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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 systeniLInet 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 €— , timerate of change of entropy, s
t g «in . FlowEffort 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 effortevery 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 1Port 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 = Ns/m
v = Cb“ (F)
Mechanical rotation r = d>(w') r = cm 6 :2 Nm—s
a) = d)" (r)
Hydraulic systems P = @(Q‘) P = RQ R = Ns/mS
Q = v” (P)
Electrical systems 8 :2 $0“) 0 n Rz’ R = VIA = 9 (ohm) i=d>"1(e) i=Ge l» P 9:21132 P
r l 2
L Q Q~X~>Q
: I .37“! Q I
7w FIGURE 3.4. The lpon inertia. (a) Bond graph symbol; (1‘)) deﬁning relation; (6‘) mprcsen
union in several physical domains. TABLE 3.3. The 1Port 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 = Nszlm
translation v 2 my‘ip) v :2 pm
Mechanical pt = (Dim) p, x M; J := Nm~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 ..._, 
$.3— IQ
e F P
M wan—uh. M
I. C v. C Q C
Air
9, 92 biadder
.3... “L. Q
s a 01 ~— 02 T
w 3 6 .P
M—ﬁ
, P
._.—.AC C
a: Q
(C) FIGURE 3.2. The 1port capacitor. (3} Bond graph symbol; ([7) deﬁning relation; (C) repre
sentation in several physmai domains. TABLE 3.2. The 1.13m Cayacitor. i c ' f=é
Genera} Linear SI Units for Linear
Relation Relation Capacitance Parameter
Generalized q = was) .9 = Ce C = qj‘e
e=¢Ez¢IqJ ezq/C lx‘Czefq
Mechanical X m (bfhr) X :z: C = min
translation F = ¢g‘(X; F = kX k m N/m
Mechanical 9 : (bar) 9 2 Cr C :: mdi~m
rotation 1' = «>31 (8) z 2 k9 k z: N«m/rad
Hydraulic v = chap) 1» ..~ cp c = {HS/N
systems P m 455’ (V) P = WC
Electrical q : (Dds) q :: Ce C : A‘s/V systcms e = Q35] (q) e = q/‘C == farm (F) 3 ' ‘ ‘1’. \/‘~'"
\ ’  F
\ _ . ,.—«~—~\__._.\_.. _J_ 113.. . f)
T 37502 Q\ Q2
"7) of
f} f2 Wax/157‘ W
8’ Zméz ) m7} :f2 ‘8! g:
fl 7": 7? f9 W W cauaww‘m TWO Port Elements 8} 32
.._._.....__.n TF ————.
f1 {2
(a) (d) FIGURE 3.8. Transfenners. (a) Bond graph; {in} ideal rigid lever; (6') gear pair: (d) elecmcal
transformer; is) hydrauhc ram. FIGURE 3.9. Gyrators. (a) Bond graph; (b) symbol for electrical gymtor; (cf; machanical
gymtor; (d) voice coil transducer. €1$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) ljuncu'on. TABLE 3.5. Summary of Bagic 3—Ports f} 8‘ How junctianﬁ I “*7 ()_ 7 e; z 62 x 83.
or (Irjunction “"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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 Spring '09
 Energy storage, Bond graph

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