非圆齿轮数控滚切加工误差分析研究
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摘要
非圆齿轮既有凸轮的优点又有圆柱齿轮的优点,广泛地应用在各种场合。常见的非圆齿轮有椭圆齿轮、Pascal(巴斯噶)蜗线齿轮等,其精密加工一直是研究的热点。滚齿是一种最为常见的非圆齿轮加工方式,随着对非圆齿轮精度要求的日益提高,非圆齿轮滚切加工误差分析显得尤为重要。对此,论文首先推导非圆齿轮滚切加工运动模型,获得机床运动轴与误差源的关系。在此基础上,分析工艺误差对非圆齿轮部分精度指标(节曲线、齿距和根切)的影响,研究插补算法对非圆齿轮滚切加工的影响。根据数控伺服进给系统模型,以椭圆齿轮节曲线为例,研究非圆齿轮节曲线轮廓误差形成机理。本文的主要研究工作和获得的结论如下：
(1)根据推导出的非圆齿轮滚切加工运动模型,获得运动轴与误差源的关系。
(2)根据非圆齿轮滚切加工运动模型,分别推导工作台转角误差和中心距误差作用下非圆齿轮节曲线误差矢量表达式,以及工作台转角误差和齿条水平移动量误差作用下非圆齿轮齿距误差的表达式,并进行实例计算。
根据齿条与工件间的相对速度和啮合方程,提出并建立非圆齿轮根切界限函数。以椭圆齿轮为例,分析非圆齿轮发生根切的极角位置和工艺误差、设计参数对椭圆齿轮根切的影响。
(3)根据推导的非圆齿轮滚切加工运动模型,获得非圆齿轮滚切插补算法,从程序段弧长和机床运动轴两方面研究不同插补算法对非圆齿轮滚切加工的加工精度和控制难易程度的影响。为对比程序段弧长的均匀性,提出程序段弧长均匀度的概念。
(4)根据数控伺服进给系统模型,以椭圆齿轮节曲线为例,分析一阶、二阶、高阶伺服进给系统动态特性对非圆齿轮节曲线轮廓误差的影响规律。借助数学推导获得非圆齿轮节曲线轮廓误差的形成机理。为减小轮廓误差,提出PID交叉耦合控制方法(cross-coupled control,简称CCC方法)。仿真结果表明交叉耦合控制能改善非圆齿轮节曲线的轮廓精度。
Noncircular gear with the advantages of cam and circular gear has been widely usded on diversified situation. The frequent noncircular gears are elliptical gears, pascal gears and so on, which precision manufacturing is always the research hotspot. As we know, CNC hobbing is the most widely used as a processing gears method, with the increase of gears machining precision, it is very important to analysis the errors of CNC hobbing non-circular gears. Thus, this thesis presents the relationship between the machine motion axis and the errors source according to the derived motion models of CNC hobbing non-circular gears. Based on the above, the influence of process errors on part precision indices (pitch curve, tooth pitch and undercut) of non-circular gears are analysed; and the effects of interpolation algorithms on hobbing noncircular gear are studied. Taking the elliptic gear as a case, the influence regularities of dynamical characteristics of one order, second order and high order servo system on the noncircular gear pitch curves contour error are analyzed. The main research contents and conclusions are as follows:
(l)The relationship between the machine motion axis and the errors source is obtained form the derived motion models of CNC hobbing non-circular.
(2)According to the mathematical model for hobbing noncircular gear, the expression of the noncircular gear tooth pitch deviation and tooth pitch accumulated error caused by the workbench rotary angle errors and the rack horizontal movement amount errors are presented. And the vector expression of the noncircular gear pitch curve errors caused by the stage rotary angle error and the centre space error are derived, respectively.
Based on the known relative velocity and meshing equation between hobbing cutter and the gear blank, the noncircular gear tooth profile undercut limit function is proposed and deduced. Then, the change regularity of undercut limit functions with the polar angle of elliptic and the design parameters are analyzed through the numerical method. The polar angle position of noncircular gear undercut can been confirmed.
(3)According to the derived mathematical model for hobbing noncircular gear, interpolation algorithms of hobbing noncircular gear are obtained. Developing regularity of different interpolation algorithms arc length of segment and machine motion axis were compared, the concept of the arc length of segment uniformity degree is proposed, thereby the uniformity of the arc length of segment can be judged.
(4)Taking elliptic gear as a case, influence regularities of dynamical characteristics of one order, second order and high order servo system on noncircular gear pitch curves contour error is analyzed based on the CNC servo system model. Also the change laws of noncircular gear pitch curves contour error is obtained by using of mathematical derivation. In order to reduce the contour error, PID cross-coupled control (CCC) is proposed, the simulation results show that CCC can improve noncircular gear pitch curves contour accuracy.
(1)根据推导出的非圆齿轮滚切加工运动模型,获得运动轴与误差源的关系。
(2)根据非圆齿轮滚切加工运动模型,分别推导工作台转角误差和中心距误差作用下非圆齿轮节曲线误差矢量表达式,以及工作台转角误差和齿条水平移动量误差作用下非圆齿轮齿距误差的表达式,并进行实例计算。
根据齿条与工件间的相对速度和啮合方程,提出并建立非圆齿轮根切界限函数。以椭圆齿轮为例,分析非圆齿轮发生根切的极角位置和工艺误差、设计参数对椭圆齿轮根切的影响。
(3)根据推导的非圆齿轮滚切加工运动模型,获得非圆齿轮滚切插补算法,从程序段弧长和机床运动轴两方面研究不同插补算法对非圆齿轮滚切加工的加工精度和控制难易程度的影响。为对比程序段弧长的均匀性,提出程序段弧长均匀度的概念。
(4)根据数控伺服进给系统模型,以椭圆齿轮节曲线为例,分析一阶、二阶、高阶伺服进给系统动态特性对非圆齿轮节曲线轮廓误差的影响规律。借助数学推导获得非圆齿轮节曲线轮廓误差的形成机理。为减小轮廓误差,提出PID交叉耦合控制方法(cross-coupled control,简称CCC方法)。仿真结果表明交叉耦合控制能改善非圆齿轮节曲线的轮廓精度。
Noncircular gear with the advantages of cam and circular gear has been widely usded on diversified situation. The frequent noncircular gears are elliptical gears, pascal gears and so on, which precision manufacturing is always the research hotspot. As we know, CNC hobbing is the most widely used as a processing gears method, with the increase of gears machining precision, it is very important to analysis the errors of CNC hobbing non-circular gears. Thus, this thesis presents the relationship between the machine motion axis and the errors source according to the derived motion models of CNC hobbing non-circular gears. Based on the above, the influence of process errors on part precision indices (pitch curve, tooth pitch and undercut) of non-circular gears are analysed; and the effects of interpolation algorithms on hobbing noncircular gear are studied. Taking the elliptic gear as a case, the influence regularities of dynamical characteristics of one order, second order and high order servo system on the noncircular gear pitch curves contour error are analyzed. The main research contents and conclusions are as follows:
(l)The relationship between the machine motion axis and the errors source is obtained form the derived motion models of CNC hobbing non-circular.
(2)According to the mathematical model for hobbing noncircular gear, the expression of the noncircular gear tooth pitch deviation and tooth pitch accumulated error caused by the workbench rotary angle errors and the rack horizontal movement amount errors are presented. And the vector expression of the noncircular gear pitch curve errors caused by the stage rotary angle error and the centre space error are derived, respectively.
Based on the known relative velocity and meshing equation between hobbing cutter and the gear blank, the noncircular gear tooth profile undercut limit function is proposed and deduced. Then, the change regularity of undercut limit functions with the polar angle of elliptic and the design parameters are analyzed through the numerical method. The polar angle position of noncircular gear undercut can been confirmed.
(3)According to the derived mathematical model for hobbing noncircular gear, interpolation algorithms of hobbing noncircular gear are obtained. Developing regularity of different interpolation algorithms arc length of segment and machine motion axis were compared, the concept of the arc length of segment uniformity degree is proposed, thereby the uniformity of the arc length of segment can be judged.
(4)Taking elliptic gear as a case, influence regularities of dynamical characteristics of one order, second order and high order servo system on noncircular gear pitch curves contour error is analyzed based on the CNC servo system model. Also the change laws of noncircular gear pitch curves contour error is obtained by using of mathematical derivation. In order to reduce the contour error, PID cross-coupled control (CCC) is proposed, the simulation results show that CCC can improve noncircular gear pitch curves contour accuracy.
引文
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