Exact Solutions for Out-of-Plane Vibration of Curved Nonuniform Beams

The governing differential equations for the out-of-plane vibrations of curved nonuniform beams of constant radius are derived. Two physical parameters are introduced to simplify the analysis, and the explicit relations between the torsional displacement, its derivative and the flexural displacement are derived. With these explicit relations, the two coupled governing characteristic differential equations can be decoupled and reduced to one sixth-order ordinary differential equation with variable coefficients in the out-of-plane flexural displacement. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the out-of-plane vibrations of the beam can be obtained. The derived explicit relations can also be used to reduce the difficulty in experimental measurement. Finally, two limiting cases are considered and the influence of taper ratio, center angle, and arc length on the first two natural frequencies of the beams are illustrated.