The Intermediate Axis Theorem

In 1985, cosmonaut Vladimir Dzhanibekov commanded a mission to repair the space station Salyut-7. During the operation, he flicked a wing-nut to remove it. As it left the end of the bolt, the nut continued to spin in space, but every few seconds, it turned over through ${180^\circ}$ . Although the angular momentum did not change, the rotation axis moved in the body frame. The nut continued to flip back and forth, although there were no forces or torques acting on it.

Flipping nut [image from Veritasium].

The Dzhanibekov effect is shown clearly in a YouTube video. The video gives an intuitive explanation of the effect, due to UCLA mathematician Terry Tao on MathOverflow in 2011. The phenomenon has been described using several terms:

The Dzhanibekov Effect
The Tennis Racket Theorem
The Intermediate Axis Theorem

According to the video, the effect observed by Dzhanibekov was classified for ten years. Allegedly, there was some concern that the effect could act on the Earth, causing its orientation relative to the Sun to change suddenly and dramatically, with catastrophic consequences.

Tossing a Cell Phone

The behaviour of a freely-rotating rigid body with three distinct principal moments of inertia ${I_1 < I_2 < I_3}$ can be illustrated using a cell phone. The phone can be spun without any wobble about axes parallel to the longest and shortest edges. These correspond to the ${X}$ -axis with minimum moment of inertia and the ${Z}$ -axis with maximum moment of inertia. However, it is impossible to spin the phone steadily about the ${Y}$ -axis, parallel to the side with intermediate length and moment of inertia.

Cell-phone with principle axes and moments of inertia ${I_1 < I_2 < I_3}$ .

Stability Analysis

A simple stability analysis shows that rotation about the ${I_1}$ and ${I_3}$ axes is stable while rotation about the intermediate or ${I_2}$ -axis is unstable. The force-free motion of a rigid body is governed by Euler’s equations

$\displaystyle \begin{array}{rcl} I_1 \dot \omega_1 = (I_2-I_3)\omega_2 \omega_3 \\ I_2 \dot \omega_2 = (I_3-I_1)\omega_3 \omega_1 \\ I_3 \dot \omega_3 = (I_1-I_2)\omega_1 \omega_2 \end{array}$

Suppose the body is spinning about the ${I_1}$ -axis so that ${\omega_2\ll\omega_1}$ and ${\omega_3\ll\omega_1}$ . Then

$\displaystyle I_1 \dot \omega_1 = (I_2-I_3)\omega_2 \omega_3 \approx 0$

so that ${\omega_1}$ is approximately constant. Then we can write

$\displaystyle \dot \omega_2 = k_2 \omega_3 \,,\qquad \dot \omega_3 = k_3 \omega_2$

where ${k_2 = (I_3-I_1)\omega_1/I_2}$ and ${k_3 = (I_1-I_2)\omega_1/I_3}$ . The equations imply

$\displaystyle \ddot \omega_2 = k_2k_3\,\omega_2 \,,\qquad \ddot \omega_3 = k_2k_3\,\omega_3$

and, since ${k_2k_3<0}$ , the solutions oscillate, remaining small.

Suppose now that the body is spinning about the ${I_2}$ -axis with ${\omega_1\ll\omega_2}$ and ${\omega_3\ll\omega_2}$ . Then, initially, ${\omega_2}$ does not change much. We have

$\displaystyle \dot \omega_1 = K_1 \omega_3 \,,\qquad \dot \omega_3 = K_3 \omega_1$

where ${K_1 = (I_2-I_3)\omega_2/I_1}$ and ${K_3 = (I_1-I_2)\omega_2/I_3}$ . We note that ${K_1K_3 > 0}$ . Thus, the equations

$\displaystyle \ddot \omega_1 = K_1K_3\,\omega_1 \,,\qquad \ddot \omega_3 = K_1K_3\,\omega_3$

imply exponential growth of ${\omega_1}$ and ${\omega_3}$ , so the motion is unstable.

The simple pendulum

We look at the behaviour of a simple pendulum near the point of unstable equilibrium. This is the point where the bob is highest. It may remain at this top point, but any disturbance, however small, will cause it to move away, and to swing in a full circle.

Phase portrait for a simple pendulum. Blue area: libration. Red area: Rotation.

The figure above shows the phase portrait of a simple pendulum. Trajectories in the blue area represent oscillating or librating motion while trajectories in the red area represent rotations. On trajectories near the boundary between blue and red regions, the pendulum bob lingers for a long time near the unstable equilibrium point.

The figure below shows trajectories of the angular momentum vector on a phase sphere for the triaxial rigid body. We see a stable equilibrium point on the ${x}$ -axis and an unstable or hyperbolic point on the ${y}$ -axis. The topology of the diagram is similar in crucial ways to the phase diagram for the simple pendulum. If the angular momentum is very close to the ${y}$ -axis, it will remain there for a long time. However, ultimately it will move away quickly to approach the antipodal point where, again, it will remain for a long time.

Trajectories on a phase sphere for a freely rotating rigid body for ${I_1 < I_2 < I_3}$ . Points on the ${x}$ -axis and ${z}$ -axis are centres (stable). Points on the ${y}$ -axis are saddle points (unstable) [image from Bender and Orszag (1978)].

The Euler equations can be solved analytically in terms of Jacobian elliptic functions. However, the character of the solution can be seen from a simple numerical integration. The graph below shows the evolution of the three components of angular momentum starting from a state where ${\omega_2}$ is large and both ${\omega_1}$ and ${\omega_3}$ are small. We see how ${\omega_2}$ remains almost constant, but flips sign from time to time.

Numerical solution of Euler’s equations for an initial rotation close to the intermediate axis.

Conclusion

The intermediate axis theorem has been known since at least the early 1800’s. However, the sudden reversals observed by Dzhanibekov were not noticed. Perhaps this is because, at the earth’s surface, there is only a short time before the body falls to the ground, and the effect is hard to observe. In space, with no forces acting on the body, it can spin undisturbed for a long time — long enough for the effect to become obvious.

The first American satellite, Explorer 1, was launched in January 1958. The satellite, which was long and slender, was spin-stabilized about its long axis, the one with the smallest moment of inertia. However, the satellite had flexible antennae which oscillated, dissipating energy and destabilising the rotation. The satellite toppled out of its stable spinning state, flipped over and began to spin about the axis with the maximum moment of inertia.

The Earth is rotating about the axis with largest moment of inertia and smallest kinetic energy, so no catastrophic overturn is likely any time soon.

Sources

${\bullet}$ Ashbaugh, M. S., C. C. Chicone and R. H. Cushman, 1991: The Twisting Tennis Racket. J.~Dynam.~and Diff.~Eqns. 3(1), 67–85.

${\bullet}$ Bender, Carl M. and Steven A. Orszag, 1978: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill Book Company. 593pp.

${\bullet}$ Marsden, Jerrold E. and Tudor Ratiu, 1999 Introduction to Mechanics and Symmetry. Springer, 582pp. ISBN: 978-0-3879-8643-2.

${\bullet}$ Poinsot, Louis, 1834: Theorie Nouvelle de la Rotation des Corps, Bachelier, Paris

${\bullet}$ Veritasium: YouTube video on the Dzhanibekov Effect.

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