An Approximate Method for Analysis of Stability and Step Response of the Nonlinear Second Order System with a Periodically Time-Varying Restoring-Force Term

Abstract

The ordinary differential equation which is expressed as ë₀+p(t)e₀=0 is called Mathieu's or Hill's differential equation according to p(t)=δ+εsint or p(t)=δ+∞Σn=1α_nsinnt respectively, and they have ever been studied in detail. However, the problem which the restoring-force term p(t)e₀ in the above differential equation is affected by some nonlinearity and exchanged for N[p(t)e₀] (where N[u] is some nonlinear characteristic for an input u and p(t)=δ+εsint) has not been yet discussed. Such a problem, for example, comes out when the servosystem with (1) an actuator whose transfer function is K₀/s(Ts+1), (2) a nonlinear element and (3) a sinusoidally time-varying gain element is studied.
The differential equation of this type, so to speak, may be regarded as a kind of nonlinearized Hill's differential equation, and it is useful to establish a convenient analytical method for the estimation of stability and step response, even if they are approximate. In this paper, (1) the equivalent nonlinear time-varying gain element for the estimation of stability, and (2) the equivalent nonlinear gain element for that of step response are defined and applied, based on the approach similar to the describing function method. The results of analysis by this method satisfactorily agreed with those by the analog computer.