Transverse-vibration responses of nonuniform, internally damped cantilever beams. II

Abstract

The Bernoulli–Euler theory of transverse beam vibration, suitably extended to take into account internal beam damping, is used to derive closed-form expressions for the mechanical driving-point impedance and force transmissibility of two types of nonuniform cantilever beams that are driven at their free ends by a sinusoidally varying point force. The two types of beams considered are ones: (1) truncated and of rectangular cross section with a linearly tapered depth and a breadth appropriately varied (hyperbolically) to maintain constant cross-sectional area, and (2) composed of three stages, each of which is uniform but may vary arbitrarily from the others in cross section and proportionate length. Representative computations of the frequency dependence of impedance and transmissibility are plotted for beams having the same length and mass as an equally long and equally massive uniform reference beam. Significant attenuation or amplification of force transmissibility is observed, depending on the proportions of the beams and on whether their depth diminishes towards their fixed or free end. The measured resonance frequencies of a set of nonuniform aluminum alloy (6061-T651) cantilever beams and the measured transmissibility of one other such beam agree closely with the predictions of theory.