Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations

A multi-degree-of-freedom vibro-impact system under white noise excitations is formulated as a stochastically excited and dissipated Hamiltonian system. The constraints are modelled as non-linear springs according to the Hertz contact law. The exact stationary solution of the system is derived under certain conditions. The approximate stationary solutions of the system are also obtained by using the stochastic averaging methods for quasi-Hamiltonian systems. It is shown that the stochastic averaging method for quasi-non-integrable-Hamiltonian systems is applicable if the non-linear forces according to the Hertz contact law take an important role in the response of the system while the stochastic averaging method for quasi-integrable-Hamiltonian systems is applicable if the non-linear forces can be neglected. An example for stochastically excited two-degree-of-freedom vibro-impact system is given to illustrate the application of the procedures in detail.