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Introduction

Power variables

Standard elements

Power directionsBond numbers

Causality

System equations

Activation

Example models

Art of creating modelsFields

Mixed-causa lled fields

Differential causality

Algebraic loops

Causal loops

DualityMulti and Vector bond graphs

Suggested readings

Generation of system equations

In this section the method of generation of system equations is discussed. From

an augmented bond graph, using a step by step procedure, system equationsmay be generated. We take the model of a simple single degree of freedom

mass-spring-damper system as the starting point.

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The differential equations describing the dynamics of the system are written interms of the states of the system. All storage e lements (I and C) correspond to

stored state variables (P for momentum and Q for displacement, respectively)

and equations are written for their time derivatives (i.e. effort and flow). These

equations are derived in four steps as described below.

Observe what the elements (sources, I's, C's, and R's) are giving to the

system and write down their equations looking at the causalities and using
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variables for strong bonds .

Write down equations for the junctions and the two-port elements for the

variables for the strong bonds.

Replace the variables, which are expressed in terms of states in other

equations. Continue sorting and replacement till the right side of the entire

set of equations are expressed in terms of states and system parameters

only.

If some equations are s till not completely reduced, there is the existence of

some kind of a loop (algebraic loop, causal loop or differential causality, as

shall be discussed later). Try solving those as a set o f linear equations e ither

through substitution or matrix inversion.

And finally, erase all trivial equations other than those for derivatives of

state variables and write them in terms of state variables.

Thus using the following steps, the equations for the above system would be

Step 1 :

e1=SE1

f3=P3/M3

e2=K2*Q2

e4=R4*f4=R4*f3 (by junction rule, f4=f3 and strong bond number is 3)

Step 2 :e1-e2-e3-e4=0 or e3=e1-e2-e4

Step 3 :

e1=SE1

f3=P3/M3

e2=K2*Q2

e4=R4*P3/M3

e3=SE1-K2*Q2-R4*P3/M3


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Step 4 :

DQ2=f2=f3

DP3=e3

where prefix 'D' stand for time derivative d/dt leads to

DQ2=P3/M3

DP3=SE1-K2*Q2-R4*P3/M3

The difference between equations derived from bond graphs and otherwise are

that there will be `N' sets of first order differential equations, where `N' is the

number o f states . The term, number o f states means the number of lumped

parameter storage elements I and C with integral causality present in a system.

The equation of motion for the system discussed above by traditional method is

:

m d2x/dt2 + r * dx/dt + k*x = F(t).

The two state equations derived from a bond graph model (dropping suffixes)

are

dP/dt = -r/m * P - k*Q + SE,

dQ/dt = 1/m*P,

where, P is momentum of m*dx/dt, Q is displacement or x and SE is F(t)).From the second equation, P=m*dQ/dt, which when replaced in the first leads to

m*d2Q/dt2 = -r*dQ/dt - k*Q + F(t).

This equation corresponds to that derived through traditional method after

rearrangement.

However, manual derivation of equations for larger systems is not all that


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simple. For instance, derivation of system differential equations for the animated

bond graph model discussed in the causality section after proper causalling

would lead to formation of the so called algebraic loops. Similarly, complexities

and errors of various types, like Causal loops, Power loops, and Differential

Causalities may exist in the model of a system. The method for derivation of

their equations is described in corresponding sections.

Example -2

Here we take another example of a system with two-ports, whose model is

shown below.


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Then we follow the normal equation derivation steps.

Step 1 :

e1 = SE1

f2 = P2/M2

e3 = R3*f3=R3*f2=R3*P2/M2

f6 = P6/M6

e11 = K11*Q11

e12 = R12*f12 = R12*f9

f13 = P13/M13

Step 2 :

f1 = f2 = f4 = f3 = P3/M3

e2 = e1 - e3 - e4 = SE1 - R3*P 2/M2 - e4

e4 = MU*f5 = MU*f6 = MU*P6/M6

e5 = MU*f4 = MU*f2 = MU*P2/M2

f5 = f7 = f6 = P6/M6

e6 = e5 - e7f8 = r*f7 = r*P6/M6

e7 = r*e8 = r*e9

e8 = e10 = e9

f9 = f8 - f10 = r*P6/M6 - f13 = r*P6/M6 - P13 /M13

f10 = f14 = f13 = P13/M13

e13 = e10 + e14 = e9 + SE14

f11 = f12 = f9

e9 = e11 + e12 = K11*Q11 + R12*f9

Step 3 :

e1 = SE1

f2 = P2/M2

e3 = R3*f3=R3*f2=R3*P2/M2

f6 = P6/M6

e11 = K11*Q11

e12 = R12*f9 = R12*(r*P6/M6 - P13/M13)

f13 = P13/M13


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f1 = f2 = f4 = f3 = P3/M3

e2 = SE1 - R3*P 2/M2 - e4 = SE1 - R3*P 2/M2 - MU*P6/M6

e4 = MU*f5 = MU*f6 = MU*P6/M6

e5 = MU*f4 = MU*f2 = MU*P2/M2

f5 = f7 = f6 = P6/M6

e6 = e5 - e7 = MU*P2/M2 - r*(e11 + e12)

= MU*P2/M2 - r*(K11*Q11 + R12*(r*P6/M6 - P13/M13))

f8 = r*f7 = r*P6/M6

e7 = r*e8 = r*e9 = r*(e11+e12) = r*(K11*Q11 + R12*(r*P6/M6 - P13/M13))

e8 = e10 = e9 = e11+e12 = K11*Q11 + R12*(r*P6/M6 - P13/M13)

f9 = f8 - f10 = r*P6/M6 - f13 = r*P6/M6 - P13 /M13

f10 = f14 = f13 = P13/M13

e13 = e10 + e14 = e9 + SE14 = K11*Q11 + R12* (r*P6/M6 - P13/M13) + SE14

f11 = f12 = f9 = r*P6/M6 - P13/M13

e9 = e11 + e12 = K11*Q11 + R12*f9 = K11*Q11 + R12*(r*P6/M6 - P13/M13)

Step 4 :

DP2 = SE1 - R3*P 2/M2 - MU*P6 /M6DP6 = MU*P2/M2 - r*(K11*Q11 + R12*(r*P6/M6 - P13 /M13))

DP13 = K11*Q11 + R12*(r*P6/M6 - P13/M13) + SE14

DQ11 = r*P6/M6 - P13/M13

In matrix form, these equations may be w ritten as d{Y}/dt = [A]{Y} + [B]{u},

where, {Y} is a vector of states (P2,P6,P13 and Q11), [A] is square matrix, {u}

is the array of sources (SE1 and SE14} and [B] is a matrix of dimension N x M; N

being the number of states and M being the number of sources. The matrices [A]

and [B] are;

A =

-R3/M2 -MU/M6 0.0 0.0

MU/M2 -R12*r2/M6 R12*r2/M13 -r*K11

0.0 R12*r/M6 -R12/M13 K11

0.0 r/M6 -1/M13 0 ,


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B =

1 0

0 0

0 1

0 0 .

Activation

Some bonds in a bond graph may be only information carriers. These bonds are

not power bonds. Such bonds, where one of the factors of the power is masked

are called Activated bonds. As an example, let the system shown in the figure

below be considered.


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Here the velocity pick-up only carries the information of velocity to the amplifier

through which an electro-magnetic exciter applies force proportional to velocity

on the mass and the exciter does not carry the information of velocity and

impose it back on the mass as an reactive force. So on the bond representing

the velocity pick-up the information of force must be masked and on the bondrepresenting the exciter the information of the flow must be masked. A full arrow

somewhere on the bonds shows that some information is masked and what

information is masked may be written near that full arrow. According to this

convention the bond graph of the system is shown to the right of the system

drawn above.

The bond number 4 in the figure is the pick-up bond where information of effort

is masked and in bond number 5, which is the exciter bond, the flow is activated.

The concept of activation is very significant to depict feedback control systems.

The term activation initially seems a misnomer. However, Paynter's idea was

based on the fact that though the information of a factor of power is masked on

one end, an activated bond on the other end can impart infinite power which is

derived from a tank circuit used for both the measurement or actuation device

(for instance, the pick-up, the amplifier and the exciter, all have external power

sources).


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In the system shown above, a D.C. motor with externally energized field is

driving an elastic shaft with two disks and with a fluctuating resistive load. The

speed, picked up by a tachometer is fed-back to control the speed of the motor

by adjusting the armature current (increasing voltage alternatively). The bond

graph for this system is drawn as shown below.

Observers

Additional states can be added for measurement of any factor of power on a

bond graph model using the Observer storage elements. An effort activated C-

element would observe the time integral of flow (and consequently flow),


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whereas a flow activated I-element would observe the generalized momentum

(and consequently effot). Activated elements are percieved conceptual

instrumentations on a model. They dont interfere in the dynamics of the system

(i.e. their corresponding states never appear on the right-hand side o f any state

equation. A system with N states, M sources and L observers would have the

state equations of the form

d/dt{Y} = [A]{Y} + [B]{u} ,

d/dy{Z} = [C]{Y} + [D] {u} ,

whre {Y} is a vector of true states, {Z} is the vector of observed states, {u} is

a vector of sources, [A] is MxM matrix, [B] is MxN matrix, [C] is ZxM matrix and

[D] is ZxN matrix.

Here is an example of a single-degree-of-freedom system with instrumentations,

whose model with observer elements is shown below.


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The conceptual instrumentations shown here, assumes a spring of zero stiffness

(thus measuring displacement and not generating any reactive effort) and an

inductive coil within which the segments of the damper move to induce emf (a

measurement of force) without any reaction on the damper segment. The

equations for this model can be de rived as shown be low.

Step 1 :

f1 = SF1

e3 = K3*Q3

e5 = R5*f5

f6 = 0

e9 = SE9

e8 = 0

e10 = 0

f12 = P12/M12


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e11 = MU*f10 = MU*f12 = MU*P12/M12

Step 2 :

e12 = e7 + e9 - e8 + e11 - e10 = e2 + SE9 + MU*P12/M12

f2 = f1 - f7 = SF1 - P12/M12

f5 = f4 - f6 = f4 = f2 = SF1 - P12/M12

e2 = e3 + e4 = e3 + e5 = K3*Q3 + R5*f5 = K3*Q3 + SF1 - P12/M12

Step 3 :e12 = K3*Q3 + R5*(SF1-P12/M12) + SE9 + MU*P12/M12

e5 = R5*(SF1 - P12/M12)

Step 4 :

The State equations are

DP12 = e12 = K3*Q3 + R5*(SF1-P12/M12) + SE9 + MU*P12/M12

DQ3 = f3 = f2 = SF1 - P12/M12

and the observer equations are

DP6 = e6 = e5 = R5*(SF1 - P12/M12)

DQ8 = f8 = f12 = P12/M12

Bond graph modelingA simple mechanical system show n in the figure below is considered here for

explaining the modeling procedure.


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The output from the pump would be flow and pressure, while the input would be

torque and angular velocity applied to the pump shaft. The input power is

provided by an electric motor. The power flow diagram is shown in the figure

below.

Two transformations are identified. First from electric power to mechanical power

through the electric motor; followed by the mechanical pow er to the fluid power

through the pump. In the next step to construct the model, the components are

to be looked into more details. In this simple analysis of the pump it has its own

moving parts (inertia); oil flow has got its own compress ibility (capacitance)

because of which it can store energy; pump has leakage due to its geometric

tolerance and friction between the moving parts (resistance) through which it

can dissipate energy. The bond graph of the system with power directions and

causality is show n below.


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A two degrees of freedom mechanical system and its corresponding Bond graph

model shown be low can be explained in the following manner.

The bottom mass velocity is same as the rate of stretching of the spring and


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The bottom mass velocity is same as the rate of stretching of the spring and

damper connected to the ground. So, these are modeled as being connected to

the 1-junction. The middle spring and damper experience components of force,

say F1 and F2, respectively such that F=F1+F2, i.e., the force felt by the bottom

mass, the spring and damper in combination and the top mass is equal.

Alternatively, the distortion of the spring and the damper is dependent on

relative motion of the masses. Hence, one can easily smell the presence of a

force pass or flow summation junction, in other words the 0-junction here.

Further, this 0 junction has three branches corresponding to three componentsof distribution of effort.

Another 1-junction appears in the right, that signifies the equal relative velocity

(i.e., rate of compression here considering the power directions) felt by the

middle spring and the damper. The 1-junction at the top depicts the coherent

motion of the forcing device w ith the top mass.

For the above system, alternative methods can be applied to arrive at a bond

graph model, that may apparently look different. However, the final equationsderived from different models lead to the same set. It may be noted that bond

graphs for mechanical systems may be drawn using the method of flow balance

(kinematics) or force transmission, or a combination of the both. Since the basic

external elements used are the same and only the energy conserving junction

structure may vary, considering one factor of power automatically takes care of

the effect of the other factor. This is perhaps the greatest contribution of the

bondgraph theory.

Art of Creating Models

Representing systems in form of models is an art. Neat and good-looking models

are arrived at following certain established practices and reduction methods.

However, ultimately the model author dese rves the credit for putting pieces

together to create good visual effect Some reduction schemes which may be


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together to create good visual effect. Some reduction schemes, which may be

used to compress or uncompress bond graphs, are presented here. This work

out in the reverse in helping to understand the bond graphs drawn by others. In

this section the philosophy of model building and reduction is presented. Most of

the matter presented here are extracts from "Lecture notes on sytem modeling"

by Prof. A. Mukherjee and Prof. R.Karmakar of the Indian Institute of Technology,

Kharagpur.

Model reduction steps

A junction with only two bonds attached to it may be replaced by a single

bond provided this does not lead to attaching any two external elements (I,

C, R, SE, SF, GY or TF) to each other without a junction. These reductions

are summarized in the table below. In this table J stands for any junction (1

or 0) and E for an external element. Reader may verify these reductions by

simply writing the junction laws for such junction.

Two neighboring identical junctions may be merged, the internal bond

between two such junction is dropped in this process (see section E and F in

table).

Any 1-junction to which a source of flow is attached which has constant zero

value and no external element is attached may be removed from the bond

graph with all bonds attached to it (see section G in table).

Any 0-junction to which a source of e ffort with constant zero value is

attached and no external element is attached may be removed from thebond graph with all bonds attached to it (see section H in table).

Junction Structure Reduced Form


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A

B

C

D

E


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F

G

H

The reductions (3) and (4) must be done after the bond graph is already

reduced due to (1) and (2). In a sense there is a hierarchy in the reduction

process. Some more elementary equivalent forms shown in the table below.


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process. Some more elementary equivalent forms shown in the table below.

These equivalent forms may be established by w riting junction laws and

accounting for the fact that a source of flow (SF) can meet any demand of effort

by the system and a source of effort (SE) can meet any demand of flow. Thus

the following rules hold.

A zero source of flow may be removed from a 0 junction.

A zero source of effort may be removed from an 1 junction.

1-junction is a distributor of SF.

0 junction is a distributor of SE.


A

B


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C

D

Two more reductions w idely used in bond graph modeling are shown below.


E


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F

Bondgraphs for mechanical systems

To create a bond graph for a mechanical system it is advisable that a good

schematic sketch of the system or a word bond graph be made. A good word

bond graph is a great aid when one deals with systems in multi energy domain.

The following three methods are most often used effectively to create bond

graphs for mechanical systems :

Method of Flow Map

Method o f Effort Map

Method o f Mixed Map

- Method of flow map

The method of flow map is based on the fact that the single port mechanical C or

Relement may be attached to a 1-junction where the relative velocity of ends of

these elements are available. Single port I elements may be attached to 1-

junctions, where absolute velocities of generalized inertias are available.


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Following steps may be followed to create system bond graph by Method of

Flow Map.

Create 1-junctions depicting the components of velocities of inertial points.

Create 1-junctions depicting the motions of end points ofC and Relements.

Relate and connect these junctions by transformer (TF)elements if a lever

relations exist be tween them. Gyroscopic relations and feedbacks may be

incorporated using gyrator (GY) elements.

Create the relative velocities between the end points ofC and Relements.

0-junction may be used for adding or subtracting the velocities.

Attach C and Relements to 1-junctions where relative velocities of their end

points are created.

Reduce the bond graph us ing reduction process discussed earlier.

Often it may be of great advantage if superscripts or subscripts are shown at 1

junctions depicting the points of the system and components of motion.

However, these superscripts or subscripts are purely for book-keeping and are

ignored during subsequent processing.

Example 1 : Let us consider a system shown in the figure below. Two 1-

junctions are created depicting velocities at mass-point and ground excitation.

Difference of these two velocities are taken using a 0-junction and relativevelocity is established at the 1-junction shown as 1mv. The velocity depicted at

1mv is d(xm)/dt - d(xv)/dt. Next, one attaches the external e lements. An

alternative bond graph may be created by making two 1mv junctions and then

attaching C to one and Rto the other. This bond graph may then be reduced to

the earlier one.


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(a) (b)

(c) (d)

Creating bond graph of a spring-mass-damper system.


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(e) (f)

Alternative bond graph for the spring-mass-damper system.

In case V(t)=0, i.e., the end of the spring is attached to the inertial frame of

observation, the final bond graph forms given above (in figs (d) and (f)) may be

reduced to a very simple form. This reduction is shown below.

(a) (b)


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(c) (d) (e)

Bond graph of a spring-mass-damper system anchored to the ground.

(a) (b) (c)

Alternative bond graph for the ground anchored spring-mass-damper system.

Example 2 : An idealized car model with rigid body and flexible suspensions with

excitations from the road is shown in the figure below. It's bond graph is

created by Method of Flow Map as shown below the schematic diagram of the

system.


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Two dimensional model of a car


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(a )


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(b)

(a) bond graph model of the car, (b) reduced bond graph

- Method of effort map

When a generalized force applied on an inertial point is such that the positive

values of the force accelerates the inertial point in a positive sense, then the

power directions on the bonds attached to the 1-junction will be as shown in

the figure below. To its right, the case of a force is shown which when positive

produces acceleration in the negative coordinate direction of motion.


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The method of effort map may be completed by the following steps .

If necessary create a 0-junction for sources of effort. Such junctions may be

treated as d istributors of efforts.

Decide whether the tensile force in C is to be taken as positive or it is the

compressive force which is to be taken positive. In linear springs such a

choice may be arbitrary.

For R elements, decide what kind o f relative velocity (compress ive or

stretching) is to be taken as positive. In linear systems this choice may be

arbitrary.

Addition of forces may be done using 1-junction.

Linear forces may be converted to couples using transformer (TF) elements.

Similarly, angular velocities may be converted to gyroscopic forces using

gyrator (GY) elements.

Reduce the bond graph using the rules discussed earlier.

Example 3 : The system of shown below in figure (a) is modeled in steps as

shown in figures (b) and (c).


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(a) (b)

(c)

In this bond graph tensile force in the spring and tractile force on the damper is

taken as positive. C and Relements are directly put on 0-junctions depicting the

forces in them. An alternative bond graph is drawn for this system in the figure

below. In this bond graph the forces due to the spring and damper are added

and then distributed using a 0-junction. The graph is then reduced to a form

shown to the right.


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Example 4 : The car model discussed in the method of flow map is drawn using

Method of e ffort map in the figure below. Here the compressive forces in

suspension are taken to be positive. An alternative bond graph is shown in the

next figure and is then reduced to a s impler form.


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(a) Model for vertical dynamics of a car


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(b) Alternative model for vertical dynamics of a car


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(c) Reduced model

- Method of mixed map

The 1-junctions on which the forces accelerating generalized inertial points are

added correspond to velocities of these inertial point in Method of Effort Map.

Now if a part of a system is modeled by Method of Effort Map, then the 1-

junctions depicting these velocities may be used to create the bond graph for

the rest of the system following Method of Flow Map.

The following example illustrates this process.

Example 5 :For the system shown below, the bond graph of the car is completed

by Method of Effort Map. The idealized passenger placed on this car represented

as mass spring system as shown in the figure is then modeled by Method of

Flow Map.


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(a) Schematic diagram of car with a passenger


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(b) Bond graph of car with a passenger

Bond graphs of electrical circuits

Following three methods are found suitable while drawing bond graphs for

electrical circuits. They can also be extended to other circuit-morphic systems in

the field of hydraulics and electronics.

Method of Gradual Uncover,

Point Potential Method,

Mixed Network Method.

- Method of gradual uncover

h h d ( ll ) l l f


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In this method one may cover (conceptually) comparatively complex parts of

circuit and then treat them as macro impedances. Make the bond graph putting

such impedances at proper places. Then uncover them and put the details on

the bond graph. However one may have sub-covers covering complex parts of

macro impedances which may then be gradually uncovered. The following

example illustrates this process.

Example 6 : An electrical circuit is shown in the figure below and to its right a

covering scheme is shown. Then in 5 steps (c to g in figure) the bond graph is

created by gradual uncovering.

(a) (b)

(c) (d)


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(e )

(f)

(g)

- Point Potential Method

Method of Gradual Uncover fails in many cases. In bridge circuits, for example, a

satisfactory scheme of creating covers becomes impossible. The alternative Point

Potential method is guaranteed to work in every case. Following are the steps

of Point Potential Method.


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Each point of circuit where ends of various elements or sub-circuits are

tagged may be made into a 0-junction.

Any element or impedance through which current flows may be attached to a

1-junction between the 0-junctions.

A suitable tag point may be grounded. This is achieved by attaching a zerosource of effort to a 0-junction. Unless grounded, the circuit remains floating.

Each individual circuit must be separately grounded, for example circuits on

primary and secondary sides of a electrical transformer are treated as two

separate circuits and must be grounded on both sides.

Reduce the bond graph.

The following example illustrates this approach.

Example 7 : Let a bridge circuit shown in the figure be low be considered. Point

Potential Method produces the bond graph shown to the right. The reduced

bond graph and rearranged graph are shown next.

(a) (b)


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(c)

(d)

- Mixed Network method

The method of point potential and gradual uncover may be often mixed to a

great advantage. The complex impedances may be covered in the first go. Once

the overall structure is produced by Point Potential Method, the uncovering may

be taken up. The ultimate bond graph may then be reduced. The following

example illustrates this method.

Example 8 : Bond graph for the bridge circuit shown earlier is developed this


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p g p g p

way. The figure given below shows the scheme of covering. The bond graph is

then developed in stages (b) to (d) shown in the figure.

(a) (b)

(c) (d)

Hydraulic circuits are drawn more or less in the same manner as electrical

circuits. Before concluding, here are two important w arnings to the reader.


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Odd power loops should be avoided: If there is a loop of junctions then two

important points should be observed.

There should be even number of bonds forming the loop.

Bonds power directed clockwise or counter clockwise must be even innumber.

Causa l loops should be avoided. In a junction loop, not a ll junctions should

have their causalities determined by internal bonds (i.e. bonds which are

part of the loop).

Proceed to the previous topic page

Proceed to the next topic page

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