Scholarlink

This chapter deals with the envelope surfaces used in gear design and in gear-machining processes, more specifically those utilizing the gear-generating principle: either when rotations of two gears are in mesh or when rotations of a gear and a gear-cutting tool are in sync. The key features of the conventional approach to generating envelope surfaces (developed for both the differential geometry of surfaces and the kinematic geometry of surfaces) are briefly outlined. The conventional approach to generating envelope surfaces is valid when no correlation exists between the parameters of the relative rolling motion of the components and between the curvature of the interacting surfaces. This chapter shows that when these correlations exist, though, the conventional approach to generating envelope surfaces is invalid. Thus, a corresponding approach for the determination of envelope surfaces in the case under consideration needs to be developed. The core of the proposed method of determining envelope surfaces in gear design and in gear-machining processes is laid out in the chapter. The problem of determining envelope surfaces in gear design and gear-machining processes was correctly formulated and then solved by the author in 2008 (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022). All three kinds of gearing—(a) parallel-axes gearing (Pa−gearing); (b) intersected-axes gearing (Ia−gearing); and (c) crossed-axes gearing (Ca−gearing)—are covered by the derived solution to the problem. (A) L. Euler (~1760) is credited with the determination of envelope-interacting tooth flanks in Pa−gearing; (B) the determination of the envelope-interacting tooth flanks in Ia−gearing was started by G. Grant (1887) (Pat. No. 407.437, (USA). Machine for planing gear teeth/G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patented: July 23, 1889), which was completed by S.P. Radzevich (~2088) (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022); and (C) S.P. Radzevich (~2088) is credited with the determination of envelope-interacting tooth flanks in Ca−gearing (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022).

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Envelopes in Gearing: A Novel Development in Classical Differential Geometry

  • 章节

  • 发表日期：

    2025年

查看全文
  • Stephen P. Radzevich

摘要

This chapter deals with the envelope surfaces used in gear design and in gear-machining processes, more specifically those utilizing the gear-generating principle: either when rotations of two gears are in mesh or when rotations of a gear and a gear-cutting tool are in sync. The key features of the conventional approach to generating envelope surfaces (developed for both the differential geometry of surfaces and the kinematic geometry of surfaces) are briefly outlined. The conventional approach to generating envelope surfaces is valid when no correlation exists between the parameters of the relative rolling motion of the components and between the curvature of the interacting surfaces. This chapter shows that when these correlations exist, though, the conventional approach to generating envelope surfaces is invalid. Thus, a corresponding approach for the determination of envelope surfaces in the case under consideration needs to be developed. The core of the proposed method of determining envelope surfaces in gear design and in gear-machining processes is laid out in the chapter. The problem of determining envelope surfaces in gear design and gear-machining processes was correctly formulated and then solved by the author in 2008 (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022). All three kinds of gearing—(a) parallel-axes gearing (Pa−gearing); (b) intersected-axes gearing (Ia−gearing); and (c) crossed-axes gearing (Ca−gearing)—are covered by the derived solution to the problem. (A) L. Euler (~1760) is credited with the determination of envelope-interacting tooth flanks in Pa−gearing; (B) the determination of the envelope-interacting tooth flanks in Ia−gearing was started by G. Grant (1887) (Pat. No. 407.437, (USA). Machine for planing gear teeth/G.B. Grant, Filed: January 14, 1887 (serial No. 224,382), Patented: July 23, 1889), which was completed by S.P. Radzevich (~2088) (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022); and (C) S.P. Radzevich (~2088) is credited with the determination of envelope-interacting tooth flanks in Ca−gearing (Radzevich SP. Theory of gearing: kinematics, geometry, and synthesis, 3rd edn., revised and expanded. CRC Press, 1208 pages, 2022).