Analyse van dynamische problemen via diskretisatie van de Hamilton functionaal

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Analyse van dynamische problemen via diskretisatie van de

Hamilton functionaal

Citation for published version (APA):

Kraker, de, A. (1976). Analyse van dynamische problemen via diskretisatie van de Hamilton functionaal. (DCT rapporten; Vol. 1976.005). Technische Hogeschool Eindhoven.

Document status and date: Gepubliceerd: 01/01/1976

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9uN,c,~ t ohowl. 1r- 1 dynavn~s~ oploss ih~ ~~~ _{C>„! 1Proi 01e}041 . ~ti crvoor b~av~ ec4, C~-0j- ~e( .~ 4. keden :

AJ . GeIprm.~tc de ~unc Ir romacx,L voor het aFle~ZclQ.r, u~t.h de dC~~erentieeal -~»<~el~~,Ng voa- het ptobleen., . 0_{MotoSSe;n vQh dete} _{d .v (Dhd<cr}

ode ge9eve„ romdli,woQ r Eïe» i&dE dan <rcL- de 9evraa9cie optossi viq . e i .h ^ particle d .v ) .

~J .Een optossin5 van cdete at, F~e~.niictGil ver'a~el~~.:~n c~ el~~.~nc~ cs <31e41j voor

r&Ck~i e~ eenvoUM5e probl~ i,h hcl- a~er,^een vno9et~~c .

Jnc~iem, de ent ab Pu,nckie vct^ de

Lb-d "rd i n acalr reta~ïe ~ ee-vwoe.cc•t-+g , al ~ ru.r,áic va, , d e

p laaE s c~r~i~ ~rc,t i v~ua E Claar enl: e 5er1 n " yole x(deM 4 Io

~5v aaml

9GOVtnebri" ihC+~ew il<.~tc~,á(+ KwtStrtAl<,t1~S, 2~h , Zou me, ~,nca l ~

ihde et~w,~l~nVr»eillocQc ~eloP.t .~.~r~,, dc. plaaEsa~lnahlcel•~4~+e^r~ ~-u.v~ disk,retiaerer, ~ waardoor e .r,4cl de Cnl•egrakte ove-r'c.Qe

L~sd ~ct ;nccxE ove.rto1~p6 .

We ~ebbe~, zod.oeMe e!e orde ~ on~ probieem ve.rt«acad •

Gbor coL vit, wear ee.n S1;a.Eio",re, waarde te- ei; S~n voor de

~t.eanc.E i on

ecd 'L

r

'~

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et

, we ee

„

9e

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n e ve,rg el'y

-1.t.i h~e.r iv~ cAc b~ de dl 3tcret i salie ~ete .oZe-vl „ vrdtie~ otsgro,den; , Ocie d• v'tS vc,ta~ Ce..wQkc.Q.i~er op te toaa~ .

Dit

i S o n

ga4

/e

s--r

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a

9

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-.

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wi is J, e ffob le,vne.n via Je jokam e.nH er+ vneth,oole tol, hu4 toe we~d lvc%epccst .

Lby~ A s~a )

.

(13)

•

C J . Zi; n h i eL aJ le m de Ve•pfaata 1nge,•, cih ~~et i e vow, d e pla<>t .t -c.O6r0inal•e.n vyWei I~ic .. Le beschrwer,, vw\aar oo4 als gu .v,ctíe vct,•, de ~~ cl ~oó r~haaL- u~ 4.ownplek da,1 h,a,,, mew, oot,- de

ver~t~a~ sS ny~, ?,owe-, ih de pl a~a b d l< Cb rr.u K«« tCe„ ) als ik] d.e ~yc,t "" naa~ t4 e s 4.vcE t's C4-c,, ovv, 2odo4t-nd e L-c l~ .o vn e,n talc

4 oyo

ec.„., ge:woon SL~ P.l. ver~ et .~4w~p,r, hel•~jee~, h i el• wi oei 1 ;-L-e lossev, irs .

Wz fc.t,wihe~n dokcArWb CQe.lke,n aGu-, dlc Wtie~ ~i od e v . wct" ;j~ over tiet- oetnete dow,P.ih ee.v, bevrado~Iw~gs ~u,v~c.ti c . lo 0sc + 1 rP.v C'^ d.oo r eCv\ aav~CaC pa.rameke.rS wordir ~el" d, als Qo1~ aav, e .e,.,

elew~P.n~e~ vneEhcxle - a~tntioje UAchpa4 wua,rb

~n sl-uhsgew;~1 bev~ac~uín5e+n wome., gelc,ahitrrt4ee4nd ~ welke voa- %el• ge.ll de dovhe;n op de- j u.i slce W~Ze wordQ .n getcAppel.cJl .

we Z.~t,ll~n (:"~de hand v ct,rn de tcncjí l-kcdi Viaal ~r ï 1(e.-id e bo.t4 de kvxiste tMcthr>de " iS c.tcre hnelaiod¢ welke

c k ih rrnyh onde.rl,oe4 heb 1-o9 epca.S b.

~~

vah de evo~Acr ~own~.~lu~ho~ ivt hc~o~~u4. ~ :

~ we

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c

AVI

_{~ 9 r}

~ Z

=

ff+4

- wu. +ZEA ~~xdt - ~CN• u)

=1

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(V-

u

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~

0(6 x x_ti

•

1;4~_{a x_a} 't

t=

o ~

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1--T x_ O ~ ~~lle ve<'del e .v , áe lengVe vld bo,t~ ~ Zn n s l- k44e.v, ~tn ,~„ ~

k4 ~e_sohowwde E~ds inFu~uJt tO,T] i.h Vw1 St- (&k4an [E h; , ~7- ] We de p i hjc re,r, e~• ~~ e! Qaw, e.,~t " als e2v, ~ ge Ibieod ~ b ero-o,l0~

er, C6e ~, ~eL J Pn I~h er dbw, áoor cle verdelrwLQe,h ( ~%', ,Q,~2 3

voor c2 .7) ^ oo i Sc,1nV c~ up.~n

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(14)

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us cta.v,s

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~h X, ev~ ~, o~9ehc~hc~ e", uah b~v d , ~ disl .ircL-P ~ p "C s; h5en w~ cle

vve Ic.u hh ~n dus

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en als va4Eor vav

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17 w

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S't

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ra~c~ wn p uts t p v ~

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2em AaUovicii re woo.roe. l•c ei ~ voo r- aIl e va.ria tï c.,

S(,( 6 wetlc,e, vuldo crr , aan de 1c,,jjnew~i sdke vc~hdvoorwacider v ï Kde ., w e '

Q~

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,3TA/oov) 5L-.bek v.ertj el~6" nge A

2ev

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ih de o,nbt4 ev,d~e,r,

2pVd Cr LA2Gt Pfb~OI~o>P." Ls oyo te IbSSPN} .

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+` de vy)c>t r-.x 0 6 ~v1~~ao.r ïs_> ddu" ov„der „-tl<e ~wMC+~i ~ i P~

e~,~, e~~ otw~ d.;gc opbss i v,~ v a h (2•9) vnug e l~ 4 es: .

S

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z~,e l le-r.

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l >c C., 6

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el in ~t - ri

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-I-cxl ~c,E Goorom haod?mk<eLs1c dc ~t,ms.t i pvrxat waai~ worri6

O

t.A~199c+cwl d~hnensiela05 l-,c mal~~~.nn~ c*-, eov, vc", de

qrod: kedc,, welke e=, , ral s pel"-*^ wwgel:94 ~e vnale~,,

(16)

•

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Voorb.e Me.n \Ii a dkt

ry,e L• koae

vo„w de d i skt'el'e I-t ~S lE on Nhck.ioyaa l

dt)o r c~ere ~ and .

_Er 2~ h v ~r-sc ~, ~ I(en áe v oorbQelc~v, ~eana tYj~d v 'c< de clzsts.rcl• e_

Haw,i

(L-on

vhck

t

ode .

pete vaorbaMen Z t.tlleo, nïe1; allcw,aoak rvort~l~ bthand .ald, i' n bej ckalde o~~alIen -ma( ~ kol ~-v,1ge in f awatic worde,,. ~"ev~ <>jar cte

a~ .~~8roh~P~, v i~~- o ~ de ver eke Y2-sultal•e." .

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I

onde,rlc,ri ~ is~-~n c~edc .w,~p ~ ~~SS~ -vee,~~ Ysl•eean , bekt s~ v~ne~ &c.c ;i Ea~ie "c>k F~ .

.ryt0rl~4.c 6Pg+v)trc*cuties :

c,li e w~ tlcc~P•c~rige ~ow,birtaGe vccin

,egiYlve.rpQ) 6C 8iw,Snc1 K-A CQ, ei~hdlvar~taaisino~ ~ ~ne~s he~~~d . - be tQSLi ng ge VAt1eM :

# s~ap ~c,w~k ie

í~ ~ta~(hto~iscó~ ~u,y~cEic

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ki s

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ie

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ele vn errE wer

a cl

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3e c~~caads ~lynovrn : bes c~,rewe,r, door ek 9

cl1shrcl:e vv plCUC~--sir-se"

U,-,"~ ;

_{L,%. 1, . . y .}

3y dii voorbftld varh een cje lerryoti vhas .3a- veersys}eem is

(17)

•

15

-We 1xsc~ouwcJn E wee s i turat i es - t .. cc 6 vr-<c~er :

Puls ~unctie :

m= 2 ~ k_ 2, T'h_

( b1nd~I~.r i t 1 S~ ) . ~ R

_0-4

~~yi~~~~táatsinc~ _ 5 , beginshethe~td=

~-~eS~~.ewc@e rL'~at3 i,nl•e.rvc,~. ~2 5ec , ve~rc~.ee[d ~~ ~ .~dew,o+,rl•4-n ~ .

~ ~ (elerv~®nt kv,9ic ea O.Ss ~ ~h : ~ ~30 ' d i~ cc

~ ~i'1 9~g1

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1~ b- diL- roblce•wn b keiC c

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be SAouw* 4htiu\/a.l iov ate Oacakl•c oplossi Ag cok 2'4 L- zZ~h •

2,-lc ode cara,Fje~ Z . _{«D<A- 9 1&)}

EB] S

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ief bie

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(18)

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'~d~ Z~h voorloeetof~„, ~ eQ,n ~in . dv vr~~k V~ -c~: ayt

C.d.é ~~ï cJ+ëntw, , Jook coë~ 4 h . vuv, de ©rvx f~an4e%t«, ~a~'~abeie .

(19)

t

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a

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f

7

-qo de-& l PiS L- e Ewee voorbmekQe;v, •za1 hie~ vr- .der wordc~n iv% 9eq accv ) .

Co hc~usie s

l/ki~ G~e reS u l~cil' e•v, verWec~~qm b~ de an«lys e ~c,rr~ ole 9ept~senl•eerde froblemo^ v~ a cQc ~ed i sl~ret S Se~de ï-iav,1 ~ 1 tan -~ c .un c~ i o v~aal. bty 1R ~ daj- deze Ic»Es1c ec ~m good gereed schcLp L:;„4 E be vormc.^ voorde

cLvk«tY s c van _{dy num'wscl% ~ on, plex c foroWew-ev .}

M ev, vnner evenas 105 st4~„1"r- p roble,w, r4 , de <irte-w+- c;r,b~ ~det~vtg i

(c3r4we~d) Oianpass0n a an dé te ve,rwc41.,ter, ~radië.nb , a" a mee

~oetrela

_~

v

_{rj ht}

de i~c

t

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aan

de

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er ~d~len~~e

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de bew~e~ ~ nc~.

J s d i t het; cjovi~kl dan bl~ Wa v-e" b b vr tatie~ wè,v,i S eo, rela~ie g -ee.nvou.c:ti9e cte ~t;ew+ _{dc ~vwgi hg} voun het; sysl•e~a+n» v'edettj~_

cjoed 6e"re,-ievi te worden .

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b4

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_{VOor9Gahde de ~t,wtab i oncz ul a~gele~d} waar m ee kekt bolale dYm+rni scA, .e tedttA9 van, e.e., sys~et.~, locsd•,re,,/e~.h. L,cth word .„

áe ~ovvtplcKilL-1-;t VCW~ ~<i 6c sc•houwo(e s~.Sl'eari. Icta,r,

c~Lc ~une~ïohctal 9 ebrLM It~ wordc,,, voor ' hcL bcpalc., va I de

otynstwii 1cC-Rne Yés Pavfs vctK e~ sVJL•cC*n ondr.r hcf aCA ldC bekcs~ivtig, bc~ lc .ov i ct i~ies e L c .

1~~ b~kk " ~ u.il: cande~t; c~a,n de ~abcl..

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pr

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V" tcu.e~bSi hc~cv, , ~- b~ 's n4ondï ~- ies

c!i 31uc~is«~ie in pk,catSCo~rd . .

e m L-~dcaórdihQl~n .

N-y, ~~ow, slel s el v9.2 ' n __~• op 2 .

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on

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t~aho~a~~ voatal 9e.ricJhE

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Zowet 9e ow+etrï Sch als i h dc i:& Ic.aM,ialne ri-oble,v~ , D, & Zot ~n k., ~ er, z~e.h al s de Llccssieke we-9 , G> de WC-5

wc t1< .r , bgepw~ wordt- i n d e v»e t hocle clew- <r,;vicUge

e

hi ~c Jooro

w

en V G1i .

Je

IMe~ izo p

Ot

:

* U vwt,c.trw, otï rtk~r dt dYna rvrCsc~ke respons van ~et sy-tocw, i.0^dorr

tpbssi

n

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