OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 4 2023 Experiments were performed at constant feed and depth of cut. The density (ρ) of a material is also constant. Therefore, let’s define k1 as a new constant, which is a product of k, ρ, f, and d. Hence, the final model to predict the power consumption under UVAHT is shown below. (3 ) (3 4) 1 . n n n c P k V F A - - = The constant “n” can be obtained by calibrating the model with the experimental power consumption values under UVAHT, obtained at different cutting conditions as depicted in table 2. 1.5987 1.4013 0.2039 0.00222 . c P V F A = (1) Modelling tool wear (Vb ) Tool wear is determined by four parameters: cutting speed (V), material hardness (H), vibrational amplitude (A), and frequency of vibration (F). Using M (mass), L (length), and T (time) as the fundamental dimensions, the dimensions of the previous values will be as follows: given that Vb = φ (V, H, A, F), for “n” is 5, and “m” is 3, and therefore n-m = 2. Thus, π1 and π2, which are two dimensionless groups, can be defined. Now, taking V, H and A as the quantities that are directly included in π1 and π2, respectively, we get 1 1 1 1 [ ] [ ] [ ] . a c b V H A Vb π = × × × Hence, 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 1 ( ) ( 1) 0 1 1 2 0 ( 2 ). 0 0 0 2 [ ] [ ] [ ] [ ] [ ]; [ ] [ ] [ ] ; a b c a b c b a b c a b M L T M L T M LT M LT M L T LT M L T L L M L T T T M L M L - - - - + + - - - - - × × × = = × = × × By equality, it can be found that a1 = 0, b1 = 0, and c1 = -1. Hence, we get: 0 0 1 1 [ ] [ ] [ ] . V H A Vb - π = × × × In similar way: 2 2 2 2 [ ] [ ] [ ] ; a b c V H A F π = × × × 2 2 2 2 2 2 2 2 2 2 2 2 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 1 1 1 ( ) ( ) ( 0 2 1) 0 0 0 1 1 2 0 0 2 [ ] [ ] [ ] [ ] [ ]; [ ] [ ] [ ]; . a b c a b c b a b c a b M L T M L T M LT M L T M L T M LT M L T L T M L T M L L T T - - - - - - + - - - - - - × × × × × × = = = By equality, it can be found that a2 = -1, b2 = 0, and c2 = 1. Hence, we get: 1 0 1 1 [ ] [ ] [ ] . V H A F - π = × × × This can now be written as: 1 2 [ ] ; n k π = π { } 1 1 1 [ ] [ ] [ ] . n A Vb k V A F - - × = × × After simplifying the term, it can be represented as: (1 ) ; n n n Vb kV A F - + = 0.1967 0.8033 0.1967 0.011336 . Vb V A F- = (2)
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