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Q. 1 In the second order di?erential equation we described the motion of the pendulum. Unfortunately, the di?erential equation is nonlinear, so at that time, we linearized about =0 which means that the solution will be good only for small angles. Recall that the di?erential equation we discussed in week 4 is

d2/dt2+ c/mL d /dt + g/L sin( ) = 0

 

For our project, we will let m = 1, L = 1, c =0.5 and recall g = 9.8. Therefore, our di?erential equation is 1) Show the steps in using the change of variables  d2/dt2 + 0.5d /dt +9.8 sin( ) = 0 x =  y = ’ 

to convert the above nonlinear, second order di?erential equation (the one with numbers) to the  first order system dx/dt = y

dy/dt = -9.8sin(x) – 0.5y

Q. 2 I used NumSysDE.xmcd to numerically solve the differential equation with the initial conditions (0) = 0 and ’(0) = 5 (i.e. x (0) = 0 and y(0) = 5) and then created the graph of vs t (keep in mind that x = ).

94_position_of_pendulum.png

Use this graph to discuss the actual motion of the pendulum in the time interval 0 ≤ t ≤ 4. That is, where is the pendulum at time t = 0? Then as time increases, is the pendulum moving clockwise or counterclockwise and for approximately how long? ETC.

Q3. I used NumSysDE.xmcd to numerically solve the di?erential equation with the given initial conditions and then created the graphs of x vs t in each case (keep in mind that x = ).

912_the-actual-motion-pendulum,.png

Using the graphs to think about the actual motion of the pendulum, describe why the graphs appear as they do. In particular, you should be able to describe why lim t→∞x(t) = 2π in the second graph.

4, a) Verify that (0, 0), (π, 0), and (2π, 0) are all critical points of the first order system (indeed, all (nπ, 0) will be critical points).

b) Linearize the nonlinear system about the critical point (0, 0) and determine stability. This  means to actually prepare down the linearized system about (0, 0) (so you will need to find partial  derivatives and evaluate at x =0 and y = 0), and then use the eigenvalues of the coe?cient matrix to determine stability.

c) Linearize the nonlinear system about the critical point (π, 0) and determine stability. This means to actually prepare down the linearized system about (π, 0) (so you will need to find partial derivatives and evaluate at x = π and y = 0), and then use the eigenvalues of the  coe?cient matrix to determine stability.

d) Linearize the nonlinear system about the critical point (2π, 0) and determine stability. This  means to actually prepare down the linearized system about (2π, 0) (so you will need to find partial derivatives and evaluate at x= 2π and y = 0), and then use the eigenvalues of the coe?cient matrix to determine stability.

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