Applied Acoustics 21 241-246 In-plane Vibrations of an Arch of Variable Cross Section Elastically Restrained Against Rotation at One End and Carrying a Concentrated Mass at the Other C. P. Filipich Universidad Nacional del Sur and Universidad Tecnologica Nacional, 8000 Bahia Blanca (Argentina) P. A. A. Laura Institute of Applied Mechanics, 8111 Puerto Belgrano Naval Base (Argentina) and V. H. Cortinez Department of Engineering, Universidad Nacional dei Sur, 8000 Bahia Blanca (Argentina) (Received 22 April 1986; accepted 9 June 1986) S UMMA R Y This study dea& with the determ&ation of upper and lower bounds for the fundamental frequency of the structural system described in the title. The upper bound is determined by approximating the fundamental mode shape by means of a polynomial coordinate function in the angular coordinate, 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. INTRODUCTION Consider the structural system shown in Fig. 1, in which the cross-sectional area and the moment of inertia are given by the following functional 241 Applied Acoustics 0003-682X/87/$03.50 © ElsevierApplied SciencePublishers Ltd, England, 1987. Printed in Great Britain