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Here's an example. This will make a lot of sense if you're familiar with driving a manual transmission car. The clutch is just an adjustable friction device between two rotating shafts. That means that the clutch generates a force proportional to the difference in the speeds of the two shafts. For rotation, force is called torque, but I use them interchangeably here to try to make things simpler, don't get confused. That proportional relationship is the friction coefficient, which is adjustable from zero (clutch fully disengaged) to some value representing the maximum force the clutch can exert (clutch fully engaged). Note that I also refer to the crankshaft as the engine's output AFTER it goes through the transmission, and the driveshaft as the shaft BEFORE the differential.
Think for a moment about a situation where the car is stopped, the clutch is disengaged, the transmission is is first gear, and the engine is idling at 1000 RPM.
Let's run the clutch update function for this timestep. The driveshaft speed is zero and the crankshaft speed is some number that corresponds to 1000 RPM in first gear. The clutch is disengaged, so even though the input speeds are different, the clutch generates zero force. Nothing changes.
Some time later, we abruptly let the clutch pedal out half way. That means the clutch is now partially engaged. Let's run the clutch update function for this timestep. The driveshaft speed is still zero and the crankshaft speed is some number that corresponds to 1000 RPM in first gear. Since the clutch is partially engaged and the input speeds are different, it will return some force. We apply the force to the driveshaft, which causes it to accelerate. We apply an equal but opposite force to the crankshaft, which causes it to decelerate. So, the next timestep, the wheels will have started to very slightly rotate. Also, the engine will have started to very slightly decrease in RPM.
If you think about what happens for the next several timesteps, you'll see that the clutch will continue to generate force to speed up the driveshaft and slow down the engine. If we give the engine some gas, it'll exert a force that wants to speed the engine up, and depending on how much gas we give it, this might cancel out the force from the clutch that tries to slow the engine down. As time goes on and the speed difference between the driveshaft and crankshaft gets smaller, less force will be generated by the clutch, which means less acceleration. Eventually we'll probably reach a state where the crankshaft speed and driveshaft speed are the same, and now we can let the clutch all the way out (fully engaged). For VDrift the clutch doesn't actually lock (as NaN mentioned); the only thing that happens is since the clutch is fully engaged, any difference between driveshaft and crankshaft speed will cause very large forces that quickly act to bring their speeds to the same value again. So if you step on the gas with the clutch engaged, the engine will get a huge boost in force from the extra gas, the crankshaft will start going faster than the driveshaft, and this speed difference causes a large force to speed up the wheels and (try to) slow down the engine (but because of the huge force from the extra gas, the engine continues to speed up, just not as fast as it would with the clutch disengaged).
Make sense?
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