In the automobile industry, the mechanical losses resulting from friction are largely responsible for various kinds of surface damage, such as the scuffing occurring in some mechanical assemblies. These scuffing processes seem to be due to a local loss of lubrication between certain mechanical elements of the same assembly, leading to a sharp increase in the friction, which can lead to a surface and volume damage in some of them, and even can cause, in the worst case, the whole destruction of the mechanical system if it has continued to operate. Predicting and checking the occurrence of this kind of undesirable phenomena, especially in some principal systems of the vehicle, represents nowadays, a crucial challenge in terms of automobile reliability and safety. This study focuses on the mechanical friction losses liable to occur in differential automobile gearboxes, which can lead in the long term to the scuffing of these mechanical systems. The friction losses involved were modeled, using a simple analytical approach, which is presented and discussed.

Although the automobile industry has contributed significantly during the last few years to increasing the CO_{2} levels polluting the atmosphere, the introduction of a Carbon Tax has been inciting car manufacturers to reduce this pollution. In line with this more environmentally friendly approach, automobile manufacturers are now attempting to decrease the mechanical friction losses occurring between some vehicle parts—causing both an increase in fuel consumption and levels of carbon dioxide emissions—while maintaining their company’s competitiveness on the market. In practice, these friction losses are frequently associated with surface damage of several kinds such as scuffing [

Generally speaking, a differential gearbox [

Differential gearbox (with its sump and the axle shafts).

(a) Sump of a differential gearbox. (b) Various components of a differential mechanism.

Components of a differential gearbox: two planetary bevel pinions, two satellite bevel pinions with their shaft and the plastic shell.

The total gearbox yield,

Upon introducing both the primary and secondary gearbox shaft yields, the ratio between the power of the differential ring gear and that of the engine is equal to the product of these shaft yields; that is,

On the other hand, the total gearbox yield can be written as the product of the primary and secondary shaft yields and the differential gearbox:

After combining (

Since a differential gearbox can both transmit and distribute the differential ring gear power (the input power) to the output gears associated with each of the driven wheels, it follows that

In the equations governing the angular velocities and the torques in the differential gearbox (Figure

Angular velocities and torques in the differential gearbox.

Adding or subtracting (

Note that from now on,

Combining (

Based on the above equations, the total differential gearbox yield is

Under straight driving conditions, the angular velocities of two satellite pinions in relation to the satellite axis are zero (

In curvilinear driving situations, the angular velocity of two satellite pinions in relation to the satellite-carrier axis is no longer zero (

If one of the driven wheels undergoes slipping [

The power absorbed in the differential gearbox is the sum of the power dissipated at the various contact points existing in the overall mechanism, which can be decomposed as follows:

In order to account for the power dissipated between the two satellite bevel pinions and the two planetary bevel pinions in the differential mechanism (giving four meshing contacts), we introduce a yield,

Based on (

Neglecting the presence of an over-centre mechanism between the satellite pinion and satellite-carrier axis (Figure

Over-centre mechanism between the satellite pinion and satellite-carrier axis;

In (

Adopting a Coulomb-type friction law, the friction torque applied to the satellite-carrier axis,

Satellite bevel pinions and satellite-carrier axis.

Forces applied to a satellite pinion.

Combining (

The power dissipated between the two bevel satellite pinions and the surrounding plastic shell,

In order to write the equations giving the equilibrium of each satellite pinion, which is meshed with the two planetary pinions associated with the driven wheels and in contact with the satellite-carrier axis as well as with the plastic shell, we use the Fundamental Principle of Statics:

Looking only at the equilibrium of the resulting force of each torsor in the direction

Geometry of a differential gearbox.

The pressure applied to the head surface of the satellite pinion

Pressure

The expression for the friction torque

Combining (

The power dissipated between the two planetary pinions and the plastic shell,

Assuming that

Using the same procedure as above, namely, meshing each of the planetary pinions with two satellite pinions, which means that the resulting force applied to the head-planetary pinion by the plastic shell,

The expressions for both the resulting force,

It is again assumed here that in a first approximation, the pressure field,

Based on the above constitutive equations, the total power dissipated in the differential gearbox can be written as follows:

Using (

In the first part of this section, we discuss the order of magnitude of the constitutive parameters of the model presented in Section

This analytical model involves thirteen parameters (

Concerning (i), although these specific parameters depend on the differential gearbox under consideration, their order of magnitude can be said to be

Concerning (ii), the parameter

The parameters that need to be investigated more closely in order to determine their influence on the mechanical losses are the various coefficients of friction (iii):

In what follows, it is proposed to test the model’s response to a given load parameter and to determine the influence of some parameters (

The values adopted for the other parameters are

Figure

(a) Yield of a differential gearbox

Friction torque ratio

A complex path of the load parameter _{2}, _{4},

Example of a load parameter

(a) Yield of a differential gearbox

Although only a sensitivity analysis was performed on the model, the results obtained show quite clearly that this model can be used to assess and predict the mechanical friction losses occurring in a differential gearbox. An experimental study shall be conducted in order to obtain more realistic values for some of the parameters, such as the friction coefficients (

In this paper, an analytical model is presented for assessing and predicting the mechanical friction losses occurring in differential gearboxes. Some of the parameters involved in this model can be determined quite easily (geometric parameters), while others, which are more delicate, depend directly on the type of friction occurring in the mechanism (and therefore on the friction coefficients), which affects the mechanical losses to a variable extent. In order to test the influence of these parameters and determine the ability of the model to predict any mechanical losses, a sensitivity analysis was conducted. After this initial numerical approach, an experimental study shall be performed in order to obtain more realistic values for some of the parameters and confirm some of the assumptions made here in the modelling procedure.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is greatly indebted to Dr. J.-L. Ligier for helpful discussions, comments, and advices. The author would like to thank also Dr. Jessica Blanc for her help with this paper.