Optimal design of full disks with respect to mixed creep rupture time

Transcription

1 Compu Aidd Opimum Dsign in Engining XII 83 Opimal dsign of full disks wih spc o mixd cp upu im K. Szuwalski & A. Uszycka Cacow Univsiy of Tchnology, Poland Absac Th mixd upu hoy o h opimizaion poblm fo h complx sss sa is usd. Th poblm of opimal shap fo h oaing full disk wih spc o mixd upu im is invsigad. Th mahmaical modl of mixd cp upu is dscibd by h sysm of fiv paial diffnial quaions. Difficuly of h poblm suls fom wo yps of nonlinaiis: gomical conncd wih h us of h fini sain hoy and physical - h maial is dscibd by h Noon s cp law, h gnalizd fo u ssss and logaihmic sains. Addiional im faco lads o subsqun complicaions. Th paamic opimizaion dscibing h iniial shap of h disk is applid. Th obaind suls a compad wih h opimal disks wih spc o ducil cp upu im. Kywods: mixd cp upu, sucual opimizaion, full disk. Inoducion Sucual lmns woking und cp condiions blong o h laivly nw banchs of sucual opimisaion, sad in h svnih yas of h las cnuy. Th poblm of im o upu valuaion is of obvious lvanc fo vaious machin pas woking und high mpau condiions. Thoical modlling of long im sngh appas o b impoan. A boad psnaion of vaious objciv funcions wih division on im-dpndn and imindpndn, was givn by Życzkowski []. To h la goup blong ciia conncd wih cp upu. Mos paps on opimal sucual dsign a basd on h bil cp upu hoy poposd by Kachanov small sain hoy). I was du o is laiv simpliciy - possibiliy of igidificaion hom WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) doi:0.495/op008 0 WIT Pss

2 84 Compu Aidd Opimum Dsign in Engining XII applicaion. Opimal soluions wih spc o bil cp upu ofn coincid wih unifom sngh sucus. In h wok publishd by Hoff [], h momn of failu of a ba und nsion is dfind as h on a which h coss-scional aa bcoms zo as a sul of quasiviscous flow. I was shown ha h calculad suls a in good agmn wih h xpimnal daa Mnl [3]). Applicaions of h ducil upu hoy, poposd by Hoff in opimizaion poblms a ah scac as i quis fini sain hoy. Fo h fis im i was usd by Szuwalski fo opimizaion of bas und nonunifom nsion [4] and fo opimisaion of Miss uss [5]. Som poblms of pismaic nsion ods w discussd by Pdsn [6]. Such an appoach inoducs no only physical nonlinaiis, conncd wih cp law usually Noon s law), bu gomical, suling fom h fini sain hoy, as wll. Addiional im faco lads o such complicaion, ha ill now, only fw paps w dvod o opimizaion wih spc o ducil upu im. Th opimal full disks wih spc o ducil cp upu im w found by Szuwalski [7], and fis amps fo annula disks w mad by Szuwalski and Uszycka [8]. Th Hoff s concp has cain limiaions. I pdics conay o obsvaions ha cp dos no sul in damag of maial. Also, his schm dos no xplain facus a small sains bil upus) and h chang of chaac of upu fom ducil o bil). H, w suggs a hoical modl fo cp dfomaion of h disk, which aks ino accoun duciliy and mbilmn of h maial. W mphasiz ha h physics of such phnomna is vy complx. Applicaion of mixd upu hoy poposd by Kachanov [9] aks ino accoun boh: gomical changs - diminishing of ansvsal dimnsions suling fom lag sains as in ducil upu) and gowh of micocacks as in bil upu). Saigh lin b Figu ) cosponds o h puly bil facu and d o h ducil facu accoding o xpimnal daa. Th cuv m showing laions fo h mixd upu, asympoically appoachs saigh lins b and d. Figu : m- mixd upu, d- Hoff s ducil cp upu, b- Kachanov s bil upu. WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

3 Compu Aidd Opimum Dsign in Engining XII 85 Th fis amp of applicaion h mixd upu hoy o shap opimizaion was mad by Szuwalski and Uszycka [0] fo bas und nonunifom nsion. In psn pap h poblm of opimal shap wih spc o mixd cp upu im fo h complx sss sa - oaing full disk is invsigad. W assum ha micocacking and diminishing of ansvsal dimnsions sa fom h vy bginning of cp pocss. Such an appoach inoducs no only physical nonlinaiis, conncd wih cp law usually Noon s law), bu gomical, suling fom fini sain hoy, as wll. Addiional im faco lads o subsqun complicaions. Th whol cp pocss mus b analyzd fom is bginning up o upu. Govning quaions Th concp of h mahmaical dscipion of mixd cp upu quis an xaminaion of h ni pocss, aking ino accoun gomical changs occuing in h cous of i. Th poblm is solvd in maial Lagangan) coodinas and all paams in iniial sa, fo im qual zo, a dnod by capial ls, whil cun valus of hs paams by h sam small ls. Du o axial symmy all quaniis will b funcions of wo indpndn vaiabls: adius R and im. Th disk oas wih consan angula vlociy and body focs conncd wih own mass of h disk a akn ino accoun Figu ). Figu : Modl of h oaing full disk. Th maial of h disk fulfills h Noon s cp low: n k ) wh dnos h ffciv sss, accoding o h Hub-Miss-Hncky hypohsis gnalizd o u ssss: ) WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

4 86 Compu Aidd Opimum Dsign in Engining XII and spcivly, is h ffciv sain a, k and n a maial consans. Th innal quilibium condiion fo plan sss sa, wih body foc, aks fom: h ) 0 h R g 3) wh sands fo cun valu of adial sss and of cicumfnial on, h fo cun hicknss, γ spcific wigh of maial and g acclaion of gaviy. Assumpion of incompssibiliy lads o: HRdR hd wh R sands fo maial coodina of h discussd poin, whil fo spaial on. Fini sains qui logaihmic sains: h ln ln ln R ; R ; z ln H 5) Th shap chang law, assumd in fom of similaiy of u ssss and vlociis of logaihmic sains dviaos accoding o Szuwalski [7] lads o: n k ) 6) Compaibiliy condiion, af som aangmns, psnd by Szuwalski [7], aks fom: [ n ) ) 6 ) ) ] [ n ) ) 4 4 ] To find h mixd upu im, h voluion quaion poposd by Kachanov [ 9] will b applid: D 8) in which D and m a maial consans. Coninuiy funcion is dfind by h aio of ffciv coss - scional aa a f o undamagd aa a : m 4) 7) a f a 9) In conas wih bil upu hoy dnos h h u sss - lad o h cun coss scion a gomical changs a akn ino accoun). WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

5 Compu Aidd Opimum Dsign in Engining XII 87 W chaaciz damag by h coninuiy funcion 0. A h iniial momn no damag):, as im gos on, i dcass. Th momn of upu cosponds o a valu 0 a which facu localizs. W dfin h upu ciion in h following fom: m 0) R : R 0, l,0 ) * 0) Tim af which h coninuiy funcion will diminish o zo will b h im of mixd upu m. Fo h sak of numical calculaions, dimnsionlss quaniis a inoducd. Boh maial and spaial coodinas a lad o h iniial xnal adius B: R R B ; B ) Th hicknss of h disk is lad o h man hicknss h m of h full disk of volum V and adius B: B H H B V ; h h ) V Radial loading a adius b of h oaing disk is suling fom mass M unifomly disibud on h ou dg. M b) p 3) h B) Dimnsionlss ssss a fd o ssss calculad using a igidificaion hom in h moionlss full plan disk subjc o nsion wih unifom pssu p 3): V i i ; i, 4) M B Consqunly dimnsionlss im is dfind: d ) 5) d ) wh: sands fo h im of ducil upu fo full plan disk Szuwalski [7]). To avoid h lag numb of maial consans in numical calculaions w inoduc h nw paam. This paam is qual o h adio of h bil upu im o h ducil upu im fo h pismaic ba subjc o nsion wih h iniial sss : s K ) p H ) p n nk s 6) m m D s WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

6 wh s is qual: V R M s 0 7) Th paam conains fou maial consans: n and k fom Noon s law ) and m and D fom voluion quaion 8). In som way i dscibs snsiiviy of maial on yp of damag: bil o ducil. Th mahmaical modl of mixd cp upu is dscibd by h sysm of fiv paial diffnial quaions in dimnsionlss fom: h h ) 4 ) ) 5 ) ) 6 n n n n d d 8) HR h m m. In psnd abov quaions w hav fiv unknowns: u ssss and, cun hicknss h, spaial adial coodina and coninuiy funcion. A h iniial momn ) 0 disk mains undfomd, hfo h iniial condiions ak fom: R R,0) ; ),0) R H R h 9) Th bounday condiions a dscibd in his fom: 0 ) 0, ; 0 0, ) 0, ) 0, 0) Th condiion a xnal adius, wh h mass M is disibud, in dimnsionlss fom may b win: ), ), h ) 88 Compu Aidd Opimum Dsign in Engining XII ISSN on-lin) WIT Tansacions on Th Buil Envionmn, Vol 5, 0 WIT Pss

7 Compu Aidd Opimum Dsign in Engining XII 89 In h fouh of quaions 6) and in h scond of iniial condiions 7), w hav h funcion H R ) dscibing h iniial pofil of h disk. I is ncssay o know his funcion in od o solv h s 6). Bcaus his funcion is bing sough in h opimisaion pocss, w shall apply paamic opimisaion. W shall look fo h bs possibl funcion H R ), lading o h longs lifim o mixd upu in assumd class of polynomial funcions. L s consid an opimaliy ciion in h fom: m)! R) f max H ) * Vcons. wh funcions f : R R, R b0 b R br... b l R, fo all agumns 0, i N 0 i is a non-ngaiv ing) and R, wh b R a consan cofficins, wh l 0,,,, i, and b 0. l 3 Numical algoihm In od o pfom an opimisaion pocdu, h upu and opimaliy ciion o h mahmaical modl is inoducd. Th numical algoihm is poposd fo h complx sss sa. W mus follow sp by sp h whol cp pocss fo ach nw gomy of h dfomd disk up o h momn of fulfilling of upu ciion in od o sablish h mixd cp upu im. Th numical algoihm consiss of wo blocks, which a squnially acivad. Th algoihm bgins by dfining h iniial gomy, consan paams and h bounday condiions fo h ssss 0). In h fis block, fo givn gomy of h disk, h u ssss disibuion is sablishd by ingaion of h fis and h scond quaions of h s 8) wih spc maial coodina R. Th Rung Kua fouh od mhod is applid. W do no know valus of ssss in h cn of h disk, so hy a assumd abiaily, bu hy mus saisfy h bounday condiion a h ou dg of h disk ). Thfo, h cunial pocdu mus b applid. Found disibuion of u ssss, wih hlp of h voluion quaion h las fom h s 8) maks i possibl o sablish h disibuion of coninuiy funcion. If i s minimal valu saisfis h upu ciion, h cp pocss finishing h im o mixd upu was found. Whn valus of coninuiy funcion a sufficinly lag, h nw changd gomy of h disk is calculad. Th ingaion of h hid of quaion 8) wih spc o im is pfomd using Eul s mhod. Th im dpndn soluions hav high snsiiviy on im sp dimnsion, so w choos h vaying im sp, dpnding on cun cp vlociy. In his way nw spaial coodinas a calculad. Finally incompssibiliy condiion fouh of s 8)) maks i possibl o find changd shap hicknss) of h disk. Fom h suls obaind fo many iniial shaps of h disk, dscibd by assumd polynomial funcion, h bs lading o h longs im o mixd cp upu l i WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

8 90 Compu Aidd Opimum Dsign in Engining XII is chosn his is h opimal disk. Consqunly, fo h nw gomy of h disk, h full cycl of h abov mniond calculaions a pad. 4 Discussion of suls A h bginning opimal soluions fo h poblm of oaing full disk wih spc o mixd cp upu im a sough in h class of lina funcions: H R; u, u ) u u R 3) 0 0 Paams u 0 and u unipaamic opimisaion), which opimal valus a sough, a linkd ogh by h condiion of givn volum V: 3 u0 ) u 4) 3 Th influnc of paam as h aio of own mass of h disk o mass unifomly disibud a h ou dg and h paam aio of h bil upu im and ducil upu im of h pismaic ba, oaing wih h consan angula vlociy ω, loadd by mass M disibud a h xnal dg, wih h ba s own mass nglcd is invsigad. Pofils of opimal disks fo unipaamic opimizaion a shown in Figu 3 as a funcion of h paam fo h diffn valus of paam Obaind soluions songly dpnd on aio and µ. Whn h mass M is vy lag in compaison wih own mass of h disk small valus of μ) opimal disks a clos o fla ons. Fo h lag valus of paam ducil maials) h hicknss of opimal disks in h viciniy of ou dg gows. Fo lag valus of paams µ small mass M a h ou adius), h mass of h disk is disibud as clos o h oaion axis as possibl. B suls may b obaind fo disks, which iniial shap is dfind by quadaic funcion: H R; b, b, b ) b b R b R, b 0 5) 0 0 Bcaus now w hav h paams, finding of hi opimal valus aks much mo im han fo unipaamic opimizaion. Fom h paams in his funcion, only wo may b ad as f ons, h hid suls fom givn volum of disk: fom which: V 0 b b b R b R Rd R 0 6) 4 b0 b. 7) 3 Pofils of opimal disks fo bipaamic opimizaion a shown in Figu 4. WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

9 Compu Aidd Opimum Dsign in Engining XII 9 A) 0, 4 B) 3 Figu 3: Opimal shaps of h disks fo unipaamic opimisaion. A) 0, 4 B) 3 Figu 4: Opimal shaps of h disks fo bipaamic opimisaion. Fo small valus of paams h own mass almos nglcd), h gowh of hicknss a h ou dg was obsvd, opimal soluions hav minimum insid h disk widh. Th lag hicknss a h ou dg woks as som kind of infocmn and hanks o i im o mixd upu may b long. Fo lag valus of paams bil maials), his ffc is small. In Figu 5A) h changd pofils of h opimal disk and in Figu 5B) h cosponding disibuion of coninuiy funcion a h sam im momns a shown. Th suls a psnd fo opimal disk fo paams: 0., n 3, m, 3, which iniial shap is dscibd by funcion H R) 3R R. W obsvd ha dspi h snghning of h ou dg of h disk, h upu ciion fo h coninuiy funcion is fulfilld h. Insid h disk h valus of funcion a qui lag. This ffc is du o limiaion only o h disks of iniial pofil dscibd by quadaic funcion 3). WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

10 9 Compu Aidd Opimum Dsign in Engining XII A) B) Figu 5: Tim coss-scion of cp pocss. Figu 6: Opimal shap of h unifom iniial sngh disks compad wih uni- and bipaamic opimisaion. W may xpc ha disks of unifom iniial sngh, in which boh adial and cicumfnial ssss a h sam fo 0 R B a clos o opimal wih spc o mixd upu im. Such disks dscibd by fomula: H us R xp R, 8) wh: - dimnsionlss qualizd iniial sss, calculad fom h condiion of consan volum:, 9) ln ) WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

11 Compu Aidd Opimum Dsign in Engining XII 93 may b slighly cocd, o obain h longs lif im o cp upu. Cocion is adopd in fom of h hid dg polynomial funcion: H co 3 p p R p R, 30) 0 wihou h lina lmn, so as o h hicknss divaiv in h middl of h disk was qual o zo. As h cocion canno chang h oal volum of h ba, only wo cofficins of q. 30) may b ad as f paams, whil h hid suls fom h consan volum condiion: 5 5 p3 p0 p. 3) 4 Fo diffn valus of hs paams w dsignad h iniial shap of h disk, H R H R H R us co 3 3) hn w inga h sysm of quaions 6) Calculaions w caid fo 0,, 3, xponn in Noon s law n 3 and xponn in Kachanov s law m. Opimal shaps of h cocd shap of unifom iniial sngh h disk compad wih obaind ali fo paamic opimizaion a psnd in Figu 6. Th opimal shaps of diffn disks a placd on h im axis a h poins cosponding o obaind ims m of mixd upu. As xpcd, h cocd shap of unifom iniial sngh disk povids h longs im o mixd cp upu. Inoducion of paabolic disk nlags his im a fo abou 4%, whil fo cocd disk of unifom sngh h gain in compaison wih conical disk is abou 70%. 5 Conclusions In conclusion, i is woh poining ou ha h mahmaical modl of mixd cp upu discussd in his pap addsss on of h fundamnal phnomna ha occu a high mpaus in h maials. Th hoical modl fo cp dfomaion of h disk wih h accoun of duciliy and mbilmn of h maial is dscibd. Applicaion of mixd upu hoy poposd by Kachanov in his modl aks ino accoun: gomical changs - diminishing of ansvsal dimnsions suling fom lag sains as in ducil upu) and gowh of micocacks as in bil upu). Th im o upu is dfind as a im whn coninuiy funcion diminishs o zo. Th pap is mbddd in h banch of sach ddicad o opimum dsign of sucus und cp condiions wih h objciv funcion in h fom of mixd cp upu im. Th poblm of opimal shap wih spc o mixd cp upu im fo h complx sss sa - oaing full disk is invsigad. Boh h own mass of h disk and a mass unifomly disibud on h ou WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss

12 94 Compu Aidd Opimum Dsign in Engining XII adius a akn ino accoun. Th Noon cp law has bn applid. Th poblm has bn solvd in h famwok of lag sains u ssss appoach. Th logaihmic masu of sains has bn adopd. Th opimal pofils of disks, saisfying h condiion of maximum im o upu, w found fo full disks and fo h consan volum of maial. Th bs pofil of h disk, lading o h longs lifim o mixd upu in assumd class of polynomial funcions was sough. Th cocd shap of unifom iniial sngh disk povids h longs im o mixd cp upu. Cocion is adopd in fom of h hid dg polynomial funcion. In compaison of conical disk nlags h im o mixd upu fo abou 70%. Th suls psnd in h pap hav fundamnal maning fo possibl applicaions in diffn chnologis. Boh h pow plans o h auomobil indusy and ship o aicaf chnologis can pofi fom dcn analysis psnd in h submid manuscip. Also, his domain of sach looks pomising fo h fuu sudis in viw of muli-scal modling applid o cp upu mchanisms. Rfncs [] Życzkowski M., Opimal sucual dsign und cp condiions, Appl. Mch. Rv., , 988. [] Hoff N.J., Th ncking and upu of ods subjcd o consan nsil loads, J. Appl. Mch. Tans. ASME 0 05, 953. [3] Mnl V., An applicaion of a phnomnological hoy of cp damag, Maials a High Tmpaus, 3, 95-00, 006. [4] Szuwalski K., Opimal dsign of bas und nonunifom nsion wih spc o ducil cp upu, Mch. Suc. Mach. 3, , 989. [5] Szuwalski K., Opimal dsign of Miss uss wih spc o im o cp upu, Engng. Tans. 4, 45 55, 994. [6] Pdsn, P., On h influnc of bounday condiions, Poisson s aio and maial non-linaiy on h opimal shap, In. J. of Solids and Sucus 383), , 00. [7] Szuwalski K., Opimal dsign of disks wih spc o ducil cp upu im, Sic. Op. 0, 54 60, 995. [8] Szuwalski K., Uszycka A., Th Influnc of Bounday Condiions on Opimal Shap of Annula Disk Wih Rspc o Ducil Cp Rupu Tim, Euopan Jounal of Mchanics, 0), in pin). [9] Kachanov L.M., Cp hoy, Fizmagiz, Moskwa, 960. [0] Szuwalski K., Uszycka A., Opimal Dsign of Bas Und Nonunifom Tnsion Wih Rspc o Mixd Cp Rupu Tim, Innaional Jounal of non-lina mchanics, 0, in pin). WIT Tansacions on Th Buil Envionmn, Vol 5, ISSN on-lin) 0 WIT Pss