A method for determining the actual coordinate system of spiral bevel gears by the measured positions of eight tooth surfaces

Abstract

Abstract The coordinate system in the measurement of spiral bevel gear (SBG) by gear measuring center (GMC) is discussed and the misalignment between the actual coordinate system and the theoretical coordinate system of the SBG is analyzed. The misalignment will cause the measurement error and cannot be neglected in the SBG precision measurement. To calculate the misalignment and accurately establish the actual coordinate system, a novel re-fitting method by the measured positions of eight tooth surfaces is proposed. First, eight sets of gear axis direction vectors and original points can be obtained from the measurement of eight tooth surfaces, by the single tooth surface matching (SSM) method. Second, the actual axis vector and original point of the gear can be obtained, by re-fitting those eight sets of gear axis vectors and original points. Then, the actual coordinate system can be established by the actual gear axis direction vector and the original point. The least square method is used to calculate the original point, and a basis vector fitting method is proposed to solve the gear axis direction vector. Finally, two groups of preset coordinate system deviation tests are carried out to verify the accuracy and feasibility of the fitting algorithm. The results show that the fitting error of the original point is far less than 1×10^(-1) μm and the fitting error of the angle is far less than 1×10^(-7) degree, which are superior to the design accuracy of the GMC. Therefore, the proposed algorithm can be applied for improving the measurement accuracy of current GMC.