Vibrating Inverted Pendulum
This Demonstration shows an inverted pendulum that vibrates sinusoidally at its base in the vertical direction at some frequency. The normalized vibration frequency is denoted by and the normalized vibration amplitude by . For some configurations of amplitude and frequency, the pendulum is relatively stable and stays inverted, while at others it starts to tumble. If or , you have a simple pendulum. The initial angle is measured from the vertical at time 0.
The inverted pendulum is modeled as a point mass on a rod of length with a vertical sinusoidal excitation of amplitude and frequency at the origin.
The position of the mass is then given by
The forces in the and direction are then
where is the acceleration of gravity.
With the force acting only in the direction of the rod, .
The equation for becomes
where , , and .
 J. P. Den Hartog, Mechanical Vibrations, New York: McGraw–Hill, 1956.