Nonlinear Torsional Vibrations of Axially Loaded Pretwisted Beam Under Primary Resonance Excitations

Thin Walled Beams are widely used in engineering applications with the minimum weight design criteria. Frequently used thin walled beams have low torsional stiffness and their torsional deformations may be of such magnitudes that it is not adequate to treat the angles of cross section rotation as small. In this paper, nonlinear torsional vibrations of thin walled beams will be investigated. The method of multiple scales will be implemented as solution method and different nonlinear phenomena will be studied. The obtained results are compared with the available results in the literature which reveals an excellent agreement between different solution methodologies. The outcomes of this study show that beam nonlinear torsional dynamics and the related phenomena could influence the linear torsional dynamic of beams under axial load, e.g. rotating beams. Forced torsional vibrations of beam with the excitation in the form of primary resonance of first and second mode have been investigated. It has been demonstrated that in the case of beam with two ends clamped boundary conditions, three-to-one internal resonance will be appeared. The primary resonance of first and second mode have been solved in two sets of boundary conditions, torsionally clamped-fixed and torsionally fixed-fixed. Nonlinear response, amplitude-phase equations, fixed points and their stability have been studied.