The video in Figure 1 of a tumbling T-handle on the International Space Station is a wonderful illustration of the instability of rotation about an asymmetric object’s intermediate principal axis. The T-handle’s rotational motion is well approximated by considering the object to be rigid and modeling its rotational behavior via a balance of angular momentum.

## Equations of motion

Suppose we parameterize the rotation of the T-handle by a 3-1-3 set of Euler angles: , , and . As illustrated in Figure 2, this sequence of rotations relates the space-fixed basis to the corotational basis associated with the T-handle’s principal axes, with the intermediate axis aligned with the object’s longitudinal axis.

The angular momentum of the T-handle about its mass center is given by , where are the principal moments of inertia (which are distinct for an asymmetric body) and are the corotational components of angular velocity: . Neglecting the small resultant moment acting on the T-handle due to the central force field while orbiting the earth, a balance of angular momentum with respect to the mass center yields , resulting in the equations of motion

For a 3-1-3 set of Euler angles, are related to the Euler angles , , and and their rates of change as follows:

Sets (1) and (2) constitute a system of first-order differential equations that solve for the T-handle’s orientation over time. These equations may be conveniently expressed in the form for numerical integration in MATLAB, where we take the state vector :

## Simulation and animation

To demonstrate instability of rotation about the intermediate principal axis, suppose the T-handle is initially spinning about this axis at a rate with small perturbations in the initial angular velocity about the other two principal axes: . For convenience, let the T-handle’s initial orientation be defined as , rad (to avoid the singularity in the 3-1-3 set of Euler angles), and . The resulting motion of the T-handle, which is very similar to the observed behavior in Figure 1, is animated in Figure 3 for a particular choice of physical and simulation parameters. It can also be shown that the simulated motion conserves both energy and angular momentum: the energy , and .

## Downloads

The animation in Figure 3 was generated using the following MATLAB code: