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<p>This paper proposes a&#xa0;novel generating method for the toroidal worm based on the meshing theory with 2‑degree-of-freedom (2DOF). During generating the hourglass worm, the rotation of the worm blank and the swinging of the generating grinding wheel are set as two independent relative motions, and thus the generating plane forms a&#xa0;family of two-parameter surfaces in the fixed coordinate system. By using third-order rotation transformation matrix, the equation of this family of surfaces and its unit normal vector are derived for computing the two meshing functions. On the basis of this, the equation of the worm helicoid is obtained through coordinate transformation, which establishes the foundation for future quantitative analysis of the meshing performance in such a&#xa0;toroidal worm gear pair. Then, the inexistence of the meshing limit line is mathematically proven. Therefore, based on the conjugate principle, it is known that the working domain of the generating plane is consistent with the range of the helical surface of the toroidal worm, which can be determined by numerically resolving the systems of nonlinear equations. The numerical outcome shows that the working domain only covers a&#xa0;small portion of the generating plane. Furthermore, from these calculations, the range of the two motion parameters can be acquired. This is conducive to designing the stroke of the toroidal worm cutting machine. In this paper, the feasibility of the proposed toroidal worm cutting scheme is verified. This study provides a&#xa0;fundamental theoretical basis for developing next-generation worm cutting machine.</p>

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Working domain of cutting meshing for plane enveloping hourglass worm with 2DOF

  • 期刊论文

  • 发表日期：

    2025年9月18日

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  • Xiaonan Zhang,
  • Yaping Zhao

摘要

This paper proposes a novel generating method for the toroidal worm based on the meshing theory with 2‑degree-of-freedom (2DOF). During generating the hourglass worm, the rotation of the worm blank and the swinging of the generating grinding wheel are set as two independent relative motions, and thus the generating plane forms a family of two-parameter surfaces in the fixed coordinate system. By using third-order rotation transformation matrix, the equation of this family of surfaces and its unit normal vector are derived for computing the two meshing functions. On the basis of this, the equation of the worm helicoid is obtained through coordinate transformation, which establishes the foundation for future quantitative analysis of the meshing performance in such a toroidal worm gear pair. Then, the inexistence of the meshing limit line is mathematically proven. Therefore, based on the conjugate principle, it is known that the working domain of the generating plane is consistent with the range of the helical surface of the toroidal worm, which can be determined by numerically resolving the systems of nonlinear equations. The numerical outcome shows that the working domain only covers a small portion of the generating plane. Furthermore, from these calculations, the range of the two motion parameters can be acquired. This is conducive to designing the stroke of the toroidal worm cutting machine. In this paper, the feasibility of the proposed toroidal worm cutting scheme is verified. This study provides a fundamental theoretical basis for developing next-generation worm cutting machine.