Calc III Lab: Nonlinear Oscillators
For this portion of the lab, you will explore a nonlinear
oscillator that models small displacements of a mass suspended between
two springs as shown in Figure 1.
Figure 1
It is possible to derive differential equations that model small
horizontal motions of this system. Assuming that the natural
unstretched length of the springs is l as shown in Figure 1,
then (after normalization) the equations modelling the motion can be
written as a system of first-order ODEs:
dx/dt= v
dv/dt= -damping v -x^3 + tau.
Notice the presence of the cubic term in the x variable. (We
are using the symbol x^3 to mean "x to the third
power.") It is this term that makes the system nonlinear. Again, the
parameter damping determines whether the mass returns to its
stable equilibrium quickly or after many oscillations. Again,
tau may be thought of as an external force; for example, the
whole experiment is mounted on a constantly accelerating cart.
At this point you should
change models to the
differential equation called Nonlinear Oscillator.
Also, if you haven't already done so,
return to the default
stopping condition (Fixed Steps).
Question #4:
- Compute the location of equilibria in phase space by
simultaneously solving the equations dx/dt=0 and
dv/dt=0. (Your answer may depend on one or more parameters.)
- How does the location of equilibria depend on the value of
the forcing parameter, tau?
You may wish to numerically verify your computation by choosing a few
specific values of tau and approximately finding the location
of the equilibrium.
Now make sure that tau=0 and damping=0.
Fix v=0 and let x=0.25. The amplitude for
this trajectory will be 0.25. Use the
event stopping abilities of DsTool to find the
period of the solution with initial condition (0.25,0).
Question #5:
Use DsTool to discover how the period and
amplitude of oscillations are related for a nonlinear oscillator with
a cubic nonlinearity.
- Conduct the experiment above for x=0.25, 0.5, 1, 1.5, 2 and
2.5. On the graph provided,
plot the period of each solution versus the
amplitude of that solution.
- Estimate the limit of the period of oscillation as x
approaches 0 from the right. Can you explain your
answer in terms of the experimental setup in Figure 1?
Question #6:
While flipping through Mademoiselle, you come across
the following advertisement:
In 500 words or less, write an essay for the general public that
describes the differences and similarities between linear and
nonlinear oscillators.
The winning entry will be published as part of a future article
entitled, "Boyfriends who won't commit: how to tell if he is
oscillating linearly or nonlinearly." The winning writer will
receive pizza. Write an essay that attempts to win the contest.
Go To
Robert E. Thurman <thurman@geom.umn.edu>
Last modified: Mon Nov 25 17:55:11 1996