In-plane vibrational analysis of rotating curved beam with elastically restrained root

This study analyzes the in-plane free vibration of a rotating curved beam with an elastically restrained root. Neglecting the effects of shear deformation and the Coriolis force, governing differential equations are derived for the coupled bending–extensional vibration of the curved beam using Hamilton's principle and a consistent linearization approach. Explicit relations are constructed to describe the correlation between the axial and radial displacements of the beam. These relations are then used to transform the coupled governing differential equations into a sixth-order ordinary differential equation expressed in terms of the radial displacement variable only. An exact closed-form fundamental solution of the transformed system is then derived. Finally, the respective effects of the arc angle, the rotational speed, the hub radius and the root spring constants on the natural frequencies and divergent instability characteristics of a curved rotating beam are systematically examined and compared with those observed for a straight cantilever beam.