FACULTE POLYTECHNIQUE DE MONS
SENSITIVITY-BASED FINITE ELEMENT MODEL UPDATING METHODS WITH APPLICATION TO ELECTRONIC EQUIPMENTS
Dissertation submitted to
the Faculte´ Polytechnique de Mons, Belgium
in Partial Fulfillment of the Requirements for the Degree of Doctor in Applied Sciences
by Yun-Xin WU
Accepted by the jury committee the 12th of July 1999 Jury Committee: Prof. D. LAMBLIN, President of the committee Prof. Y. BAUDOIN, Royal Military School, Brussels Ir. E. FILIPPI, Alcatel-ETCA Prof. S. BOUCHER, Rector Prof. C. CONTI, Dean, Thesis advisor Dr. P. DEHOMBREUX, Co-advisor Abstract
When electronic devices are introduced into astronautic structures such as rockets or satellites, the reliability becomes extremely important. Their vibration behaviour should be thoroughly investigated. Finite element analysis is the most powerful tool to predict the behaviour of the structure in the design process. Because of their complexity, the initial finite element models must be validated with experimental data. This dissertation investigates some important aspects of sensitivity-based finite element model updating procedures, and applies them to electronic structures.
The existing updating methods are shortly reviewed. The numerical correlation tech- niques, MAC, COMAC, FRAC, FDAC, are first summarised. The mode shape smoothing technique is extended to FRF smoothing.
Sensitivity-based finite element model updating equations are expressed in a generic form that is suitable for both modal and impedance updating methods. A new modal updating method, namely quasi-modeshape (QMS) updating approach, is proposed. In the proposed method, the SVD technique is employed to transform the FBM residual force vectors into quasi-modeshapes. Simulation results obtained for a wide range of structures showed that when noise is present, QMS method can produce much better results.
A TLS/QR-F model error localization method is proposed. Four steps are included in this method, which are (1) regularising the updating equations in Total-Least-Square sense with SVD technique; (2) performing a QR F forward subset selection where the pivoting
i criterion is a cos2 θ property; (3) backward excluding parameters based on their residue de- scending ratios computed via QR decomposition; (4) excluding parameters based on the subset solution. To evaluate the proposed method, a success ratio of parameter error local- ization is also proposed. This success ratio is designed to have the capability of assessing the error localization methods with statistic sense. Numerous simulations showed the advantage of the proposed TLS/QR-F method.
Perturbed Boundary Condition test technique can enrich experimental data with few extra effort. In this thesis, PBC model updating is implemented for both modal and impedance methods. The simulations showed that, besides other advantages, PBC technique can also help TLS/QR-F method to augment its error localization success ratio.
The proposed techniques are applied to validate finite element models of electronic structures, including a PCB with components, “wedge-lock” slides and a shell box. Practical considerations, such as equivalent thickness of SMT components, Bayesian estimation, are also addressed.
ii ACKNOWLEDGMENTS
First and foremost I would like to thank my advisor, Prof. C. Conti, for his continuous encouragement, stimulus and guidance throughout the preparation of my dissertation.
I would like also take this opportunity to thank Dr. P. Dehombreux, for his assistance during experimental work, and especially for his meticulous reading of the thesis draft and many recomposing suggestions.
I wish to express my gratitude to Mr. H. Algrain, for his patience to verify simulation results. Thanks also to Dr. O. Verlinden, Dr. S. Datoussaid, Mr. R. Hadjit, Mr. M. Fontaine,
Mr. J-P Devos. They have given me many helps during my study in FPMs.
Special thanks are due to my wife and my daughter. Without their emotional support
I would not have had the courage to complete this thesis.
My study in FPMs was made possible by a cooperation project between la Facult´e
Polytechnique de Mons and Central South University of Technology, for which I am very grateful.
iii iv Contents
ACKNOWLEDGMENTS iii
Nomenclature xi
1 Introduction 1 1.1Overview...... 1 1.2 Review on Electronic Equipment Modelling ...... 3 1.3ReviewonFiniteElementModelUpdating...... 7 1.3.1 The origin and the philosophy of finite element model updating . . 7 1.3.2 Uniqueness aspects of finite element model updating ...... 8 1.3.3 Matchbetweentestandanalyticalmodels...... 9 1.3.4 Correlation between measured data and analytical predictions . . . 10 1.3.5 Modeltuning...... 11 Global methods ...... 12 Local methods ...... 12 Error localization and parameter subset selection ...... 14 1.4 Outline of the text ...... 15 1.5 Research contributions and originalities ...... 17
2 Correlation and model match 19 2.1 Correlation techniques ...... 19 2.1.1 Visual comparisons ...... 20 2.1.2 Numericalcomparison...... 21 ModalAssuranceCriterion...... 21 CoordinateModalAssuranceCriterion...... 22 FRACandFDAC...... 23 2.1.3 Other comparisons ...... 24 2.1.4 Remarks...... 25 2.2Modelmatchtechniques...... 26 2.2.1 System-matrix-basedreduction/expansion...... 27 Guyanstaticreduction/expansion[23]...... 28 Kidder dynamic reduction/expansion ...... 29 2.2.2 Modeshape-basedexpansionprocedures...... 32
v EigenvectorMixingMethod...... 32 MACexpansion...... 33 ModalCoordinateMethod...... 33 System Equivalent Reduction Expansion Process (SEREP) . . . . . 34 2.2.3 FRFsmoothing...... 36 2.2.4 Remarks...... 37
3 Sensitivity-Based FE Model Updating Equations 39 3.1 Introduction ...... 39 3.2 Eigenvalue-based updating equations ...... 41 3.3 Modal updating equations ...... 46 3.3.1 FBM updating equations ...... 46 3.3.2 Quasi-modeshape (QMS) updating equations ...... 47 3.4 Impedance updating equations ...... 50 3.5Summary...... 53
4 SVD, regularization, weighting, constraints and iterations 55 4.1 Introduction ...... 55 4.2 SVD, rank-deficiency and matrix condition number ...... 57 4.3 Least-squares estimation ...... 59 4.4 Chi-squares estimation [58] ...... 63 4.5 Total Least-squares estimation ...... 65 4.6Bayesianestimationtechnique...... 69 4.7 Weighting updating equations ...... 70 4.8Quadraticprogramming(QP)approach...... 72 4.9 Iterative model updating ...... 73 4.10Summary...... 74
5 Error localization 77 5.1 Introduction ...... 77 5.2Errorlocalizationprinciples...... 78 5.2.1 Global error localization ...... 78 FBM error localization ...... 78 EMM error localization ...... 81 5.2.2 Local error localization methods ...... 86 Introduction ...... 86 QR basic solution for rank-deficient updating equations ...... 87 SVD/QR subset selection ...... 92 5.2.3 TLS-based error localization technique ...... 94 TLS-based regularising and condensing ...... 95 QR F forward subset selection ...... 99 Backward excluding updating parameters ...... 103 Parameter exclusion based on the subset solution ...... 107
vi 5.3 Approach of evaluating the efficiency of error localization methods ..... 108 5.3.1 Success ratio of parameter error localization ...... 108 5.3.2 Measurement noise influence on error localization ...... 110 5.3.3 Simulatednoisepatternincasestudies...... 111 5.3.4 POMUS and success ratio computation ...... 113 5.4Casestudies...... 117 5.4.1 Case Study 1: A Cantilever beam ...... 117 Using resonance-frequency-based updating equations ...... 117 Using modeshape-based updating equations ...... 122 Using FRF-based updating equations ...... 126 5.4.2 Case study 2: A 10-DOF stiffness-mass system ...... 128 Using modeshape-based updating equations ...... 129 Using FRF-based updating equations ...... 132 5.4.3 Casestudy3:Threedimensionbaystructure...... 134 Using modeshape-based updating equations ...... 135 Using FRF-based updating equations ...... 137 5.4.4 Case study 4: A plate structure with components ...... 141 Using modeshape-based updating equations ...... 142 Using FRF-based updating equations ...... 145 5.5Summary...... 149
6 Perturbed Boundary Condition Model Updating 153 6.1 Introduction ...... 153 6.2 PBC Testing and Total Message Amount ...... 154 6.3PBCModelUpdating...... 156 6.4Examples...... 159 6.4.1 ExampleI...... 159 6.4.2 ExampleII...... 161 6.5Summary...... 163
7 Model Updating of Electronic Structures 165 7.1 Introduction ...... 165 7.2 Methods of modelling SMT components ...... 166 7.2.1 Connection between base board and components ...... 166 7.2.2 Separately modelling electronic components ...... 168 Lumpedmassmodel...... 168 Spread mass model ...... 169 Spring-mass-dampermodel...... 169 Spring-platemodel...... 170 Plate-platemodel...... 170 Couplingmodel...... 170 Substructure model ...... 171 7.2.3 Smearing model ...... 171
vii Global mass smearing model ...... 171 Global mass/stiffness smearing model ...... 172 Local mass/stiffness smearing model ...... 172 Equivalentthicknessmodel...... 173 7.2.4 Remarks...... 174 7.3 Determining characteristics of SMT components ...... 175 7.3.1 Overview...... 175 7.3.2 Description of the card ...... 177 7.3.3 Experimentalmeasurementconfigurations...... 178 7.3.4 Dynamictestingandeigenvalueextraction...... 179 7.3.5 Finiteelementmodels...... 180 7.3.6 Initial model and correlation with test data ...... 182 7.3.7 Finite element model updating parameters ...... 183 7.3.8 RefiningFEmodelbasedonpreviousknowledge...... 184 7.3.9 Updating FE model “blindly” with all parameters ...... 188 7.3.10Errorlocalization...... 190 UsingQMSmethod...... 191 FRF-basedmethod...... 192 7.3.11Remarks...... 194 7.4 Modelling “Wedge-Lock” slides ...... 195 7.4.1 Description and modelling of “Wedge-Lock” slides ...... 195 7.4.2 Casestudy...... 196 ExampleI:virginboardgrippedtowedgelockslides...... 196 Example II: Printed circuit board gripped to Wedge-Lock slides . . 199 7.4.3 Summary...... 201 7.5 Model updating of a box-like shell ...... 202 7.5.1 Introduction ...... 202 7.5.2 Measurement configuration and modal parameter extraction . . . . 203 7.5.3 Finiteelementmodel...... 206 7.5.4 Initial correlation and updating parameter selection ...... 208 7.5.5 Modeltuning...... 210 7.5.6 Conclusion...... 213 7.6Summary...... 213
8 Conclusions and Recommendations 215 8.1Conclusions...... 215 8.2Recommendations...... 219
Bibliography 221
A Damping Treatment in FRF-based Equations 227 A.1 Introduction ...... 227 A.2 Undamped model assumption ...... 228
viii A.3 Diagonal damping model assumption ...... 229 A.4 Proportional damping model assumption ...... 231 A.5 General light damping model assumption ...... 232 A.6Illustration...... 233 A.7Summary...... 236
B POMUS scheme and GUI interface 237 B.1Programschemes...... 237 B.2MainGUIwindow...... 237 B.3Filemenu...... 241 B.4 Setting menu ...... 242 B.5Modalmenu...... 242 B.6ImpedMenu...... 243 B.7Solumenu...... 243 B.8Figuremenu...... 244 B.9Runmenu...... 244
C FE script file format 247
D Definition of the case study structures 249 D.1 Cantilever beam structure ...... 249 D.2 10-DOF stiffness-mass system ...... 250 D.33-Dbaystructure[69]...... 251 D.4Platestructures...... 251 D.5 10-DOF discrete system ...... 253 D.6 Free PCB with six components ...... 254 D.7Wedge-Lockslides...... 255
E Mode shapes of the PCB, Box structures 257
List of Tables 261
List of Figures 263
ix x Nomenclature
.x quantity from experiment .a quantity from analytical model (I) . , .i quantity of an initial FE model (U) . , .u quantity of the updated FE model .M master degree of freedom .S slave degree of freedom |···| absolute value ||···||2 Euclidean norm (2-norm) qP P ||···|| Frobenius norm; ||A|| = |a |2 F F P i j ij n ||···||∞ ∞ norm; ||A||∞ = max =1 |aij| 1≤i≤m j [···]−1 matrix inverse [···]+ pseudo inverse of a matrix [···]0 matrix transpose [···]H matrix Hermitian .Re, Re() real part of complex number .Im, Im() imaginary part of complex number th [1j] the j column of an identity matrix [0] zero vector or matrix th dRri i residue descending ratio dRrlim threshold value of the residue descending ratio dRr ∆, δ change of a quantity f(t) excitation force vector h(ω) frequency response function vector H(ω) frequency response function matrix I identity matrix K stiffness matrix λ matrix eigenvalue Λ eigenvalue matrix M mass matrix ω angular frequency (rad/sec) th th ωi the i resonance frequency or i updating frequency p physical parameter of a finite element
xi φ modal vector Φ modal shape matrix Σ singular value matrix σ singular value SR success ratio of error localization SR average success ratio of error localization σSR standard deviation of SR x(t) displace response vector Z(ω), Z dynamic stiffness matrix
Abbreviations COMAC COordinate Modal Assurance Criterion DOF Degree Of Freedom EMM Error Matrix Method FBM Force Balance Method FDAC Frequency Domain Assurance Criterion FRAC Frequency Response Assurance Criterion FE Finite Element FRF Frequency Response Function IC Integrated Circuit LS Least-Squares MAC Modal Assurance Criterion MDOF Multi-Degree-Of-Freedom PBC Perturbed Boundary Condition PC Personal computer PCB Printed Circuit Board PINV pseudo-inverse POMUS Parameter-Oriented Model Updating System PTH Plated-Through-Holes QMS Quasi-Mode Shape QR QR decomposition RF Resonance Frequency SEREP System Equivalent Reduction Expansion Process SMC Surface Mounted Component SMT Surface Mounted Technology SVD Singular Value Decomposition TLS Total Least Squares
xii Chapter 1
Introduction
1.1 Overview
Electronic devices, equipped in TV sets, computers, automobiles, aeroplanes and artificial satellites, etc., play more and more important roles in our daily life.
When electronic devices are introduced into structures such as rockets and satellites, their reliability becomes extremely important. The vibration behaviour of electronic devices is thoroughly investigated.
An electronic equipment can be subjected to, besides the severe thermal loads, many different forms of vibration over wide frequency ranges and acceleration levels, either during its normal working period of time, or during transport, or launching. Mechanical vibration is usually undesirable and in fact, for most cases, harmful.
Surface Mounted Technology (SMT) was introduced in the 1960s [41] and is now extensively used in a variety of products. SMT employs microminiature Surface Mounted
Components (SMC) that are soldered directly onto the surface of a Printed Circuit Board
(PCB). SMT provides many advantages, such as reduced PCB size, easier automation of PCB assembly, improved electronic performance, lower manufacturing costs, over the traditional
1 2 Chapter 1. Introduction
Plated-Through-Holes (PTH) technology.
However, SMT has its own issues. Comparing to PTH technology, the most critical issue may be the solder joint reliability because it is the only mechanical means of connecting the SMC to PCB. It is believed that the major cause of failure is the low-cycle thermal fatigue of the solder joints [2] though high-cycle fatigue caused by equipment vibration is also believed to contribute to fatigue damage. However, in the transition region from low- cycle to high-cycle fatigue (103–104 cycles), vibrational and thermal strain components may have comparable magnitudes and the fatigue analysis must account for both [2]. Therefore, vibration analysis for electronic devices is essential.
Over the past half century, vibration analysis techniques have been made tremendous progress. In 1947, Kennedy and Pancu presented one of the first normal mode methods and found application in the accurate determination of natural frequencies and damping levels in aircraft structures [12]. This is recognized as one of the most important contributions in experimental modal analysis. Since then, extensive work has been done. By 1970 there had been major advances in transducers, electronics and digital analysers, while the use of digital
Fast Fourier Transform analysers marked a significant milestone. Till late 1970s and early
1980s, the techniques of modal testing in the frequency domain were established [12].
A time domain modal vibration analysis technique was presented in 1973 by Ibrahim
[27]. Now it has become an alternative approach to those in the frequency domain, although not as popular as the late. Nearly parallel to modal testing techniques, the finite element method has also been developed as the most powerful tool in structure analysis during last fifty years. This idea was undoubtedly used by mathematician Courant in 1943, however, it was published in a mathematical bulletin and totally ignored by practising engineers [14]. The label “finite ele- ment method” appeared in 1960 in a paper by Clough [10]. Although in 1965, Zienkiewicz and Cheung illustrated the applicability of the finite element method to any field problem that 1.2. Review on Electronic Equipment Modelling 3 could be formulated by variational means, most finite element (FE) programs today are heav- ily oriented towards the structural analysis area. From late 1980s to present, as the memory capacity and computation speed of personal computers (PC) have been tremendously en- hanced, it has become the basic practice for structural analysts to use FE programs with
PCs.
The above two structural analysis tools were jointly applied in late 1970s. The hybrid use is nowadays referred to as dynamic finite element model updating. The basic philosophy of dynamic finite element model updating is that the modal testing results are more precise than those calculated from initial FE models which is established with analysts’ intuition, and the initial FE models can be improved according to the modal testing results.
This research attempts to improve or validate dynamic models of complex electronic devices by using finite element model updating techniques.
1.2 Review on Electronic Equipment Modelling
At first, analytical studies of electronic equipments require establishing mathematical models, either for a single lead of a component or for a complex printed circuit board, or even for a complete electronic box. The most traditional and systematic work is the “Vibration analysis for electronic equipment” written by Steinberg [63]. First published in 1973, (2nd edition in 1988) this book is mainly related to traditional modelling methods, such as multi- degree-of-freedom (MDOF) lumped mass systems, beams, frames, flat plates, etc. For design engineers, these methods have not yet been out-of-date.
However, as electronic components are more and more sophisticated and their size has been continuously growing, especially the widespread use of large size SMCs, more accurate models are required. Not absolutely isolated, we may separate the interests of mod- elling electronic devices into four groups, as shown in Table 1.1. 4 Chapter 1. Introduction
Table 1.1: Interests of modelling electronic devices local-structure global-structure Static studies Static-Local Static-Global Dynamic studies Dynamic-Local Dynamic-Global
Wong, Stevens et al. [66] have studied SMT assemblies under static bending and twisting loads. The main concern of this study is the relationship between the individual solder joint load and the overall experimental value from the package pull test. To this end a hybrid analytical/experimental analysis is employed, which included using finite element method to determine displacements of the printed circuit board. Wong’s work is a contribu- tion to the relationship between Static-Local and Static-Global studies. Lau and Rice [41] reviewed the state of the art technology for SMT solder joint fatigue, and compiled useful data on the physical and mechanical properties of bulk solder and joints. Although the related theory has not been changed, many investigations, especially case studies, have been done. Recently, Ju, Sandor, and Plesha [32] have proposed two finite element approaches incorporating fracture mechanics and continuum damage mechanics to predict time-dependent and temperature-dependent fatigue life of solder joints. The main concerns of reliability investigation are the low cycle thermal loads of solder joints. Hence these studies are of static-local interests.
Lau and Keely [42] have studied the first three resonance frequencies of four different kinds of SMC leads with finite element analysis and experimental laser Doppler vibrome- ter measurement. Soldered and un-soldered leads of SMCs are modelled and tested and compared. For all these studied cases, the results showed that the fundamental resonance frequency of a lead with good soldered joint is very high (over 67.5kHz) in the direction perpendicular to the PCB. This implied that when modelling PCB/SMC, the leads can be considered stiff and their damping can be neglected. The research found that soldered and un-soldered leads have significant difference in resonance frequencies, and suggested as a 1.2. Review on Electronic Equipment Modelling 5 judgement for solder-joint inspection during a controlled manufacturing process. This is a typical example of dynamic-local investigation for electronic packaging. Lau [40] also analysed solder joint reliability under mechanical vibration, shock and thermal variation situations. The research found that vibration perpendicular to the PCB has more effect on solder joint reliability than in plane vibration. The failure time was influenced by both the PCB resonance frequencies and the excitation frequencies.
Barker [2] has studied the combined effects of vibrational and thermal loads on solder joint fatigue. The research found that the use of a simple low cycle power law fatigue model based on thermal strains will overestimate the actual life if the assembly is subjected to random vibration, especially when solder joint life predictions are required in the transition region between low cycle fatigue and high cycle fatigue (103 to 104 cycles).
Engel [11] presented an approximate structural analysis for the bending of circuit cards with module attachments. He named this method as an “Engineering Theory” and compared with results of finite element analysis. This research is a typical example of static- global analysis.
Wang [64] performed in 1990 an experimental modal analysis for a printed circuit card (both bare PCB and with a single SMT component on). This study attempted to “iden- tify” the SMC properties by trial-and-error with structural dynamic modification technique but found that the predictions from a complex spring-mass-damper model of the SMC were almost identical with those from a lumped mass model. A finite element analysis was also performed. The IC chip was modelled in four different ways: lumped mass; lumped mass with a stand-off height; change in stiffness; lumped mass with a stand-off height plus addi- tional stiffness. However, Wang did not fully compare or correlated the finite element results with those from tests. The study concluded that accurate identification of the properties of a SMC from vibration measurements was difficult because the changes in the system dy- namics produced by a single SMC was small. Finally this study also recommended to use 6 Chapter 1. Introduction non-contact measuring devices to avoid the error introduced by transducer weight.
Pitarresi [57] presented in 1991 a technique of smearing the material properties of a printed circuit card populated with electronic modules for a finite element analysis of its dynamic behaviour. The research showed that the smearing technique significantly reduced the complexity of the finite element modelling of PCB while retained good accuracy of the frequencies and mode shapes. It is also shown in this study that only the out-of-plane translational DOFs needed be retained as master DOFs in the reduced finite element model.
For the first time, Modal Assurance Criterion was used for comparing the test results with those from FEM for PCB dynamic analysis.
Kenda [35] analysed an electronic card with SMT components. In this study, both ex- citation (magnetic) and response measurement (Doppler Laser vibrometer) of the test system are performed without contact, which absolutely avoided errors conventionally introduced by transducer mass. Many efforts had been made for the modelling of electronic SMT compo- nents, however, no significant difference had been found between calculated results from various finite element models. This indirectly supported the “smeared” property technique proposed by Pitarresi. For the first time, this research tried to use finite element model error localization technique (Error Mass Method (EMM)) to validate and improve the analytical model of PCB.
In other studies [33] [34], Kenda employed a “User Acceptance Updating Method
(UAUM)” to improve finite element model of typical electronic structures of PCB whose two bounds are fixed with “Wedge-Lock” slides. Two error localization methods, EMM and
FBM, were under investigation with simulated and test cases. The study found that when it is possible to divide a complex structure into subsystems, for which model inaccuracies can be isolated, the UAUM leads to noticeable improvements of the model.
The preceding articles exhibit the importance of reducing the adverse effects caused by mechanical vibration in electronic equipments. It is the engineers’ desire to predict and 1.3. Review on Finite Element Model Updating 7 control dynamic properties of electronic packages during the design stage. For this purpose,
finite element method is usually the first chosen tool. However, these articles also showed the difficulties to accurately model complex electronic devices for dynamic analysis. We may conclude that to obtain an accurate model and to validate it for dynamic analysis of elec- tronic packages, a hybrid analytical/experimental procedure is essentially needed. The now highlighted hybrid technique, finite element model updating, seems to be the most helpful one.
1.3 Review on Finite Element Model Updating
As the finite element model updating is a hybrid technique of modal testing and finite element method, it may be rooted to the very beginning of both of the parent techniques. The literature review on finite element model updating is restricted to topics that are closely related to the forthcoming study.
1.3.1 The origin and the philosophy of finite element model updating
In late 1960s there had been attempts to use the results of dynamic testing to identify the parameters in the equations of motion [6]. There arose a problem that the number of de- grees of freedom of the system (the order of the identified system) is usually larger than the number of available modes from practical testing. In such a situation, there are infinite ana- lytical models that will duplicate the measured response with experimental errors. One may reduce the number of DOFs of the system to the number of measured modes to force a unique solution, but such a procedure will evidently reduce the reliability of the analytical model.
Berman [6] pointed out that the ability to duplicate test data does not make the model useful in itself, while a useful model should be able to predict the results of untested loading condi- tions and the effects of changes in the mass, stiffness, or supports of the structure. Therefore, 8 Chapter 1. Introduction instead of reducing the system order, Berman proposed that the parameters (masses) used in the model should be as near to the “true” vales as possible, while the best information available as to what the “true” values are, is the approximation arrived at by the analyst. This proposal turned system identification to a priori model fitting for dynamic structure analysis.
Even though Berman did not mention finite element method in this article, this proposal has been serving as part of the philosophy of finite element model updating. Therefore, it should be thought as the origin of the art of finite element dynamic model updating.
The other half part of the philosophy of finite element model updating is that the dynamic test results are more precise than those from analytical models’ prediction. This is evident because on the contrary, FE model updating makes no sense.
1.3.2 Uniqueness aspects of finite element model updating
We may conclude directly from the principles proposed by Berman that for a linear continuous structure no unique model can be yielded via using finite element model updating technique, because the mesh is arbitrarily defined by analysts and even for the same mesh the element shape function can be different which leads to different models. In fact for a continuous structure, no matter how fine the mesh has been made, an even finer mesh is always available! Therefore, topics related to the uniqueness of finite element modelling are nearly senseless and the uniqueness aspects of finite element model updating should be discussed under such a context that an initial finite element model has been reasonably established, that is, the DOFs and the connectivity of the model are unchangeable during model tuning. Berman [6] [4] addressed incomplete mode problems and concluded that no unique solution can be drawn from system identification. The most promising method selects a solution which minimizes changes in a reasonably good analytical model that includes con- straints to force physical reality of the solution. 1.3. Review on Finite Element Model Updating 9
Janter [31] viewed uniqueness via dynamic properties of a model, i.e., eigenfrequen- cies, mode shapes and modal masses. It seems more practical to compare models with their dynamic behaviour than with physical parameters. However, the incompleteness of modes is totally ignored by Janter.
When incompleteness of modes is taken into account, the non-uniqueness has be- come the basic feature of the finite element model updating. The strategies that force model updating to a unique solution are generally highly analyst-dependent.
In a broad sense, a finite element model updating process can be viewed as a strategy that fixes the non-uniqueness problems encountered in system identification for dynamic structures when using incomplete modes. Hence, every effort paid to model updating is a contribution to uniqueness aspects of updating solutions.
1.3.3 Match between test and analytical models
To judge the differences between experimental results and analytical predictions, the model updating techniques always require a match between experimental and analyt- ical models. In practical cases, however, experimental model has much fewer degrees of freedom because of the testing costs and the accessibility. To match an experimental model with its analytical counterpart, one can either reduce the size of the FE model or expand the measured data to the same size as the FE model. Logically, it is possible to match them via an intermediate model.
Reduction and expansion are originally a pair of terms used in dynamic finite element analysis. For the solution of dynamic finite element analyses, reduction techniques reduce the cost of the analysis. Reduction affects the accuracy of the finite element results. For most of the cases, reduction and expansion are reciprocally inverse procedures as they share the same transformation matrix. The most commonly used technique for this purpose is a static reduction or condensation, which is generally referenced as Guyan reduction [23]. 10 Chapter 1. Introduction
In FE model updating processes, reduction and expansion techniques are used to solve mesh incompatibility. In this context, more information, such as eigenvalues, is ready and consequently more convenient and flexible expansion methods can be derived, such as modal coordinate expansion [49], MAC expansion [45], SEREP [54]. It should be noted that dynamic expansion, which is a reciprocal procedure of dynamic reduction [36], has become a non-iterative process as the measured resonance frequencies and mode shapes are already available.
Several surveys were made on the expansion and reduction by Gysin [24], O’Callahan [55], Freed [16], etc. Lammens [38] documented most of the expansion/reduction methods.
The principles and concepts of model match for mode shapes can be used to develop similar techniques for frequency response functions [67] [21].
1.3.4 Correlation between measured data and analytical predictions
Model match techniques, expansion/reduction techniques, solve the spatial incon- sistence between experimental and analytical models. To evaluate the degree of agreement between experimental and FE models, for example, when using eigenvalues, it is necessary to pair modes. In the frequency domain, two close resonance frequencies, one from measure- ment and the other from prediction of a finite element model, do not necessarily mean that the two corresponding modes are also close, because finite element models are not error free.
It is the mode shapes that are more reliable for judging the degree of agreement between an- alytical and test modes. When the degrees of freedom are matched with expansion/reduction techniques for test and analytical models, the similarity of the mode shapes can be expressed by the vector angles θ:
φ0 φ xi · aj cos θij = |φ | |φ | (1.1) xi aj 1.3. Review on Finite Element Model Updating 11
φ φ th th where xi and aj are the i experimental mode shape vector and j analytical one, respec- tively. Instead of θ or cos θ, a more convenient expression is extensively employed:
φ0 φ 2 |φ0 · φ |2 2 xi · aj xi aj cos θij = |φ | |φ | = φ0 · φ · φ0 · φ (1.2) xi aj ( xi xi) ( aj aj )
The above expression is now the most broadly used for comparing mode shapes.
It is named as Modal Assurance Criterion (MAC) by Allemang [1]. Following the idea of
2 calculating cos θij, many criterions have been proposed to meet different requirements, such as COMAC (Co-Ordinate Modal Assurance Criterion [44]), MoFic (Modal Filter Criterion) and FRAC (Frequency Response Assurance Criterion [38]), and FDAC [56].
2 These criterions based on cos θij have the feature that their values always lie between zero and one. A zero value represents no consistent correspondence while a one means consistent correspondence.
1.3.5 Model tuning
Finite element model updating techniques can be sorted into two categories: global methods and local methods. Global methods directly update individual components of mass and stiffness matrices. The updated matrix models usually reproduce the measured data exactly. However, the updated matrix models are generally difficult to be interpreted with physical significance. Because of this, global methods are now generally thought as obso- lete. Local methods update either the coefficients of macro elements or physical parameters or both of them. The updated models thus evidently have direct physical significance. Ac- cording to the way of composing updating equations, local methods can be divided into two groups: the modal updating techniques and impedance updating techniques. The modal updating approach minimizes the errors of the modal properties while impedance updating minimizes the response errors. 12 Chapter 1. Introduction
Global methods
Global methods are single step approaches that directly update matrix components.
They are free from iterative divergence and are usually computationally efficient. An im- portant feature of these methods is that they reproduce the testing data exactly, which means that the measurement noise directly induces analytical model corrections and hence very high quality measurements become a basic requirement.
The mostly well developed global methods are Lagrange Multiplier methods. These methods minimize weighted differences between initial and updated model matrices with constraints such as symmetry conditions and/or orthogonality conditions. Baruch and Bar- Itzhack [3] initiated these methods while Berman and Nagy [5] expanded them.
Sidhu and Ewins [62] proposed an Error Matrix Method (EMM). The error matrices
∆K and ∆M are assumed small and derived via inverse orthogonality with incomplete modal data. These methods are more often mentioned in model error localization. With the idea of inverse orthogonality, Link [47], Caesar [7] proposed matrix mixing methods which use analytical modes to make up quasi complete eigenvalues.
Local methods
Local methods are generally iterative and sensitivity-based methods which minimize the norm of a residual vector expressing the difference between the test and analytical data.
As the residual vectors can be extensively varied, local methods should be referred to as a methodology in a broad sense. The residual vectors are composed basically related to:
• the difference between analytical and test resonance frequencies;
• the difference between analytical and test mode shapes;
• the off-diagonal elements of mixed orthogonality matrices; 1.3. Review on Finite Element Model Updating 13
• the residual force vectors deduced from the free vibration equations;
• the difference between analytical and test frequency response functions.
Residues are implicit and nonlinear functions of the updating parameters. They are linearized with first order truncation of Taylor’s expansion, which produces a sensitivity ma- trix [S] with respect to the updating parameters. With the sensitivity matrix [S], an updating equation set [S][∆p]=[ε] is formed. The updating equation set is usually solved under some physical constraint conditions (usually upper and lower boundary conditions) in least squares sense. Because the sensitivity [S] is only a first order truncation, the updating process is then iteratively conducted until some convergence criterions are reached.
The calculation of derivatives of eigenvalues and eigenvectors is an ancient topic. Fox
[15], Rogers [61], Nelson [52] and many others discussed this subject. All residues, when expressed with sensitivity, are based on the first order derivatives of the system matrices:
∂M| ∂K| pi=pi0 and pi=pi0 ∂pi ∂pi
The system matrix sensitivities are most often evaluated numerically by perturbation.
Nordmann [53] used a sensitivity method to analyse large vessel and piping sys- tem. Janter [30] developed a QA model updating method (Quality-Acceptance). Wei [65] discussed some error localization problems when physical parameters are directly updated.
Hemez and Farhat [25] applied a sensitivity-based model updating method to damage de- tection. Lin, Du and Ong [46] compared eigen-sensitivity and frequency response function sensitivity-based methods.
It should be mentioned that for any local method, the involved (physical) parameters are defined by the analysts. It is of paramount importance for analysts to ensure that the updated parameters covered all main realistic model errors. It is equally important to keep
“error free” parameters away from being updated when measured data is contaminated by 14 Chapter 1. Introduction noise, because the noise may be interpreted as model errors over all parameters. Besides the intuition and confidence, error localization techniques may help analysts to select parameter subsets.
Error localization and parameter subset selection
Error localization methods aim to indicate main inaccuracies in the finite element model according to experimental data. Logically every global updating method can be de- veloped to localize main model error areas, because in global methods no special part is assumed to be error free, which leaves opportunity of revealing errors hidden in the finite element models. Based on these global methods, various residues can also be developed to form error indicating functions. We may divide error localization methods into two cate- gories: location-oriented and parameter-oriented methods.
Force Balance Method (FBM)[13] and Error Matrix Method (EMM) [62] are the most often used location-oriented error localization methods. FBM calculates a residual force vector for each measured mode. The degrees of freedom that are not in balance indi- cate errors in the finite element model. For plural measured modes, an error indicator vector
(function) may be defined. FBM is a powerful and accurate error localization method be- cause this method does not take any assumption in the calculation. However, as any other method, the need for reduction of the system matrices or expansion of test mode shapes due to the mesh incompatibility problem troubles the accurate localization for practical appli- cations. As FBM is the most direct method, the errors in measured mode shapes are also directly interpreted as model errors. EMM localization process is the same as the global
EMM updating process, however, with the result error matrices, error indicator vectors are developed to indicate inaccuracies associated with particular DOFs. The incompleteness of the test modes and the mesh incompatibility have greatly troubled the usage of this method.
Parameter-oriented error localization methods are in fact sensitivity-based subset se- 1.4. Outline of the text 15 lection methods, which are directly related to local model updating methods. Therefore, they may share every disadvantage with local updating methods. It is not ensured that the analyst-picked parameters have covered the main inaccuracies, hence, the first step is to check whether the correcting space defined by the sensitivity matrix of chosen parameters spans the residual vector. Suppose the check has been past, the error localization is then iden- tical to select, from the full parameter set, a subset that may most satisfactorily correct the initial finite element model. When test noise appears, the subset selection becomes a most difficult problem in regression. Miller [51] collected various methods on this topic, how- ever, he made it clear that his monograph is not a cookbook of recommendations. Lallement
[37] proposed an indicator function based on the forward selection method. Link [48] pro- posed a similar error indicator function based on sub-space search concept. Fritzen [19] [21] used QR decomposition with column pivoting to select the most sensitive and independent sub-space for finite element model updating/error localization. Yang [67] applied SVD/QR subset selection to choose the most robust sub-space for updating parameters. Friswell [18] summarised the “best” sub-space approach to locate errors in a finite element model of a structure.
Besides the “best” sub-space approach, Link [48], proposed energy methods. Link calculated the elastic energy caused by the difference between measured and predicted with the original analytical model displacement for each substructure. A large value indicates an erroneous substructure while a small value indicates either a small error or an insensitive substructure.
1.4 Outline of the text
This text aims to develop a Parameter-Oriented Model Updating System (POMUS) that is applicable to modelling electronic structures. The text consists of two main parts. The 16 Chapter 1. Introduction
first part, Chapter 2 to 6, discusses theoretical problems of developing POMUS and intro- duces simulated case studies to assess various updating equations. The second part, Chapter 7, discusses modelling electronic structures and refining the FE models with POMUS.
In Chapter 2, firstly, correlation techniques are reviewed. Numerical comparison techniques are summarised in a simple way of vector angle measured with cos2 θ for MAC,
COMAC, FDAC and FRAC. Following the review, the model match techniques, namely re- duction and expansion methods, are then documented with consistent notations. In POMUS, model matching is achieved in two steps: first the analytical model is reduced via Guyan reduction; then, a selection is provided, either the model is further condensed to the experi- mental DOFs with dynamic reduction method,or the experimental vectors are expanded via dynamic expansion technique. In this chapter, the mode shape smoothing technique is also extended to FRF smoothing.
The theoretical bases of setting up the updating equations are developed in Chapter
3. Firstly, a generic form of sensitivity-based updating equations is proposed. Eigenvalue- based, modal and impedance updating equations are then developed in the consistent form.
In this chapter, a new modal updating approach, namely quasi-modeshape updating equa- tions, is proposed which employs SVD technique to convert the residual force vectors into quasi-modeshapes. As errors in modeshapes are not amplified by the dynamic stiffness ma- trix, QMS updating equations are believed more noise-resistant comparing to these from
FBM. Chapter 4 documents problem solving techniques. Topics cover regularization, equa- tion weighting and iterative model updating. Least-Square estimation is the most classical approach. With the powerful SVD technique, LS estimation can be extended to a Total-Least-
Square estimation. For ill-condition problems, SVD technique is also the most transparent regularization method. Chi-squares estimation is a particular case of Maximum Likelihood estimation. It best explains the principles of weighting FE model updating. 1.5. Research contributions and originalities 17
Chapter 5 discusses error localization techniques. After a brief review of global error localization methods, A new parameter-oriented localization technique is proposed and fully developed in this chapter. A success ratio of parameter error localization is also suggested to assess various updating equations. Four case studies are introduced to illustrate the proposed methods, which include a cantilever beam, a 10-DOFs mass-spring string system, a 3D space frame and a PCB with different boundary conditions.
Chapter 6 introduces the Perturbed Boundary Condition Testing into POMUS. The
PBC technique is believed to provide some distinctive test data. In this chapter, the POMUS scheme is extended to deal with PBC enriched experimental data. Three illustrating examples are enclosed.
Chapter 7 examines modelling methods of electronic structures with POMUS. Topics cover SMT components, “Wedge-Lock” slides and an ARIANE5 BCS B shell box. A survey of the current SMCs modelling techniques is first made. Equivalent thickness model is then chosen to establish simplified FE models to be updated with POMUS updating techniques. This strategy is then applied to a PCB of six SMT components with real test data. The
“Wedge-Lock” slides are modelled with rotational springs whose stiffness values may vary in a broad range. A two-step strategy is applied to fix the unknown stiffness. Finally, an attempt is made to refine coarse FE model of a shell box by using the experimental resonance frequencies.
Finally, Chapter 8 summarises the investigation and includes recommendations for further study.
1.5 Research contributions and originalities
In this research work, a systematic theory of sensitivity-based FE model updating has been established that unifies both modal and impedance approaches. Based on it, a 18 Chapter 1. Introduction
Parameter-Oriented Model Updating System (POMUS) has been developed. As a commer- cial FE software (ANSYS) is integrated, POMUS can deal with all physical parameters in FE models to cope with industrial problems. Besides the simulated studies, applications are made for a PCB with SMT components, “Wedge-Lock” slides and an electronic box.
The original contributions to the FE model updating area have been made in the following aspects:
1. A new modal updating method is proposed, namely QMS FE model updating equa- tions. Comparing to the FBM methods, the proposed method is much more noise-
resistant and therefore is believed more robust when applied with real test data.
2. A TLS/QR-F error localization technique is proposed. The proposed technique first
regularise and condense the updating equations by SVD in TLS sense, then performs
a QR forward subset selection with a modified permutation criterion. The selected
subset is further reduced by a backward excluding procedure according to a residue
descending ratio proposed in this research work. The subset solution is served as an error indication function that is employed to identify error parameters according to
threshold values that signify errors.
3. A success ratio of parametric error localization is proposed to assess the noise influence
in FE model updating. Though the proposed success ratio is not applicable to practical
problems, it is a helpful means to assess updating methods for simulated studies when
computer-generated noise data added. Chapter 2
Correlation and model match
2.1 Correlation techniques
The degree of agreement between analytical and experimental data must be evaluated before, during and after a finite element model updating process, to assess the quality of an initial model, to decide whether a correction is needed, and to judge the acceptance of the updated model. Correlation techniques provide mechanisms of comparing two sets of data. The correlation study aims to set some generally accepted criteria to estimate the agreement. Because every technique emphasizes only a few aspects of correlation, every possible technique should be used to get a comprehensive view of the correspondence.
The mesh incompatibility and the damping discrepancy are the most troublesome problems in correlation. Though not perfect, the mesh incompatibility can be solved by model reduction/expansion techniques. For model damping, unfortunately, no method is so far commonly accepted as an accurate technique. For structural dynamic analysis, damping is generally assumed to be linear and small. Therefore, eigenvector-based correlation is usually conducted in a way that measured complex mode shapes are transformed into real ones. Many methods can be found in literature [26]. In this work the transformation is
19 20 Chapter 2. Correlation and model match conducted inside the modal analysis software ICATS [28].
Comparisons between analytical and test data can be visually or/and numerically conducted. Visual comparison gives intuitive impression while numerical comparison may help the analysts to set up some criteria.
2.1.1 Visual comparisons
If two sets of data are one-to-one corresponded, a visual comparison can be con- ducted with the so-called 45o-plots. Data from tests are plotted in a xy-plot versus the corre- sponding data from calculation. If the data are relative, such as mode shapes, they are first scaled in the same way. If all data points lie on a 45o line through the origin, the two data sets then are fully consistent. The distances from data points to the 45o line give analysts intuitive impression about the correlation between the two data sets.
o Figure 2.1: 45 fx—fa comparison 45° Line graphic for frequencies 1000 (Hz) x 800 f
600 45° Line
400
200
f (Hz) 0 a 0 200 400 600 800 1000
The resonance frequencies can be easily compared with the 45o-plot. Figure 2.1 shows a comparison of a simulated free PCB (see §5.4.4). As all experimental resonance fre- quencies are over analytical ones, we may get a simple, general impression that the stiffness is under-estimated or the mass is over-estimated. However, without comparing mode shapes the correspondence between test and analytical frequencies is not guaranteed. The resonance 2.1. Correlation techniques 21 frequency differences give a good global evaluation of correlation between analytical and test models. No local information, however, will be revealed. Animation of the mode shape is another helpful technique. With this visual tech- nique, the analysts need not even to solve the mesh incompatibility problem. However, the comparison with animation only gives an intuitive impression. It is impossible to develop a criterion from this technique.
2.1.2 Numerical comparison
Mode shapes, frequency response functions are mathematically treated as vectors.
The similarity of two vectors (v1, v2) is most intuitively expressed by the angle θ between them, which is defined as:
0 v · v2 θ =cos−1 1 (2.1) |v1|·|v2|
However, it is not necessary to calculate the angle θ directly, since cos2 θ is extensively used:
0 2 2 (v1 · v2) cos θ = 2 2 (2.2) |v1| ·|v2|
Modal Assurance Criterion
φ φ If two mode shape vectors, xi from the test and aj from the analytical calculation, are compared using (2.2), the cos2 θ is then referred to as the Modal Assurance Criterion
(MAC) [1]:
|φ0 · φ |2 xi aj MAC(i, j)= φ0 · φ · φ0 · φ (2.3) ( xi xi) ( aj aj ) 22 Chapter 2. Correlation and model match
φ MAC(i, j) measures how the experimental mode shape xi is parallel to the analytical one φ aj . For all the available mode shapes, the MAC-values compose a MAC matrix. It is evident that the MAC values, as quantities of cos2 θ, lie between 0 and 1. A zero MAC value means the two mode shapes are perpendicular each other as θ =90o, while a unit MAC value indicates a consistent correspondence as θ =0o, and the two mode shapes are parallel. A high MAC value implies a good correlation between two modes. Till now, the MAC values are the basic references for mode pairing. For an experimental mode, the corresponding analytical mode is assumed as the one that shows the highest MAC value in combination with that experimental mode.
Coordinate Modal Assurance Criterion
The MAC value correlates analytical and experimental models globally in spatial
DOFs but locally in resonance frequency. The Coordinate Modal Assurance Criterion (CO-
MAC) correlates the global set of test and analytical mode shapes for each individual degree of freedom [44]. In this context, the components of vector v1 are the elements of test mode shapes at a specified DOF for all resonance frequencies, while the analytical eigenvector v2 includes the corresponding DOFs’ components of the correlated resonance modeshape. For ith degree of freedom, the COMAC value then calculated as: P x 2 Nm =1 |φxij · φaij | j COMAC(i)= P x P x (2.4) Nm · · Nm · j=1 φxij φxij j=1 φaij φaij where: th th φxij is the i component of j experimental mode shape vector;
th th φaij the i component of j analytical mode shape vector;
x Nm the number of experimental mode shapes. 2.1. Correlation techniques 23
COMAC is very sensitive to the way mode shapes are scaled because in this con- text the components of vectors v1 and v2 are affected individually by mode shape scaling. COMAC is not used as often as MAC since the interpretation is not always clear.
FRAC and FDAC
MAC and COMAC can be easily extended into FRF databases. In a modal parameter set, a mode shape is the vector of displacements of free vibration at a resonance frequency over all DOFs. In FRF databases, a frequency response function represents the response at a specified DOF over the frequency range of interests when excited at some specified DOF. Hence in the frequency domain, local comparison in spatial DOFs may be more convenient.
The Frequency Response Assurance Criterion (FRAC) [38] is an extension of CO-
MAC in the frequency domain. As FRFs are used, the frequencies are not restricted to res- onance frequencies but can extend to the entire measured frequency range. FRAC is hence calculated: P ∗ 2 ( |h (ωl) · hxij(ωl)|) FRAC(i)= P l aij P (2.5) ∗ · · ∗ · l haij (ωl) haij (ωl) l hxij(ωl) hxij(ωl)
where hxij(ωl) and haij (ωl) describe the experimental and analytical FRFs at round fre-
th th quency ωl between i and j DOFs, respectively; subscript l denotes the measured frequen- cies; superscript ∗ denotes the conjugate complex number.
A global frequency shift due to a global over- or under-estimation of stiffness or mass can lead to low FRAC values though the receptance may look very similar and can be matched with a simple shift of frequency. The damping also troubles the FRAC very much.
Detailed discussion can be found in literature [38].
The Frequency Domain Assurance Criterion (FDAC) [56] is an extension of MAC in 24 Chapter 2. Correlation and model match the frequency domain. FDAC is calculated: