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HW1_2007_Solution_v2 - Homework 1_Solution Acoustics I Fall...

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Homework 1_Solution Acoustics I Fall 2007 A SDOF system shown in the figure below is operating on a foundation of impedance F Z % defined by 6 6 27,000 1.875 10 270 10 3000 , 3000 , 3000 F Z j j or ϖ ϖ ϖ ϖ ϖ ϖ × × = - - - % . Plot the displacement amplitude 1 X % of mass M 1 when F = 1 as a function of frequency for the range 0 500 / rad s ϖ = - for these three cases. (Solution) The impedance at the excitation point is given as: 1 1 1 j t m m m j t m m F F s js R R f Fe j Z j M j M s js x j X e R R j Z Z ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ + - = = = + = + + - + + % & % % % % % (1) For F = 1, 1 1 m X j Z ϖ = % % (2) 1 27,000 / s N m = 1 180 / m R N s m = 1 x % ( ) j t f t Fe ϖ = % 1 300 M Kg = F Z %
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Therefore, 1 X % is obtained: (1) Calculate F Z % as a function of ϖ . (2) Calculate m Z % from Eq. (1) as a function of ϖ for the calculated F Z % . (3) Calculate 1 X % from Eq. (2). Then, graphically representing the complex amplitude 1 X % is the only remaining work. (a) 27,000 3000 F Z j ϖ ϖ = - % , 300 , 27,000 / , 180 / m M kg s N m R N s m = = = 10 -1 10 0 10 1 10 2 10 3 10 -10 10 -5 10 0 amplitude (m) 10 -1
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