Harnessing Bistable Structural Dynamics

Preface xi

1 Background and Introduction 1

1.1 Examples of Bistable Structures and Systems 1

1.2 Characteristics of Bistable Structural Dynamics 6

1.2.1 Coexistence of Single-periodic, Steady-state Responses 8

1.2.2 Sensitivity to Initial Conditions 14

1.2.3 Aperiodic or Chaotic Oscillations 15

1.2.4 Excitation Level Dependence 16

1.2.5 Stochastic Resonance 21

1.2.6 Harmonic Energy Diffusion 22

1.3 The Exploitation of Bistable Structural Dynamics 24

1.3.1 Vibration Control 24

1.3.2 Vibration Energy Harvesting 27

1.3.3 Sensing and Detection 30

1.4 Outline of This Book 33

References 34

2 Mathematical Modeling and Analysis of Bistable Structural Dynamics 39

2.1 A Linear Oscillator 39

2.1.1 Free Response 39

2.1.2 Base-excited Response 44

2.2 Stability 46

2.3 A Monostable Nonlinear Oscillator 48

2.4 A Bistable Oscillator 50

2.4.1 Free Response and Stability 50

2.4.2 Base-excited Response 54

2.5 Analytical Methods for Steady-state Dynamics 57

2.5.1 Small Oscillations 58

2.5.2 Large Oscillations 62

2.6 Bifurcations of Bistable Systems 67

2.7 Multiple Degrees-of-Freedom Systems 69

2.8 An Electromechanical Bistable System 70

2.9 Summary 71

References 72

3 Vibration Control 75

3.1 Topic Review 75

3.1.1 Damping 77

3.1.2 Isolation 81

3.1.3 Absorption 83

3.1.4 Summary 86

3.2 High and Adaptable Damping Using Bistable Snap-through Dynamics 86

3.2.1 Model Formulation of the Bistable Device 87

3.2.2 A Metric for Energy Dissipation Capacity 89

3.2.3 Numerical Analysis of the Base-excited Response 89

3.2.4 Energy Dissipation Features of the Dynamic Types 90

3.2.5 Influences Due to Change in Frequency and Initial Conditions 93

3.2.6 Experimental Studies 96

3.2.7 Summary 100

3.3 Isolating Structures Under Large Amplitude Excitations Through Activation of Low Amplitude Interwell Dynamics: Criteria for Excitation-induced Stability 100

3.3.1 Governing Equation Formulation of the Bistable Oscillator 101

3.3.2 Stability of the Analytically Predicted Interwell Dynamics 102

3.3.3 Validation of the Stability Criteria Using Numerical Simulations 105

3.3.4 Experimental Validation of the Stability Criteria 109

3.3.5 Summary 115

3.4 Exploiting Excitation-induced Stability for Dual-stage Vibration Isolation 115

3.4.1 Governing Equation Formulation of a Bistable Dual-stage Vibration Isolator 116

3.4.2 Analytical Solution of the Governing Equations 118

3.4.3 Examining the Stability of Analytical Predictions 119

3.4.4 Comparison of Isolator Performance with a Counterpart Linear Design 119

3.4.5 Explanation of the Valley Response 122

3.4.6 Investigating the Design Parameter Influences 123

3.4.7 Influence of Initial Conditions 126

3.4.8 Prototype Investigations: Numerical and Experimental Validation 131

3.4.9 Summary 136

3.5 Dynamic Stabilization of a Vibration Suspension Platform Attached to an Excited Host Structure 137

3.5.1 Model Formulation of the Bistable Suspension Coupled to a Flexible Structure 138

3.5.2 Analytical Solution of the Governing Equations 140

3.5.3 Description of the Linear Suspension for Comparison 141

3.5.4 Analytical and Numerical Assessment of Key Suspension Dynamics 142

3.5.5 Excitation Condition Influences 145

3.5.6 Experimental Suspension System Platform 148

3.5.7 Experimental and Analytical Comparisons of Isolation Performance 148

3.5.8 Summary 154

3.6 Snap-through Dynamics for Vibration Absorption 154

3.6.1 Model Formulation of a Bistable Vibration Absorber 155

3.6.2 Analytical Investigation of Force Cancellation Performance 159

3.6.3 Experimental Investigation of Force Cancellation Performance 164

3.6.4 Summary 172

References 173

4 Vibration Energy Harvesting 181

4.1 Topic Review 181

4.1.1 Experimental and Numerical Developments in Energy Harvesting with Bistable Devices 183

4.1.2 Analytical Developments in Energy Harvesting with Bistable Devices 189

4.1.3 Summary 191

4.2 Effective and Straightforward Design Guidelines for High Performance Operations 191

4.2.1 Analytical Formulation of Bifurcations Associated with Achieving Snap-through 192

4.2.2 Experimental Validation of the Analytical Premise 196

4.2.3 Derivation of Criteria for Sustaining High Power Generation Performance 198

4.2.4 Evaluation of the Criteria Accuracy 199

4.2.5 Summary 200

4.3 Understanding Superharmonic Energy Diffusion in Bistable Energy Harvesters 201

4.3.1 Bistable Energy Harvester Modeling: Electromechanical Governing Equations 202

4.3.2 Amplitude Response Equations 204

4.3.3 Selection of System Parameters for Investigation 207

4.3.4 Comparison to 1-Term Harmonic Balance Solution 208

4.3.5 Effects of Varying Excitation Amplitude 210

4.3.6 Superharmonic Energy Harvesting Analysis 215

4.3.7 Experimentally Investigating the Contribution of Total Harvested Energy by the Superharmonic Component 217

4.3.8 Summary 221

4.4 Optimal and Robust Energy Harvesting from Realistic Stochastic Excitations Using the Dynamics of Structures Designed Near the Elastic Stability Limit 222

4.4.1 Modeling of Nonlinear Energy Harvester Platform 223

4.4.2 Preliminary Remarks on Accuracy, Comparisons, and Experimentation 227

4.4.3 Noise Bandwidth and Level Influences on Ideal Designs 229

4.4.4 Criticality of Design at the Elastic Stability Limit 232

4.4.5 Impact of Asymmetry on Energy Harvesting Performance 233

4.4.6 Summary 235

4.5 Amplifying the Snap-through Dynamics of a Bistable Energy Harvester Using an Appended Linear Oscillator 237

4.5.1 Coupled Energy Harvesting System Governing Equations 239

4.5.2 Analytical Formulation by the Harmonic Balance Method: 1-Term Prediction 240

4.5.3 Analytical Formulation by the Harmonic Balance Method: 2-Term Prediction 243

4.5.4 Computing the Fundamental and Superharmonic Average Power and Power Density 246

4.5.5 Roles of the Auxiliary Linear Oscillator 247

4.5.6 Roles of the Superharmonic Dynamics in the Energy Harvesting Performance 251

4.5.7 Experimental Investigations of the Multiharmonic Dynamics Enhancement via the Auxiliary Linear Oscillator 255

4.5.8 Summary 260

4.6 A Linear Dynamic Magnifier Approach to Bistable Energy Harvesting 261

4.6.1 Governing Equations for the Bistable Harvester with Linear Dynamic Magnifier 262

4.6.2 Approximate Solution by the Method of Harmonic Balance 264

4.6.3 Analytical and Numerical Investigations on the Roles of the Linear Dynamic Magnifier Stage 267

4.6.3.1 Effect of the Mass Ratio 267

4.6.3.2 Effect of the Frequency Ratio 271

4.6.3.3 Effect of the Electromechanical Coupling 272

4.6.4 Interpreting Frequency Response Characteristics of the Coupled Energy Harvester System 275

4.6.5 Experimental Validations and Investigations 277

4.6.5.1 Effect of Bistable Harvester Electromechanical Coupling 278

4.6.5.2 Effect of Bistable Harvester Mass 279

4.6.6 Summary 280

References 282

5 Sensing and Detection 291

5.1 Topic Review 291

5.1.1 Bistable Microsystems 293

5.1.2 Bifurcation-based Microsystem Applications 298

5.1.3 Summary 300

5.2 Detecting Changes in Structures by Harnessing the Dynamics of Bistable Circuits 301

5.2.1 Bifurcation-based Sensing Platform Based on Bistable Circuitry 302

5.2.2 Bistable Circuit Model Formulation and Validation 305

5.2.3 Investigation of Operational Parameters Suited for Bifurcation-based Sensing 308

5.2.4 Experimental Study of the Proposed Bifurcation-based Sensing Approach 309

5.2.5 Summary 311

5.3 Improving Damage Identification Robustness to Noise and Damping Using an Integrated Bistable and Adaptive Piezoelectric Circuit 312

5.3.1 An Integrated Bistable and Adaptive Piezoelectric Circuitry for Bifurcation-based SHM 313

5.3.2 Overview of Damage Identification Using Integrated Adaptive Piezoelectric Circuitry 316

5.3.3 Verification of Bifurcation-based Detection of Frequency Shifts 318

5.3.4 Improving the Accuracy of BB Frequency Shift Detection Through a Greater Number of Evaluations 321

5.3.5 Investigation of Noise Influences for a Mildly Damped Structure 323

5.3.6 Investigation of Noise Influences for a More Highly Damped Structure 325

5.3.7 Summary 327

5.4 Passive Microscale Mass Detection and Progressive Quantification by Exploiting the Bifurcations and Resonant Dynamics of a Two DOF Bistable Sensor 327

5.4.1 Sensor Architecture and Operational Principle Overview 330

5.4.2 Experimental Proof-of-concept Sensor Architecture 334

5.4.3 Model Formulation of the Sensor Architecture 336

5.4.4 Experimental Validation of the Model Formulation and Numerical Examinations of System Operation 337

5.4.5 Examination of Passive Quantification of Mass Adsorption via Sequential Activation of Bifurcations 340

5.4.6 Experimental Comparison of Bifurcation Triggering and Fundamental Mode Natural Frequency Reduction as Consistent Detection Metrics 342

5.4.7 Stochastic Modeling and Noise Sensitivities 344

5.4.8 Operational Parameter Influences for Passive Sensing Strategy 349

5.4.9 Sensor Embodiments and Fabrication Strategies 354

5.4.10 Summary 355

References 355

6 Emerging Themes and Future Directions 365

6.1 Emerging Themes 365

6.1.1 Vibration Control 365

6.1.2 Vibration Energy Harvesting 369

6.1.3 Sensing and Detection 372

6.2 Challenges and Future Outlooks 377

6.2.1 Theoretical Characterization of the Emerging Bistable and Multistable Structural/Material System Concepts 377

6.2.2 Application Relevance and Readiness 379

References 381

Index 387