Fast matrix exponent for deterministic or random excitations

Abstract

The solution of <i>ż=Az</i> is <i>z(t)</i>=exp(<i>At</i>)z₀=E_tz₀, z₀=z(0)</i>. Since <i>z</i>(<i>2t</i>)=E_2tz₀=E^t₂z₀</i>, <i>z</i>(<i>4t</i>)=E_4tz₀=E^2t₂z₀</i>, etc., one function evaluation can double the time step. For an <i>n</i>-degree-of-freedoms system, A is a 2<i>n</i> matrix of the <i>n</i>th-order mass, damping and stiffness matrices <i>M</i>, <i>C</i> and <i>K</i>. If the forcing term is given as piecewise combinations of the elementary functions, the force response can be obtained analytically. The mean-square response <i>P</i> to a white noise random force with intensity W(<i>t</i>) is governed by the Lyapunov differential equation: <i>Ṗ=AP+PA^T+W</i>. The solution of the homogeneous Lyapunov equation is <i>P</i>(<i>t</i>)=exp(<i>At</i>) <i>P</i>₀</i> exp(<i>A</i>^T<i>t</i>), <i>P</i>₀=<i>P</i>(0). One function evaluation can also double the time step. If <i>W(t)</i> is given as piecewise polynomials, the mean-square response can also be obtained analytically. In fact, exp(<i>At</i>) 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.