# Limited slip differential

A locking differential (or self-locking differential gear ) is a differential gear used in motor vehicles which, in contrast to the usual smooth-running ( open ) differential gear , is stiff to a certain extent. It brakes sudden increases in speed on the driven wheel that z. B. has temporarily lost his grip on the ground by jumping.

The automatic braking torque acting in the transmission between the two wheel shafts is usually generated as a function of the load and is therefore not available if one of the two wheels has hit a slippery road surface and cannot transmit any drive force to the ground. The other wheel is then not driven to compensate . For such cases a differential lock is used , the two output axes of which can be rigidly coupled to one another by manual switching. It is mainly used in commercial vehicles and is only used temporarily when driving on difficult terrain or fields. It must be possible to drive as straight ahead as possible, because both drive wheels turn at exactly the same speed and impede steering.

The self-locking differential, which is often installed in cars, generally also contains a load-independent, slight stiffness in order not to be hindered to a certain extent in progress in the event of one-sided slipperiness. This stiffness is constant and always effective, which has a slight understeer when cornering.

## Terms

• An open differential is a differential gear without a locking device.
• A differential lock is a differential gear whose compensating function can be switched off and on. The blocking effect is then 100%, i.e. H. the wheels are rigidly coupled, they always turn at the same speed.
• A limited-slip differential is a differential gear with a structurally increased frictional resistance. This restricts the compensating effect of the differential. The distribution of the torque when driving straight ahead on a homogeneous road surface is 50%: 50%.
• An axle or transverse differential is a differential gear for speed compensation between the two wheels of a driven axle.
• A longitudinal, central or center lock is a differential gear for speed compensation between two driven axles. In addition to the usual axle differentials, planetary gears are also used here, which divide the torque between the axles, for example in a ratio of 40%: 60% or 33%: 67% (in cars usually as a ratio of front: rear axle).

## function

The task of a limited-slip differential is to always supply a minimum amount of drive torque to each wheel and still allow different speeds on both wheels. It can be released with purely mechanical means, for example with friction clutches. Alternative solutions are based on mechanical friction in helical gears (e.g. in Torsen differential gears) or between sliding blocks and cams, or they are based on fluid friction in viscous couplings. Further principles are based on a freewheel. In the no-spin differential from EATON, both drive axles to the wheels are firmly connected to the middle drive part and to each other with a claw clutch each. When cornering, the outer wheel runs faster, which opens the outer clutch and the wheel turns faster than the inner one, as if released by a freewheel.

### Fixed value lock

The fixed-value lock is a differential lock that does not “lock” the two wheels firmly (100%), but allows them to rotate constantly (“fixed-value”) regardless of the load. In this simplest mechanical solution, the two output shafts are tied to the differential cage with the help of a friction clutch each. The i. d. As a rule, the contact pressure generated by springs is constant, each wheel contains a relatively small drive torque if the other should spin. But there is also a constant differential torque between the two wheels, which hinders cornering according to its value.

The locking effect is automatically generated, for example, by friction clutches whose contact force is proportional to the load (represented by the torque entering the transmission). This group also includes the Torsen limited-slip differentials.

• In normal differentials with bevel gears , the reaction forces from the toothing already act as expanding forces, which together with the differential carrier lead to frictional forces and thus to the locking effect.
• Stronger pressing forces are obtained with pressure rings, which are pushed apart by the differential bolt (shaft for the planetary bevel gears) rotating with the differential cage. The bolt slides on surfaces inclined in the circumferential direction on the pressure rings and generates axial forces on the rings, which form friction clutches with mating surfaces in the differential carrier. Variants with different levels of blocking can be produced with a different bevel angle of the contact surfaces on the pressure rings.

Transfer x times the drive torque effective on the other wheel to the wheel with the lower slip. The ratio x is called TBR ( Torque Bias Ratio , value range 1 ... , see below).

So that the wheel is driven on firm ground even when the other is freely spinning (drive torque = 0), load-dependent limited slip differentials are supported by the traction control of the ESP, whereby the reinforcement effect of the TBR has advantages over an open differential when driving.

### Speed-dependent limited-slip differentials

The locking effect depends on the difference in speed between the wheels. With viscous couplings , which are used to self-lock differential gears, shear forces arise between z. B. clutch plates and the fluid (mostly silicone oil ) that generate a locking effect through internal friction in the gearbox when there is a difference in speed between the two driven wheels.

## Torque bias ratio, locking value and efficiency

The locking effect is quantified using the torque bias ratio TBR (see above) and locking value S. Both are functions of the torques M R (right) and M L (left) on the two wheels.

Torque Bias Ratio:

${\ displaystyle TBR = {\ frac {{\ text {larger moment}} (M_ {L} {\ text {or}} M_ {R})} {{\ text {smaller moment}} (M_ {R} { \ text {or}} M_ {L})}} = 1 \ ldots \ infty}$ .

Lock value:

${\ displaystyle S = {\ frac {| M_ {L} -M_ {R} |} {(M_ {L} + M_ {R})}} = 0 \, \ ldots 1 \, \ qquad ({\ text {or:}} = 0 \, \% \ ldots 100 \, \%)}$ .

Conversion formulas:

${\ displaystyle TBR = {\ frac {1 + S} {1-S}} \ qquad S = {\ frac {TBR-1} {TBR + 1}}}$ .

The locking value describes the maximum difference between the torques distributed between the two wheels.

Frequently used values ​​for S are 25% to 50%. At 50% means that the faster wheel is driven with 1/4, the slower one with 3/4 of the input torque (3/4 - 1/4 = 1/2 or 50%), this corresponds to a TBR of 3 (3rd / 4 ÷ 1/4 = 3).

With an open differential, the torques on the right and left are ideally the same: M R = M L so that S = 0% and TBR = 1.

The braking torque acting on the friction clutches converts drive power into heat, the greater the speed deviation Δn in relation to the mean speed n. The formula for the efficiency η is:

${\ displaystyle \ eta = 1-S \ cdot {\ frac {\ Delta n} {n}} = 1-S \ cdot {\ frac {W} {2R}}}$
Δn = absolute deviation to the right and left of the mean speed
W = track width

Example: S = 0.5; W = 1.5 m; R = 10 m; η = 0.955 (95.5%)

## Differentiation from traction control

Limited slip differentials are not to be confused with traction control systems , which in turn are known as "traction control" or "traction control" or under manufacturer-specific names such as ASC or ASR.

The traction control system is now available as a pure software function of the electronic stability program (ESP) in almost every vehicle. The ESP partially mimics the effect of the limited slip differential. However, since the ESP can only brake, depending on the application, up to 50% (single-axis drive) of the engine power is braked. The ESP thus offers inexpensive help in emergencies, but it does not achieve the performance of limited-slip differentials by far and is particularly unsuitable for difficult terrain and for sporting use.

In the combination of traction control, both systems benefit from each other, but the ESP requires the use of torque-sensing limited-slip differentials for a stabilizing intervention.

## Disadvantages of simple (mechanical) limited slip differentials

Simple locking differentials react to different speeds and torques regardless of the causes. In some situations there are therefore disadvantages:

• Many designs (especially variants with friction disks) are subject to mechanical wear, so that the locking effect can decrease with increasing mileage of the vehicle.
• With some designs, a separating clutch or freewheel is required to make it compatible with an anti-lock braking system (ABS) or ESP, because otherwise the feedback between the individual wheels makes the control difficult or impossible.
• Locks react sensitively to tires with different rolling circumference (for example due to different air pressure, different tread depth) because they lead to differential speeds and result in increased wear. For example, the driving speed and driving distance with emergency bikes are limited.
• When cornering below the limit range (everyday use), they can lead to increased understeer , because here the required speed compensation is hindered when cornering.

With open axle differentials, both wheels transmit the same amount of torque, so there is no yaw moment around the vertical axis of the vehicle. With limited slip differentials, the axle drive force can also be distributed asymmetrically. A yaw moment arises, which the driver may have to compensate with counter-steering. This results in further disadvantageous effects:

• With a front axle limited-slip differential, the difference in tractive force can be felt by the driver at the steering wheel. For reasons of comfort, therefore, only a low locking value is usually selected on the front axle.
• On vehicles with rear axle differential locks, one-sided smoothness can lead to a yaw moment. Without counter-steering, the yaw moment causes the vehicle to turn towards the smoother side of the road.

## literature

• Johannes Loomann: Limited slip differentials in motor vehicles , VDI magazine 110 (1968), No. 6, pp. 209–216.
• John Loomann: gear transmission , Springer-Verlag , 1988, ISBN 3-540-18307-8 .
• Siegfried Wetzel: Technology in experiments - gearboxes , Phywe series , Industrie-Druck GmbH - Verlag, Göttingen, 1973, 4.8. Self-locking differential gear (fixed value lock) .