BOND GRAPHS AND ITS USE IN MECHATRONICS

Gunnar Künzel, René Neděla

Czech University of Life Sciences Prague Faculty of Engineering

Department of Electrical Engineering and Automation

Aims of the article

• bond graph method as a tool for modeling physical system dynamics.

• recommended procedure for bond graph construction

• examples of usage of the method: Mechanical, thermal and electromechanical, system.

Bond Graph

• allows general description of various physical systems using power and energetic quantities.

• an be transformed into block diagram in order of numerical simulation. A System of equations can be designed for dynamic properties analysis.

• Symbols used for circuit drawing is different for mechanical, hydraulic and pneumatic systems.

Bond Graph Method

• The system is created on the real object and then it is disintegrated into subsystems (decomposition) whose relative bonds are depicted in an oriented graph.

Scheme of design and usage of bond graph

Multi-ports - generalized variables • multi-port - physical system with one or more ports

• port - the place where subsystems are connected; power “flows” out or into the subsystem through it.

• power variables

– effort e(t)

– flow f(t)

their product gives a value of instant power “flowing” between two multi-ports – the power of the bond

• electrical analogy: e – voltage, f – electric current.

• energy variables:

– generalized momentum p(t) - time integral of voltage

– generalized deviation q(t), - time integral of flow

• Relation between power and energy variables e, f, p, q can be shown using so called circle diagram or using relation tetrahedron for these four variables in bond graphs.

Multi-port types

• Basic multi-ports – n-ports are junctions of power flow. • 1 Junction - the same f at all power bonds entering the

junction so that the junction describes the “effort” balance (2nd Kirchhoff’s rule).

• 0 Junction - the same e and the junction describes “flow” balance (1st Kirchhoff’s rule).

• In an electrical circuit – 1 junction represents serial wiring – 0 junctions represent parallel wiring

• In a mechanical system, junctions are assigned conversely because of duality principle.

Tetrahedron of relations

Tetrahedron of relations between e, f, p, q

variables

I, C, R symbols represent elementary generalized components of the bond graph

Table of universal value

Generalized quantity Electrical systems

e – effort u – voltage

f – flow i – current

q – deviation Q – charge

p - momentum Ф – induction flow

Mechanical translational systems Mechanical rotational systems

F – force M – twisting moment

v – velocity ω – angular velocity

x – deviation φ – angular displacement

I – impulse I – impulse momentum

General rules for bond graph construction

• Recommended procedure of bond graph construction is now shown for mechanical and electrical systems in order to simplify drawing process. After getting some experience it is possible to omit some steps.

• Construction method has got six steps:

• junctions generation

• edges generation

• power orientation marking

• reference junction elimination

• graph reduction

• causality marking

• Recommended rules for power bonds orientation and causality marking are stated in [4].

Mechanical model of a car

• K1, K2 [N.m-1] - tyre springs

• R1, R2 [N.s.m-1] - realistic shock absorbers (dumpers)

• L1, L2 [m] - distance from the point of gravity

• vi - [m.s-1] velocities

• xi - [m] deviations

• mi - [kg] masses

• J - [kg.m2] momentum of inertia

• Ω - [rad] angle of chassis displacement

Network and bond graph of plane motor-car

Thermal system

• T [K] - absolute temperature

• Φ [W] - heat flux

• R [K.W-1] - thermal resistivity

• C [J.K-1] - thermal capacity

Physical representation of thermal resistor

(a) and capacitor (b)

Thermal system

Heat transfer through the wall T temperature F heat flux

Electromechanical system

Iconic diagram of the elevator system

The circuit with generalized variables

u - voltage, w - angular velocity, v - velocity

Complete bond graph

Causal bond graph

Classical block diagram

System of equations

tf

te

rC

R

I

e

f

rC

R

C

n

I

n

I

R

e

f

dt

d

11

2

1

7

3

2

11

2

1

1

1

1

1

1

7

3

1

)(11

11

2

72

11

2

3

1

1

7

11

2

31

1

7 tfrC

Re

rC

Rf

C

ntfe

rr

Rfn

Cdt

de

)(11

1

7

1

13

1

17131

1

3 teI

eI

nf

I

RenfRte

Idt

df

Conclusions

• The method of bond graphs and n-port models allows showing individual block bonds and differential 1st order equation system can be induced from the structure together with supplementary algebraic equations.

• The method is useful for solution of mechatronical systems comprising elements of various physical natures (mechanical, electrical, pneumatic, hydraulic and thermal or their combination).

• Software available:

DYMOLA http://www.dynasim.se

20-sim v4.1 http://www.20sim.com

THAT’S ALL THANK YOU FOR YOUR ATTENTION

Gunnar Künzel, René Neděla

Czech University of Life Sciences Prague Faculty of Engineering

Department of Electrical Engineering and Automation