A fast time-domain integration method for computing non-stationary response histories of linear oscillators with discrete-time random forcing

A new approach is presented for computing displacement histories of single linear oscillators with arbitrarily light damping and general forcing—of particular use for efficient Monte Carlo simulation of modal systems with ultra-light damping and very broadband non-Gaussian excitation. Solution methods are initially presented within a state transition context, to show limitations of FFT solutions, and to establish, for long-run non-stationary stochastic analysis via fast Laplace, the need for appropriate zero-padding, high cut-off frequency, and fixed-step sampling. Truncation errors arising in single-transition time-domain convolution are then examined via the Euler–Maclaurin summation formula. Errors are shown to be minimum when transition intervals are chosen as integer multiples of the damped natural period, precisely where the O (?t2?f') error can be evaluated, and the velocity transition equation can be dispensed with. The paper shows that an optimum O (?t4?f?) integration scheme can be used for fast time-domain convolution in a two-stage algorithm. First, phased-pairs of accurate displacements are efficiently predicted at selected transition times. These are then used as boundary conditions in adaptive Chebychev polynomial solution giving continuous displacement histories for selected cycles—this considerably reduces the number of multiplications and integrations normally required. Two-stage integration turns out to be at least 100 times faster than explicit short-transition time-domain solution, and for general applications, at least as fast as the Laplace/IFFT approach. But for non-stationary probability density estimation, involving far-future history prediction the speed advantage over fast Laplace can be enormous.