The multi-degree-of-freedom system shown in Figure 4 (on next page) consists of uniform link 1 of length L and mass M, uniform link 2 of length L and mass M, and a mass m connected to the two links as shown in the figure. The system is in the vertical plane and at static equilibrium. For this system: (a) Derive the equations of motion for all masses using the coordinates shown (i.e. 0, x, and 0). Provide detailed procedure including free- body-diagrams for all masses. (b) Write the equations of motion in matrix form. (c) Vibration analysis using MATLAB: For the undamped system, obtain the natural frequencies and mode shapes of the system using MATLAB. Identify each mode shape and its corresponding natural frequency (include your MATLAB code in your single digital file submission). Assume k = k = k, k3 = 2k, M = 3m, M = 4m, d = 0.6L, and d = 0.75L2. Use L = 70 cm, L = 90 cm, m = (5 + x)/10 kg and k = 10 x (20 + x) N/m, where x is the last digit of your student number. For instance: For x = 0 m = 0.5 kg and k = 200 N/m For x = 8 m = 1.3 kg and k = 280 N/m (d) Does gravity affect the natural frequencies of this system? If no explain why, and if yes compare the natural frequencies with and without gravity and explain the differences. (e) Consider this system representing a subset of a larger system in an industrial setting. The system is subject to external harmonic excitation with frequencies in the range of 7Hz-15Hz. Does resonance happen in the system? Explain what is a safe operation frequency for the system to stay away from resonance? [18 marks] [3 marks] [7 marks] [6 marks] [6 marks] Link 1 d 0 k3 000 k 000 C1 m Figure 4 Link 2 k 000 C2 0 C3 dz