Question

2. Solve the following problems for the 2-DOF system in Problem 1 above using the order reduction in Lecture Note 24. (a) Reduce the order of the EOMs and represent them in a matrix form. In your solution procedure, use the augmented vector of z = [xTxT]T. What are the B and D matrices? (b) Find the eigenvalues and eigenvectors using the "numpy.linalg.eig" function in Python. Write the free vibration responses in terms of the eigenvalues and eigenvectors. (c) Find the expression of the forced vibration responses in terms of B and D. (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 of part (b). (e) Write a Python code to plot the free, forced, and total displacements from t = 0s to t = 0.3 s. Submit your Python code.

Fig: 1