The Duffing Oscillator
This lab is due Thursday, April 25, 2002, in class. You may find it easiest to use the software called IDE (Interactive DIfferential Equations) in the old Macintosh lab in MCS. In IDE, you should open the tool called Duffing Oscillator. You may, of course, use other differential equation solvers, but you will not see all of the images that you see in IDE.
You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to hand in hand-drawn figures, since you cannot print pictures using IDE. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.
IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly. Copying or paraphrasing another student's lab report is strictly forbidden.
Lab. In this lab you will investigate a nonlinear system of differential equations that governs the motion of the Duffing oscillator. This contraption consists of flexible steel beam attached to the top center of a box. There are magnets on either side of the bottom of the box toward which the beam is attracted. The differential equation describes the motion of the tip of this beam as the box is moved periodically from side to side. Here is the picture of the system.
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The differential equation for the Duffing oscillator is x" = -x(x2 - k) -bx' + A cos(wt) or, as a system
dx/dt = v Here A is the magnitude of the forcing function, w is the forcing frequency, b is the damping coefficient, and k is related to the stiffness of the beam. The center of the box is located at x = 0, with x > 0 to the right.
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For the entire lab, we will keep k = 1 and w = 0.6.
Before going to the lab, you should work out #1 below. In the lab you should use the IDE tool called the Duffing Oscillator to investigate this system of equations. In this tool you will see three images: the phase plane, the x(t) graphs, and the motion of the beam. You should turn off the velocity and acceleration graphs, so that you see only the (green) graph of x(t).
You should also be aware that the behavior of solutions of this system of differential equations is still not completely understood and is, in fact, the subject of considerable mathematical research. I do not expect that you will be able to completely understand this system. Your goal, especially in #6-8 below, is to view some of the complexity of this simple mechanical system.
1. You will first investigate the autonomous case, i.e., when A = 0. Find all equilibrium points for the system when A = 0. What configurations of the beam do these equilibrium points correspond to? Using linearization, determine the types of these equilibrium points. For which values of the damping constant b do you expect bifurcations to occur?
2. Use IDE as an aid to sketch the phase plane for this system when A = b = 0. Be sure to include the nullclines in your sketch. Using this sketch, explain in a brief paragraph what happens to the beam if the beam is released from position x with velocity equal to 0 for all values of x. Discuss what changes occur as you vary x.
3. Now sketch the phase plane when b > 0, say, for definiteness, when b = 0.05. What happens to the nullclines when b becomes positive? What bifurcations occur when b becomes positive?
4. In a paragraph or two, describe what now happens to the beam when it is released from position x = 0 with positive velocity v > 0 for all values of v, i.e., solution curves that begin on the positive v-axis. You should see different behaviors depending upon v. What do you expect will happen for values of v that are larger than those shown on the screen?
5. Provide a sketch of the solution curves in the phase plane that separate different types of behaviors.
6. Now turn on the forcing. Suppose A = 0.3, b = 0.2 and w = 0.6. What can you say about the ultimate behavior of the beam for various initial conditions along the x-axis? Provide sketches of the different behaviors in the phase plane and describe all possible eventual motions of the beam that you find. Be sure to let solutions run for a long time. You may use the "clear transients" button to see the eventual motion of the solution curve in the phase plane.
7. Now change b to 0.15. Do you see a bifurcation? What has changed? Why is this bifurcation called a "period doubling bifurcation?"
8. Finally let b = 0. What can you say about the ultimate behavior of solutions now?