Single Degree of Freedom, Free Undamped Torsional Vibration
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## 1 Introduction to Single Degree of Freedom, Free Undamped Torsional Vibration

Figure 1: Single DOF, 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 θ0 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]

Table 1: System Givens
Givens

Constant Inertia

Torsional stiffness is Constant

Shaft Inertia is negligible

Figure 2: Single DOF, Undamped Torsional Vibration Free Body Diagram

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

 $LaTeX: I\ddot{\theta}=-k\theta$

or

 $LaTeX: \ddot{\theta}+\left(\frac{-k}{I}\right)\theta=0$

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

 $LaTeX: \omega=\sqrt{\left(\frac{k}{I}\right)}$ rad/s

where the frequency in Hz is

 $LaTeX: \omega \left [Hz\right ]=\frac{1}{2\pi}\omega \left [rad/s\right ]$

The torsional stiffness of the shaft, k, is equal to the applied torque divided by the angle of twist. Therefore

 $LaTeX: k=\frac{GJ}{l}$

for a cylindrical shaft where

$LaTeX: G$ is the modulus of rigidity for the shaft material,
$LaTeX: J$ is the second moment of area about the axis of rotation, and
$LaTeX: l$ is the length of the shaft.

The frequency in terms of G, J, I, and l is

 $LaTeX: f=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{GJ}{Il}}$ Hz

and

 $LaTeX: \theta=\theta_{0}\mbox{cos}\sqrt{\frac{GJ}{Il}}$

when $LaTeX: \theta=\theta_{0}$ and $LaTeX: \dot{\theta}=0$ at $LaTeX: t=0$. 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 non-constant shaft diameter

A cylindrical shaft comprised of length l1 & diameter d1 and length l2 & diameter d'2 can be replaced by a length l1 of diameter d1 and a length l of diameter d1 where, for the same stiffness

 $LaTeX: \left(\frac{GJ}{l}\right)_{l_{2}/d_{2}} = \left(\frac{GJ}{l}\right)_{l/d_{1}}$

that is, for the same shaft material,

 $LaTeX: \frac{d_{2}^4}{l_{2}}=\frac{d_{1}^4}{l}$

Therefore the equivalent length le of the shaft of constant diameter d1 is given by

 $LaTeX: l_{e}=l_{1}+\left(\frac{d_{1}}{d_{2}}\right)^4 l_{2}$

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

1. Beards, pp. 15-17