Topics in Modal Analysis, Volume 10

Preface	6
Contents	8
1 An Efficient Treatment of Parameter Identification in the Context of Multibody System Dynamics Using the Adjoint Method	10
1.1 Introduction	10
1.2 The Adjoint Method in Multibody Dynamics	11
1.2.1 Adjoint Equations	11
1.2.2 Flowchart of the Adjoint Method Embedded in Multibody Systems	13
1.3 Numerical Examples	14
1.3.1 Planar Overhead Crane	14
1.3.2 Single Rigid Body Parametrized with Euler Parameters	14
1.4 Conclusions	17
References	17
2 Calibration and Validation of a Car Subframe Finite Element Model Using Frequency Responses	18
2.1 Introduction	18
2.2 Theory	20
2.2.1 A Frequency Response Calibration Metric	20
2.2.2 Damping Equalization	21
2.2.3 Surrogate Modeling	22
2.2.4 Validation and Cross-Validation	23
2.3 Calibration and Validation of the Subframe Model	24
2.3.1 Finite Element Model and Its Parameterization	24
2.3.2 Test Procedure and System Identification	26
2.3.3 Calibration and Validation Results	26
2.3.4 Computational Efficiency	27
2.4 Concluding Remarks	28
References	31
3 Subspace Identification of a 5 MW Reference Wind Turbine	32
3.1 Introduction	32
3.2 Wind Turbine Identification	33
3.2.1 Turbine Model	33
3.2.2 Subspace Identification	34
3.3 Controller Design	35
3.4 Results	35
3.5 Conclusions	37
References	38
4 Dynamic Analysis of Complex Mechanical Structures Using a Combination of Numerical and Experimental Techniques	39
4.1 Introduction	39
4.2 Class of Mechanical Systems Examined: Equations of Motion	40
4.3 Application to Laboratory Vehicle Model	41
4.3.1 Numerical Application to a Small Scale Vehicle-Like Frame Structure	41
4.3.2 Experimental Application to a Small Scale Vehicle-Like Frame Structure	44
4.4 Experimental Application to a Real Vehicle	45
4.5 Conclusions	47
References	48
5 Predicting Approximate Governing Formulafrom Experimental Observations	49
5.1 Introduction	49
5.1.1 Problem	49
5.1.1.1 Importance	49
5.1.1.2 Presence of Experimental Errors	49
5.2 Methodology	49
5.2.1 Primary Parameter	50
5.2.1.1 Calculating Error Between the Data and Random Relation	50
5.2.1.2 Choosing the Initial Temperature and Temperature Reduction Function	50
5.2.1.3 Developing the Function to be Optimized	50
5.2.2 Increasing Accuracy	50
5.2.3 Testing of Method	51
5.3 Results	51
References	52
6 Experimental Characterization and Simulation of Vibration Environmental Test	53
6.1 Introduction	53
6.2 Shaker Characterization of Closed Loop Vibration Test	54
6.2.1 Driver Voltage of Empty Shaker	54
6.2.2 Operational Deflection Shape of Empty Shakers	55
6.2.3 Operational Deflection Shape of Adapter on Top of the Shaker	56
6.3 Numerical Simulations	59
6.4 Conclusion	61
References	63
7 System Identification of an MDOF Experimental Structure with a View Towards Validation and Verification	64
7.1 Introduction	64
7.2 The Experimental Rig and Data Capture	65
7.3 System Identification Using Self-Adaptive Differential Evolution (SADE)	66
7.4 Bayesian Sensitivity Analysis	70
7.4.1 Gaussian Process Regression	70
7.4.2 Inference for Sensitivity Analysis	71
7.4.3 Results	71
7.5 Discussion and Conclusions	72
References	72
8 Some Non-conventional Boundary Conditions (From Marshmallows to Plungers: Who Would Have Guessed)	73
8.1 Introduction	73
8.2 Frame Structure with Marshmallow Supports	74
8.3 Upright Cantilever Calibration Structure on Pads	78
8.4 Shock Response Plate Fixture with Plunger Supports	80
8.5 Conclusion	84
References	84
9 Robustness of Disc Brake Systems Regarding Squeal	85
9.1 Motivation	85
9.2 Experimental Setup	86
9.2.1 Brake System	86
9.2.1.1 Excitation and Measurement Concept	86
9.3 Special Requirements for the Modal Analysis of Brake Systems	88
9.4 Results	90
9.5 Conclusion and Outlook	92
References	92
10 Measurement of Vibration Resulting from Non-contact Ultrasound Radiation Force	93
10.1 Introduction	93
10.2 Theory	93
10.3 Experimental Setup and Procedure	95
10.4 Results	95
10.5 Conclusions	98
References	98
11 The Use of Fiber Bragg Grating Sensors for Strain Modal Analysis	99
11.1 Introduction	99
11.2 Strain Modal Analysis Theory	100
11.3 FBG Measurement Principle	101
11.4 Experimental Analysis	102
11.4.1 Unsynchronized FBG Measurements	102
11.4.2 Synchronized Measurements	105
11.5 Results Analysis and Conclusion	105
References	107
12 Using Mode Shapes for Real Time ODS Animation	108
12.1 Introduction	108
12.1.1 Operating Deflection Shape (ODS)	108
12.1.2 Ways to Acquire ODS Data	109
12.2 Order-Tracked ODS	109
12.3 Variable Speed Rotating Machine	109
12.4 Multiple Measurement Sets	110
12.5 Mode Shapes of the Machine	111
12.6 Modal Participation Matrix	112
12.7 ODS Expansion	115
12.8 Modal Participation at Two Speeds	115
12.9 ODS Expansion	115
12.10 Conclusions	116
References	118
13 Removing Unwanted Noise from Operational Modal Analysis Data	119
13.1 Introduction	119
13.2 Approach	120
13.2.1 Noise Removal Scheme	120
13.2.2 Measured Data	121
13.3 Results	122
13.4 Conclusions	126
References	126
14 Adaptive-Like Vibration Control in a Three-Story Building-Like Structure with a PZT Stack Actuator	127
14.1 Introduction	127
14.2 Three Story Building-Like Structure	128
14.3 On-Line Algebraic Identification of the Harmonic Excitation	130
14.4 An Adaptive-Like Positive Position Feedback Control Scheme	131
14.5 A Multi Positive Position Feedback Control Approach	133
14.6 Conclusions	134
References	135
15 A Fast Maximum Likelihood-Based Estimation of a Modal Model	136
15.1 Introduction	136
15.2 Maximum Likelihood Estimation Based on the Modal Model (ML-MM)	137
15.2.1 The Basic Implementation of the ML-MM Estimator	137
15.2.2 A Fast Implementation of the ML-MM Estimator	140
15.3 Validations and Discussion	142
15.3.1 Seven DOF System with Mixed Noise [White and Relative Noise]	142
15.3.2 Flight Flutter Testing	146
15.3.3 Fully Trimmed Car	150
15.3.4 Acoustic Modal Analysis of a Car Cavity	151
15.4 Conclusions	155
References	158
16 An Improved Implementation of the Orthogonal Polynomial Modal Parameter Estimation Algorithm Using the Orthogonal Complement	160
16.1 The Notoriously Ill-Conditioned Rational Fraction Polynomial Algorithm	160
16.2 A Change of Basis to Orthogonal Polynomials	161
16.3 Some Basics on Orthogonal Polynomials	161
16.4 Frequency Mapping for the Orthogonal Polynomial Interval	162
16.5 Constructing the Orthogonal Polynomials	162
16.6 The Orthogonal Polynomial Companion Matrix	164
16.7 Expressing the Rational Fraction Polynomial FRF Model in Orthogonal Polynomials	166
16.8 Removing the Betas from the Problem	167
16.9 The Least-Squares Problem for the Alphas	167
16.10 Assembling the Correlation Matrix for the Least-Squares Problem	168
16.11 Accumulating the Correlation Matrix with QR Decomposition	169
16.12 How Many Orthogonal Polynomials Should Be Used?	169
16.13 The Orthogonal Complement: The New Part of the Orthogonal Polynomial Algorithm	170
16.14 Some Correlation Matrix Accumulation Time Trials	171
16.15 Concluding Remarks	172
References	173
17 An Orthogonal View of the Polyreference Least-Squares Complex Frequency Modal Parameter Estimation Algorithm	174
17.1 Introduction	174
17.2 One View of PLSCF	174
17.3 An Orthogonal View of PLSCF	175
17.4 Frequency Mapping	176
17.5 Companion Matrices	176
17.6 Characteristic Equation Normalization	177
17.7 Uneven Frequency Spacing	179
17.8 The “Reorthogonalized” PLSCF	180
17.9 A Unified View	183
17.10 Conclusion	184
References	184
18 Operational Modal Parameter Estimation from Short-Time Data Series	186
18.1 Introduction	186
18.2 Background	187
18.3 NARX Model Based OMA-MPE	188
18.3.1 Implementation of NARX Model	188
18.3.2 Selection of Model Terms	190
18.3.3 Computation of Modal Parameters	191
18.4 Test Cases	192
18.4.1 Test Case I	192
18.4.2 Test Case II	193
18.4.3 Test Case III	195
18.5 Conclusions	197
References	200
19 Order Based Modal Analysis Versus Standard Techniques to Extract Modal Parameters of Operational Wind Turbine Gearboxes	202
19.1 Introduction	202
19.2 Theoretical Background	203
19.2.1 FRF Estimation from Sine Sweep Experiments	203
19.2.2 Operational Modal Analysis	204
19.2.3 Order Tracking	204
19.2.4 Order-Based Modal Analysis	205
19.3 Numerical Validation	206
19.4 Operational Gearbox Analysis	208
19.4.1 Gearbox Signature Analysis	208
19.4.2 Shaker Data Analysis	209
19.4.3 Operational Data Analysis	212
19.5 Conclusions	214
References	216
20 Modal Identification Results of Quasi-statically Tested RC Frames at Different Damage Levels	217
20.1 Introduction	217
20.2 Test Setup and Program	218
20.2.1 Specimens and Test Program	218
20.2.2 Instrumentation	219
20.3 Damages Observed During Quasi-static Tests	220
20.4 Offline Tuning Technique to Increase Signal Fidelity	220
20.5 System Identification Methods Used for Modal Parameter Estimation	222
20.5.1 SSI-Data	223
20.5.2 NExT-ERA	223
20.5.3 EFDD	223
20.6 Modal Parameter Estimation Results	224
20.6.1 Hysteretic Damping	227
20.7 Conclusions	227
References	228
21 An Innovative Tool for Simulation and Control of a Flying-Cutting Machine	229
21.1 Introduction	229
21.2 Flying-Cutting Machines	230
21.3 The Simulation Tool	231
21.3.1 The Master	231
21.3.2 The Unwinder	231
21.3.3 The Buffer	232
21.3.4 Cutting Tools	233
21.3.5 Sliders	234
21.4 Test Case	234
21.5 Concluding Remarks	236
References	237
22 Modal Analysis and Testing of Honeycomb Sandwich Composites	238
22.1 Introduction	238
22.2 Experimental Procedure	239
22.2.1 Specimen Details	239
22.2.2 Modal Test Method	239
22.3 Results and Discussions	240
22.3.1 Experimental Results	240
22.3.1.1 Mode Shape	240
22.3.2 Frequency and Damping	241
22.3.3 Graphs	241
22.3.3.1 Frequency Variation with Varying Cell Size	241
22.3.4 Discussions	241
Conclusions	242
References	242
23 Towards an Automatic Modal Parameter Estimation Framework: Mode Clustering	243
23.1 Introduction	243
23.2 Theory	244
23.2.1 Background	244
23.2.2 Modal Observability Correlation	245
23.3 Problem Statement	247
23.4 Autonomous Modal Parameter Estimation	248
23.4.1 Library of Vibrational Modes	248
23.4.1.1 Dealing with Inconsistent Data	249
23.4.2 Repeated Random Sampling of Data	250
23.4.3 Clustering Based on Correlation Analysis	250
23.4.3.1 Dealing with Coalescent Eigenvalues	251
23.5 Numerical Evidence	252
23.5.1 Experiment with No Frequency Shift and No Measurement Noise	252
23.5.2 Experiment with Noisy Measurements	252
23.5.3 Experiment with Frequency Shift and No Measurement Noise	256
23.6 Concluding Remarks	258
References	259

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