The multi-degree-of-freedom system shown in Figure 4 (on next page) consists of uniform link 1 of...
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The multi-degree-of-freedom system shown in Figure 4 (on next page) consists of uniform link 1 of length L₁ and mass M₁, uniform link 2 of length L₂ and mass M₂, 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. 0₁, x, and 0₂). 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 k₁ = k₂ = k, k3 = 2k, M₁ = 3m, M₂ = 4m, d₁ = 0.6L₁, and d₂ = 0.75L2. Use L₁ = 70 cm, L₂ = 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 d₁ 0₁ k3 000 k₁ 000 C1 m ४ Figure 4 Link 2 k₂ 000 C2 0₂ C3 dz The multi-degree-of-freedom system shown in Figure 4 (on next page) consists of uniform link 1 of length L₁ and mass M₁, uniform link 2 of length L₂ and mass M₂, 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. 0₁, x, and 0₂). 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 k₁ = k₂ = k, k3 = 2k, M₁ = 3m, M₂ = 4m, d₁ = 0.6L₁, and d₂ = 0.75L2. Use L₁ = 70 cm, L₂ = 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 d₁ 0₁ k3 000 k₁ 000 C1 m ४ Figure 4 Link 2 k₂ 000 C2 0₂ C3 dz
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ANSWER Solving this problem requires a detailed analysis of a multidegreeoffreedom system including deriving equations of motion performing vibration analysis using MATLAB and considering the effect o... View the full answer
Related Book For
Vector Mechanics for Engineers Statics and Dynamics
ISBN: 978-0073398242
11th edition
Authors: Ferdinand Beer, E. Russell Johnston Jr., David Mazurek, Phillip Cornwell, Brian Self
Posted Date:
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