In-plane vibrations of an elastically cantilevered circular arc with a tip mass

Abstract Upper and lower bounds are determined for the fundamental frequency of in-plane, transverse vibration of the structural system described in the title in the case of constant cross-section and moment of inertia. The upper bound is determined by approximating the fundamental mode shape with a polynomial co-ordinate function in the angular co-ordinate which includes an exponential optimization parameter. The fundamental frequency equation is generated by means of the Rayleigh-Ritz method and the resulting upper bound is minimized with respect to the previously mentioned exponential parameter. The lower bound for the frequency coefficient is obtained by means of an extension of Dunkerley's method. It is felt that the methodologies developed in the present study are especially useful in the case of variable cross-section of the arch structure, presence of internal supports, etc.