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首页> 外文期刊>Journal of Engineering Mechanics >Closely spaced roots and defectiveness in second-order systems
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Closely spaced roots and defectiveness in second-order systems

机译：二阶系统中根和缺陷的分布紧密

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摘要

When two closely spaced eigenvalues merge the associated eigenvectors can either (1) form a subspace where every vector in the span is an eigenvector or (2) coalesce into a single eigenvector. In the second alternative the repeated eigenvalue is associated with a bifurcation point in the eigenvector space and the system is said to be defective. In defective systems a set of coordinates that uncouple the dynamics does not exist and the closest thing possible is the basis of eigenvectors and generalized eigenvectors (sometimes called power vectors) that lead to the Jordan form. Although true defectiveness does not occur in practice, because eigenvalues are never exactly repeated, one anticipates that the features associated with defective conditions will have a bearing on the behavior of systems that are perturbed versions of defective ones. In viscously damped second order systems with symmetric matrices the potential for defectiveness is determined by the structure of the damping. This paper focuses on identification of conditions connecting the damping matrix with defectiveness. A numerical example of a two degree-of-freedom system that varies from being classically damped, to nonclassical, to defective, depending on the position of a dashpot, is used to illustrate the features of the eigensolution as defectiveness is approached.
机译：当两个紧密间隔的特征值合并时，相关的特征向量可以（1）形成一个子空间，其中跨度中的每个向量都是特征向量，或者（2）合并为一个特征向量。在第二替代方案中，重复的特征值与特征向量空间中的分叉点相关联，并且该系统被认为是有缺陷的。在有缺陷的系统中，不存在将动力学解耦的一组坐标，并且最接近的可能是特征向量和广义特征向量（有时称为幂向量）的基础，这些特征向量导致约旦形式。尽管实际上并不会发生真正的缺陷，但是由于永远不会精确地重复特征值，因此人们预计与缺陷条件相关的特征将影响系统的行为，这些系统是缺陷系统的受干扰形式。在具有对称矩阵的粘性阻尼二阶系统中，缺陷的可能性取决于阻尼的结构。本文着重于确定连接阻尼矩阵与缺陷的条件。使用两个自由度系统的数值示例，根据阻尼器的位置，从经典阻尼到非经典阻尼，再到有缺陷的，其变化取决于发生缺陷时本征解的特征。

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