Consider the system shown in Figure 4 where the moment T is a disturbance input and...

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Consider the system shown in Figure 4 where the moment T is a disturbance input and F is the control input. Consider the case where k = 1000 N/m, k = 2000 N/m, c = 5 Ns/m, m = 2 kg, r = 2m, and J = 3 kgm. Do the following: a) Using the parameter values above, find the Transfer functions G(s) X(s) = and T(s) X(s) G(s) = You may assume the system is at rest at x = 0. F(s)' b) Draw the Bode plot for Ga(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). When Ge(s) = K, find the closed-loop transfer function X(s) 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? 000 m F Figure 4: Vibration system T T(s) Ga(s) 0 F(s) X(s) Gc(s) G(s) Figure 5: Vibration Control System Consider the system shown in Figure 4 where the moment T is a disturbance input and F is the control input. Consider the case where k = 1000 N/m, k = 2000 N/m, c = 5 Ns/m, m = 2 kg, r = 2m, and J = 3 kgm. Do the following: a) Using the parameter values above, find the Transfer functions G(s) X(s) = and T(s) X(s) G(s) = You may assume the system is at rest at x = 0. F(s)' b) Draw the Bode plot for Ga(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). When Ge(s) = K, find the closed-loop transfer function X(s) 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? 000 m F Figure 4: Vibration system T T(s) Ga(s) 0 F(s) X(s) Gc(s) G(s) Figure 5: Vibration Control System