Parametric Resonance

This demonstration illustrates parametric resonance, which is a resonance phenomenon that arises because some parameter of the system is varying periodically in time. This happens, for example, when you periodically extend and retract your legs at an appropriate frequency while sitting on a playground swing. If a pendulum consists of a mass connected to a frictionless hinge by an idealized rod of length L, then the angle θ between the rod and vertical obeys the differential equation

     θ" + (g/L) sinθ=0
If the length of the pendulum varies with time, the differential equation becomes
     θ" + 2(L'(t)/L(t))θ'+(g/L(t)) sinθ=0
In this Applet, we choose the length of the pendulum to be L(t)=1+A cos(wt) and we choose g=4π2. With this choice of g, solutions of the A=0, linearized equation θ"+gθ=0 have period 1.

At time t, the length of the blue line in the Applet below is L(t). The angle between the blue line and vertical is θ(t). For comparison purposes, the gray pendulum provides the same data for the A=0, g=4π2, linearized equation

     θ" + g θ =0 
So, at time t, the length of the gray line is 1 and the angle between the gray line and vertical is a constant (determined by the initial speed that I have chosen) times sin(2πt).

The demonstration below is an applet. Google Chrome, Firefox and Microsoft Edge no longer execute applets because of security issues with NPAPI plugins. Some browsers that still play applets are Internet Explorer, Safari and the Pale Moon browser [ windows, linux, Mac ].