A bond graph approach , simulation and modelling ( Mechatronics ), INDIA

Lectures on Bond Graph Modeling of Mechatronic Systems, July – December 2016
©JosephAnandVaz,DepartmentofMechanicalEngineering,DrBRAmbedkarNIT,Jalandhar144011,India
1
Introduction to physical systems, their
modeling and simulation: A Bond
Graph Approach
Anand Vaz
Professor
Department of Mechanical Engineering
Dr. B. R. Ambedkar National Institute of Technology
Jalandhar
Punjab 144011, India

2
Physical Systems
• What do we understand by physical systems?
• What is a system?
• entity separable from the rest of the universe by means of a
physical or conceptual boundary
• Composed of interacting parts
• Physical systems are those in which Energy or Power is transacted
between their components
• Power can have diverse forms
– Mechanical, Electrical, Electronic, Thermal, Chemical, Fluid,…
• What about cause and effect? Input and output?
• Causality is an important aspect of Physical systems
– It is the cause and effect relationship between components of the
physical system

3
Physical Systems
• Why are models needed?
– To study and understand the behaviour or response of a system
when it is subject to various conditions.
– To perform experiments or simulations and make measurements.
– For design and testing
– Control of physical systems is an important objective in
Engineering
– Systematic derivation of system equations and Computer coding
for simulation
• Modeling of Physical systems is an essential prerequisite for
studying response

4
Computer Aided Design
• Digital Computer: What are its major capabilities and features?
– Programmable
– Rapid execution of repetitive tasks
– Interface with peripherals and machines
– Affordable
• CAD
– Commercially available software: AutoDesk, I-DEAS, CATIA,…
• Drawings and material specifications
• Planning of processes
• Estimation
– Integration with Manufacturing
– Automated Inventory monitoring and control
– Costing
– Marketing
– Enterprise management

5
State determined systems
• State variables
– Minimum number of variables required to describe the behaviour
of a system completely
• System equations
– Ordinary differential equations – rates of change of states
– Algebraic equations – constraints
– outputs
• Linear systems
– Well developed theory
• Nonlinear systems
– Resort to simulation

6
Uses of dynamic models
• Analysis
– Given input future history and initial conditions, determining the
outputs
• Identification
– Given input history and output history, determining the system
• Synthesis
– Given input future history and some desired output history,
determining the system so that the input to it will produce the
desired output
• Control
– Given the system, determining the inputs to it that will produce
desired output

7
Dynamics and CAD
• Analysis of Dynamics is more important than Statics which may be
misleading
• Pictures and graphs are easier to understand than equations
• Can mathematical features of dynamics be represented pictorially?
• Can this modeling methodology be applied to any energy domain?
• Can system equations be derived easily from this pictorial model?
• How to simulate this model of a dynamic system in an easy
manner?
• Can simulation code be derived automatically?
• Can a pictorial model offer insight into analysis for design and
control?
• Bond Graph is a unified approach to the modeling of Physical
system dynamics

8
Example of an electro-mechanical system
..Eb(t)
GY
m
V(t) 1ai
R: Ra
pw
R: Rb
I: La
( ) ( )at i tτ µ=
1ωSe:V(t)
ia
( )tτ
ia ( )tω
I: Jd
( ) ( )bE t tµ ω=
( )tω
V(t)
Ra
La
Jd
Rb
ia

9
Bond graphs: A brief introduction
• Henry Paynter (MIT, 1959): The inventor of
Bond graphs.
– Pictorial grammar for Physical System Dynamics
• Power bond
– Power transaction as a product of effort and flow variables
• Elements
– Junctions, elements and connections
• Cause and effect

10
Bond graph modelling of physical system
dynamics
• Bond graph modeling – Why?
• Graphical representation of the dynamics of the system
– Offers insight into physics of the system
• Applied uniformly to multi-energy domains
• Interaction of power between the elements of the system
• Representation of Cause-effect relationships
• Algorithmic derivation of System equations in I-order state space form
– Suitable for numerical Integration. e.g. Runge-Kutta,…
– Basis for modern control theory
– Coding for simulation
• Modify the BG, append or delete part of it easily
• Helps in developing strategies for control and diagnosis, e.g.
– the impedance control strategy (Neville Hogan)
– Overwhelming or ghost control strategy (Mukherjee, …)
– Structural control properties (Genevieve Dauphin-Tanguy, Christophe Sueur, ...)
– Fault identification and diagnosis (Genevieve Dauphin-Tanguy, Bouamama,
Samantaray, Mukherjee, …)

11
Software
• CAMP-G
• 20-SIM
• SYMBOLS Shakti/Vista
• AMESim
• MATLAB/SCILAB based coding
– Easy to program
– All control with the modeler
– Easy to modify, append, etc.

12
Power transaction through a bond
effort = e(t)
flow = f(t)
Power = e(t) · f(t)
V I
F v
P Q
T
τ ω
… …
s
Variables of power: e(t), f(t)

13
• Sources
• Elements
• Junctions
• Transformer
• Gyrator
Elements
junction0
junction1
Source of effort e
S
Source of flow f
S
Inertia I
Stiffness C
Dissipation R
Transformer TF
Gyrator GY

14
Causality
effort = e(t)
flow = f(t)
cause – effect relationship
between elements/subsystems
A B
Input - Output relationship
e
f

15
Causal stroke
Effort receiving end
A B
e
f
Flow receiving end
Effort receiving end
A B
e
f
Flow receiving end

16
K·q2
Example: A simple mechanical system
K R
m
F(t)
v = x
C: K-1
R: R
( )1v t
I: m
Se: F(t)
p1
q2
1
24
3
1p v 1p
m
=
v
v
v
R·v
F

17
Example: A simple electrical system
It is the same Bond graph!!!
Bond graph can model in multi-energy domains – can’t it???
R: R
( )1i t
I: L
Se: V(t)
p1
q2
1
24
3
1p i 1p
L
=
i
i
i
R·i
V
RV(t)
L
C
i
i
2
1
q
C
C: C

18
K R
M
F(t)
v1 =x
v0 v1
F(t)
0 ( )1v t 0F
C: K-1
0 1
1 v
R: R
1 ( )1v t
I: M
Sf : v0(t) Se: F(t)
v1
p1
q2
1
2
3
4
5
8 6
7
v0 = 0x

19
Causality of Sources & Junctions
Output = effect
Se
Input = cause
Input = cause
Output = effect
Sf
Source of effort Source of flow
Junctions
1f 0e
e1
e2
e3
e4
f
f
f
f
e
e
e
e
f1
f4
f2
f3
e
f
e
f
e1 = e2 + e3 + e4 f1 = f2 + f3 + f4

20
Output = effect
Input = cause
Causality of I element
( )e t p= 
I : M
( )f p= Ψ
p
Mass M
v
1 1
( ) ( ) ( )
i i
t t
t t
p
f t p pd e d
M M M
τ τ τ=Ψ = = =∫ ∫
(effect)
(cause)
d
function
dt
=( )effect cause
i
t
t
function dτ= ∫
Natural or Integral causality for I element
;

21
Input = cause
Output = effect
I element in Derivative Causality
( )e t p= 
I : M
( )f p= Ψ
p
( )1
( ) ( ) ( )
dp d d
e t p f M f
dt dt dt
−
===Ψ = ⋅
( )effect (cause)
d
function
dt
=
Not natural causality for I element
Effect does not depend on past cause?

22
Input = cause
Output = effect
Causality of C element
( )e q= Φ 1
C: K−
f q= 
q
( ) ( ) ( )
i i
t t
t t
e t q K q K qd K f dτ τ τ=Φ = ⋅ = =∫ ∫
(effect)
(cause)
d
function
dt
=
( )effect cause
i
t
t
function dτ
 
=  
 
 
∫
Spring K
vA
A B
vB
A B
dq
q v v
dt
= = −
Natural or Integral causality for C element

23
Input = cause
Output = effect
C element in Derivative Causality
( )e q= Φ 1
C : K−
f q= 
q
1 1
( ) ( ( )) ( )
dq d d
f t q e K e
dt dt dt
− −
== = Φ = ⋅
Spring K
vA
A B
vB
A B
dq
q v v
dt
= = −
( )effect (cause)
d
function
dt
=
Not natural causality for C element
Effect does not depend on past cause?

24
Input = cause
Output = effect
Causality of R element
e
R
( )f e= Φ
Resistance R
VA
A B
VB
( ) ( )A BV V e
f i e
R R
−
= = = = Φ
Input = cause
Output = effect
( )e f= Φ
R
f ( ) ( )A Be V V R i R f f= − = ⋅ = ⋅ = Φ

25
Transformer
Power conserving transformer
1e1 2 2e
1f 2f
2 1f fµ= ⋅
1 2e eµ⇒ = ⋅
TF
1 1 2 2e f e f⋅ = ⋅
( )2 1e fµ= ⋅ ⋅
( )2 1e fµ= ⋅ ⋅
1 2e eµ= ⋅
µ

1 1,ω τ
2 2,ω τ
1
2 1
2
N
N
ω ω= −
1
1 2
2
N
N
τ τ= −
1N
2N
Ex. Gears in mesh

26
..
Hydraulic Cylinder
F(t) Q
V(t)
P
A
F(t)
V(t)
P
Q
TF
A

27
Causality for the Transformer
1e1 2 2e
1f 2f
2 1f fµ= ⋅
TF
1 2e eµ= ⋅
µ

1e1 2 2e
1f 2f
1 2
1
f f
µ
= ⋅
TF
2 1
1
e e
µ
= ⋅
µ


28
Gyrator
Power conserving gyrator
1e1 2 2e
1f 2f
2 1e fµ= ⋅
1 2e fµ⇒ = ⋅
GY
1 1 2 2e f e f⋅ = ⋅
( )1 2f fµ= ⋅ ⋅
( )2 1f fµ= ⋅ ⋅
1 2e fµ= ⋅
µ

2 1F Vµ= ⋅
Ex. Mechanical gyrator
1 2F Vµ= ⋅
Y
Z
X
V2
F2
V1
F1

29
Causality for the Gyrator
1e1 2 2e
1f 2f
2 1e fµ= ⋅
GY
1 2e fµ= ⋅
µ

1e1 2 2e
1f 2f
1 2
1
f e
µ
= ⋅
GY
µ

2 1
1
f e
µ
= ⋅

30
What does causality tell us?
1
1: mI 2: mI
1FT ( ) SEtF :
1x 2x






1
2
l
l Differential
Causality
1x
()tF
1m 2m 2x
2l1l
rigid, massless
0
2: mI
1FT ( ) SEtF :
1: mI
1x 2x






1
2
l
l
1
stKC :
2m
1x
2x
1m
()tF
2l1l
Derivative causality! Salvaging causality by BG surgery
Interpretation of the modified system

31
..Eb(t)
GY
m
V(t)
1ai
R: Ra
pw
R: Rb
I: La
( ) ( )at i tτ µ=
1ωSe:V(t) ia
( )tτ
ia ( )tω
I: Jd
( ) ( )bE t tµ ω=
( )tω
5
3
1
2
4
6 7
C: 1
8
p1
p2
q8
V(t)
Ra
La
Jd
Rb
ia

32
Electromechanical Actuator
( )v t x= 
K
R
Head mass = M
( ) ( )F t i tµ= ⋅
( )V t
( )i t

33
Voice coil motor
VOICE COIL MOTOR
Spindle motor
Disk
Magnet Primary turns
Magnetic flux
Shorted turns
ae
be
+-
-
+
aR sRaL
sLasL
ai si
( ) ( )tiktF ai=
( )tF
( ) ( )tvKte bb =
e f
f
ai
Differential
Causalitysi
1
0
YG
kb
aLI :
TBR :
YG
ki
( ) SEtea :
aRR :
ILas :
1 1
TMI :
sLI :
sRR :
( )tv
ebe

34
A simple Machine Tool
Rθ
LSθ
TPm
x
Gear
reduction
1FT
N

( ) SEtV :
aLI :
i x
YG

µ
1 1
RJI : LSLI :
FT
P







π2
1
TPmI :
slideRR :aRR :
Differential
Causality
Rotor inertia Lead screw Tool post inertia
R
θ

LS
θ


35
Example: Cam follower mechanism
A
( )
θ
θ
θ
θ
θ

∂
∂=
⋅
∂
∂=
=
rx
dt
dr
dt
dx
rx
B
B
B
A Cam-Follower (Spring Loaded)
K
B
SR
m
J
r θ
θ
Ax1
0 1
Bx1
sRR :
KC :
A Simplification
mI :
Ax1FS

0
THE BONDGRAPH
RRs : 0 0
θ∂
∂r
TF :
θ
1
KC :
JI :
Bx1
( )
FS
t


θ
θ
1
Ax1
0 1
KC :
sRR :
1
ϑ∂
∂r
TF :
JI :
( )
FS
t


θ
THE SIMPLIFIED BONDGRAPH
FS

0
mI :

36
VB VA
VA VE
KAB
RAB
MB MA ME
B A Engine
KAE
RAE
vB vA vE
VE
1 BV 0 ABF
RB
C: KAB
-1
FE
I: MB
1B AV
R: RAB
1 AV 0 AEF
RA
C: KAE
-1
I: MA
1A EV
R: RAE
1 EV
I: ME
RE
FRE
FRA
FRB

37
Deriving System Equations
Q.1. What do the elements give to the system? (Outputs of elements.)
2 2;e k q=
Q.2. What does the system give to elements with integral causality?
(Inputs to elements with integral causality.)
1 1e p= 
2 2f q= 
( )1 1
2 2
1 00
R
k
F tmp p
q q
m
 
− −      
 = +     
      
  


They can be arranged in a matrix form
The I-order state-space form!
K R
m
F(t)
v = x
K·q2
C: K-1
R: R
( )1v t
I: m
Se: F(t)
p1
q2
1
24
3
1p v 1p
m
=
v
v
v
R·v
F
1
1 ;
p
f
m
= 3 3,e R f=
1,R f=
1
;
p
R
m
=
4 2 3e e e= − − ( ) 1
2 ;
p
F t k q R
m
= − −
1f= 1
;
p
m
=
4 ( );e F t=

38
Equivalence with the classical equation?
( )1 1 2
R
p F t p k q
m
= − −
( ) ( )2 2 2
R
m q F t m q k q
m
= − − 
( )tFqkqRqm =++ 222

( )2 2 2,mq F t R q k q= − − 
Free body diagram
Applying Newton’s II law From Bond graphs
They are equivalent!!!
1
2 2 1 ,
p
f q f
m
= = =
m
F(t)
v = 2q
2R q 2k q

39
v0 v1
F(t)
0 ( )1v t 0 KRF
C: K-1
0 1
1 v
R: R
1 ( )1v t
I: M1
Sf : v0(t) Se: F(t)
v1
p1
q2
1
2
3
4
5
8 6
7
K R
F(t)
v1 =x
v0 = 0x
M1
Skate board system

40
I. What do the elements give to the system?
v0 v1
F(t)
0 ( )1v t 0F
C: K-1
0 1
1 v
R: R
1 ( )1v t
I: M1
Sf : v0(t) Se: F(t)
v1
p1
q2
1
2
3
4
5
8 6
7
3 ( )e F t=
1
1
1
p
f
M
=
4 0 ( )f v t=
2 2e K q=
5 5e R f=
5 7f f=
7 6 8f f f= −
6 1f f=
1
0
1
( )
p
R v t
M
 
= − 
 
1 1e p= 
1 3 6e e e= −
6 7e e=
7 2 5e e e= +
( )F t= 2K q− 1
0
1
( )
p
R v t
M
 
− − 
 
2 2f q= 
2 7f f=
8 4f f=
1
0
1
( )
p
v t
M
 
= − 
 
K R
F(t)
v1 =x
v0 = 0x
M1
II. What does the system give to elements
with integral causality?
Procedure for deriving System Equations

41
Simulation from Bond graphs
Using separation of variables,
f1
( )
( ) ( , , )
dx t
x t f x u t
dt
= =
1
1( , , )
dx
f x u t
dt
=
0; ( )ix t x=
1 1( , , )dx f x u dτ τ=
1
1
( )
1
( )
( , , )
i i
x t t
x t t
dx f x u dτ τ=∫ ∫
1 1 1( ) ( ) ( , , )
i
t
i
t
x t x t f x u dτ τ− =∫
1 1 1( ) ( ) ( , , )
i
t
i
t
x t x t f x u dτ τ= + ∫
x1(ti)
x1(t)
ti tft
1( )x t

42
Simulating Bond Graph models
&
More examples
Anand Vaz
Professor
Department of Mechanical Engineering
Dr. B. R. Ambedkar National Institute of Technology
Jalandhar
Punjab 144011, India

43
K·q2
K R
m
F(t)
v = x
C: K-1
R: R
( )1v t
I: m
Se: F(t)
p1
q2
1
24
3
1p v 1p
m
=
v
v
v
R·v
F

44
K·q2
Example: The Van der Pol Oscillator
system
K R
m
F(t)
v = x
C: K-1
R: R
( )1v t
I: m
Se: F(t)
p1
q2
1
24
3
1p v 1p
m
=
v
v
v
F
2
2( 1)R q v⋅ − ⋅

45
..
Eb(t)
GY
m
V(t)
1ai
R: Ra R: Rb
I: La
( ) ( )at i tτ µ=
1ωSe:V(t) ia
( )tτ
ia ( )tω
I: Jd
( ) ( )bE t tµ ω=
( )tω
5
3
1
2
4
6 7
C: 1
8
p1
p2
q8
This image cannot currently be displayed.
V(t)
Ra
La
Jd
Rb
ia

46
VB VA
VA VE
KAB
RAB
MB MA ME
B A Engine
KAE
RAE
vB vA vE
VE
1 BV 0 ABF
RB
C: KAB
-1
FE
I: MB
1B AV
R: RAB
1 AV 0 AEF
RA
C: KAE
-1
I: MA
1A EV
R: RAE
1 EV
I: ME
RE
FRE
FRA
FRB

47
V2 (t)
Bond Graph for the Fluid system
V1(t)
P
TF
A1..F1(t)
V1(t) 1
1V 0P
R1
P
Q2
C:?
I: M1
TF
1/A2..
2
1V
I: M2
F2 (t)
R2
Q1
1M
2M
1F
1x 2x
2F
Cross-sectional
area 1A
Cross-sectional
area 2A
Compressible fluid

48
..
Hydraulic Cylinder
F(t) Q
V(t)
P
A
F(t)
V(t)
P
Q
TF
A

49
.. with friction and leakage
V(t)
P
TF
A..F(t)
V(t)
1V 0P
Rfriction
P
Q
Rleakage
F(t) Q
V(t)
P
ARfriction

50
..adding piston inertia
V(t)
P
TF
A..F(t)
V(t)
1V 0P
Rfriction
P
Q
Rleakage
I: Mpiston
piston
( ) Me t p= 
F(t) Q
V(t)
P
ARfriction
F(t) Q
V(t)
P
ARfriction

51
Fluid inertia
Q
v(t)
A
PA PB
Q
x∆
0
lim
t
x
v
t∆ →
∆
=
∆
0
lim
t
x
Q A v A
t∆ →
∆
= = ⋅
∆
( )0
lim A B
t
d x
A l P P A
dt t
ρ
∆ →
∆ 
⋅ ⋅ ⋅ = − ⋅ ∆ 
Q
1Q
PA PB
Q
l
A
ρ ⋅
I:
Pp Q
Applying Newton’s II Law,
[ ]P A B
d d l
p Q P P
dt dt A
ρ ⋅ 
= = −  
l

52
Pump or motor
.. P
Q
TF
µτ (t)
ω(t)
1Q
PA
PB
τ (t), ω(t)
QPA
QPB
Pump
BG for Pump
.. P
Q
TF
τ (t)
ω(t)
1Q
PA
PB1
µ
BG for Motor

53
Fluid system
1M
2M
1F
1x 2x
2F
Cross-sectional
area 1A
Cross-sectional
area 2A
Compressible fluid

54
V2 (t)
Bond Graph for the Fluid system
V1(t)
P
TF
A1..F1(t)
V1(t) 1
1V 0P
R1
P
Q2
C:?
I: M1
TF
1/A2..
2
1V
I: M2
F2 (t)
R2
Q1
1M
2M
1F
1x 2x
2F
Cross-sectional
area 1A
Cross-sectional
area 2A
Compressible fluid

55
Hydraulic Cylinder driven Tool post
PA QA
FR
MT
( )Tv t
FRs
Area = AA
Area = AB
PB QB

56
Hydraulic Cylinder driven Tool post
PA(t)
MT
( )V t x= 
FS
PRV
PB(t)
PD
C
E
A
τ (t)
ω(t)
QB(t)
QA(t)
Area = AA
Area = AB
QE(t)

57
BG for Mechanical-Fluid system
Q
TF
µτ (t)
ω(t)
1 CQ
PD
PC
DPC
, ,
0 C A EP
0 DP
1 EQ
TF 1V
TF:R:RPRV
I: MT
R:RS
1/AA
Se:PD
AB

58
BG for Mechanical-Fluid system
Q
TF
µτ (t)
ω(t)
1 CQ
PD
PC
DPC
, ,
0 C A EP
0 DP
1 EQ
TF 1V
TF:R:RPRV
I: MT
R:RS
1/AA
Se:PD
AB
PA(t)
MT
( )V t x= 
FS
PRV
PB(t)
PD
C
E
A
τ (t)
ω(t)
QB(t)
QA(t)
Area = AA
Area = AB
QE(t)

59
String based actuation
L R 0X
0Y
1RX
1RY
1LX
1A2A1P
2P
1Rθ
1Lθ
Active finger link
Thumb link (passive)
1m (opening)
2
m (closing)
String 1
String 2

60
Bond graph for the system with
differential causality
Effective during closing
1F
0
1Rθ1 A
J:I
1:C
1R
:τe
S
1Lθ1P
J:I
1:C TF TF
TF TF
Ls1
1
Ls2
1
Rs1
1
Rs2
12F
0
1Lr-
2R
r-2L
r
1Rr
Effective during opening
12
3
8
9101213
14
15
1617192021
24 e25e
Differential
causality

61
Bond graph for the string based prosthesis
Effective during opening
Effective during closing
12
3
4 5
6 7
8
910
11
1213
14
15
1617
18
1920
21
2223
24 e25e

62
Developments in Bond graph theory
• Structural controllability and observability
• Bicausality
• Differential bond graphs
• Multibond graphs
• Fault diagnosis
• Control strategies

63
Developments at NITJ
• Multibond graph modeling of Rigid body
mechanics
• Mechanisms and machines
– Robotic systems, Crankshaft-connecting-rod-piston-cylinder, Universal joint,
Quick return mechanism, Pantograph, Power hacksaw, Epicyclic gear trains,
Differential gear mechanism, …
• Soft contact dynamics
– Rigid objects interacting with soft material
• Biomechanics
– Development of hand prosthetic systems
– Extensor mechanism of the hand, the Winslow tendon network, …

• Control systems
• Coding in MATLAB
• Learning and teaching
64
Developments at NITJ …

65
Rufus Oldenburger Award
Conferral at the Dynamic Systems and
Control Luncheon, Thursday, December 6,
1979, during the Winter Annual Meeting in
New York, N. Y. Presentation by Donald N.
Zwiep, ASME President, 1979-1980,
PROFESSOR HENRY M. PAYNTER,
Professor of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, Massachusetts

66
Henry Paynter:
The inventor of Bond Graphs
1959
At the birth of Bond Graphs
1997
Upon his election to the NAE

67

68
Summary
• Modeling of Physical System Dynamics
– Bond graphs
– Causality
– Derivation of system equations
– Simulation using Bond graphs
• Examples
• Resources:
– Books
• Karnopp, Margolis and Rosenberg
• Amalendu Mukherjee, Ranjit Karmakar and Arun Samantaray
• Jean Thoma
• F T Brown
• Wolfgang Borutzky
– Journals
• ASME JDSM&C, IEEE SMC
• Journal of the Franklin Institute - Elsevier
• Simpra – Elsevier
• Simulation - SCS
– Other
• Lecture notes: Anand Vaz, ‘Lecture notes on Introduction to physical systems, their modeling and
simulation: A Bond Graph Approach’, October 2016, DOI: 10.13140/RG.2.2.19231.56487
• P. J. Gawthrop and G. P. Bevan, "Bond-graph modeling," in IEEE Control Systems, vol. 27, no. 2, pp.
24-45, April 2007. DOI: 10.1109/MCS.2007.338279

69
Thank you!
Questions?
anandvaz@ieee.org, or
anandvaz@nitj.ac.in

A bond graph approach , simulation and modelling ( Mechatronics ), INDIA

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