Matrix Difference Equation Analysis of Vibrating Periodic Structures

The Matrix Difference Equation (MDE) method for sound transmission and forced vibration analysis of damped periodic structures is presented. A periodic structure is defined as a string of identical substructures, such as a segment of aircraft fuselage which has identical bays between circumferential frames. The finite element method is applied to the substructure to provide a mechanical impedance matrix. A matrix difference equation is derived from the impedance matrix, based upon conditions of equilibrium and compatibility at substructure boundaries. The difference equation is reduced in order by eliminating force variables and introducing substructure displacement modes. A solution is found by calculating eigenvalues and eigenvectors of a related characteristic equation. The result is a closed form expression in the longitudinal coordinate. The method is general and applicable to complex structures, because of the finite element basis. Results of an application to aircraft engine duct vibration are included.