A limited-slip differential (LSD) is a type of automotive differential gear arrangement that allows for some difference in angular velocity of the output shafts, but imposes a mechanical bound on the disparity.
In an automobile, such limited-slip differentials are sometimes used in place of a standard differential, where they convey certain dynamic advantages, at the expense of greater complexity.
It is commonly referred to as an "LSD" by today's automobile enthusiasts.
Early history 
In 1932, Ferdinand Porsche designed a Grand Prix racing car for the Auto Union company. The high power of the design caused one of the rear wheels to experience excessive wheel spin at any speed up to 100 mph (160 km/h). In 1935, Porsche commissioned the engineering firm ZF to design a limited-slip differential that would perform better. The ZF "sliding pins and cams" became available, and one example was the Type B-70 for early VWs.
The main advantage of a limited-slip differential is demonstrated by considering the case of a standard (or "open") differential in off-roading situations where one wheel has no contact with the ground. In such a case (with a standard differential), the non-contacting wheel will receive 100% of the power, while the contacting wheel will remain stationary. The torque transmitted will be equal at both wheels, and therefore, will not exceed the threshold of torque needed to move the wheel with grip. In this situation, a limited-slip differential prevents 100% of the power from being allocated to one wheel, and thereby keeping both wheels in powered rotation.
Basic principle of operation 
Automotive limited-slip differentials all contain a few basic elements. First, all have a gear train that, like an open differential, allows the output shafts to spin at different speeds while holding the sum of their speeds proportional to that of the input shaft.
Second, all have some sort of mechanism that applies a torque (internal to the differential) that resists the relative motion of the output shafts. In simple terms, this means they have some mechanism which resists a speed difference between the outputs, by creating a resisting torque between either the two outputs, or the outputs and the differential housing. There are many mechanisms used to create this resisting torque. The type of limited-slip differential typically gets its name from the design of this resisting mechanism. Examples include viscous and clutch-based LSDs. The amount of limiting torque provided by these mechanisms varies by design and is discussed later in the article.
Torque split during operation 
An open differential has a fixed torque split between the input and outputs. In most cases the relationship is:
- Trq out_1 = Trq out_2 , where 1 and 2 are typically the left and right drive wheels.
- Trq in = Trq out_1 + Trq out_2 .
Thus the wheels always see the same torque even when spinning at different speeds, including the case where one is stationary. Note, the torque split can be unequal, though 50:50 is typical.
A limited-slip differential has a more complex torque-split and should be considered in the case when the outputs are spinning the same speed and when spinning at different speeds. The torque difference between the two axles is called Trq d . (In this work it is called Trq f for torque friction). Trq d is the difference in torque delivered to the left and right wheel. The magnitude of Trq d comes from the slip-limiting mechanism in the differential and may be a function of input torque (as in the case of a gear differential), or the difference in the output speeds (as in the case of a viscous differential).
The torque delivered to the outputs is:
- Trq 1 = ½ Trq in + ½ Trq d for the slower output
- Trq 2 = ½ Trq in – ½ Trq d for the faster output
When traveling in a straight line, where one wheel starts to slip (and spin faster than the wheel with traction), torque is reduced to the slipping wheel (Trq 2 ) and provided to the slower wheel (Trq 1 ).
In the case when the vehicle is turning and neither wheel is slipping, the inside wheel will be turning slower than the outside wheel. In this case the inside wheel will receive more torque than the outside wheel, which can result in understeer.
When both wheels are spinning at the same speed, the torque distribution to each wheel is:
- Trq (1 or 2) = ½ Trq in ±(½ Trq d ) while
- Trq 1 +Trq 2 =Trq in .
This means the maximum torque to either wheel is statically indeterminate but is in the range of ½ Trq in ±( ½ Trq d ).
Several types of LSD are commonly used on passenger cars.
- Fixed value
- Torque sensitive
- Speed sensitive
- Electronically controlled
Fixed value 
In this differential the maximum torque difference between the two outputs, Trq d , is a fixed value at all times regardless of torque input to the differential or speed difference between the two outputs. Typically this differential used spring loaded clutch assemblies.
Torque sensitivity 
This category includes helical gear limited-slip differentials and clutch, cone (an alternative type of clutch) where the engagement force of the clutch is a function of the input torque applied to the differential (as the engine applies more torque the clutches grip harder and Trq d increases).
Torque sensing LSDs respond to driveshaft torque, so that the more driveshaft input torque present, the harder the clutches, cones or gears are pressed together, and thus the more closely the drive wheels are coupled to each other. Some include spring loading to provide some small torque so that with little or no input torque (trailing throttle/gearbox in neutral/main clutch depressed) the drive wheels are minimally coupled. The amount of preload (hence static coupling) on the clutches or cones are affected by the general condition (wear) and by how tightly they are loaded.
Clutch, cone-type LSD 
The clutch type has a stack of thin clutch-discs, half of which are coupled to one of the drive shafts, the other half of which are coupled to the spider gear carrier. The clutch stacks may be present on both drive shafts, or on only one. If on only one, the remaining drive shaft is linked to the clutched drive shaft through the spider gears. In a cone type the clutches are replaced by a pair of cones which are pressed together achieving the same effect.
One method for creating the clamping force is the use of a cam-ramp assembly such as used in a Salisbury/ramp style LSD. The spider gears mount on the pinion cross shaft which rests in angled cutouts forming cammed ramps. The cammed ramps are not necessarily symmetrical. If the ramps are symmetrical, the LSD is 2 way. If they are saw toothed (i.e. one side of the ramp is vertical), the LSD is 1 way. If both sides are sloped, but are asymmetric, the LSD is 1.5 way. (See the discussion of 2, 1.5 and 1 way below)
An alternative is to use the natural separation force of the gear teeth to load the clutch. An example is the center differential of the 2011 Audi Quattro RS 5.
As the input torque of the driveshaft tries to turn the differential center, internal pressure rings (adjoining the clutch stack) are forced sideways by the pinion cross shaft trying to climb the ramp, which compresses the clutch stack. The more the clutch stack is compressed, the more coupled the wheels are. The mating of the vertical ramp (80–85° in practice to avoid chipping) surfaces in a one-way LSD on overrun produces no cam effect or corresponding clutch stack compression.
2-, 1-, and 1.5-way LSD 
Broadly speaking, there are three input torque states: load, no load, and overrun. During load conditions, as previously stated, the coupling is proportional to the input torque. With no load, the coupling is reduced to the static coupling. The behavior on overrun (particularly sudden throttle release) determines whether the LSD is 1 way, 1.5 way, or 2 way.
A 2-way differential will have the same limiting torque Trq d in both the forward and reverse directions. This means the differential will provide some level of limiting under engine braking.
A 1-way differential will provide its limiting action in only one direction. When torque is applied in the opposite direction it behaves like an open differential. In the case of a FWD car it is argued to be safer than a 2-way differential. The argument is if there is no additional coupling on overrun, i.e. a 1-way LSD as soon as the driver lifts the throttle, the LSD unlocks and behaves somewhat like a conventional open differential. This is also the best for FWD cars, as it allows the car to turn in on throttle release, instead of ploughing forward.
A 1.5-way differential refers to one where the forward and reverse limiting torques, Trq d_fwd, d_rev , are different but neither is zero as in the case of the 1-way LSD. This type of differential is common in racing cars where a strong limiting torque can aid stability under engine braking.
Geared LSD 
Geared, torque-sensitive mechanical limited-slip differentials use helical gears or worm gears rather than the beveled spider gears of the clutch based differentials. As torque is applied to the gears, they are pushed against the walls of the differential housing, creating friction. The friction resists the relative movement of the outputs and creates the limiting torque Trq d .
- Torsen differential based upon the Dual-Drive Differential invented by Vernon Gleasman in 1958, then later sold to Gleason Corporation, who started marketing it in 1982;
- Quaife differential, sold under the name Automatic Torque Biasing Differential (ATB), covered by European Patent No. 130806A2.
- Eaton Corporation differential, sold under the name Eaton Detroit Truetrac.
Speed sensitivity 
Speed-sensitive differentials limit the torque difference between the outputs, Trq d , based on the difference in speed between the two output shafts. Thus for small output speed differences the differential’s behavior may be very close to an open differential. As the speed difference increase the limiting torque increases. This results in different dynamic behavior as compared to a torque sensitive differential.
The viscous type is generally simpler because it relies on hydrodynamic friction from fluids with high viscosity. Silicone-based oils are often used. Here, a cylindrical chamber of fluid filled with a stack of perforated discs rotates with the normal motion of the output shafts. The inside surface of the chamber is coupled to one of the driveshafts, and the outside coupled to the differential carrier. Half of the discs are connected to the inner, the other half to the outer, alternating inner/outer in the stack. Differential motion forces the interleaved discs to move through the fluid against each other. In some viscous couplings when speed is maintained the fluid will accumulate heat due to friction. This heat will cause the fluid to expand, and expand the coupler causing the discs to be pulled together resulting in a non-viscous plate to plate friction and a dramatic drop in speed difference. This is known as the hump phenomenon and it allows the side of the coupler to gently lock. In contrast to the mechanical type, the limiting action is much softer and more proportional to the slip, and so is easier to cope with for the average driver. New Process Gear used a viscous coupling of the Ferguson style in several of their transfer cases including those used in the AMC Eagle.
Viscous LSDs are less efficient than mechanical types, that is, they "lose" some power. In particular, any sustained load which overheats the silicone results in sudden permanent loss of the differential effect. They do have the virtue of failing gracefully, reverting to semi-open differential behavior. Typically a visco-differential that has covered 60,000 miles (97,000 km) or more will be functioning largely as an open differential; this is a known weakness of the original Mazda MX-5 (a.k.a. Miata) sports car. The silicone oil is factory sealed in a separate chamber from the gear oil surrounding the rest of the differential. This is not serviceable; when the differential's behavior deteriorates, the VLSD center must be replaced.
Gerotor pump 
This style limited-slip differential works by using a gerotor pump to hydraulically compress a clutch to transfer torque to the wheel that is rotating the slowest. The gerotor pump uses the differential carrier or cage to drive the outer rotor of the pump and one axle shaft to drive the inner rotor. When there is a difference between the left and right wheels' speed, the pump pressurizes the hydraulic fluid causing the clutch to compress. thereby causing the torque to be transferred to the wheel that is rotating the slowest. These pump-based systems have a lower and upper limits on applied pressure which allows the differential to work like a conventional or open differential until there is a significant speed difference between the right and left wheel, and internal damping to avoid hysteresis. The newest gerotor pump based system has computer regulated output for more versatility and no oscillation. Empty citation (help)
An electronic limited-slip differential will typically have a planetary or bevel gear set similar to that of an open differential and a clutch pack similar to that in a torque sensitive or gerotor pump based differential. In the electronic unit the clamping force on the clutch is controlled externally by a computer or other controller. This allows the control of the differential’s limiting torque, Trq d , to be controlled as part of a total chassis management system. An example of this type of differential is Subaru’s DCCD used in the 2011 Subaru WRX STi. Another example is the Porsche PSD system used on the Porsche 928. A third example is the SAAB XWD (Haldex Generation 4) with eLSD, it uses a common (electronically controlled via the vehicle computer network) hydraulic power pack to control both the longitudinal and transversal torque transfer of the XWD system. The same Haldex system is used on several other GM Epsilon based vehicles such as the Cadillac SRX etc.
Electronic systems: brake-based 
These systems are alternatives to a traditional limited-slip differential. The systems harness various chassis sensors such as speed sensors, anti-lock braking system (ABS) sensors, accelerometers, and microcomputers to electronically monitor wheel slip and vehicle motion. When the chassis control system determines a wheel is slipping, the computer applies the brakes to that wheel. A significant difference between the limited-slip differential systems listed above and this brake-based system, is that brake-based systems do not inherently send the greater torque to the slower wheel.
BMW's electronic limited-slip differential used on the 2012 535i is an example of such a system.
Factory names 
In the 1950s and 1960s many manufacturers began to apply brand names to their LSD units. Packard pioneered the LSD under the brand name "Twin Traction" in 1956, becoming one of the first manufacturers. Other factory names for LSDs include:
Cadillac : Controlled
- Chevrolet/GMC Positraction
- Chrysler: Sure Grip
- Dana Corporation:Trak-Lok or Powr-Lok
- Ferrari: E-Diff
- Fiat: Viscodrive
- Ford: Equa-Lock and Traction-Lok
- International: Trak-Lok (clutches only) or Power-Lok (clutch and ramping engagement process)
- Jeep: Trac-Lok (clutch-type mechanical), Tru-Lok (gear-type mechanical), and Vari-Lok (gerotor pump), Power Lok
- Oldsmobile: Anti-Spin
- Pontiac: Safe-T-Track
- Porsche: PSD (electro-hydraulic mechanical)
- Saab: Saab XWD eLSD
- Studebaker-Packard Corporation: Twin Traction
- TVR: Hydratrak
- Yukon Gear & Axle: Duragrip
- The Motor Vehicle K.Newton W.Steeds T.K.Garrett Ninth Edition pp549-550
- Chocholek, S E. "The development of a differential for the improvement of traction control". IMechE 1988 
- Deur, J. et al, "Modeling and Analysis of Active Differential Dynamics", Jour of Dynamic Systems, Measurement and Control, ASME, Nov 2010 
- Product News, GearTechnology, "Center Differential of the New Audi Quattro with Cylkro Face". GearTechnology Nov/Dec 2010 
- Donnon, Martin et al. (2004). High Performance Imports 48. Express Motoring Publications. pp. 77–80. "...being able to run the driven wheels almost fully open under deceleration. In a powerful front wheel drive scenario where torque steer is a constant enemy, this approach [1 way LSD] has some definite advantages."
- Donnon, Martin et al. (2003). Zoom 67. Express Motoring Publications. pp. 45–48. "...the gel used can quite suddenly alter with massive temperature, and lose its ability to generate torque transfer."
- Subaru, “DCCD – Driver Controlled Center Differential”
- Limited-slip differentials explained
- What is a Quaife ATB differential – R.T. Quaife Engineering Limited