Time-Domain Response of Damped Stochastic Multiple-Degree-of-Freedom Systems

Characterizing the time-domain response of a random multiple-degree-of-freedom dynamical system is challenging and often requires Monte Carlo simulation (MCS). Differential equations must therefore be solved for each sample, which is time-consuming. This is why polynomial chaos expansion (PCE) has been proposed as an alternative to MCS. However, it turns out that PCE is not adapted to simulate a random dynamical system for long-time integration. Recent studies have shown similar issues for the frequency response function of a random linear system around the deterministic eigenfrequencies. A Pade approximant approach has been successfully applied; similar interesting results were also observed with a random mode approach. Therefore, the latter two methods were applied to a random linear dynamical system excited by a dynamic load to estimate the first two statistical moments and probability density function at a given instant of time. Whereas the random modes method has been very efficient and accurate to evaluate the statistics of the response, the Pade approximant approach has given very poor results when the coefficients were determined in the time domain. However, if the differential equations were solved in the frequency domain, the Pade approximants, which were also calculated in the frequency domain, provided results in excellent agreement with the MCS results.