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.

Free Torsional Vibrations of a Geared System

          Consider a geared system as shown in Fig. a. It consists of a driving shaft C which carries a rotor A. It drives a driven shaft D which carries a rotor B, through a pinion E and a gear wheel F. This system may be replaced by an equivalent system of continuous shaft carrying a rotor A at one end and rotor B at the other end, as shown in Fig. b. It is assumed that

1. The gear teeth are rigid and do not distort under the tooth loads ( are always in contact),
2. There is no backlash in the gearing, and 
3. The inertia of the shafts and gears is negligible.

Let       d₁ and d₂  = Diameters of the shafts C and D respectively,

             l₁ and l₂ = Lengths of the shafts C and D,

             I_A and I_B = Mass moment of inertia of the rotors A and B, 

             ω_A and ω_B = Angular speed of the rotors A and B,

            d = Diameter of the equivalent shaft, 

            l = Length of the equivalent shaft, and 

           I^’_B = Mass moment of the inertia of the equivalent rotor B’.

          If the shafts are not strained beyond the limit of proportionality, each rotor in the geared system will oscillate with simple harmonic motion and there will be a node either in length l₁ or in the length l₂ .

The following two conditions must be satisfied by an equivalent system:

1. The kinetic energy of the equivalent system must be equal to the kinetic energy of the original system.
2. The maximum strain energy of the equivalent system must be equal to the maximum strain energy of the original system.

In order to satisfy the condition (1) for a given load,

K.E. of section l₁ + K.E. of section l₃ = K.E. of section l₁ + K. E. of section l₂ 

                    K.E. of section l₃ = K.E. of section l₂

In order to satisfy the condition (2)  for a given shaft diameter, 
                Strain energy of  l₁ and l₃ = Strain energy of  l₁ and l₂ 
                   Strain  energy of l₃ = Strain energy of l₂ 

Where   T₂ and T₃ = Torque on the sections l₂ and l₃, and

             θ₂ and θ3 = Angle of twist on sections l₂ and l₃.

Assuming that the power transmitted in the sections l3 and l2 is same, therefore, 

          Thus the single shaft is equivalent to the original geared system, if the mass moment of inertia of the rotor B^’ satisfies the equation (i) and the additional length of the equivalent shaft l₃ satisfies the equation (viii).

Length of the equivalent shaft, 

          Now the natural frequency of the torsional vibration of a geared system (two rotor system) may be determined as discussed below:

          Let the node of the equivalent system lies at N as shown in Fig. c. then the natural frequency of torsional vibration of rotor A, 

and natural frequency of the torsional vibration of rotor B^’,

          From the equations (x) and (xi), the value of and may be obtained and hence the natural frequency of the torsional vibrations is evaluated.