Deriving System Equations from Bond Graphs
5.4 Algebraic Loops
f10 = p6
mJ2
(5.62)
In the matrix form, the fi rst four equations can be written as:
p2
The behavior of the four independent energy storage devices is governed by the above four equations. The fi fth energy storage device (the one that is dif-ferentially causalled and, as a result, not independent) is related to the behavior of the others and can be represented through the relationship:
f10 = p6
mJ2
(5.64)
From the standpoint of system behavior and its solution, this means that the set of differential equations (Equation 5.63) will fi rst have to be solved simul-taneously and then its solution can be used to solve Equation 5.64. This is the mathematical implication of the differential causality and the dependent behavior of the device with differential causality.
5.4 Algebraic Loops
When deriving the governing equations for a system, one may encounter another problem, which is referred to as algebraic loops. Algebraic loops and derivative causality received a lot of attention from researchers during the early days of bond graph research. For more detailed discussion of this topic, refer to the works of Granda (1984), Felez and Vera (2000), Karnopp (1983), and others.
Figure 5.8 shows an example and a bond graph that illustrates a situation when an algebraic loop is encountered and how it could be handled. In Example 5.5, we will attempt to derive the governing equations in a manner similar to ear-lier examples.
EXAMPLE 5.5
Answers to Q1 (written in terms of the constitutive equations).
Q1. What do all the elements give to the system?
A circuit and its bond graph representation.
e1
0 Junction 1 1 Junction I1
4.
In matrix form these two equations become: Having gone through the process, we observe a signifi cant difference between this process and some of the earlier ones. In this case, while developing the equations for I elements associated with Bond 6, we fi nd that p6 is not only related to other quantities but to itself as well. This is typical of the situation that is known as an algebraic loop. When an alge-braic loop exists in a model, the model equation for a particular energy variable is not only a function of other quantities but also of energy vari-able itself. This does not pose a problem when the constitutive equations for the basic elements are linear. But if the basic elements have nonlinear constitutive equations, the algebraic loops may cause diffi culties in the solution of the governing equations.
The bond graph model itself can indicate the existence of an algebraic loop. If during the causality assignment we end up with more than one possible causal structure without violating any rules of causality assign-ment and assuring integral causality for storage eleassign-ments, it is a sure indication of the existence of an algebraic loop. In this particular case, the bond graph shown in Figure 5.8 is a possible causality structure as is the causal structure shown in Figure 5.9. The causal strokes of R2 and R3 can toggle without changing anything else in the model. Whenever we encounter something like this, we need to be aware that the algebraic loop will make derivation of the equation somewhat diffi cult.
I
An alternate causal structure for the model in Figure 5.9.
Problems
5.1. For all the problems in Figure P3.2a–m (for fi gure see the Problems section of Chapter 3) draw the bond graphs, assign causal strokes, and derive the governing equations for the systems.
5.2. Derive the governing equations for the bond graph models shown in Figure P5.1a–e. Use appropriate bond numbers and symbols for the elements in the bond graph.
(a)
0 Junction 1 1 Junction
I1
0 Junction 1 0 Junction 2
I1
Figure for Problem 5.2, bond graph models.
(d)
0 Junction 1 0 Junction 2
I1 I2
0 Junction 1 0 Junction 2
I see the Problems section of Chapter 3). The constitutive behavior of the resistances B1 and B2 are different from the constitutive behavior of the standard R elements. All these resistances are due to friction force and the constitutive relationship of the friction force is
Ffriction = F0Sgn(V)= F0
V V
Using this constitutive behavior model for the resistances, derive the set of governing differential equations for the behavior of these two systems.
5.4. Figure P5.2 shows a system. Derive its governing equations from the bond graph model.
B M
L1
L2
K1
K2
FIGURE P5.2
Figure for Problem 5.4, mechanical system.
5.5. Figure P5.3 shows a mechanical system. Derive its governing equations from the bond graph model.
FIGURE P5.3
Figure for Problem 5.5, mechanical system.
B
F K1
K2 M
a
b
5.6. Figure P5.4 shows a bond graph model. Will this have an algebraic loop? Develop the governing equations for this system.
1
0 0 Junction 1
1 Junction 2 1 Junction
R R1
1 R
R2 0
C C1
I I1
Se Se1
0 Junction 2
FIGURE P5.4
Figure for Problem 5.6, bond graph model.
5.7. For the system shown in Figure P5.5, develop the bond graph model of the system and then derive the system governing equation(s), fi rst with a velocity input from the right. Do the same if the input is not a velocity but a force input.
FIGURE P5.5
Figure for Problem 5.7, mechanical system.
M
B1 B3 B2
V(t)
5.8. For the system shown in Figure P5.6, develop the bond graph model of the system and then derive the system governing equation.
M J, R
K
B
F(t)
FIGURE P5.6
Figure for Problem 5.8, mechanical system.
5.9. Consider the electrical system shown in Figure P5.7. Derive its governing equations from the bond graph model.
R1
R2 C
L
V
I − +
FIGURE P5.7
Figure for Problem 5.9, electrical system.
5.10. Consider the electrical system shown in Figure P5.8. Derive its governing equation from the bond graph model.
FIGURE P5.8
Figure for Problem 5.10, electrical system.
R1
C2 R2 L
C1 I
5.11. Put together two bond graph models that could have two possible causal structures without violating any rules of causal assign-ments. Then derive the governing equations to demonstrate whether algebraic loops are encountered.
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