Free and forced response of three-dimensional waveguides with rotationally symmetric cross-sections

Abstract

The analysis of high-frequency wave propagation in waveguides of arbitrarily shaped cross-sections requires specific numerical methods. A rather common technique consists in discretizing the cross-section with finite elements while describing analytically the axis direction of the waveguide. This technique enables to account for the continuous translational invariance of a waveguide and leads to a modal problem written on the cross-section. Although two-dimensional, solving the so-obtained eigensystem can yet become computationally costly with the increase of the size of the problem. In most applications involving waveguides, the cross-section itself often obeys rules of symmetry, which could also be exploited in order to further reduce the size of the modal problem. A widely encountered type of symmetry is rotational symmetry. Typical examples are bars of polygon-shaped cross-section or multi-wire cables. The goal of this paper is to propose a numerical method that exploits the discrete rotational symmetry of the waveguide cross-section. Bloch–Floquet conditions are applied in the circumferential direction while the continuous translational invariance of the waveguide along its axis is still described analytically. Both the free and the forced response problems are considered. A biorthogonality relationship specific to the rotationally symmetric formulation is derived. Numerical results are computed and validated for the simple example of a cylinder and for the more complex test case of a multi-wire structure. In addition to reducing the computational effort, the rotationally symmetric formulation naturally provides a classification of modes in terms of their circumferential order, which can be of great help for the dynamic analysis of complex structures.