The frequency-response analysis of non-linear systems

Abstract

If a non-linear system is oscillating, either through external excitation or internal regenerative action, it may be possible to show that the waveform at the input to the non-linear elements in the system is approximately sinusoidal. In such a case the frequency-response analysis may be carried out on the assumption that all harmonic components generated by the non-linear element can be ignored.The paper discusses, with reference to feedback control systems, the results that can be obtained if this approximation is made. The amplitude, frequency and stability of steady self-excited oscillations are derived. Transient oscillations may be considered only in systems governed by second-order differential equations, but in these cases an analytic expression for the variation of frequency and damping with time, and hence for the full solution is derived. Several examples, all relating to “on-off” controllers, are given comparing the approximate and exact solutions; the accuracy obtained both for transient and steady oscillations is within about 10%.In an Appendix the necessary extension to the familiar j notation is developed and used to derive a criterion of stability.