Sensitivity-Based Finite Element Model Updating Methods with Application to Electronic Equipments

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 techniques, 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 descending ratios computed via QR decomposition; (4) excluding parameters based on the subset solution. To evaluate the proposed method, a success ratio of parameter error localization 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 element 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 heavily 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 modelling 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 contribution 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 vibrometer 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 “identify” 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 additional 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 dynamics 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 excitation (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 components, 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 electronic 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 degrees 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 analytical 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 conditions 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 constraints 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., eigenfrequencies, 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 become 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 analytical 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 measurement 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 analytical 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, respectively. 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 obsolete. 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 important 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 matrix [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 system. 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 developed 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 categories: 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 indicate 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 because 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 applications. 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 identical 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, however, 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] proposed 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 introduces 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 reduction 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 experimental 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 equations, 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 matrix, QMS updating equations are believed more noise-resistant comparing to these from

FBM. Chapter 4 documents problem solving techniques. Topics cover regularization, equation 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 equations. 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 conducted with the so-called 45o-plots. Data from tests are plotted in a xy-plot versus the corresponding 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 frequencies 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 technique, 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 context 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 resonance 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 frequencies; 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:

2 H h (ωx) · (haj (ωa) FDAC(ω ,ω ,j)= xj (2.6) x a H · · H · hxj(ωx) (hxj(ωx) haj (ωa) (haj (ωa)

where j denotes to the measured column of H matrix; ωx corresponds to the frequency at which the FRF hxj vector is measured and ωa corresponds to frequency at which haj vector is calculated; the superscript H denotes the Hermitian transpose. As ωx and ωa can be over all frequencies of interest, the size of FDAC-matrices can be tremendously larger than the size of MAC matrix, as only resonance frequencies are taken into account when calculating

MAC matrix.

2.1.3 Other comparisons

The most often referred feature of an undamped system is its modal orthogonality:

0 0 Φa · M · Φa = diag(m1,m2,... ,mn); Φa · K · Φa = diag(k1,k2,... ,kn) (2.7)

where M and K are mass and stiffness matrices of FE model; Φa is the mode shape matrix; m1,m2,... ,mn and k1,k2,... ,kn are the modal masses and modal stiffness, respectively. To assess the agreement between two data sets, eigenvectors are replaced partially or fully with test mode shapes [26]:

• Cross orthogonality is expressed (eigenvectors partially replaced):

0 0 M(a,x) = Φa · M · Φx; K(a,x) = Φa · K · Φx (2.8) 2.1. Correlation techniques 25

• The mixed orthogonality is (eigenvectors fully replaced):

0 0 M(x,x) = Φx · M · Φx; K(x,x) = Φx · K · Φx (2.9)

The off-diagonal elements of these orthogonality matrices should be small relative to the corresponding diagonal elements if a good correlation is expected. The absolute ratios are desired to be less than 0.1. The evaluation of the cross and mixed orthogonality requires normalized experimental mode shapes and either reduction of the analytical model or expansion of tested mode shapes. In practical cases the above orthogonality is quite affected by the normalization, expansion or reduction of mode shapes, and therefore, not quite often used.

Static mass check [38] may also be applied when the tested structure is isolated. The real mass of the structure, mx, is most often known or easy to capture, while the global mass of the ﬁnite element model is computed as:

0 ma = r · M · r (2.10) where r is a unitary rigid body translation.

Theoretically, the global mass moments of inertia can be compared too. But the corresponding experimental values are rather difﬁcult to measure.

2.1.4 Remarks

Correlation techniques are essential approaches to assess the validity and the quality of analytical models. Visual comparison methods provide the most intuitive ways. Analysts should conduct as much as possible visual comparison in the preliminary steps of ﬁnite element model updating, which may generally lead to intuitive reﬁnements of the analytical models. Most of the numerical comparison methods are based on evaluation of cos2 θ for 26 Chapter 2. Correlation and model match vectors. Among them, the MAC value is the most important criterion. FDAC is an extension of the MAC in the frequency domain. However, in the vicinity of resonance frequencies where the analysts should pay central attention, FDAC is affected very much by the assumed damping in the analytical models, while the damping is usually the most imprecise part of the analytical models.

Damping and mesh incompatibilities are the most troublesome problems in the correlation. In modal parameter updating methods, experimental mode shapes are normalized to real numbers, while in FRF based updating methods, damping is artificially assumed. Both approaches are not on sound bases. Mesh incompatibility is solved by either reducing finite element model or expanding test data set. Reduction destroys the connectivity of analytical model while expansion extrapolates test data using incorrect finite element model. It is possible to calculate a MAC matrix that correlates the test mode shapes to analytical “mode shapes” which are collected only from those DOFs that correspond to tested DOFs. In such a way, a MAC matrix is computed without expanding test data or reducing analytical model with specified methods.

2.2 Model match techniques

The measured degrees of freedom are usually far fewer than those of a finite element model. To simulate the continuum and complex shape of a structure, finite element analysis requires fine meshes to accurately predict dynamic behaviours. However, for measurement, the continuum and complex shape are self-guaranteed by the tested structure, therefore, it is not necessary for a test model to have the same mesh as the finite element model. Even if the test model has the same mesh as the finite element model, it is not practical or impossible to measure all degrees of freedom on the real structure because:

1. Inner DOFs are inaccessible to measurement; 2.2. Model match techniques 27

2. Rotational DOFs are notoriously difﬁcult to measure.

Unfortunately, nearly every updating method requires a one-to-one correspondence between analytical and test DOFs. The number and location of DOFs must be identical for both models. Therefore, a preliminary step is necessary for the association of the test and analytical models. This section summarises some of the reduction/expansion techniques for this purpose.

Generally speaking, the reduction and expansion procedures are reciprocally inverse processes for ﬁnite element analyses as they share the same transformation matrices. When experimental data are involved, the means of mode expansion are extended.

We catalogue the methods into two sorts: system-matrix-based and modal-shape- based reduction/expansion methods. Interpolation by applying spline functions is not referenced as its application is quite limited.

2.2.1 System-matrix-based reduction/expansion

For a linear, undamped discrete system, the governing equations of motion are expressed as:

Mx¨(t)+Kx(t)=f(t) (2.11) where: M is the mass matrix; K the stiffness matrix;

x¨(t) acceleration response vector;

x(t) displacement response vector;

f(t) the excitation force vector.

System-matrix-based reduction/expansion methods are all derived from the above 28 Chapter 2. Correlation and model match equations with special assumptions. For reduction/expansion purposes, the degrees of freedom of a vibration system are divided into master and slave DOFs while the mass and stiffness matrices are partitioned correspondingly:      

xM  MMM MMS KMM KMS x =   ; M =   ; K =   ; (2.12) xS MSM MSS KSM KSS

0 0 0 where MMS = MSM; KMS = KSM; the apostrophe denotes matrix transpose; indices M and S denote master and slave DOFs, respectively. The master DOFs usually corresponds to the measured DOFs while the slave DOFs corresponds to unmeasured DOFs when treating the model matching problems.

Guyan static reduction/expansion [23]

When dynamic (inertia) force Mx¨(t) is small enough comparing to Kx(t), equations

(2.11) are rewritten with partitioned matrices as:      

KMM KMS xM  fM    ·   =   (2.13) KSM KSS xS fS

The slave DOFs are supposed free from excitation: fS =[0]. From equations (2.13) we get:

· − −1 0 · KGuyan xM =(KMM KMSKSSKMS) xM = fM (2.14) and    

xM   I  x =   =   · xM = TGuyan · xM . (2.15) − −1 0 xS KSSKMS 2.2. Model match techniques 29

If the kinetic and potential energies of a structure are denoted as

1 1 T = x˙ 0Mx˙; V = x0Kx 2 2 and the above transformation is employed, the results are

1 1 T = x˙ 0 T0 · M · T · x˙; V = x0 T0 · K · T · x 2 M Guyan Guyan 2 M Guyan Guyan

The reduced stiffness and mass matrices are:

0 · · − −1 0 KGuyan = TGuyan K TGuyan = KMM KMSKSSKMS 0 · · MGuyan = TGuyan M TGuyan (2.16) − −1 0 − −1 0 0 · 0 − −1 0 = MMM MMSKSSKMS (KSSKMS) (MMS MSSKSSKMS)

Equation (2.15) and (2.16) are the Guyan expansion and reduction formula respectively. As it is assumed that the inertia force can be neglected, Guyan reduction/expansion is only valid for lower modes. It is practically recommended that the number of master DOFs must be greater than two times the modes of interest. This is often satisﬁed in practical problems.

Kidder dynamic reduction/expansion

The dynamic reduction/expansion method, also named as Kidder [36] method takes into account the dynamic force of the vibration system. Transforming equations (2.11) into the frequency domain, we obtain:

(K − ω2M)x(ω)=Z(ω)x(ω)=f(ω) (2.17) 30 Chapter 2. Correlation and model match where Z(ω)=[K − ω2M] denotes the dynamic stiffness matrix; x and f are displacement and excitation force vectors in the frequency domain, respectively. Applying partitioning equations (2.12) and supposing that the slave DOFs are free from excitation forces, we can rewrite equations (2.17) as:      

ZMM ZMS xM  fM    ·   =   (2.18) ZSM ZSS xS 0

The displacement vector of slave DOFs, xS, is solved from the second block of the rows as:

− −1 xS = ZSSZSMxM (2.19)

Substituting (2.19) into the ﬁrst block of rows we have the reduced dynamic equations as:

− −1 ZDyn(ω)xM =(ZMM ZMSZSSZSM)xM = fM (2.20) where

− · −1 ZDyn(ω)=ZMM ZMS ZSSZSM (2.21) and

2 2 ZMM = KMM − ω MMM; ZMS = KMS − ω MMS; (2.22) 2 2 ZSS = KSS − ω MSS; ZSM = KSM − ω MSM.

From equations (2.19) we obtain a transformation formula as:    

 xM   I  x =   = TDynxM =   · xM (2.23) − −1 xS ZSSZSM 2.2. Model match techniques 31

The dynamic reduction is frequency dependent. For this reason it is not convenient for searching system resonance frequencies in FE analysis. However, when used to expand measured modes, as the resonance frequencies and their corresponding mode shapes are ready, the expansion process becomes a direct method.

When speaking about “mode shape expansion”, the system is in a free-vibration situation, which means fM =[0] while xM and xS become mode shapes ΦM and ΦS. Because neither the analytical model Z nor the test mode shape ΦM is error free in practical cases, the redundant equations of the ﬁrst block of rows in (2.18) may also be employed to enhance the expansion. Equations (2.18) are rewritten with fM =[0] and mode shapes ΦM , ΦS as:      

 ZMM ZMS   ΦM   0    ·   =   (2.24) ZSM ZSS ΦS 0

ΦS can be solved in three different forms:

Φ − −1 Φ S = ZSSZSM M (2.25)

Φ − + Φ S = ZMSZMM M (2.26)

  +  ZMM  Φ − ·   · Φ S = ZMSZSS M (2.27) ZSM where superscript + denotes the pseudo-inverse operation of a matrix. Equations (2.25) are the same as (2.19). The Guyan and Kidder methods are the most fundamental methods. Various reduction methods based on system matrices can be found in literature [55]. 32 Chapter 2. Correlation and model match

2.2.2 Modeshape-based expansion procedures

It is not practical to develop mode shape-based reduction methods for ﬁnite element calculation, however, when mode shapes (both from test and calculation) are ready, the expansion of measured mode shapes becomes much easier. The most direct expansion approach is the Eigenvector Mixing method.

Eigenvector Mixing Method

The missing components of the ith experimental mode shape may be ﬁlled with the corresponding analytical mode shape components:    φ  φ xmi Ei =   (2.28) φ aui where: φ th Ei is the i expanded mode shape; φ th xmi,thei experimental mode shape of measured DOFs; φ th aui, the part of the i FE eigenvector of unmeasured DOFs.

Before mixing the eigenvectors, experimental modes must be correlated and scaled such that:

||φ || ||φ || xmi 2 = ami 2 (2.29)

φ th where ami is the part of the i FE eigenvector corresponding to measured DOFs. The Eigenvector Mixing is a highly inaccurate technique. It is essential to correlate the reduced mode shapes for pairing them. Only highly correlated modes may be expanded with success for this method. 2.2. Model match techniques 33

MAC expansion

This method [45] is similar to Eigenvector Mixing but uses plural mode shapes with the MAC matrix as weighting factors. The analytical modes are scaled in the same way as for the eigenvector mixing method. There is no suitable physical or mathematical explanation for this method. The unmeasured components of test mode shapes are calculated as:

Φxu = Φau · [MAC] (2.30)

The measured components of mode shapes can also be “smoothed” by:

Φ∗ Φ · xm = am [MAC] (2.31)

Modal Coordinate Method

Theoretically, modal coordinate method [49] uses the complete eigen-solutions from the FE model to expand the measured modes. The complete coordinates in the experimental model are generated by assuming that each mode is constructed from a linear combination of the analytical modes. For rth experimental mode:

      (r) φ Φ Φ (r)  xm   am1 am2   q1    =   ·   (2.32) φ Φ Φ (r) u au1 au2 q2 where subscript “1” and “2” indicate used and unused analytical modes, respectively. In practice only low modes are calculated for analytical models because of the huge calculating cost. This means q2 is set to a zero vector: q2 =[0]. However, this slight change has thoroughly destroyed the mathematical base. Consequently, more careful attention must be paid on choosing the incomplete space base, that is, the incomplete eigen-solution from the

FE model, although theoretically any number of modes could be used. The MAC matrix of 34 Chapter 2. Correlation and model match reduced mode shapes is helpful for this purpose:

2 (φ0 · φ ) MAC (i, j)= xmi amj (2.33) R |φ |2 ·|φ |2 xmi amj

With the selected analytical modes, for all test modes to be expanded, equation (2.32) is rewritten as:    

 Φxm   Φam    =   · q1 (2.34) Φxu Φau where the coefﬁcient matrix is solved with the ﬁrst block of rows as:

Φ+ · Φ q1 = am xm (2.35)

The unmeasured DOFs are expanded with:

Φ Φ · Φ · Φ+ · Φ xu = au q1 = au am xm (2.36)

The coefﬁcient matrix can also be used to smooth the measured components:

Φ∗ Φ · Φ · Φ+ · Φ xm = am q1 = am am xm (2.37)

System Equivalent Reduction Expansion Process (SEREP)

This method [54] relies on the assumption that the total modal matrix of a system can be expressed as a linear combination of components of partial modal matrix: 2.2. Model match techniques 35

 

 Φm    = TSEREP · Φm (2.38) Φu

The transformation matrix TSEREP is similar to a ﬁnite element shape function, but the “element” is now a super element which represents the whole vibration system. A Least-

Square solution of TSEREP can be obtained as:       + (1)  Φm   Φm · Φ   T  · Φ+ m SEREP TSEREP =   m =   =   (2.39) Φ Φ · Φ+ (2) u u m TSEREP

(2) When applied to test mode shape expansion, TSEREP can have two versions:

• Finite element data based:

(2) Φ · Φ+ TSEREP = au am (2.40)

• Finite element-Experimental data based:

(2) Φ · Φ+ TSEREP = au xm (2.41)

The expanded components of slave DOFs of test mode shapes are:

• Finite element data based:

Φ Φ · Φ+ · Φ xu = au am xm (2.42) 36 Chapter 2. Correlation and model match

• Finite element-Experimental data based:

Φ Φ · Φ+ · Φ xu = au xm xm (2.43)

Comparing (2.42) with (2.36) we know that Modal Coordinate Method and SEREP (1) may produce the same results. Just as other expansion techniques, TSEREP can be used to smooth measured data, and if we use ﬁnite element data based version, we obtain the exactly same smooth expression as (2.37).

2.2.3 FRF smoothing

The “raw” mode shapes can be smoothed by using equations (2.31) or (2.37). How- ever, this procedure cannot directly be used to smooth frequency response functions, because the frequency shifts are not automatically implemented between the two data sets. The following strategy can be used to solve this problem.

• The selected updating frequencies fui’s are associated to experimental resonance fre-

quencies fxi’s by an indicator vector: fui ⇒ fxi for all ui’s to xi’s;

• Experimental modes are correlated with analytical ones by assessing MAC matrix:

fxi ⇒ faj for all xi’s to their correlated aj’s;

• Let df = fui −fxi; fajxi = faj + df and compute hae for all fajxi’s, as shown in Figure 2.2;

• If FDAC value between hx(fui) and hae(fajxi) is larger than a certain minimal value:

FDAC(hx(fui), hae(fajxi)) > minF DAC,thenfui kept;

• Use hae of all fajxi’s as the base space, follow the same principle as applied in (2.37) to smooth FRF data: 2.2. Model match techniques 37

⇒ + ⇒ ∗ + hx(fx)=hae(fa)Q; Q = haehx; hx(fx)=hae(fa)hae(fa) hx(fx);

Figure 2.2: Associate hae to hx −1 faj fxi −2 fajxi fui df df −3

−4

log10(abs(h)) −5 hae h x

−6 h a

−7 150 200 250 300 Frequency(Hz)

2.2.4 Remarks

For dynamic analysis of a ﬁnite element model, as the number of degrees of freedom increases, the computational cost augments rapidly. It is not necessary and not practical to calculate all modes of analytical models. The reduction techniques then help us to reduce the cost of the analysis by decreasing the number of degrees of freedom used in the analysis.

Although a variety of reduction techniques have been proposed, the Guyan (static) reduction is still most often used. For Guyan reduction the number of master DOFs is preferably between two and ten times the number of modes that are required. This is usually satisﬁed in most practical cases.

In the model updating processes, reduction and expansion techniques are applied to solve model matching problems between analytical and experimental data. In modal- parameter-based model updating methods, expansion techniques are often used because the connectivity of the analytical model is retained. Reduction techniques are often used in FRF based model updating as the calculation of full size FRF may introduce tremendous 38 Chapter 2. Correlation and model match computation cost.

It should be aware that expansion of experimental vectors is always a sort of extrapolation. The accuracy of the expanded experimental vectors depends on the validity of the shape functions of extrapolation, which generally depends on the initial ﬁnite element model.

In order to keep the computational cost reasonable, in POMUS, it is strongly recommended to reduce the analytical model via Guyan static condensation, into a smaller model.

To match it with experimental vectors, POMUS provides two options: either the model is further condensed with dynamic reduction method, or the experimental vectors are expanded with dynamic expansion technique, because the dynamic reduction/expansion technique is known as “exact” method, which means if the model is exact, the reduction/expansion is also exact. Chapter 3

Sensitivity-Based FE Model Updating Equations

3.1 Introduction

Finite element model updating may be conducted either globally or locally.Ifthe design variables are not preselected in the tuning process, then the updating procedures globally modify all components of system matrices. Otherwise,the updating procedures locally modify only the preselected parameters. Global model tuning approaches have an important feature that they reproduce the measured data exactly. Global methods are also one step

(non-iterative) approaches and thus no divergence problem will ever arise. However, it is difficult to interpret these matrix models with physical significance as measured noise is fully transformed into model modifications. Therefore, global approaches as model tuning methods are now considered obsolete. With the non-iterative and non-preselected-design-variable features, global methods may be suitable for assessing the correlation between initial finite element models and experimental data. They may also be developed to localize main model errors.

39 40 Chapter 3. Sensitivity-Based FE Model Updating Equations

Sensitivity-based model updating methods modify preselected design variables, described by vector p. As most of the design variables describe local features of ﬁnite element models, sensitivity-based methods are generally referred to as local model updating methods.

For sensitivity-based methods, updating equations can be expressed in a generic form:

Xn ∂f ( ) ( ) i ∆p ≈ ∆f = f U − f I (i =1, 2,... ,m) (3.1) ∂p j i i i j=1 j

(U) (I) where fi denotes a dynamic quantity vector of the updated structure and fi is the counter- (U) part vector of the initial model. fi is derived from experimental data. One of the paramount key steps is selecting design variables, which will be discussed in Chapter 5. Constraints are important for successful model updating. For local methods, most of the physical constraints are imposed implicitly. For example, the connectivity of nodes is automatically kept during the model tuning process. Least-squares technique automatically imposes smallest modiﬁcation to the weighted design variables. Since experimental data are contaminated with noise and sensitivities are computed with some imprecision, some design variables may go out of their physical range during the iterative solution. Such a situation certainly causes serious numerical problems or even breaks down the updating process. Therefore, boundary constraints should be imposed to make an updating process robust.

This chapter mainly concerns the establishment of sensitivity-based updating equations and some strategies for solving the problem. Error localization techniques will be discussed in Chapter 5. 3.2. Eigenvalue-based updating equations 41

3.2 Eigenvalue-based updating equations

The resonance frequencies and mode shapes are the most considered properties of a dynamic structure. The resonance frequencies are the most easily and most accurately measured global dynamic properties. The correlation of eigenvalues between analytical model and experimental data is essentially required for all updating approaches.

To evaluate their difference, eigenvalues must ﬁrst be paired between two data sets.

Resonance frequencies, by themselves, cannot be exactly paired. The MAC matrix then served as a criterion for mode pairing. High MAC values generally (≥ 0.8) imply a good correlation between them. Assuming that the realistic structure can be represented by an updated model from a given initial ﬁnite element model, then eigenfrequencies of the updated model can be replaced by experimental resonance frequencies. When eigenfrequencies are taken as the dynamic quantity, updating equation (3.1) can be expressed as follows:

Xn ∂f ai ∆p ≈ ∆f = f − f (i =1, 2,... ,m) (3.2) ∂p j i xi ai j=1 j where:

th fxi is the i experimental resonance frequency; m, the number of total modes used for updating;

fai, the analytical eigenfrequency corresponding to fxi;

th pj,thej variable parameter of the analytical model;

∆pj, the correction of the design variable pj; n, the number of total variable parameters; ∂fai ith jth p ∂pj ,the frequency partial derivative with respect to the parameter j. 42 Chapter 3. Sensitivity-Based FE Model Updating Equations

Equation (3.2) can be expressed in matrix form as:       ∂fa1 ∂fa1 ∂fa1 ... ∆p1 ∆f1  ∂p1 ∂p2 ∂pn             ∂fa2 ∂fa2 ∂fa2       ...  ∆p2  ∆f2   ∂p1 ∂p2 ∂pn  ·         =   (3.3) ......   ...  ...        ∂fam ∂fam ... ∂fam ∆p ∆f ∂p1 ∂p2 ∂pn n m

∂fai The frequency partial derivative, ∂pj , can be easily deduced from undamped free vibration equations:

− 2 2 · φ [K 4π fai M] ai =[0] (3.4)

Differentiating (3.4) with respect to pj, φ ∂K − 2 2 · ∂M − 2 ∂fai · · φ − 2 2 · · ∂ ai 4π fai 8π fai M ai + K 4π fai M =[0] (3.5) ∂pj ∂pj ∂pj ∂pj

φ 0 φ 0 · [ − 4π2f 2 ]=[ ] φ 0 · · φ =1 ∂fai Pre-multiplying (3.5) by ai , with ai K aiM 0 and ai M ai , ∂pj is obtained:

∂fai 1 · φ0 ∂K φ − 2 2 φ0 ∂Mφ = 2 ai ai 4π fai ai ai (3.6) ∂pj 8π fai ∂pj ∂pj

In a similar way, a mode shape residue can be deﬁned as:

∆φij = φxij − φaij (3.7)

Unfortunately, the sensitivities of eigenvectors cannot be calculated directly. Mathematically, the eigenvector derivative can be expressed as a linear combination of all the eigenvectors of 3.2. Eigenvalue-based updating equations 43 the system:

φ XN ∂ ai (j)φ = βil al N is the number of total eigenvectors (3.8) ∂pj l=1

φ0 Substitute (3.8) into (3.5) and pre-multiply by ak, then:

XN φ0 ∂K − 2 2 ∂M − 2 ∂fai φ φ0 − 2 2 (j)φ ak 4π fai 8π fai M ai + ak[K 4π faiM] βil al =0 (3.9) ∂pj ∂pj ∂pj l=1

Because

φ0 φ φ0 φ 6 akK al =0; akM al =0; if k = l (3.10) φ0 φ 2 2 φ0 φ akK ak =4π fak; akM ak =1;

We have

φ0 ∂K − 2 2 ∂M − 2 ∂fai φ 2 2 − 2 2 (j) ak 4π fai 8π fai M ai +(4π fak 4π fai)βik =0 (3.11) ∂pj ∂pj ∂pj

6 (j) In the case k = i, βik can be solved from (3.11) as: φ0 · ∂K − 4π2f 2 ∂M · φ ( ) ak ∂pj ai ∂pj ai β j = ; k =6 i (3.12) ik 2 2 − 2 2 4π fai 4π fak

(j) When k = i, βii can be calculated from differentiating the mass normalization condition φ0 φ aiM ai =1:

φ φ0 ∂Mφ φ0 ∂ ai ai ai +2 aiM =0 (3.13) ∂pj ∂pj 44 Chapter 3. Sensitivity-Based FE Model Updating Equations

(j) Substitute (3.8) into (3.13), βii is obtained as:

(j) −1φ0 ∂Mφ βii = ai ai (3.14) 2 ∂pj

When the analytical eigenfrequencies are repeated, the correlation and the computation of derivatives should be conducted with great caution. Experimental data can hardly produce repeated eigenfrequencies, since the excitation is usually applied in a given direction (the mode that absorbs most energy from the excitation force will preferentially be excited). Moreover, the extraction programs may be not robust when multiple modes are present. Assuming m a a a φ repeated eigenfrequencies are f1 = f2 = ...= fm, then the associated eigenvectors ai are not unique. Any linear combination of the associated eigenvectors is also an eigenvector. If

Φ φ φ φ a =[ a1, a2,... , am] (3.15) denotes one particular set of mass normalized and orthogonal eigenvectors, then

Ψ = Φa · P (3.16) is also a set of mass normalized, orthogonal eigenvectors for any orthogonal matrix P.If φ an experimental mode xi is potentially correlating to one of the analytical multiple modes φ φ Ψ Φ aj , the MAC value must be calculated between xi and instead of a. Let one particular eigenvector as:    u1    φ ·  .  Φ · || || aj = φ ... φ  .  = a u ( u 2 =1) (3.17) a1 am  

um 3.2. Eigenvalue-based updating equations 45

φ φ u should be solved such that the MAC value between xi and aj gets a maximum value:

0Φ 0φ φ0 Φ u a xi xi au (φ , φ )= → max (|| ||2 =1) MAC xi aj φ0 φ 0Φ Φ u (3.18) xi xiu a au

It should be stressed that (3.12) is invalid when repeated eigenfrequencies appear. A recent review and detail discussion of the derivatives of repeated eigenfrequencies and their associated eigenvectors can be found in literature [17].

The use of mode shape differences has some drawbacks:

• The computation of the sensitivities is much time-consuming and can be quite inaccurate. Mathematically, all analytical modes are required in (3.8) while practically, a

limited set of modes is available.

• Individual components of experimental mode shapes are not measured with much ac-

curacy, although overall displacement shapes may be quite acceptable. According to

Eckert [9], the “reasonable” error in components of mode shapes is about 3% with

respect to maximum component. It should not be a surprise to ﬁnd some individual components having 20% errors with respect to the corresponding “true” values.

• The scaling imprecision of measurement amplitudes at different test nodes heavily

affects the updating.

• Normalization of complex experimental mode shapes must be applied and is crucial.

Few successful updating examples based on the use of individual mode shape differences have been reported in the literature. Therefore, in POMUS, the individual mode shape differences are not used to establish updating equations. 46 Chapter 3. Sensitivity-Based FE Model Updating Equations

3.3 Modal updating equations

Modal updating equations are established with modal parameters. Force Balance Method (FBM) updating equations are classical. However, FBM model updating is known as sensitive to measurement noise. A new set of modal updating equations, named as quasi- modeshape (QMS) updating equations, is developed in this section.

3.3.1 FBM updating equations

In the frequency domain, the free vibration equation for an undamped system is expressed as:

− 2 φ [K ωaiM] ai =[0] (3.19)

φ When ωai and ai are replaced with experimental data, as the FE model is not perfect, the equilibrium of the dynamic equation is broken. A residual force vector is then deﬁned as:

− 2 φ e =[K ωxiM] xi (3.20)

For an initial model, the residual force vector is:

(I) (I) − 2 (I) φ ei =[K ωxiM ] xi (3.21)

For the updated model, the residual force vector is supposed as:

(U) (U) − 2 (U) φ ei =[K ωxiM ] xi =[0] (3.22) 3.3. Modal updating equations 47

(I) (U) (I) (U) Substituting ei, ei and ei for fi, fi and fi , respectively, we can rewrite equation (3.1) as:

Xn ∂K ∂M [( − ω2 )φ ]∆p = −[K(I) − ω2 M(I)]φ (i =1, 2,... ,m) (3.23) ∂p xi ∂p xi j xi xi j=1 j j

Both resonance frequencies and corresponding mode shapes are involved in equation

(3.23). Either the expanded experimental mode shapes or the reduced analytical model can be adopted. FBM updating equations are generally overdetermined.

3.3.2 Quasi-modeshape (QMS) updating equations

However, it is important to be aware that the residues of (3.23) are components of unbalanced force vectors, which are never measured directly. Measured responses are generally accelerations or velocities. If the noise of the measured responses is normally distributed with the same standard deviation, then the noise in individual components of a mode shape should also be normally distributed with the same standard deviation since the mode shape is proportional to the vector of measured quantities. However, the standard deviations of residual force vector components are not constant since they have been disturbed by the − 2 multiplication by [K ωxiM]. The modification of standard deviations will trouble the conventional least squares regression (see §4.7). One may want to eliminate the influence by − 2 −1 pre-multiplying equation (3.23) by the inverse matrix [K ωxiM] , but unfortunately, ma- − 2 trix [K ωxiM] is near singular if the initial finite element model is not too far away from the realistic structure. − 2 −1 − 2 Instead of [K ωxiM] , a pseudo-inverse of matrix [K ωxiM] is now used to − 2 φ transform the force residual vector [K ωxiM] xi into a quasi-modeshape. Let the dynamic 48 Chapter 3. Sensitivity-Based FE Model Updating Equations

− 2 § stiffness matrix [K ωxiM] be decomposed via SVD as (see [22], also 4.2 on page 57):   0  V1  − 2 · · 0 · · [K ωxiM]=U S V =[U1 U2] diag(s1,s2,... ,sk,... ,sn)   (3.24) 0 V2 where:

U, V are unitary matrices; U ∈

n×k n×(n−k) U1, U2 partitioned matrices of U; U1 ∈< , U2 ∈< ;

n×k n×(n−k) V1, V2 partitioned matrices of V; V1 ∈< , V2 ∈< ;

S is the singular value matrix,S = diag(s1,s2,... ,sk,... ,sn); n the dimension of the square matrices M and K; k an estimator of the rank number of the dynamic stiffness matrix; then we have the pseudo-inverse matrix deﬁned as [22, p.243]:     + 0 Σ1 0 U1 − 2 + Σ+ 0 [K ωxiM] = V1 V2     = V1 1 U1; (3.25) 0 00 U2 where Σ+ = diag( 1 ,..., 1 ). 1 s1 sk

The singular values s1,s2,... ,sn are the square roots of the eigenvalues of matrix − 2 · − 2 0 [K ωxiM] [K ωxiM] .Whenωxi approaches to an analytical resonance pulsation ωai, | − 2 |≈ − 2 · − 2 0 as K ωxiM 0, matrix [K ωxiM] [K ωxiM] will have a very small eigenvalue.

Therefore, the last singular value should be excluded. If multiple modes are found at ωai, then the same number of small eigenvalues will be present. Consequently, the same number of singular values as the repeated eigenfrequencies should be excluded. 3.3. Modal updating equations 49

− 2 + Pre-multiplying equation (3.23) with [K ωxiM] , we obtain:

Xn + 0 ∂K 2 ∂M 0 V1Σ U [( − ω )φ ]∆p = −V1V φ (3.26) 1 1 ∂p xi ∂p xi j 1 xi j=1 j j

0 In the right hand side of (3.26), an incomplete space base is constructed with V1V1. φ The original modeshape xi is then projected to this incomplete space. We may refer to 0 φ 0 the projected vector V1V1 xi as a quasi-modeshape. As V is an unitary matrix, V1V1 will not very much trouble the quasi-modeshapes. When the analytical model is approaching to

0 the correct one, the incomplete space V1V1 will become more and more orthogonal to the φ φ modeshape xi. When the analytical model becomes perfect one, xi is in the range of V2, 0 that is the null space of V1V1.

0 Figure 3.1 illustrates the diagonal feature of a typical matrix V1V1 for a free PCB structure (see §5.4.4), Figures 3.2 and 3.3 anticipate the results which will be obtained in Chapter 5, illustrating the error localization success ratios (see §5.3.1, (5.65) on page 108) obtained with FBM and QMS updating equations for the PCB structure, respectively. Com- paring Figure 3.2 with 3.3, it can be seen that when measurement noise is low, both of FBM and QMS methods produce the same results, however, as the noise increases, QMS method leads to better results within a certain noise level.

0 Figure 3.1: Matrix V1V1 Figure 3.2: SR of FBM Figure 3.3: SR of QMS (with FBM method) (with QMS method) TLS/QR−F method TLS/QR−F method 100 100 1.2

1 80 80 0.8

0.6

0.4 60 60

0.2

0 40 40 Success ratios % −0.2 Success ratios % 50 20 20 40 50 30 40 20 30 20 10 0 0 10 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0 Measurement error level % Measurement error level % 50 Chapter 3. Sensitivity-Based FE Model Updating Equations

3.4 Impedance updating equations

Since modal updating equations are classically developed from the undamped free vibration equations, the damping is excluded from the model updating process. It is equally possible to directly use damped force vibration equations to establish model updating equations. For a linear, discrete and damped system, the general equation of motion in the frequency domain is:

[K + i(ωC + D) − ω2M]x(ω)=f(ω) (3.27) where: C is a viscous damping matrix;

D a structural damping matrix;

x(ω) a vector of the displacement response functions;

f(ω) a vector of the excitation forces; √ i imaginary number, i = −1.

We now deﬁne dynamic stiffness as:

Z(ω)=[K + i(ωC + D) − ω2M] (3.28)

Let the force be an unitary sinusoidal force applied to a DOF j, equation (3.27) becomes as:

Z(ω)hj(ω)=[1j] (3.29)

th th where hj(ω) is the frequency response functions when j DOF excited and [1j] is the j column vector of the identity matrix. 3.4. Impedance updating equations 51

If we deﬁne the dynamic quantity q as:

q = Z(ω)hxj(ω) (3.30)

where hxj is the experimental frequency response function, then its initial value is:

(I) (I) q = Z (ω)hxj(ω) (3.31)

If the updated model is perfect, then we have:

(U) (U) q = Z (ω)hxj(ω)=[1j] (3.32)

Therefore, equation (3.1) becomes:

Xn ∂q (U) (U) ∆pl = q − q (3.33) ∂pl l=1

And

Xn ∂Z(ω) (I) [ · hxj(ω)] · ∆pl =[1j] − Z (ω) · hxj(ω); (ω = ω1,ω2,... ,ωm) (3.34) ∂pl l=1

Equation (3.34) is generally applied to reduced ﬁnite element models since expansion of response functions is too much time consuming.

(I) It should be noticed that in (3.34) the components of residual vector [1j]−Z (ω)hxj(ω) are not directly measured. This residual vector is usually referred to as excitation residual vector while (3.34) are called “excitation residual criterion” based updating equations. As- suming the measured noise of hxj(ω) for each DOF at a given ω normally distributed with

(I) the same standard deviation, the noise of the residual vector [1j] − Z (ω)hxj(ω) for each DOF at a given ω is not necessarily characterized with the same normal distribution. This 52 Chapter 3. Sensitivity-Based FE Model Updating Equations will trouble a direct use of least squares solution (see §4.4).

Pre-multiplying (3.34) with impedance matrix Ha(ω), a receptance residual equation set is obtained:

Xn ∂Z(ω) Ha(ω) [ hxj(ω)]∆pl = haj (ω) − hxj(ω); (ω = ω1,ω2,... ,ωm) (3.35) ∂pl l=1

Equation (3.35) are referred to as “indirect-receptance-residue-based” updating equations.

It is possible to directly use frequency response functions to develop the updating equations. If one assume that the initial ﬁnite element model can be updated to represent

th (U) the realistic dynamic structure, then the j column vector of the impedance matrix, haj (ω), can be represented by the measured frequency response functions, hxj(ω):

(U) haj (ω)=hxj(ω) (3.36)

And equation (3.1) becomes:

Xn ∂haj (ω) ∆pl = hxj(ω) − haj (ω); (ω = ω1,ω2,... ,ωm) (3.37) ∂pl l=1

( ) To compute ∂haj ω , differentiating (3.29): ∂pl

∂Z(ω) ∂haj (ω) haj (ω)+Z(ω) =[0] (3.38) ∂pl ∂pl

−1 with Z (ω)=Ha(ω),wehave:

∂haj (ω) ∂Z(ω) = −Ha(ω) haj (ω) (3.39) ∂pl ∂pl 3.5. Summary 53 and (3.37) becomes:

Xn ∂Z(ω) Ha(ω) [ haj (ω)]∆pl = haj (ω) − hxj(ω); (ω = ω1,ω2,... ,ωm) (3.40) ∂pl l=1

There is a slight difference between (3.35) and (3.40). In (3.35), experimental hxj(ω)’s are used to compute the sensitivities while in (3.40) the analytical ones are employed. For impedance FE model updating methods, damping has to be treated during updating process. Damping mechanisms may vary from case to case. However, in practice, some simple damping forms, such as proportional, diagonal or modal structural damping, may be valid for general light, linear damping cases. Damping treatments are discussed in Appendix

3.5 Summary

This chapter focus on sensitivity-based model updating theory. Resonance-frequency- based updating equations, modal updating equations and impedance updating equations are developed with consistent notations. A new modal updating approach, named as QMS updating, is proposed. 54 Chapter 3. Sensitivity-Based FE Model Updating Equations Chapter 4

SVD, regularization, weighting, constraints and iterations

4.1 Introduction

The singular value decomposition (SVD) technique is a “universal” tool for matrix operations, which is especially useful when numerical rank-deﬁciency problems arise. For dynamic FE model updating applications, numerical rank-deﬁciency is often encountered.

Section §4.2 introduces this technique and the related topics: rank-deﬁciency and matrix condition number.

The measured data is always contaminated by noise, while the ﬁnite element model is established with many simpliﬁcations and assumptions. Hence, the updating equations are also spoiled by measured noise and by improper assumptions. Consequently, they are generally self-contradictory and have no solution in ordinary sense. The so-called solutions are always obtained under a certain regularization, which reforms the original equations to consistent ones.

Least-squares (LS) estimation is the most classical, and still most extensively used

55 56 Chapter 4. SVD, regularization, weighting, constraints and iterations method. It is included in section §4.3, serving as the basis and reference for the other regularization techniques. Chi-squares estimation is essentially the same LS technique. However, it reveals the weighting principle for equation regularization. It is described in section §4.4, only serving as an explanation for weighting the updating equations (§4.7).

Total Least-squares (TLS) estimation is an extension of LS technique. In LS, the data errors are assumed only in b of updating equations Ax = b, while in TLS, data errors in A are also considered. TLS is introduced in section §4.5, which will serve as a basis of TLS-based error localization (§5.2.3).

For FE model updating problems, we believe that the initial model is the best one that we can assume and the updated model should be in some extent close to it. Bayesian estimation imposes a requirement that the solution is close to a prior estimate of parameters x0. If we take the previously obtained (initial or iterative) model as the prior estimate, then this technique is just what we desired. Bayesian estimation technique is introduced in section §4.6.

Sensitivity-based model updating equations may have quite different numerical ranges in rows and in columns, due to the observation points (such as the updating frequency in FRF- based methods) and due to the various updating physical parameters (such as thickness of a plate and its Young’s module). The equations should be ﬁrst weighted for matrix operations.

Section §4.7 explains the weighting strategy based on χ-square estimation. As the updating equations are noise-contaminated and are often ill-conditioned, its solutions may cause the updating parameters out of their realistic ranges. Therefore, it is a necessary measure to impose boundary constraints. Section §4.8 describes the Quadratic programming (QP) approach, which is suitable for treating linear boundary constraints.

As the updating equations are only the ﬁrst order Taylor expansion, the FE model updating process should be performed iteratively. Section §4.9 outlines the scheme of iterative 4.2. SVD, rank-deﬁciency and matrix condition number 57

FE model updating.

4.2 SVD, rank-deﬁciency and matrix condition number

The singular value decomposition, or SVD, is a powerful set of techniques for deal- ing with sets of equations or matrices that are either singular or ill-conditioned. Because of its stability in operation, it has become an “universal” tool for matrix operations when numerically rank-deﬁciency problems must be solved [59].

If A ∈

m×m n×n U =[u1,...,um] ∈< and V =[v1,...,vn] ∈< (4.1) such that

Σ 0 [A]m×n =[U]m×m[ ]m×n[V]n×n (4.2) where

m×n Σ = diag(σ1,...,σr) ∈< r = min{m, n} (4.3)

with its singular values σ1 ≥ σ2 ≥ ··· ≥ σp ≥ 0. This matrix operation is referred to as SVD.

The rank of a matrix (rank(A)) equals the maximal number of independent rows or columns. When A is decomposed as above, then its rank number equals the number of nonzero singular values. If rank(A) < min(m, n),thenA ∈

However, when matrix A is set up with experimental data, the rounding errors and fuzzy data make rank determination a nontrivial exercise, as all of the r singular values become 58 Chapter 4. SVD, regularization, weighting, constraints and iterations non-zero’s but some of them are close to zero. Under such a situation, we may be interested in the ε-rank of a matrix which is deﬁned as

rank(A,ε)= min rank(B) (4.4) ||A−B||2≤ε

m×n for some small ε, where matrix B ∈< is close to A in the extent that ||A −B||2 ≤ ε with the minimal singular value min(σ(B)) >ε.Ifrank(A,ε) < min{m, n} with ε = u||A||2, where u is a certain small number, matrix A may be regarded as numerically rank deﬁcient.

With SVD, the ε-rank rε = rank(A,ε) is determined [22, p.73] such that

≥···≥ ≥ ≥···≥ { } σ1 σrε >ε σrε+1 σr r = min m, n . (4.5) and if necessary, a matrix B can be formed as

Xrε 0 B = σiuivi. (4.6) i=1

When a matrix is not rank-deﬁcient but nearly rank-deﬁcient, it is said ill-conditioned.

If a matrix A is ill-conditioned, the solution of algebraic equation Ax = b is very sensitive to small variations in b. For a square matrix A ∈

k(A)=||A|| · ||A−1|| (4.7) with the convention that k(A)=∞ for singular A. When 2-norm is used, the 2-norm condition number of A is expressed as:

−1 σ1(A) k2(A)=||A||2||A ||2 = (4.8) σn(A) 4.3. Least-squares estimation 59 which corresponding to the ratio between the largest to the smallest singular values. When A is not a square matrix, deﬁnition (4.7) is no longer usable. Instead, the inverse of the matrix A−1 has to be replaced with a pseudo-inverse A+. With SVD, a pseudo-inverse A+ ∈

1 1 A+ = VΣ+U0 where Σ+ = diag( ,..., , 0,...,0) ∈

Therefore, for rectangular matrix A ∈

+ σ1(A) k2(A)=||A||2||A ||2 = where r = min{m, n}. (4.10) σr(A)

A large condition number implies that a large change in the solution x may result from a small perturbation in observation b, and then the problem is said to be ill-conditioned. Conversely, the problem is said to be well-conditioned.

4.3 Least-squares estimation

The extensively employed estimation method, identiﬁed as Least-Squares (LS) estimation is the most classical approach, which may be originated from the pioneering work of Karl Friedrich Gauss at the turn of the nineteenth century [29]. As it is the basis of the other methods, LS estimation is brieﬂy described here as a reference.

Considering that errors are present in the experimental data and that assumptions are imposed in the analytical models, the sensitivity-based updating equations

[A]m×n[x]n×1 =[b]m×1 (4.11) are generally self-contradictory and have no solution. The so-called solutions are always 60 Chapter 4. SVD, regularization, weighting, constraints and iterations some particular estimation under certain regularization. In the Least-Squares estimation, the regularization is to ﬁnd a small correction vector r∗ to b, such that the original equations become consistent that have a solution in ordinary sense:

[A]m×n[x]n×1 =[b + r∗]m×1 (4.12)

The “small” correction vector r∗ is found by minimizing:

2 2 f(x)=||r||2 = ||Ax − b||2 (4.13)

The minimization of (4.13) can be obtained by requiring that

∂f(x) 0= for l =1,...,n ∂x l (4.14) ⇒ − 0=[a1l a2l ... aml](Ax b) for l =1,...,n

Equation (4.14) can be rewritten in matrix form as

A0Ax = A0b (4.15)

As long as a vector x∗ is found from (4.15), then the corresponding correction vector r∗ can be obtained as

r∗ = Ax∗ − b (4.16)

In most of the cases we do not care what the correction vector r∗ is, therefore, we may also refer to (4.15) as a LS regularised form of the original equation Ax = b.Forthe

LS regularised equation A0Ax = A0b, we can always ﬁnd its solution. However, the solution is not guaranteed to be unique and robust. In fact, the solution is totally dependent on the 4.3. Least-squares estimation 61 nature of the product matrix A0A. If an inverse matrix of A0A exists and can be robustly computed, then the LS regularised equation (4.15) can be solved as:

0 −1 0 xLS =(A A) A b (4.17)

With the powerful SVD, the LS solution can be directly obtained without forming the symmetric, square coefﬁcient matrix A0A:

Xr u0 b x = i v (4.18) LS σ i i=1 i with an equation residue as

Xm 2 || − ||2 0 2 ρLS = AxLS b 2 = (uib) (4.19) i=r+1

where ui and vi are deﬁned in (4.1); r is the rank number of matrix A. The only, and still open problem is how the numerical rank can be properly determined. As mentioned in section §4.2, an ε-rank may be more realistic when test errors pre-

−2 sented in A. If, for example, the entries in A are correct to two digits, then ε =10 ||A||∞ P n (||A||∞ = max =1 |aij|) may be used as the tolerance to determine the “numerical rank 1≤i≤m j number” rˆ of matrix A such that

σˆ1 ≥···≥σˆrˆ >ε≥ σˆrˆ+1 ≥···≥σˆr r = min{m, n} (4.20)

If σˆrˆ ε, then we can comfortably determine the numerical number rˆ. On the other hand, if {σˆ1,...,σˆn} are not split into subsets of small and large values, the determination of rˆ by this means becomes somewhat arbitrary. As the numerical rank determination is, in fact, to make the solution of the equations more robust to the measurement noise, while the 62 Chapter 4. SVD, regularization, weighting, constraints and iterations robustness of the solution is, in the most extent, determined by the condition number, we may therefore directly connect the numerical rank number with the condition number of the equations. Along with the idea of ε-related numerical rank number, an ε-related 2-norm condition number can be deﬁned as

σ1 k2(A)|ε = (4.21) σrε

When pseudo-inverse matrix deﬁned in (4.9) is calculated with the truncated SVD values

(σ1,...,σrε ), then the ε-related 2-norm condition (4.21) becomes the pseudo-condition number, which inﬂuences the solution precision of the updating equations.

The ε-related 2-norm condition number can be plotted versus trial rank number r˜’s.

If a particular small singular value σˆr˜, which is proximate to the estimated tolerance ε,sig- niﬁcantly augments the ε-related 2-norm condition number, then we may have reason not to take it into account, and consequently, the numerical rank number is determined.

Figure 4.1: k2(A)|ε vs trial rank(A, ε) (free PCB) k A with 1% simulated noise in the data 2 140

120

100

0 10 20 30 40 Trial ε−rank number

Figure 4.1 shows a simulated situation for a free PCB with QMS model updating equations (see §5.4.4 on page 141). For this particular case, it is rational to determine the 4.4. Chi-squares estimation [58] 63

ε-related numerical rank number as 41 since σ42 is sufﬁcient small comparing to σ1 and σ42 is well separated from σ41. It should be pointed out that a small singular value does not contribute much to the matrix A, as can be seen from the rewritten SVD formula:     0 σ1 0 v1      ..   0  Xr  .  v2 Σ 0     0 A = U V =[u1,..., um]    .  = σiuivi (4.22)  σ   .  i=1  r    0 00vn m×n

4.4 Chi-squares estimation [58]

Chi-squares estimation explains the principle of weighting strategies. It does not directly relate to a solution approach. After the equations are weighted based on the χ- squares principle, the updating equations can be solved by LS, TLS, quadratic programming, etc.

When we intuitively formed the overall residue (4.13), we have implicitly assumed that all of the individual residues are equally important. However, for the test data, if some observation data are more precise than the others, then we should put more conﬁdence on them. Mathematically, this can be expressed by a weighting strategy:

P m 2 2 f(x)= =1(diri) = ||D(Ax − b)||2 to be minimized i (4.23) where D = diag(d1,...,dm) is nonsingular.

There now arise questions: how is the precision of the observation data expressed and how is it related to the weighting factors di? This is a topic addressed by a more sophisticated, statistics-based estimation method—Maximum Likelihood estimation. In fact, from the point of view of maximum likelihood estimation, least-square estimation is only a case 64 Chapter 4. SVD, regularization, weighting, constraints and iterations where each observation data point has a measurement error that is independently random and distributed as normal (Gaussian) distribution around the “true” value with the same standard deviation σ. The probability of the data set b within ∆b is the product of the probabilities of each points,     P !2 Ym  n  1 bi − =1 aijxj P = exp − j  ∆b (4.24)  2 σ  i=1

Notice that there is a factor ∆b in each term in the product, which simply means a ﬁxed zone around the true “value”. Maximum likelihood estimation maximizes this probability, which is equivalent to maximizing its logarithm, or minimizing the negative of its logarithm: " P # Xm n 2 [bi − =1 aijxj] j − m · lg ∆b (4.25) 2σ2 i=1

Since m, σ and ∆b are all constants, minimizing this equation is equivalent to minimizing

(4.13).

If each observation data point has its own standard deviation, σi, then equation (4.24) is modiﬁed only by putting a subscript i on the symbol σ. And consequently the maximum likelihood estimate is obtained by minimizing the quantity

P !2 Xm n bi − =1 aijxj χ2 ≡ j (4.26) σ i=1 i

Similarly, the minimization of equation (4.26) can be easily obtained by requiring that

∂χ2 0= for l =1,...,n ∂xl ⇒ · 1 1 · − 0=[a1l a2l ... aml] diag( 2 ,... , 2 ) (Ax b) (4.27) σ1 σm for l =1,...,n 4.5. Total Least-squares estimation 65 which results in:   1  0   σ1  0 0  .  (DA) (DA)X =(DA) (Db); where D =  ..  (4.28)   1 0 σm

Comparing equation (4.28) with (4.23), it is found that the row weighting factor dl for observation data l equals the reciprocal of the standard deviation σl. And furthermore, after the row-weighting, the LS regularised equation set has the same form as (4.15).

4.5 Total Least-squares estimation

If the original equation Ax = b is row-weighted as DAx = Db based on χ-estimation technique, then, according to (4.13), the correction vector r is obtained from minimizing:

2 2 f(x)=||DAx − Db||2 = ||Dr||2 (4.29) which satisﬁes that Ax =(b + r) is not self-contradictory and has a consistent solution. If we deﬁne the range of matrix A as a vector of linear combination of the column vectors of

A [22, p.50]:

range(A)={y ∈

m min ||Dr||2 r ∈< (4.31) (b+r)∈range(A) 66 Chapter 4. SVD, regularization, weighting, constraints and iterations

In this ordinary Least-Squares problem, there is a tacit assumption that the errors are confined to the observation b. For finite element model updating problems, as both A and b are suffering from the influence of experimental data noise and model imprecision, it is natural that, besides the correction vector r to b, a correction matrix E to A is considered.

The correction matrix E and vector r should be “small” and satisfy the condition that

(A + E)x =(b + r) (4.32) is consistent.

Golub [22] proposed a way to determine E and r and named it as Total Least-squares problem:

m×n m min ||D[Er]T||F ; E ∈< , r ∈< (4.33) (b+r)∈range(A+E) where: r is the correction vector to b;

E the correction matrix to A;

D row-weighting factors: D = diag(d1, ··· ,dm);

T column-weighting factors: T = diag(t1, ··· ,tn,tn+1); qP P || || m n 2 F subscript denoting a Frobenius norm, A F = i=1 j=1 aij. If a minimizing [E0 r0] can be found for (4.33) then any x satisfying (A + E0)x =(b + r0) is called a TLS solution, xTLS. Unlike the ordinary least-square problem, total least square problems may fail to have a solution. Nevertheless, under certain conditions, a unique TLS solution can be obtained with SVD technique.

Let A, b, D and T be as above and assume m ≥ n +1,andlet

C = D[Ab]T =[C1 C2] (4.34) n 1 4.5. Total Least-squares estimation 67

0 have U CV = diag(σ1,...,σn,σn+1)=Σ where U, V and Σ are partitioned as follows:    

V11 V12 Σ1 0  U =[U1 U2]; V =   ; Σ =   (4.35) V21 V22 0 Σ2 where: m×n m×1 U1 ∈< , U2 ∈< ;

n×n n×1 1×n 1×1 V11 ∈< , V12 ∈< , V21 ∈< ; V22 ∈< ;

Σ1 = diag(σ1,...,σn), Σ2 = σn+1.

It can be shown [22, p.577, theorem 12.3.1] that if σn(C1) >σn+1(C) then the matrix

[E0 r0] deﬁned by

0 0 D[E0 r0]T = −U2Σ2[V12 V22] (4.36) solves (4.33) with

−1 −1 xTLS = −T1V12V22 T2 (4.37)

where T1 = diag(t1,...,tn) and T2 = tn+1 are partitioned column-weighting matrices of T.

To obtain an intuitive impression of difference between LS and TLS, a simpliﬁed system with single input and single output is illustrated as shown in Figure 4.2. The “measurement” data are:

a 0.98 3.01 5.10 . b 1.20 2.85 5.30

The linear system is modelled as b = αa where a is the input; b is the output and α is the coefﬁcient to be determined. With the test data and the model, the equations Ax = b are set 68 Chapter 4. SVD, regularization, weighting, constraints and iterations as:     0.98 1.20         3.01 α = 2.85     5.10 5.30

With LS estimation, αLS =1.0209, the correction vector

r∗ = Aα − b =[−0.1995, 0.2230, −0.0933]0

For TLS estimation, we here suppose that the input a and output b are measured with the same precision. Therefore, weighting matrices D and T are simply the identity matrices:

C = D[Ab]T =[Ab].

C =        0.98 1.2  0.1798 0.6321 −0.7538 8.5811 0  0       −       0.6993 0.7149 3.01 2.85 = 0.4827 −0.7244 −0.4922  00.2192         0.7149 0.6993 5.10 5.30 0.8571 0.2753 0.4354 00

The correction matrix is (D = I; T = I):

− Σ 0 0 [E0 r0]= U2 2[V12V22]=      0.6321 −0.7538  0.0991 −0.0969       0.2192   −0.7244 −0.4922   −0.7149 0.6993 = −0.1135 0.1110    0   0.2753 0.4354 0.0431 −0.0422

and the TLS estimation αTLS =0.7149/0.6993 = 1.0223. For this particular case with D = I and T = I, TLS estimation minimizes the square root of the sum of the square of the p 2 2 2 distances from test data points to the regressive line: l1 + l2 + l3 → minimized. 4.6. Bayesian estimation technique 69

Figure 4.2: Difference between LS and TLS b input a output b α Linear system α TLS

TLS LS l2 distance to the line l2 distance to the line r2 of TLS r of LS 2 r of TLS E 2 21 r2 of LS E21

4.6 Bayesian estimation technique

As the initial FE models are the best ones that we can assume with our previous knowledge of the structures under study, the updated FE models should be close to the initial ones. This requirement can be satisﬁed with Bayesian estimation technique.

In Bayesian estimation, a prior estimate of the parameters x0 is given. The new estimation is then required to be in some vicinity of the the given x0. Applied to the ﬁnite element model updating, the prior estimate of the parameters x0 should always be zeros, because we believe the initial model is the best one that we can assume. If, as in the LS estimation, we assume that the observation data are independent and normally distributed, then for Bayesian estimation, the object function (4.23) can be extended to as

P −1 || − ||2 n 2 f(y)=f(T1 x)= D(AT1y b) 2 + α j=1(βjyj) to be minimized −1 where D = diag(d1,...,dm); T1 = diag(t1,...,tn); y = {yi} = T1 x; (4.38)

α>0,βj > 0 j =1,... ,n. 70 Chapter 4. SVD, regularization, weighting, constraints and iterations and correspondingly, equation (4.28) become as

0 0 2 2 [(DAT1) DAT1 + αG]y =(DAT1) Db; G = diag(β1 ,... ,βn) (4.39)

P || − ||2 n 2 The coefficient α balances the residue D(AT1y b) 2 and the model modification j=1(βjyj) while βj’s balance the weighted parameters yj’s. In POMUS, The coefficients α and βj’s are set such that:

α =0.5 for the ﬁrst iteration, αiter =0.9αiter−1 for the other iterations (4.40) 2 th 0 βj = j diagonal component of the matrix (DAT1) DAT1

4.7 Weighting updating equations

When weighting the updating equations we balance the numerical range of rows and columns of matrix A and vector b. Profoundly, in the point of view of parameter estimation, we are actually weighting the errors in b and A, as explained in χ-squares estimation (§4.4, [58][22, p.250-251]).

Let us consider the column weighting of matrix A. Suppose that the thickness t and Young’s module E of a plate are employed to establish FRF-based receptance resid-

∂Z(ω) ual updating equations, then the corresponding columns of A are H(ω) ∂t hxj(ω) and ∂Z(ω) H(ω) ∂E hxj(ω). If we take the unit of hxj(ω) as a reference, then the two columns have 1 1 × −3 numerical scales t and E , respectively. Assume the initial values t0 =5 10 m and 9 2 −12 E0 = 207 × 10 N/m (steel), the numerical scales are 200 and 4.831 × 10 . A very small percent perturbation in the column of t will certainly spoil data in the column of E when numerical operations are performed with digital computers. Instead of directly using t and

t E E, if we use relative parameters t0 and E0 , then the scales are 1’s. This is the equivalence of weighting the columns of A with the corresponding initial parameters. Therefore, the 4.7. Weighting updating equations 71 column weighting factors are determined as:

(0) T1 = diag(t1,t2,... ,tn); tj = pj ,j=1, 2,... ,n (4.41)

(0) th where pj is the initial value of j parameter. This strategy can be explained as the distributions of errors in columns of sensitivity matrix A have the same standard deviation if we use relative parameter changes as design variables. Row-weighting balances the numerical scales of rows of A. A simple row-weighting strategy is to weight the rows of A such that they all have a unit vector length, that is, the row-weighting factors are:

1 d = p ; i =1, 2,... ,m i 2 2 2 (4.42) (ai1t1) +(ai2t2) + ...+(aintn)

This strategy has implicitly taken an assumption that data errors in rows of AT1 are proportional to row-vector length.

When TLS is applied, column-weighting factor of the residual vector Db (after row- weighting) is employed to balance errors between Db and DAT1. If we do not have knowledge of these errors, we may simply suppose their “standard deviations” are proportional to the vector-length of Db and matrix-length (measured in Frobenius norm) of DAT1:

|| || || || σDET1 = kDAT1 DAT1 F ; σDr = kDb Db F (4.43)

where E is the error matrix of A and r is the error in b; kDAT1 and kDb are coefﬁcient to express the data precision.

After column-weighting Db with tn+1, error of DAT1 and error of tn+1Db should have the same error levels: 72 Chapter 4. SVD, regularization, weighting, constraints and iterations

k || 1|| || || × || || ⇒ DAT1 DAT F kDAT1 DAT1 F = tn+1 kDb Db F tn+1 = kDb||Db||F

In practical applications, we may expect that data in matrix DAT1 is more precise, that is, kDAT1

4.8 Quadratic programming (QP) approach

The boundary-constrained least squares problems can be easily transformed to quadratic programming forms. When rank(A)=n, the boundary-constrained linear least squares problem:

2 min ||Ax − b||2 subject to xli

1 0 0 min c 2 x Hx + v x subject to xli

When rank(A)

2 2 min ||Ax − b||2 + ||x||2 subject to xli

For this situation, the equivalent quadratic programming problem is slightly changed to:

1 0 0 min c 2 x Hx + v x subject to xli

The same deduction may be applied to weighted LS problems.

So long as the boundary constraints are consistent, that is, there is a feasible area, the above constrained QP problem must lead to a unique minimization since H is positive- deﬁned. In POMUS, the quadratic programming approach is implemented with a MATLAB function QP (from optimization toolbox).

4.9 Iterative model updating

Since sensitivity-based updating equations are based on imprecise initial models, one should not expect that a single step model tuning will produce a perfectly acceptable result.

Iterative strategy is basically necessary. Let us formulate it as:

Mui = Mui−1 +∆Mui ; Kui = Kui−1 +∆Kui ; i =1, 2, 3,... (4.49) P ∂Mu ( ) P ∂Ku ( ) ∆ = n i−1 ∆p i ;∆ = n i−1 ∆p i Mui j=1 ∂pj j Kui j=1 ∂pj j

§ § § The initial model is expressed via Mu0 and Ku0 . From section 3.2, 3.3 and 3.4 together with (4.49), we know that mass and stiffness partial derivatives with respect to design variables are essentially involved in every step of sensitivity-based model updating. In POMUS, they are calculated numerically by a perturbation method:

( (i−1)+ )− ( (i−1)) ∂M M pj δp M pj | (i−1) = ∂p pj =pj δpj ( (i−1)+ )− ( (i−1)) i =1, 2,... (4.50) ∂K K pj δp K pj | (i−1) = ∂p pj =pj δpj 74 Chapter 4. SVD, regularization, weighting, constraints and iterations

th (i−1) th th where δpj is the j parameter perturbation; pj is the j parameter value before i updating iteration. After each successful updating iteration, the ﬁnite element model is modiﬁed by:

(i) (i−1) (i) pj = pj +∆pj ; j =1, 2,... ,n (4.51)

The ﬁnal parameters are calculated by:

NXiter (u) (0) (i) pj = pj + ∆pj ; j =1, 2,... ,n (4.52) i=1

4.10 Summary

When measurement noise is present, the updating equations are generally self-contradictory that have no ordinary solution. The so-called solutions are obtained from regularised equations of the original ones. The regularising principles can be explained with statistic theory, which is not a branch of mathematics according to Press [59, p.519].

Singular value decomposition is an “universal” tool for matrix operations. It is especially useful when rank-deﬁcient or ill-conditioned problems arise.

LS estimation supposes that errors are only in b of Ax = b, and regularises b by

∗ ∗ adding a correction vector r which satisﬁes ||r ||2 = ||Ax − b||2 → minimized. TLS estimation believes that errors also present in A besides in b of Ax = b and regularises both A and b by adding E and r, respectively, where E and r are “small” that satisfy

min ||D[Er]T||F . (b+r)∈(A+E) Bayesian estimation technique is not a simple regularisation. It imposes the solution x close to a prior estimate x0. Whether Bayesian estimation is better than these of other estimation techniques depends on the prior estimate x0. For FE model updating problems, we believe that the initial models are the best one we can assume, therefore, the updated 4.10. Summary 75 models should be close to them.

Weighting the equations is to balance the data range in A and b. From the point of view of regression, weighting is to balance the errors in the data. The Chi-square approach justiﬁes why the equations should be weighted. Weighting strategy inﬂuences every estimation result.

In principle, all of the estimation techniques can be interpreted as minimization process. For the mentioned estimation techniques, the solution can be directly obtained by ordinary matrix operations, such as SVD. However, the solutions may be out of their physically acceptable ranges due to measurement noise and imprecision initial FE models. Quadratic

Programming (QP) approach provides an alternative way to solve LS and Bayesian estimation with low and upper boundaries.

Finally, as the sensitivity matrix A is only the ﬁrst order Taylor expansion, the FE model updating should be iteratively performed. 76 Chapter 4. SVD, regularization, weighting, constraints and iterations Chapter 5

Error localization

5.1 Introduction

Error localization tries to locate main inaccuracies in a ﬁnite element model according to experimental data. It usually serves as a preliminary step of the model updating process, which provides some helpful information in the selection of the updating parameters.

Error localization may also be divided into global and local methods. Global methods are location-oriented while local methods are parameter-oriented. The location-oriented error localization approaches can hardly be incorporated into a parameter-oriented model updating procedure. Parameter-oriented error localization methods aim to select a smallest subset that contains main model errors, from the full design variable space. It is clear that if the full design variable space does not contain main model errors, parametric approaches will never reveal them, but select a special subset that may reproduce the same or similar effects. Section §5.2 propose and fully develops a TLS-based local error localization method.

The two most popular global error localization methods, FBM and EMM, are ﬁrstly summarized with application to an electronic structure. A brief introduction to local methods, including QR and SVD/QR, is then presented. Following the introduction, a TLS/QR-F

77 78 Chapter 5. Error localization method is developed.

Section §5.3 proposes a success ratio to evaluate the efﬁciency of the proposed error localization method. The noise pattern and program ﬂow charts are also described in this section.

Section §5.4 presents case studies which use success ratio to evaluate the proposed error localization method. Four case studies, which include a cantilever beam, a lumped mass-stiffness string, a 3-D bay structure and a PCB with components, are presented with various updating equations. Section §5.5 summarizes the results and ﬁndings.

5.2 Error localization principles

5.2.1 Global error localization

Global error localization methods are closely related to global updating methods. A global updating method can certainly be developed as a localization method, so long as a proper error indicator vector is set up. However, as a preliminary step, it is not necessary for a global error localization method to include the ability of updating the model.

Force Balance Method (FBM) and Error Matrix Method (EMM) are the most commonly used global error localization methods. FBM is simply a residual method, which produces residual force vectors by using the vibration equations in the frequency domain, while EMM is a system reconstruction methodology, which requires to reconstruct system matrices with incomplete modes.

FBM error localization

FBM identifies areas in the finite element model, especially with respect to stiffness, which is thought to be more difficult to model than inertia properties and to contribute more 5.2. Error localization principles 79 to the discrepancies between analysis and test data. In this method, the dynamic equation of the vibration system is invoked:

− 2 2 φ (Ka 4π faiMa) ai =[0] (5.1) where:

Ka is the analytical stiffness matrix;

Ma the analytical mass matrix;

th fai the i analytical resonance frequency; φ th ai the i analytical mode shape vector.

φ To correlate two data sets, fai and ai in equation (5.1) are replaced with the ex- φ perimental counterparts: fxi and xi. If the measured frequencies and mode shapes were exactly the same as the ones of the ﬁnite element model, then there would be no residuals.

In practice, there are always some errors both in ﬁnite element model and in measured mode shapes. These errors will produce force residual vector ei as [13]:

− 2 2 φ ei =(Ka 4π fxiMa) xi (5.2)

φ where ei is the residual force vector; fxi and xi are the experimental resonance frequency and the associated mode shape.

The absolute value of this residual vector |ei| can serve as a FBM error indicator. In practice, only a few resonance frequencies and mode shapes can be obtained from dynamic tests. With FBM error localization method, we can take an advantage that a single mode will produce some information about model errors.

For a free PCB card (deﬁned in §D.6), Figure 5.1 displays the FBM error indicators for the six measured modes, at each of the 104 nodes deﬁned by Figure D.9. The results 80 Chapter 5. Error localization show that the main model errors are located where the electronic are present.

Figure 5.1: FBM error localization results 6 6 x 10 Mode 1 x 10 Mode 2 2 5 | |

1 42 1.5

3 1 2

0.5 FBM error indicator |e FBM error indicator |e 1

0 0 20 40 60 80 100 20 40 60 80 100 Test points Test points

6 6 x 10 Mode 3 x 10 Mode 4 8 6

7 5 | | 63 4 4 5

4 3

3 2 2 FBM error indicator |e FBM error indicator |e 1 1

0 0 20 40 60 80 100 20 40 60 80 100 Test points Test points

6 7 x 10 Mode 5 x 10 Mode 6 12 3.5

10 3 | | 5 6 2.5 8 2 6 1.5 4 1 FBM error indicator |e FBM error indicator |e 2 0.5

0 0 20 40 60 80 100 20 40 60 80 100 Test points Test points 5.2. Error localization principles 81

EMM error localization

Basically, EMM methods have their originality from a work of Berman [6], where an incomplete model is reconstructed with limited measured modes. To avoid directly re- building mass and stiffness matrices with measured eigenvalues, an inverse strategy may be employed as following. Let ∆K be the difference between experimental and analytical stiffness matrices:

− −1 − −1 ∆K = Kx Ka = Kx(Ka Kx )Ka (5.3)

where Kx denotes the experimental stiffness matrix.

If ∆K is small compared to Kx and Ka,thenwehave:

≈ −1 − −1 ∆K Ka(Ka Kx )Ka (5.4)

−1 Now the problem becomes calculating Kx matrix. Employing the eigenvector orthogonality with respect to mass M and stiffness K matrices:

Φ0MΦ = I; Φ0KΦ = Λ (5.5)

Φ Λ 2 × 2 2 where is the eigenvector matrix; =4π diag(f1 ,... ,fn) is the diagonal eigenvalue matrix, then we have:

M−1 = ΦΦ0; K−1 = ΦΛ−1Φ0 (5.6)

In practical cases, only a small set of eigenvalues are available, therefore, the inverse matrices 82 Chapter 5. Error localization have to be rebuilt with an incomplete space base as:

−1 ≈ Φ Λ−1Φ0 −1 ≈ Φ Λ−1Φ0 Kx x x x Ka a a a (5.7)

where Φx, Λx are the matrices of measured mode shapes and the corresponding eigenvalues,

Φa, Λa are analytical eigenvector matrix and the corresponding eigenvalue matrix. In the same way we obtain a mass error matrix as:

−1 − −1 ∆M = Ma(Ma Mx )Ma (5.8) where inverse mass matrices are computed with limited modes:

−1 ≈ Φ Φ0 −1 ≈ Φ Φ0 Mx x x; Ma a a (5.9)

Uncorrelated modes should not be included in EMM error localization procedure as they will be interpreted as model errors. Suppose that the FE model is perfect and only limited experimental modes are employed to set up

−1 ≈ Φ Λ−1Φ0 Φ Λ−1Φ0 Kx x x x = xc xc xc (5.10) if uncorrelated analytical modes are introduced, then

−1 ≈ Φ Λ−1Φ0 Φ Λ−1Φ0 Φ Λ−1 Φ0 Ka a a a = ac ac ac + auc auc auc (5.11) where subscript c and uc denote correlated and uncorrelated modes, respectively. Evidently, Φ Λ−1Φ0 Φ Λ−1Φ0 Φ Λ−1 Φ0 as xc xc xc = ac ac ac, auc auc auc is interpreted as model errors. It should be aware that an analytical mode may be uncorrelated because of the limited experimental modes instead of model errors. 5.2. Error localization principles 83

It is not convenient to judge model errors directly from two-dimension error matrices.

The messages in error matrices are then gathered according to the node of the FE model mesh, as illustrated in Figure 5.2. Two error indicating vectors, one for mass and the other for stiffness, are then established as [34]:

0 0 em =[em1 em2 ... emn ] ; ek =[ek1 ek2 ... ekn ] (5.12) where: P P X |∆m | X |∆k | Pj lj Pj lj emi = ; eki = ; (5.13) |mlj| |klj| l j l j where l denotes the rows related to ith node and j the columns related to ith node.

Figure 5.2: EMM error indicators j j j j 1 2 3 c For each row, sum up the columns. ΣΣ∆ Σ ∆ j| ml1j| j| kl1j| Then, the l1 Σ ; Σ j|ml1j| j|kl1j| results are ΣΣ∆ Σ ∆ j| ml2j| j| kl2j| further l2 Σ ; Σ j|ml2j| j|kl2j| summed up.

ΣΣ|∆m | Σ |∆k | Σ j l3j ; j l3j l3 Σ Σ j|ml3j| j|kl3j|

ΣΣ|∆m | Σ |∆k | j lcj ; j lcj lc Σ Σ j|mlcj| j|klcj| Σ |∆m | Σ |∆k | Σ j lj ; Σ j lj th emi= Σ eki= Σ i node related components l j|mlj| l j|klj|

For the same free PCB card (introduced in §7.3, see also Appendix D section §D.6) as mentioned with FBM method, the mass and stiffness error indicators are shown in Figure

5.3. These results also show that the main model errors are located at the area where the 84 Chapter 5. Error localization electronic components planted.

Figure 5.3: EMM error localization results EMM, mass error indicator EMM, stiffness error indicator 0.1 1.4

0.09 1.2

0.08

1 0.07

0.06 0.8

0.05 indicator indicator m m

e e 0.6 0.04

0.03 0.4

0.02

0.2 0.01

0 0 20 40 60 80 100 20 40 60 80 100 Test point Test point

Some alternative versions assume that stiffness is much difﬁcult to be modelled and then suppose the main model errors are in the stiffness matrix. A weighted EMM method

[50] is summarised below: Φ Φ0 Assume ∆M =0,thenMx = Ma = M, weighting ∆K with x x, that is:

Φ Φ0 Φ Φ0 − Φ Φ0 x x∆K = x xKx a aKa (5.14) since:

Φ Φ Λ 0 0 Φ0 Λ Φ0 Kx x = Mx x x; Mx = Ma; Kx = Kx; Mx = Mx; xKx = x xMa (5.15) we have:

Φ Φ0 Φ Λ Φ0 − Φ Φ0 x x∆K = x x xMa x xKa (5.16) 5.2. Error localization principles 85

This equation is transposed:

Φ Φ0 Φ Λ Φ0 − Φ Φ0 ∆K x x = Ma x x x Ka x x (5.17)

Adding these two equations together, we can formulate an error matrix as:

0 0 E =(MaA +(MaA) ) − (KaB +(KaB) ) (5.18) where:

Φ Λ Φ0 0 Φ Φ0 0 A = x x x = A ; B = x x = B (5.19)

If the dominant errors are in the mass matrix, we can get a similar result as:

0 0 E =(KaB +(KaB) ) − (MaA +(MaA) ) (5.20)

An error indicator vector can also be deﬁned:

0 e =[e1 e2 ...en] (5.21) where:

X ei = |Elj| (5.22) lj where l’s are the rows related to ith node; j’s being the columns related to ith node.

For the measured free PCB card (introduced in §7.3, see also Appendix D section

§D.6), the weighted EMM error indicator is plotted in Figure 5.4. 86 Chapter 5. Error localization

Figure 5.4: Weighted EMM error localization results 7 x 10 WEMM error indicator 4.5

3.5

2.5

1.5

0.5

0 10 20 30 40 50 60 70 80 90 100 Test point

5.2.2 Local error localization methods

Introduction

Traditionally, error localization is performed globally, using system matrices of initial models. A parameter-oriented error localization approach was developed in 1990 by

Lallement [37]. The principle for the proposed method is the following.

• Let the updating equations be Ax = b. Among the columns of the sensitivity matrix

A, a column which best represents the vector b is sought. Then the combinations of

two columns, three columns, etc... which constitute the best sub-basis for the repre-

sentation of the vector b, are searched.

• Let

As be the best sub-basis of dimension s;

xs the corresponding parameters evaluated in the Least-Squares sense;

bs the best representation of b in the sub-basis of dimension s, bs = Asxs; 5.2. Error localization principles 87

(s) (s) ||b−bs|| e the distance between b and bs, e % = 100 ||b|| ;

An analysis of the error e(s) obtained with sub-space of increasing dimension,

s =1, 2, 3,... permits the selection of the most probable dominant sub-domains.

Logically, the “best sub-space” can only be veriﬁed with exhaustive evaluation [51].

The number of possible subsets of one or more variable out of n is (2n − 1). When the number of design variables is not too large, it is feasible to search all subsets. However, for

FE model updating purpose, as the updating parameter number increases, exhaustive search will become infeasible due to the required huge computation.

Fritzen [20] proposed to use the QR basic solution subset to update ﬁnite element models. As the QR with column pivoting by itself is not entirely reliable as a method for detecting near rank deﬁciency, Yang [67] employed a SVD/QR-based subset selection method.

A simple forward search approach cannot detect near rank-deﬁciency problems. In addition, it is numerically inefﬁcient, especially for modal or impedance updating equations due to the big dimension. The subset selected by QR with column pivoting has nothing to do with error localization, because this method entirely ignores the “observation” data b that represents the “observed” differences between experimental data and analytical models. Therefore, a

TLS-based forward subset selection will be proposed.

Before discussing TLS/QR-F method, QR basic solution and SVD/QR subset selection are brieﬂy reviewed in the following subsections.

QR basic solution for rank-deﬁcient updating equations

A local error localization is the effort of seeking a subset of unknowns xs which “best” solves the updating equations. The “best solution” for error localization purpose should be the one which contains minimum parameter number. 88 Chapter 5. Error localization

Let As contains only these columns of A which correspond with xs.Acolumn- reduced updating equation can be deﬁned as:

Asxs = b (5.23)

For any subset xs of x,since

min ||Asxs − b||2 ≥ min ||Ax − b||2 (5.24) a subset solution is never better than the full set solution from the point of view of residual norms. A relative norm of the residuals of the reduced updating equations is then commonly deﬁned as:

||b − A x || ε = s s 100% (5.25) ||b||

As the subset size increases, the relative norm will reduce, and when the subset becomes the full set, the relative norm reaches to its minimum. However, when the relative norm decreases, the condition number of the correspondingly reduced system will increase, which may provoke unstable estimation. Therefore, the “best” solution of the updating equations is a trade-off between reducing the subset parameter number and minimizing the residue.

Although no explicit criterion has been suggested for judging whether a “best” solution is obtained, the current subset selection methods may still be divided into sub-space searching methods and QR with column pivoting method. Sub-space searching methods have been well documented by Miller [51] in the point of view of data regression, which are indirect, trial-and-error methods, while QR with column pivoting method is direct method which seeking a most “well-conditioned” subset when the equation set is rank-deﬁcient.

Suppose A ∈

R11 R12 0   n×n m×m r×r r×(n−r) Q AP =   ; P ∈< , Q ∈< , R11 ∈< , R12 ∈< ; (5.26) 00

where Q is orthogonal; R11 is upper triangle and nonsingular; P is a permutation matrix.

The matrices Q and P are products of series of Householder matrices (see (5.53) for details of the computation of Hk) and permutation matrices respectively. The matrices Q and P can be computed as following [22].

Assume for the kth step of QR factorization we have obtained   ( −1) ( −1) R k R k k−1  11 12  (Hk−1 ···H1H0)A(P0 ···Pk−1)=R =   (5.27) (k−1) 0R22

(k−1) (k−1)×(k−1) (k−1) (k−1)×(n−k+1) (k−1) (n−k+1)×(n−k+1) where R11 ∈< , R12 ∈< , R22 ∈< ; R11 is a nonsingular and upper triangular matrix; with the convention that H0 and P0 are identity

(k−1) (k−1) (k−1) matrices, and suppose that R22 =[sk ,...,sn ] is a column partitioning, let p be the smallest index satisfying (k ≤ p ≤ n) such that

|| (k−1)|| {|| (k−1)|| || (k−1)|| } sp 2 = max sk 2,..., sn 2 (5.28) then if k − 1=rank(A), the ﬁnal result is reached.

k−1 Otherwise, ﬁrstly, the matrix R is post-multiplied by a permutation Pk that inter-

(k−1) changes the ﬁrst and the largest column in R22 . Secondly, the column-interchanged matrix

k−1 R Pk is pre-multiplied by a Householder matrix Hk that zeros all of the sub-diagonal com-

th k−1 ponents of the k column in R Pk. The operations are repeated until k − 1=rank(A). 90 Chapter 5. Error localization

Finally, the matrices Q and P are expressed as:

0 Q =(HrHr−1 ...H1) ; P = P1P2 ...Pr (5.29)

The key point of the QR with column pivoting is to take the largest column in sub

(k−1) matrix R22 . With this QR factorization, we are ready to solve the updating equations. In least- square sense, Ax = b can be multiplied with the orthogonal matrix Q0 while the parameters can be reorder to P0x:    

R11 R12  cr×1  (Q0AP)(P0x)=Q0b ⇒   (P0x)=  (5.30) 00 d(m−r)×1

   

 cr×1   yr×1  where   = Q0b.Let  = P0x, then (5.30) becomes d(m−r)×1 z(m−r)×1

R11y + R12z =[c]r×1;[0]=[d](m−r)×1 (5.31) d becomes the residual vector that can no longer be reduced. x can be expressed with   −1 R11 (c − R12z) x = P   (5.32) z where z is an arbitrary (n − r) × 1 vector. In particular, if z is set to zero, then we obtain the basic solution [22]:   −1 R11 c xB = P   (5.33) 0 5.2. Error localization principles 91

where xB is referred to as basic solution of Ax = b. The basic solution implies that, whatever the observation b is, a subset containing only r parameters can always be used to solve the updating equations. It should be aware, however, that the basic solution is not the minimal 2-norm solution unless the sub-matrix

R12 is zero. There is a relationship between the basic and the minimal 2-norm solution [22]:

q ||xB||2 −1 2 1 ≤ ≤ 1+||R11 R12||2 (5.34) ||xLS||2

The following numerical example shows the difference between LS solution and the

QR basic solution. Let the equation Ax = b be:

    0.0504 4.4522 1.9878 5.4599 10.9450     −0.6758 4.7639 3.0169 4.3587   8.3795       4.0380 −1.9781 0.0005 5.9816   13.9822   x =   (5.35)  1.6469 0.3732 −5.9917 3.6867   8.1968  −2.5746 4.2454 2.0054 −0.2060 −1.6994 −0.9051 4.4779 −1.0116 3.4383 6.4241

0 The LS solution result is xLS =[0.7990, 0.1725, 0.0038, 1.8552] . With House- holder QR decomposition with column pivoting, the Q, R, P and Q0b are obtained as:

  −0.5205 −0.2544 −0.1908 0.1772 −0.5030 −0.5861   −0.4155 −0.3558 −0.2938 −0.6798 0.3805 0.0869    −0.5702 0.5871 −0.3077 0.3406 0.2407 0.2481  Q =   −0.3514 0.1608 0.8207 −0.1056 0.2794 −0.2965  0.0196 −0.5478 −0.0324 0.6005 0.5756 −0.0816 −0.3278 −0.3716 0.3287 0.1369 −0.3673 0.7020

    −10.49 −4.684 0.1882 −2.381   −22.17      0 −7.918 −3.265 4.610  0001  2.305         00−6.581 0.084  0100 0  0.0412  R =   ; P =   ; Q b =    0000.000  0010  0.0002   0000 1000  0.0008  0000 0.0013 92 Chapter 5. Error localization

From the upper-triangular matrix R, the rank number of A is revealed as 3. which leads || || to a QR basic solution x =[0, −0.2884, −0.0064, 2.2422]0 and xB 2 =1.1151.The B ||xLS||2 LS solution involves all of the four variables, while the QR basic solution affects only three variables, the same as the rank number of A.

SVD/QR subset selection

Although QR with column pivoting is computationally efficient, unfortunately, in practical cases, the rank number of the sensitivity matrix of the updating equations is unknown. While QR with column pivoting may sometimes “discover” rank deficiency, it is not entirely reliable as a method for detecting near rank deficiency [22, p.245] when measurement noise, model imprecision and computational roundoff error are present. Since SVD is a stable and reliable way to handle near rank deficiency, the QR with column pivoting subset selection can then be improved immediately by including a SVD operation to determine a rank estimate:

• Compute the SVD A = UΣV0 and use it to determine a rank estimate r˜.

m×n • Calculate a permutation matrix P such that the columns of matrix A1 ∈< in

AP =[A1, A2] are “sufﬁciently independent”.   z • The basic solution subset is selected with permutation matrix P such that xB = P   0 r˜ and z ∈< minimizes ||b − A1z||2

The permutation matrix P may be computed in the same way as for QR with column pivoting, by using r˜ as the numerical rank number. In such a way, the SVD applied to A just helps to determine a rank estimate. However, Golub [22] showed that a heuristic permutation matrix can be obtained by applying QR with column pivoting to a sub-matrix of V0, instead of matrix A. This sub-matrix, the r˜ upper rows of matrix V0, is usually much small in size than 5.2. Error localization principles 93 matrix A, and consequently, the computation of the permutation matrix is much reduced.

Finally, a SVD/QR-based subset selection algorithm is ready:

0 • Compute the SVD U AV = diag(σ1,... ,σn) and save V;

• Determine r˜ ≤ rank(A).

0 0 • Apply QR with column pivoting: Q V (:, 1:˜r)P =[R11 R12] and set AP =[B1 B2]

m×r˜ m×(n−r˜) with B1 ∈< and B2 ∈< .   z r˜   • Determine z ∈< such that ||b−B1z||2 = min and consequently obtain xsub = P   0 where z ∈

As a numerical example, Let the equations be (obtained by adding noise to A of

(5.35)):

    0.0433 4.4481 1.9877 5.4633 10.9450     −0.6487 4.7795 3.0172 4.3456   8.3795       4.0244 −1.9859 0.0003 5.9882   13.9822   x =   (5.36)  1.6511 0.3756 −5.9916 3.6847   8.1968  −2.5985 4.2316 2.0051 −0.1944 −1.6994 −0.9106 4.4748 −1.0117 3.4409 6.4241

Matrix A may be identiﬁed now as of full-rank with QR decomposition and the unique solution is x =[0.7991, 0.1726, 0.0038, 1.8551]0. With SVD/QR method, A is ﬁrst decomposed with SVD, which produces

  0.0071 0.5047 0.3276 −0.7987 0.6269 −0.4612 −0.4266 −0.4608 V =   ; σ(A)=(12.2537, 9.5545, 5.8455, 0.05) 0.1459 −0.5514 0.8214 −0.0102 0.7653 0.4782 0.1899 0.3868 and the numerical rank number of A can be determined 3. The QR decomposition is performed on V0(:, 1:3), which produce the permutation matrix 94 Chapter 5. Error localization

  0001 0010 P =   1000 0100

The matrix B1 is therefore compounded by 3th, 4th and 2th columns of A and the solution is obtained as:

0 0 z =[−0.0064, 2.2421, −0.2884] ; xsub =[0, −0.2881, 0.0064, 2.2421].

Yang [67] employed this SVD/QR-based method in the selection of updating parameters and named this approach as a model error indicator function. SVD is a powerful tool for handling rank-deﬁciency problems while QR basic solution suggests that subset solutions which involve only as many variables as the rank number of the problem are always available for rank-deﬁciency problems. However, it should be aware that for QR or SVD/QR with column pivoting methods, the selected subset has nothing to do with the observation vector b, which is the residual vector or the model errors for model updating problems. If the model errors happens to be in this subset, the solution xsub will be a nice indicator of the main model errors. However, if some of the main model errors are not included in the selected subset, they will be redistributed over every parameter of the chosen subset. Such a situation may not be rare for QR or SVD/QR with column pivoting methods, because they do not take into account the model errors b during the subset selection.

5.2.3 TLS-based error localization technique

TLS-based error localization technique includes four steps:

• Regularising and condensing the noise-contaminated updating equations by SVD in

TLS sense; 5.2. Error localization principles 95

• Performing a forward subset selection with the condensed updating equations;

• Performing a backward exclusion according to the descending ratios of relative residual functions;

• solving the updating equations with the selected subset and determining the signiﬁcant

model errors.

TLS-based regularising and condensing

The ﬁrst order sensitivity model updating equations Ax = b, with the presence of measurement noise and model inaccuracy, are self-contradictory and can never have a perfect solution. In FE model updating problems, because matrix A is computed with the inaccurate initial FE model and the noise-contaminated test data, it is rational to regularise Ax = b by correcting both A and b, as showed in (4.32):

(A + E)x =(b + r)

To determine the “small” correction matrix E and vector r, the regularised equations are row- and column-weighted as:

−1 D(A + E)T1T1 xT2 = D(b + r)T2 (5.37)

where weighting matrices: D = diag(d1,... ,dm); T1 = diag(t1,... ,tn); T2 = tn+1. The weighted equations (5.37) can be rewritten as:

−1 −1 (DAT1)(T1 xT2)+(DbT2)(−1) + (DET1)(T1 xT2)+(DrT2)(−1)=[0] (5.38) 96 Chapter 5. Error localization and further compacted to:         −1 −1 T1 0  T1 xT2 T1 0  T1 xT2 D[Ab]     + D[Er]     =[0] (5.39) 0T2 −1 0T2 −1

Let T = diag(t1,... ,tn,tn+1), (5.39) is rewritten as   −1 T1 xT2 (D[Ab]T + D[Er]T)   =[0]. (5.40) −1

Let

C = D[Ab]T (5.41)

0 0 have SVD as C = UΣV = U · diag(σ1,σ2,... ,σn+1) · V and its estimated numerical rank number n1 ≤ n. Then we can partition U, V and Σ as following:       Σ V11 V12 0 W11 W12  1 0  U =[U1 U2]; V =   ; V =   ; Σ =   ; (5.42) V21 V22 W21 W22 0 Σ2 where: m×n1 m×(n+1−n1) n1×n U1 ∈< ; U2 ∈< ; W11 ∈< ;

n1×1 (n+1−n1)×n (n+1−n1)×1 W12 ∈< ; W21 ∈< ; W22 ∈< ; Σ Σ 1 = diag(σ1,... ,σn1 ); 2 = diag(σn1+1,... ,σn+1). Then we obtain separated matrices of C:    

Σ1 0  W11 W12 C =[U1 U2]     = U1Σ1[W11 W12]+U2Σ2[W21 W22]. (5.43) 0 Σ2 W21 W22 5.2. Error localization principles 97

Let

∗ ∗ C = U1Σ1[W11 W12]; ∆C = −U2Σ2[W21 W22]; ⇒ C = C +∆C (5.44) where C∗ denotes the signiﬁcant part of C and ∆C is the nonsigniﬁcant part, substitute (5.43) for D[Ab]T of (5.40), we have   −1 T1 xT2 (C∗ − ∆C + D[Er]T)   =[0] (5.45) −1

As C∗ represents the signiﬁcant part of C =[DA b]T, the weighted regularising matrix D[Er]T then should be

D[Er]T =∆C = −U2Σ2[W21 W22] (5.46) and equation (5.40) becomes as   −1 T1 xT2 U1Σ1[W11 W12]   =[0] (5.47) −1

−1 −1 −1 where T1 = diag(t1 ,... ,tn ).

For numerical rank number n1 of matrix C:

• If n1 = n +1,thenU1Σ1[W11 W12] is a complete space and its null space is only −1  T1 xT2  a point, which means   is a zero vector. That is, we weight observation −1 vector b with a zero factor and get a zero solution. This situation makes no sense.

0 • If n1 = n, then the null space of U1Σ1[W11 W12] is the range of [W21 W22] = 98 Chapter 5. Error localization

 

 V12   . Thus, from equation (5.47), we have: V22

    −1  T1 xT2   V12    =   α (5.48) −1 V22

−1 −1 for some coefﬁcient α.FromT1 x = V12α and −T2 = V22α, we obtain the unique TLS solution:

−1 −1 xTLS = T1V12α = −T1V12V22 T2 (5.49)

if V22 =06 (If rank(A)=n − 1,asrank(C)=n1 = n>n− 1, V22 = 0, no TLS solution exists).

• If n1

case, instead of singling out a “minimal normal” solution, we do prefer a solution that

contains minimal non-zero components for error localization purpose. To this end,

a subset selection will be conducted based on W11 and W12. Because for equation

(5.47), if the perfect solution exists, it can be rewritten as:

−1 W11(T1 xT2)=W12 (5.50)

As C = D[Ab]T, rank(C) ≥ rank(A), if the estimated numerical rank number n1 of matrix C is larger than the one of matrix A, then the observation b is out of range(A)

(see (4.30) for range(A)’s deﬁnition), which means that design variables corresponding to sensitivity matrix A cannot cover the model errors represented by b. We assume that the main model errors can be localized within the involved design variables, therefore, we can 5.2. Error localization principles 99 take the numerical rank number of matrix A as the one of matrix C. According to section

§4.3 (page 59), we use an ε-related 2-norm condition number k2A to estimate rank(A). To clarify this TLS-based regularising and condensing process, we here take the free

PCB described in section §5.4.4 (on page 141) as an example. When QMS updating equations are employed with the ﬁrst 7 modes, as 56 master DOFs are deﬁned in the mode shapes and 42 relative thickness parameters are used as updating variables, the matrix A has a dimension (7 × 56) × 42 = 392 × 42 while b has 392 × 1: A ∈<392×42; b ∈<392×1.The weighting matrices D and T are determined according to section §4.7. The numerical rank number n1 is determined as 41 by checking the ε-related 2-norm condition number k2A (see

41×42 Figure 4.1 on page 62). Then, with SVD applied on C and n1 =41, we obtain W11 ∈<

41×1 and W12 ∈< for equation (5.50).

QR F forward subset selection

If n1

(k−1) QR decomposition the ﬁrst maximum 2-norm column of R22 is chosen into the subset. This is irrational for error localization purposes. For example, for the equations:

      x1 2 132  0.9500   x2   −7259  = 2.0100 x3 3 414 4.0500 x4 the most probable error parameter should be x2 instead of x4. This is judged by evaluating the

2 2 2 2 2 cos θ between column vectors of A and b, which are [cos θ1, cos θ2, cos θ3, cos θ4]= [0.0000, 0.9998, 0.3846, 0.6075]

If, instead of the maximum norm column, we add a column that is the closest to the

ﬁtted residual vector (iteratively ﬁtted with the selected variables) into the subset, then the 100 Chapter 5. Error localization correspondingly chosen subset is somewhat the “closest to the observation” subset. Follow- ing the Householder QR with column pivoting algorithm [22, p.235], we obtain a forward subset search procedure as following. The so-called forward subset search simply means that the subset is expanded from null to a certain size under a consistent criterion.

STEP 0: initial data state.

(1) (1) (1) W = W11; [q1, q2,... ,q ]=W ; W = W12; • 11 n 11 12 (1) (1) −1 0 qn+1 = W12 ; y = T1 xT2; l =1; p =[1, 2,... ,n] . The index vector p indicates the parameter numbers associated to the columns of ma-

trix W11.

STEP l:

• Seek the ﬁrst column j ∈ [l, n] according to maximum cos2 θ criterion, such that:

0 2 (qj (l:n) qn+1(l:n)) 0 0 is maximized; (qj (l:n) qj (l:n))(qn+1(l:n) qn+1(l:n))

• Interchange column and variables between l and j:

th th (l) l and j columns of W11 interchanged; lth and jth rows of the index vector p interchanged;

(l) (l) • Perform Householder transform on W11 and W12 and replace the old ones:

(l) (l) (l) (l) W11 ⇐ HlW11 ; W12 ⇐ HlW12 (5.51)

where Hl is the Householder matrix, which zeros the all components below diagonal 5.2. Error localization principles 101

th (l) line of the l column of W11 :   (l) (l) (l)  q11 q11 ... q1n     (l) (l)  ( )  0 q22 ... q2n  H W l =   (5.52) l 11  . . . .   . . .. .    (l) (l) 0 qn12 ... qn1n

(l) where Hl is constructed with a vector u = W11 (l : n1,l):  

I(l−1)×(l−1) 0  r − 0 0 Hl =   ; Hl = I 2vv /(v v); r 0Hl (5.53)

u(2:(n1−l+1)) (1) = 1; (2 : (n1 − l +1))= || || v v (1)(1+ u 2 ) u |u(1)|

STEP k:

Suppose k − 1 steps are performed, then

• l = k;

(l) • if ||W12 (l : n1)||2 > 0;Redostepl.

FINAL RESULTS:

(l) Until ||W12 (l : n1)||2 == 0 or k = n1 − 1, we obtain ﬁnal results:     (k) (r−1) (k) (k) (k)  q11 q11 ... q1r ... q1n   q1,n+1       (k) (k) (k)   (k)   0 q22 ... q2r ... q2n   q2,n+1  W11 =   ; W12 =   ; p. (5.54)  . . . . .   .   ......   .      (k) (k) (k) 00... qrr ... qrn qr,n+1

A total permutation matrix P can be obtained by expanding the index vector p if needed: p(i) is the row number of nonzero component of the ith column of the P. 102 Chapter 5. Error localization

With MATLAB syntax conventions, this process is described as following:

Given A ∈eps, cosin2(j)=(b’*A(:,j))2 /c(j); else cosin2(j)=0; end; end; [mxcosin2,k]=max(cosin2); %ﬁnd k indicating maximum cos2 θ r=0; while (mxcosin2 >eps) r = r+1; % exchange r and k columns and the corresponding indicators. c(r) ↔ c(k); p(r) ↔ p(k); cosin2(r) ↔ cosin2(k); A(:,r) ↔ A(:,k); v(r:m) = house(A(r:m,r)); % Compute Householder vector % Compute H × A, H is the Householder matrix A(r:m,r:n) = rowhouse(A(r:m,r:n),v(r:m)); b(r:m) = rowhouse(b(r:m),v(r:m)); %Compute H × b for i = r+1:n, c(i) = c(i) - A(r,i)2; end; if (reps, cosin2(j)=(b(r+1:m)’*A(r+1:m,j))2 /c(j); else cosin2(j)=0; end; end; % ﬁnd k indicating maximum cos2 θ [mxcosin2,k]=max([zeros(1,r),cosin2(r+1:n)]); if mxcosin2eps) p(r) ↔ p(k); A(:,r) ↔ A(:,k); end; break; % break search r end; end; ns =r;p=p(1:ns); % ns is now the number of the selected parameters

The parameter selection index vector p(1 : ns) is an equivalent of the permutation matrix P. If needed, p can be expanded to P: p(i) is the row number of nonzero component in ith column of matrix P. To illustrate the parameter selection index vector p, its values are plotted vs. its row 5.2. Error localization principles 103 number. Figure 5.5 shows a typical result p for a simulated PCB structure with QMS updating method (see §5.4.4, Figure 5.42 on page 143). For parameter error localization, the horizontal axis represents the selection order during the subset search, while the perpendicular axis represents the parameter number deﬁned before error localization.

Figure 5.5: A typical parameter selection index vector p

45 The parameter index 40 number: 35 13 23 11 25 17

30 6 21 19 18 10 16 7 3 12 14 25 20 15 26 2 1 20 4 38 28 39 31 Parameter number 27 30 5 40 32 15 22 24 29 42 37 10 36 35 41 34 33 8 5

0 5 10 15 20 25 30 35 40 parameter selection order

Backward excluding updating parameters

With the TLS and QR F operations, a TLS/QR F subset and the corresponding selection order are obtained. The subset will generally have as many variables as the rank number of augmented matrix C, which is assumed the same as that of matrix DAT1, because in practice inaccuracy is present both in the initial mathematical models and in the measurement data. However, the main model error number has no direct relation with the rank number of

C, because even if there exists no model errors, matrix C has its rank number. As we have obtained the selection order from QR F operation, it is proposed to exclude some parameters backward to the QR F selection order, according to a descending ratio of the residual 104 Chapter 5. Error localization vector’s 2-norm.

With the ns ≤ n1 selected subset parameters, equation (5.47) can be rewritten as:

−1 (s) (s) (s) U1Σ1W11 (T1 x T2)=U1Σ1W12 (5.55) where superscript (s) denotes the selected (also indicating the selection order) variables and their corresponding columns.

As the forward selected updating parameter number is reduced by backward exclusion, perfect TLS solution is no longer guaranteed (less variables than needed). Therefore, for simplicity, LS solution residue of equation (5.55) is employed. Because

−1 (s) (s) (s) rs = ||U1Σ1W11 (T1 x T2) − U1Σ1W12||2 −1 (5.56) (s) (s) (s) = ||Σ1W11 (T1 x T2) − Σ1W12||2 equation (5.55) can be rewritten in LS sense as:

−1 (s) (s) (s) Σ1W11 (T1 x T2)=Σ1W12 (5.57)

This equation is QR decomposed as:

−1 (s) (s) R(T1 x T2)=br (5.58)

where R is a upper triangular matrix; br being the new residual vector. When equations (5.58) are obtained, the residual function is ready. Let us consider a 5.2. Error localization principles 105 simple numerical example:       324 2 y1  4              025 1 y2 −3           =   001 7 y3  2       

0000.5 y4 −1

Initially, if no variable is used to solve the equations, then the residue can be defined as the p 2 2 2 2 vector length ||br||2 = 4 +(−3) +2 +(−1) . If the first selected variable y1 is used to solve the equations, then the first equation is solved with y1 perfectly, while the associated p 2 2 2 residue to the whole equation set is (−3) +2 +(−1) .Wheny1 and y2 employed, the p first two equations are solve perfectly and the associated residue is 22 +(−1)2,etc.

Therefore, a relative residue function can be deﬁned as:

qP n1 2 j=i+1 brj Rri = × 100%; i =0, 1,... ,ns ≤ n1 (5.59) ||br||2

A residue descending ratio is then deﬁned as:

dRri = Rri−1 − Rri i =1, 2,... ,ns ≤ n1 (5.60)

When a main model error parameter is included, the relative residue function is expected to have a significant reduction, that is to say, the residue descending ratio dRri must have a significant value. Therefore, a threshold value for dRri may be established to perform a backward exclusion. For these parameters to be excluded, the residue descending ratio satisfied:

dRrj

where nb is the number of the retained parameters after backward exclusion; dRrlim is the threshold value of the residue descending ratio dRr. Figure 5.6 illustrates a typical residue descending ratio chart of a simulated free PCB structure (see §5.4.4). The horizontal axis represents the parameters selection order. The parameter numbers are indexed by the parameter selection index vector p, as shown in Figure

5.5.

Figure 5.6: A typical residue descending ratio Residue descending ratios of a PCB 35

0 10 20 30 40 Selected parameter’s order

The determination of dRrlim is somewhat like the determination of matrix rank, that no rule may be generally accepted. If we suppose that the main model errors are located at some few parameters of the subset, then we may expect a threshold dRrlim that is higher enough than the mean value of dRri. On the contrary, if the model errors are over most parameters of the subset, in other words, no particular parameter including the main model errors, then this threshold value may be much lower than the mean value of dRri.Asa simple rule of thumb, dRrlim maytakeavalueas: P dRri 100 dRrlim = α = α % (5.62) n1 n1 5.2. Error localization principles 107

where α is a coefﬁcient (0.5˜1); n1 is the estimated numerical rank number of the sensitivity

100 matrix; n1 % represents the average residue descending ratio of the selected parameters. Although α should be case dependent, for these case studies introduced in this chapter, α is ﬁxed to 0.8, to make it possible for the error localization program to run automatically, which is desired for statistical simulation with computer generated random noise.

Parameter exclusion based on the subset solution

After the backward exclusion, the retained subset parameters are used to solve the model updating equations. The solution values for relative parameter changes y = T1x can further be used as main model error indicators. The parameters whose solution values are below their corresponding threshold values should not be regarded as the main model error sources and therefore should be excluded from the retained subset for further iterative updating steps.

Suppose the threshold values for the relative parameter changes y = T1x have been set, then the main model errors are assumed at these parameters whose solution satisﬁed:

(0) yi >Yi i =1, 2,... ,n (5.63)

th (0) where yi is the i components of y = T1x while Yi is the corresponding threshold value. For the case studies introduced in this chapter, the threshold values for relative parameters have been determined as:

ythreshold = max(0.01, 0.2 × max(abs(y1,y2,... ,yn))) (5.64) 108 Chapter 5. Error localization

5.3 Approach of evaluating the efﬁciency of error localiza-

tion methods

5.3.1 Success ratio of parameter error localization

When measurement errors are artiﬁcially simulated, a single random shot is not quite convincible. In the literature, we would read “... the presented calculation, ... as well as results not reported here ...”. That may be an unpleasant pity. In this thesis, some random shots will be used and the results are averaged to get some statistic interpretation.

To this end, we deﬁne a success ratio of parameter error localization as:

N SR = rdet × 100%; (5.65) max(Ndet,Nerr) where:

Nerr is the number of parameters that have different values between the initial FE model and the one used to produce artiﬁcial “experimental” data;

Nrdet the number of detected errors among the errors which are introduced;

Ndet the number of parameters retained at the end of the localization procedure.

For example, a cantilever beam is simulated with 18 physical parameters, E1, E2, ...,

E18, both in the initial FE model and in the FE model used to produce artiﬁcial “experimental” data. Among the 18 parameters, E5, E6, E7 and E8 have different values between the two FE models. The error parameters detected by the error localization procedure are E5,

E6, E8, E9, E18. Among the 5 detected parameter errors, only E5, E6,andE8 are true errors. 3 × In this context, Nerr =4, Ndet =5, Nrdet =3, SR = 5 100% = 60%.

In (5.65), the denominator is decided as max(Ndet,Nerr) because of the following two situations: 5.3. Approach of evaluating the efﬁciency of error localization methods 109

• Only a part of the introduced errors are involved. For the above example, the error

Nrdet localization procedure may detect, for instance, only E5 and E6. = 100% is not Ndet to say the errors are well localized.

• All introduced errors are detected, but with error-free parameters detected as errors.

For instance, parameter E4, E5, E6, E7, E8 and E9 may be detected by the procedure.

Nrdet Nerr = 100% does not imply a perfect error localization result.

For repeated simulations with random test noise that has the same noise level, the success ratios of all random shots are used as sample data to calculate some statistics: v u 1 Xn 1 uXn SR = SR(i); σ = t (SR(i) − SR)2 (5.66) n SR n − 1 i=1 i=1 where n is the number of random shots. For case studies introduced in this chapter, this number is ﬁxed to 20.

This success ratio is only used for simulated cases, since real model errors are not available for practical applications. The average success ratio and its borders associated with the positive and negative standard deviations will be presented for different simulated noise levels.

Figure 5.7: Success ratios for a free PCB structure TLS/QR−F method 100 Solid line: the average 90 success ratios for 80

70 20 random shots.

60 Dash lines: the average 50

Success ratios % 40 success ratios

30 20 standard 10 deviation σ. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Measurement error level % 110 Chapter 5. Error localization

Figure 5.7 illustrates the success ratios of parameter error localization for a simulated free PCB structure (see §5.4.4 on page 141).

5.3.2 Measurement noise inﬂuence on error localization

Measurement noise is one of the most challenging problems in the ﬁnite element model updating. The initial ﬁnite element models are entirely independent from measurement errors. All sensitivity-based updating methods use the difference between the dynamic test data and the corresponding calculated quantity to correct the initial model, instead of the measured quantity itself. The initial models are established with analysts’ previous knowledge of the structures, therefore, they are assumed to be approximated to the structures.

Let us deﬁne:

|Qεx| Qˆx = Qx + Qεx =(1 Ex) × Qx; Ex = × 100% (5.67) |Qx| where: ˆ Qx is the measured quantity;

Qx the noise-free measured quantity;

Qεx the difference: Qεx = Qˆx − Qx;

Ex the relative errors in the measured quantity. Then the differences between measured and analytical data can be expressed as:

ˆ b = Qˆx − Qa =(Qx − Qa)+Qεx = b +∆bεx (5.68) where:

Qa is the analytical quantity; ˆ b the noise-free counterpart of b, b = Qx − Qa; ˆ ∆bεx the difference between b and b caused by test noise, ∆bεx = Qεx. 5.3. Approach of evaluating the efﬁciency of error localization methods 111

For sensitivity-based model updating (error localization), the measurement error in- ﬂuence can be expressed as:

∆bεx Qεx Eu = × 100% = × 100% (5.69) b Qx − Qa

In the model updating philosophy, we believe that the initial FE models are the best models we can assume, therefore, we can generally expect:

|Qx − Qa||Qx| (5.70) which leads to:

Eu Ex (5.71)

For example, suppose the measurement quantity is a resonance frequency, the noise- ˆ contaminated test frequency fx =95Hz; the noise-free test frequency fx = 100Hz; and with an initial FE model, the analytical frequency fa = 110Hz; then the measurement error

|Qεx| =5Hz; the noise-free difference b =10Hz; the relative measurement error Ex =5%; ˆ however, the relative error in b caused by measurement noise is Eu = 50%. Thatistosay, a reasonable measurement precision does not guarantee that the updating equations are also reliable.

5.3.3 Simulated noise pattern in case studies

For practical finite element model updating problems, neither the model errors nor the test errors are known. As the main concern here is the noise influence, then simulated case studies are used to assess the measurement error influence on parameter error localization. 112 Chapter 5. Error localization

The simulated cases are also simplified such that one-to-one correspondence exists between the errors and the finite element models. There is no realistic model for introducing test noise into analytically generated data. Basically, experimental noise consists of correlated and uncorrelated errors. Correlated errors arise from signal conditioning, transduction, signal- processing, etc. Uncorrelated errors includes these from thermal noise in electronic circuits of test system as well as unpredictable external disturbances. In this study, the artificial measurement errors are simulated in the following ways:

• Simulated errors in resonance frequency:

The measurement errors in resonance frequency are simulated such that:

p fˆ = f +∆f =(1+ × ε ) × f ∀i (5.72) i i i 100 i i

ˆ where fi is the noise-free “experimental” data and fi is the noise-contaminated value.

εi is an uniformly distributed random number in [−1, 1] and p is a percentage number.

• Simulated errors in mode shapes:

The measurement errors in mode shapes are simulated in a similar way as in resonance

frequency simulation:

p φˆ =(1+ × ε ) × φ ; ∀i, ∀j (5.73) ij 100 ij ij

th th where φij is the j mode shape’s component of i DOF while εij is an uniformly distributed random number in [−1, 1] and p is a percentage number.

• Simulated errors in FRF data:

For FRF data there exists white-noise whose level is constant over the interested frequency range. The white-noise shall have important inﬂuence if the updating frequen- 5.3. Approach of evaluating the efﬁciency of error localization methods 113

cies are chosen in those frequency ranges where FRF data are low. Nevertheless, the

white-noise impact can be limited by choosing updating frequencies with high level FRF data, such as in the vicinities of resonances. Therefore, the white-noise is not sim-

ulated in the case studies. However, in the vicinities of resonance frequencies, though

the white-noise impact has been reduced, the accuracy of FRF data is highly affected

by damping, inaccurate exciting force, etc. As no realistic model exists for introducing

simulated noise into analytically-generated data, a simple uniformly-distributed noise

model is used:

p Hˆ (ω )=(1+ × ε ) × H (ω ); ∀i, ∀k (5.74) ij k 100 ijk ij k

where Hij(ωk) and Hˆij(ωk) are respectively the error-free and noised values of FRF

th th between i and j DOFs while εijk is an uniformly distributed random number in [−1, 1] and p is a percentage number.

5.3.4 POMUS and success ratio computation

Theories described in Chapter 2, 3, 4, 5 and 6 are employed to develop a Parameter- Oriented Model Updating System (POMUS). POMUS is a menu-driven FE model updating system. The main codes are programmed in MATLAB language. Program scheme of PO-

MUS is shown in Figure 5.9. The GUI interface and more detailed information are included in Appendix B.

POMUS is designed for practical applications, not directly for evaluating the error localization success ratios where repeating random shots are required. An independent program, ERRLOCP, is developed to evaluate error localization success ratio for different methods. The program scheme of ERRLOCP is displayed in Figure 5.8. The scheme of core parts of ERRLOCP are shown in Figure 5.10. Many procedures are shared between POMUS and 114 Chapter 5. Error localization

ERRLOCP, except the noise simulation and success ratio evaluation. Required messages for both POMUS and ERRLOCP are basically the same, which can be prepared with the same GUI interface and preserved in an ASCII data ﬁle.

Figure 5.8: ERRLOCP program scheme

“Experimental” FE model

Run ANSYS program. Initial FE model

Error-free “experimental” data. ANSYS program

Set initial random shot number Mass and stiffness sensitivity

Add artificial noise

“Experimental” data

Set up mofel updating equations, weighting

Ax=b

Do Parameter Error localization

Increase No random shot Random shot number number ==20? Yes Computer (average) success ratios, etc.

Stop 5.3. Approach of evaluating the efﬁciency of error localization methods 115

Figure 5.9: POMUS program scheme The vibration structure under study

Set up the initial experimental configuration , referred to it as #1.

Set up FE model with ANSYS 1 script language. Special format required for the updating process. For all other PBC test, Conduct dynamic test. Perform The FE modal parameter extraction. models are set Assume all given parameters to up by ICATS data format required. be updated. modifing the FE model in #1 configuration #1. FE analysis with ANSYS, ErrorIndex 2 used for computing sensitivities

FRFa, fa, mode shapes; mass and FRFx, fx, mode shapes stiffness sensitivity matrices

Correlation, model match between test data and FE results 3

Set up updating equations: A1x=b1 4

Stacking all updating equations for all configurations: Ax=b. 5

-1 Weighting equation: (DAT1)(T1 xT2)=DbT2 6

No Iter==1? Yes

Do error localization 7

Solve the updating equations 8 9 Modify all To #1 No FE models Stop iteration? for all configurations To other Yes configurations STOP

1 FE script ﬁle format is described in Appendix C 2 Mass and stiffness sensitivities are computed based on (4.50) 3 Correlation and model match techniques are described in Chapter 2 4 Sensitivity-based updating equations are discussed in Chapter 3 and A 5 Stacking PBC updating equations are treated in Chapter 6 6 Weighting strategies are described in 4.4, 4.7 7 Error localization is described in 5.2.3 8 Solving methods are described in 4.3, 4.5, 4.6, 4.8 and 5.2.2 9 Updating parameters are described in 4.9 116 Chapter 5. Error localization

Figure 5.10: Error Localization scheme

The updating equations Ax=b; A, x, b have been row- and column weighted. Column weighting factor T2 needed to compute relative parameter changes from x

(1) Perform SVD on A: A=U*V’; ) ) (2) 2-norm condition number: k2(A) = 1/ r; (r=1,2,3,...,n) (3) Plot trial k2(A), interactively determine the numerical rank number n1

(1) C=[A b]; SVD C: C=U(V’; W=V’; (2) partitioning matrices according to n : ( ( ( 1 U=[U1, U2]; (=[(1, 0; 0 (2]; W=[W11, W12; W21, W22];

W11x=W12

Perform QR-F subset selection on W11x=W12

Selected.parameters that will be indexed by a vector p

Use only the selected parameters according to the selection order p, ( and take into account 1: ( (s) (s) ( 1W 11 x = 1W12 ( (s) (s) ( Perform QR decomposition on 1W 11 x = 1W12 (s) Rx =br Compute relative residue function and its descending ratio Rri , dRri. Exclude parameters reversely to the QR-F selection order p, until dRri. dRrlim

The retained parameter number

Use the retained parameters to solve equations : A(k)x(k)=b. Relative parameter changes : y = x/T2 (0) If yi >Yi , then parameter yi is identified as a main model error.

Error localization result 5.4. Case studies 117

5.4 Case studies

In the following sections §5.4.1˜§5.4.4, we ﬁrst introduce a cantilever beam where errors are simulated in Young’s module, then a lumped stiffness-mass system with both mass and stiffness errors. The third example is a 3-D bay structure whose model errors are assumed in ρ and in E. Finally, a PCB structure is simulated in free and wedge-lock boundary conditions. Detailed descriptions of the structures introduced in the following case studies are included in Appendix D.

5.4.1 Case Study 1: A Cantilever beam

The ﬁrst case is a cantilever beam structure, shown in Figure 5.11. This structure is modelled with 18 beam elements. The initial model is a uniform beam while the experimental model is simulated by reducing Young’s modules of element #5, #6, #7 and #8. All of the

18 Young’s modules of the 18 elements are taken as design variables, as listed in Table 5.2:

Table 5.2: Updating parameters of the cantilever beam (E: ×109 N/m2) Parameter number 1—4, 9—18 5—8 “experimental” px 26.68 20.00 initial FE pa 26.68 26.68 px−pa pa 0 -0.2504

Using resonance-frequency-based updating equations

Resonance frequencies contain very limited information about a structure. Generally speaking, this information is not sufﬁcient for error localization purposes. However, for parameter-oriented model updating, if the unknown number has been reduced by analyst’s previous knowledge of the structure, then, a further subset selection still makes sense.

For this cantilever beam, the first 8 resonance frequencies are used to establish the updating equations. The resonance frequencies of the “experimental” and “analytical” model 118 Chapter 5. Error localization are listed in Table 5.3. The 45o line graphic of the resonance frequencies in Figure 5.12 confirmed that analytical model’s stiffness has been overestimated. The noise-free solutions are shown in Figure 5.13. In this figure, abbreviation “ErL” denotes the simulated error level, i.e. the value of p in (5.72); “ShtIdx” denotes the shot index number for random noise simulation; “k2A 2nm C.N.” denotes the the trial ε-related 2 norm condition number.

Sub-ﬁgure a displays the 2-norm condition number of the weighted sensitivity matrix

A vs the trial ε-rank number. The definition can be found in section §4.3. Sub-figure b displays the parameter number according to the selection order during QR-F forward subset selection. For example, for this particular case, the first selected parameter is #5, the Young’s module of element #5; the second is #7, etc.

Sub-ﬁgure c shows the residual descending ratio vs the parameter selection order.

For instance, the ﬁrst selected parameter, E5, has a residue descending ratio about 60%; the second selected one, E7, has about 20%, etc. From right to left, the QR-F forward selected parameters are backward excluded if their residual descending ratios are smaller than the threshold value dRrlim, until the one whose residual descending ratio is larger than dRrlim. Sub-ﬁgure d shows the solutions with the parameters retained after backward exclusion. For those parameters not included in the main model error subset, their solutions are

ﬁlled with zero’s. Horizontal axis represents the parameter number.

Sub-ﬁgure e shows the solutions with SVD/QR subset. Zero’s are ﬁlled to those parameters out of the SVD/QR subset.

Sub-ﬁgure f shows the LS solution with all parameters. 5.4. Case studies 119

Figure 5.11: Cantilever beam structure.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 N

o Figure 5.12: 45 line for fx—fa 45o Line for frequency 1800 1600 1400 1200

1000x 800 Test f 600 400 200 0 0 500 1000 1500 Analytical f a

Table 5.3: Cantilever beam: resonance frequencies

f1 f2 f3 f4 f5 f6 f7 f8 fx 10.23 65.25 180.17 358.22 588.07 883.36 1219.84 1637.77 fa 10.69 66.99 187.58 367.56 607.66 907.99 1269.02 1691.66 |δf|% 4.4966 2.6667 4.1128 2.6073 3.3312 2.7882 4.0317 3.2905 120 Chapter 5. Error localization

Figure 5.13: Cantilever Beam Noise-free solution (Relative parameter changes). k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 6 0.1

0.05 5 0

4 −0.05

−0.1 3 −0.15 A Cond. Number

2 −0.2 k −0.25 1 Relative parameter changes −0.3

0 −0.35 1 2 3 4 5 6 7 8 5 10 15 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 15 0.2

0.1

0 10 −0.1

−0.2 5 Parameter number −0.3 Relative parameter changes −0.4

0 −0.5 1 2 3 4 5 6 7 8 5 10 15 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 60 0.1

50 0.05

40 0 30 −0.05 20 Residue descending ratio

Relative parameter changes −0.1 10

0 −0.15 1 2 3 4 5 6 7 8 5 10 15 c. Parameter selection order f. Parameter number Figure 5.13a: Trial 2-norm condition number, see §4.3 (page 59) for deﬁnition. b: Parameter number vs selection order, see Figure 5.5 on page 103. c: Residue descending ratio, see (5.60) on page 105 for deﬁnition. d: TLS/QR F solution. e: SVD/QR solution. f: LS solution with all parameters. 5.4. Case studies 121

Error localization success ratio is ﬁrst deﬁned for a single random shot in (5.65). For the solution shown in Figure 5.13, the main model error number detected Ndet’s are 4, 5 and 16 for TLS/QR-F, SVD/QR and LS methods, respectively; the detected errors that are introduced in the FE model Nrdet’s are 4, 0 and 4, correspondingly to the three solutions; the error number introduced in FE model Nerr is 4. Therefore, for this random shot, the error localization success ratios are 100%, 0% and 25% for TLS/QR-F, SVD/QR and LS methods, respectively. The simulations are repeated with 20 random shots for each simulated noise level. The average success ratio and the associated standard deviation are computed according to (5.66).

Figure 5.14 displays the average success ratios and their standard deviations vs the simulated measurement error levels. Sub-figure a shows the results obtained with TLS/QR-F method. The solid line represents the average success ratios SR. The two dash line represent SR σSR with a upper bound 100%. For example, when measurement error level is simulated as 0.5%, TLS/QR-F has about 60% average success ratio. Sub-figure b and c are the counterparts of sub-figure a, but with SVD/QR and LS methods, respectively.

SVD/QR [67] method has a zero success ratio for every simulated noise level. In fact, the subset selection has no relation to the frequency differences and unfortunately, the model error parameters are all expelled for this simulation, as shown in Figure 5.13. A simple minimal 2-norm solution redistributes errors over everywhere. Consequently, the success ratios are very low. For TLS/QR-F methods, Figure 5.14 showed that, if the measurement error levels are much lower than the model errors (see |δf%| in Table 5.3), high success ratios may be obtained, otherwise, the error localization results may not be helpful.

For systems contain more unknowns, the resonance frequency information is generally not sufﬁcient for error localization purpose. Therefore, this approach will not be further evaluated for other simulated case studies. 122 Chapter 5. Error localization

Figure 5.14: Cantilever Beam: success ratios (frequency-based)

TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

See (5.65) on page 108 for deﬁnition; See Figure 5.7 on page 109 for explanation. Figure 5.14a: TLS/QR F method; b: SVD/QR method; c: minimal 2-norm LS method.

Using modeshape-based updating equations

For modeshape-based updating equations, we can use either FBM updating equation

(3.23) or QMS updating equation (3.26). The following case studies will show that FBM updating equation (3.23) is generally much sensitive to measurement errors. Firstly, results from QMS updating equations are presented. The ﬁrst 8 modes are employed to establish the updating equations. Figure 5.15 showed the solutions without adding artiﬁcial test noise. 5.4. Case studies 123

Figure 5.15: Cantilever beam: noise-free solutions (QMS) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 35 0.1

30 0.05

0 25 −0.05 20 −0.1 15 −0.15 A Cond. Number 2

k 10 −0.2 Relative parameter changes 5 −0.25

0 −0.3 5 10 15 5 10 15 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 18 0.1

16 0.05 14 0 12 −0.05 10 −0.1 8 −0.15 6 Parameter number −0.2 4 Relative parameter changes 2 −0.25

0 −0.3 5 10 15 5 10 15 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 45 0.1

40 0.05 35 0 30 −0.05 25 −0.1 20 −0.15 15 −0.2

Residue descending ratio 10 Relative parameter changes 5 −0.25

0 −0.3 5 10 15 5 10 15 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Theoretically, the problem is of full-rank. However, when test noise is taken into account, there is a trade-off in the determination of numerical rank number. Figure 5.15a showed that, the ε-related 2-norm condition number sharply increase (from 8.8 to 31.9) if the

ε-related rank number is changed from 17 to 18. The following theorems [22, p.228, theorem

5.3.1] show how the condition number inﬂuences the precision of equation solutions. For an 124 Chapter 5. Error localization overdetermined, full rank problem, when data errors δA and δb presented in A and b of equation Ax = b,if

||δA||2 ||δb||2 σ 1 ρ = max , < n = , and sin(θ)= LS =16 ||A||2 ||b||2 σ1 k2(A) ||b||2

where ρLS = ||AxLS − b||2,then ||xˆ − x||2 2k2(A) 2 2 ≤ +tan(θ)k2(A) + O( ) (5.75) ||x||2 cos(θ)

where x and xˆ satisfy ||Ax − b||2 = min and ||(A + δA)xˆ− (b + δb)||2 = min, respectively.

For a rank deﬁcient problem, by using SVD, an equivalent underdetermined system By = d can always be established, while for By = d, the relative errors in y is slightly changed to as

(see [22, p.258, theorem 5.7.1]):

||yˆ − y||2 2 ≤ k2(B)(B min{2,n− n1 +1} + d)+O( ) (5.76) ||y||2

σ1 where k2(B)= . σn1 Therefore, we may probably say that the upper bound of relative solution error may ( )| jump up about k2 A r˜=18 =3.625 times, if the numerical rank number is determined as 18 k2(A)|r˜=17 instead of 17.

Figure 5.16 showed the error localization success ratios. For minimal 2-norm method, the numerical rank number is always set to 18, while the other results are calculated by assuming the numerical rank number as 17. It is evident that a much better result can be expected if the condition number is reduced.

Figure 5.17 showed the similar results where FBM-based updating equations are employed. Comparing it with Figure 5.16, we know that FBM updating equations are more sensitive to the noise. 5.4. Case studies 125

Figure 5.16: Cantilever beam: success ratios (QMS) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 5 10 0 5 10 0 5 10 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.17: Cantilever beam: success ratios (FBM) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.) 126 Chapter 5. Error localization

Using FRF-based updating equations

A typical receptance FRF is plotted in Figure 5.18. For FRF-based updating equations, ten updating frequencies are employed (Table 5.4).

Figure 5.18: A typical FRF at node 10 Type set to 1; C:\simunoise\beam5678\cfg1\simufrf\beama10.frf −1

"Experimental"

−2 FE model

−3

−4

−5 Modulus log10

−6

−7

−8 0 100 200 300 400 500 600 700 800 900 1000 Frequencies

Table 5.4: Updating frequencies for the cantilever beam (Hz) Index 1 2 3 4 5 6 7 8 9 10 f 56.25 87.5 168.75 193.75 337.5 375.0 568.75 612.5 868.75 906.25

Similar to modeshape-based methods, FRF-based updating equations can be either based on an excitation-residual-criterion (3.34) or an indirect-receptance-residue (3.35). Firstly, the indirect-receptance-residue-based results are presented. Then, as a comparison, excitation- residual-criterion-based success ratios are also demonstrated.

Figure 5.19 showed the noise-free solutions. The results are very much similar to the ones obtained from QMS updating equations (Figure 5.15). For this situation, the ε-related 2-norm condition number jump up about 7 times when the ε-related rank number is changed from 17 to 18.

Figure 5.20 showed the error localization success ratios. When test errors are present, properly reducing the numerical rank number may produce better localization results, even though the original problem is not rank deﬁcient. 5.4. Case studies 127

Figure 5.21 showed the similar results where excitation-residual-criterion-based updating equations were employed. This ﬁgure showed that equation (3.34) is much sensitive to measurement noise than (3.35). In fact, the test errors in hxj are ampliﬁed by the dynamic stiffness Z(I) in (3.34).

Figure 5.19: Cantilever beam: noise-free solutions (FRF-output) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 1200 0.1

0.05 1000 0 800 −0.05

600 −0.1

−0.15 A Cond. Number

2 400 k −0.2

200 Relative parameter changes −0.25

0 −0.3 5 10 15 5 10 15 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 18 0.1

16 0.05 14 0 12 −0.05 10 −0.1 8 −0.15 6 Parameter number −0.2 4 Relative parameter changes 2 −0.25

0 −0.3 5 10 15 5 10 15 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 45 0.1

40 0.05 35 0 30 −0.05 25 −0.1 20 −0.15 15 −0.2

Residue descending ratio 10 Relative parameter changes 5 −0.25

0 −0.3 5 10 15 5 10 15 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.) 128 Chapter 5. Error localization

Figure 5.20: Cantilever beam: success ratios (FRF-output) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 5 10 0 5 10 0 5 10 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.21: Cantilever beam: success ratios (FRF-input) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

5.4.2 Case study 2: A 10-DOF stiffness-mass system

The second example is a 10-DOF lumped string system, as shown in Figure 5.22. The updating parameters are shown in Table 5.5. Twenty parameters, (ﬁrst 10’s for mass and last 10’s for stiffness) are used as the updating parameters. 5.4. Case studies 129

Figure 5.22: A 10 DOF string system. Y k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 X m1 m2 m3 m4 m5 m6 m7 m8 m9 m10

Table 5.5: Updating parameters of the mass-stiffness string. Index 1 2 3 4 5 6 7 8 9 10

px(1-10) 0.9 1.0 2.0 2.25 6.0 5.0 4.6 2.0 1.0 1.0 m pa(1-10) 1.0 1.0 2.0 3.0 6.0 5.0 4.0 2.0 1.0 1.0 px−pa pa (1-10) -0.1 0 0 -0.25 0 0 0.15 0 0 0 px(11-20) 0.40 0.48 0.66 1.20 2.86 1.60 1.32 1.00 0.88 0.68 k pa(11-20) 0.40 0.48 0.60 1.20 2.20 1.60 1.32 1.00 0.80 0.68 px−pa pa (11-20) 0 0 0.1 0 0.3 0 0 0 10 0 pa(1-10), px(1-10): m1—m10 (kg); pa(11-20), px(11-20): k1—k10 (MN/m);

Using modeshape-based updating equations

To investigate the inﬂuence of test noise, we assume that all 10 modes are available and all 10 DOFs are measured.

Figure 5.23 shows the solutions where test noise is not added. TLS/QR-F methods gave perfect solutions. SVD/QR method failed to find the error source, because parameter k15 is excluded by the subset selection procedure while this error is redistributed over all of the other 19 parameters. A simple minimal 2-norm solution resulted in that all the 20 relative parameter changes were -1’s, which was not physically significant. In fact, when all of the 20 parameters are used as design variables, mode-shape-based updating equations are absolutely rank-deficient.

Figure 5.24 shows the results with 7% error level. The simulated noise are interpreted as “model errors” which are distributed over all the selected parameters. 130 Chapter 5. Error localization

Figure 5.23: 10 DOF system: noise-free solutions (QMS)

5k A 2nm C.N. ShtIdx=1, ErL(%)=0 x 10 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 6 0.4

5 0.3

0.2 4 0.1 3 0 A Cond. Number

k −0.1 Relative parameter changes 1 −0.2

0 −0.3 5 10 15 20 5 10 15 20 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 20 0.2

0.1

15 0

−0.1 10 −0.2

Parameter number −0.3 5 Relative parameter changes −0.4

0 −0.5 5 10 15 5 10 15 20 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 70 0.4

60 0.2

0 50 −0.2 40 −0.4 30 −0.6 20 −0.8 Residue descending ratio Relative parameter changes 10 −1

0 −1.2 5 10 15 5 10 15 20 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.24: 10 DOF system: solutions for noise level 7% (QMS) k A 2nm C.N. ShtIdx=20, ErL(%)=7 2 TLS/QR−F. ShtIdx=20, ErL(%)=7 300 0.3

250 0.2

0.1 200 0 150 −0.1 A Cond. Number

1002

k −0.2 Relative parameter changes 50 −0.3

0 −0.4 5 10 15 20 5 10 15 20 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=20, ErL(%)=7 SVD/QR ShtIdx=20, ErL(%)=7 20 0.2

0.1

15 0

−0.1 10 −0.2

Parameter number −0.3 5 Relative parameter changes −0.4

0 −0.5 5 10 15 5 10 15 20 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=20, ErL(%)=7 Min2nrm ShtIdx=20, ErL(%)=7 70 0.4

60 0.2

0 50 −0.2 40 −0.4 30 −0.6 20 −0.8 Residue descending ratio Relative parameter changes 10 −1

0 −1.2 5 10 15 5 10 15 20 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.) 5.4. Case studies 131

Figure 5.25 showed the error localization success ratios of QMS updating equations.

Neither SVD/QR nor 2-norm solution produces meaningful results. Figure 5.26 showed the similar results for FBM updating equations. Comparing it to

Figure 5.25, it is evident that FBM-based updating equations are much sensitive to measurement noise.

Figure 5.25: 10 DOF system: success ratios (QMS) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 5 10 0 5 10 0 5 10 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.26: 10 DOF system: success ratios (FBM) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.) 132 Chapter 5. Error localization

Using FRF-based updating equations

A typical receptance FRF is plotted in Figure 5.27. Ten updating frequencies (Table

5.6) are used to establish FRF-based updating equations.

Figure 5.27: A typical FRF at node 5 Receptance at node 5; Exciting force at 3. −3 "Experimental" Initial FE model −4

−5

−6

−7 Modulus log10 −8

−9

−10

−11 0 20 40 60 80 100 120 140 160 180 200 Frequencies

Table 5.6: Updating frequencies for the m-k string (Hz) Index 1 2 3 4 5 6 7 8 9 10 fui 20 30 60 80 90 110 120 150 170 180

A perfect solution can be obtained when simulated test errors are not added (Figure

5.28). This result also implies that the problem is not rank deﬁcient. However, from the

ε-related 2-norm condition number ﬁgure, we know that the numerical rank number may be reduced to 19 if test errors are taken into account, for doing so the condition number is half reduced. Figure 5.29 shows the error localization success ratios. This ﬁgure displays that, when test noise is low, the error localization is perfectly detected for all these methods, while with test noise, the TLS/QR-F methods provides the best results.

Similar results can be obtained from excitation-residual-criterion-based updating equations, as shown in Figure 5.30. Again, the results showed that the excitation-residual- criterion-based updating equations are more sensitive to measurement noise. 5.4. Case studies 133

Figure 5.28: 10 DOF system: noise-free solutions (FRF-output) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 45 0.4

40 0.3 35 0.2 30

25 0.1

20 0 A Cond. Number

152

k −0.1 10 Relative parameter changes −0.2 5

0 −0.3 5 10 15 20 5 10 15 20 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 20 0.3

0.2

15 0.1

0 10 −0.1

Parameter number −0.2 5 Relative parameter changes −0.3

0 −0.4 5 10 15 5 10 15 20 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 45 0.3

40 0.2 35 0.1 30

25 0

20 −0.1 15 −0.2

Residue descending ratio 10 Relative parameter changes −0.3 5

0 −0.4 5 10 15 5 10 15 20 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.29: 10 DOF system: success ratios (FRF-output) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 5 10 0 5 10 0 5 10 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.) 134 Chapter 5. Error localization

Figure 5.30: 10 DOF system: success ratios (FRF-input) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

5.4.3 Case study 3: Three dimension bay structure

The third case structure is a 19-node 3-D bay space frame modelled with 20 12-DOF 3-D beam elements (Figure 5.31 [68]). Updating parameters are shown in Table 5.7.

Figure 5.31: A 3-D bay structure (A =0.1 × 0.01 m2) Y 0.2m 0.6m 9 678VI VII VIII 0.05m 5 V IX XX IV 19 10 4 X III 18 XIX 11 3 0.4m XI II 17 XVIII 12 2 XVII XII I XV XIV XIII XVI 16 15 14 13 X 1 Z

Table 5.7: 3-D bay: Updating parameters p number 1,7,15 Others(≤ 20) 21,24,27,30,33,36,39 Others(22˜40) px 7850 7850 207 207 pa 7457.5 7850 223.56 207 px−pa pa 0.0526 0 -0.0741 0 3 9 2 pa(1-20), px(1-20): mass (kg/m ); pa(21-40), px(21-40): stiffness (10 N/m ); 5.4. Case studies 135

The experimental data are the ﬁrst 10 modes that contain only 57 translation DOFs’ messages. 40 parameters, ﬁrst 20 for relative changes of mass densities of the 20 elements and the left 20 for Young’s modules, are used as design variables.

Using modeshape-based updating equations

First 10 modes are used to establish model updating equations. Figure 5.32 showed the error free solution for QMS updating equations.

Figure 5.32: 3D bay: noise-free solutions (QMS) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 2500 0.08

0.06

2000 0.04

0.02 1500 0

−0.02 1000 A Cond. Number

2 −0.04 k −0.06

500 Relative parameter changes −0.08

0 −0.1 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 40 0.05

30 0

20 −0.05

15 Parameter number 10 −0.1 Relative parameter changes 5

0 −0.15 10 20 30 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 35 0.4

30 0.2

25 0

20 −0.2

15 −0.4

10 −0.6 Residue descending ratio Relative parameter changes 5 −0.8

0 −1 10 20 30 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

As mode shape is a relative quantity, and all of the mass densities and Young’s mod- 136 Chapter 5. Error localization ules of the ﬁnite elements are used as unknowns, the updating equations (coefﬁcient matrix

A) are of rank-deficiency. Therefore, a simple 2-norm solution(sub-figure f) does not get sense. Table 5.8 showed the numerically calculated singular values and the corresponding 2- norm condition number defined as σ1 ,j =1,... ,nfor noise-free data set. The rank number σj is determined as 39.

Table 5.8: The 3-D bay frame: singular values and 2-norm condition number

σ(1−8) 2.8309 2.2142 1.5381 1.3061 0.9915 0.7410 0.7049 0.5810

σ(9−16) 0.5477 0.5023 0.4837 0.4556 0.4455 0.4090 0.3948 0.3787

σ(17−24) 0.3514 0.3396 0.3309 0.2982 0.2820 0.2811 0.2581 0.2413

σ(25−32) 0.2238 0.2182 0.2044 0.1999 0.1818 0.1804 0.1653 0.1512

σ(33−40) 0.1394 0.1308 0.1080 0.0985 0.0933 0.0632 0.0529 0.0014

k2A(1−8) 1.0000 1.2785 1.8405 2.1675 2.8553 3.8205 4.0159 4.8729

k2A(9−16) 5.1691 5.6360 5.8523 6.2134 6.3547 6.9210 7.1704 7.4758

k2A(17−24) 8.0556 8.3368 8.5560 9.4928 10.0385 10.0694 10.9687 11.7307

k2A(25−32) 12.6505 12.9747 13.8470 14.1611 15.5695 15.6915 17.1245 18.7221

k2A(33−40) 20.3027 21.6466 26.2051 28.7452 30.3302 44.8040 53.4658 2084.4

Figure 5.33 showed the error localization success ratios. SVD/QR subset selection method does not correctly detect the model errors because in the selected subset, parameter

#1 (mass density errors at element #1) is always excluded, while this model errors will be redistributed over everywhere for minimizing the equation residue (see Figure 5.32f). A simple minimal 2-norm solution (using MATLAB pseudo-inverse function, which use a default tolerance to estimate numerical matrix rank number) neither works for error localization.

However, if the pseudo-inverse matrix is calculated with correct matrix rank number, the results become quite rational (see Figure 5.34). 5.4. Case studies 137

Figure 5.33: 3D bay: success ratios (QMS)

TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.34: 3D bay: noise-free min 2-norm solution (QMS,rk=39) Minmal 2−norm solution, noise−free with rk=39 Minimal 2−norm Method 0.08 100

90 0.06

80 0.04 70

0.02 60

0 50

Success ratios % 40 −0.02

30 −0.04 20

−0.06 10

−0.08 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Parameter number Measurement error level %

Using FRF-based updating equations

The “experimental” receptances are generated using exact ﬁnite element model, that is, all 114 DOFs are retained. However, FRFs at rotational DOFs are not used. The exciting

DOF is at node #1 in Z-direction. A typical receptance FRF is plotted in Figure 5.35. 138 Chapter 5. Error localization

Figure 5.35: A typical FRF of the 3D bay Receptance FRF α 33 −2

−3

−4

−5

−6 Modulus log10 −7

−8

−9

−10 0 50 100 150 200 250 300 350 400 Frequencies

Table 5.9: 3-D bay: Updating frequencies (Hz) Index 1 2 3 4 5 6 7 8 9 10 fui 30 70 110 135 155 180 210 230 257 280

Ten updating frequency points are used to establish updating equations. The “experimental” receptances at translation DOFs are expended into full size model with Kidder[36] dynamic expansion method. The updating equations at rotational DOFs are not used because the “observation” data b at those DOFs are artiﬁcially generated with expansion procedure.

Excitation-force-residual-criterion-based equations (3.34) do not produce a meaningful result for this simulated structure. The main reason may be that, for excitation- force-residual-criterion-based equations, the measurement errors are ampliﬁed by the pre- multiplication by the dynamic stiffness.

With the selected “test” data and based on response residual criterion, the updating equations are set up as Ax = b where A ∈<570×40 and b ∈<570. Unlike modeshape- based equations, for the FRF-based updating equations with sufﬁcient “test” data and the 40 unknowns, it is not a rank-deﬁcient problem, as the FRF data are always absolutely calibrated quantities.

Figure 5.36 showed error free solutions for “indirect receptance residue” based updating equations. The ε-related 2-norm condition number (sub-ﬁgure a) implies that a good 5.4. Case studies 139 bargain may be obtained if the numerical rank number is reduced from 40 to 38 for error localization purpose where measurement errors presented.

Figure 5.36: 3D bay: noise-free solutions (FRF-output) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 100 0.06

0.04 80 0.02

60 0

−0.02 40 A Cond. Number 2

k −0.04

20 Relative parameter changes −0.06

0 −0.08 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 40 0.06

35 0.04

30 0.02 25 0 20 −0.02 15

Parameter number −0.04 10 Relative parameter changes 5 −0.06

0 −0.08 10 20 30 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 35 0.06

30 0.04

25 0.02

20 0

15 −0.02

10 −0.04 Residue descending ratio Relative parameter changes 5 −0.06

0 −0.08 10 20 30 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.37 demonstrated a result where 2.4% error level is simulated. When test errors presented, they may be interpreted as small parameter errors over many of the design variables. However, a residue-descending-ratio-based backward exclusion can generally produce a much neat solution. Figure 5.38 displayed the error localization success ratios. For this simulation, TLS/QR-F method produces the best result. 140 Chapter 5. Error localization

Figure 5.37: 3D bay: solutions for noise level 2.4%(FRF-output) k A 2nm C.N. ShtIdx=20, ErL(%)=2.4 2 TLS/QR−F. ShtIdx=20, ErL(%)=2.4 90 0.15

70 0.1

60 0.05 50

40 0 A Cond. Number

302 k 20

Relative parameter changes −0.05 10

0 −0.1 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=20, ErL(%)=2.4 SVD/QR ShtIdx=20, ErL(%)=2.4 40 0.08

35 0.06 0.04 30 0.02 25 0 20 −0.02 15 −0.04 Parameter number 10 −0.06 Relative parameter changes 5 −0.08

0 −0.1 10 20 30 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=20, ErL(%)=2.4 Min2nrm ShtIdx=20, ErL(%)=2.4 45 0.1

35 0.05 30

25 0 20

15 −0.05

Residue descending ratio 10 Relative parameter changes 5

0 −0.1 10 20 30 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.38: 3D bay: success ratios (FRF-output) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.) 5.4. Case studies 141

5.4.4 Case study 4: A plate structure with components

A PCB structure with six components is simulated in two different boundary conditions as shown in Figure 5.39 and 5.40: (1) free PCB; (2) two opposite edges are ﬁxed with “Wedge-Lock” slides, which is modelled with uniformly distributed rotational springs.

Updating parameters are listed in Table 5.10. The rotational springs located at nodes of the

FEM mesh are determined by integrating the uniformly distributed springs.

Figure 5.39: A simulated free PCB Figure 5.40: A simulated ﬁxed PCB krotxB 162mm

c3 c6 c3 c6

178mm c2 c5 c2 c5

y y c1 c4 z c1 c4 z 2.8mm x x krotxA

Table 5.10: PCB structure: Updating parameters p number 11,13,23,25 9,27 Others(≤ 42) 43,44 px 3.2 3.2 2.8 0.50 pa 3.0 3.2 2.8 0.43 px−pa pa 0.06667 0 0 0.1628 p1—p42, thickness: mm; p43-p44, Stiffness: (Nm/rad)/m

The PCB board is simulated with shell elements, as shown in Figure 5.41. SMT components are simulated by increasing thicknesses of the covered base board elements. The thicknesses of the 42 elements are taken as design variables. For ﬁxed boundary situation, two additional variables are included for the distributed springs, each edge having its own stiffness variable. Therefore, the introduced model errors are corresponding to variable #11, 142 Chapter 5. Error localization

#13, #23 and #25, and for ﬁxed boundary case, plus #43 and #44.

Figure 5.41: FE model mesh of a simulated PCB

8 16 24 32 40 48 56

7 14 21 28 35 42 771515 23 31 39 47 55

6 13 20 27 34 41

661414 22 30 38 46 54

5 12 19 26 33 40

551313 21 29 37 45 53

4 11 18 25 32 39

441212 20 28 36 44 52 24 31 38 3 10 17 2735 43 51 3113 11 19

2 9 16 23 30 37

221010 18 2634 42 50

1 8 Y 15 22 29 36

191Z X 9179 1725 33 41 49

Using modeshape-based updating equations

Table 5.11 shows the resonance frequencies (below 1000Hz) of the two PCB structures. For the wedge-locked PCB, the third and fourth modes have very much close resonance frequencies. It must be difﬁcult to identify both of them with a single-point-exciting response data. Therefore, the fourth mode is not employed.

Table 5.11: PCB structure: Resonance frequencies Free “Experimental” 203.64 287.13 396.86 515.91 549.92 866.58 981.12 PCB FE model 200.37 282.62 391.10 509.62 543.33 857.23 970.07 Fixed “Experimental” 166.50 259.93 583.73 589.62 712.90 PCB FE model 161.77 255.28 574.70 581.11 702.25

Figure 5.42 and 5.43 show the error-free simulations with QMS updating equations.

For both boundary conditions, the introduced errors can be detected by all of the three mentioned solutions. With the residue descending ratio, TLS/QR-F method gives a clear indica- 5.4. Case studies 143 tor.

Figure 5.42: Noise free solution for the free PCB (QMS) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 150 0.08

0.06

100 0.04

0.02 A Cond. Number

2 50 k

Relative parameter changes 0

0 −0.02 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 45 0.08

35 0.06

30 0.04 25

20 0.02 15 Parameter number 10

Relative parameter changes 0 5

0 −0.02 10 20 30 40 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 35 0.08

30 0.06 25

0.04 20

15 0.02

10 Residue descending ratio

Relative parameter changes 0 5

0 −0.02 10 20 30 40 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.43: Noise free solution for the ﬁxed PCB (QMS) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 80 0.2

70 0.15 60

50 0.1 40

30 0.05 A Cond. Number 2 k 20

Relative parameter changes 0 10

0 −0.05 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 45 0.2

35 0.15

30 0.1 25

20 0.05 15 Parameter number 10

Relative parameter changes 0 5

0 −0.05 10 20 30 40 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 35 0.2

30 0.15 25

0.1 20

15 0.05

10 Residue descending ratio

Relative parameter changes 0 5

0 −0.05 10 20 30 40 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

When noise is present, the error localization results are assessed via success ratios, 144 Chapter 5. Error localization as shown in Figure 5.44 and 5.45. Comparing these two success ratio charts, we can see that when boundary parameter errors are introduced simultaneously with the PCB board errors, error localization would become much more difﬁcult. In fact, when a PCB is locked with the “Wedge-Lock” slides, within the same frequency range, fewer modes are available.

Therefore, in practical applications, these two kinds of error sources should be isolated.

Figure 5.44: Success ratios for the free PCB (QMS) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.45: Success ratios for the ﬁxed PCB (QMS) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

When FBM updating equations are directly used to do the error localization, the success ratios are much lower than the ones obtained from QMS method. Figure 5.46 and 5.4. Case studies 145

5.47 show the corresponding success ratio charts to Figure 5.44 and 5.45, respectively.

Figure 5.46: Success ratios for the free PCB (FBM) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Figure 5.47: Success ratios for the ﬁxed PCB (FBM) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 a. Measurement error level % b. Measurement error level % c. Measurement error level %

(See Figure 5.14 on page 122 for explanation.)

Using FRF-based updating equations

Typical receptance FRFs are showed in Figure 5.48 and 5.49 for free and wedge- locked PCBs. The updating frequencies are listed in Table 5.12 and 5.13, respectively. 146 Chapter 5. Error localization

Figure 5.48: FRF 43-43 of free PCB Figure 5.49: FRF 43-43 of ﬁxed PCB Typical receptance : 43−43 A typical receptance : 43−43 −1 −2

"Experimental" −2 "Experimental" FE model −3 −3 FE model

−4 −4

−5

−6 −5 Modulus log10 Modulus log10 −7

−6 −8

−9 −7

−10

−11 −8 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 Frequencies Frequencies

Table 5.12: Updating frequencies for free PCB (Hz) fu1–fu7 192.5 212.5 275 302.5 385 415 497.5 fu8–fu13 530 562.5 850 885 962.5 992.5

Table 5.13: Updating frequencies for ﬁxed PCB (Hz) fui 152.5 170 245 270 562.5 597.5 692.5 722.5

With the chosen updating frequencies, the noise-free simulation results are shown in Figure 5.50 and 5.51. For free PCB, SVD/QR and minimal 2-norm solutions are not sharp enough to distinguish the introduced error thickness T11, T13, T23 and T25 from the error-free parameter T5, T17, T18 and T19, which results in low success ratios (Figure 5.52).

However, when the PCB is wedge-locked, this problem fades. The possible reason is that, when the slides are introduced, the structure is changed such that the parameter T5, T17, T18 and T19 become much less sensitive to the structure’s dynamic behaviour, as the bending modes around axis y are vanished. This result also implies that it would be very much case- dependent whether a model updating could be achieved or not.

Figure 5.52 and 5.53 display the success ratios for the free and wedge-locked PCBs, respectively. As mentioned above, for free PCB, with SVD/QR or minimal 2-norm methods, 5.4. Case studies 147 error sources are not correctly indicated.

Figure 5.50: Noise-free simulation for free PCB (FRF-output) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 350 0.08

300 0.06 250

0.04 200

150 0.02 A Cond. Number 2

k 100

Relative parameter changes 0 50

0 −0.02 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 45 0.08

40 0.06 35 0.04 30 0.02 25 0 20 −0.02 15 Parameter number −0.04 10 Relative parameter changes 5 −0.06

0 −0.08 10 20 30 40 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 40 0.1

35 0.08 0.06 30 0.04 25 0.02 20 0 15 −0.02 10

Residue descending ratio −0.04 Relative parameter changes 5 −0.06

0 −0.08 10 20 30 40 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.)

Figure 5.51: Noise-free simulation for ﬁxed PCB (FRF-output) k A 2nm C.N. ShtIdx=1, ErL(%)=0 2 TLS/QR−F. ShtIdx=1, ErL(%)=0 800 0.2

700 0.15 600

500 0.1 400

300 0.05 A Cond. Number 2 k 200

Relative parameter changes 0 100

0 −0.05 10 20 30 40 10 20 30 40 a. Trial ε−rank number d. Parameter number

TLS/QR, ShtIdx=1, ErL(%)=0 SVD/QR ShtIdx=1, ErL(%)=0 45 0.2

35 0.15

30 0.1 25

20 0.05 15 Parameter number 10

Relative parameter changes 0 5

0 −0.05 10 20 30 40 10 20 30 40 b. Parameter selection order e. Parameter number

TLS/QR−F. ShtIdx=1, ErL(%)=0 Min2nrm ShtIdx=1, ErL(%)=0 80 0.2

70 0.15 60

50 0.1 40

30 0.05

20 Residue descending ratio

Relative parameter changes 0 10

0 −0.05 10 20 30 40 10 20 30 40 c. Parameter selection order f. Parameter number

(See Figure 5.13 on page 120 for explanation.) 148 Chapter 5. Error localization

Figure 5.52: Success ratios for free PCB (FRF-output) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level %

Figure 5.53: Success ratios for ﬁxed PCB (FRF-output) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level %

Figure 5.54: Success ratios for free PCB (FRF-output, 26 f’s) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 a. Measurement error level % b. Measurement error level % c. Measurement error level % (See Figure 5.14 on page 122 for explanation.)

It is generally believed that increasing updating frequency number may average out

some random noise in the test data. With the computer generated pseudo-random noise, this

is not evidently the case. Figure 5.54 displays a simulation result obtained by adding more 5.5. Summary 149

13 updating frequencies: 155, 222.5, 240, 280, 327.5, 355, 427.5, 490, 510, 590, 780, 815 and 902.5 (Hz). As pointed out in Chapter 3 section §3.4, for “excitation residual criterion” based updating equations, because errors in hxj(ω) were ampliﬁed by the dynamic stiffness matrix Z(I)(ω), error localization results are troubled with much low success ratios. This situation is well shown in Figure 5.55 and 5.56, in comparison with Figure 5.52 and 5.53.

Figure 5.55: Success ratios for free PCB (FRF-input) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 a. Measurement error level % b. Measurement error level % c. Measurement error level %

Figure 5.56: Success ratios for ﬁxed PCB (FRF-input) TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 a. Measurement error level % b. Measurement error level % c. Measurement error level % (See Figure 5.14 on page 122 for explanation.)

5.5 Summary

Error localization plays an important role in the process of model updating. A short review of global error localization methods has ﬁrst been made, which stressed the impor- 150 Chapter 5. Error localization tance of deﬁnition of error indicating vectors. Then, local error localization methods, which are parameter-oriented, are discussed in details. A TLS/QR-F method is then proposed, which includes the following steps:

• Regularising and condensing updating equations in total least squares sense;

• Performing a QR forward subset selection with the condensed equations;

• Performing a backward exclusion according to the descending ratio of relative residual function.

• Solving the updating equations with selected parameters. The solved relative parame-

ter adjustments are then used to judge where the main model errors located.

To assess error localization methods, a success ratio is also suggested. The presented method is then evaluated through a series of simulated case studies.

The simulated case studies showed:

• Though resonance-frequency-based updating equations contain limited information,

if the updating parameters have been reduced by analyst’s previous knowledge, then further subset selection by using the limited information may still make sense.

• For modeshape-based updating equations, QMS method is more robust to measure-

ment noise. FBM-based updating equations are more sensitive, both to model errors

and to test errors.

• Similar to modeshape-based methods, indirect-receptance-residue-based updating equa-

tions are more stable while excitation-residual-criterion-based equations are more sensitive to measurement noise.

• The simulation experience suggests that the residue descending ratio dRrj deﬁned in (5.60) may be an effective error indicator function. 5.5. Summary 151

• For all of the simulated cases, the TLS/QR-F method generally produces better success

ratios than SVD/QR method, which sometimes failed at all, especially for modeshape- based methods.

• Error localization success ratio is very much case-dependent. The simulated results

showed that, even the best results for the very simple system such as mass-spring

string and cantilever beam, when the test error level exceeds 5%, the success ratio will

decrease rapidly.

It should be noted that for the simulated case studies, the main concern is put on the influence of test noise. When other difficulties, such as expansion precision, message insufficiency, are involved, which is the usual situation in practical problems, much high test precision would be required. This seems to demand some special acquisition techniques for updating mathematical models. For example, stationary sinusoidal excitation technique, which is thought to be out-of-date for vibration mode survey testing because of its time- consuming feature, may be revived by FRF-based model updating methods, since for model updating purpose, FRF data are only required at few updating frequencies.

It should also be pointed out that, in order to obtain some statistical sense, the error localization program has been set to automatic, repeated run by assigning a ﬁxed α for residue descending ratio and a ﬁxed threshold value for relative parameter changes. For practical applications, error localization and model tuning procedures should always be integrated with known knowledge, in each decisive step. 152 Chapter 5. Error localization Chapter 6

Perturbed Boundary Condition Model Updating

6.1 Introduction

From a mathematical point of view, FE model updating is an inverse problem. It suffers from the lack of information about the structure under study. The lack of information exhibits both in temporal and in spatial ﬁelds. Frequency or time domains related information is referred to as temporal information while coordinates, geometry or physical parameters related information is referred to as spatial information.

A simply repeated measurement does not produce more information about a structure under study, though the additional experimental data may be useful for reducing the influence of measurement noise. The Perturbed Boundary Condition (PBC) testing technique, however, can provide additional messages in both temporal and spatial fields since PBC technique conducts additional testing in different configurations. So long as the perturbation does not bring new unknowns, all regression methods can profit from the enriched data from

PBC testing in one or another way.

153 154 Chapter 6. Perturbed Boundary Condition Model Updating

Lammens [39][38] applied the PBC technique to a FRF-based model updating method while Yang [67] presented some suggestions of the perturbation selection. Li [43] and Chen [8] also discussed PBC concepts and model updating. This chapter will extend PBC testing application to generalized sensitivity based ﬁnite element model updating. Since

PBC testing is more a measurement strategy than an updating tactic, this chapter will use simpliﬁed examples to demonstrate the potential capability of PBC model updating.

6.2 PBC Testing and Total Message Amount

In PBC testing a structure is tested in various conﬁgurations. In each conﬁguration, the boundary conditions of the structure under testing are changed by adding masses or stiffeners, or, if convenient, detaching mass or stiffness. Adding mass to a structure will generally bring down its resonance frequencies, which means that in the same testing frequency range, more modal information can be captured, while attaching springs to a structure will generally raise its resonance frequencies, which means less modal information presented in a certain testing frequency range.

For PBC testing technique, the aim is to enrich the total message amount from all given configurations, instead of the message from any particular configuration. Let Ωj denote the messages from jth configuration testing, if totally n configuration tests are conducted, then the total messages are expressed as:

Ω=Ω1 ∪ Ω2 ∪ ...∪ Ωn (6.1)

Let us always refer to Ω1 as the messages from the main configuration testing where no perturbation is conducted while Ω2,... ,Ωn represents messages from perturbed configuration tests. From the point of view of sensitivity-based model updating, the message Ωj for jth configuration testing can be expressed via the correspondingly created updating equa- 6.2. PBC Testing and Total Message Amount 155

tions Ajx = bj, or equally, the augmented matrix Cj =[Aj bj]. Optimal perturbation should be the one which produces an augmented matrix Cj that is as orthogonal as possible to the previously obtained augmented matrix [C1 C2 ... Cj−1]. The orthogonality of two sub- spaces can be expressed with the principal angles between sub-spaces. For two sub-spaces

F ∈

0 F = QF RF ; G = QGRG;cosΨi = σi(QF QG) (6.2)

th where F and G are matrices constructed with Cj’s; σi() denotes the i singular value. How- ever, this is not a practical way to search optimal perturbation because, if the updating equations have been created, it means the perturbed testing efforts have been done, and it is too late to say the perturbation is not efﬁcient enough.

Heuristically, the following suggestions may be helpful when choosing perturbations:

• Conduct dynamic ﬁnite element analysis with initial FE model. This will provide

nodal lines and maximum movement areas for the frequency range of interest. The perturbations should be attached to maximum movement areas, never to nodal lines,

of the resonance modes of great interest.

• The perturbations should be attached to the most uncertain areas as near as possible.

• The perturbations should be attached such that structure symmetry is destroyed.

The first suggestion is evident because masses or/and stiffeners attached to nodal points will not influence the corresponding modes for modal model updating methods. The second suggestion is not as evident. This is because the perturbation somewhat acts as a calibration to the original vibration system (see the following example). If this quasi-calibration is near to the areas where the uncertainness to be fixed, one may expect a better precision 156 Chapter 6. Perturbed Boundary Condition Model Updating since the comparison between the quasi-calibration and the uncertainness will be more direct, less affected by reduction process. The third suggestion is made because repeated mode problems may arise for symmetric structures.

It should also be aware that, for model updating purposes, the perturbation masses and stiffeners are not necessary to be small enough. In fact, so long as the conﬁguration arrangement (for attachment space) and the instrumentation allowed (for measurement scale in the same calibration), the perturbation should be big enough, to produce a more distinctive data set which will enlarge the total message amount of the PBC testing data set.

6.3 PBC Model Updating

As mentioned in the introduction, the main problem of model updating studies is the lack of accurate and independent experimental data. Since model updating process is an inverse problem, the lack of accurate and independent information will certainly invoke severe numerical problems, such as ill-conditioning or rank deficiency. Bad numerical conditioning will make the solutions quite sensitive to measurement noise. SVD technique transforms an ill numerical problem into a rank deficient one, while for rank deficient problems, the solution is not unique. For a single configuration testing, the most severe problem is that the behaviour of the structure, for some particular frequency bandwidth or for certain areas, cannot be fully measured because of the limited test bandwidth and the highly reduced test DOF set. If certain unknowns are related to the unmeasured frequency bandwidth or areas where not fully excited in the test frequency bandwidth, then naturally the experimental data will contain quite limited messages about these unknowns and the inverse solution of them is less reliable. PBC sensitivity based model updating methods aim to alter the structure under study by perturbations, such that the unknown physical parameters related substructures can be 6.3. PBC Model Updating 157 fully excited within the measured bandwidth. PBC testing also provides a mechanism to separate close or repeated resonance frequencies. Both of the above mechanisms of PBC technique will lead to enriched experimental data set.

The additional test configurations are not necessary to have direct relation with the original one which is close to its working condition, providing the new configurations must not bring new unknowns. Nevertheless, to reduce test cost, the overall configuration should not be changed too much. Assume jth configuration test results in an updating equation set:

Ajx = bj (6.3) then the total PBC model updating equations of n conﬁguration tests can be simply stacked as:     A1 b1      .   .   .  x =  .  (6.4)    

An bn

When row number of equation (6.3) is much larger than its column number, such as modal or impedance updating equations, it is possible to save computer memory via SVD technique before the equation stacking. Suppose Aj is factorised via SVD technique:

0 ∈

then equation Ajx = bj can be reduced as:

0 0 ⇒ 0 0 0 0 ∈

The regressing residues of equation (6.3) have been expelled from equation (6.6). 158 Chapter 6. Perturbed Boundary Condition Model Updating

Figure 6.1 shows the PBC ﬁnite element model updating scheme. As much more information is captured in different conﬁgurations, one can expect that the PBC model updating will be more robust and well-conditioned. And consequently, the solved physical parameters will be more realistic, which means the updated model has a better chance to accurately predict the structure’s behaviour in wider frequency ranges and is more reliable when altered to simulate totally different boundary conditions.

Figure 6.1: Perturbed Boundary Condition Finite Element Model Updating Scheme The Structure under Study, Initial Configuration as #1 Perturbed Structure from initial Perturbed Structure from initial structure, as configuration #2 structure, as configuration #Nc

Dynamic Initial Dynamic Initial Dynamic Initial Testing Model #1 Testing Model #2 Testing Model #Nc #1 #2 #Nc For all FRFs Eigen FRFs Eigen FRFs Eigen Values Values other Values PBC Eigen FRFs Eigen FRFs models Eigen FRFs Values Values do the Values same things Model/Modes Model/Modes as #1 Model/Modes match/correlation match/correlation match/correlation

Updating Equations Updating Equations Updating Equations A1x=b1 A2x=b2 Ancx=bnc Or S1*V1'*x=U1'*b1 Or S2*V2'*x=U2'*b2 Or Snc*Vnc'*x=Unc'*bnc

Stacking all updating equations of all configurations To #1 If first iteration, do error localization To Update To #2 Solving updating equations all Models ...... STOP Converged? To #Nc 6.4. Examples 159

6.4 Examples

As PBC technique is more a test strategy than an updating tactic, few additional efforts are required during updating process. The presented examples therefore are simple and just demonstrative. The “experimental” data are all simulated with ﬁnite element models.

The updating results are calculated with POMUS.

6.4.1 Example I

Perturbation acts as calibration, separates resonance frequencies and increases rank number of the updating matrix

Figure 6.2 shows a ten-DOFs mass-spring system. The problem is to ﬁnd all of the parameter values using the measured resonance frequencies. The modeshapes are assumed as unreliable and only valuable to match the modes between test and analytical calculation.

Figure 6.2: 10 DOF discrete system Table 6.1: Initial model parameters Exact model initial model k1 k8 m1 k5 m1−3 0.9 1 m6 4−5 k16 m 2.15 2 k12 m4 6−8 k2 k9 m9 m 0.9 1 m2 k14 k6 m9−10 0.05 0.05 m7 k17 k1−11 21000 20000 m5 m10 k12−15 38000 40000 k3 k10 k13 k15 k16−17 63000 60000 k7 m3 m8 Unit: mass: kg; stiffness: N/m. k4 k11 Perturbation ∆M: 0.5, 1.0 and 0.3 (kg) at m1,m5 and m9, respectively.

The exact and initial model’s parameter values are listed in Table 6.1 while the resonance frequencies are listed in Table 6.2 . As modes (#3, #4, #5) and (#6, #7) are nearly with repeated resonance frequencies respectively, from the point of view of experiment, only one of (#3, #4,#5) and one of (#6,#7) modes may be excited and measured. We also assume that 160 Chapter 6. Perturbed Boundary Condition Model Updating

Table 6.2: Conﬁguration 1: Resonance frequencies (Hz) mode #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 fx 17.26 28.20 33.64 34.27 34.46 51.98 53.23 61.39 183.7 184.1 fa 16.64 26.58 31.40 32.84 34.24 50.45 51.26 59.72 178.8 179.1

Table 6.3: Conﬁguration 2: Resonance frequencies (Hz) mode #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 fx 15.86 26.05 29.13 31.85 34.03 45.31 51.74 60.13 82.1 184.1 fa 15.34 24.71 28.34 30.52 32.51 44.97 49.35 58.37 78.7 179.1 the measured frequency range is 0— 100 Hz. Therefore, the updating process uses resonance frequencies of mode #1, #2, #3, #6 and #8.

It is impossible to calculate the parameter values only with relative values, such as resonance frequencies or even plus modeshapes. If no other calibration is provided, any one of the six parameters can be taken as a reference or if no reference is taken, the updated parameters will be most close to the initial parameters in one or another way depending on the solution criteria.

The perturbations are chosen so as, ﬁrstly to separate the repeated resonance frequencies, secondly to lower some of the high frequency mode. The repeated resonances arise because of the symmetric structure, therefore, the perturbations are applied to destroy its symmetry. The m1, m5 and m9 masses are then attached with masses 0.5, 1.0 and 0.3 (kg) respectively. For this testing conﬁguration, the “experimental” and initial model’s resonance frequencies are listed in Table 6.3.

Table 6.3 showed that the perturbed system’s resonance frequencies are well separated and frequency of mode #9 lowered to 82.1 Hz. For this testing conﬁguration ﬁrst nine modes are available and used to do model updating. The updated parameters are then listed in Table 6.4. Both of the updating cases need 4 iterations to reach convergence. Table 6.5 showed the resonance frequencies of the updated systems.

As expected, with one conﬁguration test data, for this example the updated param- 6.4. Examples 161

Table 6.4: Updated parameter values m1–3,6–8 m4–5 m9–10 k1–11 k12–15 k16–17 conf 1 0.908787 2.17108 0.0504514 21210 38373.9 60188.6 conf 1&2 0.899881 2.14849 0.0495979 20995.7 37977.4 62954.3 Table 6.5: Updated resonance frequencies (Hz) mode #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 cf 1 17.26 28.20 33.64 34.27 34.46 51.98 53.23 61.39 178.8 179.2 c1&2 17.26 28.20 33.64 34.27 34.47 51.97 53.22 61.40 184.3 184.7 Table 6.6: Condition number of the updating problems iteration #1 #2 #3 #4 conf 1 81.3361 61.9953 58.3570 58.7296 conf 1&2 27.2650 23.7665 25.7475 22.6842 eters are not necessary to converge to their “exact” values, though the selected resonance frequencies converged. For two conﬁguration test data, the updated parameters converged to their exact values, as the perturbation works as calibration. As the test data are enriched by perturbation testing, the condition number (listed in Table 6.6) of the coefﬁcient matrix of updating equations is also reduced, which means PBC updating process is more robust.

6.4.2 Example II

Though PBC is not designed for antinoise purpose, this technique seems also helpful for FE model updating when test data are noise-contaminated.

Let us perturb the free PCB discussed in Chapter 5 §5.4.4byattachingalumpedmass

∆m =0.01kg at node 56, as shown in Figure 6.3. Receptance FRFs at the exciting point are plotted in Figure 6.4. Comparing Figure 6.4 with Figure 5.48, we ﬁnd that mode 4 and 5 are separated by the perturbation. For this perturbed conﬁguration, 14 updating frequencies are chosen, as shown in

Table 6.7. Together with the initial conﬁguration data described in Chapter 5 §5.4.4, the

FRF-based updating equations are established with PBC technique. The error localization 162 Chapter 6. Perturbed Boundary Condition Model Updating success ratios are plotted in Figure 6.5. For TLS/QR-F method, the improvement obtained from PBC technique seems signiﬁcant.

Figure 6.3: Perturbed free PCB (∆m =0.01kg)

8 16 24 32 40 48 56

7 14 21 28 35 42 77151523 31 39 47 55

6 13 20 27 34 41

66141422 30 38 46 54

5 12 19 26 33 40

551313 21 29 37 45 53

4 11 18 25 32 39

441212 20 28 36 44 52 24 31 38 3 10 17 27 35 43 51 331111 19

2 9 16 23 30 37

221010 18 26 34 42 50

1 8 Y 15 22 29 36

191Z X 9179 1725 33 41 49

Figure 6.4: FRF 43-43 of the PCB Typical receptance: 43−43 −1

"Experimental" −2 FE model

−3

−4

−5 Modulus log10 −6

−7

−8

−9 0 100 200 300 400 500 600 700 800 900 1000 Frequencies

Table 6.7: Updating frequencies for the perturbed conﬁguration (Hz) fu1–fu7 165 177.5 270 285 332.5 345 447.5 fu8–fu14 465 522.5 540 752.5 767.5 895 917.5 6.5. Summary 163

Figure 6.5: Perturbed free PCB

TLS/QR−F method SVD/QR Method Minimal 2−norm Method 100 100 100

90 90 90

80 80 80

70 70 70

60 60 60

50 50 50

Success ratios % 40 Success ratios % 40 Success ratios % 40

30 30 30

20 20 20

10 10 10

0 0 0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 a. Measurement error level % b. Measurement error level % c. Measurement error level %

Thick solid lines: SR’s obtained with two configurations data; Thick dash lines: (SR σSR)’s obtained with two configurations data; Thin solid lines: SR’s obtained with initial configuration data; Thin dash lines: (SR σSR)’s obtained with initial configuration data;

6.5 Summary

From the point of view of model updating, PBC technique enriched experimental data, both in quantity and in quality. The quality of experimental data is referred to as the capacity of the structure’s messages. The PBC testing data therefore helps to obtain more reliable updating results. PBC model updating requires a little more work comparing with conventional updating process, but this additional effort can be seen as rewarding. 164 Chapter 6. Perturbed Boundary Condition Model Updating Chapter 7

Model Updating of Electronic Structures

7.1 Introduction

Besides the thermic impact, electronic devices applied to aeronautical and astronau- tical equipments suffer from heavy dynamic loads. To study their dynamic behaviour, firstly qualified dynamic models are the basic requirements. Electronic devices are usually plate- oriented structures. It is not difficult to model simple plate structures, however, it is not easy to precisely model the connections between the plates, especially between the base board and electronic components. Therefore, simplified modelling methods for electronic components are inevitably required.

This chapter tries to apply FE model updating techniques to validate mathematical models of electronic structures. The studied structures are: (1) a PCB with six SMT components (Figure 7.1); (2) wedge-lock PCB (Figure 7.2) and (3) a shell box (Figure 7.3). Firstly, a brief review of modelling techniques of electronic components is presented.

According to the review, it is decided to employ an equivalent thickness model to describe the SMT components.

With the thickness smearing technique, the FE model of the PCB is updated with

165 166 Chapter 7. Model Updating of Electronic Structures sensitivity-based methods, which includes resonance-frequency-based method, QMS method and FRF-based method. To enrich the experimental data for resonance-frequency-based updating method, a PBC test strategy is employed.

For wedge-lock slides, as their effect on the dynamic behaviour of gripped PCB depends on the contacting state and the stiffness of the PCB, two case studies are performed: (1) a thin, virgin board with 130 × 5mm2 contacting area and (2) the PCB with six components, with 162 × 7mm2 contacting area.

Finally, a shell box is studied with FE model updating technique.

Figure 7.1: Free PCB Figure 7.2: Wedge-Lock PCB Figure 7.3: Shell box krotxB

PGA JLCC c3 c6

JLCC CQFP c2 c5 178mm

y c1 c4 z CQFP PGA x krotxA

162mm

7.2 Methods of modelling SMT components

7.2.1 Connection between base board and components

Traditionally, electronic components are connected by soldering their leads to the base board through holes (Figure 7.4). Surface Mounting Technology (SMT), was invented in 1965 [63]. The SMT components (Figure 7.5) are connected by soldering their leads on to the base board surface, instead of through holes. 7.2. Methods of modelling SMT components 167

Figure 7.4: PTH component Figure 7.5: SMT component

From mechanical point of view, both classic

Component and SMT leads can be modelled using six simpliﬁed Base board Leads springs of stiffness kx, ky, kz, krotx, kroty and krotz. Y Z For a printed circuit board, generally speaking, only the rotz roty

rotx movements perpendicular to the base board will have X signiﬁcant effects on mechanical vibrations. Then the signiﬁcant vibration will be in Z,

ROT X, ROT Y directions, which in turn means only kz, krotx and kroty of the leads will take signiﬁcance in vibration analysis.

It should be pointed out that, as the component’s leads are generally distributed very densely, in practice the nodes of a ﬁnite element mesh will not certainly cover every lead.

That is to say, components’ leads will have to be grouped and integrated into some equivalent stiffness.

The leads’ equivalent stiffness depends on their shapes and materials. Engle [11] has presented an experimental method to determine them. However, as there are thousands of different components, and as electronic industry takes so rapid paces that every day some new type chips may appear, it will become extremely difﬁcult to obtain these stiffness data precisely. Therefore, simpliﬁed methods are highly required for engineering design and research. For example, instead of separately determining the leads’ properties, the effect of the whole component on the dynamic behaviour of a PCB is investigated via dynamic test,

FE model updating, etc. 168 Chapter 7. Model Updating of Electronic Structures

The following subsections will outline some practical methods into two groups: separately modelling electronic components and smearing the properties of the base board elements to represent the components.

7.2.2 Separately modelling electronic components Lumped mass model

The most simpliﬁed method is to model components as lumped masses. This method based on the following assumptions:

• The components are small enough compared with the printed circuit board;

• The encapsulating materials of the components make their bending stiffness much higher than the base board;

• The lead/solder joint stiffness perpendicular to the board is high enough.

The ﬁrst assumption is case dependent, as the component may be very large for some chips with complex functionality. The second assumption is usually true because the components are generally encapsulated with ceramic materials, which have much higher Young’s modules. However, if the components are encapsulated with plastic materials, this condition may not be met. The third assumption is almost always true because the leads are always very short. Wong and Stevens [66]’ static pull and compression tests, and Engel’s test [11] have given sound support for this assumption.

In this method, the lumped masses are attached to a base board nodes that are corresponding to components’ centres. Normally, a lumped mass attached to the base board will lower PCB’s resonance frequencies. 7.2. Methods of modelling SMT components 169

Spread mass model Instead of attaching only one lumped mass to a base board to represent a component, we may divide the component’s mass and dis- tribute it along with the nodes where the leads are soldered. However,

Wang’s study [64] shows that this model is not necessarily better than the one of lumped mass if the FE model mesh is not ﬁne enough.

Spread mass model has slightly higher resonance frequencies than the one with lumped mass model [35]. This implies that when the component size is relatively large, this model may be better than the one with lumped mass model.

Spring-mass-damper model Wang [64] has used a spring-mass-damper absorber model to syn- thesize FRFs of a PCB with one component structure. In his model the mass is ﬁxed while the stiffness and damping are determined by trial-and-error method. However, as pointed out in §7.2.2, the component’s leads have very high stiffness and therefore the spring-mass- damper subsystem has a very high resonance frequency, which in turn means the spring-mass-damper model will have little difference from the lumped mass model.

Furthermore, the damping is notoriously difﬁcult to determine. In fact, in Wang’s model, there are less than 0.1% difference in the ﬁrst four resonance frequencies between the lumped mass model and the spring-mass-damper one. Therefore, in practice, there exists not much advantage of using spring-mass-damper model to lumped mass one. 170 Chapter 7. Model Updating of Electronic Structures

Spring-plate model According to Engel’s test [11], the leads/joints’ pulling stiffness is much stronger than their bending one. Therefore the leads/joints are modelled as simple translation springs which will not transmit any moment between the base board and the component bodies while the component bodies are modelled as plates [35]. The components’ leads are grouped into ﬁnite element nodes.

Plate-plate model As a variety of spring-plate model, the leads/joints are modelled with artificial plate [35]. The artificial plates act as “curtain of legs” for the components. The Young’s module of the artificial plate is determined such that the global behaviour in traction of the component is retained. As the bending stiffness of the leads is low, the artificial plate’s thickness is assumed very thin (0.25mm).

Comparing to the spring-plate model, this plate-plate model includes some bending stiffness. However, the bending stiffness of this model is in fact arbitrary because the plate thickness is arbitrarily determined. Consequently, the prediction of the model is not necessarily better than the one of spring-plate model.

Coupling model As the component leads have strong translation stiffness and weak bending stiffness, it may be simply modelled such that it has inﬁnite translation stiffness and zero bending stiffness.

This assumption can be easily implemented in ANSYS ﬁnite element software by coupling the corresponding perpendicular DOFs of base board and the components. This 7.2. Methods of modelling SMT components 171 model is only a further simpliﬁed spring-plate one.

Substructure model

Directly modelling a complete PCB, i.e., all components, their leads and solder joints, would produce a model of tremendous complexity. Such a model may take considerable effort to be built, and require tremendous resources to be solved. Therefore, more efﬁcient techniques for detailed FE model are required. Sub-structuring technique helps to produce a detailed FE model for local regions of interests, while the global problem is solved in reduced form. In the case of electronic devices, a substructure may represent a portion of the board, leads, solder joint and component body, which may be modelled independently with great details. The substructures are condensed using Guyan reduction technique before assembled into a global model (via interface nodes). It is evident that for local region modelling, details and precise material properties are required.

7.2.3 Smearing model

A smearing model is a highly simpliﬁed ﬁnite element model of a PCB, where electronic components are not separately modelled, instead, they are accounted by smearing element properties of the base board. Smearing property models can be built in different levels: global mass smearing, global mass/stiffness smearing, or local smearing. For all of them, the PCB base board is taken as an initial prototype model which provides all necessary geometrical information.

Global mass smearing model

A most simpliﬁed global mass smearing model is built by introducing an equivalent density for the entire PCB. This equivalent density is obtained by dividing the total mass of the card, including all components, by the volume of the unpopulated PCB [2] [63]. This 172 Chapter 7. Model Updating of Electronic Structures model ignores any increase in stiffness from the components, therefore, the Young’s module is not changed from the prototype model.

Global mass/stiffness smearing model

A logical improvement to the global mass smearing is to count both the component mass and stiffness on a global scale. Besides the equivalent density deﬁned above, equivalent global stiffness may be introduced for global mass/stiffness smearing models. A common mechanism for adding stiffness is to modify the Young’s module of the prototype board.

Static force-displacement tests, such as bending tests and twist tests, are performed to determine the equivalent Young’s module, as shown in Figures 7.6 and 7.7.

Figure 7.6: A bending test Figure 7.7: A Twist test P L2 L2

t L1 L1 Twist Rigid End Block

Local mass/stiffness smearing model

A more reﬁned method of including the mass and stiffness contribution of the components is the local smearing technique, in which the analyst divides the PCB into local regions and smears the material properties region by region. This is a variety of the previous method.

The local equivalent material properties for each region are obtained in a similar way to that used for global method, which means test sample card will be cut into pieces and the tests repeated piece by piece. As mentioned, the whole card holds a unitary thickness identical to the base board. The smeared card model, which is an artiﬁcial ﬂat plate of “patch work”, is readily handled by the FE method. 7.2. Methods of modelling SMT components 173

Equivalent thickness model

For mass/stiffness smearing models, the physical properties are smeared while the geometric parameters are taken the same as the base board. On the contrary, one can smear geometric parameters, practically only the thickness, instead of the physical parameters, to represent the attachment of electronic components, as shown in Figure 7.8.

Figure 7.8: Equivalent thickness model Leads ttE

Module

PWB b b

a a

Basically, an equivalent thickness model is quite similar to a mass/stiffness smearing model, however, thickness is much more sensitive than Young’s module to the dynamic properties. For a rectangular plate, the bending behaviour is governed by a stiffness matrix of the following form [60]:

Et3 [K0] (7.1) 12(1 − ν2)

where

t is the thickness;

E the Young’s module;

ν the Poisson coefﬁcient; [K0] a matrix independent of t and E.

The Young’s module has a linear effect on the stiffness while the thickness, a cubic 174 Chapter 7. Model Updating of Electronic Structures one. In addition, an equivalent thickness model may give much more direct physical intuition than a smeared model. As the thickness is increased, the mass density must be decreased to keep the total mass unchanged. Therefore, both thicknesses and mass densities should be simultaneously adjusted during model tuning based on total mass constraints.

7.2.4 Remarks

Separately modelling electronic components gives more information about the structure while it also requires more details on the components. When the components are densely distributed over the base board, this will become a tremendously complex work. Smearing technique provides a new methodology of modelling electronic components. However, this technique is presently limited because it requires additional static tests. For global smearing method, the static test is not a problem because the test sample is always ready, but global smearing technique is not at all a ﬂexible method, because it provides only the global elastic module (or with global mass density) as variable, which means that only the fundamental resonance frequencies may be adjusted by this method. Local smearing method is much more ﬂexible to adjust the resonance frequencies of a PCB model as more variables are provided. Unfortunately, this method requires test sample structures for local regions of interest and performing static tests for all of the test samples.

In this work, instead of performing additional static tests, we propose to determine the equivalent thicknesses of local regions via model updating technique which directly uses dynamic test results. 7.3. Determining characteristics of SMT components 175

7.3 Determining characteristics of SMT components

7.3.1 Overview

The PCB to be studied has six components as shown in Figure 7.9. All of these components are sufficiently large comparing to the base board. Therefore, their stiffness effect will be significant. As remarked in §7.2.4, we decide to use local equivalent thicknesses to express the influence of the components. These equivalent thicknesses will be determined via FE model updating techniques.

As the experimental data are always noise-contaminated and the equivalent thickness model is only an approximate FE model which does not have the mechanism to precisely express the mechanical energy of the PCB structure, it is difﬁcult to set up a reliable criterion for equation residue to assess the convergence of the updating process, especially for modeshape or FRF based updating equations. For practical application purpose, the FE model updating process is stopped by one of the following three criterion:

• Maximum relative difference of the resonance frequencies is smaller then 1%:

f max(|1 − aj |×100) < 1 (7.2) j fxj

• Maximum relative parameter modiﬁcation is smaller than 0.5%:

∆p max(| j |×100) < 0.5 (7.3) j pj

• The average change of the overall resonance frequency residues of the last three itera- 176 Chapter 7. Model Updating of Electronic Structures

tion is smaller than 0.01%: v u X2 u Xm (i) 1 t 1 faj 2 |σ −1− − σ − |×100 < 0.01; σ = (1 − ) (7.4) 3 i j i j i m f j=0 j=1 xj

To conﬁne the updating process from making large modiﬁcation of the parameters,

Bayesian estimation technique is employed to set up the object functions:

(j) (j) 2 2 Fj(x)=||A x − b ||2 + αj||x||2; αj+1 =0.9 × αj; j =0, 1,... (7.5)

When mode shape or FRF based updating methods are employed, with the object q P σ = 1 m (1 − faj )2 function (7.5), the overall resonance frequency residue, m j=1 fxj does not converge, but ﬂuctuates within a certain extent. It is therefore decided that, instead of employing (7.5) as the ﬁnal object function, we just use the result as a search direction to minimize the overall resonance frequency residue: v u u 1 Xm f (β∆x) σ(β)=t (1 − ai )2 (7.6) m f i=1 xi where ∆x is the optimizer of (7.5); m is the mode number involved and β is a minimizer to be found.

As there exist no real values for the equivalent thicknesses, it is not possible to directly assess the updating results by comparing the physical parameter values. From the point of view of dynamic behaviour, it is believed that , for good correlation between the experimental data and the updated FE model, the relative difference of resonance frequencies should be smaller than 5% while the MAC values should be larger than 80%. 7.3. Determining characteristics of SMT components 177

7.3.2 Description of the card

The studied PCB consists of a base board with six electronic components (Figure

7.9). The base board is a plate whose dimension is 162×178×2.8mm3, compounded of thir- teen layers of polyamide and two layers of copper. The six SMT components placed on the base board, are of three different types (two components for each type) whose characteristics are listed in Table 7.1.

Figure 7.9: Experimental PCB

PGA JLCC Table 7.1: Components’ properties

Type V(mm3) M(g) PGA 40×40×2.2 14 × × JLCC CQFP CQFP 30 30 3 8.6 178mm JLCC 27×27×2 6.1 Properties of the base PCB: Dimension V: 162×178×2.8mm3

CQFP PGA Young’s module: 26 GPa Mass density: 2640 kg/m3

162mm

For the compound material of the base board, the manufacture provided rough estimations of the Young’s module as 30 GPa and of the Poisson’s coefﬁcient as 0.4. The

Young’s module is then reﬁned by dynamic experiments, whose value has been reported between 27.3 and 23.8 GPa for frequencies up to 2000 Hz and ﬁnally determined as 26 GPa for linear analysis [35]. 178 Chapter 7. Model Updating of Electronic Structures

7.3.3 Experimental measurement conﬁgurations

The basic conﬁguration of the dynamic experiment is shown in Figure 7.10. The

PCB is softly suspended with thin gum strips attached to the four corners of the card. The shaker below the PCB is ﬁxed (at node 17) while the non-contact laser Velocimeter head is movable. The 104 measurement points are shown in Figure 7.11.

Figure 7.10: Test conﬁguration of the PCB under free boundary condition

Pcb208

Shaker4810

LASER OFV 1100 Fiber PC/OR25 Ampli. Dytran Vibrometer PC/ICATS 2706 4114

Figure 7.11: Measurement points

104 88 66 50 28 21 14 7 97 87 75 59 49 37

103 86 74 58 48 36 20 13 6 96 65 27 57 47 35 95 85 73 102 84 64 46 26 19 12 5

94 83 72 56 45 34

101 93 82 71 63 55 44 33 18 11 4 25 92 81 70 54 43 32

100 80 62 42 24 17 10 3 53 41 31 91 79 69

99 90 78 68 61 52 40 30 16 9 2 23 89 77 67 51 39 29 76 38 22 15 8 1 Y 98 60 Z X

This basic configuration is called the original configuration or configuration #1. In this work, two additional tests are also performed, based on the concept of Perturbed Bound- ary Condition testing. The first additional test configuration, named as configuration #2, is constructed by attaching a mass (53.2g) on measurement point #6 while the second one, con-

ﬁguration #3, is formed by perturbing the conﬁguration #1 via adding the same mass (53.2g) 7.3. Determining characteristics of SMT components 179 on measurement point #100.

7.3.4 Dynamic testing and eigenvalue extraction

As shown in Figure 7.10, the workpiece is excited by a mini shaker (B&K Type 4810). The exciting source signal (synchro. Multisine) is generated by an OR25 FFT analyser and ampliﬁed by a B&K power ampliﬁer (B&K Type 2706). The exciting force is measured by a force transducer PCB 208A03/Dytran 4114. The responses are measured at

104 points in Z direction with an OFV 110 Fiber Vibrometer system. The maximum analysis frequency is set to 1000 Hz. Figure 7.12 shows a typical FRF (mobility) and the corresponding coherence function at measurement point 24. This ﬁgure indicates that below 50Hz the FRF data is noise dominated as the coherence is low, while above 50Hz, coherence data is generally high enough, except for anti-resonance and some resonance frequencies, where large coherence drops are found.

With the measured FRFs, modal analyses are performed by using ICATS software

(GRF-M method). Table 7.2 shows the resonance frequencies of the PCB in different con-

ﬁgurations. The corresponding mode shapes are shown in ﬁgures E.1˜E.3 in Appendix E.

Figure 7.12: Coherence and mobility at node 24 (conﬁguration 1) 1 −1

−1.5 0.8

−2 ) 10

0.6 −2.5 −3

0.4 Mobility (log

Coherence −3.5

−4 0.2 −4.5

0 −5 0 200 400 600 800 1000 0 200 400 600 800 1000 Frequency (Hz) Frequency (Hz) 180 Chapter 7. Model Updating of Electronic Structures

Table 7.2: An overview of experimental modal parameters (0–1000Hz) Mode Configuration one Configuration two Configuration three number f(Hz) Damp.(%) f(Hz) Damp.(%) f(Hz) Damp.(%) 1 214.16 2.82 165.3 2.42 197.53 2.22 2 302.71 1.43 262.75 1.53 247.75 2.18 3 397.92 1.06 322.14 1.09 353.72 1.9 4 531.64 1.13 480.57 1.53 474.71 1.85 5 562.69 1.16 671.21 1.44 511.2 1.59 6 837.74 0.96 771.37 1.35 716.89 1.44 7 936.78 0.073 929 0.09 842.1 1.27 8 / / / / 923.44 1.02 9 / / / / 953.1 1.08

7.3.5 Finite element models

As mentioned above, accurate models require detailed information on the components, and furthermore, such models generally need tremendous calculation efforts and the costs may become unreasonably huge. Therefore, instead of pursuing detailed models, a local equivalent thickness model is investigated in this section.

All of the 104 measured points served as the mesh nodes of the FE model. To complete the mesh, additional 25 points on the boundaries that are not measured are also added.

With the 129 mesh nodes, 112 shell elements (ANSYS SHELL63, has both bending and membrane capabilities. Both in-plane and normal loads are permitted. The element has six degrees of freedom at each node: translation in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes.) is formed, as shown in Figure 7.13. At the exciting point, a lumped mass element (0.013 kg) is added to compensate the effect produced by the force transducer. For the perturbed conﬁgurations, the perturbation element is modelled as lumped

−6 2 mass with rotation inertia ( M =0.0532kg; IX = IY =6.309 × 10 kg · m ). 7.3. Determining characteristics of SMT components 181

Figure 7.13: Element number of the PCB with 6 components

112 104 103 102 101 100 99 98

84 72 48 36 111 18 12 6 60 54 24 90 83 71 47 35

46 34 82 70 53 23 110 89 59 17 11 5 81 69 45 33

80 68 44 32 109 16 10 4 88 58 52 22 79 67 43 31

78 66 42 30 87 57 51 21 108 15 9 3 77 65 41 29 113 40 28 76 64 107 14 8 2 86 56 50 20 75 63 39 27

74 62 85 55 38 26 106 49 19 13 7 1 73 61 37 25 Y 105 97 96 95 94 93 92 91 Z X

Table 7.3: Grouped element number of the PCB with 6 components Group number Element number Group number Element number 1 62, 63, 74, 75 10 19, 20, 25, 28, 37, 2 66, 67, 78, 79 40, 49, 50, 94, 95 3 70, 71, 82, 83 11 21, 22, 29, 32, 4 26, 27, 38, 39 41, 44, 51, 52 5 30, 31, 42, 43 12 23, 24, 33, 36, 6 34, 35, 46, 47 45, 48, 53, 54 55, 56, 61, 64, 73, 13 13, 93 7 76, 85, 86, 96, 97, 14 14, 15 105, 106, 107 15 16, 17 8 57, 58, 65, 68, 77, 16 18, 100 80, 87, 88, 108, 109 17 1, 7, 91, 92 59, 60, 69, 72, 81, 18 2, 3, 8, 9 9 84, 89, 90, 103, 19 4, 5, 10, 11 104, 110, 111, 112 20 6, 12, 98, 99 182 Chapter 7. Model Updating of Electronic Structures

7.3.6 Initial model and correlation with test data

With the ﬁnite element mesh established as above, the initial parameters are set up as following: for elements not covered by SMT components, the parameters are the same as the base board, with ρ = 2640kg/m3, T =2.8mm; for elements covered by SMT components, the initial equivalent thicknesses are all set as T =3.5mm, while the mass densities are calculated according to:

Mc + ρb × Sc × Tb ρinitial = (7.7) Tinitial × Sc where:

ρinitial is the initial equivalent mass density of the elements to be updated;

Mc the mass of a SMT component to be treated;

ρb the base board’s mass density;

Sc the area of FE elements covered by the SMT component;

Tb the base board’s thickness;

Tinitial the initial equivalent thickness of the covered elements.

With these initial parameters, we get an overview of the correlation between the test data and the initial model as shown in Table 7.4. Though these initial models are not precise, the MAC values are generally high, except for test frequencies 936.0 and 479.9(Hz). The possible reason is that these frequencies are not well separated from other ones and therefore these modes are not properly extracted by modal parameter extraction software. Hence, these modes are not used in the updating process, but as result checking references. 7.3. Determining characteristics of SMT components 183

Table 7.4: Frequencies and MAC values (tests vs initial simple plate model) Configuration #1 fx#1 214.1 302.7 397.8 531.4 563.8 836.9 936.0 fa#1 196.4 275.8 384.6 494.4 530.1 822.0 949.0 ∆f% 8.27 8.89 3.67 6.96 5.98 1.78 -1.39 MAC 98.61 95.60 95.94 88.54 91.26 90.67 68.14 Configuration #2 fx#2 165.3 262.7 322.1 479.9 671.8 771.3 927.3 fa#2 156.1 253.3 314.6 456.9 656.6 754.4 890.3 ∆f% 5.79 3.83 2.18 4.79 2.26 2.19 3.99 MAC 99.51 99.39 99.26 78.63 99.07 97.90 78.80 Configuration #3 fx#3 197.6 247.2 354.1 474.3 512.4 718.9 840.1 921.9 952.8 fa#3 183.0 235.4 353.4 449.0 493.7 700.8 825.6 889.8 927.6 ∆f% 7.39 4.77 0.18 5.33 3.65 2.52 1.73 3.48 2.65 MAC 99.47 99.47 99.11 95.76 96.00 99.04 95.27 89.14 98.88

7.3.7 Finite element model updating parameters

Though the MAC values between the experimental mode shapes and the ones from initial FE model are generally high, the analytical FE models are only rough estimations from our intuitive knowledge. It is desired to reﬁne the roughly estimated models by using the measured data.

There are 112 shell elements in the FE model, therefore, the number of possible equivalent thicknesses can be up to 112. However, the experimental information is limited.

Therefore, these elements are grouped into 20 sets according to the structure feature, as listed in Table 7.3. For each group elements, they share the same equivalent thickness.

For the 20 grouped elements, according to our previous knowledge of the structure, the most uncertain equivalent thicknesses should be the ﬁrst six groups. The next six groups are those elements that encircle the ﬁrst six groups, respectively. Therefore, they are not entirely independent with regard to dynamic behaviour of the PCB. The last 8 groups should have less uncertainty. 184 Chapter 7. Model Updating of Electronic Structures

Therefore, the model reﬁning is performed according to the following three parameter selection strategies:

• The updating parameters are chosen with previous knowledge of the structure. For the

studied PCB, only the ﬁrst six group elements are involved.

• All of the grouped elements are involved by updating the 20 equivalent thicknesses.

• From the 20 equivalent thicknesses, subset parameters are chosen with parameter error

localization strategies developed in Chapter 5.

7.3.8 Reﬁning FE model based on previous knowledge

For this PCB structure, the main errors are intuitively known with our previous knowledge: they should be those parameters used to account the SMT components, i.e. the equivalent thicknesses of the ﬁrst six group elements listed in Table 7.3. Therefore, we ﬁrst update the FE model with the six variables.

With the test results obtained from conﬁguration #1, three updating methods are employed:

• Resonance frequency based FE model updating. The ﬁrst six resonance frequencies

are taken into account.

• QMS FE model updating. The ﬁrst 6 modes are used.

• FRF-based FE model updating. The selected updating frequencies are: 200, 220, 295,

310, 390, 405, 525, 540, 557.5, 570, 830 and 845 (Hz).

The updating results are shown in Table 7.5 and Figures 7.14˜7.16. A typical mobility

(for FRF-based method) is shown in Figure 7.17. These results showed: 7.3. Determining characteristics of SMT components 185

• ∆f Mathematically, resonance frequencies and MAC values are acceptable ( f < 5%, MAC> 80%);

• QMS and FRF-based methods produce similar equivalent thicknesses;

• Physically, the equivalent thicknesses are generally reasonable, however, T6 of RF-

based result and T1 of QMS and FRF-based results are not as anticipated.

To obtain more physically acceptable results, PBC updating with 3 conﬁgurations is performed. For the additional test data from conﬁguration #2 and #3, experimental data are chosen as following:

• for RF- and QMS-based methods, mode 1, 2, 3, 5 and 6 of conﬁguration #2, mode 1˜9

of conﬁguration #3 are used;

• for FRF method, updating frequencies selected from conﬁguration #2 are 155, 177.5,

255, 270, 312.5, 342.5, 467.5, 497.5, 657.5 and 702.5 (Hz); those from conﬁguration

#3 are 190, 202.5, 242.5, 255, 345, 365, 467.5, 530, 692.5, 745, 832.5 and 862.5 (Hz).

With experimental data from 3 conﬁgurations, the updating results are listed in Table 7.6.

Comparing the two sets of results obtained from 1 and 3 conﬁgurations, we ﬁnd that MAC values can generally improved with PBC technique, but resonance frequency differences are not. Physically, the equivalent thicknesses seem more reasonable, as T1 and T6 are more rational. 186 Chapter 7. Model Updating of Electronic Structures

Table 7.5: Updating results with 6 variables (Conf.#1) Index i 1 2 3 4 5 6 (1 − fai )(%) -0.77 1 -0.07 0.9 -0.51 -0.12 fxi RF-based MAC(%) 98.7 94.4 96.1 84.9 89.7 92.7 T (mm) 3.74 5.58 4.31 4.55 4.8 2.74 (1 − fai )(%) fxi -0.49 1.88 -1.45 1.5 -1.17 -0.88 QMS-based MAC(%) 98.3 94.8 94.6 95.3 95.2 90.9 T (mm) 2.82 5.57 4.83 3.69 4.46 4.4 (1 − fai )(%) fxi -0.78 1.99 -2.11 1.84 -0.98 -0.37 FRF-based MAC(%) 98.1 95.5 94.0 94.4 93.7 87.6 T (mm) 2.0 5.38 4.7 3.7 4.62 5.13 RF-based: Resonance Frequency based method; For the updating process, T ’s are conﬁned within [2mm, 10mm].

Figure 7.14: Resonance frequency residues (using Conf. #1) (1 − f / f ) × 100; (Left: RF; Middle: QMS; Right: FRF) a x 2

1.5

0.5

−0.5

−1

−1.5

−2

−2.5 1 2 3 4 5 6 Resonance frequency number 7.3. Determining characteristics of SMT components 187

Figure 7.15: MAC values (using Conf. #1) MAC × 100; (Left: RF; Middle: QMS; Right: FRF) 100

0 1 2 3 4 5 6 Mode number

Figure 7.16: Equivalent thicknesses (using Conf. #1) Thickness (mm); (Left: RF; Middle: QMS; Right: FRF) 6

0 1 2 3 4 5 6 Group element number 188 Chapter 7. Model Updating of Electronic Structures

Figure 7.17: Updated mobility at node 1 Mobility at node 1 0

Experimental Initial FE −0.5 Updated (6v)

−1

−1.5

−2 Modulus log10 −2.5

−3

−3.5

−4 0 100 200 300 400 500 600 700 800 900 1000 Frequencies

Table 7.6: Updating results with 6 variables (Conf.#1, #2 and #3) Index i 1 2 3 4 5 6 (1 − fai )(%) fxi 2.03 3.72 0.37 2.78 0.55 -0.22 RF-based MAC(%) 98.5 94.5 95.0 95.3 95.9 91.5 T (mm) 3.19 4.95 4.89 3.61 4.1 3.79 (1 − fai )(%) 2.1 3.81 0.14 2.68 0.67 -0.37 fxi QMS-based MAC(%) 98.5 94.8 95.0 95.1 95.6 91.4 T (mm) 4.06 4.5 4.21 3.94 4.07 3.89 (1 − fai )(%) fxi 2.15 4.08 -0.51 1.93 0.54 -0.95 FRF-based MAC(%) 98.5 95.7 95.0 94.7 94.7 91.3 T (mm) 3.32 4.73 4.29 3.73 3.8 4.51 RF-based: Resonance Frequency based method; For the updating process, T ’s are conﬁned within [2mm, 10mm].

7.3.9 Updating FE model “blindly” with all parameters

QMS and FRF-based methods are evaluated for “blind” FE model updating with all

20 parameters (1 test conﬁguration). The updating results are displayed in Table 7.7 and shown in Figures 7.18˜7.20. These results indicate that, though mathematically resonance frequencies and MAC values may be ﬁtted to an acceptable level, physically, the equivalent thicknesses are not necessarily rational. The results imply the necessity to select a subset parameters. 7.3. Determining characteristics of SMT components 189

Table 7.7: Updating results with 20 variables (Conf.#1) Index i 1 2 3 4 5 6 (1 − fai )(%) 1.1 0.63 -3.73 1.01 -0.03 -1.94 fxi MAC(%) 98.3 94.0 91.7 63.3 75.5 84.6 T1−6(mm) 2.68 3.84 2.77 3.77 3.88 3.81 QMS-based T7−12(mm) 3.07 3.37 2.74 2.73 3.35 2.99 T13−18(mm) 2.91 2.37 3.19 2.91 2.78 2.79 T19−20(mm) 2.95 3.65 (1 − fai )(%) fxi -0.72 -0.01 0.35 0.8 0.55 -0.19 MAC(%) 98.5 92.8 95.6 92.4 97.0 81.5 FRF-based T1−6(mm) 2.32 4.72 4.23 3.2 5.2 2.79 T7−12(mm) 2.81 3.06 2.88 2.93 2.93 2.35 T13−18(mm) 3.52 2.96 2.81 3.64 3.06 3.05 T19−20(mm) 2.81 3.61 For the updating process, T ’s are conﬁned within [2mm, 10mm].

Figure 7.18: Resonance frequency residues (20v; using Conf. #1) (1 − f / f ) × 100; (Left: QMS; Right: FRF) a x 2

−1

−2

−3

−4 1 2 3 4 5 6 Resonance frequency number 190 Chapter 7. Model Updating of Electronic Structures

Figure 7.19: MAC values (20v; using Conf. #1) MAC × 100; (Left: QMS; Right: FRF) 100

0 1 2 3 4 5 6 Mode number

Figure 7.20: Equivalent thicknesses (20v; using Conf. #1) Thickness (mm); (Left: QMS; Right: FRF) 6

0 2 4 6 8 10 12 14 16 18 20 Group element number

7.3.10 Error localization

As shown in §7.3.9, “blindly” updating a FE model may not produce physically sig- niﬁcant results. This subsection tries to conduct a parametric error localization. The experimental data used for error localization are the same as these used in §7.3.8. 7.3. Determining characteristics of SMT components 191

Using QMS method

The updating equations are established with QMS method. With test data from only one conﬁguration, the singular values and condition number of matrix A are plotted in Figure 7.21. Its numerical number is determined as 19. TLS subset solution is shown in Figure

7.23. The absolute values of this solution is plotted with gray image according to the PCB structure, as shown in Figure 7.25.

With test data from three conﬁgurations, the corresponding results are plotted in Fig- ures 7.22, 7.24 and 7.26.

Figure 7.25 and 7.26 show that main model errors are located at area where SMT components planted. It is also shown that, with PBC technique, the condition number of A is reduced and the results are more reasonable.

Figure 7.21: Svd(A) and k2A (X) Figure 7.22: Svd(A) and k2A (X) (QMS,cfg1) (QMS,cfg1,2,3) svd(A) and k A svd(A) and k A 2 2 25

20 svd(A) k2A 20

15 15

10 10

5 5

0 0 5 10 15 20 5 10 15 20 SVD order SVD order 192 Chapter 7. Model Updating of Electronic Structures

Figure 7.23: Parameter selection Figure 7.24: Parameter selection (QMS,cfg1) (QMS,cfg1,2,3) TLS/QR−F solution 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0 0.1 −0.1 0

−0.2 −0.1

−0.3 −0.2 −0.4 5 10 15 20 −0.3 Parameter number 5 10 15 20 Parameter number

Figure 7.25: Gray image of main errors Figure 7.26: Gray image of main errors (QMS,cfg1) (QMS,cfg1,2,3)

FRF-based method

When FRF-based method is employed, the updating process is more demanding because the updating frequencies are delicate. The results are arranged in the same way as for

QMS method, as shown in Figures 7.27˜7.32. These ﬁgures also give the impression of the 7.3. Determining characteristics of SMT components 193 main errors and the improvements obtained from PBC technique (for parameter #2, #20).

However, it seems that results from QMS method are more convincible.

Figure 7.27: Svd(A) and k2A (H) Figure 7.28: Svd(A) and k2A (H) (FRF,cfg1) (FRF,cfg1,2,3) svd(A) and k A svd(A) and k A 2 2 140 70

120 60

100 50

80 40 k2A 60 30

40 svd(A) 20

20 10

0 0 5 10 15 20 5 10 15 20 SVD order SVD order

Figure 7.29: Parameter selection Figure 7.30: Parameter selection (FRF,cfg1) (FRF,cfg1,2,3) TLS/QR−F solution 2 1.5 1.5

1 1

0.5 0.5 0

−0.5 0

−1 −0.5 −1.5

−2 −1 5 10 15 20 5 10 15 20 Parameter number Parameter number 194 Chapter 7. Model Updating of Electronic Structures

Figure 7.31: Gray image of main errors Figure 7.32: Gray image of main errors (FRF,cfg1) (FRF,cfg1,2,3)

7.3.11 Remarks

The FE model of the PCB with six components has been studied with various methods. For the obtained results, we make the following remarks:

• When updating parameters are conﬁned to the real error sources, FE model updating

procedures can produce physically signiﬁcant results. The correlations between FE models and experimental data can generally be improved.

• If the updating is conducted “blindly” with all parameters, the results may be only with

mathematical sense.

• Parametric error localization is helpful to assist analysts to conﬁne the updating subset

of parameters. However, for practical applications where model errors are not small

enough, error localization can only gives general impression. Analysts’ insight of the

structures is essentially required to determine the updating subset.

• When PBC technique is applied, the results can generally be improved. 7.4. Modelling “Wedge-Lock” slides 195

7.4 Modelling “Wedge-Lock” slides

7.4.1 Description and modelling of “Wedge-Lock” slides

Electronic PCBs have to be assembled to a housing box to perform their functionali- ties. Wedge-clamp-type edge guides are often used to grip the side edges of PCBs to reduce the thermal resistance for lower temperatures and to increase the resonance frequencies. A typical “Wedge-Lock” slide is shown in Figure 7.33.

Figure 7.33: A typical Wedge-Lock slide Wedge-Lock slide Figure 7.34: Spring model The chassis

The PCB

The torsional springs The chassis The PCB

The effect of the Wedge-Lock slides depends upon the contacting state between the

PCB and the chassis. The contact area and force (per unit length along the slide) are the most important factors to the Wedge-Lock effect. It is also related to the stiffness of the

PCB, and therefore to the resonance frequencies of the PCB. When a PCB is not very stiff, it needs less effort to be held rigidly to prevent rotation during the resonance condition, while on the contrary, if a PCB is very stiff, it then needs a great torque to prevent rotation when resonance occurs. It is quite difﬁcult to directly assess the inﬂuence of the contact area and force, however, it should be a basic requirement for Wedge-Lock slides that the PCB keeps contact with the chassis during the resonance condition. Therefore, the PCB sides gripped by

Wedge-Lock slides should act somewhere between simply supported (hinged) and clamped edges. This uncertainty provokes some difﬁculties in developing a ﬁnite element model of a Wedge-Locked PCB. 196 Chapter 7. Model Updating of Electronic Structures

For clamped edges, all six degrees of freedom for each node are fixed, while for hinged edges, rotational degrees around the edges are totally free, but degrees in the other five coordinate directions are fixed. Therefore, we would logically model the “Wedge-Lock” connection as “bi-clamped” boundaries, that is, for the wedge locked edges, all degrees but the rotational ones around the edge’s axis are coupled with the corresponding base DOFs, while for the uncoupled degrees, torsional springs are employed to represent the “bi-clamped” boundaries, as shown in Figure 7.34.

However, the stiffness of the springs is still to be determined with some measurements. When the torsional springs take stiffness values of zero, it just means the wedge lock slides act as hinged edges, while if the stiffness values are set to infinite, it represents a fully clamped PCB edge. That is to say, if we take the stiffness as a design variable for a model updating process, the value will be meaningful from zero through infinite. It will be difficult to get an approximate initial value. Instead of letting the stiffness be blindly updated with an arbitrary initial value, it is important to firstly analyse the two extremes: hinged and clamped wedges. It is also practical and helpful to use a trial-and- error method to determine an initial value for the torsional springs.

7.4.2 Case study

Two examples will be analysed in the following paragraphs, the ﬁrst one is a virgin board while the second example is a PCB. The PCB is gripped with more contact area and stronger contact force than the virgin board.

Example I: virgin board gripped to wedge lock slides

The measurement conﬁguration is similar to the one used for the free boundary PCB test described in previous section, while the measurement points have been reduced to 55 stations. The contact area for each locked edge is 130 × 5mm2 while the contact force 7.4. Modelling “Wedge-Lock” slides 197 acting on each edge is generated by a torque of 0.4N.m applied on the clamping screw of the wedge lock slides. A ﬁnite element model is built, which includes 216 shell elements and 26 torsional springs, as shown in Figure 7.35. The material properties of the virgin card have been determined separately, therefore, the sole unknown is the stiffness of the springs.

Figure 7.35: Virgin board’s mesh and measurement points

Y Z X

Contact area Plate thickness h=1.56mm Contact area

In the first step, the two extreme limits for the torsional stiffness, both zero and infinite, are computed. For zero stiffness which represents hinged edges, the spring elements are deleted, instead of setting the stiffness value to zero. For the case of infinite large stiffness, the rotational degrees of freedom of the locked edges are constrained, instead of setting the stiffness to a very large value. The correlations of these two extremes with the test results are shown in Table 7.8 and 7.9.

Table 7.8: Frequency differences and MAC between hinged edge model and test data

fx (Hz) 171.3 226.6 491.7 593.3 841.8 f0 (Hz) 78.9 160.5 455.9 429.1 761.1 − f0 × (1 fx ) 100 53.94 29.17 7.28 27.68 9.59 MAC (%) 93.46 92.64 69.89 89.52 96.31 Analytical mode #3 f03 = 321.8Hz has no corresponding test mode. 198 Chapter 7. Model Updating of Electronic Structures

Table 7.9: Frequency differences and MAC between clamped edge model and test data

fx (Hz) 171.3 226.6 491.7 593.3 841.8 f∞ (Hz) 183.7 241.3 503.5 595.9 891.9 − f∞ × (1 fx ) 100 -7.24 -6.49 -2.40 -0.44 -5.95 MAC (%) 94.27 93.59 70.40 90.94 97.91 Analytical mode #4 f∞4 = 508.9Hz has no corresponding test mode.

Table 7.8 shows that the resonance frequencies of the hinged edge model are very far away from the tested ones, especially for the fundamental frequency, which has a relative error over 50%. This suggests that for a wedge slide locked PCB, a hinged edge model is not reliable, even though the MAC values are fairly good. Table 7.9 indicates that, for the studied virgin card, a clamped edge model is much better than the one with a hinged edge.

However, this model should be further improved by updating the stiffness of the torsional springs, as the relative frequency differences are still quite large.

The initial value of the sole unknown, the stiffness of the torsional springs, is nearly arbitrary. With some trial-and-error calculations, we set the initial value as 1000 (Nm/rad/m) with lower and upper boundaries as from 0 to 107 (Nm/rad/m). Table 7.10 shows the correlations between the test data and the initial model.

Table 7.10: Correlation between the test data and the initial model (k=1000Nm/rad/m)

fx (Hz) 171.3 226.6 491.7 593.3 841.8 finitial (Hz) 158.22 216.59 483.9 533.97 834.29 − finitial × (1 fx ) 100 7.64 4.42 1.59 10.00 0.89 MAC (%) 94.61 93.76 70.45 91.39 97.92 Analytical mode #3 fi3 = 445.1Hz has no corresponding test mode.

To determine the stiffness of the springs, we choose the ﬁve resonance frequencies to establish updating equations. Figure 7.36 and 7.37 show the iterative updating results while

Table 7.11 gives the ﬁnal correlation of the updated model. 7.4. Modelling “Wedge-Lock” slides 199

Figure 7.36: Iterative residues (Virgin) Figure 7.37: Iterative Stiffness (Virgin) 1−−>5 Iter residues (Solid:Freq; Dash:Eq.) 3000 100

90 2500

2000 70

60 1500 50

40 1000 0 1 2 3 4 5 0 1 2 3 4 5

(j) P | j − j | j U × (j) n fxi fai The residues are deﬁned as: Rfreq = (0) 100; U = i=1 (j) U fxi (j) E(j) (j) (j) × || − ||2 and : Requa = E(0) 100; E(j)=min A x b

Table 7.11: Correlation between the test data and the updated model

fx (Hz) 170.7 226.8 488.2 591.3 844.4 fu (Hz) 172.61 230.14 493.85 566.54 862.92 Updating stiffness − fu × (1 fx ) 100 -0.77 -1.56 -0.44 4.51 -2.51 k=2751.5 Nm/rad/m MAC (%) 94.5 93.74 70.47 91.35 98.09 Analytical mode #3 fu3 = 479.3Hz has no corresponding test mode.

Example II: Printed circuit board gripped to Wedge-Lock slides

In this case, the studied PCB is exactly the same as the one described in the previous section. Two edges are gripped with Wedge-Lock slides while the other two edges are free.

Each gripped edge has a contact area of 162 × 7mm2. A torque of 0.5Nm/rad is applied to the screws of each wedge lock slide, which generates the contact forces. The experiments and modal parameter extraction are performed in a similar way as for virgin card. A ﬁnite element model is built, where the same net mesh as the case of the free boundary condition is used, and the components are treated with the updated results obtained in free boundary condition. 200 Chapter 7. Model Updating of Electronic Structures

Similar to the virgin card case, we perform analyses for the PCB with two extreme boundaries, as shown in Table 7.12 and 7.13. Again for the hinged edge model, the resonance frequency differences are very large, though most of these modes are correlated with high

MAC values.

Table 7.12: Correlation between test data and hinged edge model of the PCB

fx (Hz) 273.8 362.3 634.9 743.4 824.3 f0 (Hz) 125.3 224.7 577.0 522.4 655.1 − f0 × (1 fx ) 100 54.24 37.98 9.12 29.73 20.53 MAC (%) 92.32 96.07 87.70 77.72 88.64 Table 7.13: Correlation between test data and clamped edge model of the PCB

fx (Hz) 273.8 362.3 634.9 743.4 824.3 f∞ (Hz) 272.9 355.8 671.0 779.3 904.6 − f∞ × (1 fx ) 100 0.33 1.79 -5.69 -4.83 -9.74 MAC (%) 94.58 96.71 93.29 85.24 90.86

Table 7.13 clearly indicated that for the studied PCB, the clamped edge model is quite acceptable. For the first four resonance modes, the relative frequency differences are sufficiently low. When measurement noise is taken into account, it will not make significance to update the slide model. The following results show that if the stiffness is updated, though frequency errors in high modes reduced, for low modes, the differences become larger.

Table 7.14: Correlation between the test data and the initial model (k=104 Nm/rad/m)

fx (Hz) 273.8 362.3 634.9 743.4 824.3 fi (Hz) 244.89 326.11 642.99 712.69 835.03 (1 − fi ) × 100 fx 10.56 9.99 -1.27 4.13 -1.30 MAC (%) 94.88 97.26 93.57 84.82 90.95 7.4. Modelling “Wedge-Lock” slides 201

Figure 7.38: Iterative residues (PCB) Figure 7.39: Iterative Stiffness (PCB) 4 1−−>6 Iter residues (Solid:Freq; Dash:Eq.) x 10 100 3.5

95 3 90

85 2.5 80

75 2

70 1.5 65

60 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 See Figure 7.36 on page 199 for residue deﬁnitions.

Table 7.15: Correlation between the test data and the updated model

fx (Hz) 273.8 362.3 634.9 743.4 824.3 fu (Hz) 262.31 344.33 659.66 752.96 876.71 Updating stiffness (1 − fu ) × 100 Nm/rad/m fx 4.21 4.97 -3.89 -1.28 -6.36 k=30646 MAC (%) 94.77 96.99 93.44 85.1 91.0

7.4.3 Summary

With the above two case studies, we can summarize a two-step modelling strategy for “Wedge-Lock” slides as following:

• Analysing the correlation between the test data and the analytical models with bound-

aries of two extremes, if the clamped model is not acceptable, then a second step is

needed;

• When a clamped model is not satisﬁed, then modelling the slides as rotational springs

and performing a model updating to determine the spring stiffness. 202 Chapter 7. Model Updating of Electronic Structures

7.5 Model updating of a box-like shell

7.5.1 Introduction

In this case study, the subject is a box-like shell of an electronic package used in the ARIANE 5 launcher. The whole electronic package includes an electronic mother card, four PCB modules ﬁxed by means of “wedge lock” slides, a relay support placed on shock absorbers, and the box-like shell (Figure 7.40). This shell serves as a frame structure for the electronic PCBs.

The box-like shell is constructed with ﬁve plates:

• The base plate. It serves as the support of the other two plates. This plate is very

thick and reinforced with strong ribs. It also serves as the connection part to the main launcher.

• Left and right plates. These two side plates are directly connected to the base plate. They serve as the supports of the four PCBs, the back plate and upper plate. They are

symmetric structures and each is reinforced with three horizontal and four perpendic-

ular ribs. In addition, there are also ﬁve horizontal ribs that serve as the supports of the

four PCBs.

• Back plate. The back plate serves as the support of the electronic mother card. It is

connected to left, right and upper plates. This plate is reinforced with three horizontal

and three perpendicular ribs.

• Upper plate. The upper plate is the upper cover of the box. It is reinforced with four

ribs (two for each direction of the plate). It also serves as the support of the electronic sockets, so a lot holes are made on it.

These plate structures are shown in Figure 7.40˜7.43. The purpose of this investigation is to validate a highly simplified finite element model of the box-like shell and to 7.5. Model updating of a box-like shell 203 demonstrate POMUS’ ability to update middle size finite element models.

7.5.2 Measurement conﬁguration and modal parameter extraction

As the electronic box is rigidly attached to a large mass launcher object during its mission, therefore, instead of suspending, the box-like shell is rigidly connected to a big lump mass, which serves as the “ground” for the modal analysis. The measurement system is shown in Figure 7.44. A sweep sine signal is generated by a CF-350 FFT analyser, then it is amplified by a 2250MB power amplifier and is input into a PM-50 exciter. The exciting force is measured by a force transducer 1151V3 DYTRAN811 installed between the excitor and the shell. The response velocity is measured by a Polytec OFV 110 Fiber Vibrometer. The mobility is then calculated by the FFT analyser and transformed into a PC using PCMODAL software. 299 points are measured, which are distributed over on the four faces of the shell, as shown in figures 7.40˜7.43.

The exciting force is set near the upper-right corner of the front face of the shell box along (x) direction. The response at this exciting point cannot be measured with a laser beam. Instead of using an accelerometer, the responses of the exciting points are virtually replaced with the response at the corresponding locations on the opposite faces. This will certainly influence the mode shape data at the exciting points, and therefore influence the modal mass/stiffness. However, it will have little effect on the resonance frequencies and the overall modeshapes. In this case study the model will be updated mainly using resonance frequencies, therefore, the replacement of the exciting points’ responses should be insignificant. 204 Chapter 7. Model Updating of Electronic Structures

Figure 7.40: Back and Upper faces Figure 7.41: Upper and front faces

(Ariane 5 BCS B box) (Ariane 5 BCS B box)

Figure 7.42: Upper face Figure 7.43: Left and right faces

(Ariane 5 BCS B box) (Ariane 5 BCS B box)

An additional measurement is also performed with the exciting force acting on the right face of the shell box at the same corner, along (y) direction. However, when the exciting force acts on the right face (y), this face is then blocked from being measured with the laser beam by the excitor, which means only three faces of the shell box are measured. Therefore, the results from this additional test are not used in the model updating process, instead, they 7.5. Model updating of a box-like shell 205 are only used for comparison with the previous test results.

Figure 7.44: Measurement system for the shell box structure 1151V3 DYTRAN811 PM-50 exiter

Lens Box-like Shell

2250 MB Lump mass DYTRAN 4114 Ampli.

Polytec OFV 1100 CF-350 PC Modal; Fiber Vibrometer FFR Analyser ICATS Softeware

The modal analysis is performed using ICATS software with GRF-M method (based on the rational fractional method where the modal parameters are found N times and averaged to yield a single consistent set). Within the frequency of (0 to 1000Hz), six global modes (low frequencies) are identified. The resonance frequencies are shown in Table 7.16 and the mode shapes are displayed in Figure E.4 in Appendix E. There are still some high frequency modes in this frequency range, however, the identification of them is somewhat troublesome. One probable reason is these high frequency modes are very local, which brings difficulties to GRF-M method. Another possible reason is that the exciting energy is not sufficient to excite these high modes (no exciting force is applied in Z direction). There- fore, the model updating will be based on the first six low frequency, global modes. In the model updating, the experimental mode shapes are not directly used to tune the model, instead, they are used to pair themselves with analytical modes by evaluating

MAC-values. 206 Chapter 7. Model Updating of Electronic Structures

Table 7.16: An overview of experimental modal parameters Mode number 1 2 3 4 5 6 Frequency(Hz) 224.8 263.4 375.5 523.9 538 591.5 Damp. Ratios(%) 3.13 6.06 0.85 2.73 6.46 2.2 Mode shapes shown in Figure E.4 in Appendix D

7.5.3 Finite element model

A very coarse finite element model (Figure 7.45) of the shell box is established by using the ANSYS finite element software. The measurement geometry serves as a base of the model. However, as the ribs play so important roles in the dynamic behaviour of the shell box, some additional nodes have been added on left and right plates to exactly place the ribs. On upper and back plates, instead of adding more nodes, locations of some nodes have been adjusted to model their ribs. This will have some influence on MAC values, but it will not affect the updating precision as the MAC values are only used to pair modes. The base plate connected to the lump mass block is not modelled. All the other four plates are basically modelled with ANSYS “SHELL63” elements. SHELL63 is an elastic quadrilateral shell element with both bending and membrane capabilities. This element has six degrees of freedom at each node.

All ribs are modelled with “BEAM4” elements. BEAM4 is a uniaxial, 3-D elastic element with tension, compression, torsion and bending capabilities, which has six degrees of freedom at each node.

Many rectangular-like holes are drilled on the upper plate, which provide room for electronic connectors. These holes reduce the stiffness of the plate. As they are rectangular- like shaped, the stiffness reductions are different according to the coordinate directions.

These holes are ﬁlled with electronic plug-in’s. These holes and plug-in’s are not modelled, instead, they are accounted by reducing the plate’s Young’s module and increasing its 7.5. Model updating of a box-like shell 207 mass density. The other three plates are relatively simple. The left and right plates are lightly stiffened by overlapped plates along the connecting edges. Inside the shell box a PCB is attached on to the ribs of the back plate. This PCB is not separately modelled, but its inﬂuence is accounted by modifying the plate’s mass density and increasing the virtual height of the plate’s ribs.

Figure 7.45: Finite element Model mesh for the shell box

The connections between plates have been well designed (Figure 7.46). For translation degrees of freedom the connection should certainly be considered as rigid. For the two rotational degrees of freedom that cross the connection edges, the connection characteristic does not have significant influence on the dynamic behaviour of the box. However, for the rotational degrees of freedom around the connection edges, the connection characteristic has some great influences.

For a uniform, flat, rectangular plate that is supported on its two opposite edges, when the support acts as hinge, which corresponding to zero stiffness springs connection q π D 0 around the edges, the first resonance frequency is f = 2a2 ρ , where a is the span over Eh3 the two supported edges and ρ is the mass density per unit area, and D = 12(1−µ2) .When the supported edges become fully clamped, which corresponds to infinite stiffness springs 208 Chapter 7. Model Updating of Electronic Structures

q ≈ 3.55 D connection around the edges, the resonance frequency augments to f∞ a2 ρ . The ratio f∞ ≈ of these two frequencies is f0 2.26. For the studied shell box, the connection has been carefully designed to provide sufficient rigidity. It has been firstly modelled with rotational springs. By trial-and-error methods, it is found that these springs should have very great stiffness, therefore, the connections are finally modelled as rigid.

With the above assumption, a coarse dynamic ﬁnite element model is obtained. This model, however, should only be used to predict the global dynamic behaviour, as many local details of the structures have been omitted.

Figure 7.46: Plate connection Upper plate Left, right plate Left,right plate

Base plate Back plate

7.5.4 Initial correlation and updating parameter selection

Table 7.17 shows an overview of the correlation between the experimental modes and the analytical ones by means of resonance frequency differences and MAC values. For a coarse element model, the experimental modes seem to have reasonable correlation with the analytical ones. The fourth experimental resonance frequency is coincidentally very close to the analytical one, while for the first, third, fifth and sixth modes, the frequency differences are expected to be reduced by model updating technique. For this plate-oriented structure, parameters of the finite element are limited and the selection of the updating parameters is therefore rather straightforward. The first and sixth experimental resonance frequencies are below the analytical ones. These two modes are mainly bending modes of the upper plate around X coordinate. The frequency differences 7.5. Model updating of a box-like shell 209 are possibly caused by distributing the lumped masses of the electronic plug-in’s over the whole plate face, or by an overestimation of the equivalent Young’s module of the plate in Y coordinate direction, which is decreased by the rectangular-like holes.

Table 7.17: Resonance frequencies and MAC values (initial model)

fx (Hz) 224.8 263.4 375.5 523.9 538 591.5 fa (Hz) 240.9 252.4 326.6 523.5* 461.3* 651.3 (1 − fa ) × 100 fx -7.16 4.18 13.02 0.076 14.26 -10.11 MAC (%) 98.08 83.93 92.13 79.78 76.7 87.51 *Resonance frequencies shifted.

For the second mode, the frequency is underestimated by the ﬁnite element model.

This mode is dominated by lateral plates’ bending around X coordinate and the upper plate’s rigid translation in Y coordinate direction. The underestimation of this resonance frequency may be caused by an overestimation of upper plate’s mass density, or, by an underestimation of the bending stiffness of the lateral plates.

The third and ﬁfth experimental modes are bending modes of the back plate around Z and Y coordinates, respectively. As a PCB attached to this plate is not separately modelled, the bending stiffness is certainly underestimated, which is the reason that the calculated frequencies are below the experimental ones.

According to the above analyses, four parameters of the ﬁnite element model are selected to do model updating, which are:

1. the equivalent height of perpendicular ribs of left and right plates (hlrp);

2. the equivalent height of ribs of the upper plate in Y coordinate direction (huy);

3. the equivalent height of horizontal ribs of the back plate (hbr);

4. the equivalent mass density of the upper plate (density). 210 Chapter 7. Model Updating of Electronic Structures

These are all global parameters, therefore, the symmetry of the structure will be automatically maintained during model tuning. To check the program’s ability of parameter error localization, an additional parameter, the ribs’ height of the upper plate in X coordinate direction (hux) is also chosen. There- fore, in the ﬁrst step, ﬁve parameters are totally selected.

7.5.5 Model tuning

The first five experimental resonance frequencies are used to established model updating equations with the five selected parameters, while mode shapes are used only to evaluate the correlation and to pair the modes between the experimental data and the mathematical model. As a preliminary step to do parameter error localization, column vector norms of matrix A are checked, as shown in Figure 7.47. According to this chart, no updating parameter is found to be extra insensitive and therefore kept for further treating. Singular Value De- composition analyses are then performed on the matrix A and the augmented matrix [Ab] of the updating equations. Table 7.18 and Figure 7.47 showed these singular values.

Table 7.18: Singular values of matrix A and [A b] A 749.42 497.03 338.04 230.40 22.76 / [A b] 751.51 507.90 343.97 231.96 26.59 0.036

In this case study, according to Figure 7.48, it is obvious that both A and [A b] have the same matrix rank number as 4. Then a regularised and condensed updating equation set is obtained, which consisted of four equations with ﬁve variables. Subset selection is performed with two methods: QR forward method and QR with column pivoting method.

The subset selection results, together with the pseudo-inverse solution, are shown in Table

7.19. The QR forward subset selection is coincident with our visual and intuitive analysis of the experimental shapes, while QR with column pivoting subset selection is not: as it has 7.5. Model updating of a box-like shell 211

Figure 7.47: Norm of column of A Figure 7.48: Singular values of A and [A b] 600 800

700 500 Left: svd(A) 600 400 Right: svd([A b]) 500

300 400

300 200 200 100 100

0 0 hlrp huy hbr density hux 1 2 3 4 5 6 been pointed, it simply choose the more sensitive parameters.

Table 7.19: Subset selection results (relative parameter solution %) hlrp huy hbr density hux TLS/QR-F -0.14 -23.49 29.50 -17.33 0 SVD/QR 4.63 -13.89 27.79 0 7.11 Min 2-norm 6.06 -7.21 27.12 5.98 -2.24

After the determination of the updating parameters, model tuning is rather straightforward. One important step at each updating iteration is to pair experimental modes with the calculated ones, because when parameters are adjusted, analytical modes may shift from their originally corresponding experimental modes. This is done by analysing the MAC matrix. For this case study, initially the fourth and ﬁfth experimental modes correlate with the

fifth and fourth analytical modes, respectively. The first updating iteration changed this correlation such that the fifth and sixth experimental modes correlate with the sixth and fifth analytical modes. After the second updating iteration, all of the six experimental modes correlate with the analytical modes in their natural orders.

Figures 7.49 and 7.50 show an overview of the iterative updating results. Table

7.20 shows an overview of relative resonance frequency residues. After two iterations, the 212 Chapter 7. Model Updating of Electronic Structures resonance frequency differences are all less than 1%. As experimental data suffered from stochastic and systematic measurement errors, further iterations should not have more sig- niﬁcance.

The MAC values for the updating iterations are listed in Table 7.21. During the updating iterations, the MAC values did not change signiﬁcantly. Most of them, but mode

#4 and #5, exceed 80%, which is believed a value for a reliable model.

Table 7.20: Relative frequency residues (%) Exp. Mode mode #1 mode #2 mode #3 mode #4 mode #5 mode #6 Initial model -7.1695 4.1830 13.0066 11.9583 2.6839 -10.1125 1 iteration -0.0258 0.3462 -1.4744 -1.0897 -0.9971 -0.3412 2 iteration 0.3496 0.1321 -0.5047 -0.2733 0.3567 -0.2531 3 iteration 0.3692 0.1496 -0.4559 -0.2630 0.4063 -0.2311

Table 7.21: MAC values (%) mode #1 mode #2 mode #3 mode #4 mode #5 mode #6 Initial model 98.1 83.9 92.1 79.8 76.7 87.5 1 iteration 98.7 80.7 91.6 79.0 73.4 86.6 2 iteration 98.7 80.9 91.8 79.0 73.3 86.7 3 iteration 98.7 80.9 91.8 79.0 73.2 86.7

Table 7.22 shows the frequency differences and MAC values of the updated model.

Within this frequency range, resonance frequencies of the updated model much better correlate with the experimental data but the MAC values are not improved. For #1, #2, #3 and

#6 modes, MAC values are over 80%, which is generally thought as acceptable for a reliable model. Mode #4 has a MAC value quite close to 80%, but mode #5 has a relatively low MAC value. One possible reason of this is that this bending mode (of the back plate in Z direction) is not fully excited by the exciting force (in the front face of the box), which leads to a low precision of mode identiﬁcation. 7.6. Summary 213

Figure 7.49: Iterative residues Figure 7.50: Iterative parameters 1−−>4 Iter residues (Solid:Freq; Dash:Eq.) Y1 Y2 100 0.025 3600 hlrp(Y1) hbp(Y1) 80 0.02

60 huy(Y1) density(Y2) 0.015 40

0.01 20

0 0.005 3000 0 1 2 3 4 0 1 2 3 4 See Figure 7.36 on page 199 for residue deﬁnitions.

Table 7.22: Resonance frequency differences and MAC values

fx (Hz) 224.8 263.4 375.5 523.9 538 591.5 fu (Hz) 223.96 263.00 377.20 525.28 535.79 592.82 − fu × (1 fx ) 100 0.37 0.15 -0.46 -0.26 0.41 -0.23 MAC (%) 98.68 80.90 91.82 78.97 73.20 86.72

7.5.6 Conclusion

A very coarse finite element model of a shell box is successfully updated in the point of view of global dynamic behaviour. A shell box is a three-dimension structure. Its finite element model is quasi-three-dimension in a point of view of computation cost. When the structure is modelled in details, much finer element mesh is required, and the computation effort may tremendously increase. This case study showed that a coarse, cheap finite element model of a complex structure can be validated by using model updating technique.

7.6 Summary

This chapter presented case studies to plate-oriented electronic structures. 214 Chapter 7. Model Updating of Electronic Structures

Firstly, problems of modelling electronic components are concerned. A local equivalent thickness model is proposed and validated via FE model updating approach. This technique is successfully applied to a real PCB with six SMT components. Three different updating methods have been tried. For QMS and FRF indirect receptance updating methods, an additional constraint to the object function is applied to conﬁne the updating process.

The Perturbed Boundary Condition test technique is also applied to extend the experimental database, which leads to some encouraging results. The results seems engineering acceptable | ∆f | (MAC-value > 80%, f < 5%). Secondly, modelling problems of “Wedge-Lock” slides are discussed. In this section, a two-step modelling strategy is developed and demonstrated with two case studies. The case studies showed that, when other parameters are determined, the “Wedge-Lock” slides can be modelled as rotational springs and the springs’ stiffness can be determined by FE model updating approach.

Finally, an ARIANE 5 BCS B shell box is modelled with a coarse ﬁnite element model. The model is then successfully validated with POMUS updating technique. This case study showed a possibility of using a coarse, cheap FE model to describe the dynamic behaviour of complex structures via FE model updating approach. Chapter 8

Conclusions and Recommendations

8.1 Conclusions

This dissertation has discussed several aspects of ﬁnite element model updating and its application to electronic structures. Topics have been focused on parameter-oriented ﬁnite element model updating techniques and parameterizations of electronic structures.

Finite element model updating is the art of reconciling mathematical models of structures with experimental data acquired from modal tests. It assumes that dynamic quantities from experimental tests are more precise, and the mathematical models are ﬂexible to be tuned to more reliable ones.

Modern electronic industry has made measurement instruments more accurate and reliable. Digital-computer-based modal testing techniques have greatly improved the quality of dynamic structure identiﬁcations. However, it should be aware that not every sort of experimental quantity is in the same level of precision. It is generally agreed that, among modal parameters, the resonance frequency is the most precise and robust quantity. Experimental modal shapes are much less precise than the resonance frequencies while modal mass and stiffness are quantities with the worst measurement precision.

215 216 Chapter 8. Conclusions and Recommendations

It is difﬁcult to estimate the precision of FRFs, because it is both frequency- and location-dependent. Outside resonance frequency ranges, the measured FRFs are not com- pletely reliable because they can be dominated by noise. In the close vicinities of the resonance frequencies, the FRFs are damping dominated.

A successful model updating ﬁrst depends on the quality of experimental data. There- fore, selecting a suitable set of dynamic quantities to perform the model updating is of paramount importance.

On the other hand, a reasonable initial model is half of a success. In the early developing stage of model updating techniques, pure matrix models were employed. The updated matrix models are even able to replicate the experimental quantities. However, this does not at all mean that reliable and rational models are obtained because the experimental data are never complete for continuous structures! The incompleteness of experimental data is spatio-temporal. In the temporal domain, the frequency range is always limited. In the spatial domain, a continuous structure has inﬁnite DOFs while FE models have only the given DOFs. Even for the limited DOFs kept in FE models, unfortunately, many of them are not measured, due to unreachable for inside DOFs, or due to the lack of measurement technique for rotational DOFs.

Though spatial incompleteness can be partially solved by expansion techniques, the incomplete problem is inherent and essentially unsolvable for FE model updating. This is then left as a philosophy problem, which we are not going to solve by giving proofs, instead, we will set up a lot of constraints that we think rational, to build up mechanisms that guide the updating process to produce an acceptable model. For parameter-oriented FE model updating procedures, most of the constraints can be implicitly set up via proper parameterisation of structures.

From the point of view of model updating, the quality of experimental quantities is relative to the quality of initial ﬁnite element models and therefore can be only judged 8.1. Conclusions 217 via model updating. Parameterisation requires analysts have sufﬁcient insight of the structures. The effort of parameterisation, however, will be compromised, if the updating procedure does not support parameter updating. Therefore, a parameter-oriented model updating technique (POMUS) that provides opportunities of using various experimental quantities is developed in this study.

Chapter 2 reviewed the correlation and model match techniques. The existing numerical correlation methods have been summarized in a generic form, which is suitable for both modal parameter and FRF based comparison. Model match techniques have been classi- fied as system-matrix-based and modeshape-based reduction/expansion methods. System- matrix-based methods can directly be used for both modal and impedance updating approaches. There are some difficulties to employ modeshape-based match techniques for impedance model updating because the frequency-shift problem between analytical and experimental FRFs is not automatically solved. As Guyan static reduction can significantly reduce the computation cost of FE analysis and Kidder dynamic expansion is an exact methods, it is recommended that the FE models are condensed with Guyan method to keep only sufficient master DOFs. If necessary, the experimental data are then expanded into the size of the retained master DOFs. The POMUS is flexible to this requirement.

Chapter 3 developed the theoretical bases of sensitivity-based FE model updating. A general form of updating equations is suggested. A new modal updating approach, namely

QMS updating equations, is proposed. The new updating equations are believed more robust to measurement noise than FBM updating equations. As the errors in the experimental modeshapes are not ampliﬁed by the dynamic stiffness matrix. This is illustrated with simulated case studies in Chapter 5.

As model updating equations are usually ill-conditioned, regularization of the equations become a key step to a reliable updating result. Most popular regularization methods are documented in consistent notations in Chapter 4. SVD technique is the most powerful 218 Chapter 8. Conclusions and Recommendations tool to treat ill-condition problems, however, by itself, SVD does not determine the numerical rank number. Though no criterion is generally accepted to determine numerical rank number of a matrix, the ε-related 2-norm condition number may be a very helpful indicator in some cases.

Weighting the updating equations effectively changes the singular values. The theoretical bases of weighting strategies are the χ-square estimation technique, which is a particular case of Maximum Likelihood estimation.

For parameter-oriented model updating methods, it is important to update only the erroneous parameters, otherwise the updated models may be just artiﬁcial ones. A new parameter-oriented error localization approach is proposed in Chapter 5. This method ﬁrst cures the noise-contaminated updating equations according to SVD-based strategy developed in TLS problems. In this stage, a condensed equation set is obtained. Then, a QR forward subset selection algorithm is performed on the condensed equations with a pivoting criterion that the added parameter should most well solve the residue. The selected subset parameters are further reduced based on a residue descending ratio. With the reduced subset, the solution of the updating equations serves as the error indicator function.

To evaluate the proposed error localization method, four simulated case studies are introduced where measurement errors are artiﬁcially generated. To get some statistic sense for the simulations, a success ratio of parameter error localization is also suggested. The simulated results showed that higher success ratio can be expected comparing to SVD/QR or minimal 2-norm solution.

In Chapter 6, a simple but effective method of enhancing experimental information,

Perturbed Boundary Condition (PBC) Testing technique, is introduced into POMUS. PBC testing technique is quite independent to ﬁnite element model updating. However, this technique is regarded in the point of view of model updating. With PBC technique, more distinctive experimental data can be obtained with a few additional test efforts, consequently, a 8.2. Recommendations 219 more reliable updated ﬁnite element model can be expected.

Chapter 7 presented application case studies of electronic structures. Emphases are put on structural parameterisation. Firstly, parameterisation of SMT components is addressed with real world examples. Equivalent thickness models are examined with POMUS technique. Secondly, a two-step model strategy is developed for the parameterisation of

“Wedge-Lock” slides. Finally, an ARIANE5 BCS B shell box is modelled with a coarse

ﬁnite element model and validated with POMUS technique. The results show that POMUS are ﬂexible to treat modelling problems for electronic structures with updating approach.

8.2 Recommendations

Sensitivity-based ﬁnite element model updating methodology is extensive. Our efforts were focused on the model updating of electronic structures. Through the present investigation, several recommendations can be made regarding further research in this ﬁeld.

1. In POMUS, various techniques of setting up updating equations are accumulated sep-

arately to provide selections for users to assess the quality of experimental quantities.

However, mixing usage of these techniques is not implemented. Mathematically, there should not be problems in the mixing usage. A general way of mixing these techniques

is to set up an overall objective function with a serial of sub-functions separately es-

tablished. The difﬁculty is how to weight these sub-functions obtained from different

updating equation sets. Messages included in sub-functions are not independent each

other. The reliability of sub-functions is neither the same since the quality of experi-

mental quantities used in different updating equation sets varies. Arbitrarily weighting

sub-functions is just a strategy to lash experimental data to confess. Certainly, further investigation is necessary in order to make a mixing usage practical.

2. For impedance model updating equations, the updating frequencies are key roles for a 220 Chapter 8. Conclusions and Recommendations

success updating. In the present investigation, this topic is not fully discussed. Ziaei-

Rad [69] addressed this problem extensively, however, the measurement errors are simply assumed proportional to the corresponding amplitudes of the FRFs. For real

world structures, test errors are much more complex. As an initial model is the most

reliable FE model according to the analysts’ insight of the structures, the measured

FRFs should be compared with the predicted ones from the initial FE model. The fre-

quencies at which experimental FRF values are not correlated with analytical ones may

have been contaminated by measurement errors and should not be taken as updating frequencies. Therefore, the technique developed in smoothing FRFs may potentially

be transformed to a updating frequency selection procedure. Bibliography

[1] R.J. Allemang. Investigation of Some Multiple Input/Output Frequency Response Func- tion Experimental Modal Analysis Techniques. PhD thesis, University of Cincinnati, 1980.

[2] D. Barker, J. Vodzak, A. Dasgupta, and M. Pecht. Combined vibrational and thermal solder joint fatigue — a generalized strain versus life approach. Journal of Electronic Packaging, Transactions of the ASME, 112:129–134, June 1990.

[3] M. Baruch and I.Y. Bar-Itzhack. Optimal weighted orthogonalization of measured modes. AIAA Journal, 16(4):346–351, 1978.

[4] A. Berman. System identiﬁcation of structure dynamic models - theoretical and practical bounds. In AIAA conference, 1984. 84-0929.

[5] A. Berman and E.J. Nagy. Improvement of a large analytical model using test data. AIAA Journal, 21(8):1168–1173, 1983.

[6] Alex Berman and William G. Flannelly. Theory of incomplete models of dynamic structures. AIAA Journal, 9:1481–1487, August 1971.

[7] B. Caesar. Updating system matrices using modal test data. In Proceedings of the 5th International Modal Analysis Conference, pages 453–459, London, England, April 1987.

[8] Kun-Nan Chen, David L. Brown, and Victor T. Nicolas. Perturbed boundary condition model updating. In Proceedings of the 11th international modal analysis conference, pages 661–667, 1993.

[9] L. Eckert and B. Caesar. Inﬂuence of test data quality and completeness on modal updating results. In Proceedings of the 14th International Seminar on Modal Analysis, September 1989.

[10] R. Edward, Jr. Champion, and J. Michael Ensminger. Finite Element Analysis with Personal Computers. Marcel Dekker, Inc., 1988. ISBN 0-8247-7981-9.

[11] P. A. Engel. Structural analysis for circuit card systems subjected to bending. Electronic Packaging, 112:2–10, March 1990.

221 222 Bibliography

[12] D. J. Ewins. Modal Testing: Theory and Practice. Research Studies Press Ltd., 1984.

[13] E. Fissette, S. Ibrahim, and C. Stavrinids. Error localization and updating of analytical dynamic models using a force balance method. In Proceedings of the 6th International Modal Analysis Conference, pages 1063–1070, Kissimmee, Florida, 1988.

[14] National Agency for Finite Element Methods & Standards. A Finite Element Primer. Department of Trade and Industry, National Engineering Laboratory, East Kilbride, Glasgow G75 0QU, UK., 1986. ISBN 0 903640 17 1.

[15] R.L. Fox and M. P. Kapoor. Rates of change of eigenvalues and eigenvectors. AIAA Journal, 6:2426–2429, December 1968.

[16] A.M. Freed and C. C. Flanigan. A comparison of test-analysis model reduction methods. In Proceedings of the 8th International Modal Analysis Conference, pages 1344– 1351, 1990.

[17] M. I. Friswell. The derivatives of repeated eigenvalues and their associated eigenvectors. Journal of vibration and acoustics, pages 390–397, July 1996.

[18] M.I. Friswell, S. D. Garvey, and J.E.T. Penny. The optimum choice of parameters for single and multiple measurement sets. In Proceedings of the ISMA21, pages 1907– 1920, Leuven, Belgium, September 1996.

[19] C. P. Fritzen and S. Zhu. Updating of ﬁnite element models by means of measured information. Computer and Structures, 40:475–486, 1991.

[20] C.P. Fritzen and T. Kiefer. Localization and correction of errors in analytical models. In Proceedings of the 10th International Modal Analysis Conference, pages 1064–1071, 1992.

[21] C.P. Fritzen and T. Kiefer. Localization and correlation of errors in ﬁnite element models based on experimental data. In Proceedings of the 17th International Seminar on Modal Analysis, pages 1581–1596, K.U. Leuven, Sept. 1992.

[22] Gene H. Golub and Charles F. Van Loan. Matrix Computation. The Johns Hopkins University Press, second edition, 1989.

[23] R. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3(2):380, 1965.

[24] Hanspeter Gysin. Comparison of expansion methods of fe modelling error localization. In Proceedings of the 8th International Modal Analysis Conference, pages 195–204, Kissmimmee, Florida, 1990.

[25] F.M Hemez and C. Farhat. Locating and identifying structural damage using a sensitivity-based model updating methodology. In AIAA/ASME structures, structural dynamics and materials conference, pages 2641–2653, April 1993. Bibliography 223

[26] W. Heylen. Real modes from damped systems. In Proceedings of the 14th International Seminar on Modal Analysis, September 1989.

[27] S. R. Ibrahim and E. C. Mikulcik. A time domain modal vibration test technique. The Shock and Vibration Bulletin, Bulletin 43:part4, June 1973.

[28] ICATS. ICATS Reference Manual. Imperial College, Analysis, Testing and Software, Imperial College of Science, Technology and Medicine, Mechanical Engineering De- partment.

[29] Jerrold Isenberg. Progressing from least squares to bayesian estimation. In Proceedings of the Winter Annual Meeting (’79), December 1979. ASME 79-WA/DSC-16.

[30] T. Janter, W. Heylen, and P. Sas. Qa-model updating. In Proceedings of the 13th International Modal Analysis Seminar, K.U. Leuven, 1988.

[31] T Janter and P. Sas. Uniqueness aspects of model updating procedures. In Proceeding of the 14th International Seminar on Modal Analysis, 1989.

[32] S. H Ju, B. I Sandor, and M. E Plesha. Life prediction of solder joints by damage and fracture mechanics. Journal of Electronic Packaging, 118n:193–200, Dec 1996.

[33] R. Kenda and C. Conti. Updating methods applied to electronic structure. In Proceed- ings of the 7th International conference on computational methods and experimental measurements, Capri, 1995.

[34] R. Kenda, C. Conti, Y.X. Wu, and E. Filippi. Updating methods applied to spatial electronic equipments. In Proceedings of ISMA21, pages 1921–1931, September 1996.

[35] Renaud Kenda. Analyse dynamique d’une carteelectronique ´ avec composant SMT. TFE, Facult´e Polytechnique de Mons, 1993.

[36] R. Kidder. Reduction of structural frequency equations. AIAA Journal, 11(6):892, 1973.

[37] G. Lallement and J. Piranda. Localization methods for parametric updating of ﬁnite element models in elastodynamics. In Proceedings of the 8th IMAC, pages 579–585, 1990.

[38] Stefan Lammens. Frequency Response Based Validation of Dynamic Structural Finite Element Models. PhD thesis, Katholieke Universiteit Leuven, 1995.

[39] Stefan Lammens and David L. Brown. Model updating and perturbed boundary condition testing. In Proceedings of the 11th international modal analysis conference, pages 449–455, 1993. 224 Bibliography

[40] J. Lau and L. et Al. Power. Solder joint reliability of ﬁne pitch surface mount technology assemblies. In Proceedings of IEEE International Electronic Manufacturing Technology Symposium, pages 594–602, September 1989.

[41] John H Lau and Rice W. Donald. Solder joint fatigue in surface mount technology: State of the art. Solid State Technology, pages 91–104, October 1985.

[42] John H. Lau and Catherine A. Keely. Dynamic characterization of surface-mount component leads for solder joint inspection. IEEE Transaction on components, hybrid, and manufacturing technology, 12(4):594–602, Dec 1989.

[43] Shumin Li, Stuart Shelley, and David Brown. Perturbed boundary condition testing concepts. In Proceedings of the 13th international modal analysis conference, pages 902–911, 1995.

[44] N. Lieven and D. Ewins. Spatial correlation of mode shapes, the coordinate modal assurance criterion (comac). In Proceedings of the 6th International Modal Analysis Conference, pages 605–609, Kissimmee, Florida, 1988.

[45] N. Lieven and D. Ewins. Expansion of modal data for correlation. In Proceedings of the 8th International Modal Analysis Conference, pages 605–609, Kissimmee, Florida, 1990.

[46] R. M. Lin, H. Du, and H. Ong. Sensitivity based method for structural dynamic model improvement. Computers & Structures, 47(3):349–363, 1993.

[47] M. Link. Identiﬁcation of physical system matrices using incomplete vibration test data. In Proceedings of the 4th International Modal Analysis Conference, pages 386– 393, 1986.

[48] Michael Link. Localization of errors in computational models using dynamic test data. In European conference on structural dynamics, pages 305–313, June 1990.

[49] J. Lipkenss and Vandeurzen. The use of smoothing techniques for structural modiﬁca- tion applications. In Proceedings of the 12th International Seminar on Modal Analysis, Leuven, Belgium, September 1987.

[50] Wolfgang Luber. Error localization methods and updating of a ﬁnite element model by means of structural optimization. In Proceedings of the 15th International Seminar on Modal Analysis, pages 23–38, Leuven, Belgium, September 1990.

[51] A.J. Miller. Subset Selection in Regression. Monographs on Statistics and Applied Probability 40. Chapman and Hall, 1990.

[52] Richard B. Nelson. Simpliﬁed calculation of eigenvector derivatives. AIAA Journal, 14:1201–1205, September 1976. Bibliography 225

[53] R. Nordmann, H. Eckey, P. Kaas, and M. Steinwarz. Sensitivity of the dynamic behaviour to system parameter changes. In ASME Proceedings of the 1985 pressure vessels and piping conference, pages 53–60, 1985. [54] J. O’Callahan and P. Avitabile. System equivalent reduction expansion process (SEREP). In Proceedings of the 7th International Modal Analysis Conference, pages 29–37, Las Vegas, Nevada, 1989. [55] J.C. O’Callahan. Comparison of reduced model concepts. In Proceedings of the 8th International Modal Analysis Conference, pages 442–430, 1990. [56] R. Pascual, J.C. Golival, and M. Razeto. Testing of frf based model updating methods using a general finite element program. In Proceedings of ISMA21, pages 1933–1945, Leuven, Belgium, September 1996. [57] J. M. Pitarresi, D.V. Caletka, R. Caldwell, and D.E. Smith. The smeared’ property technique for the fe vibration analysis of printed circuit cards. Journal of Electronic Packaging, Transactions of the ASME, 113:250–257, 1991. [58] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes, chapter 14, pages 498–504. Cambridge University Press, 1986. [59] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes in C. Cambridge University Press, 1988. [60] J.S Przemieniecki. Theory of Matrix Structural Analysis. Mc Graw Hill., 1968. [61] L.C. Rogers. Derivatives of eigenvalues and eigenvectors. AIAA Journal, 8:943–944, may 1970. [62] J. Sidhu and D.J. Ewins. Correlation of finite element and modal test studies of a practical structure. In Proceedings of the 2nd International Modal Analysis Conference, pages 756–762, Orlando, Florida, 1984. [63] Dave S Steinberg. Vibration analysis for electronic equipment. John Wiley & Sons Inc., 2nd ed edition, 1988. ISBN 0-471-63301-1. [64] Gang Wang. Experimental modal analysis and dynamic response of pc boards with surface mount electronic components. Master’s thesis, Florida Atlantic University, Dec. 1990. [65] Jim J.C. Wei, Qiang Zhang, Randall J. Allemang, and Max L. Wei. Correction of finite model via selected physical parameters. In Proceedings of the 13th International Modal Analysis Seminar, K.U. Leuven, 1988. [66] Tin-Lup Wong, Karl K. Stevens, Junshi Wang, and Wayne Chen. Strength analysis of surface mounted assembles under bending and twisting loads. Journal of Electronic Packaging, 112:168–174, June 1990. 226 Bibliography

[67] Michael Ming Yang. Improved Experimental Impedance Modelling Techniques for Updating Finite Element Models. PhD thesis, University of Cincinnati, 1996.

[68] S. Ziaei-Rad and M. Imregun. On the accuracy required of experimental data for ﬁnite element model updating. Journal of Sound and vibration, 196(3):323–336, July 1996.

[69] Saeed Ziaei Rad. Methods for Updating Numerical Models in Structural Dynamics. PhD thesis, Imperial College of Science, Technology and Medicin, University of Lon- don, June 1997. Appendix A

Damping Treatment in FRF-based Equations

A.1 Introduction

In Chapter 3, we have presented the basic theory for FRF-based updating methods, but the damping property was not discussed. Unlike FBM and QMS methods, where the damping is not directly handled but bypassed by converting the complex mode shapes into real ones, FRF-based methods have to deal with the damping inside the error localization/updating process. One may simply assume that the structure is undamped and select updating frequencies away from resonance regions, however, when measurement data is noisy, data collected from these updating frequencies will be much less reliable. Damping model, just as the finite element model itself, should be established case by case. Unfortunately, limited information from test data is available to validate the damping model. Therefore, instead of pursuing a refined damping model, analysts usually find ways to bypass setting up and refining it.

This appendix will present some general strategies of bypassing damping problems

227 228 Appendix A. Damping Treatment in FRF-based Equations in the establishment of FRF-based updating equations. The strategy for diagonal damping is not suitable for “indirect receptance residue” method while the other strategies can be applied to both force residue and indirect receptance residue methods. In this appendix, we assume that the updating parameters are not directly related to damping, that is to say, the parameter ps will satisfy:

∂ZIm(ω) ∂Z(ω) ∂ZRe(ω) =0; = ; for s =1,...,n (A.1) ∂ps ∂ps ∂ps

We partition Z(ω) and H(ω) as:

Z(ω)=ZRe(ω)+iZIm(ω); H(ω)=HRe(ω)+iHIm(ω); (A.2)

A.2 Undamped model assumption

Undamped assumption is a basic mathematical expression for the other damping assumptions to be discussed. For an undamped model, the dynamic stiffness Z(ω) is real and the updating equation (3.34) is then direct. For a given updating frequency, equation (3.34) is rewritten as:

A(ω)x = b(ω) (A.3) where A(ω) is column-partitioned as:

∂ZRe(ω) ∂ZRe(ω) A(ω)= hxj(ω),... , hxj(ω) (A.4) ∂p1 ∂pn A.3. Diagonal damping model assumption 229 while b(ω) is expressed as:

Re b(ω)=[1j] − Z (ω)hxj(ω) (A.5)

Since hxj(ω) is a complex number vector, equation (A.3) is split into:     ARe  BRe    x =   (A.6) AIm BIm

These equations are repeated for all selected updating frequencies.

A.3 Diagonal damping model assumption

Diagonal damping is neither a realistic damping model for dynamic structures, however, we can usually expect that the damping matrix is diagonally dominated. Therefore, diagonal damping model assumption should be, in some extent, reasonable. With this assumption, the imaginary part of a dynamic stiffness matrix can be written as:

Im Z (ω)=diag(z1(ω),... ,zn(ω)),zi ∈< (A.7) where the damping is assumed as frequency dependent.

When damping is not ignored, equation (3.34) is rewritten as:

Im A(ω)x = b(ω) − iZ (ω)hxj(ω) (A.8) where A(ω) and b(ω) are expressed in (A.4) and (A.5), respectively. For ﬁnite element model updating, we eliminate the damping effects, instead of revealing the frequency dependent 230 Appendix A. Damping Treatment in FRF-based Equations damping. The damping related term of (A.8) can be rewritten as:            

z1 ... 0  hx1j  z1hx1j  hx1j ... 0  z1  z1               . . .   .   .   . . .   .   .   . .. .   .  =  .  =  . .. .   .  = Qj(ω)  .  (A.9)            

0 ... zn hxnj znhxnj 0 ... hxnj zn zn where

Q(ω)=diag(hx1j(ω),hx2j(ω),... ,hxnj(ω)) (A.10)

Therefore, pre-multiplying equation (A.8) with QH (ω), and taking only the real part of the product, we obtain an equation where damping is eliminated:

Re(QH (ω)A(ω))x = Re(QH (ω)b(ω)) (A.11)

However, when data precision is limited because of measurement noise or even roundoff error, the weighting matrix QH (ω) may trouble the LS regression! To reduce the row-weighting inﬂuence, instead of QH (ω), a new weighting matrix is formed as:

h∗ (ω) h∗ (ω) W(ω)=diag x1j , ..., xnj (A.12) |hx1j (ω)| |hxnj(ω)| where superscript * denotes the conjugate complex number. Therefore, (A.11) is reformed as:

Re(W(ω)A(ω))x = Re(W(ω)b(ω)) (A.13) A.4. Proportional damping model assumption 231

A.4 Proportional damping model assumption

Both proportional viscous damping and structural damping may be present in this assumption, therefore, the imaginary part of the dynamic stiffness matrix is expressed as:

ZIm(ω)=ωC + D = ω(αM + βK)+ηK = Mωα + K(ωβ + η) (A.14)

Equation (A.8) is then rewritten as:

A(ω)x = b(ω) − i · M · hxj(ω)ωα − i · K · hxj(ω)(ωβ + η) (A.15)

Since α, β and η may be frequency dependent, they are solved prior to design variables x at each updating frequency such that:

||b(ω) − i · M · hxj(ω)ωα − i · Khxj(ω)(ωβ + η)||2 ⇒ min (A.16)

After the minimization, equation (A.15) becomes:

A(ω)x = bu(ω) (A.17) where

bu(ω)=b(ω) − i · M · hxj(ω)(ωα)ω − i · K · hxj(ω)(ωβ + η)ω (A.18)

while (ωα)ω and (ωβ + η)ω are solved from (A.16). 232 Appendix A. Damping Treatment in FRF-based Equations

Equation (A.17) is then split into real and imaginary parts:     Re Re A (ω) bu (ω)   x =   (A.19) Im Im A (ω) bu (ω)

A.5 General light damping model assumption

Carefully selected updating frequencies may improve the updating results. In fact, equation (3.34) is partitioned as:

Re Im Re Im − Im Re Im Im (A (ω)+iA (ω))x = b (ω)+ib (ω) iZ (ω)hxj (ω)+Z (ω)hxj (ω) (A.20)

Re If hxj (ω)=[0], then imaginary part of equation (A.20) can be used to avoid updating damping:

AIm(ω)x = bIm(ω) (A.21)

Im If hxj (ω)=[0], then real part should be employed to avoid updating damping:

ARe(ω)x = bRe(ω) (A.22)

Unfortunately, it may be quite difﬁcult to search such frequencies, especially for MDOF system with digitalised FRF data. The following strategy may be employed as heuris-

H tic trial. Equation (A.8) is post-multiplied with hxj(ω), we obtain:

H H − · Im H A(ω)xhxj(ω)=b(ω)hxj(ω) i Z (ω)hxj(ω)hxj(ω) (A.23) A.6. Illustration 233

Take into account only real part of equation (A.23):

H H Im H Re(A(ω)Xhxj(ω)) = Re(b(ω)hxj(ω)) + Z Im(hxj(ω)hxj(ω)) (A.24) with

H Im Re0 − Re Im0 P = Im(hxj(ω)hxj(ω)) = hxj (ω)hxj (ω) hxj (ω)hxj (ω) (A.25)

As P is a one-rank matrix, to make damping effects least, we take only the tth column of (A.25) into account such that ||pt|| is the smallest one of all ||pi||’s. Therefore, (A.24) becomes as:

∗ ∗ Im ≈ ∗ Re(hxtj(ω)A(ω))x = Re(hxtj(ω)b(ω)) + Z (ω)pt Re(hxtj (ω)b(ω)) (A.26)

∗ where htj (ω) is the conjugate number of htj (ω) .

A.6 Illustration

A 3-DOF system is presented in Figure A.1 for illustrating purposes. Both viscous and structural damping are involved simultaneously. Parameters are listed in Table A.1. These parameters are used to calculate FRFs (Figure A.2 and A.3). The simulated

FRFs are then analysed with ICATS software. The modal parameters are listed in Table A.2.

The model errors are simulated by setting the initial ﬁnite element model with parameter errors on m1 and k3:

m1initial =1.3 × M; k3initial =0.7 × 7K;

To investigate the damping impact, the “experimental” data are free from artiﬁcial 234 Appendix A. Damping Treatment in FRF-based Equations

Figure A.1: A 3-DOF damping system Table A.1: Simulated parameters X1 m1=M

D C M(kg) 1 k1=K K(N/m) 10000 X2 m2=2M C* High 20

4D (Ns/m) Low 2 2C k2=4K D** High 800 X3 m3=5M (N/m) Low 80

7D *C: viscous damping 5C k3=7K **D: Structural damping.

Figure A.2: FRFs with High damping Figure A.3: FRFs with Low damping −4 x 10 |H13| |H23| |H33| |H13| |H23| |H33| −3 1 −3.5 0.8 −4

0.6 −4.5 log10 −5 0.4 −5.5 0.2 −6

0 −6.5 0 10 20 30 0 10 20 30 Frequency (Hz) Frequency (Hz)

Table A.2: Modal parameters of a 3-DOFs system mode fc0 High damping Low damping ∗ ∗ ∗ ∗ # (undamped) fx η % ζ% η% fx η % ζ% η% 1 11.435 10.97 29.02 13.92 8 11.39 2.79 1.39 0.8 2 19.194 18.41 17.29 8.29 8 19.12 1.67 0.83 0.8 3 30.735 29.48 10.80 5.18 8 30.61 1.04 0.52 0.8 ∗ ∗ fx , η : resonance frequency and structure damping extracted from “test” data with ICATS software. ζ,η: simulated viscous and structural damping values. A.6. Illustration 235

Table A.3: Updated results for high damping simulation Damping m1 k3 assumption m1u ∆m1u% k3u ∆k3u% Undamped 1.02304 -2.304 62853.8 10.209 Diagonal 1.02304 -2.304 62853.8 10.209 Proportional 1.01753 -1.753 58079.7 17.029 General 1.05337 -5.337 63889.2 8.730 Updating frequencies:9.975;17.85;29.3125 Hz

Table A.4: Updated results for low damping simulation Damping m1 k3 assumption m1u ∆m1u% k3u ∆k3u% Undamped 1.00808 -0.808 69315 0.9786 Diagonal 1.00808 -0.808 69315 0.9786 Proportional 1.00833 -0.833 69322 0.9686 General 1.00155 0.155 69425 0.8214 Updating frequencies:9.975;17.85;29.3125 Hz noise. The updating equations are based on the excitation residual criterion. Table A.3 and

A.4 show the updated results for high and low damping, respectively.

From Table A.3 and A.4, we may make the following remarks:

• Complex operations for deleting damping do not necessarily improve the results very

much comparing to undamped assumption.

• As a rule of thumb, the relative errors of the updating results caused by damping may

not exceed the damping ratios.

Therefore, when applied to engineering problems where the damping is low, undamped assumption should be acceptable for parameter updating purposes. For example, suppose the damping ratios are about 1%, then the relative errors on updating results caused by the impact of damping may not exceed 1%, while the measurement noise in FRFs will certainly cause bigger errors.

We should also be aware that, unless an explicit and accurate damping model is available, the updating program may not be smart enough to distinguish the noise from damping 236 Appendix A. Damping Treatment in FRF-based Equations effect. Because the damping impact on FRFs is only within a certain vicinity around the resonance frequencies, if updating frequencies are selected within these resonance ranges, the data themselves are not quite precise due to inaccurate exciting force. If updating frequencies are out of these resonance ranges, then the damping effect is not signiﬁcant and may be covered by measurement noise.

A.7 Summary

This appendix evaluated some general strategies of bypassing damping problems in

FRF-based model updating process. The following damping assumptions have been discussed:

• Undamped assumption;

• Diagonal damping assumption;

• Proportional damping assumption;

• General light damping assumption.

The simpliﬁed simulation showed that, for all these assumptions, the relative errors of updated parameters caused by damping do not exceed the damping ratios. The simulation also showed that, though the complex operations of deleting damping impact are quite reasonable, as the initial model itself is erroneous, they did not improve updating results very much. Therefore, for engineering problems where damping is low, the undamped model assumption will not introduce big errors in updated parameters and as the simplest way, if no other reasonably accurate and manageable damping model is provided, it can be an acceptable compromising solution. Appendix B

POMUS scheme and GUI interface

Sensitivity-based model updating techniques developed in this dissertation are implemented with POMUS (Parameter-Oriented Model Updating System). The main frame of POMUS is written in MATLAB language. Interface to ANSYS is written in FORTRAN and

C++. POMUS includes an integrated GUI program that lets the user deﬁne and conduct the model updating by menus operations, and some relatively independent utilities.

This appendix shortly describes the program schemes and the GUI interface of

POMUS.

B.1 Program schemes

The main scheme of the POMUS is illustrated in Figure B.1. Error localization scheme is shown in Figure B.2 and Equation solving scheme is presented in Figure B.3.

B.2 Main GUI window

The POMUS-program is invoked by issuing the command POMUS within the MAT- LAB command window. The main GUI window has a menu bar with seven menu items, as

237 238 Appendix B. POMUS scheme and GUI interface

Figure B.1: POMUS program scheme The vibration structure under study

Set up the initial experimental configuration , referred to it as #1.

FRFa, fa, mode shapes; mass and FRFx, fx, mode shapes stiffness sensitivity matrices

Correlation, model match between test data and FE results 3

Set up updating equations: A1x=b1 4

Stacking all updating equations for all configurations: Ax=b. 5

-1 Weighting equation: (DAT1)(T1 xT2)=DbT2 6

No Iter==1? Yes

Do error localization 7

Solve the updating equations 8 9 Modify all To #1 No FE models Stop iteration? for all configurations To other Yes configurations STOP 1 FE script ﬁle format is described in Appendix C 2 Mass and stiffness sensitivities are computed based on (4.50) 3 Correlation and model match techniques are described in Chapter 2 4 Sensitivity-based updating equations are discussed in Chapter 3 and A 5 Stacking PBC updating equations are treated in Chapter 6 6 Weighting strategies are described in §4.4, §4.7 7 Error localization is described in §5.2.3 (see also Figure B.2) 8 Solving methods are described in §4.3, §4.5, §4.6, §4.8 and §5.2.2 9 Updating parameters are described in §4.9 B.2. Main GUI window 239

Figure B.2: Error Localization scheme

The updating equations Ax=b; A, x, b have been row- and column weighted. Column weighting factor T2 needed to compute relative parameter changes from x

(1) Perform SVD on A: A=U*V’; ) ) (2) 2-norm condition number: k2(A) = 1/ r; (r=1,2,3,...,n) (3) Plot trial k2(A), interactively determine the numerical rank number n1

(1) C=[A b]; SVD C: C=U(V’; W=V’; (2) partitioning matrices according to n : ( ( ( 1 U=[U1, U2]; (=[(1, 0; 0 (2]; W=[W11, W12; W21, W22];

W11x=W12

Perform QR-F subset selection on W11x=W12

Selected.parameters that will be indexed by a vector p

The retained parameter number

Use the retained parameters to solve equations : A(k)x(k)=b. Relative parameter changes : y = x/T2 (0) If yi >Yi , then parameter yi is identified as a main model error.

Error localization result 240 Appendix B. POMUS scheme and GUI interface

Figure B.3: Equation solving scheme

Ax=b, where A, x, b are alread weighted. XU, XL, the upper and low boundaries associated to the weighted x. SolutionMethod, indicating the method for solving updating equations BayeRatio, for Bayesian estimation technique.

Yes SolutionMethod=='Total_LS' ?

TLS solution: C=[A b] 1 No [U,S,V]=svd(C). SolutionMethod == 'DIRECT_PINVSOL' Yes No

-1 xTLS = -V12V22 2 Yes SolutionMethod== LS solution by calling 'MATLAB_Backslash' MATLAB PINV function No

x = pinv(A)*b SolutionMethod== LS 'QUADRATIC_PROG' 3 No QR basic solution by using Yes MATLAB ‘\’ operator H=A’*A; v= -A’*b; c=1/sum(abs(diag(H))); 4 H = H*c; v = v*c; xQR = A\b

xQP = qp(H,v,[],[],XL,XU);

Checking the boundaries SolutionMethod==' Yes BayeQUADRATIC_PROG' No Solution x H=A’*A; v= -A’*b; H(i,i)=H(i,i)*(1+Bayeratio), for all i c=1/sum(abs(diag(H))); STOP! H = H*c; v = v*c; 5 Solution To test whether method stop the iteration invalid! xQP = qp(H,v,[],[],XL,XU); No solution

1 TLS solution. 2 LS solution via SVD. 3 QR basic solution. 4 LS solution with boundary condition via QP. 5 Bayesian estimation via QP. B.3. File menu 241 shown in Figure B.4.

B.3 File menu

File-related selections are gathered mainly in File menu. There are ten sub-menus, as shown in Figure B.5. These sub-menus are: Read IPT file: when this sub-menu is selected, a file dialogue box is presented. The user is to choose a model updating definition file to read. A model updating definition file is a ASCII file created by “Save to IPT” menu operation (see below).

Edit an IPT ﬁle: the user is prompted to choose an IPT ﬁle to be edited in a text editor.

Load .MAT file: the user is prompted to choose an MATLAB .MAT file to read in. Save to IPT: the user is prompted to give an IPT file name. POMUS tries to save the model updating definition to .IPT format.

Save Workspace: the user is prompted to give a ﬁle name. POMUS will save the whole work space to a ﬁle with the given name.

Change Work Dir: the user is prompted to select an arbitrary file, while POMUS will change the work directory to the directory of the given file. FE script file: the user is prompted to select an ANSYS FE model script file (file format is required, see appendix B)

CMP ﬁle: the user is prompted to select an ICAT .CMP ﬁle and the eigen-related indicators.

CRD ﬁle: the user is prompted to select an ICAT .CRD ﬁle and the FRF-related indicators. Exit: quit MATLAB applications. 242 Appendix B. POMUS scheme and GUI interface

B.4 Setting menu

There are 12 sub-menus in Settings menu (Figure B.6). These sub-menus include: Title: set a title to the updating problem.

DyExpansion: choose dynamic expansion method for model matches.

DyReduction: choose dynamic reduction method for model matches.

Nconﬁguration: set PBC test conﬁguration number (default value:1).

Simple Stacking: choose simple stacking method for PBC updating.

SVDRed Stacking: choose SVD Reduction Stacking method for PBC model updating. Min MAC Value: set a minimal value for MAC criterion. When MAC value is below this given value, the modes are not thought as correlated.

Iteration Number: set the maximum iteration number.

Convergence...: set a frequency convergence criterion.

Lower Boundaries: set lower boundaries of the updating parameters.

Upper Boundaries: set upper boundaries of the updating parameters. Column Weighting: set column weighting factors for the updating parameters.

B.5 Modal menu

Modal menu operation deﬁnes modal-parameter-based updating method. The four sub-menus are (Figure B.7):

Frequency Weighting: choose eigenfrequency-based model updating method, row- weighting performed such that the relative frequency differences accounted.

Frequency No Weighting: choose eigenfrequency-based model updating method; no row-weighting adopted. Quasi-ModeShape: choose QMS updating method. B.6. Imped Menu 243

FBM ModeShape: choose residual-force-based updating method.

B.6 Imped Menu

Imped menu deﬁnes impedance updating methods. The four sub-menus are (Figure

B.8): Undamping Assumption: damping is totally ignored.

Proportional Damping: proportional damping is assumed, which is eliminated by a preliminary minimization.

Diagonal Damping: diagonal damping is assumed, which is eliminated by row- weighting.

General Light Damping: general light damping is assumed. For Undamping Assumption and Proportional Damping, there are two submenus, as shown in Figure B.9 and B.10:

Input (Force) method: excitation-residual-criterion-based impedance model updating method (see (3.34)).

Output (Displ) method: indirect-receptance-residue-based impedance model updating method (see (3.35))

B.7 Solu menu

Solu menu operations deﬁne the numerical method for solving the updating equations. The available solution methods (Figure B.11) are: Total LS: total least squares solution.

Direct PINV: direct pseudo-inverse solution.

MATLAB Backslash: MATLAB backslash solution. 244 Appendix B. POMUS scheme and GUI interface

Quadratic Programming: quadratic programming solution method.

Baye QuadProgramming: Bayesian estimation solved with quadratic programming method.

B.8 Figure menu

Figure menu operations display iterative updating results. There are 13 sub-menu items (Figure B.12):

Result WkSpace: reload the iterative updating results from a .MAT ﬁle.

First set: display the initial correlation results (graphics).

Last set: display the last iteration results.

+1 set: display next (from the current) iteration results......

+10 set: display the 10th (from the current) iteration results.

-1 set: display previous iteration results.

......

-10 set: display the previous 10th iteration results.

B.9 Run menu

Run menu (B.13) operations start the main updating procedure.

Interactively Run: interactively run the main updating procedure (after each iteration, the user is prompted to decide to continue/stop the program). Aut Run And ErrLoc: automatically run the updating procedure, subset selection modules are also invoked. B.9. Run menu 245

Aut Run No ErrLoc: automatically run the updating procedure, subset selection modules not invoked.

Figure B.4: POMUS main menus Figure No. 1: POMUS main Menu File Settings Modal Imped Solu Figure Run

Figure B.5: File menus Figure B.6: Setting menus Figure B.7: Modal menus File Settings M Settings Modal Imp Modal Imped Solu Fig Frequency Weighting Read IPT file Title Frequency No Weighting Edit an IPT file DyExpansion Quasi-ModeShape Load MAT file DyReduction FBM ModeShape Save to IPT Nconfiguration Save Workspace Simple Stacking Change Work Dir SVDRed Stacking FE script file Min MAC Value CMP File Iteration Number CRD File Convergence Exit Lower Boundaries Upper Boundaries Column Weighting 246 Appendix B. POMUS scheme and GUI interface

Figure B.8: Imped menus Figure B.9: Undamping menus Imped Solu Figure Imped Solu Figure Run

Undamping Assumption Undamping Assumption Input (Force) method Proportional Damping Proportional Damping Output (Displ) method Diagonal Damping Diagonal Damping General Light Damping General Light Damping

Figure B.10: Proportional menus Figure B.11: Solu menus Imped Solu Figure Run Solu Figure Run

Undamping Assumption Total Ls Proportional Damping Input (Force) method Direct PINV MATLAB Backlash Diagonal Damping Output (Displ) method Quadratic Programming General Light Damping Baye QuadProgramming

Figure B.12: Figure menus Figure B.13: Run menus Figure Run Run Result WkSpace Interactively__Run First set Aut_Run_And_ErrLoc Last set Aut_Run__No_ErrLoc +1 set +2 set +3 set +5 set +10 set -1 set -2 set -3 set -5 set -10 set Appendix C

FE script ﬁle format

In POMUS, the analytical models are described with ANSYS model script language.

POMUS call ANSYS within each updating iteration. The communication between POMUS and ANSYS is implemented via input and output files. This is started from an ANSYS FE script file. During updating iterations, POMUS modifies this model script file, while ANSYS uses the modified FE model to recalculate the FE solutions and put them into output files, and

POMUS retrieves the FE results from ANSYS output ﬁles. For model updating purposes,

POMUS requires that the ANSYS model script ﬁle must have the following format:

/clear,nostart /filname, (jbn: job file name ) /prep7 !begin parameters’ definition = ...... (All initial values of updating parameters defined here) = !end parameters’ definition ...... commands that directly use p1, ...... , pn. (Must be present!) !begin topology ...... commands that do not directly uss p1, ...... , pn. Usually a file switch command as “/input, Filename, Ext, Dir” to short this script file. (These commands are used to construct the FE model of the first configuration. Guyan reduction method is assumed. Master degrees of freedom are also defined here.)

247 248 Appendix C. FE script ﬁle format

antype, modal, new modopt, reduc, , , ,0,off !end topology ...... commands (modifying the first configuration’s model to current one). fini /solu solve fini

In this FE model script file, the four comment lines started with “!” are used as special strings for creating new FE model script files, therefore, they must not be changed. p1, ...... , pn are variables to be updated. They are represented with any ASCII strings permitted as variables by ANSYS software. v1, ...... , vn are initial values of the parameters p1, ...... , pn, respectively. Each assigning command = must be in a separate line. The interest frequency range is defined in the command:

modopt, reduc, , , ,0,off where deﬁnes the mode number while , deﬁnes the frequency range.

If the model updating is performed with PBC testing technique, the FE models of perturbed structures are defined by modifying the one of the initial configuration. To avoid repeated model construction, commands bracketed by !begin topology and !end topology are treated only once during the whole model updating process. The database of the initial model of the first configuration is saved in a file, then for the late model construction, this database is resumed from the saved file. Appendix D

Deﬁnition of the case study structures

This appendix provides some detailed deﬁnitions for the case study structures con- tained in the thesis.

D.1 Cantilever beam structure

The cantilever beam introduced in Chapter 5 is an uniform beam structure, as shown in Figure D.1. It is modelled with 18 beam (ANSYS beam4) elements of the same length.

The initial FE model is perfect while “experimental” data are simulated with the same FE model but erroneous parameters:

Updating parameters: E : ×109 [Pa] Parameter number 1—4, 9—18 5—8 “experimental” px 26.68 20.00 initial FE pa 26.68 26.68 px−pa pa 0 -0.2504 Constant parameters for FE and “experimental” model ρ = 2089 kg/m3.

The receptance FRFs are computed by applying an exciting force at node #3 at Y direction.

249 250 Appendix D. Deﬁnition of the case study structures

Figure D.1: The cantilever beam structure

Y 360mm 2.4mm

X Fixed end Free end Z 10mm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 N

D.2 10-DOF stiffness-mass system

The 10-DOF stiffness-mass system introduced in Chapter 5 is a simple system with lumped masses, as shown in Figure D.2. The initial FE model is perfect while “experimental” data is generated by introducing parameter errors into the initial FE model:

Index 1 2 3 4 5 6 7 8 9 10

The receptance FRFs are computed by applying an exciting force at mass #3 at X direction.

Figure D.2: The 10-DOF stiffness-mass system Y k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 X m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 D.3. 3-D bay structure [69] 251

D.3 3-D bay structure [69]

The 3-D bay structure introduced in Chapter 5 is built with 7 beams of the same section (A =0.1 × 0.01 m2), as shown in Figure D.3. It is modelled with 20 3-D beam

(ANSYS BEAM4) elements, whose local y axes are parallel to the global X − Y plane.

Figure D.3: A 3-D bay structure (A =0.1 × 0.01 m2) Y 0.2m 0.6m 9 678VI VII VIII 0.05m 5 V IX XX IV 19 10 4 X III 18 XIX 11 3 0.4m XI II 17 XVIII 12 2 XVII XII I XV XIV XIII XVI 16 15 14 13 X 1 Z

Table D.1: Updating parameters p number 1,7,15 Others(≤ 20) 21,24,27,30,33,36,39 Others(22˜40) px 7850 7850 207 207 pa 7457.5 7850 223.56 207 px−pa pa 0.0526 0 -0.0741 0 3 9 2 pa(1-20), px(1-20): mass density (kg/m ); pa(21-40), px(21-40): stiffness (10 N/m );

The receptance FRFs are computed by applying exciting force at node #1 at Z axis direction.

D.4 Plate structures

The simulated plate structures introduced in Chapter 5 are shown in Figure D.4 and

D.5 for free and wedge-locked boundary conditions, respectively. The components are sim- 252 Appendix D. Deﬁnition of the case study structures ulated by increasing the corresponding thicknesses. The Wedge-Lock edges are modelled with distributed rotational springs in X axes. The parameter values of the thicknesses and rotational springs are deﬁned with respect to the numbering convention adopted for the FE model (see D.6) and are given in the table below:

p number 11,13,23,25 9,27 Others(≤ 42) 43,44 px 3.2 3.2 2.8 0.50 pa 3.0 3.2 2.8 0.43 px−pa pa 0.06667 0 0 0.1628 p1—p42, thickness: mm; p43-p44, Stiffness: (Nm/rad)/m

Figure D.4: Free PCB

162mm

c3 c6

178mm c2 c5

y c1 c4 z 2.8mm x

Figure D.5: Fixed PCB krotxB

c3 c6

c2 c5

y c1 c4 z x krotxA D.5. 10-DOF discrete system 253

The Young’s module and mass density of the base board are 26 × 109N/m2 and

2640kg/m3, respectively. The PCB board is modelled with 42 shell elements using 56 nodes, as shown in Figure D.6. The distributed rotational springs are integrated into the 14 nodes according to the length of the elements on the edges.

Figure D.6: FE model mesh of a simulated PCB

8 16 24 32 40 48 56

7 14 21 28 35 42 771515 23 31 39 47 55

6 13 20 27 34 41

661414 22 30 38 46 54

5 12 19 26 33 40

551313 21 29 37 45 53

4 11 18 25 32 39

441212 20 28 36 44 52 24 31 38 3 10 17 27 35 43 51 3113 11 19

2 9 16 23 30 37

221010 18 2634 42 50

1 8 Y 15 22 29 36

191Z X 9179 1725 33 41 49

D.5 10-DOF discrete system

The 10-DOF discrete system introduced in Chapter 6 is used to illustrate PBC model updating method. The conﬁguration and parameter values are shown in Figure D.7. The perturbation masses are attached at m1,m5 and m9 with ∆M as 0.5, 1.0 and 0.3 (kg), respectively. 254 Appendix D. Deﬁnition of the case study structures

Figure D.7: 10 DOF discrete system Initial model parameters Exact model initial model k1 k8 m1−3 0.9 1 m1 k5 m4−5 2.15 2 m6 k16 k12 m4 m6−8 0.9 1 k2 k9 m9 m2 k14 m9−10 0.05 0.05 k6 k1−11 21000 20000 m7 k17 m5 k12−15 38000 40000 m10 k3 k10 k13 k15 k16−17 63000 60000 k7 m3 m8 Units: mass: kg; stiffness: N/m. ∆M k4 k11 Perturbations : 0.5, 1.0 and 0.3 (kg) for m1,m5 and m9, respectively.

D.6 Free PCB with six components

The free PCB with six components introduced in Chapter 7 is shown in Figure D.8.

The exciting force is applied at node 17 in Z direction (perpendicular to the plate). 104 points are measured (Figure D.9). The board is modelled with 112 shell elements (Figure D.10).

Figure D.8: Experimental PCB Table D.2: Components’ properties

PGA JLCC Type V(mm3) M(g) Joint PGA 40×40×2.2 14 J CQFP 30×30×3 8.6 G* JLCC CQFP JLCC 27×27×2 6.1 J 178mm *Gullwing Properties of the base PCB: 3 CQFP PGA Dimension V: 162×178×2.8mm Young’s module: 26 GPa Mass density: 2640 kg/m3

162mm D.7. Wedge-Lock slides 255

Figure D.9: Free PCB’s test points Figure D.10: Element number of the free PCB

112 104 103 102 101 100 99 98 104 88 66 50 28 21 14 7 84 72 48 36 97 87 75 111 18 12 6 59 49 37 60 54 24 90 83 71 47 35 103 86 74 58 48 36 20 13 6 96 65 27 46 34 89 82 70 59 53 23 57 47 35 110 17 11 5 95 85 73 81 69 45 33 102 84 64 46 26 19 12 5 80 68 44 32 109 16 10 4 94 83 72 56 45 34 88 58 52 22 79 67 43 31 101 93 82 71 63 55 44 33 18 11 4 25 78 66 42 30 87 57 51 21 92 81 70 54 43 32 108 15 9 3 77 65 41 29 100 62 113 80 42 24 17 10 3 40 28 53 41 31 76 64 91 79 69 107 14 8 2 86 56 50 39 27 20 75 63 99 90 78 68 61 52 40 30 16 9 2 23 74 62 85 55 49 38 26 19 89 77 67 106 13 7 1 51 39 29 73 61 37 25 76 38 22 15 8 1 Y Y 98 60 105 97 96 95 94 93 92 91 Z X Z X

D.7 Wedge-Lock slides

The Wedge-Lock slides discussed in Chapter 7 are shown in Figure D.11. The slides are modelled as rotational springs (Figure D.12). The experimental conﬁguration is shown in Figure D.13 while the measurement points for a virgin card is displayed in Figure D.14.

Figure D.11: A typical Wedge-Lock slide Figure D.12: Spring model Wedge-Lock slide The chassis

The PCB

The chassis The PCB The torsional springs 256 Appendix D. Deﬁnition of the case study structures

Figure D.14: Measurement points Figure D.13: Wedge-Lock slides The chassis Wedge-Lock slides PCB card Y Z X

Contact area Plate thickness h=1.56mm Contact area Appendix E

Mode shapes of the PCB, Box structures

Figures E.1,E.2 and E.3 show the experimental mode shapes of the PCB with six components (Chapter 7 section §7.3).

Figure E.1: Experimental mode shapes (conﬁguration #1) Conf. 1 mode 1 f= 214.12Hz Conf. 1 mode 2 f= 302.71Hz Conf. 1 mode 3 f= 397.97Hz 10 10 10 0 0 Z 0 Z −10 −10 Z −10 200 200 200 200 200 200 100 100 100 100 100 100 Y X Y X Y X 0 0 0 0 0 0 Conf. 1 mode 4 f= 531.38Hz Conf. 1 mode 5 f= 563.84Hz Conf. 1 mode 6 f= 836.95Hz 10 10 10 0 0 0 Z −10 Z −10 Z −10 200 200 200 200 200 200 100 100 100 100 100 100 Y X Y X Y X 0 0 0 0 0 0 Conf. 1 mode 7 f= 935.98Hz

10 0 Z −10 200 200 100 100 Y X 0 0

257 258 Appendix E. Mode shapes of the PCB, Box structures

Figure E.2: Experimental mode shapes (conﬁguration #2) Conf. 2 mode 1 f= 165.30Hz Conf. 2 mode 2 f= 262.75Hz 10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 2 mode 3 f= 322.14Hz Conf. 2 mode 4 f= 479.9Hz 10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 2 mode 5 f= 671.81Hz Conf. 2 mode 6 f= 771.3Hz

10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 2 mode 7 f= 927.30Hz

10 0 Z −10 200 200 100 100 Y X 0 0 259

Figure E.3: Experimental mode shapes (conﬁguration #3) Conf. 3 mode 1 f= 197.6Hz Conf. 3 mode 2 f= 247.2Hz

10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 3 mode 3 f= 354.1Hz Conf. 3 mode 4 f= 474.3Hz

10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 3 mode 5 f= 512.4Hz Conf. 3 mode 6 f= 718.9Hz

10 10 0 0 Z −10 Z −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 3 mode 7 f= 840.1Hz Conf. 3 mode 8 f= 921.9Hz

10 10 0

0 Z Z −10 −10 200 200 200 200 100 100 100 100 Y X Y X 0 0 0 0 Conf. 3 mode 9 f= 952.8Hz

10 0 Z −10 200 200 100 100 Y X 0 0 260 Appendix E. Mode shapes of the PCB, Box structures

Figure E.4 displays the experimental mode shapes of the shell box.

Figure E.4: Test modes of the empty box List of Tables

1.1 Interests of modelling electronic devices ...... 4

5.2 Updating parameters of the cantilever beam (E: ×109 N/m2)...... 117 5.3 Cantilever beam: resonance frequencies ...... 119 5.4 Updating frequencies for the cantilever beam (Hz) ...... 126 5.5 Updating parameters of the mass-stiffness string...... 129 5.6Updatingfrequenciesforthem-kstring(Hz)...... 132 5.7 3-D bay: Updating parameters ...... 134 5.8 The 3-D bay frame: singular values and 2-norm condition number ..... 136 5.93-Dbay:Updatingfrequencies(Hz)...... 138 5.10 PCB structure: Updating parameters ...... 141 5.11 PCB structure: Resonance frequencies ...... 142 5.12 Updating frequencies for free PCB (Hz) ...... 146 5.13UpdatingfrequenciesforﬁxedPCB(Hz)...... 146

6.1 Initial model parameters ...... 159 6.2 Configuration 1: Resonance frequencies (Hz) ...... 160 6.3 Configuration 2: Resonance frequencies (Hz) ...... 160 6.4 Updated parameter values ...... 161 6.5 Updated resonance frequencies (Hz) ...... 161 6.6 Condition number of the updating problems ...... 161 6.7Updatingfrequenciesfortheperturbedconfiguration(Hz)...... 162

7.1 Components’ properties ...... 177 7.2 An overview of experimental modal parameters (0–1000Hz) ...... 180 7.3 Grouped element number of the PCB with 6 components ...... 181 7.4 Frequencies and MAC values (tests vs initial simple plate model) ..... 183 7.5Updatingresultswith6variables(Conf.#1)...... 186 7.6Updatingresultswith6variables(Conf.#1,#2and#3)...... 188 7.7Updatingresultswith20variables(Conf.#1)...... 189 7.8 Frequency differences and MAC between hinged edge model and test data 197 7.9 Frequency differences and MAC between clamped edge model and test data 198 7.10 Correlation between the test data and the initial model (k=1000Nm/rad/m) 198 7.11 Correlation between the test data and the updated model ...... 199

261 262 LIST OF TABLES

7.12 Correlation between test data and hinged edge model of the PCB ...... 200 7.13 Correlation between test data and clamped edge model of the PCB . . . . . 200 7.14 Correlation between the test data and the initial model (k=104 Nm/rad/m) . 200 7.15 Correlation between the test data and the updated model ...... 201 7.16 An overview of experimental modal parameters ...... 206 7.17 Resonance frequencies and MAC values (initial model) ...... 209 7.18 Singular values of matrix A and [A b] ...... 210 7.19 Subset selection results (relative parameter solution %) ...... 211 7.20 Relative frequency residues (%) ...... 212 7.21 MAC values (%) ...... 212 7.22 Resonance frequency differences and MAC values ...... 213

A.1 Simulated parameters ...... 234 A.2 Modal parameters of a 3-DOFs system ...... 234 A.3Updatedresultsforhighdampingsimulation...... 235 A.4Updatedresultsforlowdampingsimulation...... 235

D.1 Updating parameters ...... 251 D.2 Components’ properties ...... 254 List of Figures

o 2.1 45 fx—fa comparison...... 20 2.2 Associate hae to hx ...... 37

0 3.1 Matrix V1V1 ...... 49 3.2 SR ofFBM...... 49 3.3 SR ofQMS...... 49

4.1 k2(A)|ε vs trial rank(A, ε) (free PCB) ...... 62 4.2 Difference between LS and TLS ...... 69

5.1 FBM error localization results ...... 80 5.2 EMM error indicators ...... 83 5.3 EMM error localization results ...... 84 5.4 Weighted EMM error localization results ...... 86 5.5 A typical parameter selection index vector p ...... 103 5.6Atypicalresiduedescendingratio...... 106 5.7 Success ratios for a free PCB structure ...... 109 5.8ERRLOCPprogramscheme...... 114 5.9POMUSprogramscheme...... 115 5.10ErrorLocalizationscheme...... 116 5.11 Cantilever beam structure...... 119 o 5.12 45 line for fx—fa ...... 119 5.13 Cantilever Beam: noise-free solutions (frequency-based) ...... 120 5.14 Cantilever Beam: success ratios (frequency-based) ...... 122 5.15 Cantilever beam: noise-free solutions (QMS) ...... 123 5.16 Cantilever beam: success ratios (QMS) ...... 125 5.17 Cantilever beam: success ratios (FBM) ...... 125 5.18 A typical FRF at node 10 ...... 126 5.19 Cantilever beam: noise-free solutions (FRF-output) ...... 127 5.20 Cantilever beam: success ratios (FRF-output) ...... 128 5.21 Cantilever beam: success ratios (FRF-input) ...... 128 5.22 A 10 DOF string system...... 129 5.23 10 DOF system: noise-free solutions (QMS) ...... 130 5.24 10 DOF system: solutions for noise level 7% (QMS) ...... 130

263 264 LIST OF FIGURES

5.25 10 DOF system: success ratios (QMS) ...... 131 5.26 10 DOF system: success ratios (FBM) ...... 131 5.27 A typical FRF at node 5 ...... 132 5.28 10 DOF system: noise-free solutions (FRF-output) ...... 133 5.29 10 DOF system: success ratios (FRF-output) ...... 133 5.30 10 DOF system: success ratios (FRF-input) ...... 134 5.31 A 3-D bay structure (A =0.1 × 0.01 m2) ...... 134 5.32 3D bay: noise-free solutions (QMS) ...... 135 5.33 3D bay: success ratios (QMS) ...... 137 5.34 3D bay: noise-free min 2-norm solution (QMS,rk=39) ...... 137 5.35AtypicalFRFofthe3Dbay...... 138 5.36 3D bay: noise-free solutions (FRF-output) ...... 139 5.373Dbay:solutionsfornoiselevel2.4%(FRF-output)...... 140 5.38 3D bay: success ratios (FRF-output) ...... 140 5.39 A simulated free PCB ...... 141 5.40AsimulatedfixedPCB...... 141 5.41FEmodelmeshofasimulatedPCB...... 142 5.42 Noise free solution for the free PCB (QMS) ...... 143 5.43 Noise free solution for the fixed PCB (QMS) ...... 143 5.44 Success ratios for the free PCB (QMS) ...... 144 5.45 Success ratios for the fixed PCB (QMS) ...... 144 5.46 Success ratios for the free PCB (FBM) ...... 145 5.47 Success ratios for the fixed PCB (FBM) ...... 145 5.48 FRF 43-43 of free PCB ...... 146 5.49FRF43-43offixedPCB...... 146 5.50 Noise-free simulation for free PCB (FRF-output) ...... 147 5.51 Noise-free simulation for fixed PCB (FRF-output) ...... 147 5.52 Success ratios for free PCB (FRF-output) ...... 148 5.53 Success ratios for fixed PCB (FRF-output) ...... 148 5.54 Success ratios for free PCB (FRF-output, 26 f’s)...... 148 5.55 Success ratios for free PCB (FRF-input) ...... 149 5.56 Success ratios for fixed PCB (FRF-input) ...... 149

6.1 Perturbed Boundary Condition Finite Element Model Updating Scheme . . 158 6.2 10 DOF discrete system ...... 159 6.3 Perturbed free PCB (∆m =0.01kg) ...... 162 6.4FRF43-43ofthePCB...... 162 6.5 Perturbed free PCB ...... 163

7.1FreePCB...... 166 7.2Wedge-LockPCB...... 166 7.3Shellbox...... 166 7.4 PTH component ...... 167 LIST OF FIGURES 265

7.5 SMT component ...... 167 7.6Abendingtest...... 172 7.7ATwisttest...... 172 7.8Equivalentthicknessmodel...... 173 7.9ExperimentalPCB...... 177 7.10 Test conﬁguration of the PCB under free boundary condition ...... 178 7.11Measurementpoints...... 178 7.12 Coherence and mobility at node 24 (conﬁguration 1) ...... 179 7.13 Element number of the PCB with 6 components ...... 181 7.14 Resonance frequency residues (using Conf. #1) ...... 186 7.15MACvalues(usingConf.#1)...... 187 7.16Equivalentthicknesses(usingConf.#1)...... 187 7.17 Updated mobility at node 1 ...... 188 7.18 Resonance frequency residues (20v; using Conf. #1) ...... 189 7.19 MAC values (20v; using Conf. #1) ...... 190 7.20 Equivalent thicknesses (20v; using Conf. #1) ...... 190 7.21 Svd(A) and k2A (X)...... 191 7.22 Svd(A) and k2A (X)...... 191 7.23 Parameter selection ...... 192 7.24 Parameter selection ...... 192 7.25 Gray image of main errors ...... 192 7.26 Gray image of main errors ...... 192 7.27 Svd(A) and k2A (H)...... 193 7.28 Svd(A) and k2A (H)...... 193 7.29 Parameter selection ...... 193 7.30 Parameter selection ...... 193 7.31 Gray image of main errors ...... 194 7.32 Gray image of main errors ...... 194 7.33AtypicalWedge-Lockslide...... 195 7.34Springmodel...... 195 7.35Virginboard’smeshandmeasurementpoints...... 197 7.36 Iterative residues (Virgin) ...... 199 7.37 Iterative Stiffness (Virgin) ...... 199 7.38 Iterative residues (PCB) ...... 201 7.39 Iterative Stiffness (PCB) ...... 201 7.40 Back and Upper faces ...... 204 7.41 Upper and front faces ...... 204 7.42Upperface...... 204 7.43 Left and right faces ...... 204 7.44 Measurement system for the shell box structure ...... 205 7.45FiniteelementModelmeshfortheshellbox ...... 207 7.46Plateconnection...... 208 7.47 Norm of column of A ...... 211 266 LIST OF FIGURES

7.48 Singular values of A and [A b] ...... 211 7.49 Iterative residues ...... 213 7.50 Iterative parameters ...... 213

A.1 A 3-DOF damping system ...... 234 A.2FRFswithHighdamping...... 234 A.3FRFswithLowdamping...... 234

B.1POMUSprogramscheme...... 238 B.2ErrorLocalizationscheme...... 239 B.3Equationsolvingscheme...... 240 B.4POMUSmainmenus...... 245 B.5Filemenus...... 245 B.6 Setting menus ...... 245 B.7Modalmenus...... 245 B.8Impedmenus...... 246 B.9Undampingmenus...... 246 B.10 Proportional menus ...... 246 B.11Solumenus...... 246 B.12Figuremenus...... 246 B.13Runmenus...... 246

D.1 The cantilever beam structure ...... 250 D.2 The 10-DOF stiffness-mass system ...... 250 D.3 A 3-D bay structure (A =0.1 × 0.01 m2) ...... 251 D.4FreePCB...... 252 D.5FixedPCB...... 252 D.6FEmodelmeshofasimulatedPCB...... 253 D.7 10 DOF discrete system ...... 254 D.8ExperimentalPCB...... 254 D.9FreePCB’stestpoints...... 255 D.10 Element number of the free PCB ...... 255 D.11AtypicalWedge-Lockslide...... 255 D.12Springmodel...... 255 D.13Wedge-Lockslides...... 256 D.14Measurementpoints...... 256

E.1Experimentalmodeshapes(configuration#1)...... 257 E.2Experimentalmodeshapes(configuration#2)...... 258 E.3Experimentalmodeshapes(configuration#3)...... 259 E.4Testmodesoftheemptybox...... 260