Response of a single-degree-of-freedom nonlinear vibroimpact system to a narrow-band random parametric excitation

The resonance response of single-degree-of-freedom nonlinear vibroimpact oscillator with a one-sided barrier to narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a boundary random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the period for slowly varying inphase and quadrature responses. The averaged equations are solved exactly and algebraic equation of the amplitude of the response is obtained in the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will be reduced at large damping or large nonlinear intensity, and will be increased with large amplitudes or frequencies of the random excitations. The phenomena of stochastic jump is observed, i.e. the steady response of the system will jump from trivial solution to a large non-trivial one when the amplitude of the random excitation exceed the threshold value, or will jump from the a large non-trivial solution to trivial one when the intensity of the random disorder of the random excitation exceed the threshold value.