Mathematical Sciences: Inverse Nodal Problems and Perturbation Theory in Higher Dimensions

数学科学：高维逆节点问题和微扰理论

• 批准号: 9410700
• 负责人: Joyce McLaughlin
• 金额: $ 2.73万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-11-15 至 1996-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9410700&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Nodal; Problems

9410700 McLaughlin This project is concerned the problem of determining which properties of an elastic membrane can be deduced from knowledge of the natural frequencies and the nodal lines of the membrane. The nodal lines are places on the membrane where there is no excitation when the membrane is excited at a natural frequency. There are two major parts to the project. The first is to obtain perturbation results for the natural frequencies and mode shapes. Since the frequencies are not well spaced in higher dimensional problems, small divisor problems must be solved. The second major part of the project is related to the fact that nodal domains (connected domains in the membrane defined by the nodal lines) can be long, thin strips whose diameter is as large as the diameter of the membrane. What is required then is to define approximate nodal domains whose diameter goes to zero as the order of the eigenvalue goes to infinity. The goal of this research is to establish methods for finding properties of vibrating systems from indirect measurements, in particular, measurements of natural frequencies and nodal lines. The natural frequencies can be determined by spectral analysis of impulse response data. The nodal lines can be measured by directing a laser at the vibrating surface when the membrane is excited at a natural frequency. The lines where the Doppler shift in the backscatter is minimized are the nodal lines. From this data the amplitudes of external forces on the membrane and an expression for a (nonconstant) density of the membrane may be determined from explicit formulas for these quantities. These formulas are expected to be very useful for obtaining efficient numerical algorithms to identify the quantities in question. ***

9410700 McLaughlin 该项目涉及确定弹性膜的哪些属性可以从膜的固有频率和节点线的知识中推断出的问题。 节点线位于膜上当膜以固有频率激励时没有激励的位置。 该项目有两个主要部分。 首先是获得固有频率和振型的扰动结果。 由于在高维问题中频率分布不均匀，因此必须解决小除数问题。该项目的第二个主要部分与以下事实有关：节点域（由节点线定义的膜中的连接域）可以是长而薄的条带，其直径与膜的直径一样大。 那么需要定义近似节点域，当特征值阶趋向无穷大时，其直径趋于零。 本研究的目标是建立通过间接测量（特别是固有频率和节点线的测量）查找振动系统特性的方法。 固有频率可以通过脉冲响应数据的频谱分析来确定。 当膜以固有频率激励时，可以通过将激光对准振动表面来测量节点线。 反向散射中多普勒频移最小化的线是节点线。根据该数据，可以根据这些量的显式公式来确定膜上的外力的幅度和膜的（非恒定）密度的表达式。 这些公式预计对于获得有效的数值算法来识别相关数量非常有用。 ***