Torsional vibrations may result in shafts from following forcings:
(i) Inertia forces of reciprocating mechanisms (such as pistons in Internal Combustion engines)
(ii) Impulsive loads occurring during a normal machine cycle (e.g. during operations of a punch press)
(iii) Shock loads applied to electrical machineries (such as a generator line fault followed by fault removal and automatic closure)
(iv) Torques related to gear tooth meshing frequencies, turbine blade passing frequencies, etc.
For machines having massive rotors and flexible shafts (where system natural frequencies of torsional vibrations may be close to, or within, the source frequency range during normal operation) torsional vibrations constitute a potential design problem area.
In such cases designers should ensure the accurate prediction of machine torsional frequencies and frequencies of any of the torsional load fluctuations should not coincide with torsional natural frequencies.
Hence, determination of torsional natural frequencies of a dynamic system is very important.
Simple systems with a single disc mass: Consider a rotor system as shown in Figure 4.1. The shaft is considered as mass less and it provides the stiffness only. The disc is considered as rigid and it has no flexibility.
If we give a small initial disturbance to the disc in the torsional mode and allow it to oscillate its own, it will execute free vibrations. The oscillation will be simple harmonic motion (SHM) with a unique frequency, which is called natural frequency of the rotor system.
From the theory of torsion of shaft, we have
where, kt is the torsional stiffness of shaft, Ip is the rotor polar mass moment of inertia, J is the shaft polar second moment of area, l is the length of the shaft and q is the angular displacement of the rotor.
From free body diagram of the disc (see Figure 2)
Σ External torque on the disc
Equation (2) is the equation of motion of the disc due to free torsional vibrations. The free (or natural) vibration has the simple harmonic motion (SHM).
For the simple harmonic motion of the disc, we have
Where θ Ì‚ is the amplitude of the torsional vibration and ωn is natural frequency of the torsional vibration.
On substituting equations (3) and (4) into equation (2), we get
Hence, the torsional natural frequency is given by square root of the ratio of torsional stiffness to the polar mass moment of inertia.
A two-disc torsional system
In a two-discs torsional system as shown in Figure 3, whole of the rotor is free to rotate i.e. the shaft being mounted on frictionless bearings.
From free body diagrams of discs in Figure 4, we can write
Σ External torque on disc 
and Σ External torque on disc 
For SHM, solutions of equations (6) and (7) will take the following form
where ωn is the torsional natural frequency.
Substituting equation (8) into equations (6) & (7), we get
which can be written in the matrix form, as
Equation (9) is a homogeneous equation and the non-trial solution is obtained by taking determinant of the matrix [k] as
|k| = 0
which gives frequency equation of the following form
which can be simplified as
Roots of equation (11) are given as
From equation (9) corresponding to first natural frequency for 
, we get
From equation (12) it can be concluded that, the first root of equation (11) represents the case when both discs simply rolls together in phase with each other as shown in Figure 5 i.e. the rigid body mode, which is of a little practical significance.
From equation (9), for 
, we get
On substituting for 
in the above equation from equation (12), we get
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