The Swinging Spring System

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The Swinging Spring or elastic pendulum is a simple mechanical system that exhibits complex dynamics. It consists of a heavy mass suspended from a fixed point by a light spring which can stretch but not bend, moving under gravity. It can oscillate in two fundamentally different ways: vertically, with the elasticity of the spring acting as a restoring force or (quasi-)horizontally like a pendulum, with gravity as the restoring force.

The position of the mass is given by its three spatial co-ordinates; the system has three degrees of freedom. As there is no torque about the vertical line through the point of suspension, the angular momentum about this line must be constant. In the case of zero angular momentum, the motion takes place in a plane; there are then only two degrees of freedom.

There are just two constants of the motion (the total energy and the angular momentum about the vertical). Since the system has three degrees of freedom and only two invariants, it is not in general integrable. For small amplitudes the motion is regular. However, for larger energy levels, the motion is chaotic.

Let l0 be the unstretched length of the spring, k its elasticity or stiffness and m the mass of the bob. At equilibrium the weight is balanced by the elastic restoring force:

k(l-l0) = mg  .       (1)
We consider cartesian co-ordinates (x,y,z) centered at the point of suspension of the pendulum. The Lagrangian is
L = T-V = 1
2
 m( .
x
 
 2
 
 + .
y
 
 2
 
 + .
z
 
 2
 
 ) - 1
2
 k(r-l0)2 - mgz
where r2 = (x2+y2+z2). The equations of motion may be written:
..
x
 
 = -wz2( r-l0
r
 ) x
..
y
 
 = -wz2( r-l0
r
 ) y
..
z
 
 = -wz2( r-l0
r
 ) z - g
where wz2 = k/m. There are two equilibrium states, where the velocities vanish identically. The first is stable, with the spring hanging vertically downward ( x = 0, y = 0, z = -l), and the balance of forces is given by (1). The second is an unstable equilibrium with the mass balanced precariously above the point of suspension (x = 0, y = 0, z = 2l0-l).

The Resonant Case

There is an interesting special case when the frequency of the vertical oscillations is twice that of the horizontal oscillations:
mg
kl
 = 1
4
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. Suppose the system is excited initially in its vertical oscillation mode. Since purely vertical motion is unstable, horizontal motion soon develops. The horizontal oscillations grow to a maximum and then subside again. An alternating cycle of quasi-vertical and quasi-horizontal oscillations recurs indefinitely. Seen from above, during each horizontal excursion of several oscillations the projected motion is approximately elliptical. Experimentally and numerically one observes that between any two successive horizontal excursions the orientation of the projected ellipse rotates by the same angle, thereby causing a stepwise precession of the swing plane. The dynamics of the resonant Swinging Spring can be seen by running the Java Applet from the Swinging Spring Home Page.

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Updated March 29th, 2001.