振动梁离散模型的一类模态反问题

兰州理工大学学报 ›› 2024, Vol. 50 ›› Issue (2): 153-160.

振动梁离散模型的一类模态反问题

吴静^*1,2, 惠小健^1,2, 孙宗岐^1,2, 李文博³

1.西京学院计算机学院, 陕西西安 710123;
2.西京学院西安市智能康复人机共融与控制技术重点实验室, 陕西西安 710123;
3.西安理工大学理学院, 陕西西安 710048

收稿日期:2022-11-12 出版日期:2024-04-28 发布日期:2024-04-29
通讯作者: 吴静(1982-),男,陕西延安人,副教授.Email:specialwujing@163.com
基金资助:
国家自然科学基金(12201492),陕西省科技厅自然科学研究计划项目(2021JQ-869)

A class of modal inverse problems for discrete models of vibration beams

WU Jing^1,2, XI Xiao-jian^1,2, SUN Zong-qi^1,2, LI Wen-bo³

1. School of Computer Science, Xijing Univ., Xi'an 710123, China;
2. Xi'an Key Laboratory of Human-Machine Integration and Control Technology for Intelligent Rehabilitation, Xijing Univ., Xi'an 710123, China;
3. School of Science, Xi'an Univ. of Tech., Xi'an 710048, China

Received:2022-11-12 Online:2024-04-28 Published:2024-04-29

摘要/Abstract

摘要： 讨论了一类对系统的刚度主子阵存在一定约束的条件下,部分或缺损的模态信息构造振动梁模型刚度矩阵的模态反问题.由特征信息得到矩阵方程组,根据给定的频率数据是否属于系统,分三种情形求解.分别利用牛顿-康托洛维奇定理和线性方程组的基本理论,得到了模态反问题存在唯一解的充要条件,并给出了解的显式表达式和数值算法;证明了对应算法的收敛性和数值稳定性分析,结合工程实例,通过数值实验表明了理论的正确性和算法的有效性,并分析说明了计算过程是数值稳定的.

关键词: 振动梁, 五对角矩阵, 模态反问题, 工程实例

Abstract: The modal inverse problem of constructing the stiffness matrix of a vibrating beam model from partial or missing modal information under certain constraints on the stiffness main submatrix of the system is discussed in this paper.The matrix equations are obtained from the characteristic information and are solved in three cases according to whether the given frequency data belongs to the system. Using the Newton-Kantorovich theorem and the basic theory of linear equations, the necessary and sufficient conditions for the existence of a unique solution to the inverse modal problem are obtained. The explicit expression and numerical algorithm of the solution are given. The convergence and numerical stability analyses of the corresponding algorithm are presented and proven. Combined with an engineering example, numerical experiments indicate the correctness of the theory and the effectiveness of the algorithm. Additionally, the computational process is shown to be numerically stable through further analysis.

Key words: vibration beam, pentadiagonal matrix, inverse modal problem, engineering example

中图分类号:

O241.6

吴静, 惠小健, 孙宗岐, 李文博. 振动梁离散模型的一类模态反问题[J]. 兰州理工大学学报, 2024, 50(2): 153-160.

WU Jing, XI Xiao-jian, SUN Zong-qi, LI Wen-bo. A class of modal inverse problems for discrete models of vibration beams[J]. Journal of Lanzhou University of Technology, 2024, 50(2): 153-160.

参考文献

[1] 戴华.由谱数据数值稳定地构造实对称带状矩阵 [J].计算数学,1990,5(2):157-166.
[2] 戴华.Jacobi矩阵特征值反问题 [J].计算物理,1994,11(4):451-456.
[3] 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.
[4] 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.
[5] 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,2005,6(2):110-117.
[6] 王正盛.特征值反问题的结构探伤方法 [J].振动测试与诊断,2010,8(30):446-450.
[7] 吴静,丁小丽.实对称五对角矩阵的两类广义特征值反问题 [J].应用数学与计算数学学报,2018,30(3):408-423.
[8] 何敏,章礼华,王其申.单杠结构差分离散系统的振动反问题 [J].计算物理,2016,33(4):410-418.
[9] 徐扬生,陈仲仪.特征值反问题与振动物理参数识别 [J].应用力学学报,1985(4):83-93.
[10] 田霞,戴华.梁的离散模型的模态反问题 [J].振动与冲击,2005(6):29-31.
[11] 周硕,吕晓寰,王小雪.用三个向量对构造梁振动系统的刚度矩阵 [J].应用数学和力学,2015,36(3):303-314.
[12] 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(4):655-657.
[13] ZHANG J,WEI G S.Reconstruction of Jacobi matrices with mixed spectral data [J].Linear Algebra and its Applications,2020,591:44-60.
[14] YANG L,GUO Y X,WEI G S.Inverse eigenvalue problems for a damped Stieltjes string with mixed data [J].Linear Algebra and its Applications,2020,601:55-71.
[15] LI J C,DONG L Q,LI G.A class of inverse eigenvalue problems for real symmetric banded matrices with odd bandwidth [J].Linear Algebra and its Applications,2018,541:131-162.
[16] MAHSA R M,HANIF M,KAZEM G.On the generalized inverse eigenvalue problem of constructing symmetric pentadiagonal matrices from three mixed eigendata [J].Linear and Multilinear Algebra,2015,63(6):1154-1166.
[17] GHANBARI K,MIRZAEI M.Inverse eigenvalue problem for pentadiagonal matrices [J].Inverse Problems in Science and Engineering,2014,22(4):530-554.