# Shaft in torsion?

I have some conceptual doubts about shafts subjected to torsion. When studying Strenght of Materials, to find stress and strain in power transmission shafts, we consider that one of the ends of the shaft is fixed, with all degrees of freedom restricted, and the other one is receiving torque. I'm having trouble to understand this fixed end. If it's a rotating shaft, how can it be fixed? But if it's not fixed, we can't apply equilibrium equations to solve the problem.

Suppose that the shaft is strain free (no twist) to begin. Make a mark on each end at the same angular position. Now, twist the shaft so that the angle between the marks is phi. There is now shear stress = T*c/J in the outer fibre, OK?

Now, suppose you roll both ends through an angle theta, maintaining the angular difference phi. The torque in the shaft is the same as it was before because it depends on the relative angle between the ends, not the absolute angle.

Where you choose the reference mark, at theta = 0, theta = pi/27, or something else, it is only the relative rotation, the twist, that matters as far as the induced torque. Treating one end as fixed may be convenient, but it is certainly not necessary.

Suppose that the shaft is strain free (no twist) to begin. Make a mark on each end at the same angular position. Now, twist the shaft so that the angle between the marks is phi. There is now shear stress = T*c/J in the outer fibre, OK?

Now, suppose you roll both ends through an angle theta, maintaining the angular difference phi. The torque in the shaft is the same as it was before because it depends on the relative angle between the ends, not the absolute angle.

Where you choose the reference mark, at theta = 0, theta = pi/27, or something else, it is only the relative rotation, the twist, that matters as far as the induced torque. Treating one end as fixed may be convenient, but it is certainly not necessary.

Suppose I have a shaft supported by a pair of bearings, one at each end, and there's a motor connected to the shaft. When we start the motor, the shaft will rotate with some angular velocity. Why would the shaft twist then, if it is free to rotate because of the bearings?

If it is truly free to rotate, there will be no torque in the shaft and no shear stress. Why would you do this?

The usual reason for turning a shaft with a motor is to enable the shaft to drive some machine, to do work, and that takes a torque in the shaft to transmit power to the driven machine.

Where you choose the reference mark, at theta = 0, theta = pi/27, or something else, it is only the relative rotation, the twist, that matters as far as the induced torque. Treating one end as fixed may be convenient, but it is certainly not necessary.

Okay. So when I treat one end as fixed is just a way of simplifying things. Actually, it's not really fixed, only in relation to the other end.

That's about the size of it.