V + - K R + - - k b V R V L L J + V C M B Analogous Systems i = q. + ω = θ. C -. λ/l = q v = x F T. Se: e e(t) e = p/i R: R 1 I: I e C = q/c C = dq/dt e I = dp/dt Identical dierential equations & bond graph or Series electrical elements Parallel mechanical elements Each system has Power input, eort source S e :e = V, F, T Potential energy (o position) storage o Capacitance C:C = C, k, K o Displacement q = q, x, θ Kinetic energy (o motion) storage, o Inertance I:I= L, M, J o Momentum p = λ, p, h Power loss, resistance R:R = R, b, B Common (same) low, all elements: = i, v, ω C: C q = p /I, p = " q C " R p I + e(t) I q + Rq + 1 C q = e(t) 1
R: R i L R C S : (t) 0 ec = q/c C = dq/dt C: C I = p/i e I = dp/dt M k v I: I b p = q C, q = " q RC " p I + (t) p + 1 RC p + 1 IC p = 1 C (t) Identical dierential equations & bond graph or Parallel electrical elements Series mechanical elements Each system has Power input, low source S : = i, v Capacitance C:C = C, k Inertance I:I= L, M Power loss, resistance R:R = R, B Common (same) eort, all elements: e = V, F 2
Dynamic Systems Elements Sources: supply power, prescribe eort or low Resistance: direct relation between eort & low Energy Storage Devices (single or multiports) Inertance: kinetic energy Capacitance: potential energy IC: kinetic & potential energies Lossless (conserve power, " P j = 0) multiports bonds Junctions o 0 junction: common eort, balance lows o 1 junction: common low, balance eorts Transormers & Gyrators For Bond graphs Bond: indicates power transer between elements Hal arrow: indicates direction o postive power transer between elements 3
Sources Eort source o Prescribes eort e = e(t), eort labeled on hal arrow side o bond o Flow can be anything (rest o system determines low) o Se prescribes eort (onto A), causal stroke (short vertical bar) away (ram against A) Flow source S : (t) A o Prescribes low = (t), non-hal arrow side o Eort can be anything o S prescribes low (to A), causal stroke toward (hose squirts away) 4
S e : e e(t) A V(t) + F(t) T(t) - P(t) Figure 3.6: a) A bond graph o an eort source S e :e(t), with maniestations in the various energy domains: b) electrical voltage source V (t); c) mechanical translational applied orce F(t); d) mechanical rotational applied torque T(t); and e) luidic applied pressure P(t). 5
Flow and Flow Sources S : (t) A I(t) v(t) Ω(t) Q(t) Figure 3.7: a) A bond graph o a low source S : (t), with maniestations in energy domains: b) electrical current source I(t); c) mechanical translational prescribed linear velocity v(t); d) mechanical rotational prescribed angular velocity Ω(t); and e) hydraulic prescribed volumetric low Q(t). 6
Resistance Direct relation between eort & low: e = e() or = (e) I e vs. plots in quadrants 1 & 3, dissipates power P = e 0 2 Causality choices: Eort Controlled: i = V/R orm o Ohm s law Action: A rams R with eort e Reaction: R accepts eort e rom A, then hoses A with low = (e) 7
Flow Controlled: A e = e( ) R V = i R orm o Ohm s law Action: A hoses R with low Reaction: R accepts low rom R, then R rams A with eort e = e() I e vs. plots in quadrants 1 & 3, dissipates power P = e 0 e < 0 < 0 e e > 0 > 0 8
e R e R = e R ( R ) R R R = R (e R ) R T b Ω b F b b β p 1 i R R v b Ω b Q c + - V R F b T b p o Figure 3.10: Bond graphs o a resistance in its causal orms a) and b), with maniestations in energy domains: c) electrical resistor; d) mechanical translational linear dashpot; e) mechanical rotational rotary dashpot; and ) hydraulic low constriction or turbulence. 9
Capacitance Kinematic constraint, low C & displacement q: C = q Stores "potential" energy, energy o position or coniguration U(q) = P dt = e dt = e dq dt dt = e(q) dq => Eort-displacement dependence e = e(q) Linear capacitance: e(q) = q/c U(q) = q 2 /2C Energy variable: displacement q Note: e = e(q) = U/ q Relates eort & displacement 10
Capacitances in various power domains system kinematics displacement eort physics general = q. q e = e(q) e = e(q) electrical i = q. charge voltage Gauss law q V = V(q) C: capacitance mech. translation mech. rotation magnetic luidic v = ẋ ω = θ. displacement x angular displacement θ " = " Magnetic lux Q = v. φ luid volume v V=q/C orce F = F(x) F = kx torque T = T(θ) T = K θ Magnetomotive orce M = M(φ) M = R φ pressure P = P(v) P t = v t /(A/ρg) Spring law k: elastic stiness Torsion K: torsion stiness Magnetic R: reluctance pressure P = P(v) P t = v t /(A/ρg) 11
Capacitance e c = e c (q ) q. c = C e c = e c (q ) q. c = T c C ic C + - Vc F c F c v 2 Ω 2 v 1 T c Ω 1 A Q 2 h P = ρgh Q 1 Figure 3.9: Bond graph o capacitances in a) integral causality (preerred), b) derivative causality with maniestations in energy domains: c) electrical capacitor; d) mechanical translational linear stiness; e) mechanical rotational torsional stiness; and ) hydraulic luid 12 tank.
Multiport Capacitance e 1 1... C e k k... e m m Energy stored in ield m-ports into Capacitance, m-power lows o Flows & displacement via kinematics: k = qk. o Displacements: q k m o Power: P = Pk k = 1 o Total potential energy: m = ek k k = 1 E = Pdt = m ek k dt k = 1 = m. ek qk k = 1 dt = m ek dqk k = 1 via integral, E = E(q 1, q 2,..., q m ) 13
depends on all displacements q k Energy & Power: de m dt = P = ek k k = 1 Derivative o E = E(q 1, q 2,..., q m ), chain rule: de dt m E = qk k = 1 dqk dt m E = qk k = 1 Equate coeicients o k, in blue terms: ek = ek(q1, q2,..., qm) = E qk Eort on k th bond rom partial o energy w.r.t. displacement q k on k th bond. k 14
Inertance Physics constraint, eort e I & momentum p: e I = p inertial orce Stores "kinetic" energy T(p) = P dt = e dt = dp dt dt = (p) dp => Flow-momentum dependence = (p) Linear inertance: (p) = p/m T(p) = p 2 /2I Energy variable: momentum p Note: = (p) = T/ p Relates low & momentum e = ṗ = (p) I 15
Inertance A. e I = p I = I (p) I A. e I = p p = p( I ) I i L = i L (λ) + Ω = Ω(h) L V L = λ. v = v(p) J Q(p) - M F I = ṗ T I = ḣ P = ṗ Figure 3.8: Bond graphs o an inertance in a) integral causality, and b) derivative causality, with maniestations in energy domains: c) electrical inductance; d) mechanical translational mass inertia; e) mechanical rotational rotational inertia; and ) hydraulic low inertia. There is no magnetic or thermal inertia. 16
Table 3.3: Inertances or the various power domains used in this book. Magnetic systems, which lack inertial eects, were omitted. mechanical mechanical general electrical translation rotation luidic dynamics e I = ṗ V L = λ F I = ṗ T I = ḣ P = ṗ momentum p λ [V s] p [N s] h [N m s] p [N m 2 s] low = (p) i = i(λ) v = v(p) Ω = Ω(h) Q = Q(p) (linear I) = p/i = λ/l = p/m = h/j = p/i physics law Faraday D Alembert: Newton D Alembert:Euler Newton 17
Inertances in various power domains system type physics momentum low dependence Physics law general e = ṗ p = (p) electrical Inductor voltage V = λ. lux linkage λ current i = i(λ) mech. inertial orce linear mom. velocity v Faraday Newton translation mech. FI = ṗ inertial torque p ang. mom. v = v(p) ang.vel. ω F = ma = ṗ Euler rotation luidic TI = ḣ inertial pressure PI = ṗ h luidic momentum p ω = ω(h) luid volume Q = Q(p) T = Iα = ḣ unsteady low terms in momentum equations No magnetic or thermal inertance! (no kinetic energy in those domains)! 18
Multiport Inertance e 1 1... I e k k... e m m via integral, E = E(p 1, p 2,..., p m ) k = k (p 1, p 2,..., p m ) = E/ p k Eort on k th bond rom partial o energy w.r.t. displacement q k on k th bond. 19
IC Device Stores kinetic & potential energies in same ield E = E(p 1, p 2,..., p m, q 1, q 2,..., q n ) Ports with momenta p k & displacements q l Flows on I bonds: k = E/ p k = k (p 1, p 2,..., p m, q 1, q 2,..., q n ) Eorts on C bonds: e l = E/ q l = e l (p 1, p 2,..., p m, q 1, q 2,..., q n ) IC 20
0 & 1 Junctions No power loss or storage Power balance: n P = k=1 P in k m - i=1 P out i n = k=1 e in k in k m - i=1 e out i out i = 0 e 2 2 e 1 1 0 0 junction: common (same) eort, all bonds: e1 = e2 =... = en = em = e n in m k - out n+m i = k = 0 k=1 i=1 k=1 e 3 Flow balance 0 junction incorporates: Electrical Kircho's Current Law ( currents into = 0 ) node Mechanical kinematics (balance o velocities & rate o displacements) 3 21
Kircho s current law, n k=1 i k = 0, or electrical power domains, wherein the sum o the currents i k lowing into a circuit node must equal zero. Translation kinematics n k=1 v k = 0, or mechanical translational domains, which equates translational velocities v k along some direction across a body to zero. Rotational kinematics n k=1 Ω k = 0, or mechanical rotational domains, wherein the rotational velocities Ω k along some axis through a body must equate to zero. Continuity equation n k=1 Q k = 0,orincompressibleluidicpowerdomains,wherein the sum o the volumetric lows Q k into and out o a control volume must equate to zero. Flux rate continuity equation n k=1 φ k = 0,ormagneticpowerdomains,wherein the sum o the lux lows φ k over a node in a magnetic circuit must equal zero. 22
e 2 2 e 1 1 1 1 junction: common (same) low, all bonds: 1 = 2 =... = n = m = n e in m k - e out n+m i = ek = 0 k=1 i=1 k=1 1 junction incorporates: Electrical Kircho's Voltage Law (over loop) Mechanical D'Alembert's dynamic equilibrium e 3 3 23
Equilibrium o orces, n k=1 F k = 0, or mechanical translational domains, wherein the sum o orces F k on a body along some direction must equal zero. Equilibrium o moments, n k=1 T k = 0,ormechanicalrotationaldomains,wherein the sum o moments T k over a body along some axis must equal zero. Momentum equation, n k=1 P k = 0, or luidic power domains, wherein the sum o the pressure drops P k along a low path must equal zero. Magnetomotive orce equilibrium, n k=1 M k = 0, or magnetic power domains, wherein the sum o the magnetomotive orce drops M k along a lux path must equal zero. Note that inertial eects such as F I = ṗ arise as separate terms in these balances. 24
Causality: 0 & 1 Junctions e 2 2 0 junction common eort e 1 = e 2 =... = e n = e e 1 1 0 e 3 3 only ONE bond can set common eort e ONE ram against 0 (otherwise contradiction) Note: 0 junction has only 1 ram, but MUST have e 1 1 e 2 1 2 e 3 3 1 junction: common low 1 = 2 =... = n = only ONE bond can set common low ONE hose squirts 1 (otherwise contradiction) Note: 1 junction has only 1 hose, but MUST have 25
Transormers & Gyrators Converts power, spans domains Lossless 2 port : P1 = P2 Transormer e 1 1 TF: n e 2 2 o relates eort to eort: e 1 = n e 2 o & low to low: 2 = n 1 o conserves power: (ne 2 ) 1 = e 1 1 = P 1 = P 2 = e 2 2 = e 2 (n 1 ) Gyrator e 1 1 GY: r e 2 2 o relates eort to low: e 1 = r 2 & e 2 = r 1 o conserves power: (r 2 ) 1 = e 1 1 = P 1 = P 2 = e 2 2 = (r 1 ) 2 26
e 1 1 TF: n e 2 2 e 1 1 TF: n e 2 2 + + i 1 i 2 V 1 - F V 2 2 F 1 B n 1 n 2 l 1 l 2 - v 1 v 2 R 1 R 2 T 1 T 2 Ω 1 Ω 2 Q T, Ω A F = pa p v v = R Ω Ω R F = T/R P, Q Figure 3.11: Bond graphs o transormers in its allowed causal orms a) and b), with maniestations in energy domains: c) electrical transormer with turns ratio n = n 1 /n 2 ; d) mechanical translational lever mechanism with leverage n = l 2 /l 1 ; e) mechanical rotational gears and rollers with gear ratio n = R 1 /R 2. Transormers can also span power domains. Examples include ) translational to hydraulic piston with n = A; g) rotational to translational roller on lat, or rack and pinion with n = 1/R; and h) rotational to hydraulic positive displacement 27 pump.
e 1 1 GY: r e 2 2 e 1 1 GY: r e 2 2 + i 1 i 2 + - V 1 i + V - lines o induction B total lux φ V 2 magnetomotive orce M = n i n turn coil servo motor -V mim T, Ω B Figure 3.12: Bond graphs o gyrators in allowed causal orms a) and b), with examples: c) electrical: electrical gyrator ormed by matched pairs o ield eect transistors, or transconductance ampliiers; d) electrical-mechanical rotational: DC servo motor; e) mechanicaltranslational-mechanical rotational: gyroscope; and ) electrical-magnetic: solenoid. Gyrators oten span power domains. 28
TF & GY Causality Transormer: e1 = n e2 eort evokes eort 1 = n-1 2 low evokes low 2 choices: Gyrator: e1 = r 2 low evokes eort 1 = r-1 e2 eort evokes low 2 choices e 1 e 2 GY 1 2 29
Table 3.2: A list o bond graph elements or various energy domains, with corresponding element constants and SI units in square brackets. mechanical mechanical element general electrical translation rotation luidic eort source S e : e(t) S e : V (t) S e : F(t) S e : T(t) S e : P(t) low source S : (t) S : I(t) S : v(t) S : Ω(t) S : Q(t) capacitance C C [F] k [N m 1 ] κ [N m rad 1 ] C t [m 5 N 1 ] inertance I L [H] m [kg] J [kg m 2 ] I [N m 2 s] resistance R R [Ω= V A 1 ] b [N m 1 s] β [N m s] R c [N m 5 ] transormer TF transormer levers gears & rollers gyrator GY transconductance ampliiers 0 junction k k = 0 Kircho s kinematics: kinematics: Continuity e k = e in Current Law velocities angular velocities equation 1 junction k e k = 0 Kircho s Equilibrium: Equilibrium: Momentum k = in Voltage Law orces moments equation 30