Fast matrix exponent for deterministic or random excitations

The solution of (z)over dot = Az is z(t) = exp(At)z(0) = E(t)z(0), to = z(0 ). Since z(2t) = E(2t)Z(0) = E(t)(2)z(0), z(4t) = E(4t)z(0) = E(2t)(2)z(0), etc., one function evaluation can double the time step. For an n-degree-of -freedoms system, A is a 2n matrix of the nth-order mass, damping and stiff ness matrices M, C and K. If the forcing term is given as piecewise combina tions of the elementary functions, the force response can be obtained analy tically. The mean-square response P to a white noise random force with inte nsity W(t) is governed by the Lyapunov differential equation: (P)over dot = AP + PA(T) + W. The solution of the homogeneous Lyapunov equation is P(t) = exp(At) P-0 exp(A(T) t), P-0 = P(0). One function evaluation can also dou ble the time step. If W(t) is given as piecewise polynomials, the mean-squa re response can also be obtained analytically. In fact, exp(At) consists of the impulsive- and step-response functions and requires no special treatme nt. The method is extended further to coloured noise. In particular, for a linear system initially at rest under white noise excitation, the classical non-stationary response is resulted immediately without integration. The m ethod is further extended to modulated noise excitations. The method gives analytical mean-square response matrices for lightly damped or heavily damp ed systems without using modal expansion. No integration over the frequency is required for the mean-square response. Four examples are given. The fir st one shows that the method include the result of Caughy and Stumpf as a p articular case. The second one deals with non-white excitation. The third f inds the transient stress intensity factor of a gun barrel and the fourth f inds the means-square response matrix of a simply supported beam by finite element method. Copyright (C) 2001 John Wiley & Sons, Ltd.