Question

What is meant by torsionally equivalent length of a shaft as referred to a stepped shaft? Derive the expression for the equivalent length of a shaft which has several steps.

Answer 1

Consider a shaft of varying diameters as shown in Fig.

Let this shaft is replaced by an equivalent shaft of uniform diameter d and length l as shown in fig. These two shafts must have the same total angle of twist when equal opposing torques T are applied T are applied at their opposite ends.

Let d₁, d₂ and d₃ = Diameters for the length l₁,l₂ and l₃ respectively,

1,2 and  3 = Angle of twist for the lengths l₁,l₂ and l₃ respectively,

= Total angle of twist, and

J₁,J₂ and J₃ = Polar moment of inertia for the shafts of diameters d₁, d₂ and d₃ respectively.

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 calculations, 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 that d = d₁.

The expression gives the length l of an equivalent shaft.