International Journal of Pure and Applied Sciences and Technology

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1 t. J. Pue Appl. Sc. Techol., 4, pp teatoal Joual of Pue ad Appled Sceces ad Techology SS 9-67 Avalable ole at Reseach Pape Geetc Algothm Optmzato fo Tesog ove Double Aula Doma Masayu shhaa,*, Ho Muaam ad Yoshho Ootao 3, 3 Osaa Pefectue Uvesty, -, Gaue, aa, Saa, Osaa , Japa suzu Motos Lmted, 6-6-, Mam-O, Shagawa, Toyo 4-87, Japa * Coespodg autho, e-mal: shhaa@me.osaafu-u.ac.jp Receved: 8--3; Accepted: 9--3 Abstact: Wth the am of achevg stable opeato of ccula saws, ths study fds the soluto fo the optmzato poblem that volves choosg a set of tesog paametes a otatg ccula saw that s subjected to both a local tempeatue dstbuto ad the -plae plastc sta ove a double aula doma. The soluto fo the -plae foces s obtaed o the bass of plate bedg theoy, ad modal aalyss fo the out-of-plae behavo affected by the -plae foces s pefomed. umecal calculatos ae pefomed to vestgate the effects of tesog ove the double aula doma o the atual fequeces. The optmzato poblem to maxmze the atual fequecy of the most ctcal mode wth espect to the testes, locatos, ad wdths of tesog s solved usg a geetc algothm, ad the optmal tesog paametes ae detemed at computatoal costs that ae cosdeably lowe tha those equed fo % specto. Keywods: Ccula saw, Tesog, atual fequecy, Optmzato poblem, Geetc algothm.. toducto: Ccula saws ae dspesable tools fo machg. They ae omally subjected to a themal load esultg fom the fcto betwee the blade ad the wopece. Ths themal load geeally poduces compessve -plae foces, whch tu chage vaous dyamc chaactestcs such as atual fequeces []. Due to the chage the dyamc chaactestcs, vbatos that ae excted by the teacto betwee the teeth of the saw blade ad the wopece ted to become ustable ad degade the wog accuacy of the saw [8]. ode to ccumvet ths poblem, ccula saws ae

2 t. J. Pue Appl. Sc. Techol., 4, ofte subjected to a pestessg pocedue called oll tesog, whee the saws ae sadwched betwee two olles ude a compessve foce ad the otated slowly to duce plastc defomato ad toduce -plae teso. Plastc defomato occus aoud the olled ego. Revews o tesog studes ca be foud the lteatue [4, ]. umeous studes o tesoed saws have bee coducted [,, 3, 6, 7, 9]. As metoed the above studes, the atual fequecy of a ccula saw has a sgfcat mpact o ts stablty; geeal, the stablty ca be mpoved by ceasg the atual fequecy. Tesog seves ths pupose. Theefoe, a pevous study [5], we pefomed theoetcal aalyss of the themoelastoplastc feld ad the esultg atual fequecy of a otatg, themally loaded ccula saw that was tesoed ove a sgle aula doma, ad we vestgated the effects of tesog o the -plae foces ad atual fequeces. Moeove, we exteded the study to the case whee a saw udegoes tesog ove a double aula doma, assumg actual codtos [4]. The effcecy of tesog depeds o the cotol of the dstbuto of the appled plastc defomato, whch eques sll ad yeas of expeece o the pat of the opeato. Moeove, oce a saw s subjected to tesog, t s ealy mpossble to modfy the tesog paametes such as testes, locatos, ad wdths. Theefoe, t s mpotat to estmate the effect of tesog by pefomg smulatos befoe actual tesog s pefomed. ode to deteme the optmal combato of tesog paametes fo the assumed evomets that a patcula saw s lely to ecoute, t s ecessay to pefom umecal calculatos fo vaous combatos of may types of tesog paametes. Fo example, vaous combatos of thee paametes,.e., testy, locato, ad wdth, must be cosdeed fo a sgle tesoed saw, ad those of sx paametes,.e., two testes, two locatos, ad two wdths, must be cosdeed fo a double tesoed saw. othe wods, the umbe of dmesos the optmzato poblem ca be qute hgh. Moeove, the pevous study [4], the effects of the plae foces o the out-of-plae moto wee tae to accout, ule the case of oday plate bedg theoy. Fom the esults of ths study [4], t was foud that the effects of tesog paametes o dyamc chaactestcs such as atual fequeces exhbted hgh oleaty ad that the computatoal cost to obta the optmal dyamc chaactestcs ceased. The hgh umbe of dmesos ad computatoal cost made t mpossble to deteme the optmal combato of tesog paametes usg oday olea pogammg methods. Geetc algothms have bee wdely used fo othe optmzato poblems whch the objectve fucto exhbts hgh oleaty wth espect to may types of desg vaables [3]. The geetc algothm s oe of the heustc seach techques that mtate the mae whch ceatues adapt to the evomets ove geeatos, though geetc opeatos. The algothm s effectve solvg optmzato poblems wth a hgh degee of oleaty, as metoed above. ths study, theefoe, we vestgated the use of the geetc algothm to solve the optmzato poblem fo the tesog paametes a otatg ccula saw ude a themal load that s tesoed ove a double aula doma. The aalytcal model that we used s a otatg aula ds that s subjected to both a local tempeatue dstbuto asg fom the themal load, whch s caused by blade-wopece fcto, ad the -plae plastc sta toduced by tesog ove a double aula doma. O the bass of the aalytcal esults fo - ad out-of-plae behavos, we pefomed umecal calculatos to vestgate the effects of vaous combatos of the tesog paametes o the atual fequeces. We the used the geetc algothm to solve the optmzato poblem to maxmze the atual fequecy of the most ctcal mode wth espect to the testes, locatos, ad wdths of tesog.

3 t. J. Pue Appl. Sc. Techol., 4, Theoetcal Aalyss:.. Poblem The aalytcal model s show Fg.. Ths model cludes a aula ds wth e ad oute ad ad o, espectvely, ad thcess h ; the aula ds otates aoud the z -axs at agula velocty ω R. The mass desty, Youg s modulus, Posso s ato, ad coeffcet of lea themal expaso ae deoted by ρ, E, ν, ad α, espectvely. The cyldcal coodate system, θ, z s defed the fame of efeece of the otatg ds. Tme s deoted by t. The ds s clamped at the e m ad s fee of tacto at the oute m. The ds has a abtay tempeatue dstbuto of T T f, esultg fom the fcto betwee the blade ad the wopece, whee T T deotes a epesetatve tempeatue. Moeove, -plae plastc stas ε f ad ε f ae assumed to be duced by tesog ove a double aula doma, whee ε,, ad b, deote the testes, locatos, ad wdths of tesog, espectvely. ε f ω R θ o T T f T ε f z f Fg. : Aalytcal model.. Aalyss of -Plae Resultat Foces To smplfy the aalyss, o-dmesoal vaables ae toduced as Whee,,,, b, b,,,, b, b, t ν o, θθ,, 3 θθ Eh 4 w ν / hw, ωr ν ρo / Eh T, ε, ε ν / h αt, ε, ε ad θθ o o b f f T b Eh ν ρ ω, R 4 o o t,, deote the -plae esultat foces the adal ad hoop dectos, espectvely, ad w deotes the deflecto. a pevous study [4], the aalytcal soluto of the -plae esultat foces ude the bouday codtos stated Subsecto. was fomulated as θθ * CF + CF 3 + ν ωr / 8 [ T ft d / + ε f + ε f CF CF + 3ν ωr / 8 [ T ft d / + ft * * + ε f + f + ε f + f ] * ],,

4 t. J. Pue Appl. Sc. Techol., 4, f [ f d ]/,. The deftos of C F ad C F ca be obtaed fom * whee the pevous study [4]. Fo the dstbuto of plastc stas show Fg., f, * f, C F, ad C F ae ow calculated as f H[ b / ] H[ + b / ], * f / [ b / / ]{ H[ b / ] H[ + b + b / { H[ + b / ] H }:, wth the Heavsde ut fucto H *, ad C C F ε, / ]} CR ω R + CT T + Cε, ε, CF CRωR + CT T + Cε, ε,, 4 b + /[ + + ], C /[ + + ]:, ν ν ν ε, b ν ν ν whee the deftos of C R, C R, C T, ad C T ae the same as those peseted the pevous study [4]..3. Modal Aalyss of Flexual Vbato Modal aalyss fo the flexual vbato that s affected by the -plae foces s peseted. The pocedues of modal aalyss fo a dffeet fomula fo ad θθ the pevous study [4] ae also applcable to the peset study ad ae summazed as follows. 3 The deflecto s expessed as w φ expθ expω t, 5 whee φ deotes the mode fucto the adal decto ad, whch s efeed to as the umbe of odal dametes, deotes the umbe of susodal waves the ccumfeetal decto. By substtutg Eq. 5 to the fudametal equato fo deflecto ad usg the bouday codtos stated Subecto., we obta the goveg equatos fo mode fucto φ. The Gale method [] s used to solve the goveg equatos thus obtaed. The mode fucto s expessed as a sees of up to M tems: M φ φ f, 6 whee the mode fucto fo a plate the absece of -plae esultat foces chose as tal fucto. Thus, fom ou pevous study [4], we have f m f m m C mj ω, m + Cm ω, m + C3mY ω, m + C4mK ω, m m ad θθ m s, 7 whee J * ad Y * deote the Bessel fuctos of the fst ad secod d, espectvely, of ode, ad * ad K * deote the modfed Bessel fuctos of the fst ad secod d,

5 t. J. Pue Appl. Sc. Techol., 4, espectvely, of ode. addto, ω, m deotes the atual fequecy of the plate the absece of -plae foces ad s obtaed as the m -th smallest postve soluto of The deftos of m det[ A ]. 8 m A ad C m,,3,4 ae as show the pevous study [4]. t should be oted that m deotes the umbe of odal ccles. Afte applyg the Gale method, we obta the equato fo mode coeffcet φ m as follows: whee M { Φ } { φ j δ jω, j φ Λ [ M ]{ Φ } ω { Φ }, 9 φ d f d M } T j + θθ df d j f j f d,. Thus, by solvg the egevalue poblem gve by Eq. 9, we obta atual fequeces ω ad egevectos { Φ }, ad hece, egemodes φ though Eq. 6. Hee, ω m deotes the m -th smallest postve soluto of ω. Fom Eqs 9 ad, we fd that the atual fequeces ae depedet o -plae esultat foces ad θθ. -plae esultat foces ad θθ ae tu depedet o ε,, ad b,, as stated Subsecto.. Theefoe, the atual fequeces ae depedet o testes ε, locatos, ad wdths b, of tesog. 3. umecal Calculato: As stated Secto, the stablty of a ccula saw ca geeally be mpoved by ceasg the atual fequecy. Theefoe, ths secto pesets the umecal calculatos used to vestgate the effects of vayg the tesog paametes o the esultg atual fequeces. Moeove, ode to acheve the stable opeato of the saw, the optmzato poblem to maxmze the atual fequecy of the most ctcal mode s solved usg the geetc algothm. The codtos fo the calculatos ae as follows. Posso s ato ad the o-dmesoal e adus ae assumed to be ν. 5 ad. 3, espectvely. The saw s cosdeed to be otatg at a o-dmesoal agula velocty of ω R 3. The tempeatue s cosdeed to be dstbuted ove the oute edge as f T H.95 H, wth o-dmesoal tempeatue T. Wth egad to the atual fequeces, we focus o the modes wth the odal damete m ad odal ccles ~ 5 as the modes that would be ctcal pactce. 3.. Effect of Tesog ove Double Aula Doma Fgue shows the vaatos the atual fequeces wth the tesog locatos. As show the fgue, the vaatos ae depedet o the modes. addto, the most ctcal mode, whch mmzes the atual fequecy wth espect to the modes fo a gve combato of tesog locatos, chages depedg o the combato: fo example, Fg. a, the most ctcal mode exhbts the tasto fom though ad to 3 as ceases.

6 t. J. Pue Appl. Sc. Techol., 4, Fg., vaatos the atual fequecy of the most ctcal mode, m 5 ω, wth tesog locato ae descbed by the coected cuve composed of the cuves wth the lowest value of the odate. vew of ths, Fg. a shows that the atual fequecy of the most ctcal mode eaches the maxmum value m 5 ω 7. 3 at. 77 whle Fg. b shows that the atual fequecy eaches the maxmum value m 5 ω 7. 3 at. 59. Fgues a ad b suggest that, ode to acheve stable opeato of the saw, the combato of tesog paametes should be chose popely. Moeove, t s foud that the dffeet combatos, gve the same maxmum value m 5 ω 7. 3, whch suggests the exstece of multple solutos to the optmzato poblem teated the followg subsectos ω ω a. 6 b. 7 Fg. : Vaatos atual fequeces wth combato of tesog locatos ε 5, b.5; ε 5, b Optmzato Poblem Usg Geetc Algothm ode to acheve stable opeato of the saw, we eed to solve the optmzato poblem that maxmzes the most ctcal atual fequecy wth espect to the tesog paametes. Ths fequecy s also the mmum atual fequecy wth espect to the modes. The poblem s descbed as follows: whee Maxmze f ε,, b ; ε,, b m 5 ω subject to L ε U, L U, L b ε ε b U b,, L d ad U d d ε,, b;, deote the lowe ad uppe boudaes of desg vaables ε,, ad b, espectvely. Because the most ctcal mode depeds o the tesog paametes, as llustated Fg., vaous modes should be evewed obsevg the atual fequeces. Moeove, because the atual fequeces ae obtaed umecally usg Eqs -4 ad Eqs 7- owg to the absece of explct fomulas, a cosdeable computatoal cost s equed to calculate the fequeces fo a gve set of tesog paametes. Fgues a ad b, whch show the depedece of the

7 t. J. Pue Appl. Sc. Techol., 4, fequecy o oly tesog locato, ae obtaed by % specto,.e., by epeatg the abovemetoed method fo as may tesog locatos as possble. Howeve, t s almost mpossble to vestgate the fequecy depedece o all sx tesog paametes testes ε ad ε, locatos ad, ad wdths b ad b usg ths method. Howeve, whe a oday olea pogammg method s employed fo the optmzato poblem to avod the computato costs of % specto, thee s a potetal fo the method s esults to fall to a local soluto: fo example, Fg. a, the local maxmum of m 5 ω ca be foud ot oly at. 77 but also at. 35. ode to ovecome these dffcultes, we employed the geetc algothm fo the optmzato poblem. ths techque, the values of ε,, ad b, ae teated as a stg of bay bts, whch s efeed to as a chomosome. By deotg the dgt legths fo the bay bts fo ε,, ad b as ε,, ad b, espectvely, chomosome c s expessed as c { b b ε ε, b, b ε ε, Λ, b, Λ, b ε ε ε ε b b, b, b, Λ, b, Λ, b b b b, b b b, b, Λ, b b b b, Λ, b b b, } ε ε ε b b whee b, b, Λ, b }, b, b, Λ, b }, ad b b, b, Λ, b } ae the bay expessos of { ε { { ε,, ad b, espectvely, ad ae elated to the coespodg decmal values as ε L b L ε b Uε + U + b L ε L b ε b ε ε ε l bl, L + bl l b b b, b l l l t should be oted that desg vaables ε,, ad b ca tae espectvely. U b L l ε, l, ad The pocedue of the geetc algothm s as follows. Let us cosde a goup composed of,. 3 b vaatos, populato chomosomes C { c, c, Λ, c }, each elemet of whch s expessed the fomat show by populato Eq., the -th geeato. Hee, j,, Λ, s efeed to as a dvdual ad c j populato populato as the populato of the dvduals. The vaable c j s the tasfomed to decmal desg vaables ε,, ad b, usg Eqs ad 3. addto, the objectve fucto defed by Eq., f ε,, b ; ε,,, s evaluated by the method stated Secto as b [ ε c, c, b c ; ε c, c b c ], 4 Fj f j j j j j, whch s efeed to as ftess geetc algothm tems. The goup composed of j populato chomosomes the + -th geeato, C { c, c, Λ, c }, s geeated by a sees of + populato geetc opeatos, that s, selecto, cossove, ad mutato, ths ode. Fst, the tal goup of

8 t. J. Pue Appl. Sc. Techol., 4, chomosomes, C { c, c, Λ, }, s adomly geeated usg a ufom adom umbe c populato geeato. ext, the selecto opeato, elte pesevato ad oulette selecto ae pefomed: the dvduals wth the hghest ftess values the -th geeato ae peseved as elte; the elte emag dvduals ae selected fom all the dvduals the -th geeato populato elte accodace wth a pobablty that s popotoal to the ftess of each dvdual. The goup composed of chomosomes that s obtaed ths mae the udegoes the cossove populato opeato: elte dvduals peseved as elte ae ept uchaged; fom the emag selected dvduals, some pas of dvduals ae selected wth pobablty populato elte p cossove, ad, amog each selected pa, a stg of bay bts s exchaged potos that ae adomly selected by usg a ufom adom umbe geeato. The goup that udewet the cossove opeato s the subjected to mutato: all bts of all the chomosomes othe tha elte chomosomes ae depedetly subjected to a exchage of ad wth a slght pobablty of p. Ths method ceases oveall ftess ad geeates C { c, c, Λ, c }. These mutato pocedues ae the teated fo each geeato Vefcato of Geetc Algothm + populato ode to vefy whethe the geetc algothm fuctos popely, we fst solved the optmzato poblem descbed by Eq. oly wth espect to tesog testy, by settg L U 5, L U. 6, L U. 5, L U 5, ad L U. 5. ε ε The selected paametes ae b b ε ε b b L.35, U.975, ; populato 3, elte ; pcossove.7, pmutato.5.5 Fgue 3 shows the vaatos the dvduals dstbutos wth the geeatos, c j j,,3 deoted by 3 dots, some of whch may ovelap, fo each geeato. Ths fgue also shows the maxmum ad aveage ftess values, defed espectvely by F max max Fj, Favg populato j,, Λ, populato populato j F j. 6 Fgue 3 shows that the dvduals, tally dstbuted at adom the age L U, gathe at.766, wheeas ceases ad coveges o F 7.3 afte may max geeatos. Ths behavo agees wth the esult show Fg. a, whch f 5,.6,.5;5,,.5 m 5 ω 7.3 at 77.. Ths ageemet shows that the geetc algothm succeeds detemg the soluto fo the optmzato poblem descbed by Eq.. Moeove, the geetc algothm s seach fo a soluto s pefomed at a much lowe computatoal cost tha that of % specto. Although the soluto s obtaed fom 4 caddates of, as show Eqs 3 ad 5, the calculato the umbe of dvduals 3 multpled by the umbe of geeatos 3 to obta Fg. 3 s pefomed oly 9 tmes. Such a savg the computatoal cost s sgfcatly advatageous fo the optmzato poblem wth espect to multple desg vaables, as dscussed subsequet subsectos. Afte the covegece o.766 s substatally accomplshed, as show Fg. 3, outles sometmes appea as a esult of the mutato opeato. Because of such outles, the geetc algothm s also

9 t. J. Pue Appl. Sc. Techol., 4, expected to be applcable to a optmzato poblem whch the ftess fucto has multple local maxma Fg. 3: Vaatos dvduals ad ftess wth geeato ε 5,.6, b.5; ε 5, b Optmzato wth Combato of Tesog Locatos ext, the optmzato poblem descbed by Eq. s solved wth espect to the combato of tesog locatos ad, by settg L ε U ε 5, L b U b. 5, L ε U ε 5, ad L U. 5. The selected paametes ae b b L.35, U populato,.975, elte ; p ; L cossove.7, p.35, U mutato.5.975, ;. 7 a b

10 t. J. Pue Appl. Sc. Techol., 4, c d 3, ad coespodg ftess vaous geeatos ε 5, b.5; ε 5, b.5 Fg. 4: Dstbutos of dvduals Fgue 4 shows the dstbutos of dvduals' c, c deoted by dots fo each fgue j ad the coespodg ftess F j vaous geeatos. Fgue 5 shows the vaatos the maxmum ad aveage ftess values defed by Eq. 6 wth geeatos. Fgue 4 shows that the dvduals', that ae tally dstbuted at adom the age of {, L U, L U } gathe the doma of {,.6.7,.7.8}. addto,, the lagest legth of the dowwad les each fgue, ceases wth the umbe of geeatos. By spectg Fgs 4 ad 5, the soluto to the poblem s foud to be,.69,.7463 whch gves f 5,,.5;5,, The soluto s obtaed fom caddates of,, as show Eqs 3 ad 7; howeve, the calculato the umbe of dvduals multpled by the umbe of geeatos 3 to obta Fg. 5 s pefomed oly 6 tmes. Thus, the computatoal cost s educed to /748 6/ of the cost of % specto, whch shows a eve geate advatage fo a soluto seach usg the geetc algothm. j Fg. 5: Vaatos maxmum ad aveage ftess values wth geeato ε 5, b.5; ε 5, b.5

11 t. J. Pue Appl. Sc. Techol., 4, Optmzato wth Combato of Tesog testes ext, the optmzato poblem descbed by Eq. s solved wth espect to the combato of tesog testes ε ad ε, by settg L U. 6, L b U b. 5, L U. 7, ad L U.5. The selected paametes ae b b L ε, U populato ε, 5, elte ε ; p ; L cossove ε, U.7, p ε mutato,.5 ε ;. 8 Fgue 6 shows the dstbutos of dvduals' ε c, ε c deoted by 5 dots fo each fgue j ad the coespodg ftess F j vaous geeatos. Fgue 7 shows the vaatos the maxmum ad aveage ftess values defed by Eq. 6 wth geeatos. Fgue 6 shows that the dvduals' ε, ε that ae tally dstbuted at adom the age of { ε, ε Lε ε Uε, Lε ε Uε } gathe the doma of { ε, ε ε 3,4 ε 5}. addto,, the lagest legth of the dowwad les each fgue, ceases wth the umbe of geeatos. By spectg Fgs 6 ad 7, the soluto to the poblem s foud to be ε, ε 7.7, 437. whch gves f ε,.6,.5; ε,.7, The soluto s obtaed fom caddates of ε, ε, as show Eqs 3 ad 8; howeve, the calculato the umbe of dvduals 5 multpled by the umbe of geeatos 5 to obta Fg. 7 s pefomed oly 9 tmes. Thus, the computatoal cost s educed to / 494 5/ of the cost of % specto, whch also shows a eve geate advatage fo a soluto seach usg the geetc algothm. j ε ε ε ε a b

12 t. J. Pue Appl. Sc. Techol., 4, ε ε ε ε c 3 d 5 ε,ε ad coespodg ftess vaous geeatos.6, b.5;.7, b.5 Fg. 6: Dstbutos of dvduals Fg. 7: Vaatos maxmum ad aveage ftess values wth geeato.6, b.5;.7, b Optmzato wth Combato of Tesog testes ad Locatos the pevous subsectos, the optmzato poblems wth espect to the combatos of paametes, amely, ad ε, ε, wee solved. Because the atual fequecy of the most ctcal mode s depedet o 6 paametes ε,, b, ε,, b as descbed by Eq., such a hghe dmesoal optmzato poblem s solved ths subsecto. Oce the olles used fo tesog ae toduced a tesog ste, t s had to modfy the shape of olles,.e., tesog wdths b ad b. pactce, the vaable paametes ae the testes ad locatos ε,, ε,. Theefoe, the optmzato poblem descbed by Eq. s solved wth espect to the combato of ε,, ε,, by settg L U. 5 ad L U. 5.The selected paametes ae b b b b

13 t. J. Pue Appl. Sc. Techol., 4, L L ε ε, U, U populato ε ε,, 5, elte ε ε ; p ; L ; L cossove.35, U.35, U.7, p mutato.975,.975,.5 ; ;. 9 Fgue 8 shows the vaatos the maxmum ad aveage ftess values defed by Eq. 6 fom the th though to 6th geeato, whch suffcet covegece of the maxmum ftess value o 6 F max 7.46 s acheved. The combato of paametes to gve the coveged maxmum ftess value s foud to be ε,, ε, 54.,.7393, 46.4,.553. The soluto s obtaed fom. caddates of ε,, ε,, as show Eqs 3 ad 9; howeve, the calculato the umbe of dvduals 5 multpled by the umbe of geeatos 6 to obta Fg. 8 s pefomed oly 8 tmes. Thus, the computatoal cost s educed to 6 /37.4 8/ of the cost of % specto, whch stogly cofms the advatage of seachg fo a soluto usg the geetc algothm Fg. 8: Vaatos maxmum ad aveage ftess values wth geeato b b.5 4. Coclusos: ths study, we used a geetc algothm to solve the optmzato poblem fo the tesog paametes a otatg ccula saw ude a themal load that s tesoed ove a double aula doma. We fst peseted a aalytcal model whch otato, local tempeatue due to fcto, ad -plae plastc sta owg to tesog wee cosdeed. ext, we obtaed the aalytcal soluto fo the -plae foces ad caed out modal aalyss fo the flexual vbato. We the appled the geetc algothm to the optmzato poblem fo stable opeato of the saw to maxmze the atual fequecy of the most ctcal mode. tally, the algothm was appled oly to a optmzato poblem whee oe of the tesog locatos was vaed, ad t was foud that the geetc algothm ot oly fuctoed popely but also equed much lowe computg costs tha % specto. ext, the geetc algothm was appled to a optmzato poblem wth two tesog locatos o two tesog testes. Lastly, the algothm was appled to a eve hghe dmesoal optmzato poblem, that s, a poblem wth two tesog locatos ad two tesog testes. Optmal tesog paametes wee obtaed both the two-vaable ad fou-vaable cases; ths esulted computatoal costs that wee cosdeably lowe tha those equed fo % specto.

14 t. J. Pue Appl. Sc. Techol., 4, Refeeces [] H.. Aafat, A.H. ayfeh ad W. Fas, atual fequeces of heated aula ad ccula plates, t. J. Solds. Stuct., 44, [] C.A.J. Fletche, Computatoal Gale Method, Spge-Velag, ew Yo, 984. [3] D.E. Goldbeg, Geetc Algothms, Addso-Wesley, Massachusetts, 989. [4] M. shhaa, Y. Ootao ad. oda, Aalyss of dyamc chaactestcs of a otatg, themally loaded ccula saw subjected to tesog ove a double aula doma, J. Sold. Mech. Mate. Eg., 4, [5] M. shhaa,. oda ad Y. Ootao, Aalyss of dyamc chaactestcs of otatg ccula saw subjected to themal loadg ad tesog, J. Them. Stess., 33, [6] S. Kmua, Studes o tesog of ccula saw by ollg pessue : Tempeatue dstbuto a otatg dsc whe the themal buclg of the dsc has tae place Japaese, Mouza Gaash, 976, [7] S. Kmua ad M. Ado, Studes o tesog of ccula saw by ollg pessue Japaese, Mouza Gaash, 974, [8] C.D.J. Mote ad S. Holoye, Cofmato of the ctcal speed stablty theoy fo symmetcal ccula saws, Tas. ASME, Se B, J. Eg. d., 97975, -8. [9] C.D.J. Mote ad L.T. eh, Cotol of ccula-ds stablty wth membae stesses, Exp. Mech., 97, [] G.S. Schaje ad C.D.J. Mote, Aalyss of optmal oll tesog fo ccula saw stablty, Wood Fbe Sc., 6984, [] R. Szyma ad J. Rhemev, Latest developmets ccula saw tesog, Fo. Pod. J., 34984, [] R. Szyma ad C.D.J. Mote, Theoetcal ad expemetal aalyss of ccula saw tesog, Wood Sc. Techog., 3979, -37. [3] R. Szyma ad C.D.J. Mote, Ccula saw stffess as a measue of teso, Fo. Pod. J., 7977, 8-3. [4] R. Szyma ad C.D.J. Mote, A evew of esdual stesses ad tesog ccula saws, Wood Sc. Techog., 8974, 48-6.