In this paper, the transition probability density functions between response velocity and response displacement in nonlinear vibration systems which have the restoring force characteristic of a cubic equation are governed by the Fokker-Planck Equation. The experimental probability density functions are compared with analytical results. The analytical model of the cubic equation as Duffing Equation is proposed by the restoring force characteristic of the nonlinear vibration system with gaps in the experiments. However, a slight difference for the frequency range of the transfer function was shown by simulation results. Then, it is considered using transition probability density functions in the response characteristic. For stationary random input waves, the probability density function between the response displacement and the response velocity are easily estimated by the Fokker-Planck Equation and the Duffing Equation. The slight difference of the transfer function of the response acceleration is evaluated by the scattering of the restoring force characteristic estimated by the probability density function and self-natural frequency curve. The R.M.S. value and the transfer function of the experimental results are compared with the analytical results. It is thought that the estimation of the probability density function of the response has validity. It is thought that the evaluation of the nonlinear vibration characteristics by the probability density function is valid.
Computing rates of Markov models of voltage-gated ion channels by inverting partial differential equations governing the probability density functions of the conducting and non-conducting states
