# Basic principle of conjugate curve gear meshing

In the traditional conjugate curve gear design theory, the tooth surfaces of a pair of conjugate curve gears that can mesh normally should be a pair of conjugate surfaces. Here, conjugate tooth surface refers to a pair of tooth surfaces that transmit motion according to a given law and always maintain continuous tangency. A tooth surface can always be generated by its conjugate tooth surface through envelope motion. The condition of continuous tangency of a pair of conjugate surfaces in the meshing process is that the relative velocity of the two surfaces at the contact point should be perpendicular to the common normal of the pair of conjugate surfaces made through the point. According to this principle, select any point on the given surface 1, find out the rotation angle and displacement required for the point to enter the contact position surface, and use the coordinate transformation method or vector rotation method to obtain the position of the contact point in the fixed space, that is, a corresponding point on the meshing surface of conjugate curve gear. At the same time, the corresponding points on surface 2 can also be obtained. By solving this point by point, the whole conjugate curve gear meshing surface and the surface 2 conjugate with surface 1 can be obtained.

For the meshing theory of space conjugate curve gear, it is similar to the meshing principle of conjugate surface gear. According to the conjugate curve gear meshing theory, the conjugate curve can be described as a pair of curves that move according to a given law and always keep in contact. A curve can always be generated by the joint motion of the points on the conjugate curve and the curve. One or more normal vectors can exist at any contact point of the conjugate curve. When the conjugate curve contacts at this point, the normal vector exists in the respective normal sections of the two curves at the same time. If a section curve with the aforementioned normal vector as the normal direction is established in the normal section of any point on the conjugate curve, the continuous surface composed of all such section curves will form a pair of tooth surfaces that can continuously and stably transfer motion and force. Such a pair of tooth surfaces can transfer motion according to the given law and always maintain tangency at the conjugate curve position.

It can be seen from the above description that the basic element of conjugate surface gear meshing theory is surface, while the basic element of spatial conjugate curve gear meshing theory is curve. Because in practical application, the curve does not have bearing capacity, therefore, when applying the spatial conjugate curve gear meshing theory to the gear tooth surface design, it is also necessary to connect the curve with the surface with bearing capacity, which is a great difference between the spatial conjugate curve gear meshing theory and the conjugate surface gear meshing theory in application.

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