Random Gear Dynamics Based on Path Integration Method

Gears exhibit vibrations during operation which contain apparently random components. Due to time-varying stiffness and backlash nonlinearity, it is very hard, if not impossible, to get a closed form response of gears under combination of deterministic and random loads. This paper employs a numerical method termed as path integration to obtain the probability density of the response at discrete time instants. The random excitation is assumed to be White noise. The response of the gear is a Markovian processes under the random excitation. In order to capture the probability density evolution, discretization is applied to both space and time. The transition probability between adjacent time instants is assumed to be Gaussian and the calculation of the probability density is made on a reduced finite space. The mean and variance, which are used to construct the Gaussian distribution, are obtained through a direct numerical integration scheme with one stochastic Newmark method. Statistic linearization technique is utilized within each individual time-step to find a linear equivalent of the original system with backlash nonlinearity. Through this method the evolution of the probability distribution function of the response displacement and the velocity is calculated. Three representative cases with different levels of constant load are investigated.