System variables. Basic Modeling Concepts. Basic elements single and. Power = effort x flow. Power = F x v. Power = V x i. Power = T x w.

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1 Basic Modling Concpts Basic lmnts singl and multiport t dvics Systm variabls v m F V i Powr F x v T w Powr T x w Powr V x i P Q Powr P x Q Powr ort x low

2 Eort & low ar powr variabls Eorts t... Flows... Forc (F) Nwtons Voltag (V) Volts V Torqu (T) N-mtrs Prssur (P) N /m Vlocity (V) Currnt (i) m/s Amps Ang. vlocity(w) rad/s Volum low (Q) m / s powr ort () x low () Systm variabls Gnralizd powr and nrgy variabls hav th ollowing rlations: dq / dt q is a gnralizd displacmnt dp / dt p is a gnralizd momntum. A stat ttrahdron xplains ths rlations dp/dt p q dq/dt

3 Physical Systms Variabl Typs VARIABLE MECHANICAL TRANSLATION Eort Forc (F) (Nwtons N) MECHANICAL ROTATION Torqu (T) (N-m) ELECTRICAL Voltag (Volts V) HYDRAULIC Prssur (P) (N/m ) Flow Vlocity (v) (m/s) Angular vlocity (w) (rad/s) Currnt (i) (Amprs A) Volum low (Q) (m /s) Displacmnt Momntum Displacmnt (x) (m) Momntum (N-s) Angl (rad) Angular momntum (N-m-s) Charg (q) (A-s) Flux linkag (V-s) Volum (m ) Prssur momntum (N-s/ m ) Word bond graphs. Word bond graph : Simpl rprsntation o physical systm; using words to imply a systm componnt. Exampl: Car Startr/Solnoid battry motor bndix Th word bond graph can b drawn as battry voltag currnt motor torqu angular vlocity bndix

4 Word bond graphs Car Modl: - Powr rom ngin is d to clutch - Transmittd to gar box. Slcts gar - Powr lows to whls via a dirntial gar box and driv shat. whl ngin T clutch T gar-box T driv shat w w w T4 w4 T5 dirntial gar box T6 w5 w6 throttl control gar slction whl Eorts: Torqus (T,T,T,T4,T5,T6) Flows: Vlocitis ( w,w,w,w4,w5,w6) Concpt o causality Causality : Indicats WHO causs WHAT to WHOM A B A B in bond graph notation, in bond graph notation, A B A low () ort () ort () B low () I lmnt A imposs an ort on lmnt B, thn lmnt B rsponds back with a low or vic-vrsa,

5 Basic lmnts To convrt a word bond graph to complt bond graph w nd som basic lmnts. Basic lmnts. Sourc o ort. (SE) Sourc o low (SF) Constant ort junction () Constant low junction () Inrtia lmnt (I) Mass,inductors,luid pips Capacitiv lmnt (C) Spring,capacitor,lywhl Rsistiv lmnt (R) Damprs, rsistors Transormr lmnt (TF) Lvrs,transormrs Gyrator lmnt (GY) Cntriugal pump, gnrators With ths lmnts, bond graph modls o dynamic systms can b cratd in any nrgy domain. BOND GRAPHS and PHYSICAL VARIABLES Powr Flow Concpt powr Powr A B A B Causality Concpt powr powr A low B ort A ort low B A imposs ort on B, B rsponds with a low B imposs ort on A, A rsponds with a low

6 FUNDAMENTALS OF BOND GRAPH MODELING low powr ort powr low ort k b C 4 R m I x(t) F SE b C R k 4 m I x(t) F SE To Sum Eorts To Sum Flows A rsistiv lmnt (R). Thr is a static rlation btwn ort & low. Rsistiv lmnts ar idalization o dvics lik, damprs, rsistors, luid carrying pips. R () Linar R Non linar R Units o R Mch. translation Mch rotation Elctrical Hydraulic N-s/m N-m-s V/A (Ohms) N-s/ m

7 Rsistiv lmnt (R) Causality considrations : A rsistiv lmnt taks ithr orm R R Rlation :g () Rlation :g - () R R R lmnt R R 5 R R 5

8 Capacitiv lmnt (C). In a capacitiv lmnt a static rlation xists btwn ort & displacmnt. Ths dvics stor or dissipat nrgy without loss. Capacitiv lmnts ar idalization o dvics lik, springs, capacitors, accumulators. Cq (q) q q Units o C Linar C Non linar C Mch. translation Mch rotation Elctrical Hydraulic N/m N-m/rad arads N/ m 5 Capacitiv lmnt (C) Causality considrations : Intgral causality C dt q C C dq dt Prrrd orm or computational purposs Drivativ causality C d dq C dt dt dq d C dt dt This is not prrrd orm or computational purposs

9 C 4 C lmnt 4 C S 4 Inrtia lmnt (I). Thr is a static rlation btwn low & momntum. Ths dvics stor kintic nrgy Inrtia lmnts ar to modl inductanc cts in lctrical circuits, mass & inrtia cts in mchanical & hydraulic systms. Ip p g(p) p Units o I Linar I Non linar I Mch. translation N-s /m Mch rotation N-m-s Elctrical Hydraulic V-s/A (Hnrys) N-s /m 5

10 Inrtia lmnt (I) Causality considrations : Intgral causality I dt p I I dp dt Impuls Momntum orm Prrrd orm or computational purposs. Drivativ causality I d dp I dt dt dp d I dt dt Nwton s law orm This is not prrrd orm o causality or I lmnt. I lmnt I 4 4 I S 4

11 Th sourc lmnts SE & SF) An ort sourc : Systm/lmnt which maintains an input ort. SE s ar voltag sourcs, orcs, prssur. A low sourc : Systm/dvic which maintains a an input low. SF s ar vlocity sourcs, currnt, low sourcs V m Vhicl suspnsion systm. Road Eort Sourc Flow sourc Sourc lmnts (SE & SF) Causality considrations : Eort Sourc Flow sourc SE SF Th ort sourc imposs an ort on th connctd junction or lmnt Flow sourc imposs a low onto th systm, connctd junction or lmnt.

12 Sourc SE lmnt SE5 5 5 SE 5 Sourc SF lmnt SF5 5 5 SF 5 Transormr lmnt (TF) Two port lmnts altring magnitud o ithr low or ort ar transormr lmnts. Transormrs hav static rlation btwn input low/ort & output low/ort by mans o a transormr modulus. v F a v b a v b F a b F F v F v A A A A F P v Q A A P,Q A lvr Hydraulic Ram Ratio (a/b) & (A/A) is transormr modulus. Othr xampls: gar st, lctrical transormr, pullys.

13 Transormr lmnt (TF) Causality considrations : m.. TF m m m.. TF m m TF Incorrct causal orm, not possibl TF lmnt 4 4 TF TF TF45 5

14 TF lmnt 4 4 TF /TF /TF45 5 Gyrator lmnt (GY) Gyrators : Two port lmnts which rlat input ort to output low or vicvrsa by mans o a modulus. Typical xampls: voic coil, lctric motor, gnrator. ω F v v v i Τ ω F r v r v F F r ω i r T Gyro-scop I th rotor spins rapidly, & a small F will yild a proportional vlocity v, & vic-vrsa Motor Angular vlocity output is proportional to applid voltag

15 Gyrator lmnt (GY) Causality considrations : r.. GY r r r.. GY r r GY GY Incorrct causal orm, not possibl GY lmnt GY 5 4 GY GY45 5

16 GY lmnt 4 4 GY /GY /GY45 5 Th () junction lmnt i i i c c i -i i PPP

17 Th () junction cont. ( junction) : Is a common ort junction. All orts ar qual Th sum o th lows qual zro Th sum o th lows qual zro. Summation signs will b dtrmind by Powr Flow. Powr low and th () Junction

18 Causality and th () Junction Powr low and Causality () Junction

19 () Junction proprtis It is a common ort junction or all bonds attachd All orts ar qual Th sum o th lows qual zro. Powr consrving, powr in quals powr out Only on causal mark dtrmins th input ort and thus all othr orts will b outputs Thr can only b on bond and only on bond that sts th ort input Powr low hal arrows dtrmin how th lows will sum Th () junction lmnt V C i R N v F w Currnt through C and R is th sam. Summation o voltags Vlocity is common but sumation o orcs must ollow Nwton s law

20 Th () junction cont. ( junction) : Is a common low junction. All lows ar qual Th sum o th orts qual zro Th sum o th orts qual zro. Summation dtrmind by Powr Flow. Powr low and th () Junction

21 Causality and th () Junction Powr low and Causality () Junction

22 junction - 4 junction () Junction proprtis It is a common low junction or all bonds attachd All lows ar qual Th sum o th orts qual zro. Powr consrving, powr in quals powr out Only on causal mark dtrmins th input low and thus all othr lows will b outputs Thr can only b on bond and only on bond that sts th low input Powr low hal arrows dtrmin how th orts will sum

23 Causal bond(4) junction-c lmnt - 4 C S 4 4 C - Causal bond junction-i lmnt 4 I S 4 4 I

24 Causal bond junction R lmnt R Causal bond junction R lmnt - 4 R 4 Causal bond junction TF lmnt 4 5 TF45 - Causal bond junction TF lmnt - 4 TF45 5

25 Causal bond junction GY lmnt 4 5 GY45 - Causal bond junction GY lmnt - 4 GY45 5 Causal orms. Symbol Implid maning & variabl rlation. C I C I Prrrd intgral causal orm... dq/dt Prrrd intgral causal orm... dp/dt Drivativ causal orm...q intg ( )... not prrrd Drivativ causal orm...p intg ( )... not prrrd TF Prrrd orm or transormr lmnt. (or opposit) /m & /m OR m & m GY Prrrd orm or gyrator lmnt. (or opposit)