Method of statistical linearisation. Its application for evaluation of the response of a hysteretic system to Gaussian inputs

The paper describes a method of computing the statistical equivalent gain of a nonlinear element in the minimum r.m.s. error sense. The method is based on the facts that: (a) A band-limited Gaussian signal x(t) can be expressed as x(t)=R(t) cos (¿mt+¿) where ¿m is the midband frequency, and R and ¿ are independent random variables. (b) Ranges of these two variables are (0, ¿) and (0, 2¿), respectively. (c) Angle ¿ has uniform distribution, while R obeys a definite distribution law obtainable from the known distribution law for x(t). The method has the advantage that it is equally well applicable to single- and double-valued nonlinearities. This is illustrated by working out the equivalent gains of a number of well known nonlinear characteristics. Finally, the error-response characteristics of a hysteretic system under excitation from a Gaussian random signal are analysed by using the equivalent gain obtained in the paper. These analytical data are checked against simulated results.