Stochastic averaging of Duhem hysteretic systems

The response of Duhem hysteretic system to externally and/or parametrically non-white random excitations is investigated by using the stochastic averaging method. A class of integrable Duhem hysteresis models covering many existing hysteresis models is identified and the potential energy and dissipated energy of Duhem hysteretic component are determined. The Duhem hysteretic system under random excitations is replaced equivalently by a non-hysteretic non-linear random system. The averaged Itô's stochastic differential equation for the total energy is derived and the Fokker-Planck-Kolmogorov equation associated with the averaged Itô's equation is solved to yield stationary probability density of total energy, from which the statistics of system response can be evaluated. It is observed that the numerical results by using the stochastic averaging method is in good agreement with that from digital simulation.