Curved beams with complex geometries are vital in numerous engineering applications, where precise vibration analysis is crucial for ensuring safe and effective designs. Traditional finite element methods (FEMs) often struggle to accurately represent the dynamic characteristics of these structures due to the limitations in their shape function approximations. To overcome this challenge, the current study introduces an innovative finite element (FE)-based technique for the undamped vibrational analysis of curved beams with arbitrary curvature, employing explicitly derived interpolation functions. Initially, the exact interpolation functions are developed for circular arc elements with the force method. These functions facilitate the creation of a highly accurate stiffness matrix, which is validated against the benchmark examples. To accommodate arbitrary curvature, a systematic transformation technique is established to approximate the intricate curves with a series of circular arcs. The numerical findings indicate that increasing the number of arc segments enhances accuracy, approaching the exact solutions. The analysis of free vibrations is conducted for both circular and non-circular beams. Mass matrices are derived using two methods: lumped mass and consistent mass, where the latter is based on the interpolation functions. The effectiveness of the proposed method is confirmed through the comparisons with the existing literature, demonstrating strong agreement. Finally, several practical cases involving beams with diverse curvature profiles are analyzed. Both natural frequencies and mode shapes are determined, providing significant insights into the dynamic behavior of these structures. This research offers a dependable and efficient analytical framework for the vibrational analysis of complex curved beams, with promising implications for structural and mechanical engineering.

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