Due: Before Spring Break. More precisely, your report must be submitted by 10 a.m. on Friday, March 5.

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 second-order equations.
2. To get experience doing numerical work with differential equations.

Second-order equations: Autonomous, second-order differential equations are studied numerically by reducing them to first-order systems with two dependent variables. In this project, you will use the computer to analyze a nonlinear second-order equation with parameters. You will analyze phase planes and x(t)- and v(t)-graphs to describe the long-term behavior of the solutions for certain parameter values, and you will collect numerical evidence that suggests certain bifurcation values.

This contraption consists of a 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. The differential equation for the cantilever beam is x" + bx' + kx + x3 = 0. Here b is the damping coefficient, and k is related to the stiffness of the beam and the strength of the magnets. The center of the box is located at x = 0, with x > 0 to the right. In your report, you should describe the motion of the beam assuming certain values of b and k. (See below regarding the value of k for your group.)

Your report should discuss the behavior of the solutions in the following cases:

1. (weak magnets with no damping) First you should study the equation in the absence of damping. In other words, b=0. Examine solutions using both their graphs and the phase plane. Are the solutions periodic? If so, approximate the periods of the solutions. Describe the behavior of a few solutions that have especially different initial conditions. Be specific about the physical interpretation of the different initial conditions.

   Consider the quantity H(x,v) = (x⁴)/4 + k(x²)/2 + (v²)/2. Using the multivariate Chain Rule, show that the value of H remains constant throughout any particular motion of the beam. What does the function H tell you about the solution curves of the system?

2. (weak magnets with damping) In Part 1 above, you considered the undamped case. Now you are going to experiment with the damping parameter b and study how it affects the behavior of the solutions. First, compare your results in Part 1 with the behavior of solutions if b = 0.1. What aspects of the solutions change? What aspects are similar to those of the undamped case? Next, increase the damping dramatically by considering b = 5.0. What behavior remains the same? What changes?

   Now consider how varying the parameter b between 0.1 and 5.0 changes the behavior of the solutions. In fact, there is a particularly important b-value that separates the ``b = 0.1-like'' behavior from the ``b = 5.0-like'' behavior. This ``bifurcation'' value b₀ is difficult to locate numerically using only the computational aids that are currently available to us, but try your best. Compare the behavior of the solutions for parameter values less than b₀ and for parameter values larger than b₀.

3. (strong magnets with no damping) Once again let b=0, but now use -k in place of the parameter k. Repeat your analysis in Part 1 giving a complete description of the phase portrait.

4. (strong magnets with damping) Now bring back the damping term and determine the bifurcation value b₁ for this equation. Give a complete description of the phase portrait both for values of b such that 0<b<b₁ and for values of b such that b>b₁. What parts of the phase portrait separate the different possible behaviors of the beam?

Value of the parameter k: 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

Your report: Your report should be no longer than five 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. (Please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are worth nothing.) Illustrations are part of the five-page limit. Please insert your illustrations at appropriate places in your report rather than attaching them to the end of the report.

Your report should have a cover page, which is not counted in the five-page limit. It should list the members of your group, and it should indicate the last digits of your BU ID numbers.

Examples of good reports done here at BU in previous semesters are available for inspection during my office hours, during Mark's office hours, and during discussion section.

Asking questions: Questions by email about the project are not allowed. Instead we will try an experiment with the Blackboard Discussion Board for this course. I will check that board for questions at least once a day. The last time that I will check the discussion board will be noon on March 4. 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 CAS Academic Conduct Code. This code is designed to promote high standards of academic honesty and integrity as well as fairness. A copy of the code is available in CAS Room 105 if you cannot access it on the web, and 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 Blanchard, or with Veillette. 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. Any cases of suspected academic misconduct will be reported to the CAS Dean's Office.