When investigating the modal properties of cyclic structures (composed by a repetition of N identical sectors) such as bladed-disk assemblies, modes are often sorted by nodal diameters (or spatial harmonics) and modal families, depending on their frequency, geometrical shape and phase-shift between adjacent sectors.
It is a known phenomenon that off-axis shaft motions solely occur in 1-nodal diameter modes and that longitudinal displacements only appear in 0-nodal diameter ones. However, I have never been able to find a formal proof. Can this be shown using a purely mathematical approach?
