Subharmonic response of a single-barrier vibroimpact system to a narrow-band random excitation

Sensitivity of subharmonics to imperfect periodicity of the excitation is studied for a vibroimpact system with a single barrier slightly offset from the system's equilibrium position. The system is excited by a periodic force with random phase modulation, so that small random temporal variations of the excitation frequency are present, which make the force a narrow-band random process. Mean value of the excitation frequency is close to an even integer multiple of the system's natural frequency without a barrier. Analytical study is performed, based on the Zhuravlev transformation, which effectively removes velocity jumps from the equations of motion, combined with the Krylov-Bogoliubov averaging over the period. The reduced stochastic differential equations of motion are solved then by the method of moments for the mean square response amplitude. The solution is exact for the case of zero offset and approximate, using perturbation-based moment closure scheme for the general case. The results are verified by Monte-Carlo simulation which was used also to demonstrate possibility for incorporating moderately large impact losses through the use of previously derived equivalent viscous damping factor. The results indicate possibility for significant reduction of peak resonant amplitudes of subharmonics due to imperfect periodicity of excitation, depending on the value of the excitation/system bandwidth ratio. Applications to dynamics of moored bodies under ocean waves' excitation are discussed.