Free Vibration Analysis of Parabolic Arches in Cartesian Coordinates

The differential equations governing free vibrations of the elastic, parabolic arches with unsymmetric axes are derived in Cartesian coordinates rather than in polar coordinates. The formulation includes the effects of axial extension, shear deformation and rotatory inertia. Frequencies and mode shapes are computed numerically for arches with clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged ends. The convergent efficiency is highly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters are reported as functions of four non-dimensional system parameters: the rise to chord length ratio, the span length to chord length ratio, the slenderness ratio and the shear parameter. Typical mode shapes of vibrating arches are also presented.