Main mathematical model of non-NC ball-milling cutter

In the field of CNC machining, ball-end milling cutters play a crucial role in shaping complex and free-form surfaces. The demand for these tools is high, but their manufacturing remains challenging due to their intricate geometry. Traditionally, most ball-end cutters are produced on multi-axis CNC machines, which are costly—often reaching millions of dollars. As a result, the cost per unit is high, prompting the need for more economical alternatives. Since 1991, researchers have explored non-numerical machining methods to reduce production costs. This work led to the development of mathematical models for the rake face and flank face of ball-end mills, along with related machining techniques. Further advancements involved replacing spatial curves with planar curved profiles, significantly lowering tool production expenses. This paper aims to provide an overview of the principles and methods behind non-numerical machining of ball-end milling cutters. It summarizes key concepts from various studies and presents essential mathematical models used in the manufacturing process. Understanding these principles helps in developing efficient and cost-effective methods for producing precision cutting tools. The machining of the rake face involves a rotating cone wheel and a workpiece that rotates around its axis. The interaction between the grinding wheel and the workpiece generates the desired rake surface. Mathematical transformations are used to describe this process, enabling precise control over the final shape. For the blade curve, optimization of design parameters such as the position of the grinding wheel, angular velocity, and radius is essential to achieve the ideal "S"-shaped edge. These parameters are carefully adjusted to ensure both functionality and efficiency. When machining the flank face, the profile must align with the edge curve while maintaining sufficient clearance. This is achieved by calculating the trajectory of points along the edge, ensuring smooth and accurate grinding. The resulting model curves are not only easier to produce but also adaptable for different tool specifications. Through practical application, the proposed methods have proven effective. A series of ball-end cutters developed in collaboration with the Harbin Institute of Technology High-tech Park have been successfully commercialized. These innovations contribute to low-cost mass production and offer insights into the manufacturing of other specialized rotary cutters. While this paper outlines the fundamental framework, further research and adaptation are required to fully implement the method in industrial settings. Additional documentation and detailed models are necessary to support real-world applications. Overall, the approach represents a significant advancement in the field of non-numerical machining of cutting tools.

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