式(1)中,“ $ \pm $”取“ $ + $”、“ $ \mp $”取“ $ - $”表示凹齿面,反之表示凸齿面。图1 刀盘切削加工圆弧齿线圆柱齿轮原理Fig. 1 Principle of cutting VH-CATT by cutter disc
$\left\{ \begin{aligned}& x = [ - ( \pm u\sin \; \alpha + R \pm \frac{{\text{π} m}}{4})\cos \;\theta + R + {R_1}\varphi ]\cos\; \varphi -\\& \quad\; ({R_1} + u\cos \;\alpha )\sin\; \varphi, \\& y = [ - ( \pm u\sin\; \alpha + R \pm \frac{{\text{π} m}}{4})\cos\; \theta + R + {R_1}\varphi ]\sin \;\varphi +\\& \quad\; ({R_1} + u\cos\; \alpha )\cos\; \varphi, \\& \textit{z} = - ( \pm u\sin \;\alpha + R \pm \frac{{\text{π} m}}{4})\sin \;\theta, \\& u = \mp \frac{{\sin\; \alpha }}{{\cos \;\theta }}[\cos \;\theta (R \pm \frac{{\text{π} m}}{4}) - (R + {R_1}\varphi )]\end{aligned} \right.$(1)
根据齿面与齿轮分度圆、齿顶圆及齿根圆之间的关系,可以确定齿面在节圆上点所对应的齿面参数 $\varphi $如式(2);齿面参数 $\varphi $的取值范围如式(3)。
$\varphi = \pm \left( {\beta - \alpha + \sqrt {{{\left( {\frac{1}{{\cos \;\alpha }}} \right)}^2} - 1} } \right)$(2)
$\left\{ \begin{align}& \left| \varphi \right|\ge \beta -\alpha +\