Polynomial Chaos Expansion and Steady-State Response of a Class of Random Dynamical Systems
Résumé
The first two moments of the steady-state response of a dynamical random system are determined through a polynomial chaos expansion (PCE) and a Monte Carlo simulation that gives the reference solution. It is observed that the PCE may not be suitable to describe the steady-state response of a random system harmonically excited at a frequency close to a deterministic eigenfrequency: many peaks appear around the deterministic eigenfrequencies. It is proved that the PCE coefficients are the responses of a deterministic dynamical system-the so-called PC system. As a consequence, these coefficients are subjected to resonances associated to the eigenfrequencies of the PC system: the spurious resonances are located around the deterministic eigenfrequencies of the actual system. It is shown that the polynomial order required to obtain some good results may be very high, especially when the damping is low. These results are shown on a multidegree-of-freedom (DOF) system with a random stiffness matrix. A 1-DOF system is also studied, and new analytical expressions that make the PCE possible even for a high order are derived. The influence of the PC order is also highlighted. The results obtained in the paper improve the understanding and scope of applicability of PCE for some structural dynamical systems when harmonically excited around the deterministic eigenfrequencies.
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