A Physically Consistent Model for Forced Torsional Vibrations of Automotive Driveshafts

Abstract

The aim of this research was to design a physically consistent model for the forced torsional vibrations of automotive driveshafts that considered aspects of the following phenomena: excitation due to the transmission of the combustion engine through the gearbox, excitation due to the road geometry, the quasi-isometry of the automotive driveshaft, the effect of nonuniformity of the inertial moment with respect to the longitudinal axis of the tulip–tripod joint and of the bowl–balls–inner race joint, the torsional rigidity, and the torsional damping of each joint. To resolve the equations of motion describing the forced torsional nonlinear parametric vibrations of automotive driveshafts, a variational approach that involves Hamilton’s principle was used, which considers the isometric nonuniformity, where it is known that the joints of automotive driveshafts are quasi-isometric in terms of the twist angle, even if, in general, they are considered CVJs (constant velocity joints). This effect realizes the link between the terms for the torsional vibrations between the elements of the driveshaft: tripode–tulip, midshaft, and bowl–balls–inner race joint elements. The induced torsional loads (as gearbox torsional moments that enter the driveshaft through the tulip axis) can be of harmonic type, while the reactive torsional loads (as reactive torsional moments that enter the driveshaft through the bowl axis) are impulsive. These effects induce the resulting nonlinear dynamic behavior. Also considered was the effect of nonuniformity on the axial moment of inertia of the tripod–tulip element as well as on the axial moment of inertia of the bowl–balls–inner race joint element, that vary with the twist angle of each element. This effect induces parametric dynamic behavior. Moreover, the torsional rigidity was taken into consideration, as was the torsional damping for each joint of the driveshaft: tripod–joint and bowl–balls–inner race joint. This approach was used to obtain a system of equations of nonlinear partial derivatives that describes the torsional vibrations of the driveshaft as nonlinear parametric dynamic behavior. This model was used to compute variation in the natural frequencies of torsion in the global tulip (a given imposed geometry) using the angle between the tulip–midshaft for an automotive driveshaft designed for heavy-duty SUVs as well as the characteristic amplitude frequency in the region of principal parametric resonance together the method of harmonic balance for the steady-state forced torsional nonlinear vibration of the driveshaft. This model of dynamic behavior for the driveshaft can be used during the early stages of design as well in predicting the durability of automotive driveshafts. In addition, it is important that this model be added in the design algorithm for predicting the comfort elements of the automotive environment to adequately account for this kind of dynamic behavior that induces excitations in the car structure.