Perturbation Methods for Nonlinear Control of Lagrangian Systems

这项工作的主要目标是发展适当的正交分解（POD）作为一个可行的补充替代振动系统的传统实验模态分析。适当的正交分解已用于获得“能量模式”，而不是正常模式，从感应系统输出。如果质量分布已知，适当的正交分解确实可以在轻阻尼系统中产生正态模态。本研究旨在克服已知质量分布的要求，估计模态频率和阻尼因子并将其与正确的模态相关联，扩大对任意激励的适用性，并将POD的性能与经典的实验模态分析进行比较。辅助目标是将一维分布参数系统扩展到二维系统，以及POD在多模态非线性响应中的应用。研究方法包括理论分析、数值模拟和实验。模拟和实验将用于测试具有解析解的简单系统的理论发展，以及具有复杂几何形状的系统。如果POD得到充分发展，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。实验振动设备在全球范围内普遍存在，用于解决工业噪声、性能和安全问题。即使在这些实验室的一小部分中使用POD，总POD活性也会相当大。它对无法获得感测输入的系统特别有利。这个项目是为博士生的研究计划而设计的。该项目还将包括一名本科生研究员，将POD和传统模态分析应用于实验系统

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