用3个向量对构造梁振动系统的刚度矩阵

周硕; 吕晓寰; 王小雪

doi:10.3879/j.issn.1000-0887.2015.03.008

用3个向量对构造梁振动系统的刚度矩阵

doi: 10.3879/j.issn.1000-0887.2015.03.008

东北电力大学理学院, 吉林吉林 132012

基金项目: 国家自然科学基金（11072085）；吉林省自然科学基金（201115180）

详细信息

作者简介:
周硕（1968—），男，吉林人，教授，博士，硕士生导师(通讯作者. E-mail: zhou-shuo@163.com).

中图分类号: TH123^{+.1^{; O241.6}}
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出版历程
- 收稿日期: 2014-07-09
- 修回日期: 2014-12-14
- 刊出日期: 2015-03-15

On the Construction of Stiffness Matrices With 3 Vector Pairs for Beam Vibration Systems

College of Science, Northeast Dianli University, Jilin, Jilin 132012, P.R.China

Funds: The National Natural Science Foundation of China（11072085）

摘要

摘要: 针对梁的离散化模型的刚度矩阵是五对角矩阵,梁振动反问题的实质是实对称五对角矩阵的特征值反问题.该文利用向量对、Moore-Penrose广义逆给出了实对称五对角矩阵向量对反问题存在唯一解的条件，并结合矩阵分块讨论了双对称五对角矩阵向量对反问题解存在唯一的条件，进而计算了次对角线位置元素为负，其它位置元素均为正的实对称五对角矩阵特征值反问题.由于构造梁的离散模型需要的数据可由测试得到，故而其结果适合于模态分析、系统结构的分析与设计等方面应用.最后给出了数值算例，通过数值讨论说明方法的有效性.
- 梁振动系统 /
- 双对称五对角矩阵 /
- 向量对 /
- 反问题 /
- 刚度矩阵
Abstract: The stiffness matrix of the discrete vibrating beam model is a real symmetric 5-diagonal matrix, so the inverse problem of the vibrating beam is substantially an inverse eigenvalue problem of the real symmetric 5-diagonal matrix. The existence and uniqueness conditions for the solution to the inverse problem of the real symmetric 5-diagonal matrix vector pair were given by means of the vector pairs and the Moore-Penrose generalized inverse, and the existence and uniqueness conditions for the solution to the inverse problem of the bi-symmetric 5-diagonal matrix vector pair were discussed in combination with the partitioned matrices. Then the inverse eigenvalue problem of the real symmetric 5-diagonal matrix, of which the sub-diagonal elements were negative and the rest elements were positive, was calculated. Since the data required for the construction of discrete beam models are available through measurements, the presented method is well suited to modal analysis, or analysis and design of system structures. Furthermore, the related numerical algorithms and numerical experiments illustrate the effectiveness of the method.
- beam vibration system /
- bi-symmetric 5-diagonal matrix /
- vector pair /
- inverse problem /
- stiffness matrix

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