# 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 twelve-pointed star.
• Chaotic Motion.
The energy is much larger in this case and the bob no longer traces out a regular pattern, but follows a chaotic and unpredictable trajectory.
• Progressive Ellipse.
Here the exchange of energy between springing and swinging modes vanishes. We get a trajectory whose horizontal projection is a precessing ellipse.
• Retrogressing 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