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.