Output probability of a vibration system with an arbitrary non-linear element and random input

Abstract

In connection with various types of non-linear vibration systems, a unified theory of thestatistical treatment for the output probability distribution is newly introduced in the case when a general random process of arbitrary distribution type is passed through a time-variant system of zero-memory type or finite memory type with a non-linear feedback element, with the aid of a statistical Lagrange series expansion method. Concretely, for the purpose of finding the effect of an arbitrary non-linear feedbackelement on the output probability distribution, the explicit expressions of the probability distributions are derived in general forms of non-orthogonal expansion series, reflecting the effect of the forward linear element of the vibration system into the first term. Further, in view of the arbitrariness of the input characteristics, the possible variety ofnon-linear elements and fluctuation forms of the system parameters, and the complexity of the statistical treatment involved, the validity of the theoretical expression is experimentally confirmed by the method of digital stimulation.