Modelling and analysis of dynamical systems is a widespreadpractice as it is important for engineers to know how a givenphysical or engineering system will behave under specificcircumstances.
This text provides a comprehensive and systematic introductionto the methods and techniques used for translating physicalproblems into mathematical language, focusing on both linear andnonlinear systems. Highly practical in its approach, with solvedexamples, summaries, and sets of problems for each chapter,Dynamics for Engineers covers all aspects of the modellingand analysis of dynamical systems.
- Introduces the Newtonian, Lagrangian, Hamiltonian, and BondGraph methodologies, and illustrates how these can be effectivelyused for obtaining differential equations for a wide variety ofmechanical, electrical, and electromechanical systems.
- Develops a geometric understanding of the dynamics of physicalsystems by introducing the state space, and the character of thevector field around equilibrium points.
- Sets out features of the dynamics of nonlinear systems, such aslike 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 engineeringwill find this book a valuable resource, as will lecturers insystem modelling, analysis, control and design. This text will alsobe useful for students and engineers in the field ofmechatronics.
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About the Author
Soumitro Banerjee, Associate Professor, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, IndiaSoumitro 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.
Table of Contents
1 Introduction to System Elements.
1.2 Chapter summary.
2 The Newtonian Method.
2.1 The Configuration Space.
2.3 Differential Equations from Newtons Laws.
2.4 Practical Difficulties with the NewtonianFormalism.
2.5 Chapter Summary.
3 Differential Equations by Kirchoff’sLaws.
3.1 Kirchoff’s Laws about Current andVoltage.
3.2 The Mesh Current and Node Voltage Methods.
3.3 Using Graph Theory to Obtain the Minimal Set ofEquations.
3.4 Chapter Summary.
4 The Lagrangian Formalism.
4.1 Elements of the Lagrangian Approach.
4.2 Obtaining Dynamical Equations by LagrangianMethod.
4.3 The Principle of Least Action.
4.4 Lagrangian Method Applied to ElectricalCircuits.
4.5 Systems with External Forces or ElectromotiveForces.
4.6 Systems with Resistance or Friction.
4.7 Accounting for Current Sources.
4.8 Modeling Mutual Inductances.
4.9 A General Methodology for ElectricalNetworks.
4.10 Modeling Coulomb Friction.
4.11 Chapter Summary.
5 Obtaining First Order Equations.
5.1 First Order Equations from the LagrangianMethod.
5.2 The Hamiltonian Formalism.
5.3 Chapter Summary.
6 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.
7 Numerical Solution of DifferentialEquations.
7.1 The Basic Method, and the Techniques ofApproximation.
7.2 Methods to Balance Accuracy and ComputationTime.
7.3 Chapter Summary.
8 Dynamics in the State Space.
8.1 The State Space.
8.2 Vector Field.
8.3 Local Linearization Around EquilibriumPoints.
8.4 Chapter Summary.
9 Linear Differential Equations.
9.1 Solution of a First-Order Linear DifferentialEquation.
9.2 Solution of a System of Two First-Order LinearDifferential Equations.
9.3 Eigenvalues and Eigenvectors.
9.4 Using Eigenvalues and Eigenvectors for SolvingDifferential Equations
9.5 Solution of a Single Second Order DifferentialEquation.
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 nonlinearsystem.
11.7 Stability of limit cycles.
11.8 Chapter Summary.
12 Discrete-time Dynamical Systems.
12.1 The Poincarè 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.