An Alternative Formulation of the Dynamic Transmission Error to Study the Oscillations of Automotive Hypoid Gears

Abstract

A new modeling approach on the torsional dynamics of hypoid gear pairs is presented in this work. The current formulation is characterized by an alternative expression of the dynamic transmission error (DTE), accounting for the variation of the effective mesh position. Speed dependent resistive torque is introduced on the gear wheel, enabling the system to reach dynamic equilibrium based on realistic vehicle operating conditions. The above are supplementing past research studies, where simplifications were introduced in the calculation of DTE, while the operating angular velocity was defined a priori. The analysis is accompanied by numerical results, indicating the rich dynamic behavior captured by the new formulation. The dynamic complexity of the system necessitates the identification of the various response regimes. A solution continuation method (software AUTO) is employed to follow the stable/unstable periodic response branches over the operating range of the differential under examination.

Transient mixed non-Newtonian thermo-elastohydrodynamics of vehicle differential hypoid gears with starved partial counter-flow inlet boundary

“…The instantaneous solution is obtained through application of Runga-Kutta step-by-step integration in time steps of 10 s. The load share per meshing teeth pair is then determined, using [39,40]:…”

“…The governing equations of motion are given below, where the indices p and g refer to the pinion and gear respectively. The same methodology has been used and described in detail in [1,39,40]. Unlike parallel axes gear pairs, due to the complex geometry of hypoid gears, it is not possible to calculate the meshing stiffness, radii of curvature and contact kinematics using simple approximation methods.…”

“…The backlash non-linearity is defined as: (33) where, denotes the teeth relative displacement along the instantaneous line of approach of a pair of teeth: ( ) = ∫̇− 0 ∫0 − ( ) ( ) is the unloaded static transmission error obtained through TCA and expressed as a Fourier function of time [39,40].…”

Lubrication analysis and sub-surface stress field of an automotive differential hypoid gear pair under dynamic loading

“…As it can be observed in figure 4a, for 27 (near the primary resonance), the steady state amplitude of DTE exhibits a jump behaviour (non-linear gear dynamics). This is due to the effect of backlash (geometric) non-linearity 23,34 . When operating within the vicinty of this region, teeth separation may occur (depending on system damping and any fluctuations of the external excitations).…”

“…This relationship has been used in the hypoid gear dynamics study of Mohammadpour et al 4 , as well as in Karagiannis et al 9,22,23 . The time varying contact characteristics are calculated at each time step, using: The above equations correspond to the Fourier expansions of each of the contact data obtained by curve fitting the original TCA results, which have been produced using the commercial software CALYX 24 .…”