Many people have, although few recommend doing so for street-driven cars. There are actually two different versions of this modification - one for FWD cars and one for AWD cars.
FWD vehicles often suffer from excessive wheelspin on one drive wheel. While the one wheel is spinning, the other wheel loses power and cannot drive the vehicle forward. Winter drivers may recognize this situation from experience, as many drivers have become stuck in icy conditions when one drive wheel starts to spin. Although the other wheel might be on solid ground, it cannot get power, and the vehicle goes nowhere.
In this case, welding the front axle differential forces both front tires to rotate at the same speed, so the lost power is recovered. It is no longer possible for one wheel to spin without the other also spinning.
Of course, you don't get something for nothing - the differential is there for a reason. The main reason is that, when cornering, the wheels at the inside of the corner must travel a shorter distance than the wheels on the outside. The differential allows the wheels to rotate at different speeds. Eliminate the differential, and cornering becomes a lot more difficult, causing unusual suspension and drivetrain stresses.
AWD vehicles have a different problem. Many AWD car owners would like to measure their engine output on a chassis dyno - a big set of stationary rollers that can measure the force imparted to the wheels. The problem is that there are almost no dynos capable of handling a four-wheel-drive vehicle.
The simple solution to this problem is to remove the AWD transfer case, which distributes power between the front and rear driveshafts, and temporarily convert the car to a FWD vehicle. Alas, it is not that simple. Most AWD DSMs have a limited-slip differential installed between the front and rear axles. This component is designed to allow the front and rear driveshafts to rotate at different speeds - up to a point. If the front and rear axles start to rotate at radically different speeds, the hydraulic mechanism in the differential 'lock' the driveshafts together and force the front and rear axles to rotate in sync.
So? Well, removing the transfer case creates a situation where the front and rear axles are out of sync, all the time. The limited-slip differential 'sees' no rotation from the front driveshaft (which is now disconnected), yet there is rotation of the rear driveshaft (which is being rotated as the car moves). The limited-slip mechanism promptly locks, and stays locked. The differential was never designed to operate in this permanently locked state, and serious damage will quickly result.
To get around this problem, the limited-slip differential in the AWD DSM can be welded to ensure it is mechanically, and not hydraulically, locked. This eliminates problems with removing the transfer case, and allows an AWD DSM to be converted to FWD operation for dyno runs. Converting back is as easy as replacing the transfer case, although this isn't really that easy.
Unfortunately, it also means that the front and rear axles must always operate perfectly in sync, which is not usually the case while cornering. One or more tires in this setup with inevitably slip, as they are pushed off their direction of travel; odd handling and strange driving behavior might result. The situation is similar to the operation of manually-locked 4WD vehicles (trucks & off-road vehicles) while making tight turns - wheels will drag, and the drivetrain will 'hitch' or 'catch' as the 4WD mechanisms try to keep all four wheels rotating at the same speed while they travel different distances.
The handling problems with welded differentials usually restrict them to drag racing applications only.
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