One of the main causes for failure in the rotor machine is the presence of internal cracks in the shaft rotating at high speed. The occurrence of such cracks has a dangerous and catastrophic effect on the dynamic behavior of rotating structures and can seriously affect rotating machinery. Due to the destructiveness of the cracks in rotor system, cracks were chosen as the subject of study, and the how to formulate the dynamics equation of the rotor system with and without cracks was explained. In that, the finite element analysis method was combined with the Lagrange method to concretely interpret the equation formulation. Then, the natural frequency matrix and mode of the rotor system without crack and with one crack were determined by using the ANSYS program's tetrahedral mesh and iteration strategies of Runge–Kutta. Furthermore, how the change of the crack position and the non-dimensional crack depth influence the natural frequencies of the rotor system were analyzed in detail. And, the influence of eccentricity on the rotor system and the relationship between eccentricity and nonlinearity were obtained. The result showed that when the eccentric distance and the eccentric mass were relatively small, i.e. 0.05 m, 0.001 kg, the effect on the eccentric force is almost linear, but when the eccentric distance and the eccentric mass were relatively large, i.e. 0.1 m, 0.05 kg, the nonlinear effect of eccentric forces must be considered.

转子电机故障的主要原因之一是高速旋转的轴中存在内部裂纹。此类裂纹的出现对旋转结构的动力行为具有危险和灾难性的影响，并可能严重影响旋转机械。由于转子系统中裂纹的破坏性，选择裂纹作为研究对象，并解释了如何建立有裂纹和无裂纹的转子系统动力学方程。在这种情况下，有限元分析方法与拉格朗日方法相结合来具体解释方程式。然后，通过使用ANSYS程序的四面体网格和Runge–Kutta的迭代策略，确定了无裂纹和有一个裂纹的转子系统的固有频率矩阵和模式。此外，详细分析了裂纹位置和无量纲裂纹深度的变化如何影响转子系统的固有频率。并且，获得了偏心距对转子系统的影响以及偏心距与非线性之间的关系。结果表明，当偏心距和偏心质量相对较小时，即0.05 m，0.001 kg，对偏心力的影响几乎是线性的，而当偏心距和偏心质量相对较大时，即0.1 m， 0.05 kg时，必须考虑偏心力的非线性影响。得到了偏心距对转子系统的影响以及偏心距与非线性之间的关系。结果表明，当偏心距和偏心质量相对较小时，即0.05 m，0.001 kg，对偏心力的影响几乎是线性的，而当偏心距和偏心质量相对较大时，即0.1 m， 0.05 kg时，必须考虑偏心力的非线性影响。得到了偏心距对转子系统的影响以及偏心距与非线性之间的关系。结果表明，当偏心距和偏心质量相对较小时，即0.05 m，0.001 kg，对偏心力的影响几乎是线性的，而当偏心距和偏心质量相对较大时，即0.1 m， 0.05 kg时，必须考虑偏心力的非线性影响。