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.

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
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.

	Junction Structure	Reduced Form
A
B
C
D

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

	Junction Structure	Reduced Form
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 1_mv. The velocity depicted at 1_mv is d(x_m)/dt - d(x_v)/dt. Next, one attaches the external elements. An alternative bond graph may be created by making two 1_mv 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.

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.

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.

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).