Calculation of the inverse involute function and application to measurement over pins

Measurement over pins plays a crucial role in ensuring the quality in gear manufacturing, which requires the evolution of the inverse involute function. In this paper, an efficient algorithm for calculating the inverse involute function is proposed. Firstly, the initial approximate solution of the pressure angle near 0 rad is established by using Padé approximation and one step of Schröder iteration. Then the involute equation is transformed into another form by introducing a variable transformation, and the initial approximate solution of the pressure angle near π/2 rad is constructed by using the same method. The piecewise initial approximate solution over the entire interval of expanded angle is obtained by combining these solutions. The accuracy of the initial approximate solution can be significantly improved by using one step of Schröder iteration again. A numerical example of two-pin measurement is presented to show the superiority of our method over traditional iterative approach. The algorithm has high efficiency and accuracy, and is a useful tool for designers to accurately calculate the gear system.