PC1672 Advanced dynamics

4.9 Gyroscopes

Euler's equations are useful in situations where a body rotates under the action of a torque whose direction is fixed with respect to the body, for example: air resistance or a spacecraft with thrusters. More usually torques are due to external forces, such a gravity acting on a top or gyroscope. The full treatment of motion under an external torque requires the introduction of Euler angles (a set of three angles specifying the orientation of a rigid body with respect to space-fixed axes). Here we will focus on simple cases of a gyroscope precessing uniformly about the vertical axis, where we can avoid the need for the full formalism.

A gyroscope has an axis of symmetry ${\bf e}_3$ and so two of its principal moments of inertia are the same, . It is supported at a point on its axis and we will use this as our origin. Its centre-of-mass lies on the symmetry axis at

$\begin{displaymath}{\bf r}=r{\bf e}_3\end{displaymath}$

The gravitational force on it is

$\begin{displaymath}{\bf F}=-mg{\bf k}\end{displaymath}$

where ${\bf k}$ is the unit vector pointing vertically up.

Fast top

First we consider the case where the rate of precession is much slower than the rate of rotation about the symmetry axis. The angular velocity of the gyroscope is approximately

$\begin{displaymath}\hbox{\boldmath {$\omega$}}\simeq\omega_3{\bf e}_3\end{displaymath}$

and its angular momentum is

$\begin{displaymath}{\bf L}\simeq I_3\omega_3{\bf e}_3\end{displaymath}$

The torque about the origin is

$\begin{displaymath}\hbox{\boldmath {$\tau$}}={\bf r}\times (-mg{\bf k})\end{displaymath}$

Since ${\bf r}$ and ${\bf L}$ are parallel, we can write this in the form

$\begin{displaymath}\hbox{\boldmath {$\tau$}}\simeq {mgr\over I_3\omega_3}{\bf k}\times{\bf L}\end{displaymath}$

From the point of view of an inertial observer, the equation of motion is

$\begin{displaymath}\left({{\rm d}{\bf L}\over{\rm d}t}\right)_{\! 0}\simeq {mgr\over I_3\omega_3}{\bf k}\times{\bf L}\end{displaymath}$

This shows that ${\bf L}$ has constant magnitude and lies at a constant angle from the vertical axis ${\bf k}$ . It precesses uniformly about the vertical axis at a rate

$\begin{displaymath}\Omega\simeq {mgr\over I_3\omega_3}\end{displaymath}$

Slow top

We now turn to the general case where $\Omega$ need not be much smaller than $\omega_3$ . Uniform precession is still possible and we can describe this (without using Euler angles) by introducing a precessing coordinate system.

The precessing axes will be denoted with primes. They rotate about ${\bf k}$ at some rate $\Omega$ which we want to determine. One axis is chosen along the symmetry axis of the gyroscope, ${\bf e}_3^\prime={\bf e}_3$ , which lies at an angle $\theta$ from ${\bf k}$ . Another axis, ${\bf e}_1^\prime$ , is chosen in the same plane as ${\bf k}$ and ${\bf e}_3$ , at an angle $\theta+{\pi\over 2}$ from ${\bf k}$ . Finally ${\bf e}_2^\prime$ is perpendicular to the plane of ${\bf k}$ and ${\bf e}_3$ .

With respect to the precessing frame, the gyroscope is rotating at a rate $\omega_3^\prime$ about its axis. Its total angular velocity (with respect to an inertial frame) is the vector sum

$\begin{displaymath}\hbox{\boldmath {$\omega$}}=\omega_3^\prime{\bf e}_3^\prime+\Omega{\bf k}\end{displaymath}$

Using

$\begin{displaymath}{\bf k}=-\sin\theta{\bf e}_1^\prime+\cos\theta{\bf e}_3^\prime\end{displaymath}$

we can express this in terms of components along the primed axes as

$\begin{displaymath}\hbox{\boldmath {$\omega$}}=-\Omega\sin\theta{\bf e}_1^\prime +\omega_3{\bf e}_3^\prime\end{displaymath}$

where

$\begin{displaymath}\omega_3=\omega_3^\prime+\Omega\cos\theta\end{displaymath}$

is the total rate of rotation of the gyroscope about its symmetry axis.

Since the gyroscope is symmetric, any axis perpendicular to the symmetry axis is a principal axis with moment , including ${\bf e}_1^\prime$ (even though it is not fixed to the body). This allows us to write the angular momentum as

$\begin{displaymath}{\bf L}=-I_1\omega\sin\theta{\bf e}_1^\prime+I_3\omega_3{\bf e}_3^\prime\end{displaymath}$

From the point of view of the frame precessing with angular velocity

$\begin{displaymath}\hbox{\boldmath {$\Omega$}}=\Omega{\bf k}\end{displaymath}$

the equation of motion is

$\begin{displaymath}\left({{\rm d}{\bf L}\over{\rm d}t}\right)_{\! p}+\hbox{\boldmath {$\Omega$}} \times{\bf L}=\hbox{\boldmath {$\tau$}}\end{displaymath}$

This means that if we demand that ${\bf L}$ be constant in the precessing frame, we must have

$\begin{displaymath}\hbox{\boldmath {$\Omega$}}\times{\bf L}=\hbox{\boldmath {$\tau$}} =-mgr{\bf e}_3^\prime\times{\bf k}\end{displaymath}$

Both vector products lie in the ${\bf e}_2^\prime$ direction. For them to be equal, $\Omega$ must satisfy the quadratic equation

$\begin{displaymath}I_3\omega_3\Omega-I_1\Omega^2\cos\theta=mgr\end{displaymath}$

This has two roots

$\begin{displaymath}\Omega={I_3\omega_3\pm\sqrt{I_3^2\omega_3^2-4I_1mgr\cos\theta}\over 2I_1\cos\theta}\end{displaymath}$

The solution with the

sign describes fast precession. It is not often seen in practice. The other solution, with the

sign, is slow precession. This describes the usual behaviour of a gyroscope. In the limit where $\omega_3$ is very large it tends to the result we found above.

There is a minimum rate of rotation for uniform precession,

$\begin{displaymath}\omega_3^2\geq {4I_1mgr\cos\theta\over I_3^2}\end{displaymath}$

A gyroscope must spin about its axis faster than this, otherwise it will topple over. (This is signalled by $\Omega$ becoming complex.) If a gyroscope is suspended from above, so that $\cos\theta<0$ , uniform precession is always possible, no matter how slowly it is spinning.

Finally, if a gyroscope is set spinning fast enough with its axis vertical, it is said to ``sleep'', spinning quietly without precessing until friction slows its rate of rotation below the critical value and it stars to wobble.

Textbook references

K&K 7.3, 7.4, note 7.2
F&C 9.6
B&O 6.5 (7.10, 7.11)
M (12.10)
M&T (11.10)

Research project: spinning coin

Mike Birse
6th April 2001