Due: Your report must be submitted by 10 a.m. on Friday, April 21.

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.

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 amplitude of the solutions.

Second-order equations: Using all of the methods that we have developed in this course, you will analyze the long-term behavior of the solutions to two similar forced, second-order equations.

We begin with the forced harmonic oscillator

Here b is the damping coefficient, k is the spring constant, and w determines the frequency of the forcing.

1. (An undamped, forced linear equation) First consider the equation in the case where b=0. Using the Method of the Lucky Guess, determine a particular solution to the equation. Then using a numerical solver and/or the formulae for the solutions, estimate the amplitudes of the solutions for frequencies w in the interval 0 <= w <= 2. How do the amplitudes in the nonresonant case relate to the amplitudes in the resonant case?

2. (Damped, forced linear equations) Assume that the forcing is present along with some damping (b>0). How long does it take any given solution to get close to the steady-state solution? Using a numerical solver and/or the formulae for the solutions, estimate the maximum amplitudes of the solutions for frequencies w in the interval 0 <= w <= 2 for various values of b. Graph the maximum amplitude as a function of frequency for different values of b, all on the same set of axes. What happens to these graphs as b -> 0?

Now we return to the cantilever beam that we studied in the first project. We will assume that the magnets are strong. Consequently, the differential equation for the cantilever beam is x" + bx' - kx + x3 = A cos wt. The A cos wt is introduced to model a periodic external forcing in the system which can be considered to be the effect of someone periodically shaking the box.

3. (strong magnets with no forcing) Let A=0. Repeat your analysis of Project 1. Note that you now have the mathematical tools to compute the bifurcation value of b directly. Sketch those computations.

4. (strong magnets with forcing) Now let A=1. What aspects of the solutions carry over to the nonautonomous case? Give a physical interpretation of the "steady-state" solutions for this system.

Your report: Your report should be no longer than six 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 six-page limit. (Please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are worth nothing.) Examples of good reports done here at BU in previous semesters are available for inspection in my office.

Numerical simulation: Be aware of the fact that pplane cannot handle nonautonomous systems. Therefore, you must use one of MATLAB's built-in ode solvers when you analyze the solutions to these equations numerically. Note my FAQ answer regarding the format of MATLAB files that define systems of differential equations.

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 = 0.05 a + 0.5 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 = 0.56.

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.