Modeling of Physical Systems: Lecture Summaries Page 1

4 Motivating a bond graph approach

In this lecture, we begin to outline some of the basic concepts used in bond graph modeling.It is helpful to give some thought to how you would model certain types of systems usingwhat we’ve referred to as a classical approach. We continue to stress that this approach canoften yield a quick answer (if you have maintained familiarity/skill). This is especially truefor problems, as in the first problem set, where the system dynamics involve a single ‘energydomain’. In those cases, good results are derived by application of discipline-specific toolsand physical laws.

Modeling involves quite a bit of judgment, however, and it can be helpful to take awayas much uncertainty in the process as possible. We address this by working through someexamples. The discussion below addresses comments on a current problem set, to identifyany common issues in solving the problems, and these motivated the notes that follow.

We also initiated a discussion of bond graph based modeling, defining the power basis, powerbonds, causality, and the concept of a word bond graph. These are basic concepts that canhelp guide the modeling process as implied above. This may be why a bond graph approachappeals to many people who apply it effectively for all types of problems. While the methodswe’ll develop are especially helpful in complex, multi-energetic systems, they provide insightinto problems of any level.

4.1 Example Modeling Problems

Consider the systems summarized in Figure 1. These problems are used to motivate the needto make modeling decisions and to formulate a mathematical model that will help answerspecific questions.

The following are some comments/issues that were brought up either in class or in off-linediscussions:

1. On the reservior system problem, since the requirements are not clear it is difficult tojudge how much detail is needed in a model.

2. Trying to decide which variables to use or to solve for in different problems makes itdifficult to proceed.

3. How do we identify key elements that should be included in a given system model?

4. How do we know what lumped approximations to make, and whether they are valid?For example, a simple damper may not seem reasonable in practical cases.

R.G. Longoria, Fall 2006 UT-Austin

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Figure 1: Four systems studied in HW 1.

5. It can be difficult to determine how certain relations (e.g., the unsteady Bernoulliequation) should be applied to model a system.

We will find that some of these problems always exist in real applications that involvemodeling. However, when we adopt a bond graph approach some become non-issues.

Reference was also made to the kind of decisions you need to make in approximating asystem. The advice is added here that you should ‘choose your battles wisely’. Don’t getstuck on what the specific physical behavior of an element might be if you are first tryingto unravel the ‘interconnection’ of model elements – the physical behavior (or constitutivebehavior) can be worked out later (this will be repeated over and over again).

There was discussion on difficulties in making a lumped-parameter approximation in somecases, and whether these are valid. The example of mechanical components came up asrelates to modeling elastic effects as well as damping, especially if there is no explicit ‘spring’or ‘damper’.

This takes practice, and benefit of experience in some cases (i.e., exposure to good examples).For example, modeling viscoelastic material as a component in a vibration model demon-strates how there is never one answer to what the model should be. Figure 2 illustrateshow you can adopt one of any number of combinations of spring/damper systems [3]. Thekey is to keep in mind that there are two model decisions to make here: a) the topology orstructure, as illustrated by the different ways that springs/dampers are arranged, and b) thefunctional characteristics of each element, which depend on the specific model structure youchoose.

R.G. Longoria, Fall 2006 UT-Austin

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Figure 2: Modeling a viscoelastic material as a system component, here in a simple vibrationproblem, suggest several options in a lumped-parameter model approach.

4.2 Power-conjugate variables, signal flow, and power bonds

As a first step in building a bond graph approach, we adopt a view of a system using theconcept borrowed from thermodynamics. The energy-based system description and powerport concept is illustrated in Figure 3. In this diagram, we quantify power flow at a physicalport by power-conjugate variables u and y. Power-conjugate variables are variables whentaken together as product form power.

Figure 3: Define a power port with causal signals that quantify power.

We represent information flow by a signal bond, which has a full arrow. Note that at a portwe define u as the input variable, specified by the environment into the system, and y asthe output variable, specified by the system back to the environment. In this way, we adoptcausal relations between power-conjugate variables at a port.

Paynter conceived of the power bond to communicate this bilateral power interaction, andused a half arrow to denote the sign (or sense) of power on this power bond.

R.G. Longoria, Fall 2006 UT-Austin

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Figure 4: Power bond definition and sign. Power flows from S1 to S2.

We also adopt a way of communicating causal relations on the bond, placing a causal strokeon one end of the bond to indicate that effort is directed in that direction. The causalassignment is illustrated in Figure 5.

Figure 5: Power bonds and causality assignment.

The power bond forms the basis for quantifying interconnections that convey power. Theconcept is summarized in Figure 6.

R.G. Longoria, Fall 2006 UT-Austin

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Figure 6: Summary of power bond concept.

4.3 Bond graphs: Ideal Sources

As a first example of a bond graph element, it makes sense to introduce the effort and flowsources, which are used to model common inputs to systems. These also help to convey theconcept of causality assignment on a power bond.

Figure 7: Ideal sources of effort and flow are represented by an E or F . It is also commonto use Se or Sf for effort and flow, respectively.

The ideal effort source is used to represent an input that can provide a specified effort (force,torque, pressure, voltage) for any given flow. Note, this does not imply a constant effort.Likewise, an ideal flow source represents a system that can provide a specified flow (velocity,angular velocity, flow rate, current) for any given effort.

Review Chapter 2 of the BP notes [1] for discussions on different types of sources for differentenergy domains.

R.G. Longoria, Fall 2006 UT-Austin

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4.4 Word bond graphs

A final concept introduced in this lecture was the word bond graph (WBG). Basically, thisallows us to sketch out a high level representation of the primary components in a systemwithout detailed modeling.

Figure 8: Paynter (1949)

This is considered a very preliminaryway to convey the power flow, but noreal analysis can be done. Some times,causality can be assigned on some of thepower bonds, and this was done for asimple example.

First, historically we examine in Figure8 Paynter’s early influence in the form ofelectric power generation systems. Payn-ter shows here the use of two signals toconvey power, but we see that the no-tion of a causal restriction between thetwo power-conjugate variables is not yetformed.

Word bond graphs now use power bondsas shown in Figure 9, an example from Chapter 3 of BP [1]. Additional examples from thebook by Karnopp, et al. [2] are given in Figure 10.

Figure 9: Motor-pump Example from BP [1].

R.G. Longoria, Fall 2006 UT-Austin

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Figure 10: Examples from KMR [2].

References

[1] Beaman, J.J., and H.M. Paynter, Modeling of Physical Systems, Book in progress;notes for ME 383Q.

[2] D. Karnopp, D. Margolis and R. Rosenberg, System Dynamics, Wiley, 2000 (3rd ed.)

[3] Sun, C.T., and Y.P. Lu, Vibration damping of structural elements, Prentice-Hall PTR,Englewood, Cliffs, NJ, 1995.

R.G. Longoria, Fall 2006 UT-Austin