CA-AA242B-Hw4

CA-AA242B-Hw4 - x = 0 and attached to a spring of stiness k...

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AA 242B: Mechanical Vibrations (Spring 2010) Homework #4 Due May 19, 2010 1 Problem 1 The potential energy scalar product for a uniform bar is deFned as ( f, g ) V = Z L 0 EAf ( x ) d 2 g dx 2 dx Consider the cases where (a) the bar is Fxed at x = 0 and free at x = L and (b) the bar is Fxed at x = 0 and attached to a linear spring of sti±ness k at x = L . Discuss in each case the implication of requiring f ( x ) and g ( x ) to satisfy only the essential boundary conditions. 2 Problem 2 Use the Rayleigh-Ritz method with the assumed modes w i ( x ) = sin( x L ) i = 1 , 2 , 3 to approximate the lowest natural frequency and its corresponding mode shape for a uniform Fxed-Fxed bar of length L . 3 Problem 3 Let w i ( x ), i = 1 , ··· , 4 be linearly independent polynomials of degree four or less that satisfy the essential boundary conditions for a bar Fxed at
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Unformatted text preview: x = 0 and attached to a spring of stiness k at x = L . 1. Determine a set of w i ( x ), i = 1 , , 4 2. Use the Rayleigh-Ritz method with the functions obtained above and kL 3 /EI = . 5 to approximate the systems lowest natural frequencies and mode shapes. 1 4 Problem 4 The mode shapes of a uniform Fxed-free bar are of the form n ( x ) = sin (2 n-1) x 2 L n = 1 , 2 , 3 , Use the Rayleigh-Ritz method with i ( x ), i = 1 , 2 , 3 to approximate the lowest natural frequencies and mode shapes of a Fxed-free tapered bar of length L = 3, circular cross section of radius r ( x ) = 0 . 05 (1-. 01 x ) 2 , and with E = 2 10 11 and = 7500. 2...
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CA-AA242B-Hw4 - x = 0 and attached to a spring of stiness k...

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