In this paper, primary resonances of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) are investigated. Unbalance forces due to eccentricity and disk skew are simultaneously considered, and the shaft has initially static deflection due to the rotor's weight which causes asymmetry in the equations of motion. The rotor has large amplitude vibrations, which lead to nonlinearities in curvature and inertia. In the model, rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The method of multiple scales is applied to the discretize differential equations of motion. It is shown that the static deflection creates second-order nonlinear terms and near the primary resonances only the forward modes are excited. With considering the gravity, the inertial nonlinearities become stronger near the primary resonances. So, gravity decreases the hardening effect, and the nonlinear system tends to a linear system. It is concluded that the gravity effect has a softening effect. By using this property of gravity, a relation between weight and external forces is derived, in which by applying this relation the jumping phenomenon is eliminated. Numerical examples are presented, and the result is verified by numerical simulations.

In this paper, the combination resonance of a spinning composite shaft with geometric nonlinearity is studied by the method of harmonic balance. The full equations of motion containing the flexural-flexural-extensional-torsional coupling are employed for the analysis. The equations were discretized by both one and two modes, so two different forms of combination resonances can be analyzed. The shear deformation is neglected due to the shaft slenderness, whereas the rotary inertia and the gyroscopic effects are considered. The effects of different parameters such as external damping and the eccentricity on the response bifurcation of the shaft are investigated. The effect of the fiber orientation angle was also investigated. The results obtained for the composite shaft are compared to those of its metallic counterpart. It is shown that two geometrically identical shafts, one metallic and one composite, have different behaviors under the condition of combination resonance. It is observed that the vibration of the composite shaft occurs with smaller amplitude. This confirms the superiority of the composite shafts over metallic ones. Finally, the results were validated by the numerical simulations and there was a good agreement between the results.

In this paper, nonlinear free vibration of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) is investigated. The Rayleigh beam model is used and the rotor has large amplitude vibrations. With the assumption of inextensibility, the effect of nonlinear curvature and inertia is considered. The effect of disk mass on the dynamical behavior of the system is studied in the presence and absence of gravity (horizontal and vertical rotors). By using perturbation technique (method of multiple scales), the main focus is on the influence of gravity on equations of motion and on quantities such as amplitude and damped natural frequency. Here, a different behavior is observed due to the rotor weight. Indeed, the combination effects of gyroscopic term, nonlinearity and gravity are studied on the modal behavior of the system. It is shown that the static deflection creates second order nonlinear terms and changes the nonlinear damped natural frequency. With considering of gravity, both beat and high frequency in beat phenomenon increase. With increasing of the rotor weight, the minimum value of amplitude is extremely amplified in the direction of gravity but in the other transverse direction, amplitude of vibrations decreases. In addition, it is found that the weight has directly influence on beat frequency, while the mass ratio between disk and beam affects the high frequency.