Principle of self-assembly of abrasive balls on surface of grinding wheels into phyllotaxis arrangement
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摘要: 为实现磨料球在砂轮基体表面有序化的自组装，提出一种叶序排布磨料球砂轮的自组装制造方法。从有界空间球体最密堆积结构原理与植物组织的柱面叶序结构理论出发，设计一种磨料球自组装砂轮的实验方法。首先对磨料球自组装过程进行仿真分析，然后探讨相关尺寸参数与自组装结构之间的关系，并经过计算得到不同尺寸参数下磨料球自组装结构。结果表明：通过自组装方法能实现磨料球在砂轮基体表面的叶序排布；当磨料球直径恒定时，磨料球在砂轮基体表面形成的叶序排布结构的叶序系数随着砂轮基体直径的增大而减小；当砂轮基体直径恒定时，磨料球在砂轮基体表面形成的叶序排布结构的叶序系数随着磨料球直径的增大而增大。Abstract: Objectives: Compared with traditional abrasive random arrangement grinding wheels, structured grinding wheels offer better chip space and grinding efficiency during the grinding process. At present, the manufacturing methods for structured grinding wheels mainly include metal sintering, electroplating, mechanical dressing, laser etching, and other processing techniques. However, these processing methods generally face several issues, such as low arrangement efficiency, poor positioning accuracy of abrasive particles, and complex manufacturing processes. To address these problems, this paper proposes a method for preparing an abrasive orderly grinding wheel by the orderly accumulation of abrasive balls in a bounded space. Methods: Phyllotactic arrangement refers to the arrangement of biological tissue units in space, such as rotation, opposition, and clustering, to provide an optimal growth environment for plant grains. The spiral arrangement achieves geometric spatial complementarity and maximizes the filling effect, with the left and right spirals formed by the arrangement satisfying the mathematical law of the Fibonacci sequence or its derived series. This paper, based on the theory of phyllotactic arrangement and the study of sphere packing structures in bounded spaces, establishes the basic theory for the self-assembly of abrasive spheres into an orderly arrangement on a grinding wheel. Results: Based on the growth structure characteristics of plant primordia, the constraint barrel in the device is designed to with a hemispherical bottom and a cylindrical top. To ensure that the abrasive balls on the surface of the grinding wheel are arranged in a single layer, the distance between the cylindrical part of the constraint bucket and the grinding wheel matrix is always equal to the diameter of a spherical particle. During the self-assembly process, the motion of the constraint barrel is used to mechanically disturb the absive balls. Due to the effects of gravity, interaction forces, and friction between the grinding balls, the arrangement structure of the grinding balls in the device gradually changes from random disorder to stable order. Once the structure stabilizes, the abrasive balls are fixed to the grinding wheel. Using the evaluation method of cylindrical phyllotaxis structure, multiple parallel oblique lines can be observed, and the distance between continuously growing primordia can be determined, allowing the relevant parameters of the phyllotaxis structure to be calculated. From the cylindrical coordinate expansion diagram and the cylindrical arrangement diagram, it is evident that multiple parallel left and right diagonals resemble the phyllotactic structure. Therefore, the structural parameters were verified and calculated. Conclusions: Abrasive ball-ordered grinding wheels with different grinding wheel matrix sizes were successfully prepared through self-assembly experiments. The abrasive ball arrangement structure was measured using corresponding measuring instruments to verify the accuracy of the self-assembly motion simulation results and the feasibility of the method. The influence of size changes on the structural parameters of the abrasive ball arrangement was discussed. The results indicate that the phyllotactic arrangement of spherical abrasives on the surface of the grinding wheel matrix can be achieved using the self-assembly method. When the diameter of the spherical abrasives is constant, the phyllotaxis coefficient of the arrangement structure formed by spherical abrasives on the surface of the grinding wheel substrate decreases with the increase of the diameter of the grinding wheel substrate. Conversely, when the diameter of the grinding wheel matrix is constant, the phyllotaxis coefficient of the phyllotaxis arrangement structure formed by the spherical abrasives on the surface of the grinding wheel substrate improves with the increase in the diameter of the abrasive balls.
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Key words:
- structured grinding wheel /
- self-assembly /
- orderly arrangement /
- phyllotactic pattern /
- abrasive ball
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图 4 直径5.0 mm磨料球颗粒自组装结果与球中心位置的柱面展开图
Figure 4. Self-assembly results of 5.0 mm abrasive ball particles and cylindrical expansion of center position of the ball
图 5 直径 30 mm砂轮基体自组装结果与球中心位置的柱面展开图
Figure 5. Self-assembly results of the 30 mm grinding wheel substrate and cylindrical expansion diagram of center position of the ball
图 7 磨料球有序化自组装实验结果
Figure 7. Experimental results of ordered self-assembly of abrasive balls
表 1 磨料球与约束桶的材料特性
Table 1. Material properties of abrasice balls and confined barrels
物理参数 泊松比 μ 剪切模量 G / GPa 密度 ρ / (kg·m−3) 磨料球颗粒 0.30 105 2 400 约束桶 0.25 55 2 500 
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表 2 表面的接触特性
Table 2. Surface contact characteristics
接触参数 恢复系数 k1 静摩擦因数 k2 动摩擦因数 k3 球粒-球粒 0.10 0.30 0.01 球粒-桶 0.20 0.50 0.01
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表 3 不同砂轮直径结构参数表
Table 3. Structure parameter table of different grinding wheel diameters
砂轮基体直径 D1 / mm 叶序模式 叶序系数 r 生长角 d / (°) 30 (9,7) 0.198 9 158.220 7 35 4(4,3) 0.182 0 154.547 0 40 (5,19) 0.159 9 74.604 6 45 5(3,4) 0.145 2 71.985 6
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表 4 不同磨料球直径结构参数表
Table 4. Structure parameters of different abrasive ball diameters
磨料球直径 D2 / mm 叶序模式 叶序系数 r 生长角 d / (°) 4.5 (15,13) 0.156 6 167.173 0 5.5 (11,6) 0.227 4 62.821 5 6.5 (8,9) 0.454 1 42.924 8 7.0 (5,6) 0.480 2 63.607 4
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表 5 不同砂轮直径结构参数表
Table 5. Structure parameter table of different grinding wheel diameters
砂轮基体直径 D1 / mm 叶序模式 叶序系数 r 生长角 d / (°) 30 (7,8) 0.213 1 161.780 4 35 5(3,2) 0.193 9 154.322 6 40 (5,19) 0.178 1 79.284 8 45 (14,19) 0.144 3 76.661 5
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