Dynamics for Engineers
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.
- 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.
I: OBTAINING DIFFERENTIAL EQUATIONS FOR PHYSICAL SYSTEMS.
1. Introduction to System Elements.
1.2 Chapter summary.
2. Obtaining Differential Equations for Mechanical Systems by the Newtonian Method.
2.1 The Configuration Space.
2.3 Differential Equations from Newtons Laws.
2.4 Practical Difficulties with the Newtonian Formalism.
2.5 Chapter Summary.
3. Differential Equations of Electrical Circuits from Kirchoff’s Laws.
3.1 Kirchoff’s Laws about Current and Voltage.
3.2 The Mesh Current and Node Voltage Methods.
3.3 Using Graph Theory to Obtain the Minimal Set of Equations.
3.4 Chapter Summary.
4. The Lagrangian Formalism.
4.1 Elements of the Lagrangian Approach.
4.2 Obtaining Dynamical Equations by Lagrangian Method.
4.3 The Principle of Least Action.
4.4 Lagrangian Method Applied to Electrical Circuits.
4.5 Systems with External Forces or Electromotive Forces.
4.6 Systems with Resistance or Friction.
4.7 Accounting for Current Sources.
4.8 Modeling Mutual Inductances.
4.9 A General Methodology for Electrical Networks.
4.10 Modeling Coulomb Friction.
4.11 Chapter Summary.
5. Obtaining First-Order Equations.
5.1 First-Order Equations from the Lagrangian Method.
5.2 The Hamiltonian Formalism.
5.3 Chapter Summary.
6. Unified Modelling of Systems Through the Language of Bond Graphs.
6.2 The Basic Concept.
6.3 One-port Elements.
6.4 The Junctions.
6.5 Junctions in Mechanical Systems.
6.6 Numbering of Bonds.
6.7 Reference Power Directions.
6.8 Two-port Elements.
6.9 The Concept of Causality.
6.10 Differential Causality.
6.11 Obtaining Differential Equations from Bond Graphs.
6.12 Alternative Methods of Creating System Bond Graphs.
6.13 Algebraic Loops.
6.16 Equations for Systems with Differential Causality.
6.17 Bond Graph Software.
6.18 Chapter Summary.
II: SOLVING DIFFERENTIAL EQUATIONS AND UNDERSTANDING DYNAMICS.
7. Numerical Solution of Differential Equations.
7.1 The Basic Method, and the Techniques of Approximation.
7.2 Methods to Balance Accuracy and Computation Time.
7.3 Chapter Summary.
8. Dynamics in the State Space.
8.1 The State Space.
8.2 Vector Field.
8.3 Local Linearization Around Equilibrium Points.
8.4 Chapter Summary.
9. Solutions for a System of First-Order Linear Differential Equations.
9.1 Solution of a First-Order Linear Differential Equation.
9.2 Solution of a System of Two First-Order Linear Differential Equations.
9.3 Eigenvalues and Eigenvectors.
9.4 Using Eigenvalues and Eigenvectors for Solving Differential Equations
9.5 Solution of a Single Second Order Differential Equation.
9.6 Systems with Higher Dimensions.
9.7 Chapter Summary.
10. Linear Systems with External Input.
10.1 Constant external input.
10.2 When the forcing function is a square wave.
10.3 Sinusoidal forcing function.
10.4 Other forms of excitation function.
10.5 Chapter Summary.
11. Dynamics of Nonlinear Systems.
11.1 All systems of practical interest are nonlinear.
11.2 Vector Fields for Nonlinear Systems.
11.3 Attractors in nonlinear systems.
11.4 Different types of periodic orbits in a nonlinear system.
11.7 Stability of limit cycles.
11.8 Chapter Summary.
12. Discrete-time Dynamical Systems.
12.1 The Poincar´e Section.
12.2 Obtaining a discrete-time model.
12.3 Dynamics of Discrete-Time Systems.
12.4 One-dimensional maps.
12.6 Saddle-node bifurcation.
12.7 Period-doubling bifurcation.
12.8 Periodic windows.
12.9 Two-dimensional maps.
12.10 Bifurcations in 2-D discrete-time systems.
12.11 Global dynamics of discrete-time systems.
12.12 Chapter Summary.
Provides methodologies for modeling mechanical systems, electrical circuits, and electromechanical systems.
Outlines the methods and techniques for translating a physical problem into mathematical language.
Each chapter contains:
- Solved examples,
- A set of problems,
- A chapter summary.
Shows methods of analysing the dynamical behavior of systems through first order differential equations.
Innovative in dealing with nonlinear systems at an early stage.