﻿ 梁的广义特征值反问题及离散模型修正 Generalized Inverse Eigenvalue Problem and Model Updating for Discrete Beam

Vol.04 No.03(2015), Article ID:15826,8 pages
10.12677/AAM.2015.43029

Generalized Inverse Eigenvalue Problem and Model Updating for Discrete Beam

Zhenwei Sun, Ruru Ma, Zhigang Jia

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou Jiangsu

Received: Jul. 16th, 2015; accepted: Aug. 3rd, 2015; published: Aug. 10th, 2015

ABSTRACT

In this paper, we study the generalized inverse eigenvalue problem and the optimal model updating problem according to two given eigenpairs, while the total mass of beam is unknown. We present the general solution of the inverse generalized eigenvalue problem. Aiming at the beam model updating problem, we use the least squares method to compute the optimal quality parameter to minimize the distance between the physical parameters of the new beam system and those of the original one.

Keywords:Generalized Inverse Eigenvalue Problem, Least Squares, Matrix Norm, Model Updating, Optimal Solution

1. 引言

(1)

(2)

2. 广义特征值问题和模型修正问题

2.1. 广义特征值问题

(1)

(2)的相邻的两个分量不同时为零，若某个使得或者，则或者

(3)

(4)，且同号

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(1) 由给定的系统物理参数生成质量矩阵，刚度矩阵

(2) 用计算原系统的特征值与特征向量，任意取出两个特征对计算

(3) 如果，且同号；则由上述公式(10)，(11)和(13)计算新的系统参数

2.2. 模型修正问题

，根据分解定理[7] 知有如下分解：

,

(1) 根据与计算

(2) 计算分解；

(3) 计算

(4) 求解方程组可以得到

3. 数值算例

(1) 在总质量未知的条件下，生成新的质量和刚度参数

(2) 选取合适的总质量使得新系统的物理参数与原物理参数的误差最小。

(1) 根据梁的模型可以得到刚度矩阵和质量矩阵分别为

(2) 根据算法2，利用最小二乘算法，计算出悬臂梁的总质量为

4. 结论

Generalized Inverse Eigenvalue Problem and Model Updating for Discrete Beam[J]. 应用数学进展, 2015, 04(03): 230-237. http://dx.doi.org/10.12677/AAM.2015.43029

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