An efficient method for solving vibration equations is the basis of the vibration analysis of a cracked multistage blade–disk–shaft system. However, dynamic equations are usually time varying and nonlinear, and the time required for solving is greatly increased accompanied by an increase in the model order caused by the multistage system and the nonlinearity caused by cracks. In this article, an efficient method for solving the time varying and nonlinear vibration equation is investigated. In the proposed method, the time varying terms are transformed into constant terms, while the local nonlinear matrix of the cracked blade is separated from the assembly stiffness matrix under the constraint of proper orthogonal decomposition (POD) transformation rules. Furthermore, the POD transformations of the constant terms and the linear assembled stiffness matrix can be implemented in the pretreatment steps to achieve a more efficient POD reduction operation. This research provides a method for efficiently performing the comprehensive and rapid analysis of nonlinear vibration characteristics of rotor systems.

求解振动方程的有效方法是裂纹多级叶片-盘-轴系统振动分析的基础。然而，动力方程通常是时变和非线性的，求解所需的时间大大增加，伴随着多级系统引起的模型阶数增加和裂缝引起的非线性。本文研究了求解时变非线性振动方程的一种有效方法。该方法将时变项转化为常数项，同时在适当的正交分解（POD）转换规则的约束下，将裂纹叶片的局部非线性矩阵与装配刚度矩阵分离。此外，常数项的 POD 变换和线性组装刚度矩阵可以在预处理步骤中实现，以实现更有效的 POD 减少操作。该研究为高效全面、快速地分析转子系统的非线性振动特性提供了一种方法。