# Linear and Nonlinear Instabilities in Mechanical Systems

## Analysis, Control and Application

1. Auflage März 2021

320 Seiten, Hardcover*Wiley & Sons Ltd*

**978-1-119-06653-8**

Linear and Nonlinear Instabilities in Mechanical Systems: Analysis, Control and Application

Hiroshi Yabuno, University of Tsukuba, Japan

An in-depth insight into nonlinear analysis and control

As mechanical systems become lighter, faster, and more flexible, various nonlinear instability phenomena can occur in practical systems.

The fundamental knowledge of nonlinear analysis and control is essential to engineers for analysing and controlling nonlinear instability phenomena. The book bridges the gap between the mathematical expressions of nonlinear dynamics and the corresponding practical phenomena. Linear and Nonlinear Instabilities in Mechanical Systems: Analysis, Control and Application provides a detailed and informed insight into the fundamental methods for analysis and control for nonlinear instabilities from the practical point of view.

Key features:

* Refers to the behaviours of practical mechanical systems as aircraft, railway vehicle, robot manipulator, micro/nano sensor

* Enhances the rigorous and practical understanding of mathematical methods from an engineering point of view.

* The theoretical results obtained by nonlinear analysis are interpreted by using accompanied videos on the real nonlinear behaviors of nonlinear mechanical systems.

Linear and Nonlinear Instabilities in Mechanical Systems: Analysis, Control and Application is an essential textbook for students on engineering courses, and can also be used for self-study or reference by engineers.

References 8

1 Equilibrium States and their Stability 11

1.1 Equilibrium states 11

1.1.1 Spring-mass system 12

1.1.2 Magnetically levitated system 16

1.1.3 Simple pendulum 20

1.2 Work and potential energy 23

1.3 Stability of the equilibrium state in conservative systems 27

1.4 Stability of mechanical systems 29

1.4.1 Stability of spring-mass system 29

1.4.2 Stability of magnetically levitated system 31

1.4.3 Pendulum 32

1.4.4 Stabilization control of magnetically levitated system 32

References 34

2 Linear Dynamical Systems 35

2.1 Vector field and phase space 35

2.2 Stability of equilibrium states 40

2.3 Linearization and local stability 41

2.4 Eigenvalues of linear operators and phase portraits in a single-degree-offreedom system 44

2.4.1 Description of the solution by matrix exponential function 44

2.4.2 Case with distinct eigenvalues 45

2.4.3 Case with repeated eigenvalues 49

2.4.4 Case with complex eigenvalues 54

2.5 Invariant subspaces 60

2.6 Change of stability due to the variation of system parameters 61

References 67

3 Dynamic Instability of Two-Degree-of-Freedom-Systems 69

3.1 Positional forces and velocity-dependent forces 69

3.2 Total energy and its time-variation 71

3.2.1 Kinetic energy 71

3.2.2 Potential energy due to conservative force FK 72

3.2.3 Effect of velocity dependent damping force FD 76

3.2.4 Effect of circulatory force FN 78

3.2.5 Effect of gyroscopic force FG 81

References 83

4 Modal Analysis of Systems Subject to Conservative and Circulatory Forces 85

4.1 Decomposition of the matrix M 86

4.2 Characteristic equation and modal vector 89

4.3 Modal analysis in case without circulatory force 90

4.4 Modal analysis in case with circulatory force 97

4.4.1 Case study 1: _i are real 100

4.4.2 Case study 2: _i are complex 103

4.5 Synchronous and nonsynchronous motions in a fluid-conveying pipe (video) 114

References 115

5 Static Instability and Practical Examples 117

5.1 Two-link model for a slender straight elastic rod subject to compressive forces 117

5.1.1 Static instability due to compressive forces 117

5.1.2 Effect of a spring attached in the longitudinal direction 122

5.2 Spring-mass-damper models in MEMS 125

5.2.1 Comb-type MEMS actuator devices 125

5.2.2 Cantilever-type MEMS switch 129

References 131

6 Dynamic Instability and Practical Examples 135

6.1 Self-excited oscillation of belt-driven mass-spring-damper system 135

6.2 Flutter of wing 139

6.2.1 Static destabilization in case when the mass center is located in front of the elastic center 145

6.2.2 Static and dynamic destabilization in case when the mass center is located behind the elastic center 146

6.3 Hunting motion in a railway vehicle 149

6.4 Dynamic instability in Jeffcott rotor due to internal damping 161

6.4.1 Fundamental rotor dynamics 161

6.4.2 Effects of the centrifugal force and the Coriolis force on static stability 166

6.4.3 Effect of external damping 170

6.4.4 Dynamics instability due to internal damping 174

6.5 Dynamic instability in fluid-conveying pipe due to follower force 178

References 180

7 Local Bifurcations 183

7.1 Nonlinear analysis of a two-link-model subjected to compressive forces 184

7.1.1 Nonlinearity of equivalent spring stiffness 184

7.1.2 Equilibrium states and their stability 186

7.2 Reduction of dynamics near a critical point 190

7.3 Pitchfork bifurcation 196

7.4 Other codimension one bifurcations 197

7.4.1 Saddle-node bifurcation 197

7.4.2 Transcritical bifurcation 199

7.4.3 Hopf bifurcation 200

7.5 Perturbation of pitchfork bifurcation 204

7.5.1 Bifurcation diagram 204

7.5.2 Analysis of bifurcation point 207

7.5.3 Equilibrium surface and bifurcation diagrams 209

7.6 Effect of Coulomb friction on pitchfork bifurcation 211

7.6.1 Linear analysis 212

7.6.2 Nonlinear analysis 214

7.7 Nonlinear characteristics of static Instability in spring-mass-damper models of MEMS 217

7.7.1 Pitchfork bifurcation in comb-type MEMS actuator device 218

7.7.2 Saddle-node bifurcation in MEMS switch 220

References 222

8 Reduction Methods of Nonlinear Dynamical Systems 225

8.1 Reduction of the dimension of state space by center manifold theory 226

8.1.1 Nonlinear stability analysis at pitchfork bifurcation point 226

8.1.2 Reduction of nonlinear dynamics near bifurcation point 229

8.2 Reduction of degree of nonlinear terms by the method of normal forms 233

8.2.1 Reduction by nonlinear coordinate transformation: Method of normal forms 233

8.2.2 Case in which the linear part has distinct real eigenvaules 235

8.2.3 Nonlinear term remaining in normal form 238

8.2.4 Reduction in the neighborhood of Hopf bifurcation point 240

References 246

9 Method of Multiple Scales 247

9.1 Spring-mass system with small damping 248

9.2 Introduction of multiple time scales 251

9.3 Method of multiple scales 253

9.4 Slow time scale variation of amplitude and stability of periodic solutions 256

References 256

10 Nonlinear Characteristics of Dynamic Instability 259

10.1 Effect of nonlinearity on dynamic instability due to negative damping force 260

10.1.1 Cubic nonlinear damping (Rayleigh type and van der Pol type) 260

10.1.2 Self-excited oscillation produced through Hopf bifurcation 261

10.1.3 Self-excited oscillation by linear feedback and its amplitude control by nonlinear feedback 269

10.2 Effect of nonlinearity on dynamic instability due to circulatory force 271

10.2.1 Derivation of amplitude equations by solvability condition 272

10.2.2 Effect of cubic nonlinear stiffness on steady state response 278

References 281

11 Parametric Resonance and Pitchfork Bifurcation 283

11.1 Parametric resonance of vertically-excited inverted pendulum 284

11.1.1 Equation of motion 284

11.2 Dynamics in case without excitation 285

11.2.1 Dimensionless equation of motion subject to vertical excitation 286

11.2.2 Trivial equilibrium state and its stability 290

11.2.3 Nontrivial steady state amplitude and its stability 291

References 295

12 Stabilization of Inverted Pendulum under High-Frequency Excitation 297

12.1 Equation of motion 298

12.2 Analysis by the method of multiple scales 299

12.2.1 Scaling of some parameters 299

12.2.2 Averaging by the method of multiple scales 300

12.3 Bifurcation analysis of inverted pendulum under high-frequency excitation 302

12.3.1 Subcritical pitchfork bifurcation and stabilization of inverted pendulum 302

12.3.2 Global stability of equilibrium states 305

12.4 Experiments 307

12.5 Effects of the excitation direction on the bifurcation 308

12.5.1 Averaging by the method of multiple scales 309

12.5.2 Excitation inclined from the vertical direction and perturbed subcritical pitchfork bifurcation 310

12.5.3 Supercritical pitchfork bifurcation in horizontal excitation and its perturbation due to inclination of the excitation direction 311

12.6 Stabilization of statically unstable equilibrium states by high-frequency excitation 311

References 312

13 Self-excited Resonator in Atomic Force Microscopy (Utilization of Dynamic Instability) 315

13.1 Principle of frequency modulation atomic force microscope (FM-AFM) 316

13.2 Detection of frequency shift based on external excitation 322

13.3 Detection of frequency shift based on self-excitation 325

13.4 Amplitude control for self-excited microcantilever probe 327

References 328

14 High-Sensitive Mass Sensing by Eigenmode Shift 331

14.1 Conventional mass sensing by frequency shift of resonator 332

14.2 High-sensitive mass sensing by coupled resonators 333

14.3 Solution of equations of motion 335

14.4 Mode shift due to measured mass 336

14.5 Experimental detection methods for mode shift 337

14.5.1 Use of eternal excitation 338

14.6 Use of self-excitation 339

References 344

15 Motion Control of Underactuated Manipulator without State Feedback Control 345

15.1 What is an underactuated manipulator 345

15.2 Equation of motion 346

15.3 Averaging by the method of multiple scales and bifurcation analysis 348

15.4 Motion control of free link 352

15.5 Experimental results 354

References 355

16 Experimental Observations 359

16.1 Experiments of a single degree-of-freedom system (Chapters 2 and 6) 359

16.1.1 Stability of spring-mass-damper system depending on the stiffness k and the damping c 359

16.1.2 Self-excited oscillation of a window shield wiper blade around the reversal 362

16.2 Buckling of a slender beam under a compressive force 362

16.2.1 Observation of pitchfork bifurcation (sections 5.1 and 7.1) 362

16.2.2 Observation of perturbed pitchfork bifurcation (section 7.5) 363

16.2.3 Effect of Coulomb friction on pitchfork bifurcation (section 7.6) 364

16.3 Hunting motion of a railway vehicle wheelset (section 6.3) 365

16.4 Stabilization of hunting motion by gyroscopic damper (section 6.3) 367

16.5 Self-excited oscillation of fluid-conveying pipe (section 6.5) 368

16.6 Realization of self-excited oscillation in a practical cantilever (section 10.1.3) 369

16.7 Parametric resonance (Chapter 11) 373

16.8 Stabilization of an inverted pendulum under high-frequency vertical excitation (Chapter 12) 374

16.9 Self-excited coupled cantilever beams for ultrasensitive mass sensing (section 14.6) 375

16.10Motion control of an underactuated manipulator by bifurcation control (Chapter 15) 375

References 376

A Cubic Nonlinear Characteristics 379

A.1 Symmetric and nonsymmetric nonlinearities 380

A.2 Nonsymmetric nonlinearity due to the shift of the equilibrium state 381

A.3 Effect of harmonic external excitation 383

B Nondimensionalization and Scaling Nonlinearity 385

B.1 Nondimensionalization of equations of motion 385

B.2 Scaling of nonlinearity 389

B.3 Nondimensionalization of the governing equation of a nonlinear oscillator 391

B.4 Effect of harmonic external excitation 392

References 394

C Occurrence Prediction for Some Types of Resonances 395

C.1 Dynamics of a linear spring-mass-damper system subject to harmonic external excitation 396

C.1.1 Case with viscous damping 396

C.1.2 Case under no viscous damping 399

C.2 Occurrence prediction of some types of resonances in a nonlinear springmass-damper system 401

References 405

D Order Estimation of Responses 407

D.1 Order symbol 407

D.2 Asymptotic expression of solution 408

D.3 Linear oscillator under harmonic external excitation 409

D.3.1 Non-resonant case 410

D.3.2 Resonant case 411

D.3.3 Near-resonant case 411

D.4 Cubic nonlinear oscillator under external harmonic excitation 412

D.4.1 Large damping case ( = O(1)) 412

D.4.2 Relatively small damping case ( = O(_2=3)) 413

D.4.3 Small damping case ( = O(_)) 414

D.5 Linear oscillator with negative damping 415

D.6 Van der Pol oscillator 416

D.6.1 Large response case (_0(_) = 1) 417

D.6.2 Small but finite response case (_0(_) = o(1)) 417

D.7 Parametrically excited oscillator 418

D.7.1 Large damping case ( = O(1)) 419

D.7.2 Small damping case ( = O(_)) 420

D.7.3 Case with cubic nonlinear component of restoring force 422

D.7.4 Near-resonant case 423

References 425

E Free Oscillation of Spring-Mass System under Coulomb Friction and its Dead Zone 427

E.1 Characteristics of friction 427

E.2 Free oscillation under Coulomb friction 429

E.3 Variation of the final rest position with decrease in the stiffness 434

References 436

F Projection by Adjoint Vector 439

G Solvability Condition 441

G.1 Kernel and image of linear transformation 441

G.2 Solvability condition 443

H Effect of Contact Force on the Dynamics of Railway Vehicle Wheelset 451

H.1 A slip at the contact point of rolling disk on a plane 452