Stephenson-Ⅲ型平面六杆机构轨迹综合的代数求解  

Algebraic Solution for Path Synthesis of Planar Stephenson-ⅢSix-bar Linkage

作　　者：李学刚[1,2] 张丽娟 魏世民 李河清[2] LI Xuegang;ZHANG Lijuan;WEI Shimin;LI Heqing(School of Automation,Beijing Univ.of Posts and Telecommunications,Beijing 100876,China;College of Mechanical Eng.,North China Univ.of Sci.and Technol.,Tangshan 063210,China)

机构地区：[1]北京邮电大学自动化学院,北京100876 [2]华北理工大学机械工程学院,河北唐山063210

出　　处：《工程科学与技术》2021年第1期155-161,共7页Advanced Engineering Sciences

基　　金：国家自然科学基金项目(51375059;51605036);北京市自然科学基金-海淀原始创新联合基金项目(L172031)。

摘　　要：为弥补精确点法、优化法和数值图谱法等已有方法的不足,进一步提高Stephenson-Ⅲ型平面六杆机构连续轨迹综合的精度,提出了一种基于傅氏级数的平面六杆机构轨迹综合的代数求解新方法。通过将Stephenson-Ⅲ型平面六杆机构拆分为四杆机构和二级杆组,对设计变量进行解耦。在利用已建立综合方法得到左侧四杆机构设计参数的基础上,依据复数矢量理论,建立了含有右侧二杆组设计变量的封闭矢量方程式,将由傅氏级数表示的连杆转角函数代入封闭矢量方程中,经过消元、代换将方程转化为由设计变量、连杆转角函数谐波参数和输入转角表示的复数方程。根据复指数的性质,得到了含有机构设计变量和连杆转角函数谐波参数的表达式,确定了转角函数谐波参数与设计变量间的函数关系。根据这一函数关系建立综合设计方程,利用Groebner基代数法进行消元,将综合设计方程化简为含有机构设计参数和连杆转角函数谐波参数一元四次方程,求解得到右侧二杆组设计变量的解析解,建立了右侧二杆组设计变量计算的通用公式。利用该方法进行连杆机构轨迹综合,可以得到左侧四杆机构的12组设计参数以及右侧二杆组的4组设计参数,将所得结果进行组合,最终可以得到48组Stephenson-Ⅲ型平面六杆机构轨迹综合设计参数。通过运动仿真程序对综合结果进行验证,检验其是否存在曲柄,有无分支、顺序问题,并依据综合误差,最终可得到满足要求的设计参数值。在理论分析的基础上,进一步归纳总结明确了使用该方法进行轨迹综合的具体步骤,利用MATLAB软件编写求解程序,并通过数值实例验证该方法的有效性和可行性。结果表明:该方法实现了Stephenson-Ⅲ型平面六杆机构的连续轨迹综合,克服了精确点法受机构未知量个数限制,无法实现多点位轨迹综合的不足。与已有数值图谱In order to overcome the disadvantage of conventional precise point methods,optimization approaches and numerical atlas,and further improve the accuracy of continuous path synthesis of planar six-bar linkage,a novel analytical approach with Flourier series was presented to solve path synthesis problem for planar six-bar linkage.Firstly,the planar Stephenson-Ⅲsix-bar linkage was decomposed into a four-bar linkage and a two-bar group for the purpose of decoupling the design parameters of linkage.Then,on the basis of obtaining the design parameters of the four-bar linkage on the left by the established comprehensive method,a closed vector equation containing the design variables of the two-bar group on the right was established based on the complex vector theory.The linkage rotation angle function was formulated according to the Fourier series,and substituted into a vector loop equation.Through eliminating and substituting,the vector loop equation was changed into a complex number equation that contain the design variables of mechanism,the harmonic parameters of the linkage rotation angle function and the input angle.With regard to the properties of the complex exponent,the mathematic expression containing the design variables of mechanism and the harmonic parameters of the linkage rotation angle function was obtained.Accordingly,the relationship between the design variables and the synthetic design harmonic parameters of rotation angle function was obtained.According to this function relation,the synthetic design equation was established.Through the elimination Groebner base algebra approach,the comprehensive design equation was simplified into a unary quartic equation containing the design parameters of the mechanism and the harmonic parameters of the linkage rotation angle function,the analytical solutions for the two-bar group on the right was derived and the general formula for calculating the design variables of the right two-bar group was established.Using the method of connecting rod mechanism path synthes

关 键 词：Stephenson-Ⅲ型平面六杆机构 轨迹综合 傅氏级数 代数法 

分 类 号：TH112[机械工程—机械设计及理论]