Averaging-Based Characteristics of the Response Induced by Combined Random and Harmonic Excitation

Slender engineered structures are sensitive to unwanted vibration effects mainly due to their soft nature. With respect to physical laws, the response of the structure reaches different non-linear modes which depend on the ratio or difference of natural and excitation frequencies. The non-stationary response near the lock-in region is quasi-periodic in nature, with the frequency of beating varying with the change in distance from the locking interval. This property is naturally affected by the assumption of combined random and harmonic excitation. In the paper, several details that contribute to the probabilistic characteristics of the system on a slow time scale are described using partial amplitudes of the response. The problem is illustrated with a non-linear SDOF oscillator of the van der Pol type, where the additive excitation combines harmonic and random components. In the stochastic averaging method, the expression for the response is written in trigonometric form using partial amplitudes. The solution to the Fokker–Planck equation for the cross-PDF of the partial amplitudes is found using the Galerkin approximation.