Constructing the Physical Parameters of a Damped Vibrating System From Eigendata
Constructing the Physical Parameters of a Damped Vibrating System From Eigendata
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DOI：
10.1016/j.laa.2007.08.014
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发表时间：
2008-01
影响因子：
1.1
通讯作者：
Zhengjian Bai
中科院分区：
文献类型：
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作者：
Zhengjian Bai
本文研究了离散阻尼质量-弹簧系统的反问题，其中质量、阻尼和刚度矩阵均为对称三对角矩阵。首先证明了该模型可以由两个真实的特征值和三个真实的特征向量或两个复共轭特征对和一个真实的特征向量构成。然后，我们研究了一般的欠定和超定问题。特别地，给出了欠定问题在给定的两个真实的或复共轭特征对上有物理解的充要条件。然而，对于大的模型阶数，从这些数据的构造可能对扰动敏感。为了降低灵敏度，我们提出了欠定噪声数据的最小范数解和超定测量数据的最小二乘解。我们还讨论了所需模型的物理可实现性的正约束正则化方法的不适定欠定问题和最小二乘优化问题的不适定超定问题的正约束。最后，我们给出了简单的数值例子来说明我们的方法的有效性。
In this paper we consider the inverse problem for a discrete damped mass–spring system where the mass, damping, and stiffness matrices are all symmetric tridiagonal. We first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector. Then, we study the general under-determined and over-determined problems. In particular, we provide the sufficient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under-determined problem has a physical solution. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we propose the minimum norm solution over the under-determined noisy data and the least squares solution to the over-determined measured data. We also discuss the physical realizability of the required model by the positivity-constrained regularization method for the ill-posed under-determined problem and the least squares optimization problems with positivity-constraints for the ill-posed over-determined problem. Finally, we give simple numerical examples to illustrate the effectiveness of our methods.