TY - JOUR

T1 - Bi-orthogonality relations for fluid-filled elastic cylindrical shells

T2 - Theory, generalisations and application to construct tailored Green's matrices

AU - Ledet, Lasse Søgaard

AU - Sorokin, Sergey V.

PY - 2018

Y1 - 2018

N2 - The paper addresses the classical problem of time-harmonic forced vibrations of a fluid-filled cylindrical shell considered as a multi-modal waveguide carrying infinitely many waves. The forced vibration problem is solved using tailored Green's matrices formulated in terms of eigenfunction expansions. The formulation of Green's matrix is based on special (bi-)orthogonality relations between the eigenfunctions, which are derived here for the fluid-filled shell. Further, the relations are generalised to any multi-modal symmetric waveguide. Using the orthogonality relations the transcendental equation system is converted into algebraic modal equations that can be solved analytically. Upon formulation of Green's matrices the solution space is studied in terms of completeness and convergence (uniformity and rate). Special features and findings exposed only through this modal decomposition method are elaborated and the physical interpretation of the bi-orthogonality relation is discussed in relation to the total energy flow which leads to derivation of simplified equations for the energy flow components.

AB - The paper addresses the classical problem of time-harmonic forced vibrations of a fluid-filled cylindrical shell considered as a multi-modal waveguide carrying infinitely many waves. The forced vibration problem is solved using tailored Green's matrices formulated in terms of eigenfunction expansions. The formulation of Green's matrix is based on special (bi-)orthogonality relations between the eigenfunctions, which are derived here for the fluid-filled shell. Further, the relations are generalised to any multi-modal symmetric waveguide. Using the orthogonality relations the transcendental equation system is converted into algebraic modal equations that can be solved analytically. Upon formulation of Green's matrices the solution space is studied in terms of completeness and convergence (uniformity and rate). Special features and findings exposed only through this modal decomposition method are elaborated and the physical interpretation of the bi-orthogonality relation is discussed in relation to the total energy flow which leads to derivation of simplified equations for the energy flow components.

KW - Bi-orthogonality relations

KW - Convergence and error calculation

KW - Energy flow

KW - Modal decomposition

KW - Symmetric waveguides

KW - Tailored Green's matrices

UR - http://www.scopus.com/inward/record.url?scp=85041495657&partnerID=8YFLogxK

U2 - 10.1016/j.jsv.2017.12.010

DO - 10.1016/j.jsv.2017.12.010

M3 - Journal article

SN - 0022-460X

VL - 417

SP - 315

EP - 340

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

ER -