The Efficiency of the Rayleigh-Ritz Method Applied to In-Plane Vibrations of Circular Arches Elastically Restrained at the Two Ends and Supporting Point Masses

Abstract

The natural frequencies and mode shapes of in-plane vibration of circular arches elastically restrained against rotation at their ends are determined using the Rayleigh-Ritz method (RRM) and trial functions obtained as particular solutions of the sixth order differential equation of arch vibrations corresponding to an opening angle equal to 1 rad. The investigations are made under the classical hypotheses: the effect of shear deformation and rotary inertia are neglected, the arch axis is inextensible, and the dimensions of the cross-section are constant and small in comparison with the radius. The first eight mode shapes and natural frequencies of arches with different opening angles and torsional spring stiffnesses are determined and shown to compare well with the available literature. Arches with added concentrated masses are then examined. The effect of the rotational stiffness and the added masses on the natural frequencies and mode shapes are determined and illustrated in the joint plots. The accuracy and relative simplicity of the RRM applied in a systematic way to such complicated problems is established, making it ready to use in more complex situations, such as those of arches with more added masses, with non-uniform cross section or with one or more point supports.