Bisu Knevez Darnis Laheurte Gerard 09

7/27/2019 Bisu Knevez Darnis Laheurte Gerard 09

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International Journal of Material Forming

New method to characterize a machining system: application in turning

Claudiu F. Bisu1,2, Jean-Yves K'nevez, Philippe Darnis3, a!nald "aheur#e,3, $lain %&rard

1niversi#! P(li#ehni)a *r(+ Bu)hares#, 313 plaiul ndependen#ei, /02 Bu)hares#(u+anie e+ail4 )*5isu6g+ail.)(+2niversi#& de B(rdeau7, 381 )(urs de la "i5&ra#i(n, 3308 9alen)e-)ede7 Fran)e e+ail4 alain.gerard6u-5(rdeau71.*r3niversi#& de B(rdeau7 - 9 $ 0:/, 18 rue ;aude#, 331(r?pie)e s#i**ness, (r #he uni# #((l->(r?pie)e s#i**ness in +(delings 2D. @#hers i+p(se #he #((l a)#i(n, (r use +(re (r less)(+ple7 +(delings (* #he e**(r#s applied 5! #he #((l #a?ing a))(un# #he #((l ge(+e#r!.9hus, all #hese +(dels re+ain #>(-di+ensi(nal (r s(+e#i+es par#iall! #hree-di+ensi(nal.9his paper ai+s a# devel(ping an e7peri+en#al +e#h(d all(>ing #( de#er+ine a))ura#el! #hereal #hree-di+ensi(nal 5ehavi(ur (* a +a)hining s!s#e+ +a)hine #((l, )u##ing #((l, #((l-h(lder and ass()ia#ed s!s#e+ (* *(r)e +e#r(l(g! si7-)(+p(nen# d!na+(+e#er.

n #he >(r?-spa)e +(del (* +a)hining, a ne> e7peri+en#al pr()edure is i+ple+en#ed #(de#er+ine #he +a)hining s!s#e+ elas#i) 5ehavi(ur. $n e7peri+en#al s#ud! (* +a)hinings!s#e+ is presen#ed. Ae pr(p(se a +a)hining s!s#e+ s#a#i) )hara)#eriza#i(n. $de)(+p(si#i(n in #>( dis#in)# 5l()?s (* #he s!s#e+ A(r?pie)e-9((l-=a)hine is realized.9he 5l()? 9((l and #he 5l()? A(r?pie)e are s#udied and )hara)#erized separa#el! 5! +a#ri7s#i**ness and displa)e+en# #hree #ransla#i(ns and #hree r(#a#i(ns. 9he Cas#iglian('s #he(r!all(>s us #( )al)ula#e #he #(#al s#i**ness +a#ri7 and #he #(#al displa)e+en# +a#ri7.

$ s#i**ness )en#er p(in# and a plan (* #((l #ip s#a#i) displa)e+en# are presen#ed in agree+en#>i#h #he #urning +a)hining d!na+i) +(del and espe)iall! during #he sel* indu)ed vi5ra#i(n.9hese resul#s are ne)essar! #( have a g((d #hree-di+ensi(nal +a)hining s!s#e+ d!na+i)

)hara)#eriza#i(n presen#ed in a ne7# paper.Keywords4 e7peri+en#al +(del, displa)e+en# plan, sel*-e7)i#ed vi5ra#i(ns, #urning

1
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Nomenclature

a Dis#an)e 5e#>een displa)e+en# #ransdu)er B Bl()? 9((l

B! Bl()? A(r?pie)eCE Da+ping +a#ri7C@E C(+plian)e +a#ri7CiE Displa)e+en# #ransdu)er i 1 #( /"#B9 Bl()? 9((l Bs#i**ness )en#erD1 G(lding *i7#ure dia+e#er ++D2 A(r?pie)e dia+e#er ++HD +all displa)e+en#s #(rs(r DiI #raigh# line )(rresp(nding (* #he displa)e+en# dire)#i(n (* #he

p(in#Pi,Ii 7, !, z and I1, 2, 3diI P(in#s displa)e+en#s ve)#(rsPi,Ii 7, !, z and I1, 2, 3

D7 Dis#an)e 5e#>een #he line DiI Y(ung +(dulus ;++e7, *7 )ale *a)#(rsFi F(r)e ve)#(rs applied #( (5#ain Bs#i**ness )en#er i 7, !, zI ner#ial +(+en#KE #i**ness +a#ri7 ;+KCE #i**ness +a#ri7 (* r(#a#i(n ;+radKFE #i**ness +a#ri7 (* displa)e+en# ;+KF,B9E #i**ness +a#ri7 (* Bdispla)e+en# ;+KF,BAE #i**ness +a#ri7 (* B!displa)e+en# ;+KF,A$=E #i**ness +a#ri7 (* +a)hining s!s#e+ displa)e+en# ;+

Kerr(rs E rr(rs +a#ri7 *(r #he +a#ri7 KEKCFE #i**ness +a#ri7 (* r(#a#i(n displa)e+en# ;++KFCE #i**ness +a#ri7 (* displa)e+en# r(#a#i(n ;rad"1 G(lding *i7#ure leng#h ++"2 "eng#h >(r?pie)e ++Mi P(in# in#erse)#i(n 5e#>een s#raigh# lines DiIi 7, !, z and I1,

2, 3m Displa)e+en# +easured a# #he )harge p(in#ME =ass +a#ri7

in Plan n(r+alPi@ 9((l #ip p(in#

@C Cu5 )en#erP F(r)e ;Pi Plan in)luding #he p(in#MiPBT Displa)e+en# plan )(nsidering #((l p(in#PiI Charge p(in#s i 7, !, z and I1, 2, 3HT =e)hani)al a)#i(ns #(rs(r LE =a#ri7 eigenve)#(r K F,B9Ev1 =a#ri7 eigenvalue KF,B9E

2


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!M A(r?pie)e-9((l-=a)hine7 z Cr(ss *eed dire)#i(n! Cu##ing a7is Displa)e+en# ++

i Displa)e+en# al(ng i i1,2,3

=easured angle a# #he *(r)e p(in#i $ngular devia#i(n (* C(-planari#! 5e#>een lines DiIi 7, !, zM

and I1, 2, 3i =ini+al dis#an)e 5e#>een s#raigh# lines DiIi 7, !, zM and I1, 2,

3i (#a#i(n al(ng i i7, !, z

Introduction

=e#al )u##ing is (ne (* #he +(s# i+p(r#an# +anu*a)#uring pr()ess. 9he +(s# )(++(n)u##ing pr()esses are #urning, +illing, drilling and grinding.

During #he )u##ing pr()ess (* di**eren# +a#erials, a >h(le (* ph!si)-)he+i)al and d!na+i)phen(+ena are inv(lved. las#(-plas#i) s#rains, *ri)#i(n and #her+al phen(+ena aregenera#ed in #he )(n#a)# z(ne 5e#>een >(r?pie)e, #((l and )hip. 9hese phen(+ena arein*luen)ed 5!4 ph!si)al pr(per#ies (* #he +a#erial #( 5e +a)hined, #((l ge(+e#r!, )u##ingand lu5ri)a#i(n )(ndi#i(ns, and als( #he +a)hining s!s#e+ d!na+i) para+e#ers s#i**ness,da+ping. 9he +a)hine #((l vi5ra#i(ns are genera#ed 5! #he in#era)#i(n 5e#>een #he elas#i)+a)hining s!s#e+ and #he )u##ing pr()ess. 9he elas#i) s!s#e+ is )(+p(sed (*4 #he di**eren#par#s (* #he +a)hine #((l in +(ve+en#, #he >(r?pie)e and #he #((l. $)#i(ns (* #he+a)hining pr()ess are usuall! *(r)es and +(+en#s. 9hese a)#i(ns als( genera#e rela#ivedispla)e+en#s (* ele+en#s )(+p(sing #he elas#i) s!s#e+. 9he! ())ur *(r e7a+ple 5e#>een

#he #((l and >(r?pie)e, #he #((l devi)e and 5ed, e#). 9hese displa)e+en#s +(di*! #he)u##ing )(ndi#i(ns and in #he sa+e >a! #he *(r)es. 9hus, #he ?n(>ledge (* #he +a)hinings!s#e+ elas#i) 5ehavi(ur is essen#ial #( unders#and #he )u##ing pr()ess /E.

Cer#ain s)ien#is#s devel(ped a *ini#e ele+en# 5ea+ +(del (* spinning s#epped-sha*#>(r?pie)e #( per*(r+ s#a5ili#! anal!sis using ;!Nuis# )ri#eri(n 3OE (r #he #radi#i(nals#a5ili#! l(5e diagra+ 1/, 28E. 9his #radi#i(nal s#a5ili#! anal!sis #e)hniNue sh(>s #ha# #he)ha##er ins#a5ili#! depends (n #he s#ru)#ural da+ping in #he s!s#e+ and #he spindle speed.Chen and 9sa( presen#ed a d!na+i) +(del (* )u##ing #((l >i#h 11E and >i#h(u# #ails#()?supp(r#ed >(r?pie)e using 5ea+ #he(r! 1E. Gere, #he e**e)#s (* >(r?pie)e para+e#ers ares#udied (n #he d!na+i) s#a5ili#! (* #urning pr()ess 5! #rea#ing #he >(r?pie)e as a

)(n#inu(us s!s#e+. Carrin( e# al., OE presen# a +(del #ha# #a?es in#( a))(un# 5(#h #he>(r?pie)e de*le)#i(n and #he )u##ing *(r)e 5e#>een #((l and >(r?pie)e. 9he #hree)(+p(nen#s (* #he )u##ing *(r)e are *un)#i(n (* #he )u##ing ge(+e#r!. 9he e**e)# (* #he>(r?pie)e-#((l-+a)hine de*le)#i(ns is a shi*# (* #he >(r?pie)e )r(ss-se)#i(n and a +(ving5a)? (* #he #((l h(lder in #he radial and #he #angen#ial dire)#i(n 2D +(del.

n #hese pr()esses, #he )u##ing *(r)es +easure+en# has i+p(r#an# and #re+end(usappli)a#i(ns >i#hin indus#r! and resear)h ali?e. 9he )u##ing *(r)es es#i+a#i(n all(>s #(

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supervise #((l >ear ev(lu#i(n 3/E, es#a5lishes +a#erial +a)hina5ili#ies, (p#i+izes )u##ingpara+e#ers, predi)#s +a)hined >(r?pie)e sur*a)e Nuali#! and s#ud! phen(+ena su)h as)hip *(r+a#i(n (r vi5ra#i(ns appearan)e. e?ar and Yang pr(p(se a )(+plian# #>( degree(* *reed(+ d!na+i) )u##ing *(r)e +(del 5! )(nsidering #he rela#ive +(#i(n (* >(r?pie)e>i#h )u##ing #((l. 9((l and >(r?pie)e are +(delled as #>( separa#e single degree (*

*reed(+ spring-+ass-da+per s!s#e+s 30E.

n #he li#era#ure, #here are +an! s#udies )(n)erning #he )u##ing *(r)e +easure+en#. =an!d!na+(+e#ers *(r #his purp(se have 5een devel(ped 8, :, 12, 23E.

ndependen#l! (* #he +a)hining (pera#i(n #!pe, +e#h(ds *(r )u##ing *(r)e +easure+en#s)an 5e divided in#( #>( general )a#eg(ries. 9he *irs# )(rresp(nds #( #he )a#eg(r! #ha# uses#he )urren# (r #he v(l#age signals e+i##ed 5! #he #((l +a)hine drive +(#(r (r )(n#r(ls!s#e+s 2(r?pie)e asse+5lies2, 3er )(nsu+ed (r dire)#l! *r(+ +e#al )u##ing d!na+(+e#ersM +e)hani)al,h!drauli), pneu+a#i) (r several #!pes (* ele)#r(-+e)hani)al d!na+(+e#ers.

Kn(>ing #he )u##ing *(r)es is essen#ial #( +a)hine #((l 5uilders in )al)ula#ing p(>erreNuire+en#s and *ra+e rigidi#!. Cu##ing *(r)es a)#ing (n #he #((l +us# 5e +easured a# #hedesign #((l #ha# are s#r(ng en(ugh #( re+(ve )hip a# #he desired Nuan#i#! *r(+ #he>(r?pie)e and #( )al)ula#e p(>er (* #((l driver s!s#e+. 9he d!na+(+e#er is a5le #(+easure #hree *(r)e )(+p(nen#s4 )u##ing *(r)e Fc, *eed *(r)e Fa)and radial *(r)e Fp5u#n(# #he #(rNue a# #he #((l #ip. $7in#e e# al., 1E pr(p(se a pr()edure #( a))(un# *(r 5(#h)ali5ra#i(n and pr()ess err(rs in #he un)er#ain#! es#i+a#i(n *(r #he spe)i*i) si#ua#i(n (*single )u##ing *(r)e +easure+en#s. 9he in*luen)e para+e#ers )(nsidered in #heir >(r?,)(n#ri5u#i(n #( #he +easure+en# un)er#ain#!, >(r?pie)e, #((l and +a)hine >ere n(#)(nsidered. Perez e# al., 2OE give a +e)hanis#i) +(del *(r #he es#i+a#i(n (* )u##ing *(r)esin +i)r(+illing 5ased (n spe)i*i) )u##ing pressure. 9he +(del in)ludes #hree para+e#ers>hi)h all(> #( )(n#r(l #he en#r! (* #he )u##er in #he >(r?pie)e. 9he err(rs in #he radial and#he angular p(si#i(n (* #he )u##ing edges (* #he #((l is )(nsidered. 9he )u##ing *(r)es are)al)ula#ed (n #he 5asis (* #he engaged )u# ge(+e#r!, #he unde*(r+ed )hip #hi)?nessdis#ri5u#i(n al(ng #he )u##ing edges, and #he e+piri)al rela#i(nships #ha# rela#e #he )u##ing*(r)es #( #he unde*(r+ed )hip ge(+e#r!. 9his rela#i(n d( n(# #a?e in#( a))(un# #he elas#i)i#!(* #he +a)hining s!s#e+. n #he +easure+en# (* #he )u##ing *(r)es 38E, (nl! elas#i)de*le)#i(ns (* #he )u##ing #((l due #( #he )u##ing *(r)es >ere +easured 5! +eans (* #he l(ad)ells l()a#ed a# sui#a5le p(si#i(n (n #he )u##ing #((l.

G(>ever #he d!na+(+e#er )an +easure #hree perpendi)ular )u##ing *(r)e )(+p(nen#s and#hree #(rNue )(+p(nen#s si+ul#ane(usl! during #urning, and #he +easured nu+eri)al values)an 5e s#(red in )(+pu#er 5! da#a a)Nuisi#i(n s!s#e+ 12E. 9his d!na+(+e#er >as designed#( +easure up #( 8,N+a7i+u+ *(r)e and 38N/m#(rNue. 9he s!s#e+ sensi#ivi#! ispm0 in *(r)e andpmO in #(rNue.

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During #he )u##ing pr()ess, #he )u##ing #((l pene#ra#es in#( #he >(r?pie)e due #( #he rela#ive+(#i(n 5e#>een #((l and >(r?pie)e and #he )u##ing *(r)es and #(rNues are +easured (n a+easuring plane in #he Car#esian )((rdina#e s!s#e+. 9he )u##ing *(r)es have 5een +easured5! #he d!na+(+e#ers designed *(r di**eren# >(r?ing prin)iples as s#rain gauge 5ased 12E.9hus i# is ne)essar! #( have a g((d +e#h(d(l(g! #( +easure #he >(r?pie)e-#((l-+a)hine

rigidi#! 5e*(re +easuring *(r)es and #(rNues. 9his ne> +e#h(d(lg! is pre)isel! #he purp(se(* #his paper. n se)#i(n 2 >e presen# #he e7peri+en#al devi)e. $*#er see se)#i(n 3 >e)(n)eive #he >(r?pie)e. 9he >(r?pie)e ge(+e#r! and di+ensi(ns re#ained *(r #hese #es#-#u5es >ere sele)#ed using #he *ini#e ele+en# +e#h(d )(upled #( an (p#i+iza#i(n +e#h(d, 5!$=CFs(*#>are. n #he *(ll(>ing se)#i(n 0 a +e#h(d(l(g! 5ased (n #he vir#ual >(r?#hree #ransla#i(ns and #hree r(#a#i(ns is e7hi5i#ed #( s#ud! #he s#a#i) ai+s and #()hara)#erize #he s#a#i) eNuivalen# s#i**ness values in (rder #( iden#i*! #he #hree-di+ensi(nalelas#i) 5ehavi(ur (* #he +a)hining s!s#e+. 9he applied e**(r#s are Nuan#i*ied >i#h a *(r)esens(r. 9he #(rs(r (* s+all displa)e+en#s, 3 linears and 3 r(#a#i(ns displa)e+en#, is+easured 5! si7 displa)e+en# #ransdu)er. $ s#i**ness gl(5al +a#ri7 is dedu)ed >i#h i#svari(us resul#s. 9he su+ (* #he #>( s#i**ness +a#ri7 displa)e+en#s 5l()? #((l and 5l()?

>(r?pie)e de#er+ines #he s#i**ness +a#ri7 (* +a)hining s!s#e+ displa)e+en#. B! #heCas#iglian('s #he(re+ >e de#er+ine #he angle #ha# )hara)#erizes #he prin)ipal dire)#i(n (*de*(r+a#i(n. Be*(re )(n)luding, in se)#i(n 8 #he s#i**ness )en#er is (5#ained using #he leas#sNuares +e#h(d in #he )((rdina#e s!s#e+ 5ased (n #he #((l in @ p(in# #ha# is #he (rigin (*#he )((rdina#e s!s#e+.

General points

9(da!, +a)hines #((l are ver! rigid #here are less and less ge(+e#ri)all! *aul#!. 9hevi5ra#(r! pr(5le+s are s#r(ngl! rela#ed #( #he )u##ing. deall!, )u##ing )(ndi#i(ns are )h(sen

su)h #ha# +a#erial re+(val is per*(r+ed in a s#a5le +anner. G(>ever, s(+e#i+es )ha##er isunav(ida5le 5e)ause (* #he ge(+e#r! (* #he )u##ing #((l and >(r?pie)e. n (r?pie)e sin)e i# has a s#a#i) s#i**ness #ha# is up#( 3.2 #i+es less #han #he )u##ing #((l. in)e i# is highl! i+pra)#i)al #( ins#ru+en# #he>(r?pie)e during )u##ing #he end g(al is #( devel(p an (5server #ha# )an #rans*(r++easure+en#s +ade a# #he )u##ing #((l in#( a predi)#i(n a5(u# #he +(#i(n (* #he >(r?pie)e.Dassana!a?e 13E appr(a)hes in #he 1D )ase #he d!na+i) resp(nse (* #he #((l h(lder #( #hereNues# (* #he #((l >hi)h *(ll(>s a regenera#ive sur*a)e. 9he! )(nsider (nl! #((l +(#i(nsand disregards >(r?pie)e vi5ra#i(n. nsperger )(n#inues in #he 2D )ase ?eeping >(r?pie)erigid 5u# he #a?es in#( a))(un# #he *le7i5ili#! (* #he #((l1OE. F(r an (pera#i(n (* +illing33E, #he de*le)#i(ns (* #he +a)hine-#((l, #he #((lh(lder and #he #((lh(lder )la+ping in #he

spindle, #he #((l )la+ping in #he #((lh(lder, and #he #((l i#sel*, >ere +easurede7peri+en#all! under #he e**e)#s (* ?n(>n *(r)es. 9he resul#s (* #his s#ud! sh(> #ha# #hes#i**ness (* 5(#h #he +a)hine and #he )la+ping in #he +a)hine-spindle-#((lh(lder-#((ls!s#e+ have a si+ilar i+p(r#an)e in #he displa)e+en# (* #he #((l #ip su5Ie)#ed #( a )u##ing*(r)e #( #he de*le)#i(n (* #he #((l i#sel*. 9hus, i# is ne)essar! #( iden#i*! #he elas#i)5ehavi(ur (* +a)hine par#s /E. 9hese vi5ra#i(ns are genera#ed and sel* indu)ed 5! #he)u##ing pr()ess. $ )(nven#i(nal la#he >i#h high rigidi#! is used #( s#ud! #hese d!na+i)

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phen(+ena. 9he A(r?pie)e-9((l-=a)hine !M s!s#e+ is presen#ed (n #he *igure 1 *(ra #urning (pera#i(n.

Fig. 1 A(r?pie)e-9((l-=a)hine )(nsidering d!na+i) )u##ing pr()ess.

9he elas#i) s#ru)#ure (* !Ms!s#e+ has several degrees (* *reed(+ and has +an! spe)i*i)vi5ra#i(n +(des. 9he vi5ra#i(ns (* ea)h ele+en# (* #he s#ru)#ure are )hara)#erized 5! i#sna#ural *reNuen)! depending (n #he #i**ness +a#ri7 KE, #he =ass +a#ri7 =E and #heDa+ping +a#ri7 CE. n a *irs# #i+e, (nl! #he s#i**ness +a#ri7 KE is s#udied.

@ur e7peri+en#al appr(a)h is 5ased (n #he +a#ri7 devel(p+en# #ha# is presen#ed in 2:E. 9(iden#i*! #he !M s!s#e+ s#a#i) 5ehavi(ur, #he +a)hining s!s#e+ is divided in#( #>(5l()?s, #he 5l()? 9((l B and #he 5l()? A(r?pie)e B! *igure 2. 9hese #>( 5l()?s arerela#ed #( #he #urn 5ed supp(sed #( 5e in*ini#el! rigid.

Fig. 2 Presen#a#i(n (* #he e7peri+en#al devi)e.

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3 Components of the system WTM

3.1 Block Workpiece: BW

$s +an! >(r?ers 2, 2(r?pie)e is )h(sen. 9he B!

represen#s #he rev(lving par# (* #he !M s!s#e+M i# in)ludes #he h(lding *i7#ure, #he>(r?pie)e and #he spindle *igures 3 a, 5. 9( +a?e #he >h(le *ra+e rigid, a ver! rigid uni#>(r?pie)e, h(lding *i7#ure is )(n)eived in *r(n# (* #he !Mele+en#s *igure 0.

Fig.3 B!represen#a#i(n.

9he >(r?pie)e ge(+e#r! and his h(lding *i7#ure are sele)#ed >i#h D1 / ++, D2 12++ and "2 3 ++ )*. *igure 0. 9hese di+ensi(ns re#ained *(r #hese #es#-#u5es >ere

sele)#ed using #he *ini#e ele+en# +e#h(d )(upled #( an (p#i+iza#i(n +e#h(d 5! $=CFs(*#>are. # is ne)essar! #( de#er+ine #he h(lding *i7#ure leng#h "1#( (5#ain a signi*i)an#s#i**ness in *le7i(n. 9he (5Ie)#ive is #( +(ve a>a! #he *irs# B!vi5ra#i(n +(de (* #he la#he*unda+en#al na#ural vi5ra#i(n +(de see 3E.

Fig. 0 %e(+e#r! (* h(lding *i7#ure >(r?pie)e.


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$s 33E and (#hers #he s#i**ness is )al)ula#ed (n #he 5asis (* #he displa)e+en# *(r a given

*(r)e P 4

IE

LP

3

3 1

>i#h iner#ial +(+en# 4

I /0

0

1D 2

9he *igure 8 represen#s #he displa)e+en#s and s#i**ness values rela#ing #( #he leng#h (*h(lding *i7#ure >(r?pie)e, *(r a *(r)e P 1, ;, a Y(ung +(dulus 21.1 -8;++2

and a h(lding *i7#ure dia+e#er D1 / ++.

Fig. 8 Displa)e+en#s a))(rding #( #he h(lding *i7#ure leng#h.

$ h(lding *i7#ure leng#h 4 "1 1O ++, *(r a s#i**ness in *le7i(n (*


9/27

Fig. / $))ep#a5le aria represen#a#i(n (* #he >(r?pie)e de*(r+a#i(n.

3.2 Block Tool: BT

n #his )ase, #he Bpar# in)ludes #he #((l, #he #((l-h(lder, #he d!na+(+e#er, #he *i7ingpla#e (n #he )r(ss slide *igure een#he )r(ss slide and #he #((l-h(lder. 9his is ne)essar! #herea*#er #( +easure #he )u##ing+e)hani)al a)#i(ns. 9he s#i**ness (* Bis evalua#ed in#( #he ne7# se)#i(n.

Fig. < Bl()? #((l Brepresen#a#i(n.

4. Static characterization of machining system

9he s#a#i) s#ud! ai+s a# )hara)#erizing #he s#a#i) eNuivalen# s#i**ness values in (rder #(iden#i*! #he #hree di+ensi(nal elas#i) 5ehavi(ur (* #he +a)hining s!s#e+. %enerall!, #hes#a#i) #es#s )(nsis# in )harging 5! ?n(>n e**(r#s #he #>( 5l()?s and +easuring (nl! #he

:


10/27

ass()ia#ed displa)e+en#s )(+p(nen#s 1n e**(r#s #he #>( 5l()?s 5u# +easuring #he s+all displa)e+en#s #(rs(r i.e. #hreelinear displa)e+en#s, and #hree r(#a#i(ns. 9he applied e**(r#s are Nuan#i*ied >i#h a *(r)esens(r. 9he s+all displa)e+en#s #(rs(r is +easured 5! si7 displa)e+en# #ransdu)ers. $s#i**ness gl(5al +a#ri7 is dedu)ed >i#h i#s vari(us resul#s. # is a real 3D pa##ern. F(r

ins#an)e, Carrin( e# al., presen# a +(del #ha# #a?es in#( a))(un# 5(#h #he >(r?pie)ede*le)#i(n and #he )u##ing *(r)e 5e#>een #((l and >(r?pie)e. 9he #hree )(+p(nen#s (* #he)u##ing *(r)e are a depend (n #he )u##ing ge(+e#r!. 9he e**e)# (* #he >(r?pie)e-#((l-+a)hine de*le)#i(ns is a shi*# (* #he >(r?pie)e )r(ss-se)#i(n and a +(ving 5a)? (* #he #((lh(lder in #he radial and #he #angen#ial dire)#i(n 2D +(del OE.

4.1 Stiffness matrix

9he e7peri+en#al appr(a)h is 5ased (n #he +a#ri7 devel(p+en# presen#ed in 3E. 9hede*(r+a#i(n (* a s#ru)#ure ele+en# is represen#ed 5! displa)e+en#s (* n(des de#er+ining

#his ele+en#. 9he ass()ia#ed *(r)es )(rresp(nd #( displa)e+en#s >hi)h a)# as #hese n(des.

9he #rans*(r+a#i(n +a#ri7 >hi)h )(nne)#s generalized displa)e+en#s (* an ele+en# #( #heass()ia#ed *(r)es is #he rigidi#! +a#ri7 (r #he s#i**ness +a#ri7 (* #he ele+en#. n #he sa+e>a! #he +a#ri7 >hi)h )(nne)#s generalized displa)e+en#s (* #he s#ru)#ure #( #he appliedgeneralized dis)re#e *(r)es is #he s#i**ness +a#ri7 (* #he s#ru)#ure si+pl! na+ed as s#i**ness+a#ri7 4 KE.

9he rela#i(n 5e#>een *(r)es and displa)e+en#s is given 5! 2:E4

HT KE 7 HD 3

>here HT represen#s #he +e)hani)al a)#i(n #(rs(r, KE #he s#i**ness +a#ri7 and HD #hes+all displa)e+en#s #(rs(r.

9he general *(r+ (* #he sNuare / 7 / s#i**ness +a#ri7 KE is4

KE$, 7!zxyzACFC

FFC

KK

KK

,

0

>here KFE, KCE, KCFE and KFCE are respe)#ivel! sNuare 3 7 3 displa)e+en# +a#ri7,r(#a#i(n and r(#a#i(ns displa)e+en# and displa)e+en# r(#a#i(n e7pressed a# #he p(in# $ in

7, !, z +a)hine a7es.

4.2 Experimental determination of the stiffness matrix

9he +a#ri7 ele+en#s (* #he s+all displa)e+en# #(rs(r are iden#i*ied #han?s #( #hee7peri+en#al devi)e presen#ed in #he *igure O. 9he )(nsidered s!s#e+ is a )u5e.Displa)e+en#s are +easured 5! si7 displa)e+en# #ransdu)ers. 9>( displa)e+en#

1


11/27

#ransdu)ers are p(si#i(ned s!++e#ri)all! (n ea)h (* #he 3 dire)#i(ns. 9he *(r)e is applied #(ea)h 7, !, z dire)#i(n in #>( di**eren# levels 5! a s)re>-s>ivel s!s#e+ )(n#r(lled 5! a *(r)esens(r. a)h l(ading p(in# )((rdina#es are ?n(>n s#ar#ing *r(+ #he )u5e )en#er @ ). 9hisall(>s *(r ea)h applied *(r)e #( de#er+ine #he +(+en# and #hus #he )(+ple#e #(rs(r (*+e)hani)al a)#i(ns HT.

Fig. O 7peri+en#al devi)e *(r #he s#a#i) )hara)#eriza#i(n.

ndu)ed displa)e+en#s are s(lid 5(d! displa)e+en#s and i# is n(#ed #ha# r(#a#i(ns are l(> 1- 8rad 5u# e7is#. 9he e7is#en)e (* #hese r(#a#i(ns is i+p(r#an# and in agree+en# >i#h #he#(rNue via #he vir#ual >(r? #he(r!. 9he +easure prin)iple is presen#ed in #he *igure : and isused #( de#er+ine #he )(+p(nen#s (* #he s+all displa)e+en#s #(rs(r HD >hi)h is)(+p(sed 5! #he #hree r(#a#i(ns 7, !, zand #he #hree displa)e+en#s 7, !, z.

11


12/27

Fig. : P(si#i(n (* Displa)e+en# #ransdu)er.

9hus, #he displa)e+en#s mand r(#a#i(ns are de#er+ined, *(r ea)h l(ading dire)#i(n, 5!using #he rela#i(ns4

m2

21 mm +

, tan a

mm12

8

Fr(+ #hese rela#i(ns and )(nsidering #he si7 +easure+en#s p(in#s i# resul#s4

z

y

x

2

1

2

1....

..21

21..

....2

1

2

1

....11

11....

..11

..

aa

aa

aa

/

8

0

3

2

1

m

m

m

m

m

m

/

9he #es#s are )arried (u# >i#h spe)i*i) asse+5lies >hi)h are designed *(r ea)h dire)#i(n (*+easure+en#. 9he l(ading respe)#ivel! unl(ading is )arried (u# 5! s#ep (* 3 da; resp.

12


13/27

Q 3 da; un#il resp. *r(+ level (* 2 da;, and #his pr()edure is used *(r ea)h #es#*(ll(>ing ?n(>n dire)#i(ns. 9( )he)? #he repea#a5ili#! and a))ura)! (* iden#i*i)a#i(ns all#es#s and +easure+en#s are )arried (u# *ive #i+es and #he average is sele)#ed *(r ea)h p(in#a# *igure 1.9( e7pl(i# +easure+en#s as >ell as p(ssi5le, #he displa)e+en#s )urves are pl(##ed

depending (n #he applied *(r)e *(r ea)h l(ading dire)#i(n. $ line (* leas# sNuares is adIus#ed#( de#er+ine #he displa)e+en#s )(+p(nen#s values *(r a given *(r)e. 9hus, si7 #(rs(rs (*s+all displa)e+en#s are iden#i*ied *(r si7 l(adings )ases. 9he linear 5ehavi(ur (5served inl(ading is di**eren# *r(+ #he linear 5ehavi(ur n(#ed in unl(ading. 9his di**eren# lineari#!5e#>een l(ading and unl(ading is due #( #he e7is#en)e (* devia#i(ns and *ri)#i(n *(r)es inea)h p(in# sur*a)es (* #he asse+5l!. 9hese devia#i(ns due #( #he ins#alla#i(n (* #he par#s (*#he ass()ia#ed asse+5l! and *ri)#i(n *(r)e are di**eren# in )harge and dis)harge.

Ahen #his di**eren)e in linear 5ehavi(ur appears h!s#eresis, >e use #he line *igure 1>hi)h passes 5! #he +iddle C p(in# (* seg+en# AB )harge-dis)harge. OCis #he line>hi)h #he sl(pe, 5! assu+p#i(n, )(rresp(nds #( #he real s#i**ness. eg+en# ABrepresen#s

#he d(u5le (* #he *ri)#i(n *(r)es and devia#i(ns *(r #he de*(r+a#i(n , 20E.

Fig. 1 Diagra++a#i) represen#a#i(n (* linear 5ehavi(ur in )harge and dis)harge.

$# #his s#age, #he +a#ri7 )(lu+n (* #he s+all displa)e+en#s #(rs(rs and (* #he +e)hani)ala)#i(n #(rs(rs are ?n(>n. 9he *le7i5ili#! +a#ri7 C0E (* #he s!s#e+ is dedu)#ed 5!4

COE HT-1HD .


14/27

+easure+en#s are #a?en a# #hree p(in#s and )ap#ured 5! da#a a)Nuisi#i(n )ard ins#alled inPC. 9he s#(red da#a )an 5e re#rieved and used *(r anal!sis >hen reNuired. sing "a5vie>s(*#>are >e (5#ain #he e7peri+en#al *le7i5ili#! +a#ri7 CE. $ si+ple inversi(n gives #hes#i**ness gl(5al +a#ri7 KB9E.

Fig. 11 7a+ple (* l(ading inxBdire)#i(n.

KB9E

v

/8/

/8/


15/27

Kerr(rsE

2.12.3.13.2.1.

3.1.er )(rner are #he ele+en#s)(rresp(nding #( s#i**ness values (* r(#a#i(ns ;rad n(#ed KCE in 0,

Ri#e+ #he ele+en#s (* #he +a#ri7 3 3in #he le*# higher )(rner are #he ele+en#s)(rresp(nding #( #he )(uplings (* displa)e+en#s r(#a#i(ns n(#ed KFCE in 0,

Ri#e+ #he ele+en#s (* #he +a#ri7 3 3 in #he righ# l(>er )(rner are #he ele+en#s)(rresp(nding #( #he )(uplings (* r(#a#i(ns displa)e+en#s n(#ed KCFE in 0.

9hese #>( las# ele+en#s displa)e+en#s r(#a#i(ns KFCE, r(#a#i(ns displa)e+en#sKCFE are n(# #a?en in#( a))(un# here. @nl! #he s#i**ness par# (* displa)e+en# n(#ed

KF, B9E 5el(> is ne)essar! )(nsidering in (ur ne7# d!na+i) +(del n(# presen#ed here.

KF, B9E

xyzO,

i#h #he d!na+(+e#er )*. *igure 12.

18


16/27

Fig. 12 7peri+en#al devi)e *(r #he B!s#a#i) )hara)#eriza#i(n.

sing #his ge(+e#r!, #he B! has a ver! high rigidi#! a))(rding #( za7is and #his value isver! s+all )(+pared >i#h #he prin)ipal s#i**ness value. 9he B!5ehavi(ur is linear, >i#h anearl! null h!s#eresis. $# #he l(ading and unl(ading p(in#s, #he par# is n(# in*luen)ed 5! #he*ri)#i(n phen(+en(n (r (#her vari(us *i#s genera#ed 5! #he asse+5led ele+en#s, li?e #hespindle (r #he 5all 5earings. 9he s#i**ness +a#ri7 KF, BAE (5#ained a))(rding #( #he #hreedire)#i(ns is4

KF, BAE

xyzO,

O

s#i**ness values (* #he +a)hining s!s#e+ in #he #hree dire)#i(ns, #he elas#i)in#era)#i(n BB!is +(delled 5! s#a#i) s#i**ness >hi)h are asse+5led in parallel *igure13. 9he )hip is #he )(++(n p(in# 5e#>een de*(r+a#i(n and *(r)e. # )(nne)#s #he #>(

1/


17/27

s#i**nesses B B!. 9he su+ (* #he #>( s#i**ness +a#ri7 (* displa)e+en#s BKF,B9E andB!KF,BAE de#er+ines #he s#i**ness +a#ri7 (* +a)hining s!s#e+ displa)e+en# KF,A$=E.

Fig. 13 #a#i) s#i**ness asse+5led in parallel.

KF,A$=E

xyzO,

O//

/


18/27

Fig. 10 Diagra+ (* Kangle de#er+ina#i(n.

9he s#i**ness values KB9E and KBAE (* #he >h(le elas#i) s!s#e+, a+(ng prin)ipales a7es,are de#er+ined 5! de*(r+a#i(n energ! +ini+iza#i(n, all(>ing #hus #he diag(naliza#i(n (*#he +a#ri7 KF,B9-dE4

KF, B9-dE

xyzO,

i#h K-B9 82S and K-BA S >e (5#ain an angle K i#h K-B9 32S and K-BA S >e have an angle K /8S. @n #his dire)#i(n, #he+a7i+u+ de*(r+a#i(n (* #he s!s#e+ is (5#ained.

!otation center

Dea)u 10E and Kudin(v 20E sh(> #ha# an! +a)hine #((l, is )hara)#erized 5! #hede*(r+a#i(n prin)ipal dire)#i(ns. 9hese de*(r+a#i(n prin)ipal dire)#i(ns are *un)#i(n (* #he+a)hine s#ru)#ure, i#s ge(+e#ri)al )(n*igura#i(ns and )u##ing para+e#ers used. Ae )an(5serve ei#her a ver! s#i** 5ehavi(ur, (r ver! ru55er 5and a))(rding #( #he dire)#i(n. Gere,#he #((l is regarded as *(r+ing in#egral par# (* 5l()? B. 9he ai+ is #( de#er+ine #hes#i**ness )en#er "#B9(* elas#i) s!s#e+ B2/E. 9his s#i**ness )en#er )(rresp(nds #( #he

1O


19/27

r(#a#i(n )en#er (* #he 5l()? B)(+pared #( #he 5ed. @5#aining "#B9)(nsis#s in *inding #hep(in#s (* in#erse)#i(n (* di**eren# perpendi)ular #( displa)e+en#s.

.1 Experimental step

9he s#i**ness )en#er is (5#ained in #he )((rdina#e s!s#e+ 5ased (n #he #((l in @ p(in# #ha# is#he (rigin (* #he 7, !, z )((rdina#e s!s#e+.

9he pr()edure #( (5#ain #he s#i**ness )en#er is de#ailed in #he *igure 18M #he i+p(sed )harge*(ll(>s all #hree dire)#i(ns,Fi5eing ?n(>nM i 7, !, z.

Fig. 18 7peri+en#al pr()edure #( (5#ain s#i**ness )en#er "#B9

9he displa)e+en# ve)#(r +easure+en# under #he l(ad d i,Iis (5#ained in #>( p(in#s Pi,Iinea)h dire)#i(n i 7, !, z and I 1, 2. 9he dire)#i(n D iI is released 5eing s#raigh# line)(n#aining di,Ive)#(r #ha# in)lude #he p(in# Pi,I.

$))(rding #( #he *igure 1/, #hese lines are n(# e7a)#l! )(planar and n( se)an#. # is p(ssi5le#( *ind *(r ea)h dire)#i(n #he in#erse)#i(n p(in#, na+ed Mi, (* #he lines DiI using #he*(ll(>ing rela#i(ns, *(rxdire)#i(n4

xOM 1,1, xxx OdeOP

+ 18

sing #he +e#h(d (* leas# sNuares, >e )an +ini+ize #he dis#an)e d75e#>een #he lines D7,1and D7,2, *igure 1/M d7is )al)ula#ed using #he e7pressi(n4

1:


20/27

d72,

2,2,

x

xxx

d

dOPOM

1/

and #he angle 7)(rresp(nds (* angular devia#i(n (* )(planari#!4

7 ar))(s

1,1,

1,1,

xxx

xxx

dMP

dMP

1( lines DiI are separa#ed 5! a +ini+u+ dis#an)e i 1.O ++ and +a7i+u+devia#i(n i 2S see #he #a5le 1.

n #he ne7# s#ep, #hree +ean plansPiare de*ined 5! appr(7i+a#i(n (* #he p(in#sMiand)(n#aining #he lines DiI.

Ae )an dra> #he n(r+al in (* ea)h planPi#ha# in)ludes #he p(in#Mi*igure 1


21/27

Fig. 1< De#er+ina#i(n (* #he 5l()? #((l r(#a#i(n )en#er4 "#B9

Chargep(in#

C((rdina#e(* p(in#++

Displa)e+en#ve)#(r i#d +

hi*#dis#an)e +

$ngularshi*# iS

n#erse)#i(np(in#Mi+

C((rdina#e(* CB9+

P7,1

38 :.11-8

1.O1-3 .2O.3//

.8/-2 1.


22/27

9herea*#er is +ade a veri*i)a#i(n +easuring #he #((l p(in# displa)e+en# @ in >(r?-spa)eunder #he )harge. Ae )(nsidera#e (ur e7peri+en#al resul#s vie>ing #he #((l p(in# #ha#+(ves under (ne seg+en# (* a sphere >i#h #he )en#er "#B94 assi+ila#ed >i#h a n(r+al plan$% "#B9 *(r s+all displa)e+en#s a))(rdingl! *igure 1O.

Fig. 1O De#er+ina#i(n (* #he 5l()? #((l )en#er r(#a#i(n4 "#B9

." Comparaison #ith stiffness matri$

* >e pr()eed (n #he diag(naliza#i(n (* #he 5l()? >(r?pie)e s#i**ness +a#ri7 KF,B9E(5#ained in #he se)#i(n 0.2.1, >e (5#ain4

KF, B9-dE

xyzO,

(r?pie)e de*le)#i(nand #he )u##ing *(r)e 5e#>een #((l and >(r?pie)e OE. n 30E #((l and >(r?pie)e are+(delled as #>( separa#e single degree (* *reed(+ spring-+ass-da+per s!s#e+s. $ll #hese

+(dels are 2D, (r +(re (r less par#iall! 3D, and #hus d(es n(# all(> a real 3D +(deling (*#he )u#.

9he appr(a)h presen#ed in #his paper is *ull! di**eren# *r(+ previ(us s#udies in #he sense#ha# >e use #he si7-di+ensi(n (* #he #(rs(rs 3 *(r)e )(+p(nen#s, 3 #(rNue )(+p(nen#s.C(nseNuen#l!, #his >a! all(>s us #( 5e dire)#l! in #he +(s# sui#a5le *ra+e>(r? *(r a real3D )u##ing +(deling. 9han?s #( #he prin)iple (* vir#ual >(r?, >e de#er+ined #he )(+ple#e#(rs(r (* s+all displa)e+en#s 3 linear and 3 r(#a#i(ns displa)e+en# ass()ia#ed >i#h #he+e)hani)al a)#i(ns #(rs(r 3 *(r)e )(+p(nen#s, 3 #(rNue )(+p(nen#s via #he +a#ri7 (*rigidi#ies. 9he s#i**ness )en#er and #he r(#a#i(n )en#er >ere (5#ained e7peri+en#all!. 9he+ini+al displa)e+en# dire)#i(n >as de*ined (n #he 5asis (* e7peri+en#al +(del. sing

Cas#iglian('s #he(re+, >e de#er+ined #he angle K >hi)h )hara)#erizes #he prin)ipalde*(r+a#i(n dire)#i(n. 9hen, >e )hara)#erized 5l()? >(r?pie)e B! >as )hara)#erized 5!i#s s#i**ness values and #he diag(nal +a#ri7 (* displa)e+en#s. 9his diag(nal +a#ri7 (*displa)e+en# valida#e #he h!p(#hesis and )(n*ir+ #he g((d rigidi#! (* >(r?pie)e. Ah(les#a#i) 5ehavi(ur anal!sis (* #he s!s#e+ B&B! is #hus valida#ed. $n appli)a#i(n (* #his+e#h(d(l(g! >i#hin a *ra+e>(r? (* +a)hining reall! 3D is availa5le in 0E. 9hese resul#sare reNuired #( have a g((d 3D +(deling (* #he )u#, as 5eing presen#ed in a *(r#h)(+ingpaper.

23


24/27

!ckno"ledgements

9he au#h(rs a)?n(>ledge Jean Pierre "ariviTre, ngineer C; Cen#re ;a#i(nal de lae)her)he )in#i*iNue - Fran)e *(r #he nu+eri)al si+ula#i(n >i#h $=CF s(*#>are andPr(*ess(r =ir(n Uap)iu *(r #he help*ul dis)ussi(ns (n #his su5Ie)#.

!eferences

1. $7in#e D. $., Bellu)( A., De Chi**re ".,valua#i(n (* )u##ing *(r)e un)er#ain#! )(+p(nen#s in #urning,n#erna#i(nal J(urnal (* =a)hine 9((ls and =anu*a)#ure, '(, pp.


25/27

1. Chen C. K., 9sa( Y. =.,$ s#a5ili#! anal!sis (* regenera#ive )ha##er in #urning pr()ess >i#h(u# using #ails#()?,n#erna#i(nal J(urnal (* $dvan)ed =anu*a)#uring 9e)hn(l(g!, ),, (r?-pie)e,n#erna#i(nal J(urnal (* =a)hine 9((ls and =anu*a)#ure, '+, 1, pp. 1O-28, 2/

12. C(uard Y.,Cara)#&risa#i(n e# al(nnage des d!na+(+T#res V si7 )(+p(san#es p(ur #(rseur ass()i& Vun s!s#T+e de *(r)es,Ph. D. 9hesis, niversi#R'e B(rdeau7W1 9alen)e, 2

13. Dassana!a?e $. L., uh C. .,@n n(nlinear )u##ing resp(nse and #((l )ha##er in #urning (pera#i(n,C(++uni)a#i(ns in ;(nlinear )ien)e and ;u+er)ial i+ula#i(n, (-, 8, pp. :


26/27

21. K(enigs5erger F., 9lus#! J.,=a)hine 9((ls #ru)#ures, Perga+(n Press, 1:


27/27

32. agla+ G., Yaldiz ., nsa)ar F.,9he e**e)# (* #((l ge(+e#r! and )u##ing speed (n +ain )u##ing *(r)e and #((l #ip#e+pera#ure,=a#erials X Design, )*, pp. 388-3/, 22

33. algad( =. $., "(pez de "a)alle ".;., "a+i?iz $., =un(a J., an)hez J. $.,

valua#i(n (* #he s#i**ness )hain (n #he de*le)#i(n (* end-+ills under )u##ing *(r)es,n#erna#i(nal J(urnal (* =a)hine 9((ls and =anu*a)#ure, '., pp.

Bisu Knevez Darnis Laheurte Gerard 09

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