DEVELOPMENT OF PROPER MODELS OF ... - Semantic Scholar

Download PDF
1 downloads 0 Views 86KB Size Report
Comment
John B. Ferris and Jeffrey L. Stein ...... sponsorship of Mr. John Ferris as a doctoral student at The University of ... "Branch Mode Analysis of Vibrating Systems." J.
DEVELOPMENT OF PROPER MODELS OF HYBRID SYSTEMS: A BOND GRAPH FORMULATION John B. Ferris and Jeffrey L. Stein Department of Mechanical Engineering and Applied Mechanics The University of Michigan Ann Arbor, MI 48109-2125 [email protected] and [email protected]

ABSTRACT To improve the ability of engineers to use dynamic models as part of the design process, a model deduction procedure for developing proper bond graph models of hybrid systems is presented. An accuracy criterion and a search pattern are presented, in a bond graph format, for a model order deduction algorithm (Extended MODA) previously published. The Extended MODA process consists of systematically incrementing the rank of each component model in the system, until a set of critical system eigenvalues has been defined. Then, the rank of component models is incremented until each critical system eigenvalue has converged to within a specified tolerance. The bond graph representation is shown to provide a natural framework for assembling component models that may be modal expansions, finite segments and or bounded representations of the components. In addition, through an example the bond graph formulation is shown to have great potential utility in providing the design engineer a “visual” perspective on the dominant components (and elements of those components) in a proposed system. The example also shows that the proper system model deduced depends on both the model approximations used for the components as well as the model deduction search algorithm. A bond graph based automated modeling system using Extended MODA appears to hold much promise for improving the tools available to design engineers. Keywords: model reduction, bond graphs, dynamic, linear. 1. INTRODUCTION Early in the design process, it is important to predict the essential behavior of the proposed system and compare the predicted performance to the performance specifications. This quantitative evaluation allows for rational design changes to be posed and, because they are posed early in the design cycle, design improvements can be implemented. However, highly skilled analysts, working over an extended period of time, are currently required to develop the necessary models. Design engineers who do not have these refined skills, and who are facing rapidly approaching deadlines, do one of three things: • They use grossly oversimplified models which give inferior if not completely inaccurate results. • They use overly complicated models, perhaps taken from previous analysis projects, which provide little or no ability to see the relationship between design parameters and system performance. • They make no formal use of models as part of the design process. It is the premise of this work that more design engineers would use models as part of the design process if automated modeling tools were available. These tools would automatically deduce models of appropriate complexity, with variables and parameters that are physically meaningful to the design process (i.e., they are functions of component geometry and material properties). Wilson and Stein (1992) defined these models as Proper Models. Wilson and Stein (1992) developed a model order deduction algorithm (MODA) to automatically deduce Proper Models of linear systems containing components represented by bounded models (finite order state equations) and by unbounded models (up-to-infinite order finite segment approximations in first order form). Three key issues, not directly addressed by Wilson and Stein, are: • The accuracy of the model that is generated, • The compatibility of the various component model formats used by engineers to model components in their application area, and • The effect of the component model representation (finite segment vs. modal expansion) on the deduction of the Proper Model of the system. Ferris et al. (1994) addressed the first two issues. They developed an extension of MODA (Extended MODA) to deduce proper system

models, whose critical system eigenvalues have converged to within some user specified tolerance. They also developed a mathematical formulation, based on component mode synthesis (CMS) (Hurty 1965), to combine models developed by engineers in different application areas (e.g., structures, electromechancis, hydraulics, etc.) and implemented in different model paradigms (e.g., modal models, finite segment models, etc.). They call a system composed of component models of different formats a hybrid system. This paper will show that bond graphs provide an alternative approach to the mathematical formulation based on CMS. In addition, the third issue, the effect of component model representations, will be introduced by example and discussed. An example of a hybrid system, used in this paper for illustrative purposes, is shown in Fig 1. Component #1 is a rod fixed in a wall. Component #2 is a DC electromechanical shaker attached between components #1 and #3. Component #3 is a second rod attached to components #2 and #4 as shown. Finally, component #4 is an isolator (i.e., a coil spring with a block at each end). Assume that different engineering departments were responsible for modeling each component and each department used a different modeling technique. For example, a finite segment representation was used for component #1 (the first rod), a state space model was derived for component #2 (the shaker) from a transfer function available from the shaker manufacturer, a modal model was developed from modal data for component #3 (the second rod), and finally a model was developed from first principles (i.e., it was assumed that a single linear compliance element was sufficient to model the coil spring and a lumped mass was sufficient to model each block) was created for component #4 (the isolator). The task of the systems engineer is to deduce a Proper Model of the system. #1

#2

#3

#4

Fig 1: Schematic of a hybrid system The purpose of this paper is to address the use of bond graphs to combine component models, show how the complexity of each component bond graph model can be determined such that the Proper Model of the system is formed, and determine the effect of the component model representations (approximations) on the deduction of the proper system model with respect to the search algorithm. It is the premise of this work that better insight into the model deduction process will result from using a bond graph formulation of component models. This is posited because: First, bond graphs provide a natural means to combine component models that have been developed by engineers in different application areas and implemented in different model paradigms. Second, bond graphs provide a clear visual representation of the component models and the connectivity of the component models. This visual clarity is particularly important when the Proper Model of the system being deduced is a hybrid system, when new concepts in automated modeling, such as Extended MODA, are being presented, and when the effects of model representations are being discussed. This paper is organized as follows: Section 2 describes the background, mathematical formalization and discussion of the proper system model deduction algorithm. Section 3 describes the bond graph approach to proper system model deduction by example, followed by a Discussion. Summary and Conclusions are presented in section 4.

2. BACKGROUND: PROPER SYSTEM MODEL DEDUCTION For convenience to the reader, the non-bond graph based algorithms for deducing the Proper Model of a hybrid system, developed by Wilson and Stein (1992) and Ferris et al. (1994), is reviewed before presenting the bond graph based formulation. 2.1. Component and System Model Synthesis Ferris et al. (1994) combined component models by using a mathematical formulation taken from component mode synthesis (CMS) (Benfield and Hruda 1971; Craig and Bampton 1968; Dowell 1972; Gladwell 1964; Goldman 1969; Hou 1969; Hurty 1965). The first step involved writing the sets of component equations in a compatible form; Ferris et al. called this step Component Model Synthesis. Then they formed the system model (System Model Synthesis) by applying constraint equations which represent the connectivity of the component models. Wilson and Stein (1992) developed a model order deduction algorithm (MODA) to determine a sufficiently complex model of each component such that the system model contains all the system eigenvalues within the frequency range of interest (FROI). A new search pattern and an accuracy criterion was added to this algorithm and named Extended MODA by Ferris et al. (1994). These search algorithms used the concept of component and system rank, as defined by Wilson and Stein (1992), to specify the complexity of the component model. As the complexity of a component model increases, the rank also increases. Each component model has some minimum complexity, that is the minimum number of degrees of freedom (dof) or states, that is said to be the rank 0 model. If a modal representation of a structural component is used, then all the rigid-body modes of the component generally will comprise the rank 0 model. As each additional mode is included, the rank of the model is incremented by 1. The system rank is defined as the sum of the component ranks. 2.2. Proper System Model Deduction Once the system model has been synthesized, the Proper Model of the system can be deduced using the two stage search procedure, Extended MODA (Ferris et al. 1994). The first stage of Extended MODA is based on MODA (Wilson and Stein 1992). First the frequency range of interest (FROI) is specified by the user (e.g., FROI = the highest input frequency) and a trial system model is formed containing the rank 0 model of all the components. Then the spectral radius (i.e., the maximum absolute value of the system eigenvalues) of the trial system model, based on the rank 0 component models, is calculated. Next, a new trial system model is defined which contains the rank 0 model of all the components, except for the rank 1 model of one of the components, and the spectral radius of the new system bond graph is calculated. The spectral radius is noted and the process is repeated until the rank 1 model of each of the components has been tried. The rank 1 component model that produces the smallest spectral radius of the trial system model is then kept in the model (this is denoted by a circle around the component eigenvalue pair as shown in Fig 2). This entire procedure is repeated for the next highest rank model of each of the components, and the component that produces the smallest increase in the spectral radius of the system, when its rank is incremented, is added to the model. This continues until the spectral radius of the system bond graph exceeds the FROI (i.e., the first system eigenvalue that falls outside the FROI is included). This defines the set of critical system eigenvalues (CSE). Component s-planes

System s-plane

x

x x x x x x x x x

x x

x x

x x x x x

x x x x x x x x

x

FROI

x x x x x x x x x x

x

x

x x x x

critical system eigenvalues

Fig 2: Identifying the Component Ranks that Define the Set of Critical System Eigenvalues (CSE)

The system model can only be accurate within the FROI (for an unspecified input) if each of the critical system eigenvalues (CSE) is predicted accurately. Therefore, the second stage in the Extended MODA deduction procedure is to increment the rank of the component that produces the greatest correction in the values of the critical system eigenvalues. That is, each of the trial system models (of rank n+1) produces a set of CSE. The percent change in each of the CSE (compared to the CSE of the previous system model of rank n) is calculated, for that trial system model, and the maximum percent change is recorded. This maximum percent change is found for each of the trial system models (of rank n+1), and the trial system model that produces the largest of these maximum percent changes becomes the rank n+1 system model. The system model rank continues to be incremented until each of the CSE has converged to its true value. In practice, the process continues until some user defined convergence criteria have been satisfied. In this paper, the criterion used is a preset tolerance for the largest relative change in each of the CSE. Note that an alternative approach would be to continue using the MODA search algorithm (as previously described) beyond the FROI. It is not known at this time which approach, in general, will produce the faster convergence of the critical system eigenvalues to their true values. One of the objectives of this paper is to compare these two approaches; this is done in Section 3 for the system shown in Fig 1. The system model formulation of Ferris et al. (1994) is very powerful if the existing component models are in equation form. However, if bond graph models of the components are available, then this formulation is not necessary and, in fact, the system model formulation is simply accomplished by the natural interface features of the bond graph language. In addition, the concept behind proper system model deduction can be more readily presented and some subtleties in the definition of component rank explored. Therefore, it is a second objective of this paper to incorporate bond graph component models with Extended MODA to deduce Proper System Bond Graph Models. 3. BOND GRAPH APPROACH TO PROPER SYSTEM MODEL DEDUCTION The method used in this paper to incorporate bond graphs into Extended MODA takes advantage of the work done by Margolis (1989) and Karnopp et al. (1990). This work shows how bond graphs can be used to represent both unbounded and bounded component models, and how they can be combined to form the system model. The method developed here to deduce the proper system model, via bond graphs, will be introduced by example. 3.1. Example The hybrid system with four components, as shown in Fig 1, will be used to illustrate the deduction of a proper bond graph of the system. 3.1.1. Component and System Model Synthesis. Table 1 shows the type of model and parameter values used to describe each component model. Component #1, the first rod, is represented by a finite segment (lumped parameter) model (i.e., a series of masses and springs) where the total mass of the rod, mrod, is divided into p smaller masses, mlump = mrod/p, and the total stiffness, krod, is divided into p-1 springs, klump = (p-1) krod, shown in Fig 3. Component #2, the shaker, is represented by the bounded bond graphs in Fig 4a and Fig 4b. The models include the mass of the base of the shaker, mbase, the moving coil (armature) mass, marm, as well as the electrical resistance, r, and motor constant, k m. The rank 0 model excludes the motor winding inductance, L. Component #3 (the second rod) is represented by a modal model (see Fig 5). For the ith included mode, the contribution of each mode at the connection point to component #2 (Ψi(0)) and the connection point to component 4 (Ψi(l)) can be chosen such that Ψi(0) = (2/ρΑl)1/2, Ψi(l) = (2/ρΑl)1/2 cos(iπ), Mi (the ith modal mass) = 1 and Ki (the ith modal stiffness) = (iπ)2E/ρl2. Component #4, the isolator, is represented by a bounded model consisting of 2 masses connected by a spring (see Fig 6).

Table 1: Component model type and parameters Model Type Parameters Unbounded, density (ρ) = 7755 kg/m3 Finite Segment elastic modulus = 2.1 e10 N/m2 diameter = 0.05 m length (l) = 2 m #2: shaker Bounded, gyrator modulus (ks) = 1 N/A Lumped Parameter resistance (r) = 1 Ohm inductance (L) = 0.0005 H mass of base (mbase) = 1 kg mass of armature (marm) = 1 kg #3: second Unbounded, density (ρ) = 7755 kg/m3 rod Modal Expansion elastic modulus = 2.1 e10 N/m2 diameter = 0.05 m length (l) = 2m #4: isolator Bounded, mass 1 (m1) = 0.5 kg Lumped Parameter spring stiffness (k)=10000 N/m mass 2 (m2) = 0.5 kg

Component #1: first rod

The component models are combined to form an expandable framework for the system model. This framework is represented in the bond graph shown in Fig 7. Bonds to removable energy elements of the bound components are shown with a broken (dashed) line. The expandable structure of the unbounded components is indicated with dots. The boundary conditions are represented by sources. The left end of component #1 is fixed by a wall, and the shaker is excited by a voltage source, Se. Note that causality can only be applied when the rank of each component model has been specified. component #1

component #2

component #3

Ψ (0)

v=0: Sf

C:1/klump

C:1/k lump

0

0

1

• • •

I:m lump

0

TF 1

1

I:mlump I:m base

R:r

0

GY:r

1

0

1

I:marm

• • • Ψn(0) TF

1

Ψ0 (l) TF • • •

I:M 0 C:1/Kn

1

component #4

C:1/k

0

1

0

1

I:m1

I:m 2

Ψn (l) TF

I:L I:Mn

Se:V

Fig 7: Expandable framework for the system model represented in bond graph form.

1

• • •

0

3.1.2. Proper System Model Deduction. Three system models are deduced. First an "exact" model (a model whose critical system eigenvalues have converged to four significant digits) is generated by including: a rank 50 model of component #1, the inductance of component #2, the first 50 modes for component #3, and the compliance of the coil spring in component #4. The second model is generated using MODA (Wilson and Stein 1992)1 and the third model is obtained using Extended MODA (Ferris et al. 1994).

C:1/klump

C:1/klump

0

1

I:mlump

I:m lump

Fig 3: Lumped parameter model of component #1

1

0

1

1

0

1

I:mbase

GY:ks

I:marm

I:mbase

GY:ks

I:marm

R:r

1

R:r

1

I:L

The second and third models are deduced subject to the following proper system model deduction specifications: • A FROI of 1000 rads/sec. • A 1% tolerance on the relative change of the critical system eigenvalues (CSE). The first step in the proper system model deduction process is to form a system bond graph from the rank 0 model of each component, as shown in Fig 8.

component #1

Fig 4a: Rank 0 model of the shaker model of the shaker

0

1

Fig 4b: Rank 1

Ψ0 (l) TF

TF

0

C:1/K2 Ψ1(0) TF

1

1

I:m rod

I:M0

component #3 Ψ0 (0)

v=0: Sf

Ψ0 (0) TF

component #2

1

0

1

1

I:M0

I:mbase GY:k I:m arm s

R:r

component #4

Ψ 0(l) TF 1

1

I:m1

I:m 2

1

Se:V

Ψ1 (l) TF

Fig 8: Rank 0 system model

I:M1

Fig 5: Rank 1 model of a rod

1

0

1

I:m

C:1/k

I:m

Fig 6: Model of component #4

The spectral radius of this rank 0 system model is 0.0308 rads/sec (see Table 2). Because the spectral radius is less than the FROI, the rank 1 component model of one of the components must be used in the system bond graph, while the rank 0 models of the remaining components are retained. The 4 trial rank 1 system models are shown in Fig 9; the spectral radius of each of the trial rank 1 system models is calculated.

1Wilson and Stein only used MODA with componets represented by finite segment approximations. Here modal component models are used.

Table 2: Defining the critical system eigenvalues Component Rank Spectral Radius #1 #2 #3 #4 (rads/sec)

System Rank 0

0

0

0

0

0.0308

1

1

0

0

0

1127

1

0

1

0

0

2000

1

0

0

1

0

2430

1

0

0

0

1

143

2

1

0

0

1

1127

2

0

1

0

1

2000

2

0

0

1

1

2467

Fig. 9c: Trial rank 1 system models. Component #3 = Rank 1

component #1

component #2

Ψ0(0) TF v=0: Sf

1

I:mrod

1

0

I:mbase GY:k s R:r

component #4

component #3

1

1

Ψ0(l) TF

I:M0

I:marm

C:1/k

1

0

I:m1

1

I:m2

1

S e:V

Fig. 9a: Trial rank 1 system models. Component #4 = Rank 1 Fig 9: Trial rank 1 system models. The rank 1 component is labeled in bold.

component #1

component #2

component #3 Ψ 0(0) TF

C:1/k lump v=0: Sf

1

1

0

I:mlump

1

0

I:m lump I:mbase GY:k s

R:r

1

component #4 Ψ 0(l)

1

TF 1

1

I:M0

I:m1

I:marm

I:m 2

1

S e :V

Fig. 9a: Trial rank 1 system models. Component #1 = Rank 1

component #1

component #2

component #3 Ψ0(0) TF

v=0: Sf

1

I:mrod

1

0

I:mbase GY:k s

R:r

1

1

1

component #4 Ψ0(l) TF

I:M 0

I:marm

1

1

I:m1

I:m2

I:L

Se:V

Fig. 9b: Trial rank 1 system models. Component #2 = Rank 1

component #1

component #2

Ψ0 (0) TF v=0: Sf

1

I:mrod

1

0

I:mbase GYk s R:r

1

I:marm

component #4

component #3 1

Ψ0(l) TF 0

0

I:M0 C:1/K1 Ψ1 (0) TF

1

The spectral radius of each of the four rank 1 system models in shown in the second set of data in Table 2. Because incrementing the rank of component #4 results in the smallest increase in the spectral radius, the rank 1 model of component #4 and the rank 0 models of the remaining components define the rank 1 system model. However, because the spectral radius does not exceed the FROI, the next set of trial system models must be considered.

1

1

I:m1

I:m2

In the next set of trial system models, the rank of each component model is incremented, in turn, and combined with the rank 0 model of the remaining components, or the rank 1 model of component #4, to form the possible rank 2 system models2. Again the spectral radius of the trial system models is calculated and, because incrementing the rank of component #1 causes the smallest increase in the spectral radius (see the third set of data in Table 2), the rank 1 model of component #1 is included in the system model. The spectral radius of the system model now exceeds the FROI, and thus the critical system eigenvalues have been defined as: - 0.0308 rads/sec, - 2.411 E-4 +/- 142.5j rads/sec, - 3.081 E-2 +/- 1127j rads/sec. At this point, MODA and Extended MODA differ in their approach to determining an accurate model. For this example, Extended MODA proceeds by incrementing the rank of the component models that causes the largest correction in the critical system eigenvalues, until the accuracy criterion is satisfied (in this case, until each critical system eigenvalue has converged to within 1% of it's previous value). The result of this process is shown graphically in Fig 10. Each time the system rank is increased, the rank of one of the components is increased (represented by darker shading). Extended MODA continues to increment the rank of the component models (as previously described) until the accuracy criterion is satisfied. In this example, the accuracy criterion is satisfied when the system rank equals 5. The rank of component #1, the first rod, is equal to 4 while the rank of component #4, the isolator, is equal to 1. Only the rank 0 models of components #2 and #3 were required. For comparison purposes, the MODA search algorithm is also used to increment the rank of the component models beyond the FROI until the accuracy criterion is satisfied. This process is represented graphically in Fig 11 up to system rank 5. Again darker shading represents an increase in component model rank. Notice that a different proper model results. All four components are incremented in rank; components #1 and #3 have a rank of 2 while components #2 and #4 have a rank of 1.

Ψ1(l) TF

1 I:M 1 Se :V

2In this example component #4 is bounded, with a maximum rank of 1, therefore the set of trial system models does not include the "0 0 0 2" combination of component ranks (see Table 2).

Extended MODA

System Rank

5

4

0

0

1

4

3

0

0

1

3

2

0

0

1

2

1

0

0

1

1

0

0

0

1

0

0

0

0

0

#1

#2 #3 Component

% Error 4 3

Extended MODA 2

CSE 1

MODA 0 #1 #2 #3 Critical System Eigenvalue

#4

Fig 12: Comparison of CSE error for MODA and Extended MODA

Fig 10: Selection of proper component rank via Extended MODA MODA

System Rank

8

4

1

2

1

7

3

1

2

1

6

2

1

2

1

5

2

1

1

1

4

1

1

1

1

3

1

1

0

1

2

1

0

0

1

1

0

0

0

1

0

0

0

0

0

#2 #3 Component

#4

#1

CSE MODA

Fig 11: Selection of proper component rank via MODA 3.1.3. Accuracy Comparison: MODA vs. Extended MODA. Two different proper models are deduced by MODA and Extended MODA. In order to satisfy the accuracy criterion on the critical system eigenvalues (CSE), the rank of the MODA deduced system model is 8, as compared to the Extended MODA system model whose rank is 5. In this example, Extended MODA has produced the "more" proper model. In addition to examining the differences in component ranks for the two deduced system models, the accuracy of the CSE for system models of equivalent rank can be examined. For the given proper system model deduction specifications, there are 5 CSE. CSE #1 is real, while CSE #2 and #3 are complex conjugate pairs (see Table 3). Extended MODA produces a system model whose critical system eigenvalues are significantly (mathematically speaking) more accurate. Fig. 12 illustrates the difference in percent error in the critical system eigenvalues. As to be expected, some eigenvalues are closer to the “exact” values than others. Eigenvalue #1 and eigenvalue pair #2 are very close to the exact value and can be predicted accurately with a system model of smaller rank. It is eigenvalue pair #3 that is slow to converge and forces Extended MODA to include additional modes. Table 3: A comparison of models generated by MODA and Extended MODA for the example in Fig 1. CSE Exact MODA Extended MODA #1

-0.030814

-0.030814

-0.030813

#2

-2.4 E-04

-2.4 E-04

-2.4 E-04

+/- 143 j

+/- 143 j

+/- 143 j

2.3 E-02

2.5 E-02

3.4 E-02

+/- 1251 j

+/- 1203 j

+/- 1235 j

#3

MODA

3.2. Discussion The objective of this study is to explore both the value of the bond graph representation to the model deduction process as well as the interaction of component model approximations and the search algorithm on the proper model deduced. First we will discuss issues related to the bond graph representation. 3.2.1. Bond Graph Representations for Model Deduction. As stated previously, the premise of this work is that new insight into the model deduction process will result from using a bond graph formulation of component models. This, in part, means that potential users of this model deduction technology will more easily understand the how details of the process relate to the design . The bond graph representation of components provides visual feedback about the structure of the component models, in the context of the system, to the design engineer. And because bond graphs are a graphical representation, it is necessarily a simultaneous presentation of the information (Breedveld, 1988). That is the system model can be viewed all at once, with the distinction between the component models maintained explicitly (see Fig 7). This allows visualization of the role of each component (and each of its parameters) to the system model. The connectivity of components (junction structure) can also be seen easily. This should facilitate the design engineer in use of these models for design. This is in contrast to more cumbersome techniques such as CMS. CMS requires the constraint equations to be written explicitly and then used to couple the system together, or if the degrees of freedom corresponding to attachment points are physical, the matrices representing the components must be assembled to synthesize the system. These processes are mathematically straight forward but provide the designer nothing to help visualize the relationships between components and the system. The bond graph representation of components also conceptually simplifies what is happening in the system model deduction process. This is true due to the accessible visual nature of the graph. Connecting all the bond graph components together to form the system model is trivial. The only reformulation that must occur on each pass through the search algorithm is to add or truncate a section of the component bond graph (which can also be illustrated graphically as in Fig 7). The connection ports of the component models remain unchanged because the attachment points for component bond graphs all represent physical degrees of freedom, even when a modal expansion is used for some components. Thus the trial component bond graph models will have similar “appearance” (see e.g., Fig 9) and can be easily stored and visualized (analyzed). These “familiar” visual images will make using the information easier. In fact, due to the ease of viewing the component models, even using models not selected by MODA might be useful to explore. The accessibility of this information may be very important to the design engineer. Bond graphs also provide additional insight into the meaning of rank. For unbounded components, represented by finite segment models, higher rank simply means dividing the component mass into more segments and separating them by the respective number of springs. The ordering of the rank of a component model is not always this clear. In fact, for a general bond graph of a bounded component (whose maximum possible rank is greater than 1) the rank 1 model of the component cannot be determined a priori. The composition of the rank 1 component model, and all higher rank component models, can vary when the component model is introduced into different systems. However, this issue can be overcome. When it is time to increment the rank of the component model in question (during the formation of the trial system models), each

"unused" energy storage element in the component model is tried in the model, while the rest of the system model remains constant, until the smallest increase in the spectral radius is found. Adding the element in the component in question that causes the smallest increase in the spectral radius of the system model is, thus, the element that should be added when the component model's rank is to be increased. Also rank 0 models have been described as the rigid body model for unbounded components. However, for a general system represented by a bond graph, the definition of the rank zero model is, in general, not obvious and may require some effort to determine. This issue warrants further investigation. Finally, implementation of modal and finite segment component models can be done with the bond graph approach outlined in this paper or the CMS method (Ferris et al. 1994). However, in part, due to the ease with which the bond graph approach allows these different model paradigms to be implemented, Louca and Stein (1995) have written a program implementing the Extend MODA concept in bond graph form. 3.2.2. Modeling/Search Algorithm Interactions. The results of the example presented in this study show that the model approximations used to represent the components (finite segments versus modal approximations) in combination with the search algorithms (MODA versus Extended MODA) have an effect on the deduced proper system model. The results in Figs 10 and 11, demonstrates that the proper system model depends on the model approximation used to represent the components. With Extend MODA the complexity of component #1 is increased whereas with MODA a balance in the complexity required between component #1 and #3 is found. The common trait between the two deduced models is that the rank of component #1 is increased as much, if not more than, all other components. This is caused by the difference between representing a component with a spatial discretization (i.e., a finite segment model) versus a modal discretization (i.e., a modal model). When the rank of a component that is represented by a modal model is incremented (i.e., a mode is added) a new component eigenvalue pair is produced, but the original component eigenvalues do not change. This is in sharp contrast to a finite segment component model. When the rank of a component that is represented by a finite segment model is incremented (e.g., an additional mass-spring pair is added) a new component eigenvalue pair is produced and the original component eigenvalues can change significantly. Because the accuracy criterion taken from Ferris et al. (1994), that is to continue to increase the rank of the components until the critical system eigenvalues have converged, was used in deducing both the MODA and Extended MODA models, it is not surprising that required rank of component #1 is high. The results in Figs 10 and 11 also provide insight into the differences between Extended MODA and MODA. The main conceptual difference between the MODA search algorithm and that of Extended MODA is that MODA deduces the system model by focusing on the largest (fastest, highest frequency) eigenvalues of the trail system model. Extended MODA, on the other hand, evaluates the system based on the CSE, that is, frequency information within the FROI. One consequence of this is how modal and finite segment component representations are expanded during the deduction process. In the example, Extended MODA deduces a model by augmenting the rank of component #1 while MODA deduces a model that weights the ranks of each component approximately uniformly. This is because MODA expands the spectral radius of the system such that no system eigenvalues are "missed", that is at each step (once the CSE have been defined) it only evaluates the largest eigenvalue of the system and does not continue to evaluate the dynamics of the system within the FROI. Because incrementing the rank of both a finite segment component model and a modal model causes an increase in the spectral radius of the system, generally MODA does not favor one form over the other. On the other hand, Extended MODA chooses to augment the rank of components which cause the largest correction in each of the CSE. Because the low frequency eigenvalues of the finite segment models continue to change as more segments are added, Extended MODA naturally favors incrementing the rank of these components.

Extended MODA, was developed by Ferris et al. (1994). A bond graph formulation, due to the clarity of the representation, makes this technique easier for the design engineer to gain “visual” insight into the dominant design factors. The bond graph approach is also shown to increase the ease with which component models, when represented by bond graphs, can be combined. A comparison of results arising from the application of MODA and Extended MODA to a simple example is also presented. The results show that the proper deduced model depends on the type of model (modal versus finite segment) used to represent the components in combination with the search algorithm used to generate it. The study concludes that the bond graph formalism is a powerful way to present proper model deductions concepts and should offer a pathway to solving some remaining problems as well as being a good framework around which to implement these concepts in software. 5. ACKNOWLEDGEMENT The authors gratefully acknowledge Chrysler Corporation’s sponsorship of Mr. John Ferris as a doctoral student at The University of Michigan. 6. REFERENCES Benfield, W.A. and R.F. Hruda. 1971. "Vibration Analysis of Structures by Component Mode Substitution." AIAA Journal, vol. 9, no. 7: 1255-61. Breedveld, P.C. 1988. “Bond-Graph-Based Model Generation.” In1988 ASME Winter Annual Meeting, Proceedings of the Symposium Automated Modeling for Design. (Chicago, Il., Nov.27-Dec.2). ASME Book No. G00460. ASME, New York, NY. Craig, R.R. and M.C.C. Bampton. 1968. "Coupling of Substructures for Dynamic Analysis." AIAA Journal, vol. 6, no. 7: 1313-19. Dowell, E.H. 1972. "Free Vibration of an Arbitrary Structure in Terms of Component Modes." Journal of Applied Mechanics, vol. 39, no. 3: 727-32. Ferris, J.B., J.L. Stein and M.M. Bernitsas. 1994. “Development of Proper Models of Hybrid Systems.” 1994 ASME Winter Annual Meeting, Proceedings of the Symposium Automated Modeling for Design. (Chicago, Il., Nov.7-11). ASME, New York, NY. Gladwell, G.M.L. 1964. "Branch Mode Analysis of Vibrating Systems." J. of Sound and Vibration, vol. 1: 41-59. Goldman, R.L. 1969. "Vibration Analysis by Dynamic Partitioning." AIAA Journal, vol. 7, no. 6: 1152-4. Hou, S.N. 1969. "Review of a Modal Synthesis Technique and a New Approach." Shock and Vibration Bulletin, no. 40, part 4: 25-30. Hurty, W.C. 1965. "Dynamic Analysis of Structural Systems Using Component Modes." AIAA Journal, vol. 3, no. 4: 678-85. Karnopp, D.C.; D.L. Margolis; and R.C. Rosenberg. 1990. System Dynamics: A Unified Approach. John Wiley & Sons, Inc., New York, NY. Margolis, D.L. and D. Edeal. 1989. "Modeling and Control of Large Flexible Frame Vehicles Using Bond Graphs." SAE Trans. 892488: 47-53. Rosenberg, R.C. and D.C. Karnopp. 1983. Introduction to Physical System Dynamics. Mcgraw-Hill, New York, NY. Rosenberg, R.C. and T. Zhou. 1988. "Power-Based Model Insight." In1988 ASME Winter Annual Meeting, Proceedings of the Symposium Automated Modeling for Design. (Chicago, Il., Nov.27-Dec.2). ASME Book No. G00460. ASME, New York, NY. Stein, J.L. and L. Louca. 1995. “A Component Based Modeling Approach for System Design: Theory and Implementation.” In Proceedings of the 1995 International Conference on Bond Graph Modeling. (Las Vegas, NV, Jan.15-18). SCS, San Diego, CA.

4. SUMMARY & CONCLUSIONS

Wilson, B.H. and J.L. Stein. 1993. "Model-Building Assistant: An Automated Modeling Tool for Machine-Tool Drive Systems." In Proceedings of the 1993 International Conference on Bond Graph Modeling and Simulation. (La Jolla, CA, Jan.17-20). SCS, San Diego, CA.

This paper presents a process, Extended MODA (Ferris et al. 1994), by which Proper Models can be derived, via a bond graph formulation. This process uses a combination of two search algorithms; the first, MODA, was developed by Wilson and Stein (1992), the second,

Wilson, B.H. and J.L. Stein. 1992. "An Algorithm for Obtaining Minimum Order Models of Distributed and Discrete Systems." ASME Winter Annual Meeting, Symposium on Automated Modeling. (Anaheim, CA, Nov.8-13). ASME Book No. G00747, ASME, New York, NY.