Bifurcation & Chaos in Structural Dynamics: A Comparison of the Hilber-Hughes-Taylor-α and the Wang-Atluri (WA) Algorithms

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1 Bfurcaon & Chaos n Srucural Dynamcs: A Comparson of he Hlber-Hughes-Taylor-α and he Wang-Alur (WA) Algorhms Absrac Xuechuan Wang*, Wecheng Pe**, and Saya N. Alur* Texas Tech Unversy, Lubbock, TX, 7945 Cener for Advanced Research n he Engneerng Scences The WA algorhms (Compuaonal Mechancs, Vol. 59, No. 5, pp , 07) are rederved as beng based on opmal-feedback-acceleraed Pcard eraon, wheren he soluon vecors a any me n a fnely large me nerval + are correced by a weghed (wh a marx λ ) negral of he error from o. The 3 WA algorhms are rederved n deal, based on 3 dfferen approxmae soluons o he Euler-Lagrange equaons for λ. The nerval ( + ) n he 3 WA algorhms can be several hundred mes larger han he ncremen ( ) requred n he HHT-α, for he same sably and accuracy. Moreover, he WA algorhms-, 3 do no requre he nverson of he angen sffness marx, as s requred n HHTα. I s found ha WA algorhms-,, 3 (especally WA algorhm-) are far more superor o HHT-α and ode-45 n erms of compuaonal speed, accuracy, and convergence. Keywords: Nonlnear srucural dynamcs; bfurcaon and chaos; HHT-α algorhm; Wang-Alur algorhms * Vsng Scholar from Norhwesern Polyechncal Unversy, and Graduae Suden n Texas Tech Unversy, Correspondng Auhor, Emal: xc.wang@u.edu ** Vsng Scholar, from Behang Unversy * Drecor, Cener for Advanced Research n he Engneerng Scences

2 . Inroducon Nonlneares n he mechancs of srucures could arse from varous sources such as large deformaons, large roaons, nonlneares of he maeral, dampng, and boundary condons, ec. [] They are so common n real lfe ha he nonlnear behavors can be found everywhere rangng from cvl engneerng, auomoble manufacory, arcraf desgn, o roboc dynamcs, and on-orb maneuvers of spacecraf. Wh he developmen of lgher, more flexble and mulfunconal maerals and srucures, he effec of nonlneary wll become more sgnfcan n hese areas. Lnearzaon was once wdely used o smplfy nonlnear problems, bu n he meanme, was also realzed ha he basc prncples of lnear analyss are no vald even n some cases of weak nonlneary. The nonlnear dynamcal sysem s characerzed by some unque and complex phenomena, ncludng bfurcaon, jump phenomena, snap-hrough, and chaos, ha he lnear sysem canno exhb []. For weakly nonlnear problems, he perurbaon mehod has been successfully used o oban asympocally analycal approxmaons of he soluons. However, for srongly nonlnear dynamcal sysems wh complex forms, he soluon can only be obaned usng numercal negraon mehods, n he curren sae of mahemacs. The nvesgaon of srucural vbraon helps o enhance he relably and performance of srucures, as well as reveal poenal falures. Wh proper srucural models, one can use numercal smulaon as an alernave o expermen, and o save a lo of me and expense. In leraure, he mehods for numercally solvng a sysem of nonlnear ordnary dfferenal equaons n srucural dynamcs are roughly dvded no explc and mplc mehods. The fne dfference mehods are a broad class of explc mehods nvolvng cenral dfference mehod, Runge-Kua mehods and Taylor seres expanson schemes [3]. These mehods are very easy o mplemen, bu ofen lack sably n compuaons usng large sze seps. The mplemenaon of an mplc mehod resuls n nonlnear algebrac equaons of unknown saes, whch are usually solved wh Newon-Raphson mehods. For some ypcal mplc mehods, such as he Newmarkβ mehods [4] and he Hlber-Hughes-Taylor-α mehod [5], he sably s guaraneed only for lnear problems. However, for nonlnear problems, he sably s doubful and ofen depends on he specfc problem and he sep sze adoped n he mehod. Wh relavely large sze seps, some mplc mehods could become so unsable and naccurae ha hey are obvously unsued for numercal negraon o fnd he long-erm nonlnear dynamcal behavors [6]. Even hough, n he analyss of vbrang srucures by a fne elemen spaal dscrezaon mehod, for he negraon of he sem-dscree nonlnear me ODEs, schemes such as he cenral dfference, he Newmark and he Hlber-Hughes-Taylor-α (HHT-α ) mehods are sll wdely used n ndusral sofware whou relable proofs showng ha he nonlnear behavors ncludng bfurcaon and chaos can be accuraely predced by hese mehods. Ths paper concerns he phenomena of bfurcaon and chaos n srucural dynamcs, governed by a semdscree sysem of equaons n me, afer a spaal dscrezaon has been carred ou. Analyses of srucural dynamcs problems have been carred ou exensvely n he pas, by usng he Newmark- β and s slgh modfcaon wh added dampng, he Hlber-Hughes-Taylor- α mehods, wheren Taylor seres expansons of he dsplacemen and velocy vecors are used over a very small me sep, around he generc me. On he oher hand, he Wang-Alur algorhms, whch nvolve fnely large me nervals ( + ) were frs nroduced n [7], and appled o orbal mechancs problems n [8]. In orbal mechancs problems, he WA algorhms were proved o be he bes n leraure so far, n erms of compuaonal me, accuracy, and speed of convergence. These algorhms are renerpreed n he presen paper as beng derved from hghly opmal feedback acceleraed Pcard eraon mehods, wheren he soluon vecor conssng of dsplacemens ( x ) and veloces ( x ) a any me, +, s correced by an opmally

3 weghed (wh he weghng marx λ ) negral of he error n he soluon vecor from o. By m assumng orhogonal polynomals for he soluon vecor over ( + ), and usng collocaon pons ( m=,,..., M ) whn ( + ), we use he opmally acceleraed Pcard-eraon from m o, o reduce he sem-dscree nonlnear ODEs o algebrac equaons. We presen 3 approxmaons o solve he Euler-Lagrange equaons for he opmal wegh funcons λ. Thus we presen 3 algorhms denoed as Wang-Alur algorhms-,, 3. In hese algorhms, he me nerval ( + ) can be several hundred mes larger han he used n he HHT-α mehod, for he same sable and accurae performance. Even m m he dsance beween he collocaon pons + whn he nerval + can be several mes larger han he used n he HHT-α mehod. In addon, he WA algorhms- and 3 do no nvolve he nverson of a Jacoban (angen sffness) marx, as s he case n he HHT-α mehod. I s shown ha all he WA algorhms-,, 3 (especally he WA algorhm-) are far more superor o he HHT-α algorhm n compuaonal speed, accuracy, and speed of convergence, n he presenly consdered problems of bfurcaon and chaos n srucural dynamcs. The WA algorhms are also superor o he opmzed ODE45 algorhm n MATLAB, n erms of speed and accuracy. The WA algorhms- may be consdered as a sgnfcan mprovemen o he sae of scence n nonlnear srucural dynamcs. The nonlnear vbraon of a buckled beam s very represenave n nonlnear srucural dynamcs and has araced a number of researchers [, ]. Commonly he dynamcs of a buckled beam can be well modelled by fne elemen, boundary elemen, or meshless mehods. However, s shown n [3] ha he clamped-clamped buckled beam model can also be well dscrezed no a four-mode approxmaon usng he spaal Galerkn mehod. In hs paper, boh he Hlber-Hughes-Taylor-α and Wang-Alur algorhms are used o nvesgae he nonlnear vbraons of a buckled beam whch may exhb bfurcaon, jump phenomena, and chaos. The valdy of he four-mode buckled beam model s frs examned, and he nonlnear dynamcal behavors such as perod doublng process and chaoc moon are hen nvesgaed. The numercal resuls obaned by Wang-Alur mehod are compared wh hose by Hlber-Hughes-Taylorα mehod o evaluae he performance of hese wo mehods. I s shown ha he WA algorhms are far more superor o he HHT-α algorhms n erms of accuracy, compuaonal me, and convergence speed, n problems nvolvng bfurcaon and chaos n nonlnear srucural dynamcs.. An Elucdaon of HHT-α and WA Algorhms In srucural dynamcs modelng, he vbraon of a srucure s ofen descrbed by a sysem of second-order nonlnear ordnary dfferenal equaons afer spaal dscrezaon has already been carred ou, where x s he vecor of generalzed dsplacemens, and x are he acceleraons, M s he mass marx, C s he dampng marx, N s he vecor of nonlnear resorng forces whch may depend on x and he velocy vecor x, and F s he exernal force. Mx + Cx + N( x, x) = F (). () Eq. () can be furher rewren as a sysem of nonlnear frs order ODEs: x x =, () x M Cx M N( x, x) + M F() 3

4 where N denoes he nonlnear erms, x and x represen dsplacemen and velocy vecor of he moon. Eq. () may be consdered as he mxed-form equvalen of Eq. (), wheren x s he dsplacemen vecor and x s he velocy vecor.. HHT-α Algorhm Based on he heory of Taylor seres, he dscrezed dsplacemen x ( ) + (where s nfnesmally small) can be approxmaed [as done by Newmark (4)] by 3 x( + ) = x( ) + x ( ) + x ( ) + x ( ) + 3!, (3) 3 = x( ) + x ( ) + x ( ) + βx ( ) where β. Smlarly, usng Taylor seres expanson, x 6 ( ) + can be approxmaed [as done by Newmark (4)] by where γ. x ( ) x ( ) x ( ) x ( ) = x ( ) + x ( ) + γ x ( ) + = + + +, (5) Snce s very small, x ( ) n he las erm of Eq. (3) can be approxmaed as Subsung Eq. (6) no Eq. (3) leads o x ( ) = [ x ( + ) x ( )]. (6) x ( + ) = x ( ) + x ( ) + x ( ) + β[ x ( + ) x ( )] = x( ) + x ( ) + ( β) x ( ) + βx ( + ) Furher, x ( ) n he las erm of Eq. (5) s approxmaed by Subsung no Eq. (5) leads o. (7) x ( ) = [ x ( + ) x ( )]. (8) x( + ) = x( ) + x ( ) + γ[ x ( + ) x ( )]. (9) = x ( ) + ( γ) x ( ) + γx ( + ) 4

5 From Eq. (), we have x ( + ) = x( + ). (0) x ( + ) = x ( + ) = M Cx( + ) M N[ x( + ), x( + )] + M F( + ) By subsung Eq. (0) no Eqs. (7, 9), a sysem of nonlnear algebrac equaons abou x ( ) + and x ( ) + are obaned as M Cx( + ) M F( + ) x( + ) x( ) x ( ) ( β) x ( ) + β 0 = + M Nx [ ( + ), x( + ) ] M Cx( + ) M F( + ) ( ) ( ) ( γ) x + x x ( ) + γ + = 0 M Nx [ ( + ), x( + )] In HHT- α mehod, he proceedng algorhms of Newmark- β mehod are slghly modfed by replacng x ( ) + and x ( ) + n Eq. (0) wh x ( ) + α and x ( ) + α respecvely, where Therefore, Eq. (0) s modfed as () x( + α ) = x( + ) + α[ x( + ) x( )]. () x( + α ) = x( + ) + α[ x( + ) x( )] x ( + ) = x( + α ) ( α ) x +. (3) ( ) x + = x ( + ) = M Cx( + α ) M N + ( + α ) ( + α ) M F x Subsung Eq. (3) no Eqs. (7, 9) leads o M Cx( + α ) M F( + α ) x( + ) x( ) x ( ) ( β) x ( ) + β 0 = + M Nx [ ( + α ), x( + α ) ] (4) ( + α ) ( + α ) ( ) ( ) ( γ) M Cx M F x + x x ( ) + γ = 0 + M Nx [ ( + α ), x( + α )] α The parameers n HHT-α mehod are oponal. Normally hey are seleced as γ =, and α,0 3. α β = 5

6 Eqs. (, 4) provde a sysem of nonlnear algebrac equaons abou x ( ) + and x ( ) + ha can be solved for usng Newon-Raphson mehod, where he Jacoban marx and s nverson has o be calculaed n each eraon sep. The Newon-Raphson eraon formula for solvng Eq. (4) s wren as where R j HHT j+ j x ( + ) x( + ) j ( ) j+ = j JHHT x ( + ) x( + ) x x x x = M Cx x x x M Cx R j HHT, (5) j j ( + ) ( ) ( ) ( β) ( ) + β j j + M N x( + α ), x( + α ) j j ( ) ( ) ( γ) + ( ) + γ j j + M Nx [ ( α x ( + α ) M F( + α ) ( + α ) M F( + α ) + ), ( + α )] (6) The Jacoban marx of j R wh respec o ( ) j x + and x ( ) + s derved as j HHT J + βm ( + α)[ N x ( + α )] β( + α){ M C+ M [ N x ( + α )]} =. (7) γ ( + α)[ ( + α )] + γ( + α){ + [ ( + α )]} j j j j j HHT j j j j M N x M C M N x. Wang-Alur Algorhms Based on Opmal Error-Feedback Acceleraed Pcard Ieraon Conceps Le u and u be he ral funcons for x and x n a fnely large local me nerval [, + ]. By subsung hem no Eq. (), he error resdual funcon s obaned as R() u u = +, u + M Cu + M N( u, u) M F() [, ]. (8) To opmally correc he approxmae soluon a, a smple mechansm of an opmally weghed feedback of he error s adoped heren, whch has he expresson: u() u() u u ( τ) dτ () = () + λ + u + (, ) () c u u M Cu M N u u M F [, + ],, where he subscrp c on he lef-hand sde ndcaes he correcon. In Eq. (9), λ ( τ ), a marx, s he se of opma weghng funcons for he feedback of he error resdual R (). Eq. (9) ndcaes ha he soluon vecor [ u( ); u ( )] a any me n he nerval + s correced by an opmally weghed error resdual from me o. The eraon formula of he orgnal Pcard eraon mehod [4, 5] can be regarded as a specal case of Eq. (9), where λ ( τ ) s smply seleced as he negave un marx λ( τ ) = I. Alhough he orgnal (9) 6

7 Pcard eraon mehod may converge, s no he mos effcen approach, snce λ( τ ) = I s seleced oo roughly. The dervaon of he opmal λ ( τ ) s, however, as follows. rue rue For u and u n he neghborhood of he rue soluons,.e. u = u +δu, u = u +δu, λ s expeced o be opmally deermned so ha rue u() u () rue () =. (0) u c u () rue rue I means ha he varaon of Eq. (9) should equals o be zero f u = u, u = u, whch leads o: u u u u u u δ = δ + λδ dτ δ = () + λ 0 u + () c u u M Cu M N M F u M Cu M N M F, () Snce u and u n Eq. () are supposed o be he rue soluons, we have Therefore, Eq. () leads o: where u u + = u M Cu + M N M F() 0. () u u u u δ = δ + λδ dτ + u () c u u M Cu + M N M F, (3) u u = ( I + λ) δ + { λ + λj} δ dτ = 0 u u 0 I J = ( ) + ( ). (4) M N u M C M N u In Eq. (3), here are varaons boh nsde and ousde he negral over me, hus he varaons are made o be zero separaely by enforcng he weghng funcon marx λ o sasfy he followng condons: λ() + I = 0, and λ (,) τ + λ(,) τ J() τ = 0, τ [,] (5) rue Obvously, λ s relaed o u, u, whch are supposed o be u and u rue. However, he rue soluons are no known n advance. Consderng ha, we calculaed λ n he presen paper usng approxmaed soluons nsead of he rue soluons n he mplemenaon of he algorhms, whch are labeled as WA algorhms for convenence..3 Large Tme Inerval [, + ] Orhogonal Polynomal Collocaon 7

8 Furher, he followng weak formulaon of Eq. (9) can be esablshed, usng a marx of es funcons v (). u () u () u u v() d () dτ d () = v + + () λ. u u u M Cu + M N( u, u ) M F() + + c In hs formula, v () are es funcons, he same as hose n he classcal weghed resdual mehods. Le v () be a dagonal marx v = dag[ v, v,...], and v be Drac Dela funcon for a group of collocaon m pons, n he fne large me nerval o +. For smplcy, he collocaon pons m are smply denoed as m n he followng pars. hen Eq. (6) becomes (6) v= δ ( m ), m [, + ], m=,,..., M, (7) u( m) u( m) m u u dτ ( ) = + + ( ) m λ u c u m u M Cu + M N M F [, + ], m=,..., M. (8), m Eq. (0) may be nerpreed as he correcon of he error a each collocaon pon m, wh he error resdual beng opmally weghed by λ. By usng a se of orhogonal polynomals Φ = { φ0, φ, φ,...} T as bass funcons, he ral funcons u and u can be consruced as N u = a φ, and u, = a,, φ, (9), e, en, n n= 0 N e en n n= 0 Where u,e and u,e are elemens of u and u respecvely. a, en, and a, en, are coeffcens o be deermned. From Eq. (9), we have and where T [ u ( ), u ( ),..., u ( )] = B [ a, a,..., a ], p =,, (30) T pe, pe, pe, M pe,,0 pe,, pen,, T [ u ( ), u ( ),..., u ( )] = LB [ a, a,..., a ], p =,, (3) T pe, pe, pe, M pe,,0 pe,, pen,, 8

9 B φ0( ) φ( ) φn ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) φ ( ) + 0 N = 0 M M N M ( N ) M, LB φ 0 ( ) φ ( ) φ N ( ) φ ( ) φ ( ) φ ( ) 0 N = φ ( ) φ ( ) φ ( ) + 0 M M N M ( N ) M. (3) Normally, he number of colocaon pons M s seleced as he same as he number of bass funcons N +. I can be found ha he values of u pe, a collocaon pons are relaed wh hose of u pe, by T [ u ( ), u ( ),..., u ( )] = ( LB) B [ u ( ), u ( ),..., u ( )]. (33) T pe, pe, pe, M pe, pe, pe, M u In an analogous way, he values of dτ a collocaon pons pe, m, m =,,..., M can also be obaned from hose of u pe, hrough he followng ransformaon. where M T T upe, dτ upe, dτ upe, dτ upe, upe, upe, M = L BB, (34) [,,..., ] ( ) [ ( ), ( ),..., ( )] L B φ dτ φdτ φndτ φ dτ φdτ φ dτ 0 0 N = φ dτ φ dτ φndτ M M M 0 ( N+ ) M. (35) A schemac presenaon of he large me collocaon s gven n Fg., wheren he me nerval ( + ) can be several hundreds of mes larger han he sep sze of HHT-α algorhm. In each me nerval o +, M collocaon pons are seleced o nerpolae he approxmaed soluon. In hs paper, he wo ends of he me nerval are seleced as he frs and he las collocaon pons,.e. M =, = +. I should be noed ha even he me sep beween wo neghborng collocaon pons can be larger han he me sep used n he HHT algorhm. 9

10 + + Knowns Unknowns Unknowns m x( ) x( ) x( + ) ( ) m x x( ) x( + ) M M + May be he used n he HHT- α Algorhm Fg. Schemac presenaon of large me collocaon n he nerval +.4 Wang-Alur (WA) Algorhms.4.. WA Algorhm- Alhough we have already obaned he erave formula (8) hrough collocaon, he marx of generalzed Lagrange mulplers λ n sll reman a puzzle. The possbly of drecly solvng he consran equaons (5) for λ s unlkely for mos nonlnear cases. Forunaely, here s an approach o bypass hs dlemma. Dfferenang Eq. (9) leads o where R ( τ ) s he resdual funcon: R( τ ) d u() d u() λ () () ( τ) dτ d () = + + d () λ R R, (36) u u c u ( τ ) u ( τ ). (37) = + u ( τ ) M Cu( τ) + M N( τ) M F( τ) Accordng o Eq. (5) and usng he heory of Magnus seres, can be proved [7] ha λ() = I, and λ = J() λ. Subsung hem no Eq. (36), we have d u() u u u () dτ d () = J + c + λ. (38) u M Cu M N M F u M Cu + M N M F 0

11 u u u u Nong ha λ dτ + = u M Cu + M N M F() u c u error-feedback eraon formula (9), Eq. (38) s rewren as accordng o he opmal d u() u u u = () d () J. (39) u c M Cu + M N M F u c u Afer rearrangemen, s rewren as d u() u() u u() () () d () + J = + () J (). (40) u + c u c M Cu M N M F u Eq. (40) can be regarded as equvalen o Eq. (9). By collocang n he local me nerval [, ] algebrac erave formula s obaned as. m m m m J( m) ( ) J m ( ) ( ) ( ) ( ) u m c m + m m m m M Cu M N M F u c + u ( ) u ( ) u ( ) u ( ) + = + ( ) ( ), (4) u =, [, ] where m,,... M m +, and s a fne large me nerval. + Afer rearrangng he sequenal of he collocaon equaons and usng he relaonshp n Eq. (33), Eq. (4) s rewren as where U U U U ( E + J ) = + J, (4) U c M CU + M N M F U = u ( ) u ( )... u ( ) u ( ) u ( )... u ( )... T p p, p, p, M p, p, p, M, an, p =,. (43) The confguraon of he marces E, J, M, C, N, F are provded n Appendx C. I should be noed ha he nal condons are no ncorporaed n Eq. (4). For ha, whou loss of generaly, we usually selec he frs collocaon pon a he nal boundary, of whch he values u ( ) pe, are gven. By dong ha, Eq. (4) becomes overdeermned, hus s necessary o drop excess collocaon equaons a me. Fnally, Eq. (4) s modfed as r r r r U = ( E + J ) E + c + U U U, (44) U U U M CU M N M F The symbol [ ] r denoes he remaned marx afer he ( lm + ) h rows and columns n [ ] are dropped, l = 0,,,.... If [ ] r s a vecor, we jus need o remove he ( lm + ) h rows n [ ]. r

12 A flow dagram s presened n Fg. o llusrae he WA algorhm-. Consan Marces Consruc C, M, E, F Le U p Inalzaon be he sarng guesses: 0 U p Sae Updae U = ( U ) p p c Evaluaon of Varyng Marces The resorng force vecor: N n Eq. (4) he Jacoban marx: J n Eq. (4) Incorporang Inal Condon Drop excess collocaon equaons a n Eq. (44) Correcon Ge he correced soluon ( ) usng Eq. (44) U p c Incremen = + Correcon Norm ε = ( U ) U U p c p p Yes Soppng Creron No < max Sop No Soppng Creron ε δ, where δ s he olerance Yes Fnshed Fg. Flow char overvew of he WA algorhm-

13 .4.. WA Algorhm- Accordng o Eq. (5), he Lagrange mulplers can be approxmaed by Taylor seres as λ T T T, (45) ( τ) = 0 + ( τ ) + ( τ ) +... where T = I, 0 T = J (), formula s obaned as T J, and so on. Subsung no Eq. (8), he correconal = () u( m) u( m) m J() u u [ ( )( τ ) ( τ ) ] dτ ( ) = ( ) m I J u m c u u M Cu + M N M F (46) In mplemenaons, λ ( τ ) s commonly approxmaed by runcaed Taylor seres. The smples could be he zeroh order approxmaon or he frs order approxmaon λ( τ ) = I, (47) λ( τ) = I J ( )( τ ). (48) Hgher order approxmaons are possble, bu hey are rarely used n pracce consderng he compuaonal complexy. Wh he zeroh order approxmaon of λ ( τ ), Eq. (45) becomes u( m) u( m) m u u dτ ( ) = + ( ) m. (49) u c u m u M Cu + M N M F Wh he frs order approxmaon of λ ( τ ), Eq. (45) becomes u( m) u( m) m u u [ ( m)( τ m)] dτ ( ) = + + ( ) m I J. (50) u c u m u M Cu + M N M F =, [, ] where m,,... M nvolvng m and τ, Eq. (50) leads o m c m +, and + s a fne large me nerval. By separang he erms u( m) u( m) m m m ( ) d ( m) m ( ) d ( m) ( ) d ( ) = τ τ τ τ τ τ τ ( ) R + J R J R, (5) u u m Afer some rearrangemens, Eq. (5) can be rewren as U U = HR + JTHR JHTR, (5) U U c 3

14 where H s he ransformaon marx correspondng o negral, and T s he marx relaed wh me. The confguraons of marces H, T, J and R are provded n Appendx C. Fg. 3 shows he flow char of WA algorhm- Consan Marces Consruc H, M, C, E, T, F Le U p Inalzaon be he sarng guesses: 0 U p Sae Updae U = ( U ) p p c Evaluaon of Varyng Marces The Jacoban marx: J n Eq. (5) and he resdual marx: R n Eq. (5) Correcon Ge he correced soluon ( ) usng Eq. (5) U p c Incremen = + ε = Correcon Norm ( U ) U U p c p p Yes Soppng Creron No < max Sop No Soppng Creron ε δ, where δ s he olerance Yes Fnshed Fg. 3 Flow char overvew of he WA algorhm- 4

15 .4.3 WA Algorhm-3 Consderng Eq. (38), f he Lagrange mulpler λ s approxmaed by Taylor seres, we have d u() u u u ( ) [ 0 ( τ )...] dτ d () = J c + T T u M Cu M N F u M Cu + M N M F (53) If λ s smply approxmaed by T 0, Eq. (53) becomes d u() u u u () 0 dτ d () = J + c + T. u M Cu M N M F u M Cu + M N M F (54) Snce T = 0 I, Eq. (54) s furher rewren as d u() u u u () dτ d () = + J + c + (55) u M Cu M N M F u M Cu + M N M F By usng u ( p =, ) as ral funcons and makng collocaons, he WA algorhm-3 s obaned from Eq. (55) as p u ( ) ( ) m u m m u u = ( ) m dτ + J ( ) ( ) ( ) + ( ) m + m m + u m M Cu M N M F u M Cu M N M F c (56) where u ( ) and he negrals can be obaned as s saed n Eqs. (35, 34). Afer rearrangemens, Eq. (56) p m can be wren as U U E = + U M CU c + M N M F JHR. (57) The confguraons of marces E, M, C, N, J, H, and R are provded n Appendx C. Noe ha he nal condons are no guaraneed by Eq. (57), jus lke Eq. (4) n WA algorhm-, hus should be furher modfed as r r d r U U d U E = + JHR E U c M CU + M N M F U c, (58) The symbol [ ] r denoes he remaned marx afer he ( lm + ) h rows and columns n [ ] are dropped, l = 0,,,.... If [ ] r s a vecor, we jus need o remove he ( lm + ) h rows n [ ]. The symbol [ ] d 5

16 denoes he par of [ ] ha were dropped o oban [ ] r. The flow char of hs algorhm s provded n Fg. 4 Consan Marces Consruc H, M, C, E, T, F Le U p Inalzaon be he sarng guesses: 0 U p Sae Updae U = ( U ) p p c Evaluaon of Varyng Marces The resorng force vecor: n Eq. (57) he Jacoban marx: J N n Eq. (57) and he resdual marx: R n Eq. (57) Incorporang Inal Condon Drop excess collocaon equaons a n Eq. (58) Correcon Ge he correced soluon ( ) usng Eq. (58) U p c Incremen = + Correcon Norm ε = ( U ) U U p c p p Yes Soppng Creron No < max Sop No Soppng Creron ε δ, where δ s he olerance Yes Fnshed Fg. 4 Flow char overvew of he WA algorhm-3 6

17 3. Nonlnear Vbraons of a Buckled Beam The governng equaon of a buckled one-dmensonal Bernoull beam s wren n non-dmensonal form as, (59) v w + w + Pw + cw w w dx = F( x)cos Ω 0 BC s: w= w = 0 a x = 0 and x =, where w s he ransverse dsplacemen of he beam, and P s he axal load on he beam. F( x ) s he ransverse dsrbued load on he beam and Ω s he frequency of he appled load F( x ). The overdo denoes he dervave wh respec o me, whle he prme denoes he dervave wh respec o he spaal coordnae x. To solve he precedng paral dfferenal equaon, we frs assume he spaal modes: wx (, ) = w( x) + vx (, ) = w( x) + φ ( xq ) ( ), (60) s s n n n= where w s s he sac posbucklng dsplacemen, vx (,) s he superposed dynamc response around he buckled confguraon. φ ( x ) are he mode shapes of vbraon around he buckled confguraon, and q () n are he ampludes of φ ( x ). n N n The buckled confguraon ws ( x ) can be obaned by frs solvng he sac bucklng problem where he me dervaves and he dynamc load are removed n Eq. (59). + = 0. (6) v w Pw w w dx 0 Mahemacally, here could be varous buckled mode shapes, dependng on he correspondng load. However, n srucural mechancs, he frs buckled mode shape s mosly of mporance, from whch ws ( x ) s obaned as ws ( x) = b( cos π x). (6) The nondmensonal ransverse dsplacemen b a he mdspan of he beam s relaed o he load P va b = 4( P P) π, (63) c where P c s he crcal load correspondng o he frs Euler buckled mode, namely Pc = 4π. Subsung he assumed soluon of wx (,) no he governng equaon and droppng all he nonlnear, dampng, and forcng erms, we have he followng lnear paral dfferenal equaon ha can be ackled usng he lnear vbraon mode heory. 7

18 v 3 v + v + 4π v b π cos πx v sn πxdx = 0 0. (64) By assumng vx (,) = φ() xe ω and subsung no Eq. (64), he mode shape s obaned as φ( x) = φ + φ = ( c sn s x+ c cos s x+ c snh s x+ c cosh s x) + c cos πx, (65) h p where s, ( π 4 π ω ) / = ± + +, and 5 c should sasfy he followng equaon: h ( b π ω ) c = b π φ sn πxdx. (66) Usng he boundary condons and Eq. (66), he mode shapes φ ( x ) can be obaned. The resulng lnear vbraon mode shapes φ ( x ) are used o consruc he soluon vx (,). Usng he n n mul-mode Galerkn dscrezaon, where he weghng funcons are he same as he ral funcons φ ( x ), he paral dfferenal equaon (59) s hen reduced o a sysem of coupled Duffng equaons. N N n ωn n n nj j njk j k n, j, j, k q + q = cq + b A qq + B qq q + f cos Ω, n=,,..., N. (67) In hs paper, four modes are reaned n he reduced model. The bucklng level s seleced as b = 4. Correspondngly he naural frequences of he four lnear vbraon modes are obaned as ω = , ω = , ω 3 = 08.33, and ω 4 = The parameers A nj, and B njk are provded n Appendx A and B. 4. Numercal Resuls and Dscusson n Consderng he dscrezed nonlnear sysem (67), several ypes of nonlnear resonances of he buckled beam may occur due o he exernal harmonc excaon. The prmary resonance s mosly observed when he excaon frequency Ω s close o one of he mode frequences ω, whch ofen leads o a perodc moon of large amplude n ha mode. For he exsence of quadrac nonlneares and cubc nonlneares, he subharmonc resonances and superharmonc resonances may also occur for Ω ha has an neger relaonshp o ω. n In he followng, he resonance of he buckled beam under harmonc excaons s nvesgaed. The frequency Ω s seleced close o he naural frequency of he frs vbraon mode ω. The exernal force s supposed o be unform over he lengh of he beam, hus F( x ) n Eq. (59) s consan. Through Galerkn dscrezaon, he forces mposed on he four lnear vbraon modes are obaned as f = F, f =, f3 = F, and f 4 = 0. 0 In he numercal smulaons, he force-sweep as well as he frequency-sweep processes are used o oban an overvew of he nonlnear dynamcal behavors of he buckled beam when subjeced o a prmary n 8

19 resonance excaon of s frs vbraon mode. Consderng ha he magnudes of he coeffcens ba nj, and B njk are roughly beween force F s se o vary beween and ω n, 4 0 n he resorng forces, he amplude of he exernal F = and F = 600 n he force-sweep process. In he frequency-sweep process, F s fxed a F = 400, whle he frequency Ω s vared beween Ω= 30 and Ω= 8. There are abundan ypes of bass funcons for choce n leraure. In hs paper, he bass funcons used n WA mehod s seleced as Chebyshev polynomals of he frs knd and he collocaon pons are seleced as Chebyshev-Gauss-Lobao nodes [5]. 4. Nonlnear Dynamcal Behavor ncludng Bfurcaon and Chaos Wh he dscrezed equaons derved above, he nonlnear vbraons of a buckled beam are frs nvesgaed under unform harmonc excaons, n whch he frequency of he exernal load F( x)cos Ω s Ω= 30. A force sweepng approach s adoped heren o capure he bfurcaon phenomenon. For smplcy, a perodc moon s referred o as perod- n moon f s perod s nt, where T = π Ω. I s shown n Fg. 5 ha a perod-one moon s obaned for F = 40. Fg. 5 (a) s he Poncare map ha ndcaes he evoluon of he dsplacemen q () and he velocy q () from he nal values o he fnal sae, whch s referred o as he snk. I can be found ha he ransen process of perod-one moon s raher smple. Snce he snk s he only aracor n hs area, all he pons around are araced o asympocally. As he excaon force F ncreases, he frs perod-doublng bfurcaon occurs a abou F = 40. A ypcal perod-wo moon s presened n Fg. 6 for F = 440. In Fg. 6 (a), he Poncare maps of he ransen process are obaned usng WA algorhms of varous negrang accuracy. I s found ha he wo snks sandng for he perod-wo moon exs n a srange aracor. Ths means ha he ransen response of hs moon s chaoc-lke. For he exsence of snks, he moon wll evenually sele down o a seady lm cycle oscllaon. However, could be dffcul o deermne he exac me when he moon seles down f he ransen sage s prolonged. The accurae negraon of he sysem wll also be a challenge, because he rajecory n he chaoc regme can easly dverge even wh a very small negraon error. By furher ncreasng he force amplude, he second perod-doublng occurs a abou F = 454. A perodfour moon s presened n Fg. 7 for F = 460. I can be observed n Fg. 7 (a) ha he four snks also coexs wh he chaoc aracor, hus he ransen moon s also chaoc. By comparng he chaoc regmes n Fg. 7 (a) and Fg. 7 (a), here s no much dfference before and afer he bfurcaon occurs. Therefore, seems ha he chaoc aracor s barely affeced by he bfurcaon of perodc moons. The nex bfurcaon occurs a F = 46, leadng o he perod-egh moon. The correspondng Poncare map, phase porra and response curve are ploed n Fg. 8. The resuls n Fgs. 4-7 can be obaned by boh he HHT-α and WA algorhms. However, he me sep sze has o be seleced very small o accuraely oban he seady perodc moons by usng he HHT-α algorhm, of whch he compuaonal me wll be much more prolonged. On he conrary, he sep sze of WA algorhms can be seleced relavely very large, bu sll easly acheve hgh accuracy n predcng hese lm cycle oscllaons. I wll be shown n he nex subsecon ha he WA algorhm has a far beer performance han he HHT-α mehod n erms of accuracy, compuaonal me, and speed of convergence, n predcng he nonlnear dynamcal responses of he buckled beam nvolvng bfurcaon and chaos. 9

20 Fg. 5 (a) Fg. 5 (b) Fg. 5 (a) The Poncare map for F = 40, (b) The perod- lm cycle oscllaon. Same resuls for HHTα and WA algorhms, bu wh dfferen compuaonal performances, as s dscussed n subsecon 4.. 0

21 Fg. 6 (a) Fg. 6 (b) Fg. 6 (a) The Poncare map for F = 440, (b) The perod- lm cycle oscllaon. Same resuls for HHTα and WA algorhms, bu wh dfferen compuaonal performances, as s dscussed n subsecon 4..

22 Fg. 7 (a) Fg. 7 (b) Fg. 7 (a) The Poncare map for F = 460, (b) The perod-4 lm cycle oscllaon. Same resuls for HHTα and WA algorhms, bu wh dfferen compuaonal performances, as s dscussed n subsecon 4..

23 Fg. 8 (a) Fg. 8 (b) Fg. 8 (a) The Poncare map for F = 46, (b) The perod-8 lm cycle oscllaon. Same resuls for HHTα and WA algorhms, bu wh dfferen compuaonal performances, as s dscussed n subsecon 4.. 3

24 Fg. 9 (a) Fg. 9 (b) 4

25 Fg. 9 (c) Fg. 9 (d) Fg. 9 The same chaoc moon revealed by HHT-α (wh very small sep sze) and WA algorhms. (a) Phase porra, (b) me responses, (c) Poncare map, (d) FFT curves. 5

26 Fg. 0 (a) Fg. 0 (b) Fg. 0 Bfurcaon dagrams obaned by (a) force-sweepng, (b) frequency-sweepng process. Same resuls obaned for HHT-α (wh very small sep sze) and WA algorhms. 6

27 As he perod-doublng bfurcaon proceeds, more snks appear n he chaoc regme, and evenually he moon becomes compleely chaoc. In Fg. 9, he chaoc moon s presened hrough phase porra, response curve, Poncare map, and FFT curve. Smlar perod-doublng roue o chaos s also observed for vared excaon frequency wh he amplude fxed a F = 400. The bfurcaon dagrams obaned hrough force-sweepng process and frequency-sweepng process are ploed n Fg. 0. In predcng chaoc moon, he rajecores obaned usng he consdered wo mehods dverge afer a shor me. However, he Poncare maps and he FFT curves concdes very well, whch shows ha he same chaoc moon s revealed by hese wo mehods. 4. Comparson Beween he HHT-α and WA Algorhms The dscrezed model s solved usng boh he HHT-α and WA mehods. I s found ha boh mehods can predc he lm cycle oscllaons and chaos. However, he compuaonal performances of hese wo mehods are very dfferen. In he analyss below, varous sep szes are used o es he sably of he algorhms. I s found hrough smulaons ha he WA mehod converges for boh perodc and chaoc moons wh sep szes as large as T 5, where T s he perod of he frs vbraon mode, T = πω. The larges sep sze of HHT-α mehod depends on he nonlnear algebrac equaons (NAEs) solver. By usng Newon-Raphson mehod, he sep sze should be no greaer han T 0, oherwse he soluon of nonlnear equlbrum equaons of HHT-α may no be found. We red o reveal he seady perodc moons of he buckled beam under dfferen exernal excaons, usng HHT-α and WA mehod separaely. Afer he moon seles down, he exremes of he dsplacemen q () obaned by hese wo mehods are recorded n Table. The exac values are assumed o be provded by ode45, and hey are fully conssen wh he resuls obaned by WA mehod, wh he me sep sze beng T 5. The number of collocaon pons used n WA mehod s N = 7. Increasng he collocaon pons can furher mprove he accuracy of hs mehod. The HHT- α mehod fals o work for = T 5 and T 0, because he Newon-Raphson eraon scheme canno converge for such large seps. Wh he sep sze beng T 50, he HHT-α mehod provdes seady perodc moons, alhough hey are que erroneous compared o he exac ones. Parcularly, n he parameer regon where he perod-8 moon domnaes, he HHT-α mehod only provdes perod-4 moon, whch s due o he negrang naccuracy. By shorenng he sep sze, he accuracy s sgnfcanly mproved. However, here are sll some observable dscrepances beween he resuls of HHT-α and he exac ones, even wh he sep sze n HHT beng as small as = T 000. In he smulaons, dfferen values of α are used. In he case of α = 0., a moderae numercal dampng s ncluded, whch could fler he hgh-frequency componen of moon and avod dvergence. For α = 0, no numercal dampng s nroduced and he HHT-α mehod becomes he average acceleraon mehod. By comparng he values of exremes wh he exac ones, able ndcaes ha he perodc moons are beer predced wh α = 0 han wh α = 0.. Ths resul s reasonable snce he numercal dampng nroduced n HHT-α mehod makes he sysem behaves lke ha exra dampng exss n he smulaon. The lm cycles of perod-4 and perod-8 moons are obaned usng WA mehod wh = T 5. They are compared wh he resuls obaned from HHT- α mehod, where α = 0. and = T 50. I s shown n Fg. ha he HHT-α mehod canno predc he rue dynamcal behavor wh such a large 7

28 sep sze. In Fg. (a), an obvous dscrepancy exss beween he resuls of WA and HHT-α mehod, whle n Fg. (b), he HHT-α mehod gves a perod-4 moon a where he perod-8 moon should be revealed. In all, boh he WA and he HHT-α mehod can sably negrae he nonlnear sysem of buckled beam. However, he sep sze of HHT-α mehod should be seleced very small o oban vald soluons. Table Exremes of he seady perodc moons obaned by HHT-α and WA mehods Mehods Perod- Perod- Perod-4 Perod-8 WA ( T 5) HHT-α α = 0. α = 0 α = 0. α = 0 α = 0. α = 0 α = 0. α = 0 T 5 \ \ \ \ \ \ \ \ T 0 \ \ \ \ \ \ \ \ T \ \ T \.988 T T Fg. (a) 8

29 Fg. (b) Fg. Comparsons beween he lm cycle oscllaons obaned by he WA ( = T 5 ) and HHT-α ( = T 50, α = 0.) algorhms. Generally, he evaluaon of compuaonal accuracy s nonrval for numercal mehods, especally n nonlnear problems. Heren, he numercal soluon of MATLAB bul n ode45 funcon s used as he benchmark o esmae he compuaonal error of HHT-α and WA mehods. The absolue and he relave accuraces of ode45 are boh se as E 5. The ransen responses of perod-one, perod-wo, perod-four and perod-egh moons are obaned usng HHT-α and WA mehods respecvely. Then by comparng wh hose obaned by ode45, he dscrepances of he numercal resuls are shown n Fg.. To evaluae he hghes accuracy hese wo mehods may acheve, he seps sze of HHT mehod s seleced as = e 4, whle ha of WA mehod s = T 6 = wh 3 collocaon pons n each me sep. As s llusraed n subsecon 4., he ransen response of perod-one moon s relavely smple and nonchaoc. I s refleced by he conssen numercal dscrepances over he smulaon me n Fg. (a). As s shown, he WA mehod acheves very hgh accuracy wh respec o ode45. The dscrepances beween hem s fve magnudes lower han ha beween HHT-α and ode45. For he mulple-perod moons, he numercal dscrepances accumulae exponenally for boh HHT-α and WA algorhms before = 0, as shown n Fg. (b-d). Ths can be explaned by he fac ha chaoc regme exss around he perodc moon. I s shown ha he accuracy of WA mehod s sll much hgher han he HHT-α mehod n he negraon of ransen chaoc moons. Alhough he resuls of WA mehod dverge from ha of ode45 afer = 0, does no mean ha he WA mehod fals. Acually, he chaoc regme s so sensve o he nal sae ha even an error occurred on machne precson could blow up and cause sgnfcan dscrepancy n he fnal sae. 9

30 Fg. (a) Fg. (b) 30

31 Fg. (c) Fg. (d) Fg. The compuaonal error of he HHT-α and WA algorhms 3

32 Table lss he compuaonal me, eraon seps and sep szes of HHT-α and WA algorhms. I s found ha he WA mehod consumes only 0% of he compuaonal me requred by HHT-α mehod. Ths cos savngs are expeced o be much more dramac for large order dynamcal sysems. The sep sze used n HHT-α s 0.000, whle ha n WA s Accordng o he performances demonsraed n Fg. and Table, he WA mehod s much more accurae and effcen han HHT-α mehod, n addon o requrng only much larger sep szes. For furher comparson, he performance ndexes of ode45 are also presened n Table. I s shown ha he compuaonal effcency of he WA algorhm s far superor o ha of ode45. Nong ha he ode45 s a bul-n funcon n MATLAB ha has already been opmzed, whle he WA algorhms are mplemened n MATLAB wh a roughly desgned program. Moreover, he WA algorhms can be easly realzed usng parallel programng, whch wll furher mprove he compuaonal effcency. Table. Compuaonal cos of HHT-α and WA mehod Cases Compuaonal Tme (sec) Ieraon Seps Sep Sze HHT-α WA ode45 HHT-α WA ode45 HHT-α WA ode45 Perod e-05 Perod e-5 Perod e-5 Perod e-5 To nvesgae he energy conservaon properes of he HHT-α and he WA mehod, a smple undamped duffng equaon s used for demonsraon. The Hamlonan energy of hs sysem s 3 + =. x x x 0 H 4 x x x = +. 4 Sarng from he nal sae x (0) =.5, x (0) = 0, he sysem s negraed usng he HHT-α, he WA and he ode45 mehods. The sep sze of HHT-α mehod s = 0.0 and he smulaon s carred n he me nerval [0,000]. For he WA mehod, he sep sze s seleced as =, wh 3 collocaon pons n each sep. The absolue and relave accuraces of ode45 are boh se as E 5. The compuaonal errors of he Hamlonan are recorded and ploed n Fg. 3. I can be seen ha boh he WA mehod and ode45 are much superor o he HHT-α mehod on energy conservaon. Noably, he WA mehod behaves even beer han ode45, wh a neglgble error of Hamlonan of E 3. Among all hese mehods, he HHT-α mehod wh α = 0. s he wors on energy conservaon, as can be seen n Fg. 3 (a). Afer droppng ou he numercal dampng by seng α = 0, he performance of HHT-α mehod s much mproved, wh he error of Hamlonan beng E 5. 3

33 Fg. 3 (a) Fg. 3 (b) Fg. 3 Compuaonal error of Hamlonan usng he HHT and WA algorhms, and ode45 33

34 4.3 Dscussons of he relave performances of he 3 WA algorhms Alhough he WA algorhms-,, and 3 are derved from he same opmal error-feedback eraon concep, hey have some dfferences n mplemenaon and compuaonal performances ha he users of hem should be aware of. In Fgs. -3, he flow char overvew of hese hree algorhms are presened. In he mplemenaons of WA algorhms- and 3, he nal condons have o be ncorporaed n he eraon formula and he excess collocaon equaons need o be removed. In WA algorhm-, he nal condons are naurally sasfed, hus he mplemenaon of s more sraghforward han WA algorhms- and 3. The convergence speed of he WA algorhm- s supposed o be he fases, snce s equvalen o Eq. (), where he marx of weghng funcons λ s opmally derved. However, he WA algorhm- nvolves nverson of he Jacoban marx ( E + J ), whch s varyng durng he eraon process. For sysems wh hgh dmensons, compung he nverson of he Jacoban marx could be very me consumng. In addon, f he Jacoban marx become ll-condoned durng he eraon, he WA algorhm- may easly dverge. On he conrary, he WA algorhms- and 3 are free from nverng marces durng he eraon process. Consderng he buckled beam problem n hs paper, he performances of he WA algorhms are compared wh each oher. The comparng resuls lsed n able 3 are obaned durng he smulaon me of 00T, where T s he perod of he frs mode. Table. 3 Comparson of WA algorhms Mehod Larges Sep Sze Number of Ierave Seps Compuaonal Tme WA algorhm- T s WA algorhm- T s WA algorhm-3 T s As s shown n able 3, he larges sep szes of he WA algorhms are he same, whle he number of erave seps and compuaonal me of WA algorhm- are he leas among he hree proposed algorhms. The compuaonal errors are no provded heren because he WA algorhms behave smlar n erms of accuracy, whch s already presened n Fg.. Overall, alhough all he proposed WA algorhms can be effcenly appled o solvng he nonlnear dynamcal equaons of he buckled beam, he WA algorhm- s he mos recommended, no only because ncorporaes he nal boundary condons nherenly and s free from nverng he Jacoban marx, bu also for ha has he bes compuaonal performance among he proposed mehod. 5. Concluson The Wang-Alur Algorhms are used o solve a nonlnear dynamcal model of he clamped-clamped buckled beam. The numercal resuls show ha he proposed mehod s very accurae and effcen n predcng nonlnear dynamcal behavors ncludng bfurcaon and chaos. Through force-sweep and frequency-sweep processes, he perodc moons and he perod-doublng roues o chaos are successfully capured by he Wang-Alur algorhms. I s also found n he smulaons ha he seady mulple-perodc 34

35 moons are acually surrounded by chaoc moons n he neghborhood. I s he reason for he chaoc ransen moons o be hardly predced precsely. The robusness of Wang-Alur mehod s ndcaed by he hghly accurae negraon usng a large sep sze = T 5. Compared wh he Hlber-Hughes-Taylor-α mehod, he proposed mehod s superor on many aspecs nvolvng accuracy, effcency, sably and energy conservaon. I s shown n hs paper ha he Hlber-Hughes-Taylor-α mehod canno accuraely predc he perodc vbraons of he buckled beam unless exremely small sep szes are used. An obvous dscrepancy can sll be observed beween he exac phase porra and ha provded by Hlber-Hughes-Taylor mehod-α even when a relavely small sep sze = T 00 s used. The example of Duffng equaon also ndcaes ha he energy of he sysem s barely conserved usng Hlber-Hughes-Taylor-α mehod afer he smulaon me = 000, whle he error of energy n he resuls of Wang-Alur algorhm s neglgble. Appendx A A,, = 4.905, A,, = 0, A,,3 = 6.554, A,,4 = 0, A,, = 7.098, A,,3 = 0, A,,4 = -4.55, A,3,3 = , A,3,4 = 0, A,4,4 = ; A,, = 0, A,, = 34.95, A,,3 = 0, A,,4 = -4.55, A,, = 0, A,,3 = , A,,4 = 0, A,3,3 = 0, A,3,4 = , A,4,4 = 0 ; A 3,, = 8.768, A 3,, = 0, A 3,,3 = 84.45, A 3,,4 = 0, A 3,, = , A 3,,3 = 0, A 3,,4 = , A 3,3,3 = 64.7, A 3,3,4 = 0, A 3,4,4 = ; A 4,, = 0, A 4,, = -4.55, A 4,,3 = 0, A 4,,4 = 50.38, A 4,, = 0, A 4,,3 = , A 4,,4 = 0, A 4,3,3 = 0, A 4,3,4 = 430.6, A 4,4,4 = 0 ; Appendx B B,,, = , B,,,3 = , B,,, = , B,,,4 =., B,,3,3 = , B,,4,4 = , B,,,3 = , B,,3,4 = , B,3,3,3 = , B,3,4,4 = ; B,,, = , B,,, = -04., B,,,3 = -7.8, B,,3,3 = , B,,,4 =., B,,,4 = 369.8, B,,3,4 = , B,3,3,4 = , B,,4,4 = , B,4,4,4 = ; B 3,,, = , B 3,,, = , B 3,,,3 = , B 3,,,3 = , B 3,,3,3 = , B 3,3,3,3 = , B 3,,,4 = , B 3,,3,4 = 967.3, B 3,,4,4 = , B 3,3,4,4 = ; B 4,,, =., B 4,,, = , B 4,,,3 = , B 4,,3,3 = , B 4,,,4 = , B 4,,,4 = , B 4,,3,4 = , B 4,3,3,4 = , B 4,,4,4 = , B 4,4,4,4 =

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