Modeling & Simulation 2019Lecture 9. Bond graphs (con’d)

Claudio AltafiniAutomatic Control, ISYLinköping University, Sweden

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Lab 1 (Identification): important dates

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Summary of lecture 8

• Analogies among physical domains• Bond graphs

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Summary of lecture 8

• Analogies among physical domains• Bond graphs

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Lecture 9. More on bond graphs

Summary of today:

• Bond graphs for basic elements (recap)• Causality• Simplifications for bond graphs• Controlled elements

In the book: chapter 6

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Bond graphs

Bond graph theory

1. flowf =

{i, v, ω, Q, q

}2. effort

e ={u, F, M, p, T

}=⇒ power

e · f ={u · i, F · v, M · ω, p ·Q, T · q

}• an exchange of power is a bond

e−−−⇀f

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Bond graphs

• one-port

effort source

See−−−−⇀f

flow source

Sfe−−−−⇀f

effort storage

e−−−⇀f

I : α

flow storage

e−−−⇀f

C : β

resistive elements

e−−−⇀f

R : γ

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Bond graphs

• two-port

transformer

e1−−−−−⇀f1

TF e2−−−−−⇀f2

gyrator

e1−−−−−⇀f1

GY e2−−−−−⇀f2

• multi-port

s-junction

......................e1f

s......................

e2 f

......................

en f

p-junction

......................ef1

p......................

e f2

......................

e fn

All conserve power

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State space description

Information flow in a state space model

x = f(x, u)

• for given x and u it is possible to compute x

• in a bond graph: causality marking

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Information flow

Information flow for bond graphs of C , I, and Se, Sf types:

︸ ︷︷ ︸integral causality

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CausalityCausal strokes

• “A sets e and B sets f ” (ex: A is of C and Se types)

• “B sets e and A sets f ” (ex: A is of I and Sf types)

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Causality, cont’d

• Bonds with fixed causality strokes:1. sources

2. C and I type

3. transformers

4. gyrators

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Causality, cont’d• Causality at junctions

1. s-junctions

2. p-junctions

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Causality, cont’d

• Elements with free causality: R-type

If e = γf (or e = φ(f) invertible) then two options:

1. System computes f , R computes e

2. System computes e, R computes f

e = φ(f) f = φ−1(e)

• certain nonlinearities may constitute an exception(when φ(·) is not invertible)

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Causality propagation in a bond graph

How to mark a bond graph with causality strokes?

1. begin setting the causality strokes where it has to obey a fixed rule (oneports: Se, Sf , I, C)

2. propagate through 2-ports (TF and GY ) and multiports (s-junctions,p-junctions)

3. select one R-element with no assigned causality and give it one,arbitrarily. Then repeat from step 2.

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DC-motor with gear and torsional spring

i

v

ω1

ω2

R1 L1

Gear ratio: nTorsional spring constant: kLeft rotation axis: moment of inertia J1; friction coeff. φ1(ω1)Right rotation axis: inertia J2, friction φ2(ω2)

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DC-motor with gear and torsional spring

Causality marking of the bond graph

Se......................

R:R1

......................

s

......................

I:L1

...................... GY ......................

R:φ1(·)......................

s

......................

I:J1

...................... TF ...................... p

......................C:k−1

......................

R:φ2(·)......................

s

......................

I:J2

=⇒ causality can be “propagated” to the entire graph

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Conflict-free causality graph

If the entire bond graph can be marked with causality strokes withoutconflict in the above rules

=⇒ conflict-free causality graph

=⇒ state space description

• at C-elements: choose xk as efforts• at I-elements: choose xk as flows• at source Se: choose effort• at source Sf : choose flow

}state space variables}

input variables

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Causality and DAE models

Possible problem: causality conflict=⇒DAE (Differential Algebraic Equations)

Any bond graph directly gives an unstructured DAE: write up the equationfor each node, R, I and C elements, sources, etc.

• conflict-free causality =⇒state space description• causality conflict =⇒DAE description• often: DAE can still be converted to steady state (remember rotatingmasses example?), but calculations can be involved

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Simplification of bond graphs

Contraction of p and s junctions

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Simplification of bond graphs

A (p or s) junction with equally directed bonds can be dropped

⇔p

A junction that changes direction can be “pulled through” another one

(and similarly, if p and s swap places)

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Simplification of bond graphs

Contractions for gyrators and transformers

⇔

⇔

⇔

⇔

TFTFTF

TF

TF

TF

GYGY

GY GY

1

r

r

r

rs

rs

s

s

s rs−1

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Simplification of bond graphs

Gyrators and transformers can be “pulled through” junctions

⇔

⇔

pp

p s

TF

TF

TF

GY

GY

GY

r

r

r

r

r

r

Notice how GY converts p-junctions in s-junctions and viceversa

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Simplification of bond graphs

Contraction of parallel gyrators

⇔ ssss GY

GY

GY

r

s

t

σtt = σrr + σss

• σt, σr, σs are +1 for right-pointing bonds, −1 otherwise• If t = 0 the graph splits into two parts• Analogous formulas hold for gyrators between two p-elements and forparallel transformers.

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Merging of elements. An example

s I :

I : β

e1

e2

f

f

s I :e1 + e2

f

{α f = e1

β f = e2gives (α+ β) f = e1 + e2

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Example

• Example: Elastic shaft

• Example: Rotating shaft with gears

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Controlled elements

Elements that are regulated by other variables

Example: resistence

......................ef

R:R(θ) � θ

The normal arrow represents an information flow without power flow (unlikethe bond graph arrow)

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Modulated transformer

...............................

e1

f1TF ....................

...........e2

f2

n(θ)

?

θ

e2 = n(θ)e1

f2 =1

n(θ)f1

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Modulated element: rotating arm

F

v

θM

ω = θ` = length of the arm

M = F` sin θ

ω =v

` sin θ

Modulated transformer n = ` sin θF

v

θ

M

ω

ℓ sin θTF

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Systematic bond graph modeling

Electrical circuits:1. place p-junction in all nodes2. join p-junctions by means of s-junctions3. attach Se, Sf , C, I, R to s-junctions4. simplify

Mechanical system:1. place s-junction for every point with a well-defined velocity2. introduce s-junction for velocity differences, and p-junctions to form

these differences3. attach I and C, R4. simplify

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Bond graphs

Meaning of bond graphs:• describes various physical domains in the same way• expresses simple addition rules for effort and flow variables• illustrates the structure of a system from its components• translates physical laws into graphical interactions• systematic way to generate state space (or DAE) models• formalism prone to object-oriented programming

Claudio Altafini

www.liu.se