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MEM423Su07-08_HW6_soln_stud

# MEM423Su07-08_HW6_soln_stud - MEM 423 Mechanics of...

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MEM 423: Mechanics of Vibrations Home work #6 ( due 08/05/08) solution Relevance to a-k criteria: a, e All problems are from text, Edition IV Problems: 6.21 (use the force- displacement relation given below) 6.24 Determine only the stiffness matrix using stiffness influence coefficients. 6.61 Hint: The ”eigenvectors” (i.e., the modal vectors) satisfy ( ) ( ) 0, . T i j X X i j = M You may use a software if needed.

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Problem 6.21 Apply F 1 = 1, F 2 = 0, F 3 = 0: 3 11 1 ( 1, , 3 , , 4 ( )) 9 / (64 ) a F a l b l x l L l in E l EI = = = = = = = 3 21 2 ( 1, , 3 , 2 , 4 ( )) / (6 ) a F a l b l x l L l in E l EI = = = = = = = 3 31 2 ( 1, , 3 , 3 , 4 ( )) 13 / (192 ) a F a l b l x l L l in E l EI = = = = = = = Similarly, apply {F 1 = 0, F 2 = 1, F 3 = 0} to get {a 12 , a 22 , a 32 },and {F 1 = 0, F 2 = 0, F 3 = 1} to get {a 13 , a 23 , a 33 }. The result is
3 9 / 64 1/ 6 13/192 1/ 6 1/ 3 1/ 6 13/192 1/ 6 9 / 64 l A EI = Problem 6.24 Use the subscripts to reflect quantities from top to bottom. The force equilibrium condition applied to each mass yields:

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MEM423Su07-08_HW6_soln_stud - MEM 423 Mechanics of...

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