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Unformatted text preview: For many lightly damped situations, this is a reasonable assumption. [ K ]{ X } = − s 2 [ M ]{ X } where: • λ = −s2 General Damped Case [ M ]{ x} + [C ]{ x} + [ K ]{ x} = { f } This system of equations can be augmented by the identity shown as follows: [ M ]{ x} − [ M ]{ x} = {0} The above two equations can be combined to yield a new system of 2N equations. Note that all the matrices in the resulting equation are symmetric and the equation is now in a classical eigenvalue solution form. The notation used in the following equation is consistent with the notation used in many mathmatics and/or controls textbooks. Lecture Notes -83- 06/16/06 12:25 PM Mechanical Vibrations I [ A]{ y} + [ B ]{ y} = { f ′} where: [ 0 ] [ M ] − [ M ] [ 0] [ A] = M C [ B ] = 0 [ ] [ K ] [ ] [ ] { x} { x} { y} = { y} = { x} { x} {0} { f ′} = { f } Forming the homogeneous equation of this system equation: [ A]{ y} + [ B ]{ y} = {0} The solution of the above equation yields the complex-valued natural frequencies (eigenvalues) and complex-valued modal vectors (eigenvectors) for the augmented 2 N equation system. Note that in this mathematical form, the eigenvalues will be found directly (not the square of the eigenvalue) and the 2N eigenvectors will be 2N in length. The exact form of the eigenvectors can be seen from the associated modal matrix for the 2 N equation system. Note that the notation {φ } is used for an eigenvector in the 2N equation system and that the notation {ψ } is used for an eigenvector of the N equation system. MATLAB Solution For the general, homogeneous system equation, using either an assumed solution or transform methods, the following equation must be solved: λ [ A]{ y} + [ B ]{ y} = {0} MATLAB uses the same matrix terminology but refers to a different eigenvalue equation. A { y} = λ B { y} Therefore, using the MATLAB EIG function to solve for the eigenvalues and eigenvectors requires the following form: [ y, λ ] = EIG ( A, B ) = EIG ( B, − A) Lecture Notes -8406/16/06 12:25 PM Mechanical Vibrations I The eigenvalues of this system of equations are the same as for the original mass, stiffness and damping matrix equation. The eigenvectors of this system of equations yield the modal vectors of the original mass, stiffness and damping matrix equation through the modal matrix. The modal matrix for this system is a matrix made up of the 2 N eigenvectors. The modal matrix {φ } for this damped system can now be assembled. [φ ] = {φ }1 {φ }2 {φ }r λr {ψ }r {ψ }r {φ }2 N λ2 N {ψ }2 N {ψ }2 N [φ ] = λ1 {ψ }1 λ2 {ψ }2 {ψ }1 {ψ }2 Weighted Orthogonality Concept Proportionally Damped Case A set of weighted orthogonality relationships are valid for the system matrices [ M ] and [K ] . {ψ }r [ M ]{ψ }s = 0 T {ψ }r [ K ]{ψ }s = 0 T Modal Scaling - Proportionally Damped Case {ψ }r [ M ]{ψ }r = M r T {ψ }r [ K ]{ψ }r = K r T {ψ }r [C ]{ψ }r = Cr T General Case A set of weighted orthogonality relationships are valid for the system matrices [ A] and [ B] . {φ }r [ A]{φ }s = 0 T {φ }r [ B ]{φ }s = 0 T Lecture Notes -85- 06/16/06 12:25 PM Mechanical Vibrations I Modal Scaling - General Case {φ }r [ A]{φ }r = M A T {φ }r [ B ]{φ }r = M B T r r The terms modal A and modal B are modal scaling factors for the general case of system with damping. Note that modal A and modal B are related by the complex modal frequency for each mode. Lecture Notes -86- 06/16/06 12:25 PM...
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