Modelling of circumferential surface topography of grinding wheel with random distribution of circular truncated cone abrasive grains
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摘要: 砂轮圆周表面形貌的准确建模是研究磨削加工机理的重要基础。将刚玉磨粒形状设定为有45°锥角的圆台形，推导在砂轮圆周表面随机分布的圆台磨粒的位置坐标，构建磨粒尺寸和磨粒凸起高度符合正态分布的圆台磨粒模型，并采用碰撞检测方法检测是否存在磨粒干涉现象，最终生成符合真实工况的砂轮圆周表面形貌模型。结果表明：理论计算的磨粒数与砂轮圆周表面形貌模型的磨粒数比较，其相对误差仅为7.66%；磨粒尺寸分布和磨粒凸起高度分布与建模时设定的正态分布曲线有很好的一致性；磨粒位置呈无规则的随机分布，且砂轮圆周表面形成了无规则的块状和带状间隙区域，证明提出的建模方法可有效生成与真实砂轮具有较高相似度的圆周表面形貌。Abstract:
Objectives Due to the complexity of the grinding process, the study of the grinding mechanism has always been a research hotspot in the field of precision grinding. The establishment of an accurate surface topography model of the grinding wheel circumference is an important basis for studying the grinding mechanism. The surface topography of the corundum grinding wheel is accurately described using four parameters, namely the shape of the abrasive grain, the size of the abrasive grain, the height of the abrasive grain bulge and the number of abrasive grains per unit area. The corresponding grinding wheel model is established using Matlab. Methods To construct a circumferential surface morphology model of the corundum grinding wheel using Matlab simulation software, the geometric shape of the abrasive particles is first simplified to a truncated cone with a cone angle of 45° based on the actual shape of the abrasive particles in the corundum grinding wheel. By using the center coordinates of the large and small circular surfaces of the truncated cone abrasive particles and the radius variation relationship at different axial heights, an arbitrary truncated cone particle with size d is constructed using the cylinder function. Secondly, based on the corresponding grinding wheel circumferential surface established in the cylindrical coordinate system, the rand random function is used to randomly generate Nt grinding particle position coordinates on the circumferential surface. Based on the phenomenon that the grinding particle size and the protrusion height both follow the normal distribution law, the normrnd function is used to generate circular truncated grinding particles with randomly distributed positions on the grinding wheel circumferential surface. Then, a collision detection method is used to check whether there is interference between the abrasive grains, that is, the distance between the centers of the small circular surfaces of any two adjacent truncated cone abrasive grains should be greater than or equal to the sum of the radii of the large circular surfaces of the two abrasive grains. Finally, the real grinding wheel surface topography and the constructed grinding wheel surface topography model are compared and analyzed to determine the accuracy of the modeling method in this paper. Results A corundum grinding wheel model is constructed with a diameter and width of 20 mm and 4.5 mm, particle size code of F80, grinding wheel structure number of 7, abrasive rate of 48% and grinding particle size of 152 to 178 μm. The following results are obtained: (1) The distance between any adjacent truncated cone abrasive grains is narrow, and there is no interference between abrasive particles. The position distribution of abrasive particles on the surface of the grinding wheel is in accordance with the random distribution characteristics, and the surface topography of the grinding wheel produces blocky and narrow strip-shaped gap areas. (2) The total number of abrasive grains calculated theoretically is 9 342, while the total number of abrasive grains generated in the model is 8 626, with a relative error between the two of only 7.66%. (3) The size distribution of the abrasive grains and the protrusion height distribution of the abrasive grains in the model are statistically analyzed, and it is found that the distribution patterns are consistent with the normal distribution curve set in the modeling. Conclusions The abrasive grain shape can be set to a truncated cone with a cone angle of 45°, and the size of the truncated cone abrasive grains can be converted from spherical abrasive grains based on the principle of volume invariance, which aligns with the actual abrasive grain shapes observed in real grinding wheels during the grinding process. Compared with the traditional modeling method for grinding wheel morphology, the modeling method proposed in this paper does not require complex coordinate transformations between the grinding wheel circumferential surface and its unfolded plane, and can obtain a model of the grinding wheel circumferential surface morphology where the size of the truncated cone abrasive grains and the height of the protrusions follow the normal distribution law, and the position of the abrasive particles are randomly distributed. The model has high similarity to the distribution characteristics of real grinding wheel topography, and the grinding wheel surface forms irregular block-shaped and narrow strip-shaped gap areas. Therefore, this modeling method is suitable for the establishment of a corundum grinding wheel circumferential surface topography model. The microscopic contact mechanism between the grinding wheel and the workpiece surface can be further explored through this model, and a grinding surface topography prediction and analysis model can be established. -
图 7 随机分布的磨粒位置坐标
Figure 7. Random distribution of abrasive particle position coordinates
图 8 未施加凸起高度的砂轮表面示意图
Figure 8. Schematic diagram of grinding wheel surface without exerted protrusion heights
图 9 施加凸起高度的砂轮表面示意图
Figure 9. Schematic diagram of grinding wheel surface with exerted protruding heights
图 10 砂轮圆周表面形貌建模流程图
Figure 10. Modelling flow chart of circumferential surface topography of grinding wheel
图 11 建立的砂轮圆周表面形貌模型及其局部放大图
Figure 11. Established model of circumferential surface topography of grinding wheel and local enlarged image of model
图 13 磨粒凸起高度分布统计图
Figure 13. Statistical diagram of abrasive grain protruding height distribution
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