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这个项目与确定弹性膜的哪些特性可以从对固有频率的知识和膜的结节中推断出的问题。 节点线是膜上没有激发的地方，当膜以固有频率激发时。 该项目有两个主要部分。 首先是获得固有频率和模式形状的扰动结果。 由于频率在较高的维度问题中的间隔不佳，因此必须解决小的除数问题。该项目的第二个主要部分与以下事实有关：淋巴结结构域（由淋巴结线定义的膜中的连接结构域）可能是长的薄条，其直径与膜直径一样大。 当时需要的是定义近似淋巴结域，其直径为零，因为特征值的顺序为无穷大。 这项研究的目的是建立从间接测量的振动系统属性的方法，特别是对固有频率和节点线的测量值。 固有频率可以通过脉冲响应数据的光谱分析来确定。 当膜以固有频率激发时，可以通过在振动表面引导激光来测量节点线。 淋巴结线最小化了反向散射中多普勒移位的线。根据这些数据，可以从这些量的显式公式确定膜上外力的​​振幅和膜的（非恒定）密度的表达。 预计这些公式对于获得有效的数值算法以识别所讨论的数量非常有用。 ***