Fast matrix exponent for deterministic or random excitations

The solution of qq = Az is z(t) = exp(At)z₀ = E_tz₀, z₀ = z(0). Since z(2t) = E_2tz₀ = E_t ²z₀, z(4t) = E_4tz₀ = E_2t ²z₀, 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 stiffness matrices M, C and K. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean-square response P to a white noise random force with intensity W(t) is governed by the Lyapunov differential equation: qq = AP+PA^T+W. The solution of the homogeneous Lyapunov equation is P(t)=exp(At) P₀ exp(A^Tt), P₀ = P(0). One function evaluation can also double the time step. If W(t) is given as piecewise polynomials, the mean-square response can also be obtained analytically. In fact, exp(At) consists of the impulsive- and step-response functions and requires no special treatment. 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 method is further extended to modulated noise excitations. The method gives analytical mean-square response matrices for lightly damped or heavily damped systems without using modal expansion. No integration over the frequency is required for the mean-square response. Four examples are given. The first one shows that the method include the result of Caughy and Stumpf as a particular case. The second one deals with non-white excitation. The third finds the transient stress intensity factor of a gun barrel and the fourth finds the means-square response matrix of a simply supported beam by finite element method.