多自由度固有振动广义特征值问题的复分析证明——工科教育中的理论思维能力培养一例  

A PROOF FOR THE GENERALIZED EIGENVALUE PROBLEM OF MULTI-DEGREE-OF-FREEDOM VIBRATION SYSTEMS BASED ON COMPLEX ANALYSIS

作　　者：王怀磊[1,2] WANG Huailei(Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China)

机构地区：[1]南京航空航天大学振动工程研究所,南京210016 [2]南京航空航天大学机械结构力学及控制国家重点实验室,南京210016

出　　处：《力学与实践》2021年第3期439-441,共3页Mechanics in Engineering

基　　金：国家自然科学基金资助项目(11172126)。

摘　　要：在振动理论中,"线性系统固有振动的广义特征值问题仅具有非负实特征值"是一个基本的事实,然而现有教材对这一结论的证明一般都是基于矩阵分解理论,这对于绝大多数大学本科生而言属于超前的数学知识,因此会造成学习上的一定困难。本文针对该结论给出了一种基于复分析的较为初等的证明方法,该方法仅利用复数的基本概念和简单的矩阵代数运算而不涉及矩阵分解理论,从而既能保证理论体系的严密性,又降低了该问题的数学论证难度。It is a fundamental theorem in the theory of vibration that the inherent vibration of a linear vibration system has only non-negative real eigenvalues,and hence has real eigenvectors.This is usually referred to as the theory of the generalized eigenvalue problem.To the best knowledge of the author,most of the present textbooks on mechanical vibration theory give the proof of this problem based on the theory of the matrix factorization,which is often beyond the undergraduate students and makes it difficult for them to understand.On account of this situation,this paper presents an elementary proof for the generalized eigenvalue problem based on complex analysis with simple algebraic operations of matrix,without resorting to the theory of matrix factorization,but keeping the rigorousness and the integrity of the mathematical arguments.

关 键 词：振动理论 固有振动 矩阵广义特征值问题 复分析 矩阵分解 

分 类 号：O321[理学—一般力学与力学基础]