The diagram is also from this web site.
Also known as rebellious celt, or celt, the rattleback is a
10-centimetre-long plastic toy with a base shaped like the hull of a boat.
When you spin it one way, it turns a few times before the ends start to
rattle up and down. The more it pitches, the slower it rotates until it
stops spinning altogether. The celt then starts to spin in the opposite
direction.
Full mathematical descriptions of the motion were derived by:
Hermann Bondi, Master of Churchill College,Cambridge,
and Mont Hubbard, professor of mechanical engineering at the University of
California, Davis.
The celt's astonishing trick needs three main ingredients.
First, the curved base must have two different radii--one long radius for
the lengthwise curve and one shorter radius for the tighter curve across
its width.
Next, the axes of symmetry of the celt must be skewed slightly from its
principal axes of inertia (see Diagram). Any rigid object has three
principal axes of inertia. They sit at right angles to each other and if
you spin the object about one of them, there is no tendency to rotate about
the other two.
Finally, there must be a different distribution of mass about each of the
two horizontal axes of inertia--a long, thin shape, say.
Halfway through the celt's journey, while it is pitching up and down
friction acts horizontally--at the point of contact between the celt and
the surface--to prevent the celt from slipping. One component of this
frictional force creates a torque that tends to rotate the celt about its
vertical axis.
The point of contact is moving all the time and the torque changes. If the
inertial and symmetrical axes of the celt coincided, the average torque
over a single oscillation would be zero. But for the celt, there is a net
torque in one direction. And it is this that reverses the angular
momentum.
Another way to understand how the celt works through energy. Each direction
of spin is linked to a different mode of oscillation: if clockwise rotation
feeds the pitching oscillation, then anticlockwise spin would feed a
side-to-side, or rolling, oscillation. So, when the celt spins clockwise,
any tiny pitching oscillation grows exponentially. It feeds off the
rotational energy and so slows down the spin. But even when there is zero
spin, the torque still acts,. So the direction of spin changes. Some celts
will reverse spin directions again by trading rotational energy with the
rolling oscillation.
For more info see:
John Satterly, Rocking Experiment with Two Degrees of Freedom, AJP 21,
267-273 (1953).
John Satterly, Three Interesting Instances of Rocking, AJP 23, 14-26
(1955).
John Satterly, Vibrational Dynamics With Lenses, Mirrors, and Prisms, AJP
23, 562-581, (1955).
John Satterly, Induced Rocking, AJP 26, 625-627 (1958).
H. Crabtree, An Elementary Treatment of the Spinning Tops and Gyroscopic
Motion, Chelsea, NY., (1967).
Jearl Walker, Rattlebacks and Tippe Tops; Roundabout: The Physics Of
Rotation in the Everyday World, 33-38.
Ira B. Freeman, What Is Trevelyan's Rocker?, TPT 12, 382, (1974).
Allan J. Boardman, The Mysterious Celt; Fine Woodworking No. 53, 68-69,
(July/Aug 1985).
Robert Walgate, Tops That Like to Spin One Way, Nature Vol 323, 204, (18
Sept. 1986).
Tad McGeer and Leigh Hunt Palmer, Wobbling, toppling, and forces of
contact, AJP 57, 1089-1098 (1989).
H. Richard Crane, How Things Work: The Rattleback Revisited, TPT 29,
278-279 (1991).