The ‘damping effect’ in the dynamic response of stochastic oscillators

We consider the dynamics of linear damped oscillators with stochastically perturbed natural frequencies. When average dynamic response is considered, it is observed that stochastic perturbation in the natural frequency manifests as an increase of the effective damping of the system. Assuming uniform distribution of the natural frequency, a closed-from expression of equivalent damping for the mean response has been derived to explain the ‘increasing damping’ behaviour. In addition to this qualitative analysis, a comprehensive quantitative analysis is proposed to calculate the statistics of frequency response functions from the probability density functions of the natural frequencies. Firstly, single-degree-of-freedom-systems are considered and closed-form analytical expressions for the mean and variance are obtained using a hybrid Laplace's method. Several probability density functions, including gamma, normal and lognormal distributions, are considered for the derivation of the analytical expressions. The method is extended to calculate the mean and the variance of the frequency response function of multiple-degrees-of-freedom dynamic systems. Proportional damping is assumed and the eigenvalues are considered to be independent. Results are derived for several probability density functions and damping factors. The accuracy of the approach for both single and multiple-degrees-of-freedom systems is examined using the direct Monte Carlo simulation.