G2: The Damped Pendulum
A problem that is difficult to solve analytically (but quite easy on the
computer) is what happens when a damping term is added to the pendulum
equations of motion. Here we will use the computer to solve that equation
and see if we can understand the solution that it produces.
A viscous damping force, modeling for example the viscous damping of the oil
in the bearing at the pendulum hinge, would to a good approximation be
proportional to the angular velocity of the pendulum, with a coefficient
we'll call alpha. The damping force will tend to slow down the pendulum,
which determines the sign of the new term in the damped pendulum equation
of motion:
(G2a) Playing with the Damping
(G2a.1)
Set the damping to 0.1. How much does the amplitude decay
in the first two periods? (You'll have to increase the
"Time to Run"). For the purposes of this problem, we'll call
two periods the point where Theta reaches its maximum (the second
full hump of Theta vs Time). Measure (for the next problem) the time
at which this occurs.
The amplitude for alpha = 0.1 should decay roughly by a factor of two
after two periods. Find the value of the damping
coefficient alpha needed to make the amplitude decay to precisely
half of its value after two periods, and add it to your writeup.
Try to measure this alpha to better than two significant figures.
(G2a.2)
Take your numerical solution for the first part of this question
(with your optimal value for alpha), and make a combined plot of the
energy, the potential energy, and the kinetic energy as a function of
time, labeling each curve. Save this plot and include it in your lab
writeup. In your writeup, address the following questions:
- Is the total energy the sum of the kinetic and potential energy?
- At the initial time, is the energy coming from the potential energy
or the kinetic energy? What initial condition would make the energy
all kinetic energy?
- After two periods (as measured in the last part), is the energy
mostly potential or kinetic? The answer, in retrospect, should be obvious.
(G2b) Critical Damping
(G2b.1)
Set the damping coefficient alpha to 0.1, and run from our
standard initial conditions (Theta 0.7 and ThetaDot zero),
but for a longer time (maybe to 30, or 100 if your computer is
fast). Notice the oscillations as they slowly die out.
Set the damping coefficient to ten, and run for the same
initial conditions. The motion here is qualitatively
different from that seen for small alpha. In your writeup,
answer what's the qualitative difference?
(Hint: does the curve oscillate - ever cross zero - for alpha=10?)
Combine Theta vs Time for alpha=0.1, 1.0, and 10.0 on the same graph
(using ``Copy Graph'' and ``Steal Data'') and include it in
your writeup.
Notice that for large damping and for small damping
you needed to run for a long time to see the decay,
but for intermediate damping, the decay is quite a bit faster!
(G2b.2)
Roughly find out (to better than one significant figure) at what
value of alpha the oscillations stop. Include your value in the
lab writeup. This is called critical damping. Putting
the damping just large enough to kill the oscillations also removes
the energy as fast as possible: making the damping even larger is
counterproductive.
(G2b.3)
Imagine our pendulum is a screen door, with the ball at the end representing
the doorknob, and the pivot of the pendulum representing the hinge.
Screen doors are often attached with springs - dissipation is low, so
they are roughly described by the harmonic pendulum equation with
alpha=0. Spring-attached screens slam when they close: when the pendulum
reaches Theta=0 it hits the door frame hard. There are fancier gizmos
to pull screen doors shut, that we can model as a dashpot filled with oil
- well described by our damped equation of motion above, with alpha > 0.
What value of alpha best balances the need to shut the
door fast (avoiding mosquitos), without slamming? Are the two needs
incompatible?
Damping occurs in most physical systems. Sound is damped in
air, light is absorbed as it passes through water, guitar strings
don't vibrate forever. The wave equation we study in this course
as a model for these systems ignores this damping. This is in
many circumstances an excellent approximation, and furthermore ignoring
this complication allows us to study the fascinating concepts of wave
propogation, interference, and diffraction more simply and clearly
- just like ignoring air friction allowed Newton to understand trajectory
motion more clearly.
Links Back
- Exact vs. Approximate Solutions
- The Damped Pendulum
- Stable and Unstable Fixed Points