多項式分式法與基因演算法在模態分析的應用

Abstract

模態測試是取得機械系統的振動參數的必備工具，然而現有的商業模態測試軟體有價格昂貴和不易修改的缺點。本研究的動機是以相形低廉且高能強大的套裝軟體MATLAB為計算環境，開發模態測試軟體，以便於針對不同需求客製化調整。模態測試的核心是曲線嵌合，本研究比較多項式分式法以及多目標基因演算法兩種方式，在模態測試曲線嵌合及參數擷取上的效能差異。 多項式分式法利用頻率函數為分式的特性，應用最小平方差法求取分子、分母多項式的係數。多項分式法依照分別處理各個或是同時處理所有的頻率響應函數可分為局部與全域曲線嵌合。 多目標基因演算法則應用NSGA-II（nondominated sorting genetic algorithm-II）的優點為採用非受控排序法（nondominated sorting）及群聚距離（crowding-distance），在族群中選出較具優勢的個體，以維持基因多樣性，避免收斂至局部最佳值，同時能有較高的計算效能。 本論文開發了程式來實現這兩種方法。針對有節點、自然頻率分布接近、高阻尼比、有重根、受頻寬外模態影響等，會造成振動參數擷取困難的情形，比較這兩較方法的異同，並提出改善的方向。

Modal testing is essential for the identification of important dynamical parameters of a mechanical system. However, commercially available modal testing packages are expensive and nonflexible. This thesis aims to employ the popular and powerful package MATLAB as the environment to develop a modal testing program that can be customized to meet the user’s needs. The efficiency of a modal testing program highly depends on the curve fitting algorithm used. Two different curve fitting algorithms, the rational fraction polynomials (RFP) method and the multi-objective genetic algorithms, are adopted. The effectiveness of these two methods on parameter identification is compared. The RFP method is based on the fact that the frequency response function (FRF) of a linear time-invariant system is a rational function in frequency. The lease squares method is used to determine the coefficients of the numerator and denominator polynomials. The RFP method can be classified into two different types, called the local curve fitting and global curve fitting, according to whether the FRFs are processed sequentially or simultaneously. The non-dominated sorting genetic algorithm-II (NSGA-II) is used to realize the multi-objective optimization. This algorithm employs the non-dominated sorting and crowding distance to select elite individuals for the next generation. In this case, the genetic diversity is maintained, early convergence to a local extrema is avoided, and high computational efficiency is achieved. In this thesis, we develop programs based on RFP and NSGAII. Some benchmark tests, for example, modes with nodes, high damping ratios, and double roots, which may present difficulties for parameter identification are used to evaluate these two methods. Possible guidelines to improve these two methods are proposed.