I know that the differential gearbox is providing a torque to the wheels; as the relation in the figure indicates : Tw (wheel torque) = Td (differential torque) × (ηd×nd). Where nd:is the differential gear ratio and η:is the effeciency of the differential gearbox. Now this torque at wheels (Tw), which is provided by the differential , is also = Rw×Fx (torque from road friction ) . My question is : Won't the torque from (Fx) on the wheels accelerate (increase) the wheel's rotational speed(ωw) ? If this is the case;now ωw is increasing. And since (ωd=nd×ωw); then ωd(differential or driveshaft rotational speed) should also increase to maintain the gear ratio (nd) constant. So how would this differential/drive shaft rotational speed (ωd) increase to maintain this gear ratio (nd) constant ?
The engine, transmission, differential and wheel are all essentially locked together in a traditional transmission. There are other ways to connect the four, but let's assume they are locked. If the engine supplies more torque than required to keep the vehicle at a constant speed, the vehicle speeds up, and all the rotational speeds go up. This is strictly a function of engine load versus throttle position (or throttle position sensor input). Gear ratios have nothing to do with it, and remain constant. If we are in a turn and the wheels are spinning at different rates, the difference in speed of the two wheels is seen in rotation of the differential plentary gear.
A modern continuously variable transmission will monitor engine load and speed and change the gear ratio to maximize performance and economy. The differential gear ratio will remain constant.