This work presents group-theory-based vibration analysis to determine the highly structured, symmetry-related modal properties of symmetric systems and classify them into specific types. No eigenvalue problem solutions are required. In fact, the properties are obtained without a mathematical model. They derive only from symmetry and the degrees of freedom chosen to describe the deformation. Dynamic or static response is similarly decomposed into the sum of multiple response components associated with the mode types. One knows a priori, without any numerical computation, which mode types can or can not be excited by a given input excitation. Group theory divides the full system equations of motion and eigenvalue problem into multiple, smaller, decoupled problems associated with each mode type, which is computationally efficient. The method applies for general three-dimensional systems having any type of symmetry where components of the system are modeled using any mixture of lumped-parameter (including finite element) and elastic continuum models. Only basic elements of group theory are required, and they are introduced. Among three examples of the method, one derives the modal properties for general cyclically symmetric systems having any number of central components and substructures that can be modeled as any of rigid bodies, finite element meshes, and elastic continua.

这项工作提出了基于组理论的振动分析，以确定对称系统的高度结构化，与对称相关的模态性质，并将其分类为特定类型。不需要特征值问题解决方案。实际上，无需数学模型即可获得特性。它们仅从对称性和选择用来描述变形的自由度中得出。动态或静态响应类似地分解为与模式类型关联的多个响应组件之和。无需任何数值计算，就可以先验地知道哪种模式类型可以通过给定的输入激励来激励或不能被激励。群组理论将运动和特征值问题的完整系统方程式分为与每个模式类型相关的多个，较小的，解耦的问题，这在计算上是有效的。该方法适用于具有任何对称类型的常规三维系统，其中使用集总参数（包括有限元）和弹性连续体模型的任何混合对系统的组件进行建模。只需要组理论的基本要素，并对其进行介绍。在该方法的三个示例中，可以得出具有任意数量的中心组件和子结构的一般循环对称系统的模态特性，这些中心组件和子结构可以建模为刚性体，有限元网格和弹性连续体中的任何一个。