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Introduction

Power variables

Standard elements

Power directionsBond numbers

Causality

System equations

Activation

Example models

Art of creating modelsFields

Mixed-causalled fields

Differential causality

Algebraic loops

Causal loops

DualityMulti and Vector bond graphs

Suggested readings

Duality

Transformers and gyrators in a bond graph model may be reduced and lead to a

concise model with smaller number of elements. Such a reduction often obscuresthe physical aspects on which the original bond graph was based. However,

such reductions may sometimes reveal the physics of the system in an

alternative manner and provide deeper insight to the problem. Here, some

interesting studies on such combination of two-port elements and equivalences

are presented. The matter presented here are extracts from the from "Lecture

notes on system modeling" by Prof. A. Mukherjee and Prof. R.Karmakar of the

Indian Institute of Technology, Kharagpur.

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To start with, let us consider all possible combinations of gyrator and

transformer elements.

Gyrator and transformer combinations

Let us consider a segment of a bond graph model as shown in the figure below.

J1, J2 and J3 represent junctions (i.e., 1 or 0). The junction J2 serves the purpose

of separating two consecutive gyrators. The gyrators have causal orientations

as shown in the figure.

Let us now obtain the relations be tween the power variables in bond numbers 1

and 4.

e2 = m1*f1 and e3 = e2 = m1*f1,

f4 = 1/m2 *e3 and f4=m1/m2 * f1

Like wise

f3 = 1/m2 *e4, f2 = f3 = 1/m2 *e4

and e1=m1 * f_2, thus e1 = m1/m2 * e4.

The two gyrators are thus equivalent to a single transformer w ith modulus and

causal orientation as shown in the figure below.
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Likewise, other combinations of transformers and gyrators in various causal

postures can be reduced to simplified forms shown in the table be low.

Combination Reduced form

Combination of gyrators and sources

A gyrator converts flow to effort and effort to flow. Thus a source type and


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gyrator combination may be replaced by a dual source with suitable scaling

factor as shown in the table below.


Combination of a gyrators and transformers with storage and resiststive

elements

Let us consider a combination of a gyrator and an inertial element as shown inthe figure below. There may be an entire system model appended to the

combination. The term J in the model represents any junction (1 or 0).

The above part model has one state, namely P4. The input and output

equations may now be w ritten as follows.

DP4 = e4 = e3= m *f2 = m *f1,

e1 = e2 = m *f3 = m *f4 = m *P4/m4 = m2/m4 f1 dt.

Let us now consider another part model with same input as shown below.


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The new model has a state Q4 and its equations are

DQ4 = f1

e1 = K*Q4 = K f4 dt = K f1 dt.

Now, if we relate K = m2/m4, and Q4= P4/m, with the new state and parameter

values, the initial model is equivalent to the later simplified form.

Likewise other combinations may also be derived, as shown in the table below.

If any transformer modulus is specified in the reverse orientation as compared

to the item in the table, then in the equivalent parameter its reciprocal would

appear. It should be remembered that these equivalences are valid

only when modulii of two ports are constants.



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An example of gyrator equivalence

Let us now consider the bond graph model of an electrical circuit with a

transformer shown in the figure below. This model considers the conversion of

energy in electrical domain to magnetic domain and then back to the electrical

domain in the transformer. The transformer core losses are included in the

magnetic domain. Variables a, m and L represent the cross-sectional area,

magnetic permeability and mean length of the core.


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Let us change the 1-junction for magnetic flux f to a 0-junction by incorporating

gyrators of modulus np on all sides as shown in the figure below.


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The model shown above may now be reduced to the model shown in the figure

below.

The reduced bond graph model corresponds to a electrical system shown be low.


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The resultant system is the well-known primary referred equivalent circuit of a

transformer. The inductor and the resistance in parallel are parasitic elements

appearing in the circuit due to reluctance and eddy current losses in the

magnetic core. Ifm is very high and core resistance is very low, the total

impedance of parasitic elements would be very high. Such high impedance in

parallel can be neglected and dropped from the circuit model. Under such

conditions, the practical transformer tends to behave as an ideal transformer.

Dual Models

If a bond graph model represents a system, its dual model also represents an

admissible system. This rule can be greatly applied to derive newer dimensions

of system dynamics. For an example, let us consider a bond graph model shown

below. The dual model obtained using unitary gyrators at every junction (only

one in this case) and reduction is shown to the right.


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The corresponding systems, which may be represented by the above, are

shown below.

The mass parameter of the first model is mapped to the stiffness of the second

and vice versa. Similarly, the states are mapped in reverse, i.e. momentum of

first is mapped to displacement on the second and vice versa. Thus two systems

realized are distinctly different.

The utility of dualisation is felt significantly in control domain, where observer or

control models for the main plant may be very difficult to synthesize, whereas

the fully or partly dual model can be easily constructed. The parameters of the

system and observed variables may then be scaled back to determine states of


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the original system. However, all that is possible for linear systems only.

Dualization can be used to create a concise group of basic bond graph elements.

For instance, only one of the storage elements (I or C) may be used in the

model and the other storage element can be synthesized using a simplectic

unitary gyrator. Similarly, one needs only one junction (1 or 0) and one source

(SE or SF) type. Transformers can be equivalently represented by two gyrators

in tandem. Thus the nominal set of elements and junctions required are 5 (1 : SE

or SF, 2 : I or C, 3 : R, 4 : 1 or 0 and 5 : GY) as compared to 9 in normal notation.

Multi and vector bond graphs

When s imilarities in various sub-system components in the model morphology

can be established, they can be represented in form of a concise notation called

vector or multi bond graphs. Multi bonds are drawn as two parallel lines

augmented with power directions. The dimension of the multi-bond (number of

scalar bonds, it is composed of) is indicated between these parallel lines. Thus

multibond graphs are compact representation of large systems with identical

subsystems. Since a multibond can accept only one power direction and causal

orientation, all the subsystems represented by that multibond must have same

power and causal structure. Though multibond graphs are useful when initial

ideas are being formulated, they may obscure many physical aspects of the

system. A multi bond representation is shown in the figure below.

The multibond graph notation for single port elements (SE, SF, I, C and R) is

shown below. In the figure, m represents the bond number and n indicates the

multibond dimension. The scalar bond graph equivalents are shown to the right.


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Junction arrays in multibond graphs and its scalar equivalent are shown below.

The two port elements (TF and GY) appearing in a multibond graph are in the

form of transformation matrices. They have two ports which may have the same

or different dimensions. In a multibond transformer, the distributor is 1-junction

and the summer is 0-junction. A 3x2 multibond transformer and its scalar

equivalent with distributors and summers is shown in the figure below.


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In a gyrator both the distributor and the summer are 1 junctions. A 3x2

multibond gyrator and its scalar equivalent with distributors and summers is

shown in the figure below.

In both the previous examples, the transformer or gyrator transformation matrix

was dense (i.e., contained all non-zero elements). If these matrices are sparse

(i.e. contain some elements which are zero), then the corresponding branch can

be dropped from the scalar model. The only constraint for a sparse matrix is that

none of the summers or the distributors should be completely de-linked (i.e. any

l f th t f ti t i h ld t h ll l t l


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row or column of the transformation matrix should not have all elements equal

to zero). In such an unconstrained case, the entire bond graph collapses as that

part of the junction structure would result in a discontinuous graph.

The multibond field elements (FI, FC, FR) are multiports. Each port may or may

not have different dimensions as shown in the figure below (three multi bonds

connected with dimensions l,m and n). The field matrix is thus (l+m+n)x(l+m+n).

The Direct Sum of multibonds is a general method to

represent the composition o f multibonds, analogous to

the direct sum of vector-spaces. It is represented by a

line perpendicular to the multibonds which take part inthe summation. Power and causal orientations are

maintained, while the composition (out of scalar bonds)

and order of the multibonds may be changed. One such

direct sum is shown in the figure here.

The same transverse line notation is also used to decompose multi bonds to

scalar bonds. A composition and decomposition scheme is illustrated in the

figure be low.

A spatially anchored mass-spring damper system and its multibond graph model

are shown in the figure below.


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Let us now consider a system and its bond graph model shown in the figure

below.

The equation for effort variable in bond number 1 may be written as


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e1 = e3 = K*Q3 = K f3 dt = K (f2-f1) dt = K f2 dt - K f1 dt.

if f1 dt and f2 dt represent two states Q1 and Q2 in another model, then

e1 = e2 = -K*Q1 + K*Q2.

Now we may draw a bond graph model us ing a C field as shown below. The

coefficients of the first row of the C-field matrix are derived from the above

expression and the second row are derived from expression for -e2 (the

participation of e2 at 1-junction in earlier bond graph was negative by virtue of

its power direction). The expressions for the R-field in the model can be similarly

derived.

The new field model can now be represented in form of a multibond graph as

shown to its right.

The greatest advantage o f multibond graphs is felt in

cascaded systems. Let us consider a system shown in

the figure on the right. A bond graph model using C and

R-fields can be drawn for it. The two C-field and two R-

field e lements between three 1-junctions would be of

2x2 dimension One can then easily extend them to


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2x2 dimension. One can then easily extend them to

higher 3x3 dimension by adding a row and a column of

zeroes. The summation of these matrices would lead to

multibond fields as shown in the model below. In this

model, no extra bonds have been added as compared

to the earlier model, only the multibond dimensions

have been increased. This way, the model can be

extended to represent any large cascaded system.

Cascaded systems are common occurrence in modeling of structural members

such as beams and plates. Systems requiring spatial reticulation where

components are repeated and robotic systems ideally fit in to this scheme of

modeling.

Suggested readings (Online)

The Bond Graph Digest : An Electronic Journal for Bond Graph Research and Applications.

Introduction to Bond Graphs in "Bondgraph Modeling and Model Evaluation of Human

Locomotion us ing Experimental Data".
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Seminar presentations by Peter Gawthrop.

Bond Graph Modeling and Simulation of Electrical Machines by Sergio Junco.

Design and Implementation of a Bond Graph Observer for Robot Control.

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