Quadratic matrix polynomials are fundamental to vibration analysis. Because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters, most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials. In the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior, respecting the intrinsic structure becomes important and challenging both theoretically and practically. The issue of whether a structured inverse eigenvalue problem is solvable is problem dependent and has to be addressed structure by structure. In an earlier work, physical systems that can be modeled under the paradigm of a serially linked mass-spring system have been considered via specifically formulated inequality systems. In this paper, the framework is generalized to arbitrary generally linked systems. In particular, given any configuration of interconnectivity in a mass-spring system, this paper presents a mechanism that systematically and automatically generates a corresponding inequality system. A numerical approach is proposed to determine whether the inverse problem is solvable and, if it is so, computes the coefficient matrices while providing an estimate of the residual error. The most important feature of this approach is that it is problem independent, that is, the approach is general and robust for any kind of physical configuration. The ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported.

中文翻译：

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基于部分特征信息的振动系统参数重构

二次矩阵多项式是振动分析的基础。由于组成元素之间的预定互连性和物理参数的强制性非负性，大多数给定的振动系统都会对相应的二次矩阵多项式的系数施加一些固有的结构。在从观察到的或期望的动力学行为重建振动系统的逆问题中，尊重内在结构变得重要并且在理论上和实践上都具有挑战性。结构化特征值反问题是否可解的问题取决于问题，必须逐个结构地解决。在之前的作品中，可以在串联链接质量弹簧系统范式下建模的物理系统已经通过专门制定的不等式系统进行了考虑。在本文中，该框架被推广到任意的一般链接系统。特别是，给定质量弹簧系统中的任何互连配置，本文提出了一种系统地自动生成相应不等式系统的机制。提出了一种数值方法来确定逆问题是否可解，如果是，则在提供残差估计值的同时计算系数矩阵。这种方法最重要的特点是它与问题无关，也就是说，该方法对于任何类型的物理配置都是通用的和稳健的。