The determination of the activation energy for a Vibro-impact system under multiple excitations

Abstract In this paper, the stochastic stability of a Vibro-impact system with multiple excitation forces is studied. Due to the multiple external excitations, the probability density function (PDF) of the system is extremely difficult to solve. In addition, the existence of coexisting steady states is very common for the Vibro-impact system, and the perturbation of random noise will cause the transitions between the steady states. In this case, we are more interested in working out each attractor’s activation energy, which is specifically used to characterize the attractor’s stochastic stability, rather than the solution of the PDF. Based on the large deviation theory, the asymptotic analysis is carried out, and a time-varying Hamilton’s equation for the quasi-potential is derived. To verify the effectiveness of the theoretical analysis, two detailed examples, where an impact attractor and a non-impactor coexist in the system, are conducted. By the application of the action plot method, the activation energies and the most probable exit paths (MPEP) for each attractor are derived. Compared with the numerical simulation, it shows very good agreement. Moreover, it is found that the existence of transient chaos near the attractor could seriously deteriorate the attractor’s stability.