Free in-plane vibrations of curved nonuniform beams

By introducing two physical parameters, the analysis of free in-plane vibrations of curved non-uniform beams is simplified. The explicit relations between the flexural displacement and the longitudinal displacement are derived. With these explicit relations, the two coupled governing differential equations are reduced to a complete sixth-order ordinary differential equation with variable coefficients in the longitudinal displacement, and a limiting study from the curved beam theory to the straight beam theory is successfully revealed. Finally, the influence of taper ratio, center angle and arc length on the first two natural frequencies of the beams is illustrated.