Closed-form effects of warping and pretwist on the torsional vibration of thin-walled open-profile cantilevers

Abstract

Theoretical natural frequencies are determined for the first three modes of torsional vibration of pretwisted cantilevers with thin-walled open profiles. The equation of motion and boundary conditions are derived from a multifilament model of a pretwisted thin-walled bar undergoing warp deformation and a moderately large twist about the elastic axis. Closed-form torsional frequencies and mode shapes are derived from the linear terms of the resulting nonlinear equations. As the length-to-leg ratio ( L/b) and leg-to-thickness ratio ( b/t) of typical open-profile cantilevers are varied, the individual and collective effects of warping and pretwist on the torsional frequencies are qualitatively defined. The refined derivation and closed-form solutions presented here reveal significant destiffening of the torsional frequencies due to warping-pretwist coupling in cantilevers having pretwisted unsymmetrical open-profiles. These coupling effects, which have been ignored in thin-walled bar vibration theories heretofore, are brought out by the Wagner effect and warping shear stresses acting on the profile.