MA508 – Worksheet 2

MA508 – Worksheet 2
An overdamped pendulum on a torsional spring obeys the following differential equation
0 = ζ
˙θ + 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
d
ˆθ
dtˆ
= −βˆθ + sin ˆθ
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 − x
2
.
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 (κ  1) and you make it stronger for each subsequent
experiment. Explain what would happen.
1