Abstract
This paper deals with the in-plane free vibration of rings with a nominally elliptical centreline. Results are presented for rings of constant axial length that have a rectangular cross-section, the radial thickness of which is constant or has a simple, analytically defined circumferential variation. Additionally, and for the first time, the effects of small variations in in-plane profile, such as those arising in practical rings due to manufacturing tolerances, are considered. The problem is tackled using an approach in which the true middle surface is determined numerically from the outer and inner surface profiles, which can be defined either by exact analytical expressions or in a more general way using Fourier series. The Rayleigh–Ritz method is used to obtain the natural frequencies and mode shapes. Results are presented for a range of cases, including some that have previously been studied by other authors and some that have not. The effects on frequency splitting due to profile variations and the aspect ratio of the ellipse are emphasized. Results obtained using the developed numerical approach show excellent agreement with finite element predictions.
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