Introduction to Physical Systems Modelling with Bond Graphs

Transcription

1 Introction to Physical Systms Molling with Bon Graphs Jan F. Bronink Univrsity o Twnt, pt EE, Control aboratory PO Box, N-5 AE Ensch Nthrlans -mail: [email protected] Introction Bon graphs ar a omain-inpnnt graphical scription o ynamic bhavior o physical systms. This mans that systms rom irnt omains (c. lctrical, mchanical, hyralic, acostical, thrmoynamic, matrial) ar scrib in th sam way. Th basis is that bon graphs ar bas on nrgy an nrgy xchang. Analogis btwn omains ar mor than jst qations bing analogos: th s physical concpts ar analogos. Bon-graph molling is a powrl tool or molling nginring systms, spcially whn irnt physical omains ar involv. Frthrmor, bon-graph sbmols can b r-s lgantly, bcas bon-graph mols ar non-casal. Th sbmols can b sn as objcts; bon-graph molling is a orm o objct-orint physical systms molling. Bon graphs ar labll an irct graphs, in which th vrtics rprsnt sbmols an th gs rprsnt an ial nrgy connction btwn powr ports. Th vrtics ar ialis scriptions o physical phnomna: thy ar concpts, noting th rlvant (i.. ominant an intrsting) aspcts o th ynamic bhavior o th systm. It can b bon graphs itsl, ths allowing hirarchical mols, or it can b a st o qations in th variabls o th ports (two at ach port). Th gs ar call bons. Thy not point-to-point connctions btwn sbmol ports. Whn prparing or simlation, th bons ar mboi as two-signal connctions with opposit irctions. Frthrmor, a bon has a powr irction an a comptational casality irction. Propr assigning th powr irction rsolvs th sign-placing problm whn conncting sbmols strctrs. Th intrnals o th sbmols giv prrncs to th comptational irction o th bons to b connct. Th vntally assign comptational casality ictats which port variabl will b compt as a rslt (otpt) an consqntly, th othr port variabl will b th cas (inpt). Thror, it is ncssary to rwrit qations i anothr comptational orm is spcii thn is n. Sinc bon graphs can b mix with block-iagram parts, bon-graph sbmols can hav powr ports, signal inpts an signal otpts as thir intracing lmnts. Frthrmor, aspcts lik th physical omain o a bon (nrgy low) can b s to spport th molling procss. Th concpt o bon graphs was originat by Payntr (96). Th ia was rthr vlop by Karnopp an osnbrg in thir txtbooks (968, 95, 98, 99), sch that it col b s in practic (Thoma, 95; Van ixhoorn, 98). By mans o th ormlation by Brvl (984, 985) o a ramwork bas on thrmoynamics, bon-graph mol scription volv to a systms thory. In th nxt sction, w will introc th bon graph mtho by som xampls, whr w start rom a givn ntwork compos o ial physical mols. Transormation to a bon graph las to a omain inpnnt mol. In sction, w will introc th onations o bon graphs, an prsnt th basic bon graph lmnts in sction 4. W will iscss a systmatic mtho or riving bon graphs rom nginring systms in sction 5. How to nhanc bon graph mols to gnrat th mol qations an or analysis is prsnt in sction 6, an is call Casal Analysis. Th qations gnration an block iagram xpansion o casal bon graphs is trat in sctions an 8. Sction 9 iscsss simlation isss. In sction w rviw this chaptr, an incl som hints or rthr raing. Bon graph xampls To introc bon graphs, w will iscss xampls o two irnt physical omains, namly an C circit (lctrical omain) an a amp mass spring systm (mchanical omain, translation). Th C circit is givn in Figr. /

2 Intro Bon Graphs Jan F. Bronink, 999 U s C Figr : Th C circit In lctrical ntworks, th port variabls o th bon graph lmnts ar th lctrical voltag ovr th lmnt port an lctrical crrnt throgh th lmnt port. Not that a port is an intrac o an lmnt to othr lmnts; it is th connction point o th bons. Th powr bing xchang by a port with th rst o th systm is th proct o voltag an crrnt: P i. Th qations o a rsistor, capacitor an inctor ar: C i i t C i or t i t In orr to acilitat th convrsion to bon graphs, w raw th irnt lmnts o th lctric omain in sch a way that thir ports bcom visibl (Figr ). To this port, w connct a powr bon or bon or short. This bon nots th nrgy xchang btwn th lmnts. A bon is rawn as an g with hal an arrow. Th irction o this hal arrow nots th positiv irction o th nrgy low. In principl, th voltag sorc livrs powr an th othr lmnts absorb powr. i i _ i _ i i i C C i i Figr : Elctric lmnts with powr ports Consiring th circit o Figr, w s that th voltag ovr th lmnts ar irnt an throgh all lmnts lows th sam crrnt. W inicat this crrnt with i an connct th bons o all lmnts with this crrnt I (Figr ). Changing th lctric symbols into corrsponing bon graph mnmonics, rslt in th bon graph o th lctrical circit. Th common i is chang to a, a socall -jnction. Writing th spciic variabls along th bons maks th bon graph an lctric bon graph. Th voltag is mapp onto th omain inpnnt ort variabl an th crrnt maps onto th omain inpnnt low variabl (th crrnt always on th si o th arrow). Th - jnction mans that th crrnt (low) throgh all connct bons is th sam, an that th voltags (orts) sm to zro, consiring th sign. This sign is rlat to th powr irction (i.. irction o th hal arrow) o th bon. This smming qation is th Kirchho voltag law. /

3 Univrsity o Twnt, pt EE Intro Bon Graphs U i s i i i C i C Figr : Bon graph with lctrical symbols (lt) an with stanar symbols (right). Th stanar bon-graph symbols ar in in th sction 4 Paralll connctions, in which th voltag ovr all connct lmnts is th sam, ar not by a in th port symbol ntwork. Th bon graph mnmonic is a, th so call jnction. A jnction mans that th voltag (ort) ovr all connct bons is th sam, an that th crrnts (lows) sm to zro, consiring th sign. This smming qation is th Kirchho crrnt law. ampr v Mass Forc Figr 4: Spring Th amp mass spring systm Th scon xampl is th amp mass spring systm, a mchanical systm shown in Figr 4. In mchanical iagrams, th port variabls o th bon graph lmnts ar th orc on th lmnt port an vlocity o th lmnt port. For th rotational mchanical omain, th port variabls ar th torq an anglar vlocity. Again, two variabls ar involv. Th powr bing xchang by a port with th rst o th systm is th proct o orc an vlocity: P Fv (P Tω or th rotational cas). Th qations o a ampr, spring an mass ar (w s amping coicint α, spring coicint K s, mass m an appli orc F a ): F F F F s m a αv K s vt C v m or t orc s v vt m F t m In th sam way as with th lctrical circit, w can rraw th lmnt sch that thir ports bcom visibl (Figr 5). Th loos ns o th xampl all hav th sam vlocity, which is inicat by a v. This jnction lmnt also implis that th orcs sm p to zro, consiring th sign (rlat to th powr irction). Th orc is mapp onto an ort an th vlocity onto a low. For th rotational mchanical omain, th torq is mapp onto an ort an th anglar vlocity onto a low. This implis that orc is rlat to lctric voltag an that vlocity is rlat to lctric crrnt. /

4 Intro Bon Graphs Jan F. Bronink, 999 F v : F a F a v F s v F m v m I S : F a F a v F v F m v I : m C F s v C : C / K Figr 5: Bon graph with Mchanical symbols (lt) an with stanar symbols (right) W s th ollowing analogis btwn th mchanical an lctrical lmnts: Th ampr is analogos to th rsistor. Th spring is analogos to th capacitor; th mchanical complianc corrspons with th lctrical capacity. Th mass is analogos to th inctor. Th orc sorc is analogos to th voltag sorc. Th common vlocity is analogos to th loop crrnt. Bsis points with common vlocity, also points with common orc xist in mchanical systms. Thn orcs ar all qal an vlocitis sm p to zro, consiring th sign (rlat to th powr irction). Ths common orc points ar not as -jnctions in a bon graph (an xampl is a concatnation o a mass, a spring an a ampr: th thr lmnts ar connct in sris ). A rthr laboration on analogis can b on in th nxt sction, whr th onations o bon graphs ar iscss. Throgh ths two xampls, w hav introc most bon graph symbols an inicat how in two physical omains th lmnts ar transorm into bon graph mnmonics. On grop o bon graph lmnts was not yt introc: namly th transcrs. Exampls ar th lctric transormr, an lctric motor an tooth whls. In th nxt sction, w will iscss th onations o bon graphs. Fonations o bon graphs Analogis btwn irnt systms wr shown in th prvios sction: irnt systms can b rprsnt by th sam st o irntial qations. Ths analogis hav a physical onation: th nrlying physical concpts ar analogos, an consqntly, th rslting irntial qations ar analogos. Th physical concpts ar bas on nrgy an nrgy xchang. Bhavior with rspct to nrgy is omain inpnnt. It is th sam in all nginring isciplins, as can b concl whn comparing th C circit with th amp mass spring systm. This las to intical bon graphs.. Starting points Bor iscssing th spciic proprtis o bon graphs an th lmntary physical concpts, w irst rcall th assmptions gnral or ntwork lik scriptions o physical systms, lik lctrical ntworks, mchanical or hyralic iagrams: Th consrvation law o nrgy is applicabl. It is possibl to s a lmp approach This implis that it is possibl to sparat systm proprtis rom ach othr an to not thm istinctly, whil th connctions btwn ths sbmols ar ial. Sparat systm proprtis man physical concpts an th ial connctions rprsnt th nrgy low, i.. th bons btwn th sbmols. This ialnss proprty o th connctions mans that in ths connctions no nrgy can b gnrat or issipat. This is call powr continity. This strctr o connctions is a 4 /

5 Univrsity o Twnt, pt EE Intro Bon Graphs concptal strctr, which os not ncssary hav a siz. This concpt is call rticlation (Payntr, 96) or taring (Kron, 96). Th systm s sbmols ar concpts, ialis scriptions o physical phnomna, which ar rcognis as th ominating bhavior in componnts (i.. ral li, tangibl systm parts). This implis that a mol o a concrt part is not ncssary only on concpt, bt can consist o a st o intrconnct concpts.. Bons an Ports Th contact point o a sbmol whr an ial connction will b connct to is call a powr port or port or short. Th connction btwn two sbmols is call a powr bon or bon; it is rawn as a singl lin (Figr 6). This bon nots an ial nrgy low btwn th two connct sbmols. Th nrgy ntring th bon on on si immiatly lavs th bon at th othr si (powr continity). lmnt bon lmnt Figr 6: Ports Th nrgy low btwn two sbmols rprsnt by a bon Th nrgy low along a bon has th physical imnsion o powr, bing th proct o two variabls. In ach physical omain, thr is sch a combination o variabls, or which a physical intrprtation is sl. In lctrical ntworks, th two variabls ar voltag an crrnt. In mchanical systms, th variabl pairs ar orc an vlocity or translation an torq an anglar vlocity or rotation. In hyralics, it is prssr an volm low. For thrmoynamic systms, tmpratr an ntropy low ar s. Ths pairs o variabls ar call (powr ) conjgat variabls. In orr to nrstan th connction as stablish by a bon, this bon can b intrprt in two irnt ways, namly:. As an intraction o nrgy. Th connct sbsystms orm a loa to ach othr by thir nrgy xchang. A powr bon mbois a connction whr a physical qantity is xchang.. As a bilatral signal low. Th connction is intrprt as two signals, an ort an low, lowing in opposit irction, ths trmining th comptational irction o th bon variabls. With rspct to on o th connct sbmols, th ort is th inpt an th low th otpt, whil or th othr sbmol inpt an otpt ar o cors stablish by th low an ort rspctivly. Ths two ways o conciving a bon is ssntial in bon graph molling. Molling is start by inicating th physical strctr o th systm. Th bons ar irst intrprt as intractions o nrgy, an thn th bons ar now with th comptational irction, intrprting th bons as bilatral signal lows. ring molling, it n not b ci yt what th comptational irction o th bon variabls is. Not that, trmining th comptational irction ring molling rstricts sbmol rs. It is howvr ncssary to riv th mathmatical mol (st o irntial qations) rom th graph. Th procss o trmining th comptational irction o th bon variabls is call casal analysis. Th rslt is inicat in th graph by th so-call casal strok, inicating th irction o th ort, an is call th casality o th bon (Figr ). 5 /

6 Intro Bon Graphs Jan F. Bronink, 999 lmnt lmnt lmnt lmnt lmnt Figr : lmnt trmin th signal irction o th ort an low (W o not s th powr irction at th bons, so it is not shown hr). In qation orm, Figr can b writtn as: lmnt. : lmnt. lmnt. : lmnt. lmnt lmnt. : lmnt. lmnt. : lmnt. 4 Bon graph lmnts Th constittiv qations o th bon graph lmnts ar introc via xampls rom th lctrical an mchanical omains. Th natr o th constittiv qations lay mans on th casality o th connct bons. Bon graph lmnts ar rawn as lttr combinations (mnmonic cos) inicating th typ o lmnt. Th bon graph lmnts ar th ollowing: C storag lmnt or a q-typ variabl,.g. capacitor (stors charg), spring (stors isplacmnt). I storag lmnt or a p-typ variabl,.g. inctor (stors lx linkag), mass (stors momntm). rsistor issipating r nrgy,.g. lctric rsistor, mchanical riction. S S sorcs,.g. lctric mains (voltag sorc), gravity (orc sorc), pmp (low sorc). TF transormr,.g. an lctric transormr, tooth whls, lvr. GY gyrator,.g. lctromotor, cntrigal pmp. an jnctions, or ial conncting two or mor sbmols. 4. Storag lmnts Storag lmnts stor all kins o r nrgy. As inicat abov, thr ar two typs o storag lmnts: C lmnts an I lmnts. Th q typ an p typ variabls ar consrv qantitis an ar th rslt o an accmlation (or intgration) procss. Thy ar th stat variabls o th systm. In C lmnts, lik a capacitor or spring, th consrv qantity, q, is stor by accmlating th nt low,, to th storag lmnt. This rslts in th irntial qation q which is call a balanc qation, an orms a part o th constittiv qations o th storag lmnt. In th othr part o th constittiv qations, th stat variabl, q, is rlat to th ort ( q) This rlation pns on th spciic shap o th particlar storag lmnt. In Figr 8, xampls o C lmnts ar givn togthr with th qivalnt block iagram. Th qations or a linar capacitor an linar spring ar: q i, C q x v, F Kx C x For a capacitor, C [F] is th capacitanc an or a spring, K [N/m] is th stinss an C [m/n] th complianc. For all othr omains, a C lmnt can b in. lmnt 6 /

7 Univrsity o Twnt, pt EE Intro Bon Graphs Th ort variabl is qal whn two C storag lmnts connct in paralll with a rsistor in btwn ar in qilibrim. Thror, th omain inpnnt proprty o an ort is trmination o qilibrim. omain spciic symbols Bon-graph lmnt Eqations Block iagram xpansion Capacitor Translational spring C : C C q z q t q( ) q C otational spring Figr 8: Exampls o C lmnts In I lmnts, lik an inctor or mass, th consrv qantity, p, is stor by accmlating th nt ort,, to th storag lmnt. Th rslting irntial qation is ṗ which is th balanc qation. Th lmnt spciic part o th constittiv qations is ( p) In Figr 9, xampls o I lmnts ar givn togthr with th qivalnt block iagram. Th qations or a linar inctor an linar mass ar: λ, i λ p F, v m p For an inctor, [H] is th inctanc an or a mass, m [kg] is th mass. For all othr omains, an I lmnt can b in. Th low variabl is qal whn two I storag lmnts connct in paralll with a rsistor in btwn, ar in qilibrim. Thror, at I lmnts, th omain inpnnt proprty o th low is trmination o qilibrim. For xampl, whn two bois, moving rly in spac ach having a irnt momntm, ar bing copl (colli an stick togthr), th momntm will ivi among th masss sch that th vlocity o both masss is th sam (this is th consrvation law o momntm). omain spciic symbols Bon-graph lmnt Eqations Block iagram xpansion Inctor Mass I : I I p z p t p( ) p I Inrtanc Figr 9: Exampls o I lmnts /

8 Intro Bon Graphs Jan F. Bronink, 999 Not that whn at th two typs o storag lmnts, th rol o ort an low ar xchang: th C lmnt an th I lmnt ar ach othr s al orm. Th block iagrams in Figr 8 an 9, an also in th nxt Figrs to 6, show th comptational irction o th signals involv. Thy ar in th xpansion o th corrsponing casal bon graph. Th qations ar givn in comptational orm, consistnt with th casal bon graph an th block iagram. 4. sistors sistors, lmnts, issipat r nrgy. Exampls ar amprs, rictions an lctric rsistors (Figr ). In ral-li mchanical componnts, riction is always prsnt. Enrgy rom an arbitrary omain lows irrvrsibly to th thrmal omain (an hat is proc). This mans that th nrgy low towars th rsistor is always positiv. Th constittiv qation is an algbraic rlation btwn th ort an low, an lis principally in th irst or thir qarant. r( ) omain-spciic symbols Bon-graph lmnt Eqations Block iagram xpansion sistor Friction Friction ampr ampr Figr : Exampls o rsistors : An lctrical rsistor is mostly linar (at constant tmpratr), namly Ohm s law. Th lctrical rsistanc val is in [Ω]. i Mchanical riction mostly is non- linar. Th rsistanc nction is a combination o ry riction an viscos riction. ry riction is a constant riction orc an viscos riction is th linar trm. Somtims, also stiktion is involv, a taring loos orc only applicabl whn starting a movmnt. All ths orms o riction can b moll with th lmnt. Th viscos riction has as ormla ( in [Ns/m]: F v I th rsistanc val can b controll by an xtrnal signal, th rsistor is a molat rsistor, with mnmonic M. An xampl is a hyralic tap: th position o th tap is controll rom th otsi, an it trmins th val o th rsistanc paramtr. I th thrmal omain is moll xplicitly, th proction o thrmal nrgy shol xplicitly b inicat. Sinc th issipator irrvrsibly procs thrmal nrgy, th thrmal port is rawn as a kin o sorc o thrmal nrgy. Th bcoms an S. 4. Sorcs Sorcs rprsnt th intraction o a systm with its nvironmnt. Exampls ar xtrnal orcs, voltag an crrnt sorcs, ial motors, tc. (Figr ). pning on th typ o th impos variabl, ths lmnts ar rawn as S or S. 8 /

9 Univrsity o Twnt, pt EE Intro Bon Graphs Bsis as a ral sorc, sorc lmnts ar s to giv a variabl a ix val, or xampl, in cas o a point in a mchanical systm with a ix position, a S with val is s (ix position mans vlocity zro). Whn a systm part ns to b xcit, otn a known signal orm is n, which can b moll by a molat sorc rivn by som signal orm. An xampl is shown in Figr. omain spciic symbols Bon-graph lmnt Eqations Block iagram xpansion F Voltag sorc T b : S b b Forc sorc Torq sorc v Crnt sorc ω b : S b b Vlocity sorc Anglar vlocity sorc Figr : Exampls o sorcs Signal Powr : Ugn Voltag pls Pls Figr : Exampl o a molat voltag sorc 4.4 Transormrs an Gyrators An ial transormr is rprsnt by TF an is powr continos (i.. no powr is stor or issipat). Th transormation can within th sam omain (tooth whl, lvr) or btwn irnt omains (lctromotor, winch), s Figr. Th qations ar: n n Eorts ar transc to orts an lows to lows. Th paramtr n is th transormr ratio. to th powr continity, only on imnsionlss paramtr, n, is n to scrib both th ort transction an th low transction. Th paramtr n is nambigosly in as ollows: an blong to th bon pointing towars th TF. This way o ining th transormation ratio is stanar in laing pblications (Karnopp an osnbrg, 99; Thoma, 989; Brvl, 985; Cllir, 99). I n is not constant, th transormr is a molat transormr, a MTF. Th transormr ratio now bcoms an inpt signal to th MTF. 9 /

10 Intro Bon Graphs Jan F. Bronink, 999 omain-spciic Symbols Bon-graph lmnt Eqations Block-iagram xpansion Transormr TF.. n n n n n Cantilvr Mchanical gar TF.. n Figr : Exampls o transormrs / n / n n n An ial gyrator is rprsnt by GY, an is also powr continos (i.. no powr is stor or is issipat. Exampls ar an lctromotor, a pmp an a trbin. al li ralisations o gyrators ar mostly transcrs rprsnting a omain transormation (Figr 4). Th qations ar: r r Th paramtr r is th gyrator ratio, an to th powr continity, only on paramtr to scrib both qations. No rthr inition is n sinc th qations ar symmtric (it os not mattr which bon points inwars, only that on bon points towars an th othr points orm th gyrator). r has a physical imnsion, sinc r is a rlation btwn ort an low (it has th sam imnsion as th paramtr o th lmnt). I r is not constant, th gyrator is a molat gyrator, a MGY. omain-spciic symbols Bon-graph lmnt Eqations Block-iagram xpansion i, Motor Gnrator T,ω GY.. r r r r r T,ω Pmp Trbin p,ϕ Figr 4: Exampls o gyrators GY.. r / r / r r r 4.5 Jnctions Jnctions copl two or mor lmnts in a powr continos way: thr is no nrgy storag or issipation in a jnction. Exampls ar a sris connction or a paralll connction in an lctrical ntwork, a ix copling btwn parts o a mchanical systm. Jnctions ar port symmtric: th ports can b xchang in th constittiv qations. Following ths proprtis, it can b provn that thr xist only two pairs o jnctions: th jnction an th jnction. Th jnction rprsnts a no at which all orts o th conncting bons ar qal (Figr 5). An xampl is a paralll connction in an lctrical circit. to th powr continity, th sm o th lows o th conncting bons is zro, consiring th sign. Th powr irction (i.. irction o th hal arrow) trmins th sign o th lows: all inwar pointing bons gt a pls an all otwar /

11 Univrsity o Twnt, pt EE Intro Bon Graphs pointing bons gt a mins. (Figr X). This smmation is th Kirchho crrnt law in lctrical ntworks: all crrnts conncting to on no sm to zro, consiring thir signs: all inwar crrnts ar positiv an all otwar crrnts ar ngativ. W can pict th jnction as th rprsntation o an ort variabl, an otn th jnction will b intrprt as sch. Th jnction is mor than th (gnralis) Kirchho crrnt law, namly also th qality o th orts (lik lctrical voltags bing qal at a paralll connction). omain-spciic symbols Bon-graph lmnt Eqations Block-iagram xpansion U _ Figr 5: Exampl o a -jnction Th jnction (Figr 6) is th al orm o th jnction (rols o ort an low ar xchang). Th -jnction rprsnts a no at which all lows o th conncting bons ar qal. An xampl is a sris connction in an lctrical circit. Th orts sm to zro, as a consqnc o th powr continity. Again, th powr irction (i.. irction o th hal arrow) trmins th sign o th orts: all inwar pointing bons gt a pls an all otwar pointing bons gt a mins. This smmation is th Kirchho voltag law in lctrical ntworks: th sm o all voltag irncs along on clos loop (a msh) is zro. In th mchanical omain, th jnction rprsnts a orc balanc (also call th principl o Almbrt), an is a gnralisation o Nwton s thir law, action raction). Jst as with th jnction, th jnction is mor than ths smmations, namly th qality o th lows. Thror, w can pict th jnction as th rprsntation o a low variabl, an otn th -jnction will b intrprt as sch. omain-spciic symbols Bon-graph lmnt Eqations Block iagram xpansion i _ Figr 6: Exampl o a -jnction 4.6 Positiv orintation By inition, th powr is positiv in th irction o th powr bon (i.. irction o th hal arrow). A port that has an incoming bon connct to, consms powr i this powr is positiv (i.. both ort an low ar ithr positiv or ngativ, as th proct o ort an low is th powr). In othr wors: th powr lows in th irction o th hal arrow i it is positiv an th othr way i it is ngativ., C an I lmnts hav an incoming bon (hal arrow towars th lmnt) as stanar, which rslts in positiv paramtrs whn molling ral li componnts. For sorc lmnts, th stanar is otgoing, as sorcs mostly livr powr to th rst o th systm. A ral li sorc thn has a positiv paramtr. For TF an GY lmnts (transormrs an gyrators), th stanar is to hav on bon incoming an on bon otgoing, to show th natral low o nrgy. Frthrmor, sing th stanar inition o th paramtr at th transormr (incoming bon is connct to port an th ratio n is / ) positiv paramtrs will b th rslt. Not that a gyrator os not n sch a inition, sinc its qations ar symmtric. /

12 Intro Bon Graphs Jan F. Bronink, 999 It is possibl, howvr, that ngativ paramtrs occr. Namly, at transormrs an sorcs in th mchanical omain whn thr is a rvrs o vlocity or th sorc acts in th ngativ irction. Using th initions iscss in this sction, th bon graph inition is nambigos, implying that in principl thr is no n or consion. Frthrmor, this systmatic way will hlp rsolving possibl sign placing problms otn ncontr in molling. 4. ality an al omains As inicat in sction 4., th two storag lmnts ar ach othr s al orm. Th rol o ort an low in a C lmnt an I lmnt ar xchang. aving on o th storag lmnts (an also on o th sorcs) ot o th list o bon graph lmnts, to mak this list as small as possibl, can b sl rom a mathmatical viwpoint, bt os not nhanc th insight in physics. composing an I lmnt into a GY an a C, thogh, givs mor insight. Th only storag lmnt now is th C lmnt. Th low is only a tim rivativ o a consrv qantity, an th ort trmins th qilibrim. This implis that th physical omains ar actally pairs o two al omains: in mchanics, w hav potntial an kintic omains or both rotation an translation), in lctrical ntworks, w hav th lctrical an magntic omains. Howvr, in th thrmoynamic omain, no sch al orm xists (Brvl, 98). This is consistnt with th act that no thrmal I typ storag xists (as a consqnc o th scon law o thrmoynamics: in a thrmally isolat systm, th ntropy nvr crass). 4.8 Ovrviw W hav iscss th basic bon graph lmnts an th bons, so w can transorm a omain pnnt ial physical mol, writtn in omain pnnt symbols, into a bon graph. For this transormation, thr is a systmatic procr, which will b prsnt in th nxt sction. 5 Systmatic procr to riv a bon graph mol To gnrat a bon graph mol starting rom an ial physical mol, a systmatic mtho xist, which w will prsnt hr as a procr. This procr consists roghly o th intiication o th omains an basic lmnts, th gnration o th connction strctr (call th jnction strctr), th placmnt o th lmnts, an possibly simpliying th graph. Th procr is irnt or th mchanical omain compar to th othr omains. Ths irncs ar inicat btwn parnthsis. Th rason is that lmnts n to b connct to irnc variabls or across variabls. Th orts in th non mchanical omains an th vlocitis (lows) in th mchanical omains ar th across variabls w n. Stp an concrn th intiication o th omains an lmnts. trmin which physical omains xist in th systm an intiy all basic lmnts lik C, I,, S, S, TF an GY. Giv vry lmnt a niq nam to istingish thm rom ach othr. Inicat in th ial physical mol pr omain a rrnc ort (rrnc vlocity with positiv irction or th mchanical omains). Not that only th rrncs in th mchanical omains hav a irction. Stps throgh 6 scrib th gnration o th connction strctr (call th jnction strctr). Intiy all othr orts (mchanical omains: vlocitis) an giv thm niq nams. 4 raw ths orts (mchanical: vlocitis), an not th rrncs, graphically by jnctions (mchanical: jnctions). Kp i possibl, th sam layot as th IPM. 5 Intiy all ort irncs (mchanical: vlocity ( low) irncs) n to connct th ports o all lmnts nmrat in stp to th jnction strctr. /

13 Univrsity o Twnt, pt EE Intro Bon Graphs Giv ths irncs a niq nam, prrably showing th irnc natr. Th irnc btwn an can b inicat by. 6 Constrct th ort irncs sing a jnction (mchanical: low irncs with a jnction) accoring to Figr, an raw thm as sch in th graph. v v v v v Figr : Constrction o ort irncs (vlocity irncs) Th jnction strctr is now ray an th lmnts can b connct. Connct th port o all lmnts on at stp with th jnctions o th corrsponing orts or ort irncs (mchanical: jnctions o th corrsponing lows or low irncs). 8 Simpliy th rslting graph by applying th ollowing simpliication rls (Figr 8): A jnction btwn two bons can b lt ot, i th bons hav a throgh powr irction (on bon incoming, th othr otgoing). A bon btwn two th sam jnctions can b lt ot, an th jnctions can join into on jnction. Two sparatly constrct intical ort or low irncs can join into on ort or low irnc. /

14 Intro Bon Graphs Jan F. Bronink, 999 a i n i n i t i t b i n i n i t i t c a a c c x x b b a a c c x x b b Figr 8: Simpliication rls or th jnction strctr. (a, b) Elimination o a jnction btwn bons. (c, ) Contraction o two th sam jnctions. (, ) Two sparatly constrct intical irncs s to on irnc. W will illstrat ths stps with a concrt xampl consisting o an lctromotor by lctric mains, a cabl rm an a loa (Figr 9). Cabl rm Mains Motor oa Figr 9: Sktch o th hoisting vic. A possibl ial physical mol (IPM) is givn in Figr. Th mains is moll as an ial voltag sorc. At th lctromotor, th inctanc, lctric rsistanc o th coils, baring riction an rotary inrtia ar takn into accont. Th cabl rm is th transormation rom rotation to translation, which w consir as ial. Th loa consists o a mass an th gravity orc. Starting rom th IPM o Figr, w will constrct a bon graph sing th 8 stps mntion abov. 4 /

15 Univrsity o Twnt, pt EE Intro Bon Graphs l U sorc K Elctric omain baring otation Mchanic omain m Translation g Figr : Possibl ial-physical mol agmnt with th omain inormation o stp. Stp This systm contains: An lctric omain part with a voltag sorc (S), a rsistor (), an inctor (I) an th lctric port o th lctromotor (GY port). A rotation mchanic omain part with th rotation port o th lctromotor (GY port), baring riction (), inrtia (I), an th axis o th cabl rm (TF port). A translation mchanic omain part with th cabl o th cabl rm (TF port), th mass o th loa (I) an th gravity orc acting on th mass (S). In Figr, th omains ar inicat an all lmnts hav a niq nam. Stp Th rrncs ar inicat in th ial physical mol: th voltag, th rotational vlocity ω an th linar vlocity v. Th two vlocitis also gt a positiv orintation (i.. a irction in which th vlocity is positiv). This rslt is shown in Figr. v Figr : rncs a to th IPM. ω Stp Th othr voltags, anglar vlocitis an linar vlocitis ar soght or an ar inicat in th IPM (Figr ). Ths variabls ar rspctivly,,, ω, v. 5 /

16 Intro Bon Graphs Jan F. Bronink, 999 U ω v v ω Figr : Th IPM agmnt with rlvant voltags, vlocitis an anglar vlocitis. Stp 4 Th variabls on in stp ar pict with rspctivly jnctions in Figr, in a layot compatibl to th IPM. Th rrncs ar not rawn, bcas thy ar so to spak liminat (rrncs hav th val an o not contribt to th ynamic bhavior). ω Figr : First sklton o th bon graph: Voltags ar shown as jnctions an vlocitis as jnctions. Stp 5 Whn chcking all ports o th lmnts on in stp or voltag irncs, anglar vlocity irncs an linar vlocity irncs, only an ar intii. No vlocity irncs ar n. Stp 6 Th irnc variabls ar rawn in th bon graph, s Figr 4. Atr this stp, th jnction strctr is gnrat an th lmnts can b connct. :U :U v ω Figr 4: irnc variabls ( an ) shown in th bon graph. Stp All lmnts ar connct to th appropriat jnctions, as shown in Figr 5. Not that nonmchanical omain lmnts ar always connct to -jnctions (orts or ort irncs) an that mchanical omain lmnts ar always connct to jnctions. v 6 /

17 Univrsity o Twnt, pt EE Intro Bon Graphs : l I: I:J S.. U sorc GY.. ω K TF: / : baring v S: -mg Figr 5: Th complt bon graph. Stp 8 As last action, th bon graph ns to b simplii, to liminat sprlos jnctions (accoring to th rls givn in Figr 8). Th rslting bon graph is th otcom o th systmatic mtho. I: I:J I:m S.. GY.. TF: / U sorc K : l : baring S: -mg Figr 6: Th simplii bon graph, th rslt o th systmatic mtho. I:m Obviosly, this systmatic mtho is not th only mtho or riving bon graphs rom ial physical mols (IPMs). Anothr mtho is th so call inspction mtho, whr parts o th IPM ar rcognis that can b rprsnt by on jnction. An xampl is a sris connction in an lctrical ntwork, which is rawn as on jnction. This is th cas in th xampl abov: Th voltag sorc, inctor, lctric rsistor an lctric port o th motor ar irctly connct to on jnction. Althogh th inspction mtho is shortr than th systmatic mtho, it is rathr rror pron. 6 Casal analysis Casal analysis is th trmination o th signal irction o th bons. Th nrgtic connction (bon) is now intrprt as a bi-irctional signal low. Th rslt is a casal bon graph, which can b sn as a compact block iagram. Casal analysis is in gnral compltly covr by molling an simlation sotwar packags that spport bon graphs lik Enport (osnbrg, 94), MS (ornz, 99), CAMP (Grana, 985) an - SIM (Bronink, 99, 995, 99, 999b; Bronink an Klijn, 999). Thror, in practic, casal analysis n not b on by han. Bsis rivation o qations, casal analysis can giv insight in th corrctnss an comptncy o th mol. This last rason spcially motivats th iscssion o casal analysis in this chaptr. pnnt on th kin o qations o th lmnts, th lmnt ports can impos constraints on th connct bons. Thr ar or irnt constraints, which will b trat bor a systmatic procr or casal analysis o bon graphs is iscss. /

18 Intro Bon Graphs Jan F. Bronink, Casal constraints Fix casality Fix casality is th cas, whn th qations only allow on o th two port variabls to b th otgoing variabl. This occrs at sorcs: an ort sorc (S) has by inition always its ort variabl as signal otpt, an has th casal strok otwars. This casality is call ort-ot casality or ort casality. A low sorc (S) clarly has a low-ot casality or low casality. Anothr sitation whr ix casality occrs is at nonlinar lmnts, whr th qations or that port cannot b invrt (or xampl, ivision by zro). This is possibl at, GY, TF, C an I lmnts. Ths, thr ar two rasons to impos a ix casality:. Thr is no rlation btwn th port variabls.. Th qations ar not invrtabl ( singlar ). Constrain casality At TF, GY, an jnction, rlations xist btwn th casalitis o th irnt ports o th lmnt. Ths rlations ar casal constraints, sinc th casality o a particlar port imposs th casality o th othr ports. At a TF, on o th ports has ort-ot casality an th othr has lowot casality. At a GY, both ports hav ithr ort-ot casality or low-ot casality. At a jnction, whr all orts ar th sam, xactly on bon mst bring in th ort. This implis that jnctions always hav xactly on casal strok at th si o th jnction. Th casal conition at a jnction is th al orm o th -jnction. All lows ar qal, ths xactly on bon will bring in th low, implying that xactly on bon has th casal strok away rom th jnction. Prrr casality At th storag lmnts, th casality trmins whthr an intgration or irntiation with rspct to tim will b th cas. Intgration has prrnc abov a irntiation. At th intgrating orm, an initial conition mst b spcii. Bsis, intgration with rspct to tim is a procss, which can b ralis physically. Nmrical irntiation is not physically ralisabl, sinc inormation at tr tim points is n. Anothr rawback o irntiation occrs whn th inpt contains a stp nction: th otpt will thn bcom ininit. Thror, intgrating casality is sn as th prrr casality. This implis that a C lmnt has ort-ot casality an an I lmnt has low-ot casality at its prrnc. Ths prrncs ar also illstrat in Figr 8 an Figr 9, whn looking at th block iagram xpansion. Exampl Whn a voltag is impos on an lctrical capacitor (a C lmnt), th crrnt i is th rslt o th constittiv qation o th capacitor: i C t A irntiation is ths happning. W hav a problm whn th voltag instantly stps to anothr val, sinc th crrnt will b ininit (th rivativ o a stp is ininit). This is not th cas whn th crrnt is impos on a capacitor. Now, an intgral is s: it Th irst cas is low-ot casality (ort impos, low th rslt), an th scon cas is ort ot casality, which is th prrr casality. Frthrmor, an ort ot casality also rslts in a stat variabl with initial conition. At an inctor, th al orm o th C lmnt is th cas: low-ot casality will rslt in an intgral casality, bing th prrnc. Inirnt casality Inirnt casality is s, whn thr ar no casal constraints! At a linar, it os not mattr which o th port variabls is th otpt. Consir an lctrical rsistor. Imposing a crrnt (low) yils: 8 /

19 Univrsity o Twnt, pt EE Intro Bon Graphs i It is also possibl to impos a voltag (ort) on th linar rsistor: i Thr is no irnc choosing th crrnt as incoming variabl an th voltag as otgoing variabl, or th othr way aron. Ovrviw Th S an S hav a ix casality, th C an I hav a prrr casality, th TF, GY, an hav constrain casality, an th has an inirnt casality (provi that th qations o ths basic lmnts all ar invrtabl). Ths casal orms hav bn shown in sction 4. Whn th qations ar not invrtabl, a ix casality mst b s. 6. Casal analysis procr Th procr or assigning casality on a bon graph starts with thos lmnts that hav th strongst casality constraint namly ix casality (viation o th casality conition cannot b grant by rwriting th qations, sinc rwriting is not possibl). Via th bons (i.. connctions) in th graph, on casality assignmnt can cas othr casalitis to b assign. This ct is call casality propagation: atr on assignmnt, th casality propagats throgh th bon graph to th casal constraints. Th casality assignmnt algorithm is as ollows: a. Chos a ix casality o a sorc lmnt, assign its casality, an propagat this assignmnt throgh th graph sing th casal constraints. Go on ntil all sorcs hav thir casalitis assign. b. Chos a not yt casal port with ix casality (non-invrtabl qations), assign its casality, an propagat this assignmnt throgh th graph sing th casal constraints. Go on ntil all ports with ix casality hav thir casalitis assign.. Chos a not yt casal port with prrr casality (storag lmnts), assign its casality, an propagat this assignmnt throgh th graph sing th casal constraints. Go on ntil all ports with prrr casality hav thir casalitis assign.. Chos a not yt casal port with inirnt casality, assign its casality, an propagat this assignmnt throgh th graph sing th casal constraints. Go on ntil all ports with inirnt casality hav thir casalitis assign. Otn, th bon graph is compltly casal atr stp, withot any casal conlict (all casal conitions ar satisi). I this is not th cas, thn th momnt in th procr whr a conlict occrs or whr th graph bcoms compltly casal, can giv insight in th corrctnss an comptnc o th mol. Bor iscssing ths isss, irst an xampl will b trat. Exampl To xmpliy th casality algorithm, th sam xampl as in sction 5 is s. In Figr, th compltly casal bon graph o th hoisting vic is shown. Nmbrs at th casal stroks inicat th orr in which th bons wr ma casal. At stp a, w assign casality an. It os not mattr with which sorc w start. No casality can b propagat. Stp b is not applicabl, sinc thr ar no ports o that catgory. At stp, w start with th inctor. Propagation is rom strok ntil strok 6. Th nxt storag lmnt (prrr casality) is inrtia J. Propagation o this casality (nmbr ), complts th casality o th graph, implying that w o not n stp. Howvr, th mass o th loa os not gt his prrnc. What th consqncs ar, is sbjct o th nxt sction. 9 /

20 Intro Bon Graphs Jan F. Bronink, 999 I: I:J 5 6 S.. GY.. U sorc K TF: / : l : baring S: -mg I:m Figr : Th casal bon graph o th hoisting vic 6. Mol insight via casal analysis W iscss hr thos sitations whrby conlicts occr in th casal analysis procr or whn stp o th algorithm appars to b ncssary. Th plac in procr whr a conlict appars or th bon graph bcoms compltly casally agmnt, can giv insight in th corrctnss o th mol. Otn, th bon graph is compltly casal atr stp, withot any casal conlict (all casal conitions ar satisi). Each storag lmnt rprsnts a stat variabl, an th st o qations is an xplicit st o orinary irntial qations (not ncssarily linar or tim invariant). Whn th bon graph is compltly casal atr stp a, th mol os not hav any ynamics. Th bhavior o all variabls now is trmin by th ix casalitis o th sorcs. Ariss a casal conlict at stp a or at stp b, thn th problm is ill pos. Th mol mst b chang, by aing som lmnts. An xampl o a casal conlict at stp a is two ort sorcs connct to on - jnction. Both sorcs want to trmin th on ort variabl. At a conlict at stp b, a possibl ajstmnt is changing th qations o th ix casality lmnt sch that ths qations bcom invrtabl, an ths th ixnss o th constraint isappars. An xampl is a io or a valv having zro crrnt rsp. low whil blocking. Allowing a small rsistanc ring blocking, th qations bcom invrtabl. Whn a conlict ariss at stp, a storag lmnt rcivs a non- prrr casality. This mans that this storag lmnt os not rprsnt a stat variabl. Th initial val o this storag lmnt cannot b chosn rly. Sch a storag lmnt otn is call a pnnt storag lmnt. This inicats that a storag lmnt was not takn into accont ring molling, which shol b thr rom physical systms viwpoint. It can b libratly omitt, or it might b orgottn. At th hoisting vic xampl, th loa o th hoist (I lmnt) is sch a pnnt storag lmnt. Elasticity in th cabl was not moll. I it ha bn moll, a C storag lmnt connct to a jnction btwn th cabl rm an loa wol appar. Whn stp o th casality algorithm is ncssary, a so call algbraic loop is prsnt in th graph. This loop cass th rslting st irntial qations to b implicit. Otn this is an inication that a storag lmnt was not moll, which shol b thr rom a physical systms viwpoint. In gnral, irnt ways to hanl th casal conlicts arising at stp or stp ar possibl:. A lmnts. For xampl, yo can withraw th cision to nglct crtain lmnts. Th a lmnts can b parasitic, or xampl, to a lasticity (C lmnt) in a mchanical connction, which was moll as rigi. Aitionally aing a amping lmnt () rcs th simlation tim consirably, which is bing avis.. Chang th bon graph sch that th conlict isappars. For a stp conlict, th pnnt storag lmnt is takn togthr with an inpnnt storag /

21 Univrsity o Twnt, pt EE Intro Bon Graphs lmnt, having intgral casality. For a stp conlict, somtims rsistiv lmnts can b takn togthr to liminat th conlict. This can b prorm via transormations in th graph. Th complxity o this opration pns on th siz an kin o sbmols along th rot btwn th storag lmnts or rsistors nr concrn.. Th bon graph is not chang an ring simlation, a spcial (implicit) intgration rotin is n. Th implicit qations ar solv by mans o th itration schms prsnt in th implicit intgration mthos. In -SIM, only th BF mtho can hanl this sitation. Also, th loop can b ct by aing a on-timstp lay, which is a rathr pragmatic soltion, bt can b sl whn no implicit intgration mthos ar availabl. Th accracy might gt too low. Not that th ampliication o all lmnts in th loop mst b smallr than on to obtain a stabl soltion. Algbraic loops an loops btwn a pnnt an an inpnnt storag lmnt ar call zro orr casal paths (ZCPs). Bsis ths two kins, thr ar thr othr kins, having an incrasing complxity an rslting in mor complx qations. Ths occr or instanc in rigi boy mchanical systms (van ijk an Brvl, 99). By intrprting th rslt o casal analysis, svral proprtis o th mol can b rcognis, which col othrwis only b on atr riving qations. 6.4 Orr o st o stat qations Th casal analysis also givs inormation on th orr o th st qations. Th nmbr o initial conitions qals th nmbr o storag lmnts with intgral casality, which was also th prrnc ring casality assignmnt. This nmbr is call th orr o th systm. In th xampl (Figr 8), th orr o th systm is. Th orr o th st stat qations is smallr than or qal to th orr o th systm, bcas storag lmnts can pn on ach othr. Ths kin o pnnt storag lmnts ach hav thir own initial val, bt thy togthr rprsnt on stat variabl. Thir inpt signals ar qal, or hav a actor in btwn (Figr 8). : : C C C :C C:C C :C C:C a) b) c) Figr 8: Systm with orr o stat qations smallr than orr o th systm. a) IPM; b) casality sing intgral prrnc; c) casality sing irntial prrnc. A rcip xists to chck whthr this kin o pnnt storag lmnts show p: Prorm casal analysis again, bt chos irntial casality as prrnc. For th xampl, this is on in Figr 9. /

22 Intro Bon Graphs Jan F. Bronink, 999 I: I:J 4 9 S GY.... U sorc K 8 TF: / 6 : l : baring S: -mg 5 Figr 9: irntial prrnc casality or th hoisting vic Thos storag lmnts that gt both at irntial prrnc an at intgral prrnc thir prrr casality ar th ral storag lmnts an contribt to th stat o th systm. Th orr o th st o stat qations, is by inition th amont o storag lmnts that gt in both cass thir prrr casality. At th hoisting vic, this is (namly an J). Th storag lmnts that gt in both cass not thir prrr casality, ar th pnnt storag lmnts. Thos storag lmnts that gt only at th intgral prrr casality cas thir prrr casality ar call smi-pnnt storag lmnts, to istingish thm rom th pnnt storag lmnts. This inicats that a storag lmnt was not takn into accont ring molling, which shol b thr rom physical systms viwpoint. 6.5 Matrix orm (linar systms) I th systm is linar, w can writ th rslting st o stat qations in th stanar orm, namly, ẋ Ax B whr A is th systm matrix, an B is th inpt matrix. Th orr o th systm is th imnsion o th sqar matrix A an th orr o th st o stat qations is th rank o A. Whn pnnt stats or algbraic loops ar prsnt, th matrix scription is as ollows: Ex Ax B I:m whr E is a sqar matrix o siz qal to th amont o storag lmnts pls th amont o algbraic loops. For ach irntial storag lmnt an or ach algbraic loop, E contains on row o zros. Th stat vctor x is xtn with th algbraic loop variabls, to nabl implicit intgration mthos to solv ths kins o systms. Th hoisting vic givs sch a st o qations. Gnration o qations A casal bon graph contains all inormation to riv th st o stat qations. It is ithr a st o orinary irst orr irntial qations, OEs, whn th mol is xplicit (no casal conlicts), or a st o irntial an algbraic qations, AEs, whn th mol is implicit (a casal conlict in stp o th procr or stp is ncssary, c. sction 6.). Th procr to riv th qations is covr by bon graph sotwar lik Enport (osnbrg, 94), MS (ornz, 99), CAMP (Grana, 985) an - SIM (Bronink, 99, 995, 99, 999b; Bronink an Klijn, 999). Thror, in practic, gnration o qations n not b on by han. Howvr, w o iscss th gnration o qations on th on han to b complt an on th othr han to inicat what xactly has to b on. W s th ollowing procr to gnrat qations:. W irst writ th st o mix irntial an algbraic qations. Ths ar th constittiv rlations o all lmnts in comptational orm, or casal orm. This compriss o n qations o /

23 Univrsity o Twnt, pt EE Intro Bon Graphs a bon graph having n bons. n qations compt an ort an n qations compt a low, or rivativs o thm.. W thn liminat th algbraic qations. W can organis this limination procss by irst liminat th intitis coming rom th sorcs an jnctions. Thratr, w sbstitt th mltiplications with a paramtr, stmming rom rsistors an transcrs (TF, GY). At last, w sbstitt th smmation qations o th jnctions into th irntial qations o th storag lmnts. Within this procss, it is icint to irst mark th stat variabls. In principl, th stat variabls ar th contnts o th storag lmnts (p or q typ variabls). Howvr, i w writ th constittiv rlations o storag lmnts as on irntial qation, w can also s th orts at C-lmnts an lows at I-lmnts. I w ar going to gnrat th qations by han, w can tak th irst limination stp into accont whil ormlating th qations by, at th sorcs, irctly s th signal nction at th bon. Frthrmor, w can writ th variabl trmining th jnction along all bons connct to that jnction. Th variabl trmining th jnction is that variabl, which gts assign to bon variabls o all th othr bons connct to that jnction via th intitis o th jnction qations. At a jnction, this is th ort o th only bon with its casal strok towars th -jnction. At a jnction, this is th low o th only bon with its casal strok away rom th -jnction. In cas o pnnt storag lmnts, w hav to tak car that th accompanying stat variabl gts not liminat. Ths ar th so call smi stat variabls. Whn w mark th stat variabls, incling th smi stat variabls in this sitation, on borhan, w can prvnt th wrong variabl rom bing liminat. In cas o algbraic loops, implicit qations will b ncontr. W choos on o th variabls in ths loops as algbraic loop brakr an that variabl bcoms a smi stat variabl. S also sction 6.5. Th qation consisting th smi stat variabl o a storag lmnt gts liminat at th scon limination stp: it is a mltiplication. Th smi stat variabl itsl mst not b sbstitt.. Exampl Th st o mix irntial an algbraic qations o th hoisting vic is shown blow. Th orts an lows ar nmbr in th sam orr as thir casality was assign. W hav qations, o which compt a low an compt an ort. Sinc w want to gnrat th st o qations as irntial qations, w writ at th storag lmnts in intgral orm th qations as irntial qations an not as intgral qations. This is call rr intgration. t t K K 6 5 J sorc l 4 4 baring 8 5 mg 9 6 m t /

24 Intro Bon Graphs Jan F. Bronink, / As stat variabls, w hav an, whras is th smi stat variabl blonging to th mass having rivativ casality (it is an I lmnt). Not that th qation o th mass is writtn in casal orm: is th otpt. Atr liminating th intitis at th jnctions an sorcs, w will gt th ollowing qations. Not that th lows o th two ort sorcs ( an ) ar not s lswhr, so w lav thm ot J t K K t baring sorc l t m mg Atr sbstittion o th mltiplication with rsistors an transcrs, or st o qations rcs to th ollowing 6 qations: irntial qations, constraint qation, compting th smi stat variabl an 4 jnction qations. Blow, only stat variabls, jnction variabls an inpt variabls ar s. K J t K t baring l sorc mg t m Atr sbstittion o th smmations at th jnctions into th irntial qations an th constraint qations, th rslt looks lik: mg t m J J K t K t baring l sorc This systm is a linar systm, so w can writ th qations in matrix orm (accoring to th scon orm o sction 6.5): mg J J K K t sorc baring l

25 Univrsity o Twnt, pt EE Intro Bon Graphs Whn th mol irst was ma xplicit by aing lmnts, accoring to altrnativ o sction 6., th casal bon graph an th qations ar givn blow. W a th lasticity o th rop: A C- lmnt connct to a -jnction is insrt on th bon btwn th TF o th cabl rm an th - jnction o th payloa (s Figr ). Th orts an lows ar nmbr in th sam orr as thir casality was assign. W hav 6 qations, o which compt a low an compt an ort. I: I:J 5 6 S GY.... U sorc K TF: / : l : baring C:C lasticity S: -mg I:m Figr Casal bon graph o th Hoist with lasticity a Th mix st o irntial an algbraic qations is: t t K K 6 5 J sorc l 4 4 baring t t 6 m 8 C mg 9 lasticity As stat variabls, w hav,, an. Applying th procr o riving qations las to th ollowing st o stat qations writtn in matrix orm: 9 5 /

26 Intro Bon Graphs Jan F. Bronink, 999 t l K J C lasticity K J C baring lasticity m mg m sorc 8 Expansion to block iagrams To show that a casal bon graph is a compact block iagram, w trat in this sction th xpansion o a casal bon graph to a block iagram. Frthrmor, a block iagram rprsntation o a systm might b mor amiliar than a bon graph rprsntation. Ths this work might hlp nrstaning bon graphs. Th xpansion o a casal bon graph into a block iagram consists o thr stps:. Expan all bons to bilatral signal lows (two signals with opposit irctions). Th casal strok trmins in which irction th ort lows. Th bon graph lmnts can b ncircl to connct th signals to.. plac th bon graph lmnts by thir block iagram rprsntations (s sction 4). c th signs o th smmations o th jnctions rom th irctions o th bon arrows (hal arrows, s sction 4.5). Otn, it is icint to trmin thos signs atr all bon graph lmnts ar writtn in block iagram orm. Th block iagram is ray in principl.. raw th block iagram in stanar orm: all intgrators in an ongoing stram (orm lt to right) an all othr oprations as back loops. O cors, this is not always possibl. Blocks might b takn togthr. Sinc block iagrams rprsnt mathmatical oprations, or which commtativ an associativ proprtis apply, ths proprtis can b s to maniplat th block iagram sch that th rslt looks appaling nogh. As an xampl, w show th block iagram o th hoisting vic, sing th thr stps to constrct th block iagram. U sorc Figr Expansion o bons to bilatral signal lows o th hoisting vic 6 /

27 Univrsity o Twnt, pt EE Intro Bon Graphs I J I U sorc S - - GY K K / TF / l baring - S -mg t I m Figr Bon graph xpan to a block iagram in th layot o th bon graph (th bon graph is shown in gry) / - U sorc / K J t m - l baring -mg K Figr Th block iagram rrawn in stanar orm Th block iagram rom th casal bon graph o th Hoist with lasticity a (s Figr ) is shown in Figr 4. / -mg - U sorc / K J C lasticity /m l baring K Figr 4 Th block iagram o th Hoist with lasticity (c Figr ) /

28 Intro Bon Graphs Jan F. Bronink, Simlation Th rslting st o qations coming rom a bon graph mol is call th simlation mol. It consists o irst orr orinary irntial qations (OEs), possibly xtn with algbraic constraint qations (AEs). Hnc, it can b simlat sing stanar nmrical intgration mthos. Howvr, bcas nmrical intgration is an approximation o th actal intgration procss, it is sl to chck th simlation mol on aspcts signiicant or simlation. As a rslt, an appropriat intgration mtho can b chosn: th comptational work is minimal an th rslts stay within a spcii rror margin. Sinc at casal analysis, on can ci whthr or not to chang th bon graph mol to obtain an xplicit simlation mol, it is sl to know abot th consqncs or simlation o rlvant charactristics o th simlation mol. Th ollowing 4 aspcts o simlation mols ar rlvant or choosing an nmrical intgration mtho:. Prsnc o implicit qations. Implicit mols (AEs) can only b simlat with implicit intgration mthos. Th itration procr o th implicit intgration mtho is also s to calclat th implicit mol. Explicit mols (OEs) can b simlat with both xplicit as implicit intgration mthos. Somtims, implicit intgration mthos n mor comptation tim than xplicit intgration mthos. ring casal analysis, on can s whthr a simlation mol will b xplicit or implicit, s sction 6... Prsnc o iscontinitis. Intgration mthos with spcial provisions or vnts will prorm bst. I that is not availabl, variabl stp mthos can b s. Mltistp mthos bcom lss accrat, sinc thy n inormation rom th past, which is slss atr a iscontinity. Whil constrcting th mol, prsnc o iscontinitis can b mark.. Nmrical stinss. S(t), th stinss ratio, is a masr or th istanc btwn ral parts o ignvals, λ, namly S(t) max( (λ(t) ) / min( (λ(t) ). Sti mols (larg S) n sti intgration mthos. Th tim stp is now trmin by th stability insta o th accracy (namly, ignvals ar now s to trmin th stp siz). Whn th high rqncy parts ar a ot, thy o not inlnc th stp siz anymor. Hnc, th stp siz can grow to limits trmin by th lowr rqncis. 4. Oscillatory parts. Whn a mol has no amping, it shol not b simlat with a sti mtho. Sti mthos prorm baly or ignvals on th imaginary axis (i.. no amping) o th complx ignval plan. Eignvals can b localis in a casal bon graph, spcially whn all lmnts ar linar. Thr is a bon graph vrsion o Mason s oop rl to trmin th transr nction rom a bon graph (Brown, 9). As a si ct, th ignvals can b calclat. W will not iscss th procr to obtain ignvals rom a casal bon graph by han. Conclsion. viw In this chaptr, w hav introc bon graphs to mol physical systms in a omain inpnnt way. Only macroscopic systms ar trat, ths qantm cts o not play a signiicant rol. omain inpnc has its basics in th act that physical concpts ar analogos or th irnt physical omains. 6 irnt lmntary concpts xist: storag o nrgy, issipation, transction to othr omains, istribtion, transport, inpt or otpt o nrgy. Anothr starting point is that it is possibl to writ mols as irct graphs: parts ar intrconnct by bons, along which xchang o nrgy occrs. A bon rprsnts th nrgy low btwn th two 8 /

29 Univrsity o Twnt, pt EE Intro Bon Graphs connct sbmols. This nrgy low can b scrib as th proct o variabls (ort an low), ltting a bon b conciv as a bilatral signal connction. ring molling, th irst intrprtation is s, whil ring analysis an qations gnration th scon intrprtation is s. Frthrmor, w prsnt a mtho to systmatically bil a bon graph starting rom an ial physical mol. Casal analysis givs, bsis th comptational irction o th signals at th bons, also inormation abot th corrctnss o th mol. W prsnt mthos to riv th casality o a bon graph. In aition, procrs to gnrat qations an block iagrams ot o a casal bon graph ar prsnt. This chaptr is only an introction to bon graphs. Somtims, procrs ar jst prsnt, withot a p motivation an possibl altrnativs. It was also not th incntiv to laborat on physical systms molling. W i not iscss mltipl connctions (arrays o bons writtn as on mltibon) an mltiport lmnts (to scrib transcrs), nithr irnt casal analysis algorithms. Thos irnt casality algorithms giv slightly irnt sts o AEs spcially whn appli to crtain classs o mols (or instanc mltiboy systms with kinmatic loops).. Objct-orint physical-systms molling Not that bon graph molling is in act a orm o objct orint physical-systms molling, a trm which is crrntly otn s (Anrsson, 994; Elmqvist t al, 99; Mattson t al, 99; Åström t al, 998). This can b sn as ollows: bon graph mols ar clarativ, can b hirarchically strctr, an lly spport ncapslation ( to th non-casal way o spciying qations, an th notion o ports). Frthrmor, to allowing hirarchy, th notion o inition an s o mols ar istingish (i.. th class concpt an instantiation). Sinc bon graphs cam into xistnc bor th trm objct orint was s in th il o physical systms moling, bon graphs can b sn as an objct-orint physical systms moling paraigm avant-la-lttr. A bon-graph library was writtn in Molica (Mattson t al, 99; Åström t al, 998), th pcoming objct-orint molling langag (Bronink, 99b). Th basic bon graph lmnts an block iagram lmnts hav bn spcii in Molica, sing th ssntial objct orintation atrs inhritanc an ncapslation. Eqations hav bn spcii in an acasal ormat. Ths, it can b sai that th Molica molling concpts ar consistnt with bon graph concpts. Frthrmor, atomatic Molica co gnration rom bon graphs appar to b rathr straightorwar (Bronink, 999a).. Frthr aing For a mor thorogh analysis o bon graphs, s Payntr (96) an Brvl (984, 985), whil an xtnsiv iscssion on txtbook lvl is givn by Karnopp, Margolis an osnbrg (99). Cllir (99) wrot a txtbook on continos systm molling in which bsis bon graphs also othr molling mthos ar s. Crrnt rsarch on bon graphs is rport at th Intrnational Conrnc on Bon Graph Moling, vry two yars (Grana an Cllir, 99, 995, 99, 999). Jornals rglarly pblishing bon graph paprs ar th Jornal o th Franklin Institt, which also ha spcial isss on bon graphs (99) an th Jornal o ynamic Systms, Masrmnt an Control. Acknowlgmnts Most o th inspiration or this chaptr cam rom th tch cors matrial o Brvl an Van Amrongn (994). I sincrly acknowlg Ptr Brvl an Job van Amrongn or thir valabl sggstions an iscssions. rncs Anrsson, M., Objct orint moling an simlation o hybri systms, Ph Thsis, n Institt o Tchnology, Swn, (994). 9 /

30 Intro Bon Graphs Jan F. Bronink, 999 Åström K.J., Elmqvist, H, Mattson, S.E., (998), Evoltion o Continos tim moling an simlation, Procings o th th Eropan Simlaton Mlticonrnc (ESM 98), SCS Pblising, Manchstr UK, pp 9-8. Brvl P.C. an Amrongn, J. van (994), ynamisch systmn: molvorming n simlati mt bongran, (ynamic systms: molling an simlation with bon graphs), part -4, (in tch), tch Opn Univrsity, Hrln, Nthrlans, ISBN Brvl P.C., (98), Thrmoynamic Bon Graphs an th problm o Thrmal Inrtanc, Jornal o th Franklin Institt, vol 4, no, pp Brvl P.C., (984), Physical systms thory in trms o bon graphs, Ph thsis, Univrsity o Twnt, Ensch, Nthrlans. Brvl P.C., (985), Mltibon graph lmnts in physical systms thory, Jornal o th Franklin Institt, vol 9, no /, pp. -6. Brvl, P.C. an aphin Tangy G. (99), (s.) Crrnt topics in bon graph rlat rsarch, Jornal o Th Franklin Institt, Spcial iss on bon graph moling, Volm 8 no 5/6, Prgamon Prss. Bronink J.F. (99), Comptr ai Physical systms moling an simlation: a bon graph approach, Ph thsis, Univrsity o Twnt, Ensch, Nthrlans. Bronink, J.F. (99b), Bon graph moling in Molica, Procings o 9th Eropan Simlation Symposim, W Hahn, A hmann (s.), Passa Grmany, Oct 9-, pp -4. Bronink, J.F., (99a), Molling, Simlation an Analysis with -Sim. Jornal A, Spcial iss on CACS, vol 8 no, pp. -5, ISSN -. Bronink, J.F. (999a), Objct-orint molling with bon graphs an moica. Intrnational Conrnc on Bon Graph Moling ICBGM 99, Simlation Sris Vol no, SCS, pp Bronink, J.F., (999b), - SIM, sotwar or hirarchical bon-graph / block-iagram mols, Sbmitt to Jornal Simlation in Practic an Thory. Bronink, J.F., an C. Klijn, (999), Comptr-ai sign o mchatronic systms sing SIM., Sbmitt to WESIC'99, Workshop on Scintiic an Instrial Collaboration, Sp -, Nwport UK. Bronink, J.F., Wstink, P.B.T., (995) PC vrsion o th bon graph moling analysis an simlation tool CAMAS, Proc. Intrnational Conrnc on Bon Graph Moling ICBGM 95, Simlation Sris Vol no, SCS, pp. -8. Brown, F.T., (9), irct application o th loop rl to bon graphs, Jornal o ynamic Systms, Masrmnt an Control, Vol 94, no, pp5-6. Cllir F.E., Grana J.J., (995), (s.), Intrnational Conrnc on Bon Graph Moling ICBGM 95, Simlation Sris Vol no, Socity or Comptr Simlation, San igo, CA. Cllir, F.E., (99) Continos Systm Moling, Springr Vrlag, ISBN 8 95 ijk, J. van, Brvl P.C., (99), Simlation o Systm Mols Containing Zro-orr Casal Paths, J. Franklin Institt 8 (5/6), ixhoorn, J.J. van, (98), Bon graphs an th challng o a nii molling thory o physical systms, Progrss in molling an simlation, Cllir F.E. (.), Acamic Prss, Nw York, pp -45. Elmqvist, H., Cllir, F.E., an Ottr, M. (99) Objct orint moling o hybri systms, Proc. Eropan Simlation Symposim, lt, Nthrlans, -4, (99). Grana J.J., (985), Comptr gnration o physical systm irntial qations sing bon graphs, Jornal o th Franklin Institt, vol 9, no /, pp Grana J.J., Cllir F.E., (99), (s.), Intrnational Conrnc on Bon Graph Moling ICBGM 9, Simlation Sris Vol 5 no, Socity or Comptr Simlation, San igo, CA. Grana J.J., Cllir F.E., (999), (s.), Intrnational Conrnc on Bon Graph Moling ICBGM 99, Simlation Sris, Vol, Socity or Comptr Simlation, San igo, CA. Grana J.J., aphin-tangy G., (99), (s.), Intrnational Conrnc on Bon Graph Moling ICBGM 9, Simlation Sris Vol 9 no, Socity or Comptr Simlation, San igo, CA. Karnopp.C., an osnbrg.c., (968), Analysis an Simlation o Mltiport Systms Th bon graph approach to physical systm ynamics, MIT Prss, Cambrig MA. /

31 Univrsity o Twnt, pt EE Intro Bon Graphs Karnopp.C., an osnbrg.c., (95), Systm ynamics, a nii approach, J Wily, Nw York, NY. Karnopp.C., Margolis.., an osnbrg.c., (99), Systm ynamics, a nii approach, (n ition), J Wily, Nw York, NY, ISBN (Txtbook, a tachr s gi is also availabl). Kron G., (96), iakoptics: Th picwis Soltion o larg scal systms, Maconal & Co. t., onon. ornz F., (99), Molling Systm, srs manal, ornz Simlation, ig, Blgim Mattson, S.E., Elmqvist, H., an Bronink, J.F., (99), Molica: An intrnational ort to sign th nxt gnration moling langag, Jornal A, Spcial Iss CACS, vol 8 (), pp -5. Payntr, H.M. (96), Analysis an sign o nginring systms, MIT Prss, Cambrig, MA. osnbrg.c., Karnopp.C., (98), Introction to physical systm ynamics, McGraw Hill, Nw York, NY. osnbrg,.c. (94), A sr s gi to ENPOT 4, Wily, Nw York. Thoma, J.U. (95), Introction to bon graphs an thir applications, Prgamon Prss, Oxor. Thoma, J.U. (989) Simlation by bon graphs - Introction to a graphical mtho, Springr Vrlag. /