Calculating Maximum Acceleration and Torque

NOW TO CALCULATE THE MAXIMUM ACCELERATION OF A VEHICLE

Consider an automobile of weight W. We take it as a car, a RWD (Rear Wheel Drive). Assume a weight of W1 acts on rear wheels and W2 acts on front wheels.
W1 + W2 = W

Assume that the torque of the engine is such that it provides maximum acceleration possible – I assume a flat torque curve – we will change the assumption later.

Starting
The torque on the rear wheels is say T.
The maximum forward force that can be attained for that torque = T / R – where R is the radius of the tyre.

As explained before – provided the T / R is less than or equal to µW1.

therefore the maximum force, F = µW1

If the co-efficient of rolling friction is µr,
The car will move only if µW1 is greater than µrW – the total rolling friction on the car.

And acceleration = (µW1 - µrW) ÷ (W/g)

If the car was 4WD then the acceleration will be (µ - µr)g – as W1 and W2 acts on driving wheels.

This explains why the 4WD cars have better acceleration and Traction.


While moving
While the car is on move the wind resistance also acts on it. It can be shown as in figure 4 below.


The force due to wind will be distributed as force Fa on rear wheels and Fb on front wheels – which is in addition to the normal weight that acts on wheels. And a reaction force acts from the ground onto the wheels balancing the torque produced by the wind.

The magnitude of these forces (shown in red) acting can be found out by knowing the height at which this force is acting.

Now the total force, F = [µ (W1 + Fa) - µr W] – Wind resistance

And acceleration = F ÷ (W/g)

This shows the importance of Down force – Spoiler which is used in racing cars give down force by deflecting air, also the wind resistance is low.

ENGINE TORQUE REQD TO PRODUCE THIS ACCELERATION

Now the vehicle can be considered as having two kinds of motion – linear for the entire body + rotary for the rotating parts.

The rotating parts being the axles, propeller shaft, flywheel etc.

Torque produced by the engine = Te

Now during acceleration,
Torque reqd for the propeller shaft, Tp = I x angular acceleration.
Torque reqd for the flywheel, Tf = I x angular acceleration.

Torque reqd for other rotating parts(including gears)and for losses = To

Now torque available at the axis, Ta = Te – (Tf + Tp + To)
This should give angular acceleration to the axis + counter the torque thrust force resulting from the friction at wheels.
Ie, Ta = I x α + F x r
Where r – radius of the wheel
α – angular acceleration of axle = acceleration ÷ r.
Hence Te = Ta + Tf + Tb + To.