Basic Modeling Concepts

Basic elements single and lti t d imultiport devices

System variablesi

m

vV

F

Power = V x i

P P Q

T

Power = F x v

Pw

Power = T x w

Q

Power =P x Q

Power = effort x flow

Power = T x w

Effort & flow are power variables

Eff t •Flows•Efforts ....• Force (F) Newtons• Voltage (V) Volts V• Torque (T) N-meters• Pressure (P) N /m2

•Flows....• Velocity (V) m/s• Current (i) Amps• Ang. velocity(w) rad/s• Volume flow (Q) m3/s• Pressure (P) N /m Volume flow (Q) m s

• power = effort (e) x flow (f)

System variables

• Generalized power and energy variables have the following relations:

• f = dq / dt q is a generalized displacement• e = dp / dt p is a generalized momentum.

• A state tetrahedron explains these relations

e

p

f

q

e=dp/dt

f=dq/dt

Physical Systems Variable Types

MECHANICALROTATION

HYDRAULICVARIABLE MECHANICALTRANSLATION

ELECTRICAL

Torque (T)(N-m)

Pressure (P)(N/m2)

Effort Force (F)(Newtons N)

Voltage(Volts V)

Angular velocity (w)(rad/s)

Volume flow (Q)(m3/s)

Flow Velocity (v)(m/s)

Current (i)(Amperes A)

Angle (rad)

Volume(m3)

Displacement Displacement (x)(m)

Charge (q)(A-s)

Angular momentum Pressure Momentum Flux linkageAngular momentum(N-m-s)

momentum(N-s/ m2)

Momentum Momentum (N-s)

Flux linkage(V-s)

Word bond graphs.• Word bond graph : Simple representation of

physical system; using words to imply a system component.

• Example: Car Starter/Solenoid

motor bendixbattery

The word bond graph can be drawn as

motor bendixbatterycurrent

torque

angular velocity

voltage

Word bond graphs• Car Model: - Power from engine is fed to clutch

- Transmitted to gear box. Selects gear- Power flows to wheels via a differential

gear box and drive shaftgear box and drive shaft.wheel

engine clutch gear-box drive shaft

wheel

differentialgear box

w2

T3 T4

T5

T6

T2T1

w3 w4w6

w5

w1

control gear selectionthrottle

Efforts: Torques (T1,T2,T3,T4,T5,T6)Flows: Velocities ( w1,w2,w3,w4,w5,w6)

wheelcontrol gear selectionthrottle

• Causality : Indicates WHO causes WHAT to WHOM

Concept of causality

eeee

AA BB

ffAA BB

AA BB

ffAA BB

in bond graph notation,

effort (e)

in bond graph notation,,

effort (e) flow (f)fl (f) effort (e)effort (e) flow (f)flow (f)

If element A imposes an effort on element B, then element B responds back with a flow or vice-versa,

Basic elementsTo convert a word bond graph to “complete bond” graph we need some basic elements.

Source of effort.(SE)

Source of flow(SF)

Constant effort junction(0)

Constant flow junction(1)

Basic elements.

Inertia element(I)

Mass,inductors,fluid pipes

Capacitive element(C)

Spring,capacitor,flywheel

Resistive element(R)

Dampers, resistors

Transformer element(TF)

Levers,transformers

Gyrator element(GY)

Centrifugal pump, generators

With these elements, bond graph models of dynamic systems can be created in any energy domain.

BOND GRAPHS and PHYSICAL VARIABLESBOND GRAPHS and PHYSICAL VARIABLES

Power

A B

power

A B

Power Flow Concept

effortflow

power

A Beffort flow

power

A B

Causality Concept

A imposes effort on B,B responds with a flow

effort

B imposes effort on A,A responds with a flow

FUNDAMENTALS OF BOND GRAPH MODELINGFUNDAMENTALS OF BOND GRAPH MODELING

effortflow

effortflow

power power

CCII2233CC

II

mmx(t)x(t)kk

bb

F F mmx(t)x(t)

kk

bbF F

00RR

SESE1122

44

33

11

CC

RRSESE11

22

44

33

To Sum Efforts To Sum Flows

A resistive element (R).

• There is a static relation between effort & flow. • Resistive elements are idealization of devices like,

dampers, resistors, fluid carrying pipes.

e

f

e = Rf

i

f

ee = f(f)

liLinear R

Units of R

NN--s/ms/mMech rotationMech rotation ElectricalElectrical HydraulicHydraulic

Non linear R

NN--mm--ss V/A (Ohms)V/A (Ohms) NN--s/ ms/ m22

Mech. translationMech. translation

Resistive element (R)Causality considerations :• A resistive element takes either form

Relation :e=g (f) Relation :f=g-1(e)

RefR f

e

Relation :e=g (f) Relation :f=g 1(e)

eR

f 1=Rfe =

R element

1e fR

5

R1e f

Rf e

R5

Capacitive element (C).• In a capacitive element a static relation exists

between effort & displacement.• These devices store or dissipate energy without loss.• Capacitive elements are idealization of devices like,

springs, capacitors, accumulators. e

q

e = Cq

q

ee = f(q)

Linear C

Units of C

N/mN/mMech rotationMech rotation ElectricalElectrical HydraulicHydraulic

Non linear C

NN--m/radm/rad faradsfarads N/ mN/ m55

Mech. translationMech. translation

Capacitive element (C)

• Integral causality • Derivative causality

Causality considerations :

• Integral causality • Derivative causality

dddtdq

dtdeCf ==

dq

qC

dtfC

e == ∫11

eC fC e

f

This is not preferred form for computational purposes

Preferred form for computational purposes

dtdeC

dtdq

=∴fdtdq

=∴

C elementC

4

S1f4 e4

C1

Inertia element (I).• There is a static relation between flow & momentum.• These devices store kinetic energy• Inertia elements are to model inductance effects in

electrical circuits mass & inertia effects in mechanical &electrical circuits, mass & inertia effects in mechanical & hydraulic systems.

p

ff= g(p)f

p

f = Ip

Linear I

Units of IUnits of I

NN--ss22/m/mMech rotationMech rotation ElectricalElectrical HydraulicHydraulic

Non linear I

NN--mm--ss22 VV--s/A (Henrys)s/A (Henrys) NN--ss22/m/m55

Mech. translationMech. translation

Inertia element (I)

• Integral causality • Derivative causality

Causality considerations :

eI

ef

I f

e

edtdp

pI

dteI

f

=∴

== ∫11

dtdfI

dtdp

dtdp

dtdfIe

=∴

==

This is not preferred form of causality for I element.

Preferred form for computationalpurposes.

dtdt

Impulse Momentum form Newton’s law form

I element I4

S1e4 f4

I1

The source elements SE & SF)• An effort source : System/element which maintains an input

effort. SE’s are voltage sources, forces, pressure.• A flow source : System/device which maintains a an input

flow SF’s are velocity sources current flow sourcesflow. SF s are velocity sources, current, flow sources

Vehicle Vehicle suspensionsuspensionsystem.system.

mmV

RoadRoad

Flow sourceFlow sourceEffort SourceEffort Source

Source elements (SE & SF)

• Effort Source • Flow source

Causality considerations :

Effort Source Flow source

Fl i flTh ff t

SE fe SF

ef

Flow source imposes a flowonto the system, connectedjunction or element.

The effort source imposes an effort on the connected junction or element

Source SE element

SE5

e5

f5SE

5

Source SF element

SE5 f5

SF5 SF5

f5

e5

Transformer element (TF)• Two port elements altering magnitude of either flow or effort

are transformer elements. • Transformers have static relation between input flow/effort &

output flow/effort by means of a transformer modulusoutput flow/effort by means of a transformer modulus.

A2A1

F2a b

F1v1

v221 v

bav = 21 FF b

a =

vF

PAAF

1

2= QAAv =

1

2

P,Q

A lever

Ratio (a/b) & (A2/A1) is transformer modulus.Other examples: gear set, electrical transformer, pulleys.

Hydraulic Ram

Transformer element (TF)Causality considerations :

m..

m’..

TFf1

e1

f2

e2TFf1

e1

f2

e2

21

21

fmfeme

==

21

21

fmfeme′==′

Incorrect causal form, not possible

TFf1

e1

f2

e2

TF elemente4 e5

TF45

TFe4 e5f4 f5

TF45

f4 f5TF45

TF elemente4 e5

1/TF45

TFe4 e5f4 f5

1/TF45

f4 f51/TF45

Gyrator element (GY)• Gyrators : Two port elements which relate input effort to output

flow or viceversa by means of a modulus.• Typical examples: voice coil, electric motor, generator.

F2

v1

F1

v2

ωv

21

21

FvrvrF

==

e ωi Τ

Trire== ω

Gyro-scopeIf the rotor spins rapidly, & asmall F1 will yield a proportionalvelocity v2, & vice-versa

Tri

MotorAngular velocity output is proportional to applied voltage e

Gyrator element (GY)Causality considerations :

r..

r’..

GYf1

e1

f2

e2GYf1

e1

f2

e2

21

21

erffre

==

21

21

erffre

′==′

Incorrect causal form, not possible

e1GYf1 f2

e2

GYf1 f2

e2e1

GY elemente4

GY45e5

GYe4 e5f4 f5

f5

GY45

f4 f5GY45

GY elemente4

1/GY45e5

GYe4 e5f4 f5

f5

1/GY45

f41/GY45

The (0) junction element

f3i1 i2

f2 c2c1

f1+f2 f3

f1

i3

i1-i2=i3

e1=e2=e3P1=P2=P3

f1+f2=f3 i1 i2 i3

The (0) junction cont.• (0 junction) : Is a common effort junction.

– All efforts are equalThe sum of the flows equal zero– The sum of the flows equal zero.

03

21 0321

321

=++==

fffeee

Summation signs will be determined by Power Flow.

Power flow and the (0) Junction

32

1 32

1031

321

321

fffeee

+===

031

2 2

321

321

fffeee

=+==

03

21

03

21

0321

321

=−−−==

fffeee

0321

321

=++==

fffeee

Causality and the (0) Junction

2

03

21 0

32

1

03

21

Power flow and Causality (0) Junction

32

1 32

10

31

2 2

031

321

321

fffeee

+===

321

321

fffeee

=+==

03

21

03

21

0321

321

=−−−==

fffeee

0321

321

=++==

fffeee

(0) Junction properties

• It is a common effort junction for all bonds attached j• All efforts are equal• The sum of the flows equal zero.• Power conserving, power in equals power out• Only one causal mark determines the input effort and

thus all other efforts will be outputs• There can only be one bond and only one bond that

sets the effort input• Power flow half arrows determine how the flows will

sum

The (1) junction element

RCv

V

w

f

N Fv

i

Current through C and R is the same. Summation of voltages

Velocity is common but sumation of forces must follow Newton’s law

The (1) junction cont.• (1 junction) : Is a common flow junction.

– All flows are equalThe sum of the efforts equal zero– The sum of the efforts equal zero.

13

21 0321

321

=++==

eeefff

Summation determined by Power Flow.

Power flow and the (1) Junction

32

1 32

1131

321

321

eeefff

+===

131

2 2

321

321

eeefff

=+==

13

21

13

21

0321

321

=−−−==

eeefff

0321

321

=++==

eeefff

Causality and the (1) Junction

2 2

13

2

1 13

21

13

2

1

Power flow and Causality (1) Junction

32

132

1

2 2

131

131

321

321

eeefff

+===

321

321

eeefff

=+==

13

21

13

21

0321

321

=−−−==

eeefff

0321

321

=++==

eeefff

0 junctionf4

f1f2

+

+

1 junction

e4e12

+f2f3

+-

e4e2e3

+-

12

42

01

3

411

3

4

(1) Junction properties

• It is a common flow junction for all bonds attached All fl l• All flows are equal

• The sum of the efforts equal zero.• Power conserving, power in equals power out• Only one causal mark determines the input flow and

thus all other flows will be outputs• There can only be one bond and only one bond thatThere can only be one bond and only one bond that

sets the flow input• Power flow half arrows determine how the efforts will

sum

Causal bond(4) 0 junction-C element

+

S1f4 e4=e2=e1=e3

C1

f1f2f3

+

+-

012

3

4 C

Causal bond 1 junction-I element

f4 f2 f1e1 +

S1e4 f4=f2=f1

I1e2

e3+-

2

11

3

4 I

Causal bond 1 junction R element

e4e1e2

+

+ f41e2e3

+-

Causal bond 0 junction R element

R

f4R

f1f2f3

+

+-

e4

Causal bond 1 junction TF element

e4e1e2

+

+ e5e2e3

+-

Causal bond 0 junction TF element

TF45

f4f1f2f3

+

+-

f5TF45

Causal bond 1 junction GY element

e4e1e2

+

+ f5e2e3

+-

Causal bond 0 junction GY element

GY45

f4f1f2f3

+

+-

e5GY45

Causal forms.

Symbol Implied meaning & variable relation.

C Preferred integral causal form.... f= dq/dt

I Preferred integral causal form.... e= dp/dt

C Derivative causal form....q= integ ( f )... not preferred

Derivative causal form....p= integ ( e )... not preferredI

2TF

Preferred form for gyrator element. (or opposite)GY

Preferred form for transformer element. (or opposite)f1=f2/m & e1=e2/m OR e1=me2 & f1=mf2

1