Simultaneous resonance and stability analysis of unbalanced asymmetric thin-walled composite shafts

Abstract

The combined effects of asymmetry and geometrical nonlinearity have a serious effect on the stability of composite shafts. These effects, that should be given special attention in the designing of these structure, are revealed in this study. The asymmetry of the shaft is modeled with a rectangular cross-section that makes the shaft transverse stiffness in one plane to differ from the other. This in effect, manifests itself as a parametric excitation of the system. Moreover, the analysis also considers the large lateral amplitude that can produce geometric nonlinearity and stretching of the system. Parametrically excited equations of the motion are obtained by considering the Euler's angles, the anisotropic properties of the constituent material, and employing the extended Hamilton's principle. Non-classical effects such as gyroscopic moment, rotary inertia, and nonlinear couplings due to stretching are also taken into account. However, because of the thin-walled features of the system, the shear deformations and warping effects are ignored. After discretizing the resulting equations, they are solved by employing the method of multiple scales (MMS). To justify/evaluate the proposed model's accuracy, several comparisons are made with various authentic results in the literature. In addition, to verify the results of the MMS solutions, numerical analyses are used based on the Runge-Kutta method. The effects of damping, eccentricity, and asymmetry were examined on the stability of the system. The illustrated results indicated the occurrence of hardening nonlinearity in the system. It was proved that when the rotational speed was in the neighborhood of forward frequency, despite the presence of nonlinear couplings in the equations, axial and torsional vibrations would not be excited. Furthermore, although the asymmetry affected the amplitude and phase of the system, it did not change the frequency of the vibrations and the type of bifurcation. With a fairly good balance, the unstable response of the asymmetric shaft could be easily eliminated, while even an accurate balance could not eliminate the unstable response of the asymmetric one.