# Stochastic process method for reliability analysis of gear nonlinear vibration

Under the influence of random factors, all kinds of nonlinear vibration responses of gears can be regarded as random processes. In order to keep the generality, the Gaussian stochastic process is used to describe the nonlinear vibration response. Without considering the degradation of gear performance, the number of times that the nonlinear vibration response exceeds the specified threshold in the specified time obeys Poisson distribution.

Where: Z is the number of times that the nonlinear vibration response exceeds the specified threshold in the time period [0, t], which is a random variable; PZ (Z, t) is the probability that the nonlinear vibration response exceeds the specified threshold Z times in the time period [0, t]; ξ is the crossing rate.

For the Gaussian stochastic process g (T), the traversal rate ξ is: where φ GG · (s, G ·) and φ GG · (- s, – G ·) are the joint probability density functions of G (T) and G · (T); G is the realization value of G (t); G · is the realization value of G · (T), G · (T) is the rate of change process of Gaussian random process and also a random process; s is the safety threshold of nonlinear vibration response; μ g and μ g · are the mean values of G (T) and G · (T), respectively; σ g and σ g · are the standard deviations of G (T) and G · (T), respectively.

According to the definition and analysis framework, the calculation method of gear nonlinear vibration reliability RGV (T) is as follows:

Where: Nd, NV and Na are the times of vibration displacement, vibration velocity and vibration acceleration amplitude exceeding the specified threshold value in [0, t] time respectively; ξ D, ξ V and ξ a are the crossing rates of vibration displacement, vibration velocity and vibration acceleration respectively; PZD (nd, t), pzv (NV, t) and PZA (Na, t) are the probabilities of the vibration displacement, vibration velocity and vibration acceleration amplitude exceeding the specified threshold value for times within [0, t] time, respectively. 