Vibration analysis of pretwisted beams for the design of hybrid axial-torsional transducers

A new design for hybrid axial-torsional transducers using pretwisted beams requires the resonance frequencies of the torsional and axial vibration modes to be matched. To aid in designing such transducers, the effects of increasing pretwist and changing the cross-section geometry on the resonance frequencies are investigated analytically. The governing equations and boundary conditions for extension, torsion, and cross-sectional warping are derived using the semi-inverse method and Hamilton's principle. A general set of differential equations for the cross-sectional warping of pretwisted beams is derived. Through scaling, the warping function is shown to be locally similar to the Saint-Venant warping function when the beam is slender, low in pretwist, and torsional deformation is dominant. Using this approach, geometric and material limitations in the use of the Saint-Venant's warping function are illustrated, beyond which the simpler form may no longer be used. The simplified equations of motion are solved under the free-free boundary condition for resonance frequencies and mode shapes, and a comparison with finite element analysis illustrates the limitations