OBRABOTKAMETALLOV technology Vol. 25 No. 4 2023 where PY, PZ are the radial and tangential components of the cutting force during grinding, respectively; P∑YР, P∑ZР are the radial and tangential components of the cutting force during grinding when all grains are absolutely sharp, respectively; P∑YТ, P∑ZТ are the radial and tangential components of the cutting force from the blunting areas during grinding, respectively. 5) the equality of energy spent on the removal of the same metal volume Wall by a set of single abrasive grains and by the grinding wheel as a whole. Let us establish the relationship of energy on removing the metal volume Wall, separately for a group of grains and the wheel as a whole. According to the structure of model (12) of the cutting force PZj for a single grain, the total energy Aj (Н‧m) expended by the cutting force PZ consists of the energy AРj expended by the force PZРj and the energy AТj expended by the force PZТj, i.e.: = + . j Pj Tj A A A (20) Similarly, the total energy А∑ from the tangential component of the cutting force for the wheel consists of the energy A∑Р expended in total by the forces PZРj from all the cutting grains in the contact zone and the energy A∑Т expended in total by the forces PZТj from all the cutting grains in the contact zone, i.e.: Σ Σ Σ = + . P T A A A (21) Then, the sums of the grinding energy from the cutting forces of single grains are equal to the grinding energy of the wheel from the total cutting force. Let us find the grinding energy expended on plastic metal deformation in the shear zone by the force PZPj at a distance L of the length of the machined surface (fig. 2, a): ω = = σe = σe β 1 sin s j j Pj ZPj k k a mb A P L L L V V , (22) Σ = = = = σe σe σe = = = = ω = β ∑ ∑ ∑ ∑ 1 1 1 1 1 sin J J J J s j P Pj ZPj J k k k j j j j a mb LW L L A A L P V V V ïð , (23) where L is the length of the machined surface, m; ωj is the volume of metal removed during cutting with the j-th single grain, m3. To establish the relationship between the grinding energy А∑P expended by absolutely sharp grains to remove metal, we should find a common energy parameter inherent in single grains and the wheel as a whole. Metal removal rate Q, m3/s is an appropriate parameter for our purposes. When cutting with the j-th single grain, the metal removal rate Qj is equal to the ratio of the metal volume ωj in the shear zone to its deformation time during shear time ∆t, which is equal to a sharp grain moving across a distance hj at a speed of Vk: ω = ∆ j j Q t , (24) at e e = ∆ t , (25) where Qj is the metal removal rate when cutting with the j-th single grain, m3/s; ∆t is the deformation time of the metal volume ωj during the shear time, s. Let us take into account formulas (24) and (25) in the equation: Σ = = σe σe = ω = ∑ ∑ 1 1 J J P J J k k j j L L A Q V V , (26)
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