Dynamic simulation of the differential gear in road vehicles

The differential gear is a mechanism that is part of the kinematic chain of the vehicle’s transmission. The differential gear has the role of dividing the self-propulsion power flow received from the main drive into two branches, each transmitted to a drive wheel, giving them the possibility to rotate with different angular velocities, depending on the driving conditions. The present work presents the possibilities of computer-aided design and 3D modeling of the main drive and differential gear, respectively the bevel pinion, the crown wheel, the planet pinions and the satellites, using Autodesk Inventor Professional. The 3D model of the main drive - differential gear assembly, considered as a mechanism, was subjected to a dynamic study, using the Dynamic Simulation module. This involves the modeling of the kinematic couplings, the definition of the drive movement parameters and the external loads, respectively the graphical visualization of the kinematic and kinetostatic results for the various components of the mechanism, in real operating conditions, during a kinematic cycle. At the same time, based on the obtained results, the analysis by the finite element method of the most critical component elements, under straight-line driving and turning conditions, is presented.


Introduction
The differential is a differential planetary transmission type mechanism having two degrees of mobility (M = 2), included in the motor axle of vehicles, and has the role of dividing the flow of selfpropulsion power received from the main transmission into two branches, each transmitted to a driving wheel, at the same time giving them the possibility, depending on the self-propulsion conditions, to rotate with different angular speeds, figure 1, [1].
The main self-propulsion conditions that require the wheels to rotate at different angular velocities are as follows, [1]: -rolling on curved trajectories, when the inside wheel of the curve has a shorter distance to run than the outside wheel of the curve; -straight rolling on smooth roads, when the driving wheels have equal distances to run but for various reasons, they have unequal radii due to unequal pressure in the tires, asymmetrical distribution of the load with respect to the longitudinal axis of the car, different tires, or their different degree of wear; -rolling in a straight line on bumpy roads when, due to their random distribution, the wheels have to roll on trajectories with different lengths, [1].
The construction and working principle of the differential are presented in figure 2. Thus, when rolling in a straight line, the satellites do not rotate around their own axes, the differential becomes "locked" and will rotate around the central axis as a unitary whole, and the drive wheels will have equal angular velocities, figure 2. When rolling in turn, the differential satellites perform rotational movement around their own axes, respectively the drive wheels will have different angular velocities, figure 2.

Differential kinematics
The kinematic analysis of the differential is carried out using the method of stopping the satellite port arm (Willis method), which aims to determine the transmission ratios, respectively the angular velocities of the drive wheels in different running conditions.According to the kinematic scheme shown in figure 3, the differential consists of the main transmission formed by bevel gears 1 and 2, satellites 5 and 5', planetary pinions 3 and 4, respectively satellite arm stiffened with bevel gear 2 [4].Thus, by applying the Willis method, considering the symmetrical differential, that is, the numbers of satellites teeth 5 and 5' equals (Z5 = Z5') the expression of its transmission ratio was obtained, according to the relation 1.It is found that the sum of the angular velocities of the driving wheels is constant, which corresponds to the working principle of the differential, i12 being the transmission ratio of the main transmission [5].  , result: -When turning suddenly to the right or left, 4 0   or 3 0   result: -If the axle is not in contact with the ground, being suspended and the engine is not working, that is 1 0,   it follows that the two wheels will rotate in opposite directions:

The dynamic study of the differential
The dynamic study of the differential was carried out in two operating conditions, straight-line rolling, respectively in a turn.In order to carry out this study, the 3D model of the differential, its dynamic simulation in real operating conditions, respectively the finite element analysis of the differential components was made [6].

3D modeling of the differential
The 3D model of the differential was made in the Autodesk Inventor Professional program (AIP).In the first stage, the components of the differential were modeled, respectively the bevel gears of the main transmission, the satellites, the planetary pinions and the satellite port arm, using specific 3D modeling tools.In the second stage, the differential assembly was made, using specific geometric constraints, respectively motion constraints through which the transmission ratios of the component gears were defined.Figure 4 shows the 3D model of the differential, with the specification of its components [7], [8].

Differential dynamic simulation
The dynamic simulation was performed in the AIP program by defining the differential assembly as a mechanism.Thus, the components became kinematic elements, and the geometric assembly constraints self-converted into kinematic couples of different classes.Instead, the motion constraints between the bevel gears had to be redefined as "rolling joint -cone on cone" connections [9], [10].
As parameters of the differential drive movement, the input speed (attack pinion) was defined as a linear variation, respectively the torque (bevel pinion), as a sinusoidal variation during a kinematic cycle.The values of these parameters are presented graphically in figure 5 and figure 6.The differential loads are given by the torques corresponding to the two driving wheels, i.e. the planetary gears.Thus, for the first study case, i.e. rolling in straight line of the vehicle, the torques are equal (T1 = T2), they were defined as linear variation during a kinematic cycle, as can be seen in figure 7.For the second study case, i.e. in a turn of the vehicle, the torques are different (T1 ≠ T2), they were defined as a sinusoidal variation during a kinematic cycle, as can be seen in figure 8 and figure 9.After defining the parameters presented above, the simulation of the movement of the differential was carried out, in this sense its parameters were established, respectively the duration of the kinematic cycle (t = 1s) and the number of analysis steps (500).
Following the dynamic simulation, a correct operation of the differential was found for both cases of rolling, straight line and in a turn.The behavior of two components, the attack pinion of the main transmission, and of one of the satellites of the differential was monitored from the point of view of the loads.In this sense, the following results were obtained, in graphic form, during a kinematic cycle: -The contact force between the attack pinion and the crown wheel (main transmission), when the vehicle is rolling in a straight line (T1 = T2), shown in figure 10.
-The contact force between the satellite and the two planetary pinions, when the vehicle is rolling in a straight line (T1 = T2), shown in figure 11.In this case the two contact forces are equal.-the contact force between the attack pinion and the crown wheel (main transmission), when the vehicle is in a turn (T1 ≠ T2), shown in figure 12. -The contact force between the satellite and the two planetary pinions, when the vehicle is in a turn (T1≠T2), shown in figure 13.In this case the two contact forces are different.Analyzing the results presented above, it is found that the values of the contact forces corresponding to the two studied components are higher in the case when the vehicle is in a turn.

Finite element analysis of the attack pinion and the satellite
A natural continuation of the dynamic simulation is the finite element method (FEA) analysis of the differential components in order to determine the stress and their deformations.This can be done without going through the steps specific to an FEA analysis, because the boundary conditions and external and internal loads are automatically taken over by the analysis module [11].
In this sense, following the dynamic simulation, it is only necessary to isolate the studied component at the time step at which the loads on it are maximum and transfer it to the Stress Analysis module of the AIP component.
As an example, two components of the differential, the attack pinion and one of the satellites, were studied in the two driving conditions of the vehicle, respectively straight line rolling and in a turn.For the time step, we selected the instant t = 1s, during which the contact forces within the gears, comprising these elements, reach their peak magnitude, according to the results obtained from the simulation, presented in figures 10, 11, 12, 13.

Conclusion
Dynamic simulation is a component of computer-aided design, performed with the AIP application, being a method of studying a product in real operating conditions.It can be compared to a virtual prototyping, based on a 3D geometric model, but without the necessary expenses of making a real prototype.
Two components were studied, the attack pinion, respectively one of the satellites of the differential in straight line rolling conditions, respectively in the turn.After conducting the dynamic simulation, graphical representations of the contact forces within the gears, involving these components, were obtained.At the time step at which the contact forces acting on the attack pinion and the satellite have maximum values, their finite element analysis was performed.It was found that the stresses in both components are higher in a turn conditions, but are within allowable limits.

Figure 1 .
Figure 1.Splitting the power flow from the main transmission [2]

Figure 2 .
Figure 2. The operation principle of the differential [3]

Figure 4 .
Figure 4. 3D modeling of the differential

Figure 5 . 6 .
Figure 5. Defining the input speed Figure 6.Defining the driving torque

Figure 8 .Figure 9 .
Figure 8. Defining the load for the left drive wheel in a turn (T1 ≠ T2)

Figure 10 .Figure 11 .
Figure 10.The contact force from the main drive when rolling in straight line (T1 = T2)

Figure 12 .Figure 13 .
Figure 12.Contact force from the main drive when the vehicle is in a turn (T1 ≠ T2)

Figure 14 .
Figure 14.Von Mises stresses for the attack pinion in straight line rolling (T1 = T2)