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Contents
1 Introduction to Single Degree of Freedom, Free Undamped Torsional Vibration
Consider a system with a circular disk at the end of cylindrical shaft. The hanging disk has a mass moment of inertia I about the axis of rotation. The cylindrical shaft has a torsional stiffness k. If the mass is rotated through an angle θ_{0} and released, torsional vibration results. Typically the inertia of the shaft can be ignored.
2 Equations for Single Degree of Freedom, Free Undamped Torsional Vibration^{[1]}
Givens 

Constant Inertia 
Torsional stiffness is Constant 
Shaft Inertia is negligible 
Figure 1 presents a diagram of the general single degree of freedom, undamped torsional vibration model. When the disk is rotated by an angle θ certain forces result. The resulting forces are presented in the FBDs given in Figure 2. The equation of motion is
or
This is of similar form to Eqn. (2.1) from the translational vibration example. The motion is simple harmonic and the system has a natural frequency of
rad/s 
where the frequency in Hz is
The torsional stiffness of the shaft, k, is equal to the applied torque divided by the angle of twist. Therefore
for a cylindrical shaft where
 is the modulus of rigidity for the shaft material,
 is the second moment of area about the axis of rotation, and
 is the length of the shaft.
The frequency in terms of G, J, I, and l is
Hz 
and
when and at . If the shaft does not have a constant diameter, it can be replaced analytically by an equivalent shaft of different length but with the same stiffness and a constant diameter.
2.1 Generic Example of Torsional System with nonconstant shaft diameter
A cylindrical shaft comprised of length l_{1} & diameter d_{1} and length l_{2} & diameter d'_{2} can be replaced by a length l_{1} of diameter d_{1} and a length l of diameter d_{1} where, for the same stiffness
that is, for the same shaft material,
Therefore the equivalent length l_{e} of the shaft of constant diameter d_{}1 is given by
It should be boted that the analysis techniques for translational and torsional vibration are very similar, as are the equations of motion.
See Torsional Vibrations of a Geared System for a specific example.
3 Control of Single Degree of Freedom, Free Undamped Torsional Vibration
The equations for a torsional system are equivalent to those for the linear vibration system. See Single Degree of Freedom, Free Undamped Vibration for examples of controllers.
4 Notes
Beards, C. F. 1995 Engineering Vibration Analysis with Applications to Control Systems. ISBN 034063183X
4.1 References
 ↑ Beards, pp. 1517