modal_shapes_QA

modal_shapes_QA - Questions and Answers on Natural...

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1 Questions and Answers on Natural Frequencies and Mode Shapes Background The free response of an undamped, discrete system with N DOF s is given by: x t ( ) = f j ( ) c j cos w j t + s j sin j t [ ] j = 1 N (1) where j and j ( ) (j = 1, 2, , N) are the natural frequencies and mode shapes for the system, respectively, that are found from the solution of the eigenvalue problem : - 2 M [ ] + K [ ] [ ] = 0 (2) Question : How do I find the natural frequencies? Answer : The natural frequencies are found from the solution of the characteristic equation determined from: det - 2 M [ ] + K [ ] [ ] = 0 which is an Nth order polynomial in 2 : a N 2 N + a N - 1 2 N - 1 ( ) + ... + a 1 2 + a 0 = 0 For N > 2, we need to use a numerical solver to find these roots. Question : How do I find the mode shapes? Answer : The mode shapes are found from the solution of the algebraic equations of: - j 2 M [ ] + K [ ] [ ] j ( ) = 0 That is, for a given root j of the characteristic equation, we solve for the components i j ( ) (i = 1,2, ,N) of the corresponding vector j ( ) .
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2 Question : What does the notation f i j ( ) mean? Answer : i j ( ) represents the ith component of the jth mode shape j ( ) ; that is, the jth mode shape is made up the N components of: j ( ) = 1 j ( ) 2 j ( ) M N j ( ) Question : Why do we need to choose a value for one of the components when solving for a mode shape? Answer : Since the eigenvalue problem (1) is a set of homogeneous equations, then if is a solution, any scalar multiple of (say af ) is also a solution.
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modal_shapes_QA - Questions and Answers on Natural...

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