STOCHASTIC AND/OR CHAOTIC RESPONSE OF A VIBRATION SYSTEM TO IMPERFECTLY PERIODIC SINUSOIDAL EXCITATION

Abstract

A single-degree-of-freedom system with one-sided rigid barrier at its equilibrium position is considered. The system is subject to excitation by a sinusoidal-in-time force with random white-noise temporal variations of the excitation frequency. Whilst response analysis had been made previously for the resonant case, i.e. one with expected excitation frequency being close to an even integer multiple of the natural frequency of the system without barrier, the nonresonant case is considered in this paper which is well known to be prone to multiple bifurcations and chaos in the case of perfectly periodic excitation. A simple qualitative explanation of the corresponding phenomenon of frequency multiplication, or "breeding", is provided for this case based on a special piecewise-linear transformation of state variables, which permits to get rid of velocity jump for the transformed velocity. The influence of the imperfect periodicity is then studied by numerical (Monte-Carlo) simulation. It is shown that increasing intensity of the white-noise variations of the excitation frequency, or bandwidth of the excitation power spectral density (p.s.d.), may greatly reduce frequency content of the response with simultaneous increase of the overall response level. The excitation/system bandwidth ratio is identified as a key governing parameter.