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Question 4-Vibration Control of a 1D0F system (5 marks) Consider the system shown in Figure 4 where the moment 7 is a disturbance input and F is the control input. Consider the case where k₁ = 1000 N/m, k₂ = 2000 N/m, c = 5 Ns/m, m = 2 kg, r = 2m, and J = 3 kg-m². Do the following: a) Using the parameter values above, find the Transfer functions Ga(s) = (s) and G(s) = You may assume the system is at rest at x = 0. F(S) b) Draw the Bode plot for G (s) and use this plot to predict the amplitude of x(t) when F = 0 and T = 100 sin(30t). c) The control structure in Figure 5 is proposed to decrease the amplitude of the vibrations at w = 30 rad/s (the expected frequency of the disturbance torque). X(s) When Gc(s) = K, find the closed-loop transfer function T(S) d) Using your answer to part c), find a value of K that will decrease the amplitude of the vibrations at w = 30 (found in b)) by a factor of 10. e) Considering your answer to part c) and in the absence of control, can you suggest a modification to one of the system parameters that would provide the same effect as the proportional control law? k₁ C k₂ ele m Figure 4: Vibration system T(S) Ga(s) F(s) Gc(s) G(s) Figure 5: Vibration Control System T X(s)