The effect of a feedback operation on the probability distribution of a non-stationary non-linear vibratory system with an arbitrary random input

This paper provides a new method for the unified statistical treatment of the output probability expressions, when an arbitrarily distributed and correlated random signal is passed through a class of time-variant vibratory system having a non-linear element in the forward path and a linear element of finite memory type in the feedback path. The statistical expressions derived can reflect various effects of the feedback operation due to the linear element into the second and higher expansion terms of series solutions. The method essentially depends on an introduction of the multivariate statistical Laplace expansion method which includes the statistical Lagrange expansion method reported previously. The experimental results obtained by means of digital simulation are in good agreement with theory. The proposed method can be characterized by the feature that fewer assumptions are required about the non-linearity and input statistics to the system. The resultant probability expressions are thus obtained in the very exact solutions for the output statistics in forms closely associated with the specific purposes of analysis based on feedback operation.