Wave propagation, reflection and transmission in curved beams

Wave motion in thin, uniform, curved beams with constant curvature is considered. The beams are assumed to undergo only in-plane motion, which is described by the sixth-order coupled differential equations based on Flüggeʹs theory. In the wave domain the motion is associated with three independent wave modes. A systematic wave approach based on reflection, transmission and propagation of waves is presented for the analysis of structures containing curved beam elements. Displacement, internal force and propagation matrices are derived. These enable transformations to be made between the physical and wave domains and provide the foundation for systematic application of the wave approach to the analysis of waveguide structures with curved beam elements. The energy flow associated with waves in the curved beam is also discussed. It is seen that energy can be transported independently by the propagating waves and also by the interaction of a pair of positive and negative going wave components which are non-propagating, i.e. their wavenumbers are imaginary or complex. A further transformation can be made to power waves, which can transport energy independently. Numerical examples are given to illustrate the wave approach. The first concerns power transmission and reflection through a U-shaped connector between two straight beams while the second concerns the free vibration of finite curved beams where results are compared to other published results.