Free in-plane vibration of curved beam structures: A tutorial and the state of the art

The study of free in-plane vibration of curved beams, using different beam theories, is more challenging than that of straight beams, since the structural deformations in curved beams depend not only on the rotation and radial displacements, but also on the coupled tangential displacement caused by the curvature of structures. A critical review of the publications on the free in-plane vibration of curved beams to demonstrate the state of the art has been presented. The governing differential equations of motion for the curved beams, based on different hypotheses (including and excluding the axial extensity, rotary inertia and the shear deformation), were discussed and different approaches to solve the developed equations of motion have been identified. Finally, a systematic comparison of the dynamic properties of curved beams evaluated with various forms of curvatures based on different hypotheses were presented.