In-Plane Free Vibrations Analysis of Tapered Arches with Variable Radius Using the Rayleigh-Ritz Method

Abstract

This paper investigates the in-plane free vibrations of thin arches with variable radius and varying cross-sections employing the Rayleigh-Ritz method (RRM). A novel aspect of this study lies in the trial arc functions, which are solutions of the sixth-order differential equation governing the vibration motion of circular arches with constant cross-section. These trial functions are obtained through symbolic computation. Numerical computation is then utilized to determine the eigenvalues, representing frequency parameters, and eigenvectors, representing mode shapes. The investigation considers three different end conditions (clamped-clamped, clamped-simply supported, and supported at both ends) and five arc geometries (parabolic, catenary, spiral, circular, and cycloid) with varying opening angles, taper types, and taper ratios. The convergence study highlights the sensitivity of the results to the taper ratio, with convergence rates faster than those observed in previous RRM studies. Frequency parameters are accurately calculated and compared favourably with existing literature. Additionally, mode shapes are plotted, demonstrating the significant influence of the taper ratio on mode shapes.