Novel Empirical Relations for Accurately Estimating the Eigenfrequencies of Cantilever Beams With Linear Width Variation

Dynamic analysis of variable cross-section beams has been the focus of numerous investigations because of its relevance to aeronautical, civil, and mechanical engineering. In this article, we analyze the case of isotropic Euler-Bernoulli cantilever beams having linearly varying width, constant thickness, and classical boundary conditions. The linear width variation is characterized by a taper parameter, which can be varied between zero and unity. The free transverse vibration problem is cast as a fourth order Sturm-Liouville eigenvalue problem, and numerically solved by using the differential transform technique. A five-parameter exponential fitting model is used to develop novel empirical relations to estimate the first five eigenfrequencies as functions of the taper parameter. The proposed empirical relations are able to predict the eigenfrequencies with an error of less than 0.1% with respect to the simulated values, which make them useful for practical design applications. Using the proposed empirical relations, we then examine the sensitivity of each eigenfrequency to the variation in taper parameter near zero and near unity.