The Swinging Spring System
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
coordinates; 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 l_{0} 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:
We consider cartesian coordinates (x,y,z) centered at the
point of suspension of the pendulum.
The Lagrangian is
L = TV = 
1 2

m( 
. x

2

+ 
. y

2

+ 
. z

2

)  
1 2

k(rl_{0})^{2}  mgz 

where r^{2} = (x^{2}+y^{2}+z^{2}).
The equations of motion may be written:

.. x

= w_{z}^{2}( 
rl_{0} r

) x 


.. y

= w_{z}^{2}( 
rl_{0} r

) y 


.. z

= w_{z}^{2}( 
rl_{0} r

) z  g 

where w_{z}^{2} = 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 = 2l_{0}l).
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. 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 quasivertical and quasihorizontal 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.
Back to Swinging Spring Home Page
Updated March 29th, 2001.