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

Abstract

Abstract Measurement over pins plays a crucial role in ensuring the quality of 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. A piecewise initial approximate solution over the entire interval of expanded angles is obtained by combining these solutions. Again, the accuracy of the initial approximate solution can be significantly improved by using one step of Schröder iteration. A numerical example of a two-pin measurement is presented to demonstrate the superiority of our method over traditional iterative approaches. The algorithm is highly efficient and accurate, and is a useful tool for designers to accurately calculate gear systems.