Problem 4: (30%) Consider the system B 1+1 x, x M f(t) K M=4 kg, K...

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Problem 4: (30%) Consider the system B 1+1 x, x M f(t) K M=4 kg, K = 4 N/m, B = 1 N-s/m Mx(t) + Bx(t) + Kx(t) = f (t) with initial condition: x(0) = 0, x(0) = 0.2m (4a) The differential equation of MBK mechanical system is given above. Rewrite the differential equation in the form of K x(t) + -x(t) + M -x(t) M 1 K KM f(t) x(t)+2wx(t) + wx(t) = xf (t) and find the relationship between charateristic parameters 5, @,, xs, and the physical parameters M, B, K. (3%) = (4b) Let the physical parameters be M 4 kg, K = 4 N/m, B =1 N-s/m, find the damping ratio 5, the natural frequency @, and the pole locations of the system. Is the system overdamped? Critically damped? Underdamped? Or undamped? (3%) (4c) Let f(t)=0 and the initial ccondition be x(0) = 0, x(0) = 0.2m. That is, the mass block is initially held at the 0.2m position to the right of the equilibrium (x = 0, x = 0). Just based on the knowledge of the damping ratio 5, the natural frequency @, and the pole locations of the system without solving the differential equation, describe as detailed as possible how the block will move as a function of time if the block is released at t = = 0. (3%) (4d) Now you will use the Laplace transform approach to solve the differential equation in (4a) for the analytic solution x(t), which is the initial state response due to the initial conditions x(0) = 0, x(0) = 0.2m, with the assumption of zero input force f(t)=0. (6%) (4e) Use MATLAB "plot" or Python "plt.plot" command to plot the initial state response x(t) obtained in (4d) on a graph versus time t. (5%) (4f) Evaluate the performance of the system based on steady-state error, maximum overshoot, oscillation frequency, peak time, rise time, and settling time. (5%) (4g) Assume you do not like the performance of the current system because it is too slow or too oscillatory, What shall you do to change the physical parameters assuming that you are only allowed to redesign the damper by changing the friction coefficient B? Plot the response x(t) of your modified design. (5%) Problem 4: (30%) Consider the system B 1+1 x, x M f(t) K M=4 kg, K = 4 N/m, B = 1 N-s/m Mx(t) + Bx(t) + Kx(t) = f (t) with initial condition: x(0) = 0, x(0) = 0.2m (4a) The differential equation of MBK mechanical system is given above. Rewrite the differential equation in the form of K x(t) + -x(t) + M -x(t) M 1 K KM f(t) x(t)+2wx(t) + wx(t) = xf (t) and find the relationship between charateristic parameters 5, @,, xs, and the physical parameters M, B, K. (3%) = (4b) Let the physical parameters be M 4 kg, K = 4 N/m, B =1 N-s/m, find the damping ratio 5, the natural frequency @, and the pole locations of the system. Is the system overdamped? Critically damped? Underdamped? Or undamped? (3%) (4c) Let f(t)=0 and the initial ccondition be x(0) = 0, x(0) = 0.2m. That is, the mass block is initially held at the 0.2m position to the right of the equilibrium (x = 0, x = 0). Just based on the knowledge of the damping ratio 5, the natural frequency @, and the pole locations of the system without solving the differential equation, describe as detailed as possible how the block will move as a function of time if the block is released at t = = 0. (3%) (4d) Now you will use the Laplace transform approach to solve the differential equation in (4a) for the analytic solution x(t), which is the initial state response due to the initial conditions x(0) = 0, x(0) = 0.2m, with the assumption of zero input force f(t)=0. (6%) (4e) Use MATLAB "plot" or Python "plt.plot" command to plot the initial state response x(t) obtained in (4d) on a graph versus time t. (5%) (4f) Evaluate the performance of the system based on steady-state error, maximum overshoot, oscillation frequency, peak time, rise time, and settling time. (5%) (4g) Assume you do not like the performance of the current system because it is too slow or too oscillatory, What shall you do to change the physical parameters assuming that you are only allowed to redesign the damper by changing the friction coefficient B? Plot the response x(t) of your modified design. (5%)