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LIMITED SLIP DIFFERENTIAL BASICS
 
What is a Limited Slip Differential and why do we need one? Let's take a look at the diagram:
Forza Motorsport 3, limited slip differentials traction explained
Let's create a hypothetical situation where the front wheel drive car wants to travel the circular path shown.
The outside wheels will travel the path of the large circle and the inside wheels will travel the path of the smaller circle.
The large circle has a diameter of 45 feet and the smaller circle has a diameter of 30 feet.
The car will travel the circumference of the circles, so we use the basic formula: pi (3.14) X diameter
Large circle: 3.14 x 45 = 141.3 feet
Small circle: 3.14 x 30 = 94.2 feet
As you can see, the inside wheels travel less than the outside wheels, so the inside wheels are spinning slower and the outside wheels are spinning faster.
If both wheels were locked, and spinning at the same speed, two negative things would happen when you put the power down:
1- The car would understeer.
2- The inside driven wheel would spin, losing traction.
This is why the differential was created, to compensate for the disparity in travel between the inside and outside wheels. There are different types of LSDs. The basic concept of the differential is the same, both wheels spin at the same rate when the car is accelerating in a straight line, but have internal gears which allow the driven wheels to spin at different rates when the car is cornering.
 
THE 'OPEN' DIFFERENTIAL
The open differential is the basic form. First of all, a differential is a device that splits (distributes) engine torque (eg: from the driveshaft) in two different directions and allows each output to spin the wheels at different speeds.
The open differential distributes the torque evenly. This distribution is dependent upon power (horsepower from the engine) and traction (the interface between the tires, and road surface).
The fact that the torque is distributed evenly is the major shortcoming of the basic differential.
Forza Motorsport 3, the open differential
If you look at the first diagram, let's say one driven wheel is on ice, while the other driven wheel is on solid ground. The differential will only apply as much torque as it can without spinning either wheel, the key word being 'either'. So if one wheel has little or no traction, the same amount of torque will be applied to the opposite wheel. This means the wheel with adequate traction will practically have its power supply cut off as long as it's counterpart is spinning on the ice.
If you look at the second diagram, you'll see why an open differential is bad for for road racing. Look at the car and imagine it turning sharply in a high speed corner. As the weight of the car is being transferred to the outside wheel, the inside wheel is lifted off the ground slightly. Since the inside wheel has no traction, spinning in mid-air, the outside wheel just won't spin until the inside wheel comes back in contact with the road.
 
THE 1 WAY, 1.5 WAY AND 2 WAY LSDs
The limited slip differential was created to enhance the distribution of torque dependent upon a variety of driving/road conditions.
There are many types of LSDs (acronym), but without getting into all the specifics, we'll concentrate on how differentials work in Forza Motorsport. Basically, choosing and setting up a differential determines how, and under what driving conditions the engine torque is distributed to the driven wheels.
One common type of LSD available is a mechanical, clutch-type unit that is driveshaft torque sensitive. It is the Sport LSD named in-game in FM2. In the 'buy upgrades' menu it is ascribed to the 1.5 way unit, but a 2 way can be a clutch-type differential as well.
Semantics aside, understanding 'torque sensitive' LSDs will help you understand how to adjust either of the two tunable differentials in game.
Basically, there are three different type of torque inputs that the differential will react to:
1- LOADING (acceleration): The more throttle you give the accelerator pedal, the more the driven wheels are coupled together.  
2- NO LOADING (static): This is the degree (strength) of coupling that the LSD has built into it (preloaded) with no input.  
3- OVER RUN (deceleration): This is how much coupling the driven wheels have when you lift off the accelerator.  
How the differential operates during deceleration is what determines whether it's considered a: 1 WAY, 1.5 WAY, or 2 WAY.
The 1 WAY LSD is torque sensitive under acceleration, but when the driver lifts off the throttle, the wheels spin freely (not coupled together) and act under deceleration like an open differential. This type requires the least amount of driving skill.
The 2 WAY LSD is torque sensitive under BOTH acceleration as well as deceleration. Driving a car with a 2 way differential can be tricky, because sudden lift off the throttle (as opposed to a smooth, gradual lift) can result in the car getting away from you during cornering (understeer), and can be particularly troublesome in transitions (slaloms, or S's) where the inside wheel is still coupled to the outside wheel, while the weight of the car is being quickly transferred from one side to the other. If you're into drifting, the 2 WAY differential is preferred, to keep both wheels spinng at the same rate through the corners.
The 1.5 WAY is somewhere in between the above mentioned two, the wheels remain coupled to a lesser extent under deceleration, but not to the extent of the 2 WAY LSD.
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