In this study, the calculating principle of the meshing limit line of the conical surface enveloping conical worm pair is put forward systematically. The tooth face equations, the meshing function and the meshing limit function of a conical worm pair are all acquired. Investigating the meshing limit line is come down to solving an equivalent unary nonlinear equation, which is determined from its original equations with four variables by means of the elimination technique. Based on this, the meshing limit line characteristics are deeply researched after resolving preceding equation correctly. The numerical results declare that there may be two meshing limit lines on each helicoid of one tooth of an enveloping conical worm although not all of them have physical significance. All the significative meshing limit lines usually do not get into the worm helicoid and have no influence on its normal work. Therefore, the active length of the worm depends on the tooth face boundary of the conical worm wheel theoretically. Besides, when the center distance of the worm pair is less, the transmission ratio is larger and the number of thread of the worm is more, the meshing limit line may be closer to the little end of the conical worm e helicoid.

中文翻译：

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圆锥面包络圆锥蜗杆副的啮合极限线

本研究系统地提出了圆锥面包络圆锥蜗杆副啮合极限线的计算原理。获得了圆锥蜗杆副的齿面方程、啮合函数和啮合极限函数。研究啮合极限线归结为求解一个等效的一元非线性方程，该方程是通过消除技术从具有四个变量的原始方程确定的。在此基础上，在正确求解前面的方程后，深入研究了啮合极限线特性。数值结果表明包络圆锥蜗杆的一个齿的每个螺旋面上可能有两条啮合限制线，尽管并非所有这些线都具有物理意义。所有有意义的啮合限制线通常不会进入蜗杆螺旋面，不影响其正常工作。因此，蜗杆的有效长度理论上取决于锥形蜗轮的齿面边界。此外，当蜗杆副中心距较小，传动比较大，蜗杆螺纹数较多时，啮合极限线可能更靠近锥形蜗杆和螺旋面的小端。