updated on July, 2019
Introduction
This project is aimed to study the behaviour of a Limited Slip Differential (LSD), designed by Drexler for the Formula SAE championship.
The LSD which has been analysed is a Salisbury differential, so it’s a torquesensitive device whose locking torque (i.e. half the difference between the torque of the slower wheel and the faster one) is function of the input torque.
The Salisbury differential, also called ramp differential, may be considered as an open differential with a double clutch pack in parallel which is responsible for the torque transfer.
The input torque is transfered by the differential housing to the satellite gear pins. The pins are held inside inclined surfaces of the rings. The force on the pins due to the contact of the bevel gears causes sliding of rings along the axis that push the clutch packs. These last alternate friction plates attached on the housing and on the gears.
The locking torque is the torque amount ehnanced by the clutch pack.
Objectives
The output torques vs the input torque diagram is among the most widespread diagrams describing the behavior of the LSD. An example is shown in the next picture (Fig.2). This kind of diagram allows to know how the torque is transfered while the vehicle is on traction (positive torque) or on overunning (negative torque). One of the objective is to create this diagram.
In the diagram above, the dotted line rapresents an open differential whose input torque is equally distribuited on the driven wheels; the continuos gray line (slope equals to 1) is the maximum torque delivered on one wheel, in the case of locked differetial (= spool) the distribution depends on the vertical load; green and orange lines are output torques of the LSD; blue vertical lines show minimum and maximum applied torques.
LSDs usually have different torque bias (and in consequence slopes) depending on torque sign, this allows to adjust better vehicle handling. Also the differential been modelled has this feature.
It has to be said that this differential has a preload (a belleville spring sides the clutch pack) avoiding relative velocity between friction plates; the differential behaves like a spool until this preload is overcome by a certain torque. As a result the diagram shown before changes. The influence of the preload is neglected in this report for the sake of modelling simplicity. For better comprehension see the article [1] in the biography.
Another matter of interest is how the delivered torque on ground changes compared to angular velocity difference on wheels. A full rear axle model will be used beyond in order to see what happens.
What is expected from the last analysis can be observed on (Fig.3) and (Fig.4):
Those diagrams have been obtained by integrating on Matlab the following differential equation system, which describes the rear axle while turning:
I · d/dt(ω1) = M1 – Fx1 · re Inner Wheel Balance
I · d/dt(ω2) = M2 – Fx2 · re Outer Wheel Balance
(m1+m2) · d/dt(Ω) · R = Fx1 + Fx2 Rear Axle Balance
Where:
Fx1 = ck · (k1) Inner Longitudinal Force (Linear Magic Formula)
Fx2 = ck · (k2) Outer Longitudinal Force (Linear Magic Formula)
k1 = (ω1 · re)/(Ω · (Ra)) – 1 Inner Longitudinal Slip
k2 = (ω2 · re)/(Ω · (R+a)) – 1 Outer Longitudinal Slip
with standstill boundary conditions.
where I is wheel Inertia, ω1 and ω2 wheel angular velocites, Ω yawing velocity, re effective radius, M1 and M2 applied momenta on wheels, Fx1 and Fx2 longitudinal forces (only linear for simplicity) with ck longitudinal stiffness, k1 and k2 longitudinal slips, Ω·(R+a)) and Ω·(Ra)) wheel forward velocities expressed in terms of yaw velocity, R curvature radius and 2·a rear axle lenght, m1 and m2 wheel masses.
In this system the vertical load transfer is not taken into account and the weight is equally ripartited on wheels such as it has been modeled on the multybody software. Also drag and lift are neglected.
In (Fig.3) it is clear that even if the differential is open there is a little delivered torque on ground difference between wheels due to wheel inertias; ΔT is constant while the input torque is constant, in fact manipulating wheel balances one obtains:
ΔFx = Fx1 – Fx2 = (M1 – I · d/dt(ω1) )/re – (M2 – I · d/dt(ω2) )/re = I/re · ( d/dt(ω2) – d/dt(ω1) )
since M1 equals to M2 while dealing with open differentials. In this case ΔT is very low and it can be neglected.
In (Fig.4) a constant amount of delivered torque on ground difference will be reached after a transient behavior, with a peak due to Coulomb friction on clutches (Fig.5), that is smoother as farther is the standstill initial condition.
Of course ΔT in the LSD case will be greater than the open differential and the value of the former depends on the ramp angle.
All issues discussed above are solved using MSC Adams.
All physical quantities mentioned below are referred to SI units when it’s not specified.
The modelling problem, simulations and analysis
All the internal parts of the differential have been drawn using Adams, while the housing (main body, covers, bolts etc) has been imported as an only shell body from a *.step file (Fig.7). Its mass properties have been computed using PTC Creo mass property analisys.
The main reason why the internal parts have not been imported (the whole CAD of the differential was had to the writer) is that Adams works a lot better with parasolid models drawn in Adams instead of any other imported solid type when dealing with contacts.
Furthermore imported geometries are much slower in terms of graphic refreshing than parasolid ones.
The geometry of the differential has been modelled using booleans features.
The differential model has been simplified by drawing:
 only one satellite and one pin
 only a pair of ramps (30 and 60deg)
 only a friction plate per clutch pack (1 contact surface instead of 4)
 no preload as told before
Moreover two auxiliary body (“blocking disks”) have been inserted, which have to block the translation of the clutch plates so that the housing has the only task to transfer the torque to rings and pin by the joints and does not have any contact with other bodies.
Contacts’ Calibration
A model (Fig.8) has been built to calibrate contacts between bodies; this one contains rings, pin, friction plates and blocking disks.
Below constraints are listed:
 revolute joint between housing and ground
 translational joint between rings and housing
 cylindrical joint between friction plates (the ones attached to gears)
 fixed joint between pin and ground
 fixed joint between blocking disks and housing
Continuos Impact Modelling has been used to model the interaction of pin and rings (no friction), rings and friction plates (Coulumb friction), friction plates and blocking disks (no friction).
An external torque (1000Nm, maximum delivered torque) has been applied on the housing and revolutional motions with opposite sign and same magnitude have been applied on friction plates.
Simulation script is following:
SIMULATE/DYNAMIC, END=1.0E03, STEPS=50
SIMULATE/DYNAMIC, END=0.1, STEPS=100
Below results are shown:
 Axial force is correctly exchanged between bodies and peaks caused by impacts can be easily recognized (Fig.9)
 Forces reach a steady value much more before displacements (Fig.9) (Fig.10)
 Displacements are around millimeter tenths, this is absolutely acceptable (Fig.10)
 As expected torque is equally exchanged between housing and rings and the torque ehnanced on clutch packs has same magnitude and opposite sign (Fig.12).
After this calibration, the bevel gears have been introduced and adjusted.
Bevel gears modelling
Bevel gears have been added to the previous system by using Adams’ machinery tool. This feature allows to choose either “simplified” or “3D contact” method which are described by the software as follows:
 Simplified Method: This method calculates the gear forces and backlash between the gear pair analytically. It is useful when friction is neglected. The contact force calculation is fast because of its analytical approch.
 3D Contact Method: This method uses geometrybase contact and supports shelltoshell 3D geometry contact. It calculates true backlash based on actual working center center distance and tooth thickness. This method allows for consideration of outofplane motion within the gear pair.
Considering the fact that the gear coupling is not the principal interest, the first method matches our needs. In addiction this method analitically calculates physical quantities, that means it’s faster to be solved.
Gears have been designed as closer as possible to the CAD’s ones (Fig.13): in the model it’s been assumed pressure angle equals to 20.0 deg, mean spiral angle equals to 0.0 deg, face width equals to 1.0e2, outer cone distance equals to 3.2e2; driven gears have 14 teeth while satellite’s 9 teeth.
Steel has been chosen as gears material and the contact stiffness has been raised to 5.0e7 to avoid an exhagerate interpenetration depth unless 1000Nm input torque is overcome.
One option of machinery tool which has to be taken into account is “carrier part”. When the satellite pin is left free to rotate, the body must be chosen as the “carrier part” in place of ground otherwise the simulation is going to fail and a “joint failure” will be mentioned. The next simulation is not the case, but it will when simulating full rear axle model.
For gear calibration, joints are updated with the respect of the latter model as:
 added revolute joints between driven gears and housing
 added revulute joint between satellite and pin
 replaced cylindrical joints between friction plates and ring with translational joints between friction plates and driven gears.
Simulation is run keeping the same input torque and script while motions are transfered on gears.
Torque distribuition analysis
To analyse the torque distribution some changes are made to the model. These are:
 revolute joints between driven gears and ground (replace revolute joints between driven gears and pin)
 revolute joint between pin and housing (replaces fixed joint between pin and ground)
 motions applied on driven gears revolute joints: zero displacement on the left driven gear and 100*time on the right (note that if the transition from static to dynamic friction is neglected, the magnitude of the displament doesn’t influece torque distribution)
Output torques vs input torque diagram can be now obtained. It is also possible to determine the same diagram for an open differential by disabilitating Coulomb friction on the contacts which envolve friction plates and rings. Speaking about friction, when it is set on, static and dynamic coefficients are multiplied by 4 (remeber that only one surface out of four is modeled for the clutch pack). 4*0.23 and 4*0.16 are respectively assigned, assuming wet steel surfaces, and the other parameters are left as defualt.
To be clear, viscosity is neglected (common approch).
Last of all, an increasing linear tourque from 300Nm to 1000Nm has been applied on the differential housing.
The simulation script which has been used is:
SIMULATE/STATIC
SIMULATE/DYNAMIC, END=13, STEPS=500
GSTIFF SI2 Solver has been chosen to avoid some spikes coming out with different options (e.g. I3). Results are shown on (Fig.14)
When the applied torque is negative, the pin is touching the overrun ramp surface which is less inclined then the on power one. Slopes (distribution) and locking torque have been computed:
Open Diff  60deg  30deg  
Distribution  0.5 – 0.5  0.634 – 0.366  0.896 – 0.104 
Locking Torque  0%  13.4% (14.5% declered)  39.6% (44% declered) 
The results are very close to designer datasheet, so the simulation can be considered well done.
The last issue to be solved is how the torque distribution changes if compared to angular velocity difference between wheels when the vehicle is accelerating.
As it was mentioned before, a fullrearaxle model has been made to succed this objective.
Fullrearaxle model
Model has been updated adding following bodies and forces:
 Road: 2d_flat.rdf (file used during the course “Modelling and Simulating of Mechanical Systems” taught by professor M.Massaro)
 Left and right rear wheels: Hoosier13_80_RaceUP2019.tir has been used. Mass (whole suspension group): 8kg, Inertia (tyre, rim, hub, disc brake – remaining components neglected) Ixx=Iyy=2e1 and Izz=3.2e1
 Car Body: a solid sphere having main task to carry all the other parts and to whom is imposed a circular trajectory using a cylindrical joint. Low values have been assigned to mass and inertias. The weight has been applied by using vertical forces on the wheel (more precisly on the springs attached to the weels)
 Hubs: cylinders whose mass and inertia are very low, they have been used to carry the wheel and let it vertically translate and of course rotate.
 Axles: they haven’t been modeled as solid bodies but torsional springs attached to wheels and driven gears. Torsional stiffness has been computed considering axle its self and tripod housing as in series springs: Kt=4e2. Also damping has been added to avoid spikes.
 Suspensions: vertical axial springs attached to wheel and car body. K=33e3 and some damping.
 Gravity
Joits as well:
 Revolute joint between housing and car body
 Cylindrical joint between car body and ground; the translation axis parallel to g and centre placed 9.125m from model centre of mass and on wheel’s axis. This joint is responsable to the circular trajectory.
 Revolute joints between wheel and hub
 Translational joints between car body and hub
All motions have been removed.
In this kind of modelling, vertical load transfer in not taken into account, such as drag and lift.
More simulations have been run imposing a step function for the torque and varing its magnitude (100, 250, 500 Nm) for open differential and both 30deg and 60deg ramp surfaces.
Results reveal that this model is unstable as soon as a certain velocity is reached, even changing solvers, increasing number of steps and reducing errors. Unfortunately the unstability occours during the transient behavior in LSD cases (Fig.16) while in open differential cases after (Fig.17).
This is due probably to the cylindrical joint responsable of the circular trajectory, in fact slips are diverging and this behavior is more evident reducing curvature and applied torque. By the way GSTIFF SI2 solver is still the best choise.
In spite of this fact it’s possible to observe some interesting results before the system diverges. Three diagrams (Fig.18) have been obtained which show the difference between delivered torques on ground ‘ΔT’ and angular velocities between wheels ‘ΔΩ’.
Highligths:
 Increasing torque magnitude the slopes of those diagrams linear increase. This confirm the LSD as a torquesensitive device
 In nofriction case (i.e open differential) ΔT is very low (less then 3% applied torque) and constant.
Riassuming in table:
Open Diff ΔT 
60deg Slope 
30deg Slope 

100Nm  2.39  6.89  8.16 
250Nm  5.96  17.12  20.38 
500Nm  11.93  34.19  40.89 
Open Diff 
60deg Slope/Cin 
30deg 

100Nm  2.39%  0.0689  0.0816 
250Nm  2.39%  0.0685  0.0815 
500Nm  2.39%  0.0683  0.0818 
Conclusion
The targets of the analysis were to determine the torque distribution on axles and delivered torque on ground when different ramp angles were engaged.
As expected the locking torque is zero when friction is disabilitated on friction plates contacts in the model (open differential case), and it’s increasing as closer is the spool condition. Measured locking torque is equivalent to datasheet’s within a maximum 5% error.
The model used to achieve the second target is unstable over a certain angular velocity difference, and in the LSD case it’s not possible to see the torque difference between wheels on ground because the system diverges on transiet behavior. In open differential case, unstability still occours but it can be observed a costant torque difference (less then 3% input torque) and the predictioned behavior is confirmed. In both cases (before diverging) torque absolute values and slopes are dependent to applied input torque, so again the torquesensivity can be observed.
To obtain the delivered torque difference on ground in the LSD case it’s necessary to make the model more complex. For example it may be modelled also the front axle and leave one rotational steering degree of freedom and a PID control may be carried out in order to make the whole vehicle following the circular trajectory. Achieving this last, a minimum lap time simulation for skid pad comes with.
References
[1] Marco Gadola and Daniel Chindamo, “The Mechanical LimitedSlip Differential Revisited: HighPerformance and Racing Car Applications ”, International Journal of Applied Engineering Research, Volume 13, Number 2 (2018)