Response statistics of three-degree-of-freedom nonlinear system to narrow-band random excitation

The principal resonance of a 3-DOF nonlinear system to narrow-band random external excitations is investigated. The method of multiple scales is used to derive the equations for modulation of amplitude and phase. The behavior, stability and bifurcation of steady-state responses are studied by means of qualitative analysis. The effects of damping, detuning, and excitation intensity on responses are analyzed. The theoretical analyses are verified by numerical results. Both theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions, co-existence of two kinds of stable steady-state solutions, saturation and jump phenomena may occur. The stationary probability density function of responses for the co-existence case is obtained approximately.