Guido Sacchetti – guido.sacchetti@gmail.com Updated on July, 2019

**Introduction**

This project focuses on developing a multibody model of a Type-A Torsen differential trying to reproduce its working principles and showing its different behaviour in different scenarios.

Because of the difficulty to find out documentations about dimensions and characteristics of real built Torsen differentials it has been modelled an invented but reasonable model with the 3D CAD software Solidworks. Such model has been later imported in the multibody software MSC Adams to perform the multibody simulations.

In the end thanks to the information found out in the SAE Paper ‘’ An Evaluation of Torque Bias and Efficiency of Torsen Differential ‘’ [1] it has been possible implementing some formulas, with a MatLab code, to evaluate analytically the Torque Bias Ratio of the Torsen to perform a comparison with the results of the multibody model.

### **Objectives**

The objective of the project is simulating how the Torsen Type-A differential can split the engine input torque in different scenarios:

- Straight road with normal traction
- Curve road with normal traction
- Curve road with normal traction first and then with the inner tyre losing adherence

**The modelling problem**

**Working principles of a Torsen differential**

An open differential always split the engine torque between the two wheels of the axle with a TBR = 1. This means that whatever are the adhesion conditions of the tyres an open differential will always delieve an equal amount of torque to each of the two wheels.

In a scenario where one of the two tyres loss adhesion it is better for the performance of the vehicle to give more torque to the tyre that has the major adhesion and less torque to the tyre that has lost adhesion. The Torsen differential can give more torque to the tyre that has the major adhesion thanks to an elevate internal friction between gears and between gears and the spider gear, through some thrust washers, compared to a normal open differential. The TBR for a Torsen differential could be on average equal to 3, its means that the differential gives to the tyre with major adhesion three times the torque that it gives to the tyre that has lost adhesion.

To deepen its useful to refer to ‘’ An Evaluation of Torque Bias and Efficiency of Torsen Differential ‘’ SAE Paper [1].

**3D CAD modelling**

The 3D CAD model of the simulated Torsen differential has been made using the 3D CAD software Solidworks. To model the three types of gears needed, following ISO standards, a dedicated toolbox available in Solidworks has been used.

Helical side gears (**2 **of them that are ideally connected to the left and the right wheel respectively)

Module = 2

Number of teeth = 20

Pitch circle diameter [mm] = 40

Pressure angle [deg] = 20

Helix angle [deg] = 45

Face width [mm] = 30

Centre hole radius [mm] =10

Helical element gears (**6 **of them, 3 meshed at 120 deg with the left helical side gear and 3 meshed at 120 deg with the right helical side gear)

Module = 2

Number of teeth = 10

Pitch circle diameter [mm] = 20

Pressure angle [deg] = 20

Helix angle [deg] = 45

Face width [mm] = 55

Centre hole radius [mm] = 5

Spur gears (**10** of them that meshed together in couples to link the left 3 helical element gears with the right 3 helical element gears)

Module = 2

Number of teeth = 20

Pitch circle diameter [mm] = 40

Pressure angle [deg] = 20

Face width [mm] = 7.5

Centre hole radius [mm] = 5

**Multibody system**

The multibody system is composed of 19 rigid bodies with 19 relative joints. The Adams student edition allow using a maximum of 20 rigid bodies for a multibody system, for the purpose of modelling a Torsen differential this limit is acceptable, in fact in the Torsen multibody model developed in this work there are only two spur gears missing from the structure of the real differential. This is not a problem because the two spur gears missing don’t play an essential role for the purpose of this multibody system.

Bodies, connectors and motions

In the multibody system there are:

- 1 Spider gear with 1 revolute joint w.r.t. the ground
- 2 Helical side gears each one with a revolute joint w.r.t. the ground
- 6 Helical element gears each one with a revolute joint w.r.t the Spider gear
- 10 Spur gears each one with a fixed joint w.r.t. the relative Helical element gear

- 2 motions, one at the revolute joint of the left helical side gear and the second at the revolute joint of the right helical side gear

Forces and measures

In the multibody system there are:

- 1 Torque applied to the Spider gear
- 6 Contacts with friction between the 6 helical element gears and the 2 helical side gears
- 5 Contacts with friction between the 5 couples of spur gears

The settings of the contacts with friction between gears are:

- 2 frictions in the revolute joints of the left helical side gear and of the right helical side gear, with the aim of reproducing the friction between the helical side gears and the spider gear without modelling the thrust washers. Its settings are shown in FIG.1
- 6 frictions in the revolute joints of the 6 helical element gears, with the aim of reproducing the friction between the helical element gears and the spider gear without modelling the thust washers. Its settings are shown in FIG.2

FIG.1 FIG.2

- 2 Torque measures, one on the motion of the left helical side gear and one on the motion of the right helical side gear

Solver settings

About the solver dynamics settings, the choice has been taken for an HHT Integrator, it should work better with contacts with friction, and for an I3 formulation, the only available choosing an HHT integrator.

About the solver contacts settings, the choice has been taken for using a polygon – based interference detection package with a faceting tolerance of 800. It should be a good compromise between accuracy and velocity of the simulations.

**Simulations and analysis of results**

**Simulation in a straight road with normal traction covered at 50 km/h with a spider gear-imposed torque of 150 Nm.**

OmegaHG1 = OmegaHG2 = 43,82 Rad/s

- Without any friction: HG1 Torque (Avg) = 75,19 Nm, HG2 Torque (Avg) = 72,96 Nm

- Friction only between gears: HG1 Torque (Avg) = 74,73 Nm, HG2 Torque (Avg) = 74,16 Nm

**Simulation in a 50m radius curve road with normal traction covered at 50 km/h with a spider gear-imposed torque of 150 Nm.**

OmegaHG1 = 45,14 Rad/s (Outer wheel)

OmegaHG2 = 42,49 Rad/s (Inner wheel)

- Without any friction: HG1 Torque (Avg) = 73,19 Nm, HG2 Torque (Avg) = 74,22 Nm

- Friction only between gears: HG1 Torque (Avg) = 45,96 Nm, HG2 Torque (Avg) = 101,22 Nm

- Friction between gears and in joints: HG1 Torque (Avg) = 32,25 Nm, HG2 Torque (Avg) = 103,65 Nm

**Simulation in a 50m radius curve road with normal traction covered at 50 km/h with a spider gear-imposed torque of 150 Nm from 0.0 s to about 0.5 s. Then the inner tyre loses adherence from about 0.5 s to 1 s**

**From 0s to about 0.5s:**

OmegaHG1 = 45,14 Rad/s (Outer wheel)

OmegaHG2 = 42,49 Rad/s (Inner wheel)

**From about 0.5s to 1s:**

OmegaHG1 = 45,14 Rad/s (Outer wheel)

OmegaHG2 = 55,14 Rad/s (Inner wheel)

The parameters used for all the simulations are:

**The results** of the simulations of the Torsen differential in a **straight road condition** show how the behaviour of this differential is perfectly the same of an open differential: with the same wheel’s rotational speeds the differential has not internal relative motions and the torque delivered to each of the two wheels is the same.

**The results** of the simulations of the Torsen differential in a **curve road condition with a normal traction** show how the behaviour of this differential is the same of the behaviour of an open one only in the frictionless simulation. In the simulations with coulomb friction between gears and in joints the behaviour of the Torsen differential is completely different from the behaviour of an open one. The Torsen differential split the torque delivered to each of the two wheels also in a condition with normal traction with a little difference of rotating velocity of the two wheels. It gives the major amount of torque to the slower inner wheel and the minor amount of torque to the faster outer wheel with a ratio that is the designed TBR value.

**The results **of the simulation in a **curve road with the inner tyre losing adherence** show how the Torsen differential can change in a very fast way the amount of torque delieved to each of the two wheels. At the beginning with normal traction conditions the differential gives the major amount of torque to the inner slower wheel. When the inner slower wheel loses adherence the Torsen almost immediately starts to invert the torque splitting, it starts to give the major amount of torque to the outer wheel, now slower because it isn’t losing adherence.

The TBR derived from the simulations in a radius curve are summarized in the following table. Is shown also a comparison with the TBR values calculated thanks to the formulas of the SAE paper ‘’ An Evaluation of Torque Bias and Efficiency of Torsen Differential ‘’ [1] implemented in a MatLab code.

TBR = Thigh/Tlow (where Thigh and Tlow are the highest and lowest output torques respectively)

TBR Adams Multibody | TBR MatLab code | |

Friction only between gears | 2,20 | 1,41 |

Friction between gears and in joints | 3,21 | 3,68 |

**The last simulations **that has been done are some simulations to prove that the Torsen differential is not a speed sensitive mechanism. In these simulations the torque applied to the spider gear is kept constant at 150 Nm and the delta omega between the rotating velocities of the two wheels has been made to vary.

The results are that for delta omega from 2,5 Rad/s to 10 Rad/s the TBR is almost constant, for a delta omega of 50 Rad/s the TBR increases of about 15% from the value of the TBR with a delta omega of 2,5 Rad/s. This phenomenon is not imputable to the sliding velocities between gears because also in the condition with a delta omega of 2,5 Rad/s the sliding velocities between the gears were major than the value of Friction Transition Velocity set in the contacts settings options.

The results are summarized in the image below:

**Conclusion**

The objective of showing the behaviour of Torsen differential in different scenarios is aimed.

To clarify the Torsen differential behaviour we can say that it is a Torque Sensing mechanism. It means that the differential splits the torque between the two wheels, always with the designed TBR, in an instantaneous way as soon as a little difference of resisting torque applied by the ground on the wheels appears.

The Torsen differential is not a speed sensing mechanism because the TBR enforced by the differential should remain constant independently from the increasing of delta of rotating velocities of the two wheels.

For example, if a tyre reaches the limit of adherence and the Torsen is splitting the torque with its TBR, any additional torque provided by the engine will result in a wheel spinning because the Torsen can’t higher its value of designed TBR and of course can’t lock completely.

**About further developments **is necessary understand why in the multibody model developed in this works the TBR of the Torsen doesn’t remain almost constant with large increasing of delta between the rotating speeds of the two wheels.

** **

**References**

**[1]** SAE TECHNICAL PAPER SERIES, 2002-01-1046, ‘’ An Evaluation of Torque Bias and Efficiency of Torsen Differential ‘’, Shan Shih (ArvinMeritor, Inc.), Ward Bowerman (Zexel Torsen, Inc.)

**[2] **https://www.youtube.com/watch?v=wuBodGAVJyA