Spatial Modulation of Repeated Vibration Modes in Rotationally Periodic Structures

Abstract

When a structure deviates from axisymmetry because of circumferentially varying model features, significant changes can occur to its natural frequencies and modes, particularly for the doublet modes that have non-zero nodal diameters and repeated natural frequencies in the limit of axisymmetry. Of technical interest are configurations in which inertia, dissipation, stiffness, or domain features are evenly distributed around the structure. Aside from the well-studied phenomenon of eigenvalue splitting, whereby the natural frequencies of certain doublets split into distinct values, modes of the axisymmetric structure that are precisely harmonic become contaminated with certain additional wavenumbers. From analytical, numerical, and experimental perspectives, this paper investigates spatial modulation of the doublet modes, particularly those retaining repeated natural frequencies for which modulation is most acute. In some cases, modulation can be sufficiently severe that a mode shape will “beat” spatially as harmonics with commensurate wavenumbers combine, just as the superposition of time records having nearly equal frequencies leads to classic temporal beating. An algebraic relation and a diagrammatic method are discussed with a view towards predicting the wavenumbers present in modulated eigenfunctions given the number of nodal diameters in the base mode and the number of equally spaced model features. [S0739-3717(00)01501-4]

Measurement and Analysis of Modulated Doublet Mode Response in Mock Bladed Disks

“…When 2nOj (NB) ( j"1, 2, 2 ), the natural frequencies of the P (m, n) doublet have the same value [14]. The corresponding mode shapes are expanded in the Fourier representations…”

“…Analogous conclusions were reached by Fox et al [12,13] in their analysis of ring-like structure vibration. A criterion for predicting contamination wavenumber content was discussed by Kim et al [14] in the context of point sti!ness perturbations that were added to an otherwise axisymmetric distributed parameter system.…”

Numerical simulation of fluid added mass effect on a francis turbine runner

“…where m is the number of ND [29]. The 2ND and 3ND modes of a circular structure are shown here to illustrate the nodal diameters and the relationships between the pairs of doublet modes.…”

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

“…When structural abnormalities like manufacturing tolerances, geometric variations, and material inhomogeneities are present or an eccentric void, inclusion, or cut-out is brought in, some of the degenerate natural frequencies which are coincident in the perfectly symmetric case divide into distinct values, mode shapes become distorted, and beating among intimately spaced vibration modesmay be produced [1,2,44]. To circumvent this situation, the vibration characteristics of the axisymmetric structures with asymmetric features should be investigated carefully.…”

“…Parker and Mote [49][50][51] used perturbation analysis to investigate the eigen-solutions for plate vibration and the wave equation on annular domains with boundary shape or stiffness variations, and discussed the phenomenon of natural frequency splitting. Kim et al [44] presented a natural frequency splitting rule for general rotationally periodic structures and investigated the effects of imperfection on both repeated and split natural frequency modes. Chang and Wickert [52] studied the vibration of rotationally periodic structures and obtained a natural frequency splitting rule and a mode contamination rule for axisymmetric structures with identical, evenly spaced asymmetries.…”