The axial–torsional vibration of pretwisted beams

We investigate the coupled axial–torsional vibration of pretwisted beams. The equations of motion governing the extension, torsion, and cross-sectional warping of pretwisted beams are derived from Hamiltons principle, and the common assumptions used to simplify the equations are carefully examined through scaling analysis. Inconsistencies in previous works—such as the neglect of spatial and time derivatives of torsion in the Lagrangian functional—are remedied, giving rise to fourth-order terms in the equations of motion that are significant for higher harmonics. Furthermore, scaling analysis of the governing equations yields a set of objective criteria for checking the validity of the frequently misused assumption that the warping function under pretwist is locally identical to that of a prismatic beam; we show that this simplification is responsible for the common poor prediction of axial resonance frequencies for pretwisted beams.