Analytical modeling of natural frequencies in non-uniform beams via elastokinetic reduction and segmentation

The determination of natural frequencies in non-uniform beams is critical for applications such as rotor blade design. While structural simulations yield satisfactory results, analytical solutions remain scarce, even for uniform beams, and are particularly limited for non-uniform cases. This study proposes novel analytical formulations based on elastiokinetics to determine the natural frequencies of non-uniform beams. By focusing on the average deformation angle rather than pointwise deflection, the governing high-order partial differential equation is transformed into a second-order ordinary differential equation in the time domain. The fundamental natural frequency for diverse non-uniform cantilever types is derived analytically to demonstrate the method. Subsequently, higher-order vibration frequencies for cantilevers are obtained through a segmentation strategy applied to the theoretical model. Finally, natural frequencies for beams with other boundary conditions are derived theoretically. The segmentation approach constitutes a systematic, physics-based method for addressing higher-order modes and varied constraints. Validation against prior studies and finite element method (FEM) simulations performed in this work confirms the accuracy of the theoretical solutions. These results provide a pathway to understand non-uniform beam vibrations, with potential applications enhancing the design and operation of beam-dependent systems across multiple fields.