OBJECTIVES
i. To understand the effect of the forcing term on the behavior of the solution for different values of angular frequency in particular on the amplitude of the solution.
ii. To understand the phenomena of resonance and beats in the absence of friction.
1. What is the period of the forced oscillation?
b. In this question you are asked to modify the LAB06ex1.m in order to plot the complementary solution
2.
a. Fill in the missing parts in the M-file LAB06ex2.m and execute it. You should get a figure like. Figure L6b. Include the modified M-file in your lab report.
b. Examine the graph obtained by running LAB06ex2.m and determine for what (approximate) value of omega the amplitude of the forced oscillation.
c. Determine analytically the value of omega for which the amplitude of the forced oscillation, C, is maximal by differentiating the expression for C
d. Run LAB06ex1.m with the value of omega found in part (c)
e. Are the results affected by changes in the initial conditions?
3.
a) Explain what happens. What is the maximal amplitude? What is the value of omega yielding the maximal amplitude in the forced solution? How does this value compare to?
b) Run LAB06ex1.m with c = 0 and omega equal to the value. Comment on the behavior of the solution. Include the graph.
4. To see the beats phenomenon, set c = 0 and omega = 2.8 in LAB06ex1. Also extend the interval of simulation to 100.
a. In LAB06ex1 define the \envelope" function
b. What is the period of the fast oscillation
c. What is the length of the beats? Determine the length analytically using the envelope functions, and numerically from the graph.
d. Change the value of omega in LAB06ex1 to omega=2.9 (a value closer to omega0) and then omega = 2.6 (a value farther away from omega0).
e. If you let omega= 1:5, is the beats phenomenon still present? Why or why not?