Mathematical modelling of torque vectoring differentials

Active safety systems which distribute forces between the wheels are an integral part of a modern vehicle. The vast majority of systems are based on the brake systems, and reducing engine power, which certainly reduces the vehicle speed. However, the needs of car owners are constantly changing, there is a need for devices capable of ensuring the distribution of traction between the wheels without having to reduce speed. With this task perfectly cope with the main transmission containing the mechanisms of power distribution (MPD). Power distribution mechanisms have a greater impact on vehicle handling. More often meet term, Torque Vectoring Differential (TVD). They began to be used for the first time since the mid-1990s, and found their application in sports cars, and then in executive cars. The described main gears are an integral part of dynamic stabilization systems. During the vehicle development, these systems are developed and investigated using mathematical modeling methods. Therefore, the paper describes the design, operation principle and kinematic design of the most common. The equations systems describing the dynamics of the links and the mechanism as a whole are also composed. The mathematical modeling of vehicles equipped with TVD is modelled.


Introduction
Modern cars are equipped with various dynamic stabilization systems based on the operation of the brake system or reducing engine power [1]. However, since the mid-1990s, Mitsubishi has been using main drives with power distribution mechanisms known as Torque Vectoring Differential (TVD). Their distinctive feature compared to conventional main gears, is that they are able to transmit more torque to the high-speed wheel, thus creating an increased turning torque of the vehicle. Power distribution mechanisms, in turn, contain additional links and controls that ensure the distribution of torques [1][2][3][4][5][6][7][8][9][10].
PDM containing friction elements which activated by an electronic control system. However, to ensure effective operation of the control system, first, at the design stage, conducting the mathematical simulation of the vehicle motion [11][12][13][14][15], equipped with such a mechanism. In the development of mathematical models of multilink final drives made the following assumptions: the links of the final drive perfectly rigid, the losses in the gearing can be neglected, the ratio auxiliary links taken greater than one.
Thus, the paper analyzes the existing mechanisms, their comparison and writing of equations systems for mathematical modeling in the program MATLAB & Simulink. The final drive of Mitsubishi AYC was first used in 1996 on the Mitsubishi Lancer Evolution IV. [2] This final drive is based on using bevel differential. PDM is located on the right and is connected by one link to the differential housing and the right axle through friction clutches.  In the final drive, depending on right friction clutch CL1 engaged, part of the torque from the differential housing 3 is transmitted to the first auxiliary link 1 through a multiplier that accelerates the rotation of the right half-axis 5. The increase in torque is due to the summation of torques from the first auxiliary link 1 and the half-axial differential gear. Depending on left friction clutch CL2 engaged, part of the torque from the differential housing 3 will be transferred to the second auxiliary link 2 through the gearbox, slowing down the angular speed of the right half-axis 5. The resistance torque, which occurs on the left clutch CL2, creates a braking torque relative to the differential housing 3.

Final drive equations system
To compile an equations system revealed that the mechanism of five moving links and three gearing. Thus, it is necessary to make five dynamic equations and three kinematic equations. Let the first two equations (1, 2) describe the dynamics of the auxiliary links: ,  input input shaft angular acceleration, 2 / ; rad s , FD i final drive gear ratio; 1 , u 2 , u ratio between differential housing and first and second auxiliary links.
The equation of dynamics of the differential housing (8), reduced to the input shaft, is written below: The considered final drive equations system written bellow:

Final drive Magna
The Magna final drive was first used in 2009 on the Audi S4 [2]. It contains a symmetrical conical differential and multipliers located on the right and left.

Final drive design
Magna final drive design shown on the figure 2.
In the Magna final drive, when the right clutch CL1 is activated, part of the torque from the differential housing 3 is transmitted to the first auxiliary link 1 through the first stage of the multiplier, then through the friction right clutch CL1 to the second stage of the multiplier. This accelerates the rotation of the right half-axis 5. The increase in torque is due to the summation of moments from the first auxiliary link 1 and the right half-axis 5. In the case of activation of the left clutch CL2 part of the torque from the differential housing 3 will be similarly transmitted to the left axle 4.

Final drive equations system
The equations describing the auxiliary units dynamics do not differ from the equations (1, 2) written above. Next, dynamics equations of the half-axis are written bellow: 1 , u 2 , u accordingly ratio between right and left half-axis and first and second auxiliary links. Next, kinematic equations for auxiliary links are written bellow: , diff u the gear ratio between the differential housing and supporting links. The dynamics equation of the differential housing, reduced to the input shaft, is written, as well as (8). ;

Final drive ZF Torque Vectoring
The final drive ZF Torque Vectoring was first introduced in 2008 [2] and was installed on BMW cars. Unlike the above-mentioned mechanisms, the power distribution is due to the planetary multipliers located on the right and left.