Non-linear oscillations
Abstract
The mathematical pendulum is used to provide a survey of free and forced oscillations in damped and undamped systems. This simple model is employed to present illustrations for and comparisons between the various approximation schemes. A summary of the Liapunov stability theory is provided. The first and the second method of Liapunov are explained for autonomous as well as for nonautonomous systems. Here, a basic familiarity with the theory of linear oscillations is assumed. La Salle's theorem about the stability of invariant domains is explained in terms of illustrative examples. Self-excited oscillations are examined, taking into account such oscillations in mechanical and electrical systems, analytical approximation methods for the computation of self-excited oscillations, analytical criteria for the existence of limit cycles, forced oscillations in self-excited systems, and self-excited oscillations in systems with several degrees of freedom. Attention is given to Hamiltonian systems and an introduction to the theory of optimal control is provided.
- Publication:
-
Oxford and New York
- Pub Date:
- 1981
- Bibcode:
- 1981oxny.book.....H
- Keywords:
-
- Hamilton-Jacobi Equation;
- Nonlinear Systems;
- Optimal Control;
- Pendulums;
- Self Oscillation;
- Stable Oscillations;
- Undamped Oscillations;
- Degrees Of Freedom;
- Forced Vibration;
- Free Vibration;
- Hamiltonian Functions;
- Liapunov Functions;
- Mechanical Oscillators;
- Perturbation Theory;
- Self Excitation;
- Systems Stability;
- Vibration Damping;
- Physics (General)