Problem 3: (20%) Consider the system B X. X M 0002 K fa(t) M=4 kg, K=1N/m, B=1N-s/m Mx(t) + Bx(t) + Kx(t) = fa(t) with initial condition: x(0) = 0, x(0) = 0.2m (3a) The differential equation of MBK mechanical system is given above. Rewrite the differential equation in the form of B M K M x(t) + x(t)+ -x(t) 1 K KM (t) x(t)+2@x(t) + @x(t) = x(t) and find the relationship between charateristic parameters 5, 0, xs, and the physical parameters M, B, K. (5%) (3b) Let the physical parameters be M = 4 kg, K = 4 N/m, B=1N-s/m, find the damping ratio and the natural frequency of the system. Is the system overdamped? Critically damped? Underdamped? Or undamped? (5%) (3c) 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) = 0, x(0) = 0). Now, release the mass block and observe the motion of the block. Plot the displacement x(t) as a function of time. (5%) (3d) 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%)