The Pendulum
Part 2: The Phase Plane
- In your worksheet you will
find a definition of the gravitational constant g and starting values
for mass m of the pendulum bob, damping coefficient b, and
length L. The mass is set at 1 kilogram, the damping is 0,
and the length is 1 meter. Plot the solutions of
starting from rest with
initial values of theta at 0.5 radians and at 2 radians.
Do you see anything surprising in the plot? Explain what you see.
- Draw a direction field
and phase portrait with the trajectories of the solutions in step 1. Do
you see anything surprising in the plot? Explain what you see -- including
the relationship between the solution curves and the trajectories.
- Add modest damping to the
model, say, b = 0.2. Repeat steps 1 and 2.
- Experiment with various values of the damping coefficient. If necessary, change the time interval over which you are plotting. Describe your observations about damping.
Send comments to the
authors <modules at math.duke.edu>
Last modified: March 24,
1998