Dynamics Substructures, Volume 4

Preface	6
Contents	7
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS)	9
1.1 Introduction	9
1.2 Feedforward Control Schemes	10
1.2.1 Model-Based Dynamic Feedforward	11
1.2.2 Model-Free Inversion-Based Iterative Feedforward Control	12
1.2.3 Velocity Feedforward	13
1.3 Experimental Setup	13
1.3.1 Stewart Platform	13
1.3.2 System Identification	13
1.3.3 Benchmark Problem	15
1.3.4 Parameters Setting for the Experiments	15
1.4 Results and Discussion	16
1.4.1 Convergence of MFIIC	16
1.4.2 Comparison of the Feedforward Control Schemes	17
1.4.3 Coupling Between Directions	19
1.4.4 Discussion	19
1.5 Conclusion	21
References	21
2 Proposed 12-DOF Shaker Control of BARC Structure	23
2.1 Introduction	23
2.2 BARC Impact Tests	24
2.3 Base Input Force Definition	24
2.4 Component Rigid Body Base Input Definition	25
2.5 12-DOF Control Strategy	27
2.6 Results	28
2.7 Fixed Base Component Modes	29
2.8 Summary	30
References	33
3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes	34
3.1 Motivation	34
3.2 Modal Theory for Base Mounted Component on Fixture	35
3.3 MATV Hardware and Instrumentation	37
3.4 MATV System Level Test	38
3.5 Free Modal Test of Component and Fixture	39
3.6 Extracting the Nominal Fixed Base Modal Cross Spectra from System Level Test	40
3.7 Calculated 6 DOF Base Input Specs to Ensure Conservatism on Fixed Base Modal DOF Based on Variability	40
3.8 Typical 1 DOF SPEC Response	46
3.9 Discussion of 6 DOF and 1 DOF Test Specifications	46
3.10 Conclusion	47
References	48
4 Implementing Experimental Substructuring in Abaqus	49
4.1 Introduction	49
4.2 Background and Theory	50
4.3 Implementation	53
4.3.1 Gather Subsystem Data and Import into MATLAB	53
4.3.2 Identify Constraint DOF	53
4.3.3 Decouple the TS from the Experimental Subsystem	54
4.3.4 Form Constraint Equations for Use in Abaqus	54
4.3.5 Write Auxiliary Abaqus Input File	54
4.4 Numerical Case Study	54
4.5 Experimental Test Case	58
4.6 Conclusions and Future Work	63
Appendix A: MATLAB Function to Generate Auxillary Abaqus Input File*-10pt	64
Truncated Auxillary Abaqus Input File for Beam Case Study	66
References	66
5 Vibration Test Design with Integrated Shaker Electro-Mechanical Models	68
5.1 Introduction	68
5.2 Theory	69
5.2.1 Frequency Based Substructuring	69
5.3 Shaker Electro-Mechanical Model	70
5.4 Example of Substructuring a Shaker to a Dynamic System	72
5.5 Example of Substructuring a Shaker Model to a Measured System	73
5.6 Using the Shaker Electro-Mechanical Model to Choose Shaker Locations	74
5.7 Conclusions	77
References	77
6 Reproducing a Component Field Environment on a Six Degree-of-Freedom Shaker	78
6.1 Motivation	78
6.2 Introduction and Background	79
6.3 Experimental Results and Discussion	79
6.4 Conclusion	82
References	83
7 In-Situ Source Characterization for NVH Analysis of the Engine-Transmission Unit	84
7.1 Introduction	84
7.2 Theory	85
7.2.1 In-Situ Blocked Force TPA	85
7.2.2 Virtual Point Transformation	86
7.2.3 Procedure of the iTPA	87
Blocked Force Calculation	87
On-Board Validation	87
Cross Validation	87
7.3 Vibration Prediction from Vehicle Measurements	88
7.3.1 Force Identification	88
Discrete Speed	89
Run-up	89
7.3.2 On-Board Validation	91
Discrete Speed	91
Run-up	91
7.3.3 Discussion	92
7.4 Conclusions	96
References	96
8 Using Modal Projection Error to Predict Success of a Six Degree of Freedom Shaker Test	97
8.1 Introduction	97
8.2 Modal Projection Error Theory	98
8.3 System Configurations	98
8.3.1 BARC	98
8.3.2 Removable Component on a Rigid Fixture	98
8.3.3 Removable Component on an Aerospace Structure	99
8.4 Environment Field and Laboratory Tests	99
8.5 Results	100
8.5.1 Aerospace Structure with RC Base DOFs	100
8.5.2 Aerospace Structure with Full Field RC DOFs	102
8.5.3 BARC	103
8.6 Conclusion and Future Work	107
References	108
9 On Dynamic Substructuring of Systems with Localised Nonlinearities	109
9.1 Introduction	109
9.2 Theory	110
9.2.1 Craig-Bampton Reduction	111
9.2.2 Integration and Coupling	111
9.2.3 With Sub-cycling	113
9.3 Case Study	113
9.4 Virtual Hybrid Simulation	114
9.4.1 Reduction of the Linear Frame	115
9.4.2 Comparison of Monolithic and Partitioned Solutions	117
9.4.3 Subcycling	118
9.5 Conclusions	120
References	120
10 Source Characterization for Automotive Applications Using Innovative Techniques	121
Nomenclature	121
10.1 Background	122
10.1.1 Component-Based TPA	122
10.1.2 Virtual Point Transformation	123
10.1.3 Techniques Presented Here	123
10.2 Analysis	124
10.2.1 Rigidness Correction for Low Frequency TPA	124
10.2.2 Reciprocal FRFs for Mid-Frequency TPA Predictions	125
10.2.3 Rotational FRFs for Mid- to High-Frequency TPA	127
10.3 Conclusion	129
References	129
11 Impact of Junction Properties on the Modal Behavior of Assembled Structures	130
11.1 Introduction	130
11.2 Modelling	131
11.3 Conclusion	133
References	133
12 Quantifying Joint Uncertainties for Hybrid System Vibration Testing	134
12.1 Introduction	134
12.2 Experimental Procedure	135
12.2.1 Test Component	135
12.2.2 Experimental Setup	135
12.2.3 Test Procedure	136
12.3 Numerical Model	137
12.4 Analysis	137
12.4.1 Joint Stiffness Calibration	137
12.5 Results and Discussion	138
12.5.1 Experimentally Determined Natural Frequencies	138
12.5.2 Stiffness Uncertainty Quantification	139
12.6 Conclusions and Future Work	139
References	141
13 Damping Identification and Model Updating of Boundary Conditions for a Cantilever Beam	142
13.1 Introduction	142
13.2 Theory	143
13.2.1 Model Reduction and Modal Expansion	143
13.2.2 SEREP Modal Expansion/Model Reduction	144
13.2.3 Inverse Eigensensitivity Approach	144
13.2.4 Non-proportional Damping	145
13.2.5 Direct Damping Updating	145
13.3 Simulated Beam	145
13.3.1 Model Setup	145
13.3.2 Results	146
13.4 Experimental Beam	147
13.4.1 Model Setup	148
13.4.2 Results	148
13.5 Discussion	150
13.6 Conclusion	150
References	150
14 An Experimental Substructure Test Object: Components Cut Out From a Steel Structure	152
14.1 Introduction	152
14.2 The Test Object	153
14.3 Finite Element Analyzes of the One Piece Structure	155
14.4 Finite Element Analyzes of the Two Components	156
14.5 Future Work	157
14.6 Conclusion	159
References	159
15 Frequency Based Model Mixing for Machine Condition Monitoring	160
15.1 Introduction	160
15.2 Numerical Model	161
15.3 Modal Expansion	162
15.4 Conclusion	163
References	164
16 Using a Machine Learning Approach for Computational Substructure in Real-Time Hybrid Simulation	165
16.1 Introduction	165
16.2 System Components and Capabilities	166
16.3 Modeling Assumptions	166
16.4 Model Parameters for HS	167
16.5 Validation for RTHS with FE Model	168
16.6 Methodology for Linear Regression Algorithm	169
16.7 Methodology for Recurrent Neural Network Algorithm	170
16.8 Summary and Conclusions	173
References	174
17 On the Stability of a Discrete Convolution with Measured Impulse Response Functions of Mechanical Components in Numerical Time Integration	175
17.1 Introduction	175
17.2 Error of the Discrete Convolution	176
17.2.1 Discrete Fourier Transformation	176
17.2.2 Error Due to the Approximation of an Integral with the Trapezoidal Rule	177
17.2.3 Discrete Convolution: Error Due to Trapezoidal Rule	177
17.3 Possibilities for Stabilization	179
17.3.1 Modal Fit	179
17.3.2 Filtering in the Frequency Domain	179
17.3.3 Decreasing High Frequency Content by the Use of Artificial Mass	179
17.3.4 Systematic Stabilization Approach	181
17.4 Examples	182
17.4.1 Two-Degree-of-Freedom Oscillator	182
17.4.2 Unbalance Rotor Mounted on a Beam	183
Measurement of cdof Driving Point IRF	184
IRF Treatment for the Sake of Stabilization	185
Reference Measurement of the Complete System Beam + Unbalance Rotor	187
Simulation	187
Comparison of Measurement and Simulation	188
17.5 Summary and Conclusion	188
References	189
18 Development of an Electrodynamic Actuator for an Automatic Modal Impulse Hammer	191
18.1 Introduction	191
18.1.1 What Is an Ideal Impact?	192
18.1.2 A Comment About Sampling Rate During Impact Testing	193
18.1.3 Development Potential of the AMimpact	193
18.2 Multibody Simulation	194
18.3 Actuator Design	195
18.3.1 Physical Principle	195
18.3.2 Finite Element Simulation of the Actuator	195
18.3.3 Electrical Circuit	196
18.3.4 Mechanical Design	197
18.4 Control Strategy	198
18.4.1 Position Sensing	198
18.4.2 Control Sequence	198
18.5 Verification Measurements	199
18.5.1 High-Speed Camera	199
18.5.2 Force	199
18.6 Conclusion	200
References	201

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