Due: 9:30 a.m. on Thursday, April 30.

Group project: This is a group project with each group having either three or four members. Once a group begins work on the project, its membership cannot change. Consequently establishing your group must be your first step in this project. Each group will submit one report, and all members of the group will receive the same grade for this project.

Goals:

1. To become familiar with the long-term behavior of solutions to certain forced second-order equations.
2. To study the relationship between the frequency of the forcing and the stability of the solutions.

The pendulum:

In the first project, you studied the motion of a pendulum determined by the autonomous equation theta" + b theta' + k sin(theta) = 0, where theta is the angle that the pendulum rod makes with the vertical, b is a damping coefficient, and k is a constant that is determined by the length of the rod.

In this project, you will study a more complicated model---one that includes another term that results from periodic vertical motion of the pivot point of the pendulum. The equation of motion is

The two new parameters A and B correspond to the amplitude and frequency of the motion of the pivot point.

Your report should discuss the following items:

1. (The bifurcation value b₀) In part 3 of your first project, you estimated bifurcation values for the unforced, nonlinear pendulum (A=0) and its linear approximation. Using what you have learned since doing that project, compute these values b₀ directly.
2. (Equilibrium solutions) Determine all equilibrium solutions to the forced equation and interpret them physically.
3. (Linear approximation) Assume that the damping coefficient b=0 and approximate the sine terms with their linearizations at theta=0. In this case, the equation becomes
   theta" + (k + A B² cos(B t)) theta = 0.

   Note that this equation is linear, but it does not have constant coefficients. "Rescale" time by letting tau=B t and show that this linear equation is equivalent to the equation

   theta" + (alpha + beta cos(tau)) theta = 0.

   (The second derivative in this new equation is the second derivative of theta with respect to tau rather than t.) How do alpha and beta relate to the parameters A and B in the original equation?

   Using alphas and betas from the rectangle -.5 <= alpha <= .5 and 0 < beta <= .5, use HPGSystemSolver to identify the parameter values where all of the solutions to

   theta" + (alpha + beta cos(tau)) theta = 0

   are bounded. We call these regions the regions of stability. The complementary regions are the regions of instability.

4. (Unstable lower equilibrium position) Let b=0.1 in the forced pendulum equation. Using your results in part 3, find parameter values of the forced pendulum for which the lower equilibrium position is unstable.

5. (Stable upper equilibrium position) Let b=0 in the pendulum equation and linearize the equation around the angle theta=pi. Compare your linearization with the one you obtained in part 3. Let b=0.1 and find parameter values for which the upper equilibrium position is stable in the forced pendulum.

Your report: Your report should be no longer than four typewritten pages, and it should address all of the questions mentioned above. You may provide as many figures as you wish, but the relevance of each figure to your report must be evident. Figures are part of the four-page limit. Please insert your figures at appropriate places in your report rather than attaching them to the end of the report.

Parameter values: The value of k that you should use is determined by the last digits of the BU ID numbers of the members of your group. Use

where a is the average (accurate to two decimal places) of the last digits of all members in the group. For example, if the last digits are 0, 1, 2, and 2, then a = 1.25 and k = 1.175.

Asking questions: Questions by email about the project are not allowed. Instead we will use the Blackboard Discussion Board for this course. I will check that board for questions once a day. The last time that I will check the discussion board will be noon on April 28. Do not wait until the last minute to ask questions. Please start a new thread with each new question.

Academic Conduct: Your work and conduct in this course are governed by the Boston University Academic Conduct Code. This code is designed to promote high standards of academic honesty and integrity as well as fairness. It is your responsibility to know and follow the provisions of the code. In particular, all work that you submit in this course must be your original work. For example, you can only discuss your project with other members of your group, with Professor Blanchard, Brian, or Rob. Moreover, the computations that you do for your report as well as the text of your report must be original to your group. All group members are responsible for all aspects of the report. If you have a question about any aspect of academic conduct, please ask.