Dynamics for Engineers

Modelling and analysis of dynamical systems is a widespread practice as it is important for engineers to know how a given physical or engineering system will behave under specific circumstances.This text provides a comprehensive and systematic introduction to the methods and techniques used for translating physical problems into mathematical language, focusing on both linear and nonlinear systems. Highly practical in its approach, with solved examples, summaries, and sets of problems for each chapter, __Dynamics for Engineers__ covers all aspects of the modelling and analysis of dynamical systems. Key features: * Introduces the Newtonian, Lagrangian, Hamiltonian, and Bond Graph methodologies, and illustrates how these can be effectively used for obtaining differential equations for a wide variety of mechanical, electrical, and electromechanical systems. * Develops a geometric understanding of the dynamics of physical systems by introducing the state space, and the character of the vector field around equilibrium points. * Sets out features of the dynamics of nonlinear systems, such as like limit cycles, high-period orbits, and chaotic orbits. * Establishes methodologies for formulating discrete-time models, and for developing dynamics in discrete state space. Senior undergraduate and graduate students in electrical, mechanical, civil, aeronautical and allied branches of engineering will find this book a valuable resource, as will lecturers in system modelling, analysis, control and design. This text will also be useful for students and engineers in the field of mechatronics. Dynamics for Engineers......Page 3 Contents......Page 7 Preface......Page 13 I Obtaining differential equations for physical systems......Page 17 1.1.1 The inertial element......Page 19 1.1.2 The compliant element......Page 21 1.1.3 The resistive element......Page 22 1.1.4 The voltage source and externally impressed force......Page 23 1.1.5 The current source and externally impressed sources of flow......Page 24 1.2 Chapter Summary......Page 25 Further Reading......Page 26 2 Obtaining Differential Equations for Mechanical Systems by the Newtonian Method......Page 27 2.1 The Configuration Space......Page 28 2.2 Constraints......Page 29 2.3 Differential Equations from Newton’s Laws......Page 31 2.4 Practical Difficulties with the Newtonian Formalism......Page 32 2.5 Chapter Summary......Page 33 Problems......Page 34 3.1 Kirchoff’s Laws about Current and Voltage......Page 37 3.2 The Mesh Current and Node Voltage Methods......Page 39 3.3 Using Graph Theory to Obtain the Minimal Set of Equations......Page 44 3.3.1 Kirchoff’s laws relating to loops and cutsets......Page 45 3.3.2 Tree and co-tree......Page 46 3.3.4 The choice of the state variables......Page 47 3.3.5 Derivation of differential equations......Page 48 3.4 Chapter Summary......Page 57 Problems......Page 58 4 The Lagrangian Formalism......Page 61 4.1.2 The concept of admissible motions......Page 62 4.1.3 The generalized coordinates......Page 64 4.1.4 Dynamical equations in terms of energies......Page 66 4.2 Obtaining Dynamical Equations by Lagrangian Method......Page 69 4.3 The Principle of Least Action......Page 78 4.4 Lagrangian Method Applied to Electrical Circuits......Page 83 4.5 Systems with External Forces or Electromotive Forces......Page 84 4.6 Systems with Resistance or Friction......Page 86 4.7 Accounting for Current Sources......Page 90 4.8 Modelling Mutual Inductances......Page 91 4.9 A General Methodology for Electrical Networks......Page 93 4.10 Modelling Coulomb Friction......Page 94 Further Reading......Page 96 Problems......Page 97 5.1 First-order Equations from the Lagrangian Method......Page 101 5.2 The Hamiltonian Formalism......Page 104 5.3 Chapter Summary......Page 116 Problems......Page 117 6.1 Introduction......Page 121 6.3 One-port Elements......Page 122 6.4 The Junctions......Page 124 6.5 Junctions in Mechanical Systems......Page 126 6.7 Reference Power Directions......Page 128 6.8 Two-port Elements......Page 132 6.9 The Concept of Causality......Page 134 6.10 Differential Causality......Page 137 6.11 Obtaining Differential Equations from Bond Graphs......Page 139 6.12 Alternative Methods of Creating System Bond Graphs......Page 144 6.12.1 Electrical systems......Page 145 6.12.2 Mechanical systems......Page 146 6.13 Algebraic Loops......Page 148 6.14 Fields......Page 150 6.15 Activation......Page 154 6.16 Equations for Systems with Differential Causality......Page 157 6.17 Bond Graph Software......Page 158 6.18 Chapter Summary......Page 160 Problems......Page 161 II Solving differential equations and understanding dynamics......Page 167 7.1 The Basic Method and the Techniques of Approximation......Page 169 7.1.1 The Euler method......Page 170 7.1.2 The trapezoidal rule......Page 171 7.1.3 The fourth-order Runge-Kutta formula......Page 172 7.2 Methods to Balance Accuracy and Computation Time......Page 174 Further Reading......Page 175 Problems......Page 176 8 Dynamics in the State Space......Page 177 8.1 The State Space......Page 178 8.3 Local Linearization Around Equilibrium Points......Page 179 Problems......Page 182 9 Solutions for a System of First-order Linear Differential Equations......Page 185 9.1 Solution of a First-order Linear Differential Equation......Page 186 9.2 Solution of a System of Two First-order Linear Differential Equations......Page 187 9.3 Eigenvalues and Eigenvectors......Page 188 9.4 Using Eigenvalues and Eigenvectors for Solving Differential Equations......Page 189 9.4.1 Eigenvalues real and distinct......Page 190 9.4.2 Eigenvalues complex conjugate......Page 193 9.4.3 Eigenvalues purely imaginary......Page 196 9.4.4 Eigenvalues real and equal......Page 200 9.5 Solution of a Single Second-order Differential Equation......Page 202 9.6 Systems with Higher Dimensions......Page 205 9.7 Chapter Summary......Page 210 Problems......Page 211 10.1.1 Constant voltage applied to an RL circuit......Page 213 10.1.2 The concept of time constant......Page 215 10.1.3 Constant voltage applied to an RC circuit......Page 217 10.1.4 Constant voltage applied to an RLC circuit......Page 218 10.2 When the Forcing Function is a Square Wave......Page 219 10.3.1 First-order systems excited by sinusoidal source......Page 220 10.3.2 Second-order system excited by sinusoidal source......Page 226 10.4 Other Forms of Excitation Function......Page 229 Problems......Page 231 11.1 All Systems of Practical Interest are Nonlinear......Page 235 11.2 Vector Fields for Nonlinear Systems......Page 236 11.3 Attractors in Nonlinear Systems......Page 243 11.4 Limit Cycle......Page 244 11.5 Different Types of Periodic Orbits in a Nonlinear System......Page 245 11.6 Chaos......Page 247 11.7 Quasiperiodicity......Page 250 11.9 Chapter Summary......Page 252 Problems......Page 253 12.1 The Poincaré Section......Page 257 12.2 Obtaining a Discrete-time Model......Page 261 12.4 One-dimensional Maps......Page 264 12.5 Bifurcations......Page 271 12.6 Saddle-node Bifurcation......Page 272 12.7 Period-doubling Bifurcation......Page 274 12.8 Periodic Windows......Page 276 12.9 Two-dimensional Maps......Page 277 12.10 Bifurcations in 2-D Discrete-time Systems......Page 280 12.11 Global Dynamics of Discrete-time Systems......Page 285 12.12 Chapter Summary......Page 288 Problems......Page 289 Index......Page 293

Modelling and analysis of dynamical systems is a widespread practice as it is important for engineers to know how a given physical or engineering system will behave under specific circumstances.

This text provides a comprehensive and systematic introduction to the methods and techniques used for translating physical problems into mathematical language, focusing on both linear and nonlinear systems. Highly practical in its approach, with solved examples, summaries, and sets of problems for each chapter, Dynamics for Engineers covers all aspects of the modelling and analysis of dynamical systems.

Key features:

• Introduces the Newtonian, Lagrangian, Hamiltonian, and Bond Graph methodologies, and illustrates how these can be effectively used for obtaining differential equations for a wide variety of mechanical, electrical, and electromechanical systems.
• Develops a geometric understanding of the dynamics of physical systems by introducing the state space, and the character of the vector field around equilibrium points.
• Sets out features of the dynamics of nonlinear systems, such as like limit cycles, high-period orbits, and chaotic orbits.
• Establishes methodologies for formulating discrete-time models, and for developing dynamics in discrete state space.

Senior undergraduate and graduate students in electrical, mechanical, civil, aeronautical and allied branches of engineering will find this book a valuable resource, as will lecturers in system modelling, analysis, control and design. This text will also be useful for students and engineers in the field of mechatronics.

"This text provides a comprehensive and systematic introduction to the methods and techniques used for translating physical problems into mathematical language, focusing on both linear and nonlinear systems. Highly practical in its approach, with solved examples, summaries, and sets of problems for each chapter, Dynamics for Engineers covers all aspects of the modelling and analysis of dynamical systems." "Senior undergraduate and graduate students in electrical, mechanical, civil, aeronautical and allied branches of engineering will find this book a valuable resource, as will lecturers in system modelling, analysis, control and design. This text will also be useful for students and engineers in the field of mechatronics."--Jacket