变双曲圆弧齿线圆柱齿轮齿面曲率特性研究

齿轮副共轭齿面间的曲率关系是表征齿轮传动质量好坏的重要参数，对齿面诱导法曲率、接触区形状、接触特性、润滑特性、磨损和齿面压应力都有直接的影响。作者基于变双曲圆弧齿线圆柱齿轮的齿面方程对其曲率表达式进行推导并研究了其变化规律。首先根据大刀盘加工原理，利用坐标变换得到了变双曲圆弧齿线圆柱齿轮的齿面方程。基于得到的齿面方程，利用微分几何和空间啮合原理对变双曲圆弧齿线圆柱齿轮的主曲率、高斯曲率、平均曲率和诱导法曲率的数学表达式进行了推导。根据所推导的数学方程利用计算机软件得到变双曲圆弧齿线圆柱齿轮副的齿面数据点，并对该齿轮副的主曲率、高斯曲率、平均曲率和诱导法曲率进行了仿真分析，得到了在啮合过程中齿轮曲率的变化规律。通过仿真结果可知齿轮副的凹、凸齿面在齿线方向的主曲率变化趋势是一致的，但略有差异；在齿廓方向主曲率都在逐渐增大，但增加幅度恰好相反，并且主曲率的变化趋势与齿形的凹凸性是一致的。齿轮副的高斯曲率和平均曲率都很小，在啮合过程中变化幅度很小，没有发生突变，这就证明了该齿轮副的光滑程度很高，齿面连续。诱导法曲率在齿线方向的主值很小，基本接近于0，在齿廓方向的主值为负值，从而证明了该齿轮副在啮合过程中没有干涉现象发生。通过曲率研究证明了该齿轮传动性能很好、传动平稳，同时也为该齿轮后续的研究、开发和设计提供了一定的研究基础。

Study on the Tooth Surface and Curvature Characteristics of Cylindrical Gear with Variable Hyperbolic Arc-tooth-trace

The curvature relationship of the conjugate tooth surface of gear pair is the important parameter which is the characterization of gear transmission quality,and has a direct effect on the induced normal curvature,contact zone shape,contact property,lubrication,wear and tooth surface stress.Based on the deduced tooth surface equation of cylindrical gear with variable hyperbolic arc-tooth-trace (VH-CATT),its curvature expressions were deduced and its change regulations were studied. Firstly,based on the machining principle of the great cutter head,the tooth surface equations of VH-CATT gear were derived according to the transformation matrix of coordinates.According to the derived tooth surface equations,the principal curvature,gauss curvature,mean curvature and induced normal curvature of the VH-CATT gear were derived by using differential geometry and meshing theory.Based on the derived expressions,the tooth surface data,principal curvature,Gaussian curvature,mean curvature,induced normal curvature of the VH-CATT gear pair were simulated and analyzed by computer software.And the change regulations of the VH-CATT gear curvatures were obtained in the meshing process.It could be seen from the research results that the change trends of principal curvature of the concave tooth surface and convex tooth surface are consistent in the tooth line direction,but have the little differences.The principal curvatures gradually increase in the tooth profile direction,but the increase range is just the opposite,and the change trend of principal curvature is consistent with the convexity of profile.The Gauss curvature and mean curvature of the gear pair are very small,and the change range is also small in the meshing process,and no mutation occurs.It was proved that the tooth surfaces of VH-CATT gear pair are smooth and continuous.The main value of induced curvature is close to zero in the tooth line direction,and is the negative in the tooth profile direction.It was also proved that the interference phenomenon of gears don''t occur in the meshing process.The research results of curvature,Showed that the VH-CATT gear drive has good performance and stable transmission.