渐开线非圆齿轮的齿廓曲线数学模型

对于给定的非圆齿轮节曲线,借助于共轭齿条建立齿廓曲线的数学模型。根据渐开线齿轮齿廓成形原理,当共轭齿条分度线在非圆齿轮节曲线上作纯滚动,齿条的齿廓便在齿轮上包络出齿轮的齿廓。因此,首先建立了共轭齿条的数学模型和节曲线纯滚动模型,而后用啮合方程将这两个模型结合起来,经坐标变换后便在齿轮坐标系中获得了非圆齿轮齿廓曲线的数学模型。进一步用计算机图形仿真的方法获得椭圆齿轮的渐开线齿廓曲线图形,验证了该模型的正确性和有效性。该模型为非圆齿轮的加工刀具设计、数控加工程序设计以及轮齿强度分析等提供了基础。

MATHEMATICAL MODEL FOR TOOTH PROFILE OF NONCIRCULAR INVOLUTE GEARS

For the pitch-line of a non-circular involute gear being given, its tooth profile is modeled mathematically by using of a conjugate rack. According to the generation mechanism of involute gear tooth profile, if the pitch-line of the conjugate rack is made to roll slidelessly on the pitch-line of the non-circular gear, profile of the conjugate rack will generate the profile of the gear. The geometry model of the conjugate rackand the pure-rolling model of pitch-lines are set up separately, and then joined together with the so-called equation of mash, to obtain the mathematical model of tooth profile in the coordinate system of the non-circular gear. Based on this model, the teeth profiles of non-circular gears are described by graphic simulation to verify its correctness and effectiveness. This model can be used to design the profile of cutter for the non-circular gear, to find the locus of cutter during machining, and to analyze the teeth strength of non-circular gears, etc.