Dynamical Systems (eBook)

Preface 6
Contents 
8 
Contributors 12
Part I Nonlinear and Discontinuous Dynamical Systems 18
1 General Solution of a Vibration System with Damping Force of Fractional-Order Derivative 19
1.1 Introduction 19
1.2 Preliminaries for Sequential Fractional-Order Differential Equations 20
1.2.1 Linear Dependence and Linear Independence of Functions 21
1.2.2 Characteristic Polynomial 21
1.2.3 General Solution of SFDE 23
1.3 Analysis of the Characteristic Roots 23
1.4 The General Solution 25
References 27
2 An Analytic Proof for the Sensitivity of Chaos to Initial Condition and Perturbations 28
2.1 Introduction 28
2.2 Perturbation Equations 29
2.3 General Solutions of Perturbation Equations 30
2.4 Sensitivity to the Initial Conditions 34
2.5 Sensitivity to Perturbations 35
References 36
3 Study on the Multifractal Spectrum of Local Area Networks Traffic and Their Correlations 37
3.1 Introduction 37
3.2 Multifractal and the Spectral Parameters for Networks Traffic 38
3.3 Relationship Between Variables of Multifractal Spectrum 39
3.4 Numerical Simulation and Analysis 40
3.4.1 Multifractal Spectrum and Parameters 40
3.4.2 Relationship Between Multifractal Spectrum Parameters and Traffic Variation 42
3.5 Concluding Remarks 43
References 44
4 A Boundary Crisis in High Dimensional Chaotic Systems 45
4.1 Introduction 45
4.2 A Hyperchaotic Boundary Crisis in a Kawakami Map 46
4.3 Concluding Remarks 49
References 50
5 Complete Bifurcation Behaviors of a Henon Map 51
5.1 Introduction 51
5.2 Analysis 52
5.3 Illustrations 54
5.4 Conclusion 60
References 61
6 Study on the Performance of a Two-Degree-of-Freedom Chaotic Vibration Isolation System 62
6.1 Introduction 62
6.2 Model of the Two-Degree-of-Freedom Nonlinear Vibration Isolation System 63
6.3 Dynamical Analysis 65
6.4 Analysis of the Power Flow 69
6.5 Conclusion 70
References 73
7 Simulation and Nonlinear Analysis of Panel Flutter with Thermal Effects in Supersonic Flow 74
7.1 Introduction 74
7.2 Governing Equation 75
7.2.1 Dynamic Loads and Heating 76
7.2.2 Solution of Heat Transfer 76
7.2.3 Governing Equations 77
7.2.4 Galerkin Method 78
7.3 Numerical Results 80
7.3.1 Rf =0, Mach Number as the Bifurcation Parameter 80
7.3.2 Rf =0.056, Mach Number as the Bifurcation Parameter 82
7.3.3 M=2, Rf as Bifurcation Parameter 85
7.4 Conclusions 87
References 89
8 A Parameter Study of a Machine Tool with Multiple Boundaries 90
8.1 Introduction 90
8.2 Structured Motions by the Mapping Technique 91
8.3 Domains and Boundaries 94
8.4 Motion Switch Ability Conditions 95
8.5 Parameter Study of (e, ) 96
8.6 Numerical Prediction of Eccentricity Frequency 100
8.7 Summary and Conclusions 103
References 106
9 A New Friction Model for Evaluating Energy Dissipation in Carbon Nanotube-Based Composites 108
9.1 Introduction 108
9.2 Vibration Model 110
9.3 Results and Discussion 112
9.3.1 Excitation Frequency and Binding Strength 113
9.3.2 Excitation's Amplitude 114
9.3.3 Compliance Number M 114
9.4 Summary 116
References 116
10 Nonlinear Response in a Rotor System With a Coulomb Spline 118
10.1 Introduction 118
10.2 Mathematical Modeling 119
10.3 Analysis 121
10.4 Conclusion 133
References 133
11 The Influence of the Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing 134
11.1 Introduction 134
11.2 The Rotor/Stator Model with Cross-Coupling Effects 135
11.3 The Solution and the Stability Analysis 136
11.3.1 Synchronous Full Annular Rub Solution 137
11.3.2 Stability Analysis 138
11.4 The Cross-Coupling Effects on the Stability 140
11.4.1 Stability Domains in the Plane of - 141
11.4.2 Stability Domains in the Plane of -sr 142
11.4.3 Stability Domains in the Plane of -Msr 143
11.5 Conclusions 144
References 145
Part II Time-delay Systems 146
12 Some Control Studies of Dynamical Systems with Time Delay 147
12.1 Introduction 147
12.2 Methods of Solution 149
12.2.1 Semi-Discretization 149
12.2.2 Continuous Time Approximation 151
12.3 Spectral Properties of the Methods 152
12.4 Control Formulations 155
12.4.1 Full-State Feedback Optimal Control 155
12.4.2 Output Feedback Optimal Control 155
12.4.3 Optimal Feedback Gains via Mapping 156
12.5 Supervisory Control 157
12.6 Numerical Examples 158
12.6.1 Linear Time Invariant System 158
12.6.2 Periodic System 159
12.6.3 An Experimental Example 164
12.7 Concluding Remarks 166
References 166
13 Stability and Hopf Bifurcation Analysis in Synaptically Coupled FHN Neurons with Two Time Delays 168
13.1 Introduction 168
13.2 Stability Analysis 170
13.3 The Direction and Stability of Hopf Bifurcation 174
References 177
14 On the Feedback Controlling of the Neuronal Systemwith Time Delay 179
14.1 Introduction 179
14.2 HR Model Neuron with Time Delay 180
14.3 Influence of Synaptic Intensity on the Neuronal Discharge 181
14.4 Influence of Time Delay on the Neuronal Discharge 182
14.5 Controlling Chaotic Discharge by Feedback Control with Time Delay 182
14.6 Conclusions 183
References 184
15 Control of Erosion of Safe Basins in a Single Degree of Freedom Yaw System of a Ship with a Delayed Position Feedback 186
15.1 Introduction 186
15.2 Effects of the Time Delay on the Basin Boundary 187
15.3 Effects of the Delay on the Area of the Safe Basin 191
15.4 Conclusions 193
References 193
Part III Switching and Stochastic Dynamical Systems 195
16 On Periodic Flows of a 3-D Switching System with Many Subsystems 196
16.1 Introduction 196
16.2 Methodology for Periodic Flows 198
16.3 Analytical Predictions 203
16.4 Numerical Illustrations 207
16.5 Conclusions 207
References 207
17 Impulsive Control Induced Effects on Dynamicsof Complex Networks 209
17.1 Introduction 209
17.2 Main Results 210
17.2.1 Stability Analysis of the Presented Dynamical Networks with Different Dynamical Nodes with Impulsive Control 211
17.2.2 Impulsive Control Induced Effects on Dynamics of Complex Networks with Identical Dynamical Nodes 212
17.3 Numerical Simulations 217
17.4 Conclusion 221
References 221
18 Study on Synchronization of Two Identical Uncoupled Neurons Induced by Noise 223
18.1 Introduction 223
18.2 Numerical Results and Discussion 224
18.3 Conclusions 227
References 227
19 Non-equilibrium Phase Transitions in a Single-Mode Laser Model Driven by Non-Gaussian Noise 228
19.1 Introduction 228
19.2 Approximate Stationary Probability Distribution 230
19.3 General Analysis 232
19.4 Conclusion Remarks 234
References 235
20 Dynamical Properties of Intensity Fluctuation of Saturation Laser Model Driven by Cross-Correlated Additive and Multiplicative Noises 237
20.1 Introduction 237
20.2 Stationary Probability Distribution and Relaxation Time and Correlated Function 239
20.3 Discussion and Conclusion 244
References 252
21 Empirical Mode Decomposition Based on Bistable Stochastic Resonance Denoising 254
21.1 Introduction 254
21.2 Empirical Mode Decomposition 255
21.2.1 IMFs 255
21.2.2 The Sifting Processing 255
21.2.3 Examples 256
21.3 Stochastive Resonance 257
21.3.1 Bistable System 258
21.3.2 Examples 259
21.4 Test of EMD Based on SR Denoising 260
21.5 Conclusion 261
References 262
Part IV Classic Vibrations and Control 263
22 Order Reduction of a Two-Span Rotor-Bearing System Via the Predictor-Corrector Galerkin Method 264
22.1 Introduction 264
22.2 Modeling of the Two-Span Rotor-Bearing System 266
22.3 The Predictor-Corrector Galerkin Method 267
22.4 Numerical Results and Discussion 269
22.5 Conclusions 273
References 273
23 Stiffness Nonlinearity Classification Using Morlet Wavelets 275
23.1 Introduction 275
23.2 Response of the Nonlinear System 276
23.3 Wavelet Transform 278
23.3.1 Wavelet Transform of the Response Signal 278
23.3.1.1 Standard Morlet Wavelet 279
23.3.1.2 Complete Morlet Wavelet 280
23.4 Illustration 281
23.5 Conclusions 283
References 284
24 Dynamics of Wire-Driven Machine Mechanisms: Literature Review 285
24.1 Introduction 285
24.2 Mechanical Properties of Wire Ropes 287
24.2.1 Young's Modulus 287
24.2.2 Damping 289
24.3 Dynamics Simulation of Wire Rope Systems 291
24.3.1 Belt–Pulley Systems 292
24.3.2 Complex Wire Rope Systems 293
24.4 Conclusions 297
References 297
25 Dynamics of Wire-Driven Machine Mechanisms, Part II: Theory and Applications 299
25.1 Introduction 299
25.2 Wire Mechanics 301
25.2.1 Wire Topology 301
25.2.2 Elastic Properties of Wire and Chain Elements 302
25.2.3 Kinematics of Wire Motion 304
25.2.4 Tension Dynamics 306
25.3 Boom Mechanics 307
25.4 Fluid Power Circuit 308
25.5 System Dynamics Examples 310
25.5.1 Hydraulic Winch System 311
25.5.2 Chain Driven Elevating Boom 313
25.6 Conclusion 316
References 317
26 On Analytical Methods for Vibrations of Soils and Foundations 318
26.1 Introduction 318
26.2 Surface Response Due to Concentrated Forces 319
26.3 Dynamic Response of Foundations 321
26.3.1 Assumed Contact Stress Distributions 321
26.3.2 Mixed Boundary Value Problems 322
26.3.3 Lumped Parameter Models 329
26.3.4 Computational Methods 330
26.4 Coupled Vibrations of Foundations 331
26.5 Interactions Between Foundations 332
26.6 Experimental Studies 333
26.7 Conclusions 334
References 334
27 Inversely Found Elastic and Dimensional Properties 340
27.1 Introduction 340
27.2 Computational Overview 341
27.3 Simplifying Features 346
27.4 Graphical Relations Between the Cut-Off Frequencies and Cylinder Properties 347
27.5 Inversion Scheme 350
27.6 Illustrative Examples 351
27.6.1 Numerical Simulation 351
27.6.2 Experimental Corroboration 352
27.7 Conclusions 352
References 353
28 Nonlinear Self-Defined Truss Element Based on the Plane Truss Structure with Flexible Connector 355
28.1 Introduction 355
28.2 Finite-Element Dynamic Modeling 357
28.2.1 Introduction of the Structure 357
28.2.2 Motion Equations of the Self-Defined Truss Element 358
28.3 Nonlinear Nodal Forces 359
28.4 The Numerical Solution Method and Example 360
28.4.1 The Numerical Solution Method 360
28.4.2 Numerical Example 360
28.4.2.1 Random Excitation 361
28.4.2.2 Contact Stiffness 363
28.4.2.3 Clearance 365
28.5 Conclusions 367
References 368
29 Complex Frequency Analysis of an Axially Moving String with Multiple Attached Oscillators by Using Green's Function Method 369
29.1 Introduction 369
29.2 Formulation of the Eigenvalue Problem 370
29.3 Construction of Green's Function 372
29.4 Results and Discussions 373
29.5 Conclusion 378
References 379
30 Model Reduction on Inertial Manifolds of Navier–Stokes Equations Through Multi-scale Finite Element 380
30.1 Introduction 380
30.2 Multilevel Finite Element Method 383
30.3 The Weak Form of the Navier–Stokes Equations 385
30.4 Numerical Methods 386
30.5 Large Eddy and Small Eddy Components 387
30.6 Numerical Examples 389
30.7 Concluding Remarks 391
References 392
31 Diesel Engine Condition Classification Based on Mechanical Dynamics and Time-Frequency Image Processing 394
31.1 Introduction 394
31.2 Diesel Engine Structure Dynamics 395
31.3 Classification by Time Frequency Image 398
31.3.1 Hilbert Time-Frequency Spectrum 398
31.3.2 Euclidean Distance 399
31.3.3 Condition Classification 400
31.4 Example Analysis 400
31.4.1 Experiment 400
31.4.2 Vibration Signal Analysis 401
31.4.3 Hilbert Spectrum Recognition 401
31.4.4 ED Discriminant 403
31.5 Conclusions 405
References 405
32 Input Design for Systems Under Identification as Applied to Ultrasonic Transducers 406
32.1 Introduction 406
32.2 Input Design Formulation for ARX Models 407
32.3 System Identification Algorithm 410
32.4 Input Design for System Identification 412
32.5 Experimental Results 413
32.6 Conclusions and Recommendations 416
References 417
33 Development of a Control System for Automating of SpiralConcentrators in Coal Preparation Plants 418
33.1 Introduction 418
33.2 Technical Approach 421
33.2.1 Metal Arms and Wheatstone Bridge 422
33.2.2 Differential Amplifiers 423
33.2.3 PIC Microcontroller and Motor 423
33.2.4 Demonstration 424
33.2.5 Assembly 425
33.3 Mechanical System Design 425
33.4 The Need for a Model 428
33.5 Conclusions 431
References 432
34 On the Rough Number Computation and the Ada Language 433
34.1 Introduction 433
34.2 Some Basic Algebraic Structures 434
34.3 The Structure of Dyadic Numbers 436
34.4 The Structure of Rough Numbers 437
34.5 Hierarchical Classes and Computation 438
34.6 The Ada Language and Rough Number Computation 440
34.7 Conclusions 442
References 443
Author Index 444
Subject Index 455