Brake The Mechanics of Jointed Structures

Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics
1. Auflage 2017
ISBN: 978-3-319-56818-8
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 690 Seiten

This book introduces the challenges inherent in jointed structures and guides researchers to the still-open, pressing challenges that need to be solved to advance this critical field. The authors cover multiple facets of interfacial mechanics that pertain to jointed structures: tribological modeling and measurements of the interface surfaces, constitutive modeling of joints, numerical reduction techniques for structures with joints, and uncertainty quantification and propagation for these structures. Thus, the key subspecialties addressed are model reduction for nonlinear systems, uncertainty quantification, constitutive modeling of joints, and measurements of interfacial mechanics properties (including tribology). The diverse contributions to this volume fill a much needed void in the literature and present to a new generation of joints researchers the potential challenges that they can engage in in order to advance the state of the art. Clearly defines internationally recognized challenges in joint mechanics/jointed structures and provides a comprehensive assessment of the state-of-the-art for joint modeling; Identifies open research questions facing joint mechanics; Details methodologies for accounting for uncertainties (due both to missing physics and variability) in joints; Explains and illustrates best-practices for measuring joints' properties experimentally; Maximizes reader understanding of modeling joint dynamics with a comparison of multiple approaches.

Autoren/Hrsg.

Brake, Matthew R. W.

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;Contributors;11
4;Part I Perspectives on the Challenges of Joints Research;14
4.1;1 Introduction to Research on the Mechanics of Jointed Structures;15
4.1.1;1.1 Description of a Common Joint;16
4.1.2;1.2 Historical Development;18
4.1.3;1.3 Perspective on the Challenges;20
4.1.4;References;22
4.2;2 An Overview of the Analysis of Jointed Structures;23
4.2.1;2.1 Emergent Behavior Due to Interfaces;24
4.2.2;2.2 A Brief Introduction to the Modeling of Mechanical Joints;24
4.2.2.1;2.2.1 Analysis Levels;25
4.2.2.2;2.2.2 Strong Nonlinearities;26
4.2.2.3;2.2.3 Approaches: Both Commercial and Research;27
4.2.2.3.1;2.2.3.1 Commercial Codes;28
4.2.3;2.3 Joint Sensitivities: Reducible Uncertainty in Interfacial Mechanics;28
4.2.3.1;2.3.1 Contact Pressure;29
4.2.3.2;2.3.2 Residual Stress;29
4.2.3.3;2.3.3 Accounting for the Unknown in Modeling;31
4.2.3.3.1;2.3.3.1 Complexity Theory;31
4.2.3.3.2;2.3.3.2 Uncertainty Modeling;32
4.2.4;2.4 Perspective for a Way Forward;33
4.2.4.1;2.4.1 Designing a Better Joint;33
4.2.4.2;2.4.2 New Definitions for Better Context;33
4.2.4.3;2.4.3 Advancements in Physics for Predictive Capabilities;34
4.2.4.4;2.4.4 Advancements in Experimental Techniques;34
4.2.4.5;2.4.5 Advancements in Numerical and Stochastic Techniques;34
4.2.5;References;35
4.3;3 Are Joints Necessary?;36
4.3.1;3.1 Application Areas;37
4.3.1.1;3.1.1 Joints Within Defense Systems;37
4.3.1.2;3.1.2 Joints Within Aeroturbines;39
4.3.2;3.2 Are Joints Necessary?;40
4.3.3;3.3 Economic Considerations for Joints;41
4.3.3.1;3.3.1 The Cost of Failure;41
4.3.3.1.1;3.3.1.1 An Example of an Aerospace Failure;41
4.3.3.1.2;3.3.1.2 Examples of Civil Structure Failures;42
4.3.3.2;3.3.2 The Benefit of Saving Weight;43
4.3.3.2.1;3.3.2.1 Aircraft;44
4.3.3.2.2;3.3.2.2 Automobiles;44
4.3.3.2.3;3.3.2.3 Rocket Payload Performance;45
4.3.3.3;3.3.3 The Cost of Dynamic Testing;45
4.3.3.4;3.3.4 The Benefit of Designing Structures with Joints;45
4.3.3.5;3.3.5 The Benefit of Using Joints to Monitor Structures;46
4.3.4;3.4 Perspectives for the Economics of Jointed Structures;46
4.3.5;References;47
4.4;4 Considerations for Defining the Mechanisms of Friction;48
4.4.1;4.1 Defining the Mechanisms of Friction;49
4.4.2;4.2 Length Scale Considerations;50
4.4.3;4.3 Constitutive Model Development;51
4.4.4;4.4 Outlook for Developing a Predictive Model of Friction;52
4.4.5;References;53
5;Part II Experimental Techniques for Studying Energy Dissipation Mechanisms;54
5.1;5 Round Robin Systems;55
5.1.1;5.1 The Brake–Reuß Beam;56
5.1.2;5.2 The Square, Four-Bolt Plate;59
5.1.3;5.3 The Gaul Resonator and Dumbbell Apparatus;61
5.1.4;5.4 The Cut Beam Frictional Benchmark System;62
5.1.5;5.5 The Ampair 600 Wind Turbine;63
5.1.6;5.6 The Sumali Beam;64
5.1.7;5.7 Other Benchmark Systems;64
5.1.8;5.8 Outlook for the Adoption of Benchmark Systems;66
5.1.9;5.9 Overview of Part II;66
5.1.10;References;67
5.2;6 The Gaul Resonator: Experiments for the Isolated Investigation of a Bolted Lap Joint;69
5.2.1;6.1 Design of the Gaul Resonator;69
5.2.2;6.2 Sine Sweep Measurements;71
5.2.3;6.3 Stepped Sine Measurements;74
5.2.4;6.4 Long Duration Measurements;79
5.2.5;6.5 Perspectives for the Gaul Resonator and Potential Modeling Paths;81
5.2.6;References;82
5.3;7 The Cut Beam Benchmark System: Developing Measurement Techniques for Nonlinear Damping and Stiffness in Frictional Interfaces;83
5.3.1;7.1 Dissipation in Assembled Structures;83
5.3.1.1;7.1.1 Catalog of Previous Benchmark Systems;84
5.3.1.2;7.1.2 Experimental Methods;86
5.3.1.3;7.1.3 Purpose;87
5.3.2;7.2 Design of a Frictional Interface Benchmark System;87
5.3.2.1;7.2.1 Design of the Experimental Device;87
5.3.2.2;7.2.2 Instrumentation of the Experimental Benchmark;91
5.3.3;7.3 Measurement Techniques for Jointed Systems and Experimental Results;91
5.3.3.1;7.3.1 Excitation and Measurement Techniques;91
5.3.3.2;7.3.2 Reference Measurements;93
5.3.3.3;7.3.3 Shock Analysis;93
5.3.3.4;7.3.4 Stopped-Sine Analysis;95
5.3.4;7.4 Perspectives on the Cut Beam Benchmark System;97
5.3.5;References;98
5.4;8 The Ampair 600 Wind Turbine: An In-ContextBenchmark System;100
5.4.1;8.1 Overview of the Ampair 600 Wind Turbine;100
5.4.2;8.2 Available Research on the Ampair 600 Wind Turbine;102
5.4.2.1;8.2.1 Dynamic Substructuring;102
5.4.2.2;8.2.2 Nonlinear Identification;104
5.4.3;8.3 Perspectives on Using the Ampair 600 Wind Turbine as a Benchmark System;105
5.4.4;References;106
5.5;9 The Brake-Reuß Beams: A System Designed for the Measurements and Modeling of Variability and Repeatability of Jointed Structures with Frictional Interfaces;107
5.5.1;9.1 Design of the Brake-Reuß Beam;108
5.5.2;9.2 Experimental Evidence of Variability for the Brake-Reuß Beam;109
5.5.3;9.3 Observed Trends in Variability for the Brake-Reuß Beam;114
5.5.4;References;115
5.6;10 Considerations for Measurements of Jointed Structures;116
5.6.1;10.1 Approach for Measuring Jointed Systems;117
5.6.2;10.2 Effects of Experimental Setup;119
5.6.3;10.3 Effects of Excitation and Measurement Conditions;123
5.6.4;10.4 Effects of the Jointed Beam's Interface Conditions;129
5.6.5;10.5 Method and Results for Studying Repeatability and Variability;134
5.6.6;10.6 Best Practices for Experiments on Jointed Systems;137
5.6.7;References;140
5.7;11 Damping Due to Joints in Built-Up Structures;141
5.7.1;11.1 Sources of Damping;142
5.7.2;11.2 Damping Due to Joints;142
5.7.3;11.3 Current Approaches to Modeling Damping;145
5.7.4;11.4 Experimental Investigations of Joint Damping Sources;145
5.7.4.1;11.4.1 Apparatus 1: Two Masses and a Spring;146
5.7.4.2;11.4.2 Apparatus 2: A Chain of Bolted Joints;148
5.7.4.3;11.4.3 Apparatus 3: A Composite Beam;150
5.7.5;11.5 Discussion of Damping Experiments;152
5.7.6;References;153
5.8;12 A Survey of Contact Hysteresis Measurement Techniques;154
5.8.1;12.1 Original Motivation and Demands for Interface Hysteresis Data;155
5.8.2;12.2 Summary of Reference Set of Test Rigs;156
5.8.2.1;12.2.1 Overview;156
5.8.2.2;12.2.2 Imperial College London Rigs;159
5.8.2.2.1;12.2.2.1 Imperial College London: First-Generation Rig, Commissioned in 1998;159
5.8.2.2.2;12.2.2.2 Imperial College London: Second-Generation Rig, Commissioned in 2012;160
5.8.2.3;12.2.3 University of Oxford Rigs;160
5.8.2.4;12.2.4 Politecnico di Torino Rigs;162
5.8.2.4.1;12.2.4.1 First-Generation Torino Rig;162
5.8.2.4.2;12.2.4.2 Second-Generation Torino Rig;163
5.8.2.5;12.2.5 University of Cambridge Rig;163
5.8.2.6;12.2.6 Comments on the Rigs;164
5.8.3;12.3 Two Generations of Dynamic Friction Test Rigs at Imperial College London;166
5.8.3.1;12.3.1 Background;166
5.8.3.2;12.3.2 First-Generation Rig;167
5.8.3.3;12.3.3 Second-Generation Rig;168
5.8.3.4;12.3.4 Results;170
5.8.4;12.4 Friction Testing at the University of Oxford;171
5.8.4.1;12.4.1 Background;171
5.8.4.2;12.4.2 Results;172
5.8.4.3;12.4.3 Torsional Rig;174
5.8.4.4;12.4.4 Conclusions;175
5.8.5;12.5 Measurement of Friction Contact Properties at Politecnico di Torino;175
5.8.5.1;12.5.1 Background;176
5.8.5.2;12.5.2 Friction and Wear Test Rig for Sphere-on-Flat Contacts;176
5.8.5.3;12.5.3 Friction and Wear Test Rig for Double Flat-on-Flat Contacts;177
5.8.6;12.6 New Measurements on Dry Friction at the University of Cambridge;180
5.8.6.1;12.6.1 Background;180
5.8.6.2;12.6.2 Speed-Jump Test;181
5.8.6.3;12.6.3 Reciprocating Sliding Test;182
5.8.6.4;12.6.4 Stick-Slip Self-oscillation Tests;182
5.8.7;References;183
5.9;13 Under-Platform Damper Measurements at Politecnico di Torino;185
5.9.1;13.1 Overview of Turbine Blade Assemblies and Under-Platform Dampers;185
5.9.2;13.2 Under-Platform Damper Test Rig Description;186
5.9.3;13.3 Development of Under-Platform Force and Kinematic Relationships;188
5.9.3.1;13.3.1 Measured Force Components;188
5.9.3.2;13.3.2 Derived Force Components;189
5.9.3.3;13.3.3 Measured Kinematic Quantities;189
5.9.3.4;13.3.4 Derived Kinematic Quantities;190
5.9.4;13.4 Under-Platform Damper Measurement Uncertainties;190
5.9.4.1;13.4.1 The Improved Method to Reduce Kinematic Uncertainties;191
5.9.5;13.5 Characterization of Under-Platform Damper Behavior;191
5.9.6;13.6 Friction Repeatable Time Evolution;193
5.9.7;13.7 Beyond Experimental Results;197
5.9.7.1;13.7.1 Numerical Model;198
5.9.7.2;13.7.2 Tuning of Friction Contact Parameters;199
5.9.8;13.8 Perspectives for Under-Platform Damper Research;202
5.9.9;13.9 Methodology Details: Numerical Tuning Procedures;203
5.9.9.1;13.9.1 Piecewise-Selective Tuning;203
5.9.9.2;13.9.2 Friction Coefficients;203
5.9.9.3;13.9.3 Contact Stiffness Values;203
5.9.9.4;13.9.4 Random-Sampling Tuning Procedure;205
5.9.10;References;207
6;Part III Derivation of Constitutive Equations Based on Physical Parameters;209
6.1;14 An Overview of Constitutive Models;210
6.1.1;14.1 Modeling Considerations;210
6.1.1.1;14.1.1 Physical Parameters;211
6.1.1.2;14.1.2 Lubrication;212
6.1.1.3;14.1.3 State Dependence;212
6.1.1.4;14.1.4 Model Implementation;213
6.1.2;14.2 Friction Models;214
6.1.2.1;14.2.1 Velocity-Based Models;214
6.1.2.1.1;14.2.1.1 Coulomb Friction;214
6.1.2.1.2;14.2.1.2 Viscous Friction;214
6.1.2.1.3;14.2.1.3 Stiction;215
6.1.2.1.4;14.2.1.4 Stribeck Friction;215
6.1.2.2;14.2.2 State-Based Models;215
6.1.2.2.1;14.2.2.1 The LuGre Model;216
6.1.2.2.2;14.2.2.2 The Jenkins Element;217
6.1.2.2.3;14.2.2.3 The Masing Element;217
6.1.2.2.4;14.2.2.4 The Iwan Element;217
6.1.2.2.5;14.2.2.5 The Bouc–Wen Hysteresis Model;219
6.1.2.2.6;14.2.2.6 The Seven-Parameter Friction Model;220
6.1.2.3;14.2.3 Prediction Attempts;221
6.1.3;14.3 Outlook for Developing Friction Models;221
6.1.4;14.4 Overview of Part III;222
6.1.5;References;223
6.2;15 Assessment of Coulomb Friction in Modeling Joint Mechanics via a Parameter Study of Dissipation;225
6.2.1;15.1 Simple Lap Joint Experiments;225
6.2.2;15.2 Parameter Studies for Energy Dissipation Trends;226
6.2.2.1;15.2.1 Finite Element Model;226
6.2.2.2;15.2.2 Dependence on Clamping Load;227
6.2.2.3;15.2.3 Dependence on Friction Coefficient;229
6.2.3;15.3 Ramifications for Coulomb-Based Modeling;231
6.2.4;References;231
6.3;16 The Reduced Iwan Plus Pinning Joint Model;232
6.3.1;16.1 Overview of Iwan Modeling;232
6.3.2;16.2 Analytical Development;233
6.3.2.1;16.2.1 Pinning Force;233
6.3.2.2;16.2.2 Relation of Relative and Global Displacements for the Iwan and Pinning Forces;235
6.3.2.3;16.2.3 Four-Parameter Iwan Model Overview;235
6.3.2.4;16.2.4 Considerations for Cyclic Loading;239
6.3.2.5;16.2.5 Comparison with the Discrete Four-Parameter Iwan Model;241
6.3.2.6;16.2.6 Extension to the Five-Parameter Iwan Model;241
6.3.2.7;16.2.7 Extension to the Uniform Iwan Distribution;243
6.3.2.8;16.2.8 Extension to Other Distribution Functions;243
6.3.2.9;16.2.9 Extension to Higher Order Friction Models;244
6.3.3;16.3 Dynamic Response of the RIPP Joint Models;246
6.3.4;16.4 Discussion of the RIPP Joint Formulation;249
6.3.5;16.5 RIPP Joint Implementation in Matlab;249
6.3.5.1;16.5.1 The RIPPjoint Function;250
6.3.6;References;253
6.4;17 Modal Iwan Models for Structures with Bolted Joints;255
6.4.1;17.1 Motivation for Modal Joint Models;256
6.4.2;17.2 Nonlinear Energy Dissipation Model;257
6.4.2.1;17.2.1 Parallel-Series Iwan Model;257
6.4.2.2;17.2.2 Four-Parameter Iwan Model;258
6.4.2.3;17.2.3 Modal Iwan Model;260
6.4.2.4;17.2.4 Extracting Frequency and Energy Dissipation Data;262
6.4.2.4.1;17.2.4.1 Peak-Picking and Zero-Crossing Approach;262
6.4.2.4.2;17.2.4.2 Hilbert Transform with Polynomial Smoothing Approach;263
6.4.2.5;17.2.5 Deducing Modal Iwan Parameters;263
6.4.3;17.3 Finite Element Simulations;265
6.4.3.1;17.3.1 Two-Beam Finite Element Model;266
6.4.3.2;17.3.2 Discrete Iwan Simulations;267
6.4.3.3;17.3.3 Deducing Modal Iwan Parameters;267
6.4.3.3.1;17.3.3.1 Modal Frequency and Energy Dissipation;268
6.4.3.3.2;17.3.3.2 Comparison of Modal Time Responses;271
6.4.4;17.4 Experiments;272
6.4.5;17.5 Outlook for Modal Iwan Modeling;276
6.4.6;References;277
6.5;18 Constitutive Modeling of Contact for Elastic–Plastic Materials Engaged in Micro/Macroslip;279
6.5.1;18.1 Overview of Normal Contact Modeling;280
6.5.2;18.2 The Normal Contact Model;281
6.5.2.1;18.2.1 The Elastic–Plastic Contact Formulation;282
6.5.2.1.1;18.2.1.1 Contribution from the Elastic Contact Force;282
6.5.2.1.2;18.2.1.2 Contribution from the Plastic Contact Force;283
6.5.2.1.3;18.2.1.3 Transitionary Behavior;285
6.5.2.2;18.2.2 The Restitution Phase;290
6.5.2.3;18.2.3 Reloading;290
6.5.2.4;18.2.4 Validation of the Normal Contact Model;291
6.5.3;18.3 The Influence of Interfacial Friction;294
6.5.3.1;18.3.1 Conceptual Discretization of the Contact Patch;295
6.5.3.2;18.3.2 The Slip Regime;296
6.5.3.3;18.3.3 The Stick Regime;298
6.5.3.4;18.3.4 Stick-Slip and Slip-Stick Transitions;300
6.5.3.5;18.3.5 Oblique, Frictional Contact Validation;301
6.5.4;18.4 Calculations of Normal and Tangential Forces for a Single Time Step;307
6.5.5;18.5 Areas for Improved Contact Modeling;307
6.5.6;18.6 Implementation in Matlab of the Elastic-Plastic Contact Model;309
6.5.6.1;18.6.1 Implementation of the Normal Contact Model;309
6.5.6.2;18.6.2 Implementation of the Tangential Contact Model;314
6.5.7;References;327
6.6;19 Microslip Induced Damping in the Contact of Nominally Flat Surfaces with Geometric Defects;330
6.6.1;19.1 Historical Perspectives on Dissipation Due to Microsliding;330
6.6.1.1;19.1.1 Sources of Energy Dissipation;331
6.6.1.2;19.1.2 Perspectives on Dissipation Due to Microslip;331
6.6.1.3;19.1.3 Contact of Asperities;333
6.6.2;19.2 Tribometer and Measurements;334
6.6.2.1;19.2.1 Tribometer Set;334
6.6.2.2;19.2.2 Experimental Result;335
6.6.3;19.3 Microcontact Model;336
6.6.3.1;19.3.1 Normal Distribution of the Load on the Asperities;336
6.6.3.2;19.3.2 Distribution of Tangential Load on the Asperities;339
6.6.3.2.1;19.3.2.1 The Partial Slip Condition;339
6.6.3.2.2;19.3.2.2 The Total Slip Condition;340
6.6.4;19.4 Model Properties;341
6.6.4.1;19.4.1 Changing Phases for a Single Asperity;342
6.6.4.1.1;19.4.1.1 Oscillating Tangential Displacement with a Constant Amplitude;342
6.6.4.1.2;19.4.1.2 Oscillating Tangential Displacement with Variable Amplitude;343
6.6.4.2;19.4.2 Case of N Asperities;346
6.6.4.3;19.4.3 Relations Between the Iwan, Coulomb, and Extended Greenwood Models;348
6.6.4.4;19.4.4 Extended Greenwood Model Identification;349
6.6.4.5;19.4.5 Damping Ratio;349
6.6.5;19.5 Summary of the Extended Greenwood Model;351
6.6.6;References;352
6.7;20 Elements of a Nonlinear System Identification Methodology of Broad Applicability with Application to Bolted Joints;353
6.7.1;20.1 System Identification in the Context of Bolted Joints;353
6.7.2;20.2 Basic Elements of the Nonlinear System Identification Methodology;355
6.7.2.1;20.2.1 Analytical Slow-Flow Dynamics;355
6.7.2.2;20.2.2 Empirical Mode Decomposition: Empirical Slow-Flow Dynamics;356
6.7.3;20.3 Nonlinear System Identification Methodology;358
6.7.3.1;20.3.1 Global Aspects of Nonlinear System Identification;359
6.7.3.2;20.3.2 Local Aspects of Nonlinear System Identification;360
6.7.4;20.4 Application to Bolted Joints;362
6.7.4.1;20.4.1 Experimental Fixture and Process;362
6.7.4.2;20.4.2 Nonlinear System Identification;365
6.7.4.3;20.4.3 Nonlinear Frictional Effects;367
6.7.4.4;20.4.4 Equivalent Damping Ratios;371
6.7.5;20.5 Concluding Remarks on Nonlinear System Identification;375
6.7.6;References;375
6.8;21 Parameter Estimation via Instantaneous Frequency and Damping from Transient Ring-Down Data;378
6.8.1;21.1 Overview of Spectra Calculation Methods;378
6.8.1.1;21.1.1 The Hilbert Transform;379
6.8.1.2;21.1.2 Wavelet Transformations;380
6.8.1.3;21.1.3 The Short-Time Fourier Transform;380
6.8.2;21.2 Theoretical Development of the Parameter Estimation Technique;381
6.8.2.1;21.2.1 Instantaneous Stiffness and Damping Estimation from Ring-Down Data;382
6.8.3;21.3 Application to the Brake–Reuß Beam;384
6.8.4;21.4 Perspectives on the Short-Time Fourier Transform Method;388
6.8.5;References;389
7;Part IV Numerical Techniques for the Analysis of Jointed Structures;391
7.1;22 Historical Perspective on Numerical Techniquesfor Modeling Joints;392
7.1.1;22.1 Historical Development of Numerical Methods;393
7.1.2;22.2 Numerical Implementation;394
7.1.2.1;22.2.1 Survey of Numerical Methods for Dynamic Simulations;394
7.1.2.1.1;22.2.1.1 Craig–Bampton Formulation;395
7.1.2.1.2;22.2.1.2 Time Domain Methods;396
7.1.2.1.3;22.2.1.3 Frequency Domain Methods;400
7.1.3;22.3 Primary Challenges in Numerical Modeling;403
7.1.3.1;22.3.1 Efficiency;403
7.1.3.2;22.3.2 Accuracy;404
7.1.3.3;22.3.3 Usability;405
7.1.4;22.4 Overview of Part IV;406
7.1.5;References;407
7.2;23 A Standard Practice for Modeling Bolted Joints in a Finite Element Package;409
7.2.1;23.1 Strategy for Modeling Bolted Joints in a Finite Element Package;409
7.2.1.1;23.1.1 Nonlinear Static Analysis;410
7.2.1.2;23.1.2 Interfacial Contact Modeling;411
7.2.1.3;23.1.3 Dynamic Analysis Considerations;412
7.2.2;23.2 Overview of the Structure of the Solver Input in ABAQUS;412
7.2.2.1;23.2.1 Mesh Generation;413
7.2.2.2;23.2.2 Boundary and Initial Conditions;414
7.2.2.3;23.2.3 Material Definition;414
7.2.2.4;23.2.4 Static Analysis;415
7.2.3;23.3 Analysis of a Jointed Beam Structure in ABAQUS;416
7.2.4;Reference;420
7.3;24 Reduced Order Modeling of Nonlinear Structures with Frictional Interfaces;421
7.3.1;24.1 Key Features for Modeling Frictional Joints;422
7.3.2;24.2 Approaches Used in the Numerical Round Robin;423
7.3.2.1;24.2.1 Iwan Model Representation: Sandia's Modeling Approach;424
7.3.2.2;24.2.2 The Harmonic Balance Method: Stuttgart's Modeling Approach;426
7.3.2.3;24.2.3 FORSE: Imperial's Modeling Approach;427
7.3.3;24.3 The Benchmark Model;429
7.3.3.1;24.3.1 The Finite Element Mesh;429
7.3.3.2;24.3.2 Nonlinear Static Analysis;430
7.3.4;24.4 Comparison Between Nonlinear Elements;431
7.3.4.1;24.4.1 Calculation of Damping for Each Model;432
7.3.4.2;24.4.2 Model Comparisons;433
7.3.5;24.5 Nonlinear Dynamic Analysis;434
7.3.5.1;24.5.1 Reduced Interface Frequency Response Analysis;434
7.3.5.2;24.5.2 Full Interface Analysis;436
7.3.5.3;24.5.3 Tuning the Iwan Element Joint Model Using Amplitude-Dependent Damping;438
7.3.6;24.6 Conclusions for the Numerical Round Robin;442
7.3.7;References;443
7.4;25 The Craig–Mayes Reduction: A Craig–Bampton Experimental Dynamic Substructure Using the Transmission Simulator Method;445
7.4.1;25.1 Experimental Dynamic Substructuring;446
7.4.2;25.2 The Craig–Mayes Method;447
7.4.3;25.3 Example 1: A Two Beam System;451
7.4.4;25.4 Example 2: An Industrial Application;452
7.4.4.1;25.4.1 Description of the Transmission Simulator Model;453
7.4.4.2;25.4.2 Modal Test of Industrial Structure with Transmission Simulator;453
7.4.4.3;25.4.3 Craig–Mayes Experimental Substructure Coupled to FE Model: Comparison with Free Modal Model;454
7.4.5;25.5 Discussion of the Craig–Mayes Method;456
7.4.6;References;457
7.5;26 A Comparison of Reduced Order Modeling Techniques Used in Dynamic Substructuring;458
7.5.1;26.1 Model Reduction Theories;459
7.5.1.1;26.1.1 Transmission Simulator;459
7.5.1.2;26.1.2 Craig–Bampton;461
7.5.1.3;26.1.3 Craig–Mayes;462
7.5.1.4;26.1.4 Craig–Chang Reduction Method;464
7.5.1.5;26.1.5 Dual Craig–Bampton Method;465
7.5.2;26.2 Example 1: A Two-Beam System;466
7.5.2.1;26.2.1 The Condition of Substructuring;469
7.5.3;26.3 Example 2: Cylinder-Plate-Beam System;471
7.5.3.1;26.3.1 Model Development;473
7.5.3.1.1;26.3.1.1 Experimental Setup;473
7.5.3.1.2;26.3.1.2 Model Development;474
7.5.3.2;26.3.2 Predictions and Comparison with Experimental Truth Data;476
7.5.3.2.1;26.3.2.1 Predictions of the Transmission Simulator Method;477
7.5.3.2.2;26.3.2.2 Craig–Mayes Method;478
7.5.3.2.3;26.3.2.3 Observations and Comparison of the Transmission Simulator Method and Craig–Mayes Method;479
7.5.4;26.4 Perspective on Experimental Substructuring;481
7.5.5;References;482
7.6;27 Calculating the Dynamic Response of Jointed Structures in the Frequency Domain Using Contact Interface Elements;483
7.6.1;27.1 Modeling Assumptions for Jointed Connections;483
7.6.2;27.2 Zero Thickness Elements Theoretical Development;484
7.6.3;27.3 Contact Modeling for Zero Thickness Elements;486
7.6.4;27.4 Adaptive Harmonic Balance Analysis;488
7.6.4.1;27.4.1 Transformation of Harmonics;491
7.6.4.2;27.4.2 Criteria for Selecting Harmonics;491
7.6.4.2.1;27.4.2.1 Approach 1: Estimation of Response Displacement Harmonics;492
7.6.4.2.2;27.4.2.2 Approach 2: Estimation of Partial Derivatives;493
7.6.5;27.5 Numerical Examples;494
7.6.5.1;27.5.1 Numerical Results for Approach 1: Estimation of Response Displacement Harmonics;495
7.6.5.2;27.5.2 Numerical Results for Approach 2: Estimation of Partial Derivatives;498
7.6.6;27.6 Summary Discussion of Zero Thickness Elements;499
7.6.7;References;501
7.7;28 Application of Continuum Shell Models for Joint Dissipation;503
7.7.1;28.1 Limitations of Modeling Jointed Structures;503
7.7.2;28.2 Modal Analysis of Jointed Structures;504
7.7.2.1;28.2.1 Monolithic Structure;505
7.7.2.2;28.2.2 Jointed Structure;507
7.7.3;28.3 Distributed Interface;508
7.7.3.1;28.3.1 Continuum Interface Model;509
7.7.4;28.4 Longitudinal Example;513
7.7.5;28.5 Perspectives of Modeling Joints with Continuum Shell Elements;515
7.7.6;References;516
7.8;29 Nonlinear Modal Analysis and Modal Reduction of Jointed Structures;517
7.8.1;29.1 Overview of Nonlinear Modes;517
7.8.2;29.2 Nonlinear Modal Analysis of Jointed Structures;519
7.8.2.1;29.2.1 Dynamic Regime of Interest and Extension of the Periodic Motion Concept;519
7.8.2.2;29.2.2 Computational Procedure;521
7.8.3;29.3 Model Reduction Based on Nonlinear Modes;522
7.8.4;29.4 Application Examples;524
7.8.4.1;29.4.1 Rod with Friction Joint;524
7.8.4.1.1;29.4.1.1 Modal Characteristics;525
7.8.4.1.2;29.4.1.2 Free Decay;526
7.8.4.2;29.4.2 Bladed Disk with Under-Platform Dampers;527
7.8.4.2.1;29.4.2.1 Modal Characteristics;528
7.8.4.2.2;29.4.2.2 Forced Response;528
7.8.5;29.5 Concluding Remarks on Nonlinear Modal Analysis;529
7.8.6;References;529
7.9;30 Numerical Methods for Assessing Response Metrics;531
7.9.1;30.1 Model Reduction and Comparison for Nonlinear Systems;531
7.9.2;30.2 Theoretical Development;534
7.9.2.1;30.2.1 Reduced Order Models with Localized Nonlinearities;535
7.9.2.2;30.2.2 Nonlinear Normal Modes;536
7.9.3;30.3 Numerical Results;538
7.9.3.1;30.3.1 Nonlinear Normal Mode Convergence;540
7.9.3.2;30.3.2 Impulse Loading Verification;545
7.9.3.3;30.3.3 Random Loading Verification;548
7.9.4;30.4 Perspectives on Using Nonlinear Normal Modes to Assess Convergence;550
7.9.5;References;551
7.10;31 Predicting the Shakedown Limits of Joints Subject to Fretting and High Cycle Fatigue;553
7.10.1;31.1 Introduction to Fretting and Frictional Shakedown;554
7.10.2;31.2 Formulation;559
7.10.2.1;31.2.1 Definition of the Friction Model;560
7.10.2.2;31.2.2 Loading Regime;561
7.10.3;31.3 First Violation of the Stick Condition;561
7.10.4;31.4 The Shakedown Limit;562
7.10.4.1;31.4.1 Calculation of the Shakedown Limit;562
7.10.4.2;31.4.2 Improving Convergence;564
7.10.5;31.5 Application to Example Problem;564
7.10.5.1;31.5.1 First Violation of the Stick Condition;565
7.10.5.2;31.5.2 The Shakedown Limit;566
7.10.5.3;31.5.3 Frictional Energy Dissipation;567
7.10.5.4;31.5.4 Optimal Initial Residual Slip Displacement;569
7.10.5.5;31.5.5 Comparison of Results;571
7.10.6;31.6 Discussion of the Shakedown Calculations;572
7.10.7;31.7 Shakedown Summary;573
7.10.8;References;573
8;Part V Epistemic and Aleatoric Uncertainty in Modeling and Measurements;575
8.1;32 A Primer for Uncertainty Modeling in Jointed Structures;576
8.1.1;32.1 Epistemic and Aleatoric Uncertainty in Structural Modeling;576
8.1.2;32.2 Concepts in Uncertainty Modeling;577
8.1.2.1;32.2.1 Probability Theory;577
8.1.2.2;32.2.2 Random Variables;578
8.1.2.3;32.2.3 Maximum Entropy Distributions;578
8.1.2.4;32.2.4 Polynomial Chaos Expansion;579
8.1.2.5;32.2.5 Joint Random Variables;579
8.1.2.6;32.2.6 Random/Stochastic Processes;580
8.1.3;32.3 Application to Structural Dynamics;581
8.1.4;32.4 Challenges and Goals;582
8.1.5;32.5 Overview of Part V;583
8.1.6;References;583
8.2;33 Epistemic and Aleatoric Uncertainty in Modeling;584
8.2.1;33.1 A Geometric Problem of Model Form Error;585
8.2.1.1;33.1.1 The Experiment and Assumed Model;585
8.2.1.2;33.1.2 Comparison of Volumes from Two Models;585
8.2.1.3;33.1.3 Conclusions from the Simple Experiment;586
8.2.2;33.2 Not Even Conservative;588
8.2.2.1;33.2.1 A Problem of Nonlinear Vibration;588
8.2.2.2;33.2.2 The Truth Model;589
8.2.2.3;33.2.3 How the Parameters Compare;589
8.2.3;33.3 Discussion of Model Form Error in the Context of Bolted Joints;591
8.2.3.1;33.3.1 Observations from the Nonlinear Vibration Problem;591
8.2.3.2;33.3.2 Another Approach;592
8.2.4;33.4 Conclusions on Model Form Error;594
8.2.5;References;594
8.3;34 A Practical Application of a Maximum Entropy, Non-parametric Approach to Account for Epistemic Uncertainty Using Random Matrices;595
8.3.1;34.1 Maximum Entropy, Non-parametric Modeling for Epistemic Uncertainty;596
8.3.1.1;34.1.1 Define the Model and Truth Data;597
8.3.1.2;34.1.2 Define the Parameter and Model Variables;598
8.3.1.3;34.1.3 Sweep the Parameter Uncertainty Variable;598
8.3.1.4;34.1.4 Determine the Maximum Likelihood Function;599
8.3.1.5;34.1.5 Sweep the Model Uncertainty Variable;600
8.3.1.6;34.1.6 Determine the Model Form Dispersion Variables;602
8.3.1.7;34.1.7 Calculate the Final Distribution;602
8.3.2;34.2 One-Dimensional Example;602
8.3.2.1;34.2.1 Results and Analysis of the One-Dimensional System;603
8.3.3;34.3 Two-Dimensional Example;603
8.3.4;34.4 Alternative Matrix Generation;606
8.3.5;34.5 Special Corrections;608
8.3.5.1;34.5.1 Rigid Body Modes;608
8.3.5.2;34.5.2 High-Fidelity Models;609
8.3.6;34.6 Discussion of the Maximum Entropy, Non-parametric Modeling Approach;609
8.3.7;34.7 Implementation in Matlab of the Maximum Entropy, Non-parametric Method;609
8.3.8;References;616
8.4;35 Stochastic Iwan-Type Models for Joint Variability Modeling;617
8.4.1;35.1 Uncertainty in Bolted Joints;618
8.4.2;35.2 The Iwan Model and the Four-Parameter Iwan Model;618
8.4.3;35.3 The Sandia Bolted Joint Data and Its Modeling;620
8.4.4;35.4 Extending Iwan-Type Models;622
8.4.4.1;35.4.1 Extended Iwan-Type Models: Split Four-Parameter Iwan Model;626
8.4.4.2;35.4.2 Extended Iwan-Type Models: Five-Parameter Iwan-Type Model;627
8.4.4.3;35.4.3 Uncertainty Modeling;631
8.4.5;35.5 Stochastic Parameter Identification and Uncertainty Bands;634
8.4.6;35.6 Conclusions on Stochastic Iwan-Type Models;637
8.4.7;References;639
8.5;36 Quantifying Epistemic and Aleatoric Uncertainty in the Ampair 600 Wind Turbine;641
8.5.1;36.1 Background and Context for the Analysis of the Ampair 600 Wind Turbine;641
8.5.2;36.2 Experiments to Deduce Joint Parameters;643
8.5.2.1;36.2.1 Hub Only Test;643
8.5.2.2;36.2.2 Hub and One Blade: Testing for Nonlinearities;644
8.5.3;36.3 The Maximum Entropy Approach;645
8.5.3.1;36.3.1 Theoretical Development of the Maximum Entropy Approach;646
8.5.3.2;36.3.2 Results;649
8.5.4;36.4 An Uncertainty Analysis Using The Iwan Joint Model;650
8.5.4.1;36.4.1 Assumptions for the Parameter Distributions;651
8.5.4.2;36.4.2 Uncertainty Propagation on a Lumped Mass Model;654
8.5.4.2.1;36.4.2.1 Tuning Protocol;655
8.5.4.2.2;36.4.2.2 Influence of Parameters on Natural Frequencies;657
8.5.4.2.3;36.4.2.3 Accounting for Epistemic Uncertainty;658
8.5.4.2.4;36.4.2.4 Sensitivity Analysis;658
8.5.5;36.5 Conclusions for the Application of Epistemic Uncertainty Techniques to a Real System;661
8.5.6;References;661
9;Part VI Perspectives on the Future of Joints Research;663
9.1;37 The Next Generation of Joints Research;664
9.1.1;37.1 Outcomes from the 2015 International Workshop on Joint Mechanics;665
9.1.1.1;37.1.1 Conclusions from the Themed Sessions;665
9.1.1.1.1;37.1.1.1 Applicability;665
9.1.1.1.2;37.1.1.2 Repeatability;666
9.1.1.1.3;37.1.1.3 Predictability;667
9.1.1.1.4;37.1.1.4 Way Forward;668
9.1.1.2;37.1.2 Industrial Needs for Joints Research;669
9.1.2;37.2 A New Road Map;669
9.1.2.1;37.2.1 Strategy;670
9.1.2.2;37.2.2 Themes;670
9.1.2.2.1;37.2.2.1 Building External Consensus for Support;671
9.1.2.2.2;37.2.2.2 Experimental Investigation of Repeatability and Variability;671
9.1.2.2.3;37.2.2.3 Techniques to Characterize/Identify Nonlinearities;672
9.1.2.2.4;37.2.2.4 Constitutive Model Development;672
9.1.2.2.5;37.2.2.5 Numerical Methods for Nonlinear Dynamics;673
9.1.2.2.6;37.2.2.6 Multiscale Investigation of Interfacial Physics;674
9.1.2.2.7;37.2.2.7 Uncertainty-Based Strategies for Modeling and Experiments;674
9.1.3;37.3 Concluding Remarks and Hypotheses;675
9.1.3.1;37.3.1 On Variability;675
9.1.3.2;37.3.2 On Constitutive Modeling;676
9.1.3.3;37.3.3 On Numerical Modeling;676
9.1.3.4;37.3.4 On Experimental Techniques;677
9.1.3.5;37.3.5 On Manufacturing Systems;677
9.1.3.6;37.3.6 The Future Joint;678
9.1.4;References;678
10;Index;679