Why does a wobble board wobble?

Rod Cross, Sydney University, July 2006

The wobble board was made famous by Rolf Harris around 1960 when he used it as a backing for his song “Tie me kangaroo down”. A wobble board is just a thin, moderately stiff sheet that wobbles at a varying frequency. During one wobble burst, the frequency rises from about 100 Hz to about 500 Hz over a time duration of about 0.15 sec. Wobbling occurs when the board is driven at around 2 or 3 Hz, as shown in the wobble board movie. In the movie, a 1mm thick printed circuit board is lifted up and down while holding it between the hands A thin metal sheet can wobble at frequencies up to about 1 kHz when driven at 2 or 3 Hz. The wobble is a transient oscillation rising in pitch during part of each cycle, so it is not a discrete vibration mode or even a series of discrete modes.

If a wobble board is held vertically so that one end rests on the floor and the other end is supported lightly by one finger, then the fundamental vibration frequency will be about 100 Hz. If the top end is pushed down towards the floor so that the board bends outward by about 2cm in the middle then the fundamental frequency rises to about 200 Hz. If pushed even more firmly so the middle bows out by about 5 cm then the fundamental frequency increases to about 500 Hz. The transverse stiffness of the system therefore depends on how far the board is bent.

The board itself does not get any stiffer when it bends, but the restoring force acting on each section of the board increases the more the board is bent. It is the restoring force acting on the board that determines how fast it moves and hence it determines the vibration frequency. That force consists of stresses within the board acting on neighbouring parts of the board, plus the force needed to bend the board into a curved shape. The latter force acts directly on each end and indirectly affects the motion of all parts of the board.

The vibration frequency of anything that vibrates depends on its mass and stiffness. The stiffness of a wobble board system depends primarily on the thickness and material properties of the board itself, but it also depends on whether the board is straight or bent by some additional force. A similar thing happens with a tennis racquet or with a bow used to shoot arrows, as described in the attached paper. A string added to a bow acts to bend it further, making it stiffer and causing its vibration frequency to rise. Strings added to a tennis racquet act in the opposite direction. If an unstrung racquet vibrates at say 150 Hz, then it will become less stiff and vibrate at about 140 Hz when the strings are added. The strings don’t bend the racquet when they are installed, but if the racquet bends as it vibrates then the strings act to bend the racquet even further.

The best-known example of the effect of bending on the vibration frequency is provided by a musical saw, as shown in the attached QuickTime movie. In the movie I bounced a golf ball off the saw to get transverse vibrations started. The main trick to making a saw sing is that it needs to be bent into an S or ~ shape. A simple C-shape bend of the whole saw generates relatively weak vibrations. The S shape helps to prevent vibration dampening by the hand, by providing a point of reflection away from the hand rather than right at the end of the saw. Vibration of the saw is the result of a transverse standing wave generated by wave reflections at or near each end of the saw. Whole web sites are now devoted to musical saws and how best to play them. Listen to a sample.

The theme song from “One flew over the cuckoo’s nest”

features a musical saw as an eerie background instrument.

The movie won several oscars in 1975 (including best actor & best actress).