Vibration Analysis of an Ensemble of Structures using an Exact Theory of Stochastic Linear Systems

Abstract

An exact theory of stochastic linear systems is presented. These are made of a nominal deterministic system disturbed by a random component (itself being a deterministic disturbance weighted by a random variable whose probability density function is known). Both rank-one and multi-rank disturbances are covered, so that the theory is applicable to a wide range of situations where some parameters of a dynamic system (such as mass, stiffness, Young modulus, damping coefficient, etc. of some components) are random. Exact expressions of the statistics of the response of the stochastic system are given for any inputs and outputs, and at any frequency. An exact closed-form expression of the statistics in terms of special (error) functions is available in the case of normal variables (having a Gaussian probability density function). All expressions can therefore be evaluated precisely and efficiently. The theory is applied to a few structural dynamic systems and shown to be applicable from low to high frequencies without particular restriction in the mid-frequency range. The theory is also shown to be more precise and several orders of magnitude faster than a Monte-Carlo approach applied to the same stochastic linear systems.