Perturbation Methods for Nonlinear Control of Lagrangian Systems

本文的主要目标是发展适当的正交分解（POD）作为振动系统传统实验模态分析的一种可行的补充方法，适当的正交分解用于从检测到的系统输出中获得“能量模态”，而不是正常模态。 在质量分布已知的情况下，适当的正交分解确实可以在轻阻尼系统中产生简正模。本研究的目的是克服已知质量分布的要求，估计模态频率和阻尼因子，并将它们与正确的模式，扩大适用于任意激励，并比较性能POD与经典的实验模态分析。 辅助目标是从1-D分布参数系统扩展到2-D系统，并应用POD多模态非线性响应。研究方法将涉及理论分析，数值模拟和实验。 模拟和实验将被用来测试理论发展的简单系统的解析解，也对系统具有复杂的几何形状。 如果POD得到充分发展，它将大大提高我们提取振动参数的能力，作为传统模态分析的补充。 振动实验设施在全球范围内普遍存在，适用于噪声，性能和安全的工业问题。 即使在一小部分这样的实验室中使用POD，总POD活性也会相当大。 这对于感测输入不可用的系统将是特别有利的。 该项目是为博士生的研究计划而设计的。 该项目还将涉及一名本科生研究人员，用于在实验系统上应用POD和常规模态分析

The principal goal of this proposed work is to develop proper orthogonal decomposition (POD) as a viable complementary alternative to conventional experimental modal analysis for vibration systems.Proper orthogonal decomposition has been used for obtaining "energy modes," as opposed to normal modes, from sensed system outputs. Proper orthogonal decomposition can indeed yield normal modes in lightly damped systems if the mass distribution is known. This research aims to overcome the requirement of a known mass distribution, estimate modal frequencies and damping factors and associate them with the correct mode, expand the applicability to arbitrary excitations, and compare the performance of POD with classical experimental modal analysis. Auxiliary objectives are the extension from 1-D distributed-parameter systems to 2-D systems, and the application of POD to multi-modal nonlinear responses. The research approach will involve theoretical analyses, numerical simulations, and experiments. Simulations and experiments will be used to test theoretical developments on simple systems with analytical solutions, and also on systems with complicated geometries. If POD were sufficiently developed, it would significantly enhance our ability to extract vibration parameters as a complement to traditional modal analysis. Experimental vibration facilities are globally prevalent, geared for industrial problems of noise, performance, and safety. Even if POD were used in a fraction of such laboratories, the total POD activity would be quite large. It will be particularly advantageous to systems for which sensed inputs are not available. The project is geared for a doctoral student's research program. The project will also involve an undergraduate researcher for applying POD and conventional modal analysis on experimental systems