OBRABOTKAMETALLOV Vol. 25 No. 4 2023 technology from the workpiece has a finite value. Models also do not calculate the volume of metal removed during microcutting; the removed metal is not summed and it is unclear when the process of removing a given metal volume will end. A significant omission in the discussed models is that the cutting force does not change depending on the wheel hardness. Thus, there is no scientific approach establishing the relationship of the cutting forces and the depths of the metal cut by a single grain with the stock removal and the cutting force that occurs during grinding with a wheel as a whole. As a result, there is no analytical engineering model, which calculates the relationship between cutting force and cutting depth in flat grinding operations. It is challenging to obtain an adequate cutting force model for flat grinding operations because it is necessary to establish the interrelation between the machine parameters of the macrocutting modes (feed, cutting speed), adjusted on the machine control panel, and parameters of the microabrasive grain cutting conditions, associated with plastic metal deformation in the shear zone, the physical and mechanical properties of the metal being machined, the back rake angle and front clearance of the grain cutting edge, blunting of the cutting edge along the back edge grain surface, cutting speed, the parameters of the contact zone of the cutting tool with the workpiece, etc. For flat grinding in particular, the volume of metal removed from the workpiece and the parameters of the three machine feeds (depth, transverse, longitudinal) should be linked to microcutting parameters: the metal shear angle in the plastic deformation zone; the length and area of contact between the wheel and the workpiece; the variable depth of the metal cut by single grains of the wheel; the stochastic nature of metal being removed by an excess number of wheel grains; the geometry of the cutting grains of the grain cutting edge, the blunting area on the back edge surface; the strength of the metal being machined; as well as the total microvolume of metal removed by all grains. The purpose of this paper is to establish the relationship between the cutting force, the cutting depth, and the volume of metal removed by single grains in flat grinding. Let us consider the main modeling stages for calculating the cutting force in flat grinding based on the equality of the volume of metal removed by a set of single grains and the same volume of metal removed by the grinding wheel as a whole (the equality of the volumes of metal removed). Research methods We will use the model of the cutting force of a single abrasive grain as the foundation for modeling cutting force during flat grinding. Metal is removed by a single grain at energy costs, most often expressed in the form of grinding energy or power. The relationship between energy and power during plastic metal deformation, as established by Korchak are expressed as [1]: ω = σe ω ∫∫∫ A d , (1) ω = = σe ω ∫∫∫ dA N d dt , (2) where А is the energy expended on the deformation of the metal volume ω, N‧m (J); N is the power required to deform the metal ω, N‧m/s (W); σ is the stress intensity in the moving volume of the deformed metal, N/m2; e is the deformation rate intensity, s-1; e is the deformation degree intensity; ω is the volume of deformed metal, m3; t is the deformation time of the metal volume ω, s. The assumptions made by Korchak are not bound to a specific type of grinding; it is also valid for the developed model of the cutting force occurring during flat grinding [1]. Fig. 1 shows the calculation scheme of a metal cutoff by the cutting edge of the abrasive grain of a wheel with a blunting area lj. Korchak adopted a free cutting pattern due to the minor influence of edge effects along the length of the cutting edge, which exceeds a hundredfold the depth of cut by a single grain of a wheel [1]. It is assumed that the shear zone is a parallelogram, since the temperature and speed grinding parameters (cutting speed of 30...60 m/s
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