how limited slip differential work?

A limited slip differential has mechanisms to compensate for low grip on wheels. There are several different types. I'll list the main ones here.

Viscous Coupling - A VC diff is like the torque converter in an auto transmission. It has two sets of vanes - one set each connected to the two axles. They work by using the friction of fluid. A VC is filled with oil - similar to transmission fluid. When one wheel spins, it drives the vanes attached to its axle, which churns the fluid. This heats the fluid and increases its resistance. This creates friction which then drives the other set of vanes. VC have several downsides - because the resistance of the fluid varies with its temperature, the operation of the VC can change. Like a torque converter, you can also overheat the fluid in a VC, and it will not work.

Mechanical - Mechanical LSD's use either gears or clutches inside the diff to prevent single wheelspin. A simple clutch LSD has a set of clutch plates attached to each axle, and matching clutch plates attached to the housing. A mechanism uses the turning power applied to the LSD to push the clutches locked. The advantage of a clutch LSD is that in this manner the diffs are active whenever there is acceleration or deceleration force applied to it - even in a straight line. The downside is the same as with most LSD's. An LSD needs to have at least some resistance against both wheels or it won't function. Lift one wheel in the air and the LSD does nothing. There are ways to overcome this, however.

Electronic - There are electronic differentials out there that use various means to control power output to each wheel. These are often called 'active' differentials. The active yaw control in an EVO Lancer is one example. It uses electronics to sense turns, and physically delivers more power to the outside wheel. It works well, because even if the front wheels understeer (with the AWD), the extra power on the outside wheel actually pushes the nose around the corner.

you can watch this video for more understanding(easy to understand)

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