INC 693, 481 Dynamics System and Modelling: The Language of Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Professor
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1 INC 693, 48 Dynamics Systm and Modlling: Th Languag o Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Prossor Dpartmnt o Control Systm and Instrumntation Enginring King Mongkut s Unnivrsity o Tchnology Thonburi Thailand
2 Th Languag o Bound Graphs ˆ Lagrangian and Hamiltonian mthods can driv th dirntial quations o any givn systm. ˆ Th st o linar quations, starting rom any initial condition th tim volution o th systm can b obtaind in a closd orm by solving th quations. ˆ Most th dirntial quations ar nonlinar, which can only b solvd numrically with hlp o computrs. ˆ Human hav to driv th dirntial quations and solving thm is prormd by a computr. ˆ Pro. H. M. Payntr o MIT invntd a languag o systm rprsntation through xchang o powr and inormation by which th job o driving dirntial quations can b prormd by a computr. 2/35
3 Th Languag o Bound Graphs ˆ Th concpt o bond graphs was originatd by Pro. Payntr in 960. ˆ Th ida was urthr dvlopd by Pro. Karnopp and Rosnbrg. ˆ Thr ar a lot o sotwars supportd lik Enport, CAMP, 20-SIM. 3/35
4 Powr Variabls ˆ In bond graphs two powr conjugatd variabls ar assignd to ach dg. Th ar calld ort and low and ar dnotd by th lttr and. Powr = Eort Flow Annotating a bond with powr variabls ort and low: 4/35
5 Bond graph variabls usd in th various nrgy domains Domain Eort Flow momntum p displacmnt q Translational Forc Vlocity Momntum Displacmnt x Mchanics F v p x Rotational Angular Angular Angular Angl Mchanics Momnt Vlocity Momntum M ω p ω θ Elctro Voltag Currnt Linkag Flux Charg v i λ q Hydraulic Total Volum Prssur Volum Prssur Flow Momntum P Q P p V c 5/35
6 On-port Elmnts ˆ capacitor, rsistor, voltag sourc and currnt sourc ar on-port lmnts. ˆ it is sam or mchanical systm and hydraulic systm. ˆ th inductor has th proprty: v = L di dt or i = L t v(τ)dτ low = a constant or a unction t ort(τ)dτ ˆ th lmnt with this rlationship (also th mass in translational mchanics) will b calld an inrtial lmnt and b dnotd by I 6/35
7 On-port Elmnts ˆ th capacitor has th proprty: t or = a constant or a unction low(τ)dτ ˆ any lmnt with this rlationship (also th spring in translational mchanics) will b calld a complianc lmnt and b dnotd as ˆ th rsistanc has th rlationship ort = a constant or a unction low, low = a constant or a unction ort C 7/35
8 On-port Elmnts ˆ any lmnt with this proprty will b calld a rsistiv lmnt and will b dnotd by th symbol ˆ A voltag sourc in lctrical domain and an xtrnally imprssd orc in mchanical domain ar xampls o sourc o ort, dnotd by th symbol ˆ A currnt sourc in an lctrical circuit or a cam in a mchanical systm, th low variabl is xtrnally dtrmind and th ort variabl is dcidd by th rst o th systm. Such an lmnt is calld a sourc o low and is rprsntd by th symbol R SE SF 8/35
9 Th Junctions Junction Structur (JS) A bound graph in which bounds connct only nods that instantanously transr or distribut powr (without nrgy storag or convrsion into hat), is calld Junction Structur (JS). Thr ar two kinds o junction: ˆ 0-junction is th junction that qualizd th orts in all bonds. ˆ -junction is th junction that qualizd th lows in all bounds. 9/35
10 Th 0-Junction 0-junction A 0-junction is a multiport lmnt dind by th ollowing quations = 2 =... = n = n According to both quation abov th lmnt is also calld common ort junction. Th lowr quation has givn ris to th notion o a low junction. R E R L C SE 0 I C 0/35
11 Th -Junction -junction A -junction is a multiport lmnt dind by th ollowing quations = 2 =... = n n = 0 According to both quation abov th lmnt is also calld common low junction. Th lowr quation has givn ris to th notion o a ort junction. R R E L SE I C C /35
12 Exampls RLC L I: L C: C E C I 2 4 SE: E SF: I R L 2 R: R I: L I: L 2 C: C E L C R SE: E R: R 2 2/35
13 Exampls Mchanical Systms F (t) M SE: F (t) 2 I: M 2 C R 0 3 3/35
14 Exampls Mchanical Systms F (t) SE: F (t) 3 k M b C: k I: M 4 R: b v(t) SF: v(t) ˆ th mass M shars th sam low with th agnt applying th orc. ˆ th top and th bottom nds o th spring and th dampr shar th sam ort. 4/35
15 Rrnc Powr Dirctions ˆ Bound graph also assigns a rrnc powr dirction o ach bound. ˆ Convntions or adding th hal arrow to powr bonds ˆ Rsistor circuit xampl + v R v i R 5/35
16 Rrnc Powr Dirctions ˆ with th powr dirctions, w can stat th powr balanc at th junction as R R: R 2 E L SE: E 3 I: L C 4 C: C = 0 Sinc = 2 = 3 = 4, thn = 0 6/35
17 Rrnc Powr Dirctions ˆ with th powr dirctions, w can stat th powr balanc at th 0 junction as R: R 2 E R L C SE: E 0 3 I: L 4 C: C = 0 Sinc = 2 = 3 = 4, thn = 0 7/35
18 Two-port Elmnts Transormr lmnt : µ TF 2 2 ˆ th powr in th two sids must b qual = 2 2 ˆ th notation shown abov implis that th lows at two sids ar rlatd by 2 = µ and 2 = µ 8/35
19 Two-port Elmnts transormr systm Transormr lmnt : M F (t) a b k 2 k b M 2 I: M b/a C: k 2 SE: F TF 0 I: M 2 C: k R: b 9/35
20 Two-port Elmnts Gyrator lmnt : µ.. GY 2 2 ˆ Hr µ is th gyrator modulus and th variabls at th two sids ar rlatd by 2 = µ and = µ 2 and = 2 2 ˆ th DC machin is a typical xampl o a gyrator. Th armatur currnt and th back m in th lctrical domain ar rlatd to th spd and torqu in th mchanical domain by τ = Kϕi a and b = Kϕω r 20/35
21 Two-port Elmnts Gyrator systm Gyrator lmnt : q L a Ra I, R E + Const. q 2 k q 3 l R 2 I:L a I: I C: k I: I 2 SE: E Kϕ GY 0.. R: R a R: R R: R 2 2/35
22 Two-port Elmnts Gyrator systm Gyrator lmnt : ˆ th voltag sourc and th armatur rsistanc and inductanc shar th sam low with th back m. Thror, ths ar connctd through a junction. ˆ Th convrsion rom lctrical domain to mchanical domain is rprsntd by th gyrator lmnt. ˆ Th rotor inrtia and riction shar th sam low (rotor spd) with th coming rom th gyrator. So ths ar connctd through a junction. ˆ Th lxibl shat rprsntd by th complianc lmnt, quats th ort at both its nds, and is connctd through a 0 junction. ˆ Th load inrtia and riction shar th sam low with th right-hand nd o th shat. So ths ar connctd through a junction. 22/35
23 Causal Strok ˆ Th rlation btwn input and output variabls can b shown by using a causal strok. ˆ Th causal strok is a short lin that is prpndicular to bond. ˆ I i is an input and V is an output with V = Ri, th causal strok will b on th outsid o th bound. ˆ I V is an output and i is an input with i = RV, th causal strok will b on th sam sid lik R. V i R : R V i R : R i is an input V is an input 23/35
24 Causal Strok ˆ Considr a dampr i vlocity is an input and F = bv thn bond graph and a block diagram is F v b F v R : b ˆ i orc is an input and v = b thn bond graph and a block diagram is F v b F v R : b 24/35
25 Causal Strok ˆ Th junction qualizs th lows in th bonds connctd to it. Thror, th inormation o low must com rom only on bond, and all othr bonds must tak out th low inormation. ˆ Th 0 junction qualizs th orts in th bounds connctd to it. This implis that only on bond must bring in th ort inormation and all th othr bonds must tak this inormation out. ˆ th abov ruls o causality or th and 0 junctions ar hard ruls, and cannot b violatd in a bond graph. ˆ th bond that brings som inormation into a junction is calld strong bond whil th othr bonds taking out that inormation ar calld wak bonds. 25/35
26 Causal Strok On port lmnt: SE SF R R I C Two port lmnt: TF TF GY GY 26/35
27 Obtaining Dirntial Equations rom Bond Graphs ˆ An I lmnt th basic dirntial quation is ṗ = ˆ A C lmnt th basic dirntial quation is q = ˆ Thror th momntum associatd with a mass (or inductor) and th position o a spring (or charg in a capacitor) bcom th natural choic o gnralizd variabls in th st o irst-ordr dirntial quations. 27/35
28 Obtaining Dirntial Equations rom Bond Graphs Mchanical systm xampl F (t) M SE: F (t) k b C: k 2 I: M v(t) R: R SF: v(t) 28/35
29 Obtaining Dirntial Equations rom Bond Graphs What do th lmnts giv to th systm?. Th low 2 is givn by th mass M. Th stat variabl is 2 = p 2 /M. 2. Th ort 7 is givn by th spring k. Th stat variabl is 7 = kq Th ort 6 is givn by th dampr b and b = b 6. But 6 is not a stat variabl. Sinc 6 = R 6 = R 5 ( by junction ) = 0 ( by junction 0) 6 = R 5 = R( ) = R( ) ( by junction ) = R( p 2 M + v(t)). 29/35
30 Obtaining Dirntial Equations rom Bond Graphs What do th intgrally causal storag lmnts rciv rom th systm? 4. Th SE lmnt givs th ort = F (t) and th SF lmnt givs th low 3 = v(t). What do th intgrally causal storag lmnts rciv rom th systm?. I 2 rcivs th ort 2 rom th systm. By th proprty o th I lmnt, 2 = ṗ 2. Thn ṗ 2 = 2 = 3 ( by junction ) = 5 ( by junction 0) = 6 7 ( by junction ) ( p2 ) = F (t) R M + v(t) kq 7 30/35
31 Obtaining Dirntial Equations rom Bond Graphs What do th intgrally causal storag lmnts rciv rom th systm? 2. C 7 rciv th low 7 rom th systm. W know 7 = q 7. Th stat-spac quation is [ ] [ ṗ 2 R = M q 7 q 7 = 7 = 5 (by junction ) = ( by junction 0) = v(t) + 2 ( by junction ) = v(t) + p 2 M. M 0 ] [ k ] p 2 q 7 [ ] [ R + 0 F (t) v(t) ] 3/35
32 Obtaining Dirntial Equations rom Bond Graphs RLC circuit xampl R L 2 R: R I: L I: L 2 C: C E L C R SE: E R: R 2 th lmnts giv to th systm: SE: = E I 4 : 4 = p 4 L I 6 : 6 = p 6 L 2 C 8 : 8 = q 8 C 32/35
33 Obtaining Dirntial Equations rom Bond Graphs RLC circuit xampl R 2 : w hav R 9 : w hav 2 = R 2 = R 3 ( by strong bond 3) = R ( ) ( by junction 0) = R ( ) ( by junction ) ( p4 = R + p ) 6 L L 2 9 = 9 R 2 = 8 R 2 = q 8 R 2 C ( not q 8 = 8 C ) 33/35
34 Obtaining Dirntial Equations rom Bond Graphs RLC circuit xampl Now th thr intgrally causal storag lmnts rciv ort and low rom th systm. I 4 : I 6 : ( p4 ṗ 4 = 4 = 3 = 2 = E R + p ) 6. L L 2 ṗ 6 = 6 = 5 7 = 3 8 = 2 8 ( p4 = E R + p ) 6 q 8 L L 2 C. C 8 q 8 = 8 = 7 9 = 6 9 = p 6 L 2 q 8 R 2 C. 34/35
35 Rrnc. Wllstad, P. E. Introduction to Physical Systm Modlling, Elctronically publishd by: Banrj, S., Dynamics or Enginrs, John Wily & Sons, Ltd., Rojas, C., Modling o Dynamical Systms, Automatic Control, School o Elctrical Enginring, KTH Royal Institut o Tchnology, Swdn 4. Fabin, B., Analytical Systm Dynamics: Modling and Simulation Springr, Spong, M. W. and Hutchinson, S. and Vidyasagar, M., Robot Modling and Control, John Wily & Sons, Inc. 35/35
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