Rayleigh-Ritz Method for In-Plane Free Vibrations of Tapered Arches Having Variable Radius.

Abstract

In this paper, the in-plane free vibration of thin arches with variable radius as well as a varying cross-section is carried out with Rayleigh-Ritz method (RRM). The present work’s originality is the trial arc functions which are the particular solution of the sixth-order differential equation that governs the motion equation of vibrations of circular arches with constant cross-section. These trial functions are solved by a symbolic computation. The eigenvalues which are the frequency parameters and the eigenvectors which are the mode shapes, are determined by numerical computation. The investigation concerns three end conditions: clamped clamped, clamped simply supported and simply supported at both ends, also it concerns five arc geometries: parabolic, catenary, spiral, circular and cycloid with different opening angle, taper types and taper ratios. The convergence study is sensitive to the taper ratio. Moreover, its rate is faster than that about studies previously carried out by RRM. The frequency parameters are easily calculated and well compared to results available in the bibliography with high level of accuracy. The mode shapes are plotted. The tapered ratio significantly affects the mode shapes.