OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 5 4 3 Ta b l e 3 Dimensional analysis Parameter Representation Power consumption (Pc) (Watt) M1 L2 T-3 Material removal rate (MRR) (mm3/s) M0 L3 T-1 Density of the material (ρ) (kg/m3) M1 L-3 T0 Vibrational amplitude (A) (µm) M0 L1 T0 Frequency of vibration (F) (kHz) M0 L0 T-1 Here, “n” is 5 and “m” is 3 and hence in view of the same, (n-m = 2) i.e., π₁ and π2 are the two dimensionless groups that will be obtained. Now, Taking MRR, ρ and A as the quantities which directly go in π1, and π2 respectively, we obtain: 1 1 1 1 [ ] [ ] [ ] . a b c c MRR A P π = × ρ × × Hence, 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 3 0 0 1 0 2 3 0 0 0 3 1 1 3 1 2 3 (1 0 3 1 1 ) (3 3 2) ( 3) 0 1 0 0 [ ] [ ] [ ] [ ] [ ]; [ ] [ ]; [ . ] a b c a b c b a b c a M L T M L T M LT M L T M L T L T M L L M L T M L M L M T T L T - - - - - + - + + - - - × × × × × × = = = By equality, it can be found that a1 = -3, b1 = -1, and c1 = 4. Hence, we get, 3 1 4 1 [ ] [ ] [ ] . c MRR A P - - π = × ρ × × In similar way, 2 2 2 2 [ ] [ ] [ ] ; a b c MRR A F π = × ρ × × 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 3 1 1 3 0 0 1 0 0 0 1 0 0 0 3 1 1 3 1 0 0 1 (3 3 ) ( 1) 0 0 0 [ ] [ ] [ ]; [ [ ] [ ] ] [ ] [ ]; . a b c a b c b a b c a M L T M L T M L T M LT M L T M L T L T M L L M L T M L T M L T - - - - - - - + - - × × × × = × × = = By equality, it can be found that a2 = -1, b2 = 0, and c2 = 3. Hence, we get, 1 0 3 2 [ ] [ ] [ ] . MRR A F - π = × ρ × × This can now be written as, 1 2 [ ] , n k π = π where k and n are constants. { } 3 1 4 1 3 [ ] [ ] [ ] [ ] [ ] , n c MRR A P k MRR A F - - - × ρ × × = × × where k and n are constants. The material removal rate (MRR) is a product of a cutting speed (V), feed (f), and depth of cut (d). After simplifying the term, the power consumption can be represented as: (3 ) (3 4) ( ) . n n n c P k fd V F A - - = ρ
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