Research Article A Vibration Reliability Analysis Method for the Uncertain Space Beam Structure

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1 Hdaw Publshg Corporato Shock ad Vbrato Volume 206 Artcle ID pages Research Artcle A Vbrato Relablty Aalyss Method for the Ucerta Space Beam Structure Yayu Mo Shuxag Guo 2 ad Cheg Tag 2 Aeroautcs ad Astroautcs Egeerg College Ar Force Egeerg Uversty X a Cha 2 SceceCollegeArForceEgeergUverstyX a7005cha Correspodece should be addressed to Yayu Mo; @qqcom Receved September 205; Revsed 23 November 205; Accepted 6 December 205 Academc Edtor: Mguel Neves Copyrght 206 Yayu Mo et al Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese whch permts urestrcted use dstrbuto ad reproducto ay medum provded the orgal work s properly cted Cosderg that ucertaty s heret ad uavodable egeerg practce ad the avalable formato about the ucerta parameters s always ot suffcet the paper tres to carry out the oprobablstc vbrato relablty aalyss so as to avod resoace o ucerta structure wth bouded parameters The put ucerta-but-bouded parameters are treated as terval varables ad a terval model s adopted to descrbe bouded ucertates The a theory of oprobablstc relablty s troduced whch the dmesoless oprobablstc relablty dex ad system relablty dex are defed I order to vestgate the resoace falure wth relablty method the resoace falure domas are stated accordg to the relatoshps betwee the atural frequeces ad the exctato frequeces The the ucerta structure s modeled as a seres system ad a system relablty dex s proposed to evaluate the safety of the structure The paper also takes a frequecy aalyss o the ucerta space beam structure to get the resoace falure modes A frequecy aalyss method based o the mootocty dscrmat of the frequecy sestvty s preseted The a optmzato algorthm s troduced to verfy the valdty of the former frequecy aalyss method Two examples are provded to llustrate the effectveess ad feasblty of the preseted method Itroducto Ucertaty s heret ad uavodable almost all egeerg problems I structural egeerg ucertates may preset structural materals geometrcal propertes theoretcal modelg falure codtos appled loads ad so forth Relablty problems stem rghtly from the exstece of all these ucertates At preset structural egeerg several mathematcal theores are avalable for dealg wth ucertates for example probablty theory fuzzy set ad evdece theory terval aalyss ad covex-set models ad so forth The correspodg structural relablty method ca be developed o the bass of these mathematcal models Cosderg that the method eeds less sample data compared to the tradtoal probablstc relablty method 2 the oprobablstc relablty method based o terval model has become oe of the most wdely used relablty methods academc research ad egeerg the past few years Based o terval arthmetc a oprobablstc measure ad aalyss methodology for structural relablty computato are preseted by Shuxag et al 3 Elshakoff ad Re 4 modeled structural ucertates by both the stochastc ad terval methods to quatfy the ucertates respose quattes A terval stress ad terval stregth terferece model for structural terval relablty aalyss was vestgated by Yag ad Su 5 The vestgato was basedoaparttowhchgaveafull-scaledescrptoof comparso relatos betwee terval stregth ad terval stress A blevel robust optmzato model combg terval exteso of fucto algorthm ad a order relato of terval umber algorthm was preseted to vestgate the effects of ucertates o the objectve ad costrat fuctos for robust optmzato by Su et al 6 A ew method was developed for relablty aalyss of the ucerta structureswthbothradomadtervalvarablesbyjagetal 7 Ad the a ew oprobablstc structural relablty aalyss method based o a multdmesoal parallelepped covex model was developed by Jag et al the lterature 8 The model was costructed usg margal tervals for the varables ad correlato formato betwee ay two

2 2 Shock ad Vbrato varables A oprobablstc robust relablty method for robust cotrol of ucerta structures usg statc output feedback LQR approach was preseted systematcally by Guo ad L 9 Ad a oprobablstc relablty methodology was preseted systematcally by Guo ad Lu the lterature 0 for aalyss ad relablty-based desg optmzato of structures wth ucerta-but-bouded parameters Udoubtedly the terval oprobablstc relablty dea based o set models s creatve ad sgfcat Recetly some scholars utlze relablty method to avod resoace of the structure Zhag et al preseted the probablty of the resoace state of the structure whch was regarded as a seres system Su et al 2 carred out a vbrato relablty aalyss method for avodg the resoace ad studed the effects of radom parameters o vbrato relablty of the resoace structure The relablty methods employed the lterature 2 are both tradtoal probablstc relablty methods Cosderg that the avalable formato about the ucerta parameters may be ot suffcet egeerg practce the terval oprobablstc relablty method should be a good choce for resoace avodg I order to defe the resoace falure modes of the ucerta structure a frequecy aalyss whch ams to get theupperadlowerboudsoftheaturalfrequecess eeded It s also kow as the terval egevalue problem whch has aroused the terest of may scholars Roh 3 studed the geeralzed terval egevalue problem ad derved formulas for the terval egevalue of a symmetrc terval matrx wth a error matrx of rak oe Based o the varace propertes of the characterstc vector etres Def 4 preseted a method to compute terval egevalues for the stadard terval egevalue problem Qu et al 5 exteded Def s method to the geeralzed terval egevalue problem By vewg the devato ampltudes of the mass ad stffess matrces as perturbatos aroud the omal values of the terval matrx par a perturbato method for the soluto of the geeralzed terval egevalues problem has bee developed by Qu et al6elshakoff7proposedaprocedureforfdg the rage of egevalues due to ucerta elastc modulus ad mass desty by usg the upper ad lower stffess ad mass matrces Gao 8 proposed the terval factor method to vestgate the effects of geometrcal ad materal terval ucertates o the atural frequeces ad mode shapes of truss structures Modares et al 9 proved that the presece of ay physcally allowable ucertaty the structural stffess the solutos of two determstc egevalue problems are suffcet to obta the exact bouds of the system s fudametal frequeces wthout resort to ay combatoral soluto procedures Sof et al 20 preseted a effcet procedure for the soluto of the geeralzed terval egevalue problem arsg from vbrato aalyss of lear udamped structures wth ucerta-but-bouded parameters Theamofthspaperstoproposeaeffcetmethodfor thefrequecyaalyssofthespacebeamstructureadtakea terval oprobablstc relablty aalyss of avodg resoace o the ucerta structure The remader of ths paper s orgazed as follows I Secto 2 a seres system vbrato relablty based o the oprobablstc relablty dfferet from the tradtoal probablstc relablty s troduced A aalyss o the falure modes of resoace ucerta structure s carred out Secto 3 ad the system relablty dex of the resoace structure s obtaed I Secto 4 a frequecy aalyss o the space beam structure s carred out ad a effcet frequecy aalyss method s preseted A space frame structure ad a smplfed GARTEUR plae model are used to demostrate the preseted methods Secto 5 ad fally coclusos are gve Secto 6 2 Overvew of the Noprobablstc Relablty Method Based o Iterval Model Iterval model s oe of the most frequetly used models ucertaty aalyss of structures The model descrbes the ucerta-but-bouded parameters as terval varables X = (X X 2 X ) satsfyg X X L XU ( = 2) where X L s the lower boud of X ad X U s the upper boud of t The the terval varables ca be trasformed to stadard oes by the followg ormalzed trasformato: where X =X c +δ X X r =2 () X c = (XU +X L ) 2 X r = (XU X L ) 2 =2 ad δ X deotes the stadard terval varables correspodg to X X c sthemeavalueofx adx c s the devato of t 2MeasureofNoprobablstcRelablty I structural relablty aalyss a lmt-state fucto (LSF) Z = g(x) s usually defed by the performace or falure mode of the structure g(x) s usually a fucto of the put varables X = (X X 2 X )wthg(x) = 0 beg the lmt-state surface (LSS) whch separates the varable space to two domas: the falure doma Ω f ={X g(x) <0}ad the safe doma Ω s ={X g(x) >0} I the stuatos that all the varables X = (X X 2 X ) volved the relablty problem uder cosderato ca be bouded certa tervals ad represeted by terval varables as doe above all the terval varables aretrasformedtothestadardtervalvarablesδ X = (δ X δ X2 δ X ) by meas of trasformato () The the LSF Z = g(x) ca be trasformed to the followg ormalzed form 0: (2) Z=g(X) =G(δ X ) (3) ad the relablty dex ca be defed as follows 3: η=m { δ X G(δ X) =0} (4) where deotes the fty orm of vector

3 Shock ad Vbrato 3 δ Safe doma δ Falure doma G(δ X ) <0 G 3 (δ X ) <0 G 2 (δ X ) <0 Safe doma η sys Falure doma G (δ X ) <0 η= η> o δ 2 Support doma o Support doma η< δ 2 Fgure 2: The oprobablstc relablty of a structure system Fgure : The oprobablstc relablty based o two-dmesoal terval model Itcabeseethattheoprobablstcrelabltydexη (4) represets the mmum dstace whch s measured by the fty orm from the org to the LSS (or to the falure doma) the fte topologcal space bult by all the stadard terval varables The relablty dex η s llustrated schematcally by the two-dmesoal case Fgure Whe η>thestructurekeepssafety;wheη= the structure s the crtcal state; ad whe 0<η<part of the support doma s the falure doma the structure s cosdered to be usafe for that t ca t complete expected fucto covcgly The case whe η 0s ot cluded the defto; t wll be meagless from the pot of vew of the relablty 22 Noprobablstc Relablty Idex of a Structure System I relablty aalyss of a structure system multple LSFs whch are defed by falure modes of the structure are usually volved Assumg that there exst m potetal sgfcat falure modes whch eed to be take to accout the correspodg LSFs are represeted by Z k = g k (X) (k = 2m) where Z k < 0 dcates the kth falure mode The structural system s usually regarded as a seres system cosstg of sgfcat falure modes because the occurreceofayoeofthefaluremodesmayleadtothefalure of the structural system Suppose that the put varables X = (X X 2 X ) are depedet terval varables Trasform all the terval varables to the stadard oes δ X =(δ X δ X2 δ X ) by meas of () ad the substtute that to the LSFs The the stadardzed LSFs Z k =G k (δ X ) (k = 2m) ca be obtaed The oprobablstc relablty dex of the structural system ca be gve as follows 3: η sys = m {η k } k=2m (5) where η k s the oprobablstc relablty dex correspodg to the kth LSF The relablty dex η sys of the structure system s llustrated schematcally by the two-dmesoal case Fgure 2 It ca be see from Fgure 2 that η sys s the shortest dstace from the org to the falure surface η sys correspods to the most lkely occurred falure mode So t s reasoable to take η sys (5) as the measure of the structure system 3 Noprobablstc Relablty of the Ucerta Resoace Structure As we all kow resoace occurs whe exctato frequecy s close to the atural frequecy of a vbratg system The sgfcat pheomeo of resoace s that the vbrato ampltude ad eergy sharply crease whch may lead to the falure of the structure The falure s called resoace falure here 3 State Fucto of the Falure Resoace Resoace falure s dfferet from the frst passage problems 2; t s also a mportat falure problem vbrato aalyss Whe resoace occurs the vbrato ampltude creases obvously especallywhethedampofthestructureslowthefalureof resoace has two cases; oe of them s that the ampltude s beyod the threshold ad the other s that the ampltude does ot exceed the threshold due to the exstece of damp I the frst case the structure s surely falure; the secod case the propagato of fatgue crack s much accelerated by the resoace 22; the structure s also regarded as falure here Accordg to the relablty theory the state fucto of falure resoace ca be expressed as g j = P ω j γ =2m; j=2 (6) where P deotes the frequecy of the th exctato ω j deotes the jth atural frequecy of the structure ad γ represets the crtcal spacg betwee atural frequecy ad exctato frequecy Whe the spacg betwee atural frequecy ad exctato frequecy s less tha γthe resoace occurs To the authors kowledge the value of γ s related to the damp of the structurethe value of γ should be reasoably

4 4 Shock ad Vbrato <γ P P <γ ω j ω j o ω jc (a) P c R + o P c (b) ω jc R + P P ω j ω j o ω jc P c R + o P c ω jc R + (c) (d) P P ω j ω j o R + ω jc P c o P c ω jc R + (e) (f) Fgure 3: Relatoshps betwee P ad ω j (a)0 P c ω jc γoverlappg(b) γ P c ω jc 0overlappg(c)P c >ω jc overlappg (d) P c <ω jc overlappg(e)p c >ω jc ot overlappg (f) P c <ω jc ot overlappg defed accordace wth the effect of the ampltude o the structure ad may be obtaed based o prevous experece egeerg practce I ths paper a hypothetcal value 5 Hz s gve to γ 32 Falure Mode of Resoace Ucerta Structure For ucerta structures wth terval parameters the atural frequeces are mplct fuctos of the terval varables So the atural frequeces ω j surely are terval umbers Assumg that the exctato frequeces P are also terval umbers the falure mode ca be expressed as follows accordg to the state fucto (6): g j = P ω j γ<0 (7) where P ad ω j ca be trasformed to stadard oes as follows by meas of (): P =P c +δ P P r ω j =ω jc +δ ωj ω jr =2m; j=2 P c ad ω jc deote the mea value of P ad ω j ; P r ad ω jr deote the devato of P ad ω j respectvely;δ P ad δ ωj are stadard terval umber equal to Substtutg(8) to (7) the expresso of falure mode ca be expressed as G j = P c +δ P P r ω jc δ ωj ω jr γ<0 =2m; j=2 where G j =0deotes the two LSSs (8) (9) The falure modes are dfferet due to the dfferet relatoshps betwee P ad ω j Sx dfferet relatoshps are llustrated schematcally Fgure 3 The falure modes correspodg to dfferet cases Fgure 3 are llustrated Fgure 4 From Fgure 4 t ca be see that the falure domas of resoace are fte strp areas I Fgures 4(a) ad 4(b) the org of the stadard space s the falure doma The structure these two cases s cosdered to be falure for the absolute value of the dfferece betwee the mea value P c ad ω jc s less tha the crtcal spacg γ ad the relablty dex η j s zero I Fgures 4(c) ad 4(d) the falure doma passes through the support doma o ether sde of the org ad the relablty dex η j ca be measured by the mmum dstace (measured by the fty orm) from the org to the falure doma I Fgures 4(e) ad 4(f) the mdcourt le of the falure doma s out of the support doma ad the relablty dex η j ca also be measured by the mmum dstace from the org to the falure doma FromFgure4tcaalsobekowthattherelabltydex η j s the dstace from the org to upper boudary of the falure doma whe the mea value P c >ω jc orthedstace from the org to lower boudary of the falure doma whe themeavaluep c < ω jc The relablty dex η j ca be expressed as follows accordg to (4): η j = m { δ G j =0 δ =(δ P δ ωj )} =2m; j=2 (0)

5 Shock ad Vbrato 5 δ P Safe doma Support doma δ P Safe doma Safe doma δ P Support doma Falure doma o Gj =0 Gj 2 =0 Falure doma δ ωj o Gj =0 Support doma Gj 2 =0 Falure doma δ ωj G j =0 o η j G 2 j =0 δ ωj (a) (b) (c) δ P δ P δ P Safe doma Gj =0 η j o Support doma Gj 2 =0 Falure doma δ ωj Support doma Safe doma o η j δ ωj Falure doma Gj =0 Gj 2 =0 Falure doma Gj =0 Gj 2 =0 Safe doma η j o Support doma δωj (d) (e) (f) Fgure 4: The falure modes of resoace correspodg to dfferet relatoshps betwee P ad ω j (a)0 P c ω jc γoverlappg (b) γ P c ω jc 0overlappg(c)P c >ω jc overlappg(d)p c <ω jc overlappg(e)p c >ω jc ot overlappg (f) P c <ω jc ot overlappg where δ deotes the vector (δ P δ ωj ) ad G j =0deote the LSSs 33 Noprobablstc Relablty of the Ucerta Structure wth Resoace Falure Assumg that the ucerta structure has m atural frequeces wth exctato frequeces loadg o t the whole structure s cosdered to be falure states because the resoace mght occur whe the exctato frequecy s the vcty of ay oe of the atural frequeces Thus the structure system to whch exctato frequeces ad atural frequeces are appled to aalyze falure state s a seres system The system relablty dex s preseted as follows accordg to (5): η sys = m {η j } =2m; j=2 () System relablty dex η sys s the mmum of η j ad t represets the most dagerous case for the structure whe resoace occurs 4 Frequecy Aalyss of Ucerta Space Beam Structure 4 The Raylegh Quotet Based o Fte Elemet Aalyss For a space beam structure wth s degrees of freedom the atural frequecy ca be obtaed by solvg the followg geeralzed Raylegh quotet 23: λ t = φt t Kφ t φ T t Mφ t=2s (2) t where λ t deotes the square of the tth order atural frequecy; s deotes the umber of degrees of freedom; K deotes the global stffess matrx of the structure whch s a s order real symmetrc postve semdefte matrx; M deotes the global mass matrx of the structure whch s a s order real symmetrc postve defte matrx; φ t deotes the dmesoless tth order stadard mode Takg Euler-Beroull beam structure to cosderato space beam elemets wth m odes are obtaed by carryg a fte elemet aalyss o the structure Ad the global stffess matrx ad global mass matrx of the structure ca be expressed as follows: K = = K ele = = + E A K ele = () + G J K ele (4) E I z K ele = (2) + E I y K ele = (3)

6 6 Shock ad Vbrato M = = M ele = = ρ A (M ele () + Mele (2) + Mele (3) )+ = ρ J M ele (4) (3) where K deotes the 6m 6m global stffess matrx; M deotes the 6m 6m global mass matrx; A s the beam crosssecto area of the th elemet; I z s the z axal momet of erta; I y s the y axal momet of erta; J s the polar momet of erta; ad L s the beam elemet legth For sotropc materal G =E /(2(+μ )) holds E s the elastcty modulus of the th elemet; μ s Posso s rato ad ρ s the desty; K ele s the expaded 6m 6m elemet stffess matrx of the th elemet; M ele s the expaded 6m 6m elemet mass matrx of the th elemet; K ele () Kele (2) Kele (3) adkele (4) deote the compoet matrces of K ele adm ele () Mele (2) Mele (3) ad deote the compoet matrces of MeleThesuperscrpt M ele (4) ele stads for the beam elemet wth respect to ay matrx followg t M ele (4) The matrces K ele () Kele (2) Kele (3) Kele (4) Mele () Mele (2) Mele are gve as follows terms of fte elemet method: (3) ad () = L K ele K ele (2) = K ele 6m 6m L 3 L L 3 L L L L 2 L 2 L L 2 L 3 6 L L L L 2 L (4) = L m 6m 6m 6m

7 Shock ad Vbrato 7 K ele (3) = L 3 6 L 2 2 L 3 6 L 2 6 L 2 4 L 6 L 2 2 L 2 L 3 6 L 2 2 L 3 6 L 2 6 L 2 2 L 6 L 2 4 L 0 0 6m 6m (4) M ele (3) = L m 6m M ele () = L m 6m M ele (4) = L m 6m

8 8 Shock ad Vbrato M ele (2) = L m 6m (5) Substtutg K ad M (3) to (2) the square of the tth order atural frequecy s deduced as follows: where λ t = φt t ( = E A K ele () φ T t ( = Mele = + φt t ( = E I z K ele (2) φ T t ( = Mele + φt t ( = E I y K ele (3) φ T t ( = Mele + φt t ( = G J K ele (4) φ T t ( = Mele = E λ ()t + = E λ (2)t + + 2(+μ ) λ (4)t = E = λ ()t = φt t (A K ele () φ T t ( = Mele λ (2)t = φt t (I zk ele (2) φ T t ( = Mele E λ (3)t (6) λ (3)t = φt t (I yk ele (3) φ T t ( = Mele λ (4)t = φt t (J K ele (4) φ T t ( = Mele (7) 42 The Sestvty of Frequecy to Iterval Parameters For ucerta structures some of the parameters (6) are ucerta values Now assumg that the materal parameters E E low μ μ low ρ ρ low are ucerta ad treated as terval varables where E low μ low ad ρ low are the lower bouds ad E up μ up adρ up are the upper bouds of the terval parameters respectvely the a vestgato o the frequecy sestvty of materal parameters s expaded Take the partal dervatves wth respect to the terval varables E μ adρ (6) Cosderg that the error the computed frequecy by the Raylegh quotet s proportoal to the square of the devato of the stadard modes from ther exact values 23 the effect of the modfcato of the stadard mode o the atural frequecy s a secod-order small quatty So that the effect ca be gored whe takg the partal dervatves The the frequecy sestvtes ca be E up μ up obtaed: λ t E =λ ()t +λ (2)t +λ (3)t + λ t E = μ 2(+μ ) 2 λ (4)t λ (4)t 2(+μ ) ρ up

9 Shock ad Vbrato 9 λ t ρ = φt t (A M ele () +A M ele (2) +A M ele (3) +J M ele (4) φ T t ( = Mele = + (E λ ()t +E λ (2)t +E λ (3)t E 2(+μ ) λ (4)t) (8) 43 Frequecy Aalyss Based o Mootocty Dscrmat of the Frequecy Sestvty The terval rages of the atural frequeces ca be obtaed f the frequecy sestvtes are mootoc The mootocty aalyss of the frequecy sestvtes s carred out as follows Cosderg the stuato that the geometrc parameters are certa parameters the A I z I y J adl are all certa postve values Accordg to (6) ad (8) the global mass matrx M s postve defte ad the compoet matrces K ele () Kele (2) Kele (3) Kele (4) Mele () Mele (2) Mele (3) admele (4) are gve (4) ad (5) Based o the kowledge of lear algebra a suffcet ad ecessary codto of postve semdefte for the real symmetrc matrces s that all of the prcpal mors are oegatve Checked by calculatos (4) the ozero prcpal mors of K ele () Kele (2) Kele (3) ad K ele (4) arealloegatveumbers;(5)theozero prcpal mors of M ele () Mele (3) admele (4) are all postve umbers ad the ozero prcpal mors of M ele (2) are all oegatve umbers; therefore the compoet matrces K ele () Kele (2) Kele (3) Kele (4) Mele () Mele (2) Mele (3) admele (4) tur out to be postve semdefte Accordg to the above results of deducg we ca kow that λ ()t λ (2)t λ (3)t adλ (4)t (6) are all oegatve After careful vestgato towards (8) the values of the frequecy sestvty are obtaed where λ t / E s oegatve ad λ t / μ ad λ t / ρ are opostve As the mootocty of frequecy sestvty s obtaed the maxmums ad mmums of the frequeces ca be expressed as follows by utlzg the geeralzed Raylegh quotet (2): λ t max = φt t max K (Eup φ T t max M (ρlow μ low max max λ t m = φt t m K (Elow m φ T t m M (ρup m μ up (9) where φ t max ad φ t m are the tth order stadard modes whe λ t gets the maxmum ad mmum values respectvely λ t gets the maxmum value λ t max whe ρ μ get the lower bouds ρ low μ low ad smultaeously E gets the upper boud E up ; λ t gets the mmum value λ t m whe ρ μ get the upper bouds ρ up μ up ad smultaeously E gets the lower boud E low 44 Optmzato Algorthm to Calculate the Frequecy Iterval For ucerta structure wth bouded parameters the square of the tth order atural frequecy λ t (2) s a certa terval rage The the purpose of frequecy aalyss s to detfy the terval rages of the atural frequeces Geerally ths terval problem could be solved by a optmzato algorthm The optmzato model ca be establshed as follows: m max λ t = φt t K (X φ T t M (X) φ t λ t = φt t K (X φ T t M (X st X ={X j } X j X L j XU j j=2m (20) where X L j deotes the lower boud of X j ad X U j deotes the upper boud of t The global stffess matrx K(X) ad global mass matrx M(X) arefuctosofthetervalvarables vector X By solvg the optmzato problems the upper ad lower bouds of the atural frequeces ca be obtaed Theoretcally a approxmate value of the exact soluto couldbeobtaedvasolvgtheformeroptmzatomodel (20) The optmzato model s applcable to varous ucerta structures so t s utlzed here to verfy the correctess of the former preseted mootocty dscrmat method Partcle Swarm Optmzato (PSO) s a metaheurstc evolutoary optmzato techque whch ca be drectly appled a cotuous global space evromet ad was frst proposed by Keedy ad Eberhart 24 PSO s qute popular the swarm tellgece commuty due to ts smplest algorthmc structure less parameter use ad beg free from gradet use of a objectve fucto The PSO algorthm starts wth radomly talzed populato It works o the socal behavor of partcle to get the best soluto by adjustg each dvdual s posto whch ca respect the global best posto of the whole populato Each dvdual s adjustg by chagg the velocty accordg to ts ow experece ad by observg the posto of the other partcles by use of (2) 25 Cosder V (k+) =ω V (k) +a r (p best x (k))+a 2 r 2 (g best x (k)) x (k+) =x (k) + V (k+) =2N; k=2i (2) where ω s the erta weght 04 09; a ad a 2 are the accelerato factors 0 2 ad r ad r 2 are radom umbers 0 ; N s the populato sze ad I s the stated terato tmes; V (k) deotes the velocty of the th partcle after the kth terato ad x (k) deotes the posto of the th partcle after the kth terato; p best s the best posto

10 0 Shock ad Vbrato Partcle swarm talzato Iput the bouds of terval parameters: Xj L XU j j= 2m k = x (k) = x () (k) = () Calculate K ad M by use of FEM x (k) Iteratvely calculate the veloctes ad postos of the partcles: (k + ) = ω (k) + a r (p best x (k)) + a 2 r 2 (g best x (k)) x (k + ) = x (k) + (k + ) = 2 N; k=2 I K M k=k+ Get the best partcle ad the best prevous posto Yes No k<i Output: g best p best g best Calculate the egevalue by use of the Raylegh quotet Fgure 5: Flowchart of the PSO procedure to calculate rages of the atural frequeces 5 P m m 2 06 m Fgure 6: 2-elemet space frame structure Table : Natural tervals of the space frame structure Frequeces Sestvty mootocty method (Hz) Optmzato algorthm (Hz) ω L ωu ω L 2 ωu ω L 3 ωu ω L 4 ωu ω L 5 ωu ω L 6 ωu of the th partcle ad g best stheglobalbestpostothe populato I ths paper a stadard PSO algorthm s appled to calculate the maxmums ad mmums of the atural frequeces The flowchart of the procedure for realzato of the optmzato process of (20) s depcted Fgure 5 After terato calculato the extremums of the atural frequeces ca be worked out It also ca be see that N I tmes of fte elemet aalyss eed to be accomplshed whch may reduce the effcecy of calculato 5 Examples 5 A Space Frame Structure The space frame structure (show Fgure 6) s cosdered ths example The frame s composed of 2 space beams The sectoal area of each beam s m 2 TheelastctymodulusE Posso s rato μ ad desty ρ are terval varables where E Gpa μ adρ Kg/m 3 The ode 7 s uder trasverse load wth the frequecy P a terval rage Hz Respectvely process the frequecy aalyss o the frame structure by use of the sestvty mootocty algorthm ad the PSO The structure was dvded to 2 beam elemets wth 8 odes The the atural frequecy tervals were worked out The frst 6 order tervals are show Table Accordg to the sestvty mootocty method Secto 4 the atural frequeces get the maxmum values whe the elastcty modulus E s o the upper boud meawhle the Posso rato μ ad desty ρ are o the lower bouds ad the atural frequeces get the mmum values whe the elastcty modulus E s o the lower boud meawhle the Posso rato μ ad desty ρ are o the upper bouds From Table we ca fd that the sestvty mootocty method ad the optmzato algorthm have

11 Shock ad Vbrato Frst-order atural frequecy (Hz) Frst-order atural frequecy (Hz) Elastcty modulus (Pa) 0 μ = 029 ρ = 7760 kg/m 3 μ = 03 ρ = 7860 kg/m 3 μ = 03 ρ = 7960 kg/m 3 (a) Posso s rato E = 20e Pa ρ = 7760 kg/m 3 E = 2e Pa ρ=7860kg/m 3 E = 22e Pa ρ = 7960 kg/m 3 (b) Frst-order atural frequecy (Hz) E = 20e Pa μ = 029 E = 2e Pa μ = 030 E = 22e Pa μ = 03 Desty (kg/m 3 ) (c) Fgure 7: The chage of the frst-order atural frequecy alog wth the varables of the frame (a) The chage alog wth elastcty modulus (b) The chage alog wth Posso s rato (c) The chage alog wth desty the same results Cosderg that the PSO has ce global searchg capablty the result of the PSO s close to the exact soluto So the valdty of the sestvty mootocty method preseted ths paper s verfed I addto the sestvty mootocty method just expereced two tmes of FEM aalyss to obta the result but the optmzato algorthm Secto 44 took 20 5 tmes of FEM aalyss away (20 represets the populato; 5 represets the evoluto tmes) So the sestvty mootocty method has much hgher effcecy The chages of the frst-order atural frequecy alog wth the terval varables of the frame structure are show Fgure7Itcabeseethatthefrequecycreaseswth the creasg of elastcty modulus ad decreases wth the creasg of Posso s rato ad desty whch shows the valdty of the mootocty of frequecy sestvty Already kow are the mmal dstace γ = 5Hz exctato frequecy P Hzadtherageofatural frequeces show Table Trasform the frequeces to stadard oes The the system relablty dex η sys ca be calculated out accordg to (9) (0) ad () The value of the system relablty dex s 633 The dex s the dstace from the org to the lower boudary of the falure doma whch s formed by P ad ω 2 The value of the dex dcates

12 2 Shock ad Vbrato Total odes: 67 Total elemets: 66 (a) (b) Fgure 8: A smplfed GARTEUR plae model (a) GARTEUR plae model (b) Fte elemet model of the GARTEUR plae Table 2: Natural frequecy tervals of GARTEUR model Frequeces Sestvty mootocty method (Hz) Optmzato algorthm (Hz) ω L ωu ω L 2 ωu ω L 3 ωu ω L 4 ωu ω L 5 ωu ω L 6 ωu that the space frame structure s avodg resoace whle the exctato frequecy P s loadg o the structure 52 The GARTEUR Plae Model GARTEUR plae model was developed by the Frech Aerospace Research Isttute 995 The model has the same characterstc wth the actual arcraft such as hgh flexbltes ad modal desty 26 As show Fgure 8(a) the model has a total legth of 5 m ad a wgspa of 20 m the materal s alumum ad there s a dampg layer attached to the wg The desty elastc modulus ad Posso rato of the wg chaged due to the exstece of dampg layer 27 The whole model s smplfed to a space beam model ad there are a total of 67 odesad66beamelemets;therestrctedmodelsshow Fgure 8(b) The elastc modulus of the coecto elemet was set as terval varable E c GPa after examg the smplfcato of the coecto betwee the wg ad the fuselage Cosderg the exstece of the dampg layer the elastc modulus of the wg was set as terval varable E w GPa the desty of the wg as terval varable ρ w Kg/m 3 adposso srato of the wg as terval varable μ w Thegust frequecy loaded o the wg s assumed as P 0 5 Hz By use of the sestvty mootocty method ad the optmzato algorthm the frequecy aalyss was respectvely carred out o the GARTEUR plae model The frequecy tervals were respectvely obtaed by calculatg The frst 6 order tervals are show Table 2 From Table 2 we ca fd that the results of the two methods are early the same Cosderg that the result of the PSO s close to the exact soluto the valdty of the sestvty mootocty method s verfed I addto the sestvty mootocty method oly expereced two tmes of FEM aalyss to obta the results but the optmzato algorthm Secto 44 took 20 0 tmes of FEM aalyss away (20 represets the populato; 0 represets the evoluto tmes) So the sestvty mootoctymethodsmuchmoreeffcet The chages of the frst-order atural frequecy alog wth the terval varables of the GARTEUR plae model are show Fgure 9 It ca be see that the frequecy creases wth the creasg of elastcty modulus of the wg ad decreases wth the creasg of desty of the wg The elastcty modulus of the coecto ad Posso s rato of the wg have lttle mpact o the frst-order atural frequecy It s caused by the specal structure of the coecto ad the wg The aalyss result of Fgure 9 also shows the valdty of the mootocty of frequecy sestvty Already kow are the mmal dstace γ = 5Hz the gust frequecy P 8 4 Hz ad the rage of atural frequeces show Table 2 Trasform the frequeces to stadard oes The the system relablty dex η sys ca be calculated out accordg to (9) (0) ad () The value of the system relablty dex s 423 The dex s the dstace from the org to the upper boudary of the falure doma whch s formed by P ad ω Ths value of the dex dcates that the GARTEUR plae model s avodg resoace whle the gust frequecy s loadg o the wg 6 Cocluso Cosderg the facts that avalable formato about the ucerta parameters s ofte lmted practcal egeerg ad t s relatvely easy to costruct a coservatve terval model a seres system vbrato relablty based o the terval oprobablstc relablty s troduced to avod resoace of the structures wth ucerta-but-bouded parameters ths paper Accordg to the terval theory the falure domas smlar to fte strps are preseted ad the LSSs are obtaed correspodg to the dfferet relatoshp betwee the atural frequeces ad the exctato

13 Shock ad Vbrato 3 Frst-order atural frequecy (Hz) X: e + 0 Y: 442 X: e + 0 Y: 4348 X: e + 0 Y: 4288 X: e + 02 Y: 442 X: e + 02 Y: 4348 X: e + 02 Y: Elastcty modulus of the coecto (Pa) 0 Frst-order atural frequecy (Hz) Elastcty modulus of the wg (Pa) 0 0 E w = 675e0 Pa μ w = 029 ρ w =2700kg/m 3 E w = 68e0 Pa μ w = 03 ρ w = 2800 kg/m 3 E w = 685e0 Pa μ w = 03 ρ w =2900kg/m 3 E c =epa μ w = 029 ρ w =2700kg/m 3 E c = 55e Pa μ w = 03 ρ w = 2800 kg/m 3 E c =e2pa μ w = 03 ρ w =2900kg/m 3 (a) (b) Frst-order atural frequecy (Hz) X: 029 Y: 442 X: 029 Y: 4348 X: 029 Y: 4289 X: 03 Y: 442 X: 03 Y: 4348 X: 03 Y: 4289 Frst-order atural frequecy (Hz) Posso s rato of the wg Desty of the wg (kg/m 3 ) E c =epa E w =675e0 Pa ρ w =2700kg/m 3 E c = 55e Pa E w =68e0 Pa ρ w = 2800 kg/m 3 E c =e2pa E w =685e0 Pa ρ w =2900kg/m 3 E c =epa E w =675e0 Pa μ w = 029 E c = 55e Pa E w =68e0 Pa μ w = 030 E c =e2pa E w =685e0 Pa μ w = 03 (c) (d) Fgure9:Thechageofthefrst-orderaturalfrequecyalogwththevarablesoftheGARTEURmodel(a)Thechagealogwthelastcty modulus of the coecto (b) The chage alog wth elastcty modulus of the wg (c) The chage alog wth Posso s rato of the wg (d) The chage alog wth desty of the wg frequeces Therefore the seres system vbrato relablty dex s obtaed Although ths seres system has the same falure mechasm wth the tradtoal stochastc seres system t has a dfferet system relablty dex due to the dfferet varable types The seres system vbrato relablty dex represets the shortest dstace from the org to the LSSs whch are the boudares of the falure doma It also ca be regarded as a robust relablty dex because the structure s much more robust whe the dex s larger I order to get the falure modes of the ucerta structure a frequecy aalyss s eeded I ths paper the space beam structure wth materal parameters as the terval varables s cosdered as the object A frequecy aalyss method based o the mootocty dscrmat of the frequecy sestvty s preseted Ths method oly eeds two tmes of FEM aalyss to calculate the upper ad lower bouds of the frequeces so t s much more effcet The the global optmzato algorthm PSO s troduced to verfy the valdty of the method At last a space frame structure ad a smplfed GARTEUR plae model are provded to llustrate the valdty ad feasblty of the preseted method

14 4 Shock ad Vbrato Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper Ackowledgmet The support from the Natoal Natural Scece Foudato of Cha (Grat o 57550) s gratefully ackowledged Refereces I Elshakoff Essay o ucertates elastc ad vscoelastc structures: from A M Freudethal s crtcsms to moder covex modelg Computers ad Structuresvol56o6pp Y Be-Ham ad I Elshakoff Covex Models of Ucertaty Appled Mechacs Elsever Scece Publsher Amsterdam The Netherlads G Shuxag L Zhezhou ad F Yuasheg A o-probablstc model of structural relablty based o terval aalyss Chese Joural of Computatoal Mechacs vol8opp (Chese) 4 IElshakoffadYJReFte Elemet Methods for Structures wth Large Stochastc Varatos chapter 7 Oxford Uversty Press Oxford UK J-B Yag ad H-L Su Dscrete method for structural terval relablty aalyss Proceedgs of the Chese Cotrol ad Decso Coferece (CCDC 08) pp Yata Cha July W Su R Dog ad H Xu A ovel o-probablstc approach usg terval aalyss for robust desg optmzato Joural of Mechacal Scece ad Techologyvol23o 2 pp CJagGYLuXHaadLXLu Aewrelabltyaalyss method for ucerta structures wth radom ad terval varables Iteratoal Joural of Mechacs ad Materals Desgvol8o2pp CJagQFZhagXHaadYHQa Ao-probablstc structural relablty aalyss method based o a multdmesoal parallelepped covex model Acta Mechaca vol 225 o 2 pp S-XGuoadYL No-probablstcrelabltymethodad relablty-based optmal LQR desg for vbrato cotrol of structures wth ucerta-but-bouded parameters Acta Mechaca Scavol29o6pp S-X Guo ad Z-Z Lu A o-probablstc robust relablty method for aalyss ad desg optmzato of structures wth ucerta-but-bouded parameters Appled Mathematcal Modellgvol39o7pp Y Zhag Q Lu ad B We Quas-falure aalyss o resoat demolto of radom structural systems AIAA Joural vol 40 o 3 pp CSuYZhagadQZhao Vbratorelabltysestvty aalyss of geeral system wth correlato falure modes Joural of Mechacal Scece ad Techologyvol25o2pp J Roh Egevalues of a symmetrc terval matrx Freburger Itervall-Berchtevol87pp A S Def Advaced Matrx Theory for Scetsts ad Egeers Abacus Press Tubrdge Wells UK 99 5 Z-P Qu S-H Che ad J-X Na The Raylegh quotet method for computg egevalue bouds of vbratoal systems wth terval parameters Acta Mechaca Solda Sca vol 6 o 3 pp Z Qu I Elshakoff ad J H Stares Jr The boud set of possble egevalues of structures wth ucerta but oradom parameters Chaos Soltos & Fractals vol7o pp I Elshakoff Ed Whys ad Hows Ucertaty Modellg: Probablty Fuzzess ad At-Optmzato vol388ofcism Courses ad Lectures Sprger Vea Austra W Gao Iterval atural frequecy ad mode shape aalyss for truss structures wth terval parameters Fte Elemets Aalyss ad Desgvol42o6pp M Modares R L Mulle ad R L Muhaa Natural frequeces of a structure wth bouded ucertaty Joural of Egeerg Mechacsvol32o2pp A Sof G Muscolo ad I Elshakoff Natural frequeces of structures wth terval parameters Joural of Soud ad Vbratovol347pp Y Zhag B We ad Q Lu Frst passage of ucerta sgle degree-of-freedom olear oscllators Computer Methods AppledMechacsadEgeergvol65o 4pp L Wtek Smulato of crack growth the compressor blade subjected to resoat vbrato usg hybrd method Egeerg Falure Aalyssvol49pp WTThomsoadMDDahlehTheory of Vbrato wth Applcato chapter 2 secto Pretce-Hall Eglewood Clffs NJ USA 5th edto J Keedy ad R C Eberhart Partcle swarm optmzato Proceedgs of the 4th IEEE Iteratoal Coferece o Neural Networks pp December HDasAKJeaJNayakBNakadHSBehera Aovel PSO based back propagato learg-mlp (PSO-BP-MLP) for classfcato Computatoal Itellgece Data Mg Volume 2vol32pp46 47Sprger HDeGersemDMoesWDesmetadDVadeptte Iterval ad fuzzy dyamc aalyss of fte elemet models wth superelemets Computers & Structures vol 85 o 5-6 pp M Degeer ad M Hermes Groud vbrato test ad fte elemet aalyss of the GARTEUR SM-AG9 testbed DLR Report IB J08 996

15 Iteratoal Joural of Rotatg Machery Egeerg Joural of Hdaw Publshg Corporato The Scetfc World Joural Hdaw Publshg Corporato Iteratoal Joural of Dstrbuted Sesor Networks Joural of Sesors Hdaw Publshg Corporato Hdaw Publshg Corporato Hdaw Publshg Corporato Joural of Cotrol Scece ad Egeerg Advaces Cvl Egeerg Hdaw Publshg Corporato Hdaw Publshg Corporato Submt your mauscrpts at Joural of Joural of Electrcal ad Computer Egeerg Robotcs Hdaw Publshg Corporato Hdaw Publshg Corporato VLSI Desg Advaces OptoElectrocs Iteratoal Joural of Navgato ad Observato Hdaw Publshg Corporato Hdaw Publshg Corporato Hdaw Publshg Corporato Chemcal Egeerg Hdaw Publshg Corporato Actve ad Passve Electroc Compoets Ateas ad Propagato Hdaw Publshg Corporato Aerospace Egeerg Hdaw Publshg Corporato Hdaw Publshg Corporato Iteratoal Joural of Iteratoal Joural of Iteratoal Joural of Modellg & Smulato Egeerg Hdaw Publshg Corporato Shock ad Vbrato Hdaw Publshg Corporato Advaces Acoustcs ad Vbrato Hdaw Publshg Corporato