Multiple Eigenvalues Arising from a Class of Repetitive Substructures

Abstract

Thedynamicsubstructuremethodinstatespacewasemployedtostudyeigenvalueproblemsforstructureswitha classofrepetitivesubstructures,whichsharea common interface. Theblock propertiesoftheresulting synthesized system matrices are discussed. A very interesting result on multiple eigenvalues of the considered structures was obtained: each ® xed interface eigenvalue of the singlerepetitive substructure appeared as at least ( n i ®) multiple eigenvalues of the whole structure, where n is the number of repetitive substructures; ®is a number depending on the azimuth distributions of the repetitive substructures. It takes at most nine, and it takes three in the special cases when the repetitive substructures are oriented by rotation around a ® xed axis. The mode shapes associated with the (n i ®) multiple eigenvalues were obtained and the nondefectiveness of the obtained multiple eigenvalues is discussed. Physical explanation and numerical examples were also attempted and are given. Nomenclature C r = damping matrix of the rth substructure, symmetric f r = external force vector exerted on the rth substructure A f r = internal force vector exerted on the interface of the rth substructure G r = gyroscopic matrix of the rth substructure, skew symmetric i = indication of internal variables for substructure j = indication of interface variables for substructure K r = stiffness matrix of the rth substructure, semipositive de® nite M r = mass matrix of the rth substructure, positive de® nite m r = internal degree of freedom of the rth substructure m r j = interface degree of freedom of the rth substructure m r = number of retained Jordan blocks for the rth substructure N = degree of freedom of the whole structure with the ® nite element method discretization s r = real or complex numbers used as factors of the ® xed interface mode of rth substructure in constructing the modes for the whole structure T r = transformation between the interface coordinates in local coordinate system of the rth substructure and the global interface coordinates system A T r = transformation matrix between the local coordinate system of therth substructure and the global displacement system x r = nodal displacement vector for the rth substructure yh = state vector of the common interface in global coordinate system f r = third Euler angles used for describing the azimuth of the