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The damped pendulum with torque

Equation (1) describes the motion of a pendulum solely influenced by gravitation. In reality, the pendulum will always be subject to some friction. One generally assumes that this force is proportional to the velocity (t). The equation of motion now becomes

= - 2 sin - d ,

where d is a positive damping coefficient. The behavior of this pendulum is rather boring: eventually all oscillations die out. The position where the pendulum is hanging down vertically is a global attractor. All solutions eventually end up in this position, except the unstable equilibrium solution, where the pendulum is hanging straight up. We therefore add an external force, a torque T that can be interpreted as pushing the pendulum with a constant force perpendicular to the rod. The corresponding equation is

= -d - 2 sin + T.

Writing this as a first-order system, we get

(2)

Here, p = should be taken modulo 2. It is not hard to see that the equilibria of this system are given by points (p,0) with p such that

sin p = T / 2.

If we push with a force T larger than ² (or smaller than - ² if we push from the other side), there are no equilibria and the only solutions are oscillations where the pendulum makes full rotations. For T small enough (in absolute value) two equilibria exist, one of which is attracting and the other a saddle.

p
	T

Equilibria of equation (2) as a function of T with = 3.14 and d= 0.05.
Stable equilibria are blue, saddles are green.

Watch the pendulum oscillate by adding the following files to DsTool:

• classicdp.c
  
• classicdp.oogl
  

Written by Hinke Osinga
Last modified: Tue Mar 2 16:32:30 1999