1. A two degrees of freedom system is shown in Figure Q1, where m = 4 kg, m = 1 kg, k = 40 N/m, k = 20 N/m, k3 = 10 N/m, C = 1.2 N.s/m, C = 0.2 N.s/m and c3 = 0.3 N.s/m. The corresponding mass, damping and stiffness matrices are given by 1 = [1 kg, M = C = = 1-0.2 9 (1) m -0.21 0.5 N.s/m, K= m 92(t) 60 -201 30 fi(t) f2(t) Figure Q1. The two degrees of freedom system N/m (a) Determine the natural frequencies and corresponding mass normalised mode shapes for the undamped system. QUESTION 1 CONTINUES ON NEXT PAGE [8 marks] (b) Show that the damping is proportional, and calculate the damping ratios for the first and second modes. [4 marks] (c) The system is initially at rest with zero displacements when an impact excitation is applied to mass m. This is modelled as an initial velocity of 10 mm/s for mass m (the initial velocity of mass m is assumed to be zero). The initial displacements of both masses are zero no external force is applied to the system. Calculate the damped free response of both masses. [5 marks]