1. A 2-DOF system is shown in Figure 1. The mass of m2 is acted by the external force f(t) = fo sin(wt) and the system is also excited by x(t) xo sin(wt) through the left massless plate. The system parameters are given as m = 4m2 = 16 kg, =200 N s/m, k = 2k2 = 1 m, and w = 200 rad/s. The initial conditions are given as x(0) = 0.5 m, x2 (0) = 0 m, x (0) = 0 m/s, and x2 (0) = 20 m/s. For the given information, solve the following problems: 100 kN/m, fo = 10 kN, xo = (a) The coordinates of x and x2 are the displacements of m and m2, respectively. Draw the free- body diagrams (FBDS) of m and m2, and derive EOMs by applying the Newton's 2nd Law of Motion to the FBDs. Write the EOMs in a matrix form in terms of mass matrix, M, damping matrix, C, and stiffness matrix, K. (b) Find the free vibration responses in terms of 4 unknown coefficients, four eigenvalues, and four eigenvectors. In your solution, clearly show what the eigenvalues and eigenvectors are. Do not use an order reduction to make the eigenvalue problem to a standard form. (c) Find the expression of the forced vibration responses in terms of the mass matrix, M, damping matrix, C, and stiffness matrix, K. (d) By applying the initial conditions to the total solutions (i.e., the summation of the free and forced vibration responses), find the expression for the vector of the unknown coefficients in the free vibration responses. (e) Write a Python code to plot the free, forced, and total displacements from t = 0 stot = 0.3 s. Submit your Python code. | x(t) m1 X1 HX2 k2 m2 f(t) Massless plate Figure 1: 2-DOF System.