development of numerical and experimental ...

UNIVERSITÀ DEGLI STUDI DI BOLOGNA, FIRENZE E PERUGIA SEDI CONSORZIATE

DOTTORATO DI RICERCA IN MECCANICA APPLICATA XV CICLO

DEVELOPMENT OF NUMERICAL AND EXPERIMENTAL PROCEDURES FOR DYNAMIC ANALYSIS OF AUTOMOTIVE VEHICLES

COORDINATORE: Chiar.mo Prof. Ing VINCENZO PARENTI CASTELLI TUTORE: Chiar.mo Prof. Ing. ALBERTO MAGGIORE CORRELATORI: Chiar.mo Prof. Ing. UMBERTO MENEGHETTI Chiar.mo Prof. Ing. GIUSEPPE CATANIA TESI DI DOTTORATO DI: Dott. Ing. ALESSANDRO ZANARINI

MARZO 2003

Contents

Chapter 1 Introduction ..........................................................................................1 Chapter 2 Time-Domain Component Mode Synthesis Methods Introduction...................................................................................................................................7 2.1 Definition of Components.......................................................................................................8 2.2 Real Component Modes ..........................................................................................................9 2.2.1 Normal modes .................................................................................................................11 2.2.2 Rigid-body modes ...........................................................................................................12 2.2.3 Constraint modes.............................................................................................................13 2.2.4 Attachment modes...........................................................................................................14 2.2.5 Inertia-relief modes .........................................................................................................16 2.2.6 Residual flexibility matrix: residual attachment modes ..................................................19 2.3 Statically complete mode sets ...............................................................................................20 2.4 Dynamic component mode supersets ....................................................................................22 2.5 Coupling of CMS models......................................................................................................23 2.6 Dealing with damping ...........................................................................................................28 2.6.1 General viscous damping case ........................................................................................28 2.6.2 Proportional damping in case of modal coupling ........................................................... 30 2.7.1 Unreduced models or physical models............................................................................32 2.7.2 Modal synthesis...............................................................................................................32 2.7.3 Craig-Chang CMS...........................................................................................................33

Chapter 3 Dynamic behaviour of multibody mechanisms Introduction.................................................................................................................................41 3.1.1 Equations of motion ...........................................................................................................42 3.2 Generalised coordinates and position analysis ......................................................................46 3.3 Kinematic constraint equations .............................................................................................48 3.3.1 Basic constraints..............................................................................................................49 3.3.1.1 ORTHO1 constraint...................................................................................................49 3.3.1.2 ORTHO2 constraint...................................................................................................49 3.3.1.3 BALL constraint.........................................................................................................50 3.3.1.4 DISTANCE constraint................................................................................................50 3.3.2 Parallelism constraints.....................................................................................................51 3.3.3 Absolute constraints on a body .......................................................................................52 3.3.4 Constraints between pairs of bodies ................................................................................53 3.3.4.1 HOOKE joint.............................................................................................................53

Contents

3.3.4.2 TURNING joint.........................................................................................................54 3.3.4.3 CYLINDRICAL joint................................................................................................55 3.3.4.4 TRANSLATIONAL joint .........................................................................................57 3.3.4.5 SCREW joint.............................................................................................................58 3.4 Linearization of the equations ...............................................................................................60 3.4.1 Linearisation of the position analysis ..............................................................................61 3.4.2 Linearization of the Lagrangian function ........................................................................63 3.4.3 Linearization of the constraint equations ........................................................................65 3.5 Examples ...............................................................................................................................67 3.5.1 Crankshaft mechanism ....................................................................................................67 3.5.2 Reciprocating mechanism ...............................................................................................75 3.5.3 Serial Mechanism............................................................................................................82

Chapter 4 Frequency-Domain Component Mode Synthesis Methods: FRF Based Substructuring Introduction.................................................................................................................................85 4.1 Basic theory...........................................................................................................................86 4.1.1 Force transmissibility ......................................................................................................88 4.2 Numerical improving techniques for a better coupling .........................................................90 4.2.1 Pseudo-Inverse ............................................................................................................... 90 4.2.2 Pre-processing of FRFs: Smoothing and Simmetrisation............................................... 93 4.2.2.1 Smoothing .................................................................................................................93 4.2.2.2 Symmetrisation..........................................................................................................95 4.2.2.3 Effects .......................................................................................................................96 4.3 Determination of the FRFs ....................................................................................................96 4.3.1 Experimental approach....................................................................................................97 4.3.1.1 Reciprocity principle .................................................................................................98 4.3.2 Analytical approach.........................................................................................................99 4.3.3 Modal superposition and truncation ..............................................................................101 4.3.4 Compensation for truncated modes: static and dynamic compensation, residual attachment modes ........................................................................................................................101 4.4 Rotational dofs in FBS coupling analysis ...........................................................................104 4.5 Connectors ..........................................................................................................................106

Chapter 5 A connecting stiffness optimisation procedure Introduction...............................................................................................................................109 5.1 Description of the procedure ...............................................................................................112 5.1.1 Pre-processing the data..................................................................................................112 5.1.2 Mode pair table extraction.............................................................................................114 5.1.3 Data extraction and target function evaluation..............................................................117 5.1.4 Optimisation management.............................................................................................118 5.2 Results of the test cases.......................................................................................................119 5.2.1 Front door......................................................................................................................121 5.2.2 Back door ......................................................................................................................127 5.2.3 Front door result interpretation......................................................................................131 5.2.4 Back door result interpretation ......................................................................................132 5.3 Optimisation guidelines.......................................................................................................133 5.4 Calculation time ..................................................................................................................134 5.5 Conclusions .........................................................................................................................135

II

Contents

Chapter 6 Tire experimental modelling in vehicle dynamic analysis Introduction...............................................................................................................................137 6.1 Description of the vehicle model.........................................................................................138 6.1.1 Suspensions ...................................................................................................................139 6.1.2 Equations of motion ......................................................................................................141 6.1.3 Modal solution of the vehicle model .............................................................................142 6.2 Tire modelling.....................................................................................................................149 6.2.1 Experimental settings ....................................................................................................149 6.2.2 Modal model .................................................................................................................151 6.3 Modal coupling of the full car.............................................................................................156 6.4 Conclusion ..........................................................................................................................165

Chapter 7 Road noise modelling for NVH predictions by means of FBS approach Introduction...............................................................................................................................167 7.1 Application of FRF based substructuring to road noise NVH predictions ..........................169 7.2 Hybrid model description....................................................................................................170 7.2.1 Pre-loaded tire ...............................................................................................................170 7.2.2 Rear suspension.............................................................................................................170 7.2.3 Car body ........................................................................................................................172 7.2.4 Bushings and mounts ....................................................................................................173 7.3 Coupling specification.........................................................................................................173 7.4 Validation tests....................................................................................................................175 7.5 Comparison between the test and the hybrid model............................................................177 7.6 Conclusion ..........................................................................................................................178

Chapter 8 Conclusions ....................................................................................... 183 References............................................................................................................................ 185

III

In memory of my grandmother Nonna Lilla’s wittiness

I am especially grateful to my mother Roberta and all my Friends, at home and in Belgium, who have enriched my life.

Chapter 1 Introduction In the development of modern road vehicles the automotive industry exploits many different technologies to overtake a multitude of complex and interrelated tasks. The same basic approaches have a vast applicability, varying from civil to aerospace engineering fields. Vibrations studies have a prominent role in the design of cars, since they enhance the understanding of the phenomena related to the dynamic behaviour of the vehicle such as structural resistance and dynamics, reliability, handling and comfort (both vibrational and acoustic: air-borne, structural-borne and powertrain noise). The approaches followed in the industrial world are accelerating toward the virtual prototyping environment, where all modelling should be performed using only numerical models without any need of expensive and time-consuming tests on prototypes as modelling benchmarks. At present, the state-of-the-art of pure numerical technologies does not yet give the possibility of eliminating the testing sessions, since many problems seem nowadays too complex to be modelled by means of virtual prototyping alone. Actually, experimental modelling is still the basic approach for topics where pure numerical technologies do not give reliable results, but it has the drawback of being expensive and time-consuming. In between there exists the hybrid approach that matches the two aforementioned methodologies, giving the possibility of coupling models derived from the cheapest and most efficient technology. At the basis of all the

Chapter 1

Introduction

three approaches, virtual, experimental and hybrid, there lies the substructuring concept, which derives the behaviour of an assembled structure from the superposition of the dynamics of its single components. The study presented in this doctoral dissertation deals with an overview of the state-of-the-art technology in the field of structural and mechanisms dynamics and with applications of the proposed methodologies to automotive case studies. The overview does not intend to be exhaustive, but particularly focuses on fields related to the developed methodologies. The theoretical foundations of Chapters 2-4 offer the basis to better understand the key knowledge exploited in the development of the applications of Chapters 5-7. The reader will find that some issues are not expanded in detail: this is due to their being only instrumental to the rest of the research, where no work has been done taking advantage of their particular inner theory. This is the case of Experimental Modal Analysis that has been widely employed, but not treated in its theoretical aspects, because nothing has been developed here with the intent to enhance it. Chapter 2 reports from literature the theory that deals with the Component Mode Synthesis Methods in the time-domain, a procedure that helps in the reduction of structure models by assuming that the behaviour of a structure is expression only of a basis of component modes, and that all the physical degrees of freedom are linearly dependent on this base of orthogonal phenomena. It is a technology developed some decades ago when poor calculation resources were available to overtake the problem of modelling large scale structures such as aircrafts by finite element approach. Now it has relevance in the concurrent engineering where the single subframes models are developed by different teams and hybrid coupling is frequently requested. It has also application where time and calculation resources are badly needed by optimisations that might focus on particular

2

Chapter 1

Introduction

aspects of the design instead of dealing with the whole structure of the product, such as the optimisation of the connections between subframes. In this chapter the basic theory is outlined as well as the relation between the virtual prototyping environment and experimental measurements. Chapter 3 deals with the dynamic behaviour analysis of mechanisms with large displacements and a high number of kinematic degrees of freedom. Starting from the formulation of the motion and constraint equations for rigid bodies taken from the classical mechanics in the Lagrangian form, a linearised approach is originally developed and proposed to investigate the modal behaviour of the whole mechanism around a relevant kinematically admissible configuration. In this way it is possible to understand how the eigensolution of the system varies along large displacements in the working domain. In this chapter the basic theory is developed focusing on a library of linearised constraints of common use in mechanism engineering. The aid given by Prof. Giuseppe Catania of DIEM, Department of Mechanical Engineering, Bologna University, for the writing of the C-language code that implements the theory outlined is gratefully acknowledged by the author. Chapter 4 reports from literature the theory that deals with the Component Mode Synthesis Methods in the frequency-domain, a procedure that brings in the possibility of modelling the structures by using the frequency response functions of nodes of interest and the connection impedances. This technology is well suited for hybrid coupling, since the frequency response functions may be experimental as well as obtained from finite element models. In this chapter the basic coupling theory is outlined, as well as some techniques used to overtake some drawbacks that this methodology presented at its beginning. Particular attention is focused on the experimental approach and possibilities, since this methodology could help solve

3

Chapter 1

Introduction

the modelling problems of large damped structures, where other approaches are precluded. Chapter 5 deals with an original application of the Component Mode Synthesis theory to automotive test cases. The application develops an optimisation procedure for the updating of finite element connection models between subframes, taking experimental results as references. The optimisation function, managed by a genetic algorithm, deals with frequency differences of the modes, with the Modal Assurance Criterion values and with the differences in the frequency response functions between the finite element model and the experimental results. An algorithm has been written to implement the choice strategy about the model modes to be paired to the experimental ones. Two test cases are analysed to prove the results obtained. This work was originated by the interest of LMS International, Leuven, in developing a procedure about connection optimisation, where other strategies had proved to be ineffective in such complicated models as of cars. The aid given by Dirk Von Werne, Gaetano Fortunato, Andrea Zugna, Peter Mas and Joost Van de Peer of LMS International and by Prof. Giuseppe Catania of DIEM, Department of Mechanical Engineering, Bologna University, is gratefully acknowledged by the author. Chapter 6 deals with the coupling of flexible body models to linearised models of spatial mechanisms derived from the theoretical fundamentals developed in Chapter 3. The test case is taken from the automotive industry. The car body and its suspension sets have been modelled as a multibody mechanism with rigid bodies and springs, and the appropriate constraints. Once linearised, the car model has been coupled to the tire model derived by means of Experimental Modal Analysis performed in a constrained and preloaded testing setup. The modal solution of the coupled model has been discussed. The author gratefully acknowledges the aid given by Prof. Giuseppe

4

Chapter 1

Introduction

Catania of DIEM, Department of Mechanical Engineering, Bologna University. Chapter 7 deals with the attempt of modelling road noise induced by tire contact patch displacements in an automotive test case. The procedure proposed has been developed in the framework of the Frequency Response Function Based Substructuring methodology, outlined in Chapter 4. Experimental frequency response functions have been acquired on the car body. The suspension model has been formulated by a finite element code. The tire model has been derived from experimental measurements by means of Experimental Modal Analysis. Performing a hybrid coupling between the models obtained by different approaches, it has been possible to compare the results with the data acquired in laboratory tests on the full car. Herman Van der Auweraer and Peter Mas of LMS International, where the research was started, supported the developing of the technology here reported. The present work was completed at Katholieke Universiteit of Leuven under the supervision of Prof. Paul Sas in the framework of the European Doctorate on Sound and Vibration Studies with the project “Hybrid modelling for road noise prediction”. The support of the European Commission by the EDSVS Marie Curie Training Site programme is gratefully acknowledged. The author wish to thank all the staff of both LMS and KUL that gave their help in the development of the technology here outlined. But Prof. Alberto Maggiore, Prof. Umberto Meneghetti, Prof. Vincenzo Parenti Castelli and Prof. Giuseppe Catania of DIEM, Department of Mechanical Engineering, Bologna University, are the first addressees of the author’s gratitude for allowing him to study abroad in the academic year 2001-2002 and for their support during his doctorate course.

5

Chapter 2 Time-Domain Component Mode Synthesis Methods Introduction The finite element method was developed a few decades ago to estimate the dynamic response of structures subjected to dynamic excitation. Complete structures are frequently very complex, and major components are often designed and produced by different organisations. As the calculation speed of early computers was only a fraction of recent high-performance computers, methods have been developed which permit a structure to be subdivided into components, or substructures. These methods are now commonly termed as methods of substructure coupling. When it was previously impossible to assemble and analyse a huge finite element model of an entire structure, these component mode synthesis methods allowed to develop an approximate mathematical model of the full structural system, with much of the analysis being done on much smaller components. Component mode synthesis is a form of substructure coupling analysis in which the dynamic behaviour of each substructure is formulated as a superposition of modal contributions. Hence,

Chapter 2

Time-Domain Component Mode Synthesis Methods

component mode synthesis can be regarded as an application to complex structures of the Rayleigh-Ritz method. The present CMS method involves two basic steps. The first being the definition of a set (or sets) of component modes. Methods of CMS may be divided on the basis of the types of component modes employed in the synthesis procedure. These component modes are here briefly described. The second step being the coupling of the component models to form a reduced-order system model. Enforcing the equilibrium of forces and the compatibility of displacements along component interfaces forms this system model. Much of the material presented here is based on the work of [Craig87], [DeFonseca98] and [Hermans99]. A more comprehensive overview with strict mathematical derivations of the CMS methods can be found in books and review papers ([Hurty65], [CraigBampton68], [MacNeal71], [Hintz75], [Rubin75], [Craig-Chang76], [Craig-Chang77], [Craig81],[Duarte-Ewins96/1], [Duarte-Ewins96/2], [Hermans&al.00]). 2.1 Definition of Components The most general type of component (or substructure) is one that might be connected to one or more adjacent components by redundant interfaces. The linear equation of motion of each component in uncoupled conditions can be written as: [M ]{x}+ [C ]{x}+ [K ]{x} = {F } (2.1)

where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {x} is the displacement vector of the physical degrees of freedom (dofs) and {F} are the applied external forces.

8

Chapter 2


The dofs of each component will be referred to by using indices depending on the set to which they belong. The indices used further, as subscripts in the mathematical representations, refer to the different dofs sets: r = the reference dofs used to determine rigid-body motion, which would provide for the statically determinate (rigidbody) support of the component; e = the set of boundary dofs remaining after the reference r dofs are removed, also called excess or redundant boundary coordinates; b = the total set of physical boundary dofs (at the interface coordinates of the component, b=r+e) i = the set of omitted dofs (at the interior coordinates of the component); p = the total set of physical coordinates of the component (p=i+b). The number of dofs in each set is indicated as nx, with x being the name of the corresponding set. 2.2 Real Component Modes Neglecting damping in equation (2.1), the equation of motion for a component can be written as: [M ]{x}+ [K ]{x} = {F } (2.2)

In any CMS technique, the displacements in the physical coordinates {x} of a component are represented in terms of the component generalised coordinates {z} using the classical modal transformation: {x} = [Ψ ]{z} (2.3)

9

Chapter 2


The equations (2.2) and (2.3) yield the (undamped) component modal model: [m]{z}+ [k ]{z} = { f } (2.4)

where: [m] = [Ψ ]T [M ][Ψ ] [k ] = [Ψ ]T [K ][Ψ ] (2.5) { f } = [Ψ ]T {F }

The modal transformation matrix [Ψ] may consist of sets of several types of component modes, such as normal modes, constraint modes, attachment modes, rigid-body modes, inertia-relief modes, inertiarelief attachment modes, etc. Some of these different types of modes are defined in the next paragraph. [Craig87] suggests the following criteria for selection of the component data to be used in the CMS technique: - Accuracy and efficiency: their generation should be computationally cheap, and their approximation properties should be good as for the high accuracy of the solution obtained using these vectors. - Component independence: the data for one component should be independent of the data for other components. - Synthesis generality: the synthesis procedure should permit any type of component mode to be included. - Static solution capability: static or pseudo-static solutions, which may be required for mode-acceleration response analysis, should be obtainable using the coupled model. - Modal testing compatibility: component and coupling data should be available from modal, or other, tests.

10

Chapter 2


- Explicit boundary coordinates: boundary coordinates should be included explicitly in the final set of component generalised coordinates in order to facilitate “direct-stiffness” assembly of system matrices. 2.2.1 Normal modes The normal modes are the eigensolution of a standard eigenproblem. Normal modes can be classified according to the boundary conditions specified for the component: free-interface normal modes, fixedinterface normal modes, hybrid-interface normal modes, or loadedinterface normal modes. Free-interface normal modes, for example, follow from the solution of a standard eigenvalue problem of this type:

([K ] − ω

with

 kii  [K ] = kei   k ri

kie k ee k re

2 p

[M ]){φ p }= {0} (2.6)

kir   mii   k er  , [M ] = mei    mri k rr 

mie mee mre

mir   mer   mrr 

and {φ p }

φip      = φep  .   φrp 

(2.7)

The modes may be normalised and assembled as columns of the modal matrix. For free-interface modes the modal matrix assumes the following form:  Φin   Φ bn 

[Φ n ] = 

n P ×n p

For fixed-interface modes it becomes:

11

(2.8)

Chapter 2


Φin    0bn 

[Φ n ] = 

n P ×n I

(2.9)

Then, [Φ n ]T [M ][Φn ] = [I nn ] (2.10) [Φ n ]T [K ][Φ n ] = [Λ nn ] = diag (ω n2 ) (2.11)

The complete set of normal modes is usually truncated to a smaller set of kept normal modes (subscript k), which will be denoted as [Φk]. These normal modes are used to supplement “static” modes, which will be defined in the next paragraphs. The normal modes can be estimated on actual components by means of Experimental Modal Analysis ([Ewins86], [Heylen&al.97]), according to the boundary conditions specified for the component. Employing appropriate modal parameter extraction methods, it is possible to derive the modal model (frequencies and modeshapes) from the acquired frequency response functions. The experimental results can be the right benchmark to perform the finite element model updating, as explained later in Chapter 5. 2.2.2 Rigid-body modes Rigid-body modes (subscript r) are defined in relation to the r-set dofs, the set that restrains the component from rigid-body motion. This yields to:  kii  kei   k ri

kie kee k re

kir   Ψir  0     ker  Ψer  = 0     k rr   I rr  0

12

(2.12)

Chapter 2


Then,  kii  kei

kie   Ψir   kir    = −  kee   Ψer  ker 

(2.13)

and the rigid body modal matrix is given by  Ψir    g ii   [Ψr ] = Ψer  = −  g ei      I rr  

g ie   kir      g ee  k er    I rr 

(2.14)

where  g ii   g ei

g ie   kii = g ee  k ei

kie   k ee 

−1

(2.15)

Rigid-body modes can also be defined by geometrical considerations, or estimated by means of Experimental Modal Analysis. 2.2.3 Constraint modes A constraint mode (subscript c) is defined as the static deformation of the component when a unit displacement is applied to one coordinate of a specified set of (physical) constraint coordinates, while the other coordinates in that set are restrained, and the other (interior i-set) dofs of the component are force-free. When the e-set of dofs is taken as the set of constraint coordinates (c-set = e-set), the constraint modes satisfy the following equation:

13

Chapter 2


 kii  kci   k ri

kic kcc k rc

kir   Ψic   0      kcr   I cc  =  Rcc      k rr   0   Rrc 

(2.16)

which defines the set of constraint modes as:

[ ]

Ψc n P ×n E

−1  Ψic  − kii kic      =  I cc  =  I cc       0   0   

(2.17)

2.2.4 Attachment modes An attachment mode (subscript a) is defined as the static deformation of the component when a unit force is applied to one coordinate of a specified set of (physical) attachment coordinates, while the others in that set are force-free. When the redundant interface coordinates set (e-set) is taken as the set of attachment coordinates (a-set), the attachment modes satisfy the following equation:  kii  k ai   k ri

kia k aa k ra

kir   Ψia   0      k ar  Ψaa  =  I aa      k rr   0   Rra 

(2.18)

which defines the set of attachment modes as:

[ ]

Ψa n P ×n E

 Ψia    g    ii  =  Ψaa  =   g    ai   0  

g ia   0    g ia       g aa   I aa   =  g aa     0   0 

14

(2.19)

Chapter 2


That is, attachment modes are columns of the flexibility matrix of the component restrained at the r-set. The attachment modes can be also defined as [Ψa ] = [G ][ f a ] (2.20)

where [G] is the flexibility matrix with restrained rigid-body motions,  gii  G =  g ei   0

gie g ee 0

0  0  0

(2.21)

and [fa] is the force matrix, regardless of the [Rra] part, which is not used here. 0    [ f a ] =  I aa     0 

(2.22)

When the a-set of dofs equals the c-set and the e-set, [Craig87] proves that the columns of the attachment-mode matrix are linear combinations of the columns of the constraint-mode matrix. Since:  kii  k ei

kie   g ii  k ee   g ei

g ie   I ii = g ee   0

0  I ee 

(2.23)

it follows [gie ] = −[kii ]−1[kie ][g ee ] (2.24)

then

15

Chapter 2


−1  g ie  − kii kie      [Ψa ] =  g ee  =  I ee [g ee ] = [Ψc ][g ee ]        0   0 

(2.25)

or also, [gee] being invertible: [Ψc ] = [Ψa ][g ee ]−1 (2.26)

This indicates that both static mode sets span the same subspace. 2.2.5 Inertia-relief modes When a component has rigid-body freedom, inertia-relief modes are required to represent the complete static response (i.e. when the r-set of dofs is not empty). Two completely different definitions are provided, as follows. Inertia-relief modes denoted as [Ψm] (for mass related ones) were introduced by [Hintz75], who employs the adjective “static” as to include the rigid-body response of a free component. The inertiarelief modes are then defined as the static displacement of a component which is loaded by d'Alembert forces due to rigid-body motion and which is supported so that the stiffness matrix is not singular. By assuming that each rigid-body mode has a unit modal acceleration, the result is the following:  kii  k ei   k ri

kie k ee k re

kir  Ψim   mii    k er   0  = mei    k rr   0   mri

mie mee mre

mir   Ψir   0      mer  Ψer  +  Rer      mrr   I rr   Rrr 

The inertia-relief modes [Ψm] are given by:

16

(2.27)

Chapter 2


−1 Ψim  kii (mii Ψir + mie Ψie + mir )      [Ψm ] =  0  =  0     n P ×n R   0  0   

(2.28)

Inertia-relief attachment modes, referred as [Ψb], are based on the application of unit forces onto the entire boundary coordinate b-set and on equilibrating these unit forces with the d’Alembert inertia forces which would result from application of the same unit forces on the boundary. These inertia-relief attachment modes, which are orthogonal to the rigid-body modes, were introduced by [MacNeal71] and by [Rubin75]. They can be defined stating that the force matrix to be applied on the boundary is defined by: 0   I bb 

[ fb ] = 

(2.29)

These forces are applied to the component whose motion equation is: [M ][xb ] + [K ][xb ] = [ f b ] (2.30)

where xb is an nP x nB matrix which is a sum of rigid-body and elastic motion as given by: [xb ] = [xr ] + [xe ] = [Ψr ][ηr ] + [xe ] (2.31)

Let [µrr] be defined by [µ rr ] = [Ψr ]T [M ][Ψr ] (2.32)

Since [K][xr]=0 (stiffness matrix is singular on the dofs of r-set),

17

Chapter 2


[ f e ] = [M ][xe ] + [K ][xe ] = [ f b ] − [M ][xr ] (2.33)

defines the balanced force system applied to elastic modes. If the previous equation is premultiplied by [Ψr]T, it becomes: [Ψr ]T [M ][xe ] + [Ψr ]T [K ][xe ] = [Ψr ]T [ f b ] − [µ rr ][ηr ] (2.34)

Since [Ψr] is orthogonal to all elastic modes, the result is the following: [ηr ] = [µ rr ]−1[Ψr ]T [ f b ] (2.35)

Thus, [ f e ] = [ f b ] − [M ][Ψr ][µ rr ]−1[Ψr ]T [ f b ] = [P ][ f b ] (2.36)

where [P] is defined as: [P ] = [I ] − [M ][Ψr ][µ rr ]−1[Ψr ]T . (2.37)

Let

[Ψˆ ] be a matrix whose columns are the displacement vectors of a b

component under the inertia-force-equilibrated-boundary-force matrix [fe] and supported on the r-set. Thus,

[Ψˆ ] = [G][ f ] = [G][P][ f ] b

e

b

(2.38)

where [G] is the flexibility matrix already defined. Now let rigid-body modes be removed from

[Ψˆ ] b

to form a matrix [Ψb ] which is mass-

orthogonal to the rigid-body modes:

18

Chapter 2


[Ψb ] = [Ψˆ b ]− [Ψr ][cr ] (2.39)

[Ψr ]T [M ][Ψb ] = [Ψr ]T [M ][Ψˆ b ]− [Ψr ]T [M ][Ψr ][cr ] = [Ψr ]T [M ][Ψˆ b ]− [µ rr ]−1[cr ] (2.40)

Then, [cr ] = [µ rr ]−1[Ψr ]T [M ][Ψˆ b ] (2.41)

so [Ψb ] = ([I ] − [Ψr ][µ rr ]−1[Ψr ]T [M ])[Ψˆ b ] (2.42)

Finally [ ] = ([P ]T [G ][P ])[ f b ] = [Ge ][ f b ] (2.43)

Ψb n P ×n B

The elastic flexibility matrix [Ge] is non-singular, as opposed to the flexibility matrix [G], which is singular due to the rigid-body degrees-of-freedom of the component. Because the applied force system is chosen so that there is no rigid-body motion, these modes were called "inertia-relief modes". This might be very confusing, despite the fact that there is a clear physical difference between the inertia-relief modes of Hintz and the inertia-relief attachment modes of MacNeal and Rubin. 2.2.6 Residual flexibility matrix: residual attachment modes [Craig87] derives another expression of the elastic flexibility matrix [Ge] as the product of the elastic modes and the inverse of the diagonal matrices containing their eigenvalues, where k elastic modes

19

Chapter 2


are retained in a mode-superposition solution and d higher elastic modes are truncated from the transformation modal basis: [Ge ] = [Φ e ][Λ ee ]−1[Φ e ]T = [Φ k ][Λ kk ]−1[Φ k ]T + [Φ d ][Λ dd ]−1[Φ d ]T (2.44)

It is now possible to address the residual flexibility matrix [Gd] as follows: [Gd ] = [Φ d ][Λ dd ]−1[Φ d ]T = [P ]T [G ][P ] − [Φ k ][Λ kk ]−1[Φ k ]T (2.45)

so that a residual attachment mode set can be defined as: [ ] = [Gd ][ fb ] (2.46)

Ψd n P ×n B

The reason for using these residual (inertia-relief) attachment modes is to bypass the problem of the linear independence of the modes. For example, if a complete set of free-interface normal modes is supplemented with attachment modes, the latter are linearly dependent on the former. The residual (inertia-relief) attachment modes are, according to the construction in equation (2.45), linearly independent of the kept normal modes. 2.3 Statically complete mode sets [Hintz75] has defined two statically complete interface mode sets, which he denotes the “method of constraint modes” and the “method of attachment modes” respectively. That is, a superposition of the modes in either of these sets is sufficient to determine exactly the “static” response of the component subjected to external forces

20

Chapter 2


applied only on the boundary, together with the d'Alembert forces due to the rigid-body portion of the resulting motion. Corresponding to Hintz's method of constraint modes, a constraint-mode superset can be defined as follows:

[Ψ ] c

n P × (n B + n R )

= [Ψr

Ψc

 Ψir  Ψm ] = Ψer   I rr

Ψic Ψec

0

Ψim   0   0 

(2.47)

where the column partitions are the rigid-body modes, the redundant constraint modes (with c-set = e-set) and the inertia-relief modes (mass related ones). Corresponding to Hintz's method of attachment modes, an attachment-mode superset can be defined as follows:

[(Ψ ] a

nP × nB + nR )

= [Ψr

Ψa

 Ψir  Ψm ] = Ψer   I rr

Ψia Ψea

0

Ψim   0   0 

(2.48)

Craig introduces a third statically complete superset, called inertia-relief attachment-mode superset, using the inertia-relief attachment modes together with the rigid-body modes:

[(Ψ ] b

nP × nB + nR )

= [Ψr

Ψb ]

(2.49)

He also proves that any of the defined supersets span the same subspace. They are all capable of producing the exact static solution to a set of boundary loads and inertial loads due to rigid-body motion, thus satisfying the static solution capability criteria aforementioned.

21

Chapter 2


2.4 Dynamic component mode supersets If elastic modes are used to supplement any of the statically complete supersets, a dynamic component mode superset results. Accordingly,

[Ψ ] c

n P ×(n B + n R + n K )

= [Ψr

Ψc

Ψm

Φk ]

(2.50)

defines a dynamic constraint-mode superset, while

[Ψ ] ( a

nP × nB + nR + nK )

= [Ψr

Ψa

Ψm

Φk ]

(2.51)

defines a dynamic attachment-mode superset, and

[Ψ ] b

n P × (n B + n R + n K )

= [Ψr

Ψb

Φk ]

(2.52)

defines a dynamic inertia relief attachment-mode superset. The columns of these matrices may not be linearly independent if [Φk] includes all or a large fraction of the component normal modes. In cases where [Φk] includes only a small or moderate fraction of the total flexible modes, it is possible to define sets of linearly independent basis vectors. Employing the residual attachment-mode set, intrinsically orthogonal to both rigid-body modes and kept elastic modes, it is possible to define a linearly independent base. The dynamic residual attachment-mode superset is defined as follows:

[Ψ ] d

n P × (n B + n R + n K )

= [Ψr

Ψd

22

Φk ]

(2.53)

Chapter 2


The dynamic residual attachment mode superset has no static complement as the residual attachment modes depend on the kept normal modes. When the same set of kept normal modes is used, those four dynamic component mode sets span the same subspace. Craig suggests that only the dynamic residual attachment-mode superset produces mass and stiffness modal matrices which are likely candidates for being determined by component testing. 2.5 Coupling of CMS models Above, several possibilities were discussed for the component modes which form the transformation matrix [Ψ] given by the dynamic component mode superset. The application of the transformation results in a reduced model of that component, as outlined in § 2.2. That reduced model can be used as such for e.g. forced response analysis or correlation analysis, or, mostly, for coupling purposes with other reduced models. Applying the transformation matrices to the systems equations of uncoupled components 1 and 2 gives: − ω 2 [m1 ]{z1}+ [k1 ]{z1} = { f1}

− ω 2 [m2 ]{z2 }+ [k 2 ]{z2 } = { f 2 }

(2.54) (2.55)

In case the transformation matrix is filled up with the mode shapes of the free-interface normal modes, diagonal modal mass and stiffness matrices are obtained. The generalised coordinates then correspond to modal participation coordinates, which are expressing how much each mode shape is contributing to the response of the component.

23

Chapter 2


For rigid coupling, the assembly of both components starts with combining the transformed uncoupled component equations in one matrix equation, giving rise to the “uncoupled” assembly: m1 0   z1  k1 0   z1  Ψ1 0  −ω    +    =    0 m2   z2   0 k 2   z2   0 Ψ2 

T

2

 F1     F2 

(2.56)

Next, it is necessary to impose the compatibility conditions, stating that the interface displacements of the two substructures should be equal, and/or the continuity conditions, stating that the interface forces should cancel. Typically, only the interface displacement compatibility will be explicitly enforced. The continuity of the interface forces is implicit and does not yield additional constraints. Only in case of Craig-Chang CMS (§2.7.3 and residual flexibility method), continuity of the interface forces, which are part of the generalised coordinates, is explicitly imposed in addition. The compatibility at the interface will result in a relationship between the generalised coordinates of component 1 and component 2: [R1

 z1  R2 ]  = {0}  z2 

(2.57)

Therefore, the generalised coordinates of each component are not independent. Hence, they need to be split up in a dependent (d) and independent (i) set. The independent set will correspond to the coordinates of the assembly (as): [Rdd

 zd  Rdi ]  = {0}  zi 

24

(2.58)

Chapter 2


where [Rdd] should be a nonsingular square matrix. The number of dependent coordinates equals the rank of the relationship matrix [R]. This equation can be re-written as follows: {zd } = −[Rd ]+ [Ri ]{zi } (2.59)

where + denotes the pseudo-inversion (see § 4.2.1). In most cases, this will be a normal inversion unless the number of equations is larger than the number of dependent coordinates. Theoretically, there are many possibilities of making the split between dependent and independent coordinates. The only requirement is that the matrix [Rdd] is non-singular. Practically, for reasons of numerical robustness, a low value of the matrix condition number of [Rdd] is recommended. Then the compatibility at the interface generates an additional transformation, [Tas], on assembly level:  z1  − [Rdd ]+ [Rdas ] {zas } = [Tas ]{zas }  =   z2   I asas 

(2.60)

where {zas} represents the generalised coordinates of the assembly. Applying this compatibility transformation finally yields the following assembly equation of motion: − ω 2 [mas ]{zas }+ [k as ]{zas } = { f as }

(2.61)

with: m1

[mas ] = [Tas ]T 

 0

0 [Tas ] m2 

25

(2.62)

Chapter 2


k1

[kas ] = [Tas ]T 

 0

[ f as ] = [Tas ]

T

0 [Tas ] k 2 

Ψ1 0     0 Ψ2 

T

 F1     F2 

(2.63)

(2.64)

It needs to be stressed that two transformations are involved: • a first one on component level, to reduce the system equations of each component, prior to coupling; • a second one on assembly level, to generate the compatibility of the interface dofs and/or the continuity of the interface forces. The applied forces also need to be transformed. For flexible coupling, commonly, the generalised coordinates of the components can be kept as the coordinates of the assembly. This means that the assembly transformation matrix becomes an identity matrix: [Tas]=[I]. The connector needs to be transformed by the transformation matrices and added to the assembly. The transformed connector is given by:  mcon ,11 mcon ,12   Ψ1,c   =  mcon ,21 mcon ,22   Ψ2 ,c 

T

 k con ,11 kcon ,12   Ψ1,c   =  kcon ,21 k con ,22   Ψ2 ,c 

T

 M con ,11   M con ,21

M con ,12   Ψ1,c    M con ,22   Ψ2 ,c 

 K con ,11 K con ,12   Ψ1,c      K con ,21 K con ,22   Ψ2 ,c 

(2.65)

(2.66)

This transformed matrix has the same size as the assembly. It needs to be added to the “uncoupled” assembly given by equation (2.56). This gives the following:

26

Chapter 2


mcon ,12   z1  k1 + k con ,11 k con ,12   z1  Ψ1 0  m1 + mcon ,11   =    +  −ω    mcon ,21 m2 + mcon ,22   z2   k con ,21 k 2 + k con ,22   z2   0 Ψ2  2

T

 F1    (2.67)  F2 

Solving this equation results in the solution of the flexibly coupled system. The mode shapes of the assembly are a linear combination of the mode shapes of the uncoupled components. In general, component coupling procedures focus on the coupling of components described by a (reduced) set of generalised coordinates. This is the most intuitive way of working when using experimentally derived component models. However, when using finite element models, the entire structure might be divided into some components and a residual structure. This residual structure is not necessarily the compilation of the generalised coordinates for each component. It may still contain some physical coordinates that are not shared with one of the (reduced) components. An example of such an application of a residual structure can be one specific part of a very complex structure. Their generalised coordinates can represent the other parts of this assembly, while the part being considered is represented by its complete finite element model. The designer has then the possibility of quickly evaluating the design changes on this part, taking into account the influence of the other parts (as applied in Chapter 5). When the residual structure contains modal dofs of previously attached components together with physical degrees-of-freedom, the unity matrix can be thought of as the modal transformation matrix, as far as the physical interface degrees-of-freedom are concerned. It should be noticed here that this partitioning is a standard operation in the assembly of finite element matrices previous to the solution of the system of the unconstrained dofs equations.

27

Chapter 2


2.6 Dealing with damping The above paragraphs have outlined the coupling approaches for systems where damping is neglected. When this is not the case, a slightly different procedure has to be applied to rightly predict the assembly behaviour. The equation of motion of an uncoupled component 1 can now be written as − ω 2 [M 1 ]{δ1}+ jω [C1 ]{δ1}+ [K1 ]{δ1} = {F1}

(2.68)

where [M1], [C1] and [K1] are the mass, the damping and the stiffness matrices respectively. A different approach is required for its solution. 2.6.1 General viscous damping case For general viscous damping, instead of having purely imaginary system poles and real mode shapes, complex modes and system poles will be obtained which appear in complex conjugate pairs. The kept damped system poles, λ1,kd, and mode shapes, {Ψ1,kd}, comply with the following equation: λ12,kd [M 1 ]{Ψ1,kd }+ λ1,kd [C1 ]{Ψ1,kd }+ [K1 ]{Ψ1,kd } = 0

for kd=1…Nk (2.69)

The state space form needs to be used to solve the eigenvalue problem. This state space form is obtained by adding a dummy equation to the system of motion equations:

{}

jω [M 1 ]{δ1} = [M 1 ] δ1

(2.70)

Combining this with the equation of motion (2.69) gives the following state space form:

28

Chapter 2


  C1  jω   M   1

M 1   K1 0   δ1   F1  +    =   0   0 − M 1   δ1   0 

(2.71)

The system poles and modes shapes comply with the following eigenvalue equation: 0   Ψ1,kd   K1  C1    = − λ1,kd   0 − M 1  λ1,kd Ψ1,kd   M 1

M 1   Ψ1,kd    0  λ1,kd Ψ1,kd 

(2.72)

Pre- and post-multiplying the state space equation with the vector composed by the displacement mode shapes and the velocity mode shapes (in complex conjugate pairs) yields eq. 2.71 in the modal coordinates {q1} notation and the diagonal modal [a1] and modal [b1] matrices:

[Ψ1,kd ]  δ1   {q1}  = δ1  diag (λ1,kd )⋅ [Ψ1,kd ]

(2.73)

( jω [diag (a1 )] + [diag (b1 )]){q1} = [Ψ1,kd ]T {F1} (2.74)

It is important to stress that in case of complex mode shapes, the orthogonality of the mode shapes does not hold with respect to the mass, damping and stiffness matrices, but to the modal [a1] and [b1] matrices. For a state space CMS all equations must be formulated in state-space form. This means that the new transformation matrices introduced are compatible with this state-space form. This approach doubles the size of all matrices involved and increases the complexity of the transformation matrices. [Hermans99] indicates that this method does not really work for experimental models, although it is

29

Chapter 2


theoretically the best one for generally damped systems. Unstable poles or poles with very high damping might be often found. 2.6.2 Proportional damping in case of modal coupling Damping modelling in finite element models is very difficult. Often, the user starts with the undamped case, computes the normal modes and then adds modal damping to the system poles. This makes the system poles complex, λ=σ+jω, but the mode shapes remain real. The [M], [K] and [C] matrices are such that when pre and post multiplying them with the real mode shapes, diagonal matrices are obtained. This gives the following equation in the modal domain:

(− ω [m]+ jω [c]+ [k ]){q} = [Ψ ] {F } T

2

(2.75)

with m1  [m] =   

 − 2σ 1    , [c ] = [m]    mk 

σ 12 + ω12     , [k ] = [m]    − 2σ k  

    σ k2 + ω k2 

(2.76)

The eigenvectors can be normalised towards the mass modal matrix, which then becomes unity matrix. Note that proportionally damped systems correspond to systems with diagonal modal damping matrix. The [C] matrix is then a linear combination of the mass and stiffness matrix: [C ] = α [M ] + β [K ] (2.77)

Typically, these proportionality factors are unknown. A least-squares estimate can however be found from the identified modal mass, modal

30

Chapter 2


damping and modal stiffness matrices. For each mode, the following equation should be approximately satisfied: crr = α ⋅ mrr + β ⋅ k rr

for r=1… Nk (2.78)

Using these factors, the damping can be incorporated in the mass and stiffness diagonal matrices, where no coupling exists between the generalised dofs. This has the advantage that no complex eigenvalue solver is needed (calculation time, robustness), so that proportionally damped systems are attractive from the point of view of eigenvalue solving. The modeshapes will be real and equal to those of the undamped system, while the poles will become complex. The complex poles of this new system can be related to those of the undamped case by the following relationship: λ2r + λrα = −ω r2 , 1 + λr β

with

ω r2 =

kr mr

(2.79)

This method, however, tends to smear out the damping over the frequency range, if the assumption of proportional damping is not really fulfilled. 2.7 CMS approaches Unfortunately, there is no combination of the aforementioned sets of component modes which satisfies simultaneously all the criteria proposed by Craig. Different CMS methods, each based on a different selection of component mode sets, appear in literature. A brief overview of the approaches applied in this research will be presented here, without any intention of being exhaustive.

31

Chapter 2


2.7.1 Unreduced models or physical models An “unreduced” physical model can be considered as a special case of a CMS model. The transformation matrix equals the unity matrix, so there is no reduction at all. Considering it as a special case of CMS allows the researchers to deal consistently with such models in combination with e.g. modal models. A global CMS scheme can be implemented, allowing the coupling of physical models with modal models, [Craig-Bampton68] models, [Craigh-Chang76], etc. 2.7.2 Modal synthesis The transformation matrix is formed by the normal freeinterface modes of the structure. Applying this transformation matrix to the mass and stiffness matrix results in diagonal modal mass and modal stiffness matrices for each component. In case of general viscous damped systems, the product of the damping matrix with the normal modes might lead to a fully populated damping matrix. In case of modal damping, it will be a diagonal matrix. By applying the compatibility of the displacements at the interface dofs the coupling is performed. For rigid coupling, dependent and independent coordinates need to be selected and the transformation matrix between the dependent and independent coordinates is estimated. For flexible coupling, the connector is treated as a modification of the “uncoupled” assembly. So, the connector is transformed into the modal domain and added to the “uncoupled” assembly in modal coordinates. Continuity of the interface forces does not impose additional constraints. Note also that in case of rigid coupling, the number of generalised coordinates of the assembly is equal to the sum of the modes of both components, minus the number of interface dofs.

32

Chapter 2


The generalised coordinates actually correspond to the modal coordinates or the participation factors. The response of the coupled system can be expressed as a sum of contributions of the uncoupled modes. In many cases, however, the method seriously suffers from modal truncation errors. This can be significantly improved by including residual modes. This method is well suited for test/test coupling or hybrid coupling, as the modal mass and modal stiffness can be derived from the measured driving point frequency response function (FRF) measurements. Mode shape information in the coupling points is needed to impose the compatibility. 2.7.3 Craig-Chang CMS The transformation matrix uses the dynamic residual attachment-mode superset ([Craigh-Chang76], [Craigh-Chang77]): it is then composed of the free-interface modes (including rigid body modes) and the elastic residual attachment modes:  x1i   Φ1ik  =  x1b  Φ1bk

Ψ1id  q1k    Ψ1bd   f1b 

(2.80)

with 0 =  I bb 

[Ψd ] = [Φd ][Λ dd ]−1[Φd ]T [ fb ] = [Φ d ][Λ dd ]−1[Φ d ]T 

n P ×n B

= [Φ d ][Λ dd ]

−1

 Φid    Φbd 

T

Nd 0 Φ pr Φbr T −1   = [Φ d ][Λ dd ] [Φ bd ] = ω r2  I bb  r = k +1

(2.81)

∑

where {qk} stands for the modal coordinates of the kept normal modes, while the generalised coordinates subset {f1b} can be considered an approximation of the interface forces.

33

Chapter 2


An explanation might be found in what follows. In case all modes were available, the modal synthesis method would use the following transformation matrix: qk  Φ h ]  qh 

{x} = [Φ k

(2.82)

where h stands for the higher order modes that are normally unknown and, consequently, truncated. Applying this transformation to the equation of motion results in the following partitioned matrix equation:

( )

diag ω k2 − Iω 2   0 

 qk  ΦTk     =  {F } diag ω h2 − Iω 2  qh  ΦTh  0

( )

(2.83)

Let us assume that there are no externally applied forces. The component is just excited by the interface forces, {Fint}, due to coupling with another component. Let us then concentrate on the lower part of this matrix equation:

[diag (ω )− Iω ]{q } = [Φ ] {F 2 h

2

h

h

T

int

} (2.84)

Assuming that the frequencies of the out-of-range modes are much higher than the upper frequency of interest allows one to write that:

[diag (ω )]{q } ≅ [Φ ] {F 2 h

h

h

T

int

} (2.85)

With this assumption the out-of-range modes respond in a quasi-static manner and the inertia term is ignored. Substituting this in equation 2.82 results in:

34

Chapter 2




 0    qk     I bb    f1b 

{x} ≅ [Φ k ]{qk }+ [Φ h ][diag (ω h2 )][Φ h ]T {Fint } = Φ k [Φ h ][diag (ω h2 )][Φ h ]T  

(2.86)

This precisely corresponds to the transformation matrix of the CraigChang method. It can be seen that a subset of the generalised coordinates approximately corresponds to the interface forces (if no external loads are applied). This will have an important impact when writing down the continuity and the compatibility equations at the interface dofs. Besides compatibility of displacements, continuity of the interface forces needs to be imposed. Applying this transformation to the equation of motion of uncoupled component 1 gives:  m  − ω 2  1kk   0 

0  k1kk + m1bb   0

0   q1k  T     = [T1 ] {F1}  k1bb    f1b 

(2.87)

where [m1kk] and [k1kk] are the diagonal modal mass and stiffness matrices, [m1bb] and [k1bb] are the residual mass and stiffness matrices modelling the out-of-range effects. The terms in the above matrices can be derived as follows. If a general matrix [A] npxnp has to be transformed by the [T1] npxn(p+k) matrix, there results: 

[T1 ]T [A][T1 ] =  [0 

ΦTk

[ ]

]

  ΦTk [A]Φ k  =  2 −1 T [0 I bb ]Φ h ω h Φ h [A]Φ k 

[ ]

  [A]Φ k 

[ ]

Φ h ω h2

−1 I bb Φ h ω h2 ΦTh 

−1

 0  ΦTh    =  I bb  

     0  2 −1 T 2 −1 T [0 I bb ]Φ h ω h Φ h [A]Φ h ω h Φ h    I bb  

[ ]

ΦTk [A]Φ h ω h2

[ ]

35

−1

0 ΦTh    I bb 

[ ]

(2.88)

Chapter 2


The off-diagonal elements are equal to zero, since the free-interface kept normal modes are orthogonal to the higher modes. When the [A] matrix equals the mass matrix [M1] it follows that: Φ k [M 1 ]ΦTk = [I ]

[ ]

−1

Φ h ω h2

[ ]

ΦTh [M 1 ]Φ h ω h2

−1

(2.89)

[ ] [I ][ω ]

ΦTh = Φ h ω h2

−1

2 −1 T Φh h

[ ]

= Φ h ω h2

−2

(2.90)

ΦTh

and when the [A] matrix equals the stiffness matrix [K1] the result is:

[ ]

Φ k [K1 ]ΦTk = ω k2

[ ]

Φ h ω h2

−1

[ ]

ΦTh [K1 ]Φ h ω h2

−1

(2.91)

[ ] [ω ][ω ]

ΦTh = Φ h ω h2

−1

2 h

2 −1 T Φh h

[ ]

= Φ h ω h2

−1

(2.92)

ΦTh

Finally, it has been shown that the residual matrices are given by: [m1bb ] = [0

[ ]

I b ][Φ h ]ω h2

−2

0   I bb 

[Φ h ]T 

and [k1bb ] = [0

[ ]

I bb ][Φ h ]ω h2

−1

0   I bb 

[Φ h ]T 

(2.93)

Note that these residual mass and stiffness matrices are not coupled with the modal mass and stiffness matrices. Therefore, the modes related to these residual matrices, referred to as residual modes, are orthogonal to the free-interface normal modes. These residual modes can thus be used as extra modes in the modal synthesis calculations. Normally, the out-of-range modes are not known and equation 2.93 cannot be evaluated. However, [k1bb] is exactly equal to the residual flexibility matrix: [k1bb ] = [Ψdb ] (2.94)

and

36

Chapter 2


[m1bb ] = [Ψdb ]T [M 1 ][Ψdb ] (2.95)

The latter requires the knowledge of the mass matrix while the calculation of the residual flexibility is calculated as the inversion of the stiffness matrix (with rigid-body filtering in case of rigid-body motion) minus the contribution of the kept modes at 0 Hz. In case experimental FRF data or directly calculated FE FRF data are available for the active set of dofs (including the interface dofs), good estimates can also be found for the residual matrices from the static and dynamic compensation terms (see refernces [DuarteEwins96/1], [Duarte-Ewins96/2] and [Hermans&al.00]). The main reason is that the residual matrices actually correspond to the first 2 terms of a MacLaurin expansion of the residual FRFs:

[H

res bb

k T ( jω )] = [H bbdir ( jω )]− ∑ {Φ br2}{Φ br2} r =1

ωr − ω

N

=

∑

r = k +1

{Φbr }{Φ br }T ≅ [k ] + [m ]ω 2 + …ω 4 + … 1bb 1bb 2 2 ωr − ω

(2.96) The residual matrices can then be determined from the static and dynamic compensation terms, which can be estimated when the direct FRFs are known at least for two frequencies (see § 4.3.4):  H ibres ( jω )  H ibdir ( jω )  H ibsyn ( jω )  Ribstat ( jω )  Ribdyn ( jω ) ≅ ω 2  = − + syn dyn res dir stat     H bb      ( jω ) H bb ( jω ) H bb ( jω )  Rbb ( jω )  Rbb ( jω ) 

(2.97)

The static and dynamic compensation matrices are then good estimates for the residual matrices: [kbb ] ≅ [Rbbstat ] and [mbb ] ≅ [Rbbdyn ] (2.98)

37

Chapter 2


When coupling the Craig-Chang CMS models of component 1 with component 2, assuming that both components are rigidly coupled on their boundaries, the compatibility condition follows: {x1b}={x2b} (2.99) as well as the continuity condition needs to be formulated: {f1b}=-{f2b} (2.100) Contrary to the other CMS methods, the Craig-Chang method permits not only interface displacement compatibility constraints to be enforced, but also other constraints to be imposed, i.e. continuity of interface forces. Taking these two conditions for rigid coupling and the expressions of the displacements at the interfaces into account results in the following system of equations,  {x1b } = Φ1bk {q1k }+ Ψ1bd { f1b }  {x2b } = Φ 2bk {q2 k }+ Ψ2bd { f 2b } ⇒{ f1b } = [Ψ1bd + Ψ2bd ]−1[− Φ1bk  {x1b } = {x2b }    { f1b } = −{ f 2b }

 q1k  Φ 2bk ]  q2 k 

the transformation matrix on assembly level becomes: I 0  q1k         f1b  − kˆ [Φ1bk ] kˆ [Φ 2bk ]   q1k   q1k     = [Tas ]   =  q2 k  q2 k  I 0 q2 k        f 2b   kˆ [Φ1bk ] − kˆ [Φ 2bk ]

[]

[]

[]

[]

with

38

(2.102)

(2.101)

Chapter 2


[kˆ ] = [Ψ

1bd

+ Ψ2bd ]−1 = ([k1bb ] + [k 2bb ])−1

(2.103)

The transformation matrix between the generalised coordinates of the assembly and the physical coordinates of the components is given by:  x1i     x1b  T1 0   q1k  [Φ1k ] − [Ψ1d ] kˆ [Φ1bk ] [ ] = T     = as  q2 k   [Ψ2 d ] kˆ [Φ1bk ]  x2i   0 T2     x2b 

[]

[]

[Ψ1d ][kˆ ][Φ 2bk ]   q1k     (2.104) [Φ 2k ] − [Ψ2d ][kˆ ][Φ 2bk ] q2k 

The assembly equation in generalised coordinates then becomes:   I + [Φ1bk ]T [m ˆ ][Φ1bk ] − [Φ1bk ]T [m ˆ ][Φ 2bk ]    2 + − ω    − [Φ ]T [m T T ˆ ][Φ1bk ] I + [Φ 2bk ]T [m ˆ ][Φ 2bk ]  q1k  2 bk   T 1  [ ] = T   as   T ˆ 2  − [Φ1bk ]T kˆ [Φ 2bk ]  q2 k    ω1r + [Φ1bk ] k [Φ1bk ]    +   T T   − [Φ ] kˆ [Φ ] ω 22r + [Φ 2bk ] kˆ [Φ 2bk ]  2 bk 1bk  

[ ]

[] []

[ ]

[] []

 F1i      F1b    T2T   F2i     F2b 

(2.105)

with [mˆ] = [kˆ ]([m1bb ] + [m2bb ])[kˆ ] (2.106)

This holds for rigid coupling. Actually, what can be seen is that the residual flexibility compensation terms cause the rigid coupling to be modelled as a flexible coupling of the two components connected with a connector given by the compensation mass and stiffness matrices

[kˆ ]

and [mˆ] . This also implies that modal synthesis, without compensation for the truncated modes, will overstiffen the structure (resonance frequencies will be higher than they should be). In case of flexible coupling, the equilibrium condition of the interface forces results as follows:

39

Chapter 2


{F1b } = −{F2b } = [ω 2 M con + K con ]({x1b }− {x2b }) (2.107)

The new formulations required are very similar to those of rigid coupling. The compensation matrices must be substituted with:

[kˆ ]= ([k

1bb

] + [k2bb ] + [K con ]−1 )

−1

(2.108)

[mˆ] = [kˆ ]([m1bb ] + [m2bb ] + [M con ])[kˆ ] (2.109)

40

Chapter 2


Chapter 2 ...........................................................................................................................................7 Time-Domain Component Mode Synthesis Methods .......................................................................7 Introduction...................................................................................................................................7 2.1 Definition of Components.......................................................................................................8 2.2 Real Component Modes ..........................................................................................................9 2.2.1 Normal modes ....................................................................................................................11 2.2.2 Rigid-body modes ..............................................................................................................12 2.2.3 Constraint modes................................................................................................................13 2.2.4 Attachment modes..............................................................................................................14 2.2.5 Inertia-relief modes ............................................................................................................16 2.2.6 Residual flexibility matrix: residual attachment modes .....................................................19 2.3 Statically complete mode sets ...............................................................................................20 2.4 Dynamic component mode supersets ....................................................................................22 2.5 Coupling of CMS models......................................................................................................23 2.6 Dealing with damping ...........................................................................................................28 2.6.1 General viscous damping case ...........................................................................................28 2.6.2 Proportional damping in case of modal coupling...............................................................30 2.7.1 Unreduced models or physical models...............................................................................32 2.7.2 Modal synthesis..................................................................................................................32 2.7.3 Craig-Chang CMS..............................................................................................................33

41

Chapter 3 Dynamic behaviour of multibody mechanisms Introduction The ever-increasing speed of the machines asks for a design approach that deals with the dynamics of component mechanisms. The performance definition of broad displacement mechanisms requires the evaluation of the kinematic domain and dynamic parameters such as the overall compliance and inertia reduced to the motion of one member of the whole mechanism. Usually these characteristics are studied in static conditions for some relevant kinematic configurations. The disposition of each member and the working conditions have a relevant importance in the dynamic behaviour of spatial mechanisms, since the diversity of local vibration modes is function of the kinematic configuration. Commercial multibody software deals with non-linear motion analysis in the time domain, but it does not investigate the modal behaviour of mechanisms. Their simulations ask often for relevant time and computational resources, and are influenced by the boundary conditions imposed on the numerical calculations. For high flexibility mechanisms, having a high number of degrees of freedom (dofs), the characterisation of elastodynamic overall properties can be approached by the study of the modal behaviour of the equivalent mechanism, resulting from the

Chapter 3

Dynamic behaviour of multibody mechanisms

linearisation about a kinematic admissible configuration. It follows that the modal behaviour of the machine (frequencies and shapes of the modes) varies in the working space. It might be useful to have a methodology to help the mechanism designer in the analysis of the elastodynamic behaviour as function of spatial configurations. In this chapter a numeric approach to the problem outlined is studied and developed. The equations of the constrained motion of a multibody system, investigated by means of classical mechanics, are linearised in every kinematically admissible geometrical configuration. This brings to the definition of the eigenproblem of the equivalent mechanism linearised about a particular kinematic configuration. Once these linear differential equations are obtained, they can be eventually coupled to the modal models of other components to form the eigenproblem of an assembly. 3.1.1 Equations of motion The formulation of the equations of motion here adopted follows Lagrange’s equations for constrained systems, which implies the usage of Lagrange multipliers λ to add the m constraint functions Φ ( q, q , t ) = 0 to the system. L is the Lagrangian function and Q is the

vector of the generalised forces. The equation of motion for each of the n generalised coordinate qi is: d ∂L ∂L m − + ∑ λk Φ qk ,i = Qi dt ∂ qi ∂ qi k =1

(3.1)

where Φq is an m x n matrix whose components can be defined as Φ qk ,i =

∂Φ k ∂qi

. To solve the motion of the full system, n equations like (3.1)

42

Chapter 3


are required. In addition, it is necessary to impose m constraint equations like: Φ ( q, q , t ) = 0

(3.2)

The Lagrangian function L can be defined, assuming M as mass matrix, K as stiffness matrix and C as damping matrix, as the difference between the kinetic and the potential energy, L=T-U, where: T=

1 n n mi , j qi q j = 12 q T Mq ∑∑ 2 i =1 j =1

U elast =

1 n n ki , j qi q j = 12 qT Kq ∑∑ 2 i =1 j =1

(3.3)

(3.4)

The dissipating force generalised potential can be written, according to Raileigh, as: U dis =

1 n n ci , j qi q j = 12 q T Cq ∑∑ 2 i =1 j =1

(3.5)

and the generalised dissipating force becomes: Qidis = −

n ∂ U dis = − ∑ Ci , j q j ∂ qi j =1

(3.6)

The equation (3.1-2) of the motion of the system becomes, according to the assumption described:

43

Chapter 3


m ∂U ∂ U d is  d ∂T + + + ∑ λ k Φ q k ,i = Q i e st  ∂ qi ∂ q i k =1  d t ∂ q i  Φ ( q , q , t ) = 0 

(3.7)

and in matrix form: T  M q + C q + K q + Φ q λ = Q est  Φ ( q , q , t ) = 0 

(3.8)

3.1.2 An alternative method: the condensed stiffness matrix Instead of solving the differential algebraic system of m+n equations (3.8), it is possible to reduce the system to n differential equations containing the information of the constraints, under certain conditions. This gives a more physical meaning to the dofs of the system, especially with a large amount of rigid members. Equation (3.2) can be simplified if the constraint is holomic thus becoming: (3.9)

Φ ( q, t ) = 0

Time-deriving (3.9), the m equations of velocity (3.10) and of acceleration (3.11) are obtained: d Φ dt

(q , t ) =

Φ = Φ q q + Φ

t

= 0

d2 = Φ q + ( Φ q ) q + Φ q + Φ q + Φ = 0 Φ (q, t ) = Φ q q qt tq tt q dt 2

where:

44

(3.10)

(3.11)

Chapter 3


 ( Φ q q )q =  ∂∂q  i

 n ∂Φ k    n ∂ 2 Φ k  q j   =  ∑ q j   ∑   j =1 ∂ q j    j =1 ∂ qi∂ q j  m×n

(3.12)

 ∂ 2Φ k  Φ qt = Φ tq =    ∂ qi∂ t  m×n

Φ

tt

 ∂ 2Φ k  =   2  ∂ t  m ×1

(3.13)

(3.14)

Pre-multiplying equation (3.8) by the inverse of the mass matrix the acceleration expression for the constrained system can be obtained: q = M −1 ( Qest − Cq − Kq ) − M −1Φ Tq λ

(3.15)

By substituting (3.15) in equation (3.11), the expression of Lagrange multipliers results: Φ q  M −1 (Qest − Cq − Kq ) − M −1Φ Tq λ  + ( Φ q q )q q + 2Φ qt q + Φ tt = 0

(3.16)

Φ q M −1Φ Tq λ = Φ q M −1 ( Qest − Cq − Kq ) + ( Φ q q ) q q + 2Φ qt q + Φ tt

(3.17)

λ = ( Φ q M −1Φ Tq )

−1

(Φ M q

−1

(Qest − Cq − Kq ) + ( Φ q q )q q + 2Φ qt q + Φ tt )

(3.18)

By rewriting equations (3.8) with (3.18) it is possible to obtain a system of n differential equations containing the same information of the constrained system as the differential algebraic equations of (3.8) plus (3.9). The condensed system can be written as:

45

Chapter 3


Mq + Cq + Kq +

(3.19)

−1 +Φ Tq ( Φ q M −1Φ Tq ) Φ q M −1 (Qest − Cq − Kq ) + ( Φ q q )q q + 2Φ qt q + Φ tt  = Qest  

In the study of natural frequencies the equations (3.19) of the constrained system can be simplified noting that the constraints can be assumed as time-invariant and that there are no external driving motions, so that Φqt=0 and Φtt=0: Mq + Kq + Φ Tq ( Φ q M −1Φ Tq )  −Φ q M −1Kq  = 0

(3.20)

−1 Mq +  K − Φ Tq ( Φ q M −1Φ Tq ) Φ q M −1K  q = 0  

(3.21)

−1

Introducing the stiffness matrix of the condensed system as:

(

)

K constrained = I − Φ Tq ( Φ q M −1Φ Tq ) Φ q M −1 K −1

(3.22)

the eigenproblem of the constrained system is finally derived:

(K

constrained

− ω 2M ) q = 0

(3.23)

3.2 Generalised coordinates and position analysis In this approach the general coordinates in the threedimensional space of each rigid member of the mechanism are described by six degrees of freedom (dofs). The chosen dofs are the three displacements of the centroid from the global origin, and the three successive rotations that give the Euler angles of the local principal axes system referred to the global inertial frame. On this

46

Chapter 3


assumption, the Lagrangian function formulation becomes very simple, even in three-dimensional space. It is now necessary to define the position of each point of the components in the global inertia reference. The location of a point Pi, belonging to the i-th body and having s’i as constant vector location in the body fixed reference (if the point has no motion relative to its body), can be expressed in the global inertia reference as: Pi = xi + Ai ⋅ s 'i

(3.24)

where xi is the position vector of the mass centre and Ai is the rotation matrix. The direction ai’ of a general axis in the i-th local reference becomes in the global reference a direction vector ai, written as: ai = Ai ⋅ a 'i

(3.25)

where the apex ’ indicates that the quantities are written referring to the body fixed frame. The expression of the rotation matrix A is here written as the product of three rotation matrices, each expressing the rotation of the body reference along one local axis. Different combinations of these matrices may be employed, but it is very important to keep always the same chosen order in the product, because the final orientation is nonlinearly affected by it. Every rotation takes place starting from the previous orientation of the local reference. The rotation matrix A here chosen can be formulated as: A (φ , β ,ψ ) = Rot (ψ ) ⋅ Rot ( β ) ⋅ Rot (φ )

where:

47

(3.26)

Chapter 3


0 0  1   Rot (φ ) = 0 Cos (φ ) − Sin (φ )  0 Sin (φ ) Cos (φ ) 

(3.27)

 Cos ( β ) 0 Sin ( β )    Rot ( β ) =  0 1 0   − Sin ( β ) 0 Cos ( β ) 

(3.28)

Cos (ψ ) − Sin (ψ ) 0   Rot (ψ ) =  Sin (ψ ) Cos (ψ ) 0  0 0 1

(3.29)

3.3 Kinematic constraint equations

Figure 3.1 Spatial references

According to [Haug89] notation, the constraint equations can be formulated. Consider the pair of rigid bodies, denoted i-th and j-th body, as in Fig. 3.1. Reference points Pi and Pj and nonzero vectors ai and aj are fixed in bodies i and j, respectively. Kinematic constraints between pairs of bodies are often characterised by conditions of

48

Chapter 3


orthogonality or parallelism of pairs of such vectors. The purpose here is to derive analytical conditions with which to define a library of kinematic connections. 3.3.1 Basic constraints Four main relations can be pointed out as the skeleton of all constraint equations: ORTHO1, ORTHO2, BALL and DISTANCE. Combining them, the equations of several joints are easily obtained. 3.3.1.1 ORTHO1 constraint First, a necessary and sufficient condition for a pair of bodyfixed nonzero vectors ai and aj on bodies i and j, respectively, to be orthogonal is that their scalar product be zero; that is, ΦORTHO1 ( ai , a j ) = aiT ⋅ a j = a 'Ti ⋅ AiT ⋅ A j ⋅ a ' j = 0

(3.30)

This condition is called ORTHO1 constraint between vectors ai and aj. Thus ORTHO1 relation constrains the relative orientation of a pair of bodies, providing one constraint equation for the whole assembly. 3.3.1.2 ORTHO2 constraint The scalar product condition can also be used to prescribe orthogonality of a body fixed vector ai and a vector dij between bodies. dij is the vector that connects two points, the first on the i-th body and the second on the j-th one. It is important to point out that the orthogonality condition breaks down if dij =0. This condition is called ORTHO2 constraint between vectors ai and dij. Analytically, this scalar product becomes:

49

Chapter 3


d ij = Pi − Pj = xi + Ai ⋅ s 'i − x j − Aj ⋅ s ' j

(3.31)

ΦORTHO 2 ( ai , dij ) = aiT ⋅ d ij = a 'Ti ⋅ AiT ⋅ ( xi + Ai ⋅ s 'i − x j − A j ⋅ s ' j ) = 0 (3.32)

Thus ORTHO2 relation constrains the relative orientation of a body and a joining direction with another, once more providing one constraint equation for the whole assembly. 3.3.1.3 BALL constraint It is often required that a pair of points on two bodies coincides. A necessary and sufficient condition for point Pi and Pj to coincide is that dij = 0; this condition means that a BALL joint, also called spherical or ball and socket joint, is introduced between the bodies: Φ BALL ( Pi , Pj ) = xi + Ai ⋅ s 'i − x j − A j ⋅ s ' j = 0

(3.33)

So the BALL constraint implies three scalar equations that constrain three relative displacements. 3.3.1.4 DISTANCE constraint It is often required that the distance between a pair of points on adjacent bodies be fixed by a physical connection that may be thought of as a rod link with a ball joint at each tip. A necessary and sufficient condition that the distance between point Pi and Pj be equal to C ≠ 0 is simply the following scalar condition:

50

Chapter 3


(

)

Φ DISTANCE Pi ,Pj ,C = dijT ⋅ dij − C 2 = =

(

xiT

+ s'iT

AiT

−

xTj

−

s' Tj

ATj

)( x

i

)

(3.34) 2

+ Ai s'i − x j − A j s' j − C = 0

So the DISTANCE constraint implies only one constraint equation for the whole assembly. 3.3.2 Parallelism constraints

Figure 3.2: PARALELL1 constraints

Using ORTHO1 and ORTHO2 from the basic constraints, it is now possible to formulate two parallelism conditions. Consider the pair of bodies in Fig. 3.2, with joint definition frames located at point Pi and Pj on bodies i and j, respectively. Let the z”i and z”j axes be required to be parallel; that is, vectors hi and hj are to be parallel. The vector hj is parallel to hi if and only if it is orthogonal to fi and gi. Thus, employing two ORTHO1 constraints and obtaining two scalar equations, the condition that hi and hj are parallel is simply formulated:

51

Chapter 3


Φ

PARALLEL1

 ΦORTHO1 ( f i , h j )   0  =  ( hi , h j ) =  ORTHO1  0 g , h Φ ( ) i j    

(3.35)

This is called PARALLEL1 constraint.

Figure 3.3: PARALELL2 constraints

Finally, consider the condition that the vector along the axis on body i be parallel to the vector dij as in Fig. 3.3. Since dij ≠ 0 is parallel to hi if and only if it is perpendicular to fi and gi, the second parallelism condition, called PARALLEL2 constraint, is given when employing two ORTHO2 constraints and obtaining two scalar equations:

Φ

PARALLEL 2

 ΦORTHO 2 ( f i , d ij )   0   =   with dij ≠ 0 ( hi , dij ) =  ORTHO 2 g , d Φ ( i ij )  0  

(3.36)

3.3.3 Absolute constraints on a body Absolute constraints may be placed on the position of point Pi on the i-th body, and on the orientation of the body-fixed reference frame for body i, leading to six constraint equations on individual generalised coordinates of body i. While any constraint on the

52

Chapter 3


translational dofs is physically meaningful, the conditions on the orientation should be taken together to give the desired constraint, as explained about the fixed order used in the calculus of the rotation matrix. 3.3.4 Constraints between pairs of bodies To get the relevant constraint equations of the variety of spatial joints between pairs of bodies, most employed in the construction of mechanisms and machines, it is now sufficient to combine the basic and parallelism constraints outlined in the previous paragraphs. The BALL and DISTANCE joints have already been formulated. 3.3.4.1 HOOKE joint

Figure 3.4: HOOKE joint

A HOOKE joint between bodies i and j, shown in Fig. 3.4, is constructed with an intermediate body, or cross, that is pivoted in bodies i and j. The centre of the cross of the universal joint is fixed in bodies i and j, defined by points Pi and Pj on the respective bodies.

53

Chapter 3


Points Qi and Qj on the arms of the cross, between bodies i and j, respectively, are specified to determine the z” axes of the joint definition frames (while the others are defined at discretion), and so the unit vectors hi and hj are on the direction that connects the point Pi to Qi, Pj to Qj, respectively. The constraint equations that characterise a HOOKE joint are given by the coincidence of points Pi and Pj and by the orthogonality between hi and hj. These conditions are specified by the following constraint equation:

Φ HOOKE

  0    Φ BALL Pi ,Pj    0    =   = ΦORTHO1 h ,h    0   i j      0   

(

(

)

)

(3.37)

These four scalar equations restrict the relative position of the bodies and the rotation about the shaft of the cross. They allow two relative dofs between the bodies. 3.3.4.2 TURNING joint A TURNING joint (or revolute joint) between bodies i and j is constructed with a bearing that allows relative rotation about a common axis, but precludes relative translation along this axis, as shown in Fig. 3.5. To define the TURNING joint, the centre of the joint is located on bodies i and j by points Pi and Pj. The axis of the relative rotation is defined in bodies i and j by points Qi and Qj, and hence unit vectors hi and hj along the respective z” axes of the joint definition frames. The remaining joint definition frame axes are defined at convenience.

54

Chapter 3


The analytical formulation of the TURNING joint is obtained by the coincidence of points Pi and Pj and by the parallelism between hi and hj. These conditions are specified by the following constraint equation:

ΦTURNING

0   0  Φ BALL ( Pi , Pj )       =   0   = PARALLEL1 Φ ( hi , h j )   0    0  

(3.38)

These five scalar equations yield only one relative dof that is the rotation about the common axis of the bearing.

Figure 3.5: TURNING joint

3.3.4.3 CYLINDRICAL joint A CYLINDRICAL joint is similar to a TURNING joint in that it allows relative rotation about a common axis in two bodies. However, it does not preclude the relative translation along this axis, as shown in

55

Chapter 3


Fig. 3.6. Joint definition points Pi and Pj are located on the common axis of rotation and additional points Qi and Qj on each body are defined along the axis of relative motion, to establish the z” unit vectors hi and hj on bodies i and j, respectively. The remaining joint definition frame axes are defined at convenience.

Figure 3.6: CYLINDRICAL joint

The analytical definition of the CYLINDRICAL joint is that vectors hi and hj are co-linear. Since these vectors have points in common with dij, co-linearity is granted by the conditions that hi is parallel to both hj and dij, if dij ≠ 0. This yields: 0 Φ hi , h j )    0   (  = = PARALLEL 2   0 Φ ( hi , dij )       0    PARALLEL1

ΦCYLINDRICAL

(3.39)

Note that even when dij = 0, Pi and Pj coincide and vectors hi and hj still have a point in common. Thus, the first parallelism condition

56

Chapter 3


implies co-linearity, so eq. 3.39 brings in the right kinematic constraint of the joint. The CYLINDRICAL joint consists of four constraint equations, hence it allows two relative dofs between the connected bodies, relative rotation about the vector hi = hj and translation along this axis. 3.3.4.4 TRANSLATIONAL joint

Figure 3.7: TRANSLATIONAL joint

The TRANSLATIONAL joint, shown in Fig. 3.7, allows relative translation along a common axis between a pair of bodies, but precludes the relative rotation about this axis. It is defined, as in the case of the CYLINDRICAL joint, by joint definition points Pi, Qi, Pj and Qj along the axis of translation. The x” axes of the joint definition frames on bodies i and j are selected so that they are perpendicular, defined by vectors fi and fj. The analytical definition of a TRANSLATIONAL joint may be written using the equations 3.39 of the CYLINDRICAL joint and adding the ORTHO1 condition that vectors fi and fj are orthogonal. It becomes:

57

Chapter 3


TRANSLATIONAL

Φ

 0     0     Φ PARALLEL1 hi ,h j      0 = Φ PARALLEL 2 hi ,dij  =        0  ΦORTHO1 f , f      i j      0       

(

( (

)

) )

(3.40)

Since this joint comprises five constraint equations, only one relative translational dof exists between the bodies. 3.3.4.5 SCREW joint

Figure 3.8: SCREW joint

The SCREW joint, shown in Fig. 3.8, is a CYLINDRICAL joint between bodies i and j, with the additional condition that relative translation along the common axis of rotation is specified by a screw pitch α times the relative angle of rotation between the bodies. The relative angle θ of rotation is defined as the angle between the body

58

Chapter 3


fixed x”i and x”j axes, counterclockwise taken as positive, including the cumulative angle of rotation. This advance condition, in vector terms, becomes

(

)

(

)

Φ PITCH hi ,dij ,α ,θ 0 = ΦORTHO 2 hi ,dij − α (θ + 2π n − θ 0 ) = 0

(3.41)

where θ0 is the angle between the body fixed x” axes when Pi = Pj and n is the cumulative number of revolutions. This brings the following five scalar constraint equations for the SCREW joint:

Φ SCREW

 0     0     Φ PARALLEL1 hi ,h j      0 =  Φ PARALLEL 2 hi ,dij  =     0   Φ PITCH h ,h ,α ,θ      0  i j      0       

(

( (

) ) )

(3.42)

In order to determine the value of the relative angle θ of rotation consider what follows. From the definition of scalar product and the fact that the coordinate vectors shown in Fig. 3.8 are unit vectors, there comes: f iT

f j = Cos (θ )

(3.43)

Similarly, the definition of vector product yields: f i ∧ f j = hi Sin (θ )

(3.44)

Taking the scalar product of both sides of the previous equation with hi, there results:

59

(3.38) Dynamic behaviour of multibody mechanisms

Chapter 3

Sin (θ ) = hiT f i ∧ f j = g iT f j

as

f i ∧ hi = − g i

(3.45)

Writing the unit vectors in terms of the respective reference frames in which they are fixed, equations 3.43 and 3.45 become:

(

)

(3.46)

(

)

(3.47)

Cos (θ ) = f 'iT AiT A j f ' j = ΦORTHO1 f i , f j

Sin (θ ) = g'iT AiT A j f ' j = ΦORTHO1 gi , f j

If Sin(θ) and Cos(θ) are known, the value of θ, 0 ≤ θ < 2π, may be uniquely determined by taking the arcsine of both sides of eq. 3.41 and using the algebraic sign of eq. 3.40 to uniquely evaluate θ. Then, with −

π 2

≤ Arc sin(Sin (θ )) ≤

π

(3.48)

2

θ becomes:  Arc sin(Sin(θ ))    π − Arc sin(Sin (θ ))   θ =  π − Arc sin(Sin (θ ))   2π + Arc sin(Sin(θ )) 

if

Sin (θ ) ≥ 0 and

Cos(θ ) ≥ 0

if

Sin (θ ) ≥ 0 and

Cos(θ ) < 0

if

Sin (θ ) < 0 and

Cos(θ ) < 0

if

Sin (θ ) < 0 and

Cos(θ ) ≥ 0

(3.49)

3.4 Linearization of the equations In the study of the eigenproblem (natural frequencies and modeshapes), related to the mechanism disposition, the differential equations of motion must be linearised near each kinematically

60

Chapter 3


admissible geometrical configuration under study, in particular the expression of M, K, Φq or Kcondensed must be constant. The linearisation approach followed here consists in substituting the exact formulation with the first order term from the Taylor series expansion around the initial kinematically admissible geometrical configuration under study. This latter position must be the exact location of the mechanism and will become the reference for each local vibration mode. The motion around this layout, which results from the mode superposition, will be then studied as linear, since the displacements are not broad compared to the other mechanism movements in the working area. 3.4.1 Linearisation of the position analysis In order to achieve this formulation, first of all, it is necessary to have a linear expression of the motion of each member. As stated before, this motion can be described through the six dofs of the body fixed system reference; the translations are already three linear terms; the same does not hold for the rotation by the Euler angles, whose composition is highly non linear. The expression (see eq. 3.26) of the rotation matrix A is highly non linear because it is the product of three matrices containing trigonometric functions of the three dofs, which express the rotations of the body reference altogether. Keeping the first order term from the Taylor series expansion of the exact formulation of a the rotation matrix, there derives: Rot (γ ) Rot (γ 0 ) +

61

∂Rot (γ 0 ) (γ − γ 0 ) ∂γ

(3.50)

Chapter 3


Following the matrix order in the product of the transformations, and discarding the higher order products, it is now possible to approximate the expression of A linearly: A (φ , β ,ψ ) ( Rot (ψ 0 ) + Rot ' (ψ 0 ) δψ ) ⋅ ( Rot ( β 0 ) + Rot ' ( β 0 ) δβ ) ⋅ ( Rot (φ0 ) + Rot ' (φ0 ) δφ ) (3.51) lin

A (φ , β ,ψ ) Rot (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot (φ0 ) + + δψ Rot ' (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot (φ0 ) + + δβ Rot (ψ 0 ) ⋅ Rot ' ( β 0 ) ⋅ Rot (φ0 ) +

(3.52)

+ δφ Rot (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot ' (φ0 )

Calling: A0 (φ0 , β 0 ,ψ 0 ) = Rot (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot (φ0 ) Aφ (φ0 , β 0 ,ψ 0 ) = Rot (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot ' (φ0 ) Aβ (φ0 , β 0 ,ψ 0 ) = Rot (ψ 0 ) ⋅ Rot ' ( β 0 ) ⋅ Rot (φ0 )

(3.53)

Aψ (φ0 , β 0 ,ψ 0 ) = Rot ' (ψ 0 ) ⋅ Rot ( β 0 ) ⋅ Rot (φ0 )

A is obtained as a linear function of the variations of the rotations from the initial angular position of the body: lin

A (φ , β ,ψ ) A0 + δφ Aφ + δβ Aβ + δψ Aψ

(3.54)

Again it must be pointed out that the variations of the member orientation are evaluated starting from its initial position, and not from the global reference. In fact the linearised expression of A is composed by the initial rotation matrix A0 (calculated with the nonlinear expression) and the variations around the resulting orientation.

62

Chapter 3


Now it is also possible to write an expression of the position of point Pi as a linear function of the six variations δqi of the dofs of the body fixed reference:  s'   xi   ix  Pi  yi  + ( Ai 0 + δφi Aiφ + δβ i Ai β + δψ i Aiψ ) ⋅  s 'i y    z   i  s 'i z  lin

(3.55)

and extracting the constant term, given by the initial position, it follows:  s'   x   s'   δ xi  ix i0    ix    Pi  δ yi  + (δφi Aiφ + δβ i Ai β + δψ i Aiψ ) ⋅  s 'i y  +  yi0  + Ai 0 ⋅  s 'i y  (3.56)    z  δ z   s 'i   i  s 'iz   i0   z lin

1 0 0   lin Pi 0 1 0   0 0 1

 s'   ix  Aiφ ⋅  s 'i y     s 'i z 

 s'   ix  Ai β ⋅  s 'i y   s 'i   z

  δ xi      s '    δ yi   i x    δ zi  Aiψ ⋅  s 'i y   ⋅   + Pi 0 = Ti ( qi 0 , s 'i ) ⋅ δ qi + Pi 0     δφi   s 'iz    δβ i  (3.57)   δψ   i

3.4.2 Linearization of the Lagrangian function The linear expression of the elastic potential energy can be obtained, for the linear spring between point Pi and Pj, as the variation of the distance between the end caps of the coil along the initial direction:

63

Chapter 3

U elastij


∧

∧

T T 1 = k ( Pi − Pj ) ⋅ Pi 0 − Pj 0 ⋅ Pi 0 − Pj 0 ⋅ ( Pi − Pj ) 2 (3.58) T 1 1 T T  i  = kδ qij ⋅   ⋅ D ⋅ Ti −T j  ⋅ δ qij + const = δ qij ⋅ Kij ⋅ δ qij + const 2 2  −T j 

(

)(

)

where:

(P D=

i0

− Pj 0

Pi 0 − Pj 0

) ⋅ (P

i0

− Pj 0

)

Pi 0 − Pj 0

T

 Ti T DTi , K ij = k  T  −T j DTi

−Ti T DT j   δ qi  and = δ q  δ q  ij T jT DT j   j (3.59)

The expression of the kinetic energy for the body i can be written as: Ekini =

1 1 miδ xiT ⋅ δ xi + ω 'Ti ⋅ J 'i ⋅ ω 'i 2 2

(3.60)

where xi is the position of the centroid, m the mass of the body, J the matrix of the inertia moments, and the apex ’ indicates that the quantities are written with reference to the body fixed frame in the initial orientation. Thus, in this linear approximation, it is possible to calculate the angular velocity ω’i as:  δφi    ω 'i = Ai 0 ⋅  δβi   δψ i   

and the kinetic energy becomes:

64

(3.61)

Chapter 3

Ekini


 mi 0  1 T 0 = δ qi ⋅  2 0 0   0

0 mi 0 0

0 0 mi 0

0 0

0 0

0 0 0

0 0 0 T

Ai 0 J 'i Ai 0

0 0  0 1 T  ⋅ δ qi = δ qi ⋅ M i ⋅ δ qi 2   (3.62)  

With the process highlighted here it is now possible to write the Lagrangian function as a linear expression in δqi, because M and K are now constant matrices. 3.4.3 Linearization of the constraint equations Writing the constraint equations can become simple if a library of basic constraint functions is used, as for the nonlinear approach. In order to linearise the joint equations it is sufficient to give the approximation of ORTHO1, ORTHO2, BALL and DISTANCE constraints. The scalar equation of ORTHO1 constraint, accordingly to eq. 3.30 and 3.54, yields to: ΦORTHO1 ( ai , a j ) = aiT ⋅ a j = a 'Ti ⋅ ( Ai T0 + δφi AiφT + δβ i Ai Tβ + δψ i AiψT ) ⋅ lin

⋅ ( A j 0 + δφ j A jφ + δβ j A j β + δψ j A jψ ) ⋅ a ' j =

 a 'Ti ⋅ AiφT ⋅ A j 0 ⋅ a ' j   T T   a 'i ⋅ Ai β ⋅ A j 0 ⋅ a ' j  T T lin  a ' ⋅ A ⋅ A ⋅ a '  j  Ti iψT j 0   a 'i ⋅ Ai 0 ⋅ A jφ ⋅ a ' j   a 'Ti ⋅ Ai T0 ⋅ A j β ⋅ a ' j   T T   a 'i ⋅ Ai 0 ⋅ A jψ ⋅ a ' j 

65

T

 δφi   δβ   i  δψ i  ⋅  + const = 0 δφ j    δβ j    δψ j  

(3.63)

Chapter 3


The scalar equation of ORTHO2 constraint, accordingly to eq. 3.32, 3.54 and 3.57, yields to: ΦORTHO 2 ( ai , dij ) = aiT ⋅ d ij =  = a 'Ti ⋅ ( Ai T0 + δφi AiφT + δβ i Ai Tβ + δψ i AiψT ) ⋅  Ti  

lin

 a 'Ti ⋅ AiφT ⋅ ( Pi 0 − Pj 0 )   lin   a 'Ti ⋅ Ai Tβ ⋅ ( Pi 0 − Pj 0 )     a 'Ti ⋅ AiψT ⋅ ( Pi 0 − Pj 0 )   

  δ qi  −T j   + Pi 0 − Pj 0  =   δ qj  

T

 δφi  ⋅  δβ i  + a 'Ti ⋅ Ai T0 ⋅ Ti  δψ   i

(3.64)

 δ qi  −T j    + const = 0 δ q j 

To linearise a BALL constraint the approximation of eq. 3.57 can be usefully remembered. BALL can then be written as: Φ BALL ( Pi , Pj ) = Pi − Pj Ti lin

 qi  −T j    + const = 0  qj 

(3.65)

The linear expression of DISTANCE constraint, keeping in mind eq. 3.57, becomes:

(

)

Φ DISTANCE Pi ,Pj ,C = dijT ⋅ dij − C 2 =   TT  T  i  T T   ≅ δ qij + Pi 0 − Pj 0 Ti   −T jT      

lin 

lin

(

(

)

≅ 2 PiT0 − PjT0 Ti

)

−T j  δ qij + Pi 0 − Pj 0 − C 2 =

(3.66)

−T j  δ qij − C 2 = 0

Before expressing the linear approximation of the SCREW joint constraint, it is helpful to expand eq. 3.46-47 in Taylor series, keeping only the linear parts: Sin(θ ) ≅ Sin(θ 0 ) + Cos(θ 0 )(θ − θ 0 )

(3.67)

Cos (θ ) ≅ Cos (θ 0 ) − Sin(θ 0 )(θ − θ 0 )

(3.68)

66

Chapter 3


That yields, according to eq. 3.46, to:

(θ − θ0

ORTHO1 ( gi , f j ) − Sin (θ0 )) Sin (θ ) − Sin (θ 0 ) ) ( Φ ( = )≅ lin

Cos (θ 0 )

Cos (θ 0 )

(3.69)

The linear approximation of the last SCREW joint equation becomes:

Φ

PITCH

lin

( hi ,dij ,α ,θ0 ) ≅ Φ

ORTHO 2

( hi ,dij )

(

(

)

)

  ΦORTHO1 g , f − Sin (θ ) i j 0  −α + 2π n  = 0  Cos (θ 0 )  (3.70)  

The other constraints follow according to what aforementioned. With this approximation it is now possible to write Φq easily as constants, because the constraint equations are linear expressions of the variations of the generalised coordinates of the system components. This in fact means that, when derived, the constraint equations give only constant terms to be included in the eigensystem and that the constant parts are meaningless. 3.5 Examples Some examples are depicted to prove the effectiveness of the approach. Very simple mechanisms are analysed to illustrate the constraint equations, the equations of motion and the modeshapes. 3.5.1 Crankshaft mechanism The crankshaft mechanism of figure 3.9 is here suggested proving that the linearisation approach of the constraint equations is coherent with the linear behaviour of the mechanism around the initial location.

67

Chapter 3


Figure 3.9: Crankshaft Mechanism

Let us consider the mechanism modelled by a straight member i linked to the origin of the ground axes system by a TURNING joint. For a more general explanation, let us assume the ground as unconstrained body j. The member i is initially rotated around the zaxis by an angle of π/4 radians. Its centroid is positioned at unitary distance from the joint axis. The initial location of the i-th centroid can then be expressed by q0i vector of eq. 3.71. Cos (π / 4 )     Sin (π / 4 )      0   q0i =   0       0    π / 4 

(3.71)

With this initial location, the point Pi on the crankshaft that corresponds to the joint location is expressed by the vector s’i:  −1   s'i =  0     0 

68

(3.72)

Chapter 3


while the joint is located on the origin (Pj) of the ground axes system by the vector s’j: 0    s' j = 0    0 

(3.73)

The constraint equations of eq. 3.38 can be derived as follows. For all of them it is necessary to evaluate the rotation matrix as linear function of the rotations of each body, as in eqs. 3.74 and 3.75: i j j j j Alin HiL = j j j j j k

1 è 2 1 è 2

− +

δψi è 2 δψi è 2

−

1 è 2 1 è 2

−δβi

− −

δψi è 2 δψi è 2

δβi è 2 δβi è 2

δφi

+ −

δφi è 2 δφi è 2

1

y z z z z z z z z z { (3.74)

1 −δψj δβj y i j z j z Alin HjL = j δψ j 1 −δφ j z j z 1 { (3.75) k −δβj δφj

The linearised location of points Pi and Pj in the global reference become: i i δxi + δψ è y j z 2 j z j z δψi z j z Plin HiL = j δ yi − j è z j 2 z j z j z k δzi + δβi { (3.76)

δxj y i j z z Plin HjL = j δ yj j z j z k δzj { (3.77)

69

Chapter 3


and their difference dij is thus:

dij

lin

(

= Pi − Pj

) lin

δψ i    δ xi − δ x j + 2     δψ i  = δ yi − δ y j −  2     δ zi − δ z j + δβ i   

(3.78)

Imposing the ball joint constraints of eq. 3.33 consists in equating eq. 3.78 to zero. The parallelism equations 3.35 can be derived as follows. It is necessary to express the f’i and g’i unit vectors orthogonal to the joint axis (along which is taken h’i) for the rigid body i. Let us take them as in eq. 3.79, where they are expressed in the local reference of body i: 1    f 'i =  0    0 

0    g'i = 1    0 

0    h'i = 0    1 

(3.79)

The unit vector h’j to which h’i must be parallel is in the z-axis direction of the ground reference system. Equations 3.80-82 express fi, gi, hj in the global reference as linear function of the generalised displacements of the rigid body i and ground j.

fi

lin ,global

    =    

1 δψ i  − 2 2  1 δψ i  +  (3.80) 2 2  −δβ i  

70

Chapter 3


gi

lin,global

hj

 1 δψ i  − 2 − 2     1 δψ  = − i  (3.81) 2 2     −δφi    

lin ,global

 δβ j    =  −δφ j  (3.82)    1 

The parallelism equations 3.35 can be expressed now as the ORTHO1 products between the vectors of eq. 3.80 with eq. 3.82 and of eq. 3.81 with eq. 3.82 respectively. The following scalar equations are derived:  1 δψ i ORTHO1 f i ,h j = −δβ i + δβ j  − 2  2

  1 δψ i +  − δφ j  2  2 

 =0 

 1 δψ i ORTHO1 gi ,h j = δφi + δβ j  − − 2 2 

  1 δψ i −  − δφ j  2  2 

  = 0 (3.84) 

(

(

)

)

(3.83)

Discarding the contributions of order higher than the first, the five equations give rise to the linear system of the TURNING joint constraints:

ΦTURNING linearised

δψ i    δ xi − δ x j + 2     δψ i  0   δ yi − δ y j −  2  0       =  δ zi − δ z j + δβ i  = 0  (3.85)     δβ j δφ j  0    −δβ i + 2 − 2      0   δβ j δφ j   δφi −  − 2 2  

71

Chapter 3


As stated before, the body j is the ground, so that its generalised displacements are forced to null values. With these newly added constraints, eq. 3.85 is reduced to:

ΦTURNING linearised

δψ i  δψ i    δ xi + 2  0   δ xi = − 2         δψ i  0   δψ i  δ yi −    δ yi = −δ xi =   2    2 = = →  0    (3.86)  δ zi + δβ i      δ zi = 0   0     δβ      δβi = 0 i   0     δφ    δφi = 0 i    

that forces the centroid of the member i to move along the direction orthogonal to the line that connects it to the joint axis, and that restrains the displacement along the vertical axis, as well as the rotations around the x-axis and y-axis. There results only one dof, that is the rotation of body i around the z-axis. This is the proof that the linearisation approach respect the TURNING constraint in the local area of the initial location. If the CYLINDRICAL joint is employed instead of the TURNING joint, the first three equations of eq. 3.85 must be substituted by two ORTHO2 constraints expressing the PARALLEL2 constraint of eq. 3.36. Making the scalar product between the vectors of eq. 3.78 with eq. 3.82 and of eq. 3.81 with eq. 3.82, it follows:

(

)

δ xi

δ xj

δ yi

δ yj

− δ ziδβ i + δ z jδβ i − δβ i2 + 2 2 δ x δψ δ x jδψ i δ yiδψ i δ y jδψ i − i i + + − − δψ i2 = 0 2 2 2 2

ORTHO 2 fi ,dij =

(

)

ORTHO 2 g i ,dij = − −δψ i −

δ xiδψ i 2

+

2

δ xi 2

δ x jδψ i 2

−

+ −

2

δ xj 2

+

+

δ yi

δ yiδψ i 2

−

2 +

−

δ yj 2

δ y jδψ i 2

72

+ δ ziδφi − δ z jδφi + δβ jδφi =0

(3.87)

(3.88)

Chapter 3


Discarding the contributions of order higher than the first, the four equations give rise to the linear system of the CYLINDRICAL joint:

ΦCYLINDRICAL linearised

  δ xi δ x j δ yi δ y j − + −   2 2 2 2     0   − δ xi + δ x j + δ yi − δ y j − δψ    i  2 2 2 2   0  =  =   (3.89) δβ j δφ j   0  −δβ i + −     2 2   0    δβ δφ   δφi − j − j   2 2

Considering that the body j is the ground, eq. 3.89 becomes:

ΦCYLINDRICAL linearised

δ xi δ yi δψ     δ xi = −δ yi = − i  +    0  2 2 2       δx δy  0    δψ i i       δ yi = i =  − 2 + 2 − δψ i  =   →   (3.90) 2   0          δβ i δβ i = 0   0        0 δφ δφ = i i    

Eq. 3.90 constrains all the displacements and rotations of body i except for the displacement along the z-axis and along the direction orthogonal to the line that connects the centroid to the joint axis. This displacement is function of the rotation about the same axis. This is another proof of the effectiveness of the linearisation approach proposed. Let us study now the dynamic behaviour of this simple mechanism around its initial location. Let a vertical spring of stiffness k1 connect the centroid of the member i to the ground. If the centroid is positioned at distance r from the axis of revolution of the member i,

73

Chapter 3


with the linearised displacements assumption, the location of the centroid becomes:  rCos (ϕ 0 ) − rSin (ϕ 0 )(ϕ − ϕ 0 )    Gi =  rSin (ϕ 0 ) + rCos (ϕ 0 )(ϕ − ϕ 0 )  (3.91)   0  

The Lagrangian function (constant terms are omitted) assumes for this system the following expression: L = T− U= − 1 2

1 2

k1 Hr Sin@ϕ0D + r Cos@ϕ0D H−ϕ0 + ϕ@tDLL2 +

m1 Hr2 Cos@ϕ0D2 ϕ @tD2 + r2 Sin@ϕ0D2 ϕ @tD2L

1 2

J1 ϕ@tD2 +

(3.92)

Thus the motion equation becomes, discarding the terms of order higher than the first:

(J

1

) (

)

+ m1r 2 ϕ + k1r 2Cos 2 (ϕ 0 ) ϕ = 0 (3.93)

and the natural frequency of the crankshaft system has the following form: 1 2π

k1r 2Cos 2 (ϕ 0 ) J1 + m1r 2

(3.94)

The frequency of the crankshaft mechanism assumes the value 2.9057584 Hz, when providing the next values for the constants in eq. 3.94: k1 = 10^4 N/m, r = 1 m, φ0 = π/4, J1 = 5 kgm2, m1 = 10 kg. From eq. 3.94 it comes that the frequency depends on the initial position as in fig. 3.10. The software developed, which uses both the approaches

74

Chapter 3


of eqs. 3.8 and 3.23, calculates the frequency value as 2.905759 Hz, proving its effectiveness. The modeshape of the crankshaft in the vibration around the initial position is depicted in fig. 3.11. Freq @Hz D

Natural

frequency

4 3 2 1

1

2

3

4

5

6

ϕ@rad D

Figure 3.10: Natural frequency as function of crankshaft angle

Figure 3.11: Modeshape of the Crankshaft Mechanism

3.5.2 Reciprocating mechanism The reciprocating mechanism of fig. 3.12 is a test case where to investigate the effectiveness of the linearisation approach and of the software developed for the spatial mechanisms. Modelling this spatial system as a planar system can provide a theoretical benchmark. The

75

Chapter 3


system has only one degree of freedom: let us assume the motion as function of the crankshaft angle φ.

Figure 3.12: Reciprocating Mechanism with springs

As in the previous paragraph, the Lagrangian approach can be pursued. According to the reference axes in fig. 3.12 and to the constraints, the linearised displacements and rotations of the rigid bodies can be expressed as follows: x1lin = l + r − rg Cos@ϕ0D + rg Sin@ϕ0D H−ϕ0 + ϕ@tDL (3.95) y1lin = rg Sin@ϕ0D + rg Cos@ϕ0D H−ϕ0 + ϕ@tDL (3.96) γlin = π − ArcSinA

r Sin@ϕ0D l

E−

r Cos@ϕ0D H−ϕ0 + ϕ@tDL

x2lin = l + r − r Cos@ϕ0D − a $ 1 −

i

r Sin@ϕ0D +

k

y2lin =

a

r2

l2 $ 1 −

+

r2 Sin@ϕ0D2 l2

r2 Sin@ϕ0D2

Cos@ϕ0D Sin@ϕ0D

H− a + lL r Sin@ϕ0D l

l $1 −

r2 Sin@ϕ0D2 l2

y

l2

(3.97)

+

H−ϕ0 + ϕ@tDL

{

(3.98)

H− a + lL r Cos@ϕ0D H−ϕ0 + ϕ@tDL l

76

(3.99)

Chapter 3


r2 Sin@ϕ0D2

x3lin = l + r − r Cos@ϕ0D − l $ 1 −

i

r Sin@ϕ0D +

r2 Cos@ϕ0D Sin@ϕ0D l $1 −

k

r2 Sin@ϕ0D2 l2

y

l2

+

H−ϕ0 + ϕ@tDL

{

(3.100)

The kinetic energy of each body assumes the expression of eqs. 3.101-103, while the potential elastic energy of the two springs is formualted by eq. 3.104-105. T1lin =

T2lin =

2

2

J1 ϕ @tD2 +

1 2

m1 Hrg2 Cos@ϕ0D2 ϕ@tD2 + rg2 Sin@ϕ0D2 ϕ @tD2L

J2 r2 Cos@ϕ0D2 ϕ@tD2 2 l2 J1 −

i 1

1

m2

r2 Sin@ϕ0D2 l2

N

+

H− a + lL2 r2 Cos@ϕ0D2 ϕ@tD2 l2

k

(3.101)

i

+ r Sin@ϕ0D +

k

a r2 Cos@ϕ0D Sin@ϕ0D l2 $ 1 −

r2 Sin@ϕ0D2 l2

y2

{

(3.102) i

T3lin =

U1lin =

1 2

1 2

m3 r Sin@ϕ0D +

k


r2 Sin@ϕ0D2 l2

y2

ϕ@tD2

{

k1 Hrg Sin@ϕ0D + rg Cos@ϕ0D H−ϕ0 + ϕ@tDLL2

77

(3.103)

(3.104)

y

ϕ @tD2

{

Chapter 3


i

1

U3lin =

2

k3 l + r − r Cos@ϕ0D − l $ 1 −

i

k

r Sin@ϕ0D +

r2

Cos@ϕ0D Sin@ϕ0D

l $1 −

k

r2 Sin@ϕ0D2 l2

y

r2 Sin@ϕ0D2

+

l2

y

H−ϕ0 + ϕ@tDL ^

{

{

(3.105)

2

1

Llin = −

2

k1 Hrg Sin@ϕ0D + rg Cos@ϕ0D H−ϕ0 + ϕ@tDLL2 −

i

1 2

k3 l + r − r Cos@ϕ0D − l $ 1 −

r Sin@ϕ0D +

k

2

J1 ϕ @tD2 +

2

1 2

1 2

r2

Cos@ϕ0D Sin@ϕ0D

l $1 −

r2 Sin@ϕ0D2 l2

J2 r2 Cos@ϕ0D2 ϕ@tD2 2 l2 J1 −

i

1

l2

k

i

1

r2 Sin@ϕ0D2

m3 r Sin@ϕ0D +

k

r2 Sin@ϕ0D2 l2

N

y

y

H−ϕ0 + ϕ@tDL ^2 + {

{ +


+

r2 Sin@ϕ0D2 l2

y2

ϕ@tD2 +

{

m1 Hrg2 Cos@ϕ0D2 ϕ @tD2 + rg2 Sin@ϕ0D2 ϕ @tD2L +

i m2

i

H− a + lL2 r2 Cos@ϕ0D2 ϕ@tD2

k

r Sin@ϕ0D +

k

l2

+

a r Cos@ϕ0D Sin@ϕ0D

y2

2

l2 $ 1 −

r2 Sin@ϕ0D2 l2

78

{

y

ϕ @tD2

{

(3.106)

Chapter 3


0 = k1 rg Cos@ϕ0D Hrg Sin@ϕ0D + rg Cos@ϕ0D H−ϕ0 + ϕ@tDLL +

i

k3 r Sin@ϕ0D +

i

k


r2 Sin@ϕ0D2 l2

l + r − r Cos@ϕ0D − l $ 1 −

k

i

r Sin@ϕ0D +

k J1 ϕ @tD +

i

1 2

1 2

{

r2 Sin@ϕ0D2 l2

Cos@ϕ0D Sin@ϕ0D

l $1 −

r2 Sin@ϕ0D2 l2

J2 r2 Cos@ϕ0D2 ϕ@tD l2 J1 −

m3 r Sin@ϕ0D +

k

r2

y

r2 Sin@ϕ0D2 l2

N

y

H−ϕ0 + ϕ@tDL +

{

{

+


y

+

r2 Sin@ϕ0D2 l2

y2

ϕ @tD +

{

m1 H2 rg2 Cos@ϕ0D2 ϕ @tD + 2 rg2 Sin@ϕ0D2 ϕ@tDL +

i m2

2 H− a + lL2 r2 Cos@ϕ0D2 ϕ@tD l2

k

i

2 r Sin@ϕ0D +

k

a

r2

+

Cos@ϕ0D Sin@ϕ0D

l2 $ 1 −

r2 Sin@ϕ0D2 l2

M global − linϕ + K global − linϕ = 0

y2

y

ϕ @tD =

{

{

(3.107)

(3.108)

The Lagrangian function of the whole system can be thus assembled in eq. 3.106. From it it can be obtained the motion equation eq. 3.107 of the reciprocating mechanism with the springs. Discarding the terms

79

Chapter 3


of order higher than the first, it is possible to write the system as in eq. 3.108, where the global stiffness is expressed as in eq. 3.109 and the global inertia as in eq 3.110. Thus the natural frequency of the system can be evaluated as in eq. 3.111. i

Klin = k1 rg2 Cos@ϕ0D2 + k3 r2 Sin@ϕ0D2 1 +

k Mlin =

J1 + m1 rg2 +

i m2

J2 r2 Cos@ϕ0D2

l2 − r2 Sin@ϕ0D2

Ha − lL2 r2 Cos@ϕ0D2 l2

k

Freqlin =

i (

k

ii

r Cos@ϕ0D

l $1 −

i

r2 Sin@ϕ0D2 l2

l $1 −

k

r2 Sin@ϕ0D2 l2

a r2 Cos@ϕ0D Sin@ϕ0D

+ r Sin@ϕ0D +

l2 $ 1 −

k

{

r Cos@ϕ0D

+ m3 r2 Sin@ϕ0D2 1 +

i

y2

r2 Sin@ϕ0D2 l2

y2y

(3.109) y2 +

{

{ {

(3.110)

1 2π

i

k1 rg2 Cos@ϕ0D2 + k3 r2 Sin@ϕ0D2 1 +

kk

k

m1 rg2 +

i m2

k

J2 r2 Cos@ϕ0D2

l2 − r2 Sin@ϕ0D2

Ha − lL2 r2 Cos@ϕ0D2 l2

r Cos@ϕ0D

l $1 −

r2 Sin@ϕ0D2 l2

i

+ m3 r2 Sin@ϕ0D2 1 +

k

i

+ r Sin@ϕ0D +

k

(3.111)

80

y2y

i

ì J1 +

{ {

k

r Cos@ϕ0D

l $1 −

r2 Sin@ϕ0D2 l2

a r2 Cos@ϕ0D Sin@ϕ0D l2 $ 1 −

y2

r2 Sin@ϕ0D2 l2

+

{

y2yyyy

{ {{{{

Chapter 3


Freq @Hz D

Natural

frequency

2.4 2.2

1

2

3

4

5

6

ϕ@rad D

1.8


Freq @Hz D 16

Natural

frequency

14 12 10 8 6 4 1

2

3

4

5

6

ϕ@rad D


The frequency of the reciprocating mechanism assumes the value 2.381602527 Hz, when providing the next values for the constants in eq. 3.111: k1 = 10^4 N/m, k3 = 10^3 N/m, l = 2 m, r = 1 m, rg = 0.3 m, a = 0.7 m, φ0 = π/4, J1 = 5 kgm2, J2 = 1 kgm2, m1 = 0.5 kg, m2 = 0.8 kg, m3 = 0.4 kg. To model the translational joint between body 3 and ground, it is sufficient to impose a CYLINDRICAL constraint between the two members having joint axis along the translational direction. The other two TURNING constraints do not allow the rotation of the mechanism around the x-azis. The software developed, which uses both the approaches of eqs. 3.8 and 3.23, calculates the frequency value as 2.381604 Hz, proving again its effectiveness. From eq. 3.111 it comes that the frequency depends on the initial position as

81

Chapter 3


in fig. 3.13; introducing k3 = 5.0e+4 N/m the natural frequency behaves like in fig. 3.14. The modeshape of the reciprocating mechanism in the vibration around the initial position is depicted in fig. 3.15.

Figure 3.15: Modeshape of the Reciprocating Mechanism

3.5.3 Serial Mechanism

Figure 3.16: Serial Mechanism

Let us now analyse the dynamic behaviour of the serial mechanism of fig. 3.16. It is composed by 4 rigid members in series linked by 4 TURNING joints, the first member being connected to the ground. The i-th body is long li, has the centroid at distance rgi from the joint that connects it to the previous member. Four springs are introduced

82

Chapter 3


between the ground (in vertical direction) and the centroids of the adjacent bodies. The following values have been introduced as the input for the software developed to evaluate the modal behaviour of the whole mechanism: l1 = 1 m, l2 = 2 m, l3 = 3 m, l4 = 4 m, rg1 = 0.3 m, rg2 = 0.6 m, rg3 = 0.9 m, rg4 = 1.2 m, φ10 = π/4, φ20 = π/6, φ30 = π/9, φ40 = π/9, m1 = 10 kg, J1 = 10 kgm2, m2 = 20 kg, J2 = 20 kgm2, m3 = 30 kg, J3 = 30 kgm2, m4 = 40 kg, J4 = 40 kgm2, k10 = 10^4 N/m, k12 = 10^4 N/m, k23 = 10^4 N/m, k34 = 10^3 N/m. The four eigenmodes of the system are shown in figs. 3.17-20. The first natural frequency is calcluated at 4.055 Hz, the second at 21.981 Hz, the third at 138.871 Hz and the fourth at 158.799 Hz.

Figure 3.17: First modeshape of the Serial Mechanism

Figure 3.18: Second modeshape of the Serial Mechanism

83

Chapter 3


Figure 3.19: Third modeshape of the Serial Mechanism

Figure 3.20: Fourth modeshape of the Serial Mechanism

84

Chapter 3


Chapter 3 .........................................................................................................................................41 Dynamic behaviour of multibody mechanisms ...............................................................................41 Introduction.................................................................................................................................41 3.1.1 Equations of motion ...........................................................................................................42 3.2 Generalised coordinates and position analysis ......................................................................46 3.3 Kinematic constraint equations .............................................................................................48 3.3.1 Basic constraints.................................................................................................................49 3.3.1.1 ORTHO1 constraint.........................................................................................................49 3.3.1.2 ORTHO2 constraint.........................................................................................................49 3.3.1.3 BALL constraint...............................................................................................................50 3.3.1.4 DISTANCE constraint .....................................................................................................50 3.3.2 Parallelism constraints........................................................................................................51 3.3.3 Absolute constraints on a body ..........................................................................................52 3.3.4 Constraints between pairs of bodies ...................................................................................53 3.3.4.1 HOOKE joint ..................................................................................................................53 3.3.4.2 TURNING joint ..............................................................................................................54 3.3.4.3 CYLINDRICAL joint .....................................................................................................55 3.3.4.4 TRANSLATIONAL joint ...............................................................................................57 3.3.4.5 SCREW joint...................................................................................................................58 3.4 Linearization of the equations ...............................................................................................60 3.4.1 Linearisation of the position analysis .................................................................................61 3.4.2 Linearization of the Lagrangian function ...........................................................................63 3.4.3 Linearization of the constraint equations ...........................................................................65 3.5 Examples .........................................................................................................................67 3.5.1 Crankshaft mechanism ................................................................................................67 3.5.2 Reciprocating mechanism ...........................................................................................75 3.5.3 Serial Mechanism........................................................................................................82

85

Chapter 4 Frequency-Domain Component Mode Synthesis Methods: FRF Based Substructuring Introduction Frequency-domain Component Mode Synthesis (CMS) methods are based primarily on the use of frequency response functions (FRFs) either experimental or after conversion to modal form, i.e., after curve fitting and synthesis of the FRFs from modal parameters. The aim of frequency-domain CMS is usually to determine frequency response functions of a coupled system. But the same concepts are extensively used for a task that is generally referred to as Structural Modification. Many attempts at applying frequency-domain CMS techniques failed in the past because of the demands placed on FRFs accuracy. The reasons of failure were the inconsistencies of measured data (noise, frequency shifts), the inability to measure correctly the responses in all degrees of freedom (dofs) and a matrix inversion that could become ill-conditioned (for a large number of dofs and in the

Chapter 4

Frequency-Domain CMS Methods: FRF Based Substructuring

neighbourhood of the natural frequencies of the coupled structure). Careful studies have been developed to determine the accuracy and the types of data required (i.e. rotational coupling FRFs) in order to make frequency-domain CMS usable for vibration analysis of connected structures. With the availability of better data-acquisition hardware (multichannel data acquisition systems) and techniques (multiple input estimation of FRFs), and with the need to couple components having dense spectra, there has been a return to the use of experimental FRFs for frequency-domain CMS. The FBS technique has been widely discussed and is here briefly outlined from ([Leuridan&al.88], [Otte&al.90], [Cuppens&al.00], [Otte94], [Wyckaert&al.96], [Wyckaert&al.97/1], [Wyckaert&al.97/2]). It predicts the dynamic behaviour of a coupled system on the basis of free-interface FRFs of the uncoupled components and coupling stiffness (frequency dependent flexible joints). Making a comparison between time-domain and frequency-domain CMS methods, a duality between the Craig-Chang method of § 2.7.3 and the FRF Based Substructuring approach could be pointed out. The basic theory is here reproduced, as well as some techniques used to improve the results of the coupling methodology and consistency of the measured data. Both experimental and analytical approaches to the FRFs acquisition are also discussed. Some of the most relevant advantages of FBS technique are the good accuracy in wide frequency ranges, the computational time and the easy applicability to hybrid modelling, while noise on data is a source of failure of the methodology. 4.1 Basic theory The problem of coupling two structures is visualised in figure 4.1. The general formulation for the FRFs matrix [HC] given in § 4.1 is

86

Chapter 4


reported from ([Leuridan&al.88], [Otte&al.90], [Cuppens&al.00], [Otte94], [Wyckaert&al.96], [Wyckaert&al.97/1], [Wyckaert&al.97/2]) and it comes from the compatibility conditions for the displacements and the equilibrium conditions for the forces in the interface dofs. [K] is the stiffness matrix of the connections between A and B. The FRFs matrix [HC] is partitioned according to input dofs (ia, ib), interface dofs (ca, cb) and output dofs (oa, ob).

Figure 4.1: coupling process

[ [H ] [ [ [H ]  = [H ]  C

A oaia A caia

 

] [H ] [H ] ] [H ] [H ] = ] [H ] [H ]  [H ]  [H ] 0   [H ]    [H ] 0  −  [H ] [ H ]+ [H ]+ [K ] ]  [H ]    [H ] − [H ] 0 − [H ]

C  H oaia  C =  H caia   HC  obia

0

C oaca

C oaib

C caca

C caib

C obca

C obib

A oaca

A oaca

A caca

A caca

B obib

B obcb

A iaca

A caca

B cbcb

−1 −1

T

A caca

B ibcb

(4.1)

The response of the coupled structure in the output dofs of A, respectively B, due to the excitation on input dofs on B, respectively on A, is obtained from eq. 4.1 as eq. 4.2 or 4.3:

87

Chapter 4


{X } = [H ]{F }= [H ]⋅ ([H ]+ [H ]+ [K ] ) ⋅ [H ]⋅ {F }

(4.2)

{X } = [H ]{F }= [H ]⋅ ([H ]+ [H ]+ [K ] ) ⋅ [H ]⋅ {F }

(4.3)

C oa

C ob

ib

ia

C oaib

C obia

B ib

A oaca

A ia

A caca

B obcb

−1 −1

B cbcb

A caca

B cbib

−1 −1

B cbcb

A caia

B ib

A ia

If the query is reduced to a single scalar output, this formulation easily gives the response of the coupled structure excited on one of the subparts. As an example, the acoustical response p in subsystem A due to an operational force FB at input location j on B can be written as the multiplication of the vibro-acoustical transfer function vector H pA− ca  with the operational force:



 ([

][

]

A B p = H pA− ca ⋅ H caca + H cbcb + [K ]−1

) ⋅ {H }⋅ F −1

B cb − j

(4.4)

B j

where H pA− ca  is the row corresponding to p in the FRFs matrix between the responses on A and all the interface dofs ca, and the column corresponding to j in the FRFs matrix

A H oaca

{H } is

B H cbib between

B cb − j

all the

interface dofs cb and the inputs on B. 4.1.1 Force transmissibility This formulation allows calculating force transmissibility characteristic between possible input dofs and interface dofs. The resulting functions can be used to recalculate operational forces at the interface connections due to modifications at subsystem level. These force transmissibilities can then be combined with vibro-acoustical FRFs in order to estimate the acoustical response functions due to the coupling of A and B.

88

Chapter 4


Starting from the formulation of eq. 4.1, it is possible to write the FRFs matrix of the coupled structure C with inputs and outputs on the same subpart A, respectively on B, as in eq. 4.5 and 4.6. This easily yields to the formulation of eq. 4.7-10, where the force transmissibility matrix between the inputs and all the interface dofs is obtained.

[H ] = [H ]− [H ]⋅ ([H ]+ [H ]+ [K ] ) ⋅ [H ]

(4.5)

[H ] = [H ]− [H ]⋅ ([H ]+ [H ]+ [K ] ) ⋅ [H ]

(4.6)

[H ]= [H ]+ [H ][R ]

(4.7)

[H ]= [H ]+ [H ][R ]

(4.8)

C oaia

A oaia

C obib

A oaca

B obib

B obcb

C oaia

A caca

A caca

A oaia

C obib

A caia

−1 −1

B cbcb

A oaca

B obib

−1 −1

B cbcb

B cbib

C caia

B obcb

C cbib

[R ] = −[R ] = −([H ]+ [H ]+ [K ] ) [H ]

(4.9)

[R ] = −[R ] = −([H ]+ [H ]+ [K ] ) [H ]

(4.10)

C caia

C cbia

C cbib

A caca

C caib

−1 −1

B cbcb

A caca

−1 −1

B cbcb

A caia

B cbib

From this point of view, the force transmissibility between a single input j on system B and the interface connections cb can be written as eq 4.13:

{R }= −([H ]+ [H ]+ [K ] ) {H } C cb − j

A caca

B cbcb

−1 −1

B cb − j

(4.11)

The kernel in the equations 4.2-4 is of the form: v = x [Cc ]−1{y}

(4.12)

A B ]+ [H cbcb ]+ [K ]−1 ] is a sum of complex matrices and in which [Cc ] = [ H caca

independent of the particular pair of dofs i and j for which the

89

Chapter 4


coupling has to be evaluated. The terms x  and {y} reflect dependence on the dofs for which coupling is to be calculated: for example, in eq. 4.4, x  equals H pA− ca  and {y} equals {H cbB − j }. Some interesting interpretations can be given to the above equations. A first observation is that the matrix [Cc] should be symmetric if the structures to be measured satisfy reciprocity. Another observation is that the matrix [Cc] to be inverted may tend to be singular at some particular frequencies. At the natural frequencies of the assembling structures, the rank of [Cc] tends to be 1. On the other hand, the poles of the assembled structure will coincide with minima of the determinant of [Cc] and may cause rank deficiency. The latter is clear from intuition, observing that [Cc] is really independent of the particular set of dofs i and j, so that the poles of the coupled function should be described by [Cc]. If [Cc] is rank-deficient, its inversion, required for the solution of either of the previous equations, might be ill conditioned at the frequencies mentioned above, causing the solution to be heavily influenced by minor perturbations of the data. It can be seen that, knowing the FRFs of the uncoupled structures, the prediction quality on the assembled one strongly depends on the inversion of the matrix [Cc], for which FRFs Singular Value Decomposition, pseudo-inverse, smoothing and symmetrisation are applied numerical procedures ([Leuridan&al.88], [Otte&al.90], [Otte94]). The frequency range and resolution of the coupled FRFs depend on those of the component FRFs. 4.2 Numerical improving techniques for a better coupling 4.2.1 Pseudo-Inverse [Otte94] indicates that, by means of a Singular Value Decomposition (SVD), the symmetric matrix [Cc] can be decomposed as:

90

Chapter 4


[Cc ] = [U ][Σ][V ]H

(4.13)

Taking into account the symmetry of [Cc], there exists a pair of singular vector matrices [U] and [V] such that [U] = [V]*, so: [Cc ] = [V ]* [Σ][V ]H = [U ][Σ][U ]T

(4.14)

The kernel equation 4.12 now becomes: v = x [V ][Σ ]−1[U ]H {y}

(4.15)

Also, it can be expanded in: v = x [V ][Σ ]− 2 [Σ ]− 2 [V ]T {y} = x [U ]* [Σ ]− 2 [Σ ]− 2 [U ]H {y} 1

1

1

1

(4.16)

A matrix [Cc’] can be defined, [Cc '] = [Σ] [V ]H = [U ][Σ]

(4.17)

[Cc ] = [Cc '][Cc ']T

(4.18)

1 2

1 2

Consequently,

The kernel equation finally becomes:

(

v = x  [Cc ']T

)

−1

[Cc ']−1{y}

(4.19)

The solution of the coupling equation can now be considered as a vector scalar product:

91

Chapter 4


v = {z1}T {z2 }

(4.20)

with {z1} and {z2} solutions of an equation with [Cc’] as a coefficient matrix and {x} and {y} as observation vectors: [Cc ']{z1} = {x}

(4.21)

[Cc ']{z2 } = {y}

(4.22)

If [Cc’] is singular (then also [Cc] is singular) or close to singular, where close to singular is measured by the condition number of [Cc], then a solution can be calculated using linear least square techniques. Using a linear least squares approach, the above equations are made compatible by finding approximation of {x} and {y}. The minimum effort approximation is found by an orthogonal projection of these vectors in the vector space of [Cc’]. Mathematically, the vectors {z1} and {z2} are found by using the pseudo inverse [Cc’]+ of [Cc’], {z1} = [Cc ']+ {x}

(4.23)

{z2 } = [Cc ']+ {y}.

(4.24)

[Cc’]+ is the inverse of the reduced matrix [Cc’]r, i.e. constructed by its non-zero singular values and corresponding singular vectors. If [Cc’] is full rank, one always obtains the exact solution. In practice, the coupling solution value v is approached by directly performing an SVD on [Cc] and taking its pseudo inverse: v = x [Cc ]+ {y} = x [V ]r [Σ ]−r 1[V ]Tr {y}

92

(4.25)

Chapter 4


The orthogonal projection of {x} and {y} in the (reduced) vector space of [Cc] is expressed in the outer products of this equation. As each singular vector of [Cc], corresponding to a singular value forced to zero, can be considered as a deflection vector, forcing some smaller singular values of [Cc] to zero is equivalent to imposing a number of physical constraints to the coupling dofs. This can also be seen as stiffening the substructures in the coupling dofs. 4.2.2 Pre-processing of FRFs: Smoothing and Simmetrisation In § 4.1 it has been pointed out that the matrix [Cc] to be inverted could be ill conditioned, especially at the natural frequencies of the coupled structures. Slight perturbations on the data could therefore influence the result considerably. A reasonable approach to improve results could therefore consist of smoothing the FRFs from perturbation prior to the coupling calculations. The matrices

[H ] A caca

and

[H ] B cbcb

should be also symmetric for

structures that satisfy reciprocity. If those matrices are fully measured, then some symmetrisation prior to the coupling calculation may be helpful. 4.2.2.1 Smoothing The technique of FRFs smoothing should be such that the smoothed FRFs describe the structure dynamics and stay compatible in format with the coupling equations. The method outlined by ([Leuridan&al.88], [Otte&al.90], [Otte94]) is based on the analysis of the correlation between the FRFs at all dofs, obtained over a given frequency range. The correlation analysis is based on Principal Component Analysis (PCA).

93

Chapter 4


PCA is a statistical technique that is based on a generalisation of factor analysis [Otte94]. It evaluates the dependence structure of multinormal observations where no a priori patterns of causality are available. It consists of finding an orthogonal transformation of the original stochastic variables to a new set of uncorrelated variables, which are derived in non-increasing order of importance. These uncorrelated variables are called principal components and are linear combinations of the original variables. If the first few components account for most of the variation in the original data, the effective dimensionality of the data can be reduced. PCA can be carried out directly by means of an SVD of the observation matrix, so that the two techniques become only one. PCA can then be seen as an SVD and easily extended to complex data. Performing a Principal Component Analysis on the FRFs at all dofs, it yields to:

[… {H j (ω k )} …] = [U ][… {H ' j (ω k )} …]

(4.26)

[H ] = [U ][Σ][V ]H = [U ][H ']

(4.27)

Where,

{H j (ω k )} is the FRFs vector between input j and all response dofs, {H ' j (ω k )} is one of the principal components of [… {H j (ω k )} …]. The elements are ordered by descending variance; [U] is the unitary transformation matrix. An approximation of

[… {H j (ω k )} …], [… {Hˆ j (ω k )} …] , follows the

form:

[… {Hˆ

j

(ω k )}

]

[ {H ' j (ω k )} …]r

… = [U ]r …

94

(4.28)

Chapter 4


[Hˆ ]= [U ] [Σ] [V ] r

H r

r

= [U ]r [H ']r

(4.29)

Here the subscript r refers to a sub-vector of {H ' j (ω k )} with the r elements of largest variance, the largest principal components, and the corresponding columns of the matrix [U].

{Hˆ

j

(ω k )} should, compared to

{H j (ω k )}, describe most of the phenomena in correlation among the elements of {H j (ω k )}, therefore the response caused by structure behaviour. The contribution of the smaller principal components is rejected, therefore considered as perturbations. 4.2.2.2 Symmetrisation The symmetrisation method proposed by ([Leuridan&al.88], [Otte&al.90]) is based again on the PCA-SVD of each pair of reciprocal FRFs, as a function of frequency. The first principal component is then normalised to give a symmetrised approximation for the pair of reciprocal FRFs. So, consider a pair of reciprocal FRFs, Hij(ω) and Hji(ω). Then   hij (ω k )   …  h ji (ω k ) 

(4.30)

[H ] = [U ][Σ][V ]H = [U ][H ']

(4.31)



[H ] = …  

 …  

 u11   hij (ω k )     … =   … h'ij (ω k ) …  h ji (ω k )  u21 

(4.32)

Since [U] is unitary, one has approximately, approximation being function of the importance of the neglected principal component,

95

Chapter 4


1 hˆij (ω k ) = hˆ ji (ω k ) = h'ij (ω k ) 2

(4.33)

4.2.2.3 Effects As a conclusion of the paragraph on numerical improving techniques for a better coupling, what suggested by ([Leuridan&al.88], [Otte&al.90], [Otte94]) can be pointed out. The application of the pseudo-inverse to solve the coupling equations is justified and converges to the solution obtained with direct matrix inverse. When the data are contaminated with noise, then a significant improvement can be achieved by applying the pseudo-inverse. The pre-processing of the FRFs improves drastically the results, both when applying the direct inverse and pseudo inverse. The influence of the smoothing gives much less significant contribution for data with no explicit modal behaviour at higher frequencies. This could be in part attributed to the complexity of applying the smoothing method, requiring intricate judgement of the operator in terms of selection of sub-bands for smoothing and number of principal components to be used ([Leuridan&al.88], [Otte&al.90]). 4.3 Determination of the FRFs The FRFs can be obtained by tests on the substructures in free-free conditions, or can also come from direct calculation using the FE models or modal synthesis, allowing hybrid modelling ([Cuppens&al.00], [Wyckaert&al.96], [Wyckaert&al.97/1], [Wyckaert&al.97/2]). Depending on their origin, the FRFs may suffer higher mode truncation.

96

Chapter 4


4.3.1 Experimental approach FRFs measurements are done on the detached substructures along all the transfer paths, in between all connection points, as well as between the connection and the input/output points. Reference and response could be vibrational or acoustical, using hammer, shakers or volume velocity source excitation for the input signal, and accelerometers or microphones for the output signal. The quality of test data influences results widely, the kernel matrix being very sensitive when inverted. Reciprocal FRFs should be measured, and reciprocity should not be assumed, to have a quality check of the measurement setting. Noise should be low on the FRFs. Responses should be measured as closely to the excitation as possible. Frequency consistency between reciprocal FRFs is very important, as far as the substructuring methods use linearity as a basic assumption. With experimental FRFs, the effects of those seemingly truncated higher modes are inherent in the measurements. A high condition number of the structural FRF matrix that is used in the matrix inversion needs to be avoided. A high condition number means that small errors on the original FRF matrix (which are of course always present due to measurement noise, if not already smoothed) can lead to big errors in the inverted matrix, and thus on the final FBS coupling. Hereafter some possible physical reasons for high condition number are outlined. First, some transfer path might be closely spaced. The resultant FRF matrix has columns that are very similar, that means it will be impossible to determine in which of the transfer paths a force is present, as the response is similar. Second, some transfer paths might have a very stiff connection between them, and the effect is as in the previous point. Third, a dominant global mode might be present in the structure and the response is similar, no matter

97

Chapter 4


where it is exited. This also results in very similar FRF matrix columns. Eliminating singular values yields a matrix with a good condition number, but of a lower rank. Since the matrix rank also indicates the number of singular vectors that can be estimated, a lower matrix rank also means a reduction in the independent phenomena. 4.3.1.1 Reciprocity principle When the structure behaviour satisfies linearity, each output is the result of a linear superposition of the inputs through their transfer functions. The responses in i and j dofs of a flexible body excited in the same dofs is thus described by eq. 4.34. If one of the interfaces becomes fixed, while all forces are present, it is thus possible to express the reaction force in this dof as in eq. 4.35. When no force is acting on a dof, the free displacement of this latter can be described through eq. 4.36. The transfer function between the fixed interface forces in dofs i and j equals the reciprocal one between the free-free displacements of the same dofs if and only if eq. 4.37 is satisfied. This latter condition is realised when Hij is symmetric, that means the system is linearly behaving, as supposed at the beginning. In this situation the reciprocity principle is verified between the fixed interface force transmissibility (calculated between excitation forces and interface forces) and the free-free displacement transmissibility (between interface dofs and the former excitation dofs). Using the reciprocity principle, it is thus possible to evaluate an experimental model of a structure being able to introduce displacement and rotation excitation at the fixed interface whereas it is practically impossible to apply a direct input due to the testing set-up.

98

Chapter 4


{ } { }

 {X i } = H ii {Fi }+ H ij F j   X j = H ji {Fi }+ H jj F j

(4.34)

{X j }= {0}⇒ {F j }= − H −jj1H ji {Fi }

(4.35)

{ }

H Ffixed j Fi

{ }

{ } { }

 F j = H −jj1 X j   X j = H jj F j {Fi } = {0}⇒  ⇔ {X i } = H ij H −jj1 X j  X j = H ij F j  H XfreeX − free  i j

{ } { }

{Fi }

=

{X j }

{F j } x =0 {X i } F =0

{ } { }

(

− free ⇔ H Xfree = − H Ffixed ⇔ H −jj1H ji iX j j Fi

)

T

= H ij H −jj1

(4.36)

(4.37)

i

j

It is also possible to define an acoustic and vibro-acustic reciprocity principle as in eq. 4.38-39. The acoustic reciprocity is between the volume velocities and the pressures, while the vibroacustic reciprocity is between the fixed interface forces induced by a volume velocity source and the pressures induced by displacements in volume velocity free condition.

H

H

Pi Pj

P j Fi

=

=

Pj Pi =− Qj Qi

⇒

Pj xi =− Fi Qj

⇒

Qi Qj

Fi Qj

= Pi = 0

= xi = 0

Pj Pi

Q j =0

Pj xi

(4.38)

(4.39) Q j =0

4.3.2 Analytical approach Different ways can be followed to determine the FRFs on a component when the FE model is available. The direct calculation and the synthesis from a modal base are here quoted.

99

Chapter 4


The direct approach calculates the FRF matrix Hdir(ω) at each frequency line as in eq. 4.40, where [M], [C], [K] are respectively the mass, the damping and the stiffness matrix of the FE full dofs model. It must be noted that this calculation could be computationally relevant if the component model has a large dofs number and the frequency band of interest is wide.

[

]

H dir (ω ) = − ω 2 [M ] + jω [C ] + [K ]

−1

(4.40)

The synthesis approach calculates an approximation Hsyn(ω) of the FRF matrix by using a finite number of mode shapes and eigenfrequencies of the component. Assuming the hypothesis of proportional modal damping, the synthesis of the FRF is given by eq. 4.41, where N is the number of modes included, {Φ}i is the i-th mass normalised mode shape, ωni is the i-th natural frequency and ξi the i-th modal damping ratio. The number of modes to be calculated in the eigensolution depends on the judgement of the operator. H syn (ω ) =

{Φ}i {Φ}iT

N

∑ (ω i =1

2 ni

2

)

(4.41)

− ω + j 2ξiω ni ω

By assuming the system present general viscous damping, the synthesis of the FRF is given by eq. 4.42-43. The mode shapes are now complex conjugate (see Chapter 2). λi is the complex eigenvalue, N is the number of modes included, {Φ}i is the i-th complex mode shape, ωni is the i-th natural frequency and ξi the i-th viscous damping ratio. N

1 {Φi }{Φi }T 1 {Φi }* {Φi }*T + * jω − λi ai jω − λ*i i

∑a

(4.42)

λi = −ξiω ni ± ω ni 1 − ξi2

(4.43)

H syn (ω ) =

i =1

100

Chapter 4


4.3.3 Modal superposition and truncation The “correct” FRF [Hc(ω)] of a structural system can be expressed as a function of its modal parameters ([Wyckaert&al.97/1],[DuarteEwins96/1]), [H c (ω )] = [H k (ω )] + [H h (ω )] = [Φ k ][Λ2k − ω 2 ] [Φ k ]T + [Φ h ][Λ2h − ω 2 ] [Φ h ]T −1

−1

(4.44)

where [Hk(ω)] and [Hh(ω)] represent respectively the FRF matrices synthesised from eigenvectors and eigenvalues of kept lower modes ([Φk] and [Λk2], including rigid body modes) and from the truncated higher modes and frequencies ([Φh] and [Λh2]). This is a general expression where both the eigenvectors and the eigenvalues can be complex, so that it accommodates general damping conditions such as viscous damping and structural damping. The kept modes may not be just those appearing inside the frequency band of interest in which the FRF is calculated. The cutoff frequency for defining the kept set modes is normally higher than ωmax, the highest frequency value of the FRFs. 4.3.4 Compensation for truncated modes: static and dynamic compensation, residual attachment modes If the effects of the truncated higher modes are completely ignored, only the truncated FRF matrix that accounts for the contributions from the kept lower modes is left, [H k (ω )] = [Φ k ][Λ2k − ω 2 ] [Φ k ]T −1

101

(4.45)

Chapter 4


Assuming that the natural frequencies of those truncated modes are much higher than ωmax, a first order approximation is proposed by [MacNeal71]:

[Λ

2 h

−ω2

] ≅ [Λ ] −1

(4.46)

2 −1 h

Thus, the contributions from those higher modes will become a frequency-independent constant, [H h (ω )] = [Φ h ][Λ2h − ω 2 ] [Φ h ]T ≅ [Φ h ][Λ2h ] [Φ h ]T −1

−1

(4.47)

The definition of the static residual matrix (static compensation) follows: [R0 ] = [Φ h ][Λ2h ] [Φ h ]T −1

(4.48)

Although the modal parameters of those higher modes are assumed as unknown, this residual matrix can still be accurately obtained if the structure has no rigid body modes: [R0 ] = [H c (0)] − [Φ h ][Λ2h ] [Φ h ]T −1

(4.49)

In case rigid body modes are present, it is possible to follow the approximation proposed by [Duarte-Ewins96/1], in which [R0] is calculated at a near-zero frequency: [R0 ] ≅ [H c (ω1 )] − [H k (ω1 )]

(4.50)

The same idea can be exploited to improve the FE-derived FRF: [H s (ω )] = [H k (ω )] + ([H c (ω1 )] − [H k (ω1 )])

102

(4.51)

Chapter 4


In doing so, the “correct” FRF [Hc(ω1)] must be calculated at a selected frequency point ω1. The above discussed static compensation works well if the cutoff frequency is much higher than ωmax. If such a condition is not satisfied, a higher order approximation or a dynamic compensation becomes necessary. This is based on the MacLaurin expansion theory,

[Λ

2 h

−ω2

] = [Λ ] −1

2 −1 +ω2 h

[Λ ]

2 −2 h

[ ]

+ ω 4 Λ2h

−3

+…

(4.52)

A second order dynamic compensation can be calculated evaluating the static and dynamic compensation matrices [A] and [B] of eq. 4.53 at two distinct frequencies ωlow and ωhigh. As a rule of thumb, one can select ωlow below the first flexible mode and ωhigh at about the 75% of the highest frequency of interest. [R1 (ω k )] ≅ [A] + [B ]ω k2

(4.53)

A more general solution for the approximation of the residual compensation matrix can be based on the direct response calculation at a number of frequency lines ωi, i=1…n ([Wyckaert&al.97]). The exact solution can be approximated by a polynomial: [Rn (ω )] = [A0 ] + [A1 ]ω 2 + [A2 ]ω 4 + … + [An ]ω 2n

(4.54)

The n+1 coefficients of the polynomial derive from the solution of a set of equations:

103

Chapter 4


 [A0 ] + [A1 ]ω12 + [A2 ]ω14 + … + [An ]ω12n = [H c (ω1 )] − [H k (ω1 )]   [A0 ] + [A1 ]ω 22 + [A2 ]ω 2 4 + … + [An ]ω 2 2n = [H c (ω 2 )] − [H k (ω 2 )]    …  2 4 [A0 ] + [A1 ]ω n +1 + [A2 ]ω n +1 + … + [An ]ω n +12 n = [H c (ω n +1 )] − [H k (ω n +1 )]

(4.55)

where [H c (ωi )]i =1:n +1 is evaluated directly in the n+1 different frequency lines. In the modal synthesis, another technique tries to compensate the static contribution of all higher modes that are truncated by using residual attachment modes. These residual attachment modes are obtained as explained in § 2.2.6. This compensation method tries to approximate the static contribution of all higher modes by means of some additional higher frequency residual modes added to the normal modal base. The number of residual modes is equal to the applied number of static loads, corresponding to the number of coupling dofs. The natural frequencies of these residual modes are higher than the highest natural frequency of the normal modes, if these normal modes include rigid body modes. 4.4 Rotational dofs in FBS coupling analysis Through the QUATTRO Brite-Euram project no: BE 97-4184, the European research community aimed at solving the critical issues which today prohibit the consistent use of rotational degrees of freedom information in test-based and hybrid structural dynamics modelling. In the investigation of the dynamic behaviour of coupled or modified structures, it is of topmost importance the knowledge of the full frequency response functions matrix between all the possible inputs and the possible outputs of the structures under analysis. Since each point of a structure can theoretically be excited by forces (Fx, Fy,

104

Chapter 4


Fz) and moments (Mα, Mβ, Mγ) and move with six dofs, three translations (x, y, z) and three rotations (α, β, γ), the full FRF matrix for each considered point is a six by six matrix as in eq. 4.56. Of this matrix a three by three sub-matrix is related to forces and displacements (just translational quantities) while the other three, three by three, sub-matrices are related to moments and displacements, forces and rotations and moments and rotations respectively, thus combining translational and rotational quantities or just rotational quantities. It is easy to see that, if rotations and/or moment information are neglected, up to 75% of the FRF matrix disappears with evident negative effects on the results of the requested estimation. When the FRF matrix is inverted, the result of the inversion presents serious errors when matrix's rows or columns are null. These errors will influence the estimation of the FRF matrix of the modified structure heavily, especially for what concerns the positions and the amplitudes of the resonances and anti-resonances in the FRFs.  x   H xx     y   H yx     z   H zx     = α   H αx     β   H βx     γ   H γx

H xy

H xz

H xα

H xβ

H yy

H yz

H yα

H yβ

H zy

H zz

H zα

H zβ

H αy

H αz

H αα

H αβ

H βy

H βz

H βα

H ββ

H γy

H γz

H γα

H γβ

H xγ   Fx    H yγ   F y    H zγ   Fz    H αγ   M α    H βγ   M β    H γγ   M γ 

(4.56)

The importance of the rotational dofs data and the poor quality of the results when these data are neglected are well known. Unfortunately no quantification of the errors or suggestions how to quantify them are provided in literature. The reasons why the information about rotational dofs has been almost totally neglected, even if their importance has been many times demonstrated in

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literature (e.g. [Heylen&al.97]), depend on different factors: mostly costs, the difficulties of using arrayed sensors, the time required for accurate measurements or the simple ignorance of the problem. Even after the QUATTRO project, still a lot of basic research work has to be done in order to overtake the practical problems of measuring the rotational dofs. 4.5 Connectors The impedance of the connections is generally frequency, temperature and pre-load dependent. Numerical modelling is still a tricky approach, since non-linear contributions are given by material, boundary conditions and inner geometry. Appropriate experimental test should be performed, using driving force and measuring displacements or accelerations on the source and target side of the connection. When the connection stiffness is not large in comparison to the attached body impedance and the relative induced displacement is much greater than the noise signal, the stiffness can be evaluated with the mount stiffness method as in eq. 4.57:

(

)

K (ω ) = F (ω ) / xsource − active (ω ) − xt arg et − passive (ω )

(4.57)

This approach is not applicable when on the contrary the relative displacement is minimal. Some connections are very stiff and the displacement differences on the active side and the passive side may become very small. In this latter situation, the FRF matrix of structural transfer functions relating the response on the output dofs to the force excitation at the input dofs might be the stiffness matrix needed; numerical refinement is appropriate to avoid problems in the inversion.

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The impedance can be roughly approximated by the static stiffness when the connection rigidity is low. For very stiff joints again this approach does not give good results.

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Chapter 4 ...........................................................................................................................................1 Frequency-Domain Component Mode Synthesis Methods: ..............................................................1 FRF Based Substructuring.................................................................................................................1 Introduction ...................................................................................................................................1 4.1 Basic theory...........................................................................................................................86 4.1.1 Force transmissibility .........................................................................................................88 4.2 Numerical improving techniques for a better coupling .........................................................90 4.2.1 Pseudo-Inverse ...................................................................................................................90 4.2.2 Pre-processing of FRFs: Smoothing and Simmetrisation...................................................93 4.2.2.1 Smoothing .......................................................................................................................93 4.2.2.2 Symmetrisation................................................................................................................95 4.2.2.3 Effects..............................................................................................................................96 4.3 Determination of the FRFs ....................................................................................................96 4.3.1 Experimental approach .......................................................................................................97 4.3.1.1 Reciprocity principle .......................................................................................................98 4.3.2 Analytical approach............................................................................................................99 4.3.3 Modal superposition and truncation .................................................................................101 4.3.4 Compensation for truncated modes: static and dynamic compensation, residual attachment modes ........................................................................................................................................101 4.4 Rotational dofs in FBS coupling analysis............................................................................104 4.5 Connectors...........................................................................................................................106

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Chapter 5 A connecting stiffness optimisation procedure

Introduction In this chapter part of the theory outlined in Chapter 2 is applied to solve a problem of industrial relevance: the right determination of connections between modal models of coupled substructures. Complex structures can be modelled in a virtual prototyping environment through an accurate finite element model of each subcomponent and of their connections. The connections often consist in complex subassemblies, spatial mechanisms, rubber sealing and stoppers. Sometimes it is not efficient to describe the connections in detail in the full assembly model, due to their morphology, different materials and boundaries. In fact the accurate model of the connection might become more complex than that of the connecting substructures, especially after a component mode reduction. To achieve the best results in the finite element modelling technique it is common practice to update each substructure model separately. Comparing the eigensolution of the numeric model with the results

Chapter 5

A connecting stiffness optimisation procedure

coming from an Experimental Modal Analysis done on the actual part often does the model updating. Additional information comes from the comparison between measured and numerically calculated frequency response functions. Once the finite element models are updated, they might be processed by one of the CMS techniques of Chapter 2. This brings an efficient (from a computational standpoint) and quite accurate (depending on the amount of kept modes) model of the substructure. Determining the right properties of a connection is not always a straightforward task. Many times there is the need of running different trials varying the connection properties and checking the whole virtual assembly results against the experimental tests. Solving times might become relevant in the management of the modelling and in the choice of the approaches to follow. In order to keep the assembly models efficient (from a computational standpoint), one of the procedures followed is to run a component mode synthesis on the detailed models of substructures separately, and then assemble them with low parameters connection models. In this way, the connection models might not be based on pure physical elasto-dynamic properties and dofs, becoming virtual entities like numerical constraints between the generalised coordinates. It follows that the stiffness properties of these models are no longer necessarily a direct expression of the detailed physical description of the connection and of the boundaries, but are used to create a relatively simple numerical model of the assembly. From this point of view, updating finite element models of complex assembled structures can be a very difficult and timeconsuming task, because, if any physical link to the actual connection is lost, virtual elasto-dynamic properties have to be defined, instead of updating connection properties with a clear physical meaning. Different papers in literature ([Mottershead&al.96], [Dascotte00], [Dascotte-Schönrock99], [Schendlinski&al.98], [Schendlinski00],

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[VanLanenhove&al.02], [Görge-Link02]) are based on the sensitivity method, where frequency and modal assurance criterion (MAC) data are employed as residuals for the updating procedure. These approaches proved their effectiveness in particular when the models provided include physically well defined connections. Another approach found in literature [Heylen&al.] gains results by optimising the modification of the dynamic flexibility matrix of the connection in the FRF Based Substructuring (see Chapter 3) framework: no modeshapes are investigated, but only the FRFs. In this chapter, a procedure to update finite element models of the connections between parts of an assembled structure is introduced, also for test cases where virtual connectors are used. The whole procedure is based on the global optimisation of a target function that manages the differences between eigenvalues, eigenmodes and frequency response functions of the test and of the virtual assembly. The variables of the optimisation problem are the properties of the connection models. To extract the right information from the eigensolutions, a routine has been developed, based on manual updating actions about the comparison between frequencies, modal assurance criterion matrices and frequency response functions. A self-adaptive evolution (genetic) algorithm is used to manage the optimisation progress. Different simulations have been performed in order to trace out guidelines for the weights of the target function terms and for the variables’ ranges. The developed procedure is applied to two automotive test cases. The choice of these applications on large-scale models, even if much more complex than an academic example, has been suggested by the experience gained by LMS International consulting activity with other approaches. A comparison between manual updating, tests and the results achieved with the developed procedure is here provided.

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5.1 Description of the procedure 5.1.1 Pre-processing the data The experimental model of the assembly can be derived through well known Experimental Modal Analysis (EMA) techniques ([Ewins86], [Heylen&al.97]) using the frequency response functions (FRFs) acquired in a test session; accuracy is needed in these measurements to be able to treat them as reference for all the next connection updating. For each substructure, the finite element (FE) model must be formulated. However, the discussion about the finite element strategy to follow in the modelling is beyond the boundaries of the present work. Since the procedure finds the optimum among the results given by thousands of parameter combinations, for which the assembly model has to be evaluated each time, it is advisable to reduce the size of the model to greatly decrease the computation time of each optimisation step. A CMS method helps to reduce the detailed physical model to generalised coordinates, using the appropriate dynamic superset as transformation matrix. The dynamic properties of the models of this chapter have been evaluated according to the [Craig-Chang76] residual flexibility method, which combines freeinterface normal modes with residual attachment modes. The application here developed does not couple the substructures by using the generalised coordinates directly, but by modelling the connections by means of their physical dofs. The simple models of the connections consist of virtual spring between the physical dofs of the subframes, trying to approximate the behaviour of much more complex connector architectures. This implies that the connector properties must be transformed according to eq. 2.65-67. It is thus necessary to make the back transformation from the generalised coordinates to the physical dofs explicit for some node of interest. It

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must be here remarked that even if there is no filtering on the nodes, the reduced model brings computational advantages because during the calculations it has much fewer dofs than the original substructure, often in the measure of at least one-percent. The back transformation is necessary only on the expansion of the final results, so all the computational effort takes place on the reduced models. The test cases explained in this chapter have been processed using the commercial software MSC NASTRAN and LMS GATEWAY / LINK FOR NASTRAN. The attachment modes can be calculated in MSC NASTRAN by switching the parameter RESVEC on and by specifying the set of attachment (boundary set) coordinates on a userset with name U6. Once MSC NASTRAN has calculated the dynamic attachment mode superset, it is possible in LMS GATEWAY to filter the behaviour of a specified node set from the modes. LMS LINK FOR NASTRAN calculates the modal mass and stiffness matrices and write them in the DMIG direct input matrix formulation for NASTRAN; it also calculates the transformation equations between generalised coordinates and the filtered dofs subset, and writes them in the MPC multipoint constraint formulation for NASTRAN. The NASTRAN file written by LMS GATEWAY/LINK FOR NASTRAN contains both the model in generalised coordinates and explicitly the nodes in the subset of interest. The coupling nodes are then no longer real dofs, because they are expressed by a weighted sum (through the MPCs) of the generalised coordinates displacements. The whole procedure brings in a reduced model and physical nodes to which it is much easier to relate the connection properties. The connecting dofs and the nodes used in test measurements must be retained in the subset of the grid points when back transforming the results from the generalised coordinates models, as well as other nodes of interest to better display the behaviour of the components.

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Once the substructures’ FE models are reduced, they can be coupled with the connections’ models; the node pair table (npt) between test and virtual environment can then be written to relate the FE nodes to those of the measurements. The substructures’ updated FE models must be provided at their best quality [Chen-Ewins01], otherwise the optimisation will try to solve the deficiency of both EMA and FE models, missing its target. 5.1.2 Mode pair table extraction By observing an experienced engineer executing the correlation between test and FE models and using frequency differences and MAC matrices between eigensolutions as judgement tools, it has been possible to write an algorithm that simulates human choices. Every time a correlation is performed, it is necessary to choose the eigenvectors from the FE eigensolution to be matched with the EMA ones, writing the mode pair table (mpt). Due to the change of the connections’ properties, at each optimisation step the FE eigenmodes related to the EMA ones might be shifted and interchanged in the sequence (ordered by increasing eigenvalues): this brings to a new evaluation of the mpt. An automatic routine has been developed to provide at each optimisation step the updated mpt before the data extraction. The routine writes the mpt as the result of an information function minimisation. Starting from the first EMA eigenmode, this function is evaluated for each FE eigenvector; the FE mode that has the lowest value of the information function will be paired with the reference mode. The information function value is a weighted sum of four different contributions.

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Σinfo [i, j ] = wmi ⋅ ∆MACi [i, j ] + wmj ⋅ ∆MAC j [i, j ] + w f ⋅ ∆freq [i, j ] + wmc ⋅ MClass [ j ]

(5.1)

The first contribution in the weighted sum of eq. 5.1 is given in eq. 5.2. It is, for each EMA modal vector i, the relative difference between the maximum MAC value (reached in the correlation with all FE modes) and the MAC value with FE eigenvector j. This comes down to selecting the FE mode with the highest correlation with the given EMA mode i. ∆MACi [i , j ] =

MaxMACi [i ] − MAC [i , j ] MaxMACi [i ]

(5.2)

The second contribution in eq. 5.1, given in eq. 5.3, is composed by two terms. The first of them is the choice penalty factor given in eq. 5.4, which penalises the selection of the eigenvector j if it actually matches other EMA modes previously analysed. The choice penalty is equal to one plus the weighted (by wc) value of the complement of the relative difference between the maximum MAC value (grasped by the FE eigenvector j in the correlation with all EMA modes) and the maximum value reached by MAC when the mode j was previously chosen. If eigenmode j has already been picked with a higher MAC value, the repeated choice of it will be more penalised, because this means that a worse correlation has been encountered in this last mpt choice step. The second term in eq. 5.3 is, for each inquired FE eigenvector j, the relative difference between the maximum MAC value (as before) and the MAC value with the actual EMA mode shape i. The purpose of introducing ∆MACj[i,j], in the choice of the FE eigenmode to be paired with the EMA mode i, is to penalise the FE eigenvector j when it is not at its best correlation level because of the better MAC values shown with other EMA modes.

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∆MAC j [i, j ] = ChoicePenalty[ j ]⋅

MaxMAC j [ j ] − MAC[i, j ] MaxMAC j [ j ]

(5.3)

ChoicePenalty[ j ] =  MaxMAC j [ j ] − MaxMAC sel [ j ]   = 1 + w c ⋅ 1 −   [ ] MaxMAC j j  

(5.4)

The third contribution in eq. 5.1, given in eq. 5.5, is the absolute value of the relative difference between the frequency value of each reference EMA mode i and the actual FE eigenshape j. Target: to choose the FE mode closest in frequency to the reference one.  freqREF [i ] − freq[i, j ]   ∆freq[i , j ] = abs freqREF [i ]  

(5.5)

The fourth contribution in eq. 5.1 is given in eq. 5.6 and has been introduced to evaluate the quality of the FE mode shapes. Through the filter of the npt, only a subset of displacements is read from the full vectors. It can happen that some modes, that are very different if analysed on all their degrees of freedom (dofs), become close to one another if compared only on the nodes of the npt, losing the uniqueness and orthogonality of eigenmodes representation. The first step to evaluate MClass is to correlate all the FE modes to themselves, calculating a new MAC matrix, here named AutoMAC. The latter could not be a diagonal matrix if the npt includes only a subset of dofs, and the off-diagonal terms might be relevant in witnessing an unexpected correlation (and similarity) between the FE eigenvectors. MClass is the value of the sum of all terms in a row of the AutoMAC matrix, divided by the maximum of these summations. Modes with diagonal contribution far from one and high values of the off-diagonal quantities will be penalised, preferring those modes strictly similar to themselves only.

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m _ FE ∑ AutoMAC [i, j ] i MClass[ j ] = = 1 Max (MClass[ j ])

(5.6)

By changing the weights introduced in the information function of eq. 5.1, it is possible to increase the role of the single contributions to simulate human behavior better. ∆MACi[i,j] has proved to be the most important factor, its weight being fixed three times that of the other information. 5.1.3 Data extraction and target function evaluation Once the mpt is available, it is possible to proceed to extract the data from the full frequency table and MAC matrix, in order to build up the optimisation target function. In the latter three contributions are added: ∆freq, ∆MAC and ∆FRFs, with appropriate weights. Target function = wfreq·∆freq + wMAC ·∆MAC +wFRFs·∆FRFs (5.7) The ∆freq term in eq. 5.7 is used to evaluate the contribution of the frequency errors between the test and the FE assembly. It is the square root of the sum of all the squared relative differences between the frequency of the reference and FE modeshapes, divided by the number of modes considered (see eq. 5.8).  1 ∆freq = sqrt  n  mod es

n mod es

∑ i

 freqFEi − freqREFi     freqREFi  

2

  

(5.8)

The ∆MAC term in eq. 5.7 evaluates the correlation between the test and the FE assembly eigenmodes. It is the square root of the sum of all the squared complements of the MAC[i,mpt[i]] values, divided by

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the total number of modes considered (see eq. 5.9). MAC[i,mpt[i]] is the MAC value resulting from the correlation of the reference modeshape i and the FE eigenvector paired to it through mpt[i] information.  1 ∆MAC = sqrt   nmodes 

nmodes



i



∑ (1 − MAC[i, mpt[i ]])2 

(5.9)

∆FRFs = 2    n spectral _ lines  ModREFi − ModSynti        ⋅ + w abs   Mod     nspectral _ lines  (5.10)   i   sqrt   2    n spectral _ lines  PhaseREFi − PhaseSynti    + w ⋅   abs Ph      ⋅ 2 π n spectral lines _  i    

∑

∑

The ∆FRFs term in eq. 5.7 evaluates the difference between the measured FRFs and the synthesised ones. For each spectral line, the absolute difference in modulus and phase is calculated; summing up all the contributions through the frequency range of interest separately, yields two quantities as shown in eq. 5.10. The use of weights allows an optimal balance between amplitude and phase. The weights in the target function give the possibility to simulate the expert behaviour actions. 5.1.4 Optimisation management The target function can be managed by different kinds of global optimisation algorithms, in order to explore the full range of the variables in the connection description and to increase the probability to avoid local optima. For the test cases provided, LMS OPTIMUS 3.1 software was used to carry out all the sequences of the optimisations and to link the files

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written by the C-language ([Stallman01], [Stallman-Grath00], [Stallman-Weinberg01], [HP92/1], [HP92/2]) data extraction routines developed. A self-adaptive evolution (genetic) algorithm was chosen: its parameters were selected as the default values suggested by the software documentation [LMS01], to stress the importance of the data extraction procedure rather than of the optimisation algorithm. According to this, the number of parents (the best designs in the previous iteration) is set equal to that of the optimisation parameters. The sexuality (which describes the number of parents donating to the generation of one new offspring) is set equal to the number of parents. The population size, which defines the number of designs that are evaluated in each iteration, is chosen five times the number of variables. To control the parents’ selection, for each generation, the COMMA strategy algorithm has been selected: this means that the parents of each step are only selected from the offsprings of the previous one. The initial step width, which controls how much the design parameters may vary, is chosen half of the allowed variation. It is assumed that a small difference of the design parameters, that does not produce a large variation in the quality of the results, can be used as stopping criterion: this average stopping step width is set to the normalised value 0.01. After each iteration, the step width of the new parents is adjusted (both to a larger and smaller step width) by the step width mutation factor, here set to 1.3. 5.2 Results of the test cases The whole procedure has been tested on two automotive structures: the first is the assembly of a front door with the body in white of a car (without the mechanics and all the interior trims); the second is a back door attached to the same frame. Different kinds of connections are involved in the two examples: latches, hinges, rubber stoppers and

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sealings. Both the reduced models and the testing results here manipulated were subjected to some little inaccuracies, such as no perfect FE model updating of the subframes and no complete satisfactory consistency of the measured FRFs. For both test cases, the most significant results will be showed with the aid of MAC matrices and FRFs plots referred to some dofs of interest. The MAC matrices are displayed both from the top and side views, in order to give a better perception on the quantity and quality of modes correlated. The manual updating results are provided as a second reference, after the test one. It must be pointed out that the FRFs plots in this chapter have been selected among hundreds to stress the behaviour in some nodes of interest but to give an average of the demeanour of the full simulations too. This because there are some nodes of the assembled structure where the procedure performs much better than the expert manual updating, but some others where not. Frequency information is not given due to confidentiality reasons. To trace out the guidelines of the weights’ effects (wfreq, wMAC, wFRFs, wMod, wPh) in the target function, different simulations have been executed. For both test cases, three different values have been provided for wfreq and wMAC: 1.0, 0.33 and 0.1, while leaving wFRFs equal to one. wMod has needed two runs: in the first both wMod and wPh have been set to one; in the second run wMod has been increased to better balance the modulus and phase contributions. The design space of the optimisation variables (the connections virtual stiffness) has been first explored leaving to the optimisation manager the possibility of choosing the optimum parameters values in a wide range (10e-1 ÷ 10e9), regardless of any physical meaning. In a second run the optimisation manager has been forced to search for the best results in the range of one tenth to ten times the values obtained from a successful manual updating (performed in the past). It must be

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remarked that the procedure here presented can update the connections, when the range of the parameters is kept very close to experience values, and identify them, when the range is left wide. Tables 5.1 and 5.2 provide the labels to trace back the experiments in the plots. Nine experiments have been conducted for the front door test case, twelve for the back door one. Both test cases were modelled with MSC NASTRAN software [Reymond-Miller96]. The data exchange between models and information extraction routines has been performed through Universal Dataset 55 and 58 files [SDRC00]. In all the FRFs figures, the thin continuous black line always corresponds to the test measurement, while the thick continuous red one represents the manual updating. 5.2.1 Front door

Figure 5.1: front door response node

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In the first test case (fig. 5.1) the connection model includes 18 variables (some other properties are kept constant); in the npt 82 measurement points (with 3 translational dofs each) are paired to FE nodes; 14 reference EMA modes have been calculated from the acquired test FRFs. The car body has been reduced to a model of 169 generalised coordinates, 69 for the door. 588 test FRFs have been processed.

Front door tests

wfreq=1 wMAC=1 wFRFs=1

wfreq=0.33 wMAC=0.33 wFRFs=1


Wide range

Front_Door_3

Front_Door_4

Front_Door_5

Redued range

Front_Door_3R

Front_Door_4R

Front_Door_5R

Front_Door_4R2

Front_Door_5R2

Reduced range + Front_Door_3R2 wMod tuned

Table 5.1: labels of the simulations on front door

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Figure 5.2: top view of manual updating MAC for front door

Figure 5.3: side view of manual updating MAC for front door

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Chapter 5


Figure 5.4: top view of Front_Door_5R updating MAC for front door

Figure 5.5: side view of Front_Door_5R updating MAC for front door

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Figure 5.6: FRFs in node 1 of figure 5.1 in the outplane direction

Figure 5.7:FRFs in node 1 of figure 5.1 in the outplane direction

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Figure 5.8: FRFs in node 1 of figure 5.1 in the outplane direction

Figure 5.9: back door response nodes

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5.2.2 Back door In the second test case (fig. 5.9) the connection definition includes 17 variables (some other properties are kept constant); in the npt 110 measurement points (with 3 translational dofs each) are paired to FE nodes; 25 reference EMA modes have been calculated from the acquired test FRFs. The car body has been reduced to a model of 147 generalised coordinates, 126 for the door. 660 test FRFs have been processed.

Back door tests

wfreq=1 wFRFs=1

wMAC=1 wfreq=0.33 wMAC=0.33 wFRFs=1

Wide range

Back_Door_3


Back_Door_4

Back_Door_5

Wide range + wMod Back_Door_3E tuned

Back_Door_4E

Back_Door_5E

Reduced range

Back_Door_3R

Back_Door_4R

Back_Door_5R

Reduced range + Back_Door_3E wMod tuned

Back_Door_4E

Back_Door_5E

Table 5.2: labels of the simulations on back door

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Figure 5.10: top view of manual updating MAC for back door

Figure 5.11: side view of manual updating MAC for back door

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Figure 5.12: top view of Back_Door_5RE updating MAC for front door

Figure 5.13: side view of Back_Door_5RE updating MAC for back door

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Figure 5.14: FRFs in node 1 of figure 5.9 in the vertical direction

Figure 5.15: FRFs in node 2 of figure 5.9 in the driving direction

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Figure 5.16: FRFs in node 3 of figure 5.9 in the driving direction

5.2.3 Front door result interpretation Figure 5.2 and 5.4 show qualitatively the correlation of EMA and FE modes done in the expert manual updating and in one of the best results in MAC data by the procedure. When the maximum weight is given to the FRFs term, in the optimisation function, and the reduced range is imposed to the variables’ values the results are very similar. The comparison of figures 5.3 and 5.5 suggests that for the low and high frequency behaviour there is an improvement using the automated procedure instead of the manual updating, while for midfrequency modes the correlation decreases. Figures 5.6 through 5.8 plot the effect of raising the role of FRFs difference in the optimisation function of the automated procedure. Focusing the attention on the amplitude, it is possible to appreciate that the test fit is improved more in the mid-high frequency range than

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for low one, especially for the complete unknowledged procedure (wide range for variables). At the same time the phase contribution is enhanced. The reduction of the connections’ parameters range brings to a closer fit of manual updating results, as expected, but the balance between enhancements and droops is not clear: sometimes the procedure with reduced range parameters gives better results than the expert manual updating, sometimes not, but the trend is matched. The increased role of the amplitude over the phase information, through wMod, does not produce better results because at high frequency, where the amplitude fit is better, there is a significant drop in the phase. 5.2.4 Back door result interpretation As for the previous test case, figure 5.10 and 5.12 give a quality check of the correlation in the expert manual updating and in one of the best MAC results by the procedure; the outcome, even if poor, is quite similar. In figure 5.11 and 5.13 a decrement in the correlation of high frequency modes is displayed, while an improvement in the mid-range and a parity result in the lower modes is sketched. Figure 5.14 shows that the high weight of the FRFs information in the optimisation function brings the automated procedure results very close to the expert manual updating. It is remarkable that this happens starting from the back_door_5-5E which has no experience on the test case, meaning that the procedure is working nicely also as an identification tool. In many areas of the plot the difference between all of the updating and the acquired FRFs is still evident: some enhancement in the substructures and connections models are required to better fit the test data. Figure 5.15 gives an example of what happens when a test is not conducted with high accuracy (has witnessed). The FE assembly model is not able to match the acquired data, except when there are no

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restrictions on the variables value selection and the phase plays a stronger role in the optimisation. But this probably brings the procedure to try to solve deficiencies in the models and not to simulate correctly the connections behaviour, becoming source of error for the whole optimisation process. The analysis of figure 5.16 reveals that back_door_4RE is probably the curve that better fits the test measurements, even if the other two are quite close to the expert evaluation. At low frequency the enhancement of back_door_4RE is manifested in the matching of the first peak of the amplitude plot, while in the mid-high range the trend of test data is followed better in the sequence of peaks and valleys. This time the improvement is obtained having a medium weight for FRFs information, but still greater than the frequency and MAC differences contributions. The curve labelled back_door_5RE is the one that better fits the expert updating, especially in the low-mid frequency range, as expected from figure 5.11 and 5.13. 5.3 Optimisation guidelines Insight on information weight effect must be given, because an engineer’s judgement is now focused more here than in the connection properties selection. From the experience acquired, it can be deduced that by raising the weight of ∆MACi[i,j] information in the mpt choice brings the outcome closer to human estimation. Previous simulations performed without the ∆FRFs term, using only ∆freq and ∆MAC, have outcome to be less effective than those here commented. By increasing the weight of ∆FRFs information brings to a better fit of the test results. In some cases it has resulted that to raise this contribution too much produces a better fit of manual updating, while it deviates the models from the best solution. The

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influences of wMod have been more random but, at the same time, it suggests to keep a relevant importance of the phase data to achieve a better result. Reducing the range of variables gives a solution closer to manual optimisation results, as expected. On the other side, leaving the range wide and keeping a high contribution from FRFs, the procedure has shown identification tool behaviour. Besides the comparison of the results, it must be remarked once again that both the optimised and the manual updating could be affected by errors in the models and/or in the test accuracy. At the same time, by exploiting the results from different weightings in the optimisation function, it is possible to get a hint of what may be lacking in models and tests. Due to its generality, this procedure can be easily applied to different test cases: the requirement is that reduced models of the subframes be available, otherwise the optimisation will not be timeworthy. The procedure proposed has not been developed to substitute an engineer’s evaluation, but it can be implemented as a useful tool to support the assembly updating or identification. 5.4 Calculation time The effectiveness has been proved towards the fit of the experimental data and manual optimum, with a calculation time enhancement. In the pre-processing, the time required by adapting the LMS OPTIMUS scripts and the MSC NASTRAN connection models’ files (ASCII format) for a new test case has been about 2 hours; to this, the time required to reduce the FE models must be added. No change is needed for the data extraction routines. For each set-up in tables 5.1 and 5.2, the whole optimisation management has taken about 36 hours

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on a HP C3200 Visualise Workstation; each experiment has taken about 45 seconds, and about 3000 parameters configurations have been necessary to reach the final optimum. 5.5 Conclusions The procedure presented is at its first step and is open to future developments. From the experience acquired with the test cases analysed, some guidelines have been drawn. The right choice of the modes to be paired has brought a fairly good improvement on the optimisation: more work can be spent to enhance the mpt choice routine. The optimal tuning of the mpt weights could be included in the whole optimisation process by adding new variables to the latter; the same could be performed for the information weights in the optimisation target function. A more extended test campaign on the influences of wMod should be performed. The choice of self-adaptive evolution algorithm has proved to fulfil the avoidance of local optima in the optimisation progress and to obtain the convergence to the final parameters’ estimation. Due to the relevance of the FRFs, it could be interesting to check if choosing only a subset of measured FRFs can improve the final optimum. This in order to focus the optimisation on the response in important nodes or in frequency ranges and to avoid error sources, as explained. Appropriate comparison criteria for the quality and/or the importance of FRFs should be added as well.

135

Chapter 5


Chapter 5 .................................................................................................................... 109 A connecting stiffness optimisation procedure .......................................................... 109 Introduction............................................................................................................ 109 5.1 Description of the procedure............................................................................ 112 5.1.1 Pre-processing the data ................................................................................. 112 5.1.2 Mode pair table extraction ............................................................................ 114 5.1.3 Data extraction and target function evaluation ............................................. 117 5.1.4 Optimisation management ............................................................................ 118 5.2 Results of the test cases.................................................................................... 119 5.2.1 Front door...................................................................................................... 121 5.2.2 Back door ...................................................................................................... 127 5.2.3 Front door result interpretation ..................................................................... 131 5.2.4 Back door result interpretation...................................................................... 132 5.3 Optimisation guidelines ................................................................................... 133 5.4 Calculation time ............................................................................................... 134 5.5 Conclusions...................................................................................................... 135

136

Chapter 6 Tire experimental modelling in vehicle dynamic analysis Introduction Numerical simulation of road vehicle motion in different conditions is a useful tool to analyse the handling characteristics of a vehicle system, and many scientific and technical papers have been published on this subject ([Hegazy&al.99], [Rahnejat00]). Modelling of such systems generally requires an engineer to provide the inertia of the bodies as well as the kinematical relationships between them. Multibody dynamic analysis is able to describe the dynamic, nonlinear behaviour of very articulate systems composed of rigid bodies [Haug89]. Application of such techniques to dynamic analysis of road vehicles makes it possible to describe the kinematics of the suspensions, including any kind of linkage topology, non-linear springs, dampers and friction phenomena [Sharp-Crolla87]. Some recent papers show increasing interest in modelling the flexible properties of the bodies, especially by using the component mode synthesis technique [Ichikawa-Hagiwara96]. In this context, the study of tire properties has grown increasingly important. Two main categories of tire models can be outlined: the first one deals with side force, longitudinal force and self-aligning

Chapter 6

Tire experimental modelling in vehicle dynamic analysis

torque that occur between a tire and the road pavement [PacejkaSharp91], while the second one takes tire flexibility and inertial properties into account. Some analytical models can be found in literature, such as those dealing with a ring on an elastic foundation ([Soedel75], [Kung&al.86], [Huang92], [Huang-Su92], [HuangHsu92], [Dohrmann98], [Huang-Soedel87/1], [Huang-Soedel87/2]) or on finite element models ([Kung&al.86], [Richards91]). Both approaches outline that a tire has different mode shapes in the frequency range 0-200 Hz; ring models may be ineffective in most cases, while finite element ones may be inefficient from a computational standpoint. In this work a multibody full vehicle model is introduced according to the linearised approach of Chapter 3. An eigenproblem is solved by linearising the equations of the constrained motion in the static equilibrium position of the full car. The modal properties of the vehicle are later coupled to the modal model generated from data experimentally acquired on a standard tire. The results are then discussed. 6.1 Description of the vehicle model A simplified vehicle model (Fig. 6.1) is proposed according to the theory of linearised spatial mechanisms outlined in Chapter 3. The full three-dimensional model consists of 13 bodies, 12 CYLINDRICAL joints, 4 BALL joints and 4 linear springs; the suspension subsystems are based on the MacPherson architecture [Haug89]. All elements are rigid and flexibility is lumped in the suspension springs. In this way only the elastic modes allowed by the tire-suspension compliance can be determined, because the car body does not present all the flexible modes usually occurring at low frequencies due to the elasto-dynamic properties of the car frame.

138

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The global coordinate system has been chosen so that the x-axis is along the driving direction of the car, the y-axis is along the lateral direction and the z-axis is along the vertical direction. Each reference fixed on the rigid bodies is the principal axes system with centroidal origin. In Tab. 6.2 there are reported the initial displacements and consecutive rotations of the fixed references that identify all the bodies in the global reference.

Figure 6.1: Vehicle model

6.1.1 Suspensions Mass and inertia properties (Tab. 6.1) of each rigid body included in the suspensions are obtained according to a work previously published by other researchers [Hegazy&al.99]. Fig. 6.2 refers to the kinematic of the suspension. In each suspension, the motion of the wishbone is constrained by two CYLINDRICAL joints, one connecting it to the car body, the other to the so-called lower knuckle. Since the latter is the outer part of the shockabsorber system, a CYLINDRICAL joint leaves the relative translation and rotation with the so-called upper knuckle free. A BALL joint models the mount that connects the upper knuckle to the car body. A linear spring is linked to the upper and

139

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lower knuckles. The CYLINDRICAL joint between the wishbone and the lower knuckle restrains the steering motion of the wheel. The rigid body dofs left in the system, due to the usage of the CYLINDRICAL joints instead of TURNING ones, are filtered out numerically from the solution of the eigensystem of the vehicle. So the simple architecture adopted makes the independent motion of each suspension set possible, while wheels’ steering is not allowed.

Figure 6.2: Suspension set sketch

N.

Part name Mass (kg) Ixx (kg m2)

Iyy

Izz

1

Front wishbone left

6

0.092 0.082

0.174

2

Front lower knuckle left

11.961

0.157 0.088

0.070

3

Front upper knuckle left

2.816

4

Front wishbone right

6

0.092 0.082

0.174

5 Front lower knuckle right

11.961

0.157 0.088

0.070

6 Front upper knuckle right

2.816

7

Vehicle body

1200

483 2404

2482

8

Rear wishbone left

6

0.092 0.082

0.174

9

Rear lower knuckle left

11.961

0.157 0.088

0.070

10

Rear upper knuckle left

2.816

11

Rear wishbone right

6

0.092 0.082

0.174

12

Rear lower knuckle right

11.961

0.157 0.088

0.070

13

Rear upper knuckle right

2.816

0.033 0.033 0.00036

0.033 0.033 0.00036

0.033 0.033 0.00036

0.033 0.033 0.00036

Table 6.1- Mass and inertia properties in the vehicle model

140

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N.

x0 (m)

y0

z0

φ0 (deg)

β0

ψ0

1

-0.58

-0.38498

-0.149169

9.454

0

0

2

-0.58

-0.667593

-0.008635

-10.707

0

0

3

-0.58

-0.582182

0.264266

-6.009

0

0

4

-0.58

0.38498

-0.149169

9.454

0

180

5

-0.58

0.667593

-0.008635

-10.707

0

180

6

-0.58

0.582182

0.264266

-6.009

0

180

7

0.75

0

0.16

0

0

0

8

2.08

-0.38498

-0.149169

9.454

0

0

9

2.08

-0.667593

-0.008635

-10.707

0

0

10

2.08

-0.582182

0.264266

-6.009

0

0

11

2.08

0.38498

-0.149169

9.454

0

180

12

2.08

0.667593

-0.008635

-10.707

0

180

13

2.08

0.582182

0.264266

-6.009

0

180

Table 6.2 - Initial position of vehicle bodies

6.1.2 Equations of motion Vehicle motions are referred to a fixed inertial reference. Each rigid member Λi of the vehicle is described by six dofs qi belonging to the local reference attached to the centroid; inertia principal axes are considered. These dofs consist of the centroid’s three displacements and its three successive rotations (Euler angles) [Haug89]. Vectors qi are thus assembled in global vector

q = ∪ i qi

.

As mentioned in Chapter 3, the formulation of the equations of motion follows Lagrange’s equations for constrained systems. L=T-U is the Lagrangian function, defined as the difference between the kinetic energy T of unrestrained bodies and the elastic potential energy U related to coupling springs. Constraint functions Φ ( q ) = 0 (m equations) are used to model body coupling. The m+n equations of constrained motion are those of Chapter 3:

141

Chapter 6


d ∂L ∂L m − + ∑ λk Φqk = 0  k =1  dt ∂ q ∂ q Φ ( q ) = 0 

(3.7)

where Φq is an m x n matrix whose components can be defined as Φq k , j =

∂Φk ∂q j

, and λk is the k-th Lagrange multiplier.

The m+n equations of constrained motion are linearised around a relevant initial position expressed by q0 (as in Tab. 6.2) with the procedure highlighted in Chapter 3 and implemented into a Clanguage software1. This position corresponds to the static equilibrium location of the vehicle supported by tires. It is now possible to write the Lagrangian function as a linear expression in δqi, because M and K are now constant matrices. Introducing the stiffness matrix of the condensed system as in Chapter 3:

(

)

K constrained = I − Φ Tq ( Φ q M −1Φ Tq ) Φ q M −1 K −1

(3.22)

the eigenproblem of the constrained linear system is finally derived:

(K

constrained

− ω 2M ) q = 0

(3.23)

6.1.3 Modal solution of the vehicle model The solution of the eigenproblem of the constrained motion equations gives the frequencies and the modeshapes of the vehicle in the selected configuration. Rigid body modes may cause ill-conditioning problems of the matrices. For a more robust numerical solution some weak (10 N/m) springs are added to the mechanism to restrain the 1

The aid of Prof. Giuseppe Catania of DIEM, Mechanical Department , Bologna University, is gratefully acknowledged.

142

Chapter 6


rigid body modes of translation along the driving and lateral directions and rotation about the yaw vertical axis. By this, the rigid body modes become flexible (with very low frequency) and are not filtered out by the aforementioned procedure. The modeshapes contain the displacement of the degrees of freedom of the vehicle, which are the centroidal translations and rotations of each body. To increase the comprehension of the eigenmodes a routine has been developed so that it is possible to represent the motion of nodes rigidly attached to the body of interest. In this way a rough geometry of the mechanism can be sketched. The modeshapes have been visualised in SDRC I-deas/Test postprocessor. The data exchange between the mechanism analysis software and I-deas has been realised by using the Universal Dataset 55 format. Some incongruencies in the deformation of the wireframes may be ascribed to the visualisation approach followed by the commercial software. Table 6.3 lists the frequencies of the modeshapes of the vehicle alone unsupported by tires. It can be easily understood that in this situation six rigid-body modes are included in the eigensolution, as well as the four elastic modes of the suspensions. In figs. 6.3-12 the modeshapes are shown. MODAL PARAMETERS of the vehicle without tires SHAPE LABEL

FREQUENCY (HERTZ)

1-Flexible 2-Flexible 3-Flexible 4-Flexible 1-Rigid-body 2-Rigid-body 3-Rigid-body 4-Rigid-body 5-Rigid-body 6-Rigid-body

4.469 4.312 4.423 4.366 0.079 0.074 0.047 0.043 0.040 0.038

DAMPING (%) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

AMPLITUDE

PHASE (RAD)

4.9809E-03 2.6007E-03 2.5328E-03 5.1041E-03 2.3974E-03 3.4877E-03 3.3305E-03 2.7678E-03 1.6195E-03 2.4672E-03

-1.571 -1.571 -1.571 -1.571 -1.571 -1.571 -1.571 -1.571 -1.571 -1.571

Table 6.3 Modal parameters of the vehicle without tires

143

Chapter 6


Figure 6.3 Flexible modeshape n° 1 of the vehicle without tires


144

Chapter 6




145

Chapter 6


Figure 6.7 Rigid-body modeshape n° 1 of the vehicle without tires


146

Chapter 6




147

Chapter 6




148

Chapter 6


6.2 Tire modelling The simplest model found in literature [Levy-Wilkinson76] deals with a single dof elastic spring; inertia (lumped) properties are also taken into account. Flexibility is modelled by a linear spring orthogonal to the ground, connecting the wheel spindle to the road. The model here adopted consists of the experimentally obtained modal model of the tire. 6.2.1 Experimental settings

Figure 6.13: Experimental setting for the tire

Since the elasto-dynamic properties of tires vary with the inflating pressure and load, an appropriate setting has been built for the experiments. The tire has been considered in static conditions, which means that no rolling motion has been introduced. This assumption offers the possibility of building a much simpler testing layout, but precludes the investigation about the variability of the modal model of the tire as a function of rolling speed, as documented in literature

149

Chapter 6


([Soedel-Prasad80], [Huang87/1], [Huang87/2], [Huang92], [HuangSu92]). To simulate the behaviour in road contact conditions, a preload corresponding to a quarter car weight has been applied to the tire spindle. The modal model has been thus extracted starting from the deformed and pre-loaded configuration.

Figures 6.14-15: added spindle and wires constraints

The pre-loading setting is very simple (see figs. 6.13-15) and consists of a massive spindle bolted to the wheel rim, of metallic ropes and of an iron plate. The wires are connected to the plate, passing through the spindle. On each of them there are placed a screw joint and a force cell, the latter being realised by using strain gauges (see fig. 6.16). By rotating the screws it is possible to load the spindle with the desired force.

Figure 6.16: force cell to measure the tire pre-load

Certainly this type of setting influences the tire behaviour, since the spindle adds a lumped inertia and the ropes introduce constraining stiffness. When deriving the modal model, the behaviour of the setting must be taken into account, as explained further.

150

Chapter 6


The measurements have been performed by using rowing hammer excitation and by measuring accelerometer responses. Three translational displacements have been measured on 25 of the 26 grid nodes. Eight nodes have been placed on the tread belt (one corresponds to the contact patch between tire and ground); eight nodes have been considered on each of the two sides of the tire, while two other nodes have been positioned on the endings of the spindle. Employing a cylindrical reference system with the spindle as vertical axis, the acceleration signals in the radial, tangential and axial directions have been acquired. 6.2.2 Modal model The well-established methodology of the Experimental Modal Analysis [Ewins86] has been employed to extract the complex eigensolution from the measurements. The identified modes are complex, even if their damping is relatively low. As previously mentioned, these modes are affected by the presence of the setting constraints, i.e. the modal mass and the modal stiffness matrices of the measured structure contain the contributions of the spindle and of the wires. Before the tire model can be treated as a component of a whole vehicle, it is necessary to eliminate the influences of the experiment setting on the elasto-dynamic properties of the tire. In order to solve this problem, the contributions of the testing set-up can be seen as structural modifications of the final tire model. If the final model is coupled to the testing wires and spindle, the previously identified modal model is obtained. This means that the de-coupling of the wires and of the spindle can be obtained from the extracted model by subtracting the contribution of the equivalent inertia and stiffness in the modal [a] and [b] matrices of the phase space system (see § 2.6).

151

Chapter 6


PARISON TABLE Row Source : Column Source :

Test1: tire_constrained Test2: tire_decoupled

Test1/Test2 Freq 1 Freq 2 % Change MAC --------------------------------------------------------1 1 16.315 1.553 -90.482 1.000 2 2 31.918 1.111 -96.520 1.000 3 4 35.935 24.347 -32.246 1.000 4 3 41.569 3.831 -90.785 1.000 5 5 44.418 46.465 4.608 0.965 6 6 82.656 81.936 -0.872 0.995 7 7 90.195 100.504 11.430 0.886 8 0 119.926 ------9 0 127.570 ------10 9 135.654 134.676 -0.721 0.640 11 10 147.114 147.969 0.581 0.994 12 12 162.178 162.647 0.289 0.844 13 13 178.343 178.386 0.024 0.999 14 16 188.762 209.635 11.058 0.968 15 14 193.269 193.231 -0.019 0.794 16 15 207.654 207.919 0.127 0.966 17 17 228.013 228.040 0.012 0.997 18 0 233.623 ------19 19 252.080 252.214 0.053 0.994 20 20 261.724 261.803 0.030 0.994 21 21 287.923 288.055 0.046 0.998 22 22 303.882 304.118 0.078 1.000 23 23 323.946 324.114 0.052 1.000 24 24 335.068 335.892 0.246 0.987 25 25 379.756 382.942 0.839 0.990 0 8 --131.702 ----14 11 188.762 154.731 -18.029 0.669 14 18 188.762 235.537 24.780 0.569 Table 6.4: Comparison between tire models: constrained and de-coupled

This procedure results in a coupling with a connector having negative elasto-dynamic properties. 25 modes have been identified for both models in the frequency range 0-380 Hz. For the de-coupled tire the rigid body modes have been included. In table 6.4 a comparison between the two tire models (constrained and de-coupled) is reported: the first two columns are the mode pair table, the third and fourth columns contain the frequencies of the paired modes, the fifth is the difference in frequency and the sixth is the Modal Assurance Criterion value between the paired modes. It can be easily seen that the test setting is reflected rather in a frequency shift (especially for low-order modes) than in a change of

152

Chapter 6


the modeshapes, since the frequency differences are sometime marked, while the MAC value is almost close to one. The first four modes are the most affected ones by this change, because they involve the rigid body motion of the tire. Some modes (respectively 8, 9 and 18) of the constrained tire do not appear in the de-coupled model, neither as the eighth modeshape of the de-coupled tire has any pair in the constrained one. In figs. 6.17-24 some of the modeshapes of the tire with the constraining testing set-up are shown. Figs. 6.25-32 report the paired modes of the de-coupled tire.

Figure 6.17-18 Modeshapes n° 1-2 of the constrained tire


153

Chapter 6




Figure 6.25-26 Modeshapes n° 1-2 of the de-coupled tire

154

Chapter 6





155

Chapter 6


Since the colours represent the magnitude of the shape displacement (increasing from blue to green to yellow and then to red), it can be observed that the displacement of the spindle dofs decreases when the mode frequency rises, in the same way as all displacements become more localised instead of witnessing a global motion. 6.3 Modal coupling of the full car Once the models of both vehicle and tires are obtained, the flexible coupling approach has been followed to assemble the full car model. By introducing stiff connectors between the spindle dofs and the knuckle dofs it is possible to couple the tire to the suspension of the car. As outlined in Chapter 2, the structural modifications (because of the connectors added) must be transformed from the physical domain to the generalised coordinates of the modal models; then they must be added to the modal [a] and [b] matrices in the phase space formulation of the motion equations of the full vehicle. The eigensolution is complex, both in eigenvalues and eigenvectors. Table 6.5 reports the list of modeshapes for the full car. Figs. 6.33-52 some modeshapes are shown. MODAL PARAMETERS SHAPE LABEL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

FREQUENCY (HERTZ) 1.662 1.821 2.006 18.278 20.683 28.207 28.627 31.126 35.423 40.220 116.192 116.196 116.204 116.246 124.642 124.702

DAMPING (%) 0.130 0.190 0.327 5.442 6.332 5.269 5.696 2.653 2.835 2.321 2.424 2.421 2.419 2.423 3.082 3.078

156

AMPLITUDE 7.6154E-03 7.6155E-03 7.6154E-03 7.6154E-03 7.6154E-03 7.6154E-03 7.6154E-03 7.6154E-03 7.6155E-03 1.0142E-02 1.6856E-01 1.6856E-01 1.6758E-01 1.6465E-01 6.3288E-01 6.3345E-01

PHASE (RAD) -1.571 -1.572 -1.573 -1.633 -1.615 -1.688 -1.625 -1.093 -1.519 -1.569 -1.653 -1.727 -1.650 -1.770 -1.581 -1.594

Chapter 6


17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

124.846 125.073 138.965 138.966 138.984 138.986 160.346 160.346 160.559 160.633 162.786 162.787 162.844 162.857 178.360 178.360 178.360 178.360 190.418 190.426 190.624 190.930 197.499 197.526 197.743 197.965 227.897 227.899 227.899 227.900 234.007 234.008 234.010 234.011 248.772 248.790 248.860 248.924 261.577 261.577 261.577 261.577 281.040 281.143 281.323 281.540 298.708 298.770 298.846 298.984 315.925 315.971 316.025 316.142 330.818 330.834 330.853 330.900 374.784 374.855 374.963 375.194

3.082 3.098 3.644 3.644 3.644 3.645 3.285 3.285 3.284 3.289 3.254 3.253 3.255 3.257 3.408 3.408 3.408 3.408 2.925 2.923 2.942 3.003 4.102 4.104 4.118 4.133 3.582 3.582 3.582 3.582 0.315 0.314 0.314 0.313 3.653 3.652 3.656 3.661 3.249 3.249 3.249 3.249 4.211 4.220 4.249 4.264 4.151 4.153 4.161 4.164 3.120 3.116 3.116 3.105 2.575 2.575 2.573 2.571 3.370 3.381 3.387 3.410

6.3741E-01 6.7881E-01 2.7174E+00 2.7124E+00 2.7301E+00 2.6816E+00 1.4104E+01 1.4240E+01 1.3694E+01 1.4369E+01 2.9448E+01 2.9330E+01 2.2629E+01 2.3081E+01 3.3866E+01 3.4066E+01 3.4614E+01 3.4609E+01 2.0155E+02 1.9020E+02 2.1411E+02 3.5674E+02 4.6400E+01 4.3833E+01 4.3869E+01 4.9492E+01 8.1965E+01 8.0822E+01 8.1699E+01 8.3588E+01 9.3372E-01 9.3132E-01 9.3172E-01 9.3274E-01 3.0161E+01 2.8621E+01 3.0869E+01 3.2790E+01 4.0572E+02 4.0077E+02 3.9973E+02 4.0944E+02 7.2558E+01 6.8435E+01 7.6099E+01 7.6875E+01 4.0353E+02 3.8255E+02 3.9826E+02 4.2833E+02 4.8008E+02 4.5111E+02 4.4847E+02 4.5130E+02 3.4761E+02 3.3598E+02 3.2257E+02 3.3458E+02 5.4427E+02 5.3683E+02 4.9658E+02 5.4261E+02

-1.599 -1.625 -1.640 -1.663 -1.798 -1.882 -1.555 -1.785 -1.646 -1.648 -1.491 -1.523 -1.554 -1.549 -1.610 -1.613 -1.606 -1.599 -1.438 -1.470 -1.483 -1.477 -1.374 -1.374 -1.386 -1.527 -1.617 -1.657 -1.454 -1.607 -1.640 -1.628 -1.588 -1.641 -1.558 -1.601 -1.553 -1.674 -1.387 -1.983 -1.618 -2.124 -1.552 -1.591 -1.462 -1.541 -1.841 -1.890 -1.973 -1.802 -1.451 -1.501 -1.682 -1.423 -1.714 -1.645 -1.702 -1.710 -1.616 -1.601 -1.614 -1.625

Table 6.5: Modal parameters of the full car

157

Chapter 6


Figure 6.33 Modeshape n° 1 of the full car



158

Chapter 6





159

Chapter 6





160

Chapter 6





161

Chapter 6





162

Chapter 6





163

Chapter 6




From the pictures it can be easily understood that the first 6 modes of the full car correspond to the constraining of the rigid body motion of the vehicle body. The modes 7-10 are linked to those of the suspensions, while the others involve the displacements of the tire nodes much more than those of the other dofs of the car. The modes of the suspensions are of course influenced by the presence of the tires, which provides extra inertia and stiffness. The car body (rigid mechanism) is less influenced by the high frequency modes of the tires, probably because they involve localised displacement along the tire belt and not global motions of the spindle.

164

Chapter 6


6.4 Conclusion The procedure followed here has thus proved to be effective in the dynamic analysis of the full car around the static equilibrium position. By changing the initial position of the vehicle-mechanism, it is possible to investigate the variability of the eigensolution as a function of the location of its members, but this goes well beyond the limits of this application. The author wishes to acknowledge D. Da Re and F. Mancosu of Pirelli Pneumatici S.p.A., Milano, Italy for the testing tire.

165

Chapter 6


Chapter 6 .......................................................................................................................................137 Tire experimental modelling in vehicle dynamic analysis ............................................................137 Introduction ...............................................................................................................................137 6.1 Description of the vehicle model.........................................................................................138 6.1.1 Suspensions ......................................................................................................................139 6.1.2 Equations of motion .........................................................................................................141 6.1.3 Modal solution of the vehicle model ................................................................................142 6.2 Tire modelling .....................................................................................................................149 6.2.1 Experimental settings .......................................................................................................149 6.2.2 Modal model.....................................................................................................................151 6.3 Modal coupling of the full car .............................................................................................156 6.4 Conclusion...........................................................................................................................165

166

Chapter 7 Road noise modelling for NVH predictions by means of FBS approach Introduction Road noise modelling for NVH (Noise, Vibrations & Harness) predictions is becoming a challenging topic among all the targets of modern vehicle design. The contact between tires and road was studied in the past [Pacejka-Sharp91] mainly to model the generalised forces that act in the handling analysis of the vehicle, discarding the elasto-dynamic properties of tires at mid-low frequencies. Modern vehicle manufacturers are now paying more and more attention to vibro-acoustic targets for passengers’ comfort: road noise is thus playing a strategic role in the modelling of the acoustic quality of a vehicle, and studies were carried out about tire influences on vehicle comfort [Iwao-Yamazaki96]. At the same time the suspension sets, treated as elastic bodies, as well as the trim body of the car, contribute to the vibro-acoustic tuning of the whole structure. The high modal density of the trim body is generally a limit for some analysis

Chapter 7

Road noise modelling for NVH predictions by means of FBS approach

techniques when the investigations are extended to the mid frequency range. In this chapter a hybrid modelling approach based on FRF Based Substructuring (FBS) is suggested to predict noise and vibrations on a car due to tire contact excitations in the mid-low frequency range [0÷500 Hz]. The well-established FBS technique makes it possible to extend the analysis to this wide frequency range, giving the possibility of using the measured FRFs that can not be easily obtained from computational models. At higher frequencies the trim body modal density and modal overlap effect become too relevant to be modelled with accuracy by means of finite element (FE) or modal analysis approach, which would require a finer mesh than the wave lengths, not efficient either for computational or experimental modelling. The structure modelled in this work is made essentially of three components: the trim body of the car, the suspensions and the tires, plus the connections between them. Different techniques are employed to achieve an accurate modelling of each subframe. The vibro-acoustic FRFs of the trim body are computed from data measured in the experimental transfer path analysis: acceleration and pressure signals on points of interest (including acoustic interior cavity and coupling points to suspensions) excited by hammer or volume velocity source. FE modelling is preferred for the suspension assembly, this latter having a low modal density in the frequency range of interest due to its stiff constituent parts; test validation for the suspension parts is performed to assure a good tuning of the FE model. The model of the tire is derived by means of experimental modal analysis and the reciprocity principle: in the measurement setup the tire is static (non-rolling) and preloaded on an instrumented platform. The suspension and tire models are then coupled by modal coupling techniques; FRFs are then synthesised between tire contact patch and some points of interest on the suspension subparts and car

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frame coupling points. The model of the whole structure is realised by coupling the trim body to the tire-suspension system in the frequency domain, using frequency dependent mounts and bushings properties in each connection. A validation reference has been built using the reciprocity principle on the FRFs derived from a hammer and volume velocity source excitation on the full vehicle, while having the tire on the instrumented platform. A comparison between the test and the hybrid model FRFs in some points of interest is commented upon to prove the results achieved with the early step of this approach. 7.1 Application of FRF based substructuring to road noise NVH predictions Road noise predictions in the vibro-acoustic design of modern vehicles implies many modelling difficulties which have to be overtaken. A real car is usually manufactured by using materials with very different dynamic impedance, non-linear behaviour, and boundary conditions sometime too complex to be modelled with accuracy. The configuration of the vehicle also needs quite computationally relevant models because most subframes must be analysed as flexible bodies for NVH studies. Furthermore the acoustic interior cavity needs to be modelled. Modelling the vehicle by means of finite-boundary element techniques forces the analysis not to go over the threshold of 200 Hz. Even the experimental modal analysis does not give the desired accuracy at higher frequencies for the highly damped structural model of the trim body, where modal density and overlap effect require a finer response mesh. To let the analysis be extended to mid frequencies even for such a complex structure as a full real vehicle, the FBS technique is here experimented as hybrid coupling approach.

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7.2 Hybrid model description The analysed vehicle consists mainly of three components: tires, suspensions and the car body. For each of them a different modelling technique has been adopted to get the necessary FRFs for the final FBS coupling. Also the connection impedances have been modelled in the frequency domain. 7.2.1 Pre-loaded tire Numerical tire models are generally available from tire manufacturers for the use in full vehicle dynamic models. These tire models are often accurate with respect to the predicted resonance frequencies. However they are not able to predict accurately the dynamic response of the tire using road input at the interface patch between tire and ground. Therefore LMS1 has developed a technique to extract the dynamic behaviour of the tire based on measurements on a preloaded nonrolling tire, being able to input displacement and rotation at the interface patch. The procedure generates a modal model of the tire, which can be used in a CAE environment e.g. LMS Virtual.Lab or MSC Nastran. 7.2.2 Rear suspension Only the rear suspension has been analysed, since the experimental tests on the trim body and the final validation test have been done on the car without front mechanics. Bungees have hung the car at the front frame, while the rear wheels have been positioned on the ground, one on the instrumented base, the other on the ground.

1

The experimental model was developed at LMS International by Koen Cuppens and Peter Mas of the Engineering Services – Hybrid CAE/Test Division

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The suspension system is the assembly of the cast iron knuckle, the shock absorber and spring column, two parallel links and a guiding link as in fig. 7.1. Between the knuckle and the links rubber bushings are placed. To model the suspension in free-free condition, FE approach has been followed, approximating the real configuration. The geometry has been manually drawn from the real structure, with accurate dimensioning and weighing but without the possibility of cutting off anything, so that the inner parts still remain unknown (especially as regards the shock absorber).

Figure 7.1: rear left suspension assembly

It can be assumed that the knuckle-braking system assembly might be modelled like a rigid body (with accurate inertia properties) in the frequency range of interest, as revealed by a FE analysis on the flexible modes. The shock absorber column has been modelled using straight beam elements for the outside structure, shell elements for the spring case and curved beam elements for the spring, that is with a varying

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diameter and pitch. Two sliding bearings have been supposed to be inside the column so as to leave only the longitudinal relative motion free between the head and the base of the mount; they have been modelled by two multpipoint constraints, which leave free the relative translation and rotation about the shockabsorber axis. The links have also been modelled by means of straight beam elements. A test has been executed by hammering the links and the spring in free-free conditions and by acquiring some acceleration FRFs. The FE models have been updated for the mass conservation and the natural frequencies. For the shock absorber column this has not been possible, because the high pre-load of the spring drastically changes the boundary conditions of the longitudinal motion; it has only been possible to update the total mass by discrete mass distribution. For the bushings, while the mass is added to the connected links, test and the best final FBS coupling have determined the stiffness. The full model of the suspension has been translated from SDRCIdeas to MSC/Nastran code. 7.2.3 Car body An experimental approach, as well as the suggestions of Chapter 4, has been followed to get the FRFs on the trim body of the car suspended in free-free condition by bungees. Hammer excitation has been adopted in the suspension connection dofs, while volume velocity source excitation has been used for the input signal in the interior cavity. Only translational accelerations have been acquired. A microphone has been placed for the acoustic output at the passenger’s head position on the right back seat of the car. As shown in § 7.2.2, the engine system, the gearbox, the transmission and the front suspensions have been taken out of the car

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under test. All the other parts are left where they are in normally sold cars. This experimental model was developed at LMS International by Peter Mas, Engineering Services – Hybrid CAE/Test Division. 7.2.4 Bushings and mounts As explained in § 4.5, the identification of the connection impedance should be performed by appropriate experimental tests, using driving force and measuring displacements or accelerations on the source and target sides of the connection. When the connection stiffness is high in comparison with the attached body impedance and the relative induced displacement is minimal, the FRF matrix between structural response on the output dofs due to force excitation at the input dofs may be the stiffness matrix needed. The mount data have been derived experimentally by means of this methodology. The impedance can be roughly approximated by the static stiffness when the connection rigidity is low, as in the guiding link bushings. For very stiff joints once again this approach does not give fine results: the trailing link bushing stiffness has been selected as the constant trial value that gives the best final FBS coupling. 7.3 Coupling specification Before proceeding to the final FBS coupling, the FRFs between suspension connection responses and tire patch input must be evaluated. First of all the tire model has been coupled in the physical domain to the suspension model in the FE formulation by MSC/Nastran [Reymond-Miller96] (see fig. 7.2). Right and left suspension–tires have been joined in the same FE model. The coordinate system is with

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X axis as the driving direction, Y as the lateral and Z as the vertical ones. To restrain the rotational rigid modes of the links, very weak springs have been introduced.

Figure 7.2: tire and suspension model coupling

In order to calculate an appropriate modal basis (see Chapter 2) for the FRFs synthesis, the dynamic residual attachment mode superset has been chosen. To get the residual attachment modes in the suspension+car connections and tire patch dofs it is possible to follow the appropriate calculation procedure in MSC/Nastran by specifying the RESVEC parameter (see also § 5.1.1). 60 residual attachment modes have been added to the normal mode basis, one for each dof (rotational dofs included) of the connecting nodes between the suspensions (left and right), the ground and the car body. The FRFs have been synthesised in LMS/Gateway using the dynamic residual attachment mode superset just calculated. Synthesis of the FRFs has been done in the car-connected dofs, in the ground dofs and in the verification test dofs. Modal damping has been set to different values during the several hybrid couplings.

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For the final FBS coupling the LMS CADA-X software has been used. As in eq. 4.1-3, substructure A is the tire and suspension model, while B is the car body; [K] matrix has been written as related in § 7.2.4.

C C H oaia and H obia matrices

have been calculated. The inputs ia are

the dofs displacement on the tire patches, while the outputs oa are the verification test dofs displacement on the suspension parts and ob is the ear pressure at the microphone inside the interior cavity; the coupling dofs ca=cb are the translations in the links and mounts connections between the suspension and the car body. 7.4 Validation tests As said in § 7.2.2, the testing set-up at LMS laboratory consists of the car without the front mechanics hung by bungees at the bumper and resting on the ground on its rear tires, the left one on the instrumented platform, the right one on the concrete floor. The force cells placed under the rear left tire are positioned in a triangular shape: this offers the possibility of deriving the torque in RX and the force in Z dof, from the force measurements in vertical and longitudinal direction at the interface between the tire patch and the ground. Two tests have been made. In the first the excitation has been carried out by using a roving hammer technique at different points on the suspension parts and by measuring the forces at the interface (see tab. 1). Using the reciprocity principle, as in eq. 4.37, these force/force measurements can be interpreted as displacement/displacement measurements. This gives the vibration FRFs matrix

C H oaia between

the

inputs on substructure A and outputs also on A for the final FBS coupling validation. It should be remembered that the force cell signal must be derived twice in order for it to be compared to the hammer signal, because of the difference between displacement and acceleration measured by the two different gauges or viceversa. In the

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Label Description FRLI1 Front link close to hub FRLI2 Front link close to subframe (body) HUB hub RELI3 Rear link close to hub RELI4 Rear link close to subframe (body) SARL Shock absorber close to car body TRAI5 Trailing link close to hub TRAI6 Trailing link close to car body ti_l:INP Left tire input patch mic:reri microphone rear right Table 7.1: points used in the validation tests

second test a volume velocity source instrument positioned inside the car cavity has provided the excitation, while the force cells under the platform and the accelerometers on the suspension parts have measured the responses (see tab. 7.1). The omni-directional source has sufficient power to excite the cavity structure and to work in a free from acoustic boundary situation. The vibro-acoustic reciprocity of eq. 4.39 leads to interpret the measurements as tire patch displacement /interior pressure FRFs. This gives the vibro-acoustic FRFs matrix C H obia between

the inputs on substructure A and outputs on B for the

final FBS coupling validation. The accuracy of both tests has not resulted as completely satisfactory. Repeating the tests, after having moved the tire on the platform, has brought to fairly different results. This is probably caused by non-linear friction condition in the shock absorber column, links bushings and tires. It is important to keep in mind that, when the reciprocity principle is used together with the volume velocity source, the energy input in the system might have a different distribution from that of a directly excited tire test. Especially this energy might be rather less and so the structure might be closer to displacement domains where the non-linearity is more evident. At the same time these two tests have offered an approach to build the first validation file for the FBS model.

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7.5 Comparison between the test and the hybrid model Different FBS couplings have been performed to find the proportional damping ratio to be introduced into the FRFs synthesis for substructure A and the stiffness matrix of the very stiff suspension bushings. The trials have revealed no clear evidence of the direct influence of each variable; especially for the calculus of the transmissibility between the tire input patch displacements and pressure at the microphone inside the cavity, the results are not much affected by the changes of the parameters values. For the test case here reported, there have been adopted a damping ratio of 2.5% for each mode of substructure A, a stiffness of 10e+6 N/m for the bushings of the parallel links, and a stiffness of 10e+5 N/m for the guiding link (no cross coupling between the three directions has been assumed; no rotational dofs have been considered). An optimisation procedure could help in the identification of these parameters, even if, as reported in the previous paragraph, the reference validation test should be treated with caution. The reciprocity principle has been widely applied in deriving the final validation measurements and the experimental model of the tire in free-free conditions. The assumption of its validity has brought the possibility of modelling the structure with the displacements at the tire contact patch without having a direct excitation on it, keeping the measurement system simple and affordable. Nevertheless, for this kind of structure, there could be a non-linear response dependence on the amount of energy deriving from the excitations. From the pictures proposed, where the y-axes of the graphs are in logarithmic scale, it can be noted that the model (blue) fairly follows the validation data (red) in the frequency information, while there is sometimes a marked difference in the amplitude. The predictions on the stiff links (figs. 7.3 and 7.4) seem to be better than those on the

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suspension column (figs. 7.5 and 7.6), except in the 200-280 Hz frequency band. This might be due to a faulty quality in the suspension modelling and also in the damper as shown in § 7.2.2, rather than to errors in bushing data. The pressure prediction (figs. 7.7 and 7.8) suffers the same marked error between 200 and 260 Hz, while in the other part of the frequency range it follows the trend of the validation test quite well. It is important to point out that the best calculated FRFs are those in which the rotational displacement RX at the tire patch is given, while lower correlation is reached with vertical Z displacement. The whole structure seems to be better modelled for rotations than for vertical motion: this could be explained partly by saying that the interior part of the shockabsorber does not play an important role when the column bending is excited, the contrary when the damper is compressed. Once again a better knowledge of the inner suspension could help the whole modelling. 7.6 Conclusion This chapter has discussed a first attempt to predict the road noise in a car by means of FBS technology combined with a hybrid CAETEST approach. The modelling approach for each substructure has been described and commented in detail. In the framework of this technology experimental and numerical modelling have been exploited whenever the choice of either approach grants a reliable and fast result. In this study the reciprocity principle plays a central role to keep the experimental settings as simple as possible. This research work has suggested that the hybrid FBS approach to road noise modelling looks very promising. Some further steps will be taken to improve the accuracy of the model prediction including a flexible suspension, an improved experimental tire model and a better estimate of the suspension bushings. Additional measurements will

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also be done on the full vehicle using controlled displacement input at the tire patch.

Figure 7.3: vibrational FRF FRL1:+X vs ti_l:INP:+Z

Figure 7.4: vibrational FRF FRL1:+X vs ti_l:INP:+RX

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Figure 7.5: vibrational FRF SARL:+Z vs ti_l:INP:+Z

Figure 7.6: vibrational FRF SARL:+Z vs ti_l:INP:+RX

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Figure 7.7: vibro-acoustical FRF mic:reri:S vs ti_l:INP:+Z

Figure 7.8: vibro-acoustical FRF mic:reri:S vs ti_l:INP:+RX

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Chapter 7 .......................................................................................................................................167 Road noise modelling....................................................................................................................167 for NVH predictions......................................................................................................................167 by means of ...................................................................................................................................167 FBS approach ................................................................................................................................167 Introduction ...............................................................................................................................167 7.1 Application of FRF based substructuring to road noise NVH predictions ..........................169 7.2 Hybrid model description ....................................................................................................170 7.2.1 Pre-loaded tire ..................................................................................................................170 7.2.2 Rear suspension ................................................................................................................170 7.2.3 Car body ...........................................................................................................................172 7.2.4 Bushings and mounts........................................................................................................173 7.3 Coupling specification.........................................................................................................173 7.4 Validation tests ....................................................................................................................175 7.5 Comparison between the test and the hybrid model ............................................................177 7.6 Conclusion...........................................................................................................................178

182

Chapter 8 Conclusions This doctoral dissertation has outlined the theoretical aspects of some tools available in literature, and other originally developed here, in the field of structural and mechanism dynamics for modern design of large scale structures and mechanisms, focusing in particular on their application to automotive test cases. For each technology an appropriate procedure has been developed. All material presented shows that it is always necessary to keep a test as a reference, or even to include the experimental models in a more comprehensive hybrid model, when the virtual prototyping approach can not model the phenomena with accuracy. In fact all the approaches presented in Chapters 2-4 define the possibility of performing a hybrid coupling, and Chapters 5-7 outline the necessity of experimental approach when numerical models need updating and when complex structures such as composite tires or the trim body of a car are at issue. Both Chapters 5 and 7 have revealed the need of a deeper study of connection problems, since the determination of connection models seems to be still unsatisfactory. The identification or updating of connection models is feasible of future developments, especially in the direction of an enhanced approach towards FRFs data. The experimental models, and so the hybrid models, have underlined the necessity of further studies on the acquisition of rotational degrees of freedom signals. On the contrary, finite element

Chapter 8

Conclusions

models can manage the rotational degrees of freedom without difficulties, but they can not be easily updated because of the lack of reliable experimental references. This aspect is also deeply linked to the description of the connection models. In Chapters 6 and 7 the need for a more evolved tire model is felt, since in both applications the tire is considered as static, while the literature has shown that tire properties change when it is rolling. Future developments could be in the direction of overtaking the experimental testing difficulties. The development of the linearised mechanism theory and application of Chapters 3 and 5 have proved their effectiveness and original contribution by the author. They have furthermore revealed an interesting opportunity for enhancing the tool sketched there. From the library of constraint equations it is now possible to build more complex joints and to study the behaviour of complex mechanisms such as serial and parallel robot manipulators, where the multibody commercial codes are not able to give precise information about the varying eigensolution in all working configurations. Future developments of this research with the implementation of software with automatic constraint library and graphic interface are awaited with great interest.

184

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