Question 1. (a) What is the period of the forced oscillation? What is the numerical value (modulo 2Π) of the angle α defined by?

(b) In this question you are asked to modify the file LAB06ex1.m in order to plot the complementary solution of (L6.1), that is, the first term in (L6.2). First define in the file the angle α (alpha) using (L6.4), then evaluate the complementary solution yc by subtracting the quantity C cos(ωt - α) from the numerical solution y. Plot the resulting quantity. Does it look like an exponentially decreasing oscillation?

Why or why not? Include the modified M-file and the corresponding plot.

Question 2. We now consider C as a function of ω. We use again ω₀ = 2, c = 1 and y(0) = y′(0) = 0. The previous problem determined C for a specific value of ω. Here we consider a range of values for ω and determine numerically the corresponding amplitude C. We then plot the result as a function of ω, together with the theoretical amplitude from (L6.3). You may need the following MATLAB program.

function LAB06ex2 omega0 = 2; c = 1;
OMEGA = 1:0.02:3;
C = zeros(size(OMEGA)); Ctheory = zeros(size(OMEGA));
t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50; t1 = 25;
for k = 1:length(OMEGA) omega = OMEGA(k);
param = [omega0,c,omega];
[t,Y] = ode45(@f,[t0,tf],Y0,[],param);

i = find(t>t1);
C(k) = (max(Y(i,1))-min(Y(i,1)))/2;
Ctheory(k) = ??; %FILL-IN
end figure(2)
plot(??); grid on; %FILL-IN
xlabel('\omega'); ylabel('C');
---------------------------------------------------------
function dYdt = f(t,Y,param) y = Y(1); v = Y(2);
omega0 = param(1); c = param(2); omega = param(3);

dYdt = [ v ; cos(omega*t)-omega0^2*y-c*v ];

(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 ω the amplitude of the forced oscillation, C, is maximal. This value of ω is called the practical resonance frequency. Give the corresponding maximum value of C.

(c) Determine analytically the value of ω for which the amplitude of the forced oscillation, C, is maximal by di?erentiating the expression for C in (L6.3) as a function of ω. Compare the value you find with the value obtained in part (b).

(d) Run LAB06ex1.m with the value of ω found in part (c) (include the graph). What is the amplitude of the forced oscillation? How does it compare with the amplitude of the forced oscillation in problem 1.? If you run LAB06ex1.m with any other value of ω, how do you expect the amplitude of the solution to be?

(e) Are the results a?ected by changes in the initial conditions? Answer this question both nu- merically (by modifying the initial conditions in LAB06ex2.m) and theoretically (by analyzing the expression for C in (L6.3)). Note that the initial conditions for the DE are y₀ and v₀.

Question 3. Set c = 0 in LAB06ex2.m.

(a) Explain what happens. What is the maximal amplitude? What is the value of ω yielding the maximal amplitude in the forced solution? How does this value compare to ω₀?

(b) Run LAB06ex1.m with c = 0 and ω equal to the value found in part (a). Comment on the behavior of the solution. Include the graph.

Question 4. To see the beats phenomenon, set c = 0 and ω = 1.8 in LAB06ex1. Also extend the interval of simulation to 100.

(a) In LAB06ex1 define the "envelope" function A = 2C sin ( 1/2(ω₀ - ω)t) with C = 1/|ω₀² - ω²|.

Plot A in red and -A in green, together with the solution. You should obtain Figure L6c. Include the modified M-file.

(b) What is the period of the fast oscillation (that is, the period of sin ( 1/2(ω₀ + ω)t))? Confirm your answer by zooming in on the graph of the solution. Include a graph to support your answer.

(c) What is the length of the beats? Determine the length analytically using the envelope func- tions, and numerically from the graph.

(d) Change the value of ω in LAB06ex1 to 1.9 (a value closer to ω₀) and then ω = 1.6 ( a value farther away from ω₀). Include the two graphs. For each of these two values of ω find the period of the fast oscillation and the length of the beats. How do the periods change compared to parts (b) and (c)?

(e) If you let ω = 0.5, is the beats phenomenon still present? Why or why not?

Attachment:- Forced Equations and Resonance.pdf