Nonlinear longitudinal–bending–twisting vibrations of extensible slowly rotating beam with tip mass

Abstract

Nonlinear longitudinal–bending–twisting vibrations of a slowly rotating extensible beam are studied in the paper. The beam model is based on extended Euler–Bernoulli theory with shear deformation neglected. Geometrical nonlinearities, caused by large oscillations, couple longitudinal vibrations, bending in two directions and twist. The beam is fixed to a rigid hub at a given preset angle and carries mass attached at the beam’s tip. The governing partial differential equations and dynamical boundary conditions are derived from the extended Hamilton principle. The model is solved analytically up to the third order perturbation by the multiple time scale method applied directly to the partial differential equations. The influence of angular velocity of the hub, tip mass and other structural beam parameters on nonlinear vibrations around selected resonance zones is presented.