On the application of bi-orthogonality relations for analysis of linear dynamic systems

This paper derives a new class of modal decompositions of Somigliana's identities for a certain class of problems displaying symmetry and uniformity such as, for instance, elastic vibration problems including the effect of fluid loading. The decompositions are shown to lead to a number of new analytical results relating to homogeneous and inhomogeneous boundary value problems in whichever realm of physics, for example, free and forced vibration problems in a bounded domain, and also to unify many known results which have previously been obtained by other methods. At every stage, the mathematical analysis is shown to correspond to physical concepts relating to standing waves, transmission of energy and bi-orthogonality. Details are given for a number of examples in elastodynamics and vibro-acoustics, of which the fluid-loaded membrane provides particularly clear demonstrations of the power of the method in a non-trivial setting.