Answer (2-e).

Consider the spring-mass-damper system (SMD) mounted on a massless cart as shown in Figure 1. The mathematical model of the system is given by: mdt2d2y+bdtdy+ky=bdtdx+kx Where m is the mass of the cart, x=u (input of the system), b is damping coefficient, and k is spring constant. If m=1kg,b=2Ns/m, and k=10N/m. Answer the following questions: (2-a) If x(t) equals zero, solve the differential equation (i.e, find the general solution of the resulted homogeneous differential equation). (2-b) If the input x(t)=2t+40sin(t), solve the differential equation (i.e, find the general solution of the resulted non-homogeneous differential equation). What will be the solution assuming zero initial conditions? (2-c) Derive the transfer function of the system (X(s)Y(s)) assuming zero initial condition. (2-d) If the input x(t)=2t+40sin(t), what is X(s) ? what is Y(s) ? and what is y(t) ? compare with the result from 2-b. (2-e) Plot time response y(t) of the system from part 2-d using MATLAB. Produce another plot of the response by modelling the system in SIMULINK. Compare critically, and comment on your findings in terms of steady state and transient states.|