Investigation of stationary trajectories with associated milling by spur gears

Abstract

When milling in a steady state, unlike, for example, turning, there are periodic elastic deformation displacements of the tool relative to the workpiece along the machining path. Instead of an equilibrium point, we consider a certain closed trajectory of elastic deformations. This is the trajectory to which all the trajectories approach asymptotically, while forces and deformations that mutually affect each other through the mechanism of changing the area of the cut-off layer are redistributed. The article proposes a mathematical apparatus and algorithms for calculating the trajectory of establishing a stationary state. The case of milling with spur mills is considered. A mathematical description of the “basic” dynamic model of the associated milling by the side teeth of the end mills for the complete non-stationary nonlinear connection formed by the milling process is presented. It differs from known models by considering spatial oscillations, taking into account the dependence of forces on the cutting speeds, taking into account the rates of elastic deformation displacements, taking into account the retardation of forces with respect to deformation displacements, and also by nonlinear damping caused by forces acting on the trailing edge of the tool. In addition, periodic changes in the parameters and the formation of the surface at the previous contact of the tooth are taken into account.