Study on geometric theory of involute gear

The geometric configuration of involute gear teeth has a great influence on the performance of involute gear transmission system. The use and transmission performance of involute gears are different with different tooth shapes. For the tooth profile configuration of involute gear, the conjugate surface theory is more mature at present. Based on the conjugate surface theory, the tooth profiles of involute gears such as involute and cycloid have been developed. The theory of conjugate surfaces has been comprehensively discussed by domestic scholars. Wu Daren and Wu Xutang expounded the conjugate surface theory comprehensively and in detail, and gave the algorithms of meshing equation, conjugate surface, meshing limit position, induced normal curvature and so on. Involute gears based entirely on conjugate surface theory are in line contact during meshing, while line contact involute gear pairs are prone to edge contact, and are usually sensitive to errors. Therefore, since the last century, scholars have been committed to the research of point contact involute gear pairs.

At present, the development is relatively mature and widely used. The point contact involute gear is mainly circular arc gear. Circular arc gear is a kind of cylindrical gear. Wildhaber proposed the tooth profile of circular arc gear in 1926. In 1955, Novikov of the Soviet Union put forward the point meshing transmission theory using the end face circular arc tooth profile. This involute gear transmission is in point contact along the tooth width direction or tooth height direction, and its bearing capacity and efficiency are high. However, this end face circular arc tooth profile was difficult to manufacture at that time, so it was not applied in industrial production. In fact, circular arc gears are widely used. After kudrjavcjsev of the Soviet Union proposed and solved the problem of using hobs to process circular arc gears according to the hobbing method, the normal tooth profile of the helical rack tool corresponding to this circular arc gear (i.e. the basic tooth profile of involute gear) is circular arc, and the end face of involute gear machined by this circular arc hob is not circular arc tooth profile, Instead, the normal tooth profile passing through the center line is a circular arc. Litvin et al. Designed a new Novikov wildhaber involute gear. Compared with the original involute gear pair, the new involute gear pair can significantly reduce the vibration and noise caused by assembly error, and its transmission error is in the form of parabola. At the same time, the involute gear pair can be processed by cutter head or worm. In addition, The author also uses the finite element method to simulate the new involute gear pair. The results show that the involute gear pair can be used in high-speed transmission.

In addition to the traditional point meshing gear transmission based on circular arc gear, many new point meshing involute gear pairs have also been developed. In foreign countries, Litvin et al. Studied the linear discontinuous transmission error caused by assembly error, and proposed a method to absorb the assembly error by using the preset parabola transmission error after trimming the line contact involute gear pair into the point contact involute gear pair, This paper compares two types of face gear transmission: the meshing of cylindrical gear with involute profile and its conjugate face gear, and the meshing of cylindrical gear with non involute profile (parabola) and its conjugate face gear. The research shows that the new face gear transmission has the following advantages: longitudinal support contact, avoiding edge contact and reducing contact stress. Using non-uniform rational B-spline as the error function, Yeh and others in the United States designed a point line meshing gear. This involute gear can achieve higher tooth surface isomorphism, and the contact stress and bending stress of the tooth surface are significantly reduced. Compared with the involute gear under the same size, it can bear greater load. Japanese scholar Xiao Shoumian proposed a new type of involute gear called Logix tooth profile. The tooth profile is connected by many micro segment involutes, adopts symmetrical convex concave meshing form, and makes the relative curvature of the joint point of micro segment involute zero during meshing; Logix gear overcomes some shortcomings of circular arc gear and involute gear, basically realizes rolling friction, enhances the contact fatigue strength of involute gear surface, and can be designed as a pinion with very few teeth.

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