A Lagrangian approach for the electromechanical model of single-stage spur gear with tooth root cracks

Abstract

This paper presents a unified electromechanical modeling framework of a single-stage spur gear with cracks at tooth root. The model has been developed following modified Lagrangian approach with the consideration of Rayleigh dissipative potential. The coupled electromechanical equations of motion represent a non-autonomous system. Improved analytical model based time-varying mesh stiffness (IAM-TVMS) at different crack depths is incorporated in the developed model to examine the impact of cracks on the dynamic behavior of the system. The IAM-TVMS model considers the effect of misalignment between base and root circle, accurate transition curve, nonlinear Hertzian contact stiffness, revised fillet foundation stiffness with the structural coupling effect when nearby tooth is loaded and improved crack model which considers the straight line for crack path and parabolic curve for limiting line. The current and vibration responses are taken after solving the coupled equations of motion through fourth order Runge–Kutta method. In this study, the impact of crack depths are effectively visualized using stator current as a primary source of signal. Even when the crack depth is significantly small, the residual signal clearly depicts the presence of crack in the cepstrum analysis. The cepstrum analysis of residual stator current signal is also compared and verified with the residual vibration signal for different crack depths. The developed analytical model is one of the first attempt of its kind in modeling approach; the model successfully shows the impacts of crack on the system with motor current signal.