Stochastic Structural Dynamics Using Frequency Adaptive Basis Functions

A novel Galerkin subspace projection scheme for structural dynamic systems with stochastic parameters is developed in this chapter. The fundamental idea is to solve the discretised stochastic damped dynamical system in the frequency domain by projecting the solution into a reduced subspace of eigenvectors of the underlying deterministic operator. The associated complex random coefficients are obtained as frequency-dependent quantities, termed as spectral functions. Different orders of spectral functions are proposed depending on the order of the terms retained in the expression. Subsequently, Galerkin weighting coefficients are employed to minimise the error induced due to the reduced basis and finite order spectral functions. The complex response quantity is explicitly expressed in terms of the spectral functions, eigenvectors and the Galerkin weighting coefficients. The statistical moments of the solution are evaluated at all frequencies including the resonance and antiresonance frequencies for a fixed value of damping. Two examples involving a beam and a plate with stochastic properties subjected to harmonic excitations are considered. The results are compared to direct Monte Carlo simulation and polynomial chaos expansion for different correlation lengths and variability of randomness.