非线性方程组的混沌求解方法及在机构学中的应用  

Chao Solution of Non - linear Equations and Its Application to Mechanism Synthesis

作　　者：张龙庭[1] 董湘怀[1] 

机构地区：[1]华中科技大学材料科学与工程学院,武汉430074

出　　处：《机床与液压》2003年第1期165-169,共5页Machine Tool & Hydraulics

摘　　要：机构学问题的数学模型常可化为多元非线性方程组,一般求解多元非线性方程组需要初始值,而初始值的选择是相当困难的,同伦方法不需初始值就能求出全部解,为求解这一问题提供了可行的方法,但需要编写专用的程序,且计算量比较大,本文结合MATLAB6.1高级程序设计语言采用简单的牛顿迭代法迭代,并将非线性方程视为非线性的动力学系统,利用使得系统产生混沌的Julia集的点求解方程的全实数解,而Julia集的点在Jocobi矩阵为零的解集的邻域内,因而求得|J|=0的解,再在其邻域内求解即可。运用该算法编写了MATLAB程序,对平面四杆机构的函数发生器综合问题进行了研究,从而找到了实现最大精确点时该问题的全部的解,为实际机构的设计提供了多种选择方案,为机构学设计提供了全新的方法。The mathematical model of mechanism problems can be translated a multi - variables non - linear equations, and, it is a difficult problem to give the initial value for the no - linear equation. The homology continuation algorithm has resolved this difficult problem without given initial value and can find all solution, but a special programming is compiled and the calculating workload is very larger at the same time. A numerical method is be found, taking the iteration of Newton method as a nonlinear dynamitic system in which Julia sets leads to Chaos . Those points of Julia set is in the neighborhood of those points where belong to the determinant of satisfies Jacobian matrix is equal to zero. The problem of function generation for planar four - link mechanism is solved by this method, and thus all solutions for this problem with maximum precision positions are obtained. This provide a simple realization method for homology continuation method.

关 键 词：非线性方程组 机构学 机构尺度综合 混沌算法 牛顿迭代法 MATLAB程序设计语言 

分 类 号：TH112[机械工程—机械设计及理论]