The Swinging Spring
This simple system looks like a toy at best, but
its behaviour is astonishingly complex, with
many facets of more than academic lustre.
(Breitenberger and Mueller, 1981)
The Swinging Spring is an elastic pendulum, a heavy mass hanging
from a spring suspended at a pivot. The mass is free to swing (and spring) in
three dimensions. The system is simple but its dynamics are deliciously complex.
It exhibits both regular and chaotic motion.
A photograph of the Swinging Spring, and a description of the basic dynamics,
appear on the System Page.
The Resonant Case
There is an interesting special case when the frequency of the vertical oscillations
is twice that of the horizontal oscillations.
This is the case of resonance and in this case energy is
transferred back and forth between vertical or springing oscillations
and horizontal or swinging oscillations. Each horizontal excursion is in a
different plane, and the precession of the swing-plane is one of the
characteristic and fascinating aspects of the system.
Swinging Spring Java Applet
A Java Applet (the Lagrange Applet)
has been written by Peter Selinger of Dalhousie University.
Information on the application of the applet to calculate
solutions of the Swinging Spring is given on
the Applet Page.
Some examples of the applet are given here
[you must have a Java-enabled browser]:
Regular Motion: Star-Pattern.
This shows a case of low-energy
motion, in which the bob traces out a regular pattern whose horizontal projection is a
The energy is much larger in this case and the bob no
longer traces out a regular pattern, but follows a chaotic and unpredictable
Here the exchange of energy between springing and swinging modes
vanishes. We get a trajectory whose horizontal projection is a precessing ellipse.
In this case the ellipse moves in the
opposite direction to that in which the bob is revolving.
Here is a list of
Publications on the Swinging Spring
available at this site
Back to Peter Lynch's Home Page
Updated 3rd December, 2002.