Design and Identification of Dissipative Bodies

该项目追求的是如何最好地抑制振动体的平行问题，以及如何从振动体的观察中识别作用在其内部和其上的耗散力。阻尼可以通过各种机械手段来实现，对于每一种机械手段，人们都在寻找方法来分配材料特性以最佳地抑制振动。 然而，最优性可以用几种方式来表征，例如最坏情况下的振动所引起的总能量，所有振动的平均总能量，以及能量衰减的渐近速率。 这些不同的度量中的每一个都需要不同的数学工具和计算技术。 为了解决这些问题，本项目将借鉴和增强力学，功能分析和数值分析的工具。 找到最佳阻尼的任务是一个频谱优化问题，这是特别具有挑战性的，因为这些振动体的数学模型是非自伴的，并具有可变系数。 因此，它们代表了流体动力学等应用领域中的广泛的一类重要问题，因此从这项工作中获得的见解可能会在广泛的一类问题上取得进展。 除了优化设计外，该项目还将开发基于振动测量表征物体材料特性的技术。 特别感兴趣的是包括一起振动的若干一维实体的模型。 在宏观层面上，这样的模型描述了小提琴弦组成的一个弹性芯包围在一个耗散鞘下面的外部弹性绕组，而在微观层面上，类似的模型已经提出了模型DNA作为两个相互缠绕的杆弹性粘在一起。

This project pursues the parallel problems of how best to dampen a vibrating body and how to identify, from observations of a vibrating body, the dissipative forces acting within it and upon it. Damping can be achieved through various mechanical means, and for each of these, one seeks ways to distribute material properties to optimally stifle vibrations. Yet optimality can be characterized in several ways, such as the total energy caused by a worst-case vibration, total energy averaged over all vibrations, and the asymptotic rate of energy decay. Each of these different metrics requires different mathematical tools and computational techniques. To address these problems, this project will draw upon, and augment, tools from mechanics, functional analysis, and numerical analysis. The task of finding the best damping is a spectral optimization problem, which is particularly challenging because the mathematical models for these vibrating bodies are non-self-adjoint and have variable coefficients. As such, they are representative of a broad class of important problems in application areas such as fluid dynamics, and thus insights gained from this work may suggest advances on a broad class of problems. In addition to design optimization, this project will also develop techniques for characterizing the material properties of a body based on measurements of vibration. Of particular interest are models comprising several one-dimensional entities vibrating together. At the macroscopic level, such models describe violin strings composed of an elastic core enclosed in a dissipative sheath below an outer elastic winding, while at the microscopic level, similar models have been proposed to model DNA as two intertwined rods glued together elastically.