lecture 2

# lecture 2 - FUNDAMENTALS OF BOND GRAPH'MODELING METHOD...

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Unformatted text preview: FUNDAMENTALS OF BOND GRAPH'MODELING METHOD Conservation of Energy is key (note power and energy are scalar quantities): 6E ZPinJ = :3? + ZPouLj Net power into system=rate of change of energy in systeniLI-net power out of system Deﬁne Power Conjugates variables: Translation F, force V, velocity Rotation 1, torque (2 , angular velocity ' Fluid/Hydraulic P, pressure Q, ﬂow rate Electrical _ e, voltage i, current H Thermal T, temperature €— , time-rate of change of entropy, s t g «in .- Flow-Effort variables (developed by James C. Maxwell) Effort variables “e”: ‘pushing’: F, T, P, e, T - . . ds ' F1 b1 :”’:‘ t ’2V9 '— ow varia es f mo 10n , ,Q, 1, dt I. Power, P=ef It Energy, E= I}.ng = J'efdt Deﬁne energy variables: Generalized momentum: p = Jedt Generalized displacement: q = det Energy of a system can be rewritten as: E = Ifdp “kinetic” energy E = jedq “potential” energy we: m5 BOND GRAPHS Bond graph modeling elements: 0 Energy storage elements (C, I) 0 Energy dissipation elements (unavailable forms) (R) 0 Energy transforming elements (T, G) '0 Junction structures (1, 0) ‘ 0 Power sources and sinks (Se, Sf) Port: place at which power can be exchanged For example: at sources/sinks, energy storage elements, Ports are represented by short line segments: effort is written above or to the leﬁ of the line, while ﬂow is written below or to the right of the line. Half arrows on the segments point in the direction of positive power ﬂow. For C and I elements, power can ﬂow in or out, so the half arrows can point in either direction. R elements always remove power, so the half arrow is always directed toward R. ’ Se and Sf : Sources supply power to a system and the half arrow points away from the source and toward the system. Sinks drain power from a system and the half arrow points toward the sink and away from the system. Junctions are used to connect the basic elements together. 0 junction: common effort-every port connected to it has the same effort 2 fl. = O Flows divide (Kirchoffs Current Law) [:1 1 junction: common ﬂow: every port connected to it has the same ﬂow m 2 el. = 0 Efforts divide (Kirchoffs Voltage Law) [:1 TABLE 3‘4 M Bond Graph Symbol Deﬁning Relation Generalized variables 54.— 8(1) givcn‘ f (I) arbitrary Sf—a fm given. w) arbitrary Mechanical translation Sf—“ Hz) given, 31(1) arbitrary Sv—ﬁ VU} given. Fm arbitrary Mechanical rotan'on St #- rm gwcn. mm arbitrary Sm~ mm given, 1(1) arbitrary Hydraulic systems 5p~ PU) given, Qtr} arbitrary SQ- Qm given, PU) arbitrary Electrical systems Se ~ e (t) given, i (t V) arbitrary 5, ~ in) given. 90) arbitrary One Port Elements F P ..__.A.e “R ._._AR 1. R V Q (c) FIGURE 3.1. The i—purt resistor. (43) Bond graph symbol; ([2) deﬁning relation; (6) represen- anions in several physical domains. TABLE 3.1. The 1-Port Resistor, i R I General Linear SI Units for Linear Relation Relation Resistance Parameter Generalized variables e 2- (hmf) e = Rf R = e/f fxcbglie) fnGe=efR Mechanical translation F = (bf V) F = IN 27 = N-s/m v = Cb“ (F) Mechanical rotation r = d>(w') r = cm 6 :2 N-m—s a) = d)" (r) Hydraulic systems P = @(Q‘) P = RQ R = N-s/mS Q = v” (P) Electrical systems 8 :2 \$0“) 0 n Rz’ R = VIA = 9 (ohm) i=d>"1(e) i=Ge l» P 9:21-132 P r l 2 L Q Q~X~>Q : I .37“! Q I 7w FIGURE 3.4. The l-pon inertia. (a) Bond graph symbol; (1‘)) deﬁning relation; (6‘) mprcsen- union in several physical domains. TABLE 3.3. The 1-Port Inertia, ‘3‘? I I General Linear SI Units for Linear Relation Rcia’mm lnertance Paramemr Generaiizcd p = dang”) p m 1f 1 = pf'f vaﬁabies f = dﬁ’gp) - r. pg; 1,; a 3, Mechanical p = d), (v; p s m v m = N-szlm translation v 2 my‘ip) v :2 pm Mechanical pt = (Dim) p, x M; J := N-m~52 rotation w=¢flipd w==PEH Hydraulic p,, = «mg; pp = IQ I .~. Nuszlm5 systems Q = mrln'pp) Q = pw metrical A = mm} .3. = Li L = v.5/A systems 1‘ z a); 1 (A) i 2 UL = bemys (H) q 3: QC (9’ e a: 65%) e MC f q Ef fit (a) (5} 1' V, = X FP— X », 1g T1 r i E F P e «cum—H ..._, - \$.3— IQ e F P M wan—uh. M I. C v. C Q C Air 9, 92 biadder .3... “L. Q s a 01 ~— 02 T w 3 6 .P M—ﬁ , P ._.—.AC C a: Q (C) FIGURE 3.2. The 1-port capacitor. (3} Bond graph symbol; ([7) deﬁning relation; (C) repre- sentation in several physmai domains. TABLE 3.2. The 1.13m Cayacitor. i c ' f=é Genera} Linear SI Units for Linear Relation Relation Capacitance Parameter Generalized q = was) .9 = Ce C = qj‘e e=¢Ez¢IqJ ezq/C lx‘Czefq Mechanical X m (bfhr) X :z: C = min translation F = ¢g‘(X; F = kX k m N/m Mechanical 9 : (bar) 9 2 Cr C :: mdi~m rotation 1' = «>31 (8) z 2 k9 k z: N«m/rad Hydraulic v = chap) 1» -.-.~ cp c = {HS/N systems P m 455’ (V) P = WC Electrical q : (Dds) q :: Ce C : A‘s/V systcms e = Q35] (q) e = q/‘C == farm (F)- 3 ' ‘ ‘1’. \/‘~'" \ ’ - F \ _ . ,.—«~—~\__._.\_.. _J_ 113.. . f) T 37502 Q\ Q2 "7) of f} f2 Wax/157‘ W 8’ Zméz ) m7} :f2 ‘8! g: fl 7": 7? f9 W W cauaww‘m TWO Port Elements 8} 32 .._._.....__.n TF -————-. f1 {2 (a) (d) FIGURE 3.8. Transfenners. (a) Bond graph; {in} ideal rigid lever; (6') gear pair: (d) elecmcal transformer; is) hydrauhc ram. FIGURE 3.9. Gyrators. (a) Bond graph; (b) symbol for electrical gymtor; (cf; machanical gymtor; (d) voice coil transducer. €1\$rfz rfiZQ’Z- Three Port Elements ‘13 .40“. I 2 “E3 jihad“ =i=y =V Hm V1 V2 V3 FIGURE 3.1]. Basic 31mm in various physical domains. {a} @jnnction; (b) l-juncu'on. TABLE 3.5. Summary of Bagic 3—Ports f} 8‘ How junctianﬁ I “*7 ()_ 7 e; z 62 x 83. or (Ir-junction “"3 f1 + 154* f3 = D . . E ,. .- Effortjuncuon. M I f1 = f: = \$3. . . a w W, or lajuncuon ' UL 1“ e; 4» 9» + as = O ...
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lecture 2 - FUNDAMENTALS OF BOND GRAPH'MODELING METHOD...

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