Introduction to Bond Graph Theory - II

7/24/2019 Introduction to Bond Graph Theory - II

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Introduction to bond

graph theory

Second part: multiport field andjunction structures, and

thermodynamics


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Multiportfi

eldsWe will look at multiport generalizations ofC, I and R elements.

Ce1

e2

en

f1

f2

fnCn

e

f

C-fields

Allows easier displayingof individual causalities

State variables q1, q2, . . . , q n

Used mainly for sets of bondswith geometrical properties.

q1 = f1

q2 = f2...

qn = fn

e1 = 1(q1, . . . , q n)

e2 = 2(q1, . . . , q n)

...

en = n(q1, . . . , q n)


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Energy is computed as

H(t) =H(t0) +

Z tt0

n

Xk=1ek()fk() d

Changing t qyields the line integral

H(q) =H(q0) + Z

e(q) dq is any curve connecting q0 and q

However, this must be independent of theparticular curve connecting q0 and q!

Barring topological obstructions, this is equivalent to

i

qj=j

qi, i, j = 1, . . . , n

Maxwell reciprocity condition

exactness of the 1-form given by e

e= d


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q=CeLinear case:

stiffness form compliance form

e=kq all differentialall integral

Mixed forms are also possible, but for a given systemsome of the forms, including the compliance one, may not exist.

The above nomenclature extends to the nonlinear case as well.

In the linear case, exactness ofe implies thatthe matrices k and C, if the latter exists, are symmetric.

The available forms determine which causal patterns are admissible.

k=

2 0 20 1 1

2 1 3

C

e1

e2

f1

f2

e3

f3

all-integral is possibleall-differential is notdet k= 0

furthermore . . .

Ce1

e2

f1

f2

e3f3

q1q2e3

=

1

2 0 1

0 1

11 1 0

e1

e2

q3

is possible


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C-fields given from the beginning as a set of

effort-displacement relations at n ports are calledexplicit.

Implicit C-fields are obtained when several C-elementsare assembled by way of a power continuous network.

Implicit C-fields can be reduced to implicit form. In the process, someelements with differential causality may be hidden from the port interface.

C2C1

C3

P1 P2


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P2C1

C3

C2

00 1

CC C

P2P1

:C1 :C2:C3

1 2

3 45

6 7

P1

e1=e6 =e3 e4=e7 =e2f6=f5=f7

f3=f1+f6 f4=f7+f2e5= e6 e7

q5=C3e5q3=f3 q4 =f4 f1 =i1

e3= 1

C1q3 e4=

1

C2q4f5= q5 f2 =i2

q5 = C3

q3C1

+ q4C2

q5 =C3e5 =C3(e6 e7) = C3(e3+e4) = C3

q3

C1+

q4

C2


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=i1 C3 q3C1

+ q4

C2q3=f3=f1+f6=i1+f5=i1+ q5

=i2 C3

q3

C1+

q4

C2

q4=f4=f2+f7=i2+f5=i2+ q5

1 + C3C1 C

3

C2C3C1

1 + C3C2

! q3q4

=

i1i2

q3q4

=

1

C1C2+C2C3+C1C3 C1C2+C1C3 C1C3

C2C3 C1C2+C2C3 i1

i2

We introduce new state variables q1, q2 such that q1 =i1=f1, q2 =i2 =f2.

Using q3=C1e3 =C1e1, q4 =C2e4 =C2e2, and integrating in time:

e1e2 =

1

C1C2+C2C3+C1C3 C2+C3 C3C3 C1+C3

q1q2


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e1e2

=

1

C1C2+C2C3+C1C3 C2+C3 C3

C3 C1+C3 q1

q2

kThis is a 2-port C-field in stiffness form.

Ce1

f1

e2

f2

They are just a convenientparametrization of the R3 surface

q5= C3

q3C1

+ q4C2

The state variables q1, q2 do notcorrespond to physical charges.

The dependent state variable has been hidden away from the port interface.


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I-fields e1e2

en

f1

f2

fnI

n

e

f I

State variables p1, p2, . . . , pn

p1 = e1

p2 = e2...

pn = en

f1 = 1(p1, . . . , pn)

f2 = 2(p1, . . . , pn)

...

fn = n(

p1, . . . , pn)

Energy:

H(p) =H(p0) +Z

f(p) dp

i

pj

=j

pi

, i, j = 1, . . . , n

independence of


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CM

F1

F2

V1

V2Rigid bar with mass m, length L and

moment of inertia Jrespect to the CM.

We consider only vertical displacementsandsmallrotations around CM.

Under these assumptions, this can be described as anexplicitI-field, with constitutive relation

V1V2

= 1

m + L2

4J1

m L2

4J1

m

L2

4J1

m+ L

2

4J

! p1p2


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IC-fields p1 = e1

...

pI = eI

q1 = f1...

qC = fC

f1 = 1(p1, . . . , pI, q1, . . . , q C)...

fI = I(p1, . . . , pI, q1, . . . , q C)

e1 = 1(p1, . . . , pI, q1, . . . , q C)

...

eC

= I

(p1, . . . , pI

, q1, . . . , q C

)

e1

f1IC

fI

eI

eC

fC

e1f1

Maxwell reciprocity equations

iqj

= jqi

, i, j = 1, . . . , C ipj= j

pi, i, j = 1, . . . , I

i

qj =

j

pi, i

= 1, . . . , I, j

= 1, . . . , C


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A typical example of an IC-field is an electrical solenoid transducer.

A more academic example is

P20

IC

0 GYP1

:L:C

..

m

1

2

3 4

5

6

q2 = f1 1

me6

p5 = e6

To get an explicit IC-field, definestate variables q, p such that f1= q, e6 = p

p=p5, q=q2+ 1mp5

e1 = 1

Cq

1

mCp

f6 = 1

mCq+

1

L+

1

m2C

p

Maxwell condition


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R-fi

elds Onsager forms

e1

e2

en

f1

f2

fnR

resistanceform e= (f)

f= 1

(e)conductanceform

Mixed causality forms may also be possible

In the linear case, implicit R-fields without gyratorsor sources have Onsager forms with symmetric matrices.

If some (e, f) pairs are interchanged in their causality froman Onsager form, the corresponding matrix adquires antisymmetric

terms. Such contitutive relations are said to be in Casimirform.


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0

1P1

1P2

R

GY R

R

:m

:R3

:R4

:R5

e1

f1

e2

f2

Several forms are possible by

switching the causality around.

e1e2

=

R3+R4 m+R4m+R4 R4+R5

f1f2

This Onsager form is not symmetric, due to the presence of a gyrator.


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Junction structuresAssemblages of 0, 1, TF and GY elements which switch energy around.Limiting cases ofR-fields (without sources)which do not dissipate.

Unless modulated TF or GY elements are present, effort/flowconstitutive relations in a junction structure are always linear.

With an all-input power sign convention, the matrixrelating inputs to outputs must beantisymmetric.

Causality patterns are more restricted, though.

Junction structures without gyrators cannot acceptconductance or resistance causality on all ports.


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0

1

0

TF

GY

P2

P3

P1

m:

:

r f1e2f3

= 0 m m

rm 0 m

mr

m 0

e1f2e3

Pure conductance or resistanceforms are not possible.

Multiport transformers are an special case of junction structures.

TF 0

1 0

P4

P1

TF

P3

1P2

:m1

..

m2

Through-powerconvention

With an all-input powerconvention, (e1, e2, f3, f4) would beobtained from (f1, f2, e3, e4) with

an antisymmetric matrix.


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With the through-power convention, the matrix is symmetric

and can be decomposed into two matrices wich are transpose:

e1e2 =

1 1m1 m2

e3e4

f3f4 =

1 m11 m2

f1f2

TF

M

.. The fact that the flow

transformation is given by M

T

ensures the power continuity.

Multiport transformers need not havethe same number of inputs ans outputs.

Example: abc dqtransformation in induction machines.

Junction structures are also necessary to connect the bond graph

formalism with port Hamiltonian and Dirac structure concepts.


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Thermodynamics fromthe bond graph point of viewPure substance with no motion, constant mass and

no electromagnetic or surface-tension forces:

u=u(s, v)internal energyper unit massvolume

per unit massentropy

per unit mass

du=T ds p dvGibbs equation:

absolute temperature pressure


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T =u

sT

v =

(p)

sdu=T ds p dv

p=u

v

Maxwell relation for a 2-port C-fieldwith a power-through convention

T and p areefforts

T =T(s, v)

p=p(s, v)

constitutive

equations

s and v are the corresponding flows

for all-integral causality

CT p

s v


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In thermodynamics, mixed and all-derivative causality

is implemented by means ofLegendre transformations.

entalphy h Gibbs equation

h=u+pv dh= du+p dv+v dp =T ds+v dp

h=h(s, p)

constitutive equations Maxwell conditionT =

h

s

v = h

p

T = T(s, p)

v = v(s, p)

T

p =

v

s

C

T p

s vmixed causality


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Helmholtz free energy fGibbs equation

f=u T s df= du T ds s dT = s dTp dv

f=f(T, v)

constitutive equations Maxwell conditions =

f

T

p = f

v

s = s(T, v)

p = p(T, v)p

T =

s

v

CT p

s vmixed causality


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Any of the four formulations gives constitutive equations

which guarantee conservation of energy.

The computation path depends on the causality pattern.

For instance, assume the entalphy is given, h=h(s, p)

CT p

s v

s=Rs dt

h=h(s, p)T =

h

s

v = h

p

v=2h

p2 p +

2h

sps

s

p

v

T


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One can also give constitutive equations without

using any of the energy functions u, h, f or .

However, this can easily give impossiblesubstances, whichviolate the First Principle of Thermodynamics (energy conservation).

Ideal gas:pv =RT

Since this is a pure substance of the type considered, anotherconstitutive equation is needed to specify the 2-port.

The remaining equation is related to the gas being mono- or diatomic.

Giving this second equation arbitrarily runs into the above problem.

It is better to start with two other relations and build

an energy function from them, incorporating pv =RT.


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specific heat at constant pressure

this makes sense since p is aninputin the h(s, p) formulationcp =

h

T

specific heat at constant volume

cv = u

T

this makes sense since v is an

inputin the u(s, v) formulation

Tis not a natural variable ofhoru.

Together with pv=RT, we assume that cv

is aconstant, determined by the particular ideal gas.

pv = RT

h = u+pvcp =cv+R


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cv constant

cv =

u

T referencetemperature

u=cv(T T0)

h is also functionofT alone

h=u+RT

h=cp(T T0) +RT0

cp= h

Tcp constant

due to cp=cv+R

ds= duT

+p dvT

=cv dT

T +R dv

vdu=T ds p dv

T =T0escv v

v0

Rcv

s=cvlog T

T0+R log

v

v0integration ofthe 1-form


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ds=

dh

T

v

dp

T =cp

dT

T

v

dp

Tdh=T ds+v dp

=cp1

RT(p dv+v dp)

v

Tdp =cp

dv

v +

v

T cp

R 1

dp= cp

dv

v +cv

dp

p

p=p0escv

v

v0

cp

cv

s=cplog v

v0+cvlog

p

p0integration of

the 1-form

p=p0escv

v

v0

cp

cv constitutive equations fora perfect gas in all-integral form

s, v

T, pR, cvT =T0e

scv

v

v0

Rcv


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Exercise: compute the constitutive equations

for the other three causality patterns.

Chemical engineering:transport phenomena

quantities of substances vary with time

Requires an extension of the basicthermodynamic bond graph framework

Universit Claude Bernard Lyon:Bernhard Maschke& Christian Jallutstirred reaction tanks fuel cells
http://www.geoplex.cc/

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Introduction to Bond Graph Theory - II