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,l2and 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.
