Physics review & bond graphs

Causality SourcesDamping: Resistance RInertanceCapacitance

Lossless MultiportsCausality: 0 & 1 Junctions

Power/Energy conversion: Transformers & GyratorsTF Example: Electrical transformerTF Example: Traction RollerGyratorGY Example: Solenoid CoilGY Example: Gyrator & Capacitance synthesizes InertanceTF & GY Causality

Bond Graph Causality Assignments: Procedure & Priorities

Multi-Port Storage Elements

EXAMPLE: RLC circuit

Causality

• Cause & Effect (action/reaction): Does effort e evoke flow f, or vice versa ?

• Not contained in Power P = e f ; Causality independent of arrow direction

• Mechanical: equivalent to Newton's action/reaction + Free Body Analysis

BAe

f

• A impresses B with effort e ⇒ flow f from B to A

• B impresses A with flow f ⇒ effort e from A onto B

• Memory Aid

BA

Battering Ram

B

Fire Hose

A

A pushes (exerts effort e on) B B squirts (exerts flow f on ) A

Sources

Effort

AS : eee(t)

• Prescribes effort e = e(t)

• Flow can be anything

• Since Se prescribes effort (onto A), causal stroke away (ram against A)

Flow

AS : ff f(t)

Prescribes flow f = f(t)

Effort can be anything

Since Sf prescribes flow (to A), causal stroke toward (hose squirts away)

Causal stroke choices

Sources prescribe e or f ⇒ single causality

Se

e(t)

e = output (can not impress effort e on Se )

Sff(t) f = output (can not impress flow f on Sf )

Resistance R

Direct relation between effort & flow: e = e(f) or f = f(e)

Dissipates power P = e f if curve in 1st or 3rd quadrants

Energy variable(s) from Independent Energy Storage Devices

2 Causality choices:

Effort Controlled:

e

f = f (e)RA

Action: A rams R, i.e., applies effort e onto RReaction: R accepts effort e from A, then hoses A with flow f = f(e)

Flow Controlled:A

f

e = e( f )R

Action: A hoses R, i.e., applies flow f onto RReaction: R accepts flow f from R, then R rams A with effort e = e(f)

Resistance choices

f

eR f ( ).

f

e

e

f

e = ΦG(f)

f = ΦR-1(e)

R accepts effort e , then hoses with flow f = f(e) = Φ-1R (e) .

Instantaneous local conductance: G = dfde

f

eR

f

e

e ( ).

e

f

e = ΦR(f)

f = ΦG-1(e)

R accepts flow f, then R rams with effort e = e(f)= ΦR(f) .

Instantaneous local resistance: R = dedf

Inertance

Ie = p

.

f = f (p) p

f

f = ΦI-1(p)

p = ΦI(f)

Effort from dynamics: e = p.

Stores "kinetic" energy

E = T(p) = ⌡⌠

P dt = ⌡⌠

f e dt = ⌡

⌠

f p

. dt = ⌡

⌠

f dpdt dt = ⌡⌠

f(p) dp

Energy variable = momentum p

Flow dependence (most desirable): f= f(p) = Φ-1I (p) .

Instantaneous local inertance: I = dpdf

Inertances in various power domainssystem type dynamics momentum flow dependence law

general e = p.

p f= f(p)

electrical V = λ.

flux linkage

λcurrenti = i(λ)

Faraday

mech.

translation

inertial force

FI = p.

linear mom.

p

velocity v

v = v(p)

Newton

F = ma = p.

mech.

rotation

inertial torque

TI = h.

ang. mom.

h

ang.vel. ω

ω = ω(h)

Euler

T = Iα = h.

fluidic inertial

pressure

PI =�p....

fluidicmomentum

p

fluid volume

Q = Q(p)

unsteadyflow terms

inmomentumequations

Capacitance

Cq.

f =

e = e(q)

q

e

e = Φc-1(q)

q = Φc(e)

Flow from kinematics: f = q.

Stores "potential" energy

E = U(q) = ⌡⌠

P dt = ⌡⌠

f e dt = ⌡

⌠

e q

. dt = ⌡

⌠

e dqdt dt = ⌡⌠

e(q) dq

Energy variable = displacement q

Effort dependence (most desirable): e = e(q) = Φ-1C (q)

Instantaneous local capacitance (compliance): C = dqde

Capacitances in various power domainssystem type kinematics displacement effort dependence

general f = q. q e = e(q)

electrical i = q.

chargeq

voltageV = V(q)V=q/C

mech.

translationv = x

.

displacementx

forceF = F(x)F = kx

mech.

rotationω = θ

.

angulardisplacement

θ

torqueT = T(θ)T = κ θ

fluidicQ = v

.

fluid volumev

pressureP = P(v)

Pt = vt/(A/ρg)

Causality: Energy Storage Elements

Capacitance CIntegral DerivativeCausality Causality

constitutive relations e = e( q ) q = q ( e )

kinematics f = q. ⇒ q = ⌡⌠ f dt

Inertance I

constitutive relations f = f( p ) p = p ( f )

dynamics e = p. ⇒ p = ⌡⌠ e dt

Integral (independent) Causality Derivative (dependent) Causality

Capacitance C

Cf = d q(e)

dt

e

Inertance I

Ie = p.

f = f (p)

e = d p(f)dt If

Preferred Causality Stuck with (dependent) Causality

Integral Causality Derivative Causality

Capacitance C

f

eC

f

e

∫( ) dt.q

e(t) = e [ ∫ f( t ) dt ]t

e ( ).

f

eC

f

e q ( ).q

f(t) = q[ e(t) ]ddt

ddt

Inertance I

f

eI

f

e ∫( ) dt.p

f(t) = f [ ∫ e( t ) dt ]t

f ( ).

f

eI

f

e ddt

p ( ).p

e(t) = p[ f(t) ]ddt

Independent Energy Storage Dependent ESE (DESE)Element (IESE)

Energy stored over time Energy storage instantaneous

Integral causality for C

Ec = PE = ⌡⌠t e[ q(t) ] f(t) dt = ⌡

⌠t

e[ ⌡⌠t f(τ) dτ ] f(t) dt

depends on history of flow f(t) into C, not instantaneous state(s)of other C's & I's

Derivative causality for C

Ec = ⌡⌠t e(t) f(t) dt = ⌡

⌠t

e(t) ddt q[ e(t) ] dt

= e(t) q[ e(t) ] |t - ⌡⌠t q[ e(t) ]

de(t)dt dt

1442443 144424443

instantaneous history

depends on instantaneous state(s) of system, including other C's & I's

Prefer integral causality for ESE (I's & C's), not always possible!

Junctions

Pseudo-elements: no direct correspondence to physical components

⇒ No power losses & no energy storage. Power balance:

Ptotal = ∑k=1

n

Pink -∑

i=1

m

Pouti = ∑

k=1

n

eink f

ink -∑

i=1

m

eouti f

outi = 0

0e

1

e2

e3

f1

f2

f3

0 junction: common (same) effort / all bonds: e1 = e2 = ... = en = em = e

∑k=1

n

fink -∑

i=1

m

fouti = ∑

k=1

n+m fk = 0

Incorporates into Bond Graph• Electrical: Kirchoff's Current Law ( ∑

node currents into = 0 )

• Mechanical: kinematics (balances derivative of displacements)

1e

1

e2

e3

f1

f2

f3

1 junction: common (same) flow/ all bonds: f1 = f2 = ... = fn = fm

∑k=1

n

eink -∑

i=1

m

eouti = ∑

k=1

n+m ek = 0

Incorporates into Bond Graph

• Electrical: Kirchoff's Voltage Law (over loop)• Mechanical: D'Alembert's dynamic equilibrium

Causality: 0 & 1 Junctions

0

e1

e2

e3

f1

f2

f3

0 junction: common effort e1 = e2 = ... = en = e⇒ only ONE bond can set common effort e ⇒ SINGLE ram against 0

(otherwise contradiction of common effort)Note: a 0 junction can have only one ram, but it MUST have a ram

(otherwise no common effort)

1e

1

e2

e3

f1

f2

f3

1 junction: common flow f1 = f2 = ... = fn = f⇒ only ONE bond can set common flow f ⇒ SINGLE hose squirts 1

(otherwise contradiction of common flow)Note: a 1 junction can have only one hose, but it MUST have a hose

(otherwise no common flow)

Power/Energy conversion: Transformers & Gyrators

Lossless 2 port : P1 = P2

Transformer

TF: ne2e1

f2f1relates

input effort to output effort: e1= n e2

lossless: e1f1 = P1 = P2 = e2 f2 = n e2f1

⇒ f1 = n-1 f2

TF Example: Electrical transformer

V2V1

+ +

i2i 1

--

n1 primary windings, n2 secondary windingssame flux in all windings

= dφ dt =

V2n2

⇒ V1 = n1n2

V2 ⇒ n = n1n2

TF: ne2e1

f2f1NOTE: Electrical TF converts: elect. power ⇒ magnetic power ⇒ elect. power

TF Example: Traction Roller

vω R

Kinematics: v = R ω

Equilibrium: T = R F

T

ωTF: R

F

v

NOTE: This TF converts rotational power to translational power

Gyrator

GY: re2e1

f2f1

relates input effort to output flow: e1 = r f2

lossless: P1 = e1 f1 = r f2 f1 = P2 = e2 f2

⇒ f1 = r-1 e2

21

GY Example: Solenoid Coil

+i

-

V

B

n turn coil

total flux φ

magnetomotive

force M = n i

lines of magnetic

induction B

n windings ⇒ flux linkage

€

V = ˙ λ =dλdt

= n dφdt

= n ˙ φ ⇒ r = n

also M = n I

NOTE: This GY converts electric power to magnetic power

GY Example: Gyrator & Capacitance synthesize Inertance

e1

f1

GY: r(q)e2

f2q.

=C

e1 = r f2 = r q. f1 = r-1 e2(q)

Define p = r q, then q = p r-1 and

e1 = d(r q)

dt = p. f1 = r-1 e2(p r-1) = ƒ(p)

Compare to inertance

TF & GY Causality

Transformer: e1 = n e2 effort evokes effort

f1 = n-1 f2 flow evokes flow

⇒ 2 choices:

TFe1

f1 f2

e2

e1

f1 f2

e2TF

Gyrator: e1 = r f2 flow evokes effort

f1 = r-1 e2 effort evokes flow

⇒ 2 choices

e1

f1 f2

e2GY

e1

f1 f2

e2GY

Bond Graph Causality Assignments: Procedure & Priorities

1. Sources 1st: effort ⇒ ram, flow ⇒ hosePropagate ramifications through BG

single RAM against 0single HOSE squirts 1one of 2 choices: TF & GYR: no restrictions, assign to make rest of BG happy

2. Attempt integral causality on C 's & I 's, one by onePropagate ramifications

3. Choose causality for an unassigned RPropagate ramifications

NOTE:derivative causality may be forced on some C 's or I 'sphysical significance: states for these C 's & I 's depend on other C 's & I 's

Example:translation of 2 dumbbell masses attached via rigid barmomenta of one dependent on other

EXAMPLE: RLC circuit

+

-

V

R

C

L

+

-

+ -

+-

1S : Ve

R: R

C: C

I: L

V(t)

q.

q / C

λ.

λ / L

1S : Ve

R: R

C: C

I: L

V(t)

q.

q / C

λ.

λ / L

BG: power flows shown. Efforts Effort source gets RAM.& flows labeled with energy variables. No ramification with 1 junction.

1S : Ve

R: R

C: C

I: L

V(t)

q.

q / C

λ.

λ / L

1S : Ve

R: R

C: C

I: L

V(t)

q.

q / C

λ.

λ / L

Integral causality assigned to I. Note Propagate ramifications1 junction now has its HOSE. mandated by 1 junction.

1S : Ve

R: R

C: C

I: L

V(t)

q.

q / C

λ.

λ / L

λ R / L λ / L

Causality ⇒ R accepts flow λ/L from 1---dictated byflow on input (hose) bond;

R responds with effort (λ/L ) R.

Multi-Port Storage Elements

Ce

1

ke

em

......

fm

f1

kf

Multi-Port Capacitance Cm

Flow from kinematics: fk = qk.

Stores potential energy

E = ⌡⌠

∑k = 1

m Pk dt =

⌡⌠

∑k = 1

m ek fk dt =

⌡⌠

∑k = 1

m ek qk

. dt =

⌡⌠

∑k = 1

m ek dqk

= E(q1 , q2 , ..., qm )

Energy variables = displacements: qk

Power flowdEdt = Ptotal = ∑

k = 1

m Pk = ∑

k = 1

m ek fk

Time derivative of Energy / Apply chain rule:

dEdt = ∑

k = 1

m

∂E∂qk

dqkdt = ∑

k = 1

m

∂E∂qk

fk

Compare: ek = ∂E

∂qk = ek(q1, q2, ..., qm)