collecting like terms b b a a f t

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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 eigenvalue-eigenvector 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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