Dynamics of Honing of Deep Cylindrical Holes with Memory Effects in the Frictional Interaction

In this paper a semi-analytical methodology is presented of the mathematical modelling of abrasive machining of deep cylindrical holes. The modelling accounts for the memory effects in the frictional interaction that governs the process. The mathematical model is presented in the form of a non-autonomous system of differential-difference equations of variable structure. The model is investigated using mapping of the Poincare surface, the boundaries of which vary in time in correspondence with the functional dependence of the coefficient of static friction. To this end an original semi-analytical approach is developed for the identification of the fixed points that correspond to periodic motions of arbitrary complexity including chaotic one. Obtained with the help of developed software package in a high-level language, the bifurcation diagrams enabled an in-depth analysis of the main regimes of abrasive machining in the presence of memory effects in the frictional interaction. These regimes were previously unknown when using frictional models without memory effects. In particular, the paper shows that imperfect smoothness of surfaces after honing is due to multiple temporary relative rests of the hone and the workpiece.