Proceedings of the ASME th Biennial Conference on Engineering Systems Design and Analysis ESDA2010 July 12-14, 2010, Istanbul, Turkey

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1 Pcings th ASME th Binnial Cnnc n Engining Sstms Dsign an Analsis ESDA Jul 4,, Istanbul, Tuk ESDA49 EXPERIMENTAL AND ANALYTICAL INVESTIGATION ON A LIQUID BALANCE RING FOR AUTOMATIC WASHING MACHINES Lna UbilaSt Mab Tchnlg & Pjcts, Quta, Quta, 76, Mic lna.ubila@mab.cm.m Macl LpPaa Pss, Dpatmnt Mchanical Engining Natinal Autnmus Univsit Mic (UNAM) Mic Cit, D.F., 45, Mic ABSTRACT A highsp cama an a Paticl Imag Vlcimt (PIV) tchniqu a us n a tanspant liqui balancing vic tpical mn washing machins. Epimntal sults inicat that th bals plac insi th ing cavit intact with th lui it t vlp ms vibatin cicumntial an aial tp, thus ning a cmpl swil lw insi th annula cavit cmpis tw intial wavs. On such wavs tavls backwas lativ t th igi b mtin. As as a sult, a amping ct is inuc an a phas lag with spct t th citing unbalanc ccus. An analtical namic ml th unbalanc spns is iv an clat with lui lw visualiatin pimnts an vibatin masumnts. NOMENCLATURE p Liqui pssu il, N/m Spin at, a/s Natual qunc lui, a/s m,n Cicumntial an latal ms vibatin, Catsian cinats, m F Flui balancing c, N a Rt man bit aius ccnticit, m v Balanc ing vlum, m 3 v Flui vlum, m 3 m Rt mass, kg m t Whiling mass, kg m u Unbalanc mass, kg Cmpl t mtin ccnticit, m Cmpl tub whiling mtin, m Angula cinat, a Unbalanc spns phas angl, a INTRODUCTION Th a th imptant cmpnnts achiving a washing machin sign with g namic pmanc, naml; th suspnsin, th autbalancing vics, an th cabint stuctu. Dpning n th passiv mchanism mpl, th autbalancing vics can b classii int th liqui an ball balanc ing tp. Th balanc ing us in mn autmatic washing machins mainl cnsists an annula cavit patl ill with a bin slutin. Th hllw ing accunts with a numb aial bal bas quiangulal spac. With ga t thi lcatin n th um t, als call tating baskt, th autbalancing vics can b ivi int tp balanc ing, i it is plac n th upp pat th baskt an bttm balanc ing, whn situat n th lw pat th baskt. Liqui ma b in th inti a t b sign, as in th cas cntiugs, ctain bla cling sstms, in balanc ing vics mpl in washing machins. Als, liquis can b cntain in tating bis b accint, as in th cas stating tubins with tapp cnnsat insi. Th litatu viw will b ivi int tw main aas sach: () th instabilit cts tapp luis in hllw ts, an () liqui balanc ings mpl cntlling th vibatin washing machins. TRAPPED FLUIDS IN HOLLOW ROTORS Th subjct call ist th attntin in th instabilit pblm caus b th tapp tatinal lui in hllw ts. It has bn pt in a liquicl gas tubin b Kllmann [], a jt ngin b Ehich [] an in a highsp gatp cupling b Kik t al. [3]. Accing t Wihs [4] tw kins liqui scillating mtins a un in a patiall ill spinning clin, naml suac an intial (intnal) wavs. O Cspning auth Cpight b ASME

2 ths tw, th intial wav m is th mst imptant bcaus th snant qunc such m is in th ang that can b cit b nutatin whiling. F suac wavs a simila t th slshing that ccus in nnspinning tanks. Th nutatin mtin pucs an scillat isplacmnt th liquis in th tanks. This liqui mtin issipats kintic ng as a sult th viscus stsss caus b th mtin. Intial wavs a scillatins th liqui inti an can ccu vn in th absnc a liqui suac. F suac wavs a simila t th slshing that ccus in nnspinning tanks. In a smmtical tank spinning abut its smmt ais, th th shws, ampl, that th liqui pssu il (i t) p (,,) cspning t an scillatin qunc is gvn b: p p 4 p Whn th scillatin qunc >, this intial quatin is lliptic m an th scillatins a th suac slshing scillatins mntin abv. But whn <, th quatin is hpblic natu an th scillatins a intial wavs [4]. Appimat analtical mls inicat that all th intial wav snancs ccu in th qunc ang btwn an, an th mst pminnt snancs hav quncis lss than. Mils an Tsch [5] cnsi th thimnsinal scillatins th tating tank sstm patiall ill with liqui shwn in Fig.. Th shw that th istub mtin th lui suac cul nt main itatinal u t a vtical gaint vlcit that ncssail gnats vticit. With nc t shallw wat in cicula tanks, th bsv that th tank tatin sults in splitting th pais natual quncis that wul b qual magnitu an ppsit sign in th absnc tatin. Mils an Tsch hav utlin an act analtical appach tmining th slsh quncis in spinning clins patiall ill with liqui. Th pt that th natual qunc th suac can b cmput m Eq. (). h Z X Figu. Rtating tank patiall ill with liqui, m [3]. Y v u X () Tw quncis that i in magnitu as wll as sign a un. m, () m m m LIQUID BALANCE RINGS Th limit inmatin ma public can b un n a numb patnts an a w tchnical paps an thsis wk. Lblanc [6] intuc a passiv namic balancing vic tubin ts cnsisting an annula cavit patiall ill with a liqui high viscsit. It is bliv that th ist balanc ing vic mpling wat an aapt int a washing machin was intuc in 945 as patnt b D [7]. Cna [8] anal a multichamb balanc ing with succssiv cncntic annula cavitis t incas th balancing capabilit lui. Cna aiv t a namic ml picting th unbalanc spns th washing machin, which is in natu quit simila t th n btain b Ehich [] a tubin ngin hllw t with tapp lui. Siichiu [9] vlp a numical ml th washing machin psnting th liqui balanc as lump masss. Th lump masss w link t th cnt lin th spinning baskt with vlut jints an tatinal spingamps, which w libatl tun t clat numical ata with pimntal isplacmnt masumnts. Mi t al. [] us th llwing pssin th balanc ing c: F h Wh th guss valu is btain with a init lmnt ml ma using cmmcial stwa. Th sulting c is analgus t a back cntl sstm. Th ml is thn slv qunc spns. This quatin vstimats th lui balancing c in sm imptant masu, this can b sn in Mi s pimntal stain valu th tating stuctus, which a cnsistntl lw cmpa t th init lmnt slutin. Ba t al. [], in a v nat wk, stablish th intial quatins mtin a mn vticalais washing machin with pnula suspnsin sign. H un th lui cnt mass b intgating th tp viw aa cv b th liqui insi th cavit un a pscib ccntic cnitin, an thn h calculat th lui c, which h valiat with pimntal masumnts. This c was lat input int his ml t slv th stastat unbalanc spns. H aiv t a simila quatin un pviusl b Cna [8]. Ubila [] withut having ha accss t Cna s an Ba s wk als pt th balanc lui c t b: F ill h i (3) (4) Cpight b ASME

3 Wh th lui inn suac aius is givn b ill. 5 i Jung t al. [3] ppss th installatin a mil plat accunting with a numb hls t avi lui slshing. This mil plat smbls th lat ing bai us in clinical tanks ist stui b Kulgan [4] an Mils [5] t pvnt lui slshing. A CFD mth was mpl t vi its ctivnss. H us th balanc ing c ml m Ba []. Rcntl, Ubila t al. [6] intuc th ist balanc ing sign with nnstaight bals. Ubila s sign cmpiss altnativl isps psitiv an ngativ cuv blas, as wll as psitiv cuv an staight blas, that in cmbinatin with blas int hights, uc th tim an th amplitu th vibatins th tansint stat. Als, this nvl sign nabls th um t spin at high spin vlcitis. Th incnvninc pvius sach ts lis n th act that th ccnticit in stastat ns t b knwn a pii, usuall b isplacmnt masumnts, subsquntl th lui c can b stimat an input int a namic ml t pict th unbalanc spns at th at sp, th unbalanc spns acss a qunc ang can nt b pict t thugh. (5) A maj isavantag liqui balancing vics is t a mass t th unbalanc whil pating blw th ist citical sp [8]. Hnc, an sstm bing cnsi this tp balancing ns t hav a lativl lw snant qunc. Un mat unbalanc la, this pblm ma b allviat b inucing a ast amp up as is th cmmn pactic hm applianc manuactus. Epinc ictats that th statup amps shul spciicall minimi th tim qui t pass thugh all snancs. Mst washing machins hav thi ist buncing an cking natual quncis in th ang m 5 pm, whil th lui, as will b shwn lat, attachs t th ut wall at 5 pm. Thn, th lui all s nt hav much pptunit t a cnsiabl unbalanc at lw sps. Actuall, th natual qunc is tavs in th ist t 3 scns, whil th um t bal cmplts 6 t tuns, s th machin is nt allw t wll within th banwith th sstm snanc. Hwv, lag unbalanc las, th amp up is tn nt nugh t attain a smth statup an a lui balanc ing with btt pmanc th tansintstat is si t avi tubcabint cllisin. Th assum phsical pincipl that this balancing vic wks un is that, an lasticall munt t will isplac 8 ut phas th appli unbalanc abv its ist citical sp. Hwv, this is nt ncssail tu u t lui amping cts. EXPERIMENTAL APPARATUS Th imag n Fig. psnts a cutawa viw a mn washing machin, which cmpiss a tub that cntains a washing baskt, th lat capp b a liqui balanc ing. An unbalanc mass is plac in th baskt t simulat its unbalanc cnitin imps b th gamnts bing wash, sinc th nt sttl uniml n th bttm th baskt at th washing ccl has inish, Knuckl jint R Tub Mt Liqui balancing Lss lui Unbalanc mass Divshat Pull an blt M lui Figu. Washing machin cutawa viw. In t icintl th bjcts bing wash, cntiugal mvmnt at high angula vlcitis is qui, thus th mt th washing machin will b ngi lss tim, which will puc a sht cntiugal ccl, an in tun will sav ng u basicall t tw cncpts: ist, th mt th washing machin is ngi lss tim; an scn an m imptant, thanks t th high cntiugal c, m wat is mv m th gamnts an th bjcts a liv n a cnitin. Th, whn a ing machin is us subsquntl ing th maining humiit n th clths, such ing machin will qui lss cnsumptin lctic pw cmbustibl gas. Hwv, as th vlcit incass th citatin quncis might cat gat amplitu vibatin, which wul caus kncking btwn th tub an th cabint stuctu, ith making th washing machin t "walk" causing th subsqunt amags t th machin an/ us annanc u t vibatin an nis. Th washing unit is munt n a igi am pming th vibatin pimnts. Th thgnal cmpnnts th vibatin w masu at th balanc ing an tub using Bann LG sis las shtang pcisin isplacmnt tansucs with a slutin m, a schmatic is shwn n Fig. 3 Las Balanc ing Las 3 Las Las 4 Tub Figu 3. Las isplacmnt tansucs aa. 3 Cpight b ASME

4 Th sstm was pat at a sp 78.5 a/s (75 pm) b a vaiabl qunc mt that cul iv th t at an givn acclatin at maintain an sta tatinal sp up t 4.7 a/s ( pm). Th balanc ing mpl mainl cnsists an annula cavit 3.79E3 m 3 accunting a numb ight aial bals quiangulal spac. A bnchmak stu inicats that th liqui vlum actin man washing machins balanc ing appachs 5 t 8 pcnt th ttal ing vlum. Duing th pimnts a ill ati 8% b sium chli with a nsit 3 kg/m 3 was mpl. Figu 4 shws th balanc ing un stu cnsisting a thmm Plthln Tphthalat (PET) ing assmbl b tw pats. Th bttm pat huss th bals, which a put in plac b bing inst in machin guiing gvs. Th upp pat wks as a tp cv blt t th bttm ptin. In btwn, th is an pn cll gaskt aim t pvnt lui lakag. Th balanc ing assmbl is igil munt n tp th tating baskt thugh a st backts. Figu 4. Tanspant balanc ing; tp viw, an aial bal, imnsins in mm. Impact tsting tchniqu was us t masu natual qunc t in th an ictins. Rt stinss masumnts w pm appling static las thugh a nammt an cing th assciat lctin inicat b a ial gag. Suspnsin cil sping stinss masumnts w n with th aim a nammt tst stan. Owing t th clsl vtical intatin th suspnsin, th nl appciabl amping in th plan cms m gas lubicat knuckl jints at bth ns th suppting s. Th amping was masu using th washing unit pnula mtin an masuing th lgaithmic cmnt spns. Tabl pts th pimntal paamts th iniviual cmpnnts namic chaactiatin. An unbalanc.34 kg was mpl uing th pimnts. A highsp cama at ams/s an a st whit light sucs Xnn lamps tp w us ict lui lw visualiatin. It was bsv that th lui attachs t th ut wall at 5.7 a/s (5 pm) an that a ptin lui tavls backwas lativ t th igi b mtin. Th PIV tchniqu mpl pmitt th visualiatin an quantiicatin biimnsinal vlcit ils th lui thugh th s paticls alumina whs iamt angs m 5 5 m. Figu 5 shws th pimntal aa. Th lw was lightn thugh a las bam N:YAG at 53 nm. Th light spa b th 3.8 i h paticls achs a Chag Cupl Dvic (CCD) imag sns cama, which in tun gnats cnscutivs imags th lw. Sinc th las bam is v high qunc natu (pulss 5 ns), it is pssibl t tw cnscutiv lui mtin ams an m thm, buil th vlcit vcts. Th plant viw th balanc ing was ma n an aa 965 mm, whil th latal viw an aa 47 mm was mpl. Tabl. Epimntal sstm paamts. Paamt Dsciptin Amunt Units/Chaactistics Out ing aius.59 m 3 i Inn ing aius.5 m 3 h Ring hight.74 m 3 Fill Cavit lui ill ati.8 Dimnsinlss Bals Raial ibs 8 Staight quiangulal istibut Flui nsit 3 kg/m 3 (sium chli) m Flui mass kg c Man citical sp t 94.5 a/s assmbl k Rt stinss N/m at Suspnsin stinss 4343 N/m Angl btwn suspnsin an th 8 g vtical c Suspnsin pnula amping 85. Ns/m n Suspn washing unit.74 a/s natual qunc m u Maimum Unbalanc t puc a tubcabint stik.34 kg Du t limitatins th pttp, th maimum attain sp lui lw visualiatin tsting was 5.3 a/s (49 pm) sinc abv this sp imminnt ubbing btwn th t an tub ccu. This shtcming mainl as bcaus th cnsqunt tub lst igiit whn its upp pat was cut t allw th tanspant balanc ing si viw. Th assciat maimum tangntial vlcit lui was thn 3. m/s. Epimntal aa plant an si viws a psnt b tt an sli lins, spctivl. Th gn lins inicat th light ictin, whas th lins shw th ictin th cama. Th an gn cicls inicat th psitin th cama an light, spctivl. Th las bam was thgnall int with spct t th ictin th bsving cama b using an aticulat am sstm. FLUID FLOW VISUALIZATION AND PIV RESULTS Th mval bals insi th cavit n a bulklw bhavi lui as shwn n Fig. 6. Whil th aitin bals, incas th natual qunc lui an inuc a 4 Cpight b ASME

5 cmpl lui m vibatin. At lw sp, th lui c acting n th ing cnsists tw cmpnnts, n which is in phas with th unbalanc spns, an th th is ut phas b 9 with it, such that whn th t is vibating, th lat cmpnnt has a phas lag 9 with th unbalanc spns. This tangntial c is u t th lui wav tavling backwas lativ t th igi b mtin. Hwv, at high sp, th is a balancing lui c acting t th isplacmnt an pps b 8 t it. Th is als a tangntial c 9 phas lagg with spct t th unbalanc spns u t lui natual qunc shwing as a backwa wav, s Fig. 7. ictin. Futhm, a cmpaisn with th latal ms vibatin a sting th ms las t asctain that th ighth latal m vibatin maks a paalll with th aial wav inuc b th ballui intactin. Cupling th uth cicumntial m vibatin a ing with th ighth latal m vibatin a sting, a cmpl 3D wav shwn schmaticall n Fig. 9 is btain. Rigi b mtin Rigi b mtin backwa tavling wav Backwa tavling wav Figu 7. Backwa tavling wav, tp viw, an si viw, cicl an inicat a cst an vall, spctivl. m = m = m = m = 3 m = 4 Figu 5. Flui lw visualiatin; highsp cama tp viw, lui lw attach t ut wall at 5.7 a/s (5 pm), an (c) PIV pimntal aa. Rigi b mtin (c) Rigi b mtin n = n = n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 Figu 8. Ms vibatin; cicumntial ms vibatin a thin wall ing, an latal m shaps a sting. Backwa tavling wav Backwa tavling wav Figu 6. Flui bulklw in a hllw ing with n bals; tp viw, an si viw. Establishing an analg with th cicumntial ms vibatin a ing with mms, it can b sn m Fig. 8 [7] that th uth m is cmpaabl t th lui wavs inuc b th ballui intactin in th cicumntial Th pimntal wk shw that th lui is nt statina lativ t th igi b. Th mtin in th tating b is tansmitt t th cntacting liqui b sha stss. This suggsts that wavs a ccuing in th lui. Sinc th spin ais s nt cinci with th angula mmntum vct, th um t appas t scillat abut its tansvs an as. This scillatin whiling causs th liqui in th ing t mv lativ t th igi b. I th um t spins abut a maj mmntintia ais, th liqui mtins tn t amp th whiling. Th ng issipat b th scillat mtins is tml lag whn th whiling mtin cits th liqui 5 Cpight b ASME

6 Unbalanc Rspns Flui natual qunc [a/s] Unbalanc Rspns int snanc. F that asn, th natual quncis a liqui in a spinning igi b a much intst. This mtin s nt lat t suac slshing. In act, th mtin ccus thughut th liqui inti an is thus call an intnal wav intia scillatin. Givn that th liqui pth is init an that th ing hight is v small cmpa t its iamt, th lui wavs can b appach llwing th D th vlp b Mils an Tsch [5]. Futhm, sinc th bals is m th bttm th ing allwing nl a claanc with th tp cv lui ciculatin, thn as th sp incass an th lui shap appachs a clin, th aial m vibatin is cnstain t vlp n th intac with th tp cv. Th, it can b sai that th aial m vibatin has littl n ct n th wavs vlp b th lui. Th pblm can thn b tat as D lui lw phnmna with nugh accuac. Th th shws that th a man intia wav ms. Th liqui mtin assciat with ths ms s nt ca pnntiall with pth blw th suac, as wul a suac slshing m, but psists all th wa thugh th liqui vlum. Ths a intia ms an th can ccu nl a spinning tank accing t Mils an Tsch. Cmputing th natual quncis th suac as scib b Eq. (), th pai lui natual quncis is pict n Fig. 9. On qunc is psitiv an gat than th t angula vlcit (ast wav) an th th is ngativ, i.. tavling backwas lativ t th igi b, an is slw than th t sp (slw wav). m = 4 & n= bw 5 Running qunc [a/s] Figu 9. Cmpl 3D wav; si viw with cst an vall, an cupl ms vibatin; m = 4 an n = 8. Hwv, PIV shw th ast wav nt t b v utstaning, but ath bhaving lik a sli. Hnc, nl th slw ngativ wav is imptanc. Fm Fig. 9, it can b appciat that th lui thtical uth cicumntial natual qunc was un t b.95, i.. almst snchnus with th unning sp. Th lui vlcit map was built b suppsitin PIV ams int ptins th balanc ing as illustat n Fig.. Th pimntal lui natual qunc ati is.93 ; cniming that an imptant actin lui mass tavls backwas lativ t th igi b mtin. 5 5 w m/s 3 m/s Epimntal lui natual qunc ati: /3 =.93 Lw vlcit n High vlcit Backwa n (5 ) Lw vlcit Fwa n High vlcitbackwa n (5 ) Lw vlcitwa n Figu. Paticl imag vlcimt; vlcit histgam, an balanc ing vlcit map. Watall plts balanc ing vibatin shwn n Fig. vinc stng snchnus mtin with an n subsnchnus activit. [mm] [mm] Figu. Watall vibatin plt balanc ing vibatin; ais, an ais. m/s 6 Cpight b ASME

7 Flui cnti [mm] Flui Fc F [N] Th ctiv mass (m ) tavling backwas is a actin th ttal lui mass givn b Eq. (5) 5 8 High Vlcit Zns Cicumnc 3 Th, th ctiv lui amping mass is n thi th ttal lui mass m. 33 (6) m This mass inucs a tangntial c 9 phas lagg with th sli b mtin, thus bhaving as a amping ct. DYNAMIC MODEL Fiv cass can b intii abut th lui istibutin insi th balanc ing cavit as illustat n Fig.. Cas I Cas III Cas V O M F C O t uni O u u M β C (c) M β F C F t ma t ma Cas II Cas IV IIIIII IV M u β F O C u M β F O C () V t t ma ma Gap tc * F ma Rt mtin [mm] () () Figu. Flui istibutin; cas I, cas II, (c) cas III, () cas IV, () cas V, an () lui cnt mass lcatin (sli an tt lin) an lui c (pints) vs t mtin. (5) Cas I cspns t c ccnticit an unim lui istibutin insi th cavit, thus cnming a lui ughnut with cnstant lui ilm (t uni ) an with nt balancing c. This cas is an ial situatin wh th is n unbalanc la insi th baskt, an th unlikl t ccu. Th cnt Mass M, cnt tatin C, cnt whiling O an lui cnt mass F a all cincint as shwn n Fig.. Cas II taks plac whnv a nn unbalanc laing cnitins ists, th cnt mass M lcats at an st u m th cnt tatin C an an angl m th ais, whas th lui cnt mass F is plac n th ais at a istanc m th cnt whiling O. A nn ccnticit vlps an th lui istibuts unvnl aun th ing cavit, having a ilm thicknss t at =. Cas II is th m cunt lui istibutin cnitin. Cas III happns un lag unbalanc cnitins, whn th lui ilm thicknss aun th ing quats t i at = an amunts t at = 8 as shwn n Fig.. Th ccnticit in such cas is givn b Eq. (7) an is th maimum t mtin th liqui balanc ing can ctivl cmpnsat, as shwn n Fig. (). Th maimum availabl balancing c m th lui is thn pict b Eq. (8). i ma (7) i F ma ill i h (8) An t mtin bn ma inuc b a v lag unbalanc la wul shw a lui istibutin simila t cas IV n Fig. (), wh th lui ilm n lng wts th nti ing cavit. Cas V ma vlp un tml lag unbalanc cnitins, wh th lui is ttall plast at n si th balanc ing cavit as pict n Fig. (). Epimntal lui lw visualiatin shwn n pictus Fig. 6 smbls this cas. Wh, a cla suac nt wav can b appciat. Th lui mv lik a bulk an isplac backwas lativ t th igi b tatin. Th unbalanc baskt pimnts withut bals in th ing vinc a v angus t mtin, bcaus at lw sp, th unbalanc la plac insi th baskt, incas b th a lui mass in th ing, thus inucing vilnt vibatin. Th iving mt was stpp t avi stuctin th tst ig. Hwv, th aitin bals in th ing, pvnt m vilnt liqui slshing an thughut all pimnts pm, th lui istibutin was alwas sn t b simila t cass I t III, a tpical cas is shwn n Fig. 5. Th istanc m th lui cnt mass t th cnt whiling is givn b Eq. (9) ill i ill cs i 3 sin( ill ) i ill i (9) 7 Cpight b ASME

8 In all vnts, u t th psnc th bals, th lui cnt mass lcatin ma b pict as linal pnant n th t mtin accing t Eq. (). () Assuming paticula unctin slutins th m i i t t () () On th th han, th ing ill ati t maimi th balancing c is givn b Eq. (), which th balanc ing un stu is abut 5%. ill pt ( i ( ) ) ma i Figu 3 pvis a namic ml th washing machin. Using cmpl ntatin, an can b pss as gnal cmpl unctins. i i PIV masumnts an lui natual qunc analsis allw assuming snchnus mtin, i.. with nugh accuac. Thn, th quatin mtin th um t is givn b m k it m m m Nt that a amping tm m has bn a, which can b thught mainl u t viscus issipatin in th bulklw. In th ws, this aitinal amping is u t th mass actin mving backwas lativ t th igi b, which wav qunc is snchnus with. On th th han, th quatin mtin th whiling tub is in as c mt k k Dining th stinss, stinss ati, amping ati, mass ati, an qunc ati as Eqs. (6), spctivl u () () (3) (4) (5) m k O k U = m u c k T M u k( ) C m t m t m F c β m m Figu 3. Dnamic ml; isplacmnt vct an c iagam. Nw, substituting atis, assum slutins, iviing b m it an c, an aanging, Eq. (5) ns Wh D D D c Eq. (3) is substitut int (4) t btain m (3) (4) k 4 at cs ( ) k k c m mt m c c (6) (7) (8) (9) () Dining h m u c mu u m m m m u (5) (6) 8 Cpight b ASME

9 Tansmissibilit ati Amplitu vibatin [mm] Phas angl [g] Th phas lag is givn b tan D c D D ill( ) m c m h m Thn, slutin is givn b i Finall, having in ξ, th tansmissibilit ati t th stuctu is as llws 4 n TR 4 n n Wh n is th natual qunc th tubsuspnsin assmbl, givn b n 4 at m m t ( ) ( ) TR( ) (7) (8) (9) (3) Fm Eq. (5), it can b appciat that th quivalnt mass th balanc ing is th ull isk, nt th lui annulus actuall psnt. Eqs. (5, an 33) a gaph aminatin n Fig. 4. Ml clats v wll th tansint an stastat thughut th qunc ang tst. Futhm, slutin inicats that an angl 53 btwn th unbalanc an maimum isplacmnt lcatin ist, which can b cnim n Fig. 5 b th 3 angl btwn th unbalanc vct an th thinnst lui n. Owing t th act that, th lui cnt mass is align with th isplacmnt vct, thn th is a cmplmnta angl 5 btwn th unbalanc an th lui cnt mass. In th ws, th unbalanc vct U is nt actl pps b th lui balancing c vct F, but b a actin it, cmput as F cs(3 ). Epimntal visualiatin th unbalanc mass an lui istibutin allws g ml clatin. As mntin b, a maj isavantag th liqui balanc is t a mass t th unbalanc whil pating blw th ist natual qunc. Hnc, an sstm bing cnsi this tp balancing ns t hav a lativl lw snant qunc, as is th cas th washing machin. Fig. 6 picts th unbalanc spns a washing machin withut lui balanc ing. Nt that th vibatin at lw qunc gatl ucs. On th th han, vibatin at stastat appimatl ubls u t absnc th balancing c m th liqui. Futhm, th vibatin tansmissibilit ungs tw shap paks, bing th scn n maj cncn, sinc it lcats in th high qunc n, wh th 3 citing unbalanc cs a gat magnitu 3 u t cntiugal ct. Vibatin tansmitt t th cabint 3 stuctu is magnii laing t cabint shaking an imminnt walking. 5 ( ) ( ) ( ( ) ) ( ) ill.8 TR ( ( ) ) 5 5 TR 6 6TR( ) m u TR( ) masu Running sp [pm] Figu 4. Epimnts vs analtical slutin th (3) (3) (33) washing machin namic unbalanc spns at a ill ati.8. Unbalanc Rigi b mass mtin Unbalanc c U = m u Rigi b mtin Backwa Displacmnt tavling vct wav Flui balancing c F = m 3 Backwa tavling wav Figu 5. Flui istibutin lativ t unbalanc mass lcatin; unbalanc mass lcatin, lativ lcatin thinnst lui n t unbalanc mass lcatin. A balanc ing ill ati.5 is cnsi n Fig. 7. Un this situatin, th unbalanc spns acss th qunc ang is lw than that th.8 ill ati cas. Hwv, th tansmissibilit ati ins an unsiabl pak at th unning qunc. Th, inicating th n th a mass pvi b th.8 ill ati cas. 9 Cpight b ASME

10 Tansmissibilit ati Amplitu vibatin [mm] Phas angl [g] Tansmissibilit ati Amplitu vibatin [mm] Phas angl [g] ( ) 8 8 ) 3 3 ( ) ill.8 ( ) 8 ) ill.8 ( ) m u.3 ( ) ill.8 ( ) ( ) TR ( ) m 5 u.3 ( ) ill TR( ) m u.3 ( ) TR 6 6 m 5 u.3 ( ) 4 4 TR( ) masu Ingniuws., 8, pp Running 6 6 sp [pm] Figu 6. Epimnts vs 4 analtical slutin 3 th 3 washing machin namic 4 6 unbalanc spns 3 at a ill 5 ati. pp ( ) ( ) ill.8 ( ) 8 ill.8 ( ) m u.3 ( ) ill.8 ( ) ( ) TR ( ) m 5 u.3 ( ) ill.8 8, TR( ) m u.3 ( ) TR m 5 u.3 TR( ) Bis, US Patnt,59, ,375, Cnstaints, Ph.D. Thsis, Puu Univsit Running sp [pm] Figu 7. Epimnts vs analtical slutin th washing machin namic unbalanc spns at a ill ati.5. CONCLUSIONS Th lack bals ns unvn istibutin th lui insi th cavit with stng unstabl bhavi. On th cnta, th aitin ight bals hlps vlp a snchnus lui tangntial vlcit with t unning sp. A numb ight bals cit th lui uth cicumntial m vibatin an th ight aial m vibatin. Th lui is un t vlp a backwa intia wav lativ t th igi b mtin. Th ctiv mass tavling backwas amunts t.33 lui ttal mass. This backwa wav acts as a amping c 9 phas lagg with spct t t mtin, thb causing th phas angl btwn th unbalanc an th lui cnt mass lcatin t chang m 8 t 5. ACKNOWLEDGEMENTS Thanks t mab Cmpan ganting pmissin this publicatin an t Natinal Autnmus Univsit Mic (UNAM) an th Mican Scinc an Tchnlg Cuncil (Pct SEPCONACYTCincia Básica N. 8339) ganting suppt psnting this pap. REFERENCES [] Kllmann, F. G., 96, Epimntll un Thtisch Untsuchungn Ub i Kitischn Dhahln Flussigkitsgullt Hhlkp, Fsch. Gb. Ingniuws., 8, pp. 5 3; 96, Fsch. Gb. [] Ehich, E. F., 967, Th Inlunc Tapp Fluis n High Sp Rt Vibatin, ASME J. Eng. In., 89, pp. [3] Kik, R. G., Mn, M. E., an Muph, R. C., 984, Th an Guilins t Pp Cupling Dsign Rt Dnamic Cnsiatins, ASME J. Vib. Acust., Stss, Rliab. Ds., 6, pp [4] Wihs, D. an Dg, F.T., 99, Liqui Mtins in Nnaismmtic, Patiall Fill Cntains Rtating at Z Gavit, AIAA J. Spaccat an Rckts, 8, [5] Mils, J. W., an Tsch, B. A., 96, Suac Oscillatins a Rtating Liqui, ASME J. Appl. Mch., [6] LBlanc, M., 94, Autmatic Balanc Rtating [7] D, J., 945, Dmstic Applianc, US Patnt [8] Cna, D.C., 994, On Th Funamntals Autmatic Washing Machin Dsign Bas Upn Dnamic [9] Siichiu, S.,, Stu n th Dnamic Bhaviu an Autmatic Washing Machin, Ka ADAMS Us Cnnc,. 89, 6. [] Mi, M., an Yutaka, N.,, Washing Machin Dhatin Dnamics Analsis, Nihn Kikai Gakkai Nnji Taikai Kn Rnbunshu, 5, 96. [] Ba, S. L, J. M., Kang, J. S. an Yun, J.R.,, Dnamic Analsis an Autmatic Washing Machin with a Haulic Balanc, Junal Sun an Vibatin, 57, 38. [] Ubila, L., 5, Cálcul ls Ects Incials (Raials Tangncials) l Flui l A Balanc, Intnal Rsach Rpt, mab TP. [3] Jung. C.H., Kim, C.S., an Chi, Y.H., 8, A Dnamic Ml an Numical Stu n Th Liqui Balanc Us in an Autmatic Washing Machin, Junal Mchanical Scinc an Tchnlg,, [4] Kulgan, G.H.; an Capnt, L.H., Fcs n Clins an Plats in an Oscillating Flui, J. Rs. Natl. Bu. St., vl. 6, n. 5, Ma 958, pp [5] Mils, J.W., Ring Damping F Suac Oscillatins in a Cicula Tank, J. Appl. Mch., vl. 5, n., Jun 958, pp [6] Ubila, L., Otga, M., Thmpsn, A., Hnamic Balanc Ring Cntiugal Rtatin Machins, 9, US Patnt Applicatin [7] Timshnk, S., Yung, D. H., an Wav, W. JR., Vibatins Pblms in Engining, 974, Jhn Wil & Sns. Cpight b ASME