Background

This model describes a cantilevered steel beam suspended between two magnets in a rigid frame. The two magnets are fixed to the frame, equidistant from an imaginary vertical line extended down from where the beam is fixed to the frame. T he tip of the beam is attracted to the magnets so the beam will bend slightly so that the tip of the beam will be closer to one of the two magnets. The whole frame is set up to be shaken from side to side by a sinusoidal exciting force.

The model that describes this system is given by the following differential equation:

x'' + dx' - x + x3 = gcos(wt)

where x(t) is the displacement of the tip of the beam from the centerline, delta is the damping coefficient, gamma is the amplitude of the forcing term and omega is the forcing frequency. All of these parameters are non-negative and delta is non-zero.

Since this system is non-autonomous, the analysis of the autonomous case,

x'' + dx' - x + x3 = 0,

is important in determining the steady states of this system. The damping coefficient, delta, determines how quickly the trajectories decay into one of the steady states.