Dynamic modal characteristics of transverse vibrations of cantilevers of parabolic thickness

This paper deals with free transverse vibrations of nonuniform homogeneous beams. Cantilevers of rectangular (or elliptical) cross-section with parabolic thickness variation, and cantilevers of circular cross-section with parabolic radius variation, are considered. Factoring their fourth order differential equations of transverse vibrations into a pair of second order differential equations leads to general solutions in terms of hypergeometric functions. Exact natural frequencies and exact mode shapes are reported for sharp parabolic cantilevers of various dimensionless lengths.