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     d₁, d₂ 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 d₁, d₂ and d₃ respectively.

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.