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.

中文翻译：

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重力作用下悬臂转子的非线性自由振动分析

在本文中，研究了在其自由端（悬臂转子）带有刚性盘的悬臂软轴的非线性自由振动。采用瑞利梁模型，转子有大振幅振动。在不可扩展的假设下，考虑了非线性曲率和惯性的影响。在存在和不存在重力（水平和垂直转子）的情况下，研究了盘质量对系统动力学行为的影响。通过使用微扰技术（多尺度方法），主要关注重力对运动方程的影响以及对振幅和阻尼固有频率等量的影响。在这里，由于转子重量而观察到不同的行为。实际上，研究了陀螺项、非线性和重力的组合效应对系统模态行为的影响。结果表明，静态偏转会产生二阶非线性项并改变非线性阻尼固有频率。考虑到重力，节拍现象中的节拍和高频都增加了。随着转子重量的增加，振幅的最小值在重力方向上极大地放大，而在另一个横向上，振幅减小。此外，发现重量直接影响拍频，而盘和梁之间的质量比影响高频。振幅的最小值在重力方向上被极大地放大，但在另一个横向上，振动的振幅减小。此外，发现重量直接影响拍频，而盘和梁之间的质量比影响高频。振幅的最小值在重力方向上被极大地放大，但在另一个横向上，振动的振幅减小。此外，发现重量直接影响拍频，而盘和梁之间的质量比影响高频。