基于数字法的成形砂轮廓形计算及包络面仿真

Calculation of Form Grinding Wheel Profile and Simulation of Envelope Surface Based on Digital Method

成形磨齿过程中,砂轮的廓形精度直接影响到最终齿轮的加工精度。针对成形磨削时砂轮廓形精确计算的需求,提出了一种基于点矢量包络的数字法,用于砂轮廓形计算及包络面仿真。将齿轮的端截面廓形离散为一系列点矢量,利用点矢量完整地描述廓形离散点处的几何特征,将曲面的包络转换为点矢量沿齿面螺旋线的运动包络;通过坐标变换及旋转投影,将点矢量包络运动形成的空间点矢量族投影到选取的计算平面上形成点矢量族;建立点矢量逼近算法,求取计算平面上各点矢量族的包络点,所有的包络点通过拟合的方式构成砂轮廓形。利用点矢量在包络过程中的空间映射关系,确定砂轮廓形点对应的空间接触点,所有的接触点构成砂轮与齿轮的接触线;在确定的砂轮廓形及磨削路径前提下,采用点矢量包络法计算不同磨削位置处的接触线,利用所有的接触线完成对砂轮包络面的仿真,并提取包络面的端截面廓形进行误差显化和预测。最后以一种齿轮为例,完成了对成形砂轮廓形的计算和包络面的仿真,并通过标准齿轮磨削试验验证了点矢量包络方法的计算精度,通过齿向修形齿轮磨削实验验证了包络面仿真结果的准确性。

In the process of forming grinding, the profile accuracy of the grinding wheel directly affects the machining precision of the gear. Aiming at meeting the requirement of accurate calculation of the grinding wheel profile, this paper propose a digital method based on the point-vector envelope to calculate the profile of the wheel and simulate the envelope surface. Firstly, the transverse profile of the gear is divided into a series of point-vectors and the geometrical features of the discrete points are completely described by the point-vector. The envelope of the surface is converted into the motion envelope of the point-vector along the helix. Secondly, the point-vector group formed by the envelope motion of the point-vector is projected onto the selected plane by coordinate transformation and rotation projection. A point-vector approximation algorithm is established to calculate the envelope point of the point-vector group on the calculation plane and the profile of the grinding wheel is formed by fitting all the envelope points. Thirdly, by using the space mapping of the point-vector in the enveloping process, the spatial contact line between the grinding wheel and the gear is determined. The contact lines at different grinding positions are calculated by the point-vector envelope method and the envelope surface of grinding wheel is simulated by all the contact lines, and the transverse profile of the envelope surface is extracted for error manifestation and prediction. Lastly, a kind of gear is taken as an example to complete the calculation of the grinding wheel profile and the simulation of the envelope surface. The accuracy of the point-vector envelope method is verified by the grinding experiment of standard gear, and the accuracy of the simulation results of the envelope surface is verified by the grinding experiment of lead modification gear.