Glider Project

Due date: Your report must be submitted by 12:30 p.m. on Thursday, December 7.

Group project: This is a group project. The groups will be announced in class on November 21. Each group will submit one report, and all members of the group will receive the same grade for this project.

Goals:

  1. To apply what you have learned in this course to an example that is somewhat more complicated than the types of examples we discuss in class and study in the homework.
  2. To get more experience doing numerical work with differential equations.

The Glider: One of my favorite tools in IDE is the glider tool, which is based on the system of equations

dv/dt = -sin(theta) - Dv2
d(theta)/dt = (v2 - cos(theta))/v
where v>0. The parameter D is a coefficient for the drag force on the glider. Also, note that the equations are periodic in theta. Actually these equations are not the true equations of motion but convenient "rescalings" of those equations.

Basic analysis: Start your analysis of this system by identifying the different types of motions of the glider both in the case of no drag (D=0) and when drag is present (D>0). What do the variables represent? Does the system have equilibrium points? What is the physical interpretation of the equilibria, both in the D=0 and the D>0 cases?

If D=0, the tool also displays a value of a quantity that is denoted by E. In fact, E=v3 - 3*v*cos(theta). Using the Chain Rule, show that the value of E remains constant during any particular flight of the glider. (The function E is called a conserved quantity of the system.) What is the most negative value of E? What does the function E tell you about the solution curves of the system? Are the solutions periodic?

Now assume the presence of some drag (0< D< 4) and use pplane to study the behavior of various solutions. What are the different possible behaviors of solutions? Are the solutions periodic? What eventually happens to the flight of the glider?

Linearization about the equilibria: Apply the theory of linearization to classify all equilibria for all values of D in the interval 0 <= D <=4, and determine the bifurcation values of D. That is, determine the values of D where nearby D's lead to "different" behaviors in the phase portrait. For example, D = 0 is a bifurcation value because for D = 0 the long-term behavior of the solutions is dramatically different than the long-term behavior of the solutions if D is slightly positive. (Hint: If the type of an equilibrium point changes at a certain value of D, then that value of D may be a bifurcation value.)

Reconstructing the motion of the glider from the equations: Given a value of D and an initial condition (v0,theta0), then the the motion of the glider is determined from the equations. Show how one can start with values of D and (v0,theta0) and obtain the path of the glider. Be precise. (One good way to do this part of the project is to write the code that you would need to draw the path of the glider in some convenient programming language such as MATLAB.)

The phase cylinder: Why is it more natural to talk about the phase "cylinder" rather than the phase plane for this system? What changes if you analyze the system using a phase cylinder rather than a phase plane?

Your report: Your report should be no longer than five typewritten pages, and it should address all of the questions raised in the project. 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 five-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.

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.