An overdamped pendulum on a torsional spring obeys the following differential equation
0 = ζθ˙ + κθ − mg`sin(θ)
where θ(t) is the angle of the pendulum (with θ = 0 being straight up), ζ is the torsional damping coefficient, κ 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 tˆ and β in terms of the dimensional variables and parameters (t,θ,ζ,κ,m,g,`)?
2a) Sketch a phase portrait for the non-dimensionalized equation, for the case β = 0.1 [Note: “phase portrait” is a generic term for what I’ve been calling the phase line in class].
In your diagram:
i. 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 θ(t) when θ(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:
i. 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)
v. Clearly identify and label any saddle-node bifurcation(s)
4) If you’ve done part 3 correctly, you found a bifurcation at β = 1, θ = 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.
5) Suppose you have a pendulum whose stiffness, κ, 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.
