Private Rayleigh and Private Dunkerley are marching up and down the parade square which has a...
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Private Rayleigh and Private Dunkerley are marching up and down the parade square which has a flagpole located in one corner. They notice that after strong gusts of wind that the flagpole oscillates, bending from side to side. They both believe that they can estimate the fundamental lateral frequency of vibration for the flagpole using the knowledge they have gained in their officer training course named Divide and Conquer. Although they both plan to use different approaches, both involve considering an equivalent system with the flagpole mounted horizontally and to consider the deflections due to gravity on the masses. (They learnt this in the course United We Stand, Divided We Fall.) They both plan to model the pole as three equal length sections with three equally spaced masses, .. at the centre of each section, on a massless cantilever spring and assume that the flag has negligible mass. The flagpole is of length, 1 = 4.5 m, with geometric and material properties: Mass, M = 30 kg Second moment of area, I = 1E-7 m^4 Young's modulus, E = 2.09E11 Pa Consider the clamped end of the cantilver as x = 0 and the free end as x = 1. Remember to retain high accuracy throughout your calculations. Using the approach appropriate to each Private, firstly calculate the displacements that Dunkerley should use, namely: The deflection at the location of Mass 1 due to Mass 1 only is: mm Your answer should be accurate to within +/- 0.1. The deflection at the location of Mass 2 due to Mass 2 only is: mm Your answer should be accurate to within +/- 0.1. The deflection at the location of Mass 3 due to Mass 3 only is: mm Your answer should be accurate to within +/- 0.1. In addition Rayleigh needs to calculate the displacements at the other locations due to the actions of the masses, namely: The deflection at Mass 1's location due to Mass 2: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 1's location due to Mass 3: mm Your answer should be accurate to within +/- 0.1. Furthermore Rayleigh must calculate: The deflection at Mass 2's location due to Mass 1: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 2's location due to Mass 3: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 3's location due to Mass 1: mm Your answer should be accurate to within +/-0.1. The deflection at Mass 3's location due to Mass 2: mm Your answer should be accurate to within +/- 0.1. Calculate the first transverse natural frequency of vibration in rad/s using Dunkerley's Equation: rad/s Your answer should be accurate to within +/- 0.01. Calculate the first transverse natural frequency of vibration in rad/s using Rayleigh's Method: rad/s Your answer should be accurate to within +/-0.01. Private Rayleigh and Private Dunkerley are marching up and down the parade square which has a flagpole located in one corner. They notice that after strong gusts of wind that the flagpole oscillates, bending from side to side. They both believe that they can estimate the fundamental lateral frequency of vibration for the flagpole using the knowledge they have gained in their officer training course named Divide and Conquer. Although they both plan to use different approaches, both involve considering an equivalent system with the flagpole mounted horizontally and to consider the deflections due to gravity on the masses. (They learnt this in the course United We Stand, Divided We Fall.) They both plan to model the pole as three equal length sections with three equally spaced masses, .. at the centre of each section, on a massless cantilever spring and assume that the flag has negligible mass. The flagpole is of length, 1 = 4.5 m, with geometric and material properties: Mass, M = 30 kg Second moment of area, I = 1E-7 m^4 Young's modulus, E = 2.09E11 Pa Consider the clamped end of the cantilver as x = 0 and the free end as x = 1. Remember to retain high accuracy throughout your calculations. Using the approach appropriate to each Private, firstly calculate the displacements that Dunkerley should use, namely: The deflection at the location of Mass 1 due to Mass 1 only is: mm Your answer should be accurate to within +/- 0.1. The deflection at the location of Mass 2 due to Mass 2 only is: mm Your answer should be accurate to within +/- 0.1. The deflection at the location of Mass 3 due to Mass 3 only is: mm Your answer should be accurate to within +/- 0.1. In addition Rayleigh needs to calculate the displacements at the other locations due to the actions of the masses, namely: The deflection at Mass 1's location due to Mass 2: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 1's location due to Mass 3: mm Your answer should be accurate to within +/- 0.1. Furthermore Rayleigh must calculate: The deflection at Mass 2's location due to Mass 1: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 2's location due to Mass 3: mm Your answer should be accurate to within +/- 0.1. The deflection at Mass 3's location due to Mass 1: mm Your answer should be accurate to within +/-0.1. The deflection at Mass 3's location due to Mass 2: mm Your answer should be accurate to within +/- 0.1. Calculate the first transverse natural frequency of vibration in rad/s using Dunkerley's Equation: rad/s Your answer should be accurate to within +/- 0.01. Calculate the first transverse natural frequency of vibration in rad/s using Rayleigh's Method: rad/s Your answer should be accurate to within +/-0.01.
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