Fundamental natural frequencies of thin-walled elliptical composite cylinders

Abstract

Hamilton’s principle coupled with the Rayleigh–Ritz technique is used to compute the fundamental frequencies of simply supported thin-walled fiber-reinforced composite cylinders with elliptical cross sections. Owing to the decreased geometric stiffness resulting from less curvature, it is expected that the normal displacement component of the vibratory motion will be larger in the flatter regions of the cross section than that in the more curved regions. Accordingly, in the Rayleigh–Ritz formulation, the normal displacement component of the vibratory motion is modulated with circumferential location to represent this characteristic by using a so-called shape factor. A number of simplifications in the analysis lead to a hierarchy of expressions for the fundamental frequency, including the one termed Lo’s approximation. The so-called large and small cylinders, as measured by cylinder circumference and with wall laminates [±θ/0/90]2S and[±θ/0/90]S, respectively, θ in the range of 0 to 90°, are considered. It is demonstrated that the comparisons with finite element calculations are good, particularly for Lo’s approximation. Then, parameter studies using Lo’s approximation are conducted to illustrate the dependence of the fundamental frequency on fiber angle θ, cross-sectional geometry, cylinder circumference, and cylinder length. It is shown that for cylinders of the same circumference, an elliptical cylinder has a lower fundamental frequency than a circular one and that difference is quantified. However, the dependence of the fundamental frequency on other geometric parameters and fiber angle is much the same for cylinders with elliptical cross sections as for circular cylinders.