Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
DEFORMATION PROCESSES IN 18 BULK DEFORMATION METALWORKING Review Questios 18.1
Wht re the the resons resons !hy !hy the bul" defor#tion defor#tion pro$esses pro$esses re i#portnt i#portnt $o##er$il $o##er$illy ly nd nd te$hnolo%i$lly& Aswe!. 'esons !hy the bul" defor#tion pro$esses re i#portnt in$lude the follo!in% (1) they re $pble of si%nifi$nt shpe $hn%e !hen hot !or"in% is used, (2) they hve positive effe$t on prt stren%t stren%th h !hen $old !or"in% !or"in% is used, used, nd () () #ost #ost of the pro$esses pro$esses produ$e produ$e little little #teril #teril !ste* !ste* so#e re net shpe pro$esses.
18.2 18.2
+#e +#e the the four four bsi bsi$$ bul" bul" defo defor#t r#tion ion pro$es pro$esses ses.. Aswe!. he four bsi$ bul" defor#tion pro$esses re () rollin%, (2) for%in%, () e-trusion, nd () !ire nd br dr!in%.
18. 18.
Wht Wht is rolli rollin% n% in the the $onte-t $onte-t of of the bul" bul" defor# defor#ti tion on pro$es pro$esses ses&& Aswe!. 'ollin% is defor#tion pro$ess in !hi$h the thi$"ness of the !or"pie$e is redu$ed by $o#pressive for$es e-erted by t!o opposin% rolls. he rolls rotte, thus pullin% nd si#ultneously s/ueein% the !or"pie$e bet!een the#.
18.
n rollin% rollin% of steel, steel, !ht !ht re re the differen$es differen$es bet!een bet!een bloo#, bloo#, slb, nd billet& billet& Aswe!. bloo# is rolled steel !or"pie$e !ith s/ure $ross se$tion of bout 150 ## by 150 ##. he strtin% !or" unit for bloo# is n in%ot heted in so"in% pit. slb is rolled fro# n in%ot or bloo# nd hs re$tn%ulr $ross se$tion of bout 250 ## by 0 ##. billet is rolled fro# bloo# nd hs s/ure $ross se$tion of bout 0 ## by 0 ##.
18.5 18.5
3ist 3ist so#e so#e of the produ$ produ$ts ts prod produ$e u$ed d on rollin rollin% % #ill. #ill. Aswe!. 'olled produ$ts in$lude flt sheet nd plte sto$", round br nd rod sto$", rils, stru$turl shpes su$h s 4be#s nd $hnnels.
18. 18.
Wht Wht is is drft drft in roll rollin in% % oper opert tio ion& n& Aswe!. 6rft is the differen$e bet!een the strtin% thi$"ness nd the finl thi$"ness s the !or"pie$e psses bet!een the t!o opposin% rolls.
18.7 18.7
Wht Wht is is sti$" sti$"in% in% in hot hot rollin rollin% % oper operti tion& on& Aswe!. Sti$"in% is $ondition in hot rollin% in !hi$h the surf$e of the !or"pie$e dheres to the rolls s the pie$e psses bet!een the rolls, $usin% severe defor#tion of the #etl belo! the surf$e in order to llo! pss%e throu%h the roll %p.
18.8 18.8
denti dentify fy so#e so#e !ys !ys in !hi$h !hi$h for$e for$e in in flt flt rollin rollin% % $n be be redu$ed redu$ed.. Aswe!. Wys to redu$e for$e in flt rollin% in$lude (1) use hot rollin%, (2) redu$e drft in e$h pss, nd () use s#ller di#eter rolls.
18. 18.
Wht Wht is t!o t!o4h 4hi% i%h h rol rolli lin% n% #ill #ill&& Aswe!. t!o4hi%h rollin% #ill $onsists of t!o opposin% rolls bet!een !hi$h the !or" is $o#pressed.
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
18.10 18.10 Wht Wht is rever reversin sin% % #ill #ill in rolli rollin%& n%& Aswe!. reversin% #ill is t!o4hi%h rollin% #ill in !hi$h the dire$tion of rottion of the rolls $n be reversed reversed to to llo! llo! the !or" !or" to pss throu%h throu%h fro# either either side. side.
18.11 18.11 :esides :esides flt rollin% nd shpe shpe rollin%, rollin%, identify so#e dditionl dditionl bul" bul" for#in% pro$esses pro$esses tht use rolls rolls to effe$t the defor#tion. Aswe!. So#e other pro$esses tht use rolls re rin% rollin%, thred rollin%, %er rollin%, roll pier$in%, pier$in%, nd roll for%i for%in%. n%.
18.12 18.12 Wht Wht is for% for%in in%& %& Aswe!. ;or%in% is defor#tion pro$ess in !hi$h the !or"pie$e is $o#pressed bet!een t!o dies, usin% i#p$t or %rdul pressure to for# the prt.
18.1
18.1 Why is is flsh desirble desirble in i#press i#pression ion die die for%in%& for%in%& Aswe!. :e$use its presen$e $onstrins the #etl in the die so tht it fills the detils of the die $vity.
18.15 Wht is tri##in% tri##in% opertio opertion n in the $onte-t $onte-t of i#pressio i#pression n die for%in%& for%in%& Aswe!. ri##in% is sherin% opertion used to re#ove the flsh on the !or"pie$e fter i#pression die for%in%.
18.1 Wht re re the t!o t!o bsi$ bsi$ types types of for%in% for%in% e/uip#ent e/uip#ent&& Aswe!. he t!o types of for%in% #$hines re h##ers, !hi$h i#p$t the !or" prt, nd presses, !hi$h pply %rdul pressure to the !or".
18.17 18.17 Wht Wht is isothe isother# r#ll for%i for%in%& n%& Aswe!. sother#l for%in% is hot for%in% opertion in !hi$h the die surf$es re heted to redu$e het trnsfer fro# the !or" into the toolin%.
18.18 18.18 Wht Wht is is e-tr e-trus usio ion& n& Aswe!. 9-trusion is $o#pression for#in% opertion in !hi$h !or"pie$e is for$ed to flo! throu%h die openin%, thus t"in% the $ross4se$tionl shpe of the die openin%.
18.1 6istin%uis 6istin%uish h bet!een bet!een dire$t nd indire$t indire$t e-trusion. e-trusion. Aswe!. n dire$t e-trusion, lso "no!n s for!rd e-trusion, #etl billet is loded into $ontiner, nd r# $o#presses the #teril, for$in% it to flo! throu%h die openin% t the opposite end of the $ontiner. n indire$t e-trusion, lso "no!n s b$"!rd e-trusion, the die is in$orported into the r#, nd s the r# $o#presses into the #etl billet, the #etl is for$ed to flo! throu%h the die openin% in dire$tion tht is opposite (b$"!rds) of the r# #otion.
18.20 +#e so#e so#e produ$t produ$tss tht tht re produ$ed produ$ed by e-trusion. e-trusion. Aswe!. Produ$ts produ$ed by $ontinuous e-trusion in$lude stru$turl shpes (!indo! fr#es, sho!er stlls, $hnnels), tubes nd pipes, nd rods of vrious $ross se$tions. Produ$ts #de by dis$rete e-trusion in$lude toothpste tubes, lu#inu# bever%e $ns, nd bttery $ses. 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
18.21 Why is fri$tion fri$tion f$tor in deter#inin deter#inin% % the r# for$e in dire$t dire$t e-trusion e-trusion but not f$tor in indire$t indire$t e-trusion& Aswe!. ;ri$tion is f$tor in dire$t e-trusion be$use the !or" billet is s/ueeed %inst the !lls of the $ontiner so tht fri$tion resists the #ove#ent of the billet to!rd the die openin%. n indire$t e-trusion, the billet does not #ove reltive to the $ontiner !lls, nd thus there is no fri$tion.
18.22 Wht does the the $enterburst $enterburst defe$t in e-trusion e-trusion hve hve in $o##on !ith !ith the roll pier$in% pier$in% pro$ess& pro$ess& Aswe!. hey re both e-#ples of ho! $o#pressive stresses pplied to the outside surf$e of solid $ylindri$l $ross se$tion $n $rete hi%h tensile stresses in the interior of the $ylinder.
18.2 18.2 Wht Wht re !ire !ire dr!i dr!in% n% nd br br dr!in dr!in%& %& Aswe!. Wire nd br dr!in% re bul" defor#tion pro$esses in !hi$h the $ross se$tion of !ire or br is redu$ed by pullin% (dr!in%) it throu%h die openin%.
18.2 lthou%h lthou%h the !or"pie$e !or"pie$e in !ire dr!in% opertion opertion is obviously obviously sub=e$ted sub=e$ted to tensile tensile stresses, stresses, ho! do $o#pressive stresses lso ply role in the pro$ess& Aswe!. >o#pressive stresses re present in !ire dr!in% be$use the strtin% #etl is $o#pressed s it is for$ed throu%h the ppro$h of the die openin%.
18.25 n !ire dr!in% dr!in% opertion, opertion, !hy #ust the dr!in% dr!in% stress stress never e-$eed e-$eed the yield stren%th stren%th of the !or" #etl& Aswe!. :e$use if the dr!in% stress e-$eeded the yield stren%th, the #etl on the e-it side of the dr! die !ould stret$h rther thn for$e #etl to be pulled throu%h the die openin%.
P!o"#e$s ns!ers to proble#s lbeled %A& re listed in n ppendippendi- t the b$" of the boo". Ro##i'
18.1
%A& (S units) 04##4thi$" plte #de of lo! $rbon steel is to be redu$ed to 25 ## in one pss in rollin% opertion. s the thi$"ness is redu$ed, the plte !idens by ?. @ield stren%th of the steel A 17 MP, nd tensile stren%th A 20 MP. he entrn$e speed of the plte A 77 #B#in. 'oll rdius A 00 ##, nd rottionl speed A 5 revB#in. 6eter#ine () the #ini#u# re/uired $oeffi$ient of fri$tion tht !ould #"e this rollin% opertion possible, (b) e-it velo$ity of the plte, nd ($) for!rd slip. 2 So#utio( () M-i#u# drft d ma" ma" A # Given tht d A A t o 4 t f A A 0 4 25 A 5 ## 2 A 5B00 A 0.017 A A (0.017) 0.5 A )*1+,
(b) Plte !idens by ?. t owo$o A t f w f $ f w f A A 1.0 wo 0(w 0(wo)(12) A 25(1.0w 25(1.0 wo)$ f $ f A A 0(w 0(wo)(77)B25(1.0w )(77)B25(1.0 wo) A 210B2 A 88*8- $.$i ($) $r A A 2%r& 2%r& A 2% 2% (00 (00 - 10 4)(5) A 8.82 #B#in s A s A ($ ($ f 4 4 $r )B$ )B$r A A (88.85 C 8.82)B8.82 A )*)/018.2
(DS>S units) units) 4in4thi$" 4in4thi$" slb slb is 12 in in !ide nd nd 15 ft lon%. lon%. hi$"n hi$"ness ess of the the slb is redu$ed redu$ed in three three steps in hot rollin% opertion. n e$h step, thi$"ness is redu$ed by 20? nd !idth in$reses by
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
?. f the entry speed of the slb in the first step is 0 ftB#in, nd roll speed is the s#e for the three steps, deter#ine () len%th nd (b) e-it velo$ity of the slb fter the finl redu$tion. A (0.80)(0.80)(0.80)(.0) A 1.5 in So#utio( () fter three psses, t f A w f A A (1.0)(1.0)(1.0)(12) A 1.11 in t owo ' 'o A t f w f ' f ()(12)(15 - 12) A (1.5)(1.11) ' (1.5)(1.11) ' f ' f A A ()(12)(15 - 12)B(1.5)(1.11)A 21.7 in A +*81 2t (b) Given tht roll speed is the s#e t ll three stnds nd tht t owo$o A t f w f $ f , Step 1 $ f A A ()(12)(0)B(0.8 - )(1.0 - 12) A 8.5 ftB#in Step 2 $ f A A (0.8 - )(1.0 - 12)(0)B(0.8 2 - )(1.0 2 - 12) A 8.5 ftB#in Step $ f A A (0.82 - )(1.0 2 - 12)(0)B(0.8 - )(1.0 - 12) A /8*-/ 2t.$i 18.
(S units) units) series series of $old rollin% rollin% opertio opertions ns re to be be used to redu$e redu$e the the thi$"ness thi$"ness of of #etl plte plte fro# 50 ## to 20 ## in reversin% t!o4hi%h #ill. 'oll di#eter A 00 ##, nd $oeffi$ient of fri$tion bet!een rolls nd !or" A 0.15. he spe$ifi$tion is tht the drft is to be e/ul on e$h pss. 6eter#ine () #ini#u# nu#ber of psses re/uired, nd (b) drft for e$h pss& 2 # A (0.15)2 (00) A .75 ## So#utio( () M-i#u# drft d ma" ma" A # A Mini#u# nu#ber of psses A (t ( t o 4 t f )Bd )Bd ma" ma" A (50 4 20)B.75 A . rounded up to - 34sses
(b) 6rft per pss d A A (50 4 20)B5 A *) $$ 18.
(S units) units) n the the previous previous proble proble#, #, suppose suppose tht tht the per$ent per$ent redu$tion redu$tion rther rther thn thn the the drft !ere spe$ified to be e/ul for e$h pss. () Wht is the #ini#u# nu#ber of psses re/uired& (b) Wht is the drft nd e-itin% sto$" thi$"ness for e$h pss& 2 So#utio( () M-i#u# possible drft o$$urs on first pss d ma" # A (0.15)2 (00) A .75 ## ma" A # A his $onverts into #-i#u# possible redu$tion " A " A .75B50 A 0.15 3et " 3et " A A fr$tion redu$tion per pss, nd n A nu#ber of psses. he nu#ber of psses #ust be n inte%er. o o redu$e fro# t o A 50 ## to t o A 20 ## in n psses, the follo!in% reltionship #ust be stisfied 50(1 4 " 4 "))n A 20 (1 4 " 4 "))n A 20B50 A 0. 1Bn (1 4 " 4 ")) A 0.1Bn ry n A 5 (1 4 " 4 ")) A (0.)1B5 A 0.825 " A " A 1 4 0.825 A 0.17, !hi$h e-$eeds the #-i#u# possible redu$tion of 0.15. ry n A (1 4 " 4 ")) A (0.)1B A 0.858 " A " A 1 4 0.858 A 0.11, !hi$h e-$eeds the #-i#u# possible redu$tion of 0.15. ry n A 7 (1 4 " 4 ")) A (0.)1B7 A 0.877 " A " A 1 4 0.877 A 0.1227, !hi$h is !ithin the #-i#u# possible redu$tion of 0.15.
(b) Pss 1 d A A 50(0.1227) A *15- $$, t f A A 50 4 .15 A /5*8 $$ Pss 2 d A A .8(0.1227) A -*58 $$, t f A A .8 4 5.8 A 58*/8 $$ Pss d A A 8.8(0.1227) A /*0+ $$, t f A A 8.8 4 .72 A 55*0 $$ Pss d A A .7(0.1227) A /*1/ $$, t f A A .7 4 .1 A +,*+ $$ Pss 5 d A A 2.2(0.1227) A 5*5 $$, t f A A 2.2 4 . A +-*,, $$ Pss d A A 25.(0.1227) A 5*1, $$, t f A A 25. 4 .1 A ++*8) $$ Pss 7 d A A 22.80(0.1227) A +*8) $$, t f A A 22.80 4 2.80 A +)*)) $$ 18.5
(S units) units) $ontinu $ontinuous ous hot rollin% rollin% #ill hs hs t!o stnds. stnds. hi$"ness hi$"ness of the the strtin% strtin% plte plte A 25 ## nd nd !idth A 00 ##. ;inl thi$"ness A 1 ##. 'oll rdius t e$h stnd A 250 ##. 'ottionl speed t the first stnd A 20 revB#in. 9/ul drfts of ## re t"en t e$h stnd. he plte is !ide enou%h
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
reltive to its thi$"ness tht no in$rese in !idth o$$urs. Dnder the ssu#ption tht the for!rd slip is e/ul t e$h stnd, deter#ine () roll speed t e$h stnd nd (b) for!rd slip. ($) lso, deter#ine the e-itin% speeds t e$h rollin% stnd if the enterin% speed t the first stnd A 2 #B#in. So#utio( () 3et t o A enterin% plte thi$"ness t stnd 1. t o A 25 ##. 3et t 1 A e-itin% plte thi$"ness t stnd 1 nd enterin% thi$"ness t stnd 2. t 1 A 25 4 A 1 ## 3et t 2 A e-itin% plte thi$"ness t stnd 2. t 2 A 1 4 A 1 ## 3et $o A enterin% plte speed t stnd 1 3et $1 A e-itin% plte speed t stnd 1 nd enterin% speed t stnd 2 3et $2 A e-itin% plte speed t stnd 2 3et $r 1 A roll speed t stnd 1. $r 1 A (& r r A A (2 (2 - 250)(10 4)(20) A 51*/+ $.$i 3et $r 2 A roll speed t stnd 2. ;or!rd slip s slip s A A ($ ($ f 4 4 $r )B$ )B$r s$r A A $ f 4 4 $r (1 E s E s))$r A A $ f t stnd 1, (1 E s E s))$r 1 A $1 (9/. 1) t stnd 2, (1 E s E s))$r 2 A $2 (9/. 2) :y $onstnt volu#e, t owo$o A t 1w1$1 A t )w)$) Sin$e there is no $hn%e in !idth, wo A w1 A w2 herefore, t o$o A t 1$1 A t 2$2 25$ 25$o A 1$ 1$1 A 1$ 1$2 $2 A (1B1)$ (1B1) $1 A 1.$ 1.$1 (9/. ) >o#binin% (9/s. 2 nd ), (1 E s E s))$r 2 A $2 A 1.$ 1.$1 Substitutin% (9/. 1), (1 E s E s))$r 2 A 1.(1 E s E s))$r 1 , thus $r 2 A 1.$ 1.$r 1 $r 2 A 1.(1.2) A /-*, $.$i
(b) 25$ 25$o A 1$ 1$1 $1 A 25(2)B1 A .2 #B#in (9/. 1) (1 E s E s))$r 1 A $1 (1 E s E s)(1.) )(1.) A .2 (1 E s E s)) A .2B1. A 1.08 s 6 )*)8, ($) $1 A 5/*+ $.$i7 previously $l$ulted in (b) $2 A 1.$ 1.$1 A 1.(.2) A /,*, $.$i 18.
(DS>S units) units) $ontinuous $ontinuous hot hot rollin% rollin% #ill hs hs ei%ht ei%ht stnds. stnds. he thi$"n thi$"ness ess of the the strtin% strtin% slb A .0 in, its !idth A 15.0 in, nd its len%th A 10 ft. he finl thi$"ness A 0. in. 'oll di#eter t e$h stnd A in, nd rottionl speed t stnd nu#ber 1 A 0 revB#in. t is observed tht the speed of the slb enterin% stnd 1 A 20 ftB#in. ssu#e tht no !idenin% of the slb o$$urs durin% the rollin% se/uen$e. Per$ent redu$tion in thi$"ness is to be e/ul t ll stnds, nd it is ssu#ed tht the for!rd slip !ill be e/ul t e$h stnd. 6eter#ine () per$ent redu$tion t e$h stnd, (b) rottionl speed of the rolls t stnds 2 throu%h 8, nd ($) for!rd slip. (d) Wht is the drft t stnds 1 nd 8& (e) Wht is the len%th nd e-it speed of the finl strip e-itin% stnd 8& A 0. in over 8 stnds, .0(1 4 " 4 "))8 A 0. So#utio( () o redu$e fro# t o A .0 in to t f A (1 4 " 4 "))8 A 0.B.0 A 0.10 (1 4 " 4 ")) A (0.10)1B8 A 0.78 6 )*+-)1 6 +-*)1 4t e49: st4;* " A " A 1 4 0.78 A r 6 (b) ;or!rd slip s A ($ ( $ f 4 4 $r )B$ )B$r s$r A A $ f 4 4 $r (1 E s E s))$r A A $ f
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
t stnd 1 (1 E s E s))$r 1 A $1 , !here $r 1 A roll speed, $1 A e-it speed of slb. t stnd 2 (1 E s E s))$r 2 A $2 , !here $r 2 A roll speed, $2 A e-it speed of slb. 9t$. t stnd 8 (1 E s E s))$r 8 A $8, !here $r 8 A roll speed, $8 A e-it speed of slb. :y $onstnt volu#e, t owo$o A t 1w1$1 A t 2w2$2 A . . . A t 8w8$8 Sin$e there is no $hn%e in !idth, wo A w1 A w2 A . . . w8 herefore, t o$o A t 1$1 A t 2$2 A . . . A t 8$8 t o A .0 in $o A (1 4 r )$1 A (1 4 r )2$2 A . . . (1 4 r )8$8 , !here r A A 0.2501 s deter#ined in prt (). Sin$e s Sin$e s is is $onstnt, $r 1 $r 2 . . . $r 8 A $1 $2 . . . $8 Given tht & tht & r r 1 A 0 revB#in, $r 1 A (& r r1 A (2 (2 - 18B12)(0) A 282.78 ftB#in n %enerl & %enerl & r r A A (0B282.78) A 0.100$ 0.100 $r & r r 2 A 0.100 - 282.78B(14r 282.78B(14 r ) A 0.100 - 282.78B(140.2501) A /) !ev.$i & r r A 0.100 - 282.78B(14r 282.78B(14 r )2 A -5*5 !ev.$i & r r A 0.100 - 282.78B(14r 282.78B(14 r ) A 01*1 !ev.$i & r r 5 A 0.100 - 282.78B(14r 282.78B(14 r ) A ,/*, !ev.$i & r r A 0.100 - 282.78B(14r 282.78B(14 r )5 A 1+*,*5 !ev.$i & r r 7 A 0.100 - 282.78B(14r 282.78B(14 r ) A 18*- !ev.$i & r r 8 A 0.100 - 282.78B(14r 282.78B(14 r )7 A ++/*, !ev.$i ($) Given $o A 20 ftB#in $1 A 20B(14r 20B(14 r ) A 20B0.78 A 20 ftB#in $2 A 20B0.78 A 2.8 ftB#in ;ro# e/utions for for!rd slip, (1 E s) s)$r 1 A $1 (1 E s E s)(282.78) )(282.78) A 20 s 6 )*15+ (1 E s E s)) A 20B282.78 A 1.12 >he$" !ith stnd 2 %iven $2 A 2.8 ftB#in fro# bove & r r 2 A 0.100$ 0.100$r 2 'errn%in%, $r 2 A & A & r r2 B0.100 A .2 & .2 & r r 2 A 0.2(0) A 77.0 ftB#in (1 E s E s)(77.0) )(77.0) A 2.8 (1 E s E s)) A 2.8B77.1 A 1.12 s A s A 0.12, s before (d) 6rft t stnd 1 d 1 A .0(0.2501) A )*0-)5 i 6rft t stnd 8 d 8 A .0(1 4 0.2501) 7(0.2501) A )*1))) i (e) 3en%th of finl strip ' strip ' f A ' A '8 t owo ' 'o A t 8w8 ' '8 Given tht wo A w8, t o ' 'o A t 8 ' '8 .0(10 ft) A 0. '8 t owo$o A t 8w8$8 t o$o A t 8$8 $8 A 20(B0.) A +/)) 2t.$i 18.7
L8 6 1)) 2t
%A& (S units) lo!4$rbon steel plte is 00 ## !ide nd 25 ## thi$". t is redu$ed in one pss in t!o4hi%h rollin% #ill to thi$"ness of 20 ##. 'oll rdius A 00 ##, nd roll speed A 0 #B#in. Stren%th $oeffi$ient A 500 MP, nd strin hrdenin% e-ponent A 0.25 (ble .). 6eter#ine () roll for$e, (b) roll tor/ue, nd ($) po!er re/uired to perfor# the opertion. So#utio( () 6rft d A A 25 4 20 A 5 ## >ont$t len%th ' len%th ' A A (00 - 5) .5 A 5.77 ## rue strin A A ln(25B20) A ln 1.25 A 0.22 * f A 500(0.22) 0.25B1.25 A 27. MP
9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
'ollin% for$e + for$e + A A 27.(00)(5.77) A /7-17/81 N (b) or/ue , A A 0.5(,51,81)(5.77 - 10 4) A 1+578/ N<$ ($) & ($) & A A (0 #B#in)B(2 - 0.00) A 7. revB#in A 0.1 revBs Po!er - Po!er - A A 2 2 (0.1)(,51,81)(5.77 (0.1)(,51,81)(5.77 - 10 4) A 20,10 +4#Bs A 20,10 W A +) =W 18.8 18.8
(S unit units) s) Solve Solve Prob Proble# le# 18.7 18.7 usin usin% % roll roll rdiu rdiuss A 00 00 ##. So#utio( () 6rft d A A 25 4 20 A 5 ## >ont$t len%th ' len%th ' A A (00 - 5) .5 A 8.7 ## rue strin A A ln(25B20) A ln 1.25 A 0.22 * f A 500(0.22) 0.25B1.25 A 27. MP 'ollin% for$e + for$e + A A 27.(00)(8.7) A 571,5708) N
(b) or/ue , A A 0.5(,1,780)(8.7- 10 4) A 178/8 N<$ ($) & ($) & A A (0 #B#in)B(2 - 0.00) A 15. revB#in A 0.25 revBs Po!er - Po!er - A A 2 2 (0.25)(,1,780)(8.7(0.25)(,1,780)(8.7- 10 4) A 205,58 +4#Bs A 205,58 W A +)-*, =W Note tht the for$e nd tor/ue re redu$ed s roll rdius is redu$ed, but tht the po!er re#ins the s#e (!ithin $l$ultion error) s in the previous proble#.
18.
(S units) units) Solve Solve Proble# Proble# 18.7 18.7 usin% usin% $luster $luster #ill #ill !ith !ith !or"in% !or"in% rolls rolls of rdius A 0 ##. >o#pre the results !ith the previous t!o proble#s, nd note the i#portnt effe$t of roll rdius on for$e, tor/ue, nd po!er. So#utio( () 6rft d A A 25 4 20 A 5 ##, >ont$t len%th ' len%th ' A A (0 - 5) .5 A 17.2 ## rue strin A A ln(25B20) A ln 1.25 A 0.22 * f A 500(0.22) 0.25B1.25 A 27. MP 'ollin% for$e + for$e + A A 27.(00)(17.2) A 17/+87+-/ N
(b) or/ue , A A 0.5(1,28,25)(17.2 - 10 4) A 1+75, N<$ ($) & ($) & A A (0 #B#in)B(2 - 0.00) A 7.58 revB#in A 1.2 revBs Po!er - Po!er - A A 2(1.2)(1,28,25)(17.2 - 10 4) A 20,15 +4#Bs A 20,15 W A +)*1- =W Note tht this is the s#e po!er vlue (!ithin $l$ultion error) s in Proble#s 18.7 nd 18.8. n f$t, po!er !ould probbly in$rese be$use of lo!er #e$hni$l effi$ien$y in the $luster type rollin% #ill.
18.10 (DS>S units) units) 4in4thi$" 4in4thi$" lu#inu# lu#inu# slb is 10 in !ide !ide nd 100 in lon%. t is redu$ed redu$ed in one pss in t!o4hi%h rollin% #ill to thi$"ness A .5 in. he roll rottes t speed A 12 revB#in nd hs rdius A 20 in. he lu#inu# hs stren%th $oeffi$ient A 25,000 lbBin 2 nd strin hrdenin% e-ponent A 0.20 (ble .). 6eter#ine () roll for$e, (b) roll tor/ue, nd ($) po!er re/uired to $$o#plish this opertion. So#utio( () 6rft d A A .00 C .5 A 0.5 in, >ont$t len%th ' len%th ' A A (20 - 0.5) 0.5 A .1 in rue strin A A ln(.0B.5) A ln 1.1 A 0.15 * f A 25,000(0.15) 0.20B1.20 A 1,28 lbBin 2
'ollin% for$e + for$e + A A * f w' A w' A 1,28(10)(.1) A //)71+- #" (b) or/ue , A A 0.5 +' A +' A 0.5(0,125)(.1) A 08,70)) i<#"* 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
1847
Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
($) & ($) & A A 12 revB#in Po!er - Po!er - A A 2(12)(0,125)(.1) A 10,8,7 in4lbB#in .- A A (10,8,7 in4lbB#in)B(,000) A +- :3 18.11 18.11 (S units) sin%le4pss sin%le4pss rollin% rollin% opertion opertion redu$es 204##4t 204##4thi$" hi$" plte plte to 18 ##. he strtin% strtin% plte is 200 ## !ide. 'oll rdius A 250 ## nd rottionl speed A 12 revB#in. he !or" #etl hs stren%th $oeffi$ient A 00 MP nd strin hrdenin% e-ponent A 0.22. 6eter#ine () roll for$e, (b) roll tor/ue, nd ($) po!er re/uired for this opertion. So#utio( () 6rft d A 20 4 18 A 2.0 ##, >ont$t len%th ' len%th ' A A (250 - 2) .5 A 11.18 ## A 0.0112 # rue strin A A ln(20B18) A ln 1.111 A 0.105 * f A 00(0.105) 0.22B1.22 A 00 MP 'ollin% for$e + for$e + A A 00(0.0112)(0.2) A )*0+ MN 6 0+7))) N
(b) or/ue , A A 0.5(72,000)(0.0112) 6 570+) N<$ ($) Given tht & tht & A A 12 revB#in Po!er - Po!er - A A 2(12B0)(72,000)(0.0112) A 507,0 W 6 50*0 =W 18.12 %A& (DS>S units) hot rollin% opertion is perfor#ed usin% rolls of di#eter A 2 in. he rollin% stnd $n e-ert #-i#u# for$e of 00,000 lb, nd the #ill hs #-i#u# horsepo!er of 100 hp. t is desired to redu$e 1.54in4thi$" plte by the #-i#u# possible drft in one pss. he strtin% plte is 10 in !ide. !ide. n the the heted heted $onditio $ondition, n, the !or" !or" #teril #teril hs hs stren%th stren%th $oeffi$ient $oeffi$ient A 20,000 20,000 2 lbBin nd strin hrdenin% e-ponent A ero. 6eter#ine () #-i#u# possible drft, (b) sso$ited true strin, nd ($) #-i#u# speed of the rolls for the opertion. So#utio( () ssu#ption #-i#u# possible drft is deter#ined by the for$e $pbility of the rollin% #ill nd not by $oeffi$ient of fri$tion bet!een the rolls nd the !or". 6rft d A A 1.5 4 t f >ont$t len%th ' len%th ' A A (12d (12d )0.5 * f A 20,000( 20,000( )0B1.0 A 20,000 lbBin 2 ;or$e + ;or$e + A 20,000(10) (12d (12 d )0.5 A 00,000 (the li#itin% for$e of the rollin% #ill) (12d (12d )0.5 A 00,000B200,000 A 2.0 12 d A A 2.02 A d 6 6 /.1+ 6 )*555 i
(b) rue strin A A ln(1.5Bt ln(1.5Bt f ) t f A A t o 4d 4d A A 1.5 4 0. A 1.17 in A A ln(1.5B1.17) A ln 1.285 A )*+-1 ($) Given #-i#u# possible po!er .- po!er .- A A 100 hp A 100 - 000 (in4lbB#in)Bhp A ,00,000 in4 lbB#in >ont$t len%th ' len%th ' A A (12 - 0.) 0.5 A 2.0 in - A A 2 & (00,000)(2.0) (00,000)(2.0) A 5,02,58+ in4lbB#in 5,02,58 & A A ,00,000 & A A 7.88 revB#in $r A A 2 #& A A 2(12B12)(7.88) A /,*- 2t.$i 18.1 (DS>S units) units) Solve Proble# Proble# 18.12 e-$ept e-$ept tht the opertion opertion is !r# rollin% rollin% nd the strin4h strin4hrdeni rdenin% n% e-ponent is 0.18. ssu#e the stren%th $oeffi$ient re#ins t 20,000 lbBin 2.
9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
So#utio( () ssu#ption (s#e s in previous proble#) #-i#u# possible drft is deter#ined by the for$e $pbility of the rollin% #ill nd not by $oeffi$ient of fri$tion bet!een the rolls nd the !or". 6rft d A A 1.5 4 t f >ont$t len%th ' len%th ' A A (12d (12d )0.5 A A ln(1.5Bt ln(1.5B t f ) * f A 20,000( 20,000( )0.18B1.18 A 1, .18
+ A A
@
f
(10)(12d (10)(12d )0.5 A .1 * f (d (d )0.5 A 00,000 (s %iven)
* f (d (d )0.5 A 00,000B.1 A 11,57
+o! use use tril4nd4 tril4nd4error error to to vlues vlues of * f nd d tht tht fit this e/ution. ry d A A 0. in, t f A A 1.5 4 0. A 1.2 in A A ln(1.5B1.2) A ln 1.25 A 0.22 * f A 1,(0.22) .18 A 1,1 lbBin 2. (d )0.5 A 11,57B1,1 A 0.871 d A A 0.77, !hi$h does not e/ul the initil tril vlue of d A A 0. ry d A 0.5 in, t f A A 1.5 4 0.5 A 1.0 in A A ln(1.5B1.0) A ln 1.50 A 0.055 * f A 1,(0.055) .18 A 1,58 lbBin 2. (d )0.5 A 11,57B1,58 A 0.72 d A A 0.1, !hi$h does not e/ul the tril vlue of d A A 0.5 ry d A A 0. in, t f A A 1.5 4 0.A 0. in A A ln(1.5B0.) A 0.5108 * f A 1,(0.5108) .18 A 15,120 lbBin 2. (d )0.5 A 11,57B15,120 A 0.77 d A A 0.58, !hi$h is too #u$h $o#pred to d A A 0. ry d A A 0.58 in, t f A A 1.5 4 0.58 A 0.2 in A A ln(1.5B0.2) A ln 1.57 A 0.8 * f A 1,(0.8) .18 A 15,007 lbBin 2. (d )0.5 A 11,57B15,007 A 0.7 d A A 0.52, !hi$h is $lose but still bove the tril vlue of d A A 0.55 ry d A A 0.585 in, t f A A 1.50 4 0.585 A 0.15 in A A ln(1.5B0.15) A 0. * f A 1,(0.) .18 A 15,0 lbBin 2. (d )0.5 A 11,57B15,0 A 0.78 d A A 0.50, !hi$h is $lose but still bove the tril vlue of d A A 0.585. ry d A A 0.588 in, t f A A 1.50 4 0.588 A 0.12 in A ln(1.5B0.12) A 0.8 A * f A 1,(0.8) .18 A 15,05 lbBin 2. (d )0.5 A 11,57B15,05 A 0.77 6 )*-88* d A A 0.588, !hi$h is l#ost the s#e s the tril vlue of d 6 (b) rue strin A A ln(1.5B0.12) A )*/,8 ($) Given #-i#u# possible po!er FP A 100 hp A 100 - 000 (in4lbB#in)Bhp A ,00,000 in4lbB#in 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
184
Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
>ont$t len%th ' len%th ' A A (12 - 0.588) 0.5 A 2. in - A A 2 & (00,000)(2.) (00,000)(2.) A ,85,000+ in4lbB#in ,8,000 & A A ,00,000 & A A 5.2 revB#in $r A A 2 #& A A 2(12B12)(5.2) A 50*+ 2t.$i Fo!'i'
18.1 %A& (S units) $ylindri$l prt is upset for%ed in n open die. he strtin% di#eter A 50 ## nd hei%ht A 0 ##. Fei%ht fter for%in% A 0 ##. >oeffi$ient of fri$tion t the die4!or" interf$e A 0.20. @ield stren%th of the lo!4$rbon steel A 105 MP, nd its flo! $urve is defined by stren%th $oeffi$ient A 500 MP nd strin4hrdenin% e-ponent A 0.25 (ble .). 6eter#ine the for$e in the opertion () =ust s the yield point is re$hed (yield t strin A 0.002), (b) t hei%ht of 5 ##, nd ($) t hei%ht of 0 ##. Dse of spredsheet $l$ultor is re$o##ended. A (2 'B 'B A (50)2(0)B A 78,50 ## So#utio( () / A Given A A 0.002, * f A A 500(0.002) 0.25 A 105.7 MP, nd h A 0 4 0(0.002) A .2 ## A A A A /h A /h A 78,50B.2 A 17 ## 2 f A A 1 E 0.(0.2)(50)B.2 A 1.10 + A A 1.10(105.7)(17) A ++870/- N (b) Given h A 5, A A ln(0B5) A ln 1.1 A 0.15 0.25 A 500(0.15) A 02.2 MP * f A / A A 78,50 ## fro# prt () bove. t h A 5, A 5, A A A /h A /h A 78,50B5 A 22 ## 2 >orrespondin% ( >orrespondin% ( A A 5.5 ## (fro# A (fro# A A A (2B) f A A 1 E 0.(0.2)(5.5)B5 A 1.122 + A A 1.122(02.2)(22) A 0)7,8, N ($) Given h A 0, A A ln(0B0) A ln 1. A 0.2877 0.25 * f A A 500(0.2877) A .2 MP / A A 78,50 ## fro# prt () bove. t h A 0, A 0, A A A /h A /h A 78,50B0 A 218 ## 2 >orrespondin% ( >orrespondin% ( A A 57.7 ## (fro# A (fro# A A A (2B) f A A 1 E 0.(0.2)(57.7)B0 A 1.15 A 1.15(.2)(218) A 171)751- N + A 18.15 (DS>S units) units) $ylindri$l $ylindri$l !or" !or" prt !ith di#eter di#eter A 2.5 in, nd hei%ht A 2.5 in is upset upset for%ed for%ed in n open die to hei%ht A 1.5 in. >oeffi$ient of fri$tion t the die4!or" interf$e A 0.10. he !or" #etl hs stren%th $oeffi$ient A 0,000 lbBin 2 nd strin hrdenin% e-ponent A 0.15. @ield stren%th A 15,750 lbBin 2. 6eter#ine the instntneous for$e in the opertion () =ust s the yield point is re$hed (yield t strin A 0.002), (b) t hei%ht h A 2. in, ($) h A 2.1 in, (d) h A 1. in, (e) h A 1.7 in, nd (f) h A 1.5 in. Dse of spredsheet $l$ultor is re$o##ended. So#utio( () / A A (2 'B 'B A (2.5)2(2.5)B A 12.27 in Given A A 0.002, * f A A 0,000(0.002) 0.15 A 15,78 lbBin 2 nd h A 2.5 4 2.5(0.002) A 2.5 A A A A /h A /h A 12.27B2.5 A .2 in 2 f A A 1 E 0.(0.1)(2.5)B2.5 A 1.0 + A A 1.0(15,78)(.2) A 8)7-0, #"
(b) Given h A 2., A A ln(2.5B2.) A ln 1.087 A 0.08 * f A A 0,000(0.08) 0.15 A 27,55 lbBin 2 / A A 12.27 in fro# prt () bove. 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
t h A 2., A 2., A A A /h A /h A 12.27B2. A 5. in 2 >orrespondin% ( >orrespondin% ( A A 2.1 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.1)B2. A 1.05 + A 1.05(27,55)(5.) A 1-578++ #" ($) Given h A 2.1, A A ln(2.5B2.1) A ln 1.11 A 0.17 * f A A 0,000(0.17) 0.15 A 0,780 lbBin 2 / A A 12.27 in fro# prt () bove. t h A 2.1, A 2.1, A A A /h A /h A 12.27B2.1 A 5.8 in 2 >orrespondin% ( >orrespondin% ( A A 2.7 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.7)B2.1 A 1.052 + A 1.052(0,780)(5.8) A 18,7+5 #" (d) Given h A 1., A A ln(2.5B1.) A ln 1.1 A 0.27 * f A A 0,000(0.27) 0.15 A 2,8 lbBin 2 / A A 12.27 in fro# prt () bove. t h A 1., A 1., A A A /h A /h A 12.27B1. A . in 2 >orrespondin% ( >orrespondin% ( A A 2.87 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.87)B1. A 1.00 + A A 1.00(2,8)(.) A ++-7,- #" (e) Given h A 1.7, A A ln(2.5B1.7) A ln 1.71 A 0.8 0.15 * f A A 0,000(0.8) A ,7 lbBin 2 / A A 12.27 in fro# prt () bove. t h A 1.7, A 1.7, A A A /h A /h A 12.27B1.7 A 7.22 in 2 >orrespondin% ( >orrespondin% ( A A .0 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(.0)B1.7 A 1.071 + A 1.071(,7)(7.22) A +8710 #" (f) Given h A 1.5, A A ln(2.5B1.5) A ln 1.7 A 0.511 * f A A 0,000(0.511) 0.15 A ,1 lbBin 2 / A A 12.27 in fro# prt () bove. t h A 1.5, A 1.5, A A A /h A /h A 12.27B1.5 A 8.182 in 2 >orrespondin% ( >orrespondin% ( A A .2 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(.2)B1.5 A 1.08 + A A 1.08(,1)(8.182) A 5+1750, #" 18.1 (DS>S units) units) $ylindri$l $ylindri$l !or" !or" prt hs di#eter A 2.5 in nd hei%ht hei%ht A .0 in. t is upset for%ed for%ed to hei%ht A 2.75 in. >oeffi$ient of fri$tion t the die4!or" interf$e A 0.10. he !or" #teril hs flo! $urve !ith stren%th $oeffi$ient A 25,000 lbBin 2 nd strin hrdenin% e-ponent A 0.22. 6eter#ine the plot of for$e vs. !or" hei%ht. Dse of spredsheet $l$ultor is re$o##ended. So#utio olu#e of $ylinder / A A (2 'B 'B A (2.5)2(.0)B A 1.5 in We $o#pute the t he for$e + for$e + t t sele$ted vlues of hei%ht h h A () .0, (b) .75, ($) .5, (d) .25, (e) .0, (f) 2.75, nd (%) 2.5. hese vlues $n be used to develop the plot. he shpe of the plot !ill be si#ilr to ;i%ure 18.12 in the te-t.
t h 6 /*), !e ssu#e yieldin% hs =ust o$$urred nd the hei%ht hs not $hn%ed si%nifi$ntly. si%nifi$ntly. Dse A 0.002 (the ppro-i#te yield point of #etl). t A A 0.002, * f A A 25,000(0.002) 0.22 A ,70 lbBin 2 d=ustin% the hei%ht for this strin, h A .0 4 .0(0.002) A .2 A A A A /h A /h A 1.5B.2 A .2 in 2 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
f A A 1 E 0.(0.1)(2.5)B.2 A 1.025 + A A 1.025(,70)(.2) A 5+71+- #" t h 6 5*0-, A A ln(.0B.75) A ln 1.07 A 0.05 * f A A 25,000(0.05) 0.22 A 1,80 lbBin 2 / A A 1.5 in $l$ulted bove. t h A .75, A .75, A A A /h A /h A 1.5B.75 A 5.2 in 2 >orrespondin% ( >orrespondin% ( A A 2.582 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.582)B.75 A 1.028 + A A 1.028(1,80)(5.2) A 057)1 #" t h 6 5*-, A A ln(.0B.5) A ln 1.1 A 0.15 * f A A 25,000(0.15) 0.22 A 1,05 lbBin 2 t h A .5, A .5, A A A /h A /h A 1.5B.5 A 5.1 in 2 >orrespondin% ( >orrespondin% ( A A 2.7 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.7)B.5 A 1.01 + A A 1.01(1,05)(5.1) A ,+78)8 #" t h 6 5*+-, A A ln(.0B.25) A ln 1.21 A 0.207 * f A A 25,000(0.207) 0.22 A 17,1 lbBin 2 t h A .25, A .25, A A A /h A /h A 1.5B.25 A .02 in 2 >orrespondin% ( >orrespondin% ( A A 2.77 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.77)B.25 A 1.0 + A A 1.0(17,1)(.02) A 11)7-58 #" t h 6 5*), A A ln(.0B.0) A ln 1. A 0.287 * f A A 25,000(0.287) 0.22 A 1,00 lbBin 2 t h A .0, A .0, A 2 /h A 1.5B.0 A .55 in 2 /h A >orrespondin% ( >orrespondin% ( A A 2.887 (fro# A (fro# A A A (2B) f A A 1 E 0.(0.1)(2.887)B.0 A 1.08 + A A 1.08(1,00)(.55) A 1+,718+ #" t h 6+*0-, A A ln(.0B2.75) A ln 1.55 A 0.77 * f A A 25,000(0.77) 0.22 A 20,1 lbBin 2 / A A 1.5 in $l$ulted bove. t h A 2.75, A 2.75, A A A /h A /h A 1.5B2.75 A 7.10 in 2 >orrespondin% ( >orrespondin% ( A A .015 (fro# A (fro# A A A (2B) A 1 E 0.(0.1)(.015)B2.75 A 1.0 f A + A A 1.0(20,1)(7.10) A 1-)715 #" 18.17 (S units) units) $old hedin% hedin% opertion opertion for#s the hed hed on steel nil. he he stren%th $oeffi$ $oeffi$ient ient for the steel A 500 MP, nd the strin hrdenin% e-ponent A 0.22. >oeffi$ient of fri$tion t the die4!or" interf$e A 0.1. he di#eter of the !ire sto$" for the nil A .0 ##. he nil hed di#eter A 10 ##, nd its thi$"ness A 1.5 ##. he finl len%th of the nil A 0 ##. () Wht len%th of sto$" #ust pro=e$t pro=e$t out of the die in order to to provide provide suffi$i suffi$ient ent volu#e volu#e of !or" !or" #teril #teril for for this this upsettin% upsettin% opertion& (b) >o#pute the #-i#u# for$e tht the pun$h #ust pply to for# the hed in this open4die opertion. A ( f 2h f B B A (10)2(1.5)B A 117.8 ## . So#utio () olu#e of nil hed / A >ross4se$tionl re of !ire sto$" A sto$" Ao A (o2B A (.0)2B A 12.57 ## 2 ho A /Ao A 117.8B12.57 A ,*50 $$ (b) A A ln(.7B1.5) A ln .25A 1.8 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
* f A A 500(1.8) 0.22 A 571 MP re of hed A hed A f A A (10)2B A 78.5 ## 2 f A A 1 E 0.(0.1)(10B1.5) A 1.7 + A A 1.7(571)(78.5) A 17--8 N 18.18 l$ultions for the dt developed by the student re si#ilr to those in the pre$edin% proble#.
18.1 %A& (S units) hot upset4for%in% opertion is perfor#ed in n open die. nitil di#eter of the $ylindri$l !or" prt A 0 ##, nd its hei%ht A 50 ##. he prt is upset to n ver%e di#eter A 5 ##. he !or" #etl t this elevted te#perture yields t 0 MP. >oeffi$ient of fri$tion t the die4!or" interf$e A 0.0. 6eter#ine () the finl hei%ht of the prt nd (b) the #-i#u# for$e in the opertion. So#utio () / A A (o2hoB A (0)2(50)B A 5, ## A f A A ( f 2B A (5)2B A 150 ## 2 h f A A /A f A A 5,B150 A ++*+ $$
(b) A A ln(50B22.2) A ln 2.25 A 0.811 Given hot for%in% opertion, !e ssu#e the stren%th $oeffi$ient is e/ul to the yield stren%th nd the strin hrdenin% e-ponent is ero. * f A A 0(0.811) 0 A 0 MP ;or$e is #-i#u# t lr%est re vlue, A f A A 150 ##2 ( A ( f A 5 ## f A A 1 E 0.(0.)(5B22.22) A 1.2 + A A 1.2(0)(150) A 18,7/, N 18.20 (S units) units) hydruli$ hydruli$ for%in% for%in% press is $pble $pble of e-ertin% e-ertin% #-i#u# for$e for$e A 1,000,000 +. +. $ylindri$l !or" prt is to be $old upset for%ed. he strtin% prt hs di#eter A 0 ## nd hei%ht A 0 ##. he flo! $urve of the #etl is defined by stren%th $oeffi$ient A 00 MP nd strin hrdenin% e-ponent A 0.20. 6eter#ine the #-i#u# redu$tion in hei%ht to !hi$h the prt $n be $o#pressed !ith this for%in% press if the $oeffi$ient of fri$tion A 0.1. Dse of spredsheet $l$ultor is re$o##ended. So#utio olu#e of !or" / A A (o2hoB A (0)2(0)B A 21,20 ## . ;inl re A re A f A A 21,20Bh 21,20Bh f A A ln(0Bh ln(0B h f ) * f A A 00 00 0.2 A 00(ln 0Bh 0Bh f )0.2) f A A 1 E 0. 0. ( ( ( f Bh f ) A 1 E 0.(0.1)( ( f Bh f ) ;or%in% for$e + for$e + A A f * * f A f A A (1 E 0.0 ( f Bh f )( )( 00(ln 0Bh 0B h f )0.2)( 21,20Bh 21,20Bh f )
'e/uires tril nd error solution to find the vlue of h f tht tht !ill #t$h the for$e of 17)))7))) N. (1) ry h f A A 20 ## A f A A 21,20B20 A 100. ## 2 A A ln(0B20) A ln 1.5 A 0.05 * f A A 00(0.05) 0.2 A . MP ( f A A ( - 100.B )0.5 A .7 ## 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
f A A 1 E 0.0(.7B20) A 1.07 + A A 1.07(.)(100.) A 58)7)-) N oo lo!. ry s#ller vlue of h f to to in$rese + in$rese + . (2) ry h f A A 10 ##. A f A A 21,20B10 A 2120. ## 2 A A ln(0B10) A ln .0 A 1.0 * f A A 00(1.0) 0.2 A 07. MP ( f A A ( - 2120.B )0.5 A 51. ## f A A 1 E 0.0(51.B10) A 1.208 + A A 1.208(07.)(2120.) A 17)/57,,8 N Sli%htly hi%h. +eed to try vlue of h f bet!een bet!een 10 nd 20, $loser to 10. () ry h f A A 11 ## A f A A 21,20B11 A 127.8 ## 2 A A ln(0B11) A ln 2.727 A 1.00 * f A A 00(1.00) 0.2 A 00. MP ( f A A ( - 127.8B )0.5 A .5 ## f A A 1 E 0.0(51.12B11) A 1.18 + A A 1.18(00.)(127.8) A ,1)7-5 N () :y liner interpoltion, try h f A A 10 E (B1) A 10. ## 2 A f A A 21,20B10. A 2052.8 ## A A ln(0B10.) A ln 2.02 A 1.0 * f A A 00(1.0) 0.2 A 05.1 MP ( f A A ( - 2052.8B )0.5 A 51.12 ## f A A 1 E 0.0(51.12B10.) A 1.18 + A A 1.18(05.1)(2052.8) A ,,75/ N (5) :y further liner interpoltion, try h f A A 10 E (B8)(0.) A 10.0 2 A f A A 21,20B10.0 A 2058.8 ## A A ln(0B10.0) A ln 2.1 A 1.0 * f A A 00(1.0) 0.2 A 05.8 MP ( f A A ( - 2058.8B )0.5 A 51.2 ## f A A 1 E 0.0(51.2B10.) A 1.1 + A A 1.1(05.8)(2058.8) A 17)))7--5 N >lose enou%hH M-i#u# hei%ht redu$tion A 0.0 4 10. A 1,*0 $$ Dsin% spredsheet $l$ultor, the uthorIs pro%r# (!ritten in 9-$el) obtined vlue of h A 1.0 ## to $hieve for$e of 1,000,000 lb !ithin one pound. 18.21 (S units) units) $onne$tin% $onne$tin% rod is hot for%ed for%ed in n i#pression i#pression die. die. he pro=e$ted pro=e$ted re of the prt A ,50 2 ## . he desi%n of the die $uses flsh to for# durin% for%in%, so tht the re, in$ludin% flsh, A ,120 ##2. he prt %eo#etry is $o#ple-. s heted, the !or" #etl yields t 70 MP nd hs no tenden$y to strin hrden. 6eter#ine the #-i#u# for$e re/uired to perfor# the opertion. So#utio Sin$e the !or" #teril hs no tenden$y to !or" hrden, n A 0 ;ro# ble 18.1, $hoose $hoose f A8.0 A8.0 + A A 8.0(70)(120) A 57/+07+)) N
18.22 %A& (DS>S units) prt is hot for%ed in n i#pression die. he pro=e$ted re of the prt, in$ludin% flsh, A 17. in 2. fter tri##in%, the prt hs pro=e$ted re of 12.5 in 2. Prt %eo#etry is $o#ple-. s heted the !or" #etl yields t 1,000 lbBin 2 nd no strin hrdenin% o$$urs. t roo# 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
te#perture, the #etl yields t 0,000 lbBin 2. 6eter#ine the #-i#u# for$e re/uired to perfor# the for%in% opertion. So#utio Sin$e the !or" #teril hs no tenden$y to !or" hrden, n A 0 ;ro# ble 18.1, $hoose $hoose f A A 8.0 + A A 8.0(1,000)(17.) A 170,,7+)) #" E>t!usio
18.2 %A& (S units) $ylindri$l billet is 150 ## lon% nd 75 ## in di#eter. di#e ter. t is redu$ed by indire$t e-trusion to 0 ## di#eter. 6ie n%le A 0 . n the Johnson e/ution, a A 0.8 nd b A 1.. n the flo! $urve for the !or" #etl, stren%th $oeffi$ient A 800 MP nd strin hrdenin% e-ponent A 0.15. 6eter#ine () e-trusion rtio, (b) true strin (ho#o%eneous defor#tion), ($) e-trusion strin, (d) r# pressure, nd (e) r# for$e. A AoB A A f A ( A (o2B ( ( f 2 A (75)2B(0)2 A *+So#utio () r " A A (b) A A ln r " A ln .25 A 1*855 ($) " A a E b ln r " A 0.8 E 1.(1.8) A 5*5 (d) * f A 800(1.8) 0.15B1.15 A 72 MP p A p A 72(.) A +-/ MP4 (e) A (e) Ao A (o2B A (75)2B A 18 ## 2 + A A 25(18) A 1175+070-+ N 18.2 (DS>S units) units) 4in4lon 4in4lon% % $ylindri$l $ylindri$l billet !hose !hose di#eter A 1.5 in is redu$ed by indire$t indire$t e-trusion e-trusion to di#eter A 0.75 in. 6ie n%le A 0 . n the Johnson e/ution, a A 0.8 nd b A 1.5. he !or" #etl is lo!4$rbon steel !ith stren%th $oeffi$ient A 75,000 lbBin 2 nd strin hrdenin% e-ponent A 0.25 (ble .). 6eter#ine () e-trusion rtio, (b) true strin (ho#o%eneous defor#tion), ($) e-trusion strin, (d) r# pressure, (e) r# for$e, nd (f) po!er if the r# speed A 20 inB#in. So#utio () r " A A A AoB A A f A ( A (o2B ( ( f 2 A (1.5)2B(0.75)2 A 2 A 1*)
(b) A A ln r " A ln 1 A +*005 ($) " A a E b ln r " A 0.8 E 1.5(2.77) A /*,-, (d) * f A 75,000(2.77) 0.25B1.25 A 77,2 lbBin 2 p A p A 77,2(.5) A 5857,5/ #".i+ (e) A (e) Ao A (o2B A (1.5)2B A 1.77 in 2 A (8,)(1.77) A 087/11 #"* + A (f) - (f) - A A 78,11(20) A 157-87++8 i<#".$i .- A A 1,58,228B,000 A 5/*+ :3 18.25 (S units) units) lo!4$rbon lo!4$rbon steel steel billet is 75 ## lon% !ith !ith di#eter di#eter A 5 ##. t is dire$t e-truded to di#eter A 20 ## usin% n e-trusion die n%le A 75 . he steelIs stren%th $oeffi$ient A 500 MP, nd its strin hrdenin% e-ponent A 0.25 (ble .). n the Johnson e-trusion strin e/ution, a A 0.8 nd b A 1.. 6eter#ine () e-trusion rtio* (b) true strin (ho#o%eneous defor#tion)* ($) e-trusion strin* nd (d) r# pressure nd for$e t ' A ' A 70, 50, 0, nd 10 ##. Dse of spredsheet $l$ultor is re$o##ended for prt (d). A AoB A A f A ( A (o2B ( ( f 2 A (5)2B(20)2 A 5*)+So#utio () r " A A (b) A A ln r " A ln .025 A 1*11, 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
($) " A a E b ln r " A 0.8 E 1.(1.11) A +*50 (d) * f A 500(1.11) 0.25B1.25 A 11. MP Ao A (5)2B A 2.1 ## 2 t is pproprite to deter#ine the volu#e of #etl $ontined in the $one of the die t the strt of the e-trusion opertion, to ssess !hether #etl hs been for$ed throu%h the die openin% by the ti#e the billet billet hs been been redu$ed redu$ed fro# fro# ' ' A A 75 ## to ' to ' A A 70 ##. ;or $one4shped die !ith n%le A 75 , the hei%ht h of the frustu# is for#ed by #etl bein% $o#pressed into the die openin% he t!o rdii re #1 A 0.5 (o A 17.5 ## nd # nd #2 A 0.5 ( f A 10 ##, nd h A ( # #1 4 # 4 #2)Btn 75 A 7.5Btn 75 A 2.01 ##. 2 2 ;rustu# volu#e / A A 0. h( #1 E # E #1 # #2 E # E #2 ) A 0. (2.01)(17.5 2 E 10 - 17.5 E 10 2) A 122. ## . >o#pre this !ith the volu#e of the portion of the $ylindri$l billet bet!een ' A ' A 75 ## nd ' nd ' 2 2 A 70 ##. / A A (o hB A 0.25(5) (75 4 70) A 810. ## . Sin$e this volu#e is %reter thn the volu#e of the frustu#, this #ens tht the #etl hs e-truded throu%h the die openin% by the ti#e the r# hs #oved for!rd by 5 ##. L 6 L 6 0) $$ pressure p pressure p A A 11.(2.7 E 2 - 70B5) A ;or$e + ;or$e + A A 21.(2.1) A +7-+)71), N pressure p A A 11.(2.7 E 2 - 50B5) A L 6 L 6 -) $$ pressure p ;or$e + ;or$e + A A 21.2(2.1) A +7)070-0 N L 6 L 6 5) $$ pressure p pressure p A A 11.(2.7 E 2 - 0B5) A ;or$e + ;or$e + A A 17.0(2.1) A 171-7/)- N pressure p A A 11.(2.7 E 2 - 10B5) A L 6 L 6 1) $$ pressure p ;or$e + ;or$e + A A 1208.(2.1) A 17157)-5 N
+1,*/ MP4 +1/,*+ MP4 10,*) MP4 1+)8*, MP4
18.2 (DS>S units) units) 2.04in4lon% 2.04in4lon% billet billet !ith di#eter di#eter A 1.25 in is dire$t e-truded e-truded to di#eter di#eter A 0.50 in. he e-trusion die n%le A 0 . he !or" #etl hs stren%th $oeffi$ient A 5,000 lbBin 2 nd strin hrdenin% e-ponent A 0.20. n the Johnson e-trusion strin e/ution, a A 0.8 nd b A 1.5. 6eter#ine () e-trusion rtio, (b) true strin (ho#o%eneous defor#tion), ($) e-trusion strin, nd (d) r# pressure pressure t ' t ' A A 2.0, 1.5, 1.0, nd 0.5 in. Dse of spredsheet $l$ultor is re$o##ended for prt (d). A AoB A A f A ( A (o2B ( ( f 2 A (1.25)2B(0.5)2 A *+So#utio () r " A A (b) A A ln r " A ln .25 A 1*85+ ($) " A a E b ln r " A 0.8 E 1.5(1.82) A 5*-/, (d) * f A 5,000(1.82) 0.20B1.20 A 2,0 lbBin 2 Ao A (1.25) 2B A 1.227 in 2 Dnli"e the previous proble#, the die n%le A A 0, so #etl is for$ed throu%h the die openin% s soon s the billet strts to #ove for!rd in the $h#ber. L 6 L 6 +*) i pressure p pressure p A A 2,0(.5 E 2 - 2.0B1.25) A +8-700 #".i+ ;or$e + ;or$e + A A 285,77(1.227) A 5-)7-0, #" pressure p A A 2,0(.5 E 2 - 1.5B1.25) A +-17815 #".i+ L 6 L 6 1*- i pressure p ;or$e + ;or$e + A A 251,81(1.227) A 5),7)++ #" L 6 L 6 1*) i pressure p pressure p A A 2,0(.5 E 2 - 1.0B1.25) A +107,-) #".i+ ;or$e + ;or$e + A A 217,50(1.227) A +07/- #" pressure p A A 2,0(.5 E 2 - 0.5B1.25) A 18/7)8 #".i+ L 6 L 6 )*- i pressure p ;or$e + ;or$e + A A 18,08(1.227) A ++-7,)8 #" he pressure nd for$e for vlue of ' A ' A 0 $ould be $l$ulted, but !ould not be relisti$ be$use of the in$rese in pressure nd for$e due to the butt re#inin% in the e-truder $ontiner t the end of the opertion. 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
18.27 (DS>S units) units) dire$t dire$t e-trusion e-trusion opertion opertion is perfor#ed on $ylindri$l $ylindri$l billet billet !ith initil initil di#eter A 2.0 in nd len%th A .0 in. 6ie n%le A 0 , nd die orifi$e di#eter is 0.50 in. n the Johnson e-trusion strin e/ution, a A 0.8 nd b A 1.5. he opertion is $rried out hot, nd the hot #etl yields t 1,000 lbBin 2 nd does not strin hrden. () Wht is the e-trusion rtio& (b) 6eter#ine the r# position t the point !hen the #etl hs been $o#pressed into the $one of the die nd strts to e-trude throu%h the die openin%. ($) Wht is the r# pressure $orrespondin% to this position& (d) lso deter#ine the len%th of the finl prt if the r# stops its for!rd #ove#ent t the strt of the die $one. So#utio () r " A A A AoB A A f A ( A (o2B ( ( f 2 A (2.0)2B(0.5)2 A 1*)
(b) he portion of the billet tht is $o#pressed into the die $one for#s frustu# !ith #1 A 0.5 (o A 1.0 in nd # nd #2 A 0.5 ( f A A 0.25 in. in. he hei%ht of the the frustu# h A ( # #1 4 # 4 #2)Btn 5 A (1.0 4 0.25)Btn 0 A 0. in. he volu#e of the frustu# is / A A 0. h( # #12 E # E #1 # #2 E # E #22) A 0. (0.)(1.0 2 E 1.0 - 0.25 E 0.25 2) A 0.55 in he billet hs dvn$ed $ertin distn$e by the ti#e this frustu# is $o#pletely filled nd e-trusion throu%h the die openin% is therefore initited. he volu#e of billet $o#pressed for!rd to fill the frustu# is %iven by / A A #12( ' 'o 4 ' 4 '1) A (1.0)2( ' 'o 4 ' 4 '1) Settin% this e/ul to the volu#e of the frustu#, !e hve 'o 4 ' 4 '1) A 0.55 in ( ' ( ' 'o 4 ' 4 '1) A 0.55B A 0.18 in '1 A .0 4 0.18 A 5*811 i ($) A A ln r " A ln 1 A 2.772 " A a E b ln r " A 0.8 E 1.5(2.772) A .5 * f A 1,000(2.772) 0B1.0 A 1,000 lbBin 2 p A p A 1,000(.5 E 2 - .811B2.0) A 11/7))) 11/7))) #".i+ (d) 3en%th of e-truded portion of billet A .811 in. With redu$tion r " A 1, the finl prt len%th, e-$ludin% the $one shped butt re#inin% in the die is ' A ' A .811(1) A )*,0 i* 18.28 (DS>S units) units) n indire$t indire$t e-trusion e-trusion opertion opertion is perfor#ed perfor#ed on n lu#inu# lu#inu# billet billet !ith di#eter di#eter A 2.0 in nd len%th A 8.0 in. ;inl shpe fter e-trusion is re$tn%ulr $ross se$tion tht is 1.0 in by 0.25 in. 6ie n%le A 0 . he lu#inu# hs stren%th $oeffi$ient A 2,000 lbBin 2 nd strin4 hrdenin% e-ponent A 0.20. n the Johnson e-trusion strin e/ution, a A 0.8 nd b A 1.2. () >o#pute the e-trusion rtio, true strin, nd e-trusion strin. (b) Wht is the shpe f$tor of the e-trudte& ($) f the butt left in the $ontiner t the end of the stro"e is 0.5 in lon%, !ht is the len%th of the e-truded se$tion& (d) 6eter#ine the r# pressure in the pro$ess. So#utio () r " A A A AoB A A f 2 2 Ao A (o B A (2) B A .12 in 2 A f A A 1.0 - 0.25 A 0.25 in 2 r " A .12B0.25 A 1+*- A A ln 12.5 A +*-51 " A 0.8 E 1.(2.51) A /*),
(b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A A0.25 A A0.25 in2. he rdius of the $ir$le is # is # A A (0.25B )0.5 A 0.282 in, C c A 2(0.282) A 1.772 in he peri#eter of the e-truded $ross se$tion C " A 1.0 E 0.25 E 1.0 E 0.25 A 2.50 in " A 0.8 E 0.02(2.50B1.772) 2.25 A 1*)+5 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
($) Given tht the butt len%th A 0.5 in o#pute the #-i#u# for$e re/uired to drive the r# for!rd t the strt of e-trusion. A AoB A A f So#utio () r " A A 2 Ao A (100) B A 785 ## 2 A f A A 2(12 - 50) A 1200 ## 2 r " A 785B1200A *-/(b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A A A A 1200 ##2. he rdius of the $ir$le is # is # A A (1200B )0.5 A 1.5 ##, C c A 2(1.5) A 122.8 ##. he peri#eter of the e-truded $ross se$tion C " A 2 E 50 E 12 E 8 E 50 E 12 A 22 ## " A 0.8 E 0.02(22B122.8) 2.25 A 1*)-0 ($) otl ori%inl volu#e / A A 0.25(100)2(500) A ,2,1 ## he finl volu#e $onsists of t!o se$tions (1) butt, nd (2) e-trudte. he butt volu#e / 1 A 0.25(100)2(25) A 1,50 ## . he e-trudte hs $ross4se$tionl re A f A A 1200 ##2. ts volu#e / 2 A 'A A 'A f A A ,2,1 4 1,50 A ,70,1 ## . hus, len%th ' len%th ' A A ,70,1B1200 A 51)8*, $$ 6 5*1), $ (d) A A ln 5.08A 1.2 " A 0.8 E 1.5(1.2) A .2 * f A 20(1.2) 0.15B1.15 A 22. MP M-i#u# r# for$e o$$urs t be%innin% of stro"e !hen len%th is #-i#u# t ' A ' A 250 ## p A p A " * f ( ( " E 2 'B 'B ( (o) A 1.057(22.)(.2 E 2(250)B100) A 15 MP A pAo A 15(785) A 1-755878+ N 6 1-*5/ MN + A pA 18.0 (S units) units) $up4shped $up4shped prt is b$"!rd b$"!rd e-truded e-truded fro# n lu#inu# lu#inu# slu% tht is 50 ## in di#eter di#eter.. he finl di#ensions of the $up re outside outside di#eter A 50 ##, inside di#eter A 0 ##, hei%ht A 100 ##, nd thi$"ness of bse A 5 ##. 6eter#ine () e-trusion rtio, (b) shpe f$tor, nd ($) hei%ht of strtin% slu% re/uired to $hieve the finl di#ensions. (d) f the #etl hs flo! $urve pr#eters pr#eters A A 00 MP nd n A 0.25, nd the $onstnts in the Johnson e-trusion strin e/ution re a A 0.8 nd b A 1.5, deter#ine the e-trusion for$e. So#utio () r " A A A AoB A A f 2 Ao A 0.25(50) A 1.75 ## 2 A f A A 0.25(502 4 02) A 70.8 ## 2 r " A 1.75B70.8 A +*008 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
(b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A A A A 70.8 ## 2. he rdius of of the $ir$le is # is # A A 0.5 (70.8B) A 15 ##, C c A 2(15) A .25 ##. he peri#eter of the e-truded $ross se$tion C " A (50 E 0) A 0 A 282.7 ##. " A 0.8 E 0.02(282.7B.25) 2.25 A 1*+10 ($) olu#e of finl $up $onsists of t!o %eo#etri$ ele#ents (1) bse nd (2) rin%. (1) :se t A A 5 ## nd ( nd ( A A 50 ##. / 1 A 0.25(50)2(5) A 817.5 ## (2) 'in% <6 A 50 ##, 6 A 0 ##, nd h A 5 ##. / 2 A 0.25(502 4 02)(5) A 0.25(2500 4 100)(5) A 7,151.5 ## otl / A A / 1 E / 2 A 817.5 E 7,151.5 A 7, ## olu#e of strtin% slu% #ust be e/ul to this vlue / A A 7, ## / A A 0.25(50)2(h) A 1.5h 1.5 h A 7, ## h 6 5,*+ $$ (d) A A ln 2.778 A 1.0218 " A 0.8 E 1.5(1.0218) A 2. * f A 00(1.0218) 0.25B1.25 A 21.7 MP p A p A " * f " A 1.217(21.7)(2.) A 12. MP Ao A 0.25(0)2 A 125. ## 2 + A A 12.(125.) A 171/7/5) N 18.1 (S units) units) 6eter#ine 6eter#ine the shpe f$tor f$tor for e$h of the e-trus e-trusion ion die orifi$e orifi$e shpes in ;i%ure ;i%ure P18.1. P18.1. So#utio () A () A " A 20 - 0 A 1200 ##, C " A 2(20 E 0) A 10 ## 2 Ao A # A 1200 #2 A 1200B A 81.7, # 81.7, # A A 1.5 ##, C c A 2 # A # A 2(1.5) A 122.8 ## 2.25 " A 0.8 E 0.02(10B122.8) A 1*)1
(b) A (b) A " A #o2 4 #i2 A (252 4 22.52) A 7.0 ## 2 C " A (o E (i A (50 E 5) A 28.5 ## #2 A 7.0B A 118.75, # 118.75, # A A 10.87 ##, C c A 2 # A # A 2(10.87) A 8.7 ## " A 0.8 E 0.02(28.5B8.7) 2.25 A 1*-5 ($) A ($) A " A 2(5)(0) E 5(0 4 10) A 00 E 250 A 550 ## 2 C " A 0 E 0 E 0 E 5 E 25 E 50 E 25 E 5 A 20 ## Ao A #2 A 550, # 550, #2 A 550B A 175.07, # 175.07, # A A 1.2 ## C c A 2 # A # A 2(1.2) A 8.1 ## " A 0.8 E 0.02(20B8.1) 2.25 A 1*100 (d) A (d) A " A 5(55)(5) E 5(85 4 5-5) A 175 ## 2 C " A 2 - 55 E 1 - 25 E 8 - 15 E 10 - 5 A 80 ## Ao A #2 A 175, # 175, #2 A 175B A 5.17, # 5.17, # A A 2.0 ## C c A 2 # A # A 2(2.0) A 15.08 ## " A 0.8 E 0.02(80B15.08) 2.25 A 1*+ 18.2 (S units) units) dire$t e-trusio e-trusion n opertion opertion produ$es produ$es the $ross se$tion se$tion sho!n sho!n in ;i%ure P18.1() P18.1() fro# brss billet billet !hose !hose di#eter di#eter A 125 125 ## nd nd len%th len%th A 500 500 ##. he flo! flo! $urve $urve pr#eters pr#eters of of the brss brss re re A A 700 MP nd n A 0.5 (ble .). n the Johnson strin e/ution, a A 0.7 nd b A 1.. 6eter#ine () the e-trusion rtio, (b) the shpe f$tor, nd ($) the for$e re/uired to drive the r# for!rd durin% e-trusion t the point in the pro$ess !hen the billet len%th re#inin% in the $ontiner 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
A 00 ##. (d) lso, deter#ine the len%th of the e-truded se$tion t the end of the opertion if the volu#e of the butt left in the $ontiner is 250,000 ## . A AoB A A f So#utio () r " A A 2 Ao A (125) B A 12,272 ## 2 A f A A A A " A 20(0) A 1200 ## 2 r " A 12272B1200 A 1)*+5 (b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A f A A 1200 ##2. he rdius of the $ir$le is # is # A A (1200B )0.5 A 1.5 ##, C c A 2(1.5) A 122.8 ##. he peri#eter of the e-truded $ross se$tion C " A 2(20 E 0) A 10 ## " A 0.8 E 0.02(10B122.8) 2.25 A 1*)1 ($) A A ln 10.2 A 2.25 " A 0.7 E 1.(2.25) A .55 * f A 700(2.25) 0.5B1.5 A . MP p A p A " * f " A 1.01(.)(.55 E 2(00)B125) A 1. MP + A pA A pAo A 1.(12,272) A 07+,-7+)) N 6 0*5 MN (d) otl ori%inl volu#e / A A (125)2(500)B A ,15,2 ## he finl volu#e $onsists of t!o se$tions (1) butt, nd (2) e-trudte he butt volu#e s %iven / 1 A 250,000 ## he e-trudte hs $ross4se$tionl re A f A A 1200 ##2 fro# prt () ts volu#e / 2 A 'A A 'A f A A ,15,2 C 250,000 A 5,885,2 ## hus, len%th ' len%th ' A A 5,885,2B1200 A /,)- $$ 6 /*,)- $ 18. (S units) units) n dire$t e-trusion e-trusion opertion, opertion, the $ross $ross se$tion sho!n sho!n in ;i%ure ;i%ure P18.1(b) P18.1(b) is produ$ed fro# $opper billet !hose di#eter A 100 ## nd len%th A 00 ##. n the flo! $urve for $opper, the stren%th $oeffi$ient A 00 MP nd strin hrdenin% e-ponent A 0.50 (ble .). n the Johnson strin e/ution, a A 0.8 nd b A 1.5. 6eter#ine the () e-trusion rtio, (b) shpe f$tor, nd ($) for$e re/uired to drive the r# for!rd durin% e-trusion t the point in the pro$ess !hen the billet len%th re#inin% in the $ontiner A 50 ##. (d) lso, deter#ine the len%th of the e-truded se$tion t the end of the opertion if the volu#e of the butt left in the $ontiner is 200,000 ## . A AoB A A f So#utio () r " A A 2 Ao A (100) B A 785 ## 2 A f A A A A " A (50)2B 4 (5)2B A 1.5 C 150. A 7.1 ## 2 r " A 785B7.1 A +1*)(b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A " A 7.1 ##2. he rdius of the $ir$le is # is # A A (7.1B )0.5 A 10. ##, C c A 2(10.) A 8.5 ##. he peri#eter of the e-truded $ross se$tion C " A (50) E (5) A 28.5 ## " A 0.8 E 0.02(28.5B8.5) 2.25 A 1*-5 ($) A A ln 21.05A .07 " A 0.8 E 1.5(.07) A 5.7 * f A 00(.07) 0.50B1.50 A .1 MP p A p A " * f " A 1.5(.1)(5.7 E 2(50)B100) A 775. MP + A pA A pAo A 775.(785) A )7+8+710, N 6 )*5MN 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
18420
Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
(d) otl ori%inl volu#e / A A (100)2(00)B A ,11,5 ## he finl volu#e $onsists of t!o se$tions (1) butt, nd (2) e-trudte. he butt volu#e s %iven / 1 A 200,000 ## . he e-trudte hs $ross4se$tionl re A f A A 7.1 ##2 fro# prt () ts volu#e / 2 A 'A A 'A f A A ,11,5 C 200,000 A 2,1,5 ## . hus, len%th ' len%th ' A A 2,1,5B7.1 A 0788/ $$ 6 0*88/ $ 18. (S units) units) dire$t e-trusio e-trusion n opertion opertion produ$es produ$es the $ross se$tion se$tion sho!n sho!n in ;i%ure P18.1($) P18.1($) fro# n lu#inu# lloy billet !hose di#eter A 150 ## nd len%th A 500 ##. he flo! $urve pr#eters for the lu#inu# re re A A 20 MP nd n A 0.15 (ble .). n the Johnson strin e/ution, a A 0.8 nd b A 1.2. 6eter#ine () the e-trusion rtio, (b) the shpe f$tor, nd ($) the for$e re/uired to drive the r# for!rd durin% e-trusion t the point in the pro$ess !hen the billet len%th re#inin% in the $ontiner A 00 ##. (d) ;ind the len%th of the e-truded se$tion t the end of the opertion if the volu#e of the butt left in the $ontiner is 50,000 ## . A AoB A A f So#utio () r " A A 2 Ao A (150) B A 17,71.5 ## 2 A f A A A A " A 0(5) E 2(25)(5) A 00 E 250 A 550 ## 2 r " A 17,71.5B550 A 5+*1 (b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A " A 550 ##2. C " A 0 E 0 E 0 E 5 E 25 E 50 E 25 E 5 A 20 ## Ao A #2 A 550, # 550, #2 A 550B A 175.07, # 175.07, # A A 1.2 ## C c A 2 # A # A 2(1.2) A 8.1 ## " A 0.8 E 0.02(20B8.1) 2.25 A 1*100 ($) A A ln 2.1A .7 " A 0.8 E 1.2(.7) A . * f A 20(.7) 0.15B1.15 A 251.5 MP p A p A " * f " A 1.177(251.5)(. E 2(00)B150) A 07 MP + A pA A pAo A 07(17,71.5) A -578//7810 N 6 -5*8 MN (d) otl ori%inl volu#e / A A (150)2(500)B A 8,85,750 ## he finl volu#e $onsists of t!o se$tions (1) butt, nd (2) e-trudte. he butt volu#e s %iven / 1 A 50,000 ## . he e-trudte hs $ross4se$tionl re A f A A 550 ##2 fro# prt () ts volu#e / 2 A 'A A 'A f A A 8,85,750 C 50,000 A 8,85,750 ## . hus, len%th ' len%th ' A A 8,85,750B550 A 1-7+/0 $$ 6 1-*+/0 $ 18.5 (S units) units) dire$t e-trusio e-trusion n opertion opertion produ$es produ$es the $ross se$tion se$tion sho!n sho!n in ;i%ure P18.1(d) P18.1(d) fro# n lu#inu# lloy billet !hose di#eter A 150 ## nd len%th A 00 ##. he flo! $urve pr#eters for the lu#inu# re re A A 20 MP nd n A 0.15 (ble .). n the Johnson strin e/ution, a A 0.8 nd b A 1.5. 6eter#ine the () e-trusion rtio, (b) shpe f$tor, nd ($) for$e re/uired to drive the r# for!rd durin% e-trusion t the point in the pro$ess !hen the billet len%th re#inin% in the $ontiner A 800 ##. (d) lso, deter#ine the len%th of the e-trudte t the end of the opertion if the volu#e of the butt left in the $ontiner is 500,000 ## . So#utio () r " A A A AoB A A f 2 Ao A (150) B A 17,71 ## 2 A f A A A A " A 5(55)(5) E 5(85 C 5(5)) A 175 ## 2 r " A 17,71B175 A 1)*-9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
18421
Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
(b) o deter#ine the die shpe f$tor !e need to deter#ine the peri#eter of $ir$le !hose re is e/ul to tht of the e-truded $ross se$tion, A " A 175 ##2. C " A 2 - 55 E 1 - 25 E 8 - 15 E 10 - 5 A 80 ## Ao A #2 A 175, # 175, #2 A 175B A 5.17, # 5.17, # A A 2.0 ## C c A 2 # A # A 2(2.0) A 15.08 ## " A 0.8 E 0.02(80B15.08) 2.25 A 1*+ ($) A A ln 10.55A 2. " A 0.8 E 1.5(2.) A . * f A 20(2.) 0.15B1.15 A 27. MP p A p A " * f " A 1.2(27.)(. E 2(800)B150) A 578 MP + A pA A pAo A 578(17,71) A 1)+7+,87-5/ N 6 1)+*5 MN (d) otl ori%inl volu#e / A A (150)2(00)B A 15,0,1 ## he finl volu#e $onsists of t!o se$tions (1) butt, nd (2) e-trudte. he butt volu#e s %iven / 1 A 500,000 ## . he e-trudte hs $ross4se$tionl re A f A A 175 ##2 fro# prt () ts volu#e / 2 A 'A A 'A f A A 15,0,1 C 500,000 A 15,0,1 ## hus, len%th ' len%th ' A A 15,0,1B175 A ,71,0 $$ 6 ,*1,0 $ D!4wi'
18. %A& (S units) spool of $opper !ire hs strtin% di#eter of 2.5 ##. t is dr!n throu%h die !ith n openin% tht is 2.1 ##. he entrn$e n%le of the die A 18 . >oeffi$ient of fri$tion t the !or"4die interf$e is 0.08. he pure $opper hs stren%th $oeffi$ient A 00 MP nd strin hrdenin% $oeffi$ient A 0.50 (ble .). he opertion is perfor#ed t roo# te#perture. 6eter#ine () re redu$tion, (b) dr! stress, nd ($) dr! for$e re/uired for the opertion. So#utio () r A A ( A Ao 4 A 4 A f )B )B A Ao 2 Ao A 0.25(2.50) A .1 ##2 A f A A 0.25(2.1)2 A . ##2 r A A (.1 C .)B.1 A )*+,/
(b) 6r! stress d d A A ln(.1B.) A ln 1.17 A 0. * f A 00(0.) 0.5B1.5 A 118.2 MP A A 0.88 E 0.12( ('c) ( A ( A 0.5(2.5 E 2.1) A 2.0 'c A 0.5(2.5 4 2.1)Bsin 18 A 0.7 A A 0.88 E 0.12(2.0B0.7) A 1.1 d d A A * f (1 E Btn Btn ) (ln A (ln Ao A A f ) A 118.2(1 E 0.08Btn 18)(1.1)(0.) A 0*5 MP4 ($) 6r! for$e + for$e + A A f d d A A .(7.) A +55 N + A A 18.7 (DS>S units) units) lu#inu lu#inu# # rod sto$" !ith strtin% strtin% di#eter di#eter A 0.50 in is dr!n throu%h throu%h dr! die !ith n entrn$e n%le A 1 . he finl di#eter of the rod is A 0.75 in. he #etl hs stren%th $oeffi$ient A 25,000 lbBin 2 nd strin hrdenin% e-ponent A 0.20 (ble .). >oeffi$ient of fri$tion t the !or"4die interf$e A 0.1. 6eter#ine () re redu$tion, (b) dr! for$e for the opertion, nd ($) horsepo!er to perfor# the opertion if the e-it velo$ity of the sto$" A 5 ftBse$. So#utio () r A A ( A Ao 4 A 4 A f )B )B A Ao 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
Ao A 0.25(0.50)2 A 0.1 in 2 A f A A 0.25(0.5) 2 A 0.110 in 2 r A A (0.1 4 0.110)B0.1 A )*/50(b) 6r! for$e + for$e + A A ln(0.1B0.110) A ln 1.778 A 0.575 * f A 25,000(0.575) 0.20B1.20 A 18,5 lbBin 2 A A 0.88 E 0.12( ('c) ( A ( A 0.5(.50 E 0.75) A 0.8 'c A 0.5(0.50 4 0.75)Bsin 1 A 0.2778 A A 0.88 E 0.12(0.8B0.2778) A 1.0 + A A A A f * f (1 E Btn Btn ) (ln A (ln Ao A A f ) + A A 0.110(18,5)(1 E 0.1Btn 1)(1.0)(0.575) A 181- #" ($) - ($) - A A 1815(5 ftBse$ - 0) A 5,500 ftBlbB#in .- A A 5,500B,000 A 1*- :3 18.8 (S units) units) :r sto$" of of initil initil di#eter di#eter A 0 ## is dr!n !ith !ith drft A 15 ##. he dr! die die hs n entrn$e n%le A 18 , nd the $oeffi$ient of fri$tion t the !or"4die interf$e A 0.08. he #etl behves behves s perfe$tl perfe$tly y plsti$ plsti$ #teril #teril !ith !ith yield yield stress stress A 105 MP. MP. 6eter#ine 6eter#ine () () re redu$tio redu$tion, n, (b) dr! stress, ($) dr! for$e re/uired for the opertion, nd (d) po!er to perfor# the opertion if e-it velo$ity A 1.0 #B#in. So#utio () r A A ( A Ao 4 A 4 A f )B )B A Ao 2 Ao A 0.25(0) A 1.7 ## 2 ( f A ( A (o 4 d A A 0 4 15 A 75 ##, A f A A 0.25(75)2 A 17. ## 2 r A A (1.7 4 17.)B1.7 A )*5)-
(b) 6r! stress d d A A ln(1.7 B17.) A ln 1.0 A 0. * f A k A A 105 MP A A 0.88 E 0.12( ('c) ( A ( A 0.5(0 E 75) A 82.5 ## 'c A 0.5(0 4 75)Bsin 18 A 2. ## A A 0.88 E 0.12(82.5B2.) A 1.288 d d A * f (1 E Btn Btn ) (ln A (ln Ao A A f ) A 105(1 E 0.08Btn 18)(1.288)(0.) A 1*/- MP4 ($) + ($) + A A A A f d d A A 17. (1.5) A +017/0- N (d) - (d) - A A 271,75(1 #B#in) A 271,75 +4#B#in A 52. +4#Bs A /-+/* W 18. (DS>S units) units) Wire Wire sto$" of initil initil di#eter di#eter A 0.125 in is dr!n throu%h throu%h t!o dies dies e$h providin% providin% 2 0.20 re redu$tion. he strtin% #etl hs stren%th $oeffi$ient A 0,000 lbBin nd strin hrdenin% e-ponent A 0.15. 9$h die hs n entrn$e n%le of 12 , nd the $oeffi$ient of fri$tion t the !or"4die interf$e is esti#ted to be 0.10. he #otors drivin% the $pstns t the die e-its $n e$h deliver 1.50 hp t 0? effi$ien$y. 6eter#ine the #-i#u# possible speed of the !ire s it e-its the se$ond die. dr! (o A 0.125 in, A in, Ao A 0.25(0.125)2 A 0.01227 in 2 0081 in 2 So#utio ;irst dr! ( A A ln(0.01227B0.0081) A ln 1.250 A 0.221 r A A ( A Ao 4 A 4 A f )B )B A Ao, A f A A A Ao(1 4 r ) A 0.01227(1 4 0.2) A 0.008175 9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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Solutions for Fundamentals of Modern Manufacturing , 5e (published by Wiley) MPGroover 2012
* f A 0,000(0.221) 0.15B1.15 A 27,775 lbBin 2
A A 0.88 E 0.12( ('c) ( f A A 0.125(1 4 r )0.5 A 0.125(0.8) .5 A 0.1118 in ( A ( A 0.5(.125 E 0.1118) A 0.118 'c A 0.5(0.125 4 0.1118)Bsin 12 A 0.017 A A 0.88 E 0.12(0.118B0.017) A 1. + A A A A f * f (1 E Btn Btn ) (ln A (ln Ao A A f ) + A A 0. 008175(27,775)(1 E 0.1Btn 12)(1.)(0.221) A 11 lb 1.5 hp t 0? effi$ien$y A 1.5 - 0.0(,000 ft4lbB#in)B0 A 72.5 ft4lbBse$ - A +$ A +$ A A 11$ 11$ A 72.5 $ A 72.5B11 A *+/ 2t.se9 Se$ond dr! ( dr! (o A 0.1118 in, A in, Ao A 0.25(0.1118) 2 A 0.0081 in 2 r A A ( A Ao 4 A 4 A f )B )B A Ao, A f A A A Ao(1 4 r ) A 0.0081(1 4 0.2) A 0.007855 in 2 A A ln(0.0081B0.007855) A ln 1.250 A 0.221 otl strin e-perien$ed by the !or" #etl is the su# of the strins fro# the first nd se$ond dr!s A A 1 E 2 A 0.221 E 0.221 A 0.2 * f A 0,000(0.2) 0.15B1.15 A 0,818 lbBin 2 A A 0.88 E 0.12( ('c) ( f A A 0.1118(1 4 r )0.5 A 0.1118(.8) .5 A 0.100 in ( A ( A 0.5(0.1118 E 0.100) A 0.105 'c A 0.5(0.1118 4 0.100)Bsin 12 A 0.02 A A 0.88 E 0.12(0.105B0.02) A 1.5 + A A A A f * f (1 E Btn Btn ) (ln A (ln Ao A A f ) + A A 0.007855(0,818)(1 E 0.1Btn 12)(1.5)(0.2) A 21 lb. 1.5 hp t 0? effi$ien$y A 72.5 ft4lbBse$ s before in the first dr!. - A +$ A +$ A A 21$ 21$ A 72.5 $ A 72.5B21 A 5*/0 2t.se9 Note he $l$ultions indi$te tht the se$ond dr! die is the li#itin% step in the dr!in% se/uen$e. he first opertion !ould hve to be operted t !ell belo! its #-i#u# possible speed* or the se$ond dr! die $ould be po!ered by hi%her horsepo!er #otor* or the redu$tions to $hieve the t!o st%es $ould be rello$ted to $hieve hi%her redu$tion in the first dr!in% opertion.
9-$erpts fro# this !or" #y be reprodu$ed by instru$tors for distribution on not4for4profit bsis for testin% or instru$tionl purposes only to students enrolled in $ourses for !hi$h the te-tboo" hs been dopted. Any other reproduction reproduction or translation of this work work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyriht Act without the permission of the copyriht owner is unlawful!
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