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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

Fields

So far the external elements like C, I and Rwere connected to a single bond like

-C, -I and -R. If the parameter (spring constant or capacitance) of the Celements is any nonlinear or linear function of displacement only then C element

will be conservative. It w ill release stored energy when brought back to a given

state. The single port C, I and R elements may be generalized to represent

higher dimensions. To start with, let us consider the multi port generalization of

element C. Whenever the efforts in a set of bonds are determined by

displacement in the bonds of the same set as following,

ei=Sjn=1 Kij Qi, i=1..n;

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the relation may be represented by a multiport C element called a C field.

Similar relationship can be established for I and R elements as well. Fields are

always referred enclosed w ithin square braces ([C], [I] and [R]).

When the field matrix is diagonal, i.e., cross-couplings are not present, the field

may be dissociated into a set of one port elements. In the example system

shown above, when the set of equations are written with reference to the X'-Y'

coordinate axes through use of rotation matrices, 2x2 [C] and [R] fields are the

natural outcome.

Occurrences of [C] fields are common in analysis o f beam vibration problems,

where the bas ic beam element is represented by a 4x4 stiffness matrix. This

matrix relates the two sets o f bending moments and shear forces at both endsof the infinitesimal mass less element to the corresponding set of angles and

displacements.

[C] fields are a common feature in modeling of thermodynamic systems. For

instance, a collapsible chamber in an engine or a compressor chamber can store

energy through interaction of three modes, viz. the mechanical port associated

with the piston, thermal port for the heat transfer and the chemical work done

by mass transfer and combustion. The basic equations of force and energy of a
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single port C element, for instance a spring, are as follows.

F= K x, E=-t F dx = K x

2.

For the thermal domain, the differential equations of the internal energy(U) can

be expressed as follows.

dU= -P dV + T ds + m dN,

where P, dV, T, ds, m and dN represent pressure, volume flow rate,

temperature, rate of change of entropy, chemical potential, and mole flow rate,

respectively (prefix d stands for time derivative). An alternative expression may

be written using enthalpy and mass flow rate for the chemical work. Thus the

representation of this thermodynamic process (due to Breedveld) may be given

as follows .

It can be easily observed from theanalysis that P, T and m are effort

variables and the corresponding flow

variables are rates o f V, s and N,

respectively. The coefficients of C-field

or its equivalent representations in

terms of sources can be derived with

assumption of a particular

thermodynamic process. The three

independent ways of energy exchange

are depicted by three ports of the field.

The other type of commonly occurring field element is the [R] field. It is mostly

encountered in modeling of transistors and other electronic devices, and

problems involving heat transfer. Occurrence of [I] fields is not so common, as

compared to the other two. The inertia field is mostly encountered in modeling of

rigid body dynamics, gyro motions, etc., as in problems of robotic manipulators,

or can be artificially synthesized through co-ordinate transformations. Integral


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causality to field elements are given in similar manner as in the case of one port

elements.

The problem of linear heat conduction through a flat plate can be posed as

dQ/dt= T1 dS1/dt = T2 dS2/dt

dQ/dt = H (T1 - T2),

where T1 and T2 are temperatures on both sides of the plate, S1 and S2

represent entropy, and H is the overall heat transfer coefficient. Thus, the

equations for entropy flow rate can be written as

dS1/dt = H (T1-T2)/T1,

dS2/dt = H (T1-T2)/T2.

Identifying entropy flow rates as the flow variables and temperatures as effort

variables, the constitutive equations represent an R-field as follows.

where,

R* = H/T1 -H/T2

H/T2 -H/T1 .


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It ma y be noticed that the R field is in conductive causa lity and the m atrix written a bove

describes the conductance matrix. This m atrix is not invertible, which implies e ntropy flow

rates can be functions o f tempe rature, but no vice versa. This a lso imp lies that entropy

generation is due to temperature and not vice-versa.

State equations fo r mode ls with field elem ents are written in similar ma nner as the one-

port elements. The integrally causalled storage fields (C- and I-fields) are described by

following relations, respectively.

{e}=[K]{Q}, where {e} is the effort vector and {Q} is the generalized displacement vector.

{f}=[M]-1{P}, where {f} is the flow vector and {P} is the generalized momentum vector.

Let us consider a system shown be low, which is described by a bond graph

model shown to its right.


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The stiffness matrix in non-principal co-ordinates X'-Y' is obta ined by rotation

matrix as follows.

Kx'x' = Kxx cos2q + Kyy sin

2q

Kx'y' = Ky'x' = (-Kxx + Kyy) cosq sinq

Ky'y' = Kxx sin2q + Kyy cos

2q

Similarly, the damping matrix is obtained in X'-Y' co-ordinates. The state

equations can then be derived as shown below.

The state variables are P1, P2, Q3 and Q4.

The constitutive relations are :

f1 = P1/m1

f2 = P2/m2

e3 = K3_3*Q3 + K3_4*Q4

e4 = K4_3*Q3 + K4_4*Q4

e5 = R5_5*f1 + R5_6*f2 = R5_5*P1/M1 + R5_6*P2/M2

e6 = R6_5*f1 + R6_6*f2 = R6_5*P1/M1 + R6_6*P2/M2


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The state equations are :

dP1 = e1 = (e3 + e5) = K3_3*Q3 + K3_4*Q4 + R5_5*P1/M1 + R5_6*P2/M2

dP2 = e2 = (e4 + e6) = K4_3*Q3 + K4_4*Q4 + R6_5*P1/M1 + R6_6*P2/M2

dQ3 = P1/m1

dQ4 = P2/m2

Storage fields are not always conservative, even when the system is linear.

Cons ider a 2x2 C-field, whose s tiffness matrix is such that it cannot bediagonalized using any rotation. Such a field is then represented as sum of two

matrices, where one is a conservative part with symmetric cross-stiffnesses and

the other is the non-conservative part with anti-symmetric cross-stifnesses. The

field is then called a Non-Potential C-field.

Resistive fields are always symmetric and thus can be diagonalized using

suitable transformations. The symmetry of R-fields is a fundamental principle

established through Onsager's principle.

Mixed causalled fields

Consider the case of a field of storage elements (I or C), where some of the

bonds connected to it are not integrally causalled. Such fields give rise to

complex equations where differential causalities on field elements requireinversion of matrix derivatives. For instance, consider a 2x2 I-field whose one

port is differentially causalled and the other is in integral causality. Say these

two ports are numbered 1 and 2, respectively. Then the equations would be

e1 = d(f1*M11)/dt + P2 /M12,

f2 = d(f1*M21)/dt + P2 / M21.

P2 is the state variable corresponding to integrally causalled port and


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M11,M12,M21,M22 are components of the mass matrix for the I-field. These

equations are simple for this 2x2 field. However, for higher orders, partial

inversion of field matrices followed by derivatives is required to arrive at the

state equations. In those cases, where the matrix elements are non-linear, the

process becomes further complicated.

Thus, only the case of [R] fields is discussed here. Three types o f causal

patterns are possible in a [R] field, as shown in the figure below.

(a) (b) (c)

The first type of causal pattern shows all the bonds causalled with resistive

causality. For such a case, the equations may be written as

When the field is in conductive causality completely, then equation for output

variables may be written as


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When the field is mixed causalled, then the process of writing the equations

is a bit different. Let [RO] be a unit matrix, [RI] be a matrix containing the

elements of [R] field, i.e.,


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Without considering the deta iled mathematical backgrounds, one may

proceed as follows .

Interchange those columns of [RO] and [RI] with a negative sign, which

correspond to conductive causality. Then, the equivalent [R] that relates

input vectors to output (cause and effect) may be written as

[R]equiv = [RO]-1 [RI].

Thus, for the mixed causalled case shown in figure (c),

It may be noted that, in case of complete resistive causality, [RO]=[I],

[RI]=[R] and hence [R]equiv=[R]. In the other extreme case of complete

conductive causality, [RO]=-[R] and [RI]=-[I], thus implying [R]eqiv=[R]-1

.These two cases satisfy the equation derived earlier for fist two types of

causality patterns.

Differential Causality


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The cause and implications of differential causality in a system mode l has

already been discussed in the section on causality.

In presence of differential causalities, the order of the set state equations is

smaller than the order of the system, because storage elements can depend on

each other. These kind of dependent storage elements each have their own

initial value, but they together represent one state variable. Their input signals

are equal, or related by a factor, which may not be necessarily constant.

Let us consider a system and it's bondgraph shown below.

The equations of motion may then be derived as follows (assuming m1,m2, a

and b as constants).

e1 = SE1

f2 = P2/m1

e3 = K3*Q3

f5 = -b/a * f4 = -b/a * f2 = -b/a * P2/m1

e4 = -b/a * e5

e5 = d (m2*f5)/ dt = -m2 * b/a /m1 * d(P2)/dt = -m2/m1 * b/a * e2

e2 = e1 -e3 -e4 = SE1 - K3*Q3 + b/a *( -m2/m1 * b/a * e2)

After reduction and solving out e2 algebraically, the state equations are

DP2 = e2 = (SE1 - K3*Q3) / (1 + m2/m1*(b/a)2)
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DQ3 = f3 = f2 = P2/m1.

Though the equation could be derived properly, it would be better to make the

model integrally causalled using the so-called pad elements. Pad elements are

normally representation of missing or unknown stiffnesses in the system. In this

case, it may be the flexibility of the lever segment, which may be set to a very

high value during simulation. A padded model would then be as shown below.

Padded models though ensure integral causality in the model, may turn out very

stiff during numerical solution due to high frequency oscillations in the pad

region. Differential model models however produce are very fast simulation.

Let us now consider a simple mechanical system and its electrical equivalent asshown below.


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An integrally causalled model of these systems is shown in the left and another

with a preferred differential causality is shown to the right.

The equations for the first model can be eas ily derived. There are two state

variables Q1 and Q2, each of which can be assigned different initial conditions

separately. However, in the second case, there is only one state-variable Q1and initial conditions can be ass igned to it only. Assigning initial value to Q2

(which is not a state) does not affect equations and dynamics of the model

derived from second model, since only the rate of deformation is considered in

equations and not Q2 it self as a state. Let us now proceed to derive the state

equations for the second model with a preferred differential causality (knowing

very well that a well causalled integral model exists) and find out the pit falls of

differential causality, especially the preferred cases.


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e3 = SE3

e1 = K1 * Q1

e3 = R3 * f3 = R3*f2

f2 = d(e2/K2)/dt = 1/K2 d(e3 -e1 -e4)/dt = 1/K2*d(SE3)/dt - K1/K2 *f1 -R3/K2 *

d(f2)/dt

The above equation is derived assuming K2 and R3 are constants. This equation

cannot be further resolved algebraically. Let us assume the forcing function is aconstant. Then for f1=f2,

f2 = -R3*K2/(K1+K2) * d(f2)/dt

The above equation is a differential equation, solution to which is of the kind

f2 = e-R3*K2*t/(K1+K2) + C, where C is a constant and t is time.

The above solution is not dependent on any initial conditions and is amonotonically decreasing function of time. This obviously is not the case, since

when we compare it to the integrally causalled model, there are gross

anomalies.

Algebraic solution of state equations almost always fails when causal

coupling of preferred differentially causalled elements takes place with

resistive elements at strong bonds.

An alternate bond graph model for the system can be drawn by merging the

mechanically parallel and electrically in se ries, storage elements, as shown

below.


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However, such a model is incongruent with the system morphology. This model

cannot take different initial conditions for two different system components,

since they are represented by a s ingle storage e lement in the model. Consider a

case, where one of the springs in the system is in pre-tension and the other in

precompression, so that K1*Q1t=0+K2*Q2t=0 = 0, and the system is in

equilibrium. This locked up mode cannot be represented in the merged state

model.

Algebraic Loops

Often during derivation of state equations, the entire set of equations cannot be

expressed in terms of system parameters, state variables and excitations,

through s imple substitutions. Some components of the equation need to be

solved as a set of linear equations. These cases are termed as algebraic loopsand the minimal set of linear equations to be solved to completely resolve the

set of equations is termed the order of the loop. Algebraic loops normally appear

in models where resistive e lements are on the strong bonds and/or in presence

of internal strong bonds (internal bonds refer to bonds between junctions).

Differentially causalled storage elements in system models also lead to algebraic

loops.

Let us consider a electrical circuit and it's bond graph model as shown in the

figure below


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figure be low.

The state variables corresponding to integrally causalled storage elements are

Q3, P7 and Q9. The constitutive relations are

e3 = K3*Q3, f7 = P7/m7, e9 = K9*Q9, f2 = e2/R2, e6 = R6*f6 and f8=e8/R8.

The equations for strong bonds (junction a lgebra) are

e2 = e1 - e3 -e4 = SE1 - K3*Q3 - R6*f6

f6 = f4 - f5 -f7 = f2 - f8 - P7/m7 = e2/R2 - e8/R8 - P7/m7

e8 = e5 - e9 = e6 - e9 = R6*f6 - K9*Q9

Now these expressions are interwoven functions of each other (a third order

algebraic loop) and need to be solved out algebraically as follows.

Let us substitute the expression for e2 in that for f6, which leads to

f6 = (SE1 - K3*Q3 - R6*f6)/R2 - e8/R8 - P7/m7,

or, (1+R6/R2) * f6 = (SE1 - K3*Q3)/R2 - e8/R8 - P7/m7.

Let ID1 be a dimensionless terms defined as ID1 = 1+R6/R2. Then

f6 = (SE1 - K3*Q3 - R6*f6)/R2/ID1 - e8/R8/ID1 - P7/m7/ID1,

Substitution of f6 in expression for e8 leads to


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Substitution of f6 in expression for e8 leads to

e8 = R6 * (SE1-K3*Q3)/R2/ID1 - R6*e8/R8/ID1 - R6*P7/m7/ID1 - K9*Q9,

or, (1+R6/R8/ID1)*e8 = R6 * (SE1-K3*Q3)/R2/ID1 - R6*P7/m7/ID1 - K9*Q9.

Let ID2 be a dimensionless terms defined as ID2 = 1+R6/R8/ID1. Then

e8 = R6*(SE1-K3*Q3)/R2/ID1/ID2 - R6*P7/m7/ID1/ID2 - K9*Q9/ID2.

So, e8 is now fully resolved and can be back substituted in expressions for f6.

The resolved expression for f6 has to be then back substituted in expression for

e2. This leads to the following state equations.

DP7 = e7 = (((SE1-K3*Q3)/R2-P7/m7)/ID1 - ((((SE1-K3*Q3)/R2 -P7/m7)/ID1)/R6 -

K9*Q9)/ID2/R8/ID1)*R6.

DQ3 = f2 = (SE1-K3*Q3 - (((SE1-K3*Q3)/R2 -P7/m7)/ID1 - ((((SE1-K3*Q3)/R2 -

P7/m7)/ID1)/R6 - K9*Q9)/ID2/R8/ID1)*R6)/R2.

DQ9 = f8 = ((((SE1-K3*Q3)/R2-P7/m7)/ID1)*R6 - K9*Q9)/ID2/R8.

Complex systems with algebraic loops may lead to very long equations. Thus it

is always better to break the large loops using realization of some neglected

storage elements at causally indeterminate junctions (i.e, junctions determined

by resistive elements, differentially causalled elements or internal bonds as

strong bonds). If, however that is not possible, a numerical solution of the loops

using matrix inversion may be carried out instead of formally resolving theequations beforehand.

Causal Loops

When there is a loop of junctions connected to each other sequentially by bonds


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When there is a loop of junctions connected to each other sequentially by bonds

all of which are strong bonds for at least one junction on their both ends, the

resulting junction structure is said to form a causal loop. Such forms lead to an

irresolvable set of equations, and the state equations cannot be derived in

terms of states. Causal loops may also be outcome of hidden differential

causalities in the mode l, which apparently do not show up in system-morphic

bond graphs.

Let us consider a contraption shown below.

The two alternative bond graph models for the system are shown here. All the

damping in the system are neglected. The transformers in these models

represent the ratio of cross-sectional areas of the frame and the plug.


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In the first model, all the storage elements are integrally causalled, whereas in

the second model two storage elements are differentially causalled. The first

model contains a causal loop and equations for it cannot be derived. The second

model though contains two differentially causalled storage elements, is the valid

representation of the system.

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