Dynamics For Engineers

Thoroughly class-tested, this book focuses on the modelling and analysis of linear as well as nonlinear dynamical systems with emphasis on electrical, mechanical, and electromechanical systems. The readers are first introduced to the methods and techniques for translating a physical problem into mathematical language by formulating differential equations.Nowtonian, Lagrangian, and Hamiltonian methods are introduced, along with the method of bond graphs. Solutions of differential equations, both by analytical as well as numerical methods then follow, with an emphasis on developing a geometric understanding of dynamics in the state space. As well as drawing from the methods used in the study of nonlinear dynamics, this text also illustrates the application of these methods in the context of engineering systems. This book also covers discrete-time dynamical systems and methods of analyzing discrete dynamics. Each chapter contains numerous solved examples, a summary, and a set of problems.

Soumitro Banerjee, Associate Professor, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, India
Soumitro Banerjee has been at the Indian Institute of Technology, in the Department of Electrical Engineering since 1985. He currently teaches courses on 'Dynamics of Physical Systems', 'Signals and Networks', 'Energy Resources and Technology', 'Fractals, Chaos and Dynamical Systems' and 'Nonconventional Electrical Power Generation'. His research interests include bifurcation theory and chaos, and he has written and co-written over 43 papers on these subjects.

Preface

xi

I Obtaining differential equations for physical systems

1

(150)

1 Introduction to System Elements

3

(8)

1.1 Introduction

3

(6)

1.1.1 The inertial element

3

(2)

1.1.2 The compliant element

5

(1)

1.1.3 The resistive element

6

(1)

1.1.4 The voltage source and externally impressed force

7

(1)

1.1.5 The current source and externally impressed sources of flow

8

(1)

1.2 Chapter Summary

9

(1)

Further Reading

10

(1)

2 Obtaining Differential Equations for Mechanical Systems by the Newtonian Method

11

(10)

2.1 The Configuration Space

12

(1)

2.2 Constraints

13

(2)

2.3 Differential Equations from Newton's Laws

15

(1)

2.4 Practical Difficulties with the Newtonian Formalism

16

(1)

2.5 Chapter Summary

17

(1)

Further Reading

18

(1)

Problems

18

(3)

3 Differential Equations of Electrical Circuits from Kirchoff's Laws

21

(24)

3.1 Kirchoff's Laws about Current and Voltage

21

(2)

3.2 The Mesh Current and Node Voltage Methods

23

(5)

3.3 Using Graph Theory to Obtain the Minimal Set of Equations

28

(13)

3.3.1 Kirchoff's laws relating to loops and cutsets

29

(1)

3.3.2 Tree and co-tree

30

(1)

3.3.3 The independent KCL and KVL equations

31

(1)

3.3.4 The choice of the state variables

31

(1)

3.3.5 Derivation of differential equations

32

(9)

3.4 Chapter Summary

41

(1)

Further Reading

42

(1)

Problems

42

(3)

4 The Lagrangian Formalism

45

(40)

4.1 Elements of the Lagrangian Approach

46

(7)

4.1.1 Motivation

46

(1)

4.1.2 The concept of admissible motions

46

(2)

4.1.3 The generalized coordinates

48

(2)

4.1.4 Dynamical equations in terms of energies

50

(3)

4.2 Obtaining Dynamical Equations by Lagrangian Method

53

(9)

4.3 The Principle of Least Action

62

(5)

4.4 Lagrangian Method Applied to Electrical Circuits

67

(1)

4.5 Systems with External Forces or Electromotive Forces

68

(2)

4.6 Systems with Resistance or Friction

70

(4)

4.7 Accounting for Current Sources

74

(1)

4.8 Modelling Mutual Inductances

75

(2)

4.9 A General Methodology for Electrical Networks

77

(1)

4.10 Modelling Coulomb Friction

78

(2)

4.11 Chapter Summary

80

(1)

Further Reading

80

(1)

Problems

81

(4)

5 Obtaining First-order Equations

85

(20)

5.1 First-order Equations from the Lagrangian Method

85

(3)

5.2 The Hamiltonian Formalism

88

(12)

5.3 Chapter Summary

100

(1)

Problems

101

(4)

6 Unified Modelling of Systems Through the Language of Bond Graphs

105

(46)

6.1 Introduction

105

(1)

6.2 The Basic Concept

106

(1)

6.3 One-port Elements

106

(2)

6.4 The Junctions

108

(2)

6.5 Junctions in Mechanical Systems

110

(2)

6.6 Numbering of Bonds

112

(1)

6.7 Reference Power Directions

112

(4)

6.8 Two-port Elements

116

(2)

6.9 The Concept of Causality

118

(3)

6.10 Differential Causality

121

(2)

6.11 Obtaining Differential Equations from Bond Graphs

123

(5)

6.12 Alternative Methods of Creating System Bond Graphs

128

(4)

6.12.1 Electrical systems

129

(1)

6.12.2 Mechanical systems

130

(2)

6.13 Algebraic Loops

132

(2)

6.14 Fields

134

(4)

6.15 Activation

138

(3)

6.16 Equations for Systems with Differential Causality

141

(1)

6.17 Bond Graph Software

142

(2)

6.18 Chapter Summary

144

(1)

Further Reading

145

(1)

Problems

145

(6)

II Solving differential equations and understanding dynamics

151

(126)

7 Numerical Solution of Differential Equations

153

(8)

7.1 The Basic Method and the Techniques of Approximation

153

(5)

7.1.1 The Euler method

154

(1)

7.1.2 The trapezoidal rule

155

(1)

7.1.3 The fourth-order Runge-Kutta formula

156

(2)

7.2 Methods to Balance Accuracy and Computation Time

158

(1)

7.3 Chapter Summary

159

(1)

Further Reading

159

(1)

Problems

160

(1)

8 Dynamics in the State Space

161

(8)

8.1 The State Space

162

(1)

8.2 Vector Field

163

(1)

8.3 Local Linearization Around Equilibrium Points

163

(3)

8.4 Chapter Summary

166

(1)

Problems

166

(3)

9 Solutions for a System of First-order Linear Differential Equations

169

(28)

9.1 Solution of a First-order Linear Differential Equation

170

(1)

9.2 Solution of a System of Two First-order Linear Differential Equations

171

(1)

9.3 Eigenvalues and Eigenvectors

172

(1)

9.4 Using Eigenvalues and Eigenvectors for Solving Differential Equations

173

(13)

9.4.1 Eigenvalues real and distinct

174

(3)

9.4.2 Eigenvalues complex conjugate

177

(3)

9.4.3 Eigenvalues purely imaginary

180

(4)

9.4.4 Eigenvalues real and equal

184

(2)

9.5 Solution of a Single Second-order Differential Equation

186

(3)

9.6 Systems with Higher Dimensions

189

(5)

9.7 Chapter Summary

194

(1)

Further Reading

195

(1)

Problems

195

(2)

10 Linear Systems with External Input

197

(22)

10.1 Constant External Input

197

(6)

10.1.1 Constant voltage applied to an RL circuit

197

(2)

10.1.2 The concept of time constant

199

(2)

10.1.3 Constant voltage applied to an RC circuit

201

(1)

10.1.4 Constant voltage applied to an RLC circuit

202

(1)

10.2 When the Forcing Function is a Square Wave

203

(1)

10.3 Sinusoidal Forcing Function

204

(9)

10.3.1 First-order systems excited by sinusoidal source

204

(6)

10.3.2 Second-order system excited by sinusoidal source

210

(3)

10.4 Other Forms of Excitation Function

213

(2)

10.5 Chapter Summary

215

(1)

Further Reading

215

(1)

Problems

215

(4)

11 Dynamics of Nonlinear Systems

219

(22)

11.1 All Systems of Practical Interest are Nonlinear

219

(1)

11.2 Vector Fields for Nonlinear Systems

220

(7)

11.3 Attractors in Nonlinear Systems

227

(1)

11.4 Limit Cycle

228

(1)

11.5 Different Types of Periodic Orbits in a Nonlinear System

229

(2)

11.6 Chaos

231

(3)

11.7 Quasiperiodicity

234

(2)

11.8 Stability of Limit Cycles

236

(1)

11.9 Chapter Summary

236

(1)

Further Reading

237

(1)

Problems

237

(4)

12 Discrete-time Dynamical Systems

241

(36)

12.1 The Poincaré Section

241

(4)

12.2 Obtaining a Discrete-time Model

245

(3)

12.3 Dynamics of Discrete-time Systems

248

(1)

12.4 One-dimensional Maps

248

(7)

12.5 Bifurcations

255

(1)

12.6 Saddle-node Bifurcation

256

(2)

12.7 Period-doubling Bifurcation

258

(2)

12.8 Periodic Windows

260

(1)

12.9 Two-dimensional Maps

261

(3)

12.10 Bifurcations in 2-D Discrete-time Systems

264

(5)

12.11 Global Dynamics of Discrete-time Systems

269

(3)

12.12 Chapter Summary

272

(1)

Further Reading

273

(1)

Problems

273

(4)

Index

277

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Dynamics For Engineers

Summary

Author Biography

Table of Contents

Supplemental Materials

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Dynamics For Engineers

Summary

Author Biography

Table of Contents

Supplemental Materials

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