The dynamic behavior of a single-stage planetary gear set with helical gears of multishaft automotive automatic transmissions has been studied, in which one component of the planetary gear set is imposed by additional external vertical and axial loading from countershaft gear pair in addition to the moment. Under these combined loading conditions, the contributions of the deflections of the ring gear and the carrier cannot be neglected. A three-dimensional nonlinear time-variant dynamic model considering not only the transverse, torsional, axial, and rotational motions of the gears but also the elasticity of the mounted shafts has been developed by combining the lumped parameter method with finite element method. The natural modes and the forced vibration responses due to static transmission errors have been obtained. The proposed dynamic model is employed to describe the effects of the combined external loading condition and positioning on the dynamic behavior of a four-planet system.
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