共轭曲线的奇异点和根切界限点

在齿轮加工和设计中常遇到齿轮的根切和干涉问题。目前在平面啮合理论中对根切(或干涉)的一般条件并未完全弄清楚,[1]中未加证明地认为共轭曲线的奇异点就是根切界限点,实际上并不完全是这样。本文分析了根切现象的几何本质,从而得出根切(或干涉)的一般条件,指出除了通常熟知的第一类根切界限点外,还存在第二类根切界限点,后者尚未见诸文献。此外,文中讨论了解析曲线上奇异点曲率半径的情况,找出共轭曲线     

A matrix method of displacement analysis of the general spatial 7R mechanism

An input-output equation of the general spatial 7R mechanism is derived in this paper by using the method in [1] and applying the rotation matrices. The result is the same as [2], but the process of derivation is simpler. Applying the character of rotation matrices, it is not difficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors, to calculate the scalar products and triple products of these unit vectors, and to derive the 6th constraint equation. Moreover, an algorithm, which consists of successive applications of row transformation and expansion based on Laplace’s Theorem, is given to evaluate the 16 × 16 determinant according to its characteristic, so that the evaluation is much simplified.     

一般空间7R机构的运动分析

本文对一般空间7R机构进行位移、速度和加速度分析.在文[2]所述方法的基础上,导出输入输出16次代数方程,推导和计算比较简便.文中利用人工智能语言确定位移方程的系数,大大减轻人工劳动量.此外,在位移分析的基础上进行速度和加速度分析.文末以数字实例验证了计算结果.本文和文[5]的结果,为进一步应用人工智能语言,建立空间机构运动分析专家系统打下了基础.     

KINEMATIC ANALYSIS OF THE GENERAL SPATIAL 7R MECHANISM

The displacement,velocity and acceleration analysis of the general spatial7Rmechanism is discussed in this paper.Based on the method proposed in Ref[2].an input-output algebra equation of the16th degree in the tan-half-angle of the output angulardisplacement is derived.The derivation process and computation are considerably simple.Aprogram written in Al language is used to derive the coefficients of displacement equations:therefore the amount of manual work is greatly decreased.The results are verified by anumerical example.The researches of this paper and Ref[5]found a base for establishingan expert system of spatial mechanism analysis in the future.     

Kinematic analysis of the general spatial 7R mechanism

The displacement. velocity and acceleration analysis of the general spatial 7R mechanism is discussed in this paper. Based on the method proposed in Ref. [2], an input-output algebra equation of the 16th degree in the tan-half-angle of the output angular displacement is derived. The derivation process and computation are considerably simple. A program written in AI language is used to derive the coefficients of displacement equations: therefore the amount of manual work is greatly decreased. The results are verified by a numerical example. The researches of this paper and Ref.[5] found a base for establishing an expert system of spatial mechanism analysis in the future.     

一般空间7R机构位移分析的矩阵法

本文利用文[1]的方法,并以旋转矩阵作为主要数学工具,进行一般空间7R机构的位移分析,得出和文[2]相同的结果,而推导和计算显著简化。文中利用旋转矩阵的性质[3],容易得出各坐标轴单位矢量的方向余弦的递推公式,求出这些单位矢量的标积和混合积,并且可以方便地导出第六个约束方程,推导颇为简捷。此外,文中根据所得16阶行列式的特点,采用先作行变换再按Laplace定理展开的方法进行计算,使计算工作量大为减轻。     

A MATRIX METHOD OF DISPLACEMENT ANALYSIS OF THE GENERAL SPATIAL 7R MECHANISM

An input-output equation of the general spatial 7R mechanism is derived in this paperby using the method in [1] and applying the rotation matrices.The result is the same as[2],but the process of derivation is simpler.Applying the character of rotation matrices,it is notdifficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors,tocalculate the scalar products and triple products of these unit vectors,and to derive the 6thconstraint equation.Moreover,an algorithm,which consists of successive applications ofrow transformation and expansion based on Laplace’s Theorem,is given to evaluate the16×16 determinant according to its characteristic.so that the evaluation is muchsimplified.