The New Jersey Institute of Technology's
Electronic Theses & Dissertations Project

Title:	Analysis of oscillations in nonlinear systems using multiple input describing functions
Author:	Swern, Frederic Lee
View Online:	njit-etd1981-009 (v, 138 pages ~ 4.5 MB pdf)
Department:	Department of Electrical Engineering
Degree:	Doctor of Engineering Science
Program:	Electrical Engineering
Document Type:	Dissertation
Advisory Committee:	Meyer, Andrew Ulrich (Committee chair) Blackmore, Denis L. (Committee member) Cohen, Edwin (Committee member) Ball, W. H. Warren (Committee member)
Date:	1981
Keywords:	Nonlinear oscillations. System analysis.
Availability:	Unrestricted
Abstract:	Sufficient conditions for oscillation as well as absence of oscillations are presented for a class of systems containing one lumped linear element and a differentiable nonlinearity. The results are obtained by estimating the error inherent in using a describing function analysis. Contraction type arguments are used to show common topological properties of the describing function solution and the balance of first harmonic terms of the system. After discussing the describing function method, two theorems are presented regarding existence or nonexistence of oscillations from homotopic considerations. A graphical method for examining systems with power law nonlinearities is given using a parameter plane of frequency and amplitude. As an example, the method is applied to the Van der Pol oscillator when the linear element is sufficiently low pass. An analytical method is derived that is particularly easy to apply in the design of power law oscillators. It is shown that multiple input describing functions may be used in some cases for which the describing function method is inconclusive. The results obtained in estimating the amplitude and frequency of oscillation using dual input describing functions are compared to their single input counterparts for a number of examples. The class of nonlinearities for which the methods may be applied includes polynomial functions. It is shown that one can also apply similar techniques to systems containing jump discontinuities when the nonlinear element can be approximated arbitrarily closely by a continuous function. One can say that such a function is almost continuous. An ideal relay, a relay with deadzone and a staircase function are analyzed in this manner. In some systems, improved results are obtained by representing the nonlinearity as the sum of a bounded almost continuous function and a polynomial. All of the methods developed have been computerized. Numerous examples are presented to illustrate the application of the methods.