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,

e_i=S_jⁿ₌₁ K_ij Q_i,    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=ò_-^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 + m dN,

where P, dV, T, ds, m and dN represent pressure, volume flow rate, temperature, rate of change of entropy, chemical potential, and mole flow rate, respectively (prefix d stands for time derivative). An alternative expression may be written using enthalpy and mass flow rate for the chemical work. Thus the representation of this thermodynamic process (due to Breedveld) may be given as follows.

It can be easily observed from the analysis that P, T and m 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* =	é	H/T1	-H/T2	ù
	ë	H/T2	-H/T1	û	.

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.

K_x'x' = K_xx cos²q + K_yy sin²q
K_x'y' = K_y'x' = (-K_xx + K_yy) cosq sinq
K_y'y' = K_xx sin²q + K_yy cos²q

Similarly, the damping matrix is obtained in X'-Y' co-ordinates. The state equations can then be derived as shown below.

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

The constitutive relations are :

f1 = P1/m1
f2 = P2/m2
e3 = K3_3*Q3 + K3_4*Q4
e4 = K4_3*Q3 + K4_4*Q4
e5 = R5_5*f1 + R5_6*f2 = R5_5*P1/M1 + R5_6*P2/M2
e6 = R6_5*f1 + R6_6*f2 = R6_5*P1/M1 + R6_6*P2/M2

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*Q1_t=0+K2*Q2_t=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/m7
e8 = e5 - e9 = e6 - e9 = R6*f6 - K9*Q9

Now these expressions are interwoven functions of each other (a third order algebraic loop) and need to be solved out algebraically as follows.

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

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

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

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

Substitution of f6 in expression for e8 leads to

      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.