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本文提出了齿轮理论中的一个新命题——最优啮合(即最优齿型)问题,并在理论上对它作了一些初步探讨。首先,建立了啮合基本微分方程,然后,以之为基础进而对一种效率最高的最优啮合齿轮作了具体研究,建立并求解了相应的变分问题。结果表明,已在钟表中长期广泛应用着的一种摆线齿轮(其下半齿廓是纯径向线)实际上是效率最高的实用齿型。作为一个副产品,文中还提出了求解齿轮啮合理论中其他三个基本命题——正命题、反命题、杂交型命题——的简便而通用的解析方法。本文的理论不难推广到斜齿轮上去。
In this paper, we propose a new proposition in gear theory, the optimal meshing (the optimal tooth shape), and make some preliminary discussions on it in theory. First of all, the basic differential equation of meshing is established. Then, based on it, a most efficient and optimal meshing gear is studied. The corresponding variational problems are established and solved. The results show that a cycloid gear that has been widely used in watches for a long time (its lower half-tooth profile is a pure radial line) is actually the most efficient practical tooth profile. As a byproduct, a simple and general analytical method for solving three other basic propositions in the theory of gear meshing is also presented: positive proposition, inverse proposition, hybrid proposition. The theory of this article is not difficult to generalize to helical gears.