Principles of

Engineering System Design

Dr T Asokan

asok@iitm.ac.in

Principles of

Engineering System Design

Dr T Asokan

asok@iitm.ac.in

Bond Graph Modelling of Dynamic systems

Physical System Modelling

• Bond Graph Method• The exchange of power between two parts of a

system has an invariant characteristic.• The flow of power is represented by a Bond• Effort and Flow are the two components of

power.

Physical System

Engineering Model

Differential Equations

Output

Simulation Language

Block Diagrams

Physical System

Engineering Model

Bond Graph

Output

Software(Computer Generated Differential Equations)

Classical approach for modeling of physical system Bond Graph Modeling

Generalised Variables

p = e · dt

q = f · dt

Power variables: Effort, denoted as e(t); Flow, denoted as f(t)Energy variables: Momentum, denoted as p(t); Displacement, denoted as q(t)

The following relations can be derived:Power = e(t) * f(t)

October 2, 2008

Energy FlowThe modeling of physical systems by means of bond graphs

operates on a graphical description of energy flows.

The energy flows are represented as directed harpoons. The two adjugate variables, which are responsible for the energy flow, are annotated above (intensive: potential variable, “e”) and below (extensive: flow variable, “f”) the harpoon.

The hook of the harpoon always points to the left, and the term “above” refers to the side with the hook.

e

f P = e · f e: Effortf: Flow

Modeling: Bond Graph Basics

• effort/flow definitions in different engineering domains

Effort e Flow f

Electrical Voltage [V] Current [A]

Translational Force [N] Velocity [m/s]

Rotational Torque [N*m] Angular Velocity[rad/sec]

Hydraulic Pressure [N/m2] Volumetric Flow[m3/sec]

Chemical Chemical Potential[J/mole]

Molar Flow[mole/sec]

Thermodynamic Temperature[K]

Entropy FlowdS/dt [W/K]

• I for elect. inductance, or mech. Mass

• C for elect. capacitance, or mech. compliance

• R for elect. resistance, or mech. viscous friction

• TF represents a transformer

• GY represents a gyrator

• SE represents an effort source.

• SF represents a flow source.

Modeling: Bond Graph Basic ElementsI

C

R

TFm

e1f1

e2f2

e2 = 1/m*e1f1 = 1/m*f2

GYe1f1

e2f2d

f2 = 1/d*e1f1 = 1/d*e2

SESF

Modeling: Bond Graph Basic Elements• Power Bonds Connect at Junctions.• There are two types of junctions, 0 and 1.

0 11

2

3

45

11

12

13

Efforts are equale1 = e2 = e3 = e4 = e5

Flows sum to zerof1+ f2 = f3 + f4 + f5

Flows are equalf11 = f12 = f13

Efforts sum to zeroe11+ e12 = e13

Causal Bond Graphs • Every bond defines two separate variables, the effort e and

the flow f.• Consequently, we need two equations to compute values

for these two variables.• It turns out that it is always possible to compute one of the

two variables at each side of the bond.• A vertical bar symbolizes the side where the flow is being

computed.e

f

•Mandatory Causality ( Sources, TF, GY, 0 and 1 Junctions)•Desired Causality (C and I elements)•Free Causality (R element)

“Causalization” of the Sources

U0 = f(t)

I0 = f(t)

U0

iSe

Sfu

I0

The source computes the effort.

The flow has to be computed on the right side.

The source computes the flow.

The causality of the sources is fixed.

“Causalization” of the Passive Elements

u

iR

u = R · i

u

iR

i = u / R

u

iC

du/dt = i / C

u

iI

di/dt = u / I

The causality of resistors is free.

The causality of the storage elements is determined by the desire to use integrators instead of differentiators.

Integral Causality (desired Causality)

e

fI

e

f sI

1

fe C

f

e

sC

1

Integral causality is preferred when given a choice.

edtI

f1

maF

fdtC

e1

“Causalization” of the Junctions

0e1

e2

e3f1

f2

f3

e2 = e1

e3 = e1

f1 = f2 + f3

1e1

e2

e3f1

f2

f3

f2 = f1

f3 = f1

e1 = e2+ e3

Junctions of type 0 have only one flow equation, and therefore, they must have exactly one causality bar.

Junctions of type 1 have only one effort equation, and therefore, they must have exactly (n-1) causality bars.

Modelling Example Mechanical Systems

R

C

mx 0x

MF

FKxxRxM mmm

Equation Governing the system F

FR

Fm

Fk

Mass, Spring and Damper Syetms

C

R1Se

I

Final Bond Graph

e1

em

ec

f1

fmeR

fcfR mc

mmcc

cRm

KxF

xdtxKdtfC

e

FFFF

1

System Equations

Bond Graph model

Sf1

C

1

01Se

R

IReference Velocity=0 for this case

Velocity Junction

DamperSpring

Mass

Simulation-Second order system

Modelling of underwater Robotic systems

Euler angles

Linear velocity of the base point w.r.t Inertial frame

TzTy

Tx

Ixx+Iax

wx*(Ixx+Iax)

wz*(Izz+Iaz)wy*(Iyy+Iay)

wz

wx

wyIzz+IazIyy+Iay

PV TFMV TFMV

Body fixedangularvelocity

Body fixedlinearvelocity

m+max

m+may m+maz

Euler angleTransformationmatrix

wx*(m+max)

wy*(m+may)wz*(m+maz)

Vx

Vy Vz

Angular velocity to first link of themanipulator

Tbx

TbyTbz FzFy

Fx Fbx

FbzFby

MTF0MTF MGY MGY

MGYI I

I

11

1

MSe

MSeMSe

MTF

MSe

MSe

MGY MGY

MGYI I

I

11

1

MSe

MSeMSe

MSe

1

MTF

1

1

ò

MSeMSeMSe

1

MR

MRMR

MR

MRMR

1

Angular velocityfrom previous link

Joint velocity

angular velocityof the manipulator

PV1 TFM1

TFM1PVM1

Ix1+Iax1Iy1+Iay1

Iz1+Iaz1

m1+ma1

Tip velocityof link1

TF1

TF2

TF3

Pad

Pad

Euler angles

Linear velocity of the base point w.r.t Inertial frame

TzTy

Tx

Ixx+Iax

wx*(Ixx+Iax)

wz*(Izz+Iaz)

wy*(Iyy+Iay)wz

wx

wyIzz+IazIyy+Iay

PV TF TF

Body fixedangularvelocity

Body fixedlinearvelocity

m+max

m+may m+maz

Euler angleTransformationmatrix

wx*(m+max)

wy*(m+may)wz*(m+maz)

Vx

Vy Vz

Angular velocity to first link of themanipulator

Tbx

TbyTbz FzFy

Fx Fbx

FbzFby

Wx1*(Ixx1+Iax1)

Wz1*(Izz1+Iaz1) Wy1*(Iyy1+Iay1)

Wz1

Wx1

Wy1

Angular velocityfrom previous link

Joint velocity

angular velocityof the manipulator

PV2 TFM2

TFM2PVM2 m2+ma2

Tip velocityof Link2

TF1

TF2

TF3Pad

Pad

Joint velocityPV3 TFM3

TFM3PVM3 m3+ma3

Tip velocityof the manipulator

TF1

TF2

TF3

Pad

Se 1

R

I

1

TF

TF

TF

0

1MGY MGY

MGY

I I

I

11

1

1

MTF MTF

MTF MTF

0

0 I0

MTF

1

1

Se

SeMR

MR

MR

MR

MTF0MTF MGY MGY

MGYI I

I

11

1

MSe

MSeMSe

MTF

MSe

MSe

MGY MGY

MGYI I

I

11

1

MSe

MSeMSe

MSe

1

MTF

1

1

ò

MSeMSeMSe

MR

MRMR

MR

MRMR

1 1

Se 1

R

I

1

TF

TF

TF

0 1 1

MTF MTF

MTF MTF

0

0 I0

MTF

1

1

Se

SeMR

ADL2

Se 1

R

I

1

TF

TF

TF

0 1 1

MTF MTF

MTF MTF

0

0 I0

MTF

1

1

Se

SeMR

Pad

ADL3

Link1 1111asffa11

Link 2 221

Link 3 221

Advantages and disadvantages of modelling and simulation

Advantages• Virtual experiments (i.e. simulations) require less resources• Some system states cannot be brought about in the real

system, or at least not in a non-destructive manner ( crash test, deformations etc.)

• All aspects of virtual experiments are repeatable, something that either cannot be guaranteed for the real system or would involve considerable cost.

• Simulated models are generally fully monitorable. All output variables and internal states are available.

• In some cases an experiment is ruled out for moral reasons, for example experiments on humans in the field of medical technology.

Disadvantages:

•Each virtual experiment requires a complete, validated and verified modelling of the system.

•The accuracy with which details are reproduced and the simulation speed of the models is limited by the power of the computer used for the simulation.

SUMMARY Modelling and simulation plays a vital role in various stages of the system design

Data Modelling, Process Modelling and Behavior modelling helps in the early stages to understand the system behavior and simulate scenarios

Dynamic system models help in understanding the dynamic behavior of hardware systems and their performance in the time domain and frequency domain.

Physical system based methods like bond graph method helps in modelling and simulation of muti-domain engineering systems.