Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

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1 695 Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov Tie-Discreizaio Meho Absrac I his paper we prese a ew uerical eho for oliear vibraioal aalysis of Euler-Beroulli beas Our approach is base o he coiuous Galeri-Perov ie iscreizaio eho The Euler-Beroulli bea equaio which govers is vibraios is rasfore io se of oriary iffereial equaios a he presee eho is eploye i orer o ivesigae he vibraioal respose A copariso is ae bewee prese eho a iffere oher ehos available i lieraure I is observe ha he obaie resuls are i srog agreee wih oher resuls i lieraure We coclue ha he prese eho has a grea poeial o eal wih oliear vibraio aalysis probles of beas a relae srucures lie ros a shafs Keywors Noliear vibraio; Euler-Beroulli beas; Tie iscreizaio; Nuerical eho M Sabeel Kha a * Kaeez a a Depare of Applie Maheaics Isiue of Space Techology Islaaba Paisa * uhaasabeel@iseup hp://oiorg/59/ Receive 986 I revise for 7 Accepe 67 Available olie 77 INTRODUCTION I he esig a fabricaio of ay egieerig srucures a achies vibraioal aalysis a yaical respose of srucures lie beas (or ros or shafs) are ipora facor o ivesigae i orer o icrease he perforace of hese srucures For he aalysis of cople egieerig srucures lie briges all builigs vehicle guie-ways huge craes urbies a copressor blaes beas ca be use as siple oel The yaical respose of beas is govere by liear as well as oliear iffereial equaios boh i space a ie To suy heir behavior i is herefore ipora o esig ehos for he uerical soluios of hese parial iffereial equaios I his respec recely soe approiae aalyical echiques (see for isace Barari e al ; Baya e al ; Baghari e al ; Jafari e al ; Baya e al ; Gai ; e 999; e 6; Kha e al ; Liu a Gurra 9; Mirgolbabaei e al ; Sfahai e al ) as well as uerical ehos by Lai e al (8) (see also Bouhalfa e al ; Da Silva

2 696 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov e al 9; Gai e al ) a refereces herei have bee propose o ivesigae he oliear vibraios of beas for esigig a fabricaio purpose The heory of Euler-Beroulli beas which is base o he assupio ha bea curvaure is relae o is beig oe provies a goo eplaaio of log isoropic bar s beig behavior Several ehos have bee propose for obaiig soluio o he oliear equaios of oios goverig he Euler-Beroulli bea s respose Noliear vibraios of he Euler-Beroulli bea has bee ivesigae by Barari e al () usig he Variaioal ieraio a paraeerize perurbaio ehos Niar e al () suie he oliear vibraio respose of Euler-Beroulli bea by approiae aalyical echiques where hey uilize Variaioal e s approach a Laplace ieraio echique i orer o solve he respecive oliear goverig equaios Paar a Baya (3) applie a a-i approach also by e o ivesigae he oliear vibraioal respose of Euler-Beroulli beas subece o a aial loa Noliear behavior of Euler-Beroulli bea wih iffere e coiios has also bee suie by Rafieipour e al () usig Laplace ieraio eho Lise-Poicare echiques for o liear vibraioal aalysis have bee eploye o ouble-clape a siply-suppore beas subece o a aial loa by Ahaia e al (9) Javaar e al (3) use e s Eergy eho i orer o ge soluio of oliear vibraio proble of Euler-Beroulli bea subece o aial loas A siilar suy has bee perfore earlier by Pirboaghi e al (9) bu wih hooopy aalysis eho The oliear equaio of oio for large apliue free vibraioal aalysis of Euler-Beroulli bea resig o variable elasic fouaio was solve by Mirzabeigy a Maolia (6) wih he applicaio of seco orer hooopy perurbaio eho Moreover he oliear vibraioal respose of Euler- Beroulli beas wih geoeric oliariy a subece o aial loas was copue by Johso e al S () where hey use a iffereial rasfor a wo auiliary paraeer base hooopy aalysis echiques ere i his aricle we prese a ew approach o solve equaios of oios goverig he yaics of Euler-Beroulli clape-clape-bea which is fie fro oe e This aricle is orgaize as follows I Secio II a aheaical fraewor goverig he oliear vibraios of Euler-Berolulli bea is presee I Secio III coiuous Galeri-Perov ie-iscreizaio eho is evelope I Secio IV he evelope eho aferwars applie o he goverig equaios of Euler-Beroulli bea presee i Secio II I Secio V siulaios aa a uerical resuls are iscusse I las Secio coclusios are raw base o he obaie siulaio resuls i Secio V MATEMATICAL FORMULATION Le us cosier a sraigh bea copose of hoogeeous aerial havig legh L place o a elasic fouaio wih oulus of elasiciy as E The bea is eperiecig a aial force F as show i Figure below Le A be he cross-secioal area of bea which is assue o be uifor hroughou is legh Furher suppose ha here is o i plae eforaio ie he plaes of he cross secio reais plae afer eforaio Now by igorig he rasverse oral a shear srais oe ca wrie he equaio of oio of he Euler-Beroulli bea icluig he i-plae srechig effec as follows Lai Aerica Joural of Solis a Srucures (7)

3 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 697 Lai Aerica Joural of Solis a Srucures (7) ) q( L EA K F EI L () where is he coefficie of viscous apig K is he siffess of fouaio a ) q( is he rasverse irecioal loa I he absece of o-coservaive forces above equaio () reuce o L L EA K F EI () I orer o uiesioalize he above equaio we choose he followig variables ~ a ~ ~ ~ ~ EI KL K EI FL F L EI r L X X G (3) where G r is he gyraio raius of he cross-secio of he bea Usig equaio (3) a oiig he ile sig fro he variables equaios () hus ca be wrie as follows K F () Figure : A scheaic represeaio of a Euler-Beroulli bea fie a oe e a subece o aial loa Now by assuig a prouc soluio ) ( ) ( ) ( where ) ( is he bea s eige oe a by applyig he Galeri eho a equaio goverig he bea yaics is obaie as ) ( ) ( ) ( 3 (5) Where a ca be eerie by he relaios ) ( F K iv a

4 698 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov Moreover i orer o coplee he above proble escripio he bea is subece o followig iiial coiios ( ) A () (6) where A is he apliue of oscillaio Equaio (5) alog wih iiial coiios i Equaio (6) govers he oliear vibraioal respose of Euler-Beroulli beas 3 DESCRIPTION OF CONTINOUS GALERKIN-PETROV TIME-DISCRETIZATION METOD I his secio we prese a ie iscreizaio eho o solve oliear equaios goverig he yaics of Euler-Beroulli beas I is easy o see ha Equaio (5) ca be rasfore io a syse of oliear iffereial equaios of he followig for U ( ) f( U()) (7) T Where U ( ) ( ) a f eoes he vecor of uow-sae variables a vecor coprisig o-liear fucios i boh ie a space respecively ere is he firs orer ie erivaive of I orer o evelop a uerical eho o solve such yaical syses here we use he cocep of a Galeri-ype forulaio a ie-iscreizaio Le S be a space of all possible soluios o se of oliear iffereial equaios i Equaio (7) We search a vecor of uow saes U :[ ] S such ha U ( ) f(u()) ( ) U ) U ( where (8) By choosig as a es space he proble i Equaio (8) ca be forulae as follows: Fi U S such ha ( ) U ( ) V T Wih ( ) U V T f U V U (9) Equaio (9) is he wea for of proble i Equaio (8) Now o solve i over he ie ierval I le us iscreize he ierval I io furher sub-iervals wih each sub-ierval I - where belogs o he se N} Now each ierval is havig he followig propery Moreover i also saisfy N Lai Aerica Joural of Solis a Srucures (7)

5 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 699 {} for all N I orer o copue he vecor of sae variables U () a ie ierval I we choose Ρ Ι as a se of basis fucios ere Ρ Ι represes he -h egree polyoials space efie o ierval I by Ρ Ι S : U : Ι S ; U u for all Ι u S () The vecor of sae variables U() i Equaio (5) is hus approiae by U U ( ) for all I u () Where u are vecor elees i ilber space S Global vecor of sae variables U ( ): Ι S is i he iscree-soluio space S S where S wih Ι Ι CΙ S : U Ρ Ι S ;for all : U N The sybol σ above represes a iscreizaio paraeer The es fucio V() is ae fro iscree es space Ι where V L Ι S : V Ρ Ι S ; N : Ι - By eoig V σ () as a iscree approiaio of he es fucio V() he he wea proble i Equaio (9) aes he followig Variaioal for which saes: Fi U o S such ha Above U s : T T V U V f U ; V () So S S wih zero as iiial coiio Defie a polyoial Ι eoes he subspace of i Ρ - o he ie ierval I zero everywhere i he se I ecep a he ie ierval Ī = [ ] a v S as a arbirary scalar fucio he he variaioal proble o ie ierval I i Equaio () reas Ι v T T U v f U ; vs i i (3) Ι Or equivalely we ca wrie by usig Equaio () Lai Aerica Joural of Solis a Srucures (7)

6 7 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov I v T i T u v f u ; v S i I () To efie Basis fucios i Equaio () a appig ie ierval I [ ] I here eiss Ι such ha M I Ι : is use which aps he fro aural ie ierval I o physical ie ierval Ι Now for every M ˆ : - ˆ Ι ; N The basis fucios efie o physical ie ierval Ι ca ow be calculae i aural ie ierval as Where Ρ Ι ˆ a ˆ Ιˆ i Ρ - M ; M ; i - : : ˆ i i are saisfyig he propery ˆ Deoe he Kroecer s ela To ge a seco-orer approiaio of iscree vecor of sae variables () U we require o fi he coefficies u ; (5) By usig propery of basis fucio fro Equaio (5) a wih he applicaio of iiial coiio he coefficies u ca be eerie fro he followig U u - u (6) Sice i is easy o eal wih aural-ie ierval Î hus Equaio () is recosruce io where - T T ˆ f M ˆ ˆ ˆ ˆ ˆ i v u i ; S v u v ˆ (7) i a ˆ ˆ ˆ ˆ ˆ i : i ˆ (8) ˆ Now wih he applicaio of -poi Gauss-Lobao forula i Equaio (8) we wrie Lai Aerica Joural of Solis a Srucures (7)

7 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 7 l are roos of he Le- Where l gere polyoial T T ŵ f M ˆ ˆ ˆ ˆ ˆ i v u l l l i l ; S v u v l (9) ŵ represes he weighs - ˆ a ˆ Ιˆ; l Ρ - Usig he abbreviaio ˆ ˆ : ŵ ˆ ˆ a : ˆ ˆ : g l i l () l l l l l ece we fially arrive a he followig syse of couple equaios æ æ ö ö = f ; Î S i = - a = () çè è øø T s T ( ) åai v u åb il l h v i = ç åu v l= ç = wih iiial coiios as escribe i Equaio (6) 3 Copuaio of Paraeers i i a l To copue uow paraeers i Equaio () le us selec he followig se of es fucios 3 ˆ ˆ ˆ ˆ ˆ ˆ i i for all i ˆ Where each ˆ Ρ Ι Ρ Ι ˆ ˆ ˆ a ˆ - ˆ a ˆ ˆi - es fucios i i a l are calculae a are give as uer 3 i for all i } i APPLICATION TO GOVERNING EQUATIONS OF EULER-BERNOULLI BEAMS By his choice of The eho presee i Secio 3 is applie o oio equaio i Equaio (5) a he obaie iscree se of equaios are solve usig uerical algorih oulie as uer Nuerical Algorih: Sep : Iiialize he ow paraeers a Sep : Choose a ie sep size Lai Aerica Joural of Solis a Srucures (7)

8 7 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov Lai Aerica Joural of Solis a Srucures (7) Sep 3: Se he iiial coiios a ie as A Sep : Firs solve he followig equaios for uows Sep 5: Usig he copue values for a fro Sep Solve he followig oliear equaios for a Sep 6: Use he copue values of a fro Sep 5 o ge he uerical values of a fro Sep by bac subsiuio Sep 7: Copue he soluio a ie sep by equaio () Sep 8: Upae he soluio for he e ie ieraio by Sep 9: Upae he ie sep Sep : Go o Sep 5 RESULTS AND DISCUSSIONS I his Secio oliear vibraioal respose of a clape-clape Euler-Beroulli bea fie a oe e is presee by a ew uerical eho The resuls copue by he presee eho are copare wih he oher eisig ehos i lieraure (for isace wih he resuls of Barai e al Niar e al a Johso e al ) For he uerical copuaios a ie sep size of 5 is chose for he iscreizaio of ie oai by he prese eho The oiesioal eflecio ) ( is calculae by he presee eho a he ceer of he bea a is show i Figure wih he variaio of o-iesioal aiu apliue of oscillaio over a wie rage of ie

9 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 73 Figure : No-iesioal eflecio wih varyig values of apliue A a wih paraeric values I Table a Table a copariso is ae bewee he presee eho a four iffere ehos wih sei-aalyical approach by Barari e al () wih a sei-aalyical eho by Niar e al () wih sei-aalyical approach by Johso e al () a wih uerical eho RK- I is see ha he copue values of eflecio a he ceer of he bea for iffere values of aiu apliue of oscillaio are i srog agreee wih he resuls by hese ehos I Table 3 relaive errors are provie which are copue by he forula (Soluio by Prese Meho - Soluio by Eisig Meho ) % Soluio by Eisig Meho Wherei Error- is calculae by aig RK- soluio as a eisig soluio I Error-II he eisig soluio is ae fro he sei-aalyical approach by Barari e al () Error-III is obaie by aig he eisig soluio for he eho by Niar e al () I all of hese calculaios he errors are show for hree iffere apliue of oscillaio of bea over a wie rage of ie Fro Table 3 i ca be clearly see ha he resuls by prese eho coverges closely o oher resuls i lieraure a are i srog agreee A RK- Barari e al () Niar e al () Prese Table : Copariso of ie archig soluio of equaio (5) wih paraeric values Lai Aerica Joural of Solis a Srucures (7)

10 7 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov A Niar e al() prese Johso e al() prese RK prese Bararie al() prese Table : Differeces bewee he ie archig soluio of equaio (5) by presee eho a iffere oher schees i lieraure The paraeers are chose o be A ie Error-I Error-II Error-III Table 3: Relaive errors bewee he ie archig soluio of equaio (5) by presee eho wih iffere oher schees i lieraure The paraeers are chose o be I Figure 3 ie archig eflecio respose over a wie rage of ie is show where he resul obaie fro prese eho is graphically copare wih he resul copue fro seiaalyical approach by Niar e al () I Figure he ie archig respose of eflecio by prese eho is copare wih he soluio of eflecio respose of ceer of he bea by Barari e al () I ca be observe fro hese wo Figures ha he ceer of he bea shows sae eflecio behavior for a large ie perio as epice by he sei-aalyical approaches i lieraure A paraeric suy is carrie ou o observe ie archig respose of he ceral eflecio of bea I Figure 5 a Figure 6 he ceral eflecio of bea is observe wih varyig he paraeers a The iiial aiu apliue of vibraio was chose o be a a ie sep size of 5 for he calculaio by he presee eho By varyig he paraeers a a icrease i he Lai Aerica Joural of Solis a Srucures (7)

11 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 75 aiu apliue of vibraio is see as he ie arches I Table a copariso is ae bewee he accuulae errors by classical uerical schee RK- a prese eho Also he accuracy of soluio by boh of he uerical ehos is show for wo iffere ie sep sizes I all he copuaios of accuulae errors i Table he referece soluio by Johso e al () is ae io cosieraio I is observe ha our prese uerical eho is ore accurae i preicig he soluio over he classical RK- eho This is highly ue o he fac ha prese eho oes o ivolve ieraive ie iegraio a herefore have less accuulae errors as show i Table Moreover i is observe ha ha he accuulae error ecreases wih a icrease i he ie sep size for boh he classical RK- a prese eho 8 6 Prese Niar e al () η() Figure 3: A copariso of ie hisory of ceral eflecio curve of bea The ie sep use for his siulaio is 5 a he paraeers A 8 Prese Barari e al () 6 η() Figure : A copariso of ie hisory of ceral eflecio curve of bea The ie sep use for his siulaio is 5 a he paraeers A Lai Aerica Joural of Solis a Srucures (7)

12 76 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 5 η() 5 ε = 5 ε = 5 ε = 5 ε = Figure 5: Tie-hisory of ceral-eflecio of bea wih varyig The paraeric value of is chose o be Tie - sep size Nuber of ie seps Accuulae error by RK- Accuracy (i perce) by RK- Accuulae error by prese eho Accuracy (i perce) by prese eho E E E E+ 6973E E E E E E E E E E E E E E E E E E E E E E E- 7387E E E E E E E E-5 356E E E E-5 87E E E+ 3987E E E E+ 6686E E E E E E E E+ 9769E E+ Table : A copariso of accuulae errors a accuracy for RK- a prese eho Lai Aerica Joural of Solis a Srucures (7)

13 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 77 5 λ = 5 η() 5 λ = 5 λ = 5 λ = Figure 6: Tie-hisory of ceral-eflecio of bea wih varyig The paraeric value of is chose o be 6 CONCLUSIONS A ew uerical eho is presee for oliear vibraioal aalysis of Euler-Beroulli beas The o-iesioal ceral eflecio respose of a clape-clape-euler-beroulli bea (fie fro oe e sie) is ivesigae uer iffere iiial vibraio apliues usig his uerical echique A copariso of copue resuls by prese eho is ae wih he resuls by oher sei-aalyical schees i lieraure a he uerical values of ceral eflecio of bea are abulae Relaive errors bewee he prese eho a oher sei-aalyical approaches i lieraure have bee show o observe he covergece of soluio obaie by prese eho Moreover a paraeric suy is carrie ou o ivesigae effec of iffere paraeers o bea s ceral eflecio I is worh eioig ha he obaie resuls by presee eho are i srog agreee wih resuls by sei-aalyical approaches (for isace Barari e al Niar e al a Johso e al ) i lieraure a also wih classical uerical schee RK- I is worhy of oig ha he prese schee oes o ivolve ieraive ie iegraio lie oher ehos i lieraure a hus o associae accuulaive errors Therefore i is capable o provie accurae resuls wih high accuracy a covergece characerisics as ca be see by he presee resuls a hus ca be uilize o copue yaical respose of srucures lie ros a shafs We coclue ha he prese eho has grea poeial i ealig wih oliear vibraioal respose of beas a siilar srucures lie ros a shafs Acowlegee The auhors are graeful o he aoyous reviewers for heir careful reaig a cosrucive suggesios i orer o iprove he qualiy of his research aricle Lai Aerica Joural of Solis a Srucures (7)

14 78 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov Refereces Ahaia MT Moahei M Moeefar (9) Free vibraio aalysis of a oliear bea usig hooopy a oifie Lise-Poicare Mehos Joural of Soli Mechaics 9-36 Bagheri S Niar A Ghaffarzaeh () Suy of oliear vibraio of Euler-Beroulli beas by usig aalyical approiae echiques La A J Solis Sruc : Barari A Gaavi B Ghabari J M Doairry G () Assesse of wo aalyical ehos i solvig he liear a oliear elasic bea eforaio probles Joural of Egieerig Desig a Techology 8():7 5 Barari A Kalii D Ghaii M Doairry G () No-liear vibraio of Euler-Beroulli beas La A J Solis Sruc 8: 39-8 Barari B Kiiaeifar A Doairry G Moghii M () Aalyical evaluaio of bea eforaio proble usig approiae ehos Soglaaari Joural of Sciece a Techology 3(3):7 36 Baya M Paar I Baya M () Aalyical suy o he vibraio frequecies of apere beas La A J Solis Sruc 8():9-6 Baya M Shahii M Barari A Doairry G () O he approiae aalysis of oliear behavior of srucure uer haroic loaig Ieraioal Joural of Physical Scieces 5(7):7 8 Bouhalfa A aoui A () Free vibraio aalysis of a ebare roaig coposie shaf usig he hpversio of he FEM La A J Solis Sruc 7:5 Da Silva J C R Á Bec A T a Rosa E (9) Soluio of he sochasic bea beig proble by Galeri eho a he Asey-Wieer schee La A J Solis Sruc 6():5-7 Gai D D () A sei-aalyical echique for o-liear selig paricle equaio of Moio Joural of yro-eviroe Research 6(): Gai SS Barari B Gai DD () Approiae aalyses of wo ass-sprig syses a buclig of a colu Copuers & Maheaics wih Applicaios 6():88 95 e J (6) Soe asypoic ehos for srogly oliear equaios Ieraioal Joural of Moer Physics B ():-99 e J (999) Variaioal ieraio eho a i of o-liear aalyical echique: Soe eaples Ieraioal Joural of Noliear Mechaics 3: Jafari SS Rashii MM Johso S () Aalyical soluio of he oliear vibraio of euler-beroulli beas via he hooopy aalysis eho a iffereial rasfor eho I J of Appl Mah a Mech (9): 96- Javaar M Baya M Araai A (3) Noliear vibraio of Euler-Beroulli beas resig o liear elasic fouaio Seel Cop Sruc 5(): Jia-She Peg J a Liu ag J () A sei-aalyical eho for oliear vibraio of Euler- Beroulli beas wih geeral bouary coiios Mah Probl Eg Kha Taghipour R Fallahia M Niar A () A ew approach o oifie regularize log wave equaio Neural Copuig a Applicaios oi:7/s Lai su JC Che CK (8) A iovaive eigevalue proble solver for free vibraio of Euler- Beroulli bea by usig he Aoia Decoposiio Meho Copuers a Maheaics wih Applicaios 56: 3-3 Liu Gurra C S (9) The use of e s Variaioal ieraio eho for obaiig he free vibraio of a Euler-Bea bea Maheaical a Copuer Moelig 5:55-55 Mirgolbabaei Barari A Ibse LB Sfahai MG () Nuerical soluio of bouary layer flow a covecio hea rasfer over a fla plae Archives of Civil a Mechaical Egieerig : 5 Lai Aerica Joural of Solis a Srucures (7)

15 MS Kha a Kaeez / Noliear Vibraio Aalysis of Euler-Beroulli Beas by Usig Coiuous Galeri-Perov 79 Mirzabeigy A Maolia R (6) Large apliue free vibraio of aially loae beas resig o variable elasic fouaio Alearia Egieerig Joural 55 7 Paar I Baya M (3) A aalyical suy of oliear vibraios of bucle Euler Beroulli beas Aca Physica Poloica A 3: -5 Rafieipour Tabaabaei SM Abbaspour M () A ovel approiae aalyical eho for oliear vibraio aalysis of Euler Beroulli a Rayleigh Beas o he oliear elasic fouaio Arab J Sci Eg 39: Sfahai MG Barari A Oivar M Gai SS Doairry G () Dyaic respose of ieesible beas by iprove Eergy Balace Meho I Proceeigs of he Isiuio of Mechaical Egieers Par K: Joural of Muli-boy Dyaic 5(): Lai Aerica Joural of Solis a Srucures (7)