Ch4-2(3)c - 1 MAE351 Mechanical Vibrations Lecture 16...

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1 1 MAE351 Mechanical Vibrations Lecture 16 (Chap. 4.3) 2 Review: Window 4.2- Normalization Orthonormal Vectors similar to the unit vectors of statics and dynamics T 1 2 11 22 1 2 and are both if 1 and 1; if x x 0 0 if This is abbreviated by 1i f TT T ij i j normal orthogonal  x x xx 1 if A set of vectors are set to be if T ii j n orthonormal x and . ij  1) Find the missing component by fixing it, then computing x x T x x T x 1 2) Scaling with respect to the mass: 2 where = and T iii MM K    uu u w u w w w w 1 1 2 2 () 1 T i i i i M M vv u u u 3) Consider the transformation used here and compute v i M 1 2 u i u = modeshape vector; v = eigenvector
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2 3 4.3 - Modal Analysis Physical coordinates are not always the easiest to work in Eigenvectors provide a convenient transformation Eigenvectors provide a convenient transformation to modal coordinates – Modal coordinates are linear combination of physical coordinates – Say we have physical coordinates x and want to transform to some other coordinates u u 1 x 1 3 x 2 u 2 x 1 3 x 2 2 1 2 1 3 1 3 1 x x u u 4 Review of Eigenvalue Problem 11 22 00 Start with ( ) & initial conditions and . Rewrite as Mt K MM K  xx 0 x x 0  Let (coord. trans. #1) q 1 1 1 2 2 2 Premultiply by M M K M K I   qq q q 0  • Now, we have obtained a real symmetric matrix • This guarantees the real eigenvalues and the distinct, mutually orthogonal eigenvectors
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3 5 Eigenvectors = Mode Shapes?
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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Ch4-2(3)c - 1 MAE351 Mechanical Vibrations Lecture 16...

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