[ ] [ ] 3.3 Given: turning corner radius, r ε = 0 mm lead angle, ψ r = 15 back rake angle, γ p = 5 side rake angle, γ f = 5

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Dwayne Powell
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1 33 Given: tuning cone adius, ε = 0 mm lead angle, ψ = 5 back ake angle, γ p = 5 side ake angle, γ f = 5 initial wokpiece diamete, D w = 00 mm specific cutting and thust enegy models feed ate, f = 020 mm/ev depth of cut, d = 075 mm spindle speed, n s =000 pm a) he chip aea is appoximated as the poduct of the feed and depth, ignoing the actual absence of the tiangula cusp left on the machined suface Substituting known values, the final esult (in mm 2 ) is a = (0020)(075) = 05 he mateial emoval ate, by definition, is the poduct of the chip aea and the cutting speed he cutting speed is the poduct of the spindle speed and the wokpiece cicumfeence he aveage cutting speed (aveaged acoss the tooth pofile contact), (in m/min) is ( ) [ π ] V = ns π Dw d = 000 (00 075) = 38 Of couse, since the depth of cut is vey small compaed to the wokpiece diamete, the suface speed at the oute diamete (in m/min), V n πd = 000 π(00) = 342, s w could just as soon be used as a good appoximation onveting cutting speed fom 38 m/min to 3,800 mm/min, the final esult (in mm 3 /min) is v = (05)(3,800) = 46, 770 (05)(34, 200) = 47,30 b) he cutting and thust foces ae each the poduct of the espective specific enegy and the chip aea found above omputing the specific enegies fom the empiical models given equies uncut chip thickness, cutting speed (found above fo mateial emoval ate calculation) and nomal ake angle he uncut chip thickness fo a zeo cone adius is h = f cos ψ, f = f ev Substituting known values, the esult (in mm) is h = 05cos(5 ) = 0932 he nomal ake angle is γ = tan tanγ cos λ, n whee the othogonal ake angle is γ o = tan tanγ f + tanγ p sinψ and the inclination angle is λ = tan tanγ p tanγ f sinψ Substituting known values, the inclination angle is λ = tan tan( 5)cos(5) tan( 5)sin(5) = 354 and, subsequently, the othogonal ake angle is = tan tan( 5)cos(5) + tan( 5)sin(5) = 62 γ o heefoe, the nomal ake angle is = tan tan( 62 )cos( 354 ) = 6 γ n o

2 Substituting known values (h in mm, V in m/min and γ n in adians ( 0066 ad)) into the specific enegy models, the specific enegies (in N/mm 2 ) ae ( 0066) u = 350 (0932) (38) e = 2638 and ( 0066) u = 75 (0932) (38) e = 555 Using these and the chip aea computed ealie, the final esults (in N) ae F = u a = (2638)(05) = 396 and F = u a = (555)(05) = 233 c) he tansfomations fom the edge-local cutting and thust foces to the tooth-local tangential, longitudinal and adial ae given in the text as Fan = F, FLon = F and FRad = F sinψ Substituting known values, the final esults (in N) ae F = 396, F = 233cos(5 ) = 225 and F = 233sin(5 ) = 603 an Lon d) Intoducing a non-zeo cone adius would decease the uncut chip thickness used in the specific enegy models, which would incease the specific enegies, moe so in the thust diection A non-zeo cone adius would also damatically change how the thust foce is oiented between the adial and longitudinal diections heefoe, intoducing a non-zeo cone adius would be expected to affect the foce components would be affected in the following ways: Foce omponent Magnitude Diection utting Incease vey slightly None, diection is by definition hust Incease slightly moe than F Rad Much moe towads the adial diection angential Identical to F None, diection is by definition Longitudinal Radial Likely decease since the diection of F is moe stongly moved away fom the longitudinal diection than the magnitude of F is deceased Inceases since both the incease in magnitude of F and the change in diection of F both seve to incease in the adial foce None, diection is by definition None, diection is by definition e) he final wokpiece diamete is the oiginal diamete less twice the depth of cut Substituting known values, the final esult (in mm) is D = 00 2(075) = 985 wf 34 Given: tuning cone adius, ε = 0 mm lead angle, ψ = 5 back ake angle, γ p = 5 side ake angle, γ f = 5 initial wokpiece diamete, D w = 00 mm specific cutting and thust enegy models feed ate, f = 020 mm/ev depth of cut, d = 075 mm spindle speed, n s =000 pm

3 a) he exact chip aea is the poduct of the feed and depth less the cusp left on the machined suface When the cone adius is not zeo, the exact chip aea is /2 2 2 f 2 f ε f a = f d 2 2 ε ε sin 4 4 ε Substituting known values, the final esult (in mm 2 ) is / (05) (0) 05 a = (0020)(075) 2 2(0) (0) sin = 4 2 2(0) he mateial emoval ate, by definition, is the poduct of the chip aea and the cutting speed he cutting speed is the poduct of the spindle speed and the wokpiece cicumfeence he aveage cutting speed (aveaged acoss the tooth pofile contact), (in m/min) is ( ) [ π ] V = ns π Dw d = 000 (00 075) = 38 Of couse, since the depth of cut is vey small compaed to the wokpiece diamete, the suface speed at the oute diamete (in m/min), V n πd = 000 π(00) = 342, s w could just as soon be used as a good appoximation onveting cutting speed fom 38 m/min to 3,800 mm/min, the final esult (in mm 3 /min) is v = (049666)(3,800) = 46,666 (049666)(34,200) = 47,025 b) he expession fo equivalent lead angle unde olwell s method is y2 y ψ = tan x x2 he depth of cut is geate than the tansition depth of cut d = ε sin ψ = (0) sin(5 ) = 074, t ( ) ( ) baely! heefoe, the expessions fo the x and y tems ae Substituting known values, ( ) /2 ε x =+ f, y = f 2, 4 ε x2 = ε d, y2 = + ( d ε ) tanψ ( ) /2 x =+ (0) (05) 4 = 0995, y = 05 2 = 0, 0 x 2 = = 025, y 2 = + (075 0) tan(5 ) = 0968 cos(5 ) Substituting these into the equivalent lead angle expession, the final esult is 0968 ( 0) ψ = tan = c) By definition of equivalency, the aveage uncut chip thickness is the chip aea divided by the equivalent width of cut he chip aea was found above; the equivalent width of cut, noting again that the depth of cut is geate than the tansition depth, is

4 ( ψ ) d ε sin w= ε sin ( f 2ε) π 2 ψ + + Substituting known values, the esult is 2π 075 (0) sin(5 ) w = (0) sin ( 05 2(0) ) + π 2 (5 ) cos(5 ) = = 483 Subsequently, the final esult (in mm) is ( ) h = = d) he cutting and thust foces ae each the poduct of the espective specific enegy and the chip aea found above omputing the specific enegies fom the empiical models given equies aveage uncut chip thickness (found above), cutting speed (found above fo mateial emoval ate calculation) and nomal ake angle, whee it is computed using the equivalent lead angle he nomal ake angle is γ = tan tanγ cos λ, n whee the othogonal ake angle is γ o = tan tanγ f + tanγ p sinψ and the inclination angle is λ = tan tanγ p tanγ f sinψ Substituting known values, the inclination angle is λ = tan tan( 5 )cos(55 ) tan( 5 )sin(55 ) = 24 and, subsequently, the othogonal ake angle is = tan tan( 5 )cos(55 ) + tan( 5 )sin(55 ) = 695 γ o heefoe, the nomal ake angle is = tan tan( 695 ) cos(24 ) = 695 γ n Substituting known values (h in mm, V in m/min and γ n in adians ( 023 ad)) into the specific enegy models, the specific enegies (in N/mm 2 ) ae o ( 023) u = 350 (0055) (38) e = 3092 and ( 023) u = 75 (0055) (38) e = 2289 Using these and the chip aea computed ealie, the final esults (in N) ae F = u a = (3092)(049666) = 463 and F = u a = (2289)(049666) = 343 he tansfomations fom the edge-local cutting and thust foces to the tooth-local tangential, longitudinal and adial ae given in the text as Fan = F, FLon = F and FRad = F sinψ Substituting known values, the final esults (in N) ae F = 463, F = 343cos(5 ) = 96 and F = 343sin(55 ) = 28 an Lon e) he change in chip aea is negligible; but, since the aveage uncut chip thickness is substantially (45%) less than f in the pevious poblem, the specific enegies ae substantially highe (7% fo the cut- Rad

5 ting diection, 47% fo the thust diection) hee; theefoe, F and F ae highe by the same pecentages In addition, since the equivalent lead angle is much (267%) geate than the lead angle, F Rad inceases enomously (366%) while F Lon deceases slightly (3%), even though F has inceased

Chapter 2: Basic Physics and Math Supplements

Chapter 2: Basic Physics and Math Supplements Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate