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In this paper a critical brief review of the free dynamics of circular arches is presented, under the classical hypotheses that (1) the dimensions of the cross-section are small in comparison with the radius, (2) the axis is inextensible and (3) shear forces and rotatory inertia are neglected. A simple—yet efficient—method is proposed, which allows one to take into account quite naturally the cross-section variations and the presence of flexible supports. A number of other approaches are also briefly illustrated, from the Ritz and Galerkin methods to the finite element techniques, and their numerical performances are compared for a wide range of examples. Uniform arches with rotationally flexible and axially flexible supports have been extensively examined, together with three different kinds of stepped arches and two different arches with linearly varying cross-section. Whenever possible, a comparison with available exact solutions is also performed. The agreement among all these methods is generally good, even if there are some theoretical discrepancies to be pointed out. Copyright 1994, 1999 Academic Press

The free vibration frequencies of a beam composed of two tapered beam sections with different physical characteristics with a mass at its end can be determined by using either the exact procedure, for which purpose the solution to the problem can be expressed using Bessel functions, or the approximate Rayleigh–Ritz treatment, with the assumption of orthogonal polynomials as test functions. The results and the numerical comparison between the two methods are provided in diagrams and tables. The effect of different materials on vibrations modes is also provided.

An exact analysis of free vibrations of cantilever tapered beams with a mass at the tip and flexible constraint is presented. The rotary inertia of the concentrated mass is considered, with its eccentricity. The non-dimensional frequency coefficients are given in tabular form at the end of the paper. Some comparisons with other results from the literature are presented.