The motion of one of the simplest dynamical systems, a torqued, damped, nonlinear **pendulum**, can be **infinitely** complicated.

Consider a simple pendulum of length l and mass m rigidly connected to an axle of radius r wrapped by a rope that hangs down one side with a mass M climbing up and down it, as in the attached animation.

If the climber’s height varies **sinusoidally** (relative to the axle)

x = x_0 + \frac{a}{\omega^2} \sin \omega t,
then its acceleration also varies sinusoidally

\ddot x = - a \sin \omega t,
so the total force on the climber

M\ddot x = \sum_\text{down} f = Mg-T
implies upper rope tension

T = Mg + ma \sin \omega t,
where 0 < a < g. If the axle and rope have negligible inertia, then the total torque on the axle

m l^2 \ddot \theta = \sum_\text{CCW}\tau = - mgl \sin\theta + rT - \gamma \dot\theta,
where \gamma is the axle **viscosity**. The full motion equation

m l^2 \ddot \theta = - mgl \sin\theta + rMg + rma \sin \omega t - \gamma \dot\theta
reduces to

\ddot \theta = - \sin\theta + 0.7155 + 0.4 \sin 0.25 t - 0.75 \dot\theta
for parameters that describe the animation **chaotic** motion.

Green mass sinusoidally climbs up and down brown rope torquing blue pendulum into chaotic motion.

Upper rope tension is largest with climber lowest and smallest with climber highest.

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About John F. Lindner

John F. Lindner was born in Sleepy Hollow, New York, and educated at the University of Vermont and Caltech. He is an emeritus professor of physics and astronomy at The College of Wooster and a visiting professor at North Carolina State University. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and NCSU. His research interests include nonlinear dynamics, celestial mechanics, and neural networks.