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Abstract
In this paper, the stochastic stability of a vibro-impact system with multiple excitation forces is studied. Due to the multiple external excitations and the coexistence of metastable states, the solution of each attractor’s activation energy, which is specifically used to characterize the attractor’s stochastic stability, is much more suitable for the stability analysis rather than the solution of the probability density function. Based on the large deviation theory, the asymptotic analysis is carried out, and a time-varying Hamilton’s equation for the quasi-potential of the vibro-impact system 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 for each attractor are derived. Compared with the numerical simulation, the results show very good agreement. Moreover, it shows that the existence of transient chaos near the attractor could seriously deteriorate the attractor’s stability.
Highlights
Vibro-impact systems are usually of many engineering applications, such as vibratory pile drivers, pie placers, heat exchange tubes in nuclear reactors [1]
We review from large deviation theory that the asymptotic analysis for our system’s response in weak noise limit is carried out, and the activation energy and the most probable transition path for each attractor are obtained theoretically[25]
For the system excited by multiple external forces, including a constant force, a harmonic force, and a random perturbation, the probability density function (PDF) of the system is extremely difficult to solve
Summary
Vibro-impact systems are usually of many engineering applications, such as vibratory pile drivers, pie placers, heat exchange tubes in nuclear reactors [1]. We often coincide with such a situation that a machine that runs smoothly suddenly becomes noisy; after a while, it again runs smoothly, the motion state is alternatively between smoothy and noisy, and there are no changes in the system’s parameters in the whole procedure This is usually the case that the noise induces transitions between the steady non-impact and steady-impact states. At those parameters, where the system has multiple coexisting attractors in the state space, we perturb the system with weak random noise.
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