Page 153 - The Mechatronics Handbook

modeled Resistive load
e
R 2 effort-flow R 2 e 3 curve
characteristic
operating
point
2 2
e e
S e 1 S e 1 R 3
1 3 f f 1 3 f f 3
(a) (b) (c) (d)
FIGURE 9.26 Algebraic loop in a simple source-load model.
Note on Some Difficulties in Deriving Equations
There are two common situations that can lead to difficulties in the mathematical model development.
These issues will arise with any method, and is not specific to bond graphs. Both lead to a situation that
may require additional algebraic manipulation in the equation derivation, and it may not be possible to
accomplish this in closed form. There are also some ways to change the model in order to eliminate these
problems, but this could introduce additional problems. The two issues are (1) derivative causality, and
(2) algebraic loops. Both of these can be detected during causality assignment, so that a problem can be
detected before too much time has been spent.
The occurence of derivative causality can be described in bond graph terms using Table 9.7. The issue
is one in which the state of an energy-storing element (I or C) is dependent on the system to which it
is attached. This might not seem like a problem, particularly since this implies that no differential equation
need be solved to find the state. It is necessary to see that there is still a need to compute the back-effect
that the system will feel in forcing the element into a given state. For example, if a mass is to be driven
by a velocity, V, then it is clear that we know the energy state, p = mV, so all is known. However, there
p ˙
˙
is an inertial force computed as = m = ma. Many times, it is possible to resolve this problem by
V
performing the algebraic manipulations required to include the effect of this element (difficulty depends
on complexity of the system). Sometimes, these dependent states arise because the system is not modeled
in sufficient detail, and by inserting a compliance between two gears, for example, the dependence is
removed. This might solve the problem, costing only the introduction of an additional state. A more
serious drawback to this approach would occur if the compliance was actually very small, so that
numerical stiffness problems are introduced (with modern numerical solver routines, even this problem
can be tolerated). Yet another way to resolve the problem of derivative causality in mechanical systems
is to employ a Lagrangian approach for mechanical system modeling. This will be discussed in section 9.7.
Another difficulty that can arise in developing solvable systems of equations is the presence of an
algebraic loop. Algebraic loops are relatively easy to generate, especially in a block diagram modeling
environment. Indeed, it is often the case that algebraic loops arise because of modeling decisions, and
in this way a bond graph’s causality provides quick feedback regarding the system solvability. Algebraic
loops imply that there is an arbitrary way to make computations in the model, and in this way they reveal
themselves when an arbitrary decision must be made in assigning causality to an R element. 3
As an example, consider the basic model of a Thevenin source in Fig. 9.26(a). This model uses an
effort source and a resistive element to model an effort-flow (steady-state) characteristic curve, such as
a motor or engine torque-speed curve or a force-velocity curve for a linear actuator. A typical charac-
teristic is shown in Fig. 9.26(b). When a resistive load is attached to this source as shown in Fig. 9.26(c),
the model is purely algebraic. When the causality is assigned, note that after applying the effort causality
on bond 1, there are two resistive elements remaining. The assignment of causality is arbitrary. The solution

3
to energy-storing elements. An “arbitrary” decision to assign integral causality on an energy-storing element leads to
The arbitrary assignment on an R element is not unlike the arbitrariness in assigning integral or derivative causality
a requirement that we solve a differential equation to find a state of interest. In the algebraic loop, a similar arbitary
decision to assign a given causality on an R element implies that at least one algebraic equation must be solved along
with any other system equations. In other words, the system is described by differential algebraic equations (DAEs).

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