HW2_2007 - Figure 1 The time history of the stress at...

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Homework 2 Acoustics I due date: October 15, 2007 Shown in Figure 1 is a 1-D bar made of steel ( 3 7700 / Kg m ρ = , 10 19.5 10 E Pa = × , L = 0.2 m ). The equation of motion of the bar is given as follows; 2 2 2 2 u u E t x = (1) where, u is displacement of the bar from rest. The stress-displacement relationship is: u E x σ = (2) (1) Comparing Eq. (1) with the acoustic wave equation, find the expression and value of the wave speed in the bar. Figure 1: One-Dimensional Bar Because u E x = , it can be easily show that the wave equation can also be written as; 2 2 2 2 E t x = (3) The boundary conditions are 0 u at x L = = and 0 = at x=0 . (2) An impact produces a right traveling stress from the left end of the bar as shown in
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Unformatted text preview: Figure 1. The time history of the stress at Stress wave will be reflected with its sign reversed at x = 0 (free end) and with its sign unchanged at x=L (fixed end). Displacement wave will be reflected with its sign unchanged at x= 0 and with its sign reversed at x=L . Explain the reasons why. (3) Sketch the snap shot of the stress wave at t = 7T/8. (4) Sketch time histories of the stress wave as in Fig.2 at x= L for the duration of t=0, 2T. ( , ) x t , u x L Figure 2: Time history of the stress wave at x= 0 . 1 Mpa σ = t 1 , where 4 L T T c =...
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This note was uploaded on 10/19/2009 for the course MECH 666 taught by Professor Kim during the Fall '07 term at University of Cincinnati.

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HW2_2007 - Figure 1 The time history of the stress at...

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