For this problem, take a look at Figure 2. Assume that the rod is massless, perfectly...

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For this problem, take a look at Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y = 0, and the spring is in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. k a a a F Schen The equation of motion for the system can be derived to be: 4aM0+ ak0=-F-2Mg M T rabiem 2 Your tasks: A. Transform the rotational equation of motion, which is in 0, given above, to another variable, p, which is zero at the static equilibrium position. ( B. Represent the mechanical system in state space form. Using MATLAB or a calculator (just say what you used), calculate the eigenvalues of your A matrix for the following system parameters: a = 0.25 [m], M = 1 [kg], k = 16 [N/m], and F = 0 [N]. What do the eigenvalues say about the stability of the system? C. Derive the response of the system in the Laplace (s) domain. Use the static equilibrium value found in part A (st) as the initial value, (0), for the problem. Assume (0) and the force, F, are both zero. You may treat gravity as g 10 [m/s] for ease of calculation. For this problem, take a look at Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y = 0, and the spring is in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. k a a a F Schen The equation of motion for the system can be derived to be: 4aM0+ ak0=-F-2Mg M T rabiem 2 Your tasks: A. Transform the rotational equation of motion, which is in 0, given above, to another variable, p, which is zero at the static equilibrium position. ( B. Represent the mechanical system in state space form. Using MATLAB or a calculator (just say what you used), calculate the eigenvalues of your A matrix for the following system parameters: a = 0.25 [m], M = 1 [kg], k = 16 [N/m], and F = 0 [N]. What do the eigenvalues say about the stability of the system? C. Derive the response of the system in the Laplace (s) domain. Use the static equilibrium value found in part A (st) as the initial value, (0), for the problem. Assume (0) and the force, F, are both zero. You may treat gravity as g 10 [m/s] for ease of calculation.