OBRABOTKAMETALLOV technology Vol. 25 No. 1 2023 Despite the wide variety of models describing the processes of abrasive diamond processing, there are practically no scientifically-based recommendations in the modern literature that make it possible to guarantee the receipt of the specified product quality parameters in non-stationary conditions of the technological process [8–12]. Therefore, a comprehensive study of the regularities of the surface geometry generation processes, the development of mathematical models will ensure the creation of highly efficient technological processes and optimal designs of abrasive tools on this basis. The analysis of works in the field of grinding theory allows concluding that all existing models of abrasive-diamond machining processes can be divided into two classes. The first class (pulse models) includes mathematical dependencies that simulate the impact of single abrasive grains on the workpiece. The processed surface is formed as a set of grain traces, which in a section perpendicular to the direction of the cutting speed are identical to the profile of the radius of the vertex of the abrasive grain, for example, mathematical models, developed by I.M. Brozgol, D.V. Korolev, E.N. Maslov, Yu.K. Novoselov, V.A. Nosenko and others [13–17]. The second class (geometric models) includes mathematical dependencies that simulate the effect on the workpiece by a set of elementary cutting profiles. On this basis, work has been carried out on the mechanisms of surface roughness formation, for example, mathematical models developed by Yu.R. Witenberg, Yu.V. Linnik, S.A. Popov, V.A. Shchegolev, A.P. Husu and other scientists [18–23]. In real production conditions, the technological modes, recommended in the above-reviewed works, and reference literature do not reflect the declared qualities, due to the fact that it does not take into account many factors inherent in the process of finishing grinding, for example, its stochastic nature, changes in its dynamic properties, an increase in mutual vibrations of the tool and the workpiece, which appear due to changes in the state of the technological system, for example, an increase in machine vibrations due to uneven tool wear, etc. All previously developed models have a limited scope of application, it does not take into account the fact that the appearance of vibrations leads to fluctuations in the depth of grinding, with accidental grains contact with the material being processed, where one group of grains cuts off the material, the other falls into the trace of scratches left by previous grains, etc. This leads to changes in the values of material removal, surface roughness and other parameters of the technological system, which directly affects the accuracy of processing and the quality of the processed surfaces. To compensate for calculation errors in real production conditions, various technological approaches are used, for example, tools with soft binders are used, feed rates are reduced and other methods are used, which reduce the efficiency of the operation and increases the cost of manufactured products. Advanced approach to problem is to continue research of grinding operations, (in particular internal), in the course of which it is necessary to identify and describe the relationship between input factors and output indicators of the process. Based on the established relationships, it is necessary to build mathematical models that adequately simulate the grinding process, taking into account the mutual vibrations of the tool and the workpiece. To date, one of the most time-consuming technological processes is the grinding operation. The amount of products where internal grinding was used as a finishing machining is not inferior to the amount of products, processed by the external method. However, internal grinding is more difficult due to the heavy flow of the machining process and the lower rigidity of the cutting tools. In connection with the above, the purpose of this paper is to develop mathematical models that establish the relationship between the working modes and the current parameters of the contact zone during fine grinding of precise holes, taking into account the mutual vibrations of the tool and the workpiece. Research methodology The scheme of the finishing process of the hole (internal grinding) is shown in Fig. After inserting a workpiece into a chuck, the tool and the workpiece are set to rotate at a circumferential speed Vu and Vk accordingly. When moving the grinding head in the direction of radial feed Sy, the difference between the radius vectors of the workpiece and the tool becomes less than the center distance Ai, and an area of interpenetration of the tool into the workpiece material – the contact zone – is formed [24].
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