Name:____________________Student No.:________ME143 2nd Long ExamSeptember 6, 2012

1. For the three degree of freedom system shown, derive the equations of motion and express in matrix form. Determine the fundamental frequencies of the system and the mode shapes for each natural frequency for m1 = m2 = m3 = 1 kg and k1 =k2 = k3 = k4 = 20 N/m.

2. For the two degree of freedom system shown, derive the equations of motion and express in matrix form. Determine the fundamental frequencies of the system and the mode shapes for each natural frequency for m1 = m2 = 1 kg and all k = 20 N/m. Find the system response x1(t) and x2(t) for the following initial conditions: x1(0) = 1.5, x2(0) = 2, 1(t) = 0 and 2(t) = 0.