采用分形理论与微分几何的摆线针轮形貌模型  

Surface Morphology Models of Cycloid and Pin Teeth Based on Fractal Theory and Differential Geometry

作　　者：韩炬 李威[1] 王志军 HAN Ju;LI Wei;WANG Zhijun(School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China;School of Mechanical Engineering, North China University of Science and Technology, Tangshan, Hebei 063210, China)

机构地区：[1]北京科技大学机械工程学院,北京100083 [2]华北理工大学机械工程学院,河北唐山063210

出　　处：《西安交通大学学报》2017年第11期97-105,136,共10页Journal of Xi'an Jiaotong University

基　　金：国家自然科学基金资助项目(51505124)

摘　　要：根据分形理论与微分几何,应用Weierstrass-Mandelbrot函数与矢量函数构建了摆线轮齿与针齿的表面形貌模型,该模型综合体现了摆线轮齿与针齿的宏观特征与微观特征;应用MATLAB绘制了各向同性粗糙针齿及单个摆线轮齿的二维截面图,构建了摆线轮齿与针齿的接触比例系数。通过分析可知:接触比例系数随啮合点的变化而变化,且针齿与摆线轮齿内凹部分接触时的接触比例系数远大于针齿与摆线轮齿外凸部分接触时的值;接触比例系数随分形维数的增加而减小,随摆线轮与针轮中心距的增加先增大后减小,随特征尺度、针齿半径和针轮齿数的增加而增大。The key point to analyze the contact characteristics of cycloid and pin wheels is to establish a contact model with comprehensive embodiment of the gross feature and microcharacteristic.A surface topography model of rough surfaces is established based on WeierstrassMandelbrot function and differential geometry.The function of cycloid is determined by using a vector function with modification being taken into account.The unit normal vector is calculated with the application of differential geometry.The 2 Dsections of a pin tooth and a cycloid tooth which is assumed as isotropic are plotted by MATLAB.The contact model between cylinder and rigid flat is established,aparameter called equivalent contact angle is proposed,and the relations of the angle with fractal dimension,characteristic scale and radius of cylinders are studied.Furthermore,the contact proportionality coefficient that is used to describe the contact area between cycloid and pin tooth surfaces is given.The coefficient is less than 1 in any case,and the value for the area between pin tooth and inter part of cycloid is much larger than that for the area in outer part of cycloid.It is indicated by analysis that the value of the coefficient decreases with the increase of the fractal dimension,increases at first and then decreases with the increase of the center distance,and increases with the increase of pin tooth radius,characteristic scale,and number of pin teeth.

关 键 词：摆线针轮 分形理论 微分几何 Weierstrass-Mandelbrot函数 矢量函数 表面形貌模型 

分 类 号：TH114[机械工程—机械设计及理论]