Bond Graphs

About Bond graphs - The System Modeling World

About Bond Graphs

1. Introduction2. Power variables3. Standard elements4. Power directions5. Bond numbers6. Causality7. System equations8. Activation9. Example models

10. Art of creating models11. Fields12. Mixed-causalled fields13. Differential causality14. Algebraic loops15. Causal loops16. Duality17. Multi and Vector bond graphs18. Suggested readings

1. Introduction

Bond graph is an explicit graphical tool for capturing the common energy structure of systems. It increases one's insight into systems behavior. In the vector form, they give concise description of complex systems. Moreover, the notations of causality provides a tool not only for formulation of system equations, but also for intuition based discussion of system behavior, viz. controllability, observability, fault diagnosis, etc.

In 1959, Prof. H.M.Paynter gave the revolutionary idea of portraying systems in terms of power bonds, connecting the elements of the physical system to the so called junction structures which were manifestations of the constraints. This power exchange portray of a system is called Bond Graph (some refer it as Bondgraph), which can be both power and information oriented. Later on, Bond Graph theory has been further developed by many researchers like Karnopp, Rosenberg, Thoma, Breedveld, etc. who have worked on extending this modeling technique to power hydraulics, mechatronics, general thermodynamic systems and recently to electronics and non-energetic systems like economics and queuing theory.

By this approach, a physical system can be represented by symbols and lines, identifying the power flow paths. The lumped parameter elements of resistance, capacitance and inertance are interconnected in an energy conserving way by bonds and junctions resulting in a network structure. From the pictorial representation of the bond graph, the derivation of system equations is so systematic that it can be algorithmized. The whole procedure of modeling and simulation of the system may be performed by some of the existing software e.g., ENPORT, Camp-G, SYMBOLS, COSMO, LorSim etc.

Bond graph bibliography by F.E.Cellier and Bondgraphs.com >> Bibliography give a through account of the work carried out in the particular field. The following sections will briefly explain the reader to understand the mnemonics of the bond graph. However, the explanation given here is not a complete one; it is only limited to the extent of initiating new bond graphers.

2. Power variables of Bond Graphs

The language of bond graphs aspires to express general class physical systems through power interactions. The factors of power i.e., Effort and Flow, have different interpretations in different physical domains. Yet, power can always be used as a generalized co-ordinate to model coupled systems residing in several energy domains. One such system may be an electrical motor driving a hydraulic pump or an thermal engine connected with a muffler; where the form of energy varies within the system. Power variables of bond graph may not be always realizable (viz. in bond graphs for economic systems); such

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factual power is encountered mostly in non-physical domains and pseudo bond graphs.

In the following table, effort and flow variables in some physical domains are listed.

Systems Effort (e) Flow (f)

MechanicalForce (F) Velocity (v)

Torque (τ) Angular velocity (ω)

Electrical Voltage (V) Current (i)

Hydraulic Pressure (P) Volume flow rate (dQ/dt)

ThermalTemperature (T) Entropy change rate (ds/dt)

Pressure (P) Volume change rate (dV/dt)

ChemicalChemical potential (µ) Mole flow rate (dN/dt)

Enthalpy (h) Mass flow rate (dm/dt)

Magnetic Magneto-motive force (em) Magnetic flux (φ)

3. Bond Graph Standard Elements

In bond graphs, one needs to recognize only four groups of basic symbols, i.e., three basic one port passive elements, two basic active elements, two basic two port elements and two basic junctions. The basic variables are effort (e), flow (f), time integral of effort (P) and the time integral of flow (Q).

Basic 1-Port elementsA 1-port element is addressed through a single power port, and at the port a single pair of effort and flow variables exists. Ports are classified as passive ports and active ports.

The passive ports are idealized elements because they contain no sources of power. The inertia or inductor, compliance or capacitor, and resistor or dashpot are classified as passive elements.

R-Elements :The 1-port resistor is an element in which the effort and flow variables at the single port are related by a static function. Usually, resistors dissipate energy. This must be true for simple electrical resistors, mechanical dampers or dashpots, porous plugs in fluid lines, and other analogous passive elements. The bond graph symbol for the resistive element is shown below.

The half arrow pointing towards R means that the power i.e., product of F and V (or e * f) is positive and flowing into R, where e, represents effort or force, and f, represents flow or velocity. The constitutive relationship between e, f and R is given by :

e = R * f Power = e * f = R * f 2



C-Elements : Consider a 1-port device in which a static constitutive relation exists between an effort and a displacement. Such a device stores and gives up energy without loss. In bond graph terminology, an element that relates effort to the generalized displacement (or time integral of flow) is called a one port capacitor. In the physical terms, a capacitor is an idealization of devices like springs, torsion bars, electrical capacitors, gravity tanks, and accumulators, etc. The bondgraphic symbol, defining constitutive relation for a C-element are shown below.

In a spring, the deformation (Q) and the effort (e) at any moment is given by,

Q = ∫ −t∞ f dt, e = K ∫ −t∞ f dt.

Here, flow is the cause and deformation (and hence effort) is the consequence. In a capacitor, the charge accumulated on the plates (Q) or voltage (e) is given by,

Q = ∫ −t∞ i dt, e = C -1 ∫ −t∞ i dt.

Here, the current is the cause and the total charge (and hence voltage) is the consequence.

I-Elements : A second energy storing 1-port arises if the momentum, P, is related by a static constitutive law to the flow, f. Such an element is called an inertial element in bond graph terminology. The inertial element is used to model inductance effects in electrical systems and mass or inertia effects in mechanical or fluid systems. The bond graph symbol for an inertial element is depicted in the figure given below.

If the mechanics of mass point is examined by considering the impulse-momentum equation, then we have;

P = ∫ −t∞ e dt, f = m-1 ∫ −t∞ e dt.

Here, effort is the cause and velocity (and hence momentum) is the consequence. Similarly the current in an inductor is given by;

i = L -1 ∫ −t∞ e dt.

Effort and Flow Sources :The active ports are those, which give reaction to the source. For, example if we step on a rigid body, our feet reacts with a force or source. For this reason, sources are called active ports. Force is considered as an effort source and the surface of a rigid body gives a velocity source. They are represented as an half arrow pointing away from the source symbol. The effort source is represented by



and the flow source is represented as shown below.

In electrical domain, an ideal shell would be represented as an effort source. Similarities can be drawn for source representations in other domains.

Basic 2-Port elements

There are only two kinds of two port elements, namely ``Transformer'' and ``Gyrator''. The bond graph symbols for these elements are TF and GY, respectively. As the name suggests, two bonds are attached to these elements.

The Transformer :The bondgraphic transformer can represent an ideal electrical transformer, a mass less lever, etc. The transformer does not create, store or destroy energy. It conserves power and transmits the factors of power with proper scaling as defined by the transformer modulus (discussed afterwards).

The meaning of a transformer may be better understood if we consider an example given here. In this example, a mass less ideal lever is considered. Standard and bondgraphic nomenclature of a lever are shown in the figure below. It is also assumed that the lever is rigid, which means a linear relationship can be established between power variables at both the ends of the lever.

From the geometry, we have,

V2 = (b/a) V1

The power transmission implies

F2 = (a/b) F1 , so that V2 F2 = V1 F1.

In bondgraphs, such a situation may be represented as shown in the above figure.

The 'r' above the transformer denotes the modulus of the transformer, which may be a constant or any expression (like 'b/a'). The small arrow represents the sense in which this modulus is to be used.

fj = r fi , and ej = (1/r) ei.

Thus the following expression establishes the conservation of power,

ej fj = ei fi .

The Gyrator : A transformer relates flow-to-flow and effort-to-effort. Conversely, a gyrator establishes relationship between flow to effort and effort to flow, again keeping the power on the ports same. The simplest gyrator



is a mechanical gyroscope, shown in the figure below.

A vertical force creates additional motion in horizontal direction and to maintain a vertical motion, a horizontal force is needed. So the force is transformed into flow and flow is transformed into force with some constant of proportionality. In this example, Izz stands for moment of inertia about z axis. wx , wy and wz stand for angular velocities about respective axes; Tx , Ty and Tz represent torque acting about the corresponding axis.

Tx = Izz wzwy .

The power transmission implies

Ty = Izz wz wx , so that Txwx = Ty wy .

Such relationship can be established by use of a Gyrator as shown in the figure above.

The µ above the gyrator denotes the gyrator modulus, where µ = Izz wz . This modulus does not have a direction sense associated with it. This modulus is always defined from flow to effort.

ej = µ fi , ei = µ fj .

Thus the following expression establishes conservation of power, ei fi = ej fj .

In the electrical domain, an ideal DC motor is represented as an gyrator, where the output torque is proportional to the input current and the back emf is proportional to the motor angular speed. In general, gyrators are used in most of the cases where power from one energy domain is transferred to another, viz. electrical to rotational, electrical to magnetic, and hydraulic to rotational.

The 3-Port junction elements The name 3-port used for junctions is a misnomer. In fact, junctions can connect two or more bonds. There are only two kinds of junctions, the 1 and the 0 junction. They conserve power and are reversible. They simply represent system topology and hence the underlying layer of junctions and two-port elements in a complete model (also termed the Junction Structure) is power conserving.

1 junctions have equality of flows and the efforts sum up to zero with the same power orientation. They are also designated by the letter S in some older literature. Such a junction represents a common mass point in a mechanical system, a series connection (with same current flowing in all elements) in a electrical network and a hydraulic pipeline representing flow continuity, etc. Two such junctions with four bonds are shown in the figure below.



Using the inward power sign convention, the constitutive relation (for power conservation at the junctions) for the figure in the left may be written as follows;

e1 f1 + e2 f2 + e3 f3 + e4 f4 = 0.

As 1 junction is a flow equalizing junction,

f1 = f2 = f3 = f4 .

This leads to, e1 + e2 + e3 + e4 = 0.

Now consider the above bond graph shown on the right. In this case, the constitutive relation becomes,

e1 f1 - e2 f2 + e3 f3 - e4 f4 = 0 , and, f1 = f2 = f3 = f4 .

Thus, e1 - e2 + e3 - e4 = 0.

So, a 1 junction is governed by the following rules:

The flows on the bonds attached to a 1-junction are equal and the algebraic sum of the efforts is zero. The signs in the algebraic sum are determined by the half-arrow directions in a bond graph.

0 junctions have equality of efforts while the flows sum up to zero, if power orientations are taken positive toward the junction. The junction can also be designated by the letter P. This junction represents a mechanical series, electrical node point and hydraulic pressure distribution point or pascalian point.

In case of the model in the left, the constitutive relation becomes,

e1 f1 + e2 f2 + e3 f3 + e4 f4 = 0.

whereas, the model in the right is governed by the following relation,

e1 f1 - e2 f2 + e3 f3 - e4 f4 = 0.

As 0 junction is an effort equalizing junction,

e1 = e2 = e3 = e4 .



This leads to, f1 + f2 + f3 + f4 = 0 and f1 - f2 + f3 - f4 = 0, for the left and the right models, respectively.

So, a 0 junction is governed by the following rules:

The efforts on the bonds attached to a 0-junction are equal and the algebraic sum of the flows is zero. The signs in the algebraic sum are determined by the half-arrow directions in a bond graph.

4. Power directions on the bonds

When one analyses a simple problem of mechanics, say, the problem of a single mass and spring system as shown in the figure below, one initially fixes a co-ordinate system.

One may take positive displacement, x, towards right and all its time derivatives are then taken positive towards right. The force acting on the mass may also be taken positive towards right. The system, however, in the course of motion may attain such a state that when it is displaced towards the right, the force on the mass happens to be towards the left.

This phenomenon may be interpreted from the results obtained by solving the system of equation(s) when a positive value of displacement and a negative value of force are seen. So, the initial fixing of a positivity is arbitrary. However, further analysis is related to this fixation. In practice, bond graphs are drawn for general systems. Thus left and right, up and down, clockwise or counter-clockwise, etc., may not be of general relevance. One has to then create a view point which is general and any particular system interpretation should be easily derivable. This is done by assigning the bonds with Power directions. This may be as arbitrary as fixing co-ordinate systems in classical analysis. Say in a bond graph, the power is directed as shown in figure below, where

J : junction, E : element, half arrow : direction of power.

This assignment means, such variables are chosen for effort and flow, so that whenever both these variables acquire positive values, then the power goes from J to E. However, for mixed signs of the variables, the power direction is reversed.

As has been discussed earlier, the interpretation of the relative orientation of positive effort and flow may be subjective depending on whether the analysis is carried out from the stand point of J or E. For instance, we can say the downward motion is positive for a particular mass and the upward motion is positive for another mass according to our co-ordinate system; but for a spring, upward and downward motion do not convey any meaning. In such cases, either the compression or the tension of the spring must be identified as a co-ordinate.

Consider, the following example which belongs to the field of mechanics. The mechanical system with its bond graph model are displayed in the figure below.



Here the variables are so chosen that for positive force and velocity, the power goes to the spring. This may be as per the instrumentation arranged as shown in the two illustrations given below.

If one assumes the stand point of the junction 1, then characteristic of the spring should be as determined by the arrangement as shown above, and may appear as plotted in the right.

5. Assigning numbers to bonds

The bonds in a bond graph may be numbered sequentially using integers starting with 1. However, one need not follow any fixed rule. Assignment of bond number also fixes the name of the elements or junctions. This is the best bookkeeping technique adopted by most of the existing software products. Some software though follow numbering of elements according to their instance. However, in models using fields, where many bonds are connected to an element, such a nomenclatures cause difficulty in book-keeping.

For example, the two 1-junctions in the bond graph shown in the right can be uniquely identified as (S 1 2 3 4) and (S 5 8 9); similarly symbols like C3, C9 can be used to identify a particular element.

The use of characters S and P instead of numerals 1 and 0 allows to represent models in tele-type code in the list form, which can be exchanged across platforms. The power direction and causal information may also be appended to the list form to create a globally adoptable model code, which may read something like (S +-1 --2 --3 -+4)(P +-4 ....



6. Causality

Causality establishes the cause and effect relationships between the factors of power. In bondgraphs, the inputs and the outputs are characterized by the causal stroke. The causal stroke indicates the direction in which the effort signal is directed (by implication, the end of the bond that does not have a causal stroke is the end towards which the flow signal is directed).

To illustrate the basic concept of the causality, let us assume that there is a prime mover with a smart speed governor as shown in the figure below.

The prime mover is driving the load i.e., the power is going from the prime mover to the load. Apart from sending the power, the prime mover also decides that the load should run at a particular speed depending on the setting of the governor. So, it may be said that from the prime mover, the information of flow is generated which goes to the load. The load generates the information of torque (effort) which the prime mover receives and adjusts the inner mechanisms to compensate for it. The wavy lines in the following figure indicate the direction of flow of the particular information.

The selected causality is generally indicated by a cross bar or causal bar at the bond end to which the effort receiver is connected. In expressing causal relations between the effort and the flow, the choice of causality has an important effect. In the following section, the causality of the different elements are discussed.

● For inertance I type storage elements, the flow (f) is proportional to the time integral of the effort.

f = m-1 ∫ −t∞ e dt

An example of this would be a mass subjected to a force, causing it to accelerate.

v = m-1 ∫ −t∞ F dt

In the above relationship effort history is integrated to generate contemporary flow. Therefore, an I element receives effort (cause) and generates flow (effect).

● For capacitive C type storage elements, the effort (e) is proportional to the time integral of the flow (f).

e = k ∫ −t∞ f dt

Examples of this would be a spring being deformed by a force or a capcitor being charged.

F = K ∫ −t∞ v dt

V = C-1 ∫ −t∞ i dt



The above integral relationships show that a C element receives flow and generates effort.

● The resistive or dissipative elements do not have time integral form of constitutive laws.

e = R f or f= e/R

The flow and the effort at this port are algebraically related and can thus have any type of causal structure, either with an open-ended bond (causal stroke is away from the element, i.e., at the junction end) indicating a resistive causality or a stroke ended bond indicating a conductive causality.

As per the above discussions, the causal strokes for I, C and R elements are shown in the figure below.

● Sources impose either an effort or a flow on a system, but not both.

The bondgraphic sources are assumed to be robust, i.e., as providers of active and infinite energy. The effort source (SE), imposes an effort on the system which is independent of flow. An Example of this would be an electrical cell that decides the terminal voltage and the attached load decides the current that the cell has to take and adjust its chemical reactions to maintain the rated terminal voltage.

The flow source (SF), imposes a flow on the system independent of the effort. Examples of this would be cams, constant displacement hydraulic pumps, road excitation, etc. Thus the causality of the source elements are mandatory as shown in the figure below.

The elements I, C, R, SF, and SE are classified as single port, since they interact with the system through one bond only. However, the I, C and R elements can be connected to many bonds to represent tensorial nature, such as the spatial motion of a free body or the stress-strain relationships in a compressible material; in which case they are termed as field elements.

The transformer, by its elemental relation, receives either flow or effort information in one bond and generates the same in its other bond. Thus, one of its port is open-ended with the other end stroked as shown in the figure below.

For the first case, the constitutive equations would be,

fj = r fi and ei = r ej ,

whereas; for the second, the relations are

fj = 1/r fi and ei = 1/r ej .

As mentioned earlier, the gyrator relates flow to effort and effort to flow, therefore, both of its ports have either open-ended or stroke-ended causality as shown below.



For the first case, the constitutive equations would be

ej = r fi and ei = r fj ,

whereas; for the second, the relations are

fj = r ei and fi = r ej .

At a 1 junction, only one bond should bring the information of flow; i.e., only one bond should be open ended and all others should be stroked as displayed in the figure on the right. This uniquely causalled bond at a junction is termed as Strong bond. In any case two bonds cannot be causalled away from the 1-junction, since this would lead to violation of rules of information exchange (the two bonds may not impart equal flows).

Similarly at a 0 junction, only one bond should be stroked nearer to the junction. This strong bond determines the effort at the junction, which the weak bonds (other bonds besides the strong bond) carry around.

The proper causality, for a storage element (I or C), is called Integral Causality, where the cause is integrated to generate the effect. For example, in C element, a study of the constitutive equations reveal that flow is integrated and multiplied with the stiffness to generate effort.

Sometimes the causal strokes will have to be inverted, which means the constitutive relationship for the corresponding element is written as a differential equation. For example, flow in a spring is the time derivative of the ratio of effort and stiffness. Such causality pattern is called Differential Causality.

Implications of an Integral causality means the past data of cause or history function is integrated to arrive at the effect felt at present, whereas, differential causality needs differentiation of the cause at present (that cannot be found properly, since the future is not known) to arrive at the effect. Hence differential causality makes system dependent on future, as if the system is being dragged towards a predestined configuration or in other terms adds specific constrains on the dynamics of the system.

Genuine differential causality is not commonly encountered during system modeling except in certain cases of modeling mechanisms, robotics, etc., where link flexibilities or other aspects are neglected in the model. Such causality is a common occurrence owing to certain direct manipulations in the model. However, spurious occurrences are not ruled out in certain cases such as modeling of a superfluous element (try bond graph of a single degree freedom system, where another mass is placed directly in contact with the main mass), or leaving out certain important element from system (neglecting contact point stiffness etc.).

The occurrence of differential causalities in a system may indicate serious violations of principles of conservation of energy, as illustrated in the following example.

Let us consider the charging of a capacitor through a constant voltage battery source. The system and its bond graph model are shown below.



From the causalled bond graph model, we observe the differential causality in the C-element. If we consider the energy exchange, the energy (E) stored in the capacitor after complete charging to a voltage V is

E = Q2 / 2C , where C is the capacitance and Q is the charge stored in the capacitor.

Also Q = ∫ −t∞ i dt and V = Q / C .

The energy spent by the shell during charging is

E = ∫ −t∞ V i dt = V ∫ −t∞ i dt = Q2 / C .

The later result is anomalous with the energy stored in the capacitor. The loss of half the energy is un accounted for. This can be attributed as one of the implications of the differential causality. The half energy lost is always through dissipation in the system which wrongly has been neglected. If we introduce the resistance in the model, the causality problem is automatically corrected and proper energy conservation is obtained as shown in the figure on the right.

Causality Assignment Procedure:

1. Assign fixed causalities to sources.

2. Propagate the causality through junctions, if possible, i.e. if any bond has got a causality such that it has become the strong bond for a junction, the causality for all other bonds (week bonds) is determined by laws for causality of junctions and if all other bonds of junction are causalled, the last bond should be the strong bond. Similarly, if any port of a two port is causalled (of TF and GY), the causality of the other can be assigned.

3. Assign integral causality to one of the storage elements and propagate the causality through junctions. Continue the procedure with other storage elements. This should normally result in complete causalling of the graph.

4. If the graph is not completely causalled yet, start assigning a resistive causality to a R-element and propagate it. Continue till the entire graph is causalled. In cases, where the model is determined through causalities of R-elements, there may be several possible causal models. It is always advisable to maximize resistive causalities and minimize the conductive causalities in R-elements.

5. If the system develops differential causalities in some storage elements, try minimizing its number of occurrence through assigning initial integral causalities to other storage elements than those selected before.

6. Try to avoid differential causalities by suitable changes to the model, such as introducing some compliance or resistance or both.

7. Discard all models, which result in a causal structure, that violates junction causality rules.



Two animated examples of causalling a model in different ways are illustrated below.

7. 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 equations may be generated. We take the model of a simple single degree of freedom mass-spring-damper system as the starting point.

The differential equations describing the dynamics of the system are written in terms of the states of the system. All storage elements (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.

1. 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 variables for strong bonds.

2. Write down equations for the junctions and the two-port elements for the variables for the strong bonds.

3. 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.

4. If some equations are still 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 of linear equations either 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

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 of states. The term, number of 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 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.



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*P2/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 - e7 f8 = 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 f1 = f2 = f4 = f3 = P3/M3 e2 = SE1 - R3*P2/M2 - e4 = SE1 - R3*P2/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*P2/M2 - MU*P6/M6 DP6 = 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 written 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 ,

B =

1 0

0 0

0 1

0 0 .

8. 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.

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 bond representing 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).

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), whereas a flow activated I-element would observe the generalized momentum (and consequently effot). Activated elements are percieved conceptual instrumentations on a model. They don’t interfere in the dynamics of the system (i.e. their corresponding states never appear on the right-hand side of 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.



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 derived as shown below.

Step 1 : f1 = SF1 e3 = K3*Q3 e5 = R5*f5 f6 = 0 e9 = SE9 e8 = 0 e10 = 0 f12 = P12/M12 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

9. Bond graph modeling

A simple mechanical system shown in the figure below is considered here for explaining the modeling procedure.



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 power 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 compressibility (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 shown below.

A two degrees of freedom mechanical system and its corresponding Bond graph model shown below can be explained in the following manner.



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 components of 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 with 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 equations derived 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.

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About Bondgraphs - The System Modeling World

About Bond Graphs



10. Art of Creating BG 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 deserves the credit for putting pieces 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

1. 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.

2. 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).

3. 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).

4. Any 0-junction to which a source of effort with constant zero value is attached and no external element is attached may be removed from the bond graph with all bonds attached to it (see section H in table).

Junction Structure Reduced Form

http://www.bondgraphs.com/about1.html (1 of 16)9/1/2005 5:54:17 PM









A

B

C

D

E

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. These equivalent forms may be established by writing 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.

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

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

3. 1-junction is a distributor of SF.

4. 0 junction is a distributor of SE.


A

B

C



D

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


E

F

10.1 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 :

1. Method of Flow Map

2. Method of Effort Map

3. Method of Mixed Map



10.1.1 Method of flow map

The method of flow map is based on the fact that the single port mechanical C or R element 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. Following steps may be followed to create system bond graph by Method of Flow Map.

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

2. Create 1-junctions depicting the motions of end points of C and R elements.

3. Relate and connect these junctions by transformer (TF)elements if a lever relations exist between them. Gyroscopic relations and feedbacks may be incorporated using gyrator (GY) elements.

4. Create the relative velocities between the end points of C and R elements. 0-junction may be used for adding or subtracting the velocities.

5. Attach C and R elements to 1-junctions where relative velocities of their end points are created.

6. Reduce the bond graph using 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 relative velocity 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 elements. An alternative bond graph may be created by making two 1mv junctions and then attaching C to one and R to the other. This bond graph may then be reduced to the earlier one.

(a) (b)



(c) (d)

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

(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)



(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.

Two dimensional model of a car



(a)

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

10.1.2 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.



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

1. If necessary create a 0-junction for sources of effort. Such junctions may be treated as distributors of efforts.

2. 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.

3. For R elements, decide what kind of relative velocity (compressive or stretching) is to be taken as positive. In linear systems this choice may be arbitrary.

4. Addition of forces may be done using 1-junction.

5. Linear forces may be converted to couples using transformer (TF) elements. Similarly, angular velocities may be converted to gyroscopic forces using gyrator (GY) elements.

6. 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).

(a) (b)



(c)

In this bond graph tensile force in the spring and tractile force on the damper is taken as positive. C and R elements 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.

Example 4 : The car model discussed in the method of flow map is drawn using Method of effort 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 simpler form.

(a) Model for vertical dynamics of a car



(b) Alternative model for vertical dynamics of a car

(c) Reduced model

10.1.3 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.



(a) Schematic diagram of car with a passenger

(b) Bond graph of car with a passenger

10.2 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.

1. Method of Gradual Uncover,

2. Point Potential Method,

3. Mixed Network Method.

10.2.1 Method of gradual uncover

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)

(e)

(f)

(g)



10.2.2 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.

1. Each point of circuit where ends of various elements or sub-circuits are tagged may be made into a 0-junction.

2. Any element or impedance through which current flows may be attached to a 1-junction between the 0-junctions.

3. A suitable tag point may be grounded. This is achieved by attaching a zero source 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.

4. Reduce the bond graph.

The following example illustrates this approach.

Example 7 : Let a bridge circuit shown in the figure below 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)

(c)



(d)

10.2.3 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 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 warnings to the reader.

1. 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 in number.

2. Causal loops should be avoided. In a junction loop, not all junctions should have their causalities determined by internal bonds (i.e. bonds which are part of the loop).

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About Bond Graphs



11. Fields

So far the external elements like C, I and R were connected to a single bond like -C, -I and -R. If the parameter (spring constant or capacitance) of the C elements is any nonlinear or linear function of displacement only then C element will be conservative. It will 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=Σjn=1 Kij Qi, i=1..n;

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 within 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 of beam vibration problems, where the basic beam element is represented by a 4x4 stiffness matrix. This matrix relates the two sets of bending moments










and shear forces at both ends of 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 single port C element, for instance a spring, are as follows.

F= K x, E=∫ −t∞ F dx = K x2.

For the thermal domain, the differential equations of the internal energy(U) can be expressed as follows.

dU= -P dV + T ds + µ dN,

where P, dV, T, ds, µ 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 the analysis that P, T and µ are effort variables and the corresponding flow variables are rates of 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 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 .

It may be noticed that the R field is in conductive causality and the matrix written above describes the conductance matrix. This matrix is not invertible, which implies entropy flow rates can be functions of

temperature, but no vice versa. This also implies that entropy generation is due to temperature and not vice-versa.

State equations for models with field elements are written in similar manner 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 below, which is described by a bond graph model shown to its right.

The stiffness matrix in non-principal co-ordinates X'-Y' is obtained by rotation matrix as follows.

Kx'x' = Kxx cos2θ + Kyy sin2θ Kx'y' = Ky'x' = (-Kxx + Kyy) cosθ sinθ Ky'y' = Kxx sin2θ + Kyy cos2θ

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*Q4e4 = K4_3*Q3 + K4_4*Q4e5 = R5_5*f1 + R5_6*f2 = R5_5*P1/M1 + R5_6*P2/M2e6 = R6_5*f1 + R6_6*f2 = R6_5*P1/M1 + R6_6*P2/M2

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. Consider a 2x2 C-field, whose stiffness matrix is such that it cannot be diagonalized 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.

12. 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 require inversion 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 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 of 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

● 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.,

Without considering the detailed 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.

13. Differential Causality

The cause and implications of differential causality in a system model 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) 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 as shown below.

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 easily 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 Q1 and initial conditions can be assigned 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.

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 a constant. 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 a monotonically 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 series, storage elements, as shown below.

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 single storage element 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.

14. 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 simple substitutions. Some components of the equation need to be solved as a set of linear equations. These cases are termed as algebraic loops and 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 elements 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.



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 algebra) are

e2 = e1 - e3 -e4 = SE1 - K3*Q3 - R6*f6 f6 = f4 - f5 -f7 = f2 - f8 - P7/m7 = e2/R2 - e8/R8 - P7/m7e8 = 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

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 the equations beforehand.

15. Causal Loops

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 model, 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.

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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About Bond Graphs



16. 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 obscures the 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.

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

16.1 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 between the power variables in bond numbers 1 and 4.

e2 = µ1*f1 and e3 = e2 = µ1*f1, f4 = 1/µ2 *e3 and f4=µ1/µ2 * f1

Like wise

f3 = 1/µ2 *e4, f2 = f3 = 1/µ2 *e4

and e1=µ1 * f_2, thus e1 = µ1/µ2 * e4.

The two gyrators are thus equivalent to a single transformer with modulus and causal orientation as shown in the figure below.










Likewise, other combinations of transformers and gyrators in various causal postures can be reduced to simplified forms shown in the table below.

Combination Reduced form

16.2 Combination of gyrators and sources

A gyrator converts flow to effort and effort to flow. Thus a source type and gyrator combination may be replaced by a dual source with suitable scaling factor as shown in the table below.


16.3 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 in the 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 written as follows.

DP4 = e4 = e3= µ *f2 = µ *f1, e1 = e2 = µ *f3 = µ *f4 = µ *P4/m4 = µ2/m4 ∫ f1 dt.

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

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 = µ2/m4, and Q4= P4/µ, 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.


16.4 Combination of gyrators and junction elements

Gyrators convert effort to flow and flow variable to effort variable. Thus they may be used to alter the junction types. Some such equivalences, which are independent of power and causal orientations, are shown in the table below. All the gyrators shown in the table are normally unitary, i.e. the modulus µ=1.



Junction Equivalent form

16.5 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, µ and L represent the cross-sectional area, magnetic permeability and mean length of the core.



Let us change the 1-junction for magnetic flux φ to a 0-junction by incorporating gyrators of modulus np on all sides as shown in the figure below.

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 below.

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. If µ 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.

16.6 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.

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 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.

17. Multi and vector bond graphs



When similarities 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.

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.



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 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 of multibonds, analogous to the direct sum of vector-spaces. It is represented by a line perpendicular to the multibonds which take part in the 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 below.

A spatially anchored mass-spring damper system and its multibond graph model are shown in the figure below.



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

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 using 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 of 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 elements between three 1-junctions would be of 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.

18. Suggested readings (Online)

The Bond graph method by Benjamin Quincy Cabell V, Molly F. Usilton, and Professor Mustapha S. Fofana. **

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 using Experimental Data". **

Seminar presentations by Peter Gawthrop. ****

Introduction to Physical Systems Modelling with Bond Graphs. *****

Introduction to the Energetic Description. *

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

Modélisation Bond Graph des Systèmes en Commutation Application aux Sytèmes Electriques by Jean


http://www.cabell.org/Quincy/Documents/Bondgraph/

http://www.ece.arizona.edu/~cellier/bgd_1_1.html

http://www.me.uwaterloo.ca/~jph/pubs/bondgraph.pdf

http://www.mech.gla.ac.uk/~peterg/seminars/BondGraphs/

http://www.rt.el.utwente.nl/bnk/papers/BondGraphsV2.pdf

http://java.sch.bme.hu/applets/verseny/ca4d/elembeha.htm

http://www.fceia.unr.edu.ar/fceia1/publicaciones/numero3/articulo4/Paper20SJ.htm

http://www.irccyn.ec-nantes.fr/gdr/Autrans/Session15c.pdf


Buisson. **

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

KaliBond - A Tool For Teaching Bond Graph Modeling. *

Modern Control of Physical Systems by S.Stramigioli. **

TPA-Tecniche di Misura e Simulazione by Domenico de Falco. **

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http://www.mech.gla.ac.uk/Research/Control/Publications/Rabstracts/abs95004.html

http://www.impa.tuwien.ac.at/forschung/publikationen/ace97_kali.pdf

http://dutera.et.tudelft.nl/~et4102/mechatronics/notes/Modern%20Control%20of%20Physical%20Systems.pdf

http://defalco.dime.unina.it/elasis/1/Elasis1.pdf












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