Exact out-of-plane natural frequencies of curved Timoshenko beams

A powerful and efficient method is presented for finding exact out-of-plane natural frequencies of plane structures composed of curved Timoshenko beams. Initially, exact dynamic stiffnesses are derived from the governing differential equations of motion in a form that can be used directly in the stiffness method of analysis. This enables any appropriate structure to be modeled according to standard techniques, which, in this case, yield a transcendental eigenvalue problem. Then it is shown how any desired natural frequency may be obtained with certainty by employing a modification to a well-established algorithm, which ensures that no natural frequencies can be missed and avoids the usual approximations associated with traditional finite elements. Finally, comparisons are made with published results and an example shows how the natural frequencies of a continuous curved beam are altered when the effects of shear deflection and rotary inertia are considered.