This paper strives to derive the analytical solutions for the dynamic analysis of stepped multi-span rotor system, where the rotating shaft's internal damping effect is also considered. Specifically, the steady-state whirl analysis and free vibration analysis is given herein. For the steady-state whirl analysis, firstly, the stepped shaft is spilt to several uniform segments and their governing equations are established by Hamilton principle. Subsequently, by applying variable separation method and Laplace transform method to each segment's governing equation, their steady-state responses in terms of determinate interpolation functions multiplied by unknown boundary constants are derived. Then based on the transfer matrix method, each segment's boundary constants are determined with consideration of the compatibility conditions of each two adjacent segments and the end boundary conditions. For the free vibration analysis, one only need to remove the forced term in the steady-state form of transfer matrix and the rotor's characteristic equation will be obtained. Through the theoretical derivation, it is found that this analytical approach can be generalized to any isotropic elastic boundary conditions. To validate the proposed method, two case studies are proposed, where the finite element method is used as benchmark method. For the first one, the influence of internal damping on a uniform rotor's damped whirl characteristics and its stability is analyzed. The damped whirl characteristics analysis reveals that viscous internal damping result in the destabilization of forward modes as long as the spinning speed becomes higher than the critical speed. This phenomenon is also demonstrated by the time responses analysis. For the second one, the dynamic responses of a two-span three steps rotor-bearing system with isotropic viscoelastic boundary conditions are determined. All of simulated results show a great agreement between analytical results and FEM results.

本文力求推导出阶梯式多跨转子系统动力学分析的解析解，其中还考虑了旋转轴的内部阻尼效应。具体地，本文给出了稳态涡旋分析和自由振动分析。对于稳态涡旋分析，首先将阶梯轴溢出到几个均匀段，并根据哈密顿原理建立它们的控制方程。随后，通过将变量分离法和拉普拉斯变换法应用于各段的控制方程，推导出它们在确定插值函数乘以未知边界常数方面的稳态响应。然后基于传递矩阵的方法，每段' s 边界常数的确定考虑了每两个相邻段的兼容性条件和末端边界条件。对于自由振动分析，只需去掉传递矩阵稳态形式中的受迫项，即可得到转子的特征方程。通过理论推导，发现该解析方法可以推广到任何各向同性弹性边界条件。为了验证所提出的方法，提出了两个案例研究，其中有限元方法被用作基准方法。第一个分析了内部阻尼对均匀转子阻尼涡动特性及其稳定性的影响。阻尼涡流特性分析表明，只要旋转速度高于临界速度，粘性内部阻尼就会导致前向模式的不稳定。时间响应分析也证明了这种现象。对于第二个，确定了具有各向同性粘弹性边界条件的两跨三步转子轴承系统的动态响应。所有的模拟结果都表明分析结果和有限元结果之间有很大的一致性。