In-plane free vibrations of curved timoshenko beams with the rayleigh-ritz method

Curved beams are frequently used structural elements in traditional and emerging fields of civil and mechanical engineering with usual requirements of vibration and deformation analyses with the beam theory and numerical methods. There are challenges in such analyses due to complex equations with the curved beams and numerical methods in case analytical solutions are not available. In this study, curved beams with commonly encountered arc types are studied with the Rayleigh-Ritz method used with the polynomial functions as the deformation for the calculations of strain and kinetic energies with the Timoshenko beam theory, and accurate frequencies and mode shapes are obtained from convergent and verified solutions. It is the objective of this study that the method and procedure will be extended to a short, curved, and periodic beam for its free vibrations for in-depth understanding of such unusual but widely encountered structures from recent 3D-printing technology. It is found that for the sinusoidal-shaped beams, we need the deformation in sine series up to the 25 th -order, resulting a large size matrix equation for the eigenvalue extraction. Clearly, it is worth to try other types of deformation functions to improve the computing efficiency for curved beams. Eventually, the study will be extended to composite structures of periodic beams with shorter unit cells from the 3D-printing technology showing up in broad applications today. Furthermore, additional considerations of higher-order theories and couplings of modes can also be included for more practical and robust analyses.