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工程科学与技术:2021,53(1):155-161
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Stephenson-Ⅲ型平面六杆机构轨迹综合的代数求解
(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)
Algebraic Solution for Path Synthesis of Planar Stephenson-Ⅲ Six-bar Linkage
(1.School of Automation, Beijing Univ. of Posts and Telecommunications, Beijing 100876, China;2.College of Mechanical Eng., North China Univ. of Sci. and Technol., Tangshan 063210, China)
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投稿时间:2019-12-10    修订日期:2020-01-10
中文摘要: 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。与已有数值图谱法和优化法相比,该方法不需要预先建立数值图谱库,也不需要提供优化初值,其通过方程求解得到综合设计结果,具有求解精度高、计算速度快、可重复性强的优点,更加便于计算机编程,为机构综合软件的开发提供了理论基础。
Abstract: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-Ⅲ six-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 synthesis, 12 groups of design parameters of the four-bar linkage on the left and 4 groups of design parameters of the two-bar linkage on the right can be obtained. By combining the results, 48 sets of integrated design parameters of planar Stephenson-Ⅲ six-bar linkage path synthesis can be obtained. Through the dynamic simulation program, the comprehensive results were verified to check whether there were cranks, branching problems and order problems. And the design parameter value which met the requirements can be finally obtained according to comprehensive error. Based on the aforementioned theory, the procedure of solving path synthesis problem by the proposed method can be obtained. The computer programs have been developed for the proposed method by MATLAB for solving the problem. An example was provided to verify the validity and feasibility of the proposed method. Verification results showed that the proposed approach can overcome the shortage of multi-point path synthesis method and to solve path synthesis problems for planar Stephenson-Ⅲ six-bar linkage with no limitations on the number of precision points. Compared with numerical atlas and optimization approaches, the proposed method avoided the use of extensive numerical atlas databases and optimal initial solutions, and obtained the results by solving the equation. Therefore, this approach has the characteristics of high accuracy, fast solution velocity and high repeatability, and is suitable for computer programming. The research of this approach provided the theory basis for development of synthesis software.
文章编号:201901183     中图分类号:TH112    文献标志码:
基金项目: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)
作者简介:第一作者:李学刚(1979-),男,副教授,博士.研究方向:机构学与机器人技术.E-mail:hblgyjs@126.com
引用文本:
李学刚,张丽娟,魏世民,李河清.Stephenson-Ⅲ型平面六杆机构轨迹综合的代数求解[J].工程科学与技术,2021,53(1):155-161.
LI Xuegang,ZHANG Lijuan,WEI Shimin,LI Heqing.Algebraic Solution for Path Synthesis of Planar Stephenson-Ⅲ Six-bar Linkage[J].Advanced Engineering Sciences,2021,53(1):155-161.