This paper proposes a new analytical model for helical gears that characterizes the contact plane dynamics and captures the velocity reversal at the pitch line due to sliding friction. First, the tooth stiffness density function along the contact lines is calculated by using a finite element code. Analytical formulations are then derived for the multidimensional mesh forces and moments. Contact zones for multiple tooth pairs in contact are identified, and the associated integration algorithms are derived. A new 12-degree-of-freedom, linear time-varying model with sliding friction is then developed. It includes rotational and translational motions along the line-of-action, off-line-of-action, and axial directions. The methodology is also illustrated by predicting time and frequency domain results for several values of the coefficient of friction.
Harmonic Green's functions for a thin semi-infinite plate with clamped or free edges are developed starting from either simply supported or roller supported solutions and applying corrections to account for boundary excitation. This is achieved by connecting the solutions in terms of polar coordinates with the solutions in Cartesian coordinates. The formal solutions in the form of improper wave-number integrals are numerically evaluated using adaptive Clenshaw-Curtis integration. Alternate solutions obtained for each boundary condition compare well. Only harmonic point loads are considered in this article but the methodology may be extended to moment excitation and distributed loads. The methodology developed here will form the basis for advancing the ray tracing technique for vibration analysis of finite plates.
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