Due: Before Patriots Day long weekend. More precisely, your report must be submitted by 10 a.m. on Friday, April 12.

Group project: Your group is limited to at most 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.

In the past I have observed that groups with only one or two members do not do as well as groups with four members. Consequently, you will need my permission to work in a group with fewer than three members. Before I grant such a request, I will try to help you make contact with other students in the class who might be compatible group members. If you are not in a group with at least three members, you must send me email by April 3.

Goals:

1. To become familiar with the long-term behavior of solutions of 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.

In your report, you will study the motion of this forced pendulum assuming certain values of b and k. (See below regarding the value of k for your group.) The effects of A and B on the motion of the pendulum are complicated, so your analysis will not be complete. Rather, you will identify parameter values that produce surprising solutions.

Your report should discuss the following items:

1. (Equilibrium solutions) Determine all equilibrium solutions to the forced equation and interpret them physically.

2. (Piecewise linear approximation) High frequency oscillations of the pivot point produce surprising results. However, before you tackle the complicated pendulum equation, you should study the piecewise linear equation
   theta" plus/minus k theta = 0,

   where the sign switches from plus to minus and back with a given frequency. This is analogous to assuming that the pivot point is subject to periodic but constant acceleration up and down. Analyze the stability of solutions as a function of the frequency of oscillation. (In this case, solutions are stable if they do not go off to infinity.) You might be wondering how you can enter a function that periodically jumps from +k to -k (and back) in HPGSystemSolver. Think about the behavior of

   2*step(cos(B*t))-1.

3. (Producing a counterintuitive but stable solution) In this part of the project, you should return to the true pendulum equation
   theta" + b theta' + k sin(theta) = - A cos(B t) sin(theta).

   Your goal is to find values for the parameters A and B that lead to the pendulum standing straight up even if it is slightly deflected from the vertical. To get some idea of what I mean, first try the equation

   theta" + 0.1 theta' + 10 sin(theta) = - 110 cos(18 t) sin(theta).

   (What happens in this case should not be part of your report.)

   Return to your equation with your k value. Find values of A and B that produce the desired stable solution in the situation where there is no damping (b=0). Then add some damping by setting b=0.1. What changes?

   Compare your parameter results with the results that you obtained in the piecewise linear case and explain what you observe.

   Are there other interesting solutions to this equation?

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 illustrations from the computer as you wish, but the relevance of each illustration to your report must be evident. Illustrations are not included in the four-page limit. (Please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are worth nothing.) Please insert your illustrations 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 k = 0.1 a + 1.05 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.

Academic Conduct: Your work and conduct in this course are governed by the CAS Academic Conduct Code. This code is designed to promote high standards of academic honesty and integrity as well as fairness. Copies of the code are available in CAS Room 105, and it is your responsibility to know and follow the provisions of that code. In particular, all work that you submit in this course must be your original work. For example, 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. Any cases of suspected academic misconduct will be referred to the CAS Student Academic Conduct Committee.