Exact solution for the transverse vibration of a beam a part of which is a taper beam and other part is a uniform beam

In this paper, the free-vibration frequencies of tapered beams in the most general possible boundary conditions, are determined by means of direct method. The study is extended to beams, made up of two sections with different cross-sectional variations: the equations of motion can be solved by means of the well-known Bessel function. The natural frequencies were determined using the false position method and symbolic program as roots of the corresponding characteristic equations. Finally an extensive number of structural system previously studied both exactly and approximately by other authors, mentioned in the bibliography, confirms the exactness of the results obtained. (C) 1997 Elsevier Science Ltd.