Single Degree of Freedom, Free Undamped Torsional Vibration


Jump to: navigation, search
Single Degree of Freedom, Free Undamped Torsional Vibration
Green carrot left.gif
Single DOF, Undamped Translational Vibration Modeling
Green carrot.jpg
In order to prevent spam, users must register before they can edit or create articles.

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

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


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


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