Applied Asymptotic Methods in Nonlinear Oscillations
Many dynamical systems
are described by differential equations that can be separated into one part, containing linear terms with constant coefficients,
and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly
nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called
quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic
oscillators in the absence of perturbations. This book is devoted primarily to applied asymptotic methods in nonlinear oscillations
which are associated with the names of N. M. Krylov, N. N. Bogoli- ubov and Yu. A. Mitropolskii. The advantages of the present
methods are their simplicity, especially for computing higher approximations, and their applicability to a large class of
quasi-linear problems. In this book, we confine ourselves basi- cally to the scheme proposed by Krylov, Bogoliubov as stated
in the monographs [6,211. We use these methods, and also develop and improve them for solving new problems and new classes
of nonlinear differential equations. Although these methods have many applications in Mechanics, Physics and Technique, we
will illustrate them only with examples which clearly show their strength and which are themselves of great interest. A certain
amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate
course on nonlinear oscillations.
Preface. 1. Free Oscillations of Quasi-Linear Systems. 2. Self-Excited Oscillations.
3. Forced Oscillations. 4. Parametrically-Excited Oscillations. 5. Interaction of Nonlinear Oscillations. 6. Averaging Method.
Appendix 1: Principal Coordinates. Appendix 2: Some Trigonometric Formulae Often Used in the Averaging Method. References.
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