Approximate fundamental frequency formula for cantilevers with weakly non-uniform sections and verification with experiments and other studies in the literature

The vibrational frequencies of beams with uniform cross-sections are easily derived from the analytical solutions of the Euler-Bernoulli equation (EBE). However, for beams with non-uniform cross-sections, complex and time-consuming numerical solutions are typically required for each cross-sectional geometry. This paper presents a formula that accurately predicts the vibrational frequencies in the presence of weak heterogeneities, regardless of how the cross-sectional shape varies along the beam's length. Experimental verification and comparisons with literature confirm that the formula reliably predicts frequency shifts in cases where a thin heterogeneous film is coated on a uniform beam, thin heterogeneous layers are removed from a uniform beam, and when a moving point load affects the vibrational frequency. The average absolute percentage error between the experimentally measured and calculated frequencies was <0.17 %. As such, this formula serves as a practical tool for various engineering applications involving weakly heterogeneous beams.