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BK + KJ BC KC + KJ a cot y + a tan (f - a) a (e - cot y) a cos A
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Fig 314: Strain geometry in metal cutting
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e = 2 cot 2y = cot f + tan (f - a) (4)
Solving this equation,
Precision Engineering
337 Piispanen s Shear Strain Model (Card Model)
Shear strain = =
s y
AB AD BD = = DC CD CD
= tan (f + g ) + cot f
Fig 315: Piispanen s shear strain model
338 A Graphical Method to Construct Merchant s Circle
Merchant s Force Circle is a method for calculating the various forces involved in the cutting process This will be first explained with vector diagrams, which in turn will be followed by a few formulas The procedure to construct a Merchant s force circle diagram (using drafting techniques/ instruments) is as follows:
Mechanics of Materials Cutting
1 Set up an x y-axis labelled with forces, and the origin at the centre of the page The scale should be selected so that it is enough to include both the measured forces The cutting force (Fc) is drawn horizontally, and the tangential force (Ft) is drawn vertically (These forces will all be in the lower left hand quadrant) (Note: it is essential to have a square graph paper and equal x & y scales) 2 Draw the resultant (R) of Fc and Ft 3 Locate the centre of R, and draw a circle that encloses vector R If done correctly, the heads and tails of all three vectors will lie on this circle 4 Draw the cutting tool in the upper right hand quadrant, taking care to draw the correct rake angle (a) from the vertical axis 5 Extend the line that is the cutting face of the tool (at the same rake angle) through the circle This now gives the friction vector (Ff ) 6 A line can now be drawn from the head of the friction vector to the head of the resultant vector (R) This gives the normal vector (FN) Also add a friction angle (t) between vectors R and N As a side note recalls that any vector can be broken down into its components Therefore, mathematically, R = Fc + Ft = Ff + FN 7 We next use the chip thickness, compared to the cut depth to find the shear force To do this, the chip is drawn before and after the cutting is done Before drawing, select some magnification factor (eg, 200 times) to multiply both values by Draw a feed thickness line (t1) parallel to the horizontal axis Next draw a chip thickness line parallel to the tool cutting face 8 Draw a vector from the origin (tool point) towards the intersection of the two chip lines, stopping at the circle The result will be a shear force vector (F s) Also, measure the shear force angle between Fs and Fc 9 Finally, add the shear force normal (Fn) from the head of Fs to the head of R 10 Use a scale and a protractor to measure all distances (forces) and angles The resulting diagram is shown in Figure 316
Fig 316: Geometric solution of Merchant s circle
Precision Engineering Sample calculation Orthogonal machining Cutting tool : HSS, 18-4-1 with 10% cobalt Workpiece : 04% C steel AISI 1040 Cutting condition : Vc = 30 m/min; Feed = 04 mm/rev; DOC = 5 mm Cutting forces : Fc 3550 N; Ft = 1030 N Tool geometry : g = 25 ; a1 = 6 ; a2 = 6 ; K = 0 Thus, Friction force, Ft = Ft cos g + Fc sin g = 1030 cos (25 ) + 3550 sin (25 ) = 2430 N Normal force, FN = Fc cos g Ft sin g = 3550 cos (25 ) 1030 sin (25 ) = 2790 N Coefficient of friction, m = Friction angle, Chip ratio, Shear angle,
F 2430 = 087 = N 2790 t = tan 1 m = tan 1 (087) = 41 2
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