Random nonlinear dynamics of a spur gear pair by path integration

The research investigates the random nonlinear vibration of a spur gear pair subjected to both deterministic and random loads by path integration method. Different models and approaches to apply the path integration method are presented in Chapters 2– 4. Backlash nonlinearity and time-varying mesh stiffness in gear systems are both considered in Chapters 2 and 3. In Chapter 2, the time-varying mesh stiffness is modeled as a constant plus a cosinusoidal component, and the discontinuous backlash nonlinearity is approximated with a cubic polynomial through curve fitting. Then Gaussian closure procedure is employed to obtain the mean and variance of transition probability density function (PDF). In Chapter 3, the time-varying mesh stiffness is approximated with a square wave function. The variance of the responses is calculated and expressed as closed forms for two different cases in gear systems. In Chapter 4, the gear rattling model which only considers backlash is presented. A degenerate Gaussian distribution is employed as transition PDF. The path integration results are compared with deterministic results (Chapters 2 and 3) and Monte Carlo simulation results (Chapters 3 and 4). Good agreement is found between them, which could verify the accuracy of path integration method in the study of random gear dynamics