[Solved] MA508 Worksheet 2

An overdamped pendulum on a torsional spring obeys the following differential equation

0 = + mg`sin()

where (t) is the angle of the pendulum (with = 0 being straight up), is the torsional damping coefficient, is the torsional spring constant, m is the mass of the pendulum, g is the gravity constant and ` is the length of the pendulum.

The equation can be non-dimensionalized to

1. In the non-dimensionalized form, what are t and in terms of the dimensional variables and parameters (t,,,,m,g,`)?

2a) Sketch a phase portrait for the non-dimensionalized equation, for the case = 0.1 [Note: phase portrait is a generic term for what Ive been calling the phase line in class].

In your diagram:

1. Indicate stable fixed points with a filled circle and unstable ones with a hollow circle. ii., Indicate flow directions with arrows on the horizontal axis.

2b) For this case, sketch (t) when (0) is a small positive number (the pendulum is initially pointing almost straight up).

3) Sketch a bifurcation diagram for the non-dimensionalized equation. In your diagram:

1. Indicate stable fixed points with a solid line and unstable fixed points with a dashed line ii. Show your calculations for how you determined the fixed points iii. Explain how you determined stability and/or show your calculations iv. Clearly indicate any bifurcation(s) (if they exist)
2. Clearly identify and label any saddle-node bifurcation(s)

• If youve done part 3 correctly, you found a bifurcation at = 1, = 0. This is a new kind of bifurcation, called a transcritical bifurcation. By doing a Taylor expansion about this point, show that transcritical bifurcations (including this one) have the normal form x = ax x².
• Suppose you have a pendulum whose stiffness, , can be tuned. You perform a series of experiments, where the pendulum starts nearly vertical and then is released. For the first experiment, the spring is very weak ( 1) and you make it stronger for each subsequent experiment. Explain what would happen.