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}}
计量
- 文章访问数: 1182
- HTML全文浏览量: 101
- PDF下载量: 810
- 被引次数: 0
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

HTML全文

参考文献(17)

[1]	Gladwall G M L.Inverse Problems in Vibration [M]. Martinus Nijhoff Publishers, 1986.
[2]	田霞, 戴华. 梁的离散模型的模态反问题[J]. 振动与冲击, 2005,24(6): 29-31, 135.(TIAN Xia, DAI Hua. Inverse mode problems for the discrete model of beam[J].Journal of Vibration and Shock,2005,24(6): 29-31, 135.(in Chinese))
[3]	周硕, 王霖, 韩明花. 约束矩阵方程的中心对称解及其在振动理论反问题中的应用[J]. 应用数学和力学, 2013,34(3): 306-317.(ZHOU Shuo, WANG Lin, HAN Ming-hua. Centrosymmetric solutions of constrained matrix equation and its application to inverse problem of vibration theory[J].Applied Mathematics and Mechanics,2013,34(3): 306-317.(in Chinese))
[4]	常晓通, 闫云聚, 刘鎏. Landweber迭代正则化方法在动态载荷识别中的应用[J]. 应用数学和力学, 2013,34(9): 948-955.(CHANG Xiao-tong, YAN Yun-ju, LIU Liu. Applications of Landweber interation regularization method in dynamic load identification[J].Applied Mathematics and Mechanics,2013,34(9): 948-955.(in Chinese))
[5]	周硕, 韩明花, 孟欢欢. 用试验数据修正振动系统的双对称阻尼矩阵与刚度矩阵[J]. 应用数学和力学, 2014,35(6): 697-711.(ZHOU Shuo, HAN Ming-hua, MENG Huan-huan. Bisymmetric damping and stiffness matrices calibration with test data of vibration systems[J].Applied Mathematics and Mechanics,2014,35(6): 697-711.(in Chinese))
[6]	戴华. Jacobi矩阵和对称三对角矩阵特征值反问题[J]. 高等学校计算数学学报, 1990,12(1): 1-13.(DAI Hua. Inverse eigenvalue problems for Jacobian and symmetric tridiagonal matrices[J].Numerical Mathematics—A Journal of Chinese Universities,1990,12(1): 1-13.(in Chinese))
[7]	王正盛. 实对称五对角矩阵逆特征值问题[J]. 高等学校计算数学学报, 2002,24(4): 366-376.(WANG Zheng-sheng. Inverse eigenvalue problem for real symmetric five-diagonal matrix[J].Numerical Mathematics—A Journal of Chinese Universities,2002,24(4): 366-376.(in Chinese))
[8]	田霞, 张方春. 全对称五对角阵的一类特征值反问题[J]. 山东大学学报(工学版), 2002,32(6): 529-532, 585.(TIAN Xia, ZHANG Fang-chun. A kind of inverse eigenvalue problems for bi-symmetric pentadiagonal matrices[J].Journal of Shandong University(Engineering Science ), 2002,32(6): 529-532, 585.(in Chinese))
[9]	林秀丽, 卢琳璋. 双对称五对角矩阵逆特征问题[J]. 厦门大学学报(自然科学版), 2004,43(3): 288-292.(LIN Xiu-li, LU Lin-zhang. Inverse eigenvalue problem for doubly symmetric five-diagonal matrix[J].Journal of Xiamen University(Natural Science ), 2004,43(3): 288-292.(in Chinese))
[10]	王正盛. 实对称带状矩阵逆特征值问题[J]. 高校应用数学学报A辑, 2004,19(4): 451-459.(WANG Zheng-sheng. Inverse eigenvalue problems for real symmetric banded matrix[J].Applied Mathematics—A Journal of Chinese Universities(Series A), 2004,19(4): 451-459.(in Chinese))
[11]	易福侠, 王金林, 孟旭东, 周宁. 由三个特殊次序向量对构造三对角对称矩阵[J]. 数学的实践与认识, 2011,41(19): 185-191.(YI Fu-xia, WANG Jin-lin, MENG Xu-dong, ZHOU Ning. On the construction of tridiagonal symmetric matrices from three special ordered vector pairs[J].Mathematics in Practice and Theory,2011,41(19): 185-191.(in Chinese))
[12]	李承宽, 王金林. Jacobi矩阵的一类广义特征值反问题[J]. 南昌航空大学学报, 2010,24(1): 64-68.(LI Cheng-kuan, WANG Jin-lin. An inverse problem of generalized eigenvalue for Jacobi matrix[J].Journal of Nanchang Hangkong University,2010,24(1): 64-68.(in Chinese))
[13]	钱爱林, 吴又胜. 五对角矩阵的特征值反问题[J]. 数值计算与计算机应用, 2005,26(2): 111-116.(QIAN Ai-lin, WU You-sheng. Inverse eigenvalue problem for real symmetric five-diagonal matrix[J].Journal of Numerical Methods and Computer Applications,2005,26(2): 111-116.(in Chinese))
[14]	周硕. 缺损特征对的梁振动反问题[J]. 吉林大学学报(理学版), 2008,46(4): 655-657.(ZHOU Shuo. Inverse problem for the vibrating beam from its defective eigen-pair[J].Journal of Jilin University(Science Edition ), 2008,46(4): 655-657.(in Chinese))
[15]	马陆陆, 黄敬频. 由两个右特征对构造三对角四元数矩阵[J]. 数学的实践与认识, 2012,42(12): 209-214.(MA Lu-lu, HUANG Jing-pin. On the construction of tridiagonal quaternion matrices from two right eigenpairs[J].Mathematics in Practice and Theory,2012,42(12): 209-214.(in Chinese))
[16]	陈景良, 陈向晖. 特殊矩阵[M]. 北京: 清华大学出版社, 2001.(CHEN Jing-liang, CHEN Xiang-hun.Special Matrices [M]. Beijing: Tsinghua University Press, 2001.(in Chinese))
[17]	谢冬秀, 张磊, 胡锡炎. 一类双对称矩阵反问题的最小二乘解[J]. 计算数学, 2000,22(1): 29-40.(XIE Dong-xiu, ZHANG Lei, HU Xi-yan. Least-square solutions of inverse problems for bisymmetric matrices[J].Mathematica Numerica Sinica,2000,22(1): 29-40.(in Chinese))