On the dynamics of lubricated hypoid gears

Abstract

The torsional dynamics of a vehicle differential hypoid gear pair is investigated. The model comprises applied torque, representing transmitted engine power, including engine order vibration. A number of gear teeth pairs transmit the applied torque through their lubricated conjunctions. Tooth contact analysis (TCA) is used to obtain the appropriate geometrical, kinematic and meshing parameters. These enable the evaluation of contact loads, film thickness and friction for conjugate teeth pairs, which are subject to mixed thermo-elastohydrodynamic regime of lubrication. It is shown that the lubricant undergoes non-Newtonian shear in line with the Eyring regime of traction. The inclusion of combined thermal non-Newtonian shear and boundary interactions has not hitherto been reported for the tribo-dynamics of hypoid gear pairs. When rate of change of gear teeth contact radii is included in the analysis more complex system dynamics (loss of teeth contact) result, particularly at higher speeds. The stated features constitute the main contributions of the current work, which have not hitherto been reported in literature. It is also shown that teeth contact separation ensues when resonant conditions are noted. This is regarded as the main root cause of a noise and vibration phenomenon, known as axle whine.

Introduction

Gears have been studied extensively for a very long time, initially because of their inherent unreliability due to poor lubrication and, now in addition, as a noise and vibration concern. There are a multitude of reasons underlying gear vibration; including backlash and errors in the form and finish of mating gear teeth pairs, defined as kinematic transmission error. These errors and misalignment of gear pairs and their supporting shafts are important causes of vibration and noise, as well as poor lubrication, friction and wear. Therefore, besides studying the effect of machine and cutter tool settings to reduce mal-form and finish, the dynamic response of gear sets, when in situ, has also been extensively researched. The latter area is due to the effect of various system non-linearities. Consequently, an array of modelling techniques is used, depending on the conditions pertaining to a defined problem [1].

A significant number of models have been proposed to obtain the dynamic response of parallel axes gears in order to ascertain the extent of system stability and attainment of desired periodic motions. Implementation of lumped parameter models is a common practise, followed by analytical expressions for time varying parameters, such as for the meshing stiffness. The main source of non-linearity in these formulations is the presence of backlash, promoting impacts which can lead to impulsive actions and potential chaotic behaviour [2], [3], [4], [5], [6], [7], [8], [9], [10]. Similar analyses on crossed-axes gear sets have shown more complex responses. For example, hypoid gears, used in a wide range of applications, present complex meshing geometry. Consequently, there is a lack of analytical expressions to quantify the effect of their underlying governing parameters. Prior to the development of Tooth Contact Analysis (TCA) tools, experimental and empirical formulations were commonplace [11], [12]. These early models precluded the exact meshing geometries. Instead, they were based on simplifications to the meshing force vector used in purely torsional dynamic analyses.

The first attempt to build a hypoid gear vibration model, based on exact geometry, was made by Cheng and Lim [13]. Generation of gear pair surfaces and the discretisation of the elliptical contact area resulted in the development of a three-degree of freedom (DOF) model. The significance of this approach was in relating the meshing parameters to the actual gear geometrical characteristics. Moreover, it allowed for the transmission of mesh load to the structural components of the differential unit [13]. A further study [14] included backlash non-linearity and time-dependent meshing parameters, enabling the identification of resonant modes and, therefore, the study of the effect of load torque on system dynamics. Wang et al. [15] focused on a hypoid gear pair, describing the dependence of meshing parameters' variation with the dynamic response of the system. An original two-DOF system was reduced to a single DOF and its dynamic response was computed using two different models. The first model only included the fundamental harmonics of the meshing parameters, whilst the second one imported their exact values. The generation of impact phenomenon was discussed, as well as the transition of system response from a periodic motion to a chaotic state with the variation of load torque and introduction of damping.

As already noted, friction generated in gearing systems is also an important area of investigation as this determines the efficiency of transmission and differential systems, as well as affecting their dynamic response. Confining oneself to some representative studies, it is important to note that gear teeth are often only partially lubricated as many fore-running contributions have shown [16], [17]. This is known as mixed elastohydrodynamic regime of lubrication, where the mechanisms contributing to friction are viscous shear of a thin lubricant film and interaction of asperities on the contiguous surfaces with an insufficient film thickness. This is referred to as boundary friction and is prevalent in gear teeth interactions [18]. Vaishya and Singh [19], [20], [21] developed a dynamic model with viscous friction on gear flanks. Neglecting the effect of backlash and simplifying the derivation of coefficient of friction, they were able to perform a stability analysis for gear pairs using the Floquet theory [22]. Another study by He et al. [23] concentrated on the effect of sliding friction upon the dynamic response of a gear set, while assuming empirical formulae for the coefficient of friction. Similar analyses were conducted for helical gears by Velex and Cahouet [24], Velex and Sainsot [25], as well as Kar and Mohanty [26]. The common approach in the above studies was the dependence of friction on the variation of length of line of contact. A method based on TCA was introduced by He et al. [27] for modelling the bearing force on a twelve-DOF system.

For helical gear pairs, De la Cruz et al. [18], [28] have reported models for a transaxle transmission system, where a combined tribological and vibration analysis was carried out. In their model the unselected loose gear teeth pairs were modelled as lightly loaded thermo-hydrodynamic conjunctions with viscous friction, whilst the engaged gear pairs were subject to a mixed thermo-elastohydrodynamic regime of lubrication. They showed that thermal effects in the gear teeth pair contacts significantly reduce lubricant film thickness. An analytic solution to energy equation for the determination of lubricant temperature in the contact was used [29], whilst Grubin's [30] analytical solution was employed for the loaded elastohydrodynamic conjunctions of teeth pairs of engaged gears. This provided a quasi-steady solution, one which does not take into account the enhanced load carrying capacity of the teeth pair contact conjunctions due to any lubricant squeeze film effect. The approach, including the squeeze film effect was advocated by Rahnejat [31], [32] subject to various regimes of lubrication, including isothermal elastohydrodynamic conditions. It was shown that the lubricant film behaviour is frequency dependent. An extension of this work by Mehdigoli et al. [33], representing a pair of gears as wavy surfaced discs showed that fluid film lubrication possesses insignificant damping under elastohydrodynamic conditions, which verified the earlier experimental findings of Johnson and Gray [34]. However, these studies did not include the effects of viscous or boundary friction, nor shear thinning of the lubricant in a thermal contact. Another numerical quasi-steady mixed isothermal EHL solution, combined with torsional vibration of gear pairs, was highlighted by Li and Kahraman [35] for line contact condition, applicable to spur gears and as an approximation for helical gears.

A few tribodynamics' studies have been reported for hypoid gears; including the effects of viscous and boundary friction. Geometrical complexities of hypoid gears in mesh and the need to determine the instantaneous area of contact necessitates use of numerical methods, rather than the simpler analytical approaches. The sliding velocities of mating gear teeth pairs and the sense of application of friction cannot be calculated analytically due to the time varying nature of the mesh vector. As already noted, an approach to obtain the friction vector has been reported by Cheng and Lim [14], based upon a simulated geometry, whereas the derivation of kinematic contact properties was described by Xu and Kahraman [36]. Authors validated the various empirical formulae for representation of coefficient of friction against an elastohydrodynamic lubrication model. Good agreement with experimental results was shown and a formula describing the results was obtained.

A number of other researchers have also focused on transient EHL representation of the contact zone between the gear flanks. These include the works reported by Holmes et al. [37], [38], [39], who treated the contact zone of pairs of hypoid gear teeth as a point contact problem, which is a reasonable approximation. Isothermal solution of gear lubrication problem has received more attention than those including thermal effects. An investigation of thermal effects was also reported by Handschuh and Kircher [40], who calculated the temperature distribution inside the contact zone due to heat generation.

In this work, TCA is used to determine the kinematic and geometrical properties of the hypoid gears, necessary for the thermo-elastohydrodynamic analysis. The rate of change of the teeth contact radii has been considered in the analysis. This is the main contribution of the current work compared with those previously published. It reveals a more pronounced dynamics, characterised by teeth separation near 1:1 resonant conditions. Asperity contribution to friction is also included by characterisation of the tooth flank topography and use of Greenwood and Tripp [41] friction model. Therefore, the analysis in this paper is that of a quasi-steady mixed thermo-elastohydrodynamics of hypoid gear teeth pairs, and its effect upon the dynamics of a hypoid gear pair of a vehicle differential unit. Such an approach has not hitherto been reported in literature.