Introduction
A differential is a device, usually, but not necessarily, employing gears, which is connected to the outside world by three shafts, through which it transmits torque and rotation. The gears or other components make the three shafts rotate in such a way that a=pb+qc, where a,  b, and c are the angular velocities of the three shafts, and p and q are constants. Often, but not always, p and q are equal, so a is proportional to the sum (or average) of  b and c.
In automobiles and other wheeled vehicles, a differential allows the driving road wheels to rotate at different speeds. This is necessary when the vehicle turns, making the wheel that is travelling around the outside of the turning curve roll farther and faster than the other. The engine is connected to the shaft rotating at angular velocity a. The driving wheels are connected to the other two shafts, and p and q are equal. If the engine is running at a constant speed, the rotational speed of each driving wheel can vary, but the sum (or average) of the two wheels' speeds cannot change. An increase in the speed of one wheel must be balanced by an equal decrease in the speed of the other.

Types of differential
Ø   Open differential
Ø  LSD(limited slip differential)
Ø  Locking differential
Ø  Auto LSD(limited slip differential)

Open differential
Are the conventional type of differential usually consist of :



simple TAIL PINION (as shown below)

A Crown gear (as shown below)

Sun & Planet gear 



When road is straight wheel moves in a straight line direction, Pinion rotates the crown(pinion is connected by the engine by the means of propeller shaft).Crown with its housing consist of BEVEL gears i.e sun & planet gears. When vehicle moves straight there is no speed difference between two wheels hence when pinion rotates the crown gear rotates and in housing bevel gear rotates but sun gears does not rotate.(more clearly explain in video) thus providing constant speed to both wheels, but in case of turn both wheels travel different distance inner-one has to travel lesser distance as compared to outer, therefore the difference in speed is compensated by bevel gears they relatively increase  speed of one side of axle & reduce the speed of other. Assume on straight line both wheels are moving with speed 10km/h. but on turn the wheels will move with different speed the bevel gears increase the speed of outer wheel say 12km/h & will reduce the speed of inner one by 8km/h.

AXLE RATIO : Is defined as in how much no. of pinion turns , crown turns one revolution.
=  No. of turns by pinion/ No. of turns by crown

IT’S DRAWBACK
If one get off road or enter in low traction area whole power will transmitted to low traction area. Hence wheel will rotate there with full R.P.M but will not come out due to low griping or friction b/w tyre surface & road surface.
This generally happen when one wheel struck in mud, sand or watery surface.  


LSD(limited slip differential)
Like open differential drive LSD also work on same principle but in this case we have little different assembly. LSD is manufactured to remove the drawback of open differential (when one wheel lose traction vehicle lose traction i.e. one wheel spin freely over low friction area because whole torque is diverted  over that wheel). Therefore in this case when such condition occurs the major power here transmitted to that wheel but some amount of necessary torque is given to other wheel so that vehicle moves & one wheel which has traction pulls the other wheel which has no traction.

CONSTRUCTION
Pressure rings with V-grooves are assembled in the case of the mechanical differential. The four
lugs (protrusions) on the periphery of the pressure rings fit in the grooves provided inside the differential
case when the pressure rings are assembled, and the pressure rings can slide in axial directions.
The tips of the pinion shaft are chamfered to assemble the pinion shaft into the V-grooves of the
pressure rings, forming V-shaped cams. Similar to the pressure rings, the friction plates and spring
plates also have four lugs (protrusions) on the periphery and these lugs fit in the grooves provided
inside the differential case when the plates are assembled.
On the internal circumference of the friction disks and spring plates there are 4 or 6 lugs (protrusions)
which fit in the grooves provided on the outer circumference of the side gears, and the friction
disks and spring plates can slide in axial directions.
These friction plates, friction disks, spring plates, spring disks are alternately arranged and form a
clutch as a whole.
The spring plates and spring disks act as Belleville springs and provide initial torque. Moreover,
thrust washers are inserted between the side gears and differential case halves to adjust clearance
in the axial direction and prevent thrust from the side gears to work against the pressure rings.

    

IT’S DRAWBACK
It has one dis-advantage that is most of power is transmitted to low traction wheel. Here most of power is wasted due to transferring of more power to low traction wheel.



Locking differential
To overcome the fault or dis-advantage of open-differential & Limited slip differential (LSD), we have applied a mechanical or electronic or hydraulic torque provider unit which lock two axle providing with same torque or divides 1 torque coming from propeller to ½ & ½ torque on each axel & from the to wheel.
So when one wheel loses the traction the other wheel will be having same torque as wheel which loses the traction. 

IT’S DRAWBACK
The dis-advantage of locking differential is that at times of turning we have to un-lock this process of dividing the torque because at times of turning if both wheel will turn at same speed it will be difficult to turn. 



AUTO LSD or BREAK ACTIVATED LSD
To overcome limitations of open, lsd, locking differential a new type of differential is used they are electronic operated.
They work on basic principle of open differential but electronic operated which result when one wheel loses its traction the break are automatically activated to that  side wheel resulting in providing more resistance to that wheel which result in transferring most of the power to other wheel.
In this type breaks are selectively applied to those wheels which have least traction & power transfer to least resistance path. Hence will take out vehicle if it struck somewhere in low traction area.  

Refer this video for interactive learning




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