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Unformatted text preview: .3456 j 1.0000 0.5772 −0.9933 −1.6417 − 18.0458 j −0.9394 − 10.3255 j 1.6307 + 17.9246 j 1.0000 1.5435 1.8763 −0.1848 − 6.0760 j −0.2852 − 9.3783 j −0.3467 − 11.4003 j 1.0000 1.5435 1.8763 −0.1848 + 6.0760 j −0.2852 + 9.3783 j −0.3467 + 11.4003 j 1.0000 0.5772 −0.9933 −1.6417 + 18.0458 j −0.9394 + 10.3255 j 1.6307 − 17.9246 j −0.7863 0.3105 −3.6795 + 26.8767 j 2.8932 − 21.1335 j −1.1425 + 8.3456 j 1.0000 Therefore:
α1 0.2567 + 0.2265 j α 2 0.1890 + 0.2246 j α 3 0.0584 + 0.1902 j = α 4 0.0584 − 0.1902 j α 5 0.1890 − 0.2246 j α 6 0.2567 − 0.2265 j Complementary Solution Approach  Adjoint Matrix
An alternative approach to determining the modal vectors in the complementary solution involves evaluating the adjoint matrix of the system impedance matrix at the modal frequencies ( λr ). This method has two advantages: 1) Formulating the adjoint matrix effectively solves the set of linear equations one time rather than one set of linear equations for each modal frequency; 2) Evaluating the adjoint matrix handles the rank deficient problem in a uniform way (no assumption of 1.0 for a modal coefficient). The development is as follows: s 2 [ M ] + s [C ] + [ K ] { X } = {0} Define: Lecture Notes 81 06/16/06 12:25 PM Mechanical Vibrations I [ B ( s ) ] = s 2 [ M ] + s [ C ] + [ K ] where: • [ B( s ) ] = System Impedance Matrix [ B( s)][ B( s)]
−1 −1 = [I ]
A [ B( s)] [ B( s)] = [ B( s)]
where: A • [ B ( s ) ] is the adjoint of matrix [ B( s ) ] . [ B( s)][ B( s)]
Note: [ B(λr ) ] = 0 . Evaluating at s = λr gives: A = [ B( s ) ][ I ] [ B(λr )][ B(λr )]
A A = [ 0]
A Using any column of [ B(λr ) ] , the i th column for example { B (λr )}i . Therefore: [ B(λr )]{B(λr )}i = {0} λr2 [ M ] + λr [C ] + [ K ] {ψ r } = {0} A Note that the columns of the adjoint matrix [ B(λr ) ] are all proportional to the r th modal vector.
A When the mass, damping and stiffness matrices are symmetric (when absolute coordinates are used), the system impedance matrix [ B( s ) ] is symmetric. Therefore, in this case, the adjoint matrix of [ B(λr ) ] is also symmetric. Thus, in this case, the rows of the adjoint matrix are also proportional to the modal vector. Lecture Notes 82 06/16/06 12:25 PM Mechanical Vibrations I Complementary Solution Approach  Eigenvalue/Eigenvector Method
The homogeneous form of the Laplace domain model can be used as a general representation of the matrix relationship that yields the system modal characterisitics:
s 2 [ M ]{ X } + s [C ]{ X } + [ K ]{ X } = {0} Generally, this problem is solved using eigenvalueeigenvector solution methods once the problem is put in the standard eigenvalue form:
[ A] − λ [ I ] { X } = {0} [ A]{ X } = λ { X } [ A]{ X } = λ [ B ]{ X }
Undamped Case:
In order to manipulate the system equations into a standard eigenvalue/eigenvector equation form, one approach is to assume that the undamped case is a reasonable approximation of the damped case....
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 Fall '11
 Regalla

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