## Tuesday, 16 August 2016

### Torsionally Equivalent Shaft

Definition:
Torsionally equivalent shaft is a shaft of uniform diameter which twists through the same angle as the actual shaft of different diameters and different lengths, when equal and opposite torques of given amount are applied.

In the previous articles, we have assumed that the shaft is of uniform diameter. But in actual practice, the rotors are fixed to a shaft may have variable diameter for different lengths. To find the frequency of such a system, such a shaft may, theoretically, be replaced by an equivalent shaft of uniform diameter.
Consider a shaft of varying diameters and varying lengths as shown in Fig. (a). Let this shaft is replaced by an equivalent shaft of uniform diameter 'd' and length 'L' as shown in Fig. (b). These two shafts must have the same total angle of twist when equal opposing torques 'T' are applied at their opposite ends.
Let     d1d2 and d3 = Diameters for the lengths l1, l2 and l3 respectively,
θ1,  θ2 and θ3 = Angles of twist for the lengths l1, l2 and l3 respectively,
θ = Angle of twist for the diameter 'd' and length 'l',
J1, J2 and J3 = Polar moment of inertia  for the shaft of diameters d1dand drespectively.

We know that torsional equation as

Since the total angle of twist of the shaft is equal to the sum of the angle of twists of different lengths, therefore

In actual practice, it is assumed that the diameter 'd' of the equivalent shaft is equal to one of the diameter of the actual shaft. Let us assume d = d1.
This expression gives the length (L) of the Equivalent shaft.