Premium
Fast matrix exponent for deterministic or random excitations
Author(s) -
Leung A. Y. T.
Publication year - 2000
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/1097-0207(20010120)50:2<377::aid-nme29>3.0.co;2-e
Subject(s) - mathematics , piecewise , mathematical analysis , white noise , lyapunov exponent , matrix (chemical analysis) , square (algebra) , function (biology) , differential equation , physics , quantum mechanics , nonlinear system , geometry , materials science , composite material , statistics , evolutionary biology , biology
The solution of ż = Az is z ( t )=exp( At ) z 0 = E t z 0 , z 0 = z (0). Since z (2 t )= E 2 t z 0 = E 2 tz 0 , z (4 t )= E 4 t z 0 = E 2 2 tz 0 , etc., one function evaluation can double the time step. For an n ‐degree‐of‐freedoms system, A is a 2 n matrix of the n th‐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: Ṗ = 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 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. Copyright © 2001 John Wiley & Sons, Ltd.