Abstract
Nonstationary vibrations at the major critical speed of a rotating shaft with nonlinear spring characteristics are discussced. Firstly, the first order approximate solutions of steady-state and nonstationary oscillations are obtained by the asymptotic method. The relations between these approximate solutions and the nonlinear components in the polar coordinate expression are investigated. It is clarified that, similar to the case of the stationary oscillations, only the isotropic nonlinear component has influence on nonstationary oscillations in the first order approximation. Secondly, the complex-FFT method where non-stationary time histories obtained by numerical integrations of the equations of motion are treated as complex numbers in the complex plane which coincides with the whirling plane are proposed. By this method, the amplitude variation curves of each vibration component are obtained. From the comparison of the amplitude variation curves of the first approximation of the asymptotic method, the solution of the complex-FFT method, and direct numerical integration, it is clarified that, although all these solutions coincide well in the case of stationary solutions, the first approximation of the asymptotic method has comparatively large quantitative error in the case of nonstationary solutions. In addition, the influences of the anisotropic nonlinear components which do not appear in the first approximation of the asymptotic method are investigated.