Dynamics for Engineers

1   Introduction to System Elements.

1.1   Introduction.

1.2   Chapter summary.

2   The Newtonian Method.

2.1   The Configuration Space.

2.2   Constraints.

2.3   Differential Equations from Newtons Laws.

2.4   Practical Difficulties with the Newtonian Formalism.

2.5   Chapter Summary.

3   Differential Equations by 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   The Language of Bond Graphs.

6.1   Introduction.

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.

6.14 Fields.

6.15 Activation.

6.16 Equations for Systems with Differential Causality.

6.17 Bond Graph Software.

6.18 Chapter Summary.

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   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.

11.5  Chaos.

11.6  Quasiperiodicity.

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.

12.5  Bifurcations.

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.