Influence of warping-pretwist coupling on the torsional vibration of centrifugally-stressed cantilevers with thin-walled open-profiles

A nonlinear torsional vibration theory for multifilament models of centrifugally-stressed and pretwisted cantilevers with thin-walled open-profiles is developed from D'Alembert's principle. Warp deformation and a moderately large twist about the elastic axis are assumed in describing the dynamical torsional motion. The extensional strain is eliminated by invoking equilibrium of extensional stresses along the elastic axis. The refined derivation here reveals significant warping-pretwist coupling terms resulting from the Wagner effect and warping shear stresses, which have been ignored in thin-walled bar vibration theories hitherto. Approximate torsional frequencies are drawn from the linear terms of the resulting nonlinear boundary-value equations using a finite difference procedure with first- and second-order central differences. 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 compared to exact solutions. Finally, the relative importance of centrifugal stiffening and/or softening terms, warping-pretwist coupling terms, and inertial warping-tension coupling terms is pointed out.