Patent ID: 7188039

Claim:
A method of solving a first equation associated with a structure, the first equation being of the form {−ω 2 M R +iωB R +[(1+iγ)K R +i(K 4R )]}Y=F R , wherein ω is a time-harmonic excitation frequency, wherein M R is a reduced form of a symmetric mass matrix M, wherein B R is a reduced form of a viscous damping matrix B, wherein γ is a scalar global structural damping coefficient, wherein K R is a reduced form of a symmetric stiffness matrix K, wherein K 4R is a reduced form of a symmetric structural damping matrix K 4 representing local departures from γ, and wherein F R is a reduced form of a matrix F including a plurality of load vectors acting on the structure, the method comprising: factoring the reduced matrix M R =L M L M T , wherein L M is a lower triangular matrix; defining an intermediate matrix C=(1+iγ)L M −1 K R L M −T +iL M −1 K 4R L M −T ; computing a plurality of eigenvalues and eigenvectors for a second equation CΦ C =Φ C Λ C , wherein Φ C is a matrix including eigenvectors of the matrix C, and wherein Λ C is a matrix including eigenvalues of the matrix C; computing a matrix P=Φ C T L M −1 U R and a matrix R=V R T L M −T Φ C , wherein U R and V R are matrices which satisfy a singular value decomposition B R =U R Σ R V R T , and wherein Σ R is diagonal, conformal with U R and V R , and includes singular values; solving a third equation of the form (D(ω)+PQ(ω)R)Z=Φ C T L M −1 F R for Z, wherein D(ω)=−ω 2 Φ C T Φ C +Φ C T CΦ C and Q(ω)=iωΣ R ; forming a product Y=L M −T Φ C Z, wherein Φ comprises eigenvectors satisfying the eigenvalue problem KΦ=MΦΛ, wherein Λ=Φ T KΦ, and wherein a matrix Φ multiplied by the product Y is approximately equal to a matrix X including a plurality of displacements of the structure; and communicating an approximation of a vibration-induced displacement of the structure, including at least a portion of a product comprising the matrix Φ multiplied by the product Y, to a user interface.