intensive and inappropriate because responses are compared with test data at a few locations only. This justifies the use of model reduction techniques. In Chapter V, two methods of model reduction based updating are presented: (1) MR with the Root Mean Square (RMS), in which the full model parameters are updated using a standard least square optimization to minimize the error between the experimental and simulated response histories, and (2) MR with the Singular Value Decomposition (SVD), in which the updating optimization is performed by minimizing the squared error in the SVD parameters (as in 2.26) of the simulated and measured responses. The quality of these two methods will be studied through examples.
2.3.3 Summary of Model Updating and Damage Detection Damage Ritz vector (DRV) is introduced and is given by equation (2.21) (r} = [K]-'{b} where {bj is the loading pattern. Ritz vector based model updating for linear systems in the frequency domain is an iterative method to identify the changes in stiffness due to damage, AK-alblfb}"^. The damage Ritz vector jr} is combined with the linear undamaged mode shapes to form a basis to express the measured shapes of the altered (damaged) system. These expanded
mode shapes are combined with an iterative scheme for estimating [AK]. This method, which is based on minimization of errors in the modal residual that is obtained from the difference between simulated and experimental data, identifies local changes in the stiffness matrix due to damage. The SVD method is presented as an effective way for model updating for nonlinear systems in the time domain. The main features of the SVD updating method are the following: (1) the SVD method uses time series data, so that it is applicable to both linear and nonlinear systems, (2) the SVD method is useful as a noise reduction technique, and (3) SVD reduces the dimension of the complete system because fully measured data sets are not needed to do the SVD.
2.4 Current State-Of-The-Art The preliminary discussion provides the current understanding of this research and motivation to examine several case studies.
The main current issues to be
investigated are the following: 1. The Ritz reduction method has been introduced as a way to improve modal analysis, particularly for systems with non-proportional damping or damage. It is shown that the Ritz vector approach accurately characterizes localized effects to account for physical changes in the eigenproperties induced by dampers or damage. Questions of interest include the following: a. How accurate is this method in approximating eigensolutions ol the complete system compared to the Guyan and the modal truncation methods ?
29
b. How does this method improve the low mode approximations? c. How should the Ritz vectors be formed, if more discrete dampers are added to the system or multiple damages exist in the system? d. How should the Ritz vectors be formed for a finite element beam model, for which multiple Ritz vectors are needed? 2. It is understood that the damage Ritz vector can be combined with the lineaiundamaged mode shapes to form a basis to express the expanded mode shapes of the altered (damaged) system. These expanded mode shapes are used in an iterative scheme for estimating [AK]. Two methods are introduced: (1) a model updating based on expanded mode shapes, and (2) a model updating technique based on model reduction. Questions of interest include the following: a. How are expanded measured mode shapes calculated from using Ritz vectors, when the number of sensors is less than the number of DOF? b. How is the initial expanded mode shape refined in the updating process? c. Do both model updating methods exhibit adequate convergence? d. Are the methods sensitive to measurement noise? e. What basic elements need to be considered when the methods are applied to experimental data? 3. Two methods are presented for obtaining reduced models for a nonlinear system: (1) a "linear-based" reduction, in which the exact coordinate transformation for the linear case is used to describe the leading nonlinear terms, and (2) a nonlinear normal mode or "NNM-based" reduction, in which the manifolds defining the nonlinear normal modes
30
associated with the m lowest frequency linear modes are calculated. Questions of interest include the following: a. What are the computational aspects of the two methods? b. What are the strengths and weaknesses of the two methods? c. Do these methods have ability to preserve the essential nature of non-anah tic nonlinearity such as deadband and discontinuous nonlinearity in the reduced model? 4. It is understood that the SVD method is an effective technique for model updating with nonlinear systems in the time domain. Two model updating methods are formulated in the time domain using the following cost functions: (1) the root mean square error between test and analysis data, and (2) the difference between principal component analyses of test and analysis data. The model reduction technique we employ in both cases is the linear-based model reduction. Questions of interest include the following: a. How does the selection of master DOF for model reduction affect the results of an update ? b. Are these methods sensitive to measurement noise ;* c. Is model updating sensitive to the choice of the initial condition used to integrate the equations in the time domain? d. How does the nonlinearity or the amplitude of the initial condition affect the result of an update? e. Is there an advantage of using one method over the other?
31
CHAPTER III RITZ VECTOR BASED LINEAR MODEL REDUCTION AND MODEL UPDATING
This chapter presents the application of Ritz vectors for the study of isolated effects such as damage, or structural alteration in structural systems, when the number of measurements is fewer than the number of full DOF. First, the ability of the Ritz vector based model reduction to retain eigen-structure in reduced models is presented in section 3.1. The Ritz vector based mode shape expansion, model updating, and the damage detection are presented in sections 3.2, 3.3, and 3.5. The model updating and damage detection techniques based on model reduction are presented in section 3.4. Lastly, in section 3.6 the Ritz vector based model updating and damage detection methods are applied to experimental data for validation.
3.1 Ritz Vector Based Model Reduction In this section we introduce the basic problem, and show how Ritz vectors can be used effectively to characterize the changes in the system eigen-propeities due to nonproportional damping effects and due to localized damage effects or due to structural alterations. The formulation of Ritz vectors and application of model reduction using Rit/ vectors are illustrated through examples that feature linear structural systems. The results obtained from the reduced models are compared to results from the full s\ stem.
3.1.1 A Three DOF Spring-Mass System with Non-Proportional Damping In order to illustrate the model reduction problem for systems with isolated nonproportional damping, we consider the simple, 3 DOF system shown in Figure 3.1. For the undamped version of this system, the three natural frequencies and mode shapes are as follows: coi =0.76536
{(t)}, = [l, 1.414, i r
032= 1.41421
{(t)}2 = [-l,0. i r
(03= 1.84776
{(t)}3 = [1,-1.414,1] T
A damper is placed between the wall and mass 1 to provide a non-proportional and localized damping effect in the system. The addition of the non-proportional damping destroys the classical normal mode solution and results in complex eigen-values
Wall
Wall
y y y y x x x x Figure 3.1 System with non-proportional damping (c =2.5). Unit values arc assigned to all the mass and spring elements.
^^
and eigen-vectors. The eigen-values for each mode are expressed as ?i = -cco±ico(l - c " ) ' - , where the coj and ^i denote natural frequencies and modal damping factors. The s\ stem in Figure 3.1 admits three eigen-values (taking only positives of complex conjugates): Xl = -0.10896 + 0.97973i
X2 = -1.10833 + 0.36367i
CO, =0.98577
C02= 1.16647
C03= 1.73339
^1 =0.11053
^2= 0.95016
^3= 0.01881
0.061-0.44921 {^}i = 1.052-0.21351 1
m-.
7.93622-4.99181 3.09614-0.806131 1
;.3 = -0.03271 + 1.739031
{^}.3 =
0.03389 + 0.232821 -1.02315-0.113781 1
The reduced model which results from using the first two undamped mode shapes is given below (after pre-multiplying by the inverse of the reduced mass matrix): Yi
y2
^-1-
0.625 -1.25
-0.625 Yi 0.58579 0 Yi >+ 1.25 .y2j 0 2 y?
(3.1)
where yi and y2 are modal coordinates. The eigen-values and eigen-vectors (mode shapes) of this reduced system (3.1) are Xl = -0.12997 + 1.01509i
Mr'=
2).
3.6 Experimental Validation In this section, experimental results obtained from the eight DOF structural testbed shown in Figure 3.9 are used to validate the DRV mode shape expansion based model updating method. Model updating based on model reduction is performed. The objective is to apply the two updating methods to actual experimental data in order to validate the methods for an actual physical system.
3.6.1 Introduction Shown in Figure 3.9 is the 8 DOF system which has eight translating masses connected with seven springs. The masses may slide on a highly polished steel rod that supports the masses and constrains them to translate along the rod. The masses are connected via coil springs. The motion of the system is induced by a hammer excitation.
68
r|
^g*^ 1^^ j ^ ^ ^^^ j , , , ^ .
-^
^
- ^i«»~
Figure 3.9 An 8 DOF spring-mass system.
The hammer is applied to the mass 1 (end). Accelerometers are mounted on each mass. The boundary conditions are free-floating, so that the system admits a rigid body mode. This testbed was fabricated by Los Alamos National Lab, Engineering Sciences and Applications-Engineering Analysis (ESA-EA). In the undamaged configuration of the system, all springs are identical and behave linearly. The damage is introduced as a change in the stiffness characteristics of the system. The damage is simulated by replacing an original spring with another linear spring, which has a spring constant less than that of the original spring. In this experiment, the replacement spring is located at spring 5 and the replacement spring has 14% stiffness reduction to simulate damage.
69
The test equipment used in this study and shown in Figure 3.10 was a HewlettPackard 3566A data acquisition system, composed of a model 35650 mainframe, 35653 source module, 4 35653A-8 channel input modules and a signal processing module that performed the FFT calculations. A laptop computer was used for data storage and as the platform for the software for controlling the data acquisition system. A detailed description of the experiment is provided in the report of Rhee (1998) and Duffey et al. (2000).
Figure 3.10 Experimental apparatus.
3.6.2 Experimental Results Table 3.4 lists the undamaged mass and stiffness properties of the system. The first mass is larger than the others because of the added mass of the support structure.
70
which is used as the driving point of the hammer excitation. For the undamaged s> stem, the stiffness constants are all identical. The damaged spring stiffness is 14% less than that of the nominal spring (322 Ibf/in), and it is replaced at spring 5 to simulate a linear damage. Table 3.5 lists the frequencies identified from the experiment. The rigid body mode is not useful and it not considered further. In Table 3.5, the damaged frequencies are less than the undamaged frequencies, since there is stiffness reduction (about 14%) in spring 5. Similar trends are observed in Table 3.6, which lists the calculated frequencies from the mathematical model. The calculated frequencies in Table 3.6 are different from those in Table 3.5. For the first mode, a 3.7% frequency difference is observed between experimental results and finite element calculations. For the second mode, about 4.2% frequency difference is observed. Prior to attempting damage detection, it is necessary to bring the undamaged model into conformance with the undamaged experimental results. Table 3.7 lists the
Table 3.4 Mass properties of the undamaged system Location 1 2 3 4 5 6 7 8
Mass (Kg) 0.5594 0.4195 0.4192 0.4195 0.4194 0.4194 0.4192 0.4195
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Mass (Ibf sec" / in) 3.231E-03 2.423E-03 2.421E-03 2.423E-03 2.422E-03 2.422E-03 2.421E-03 2.423E-03
Table 3.5 Identified frequencies from experiment. Mode
1 2 3 4 5 6 7
Undamaged Frequency (Hz) 22.6 44.5 65.9 86.6 99.4 113.0 133.2
Damaged Frequency (Hz) 22.3 43.9 64.8 85.9 99.7 113.2 131.9
Table 3.6 Calculated frequencies from the mathematical model Mode
1 2 3 4 5 6 7
Undamaged Frequency (Hz) 21.8 43.0 63.0 80.8 95.6 106.8 113.7
Damaged Frequency (Hz) 21.4 42.5 62.9 79.2 95.2 105.8 122.2
initial updated mass and stiffness properties with respect to the undamaged experimental results. Note that there are large increase in the updated stiffness properties compared to the original properties for the undamaged system model (320 Ibf/in for all the springs). Table 3.8 lists the updated frequencies for the undamaged system. The parameter values in Table 3.7 were taken as the undamaged system model values.
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Table 3.7 Mass and stiffness properties of the initial updated undamaged system Mass Location 1 2 3 4 5 6 7 8
Mass (Ibf sec^2 / in) 3.231E-03 2.423E-03 2.421E-03 2.423E-03 2.423E-03 2.423E-03 2.422E-03 2.423E-03
Spring Locations 1 2 3 4 5 6 7
Stiffness (Ibf/in) 391.62 325.93 349.43 331.69 338.16 338.62 349.34
Table 3.8 Frequency results Mode
1 2
Undamaged Experimental Frequency
Damaged Experimental Frequency
Initial Updated Undamaged Model Frequency
(COex)j
(«dex)j
(Wadj)j
(Hz) 22.6 44.5
(Hz) 22.3 43.9
(Hz) 22.4 44.5
3.6.3 Model Updating Based on Mode Shape Expansion These results in Tables 3.7 and 3.8 are to be used in the mode shape expansion and model updating using Ritz vectors. Note that the initial updated result for the second mode is identical to the identified undamaged frequency from the experiment, while the initial updated result for the first mode produces about 1% error compared to the identified frequency from the experiment. This observation leads us to use the updated undamaged properties for the second mode in the Ritz vector based mode shape expansion and model updating. 73
The initial updated undamaged frequencies and mode shapes are the base line for the model updating. The results from the undamaged model are stated below: {(|)adj}i = [-0.4789 -0.4007 -0.2478 -0.0711 0.1252 0.3001 0.4322 0.5010]'^ {(l)adj}2 = [-0.4083 -0.1447 0.2563 0.4913 0.4580 0.1686 -0.2149 -0.4670]"^ coadji = 22.4 (Hz)
C0adj2 = 44.5 (Hz).
The identified frequencies and mode shapes for the damaged system from the experiments are as follows: {(|)dex}i = [-0.4839 -0.4098 -0.2628 -0.0892 0.0987 0.3206 0.4138 0.4869]"^ {(l)dex}2 = [-0.4471 -0.1513 0.2634 0.50441 0.4636 0.1347 -0.1986 -0.4247]"^ COdexl = 2 2 . 1 ( H z )
C0dex2 = 4 3 . 9 ( H z ) .
We compute the residuals for each mode as follows: {R}j=([K]-COdexj'[M]){(l)dex}j
(3.41)
{R}i = [0.170 0.056 -0.022 0.254 -1.738 2.821 -1.364 0.237]'^*10^ {R}2=[-0.601 0.879 0.244 0.477
1.222 -2.319 0.273 -0.071] ^* 10 ^
The vector R\ does not indicate the damage location, because the 5'^, 6'^, 7'^ entries ha\ e the same order of magnitude. The vector R2 shows larger entries at the 5'^ and 6'^ slots, with different signs. The vector R2 indicates clearly that the damage is associated with spring 5. From the vectors Ri and R2, we expect the model updating for second mode to produce a better result in identifying damage in the system. The numerical results are presented here to identify the damage. 14% reduction oi' spring stiffness in spring 5, for the 8 DOF spring-mass testbed. Assume that the damage location is known ahead of time. Identification of the damage location has been studied
74
for this testbed by Duffey et al. (2000). We assume that the three measured coordinates are 2, 5, and 7. We follow the same procedure to compute the initial expanded mode shape and the model updating/damage detection that is described in section 3.2 through 3.4. Calculations for the second mode are shown here. Table 3.9 hsts summaries of the mode shapes and residual. Using the initial expanded mode shape {vj/adj}2, one obtains the initial estimate a2 = 0.203, which is higher than the actual value a = 0.14. The second damage estimate a2= 0.1708 is obtained. This estimate is reasonably close to the exact solution. At this stage, the updated residual {Radj}2 ^ ^ clearly showed that the damage is associated with the 5 and 6 entries which have nonzero values. This solution converged, and no further improvement resulted for the damage estimate. Thus, the result is close to the exact value, but not exact.
Table 3.9 . Updated results of the 8-DOF testbed for mode 2. The 2, 5, and 7'^ entries are corresponding to sensor locations. The damage is simulated with 14% spring stiffness reduction at spring 5. Identified Experimental Mode Shapes
Initial Expanded Mode Shape
Initial Residual
{V|/dex}2
{^adjlz
{ Radj} 2
-0.4471 -0.1513 0.2634 0.5044 0.4636 0.1347 -0.1986 -0.4247
-0.4173 -0.1513 0.2525 0.4910 0.4636 0.1828 -0.1986 -0.4517
-159.06 46.09 17.53 19.48 3.968 4.891 -41.40 -5.175
75
Updated Residual
Updated Mode Shapes
{Radj}2
{M/adj}2*''
0 0 0 0 193.2 -193.2 0 0
-0.4196 -0.1562 0.2486 0.4951 0.4797 0.1461 -0.2000 -0.4430
3.6.4 Model Updating and Damage Detection Based on Reduced Model In this section we present results using the model reduction based model updating method described in section 3.4. The numerical results presented here are used to identif\ a damage of 14% reduction in spring stiffness in spring 5, for the 8 DOF spring-mass testbed. We assume that the damage location is known ahead of time. Three measured sensor locations (master coordinates) 6, 7, and 8 are assumed. Calculations for the second mode are presented. The initial estimate is a2 = 0.1737. This estimate leads to an estimate for [AK] and the new transformation [T] and reduction [R] matrices. The second estimate is a2 = 0.1710. Compared to the exact damage (0.14), the error produced is about 22%. Note that this estimate (a2= 0.1710) is about the same as the result (a2= 0.1708) obtained from the model updating based on the expanded mode shape. Thus, both methods produce a reasonable model updating.
3.7 Chapter Summary 1. The use of the static Ritz vector to improve model reduction has been shown to be very efficient for systems in which the effect to be characterized is localized in nature. The Ritz reduction technique has the ability to preserve the eigenstructure of the complete model in the reduced model. This method improves the modal parameter estimates for the lower modes, since the Ritz vectors are calculated from the static loading pattern that is closely related to the structural change.
76
2. The response that results from using the Guyan reduction techniques may be inaccurate. This is due to the exclusion of inertial terms in the transformation matrix, resulting in incomplete retention of information from the complete model. Therefore, errors in the eigenvalues and eigenvectors result in the reduced model. Significant errors can occur in the calculated response and in derived quantities such as reaction forces. 3.
It is shown that the undamped modal truncation method does not predict the actual response well when non-proportional damping in the system is large. In this case the non-proportional damping destroys the normal modes and leads to the reduced model being non-diagonal. Errors are produced in the reduced model eigenvalues and eigenvectors, and the capability to predict the actual response of systems with nonproportional heavy damping significantly deteriorates.
4.
The damage Ritz vector (DRV) has been shown to characterize the property change due to localized damage. The mode shape expansion method based on the DRV is an effective way to overcome the situation where the number of measurements is less than the full DOF.
5. An iterative model updating method based on mode shape expansion is presented to identify the structural damage [AK], and this method is shown to be effective. 6.
A model updating method based on the reduced model eliminates the use of the expanded mode shape, but requires calculations of new reduced mass and stiffness matrices for each iteration. This method is also effective.
77
7.
The number of Ritz vectors required to be determined for the damaged system is equal to the rank of the element stiffness matrix. For a number "m" of damages in the finite element beam model, "2m" Ritz vectors are required, since the rank is 2m.
8.
For any number of damages in the structural systems, the mode shape expansion can be accurately obtained as long as Ritz vectors are correctly defined for the structural change due to the damage.
9.
The application of the updating methods presented is limited to a single damage location for the beam model. The drawback of the updating methods is that it may be difficult to use the updating methods on a system with more than two damages.
10. Prior to the application of the updating method to experimental results, the finite element model must be accurately represented, because the updating method is sensitive to the measurement errors in frequency (but not as sensitive to errors in the measured mode shapes).
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CHAPTER IV NONLINEAR MODEL REDUCTION
This chapter presents model reduction methods for a two or four degrees of freedom spring and mass oscillatory system (DOF). Numerical results are presented to show the efficiency of two model reduction methods: linear based reduction and NNM based reduction.
4.1 Model Reduction Methodology We present the general model reduction methodology which reduces the nonlinear problem of equation (1.2) to the reduced model of equation (2.19). The procedure is essentially the same as presented by Burton and Young (1994) and by Burton (1997). The general results presented here enable one to calculate the NNM's and the NNM-based reduction by first calculating the needed transformation, in which the secondary states are expressed as nonlinear functions of the master states. The linear kernel of this nonlinear state transformation forms the basis of the "linear-based" reduction. The calculation is described in the ensuing paragraphs. Once the master coordinate vector (Xm) has been defined, the nonlinear secondary/master transformation is assumed in the form {x3}=[T]{x^}-F{g(x^,x,J} where the (sxm) matrix [T] defines the linear kernel and where the s-vector {g} is a nonlinear function of the master states. [T] and (g) are to be determined; if the
79
(4.1)
nonlinearities appearing in {N(x)} in equation (1.2) are of polynomial type, then {g} will generally consist of infinite series, of which the leading polynomial terms will be used to define the NNM-based nonlinear model reduction. [T] and {g} are found as follows: first, differentiate equation (4.1) twice to obtain the secondary accelerations as {x3} = [T]{x^} + {g(x^,x^)}
(4.2)
where the original equations (1.2) of motion are used to rewrite accelerations appearing in {g} in terms of the master states {Xm} and {x^^}. Next, substitute equation (4.2) into equations (2.17) and (2.18) to eliminate the secondary acceleration terms in equations (2.17) and (2.18). Solve equation (2.18) for the master accelerations jx^ } as functions of the secondary and master coordinates, resulting in the following form, with matrix-vector notation dropped: X m M m ^ J + m^^J-' [k^^x, +k,„^x^ -f-n,^ +"^n..g]ms
(4.3)
Then substitute equation (4.3) into equation (2.17) and solve the result for the secondary coordinate vector Xs, resulting in the following equation, with matrix-vector notations dropped: Xs=[kss -(mssT + m , ^ ) ( m ^ J - H m ^ ^ ) " ' k ^ J " ' -i-i^sm -(mssT + m,^)(m^,T-Fm^,J-'k^^ ]x^
(4.4)
+(m33T-Hm,,J(m^,T-hm„^)-'(n^ + m,„J) - m ^ J - n J . Comparing equations (4.4) and (4.1), we identify [T] and {g} as follows, with matrixvector notation dropped:
80
T = -[k,3 - ( m , J - h m , J ( m , J - h m , ^ ) - ' k , J - ' -1,
,
(4.5)
•[ksm-(mssT-^m3^)(m,^3T-hm^^)-^k^^] g = [k,, - ( m , J + m,^)(m^J + m,^)"'k^,]"' _,,
_
•tKsT + m , ^ X m ^ J - h m ^ ^ ) - ' ( n ^ + m ^ J ) - m , , g - n J
(4.6)
Note that in the special case for which a lumped mass model is used, so that [M] is diagonal, one may then easily multiply equation (1.2) by [M]'', so that the mass matrix becomes the identity matrix. In this special case, the preceding results reduce to the following, with matrix-vector notation dropped: T = -(k,,-Tk,J-'(k,,-Tk,,J
(4.7)
g = Tn^ - n , -(k,, - T k ^ J g .
(4.8)
We refer to Young (1994) for the detailed derivation and the proof of the existence for the inverse matrices which appear in equations (4.3) through (4.7). In practice, the nonlinear components of the s-vector g are determined as polynomials by assuming a polynomial solution to equation (4.6) with undetermined coefficients. One substitutes the polynomial form into equation (4.6), uses the original equations of motion to eliminate master accelerations, and one then equates coefficients of like polynomial terms. The result is a set of linear algebraic equations from which the polynomial coefficients defining g are found. Shaw and Pierre (1991, 1993) have discussed this solution process in detail for the special case m = 1, that is, a single master coordinate is selected in order to find a single NNM.
Using the aforementioned results as a basis, we now describe the two model reduction methods defined in this work.
4.1.1 Linear-Based Reduction Once the solution to Equation (4.5) for T has been found iteratively. the reduced model is obtained by forming the "reduction matrix" R, defined as [R]^
[T]
(4.9)
[Iml where [R] is (n x m) and [Im] is the (m x m) identity matrix. The reduction matrix [R] expresses the complete coordinate vector x in terms of the master coordinates x,n as {x} = [R]{x^}.
(4.10)
Substitution of equation (4.10) into the nonlinear model defined by equation (1.2), followed by pre-multiplication by [R]^, produces the desired reduced model which contains the m master degrees of freedom, [m]*{x,J+[k]*{x,J+[T]Mn,(Tx,,,x,)} + {n,(Tx,,,x,)}={0}
(4.11)
where the reduced mass and stiffness matrices are defined by [m]* = [R]^[M][R] [k]* = [R]''[K][R].
(4.12)
Recalling that the system nonlinearities in equation (1.2) were assumed to be static, an important characteristic of the "linear-based" reduced nonlinear model of equation (4.1 1 is that all of the nonlinear terms which appear will be nonlinear functions of the master coordinates, but will not be functions of the master velocities. On the other hand, the
82
"NNM-based" reduced nonlinear model will generally contain nonlinearities which are functions of both the master coordinates and the master velocities. This feature is one of the main differences in the two types of nonlinear model reduction.
4.1.2 NNM-Based Model Reduction Once the linear transformation matrix [T] and the nonlinear transformation vector {g} have been found, the NNM-based reduced nonlinear model is obtained by substituting equations (4.1) and (4.2) into equation (2.18) to obtain [m^J+m^^]{x^}+[k^J-hk,„^]{x^}+[m^J{g(x^,x^)l + [kmsJ { g ( X m ' ^ m ) } + l n n i ( X s ' X m ) l = ( 0 } -
In this model the nonlinearities will generally be functions of both the master coordinates and the master velocities. Generally, the nonlinear state transformation vector g will be comprised of an infinite series of polynomial terms. In this work we will retain only the leading nonlinear terms, so that our NNM-based reduced models will actually be approximate versions of the exact the NNM-based reduced models one would obtain if the complete infinite series were used to define the nonlinear modal manifolds.
4.2 Features of the Two DOF to One DOF Reduction Prior to the comparison of the two model reduction methods for specific systems, some features of the nonlinear model reduction process are illustrated through consideration of a two degree of freedom nonlinear system in the (modal) form considered by Nayfeh and Mook (1979):
83
ii +coJ u+ttj u"* -i-a2 u" V + a3 u v^ -i-a^ v"* = 0 v-i-co2y + PiU^ -hP2u\-hP3Uv^ -hp4V^ = 0 .
(4.14) (4.15)
Here u and v are the (linearly uncoupled) modal coordinates and the nonlinearity is of cubic type, with coi < C02 assumed. In order to determine the first nonlinear modal manifold, we assume that v is defined in terms of u and u as y = buVcuii^+---H.O.T.
(4.16)
where H.O.T. stands for terms of higher degree than cubic, and where it turns out that the cubic terms proportional to u^ u and u^, if included in equation (4.16), are identically zero to preserve the conservative nature of the system. Calculating b and c according to the procedure outlined in section 4.1, we obtain the result b=-p,(co;-7(or)/A c=6p,/A
(4.17)
A = (9a)f - CO2 )(C0i" - CO2).
Then the NNM based reduced model is obtained by substituting equation (4.16) into equation (4.14) to obtain u-hcofu-^a,u'+a2u'[bu-Ucuu']+H.O.T.(7) = 0
(4.18)
where H.O.T.(7) stands for terms in u and u of degree 7 and higher. For comparison to the NNM based reduced model of equation (4.18), the linear based model reduction is obtained using in equation (4.14) the relation v =0, which defines the first linear mode, yielding the following for the projection of the first NNM onto the u, u plane: u +(o;^ u + a , u"* = 0 .
84
(4.19)
Based on the preceding results, we make the following observations: 1. Through the cubic terms the two reduced models are the same; thus, the effect of the nonlinearity on the frequency-amplitude dependence of the periodic motion in the first nonlinear modal manifold will be accurately represented by the linear based reduction. This is not surprising, as it is well known that the first order nonlinear correction to the frequency-amplitude dependence in a cubically conservative nonlinear oscillator is defined by the zeroth order linear solution. 2. In equation (4.18), we observe that calculation of the nonlinear transformation of equation (4.16) through the cubic terms will result in an NNM based reduced model (equation (4.18)) exact through the quintic terms, that is, exact to one higher level of approximation (see also comment 1). Thus, one would expect the NNM based reduction to be valid for a wider range of amplitudes than is the linear based reduction. 3. The constants b and c which define the leading terms of the nonlinear state transformation (equation ((4.17)) approach infinity as the determinant A^> 0. This occurs in the presence of a 3-to-l or a 1-to-l internal resonance (COT = 3coi, co: = coi, respectively). In this case the nonlinear modal manifold calculation breaks down. As shown by Nayfeh, Chin, and Nayfeh (1995), in this case one cannot reduce the model to a state space dimension less than four (that is, the two interacting linear modes must both be represented in the reduced model). In the examples presented in subsequent sections, the NNM-based reduction is shown to exhibit significant degradation in accuracy for near (and not so near) internal resonant conditions. 1 his
85
feature of the NNM-based reduction certainly impacts utility of the method. In the subsequent sections several simple examples are considered. The intent is to compare the efficacy of the linear-based and the NNM-based reduction methods.
4.3 Results for Systems with Analytic Nonlinearity In this section, both the linear-based and the NNM-based reduction methods are applied to simple spring-mass systems with analytic nonlinearity. Solutions obtained from both methods will be compared to the exact solution. All results are obtained from simulations via numerical integration. By "analytic" nonlinear system we mean that the system response function is smooth and continuous, so that Taylor series expansion is possible at any point on the response curve. The "non-analytic" nonlinear system does not satisfy the conditions described by the analytic nonlinear system. Examples of nonanalytic nonlinearity would be spring-mass systems with deadband nonlinearity or discontinuity. Due to the nature of non-analytic nonlinearity, the application of the NNM method to these systems is very difficult, while the linear-based method easily handles non-analytic nonlinear systems. The systems to be considered in this section are two or four DOF spring-mass systems with a static cubic nonlinearity. The efficacy of each of the two nonlinear model reduction techniques is now illustrated through examples.
86
4.3.1 A Two DOF System with a Cubically Nonlinear Spring Shown in Figure 4.1 is a two degree of freedom system with three linear springs and one cubically nonlinear spring connecting mass 2 to the right hand ground. This system has been considered by Shaw and Pierre (1993). The equations of motion, with all masses and linear springs having unit values, are X, -f-2 X1 - X 2 = 0
(4.20: x^-x,-i-2x2+£X2=0
The linear natural frequencies and mode shapes are coi =1, (02 = V3 , (j)! = Ll iX'\>2 = LI - i j . Selecting X2 as the master coordinate, the linear-based reduced model using equation (4.11) is Xj + X2+
(4.21)
i 8 X2 = 0
X, >
>
v/WW
A/VWNA
Figure 4.1. A two degree of freedom (DOF) oscillator system with a cubically nonlinear spring (8).
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which represents the projection of the first modal dynamics onto the X2, X2 plane. This is a Duffing equation with hardening nonlinearity (s > 0). The calculation of the first nonlinear modal manifold according to equation (4.1) produces the following result through the cubic terms: X, = X 2 - ^ 1 8 X 2 - f - i 8X2 X2-!-•••
(4.22)
Substitution of equation (4.22) into equation (4.20) results in the NNM-based reduced modal, exact through cubic terms, as X2-^X2 +8 [ | x 2 - j x , x ; J = 0 .
(4.23)
Notice that this NNM-based reduced model contains both hardening (-8X2) and softening ( - ^ s X2x? ) nonlinearities. There appears a nonlinear term that is a function of the velocity X2 ; this type of term cannot appear in the linear-based reduced model. The harmonic balance (HB) method (Nayfeh and Mook (1979); Burton (1994)) has been used to approximate the frequency-amplitude dependence for the two reduced models; this is done by substituting an assumed linear harmonic solution (x = a cos(cot)) into equation (4.21) or (4.23). If one uses the harmonic balance method to calculate the leading approximation to the frequency-amplitude dependence for the two reduced models, one finds the following results: for the linear-based (subscript LB) reduction. 2 cor« = l+|8a'LB = ^'T
and for the NNM-based reduction (subscript NNM),
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(4.24)
^ l-f^8a^ ^NNM = , : 2
(4-25)
where a is the amplitude of the periodic motion (for example, with X2(0) = a, XT (0) = 0). If one expands equation (4.25) in a power series in er, the leading term in the result is the same as that for the linear-based reduction, appearing in equation (4.24). As noted previously, this is expected because this leading correction term in such an approximation is determined by the solution to the linear version of the system model. Shown in Figure 4.2 - Figure 4.4 are comparisons of the linear-based and NNMbased solutions obtained via numerical integration of equations (4.20), (4.21), and (4.23). Figure 4.2 shows histories of X2 (t) for an amplitude a = 1.0 (X2(0) = a, X2 (0) = 0). For reference, the solution from the original two DOF model, equations (4.20), is also shown. The initial conditions for Xi and Xj for the original 2 DOF system (4.20) were obtained iteratively so as to place the initial state (xi, x, ,X2, X2) exactly on the first nonlinear modal manifold. This condition was realized when the responses for xi and X2, obtained via numerical integration of equation (4.20), were "exactly" periodic, (slightly "offmanifold" initial conditions generally produced aperiodic response, as exhibited in Figure 4.5, which is intended to show the effect of slightly "off-manifold" initial conditions on the response.) The exact initial conditions for the full model differ from those which would be obtained from equation (4.22), because equation (4.22) is a leading term approximation, rather than an exact result. One notes in Figure 4.2 that the full model and NNM-based reduced model results are nearly indistinguishable, whereas the linear-based reduced model exhibits a frequency higher than the actual one. Simulations similar to
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this one were conducted throughout the amplitude range a = 0 to a = 2.5. The resulting frequency co of the periodic response is shown in Figure 4.3 for the NNM-based and linear-based reduced models. Also shown for reference is the exact frequency obtained from the full, two DOF model of equations (4.20), with initial conditions X2(0) = a. x. (0) = 0, xi(0) =b, X, (0) = 0, with a specified and b determined iteratively to place the initial state exactly on the first modal manifold. Observe that for a < 0.6, both reduced models correctly determine the change in frequency with amplitude. For larger a, however, the NNM-based model is much superior to the linear-based reduced model. The linear-based reduction predicts a monotonic increase in co(a), equation (4.24), whereas the NNMbased model is hardening for small amplitudes, with a softening effect becoming important as the amplitude is increased. Shown in Figure 4.4 are various projections of the first nonlinear modal manifold onto the Xi, X2 plane. The "exact" manifold was obtained iteratively in the manner described previously. The "cubic" result is from equation (4.22). If terms through degree five (quintic approximation) were included in the manifold calculation, equation (4.22) is amended to the following: X, = X2 + 7 8 X 2 -I-T 8 X2 X2 + 0 . 0 6 0 6 1 8^ X2
-H0.272727 8' X2 x^ +0.0113636 8" x^ x', and this is also shown in Figure 4.4. Finally, we define an "implied cubic " manifold by determining the function X, = X2 + a x 2 + p X 2 X2
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(4.27)
which, when substituted into equation (4.20b), produces the linear-based reduced model of equation (4.21). In this case (4.28)
X 1 — X 2 + -J 8 X 9 .
Figure 4.4 shows that the manifold based on NNM reduction is close to the exact manifold, while the linear manifold is far from the exact manifold. In summary, for this example system, the NNM-based reduced model is superior to the linear-based reduced model. The linear-based reduced model, nevertheless, does correctly describe the leading nonlinear effect on the frequency-amplitude dependence of the periodic motion in the first modal manifold.
1.5
.>
o
.2 OL
Q
0
-0.5
-1 10 15 Time in S e c o n d s
20
25
Figure 4.2. Comparison of reduced model simulations: NNM-based reduced model, equation (4.23): "dashed line"; Linear-based reduced model, equation (4.21): "dotted line"; Original two DOF model, equations (4.20): "Solid line". Amplitude a = 1. Initial conditions for equations (4.20): X2(0) = 1, X2 (0) = 0, xi(0) = 1.3848, x,(0) = 0.
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Mode 1
1.8 -e— E x a c t o Lin—Based -^^^ N N M C u b i c
1.7
1.6
-3- "IS
tz
O) ID CT (D
Lt 1 .3
1.2
1.1 10
2.5
Figure 4.3. Frequency-amplitude dependence of periodic motion of the system of Figure 4.1 obtained by numerical integration: "Linear-Based" is from equation (4.21); NNMbased is from equation (4.23); Exact is from equation (4.20).
Mode 1 15
Exact Linear N N M Cubic Implied Cubic N N M Quintic
10
M 5 CO
o»o
0.5
1 1.5 M a s t e r D i s p l a c e m e n t , x^
2 5
Figure 4.4. First modal manifold projection onto xi, X2 plane: linear: X2 = xr. Cubic: equation (4.22): Quintic: equation (4.26); Implied cubic: equation (4.27); Exact: equation (4.20).
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r
1.5 Exact IC's Near Exact IC's
0 5
I o _o a>
>
-0.5
-1
-1.5 -1.5
-1
-0.5
D i s p l a c e m e n t (x,)
0.5
1.5
Figure 4.5. First modal manifold projection onto X2, X2 plane: Solid line based on exact IC's (X2(0) = 1 , X2 (0) = 0, xi(0) = 1.3848, x, (0) = 0): Dashed line based on IC"s initiated in vicinity of exact IC's (X2(0) = 1, X2 (0) = 0, xi(0) = 1.30, x, (0) = 0).
4.3.2
A Variation of the System of Example 4.3.1 Here we consider the system shown in Figure 4.6, which has two equal linear
springs and a cubic nonlinear spring connected from the right-hand ground to the second mass. Parameter values are m = k = 1.0, in appropriate units. This is the system of example 4.3.1, where the right-hand linear spring has been removed. The equations of motion for this two degree of freedom oscillator are the following: X, + 2 X, - X
0 (4.29.:
X O - X | + X T + 8 X T = 0 .
The first linear natural frequency is coi = 0.618034. and the first mode shape is proportional to (\>\^ = L0.6I8 i j . The second linear natural frequency is C02 =1.618033,
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and the second mode shape is proportional to 2^ = L-1.61957 i j . Note that the linear frequency ratio CO2/CO, = 2.618 is reasonably close to the 3:1 internal resonance condition. For this system with X2 again selected as master coordinate, calculation of the first nonlinear modal manifold through quintic terms yields the following: Xl = 0.618034 X2 + 0.0187917 s x^ + 2.02323 8 x, x ; +1.02336 8" x'^
(4.30)
-12.6889 8' xl x^ -20.22518^ X2 x^ X, =0.618034x,-1.489248x5 x,+10.83988'x:; x.
(4.31)
2.5
+2.02323 8x^2 +1-02142 8-x^ x^ -20.22518"x',.
If one substitutes the cubic version of equations (4.30) and (4.31) into equation (4.29) to eliminate the secondary coordinate, one obtains the approximate projection of the first NNM onto the (X2, X2) plane as X, + 0.381966 X, + 0.9812 8 x^ -2.02323 8 x, x; = 0
X
X >
>
S/S/\/\/V
(4.32)
/^/\/\/\/^s/^
/%/N^^
