Consider the following mechanical system: X3 m1 = 2 kg m2 = 4 kg F k1 k k3 m3 = 3 kg m m12 m3 k = 3 N/m k = 2 N/m k3 = 1 N/m The goal of this assignment is to observe responses for different initial conditions. 1. Find the equations of motion for the given system and represent them in a matrix-vector form. Clearly identify mass and stiffness matrices. 2. Find the natural frequencies and corresponding modes. Normalize the mode shapes such that the first element is equal to 1. 3. For the following sets of initial conditions, plot the amplitudes of free responses (actual response as well as for each mode). Case 1 x(0) = 1 x(0)=0 X3 (0)=0 *(0)=0 x2(0)=0 x3(0)=0 Case 2 Case 3 x(0)=0 x(0) = 1 x(0)=0 x(0) = 1 x3(0)=0 x3(0) = 1 x(0)=0 x(0)=0 x(0) = 1 |x3(0)=0 x2(0)=0 x3(0)=0 4. Discuss the results. Clearly present all your work step by step, and upload it on Canvas. I will not accept any submission via email. Assignment scores will be posted on Canvas (A = 5,B=4, C3, D = 2, F = 1). Feel free to use MATLAB for computing and plotting. Attach the MATLAB scripts you generated at the end.