A Novel Analytical Explicit Method to Calculate Formed Wheel and Tooth Flank of Involute Gears in Profile Grinding Process

Gear drive is a common and efficient way to transfer power and motion. To ensure the machining accuracy of gears, the tooth flanks are formed by profile grinding technology in some cases. In the profile grinding process, the calculation of wheels using the information of gears named as the forward-calculation process and obtaining gears based on wheels (the backward-calculation process) traditionally adopt numerical ways. It is always time consuming and large code quantity. To conquer these drawbacks, this article presents an analytical method using the envelope theory to compute the contacting curves that are the basis of getting tooth flanks or wheels in the forward- or the backward-calculation process. For the forward-calculation process, the tooth flank is expressed in the form of an extended straight-line surface that can be taken as the generating line moving along the helix curve. The normal vector for an arbitrary point on the generating line is the same. By using this characteristic, the contacting curve can be explicitly gained as the function of only one parameter. Similarly, in the backward-calculation process, the formed wheel is expressed by a cross section rotating about its axis. For this type of surface, the guide curve is a circle, and the normal vectors of points on the guideline insect with the axis at the same point. Taking advantage of this principle, the contacting curve can be analytically expressed by only one unknown parameter. To verify the validity of the proposed method, some examples and comparative experiments are performed. The results show that the presented method is correct. When compared with the classical numerical way, the time span for the proposed method is 15 times less than that for the numerical way. When compared with the practical grinding wheel and the practical gear, the maximum errors are 0.18 mm and 0.0099 mm, respectively. The proposed method can be served as one of the universal ways to generate formed wheels or involute gears in the profile grinding process.