Mathematical Modeling of the Accuracy of Gear Cutting at the Gear-Shaping Machines

Abstract

This paper examines the impact of the geometric and kinematic errors of gear shaping machines for accuracy of processed wheels. The equation of the machined surface is proposed based on coordinate transformation matrices and equations of involute surface of the tool in the form of a matrix equation. Impact of errors of the technological system elements is displayed by a matrix of variations, components of which are the projection of errors on the axis. The errors of machine units are determined by testing for geometric, kinematic accuracy, as well as for stiffness and thermal deformation. By combining the variation matrices of each unit and the matrix equation of the surface being processed, the equation of the output error of the machine in the form of a column matrix is displayed. The components of this matrix are brought to the machined surface through a projection on the line of machining action. As a result, the equation of deviations of the real shape of the gear from the nominal shape is displayed. The components of this equation are brought to the tangential and to the radial components. Next, the obtained values are translated into a standardized deviation of the gear accuracy. This makes it possible to predict the possibility of obtaining products specified accuracy on this machine, as well as to control the specific parameters of the technological system to ensure the accuracy and stability of the process.