A natural classification of vibration modes of polygonal ducts based on group theoretic analysis

In this work, the natural frequencies and mode shapes of polygonal ducts are obtained by a finite element analysis via a group theoretic approach. We show that the group theoretic techniques provide a natural setting for the solution of the problem via optimum utilization of the symmetry in the problem. The simplifications and insights due to the inherent symmetry is intuitively obvious in simple symmetric structures. However, a systematic exploitation of this symmetry in more complicated structures is less obvious and is possible perhaps only via useful theorems from group representation theory and the associated projection operator theory. We illustrate all the main steps with the help of an example of dihedral groups. This example is directly applicable to the problem addressed in this paper. oup theoretic approach splits up the original problem into independent subproblems and thereby affects significant computational savings. More importantly, the approach captures the role of symmetry in the problem and provides insights which are otherwise not obvious. The multiplicity of the natural frequencies, arising out of the inherent symmetry is determined a priori and neatly separated into different subproblems. This suggests a neat physical classification of the mode shapes on a symmetry basis which is unique to this approach. The values of the natural frequencies and the mode shapes for triangular, square and pentagonal ducts are shown to be in good agreement with the existing results reported in the literature. The classification of mode shapes on this symmetry basis is used in the analysis of the polygonal ducts with number of sides varying up to 16.