1 Computer Aided Engineering Design and Product Development, The Faculty of Engineering and Science (ENG), Aalborg University, VBN2 Department of Mechanical and Manufacturing Engineering, The Faculty of Engineering and Science, Aalborg University, VBN3 Solid and Computational Mechanics, The Faculty of Engineering and Science, Aalborg University, VBN4 The Faculty of Engineering and Science (ENG), Aalborg University, VBN
The bi-orthogomality conditions in terms of generalised forces and displacements are derived from the reciprocity relations for a hierarchy of elastic waveguides, which support several travelling and evanescent modes (free waves). In the simple cases of waves in a straight beam and axisymmetric waves in a thin elastic cylindrical shell, these conditions are formulated as identities in an explicit form via wavenumbers. The forced vibrations of these waveguides under localised excitation are also considered with these identities being employed. The bi-orthogonality conditions in more advanced cases, specifically, for non-axisymmetric waves in an elastic cylindrical shell and for waves in an elastic helical spring, are derived, but not presented in an explicit form via wavenumbers. The results obtained for the hierarchy of waveguides are discussed in view of the classical bi-orthogonality conditions for an elastic layer. The grouping of generalised forces and displacements in these conditions is explained by inspection of the response of a given waveguide to the localised excitation.
Journal of Sound and Vibration, 2013, Vol 332, Issue 21