Natural Frequencies of Layered Beams Using a Continuous Variation Model

This work involves the determination of the bending natural frequencies of beams whose properties vary along the length. Of interest are beams with different materials and varying cross-sections, which are layered in cells. These can be uniform or not, leading to a configuration of stacked cells of distinct materials and size. Here the focus is on cases with two, or three cells, and shape variations that include smooth (tapering) and sudden (block type) change in cross-sectional area. Euler-Bernoulli theory is employed. The variations are modeled using approximations to unit step functions, here logistic functions. The approach leads to a single differential equation with variable coefficients. A forced motion strategy is employed in which resonances are monitored to determine the natural frequencies. Forcing frequencies are changed until large motions and sign changes are observed. Solutions are obtained using MAPLE®’s differential equation solvers. The overall strategy avoids the cumbersome and lengthy Transfer Matrix method. Pin-pin and clamp-clamp boundary conditions are treated. Accuracy is partially assessed using a Rayleigh-Ritz method and, for completeness, FEM. Results indicate that the forced motion approach works well for a two-cell beam, three-cell beam and a beam with a sinusoidal profile. For example, in the case of a uniform two-cell beam, with pin-pin boundary conditions, results differ less than 1 %.