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Over the past few years there has been an increasing realization that most of the physical systems are nonlinear, and linearity is a very special case. Most of the systems that an engineer has to deal with are nonlinear, and nonlinear dynamics pervades the engineer's workplace. But the training of the engineer often renders him/her hopelessly short of the challenges. This book is aimed to address this problem, first by introducing those methodologies of system modeling that make no reference to linearity, and then by developing an understanding of dynamics where linear system description is put in proper perspective --- as local linear approximation in the neighborhood of an equilibrium point.
The book is divided into two parts. In the first part, the methods and techniques for translating a physical problem into mathematical language by formulating differential equations are introduced. Some part of it draws from classical mechanics, but instead of working with particles and groups of particles as in classical mechanics textbooks, I have dealt mainly with electrical and mechanical systems so that the ideas developed can easily be applied in engineering.
The basic methodology of deriving differential equations follows from the Newton's laws for mechanical systems and from the Kirchoff's laws for electrical circuits. However, the Newtonian method in its pure form is not suitable for handling practical systems. In Chapter 2, the Newtonian formalism is introduced, and the practical problems of this method are illustrated. In Chapter 3, the application of the Kirchoff's laws in derivation of dynamical equations for electrical circuits is considered. The mesh current method, the node voltage method, and the more general graph theoretic methods are introduced. This chapter is particularly useful for students of the electrical sciences, and may be skipped by those of the other disciplines without breaking the continuity of exposition.
The Lagrangian method is introduced in Chapter 4, and its application in handling electrical, mechanical, and electromechanical systems is illustrated. To show the basic unity of dynamical systems, an equivalence of the mechanical and the electrical systems is shown at relevant places. In order to make the model amenable to the solution techniques to be introduced in the later part of the book, first order differential equations must be obtained. Chapter 5 shows how the Lagrangian equations can be used to obtain the equations in first order through the definition of conjugate momenta. The Hamiltonian formalism --- which allows one to obtain the first-order equations directly --- is generally applied to conservative systems. I have shown how this approach, with the inclusion of the dissipative term, can be made useful in handling engineering systems also.
Then I have introduced the Bond Graph methodology --- a powerful technique for obtaining first order differential equations for a wide variety of physical and engineering systems. This method algorithmizes the process of obtaining equations, so that once the bond graph of a system is formulated, computer programs can handle the job of obtaining equations and simulating the model. This method is widely applicable to engineering problems, but has not yet entered the mainstream engineering curriculum. Engineering students are often found to put off learning this powerful technique for some day ``when there is time at hand.'' In this book the basic elements of this method are introduced in a span of only 40 pages without getting into the nitty-gritty of modeling complicated systems, so that the reader can get a feel of bond graph based system modeling without spending too much time on it.
The main advantage of the methods introduced in Part-1 is that they are equally applicable to both linear and nonlinear systems. I believe, these will constitute the basic modeling tools in the hands of the scientist and the engineer in future.
After the differential equations are obtained, one has to solve them, and from the solutions, one has to understand how a given system is going to behave in specific circumstances. The second part of the book is aimed at developing an intuitive understanding of the dynamics of physical systems. For this, the concepts of state space and vector field are introduced, and a geometric view of the dynamics in the state space is provided --- since with the availability of computers and computer graphics, that viewpoint has become visualizable and intuitively appealing.
The method of locally linearizing a nonlinear system through the Jacobian matrix is introduced, and the dynamics of linear systems are then analyzed. There are many approaches to solving linear differential equations, and I have chosen that one which allows the relationship of the eigenvalues and eigenvectors with system dynamics to be highlighted. This gives a geometric view of the dynamics, and facilitates a smooth transition to the understanding of the dynamics of nonlinear systems. The special features of the dynamics of nonlinear systems, like limit cycles, high-period orbits and chaotic orbits, are then discussed.
Even though limit cycles find wide application in engineering wherever some oscillatory behavior is desired (as in oscillators, power electronics, etc.), treatment of the stability of limit cycles is rarely found in dynamics or control textbooks. An approach to this problem, developed in nonlinear dynamics, has now reached sufficient maturity to deserve being taught at the undergraduate level. In this book I have illustrated the powerful method of obtaining discrete-time models or ``maps,'' developed by the French mathematician Henri Poincare, and discussed the various ways a limit cycle can lose stability. In that process, I have included an exposition on the dynamics of discrete-time systems that is finding increasing application in engineering.
The content of this book is suitable for teaching undergraduate students of all branches of engineering at the second- or third-year level. Though the scope of the topic is much wider, the material presented in this book is limited to the extent that can be taught in a one-semester course --- which, I feel is a proper supplement of a control systems course. Some parts of it can also be integrated in an existing course on control theory.
The book addresses dynamics problems coming from a wide range of engineering disciplines, and can be used by mechanical engineers, electrical engineers, aeronautical engineers, civil engineers etc. For example, an electrical engineer who is not interested in mechanical systems may read Chapters 1 and 3 to pick up the methods of obtaining differential equations specific to electrical systems, and then may proceed on to Chapter 7 onward to develop ideas of dynamics. For those who have to deal with electromechanical systems like relays, motors and other electrically actuated mechanical systems, Chapters 4, 5, and 6 will be particularly useful. The students of mechanical, aeronautical and allied engineering disciplines, on the other hand, may skip Chapter 3.
This book can also be used by those who are primarily interested in linear systems, because the methods of obtaining differential equations are equally applicable to linear systems. Moreover, Chapters 9 and 10 specifically deal with solving linear differential equations and developing a visual impression of linear dynamics. Courses with such leaning may leave Chapters 11 and 12 as options or for a later study.
It should be kept in mind that this book is not meant to be an exhaustive treatise on system modeling. The purpose of Part-1 is to acquaint the engineering students with the general modeling approaches, which they can later pursue to any level of detail following the lead provided at the end of each chapter. This is also not a typical nonlinear dynamics book, and includes only those aspects of nonlinear dynamics which a 21st century engineer should be exposed to. I have deliberately chosen not to burden the students with too much material.