1. A two degrees of freedom system is shown in Figure Q1, where m = 4...
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1. A two degrees of freedom system is shown in Figure Q1, where m₁ = 4 kg, m₂ = 1 kg, k₁ = 40 N/m, k₂ = 20 N/m, k3 = 10 N/m, C₁ = 1.2 N.s/m, C₂ = 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 9₁ (1) m₁ -0.21 0.5 N.s/m, K= m₂ 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 m₁. This is modelled as an initial velocity of 10 mm/s for mass m₂ (the initial velocity of mass m₂ 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] 1. A two degrees of freedom system is shown in Figure Q1, where m₁ = 4 kg, m₂ = 1 kg, k₁ = 40 N/m, k₂ = 20 N/m, k3 = 10 N/m, C₁ = 1.2 N.s/m, C₂ = 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 9₁ (1) m₁ -0.21 0.5 N.s/m, K= m₂ 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 m₁. This is modelled as an initial velocity of 10 mm/s for mass m₂ (the initial velocity of mass m₂ 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]
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a Natural Frequencies and Mode Shapes for the Undamped System The natural frequencies n for an undam... View the full answer
Related Book For
Vector Mechanics for Engineers Statics and Dynamics
ISBN: 978-0073212227
8th Edition
Authors: Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell
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