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This is to certify that the
thesis entitled
User-Guided Reduction of Certain Storage Fields

in Multiport Systems

presented by

John Michael McCalla

has been accepted towards fulfillment
of the requirements for

M o S . degree in M o E o

    

Major professor

Date M

0.7 639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

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Michigan State
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TO AVOID FINES return on or before date due.

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MSU is An Affirmative Action/Equal Opportunity Institution
WW1

 

 

USER-GUIDED REDUCTION OF CERTAIN STORAGE FIELDS
IN MULTIPORT SYSTEMS

By

John Michael McCalla

ATHESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1991

555- 77:»

ABSTRACT

USER-GUIDED REDUCTION OF CERTAIN STORAGE FIELDS
IN MULTIPORT SYSTEMS

By

John Michael McCalla

A methodology is developed which replaces linear, dependent storage
elements in a bond graph with an equivalent element. This
substitution, which utilizes a causal path variable trace, restores
integral causality to the bond graph. The procedure reduces the
number of energy variables, thereby simplifying the system model,
and facilitates the formulation of explicit state equations. This
algorithm is implemented in the ENPORT software and applied to
several examples. Simulation results demonstrate a new option for

the engineer in designing certain types of systems.

for Mom and Dad

ACKINDWLEEXSEMENTS

Thanks to Dr. Rosenberg, whose guidance kept me inspired.

iv

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF SYMBOLS

Chapter 1. Introduction

Chapter 2. Identification of Derivative Causality

2.1 Identifying Derivative Causality
2.2 Solution Options for Derivative Causality

Chapter 3. The Storage Field Reduction Technique
3.1 Identifying a Storage Field
3.2 The Reduction Technique
3.3 Example Problems
Chapter 4. Algorithm implementation
4.1 Design of Implementation
4.2 Examples of Use
4.3 Description of Software

Chapter 5. Conclusions

5.1 Summary
5.2 Recommendations for Further Development

APPENDIXA Dynamic Augmentation of Roller Coaster Model
APPENDIXB Dynamic Augmentation of Robot Arm Model
APPENDIXC The FORTRAN Algorithm

LIST OF REFERENCES

vi

vii

11
16

19
19
21
26
28

28
29

31

33

36

70

Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
Figure 9.
Figure 10.
Figure 11.
Figure 12.
Figure 13.
Figure 14.
Figure 15.
Figure A-1.
Figure 8-1.

LIST OF FIGURES

Derivative and Integral Forms of C and I elements
Bond Graph of Roller Coaster Model

C-Subsystem Required for Dynamic Augmentation
Augmented Bond Graph Model

Inertial Storage Field

Bond Graph Model with Derivative Causality
Reduced Bond Graph Model

Bond Graph Model with Derivative Causality

Flow Chart of Methodology

Lever Arm System

Bond Graph Model of Lever Arm System

Reduced Form of Bond Graph Model

Bond Graph of Field with GY Element

Reduced Form of Bond Graph

Flow Chart of Subroutines

Roller Coaster Model

Robot Arm in Vertical Plane

vi

20

21

22

24

24

25

27

31

34

LIST OF FIGURES, CONTINUED

Figure 8-2. Bond Graph Model of Robot Arm

Figure 33. Augmented Bond Graph Model of Robot Arm

vii

35

36

d/dt:

get

913

LIST OF SYMBOLS

compliance

time derivative
effort (force)
flow (velocity)
efion ganr

flour gann

inertia coefficient
mass

momentum
displacement

time

viii

Chapter 1. Introduction

One purpose of constructing a bond graph is to obtain a set of
differential equations that describe the motion of a physical system.
Through the use of power bonds and elements that represent various
pieces of a physical system, an engineer can construct a bond graph
model that closely represents an actual system. State equations for
the system can be generated by using a straightforward procedure

that follows the details of the bond graph.

This methodology produces explicit state equations, x=E(x,t), when
the bond graph model contains only integral causality. Implicit,

24mm). state equations arise when the model contains derivative

causality. Causality provides this information through its inherent
representation of the constitutive relationships for each energy
variable. These energy variables are identified through energy
storing elements inertia and capacitance. Thus, even if the model
contains only one energy variable with derivative causality, the
entire system of state equations will be implicit and the solution of
the equations will require a robust, implicit integrator. For a
discussion of causality in bond graphs see, 9.9., Rosenberg and

Karnopp. [1]

A methodology has been developed in which a storage field reduction
takes place. The technique entails following a causal path that

leads to the source of the derivative causality for the dependent

2

energy variable. This methodology reduces the number of energy
variables by combining the effects of the dependent variable with an
independent variable. Thus the technique restores integral causality
to the bond graph and allows an explicit integrator to solve the
system of equations. This reduction decreases the complexity of the
system state equations by providing only those energy variables that

are state variables.

This project implemented the storage field reduction technique as
an algorithm in the dynamic software simulation package ENPORT,
presented by ROSENCDDE Associates in [2]. The user of this package
now has the option to allow the program to make an attempt at
solving any derivative causality problem. The algorithm, which will
be presented in Chapter 3, determines if a particular model can be
corrected using this storage field reduction technique. Once deemed
eligible, the bond graph is adjusted in a user-interactive manner.
The experienced system modeler will find this option beneficial due
to the luxury of rapid equivalent coefficient computation by the

algorithm.

The details of the algorithm from initial development stages to the
final process, its implementation in ENPORT, and several examples
are presented in Chapters 2, 3, and 4. Finally, Chapter 5 summarizes
the methodology and gives a set of recommendations for further

development.

Chapter 2. Identification of Derivative Causality

2.1 Identifying Derivative Causality

Derivative causality presents dependent energy storing variables to
the engineer attempting to write a system of state equations. The
system equations cannot be reduced to explicit state-space form, in
general, when derivative causality is present in the model.
Identifying derivative causality is thereby the initial step in
forming explicit state equations. This identification can take place
after causality has been placed on the bond graph model according to
the Sequential Causality Assignment Procedure (SCAP). [1] The
number of energy storing variables can be found at this point by
noting the number of energy storing elements in the model, assuming
they are all 1-port. The dependent energy variables are evidenced by
the causal stroke on the bond connecting the C or i element to the
model. Figure 1 shows the derivative and integral forms for both the

inertial and the capacitance elements.

 

 

Integral Derivative

Figure 1. Derivative and Integral Forms of C and l elements

An example of a system model with derivative causality is shown in

Figure 2, which is a bond graph model of a roller coaster originally

4

studied by Emery. [3] The physical system model is shown in Figure
A-1, Appendix A.

35—41. l—\ TF l—AIB
(F9) | (GWUX) '1;

l1 '2
[m1] ["121

Figure 2. Bond Graph of Roller Coaster Model

Note the derivative causality resulting on the l2 element. The
formulation of state equations using this bond graph model would
produce a system of nonlinear, implicit equations. The general form
of the state equations arising from both integral and derivative
causality will be detailed in the remaining portions of this section,

where we restrict our attention to linear elements as noted.

The constitutive law for a linear capacitance element in a bond
graph always relates effort (e.g., force) to displacement (e.g.,
deflection). Displacement is the integral of flow (e.g., velocity).
The constitutive equation for a linear capacitance element with

integral causality is as follows:

Itfdt
=7 “I

Olin

4

studied by Emery. [3] The physical system model is shown in Figure
A-1, Appendix A.

SEA 1A I"_'3 TF I—AIB
(Pg) I (dy/dx) I:

n '2
[m1] [m2]

Figure 2. Bond Graph of Roller Coaster Model

Note the derivative causality resulting on the l2 element. The
formulation of state equations using this bond graph model would
produce a system of nonlinear, implicit equations. The general form
of the state equations arising from both integral and derivative
causality will be detailed in the remaining portions of this section,

where we restrict our attention to linear elements as noted.

The constitutive law for a linear capacitance element in a bond
graph always relates effort (e.g., force) to displacement (e.g.,
deflection). Displacement is the integral of flow (e.g., velocity).
The constitutive equation for a linear capacitance element with

integral causality is as follows:

tfdt
e=S=——— u)
C C

5

However, the constitutive equation for a C element with derivative

causality is quite different, namely:

(1 (1 de
f=— =—c =c— 2
<11;q (it 6 dt ( )
Note that these two sets of equations are simply two ways of
expressing the same constitutive law. Each causality indicates how
to use the relation and whether integration or differentiation is

implied.

The same ideas apply to the linear inertial element. The inertial
element relates flow to momentum, and momentum is the time
integral of effort. The constitutive equation for a linear inertial

element with integral causality is as follows:

Itedt
I

 

f=3
I

(3)

The constitutive equation for an inertial element with derivative

causality is shown below:
d
e=—p=—If=I—— (4)

These two sets of expressions represent the same constitutive law

in different forms.

2.2 Solution Options for Derivative Causality
When derivative causality arises in a model to be simulated, three
solution options exist:

1. use robust, implicit integrator

2. use Dynamic Augmentation

3. use Storage Field Reduction
The first option entails finding a robust, implicit integrator which
would provide a means for numerically solving the implicit system
of differential equations. This option is an issue involving
numerical integration methodology and software implementation It

will not be discussed in further detail in this work.

Dynamic augmentation of a bond graph model involves the
introduction of additional energy variables to facilitate dynamic
independence of all the energy variables. For some examples in
mechanical systems see [5], [6], and [7]. This procedure often
requires the insertion of capacitance elements into the bond graph

model. These insertions take the form shown in Figure 3.

 

c/ Io
Figure 3. C-Subsystem Required for Dynamic Augmentation

The insertion of these additional elements, and the state variables
they represent, takes place at strategic points in the model. The
capacitance elements represent stiff springs in mechanical systems

and are inserted at connection points in the model. Thus, the

6

7

coefficients for these elements are large, k - 1/C, in order to
represent actual local material stiffness. The ideal location for

insertion of the element remains a model specific issue.

The dynamic augmentation methodology provides explicit state
equations without eliminating any of the energy variables which may
be of interest. However, the computational problems associated
with dynamic augmentation are significant. A variable step
integrator capable of dealing with stiff systems is required for
numerical solution. The inserted variables represent the error
associated with this modelling technique and provide insight into

error control during simulation.

Dynamic augmentation can be applied to the roller coaster bond
graph model shown in Figure 2. The augmented model is shown in

Figure 4.

eel 1A l——>o ATP—41.

(F9) ‘1; (dy/dx)
l1 C3 '2
[m1] [m2]

Figure 4. Augmented Bond Graph Model

Note the elimination of derivative causality with the insertion of
the stiff spring, CS. Further discussion of this model may be found

in Appendix A. An additional example of the dynamic augmentation

8

procedure can be found in Appendix B, which details the development

of a robot arm in the vertical plane.

The storage field reduction of a bond graph is the third option for
dealing with derivative causality. The application of causality, the
crucial step in the construction of a bond graph, leaves a causal
path. This path is used to locate the source of derivative causality
and to eliminate it from the bond graph. This methodology provides
explicit state equations by reducing the effects of a dependent
energy variable into the parameter of a independent energy variable.
Nonlinear and linear storage fields are both conceivably eligible for
this reduction. The remaining chapters of this thesis will develop
an algorithm for linear storage fields whose derivative causality is
a direct result of the application of integral causality on a one port

inertial or capacitance element during SCAP.

Chapter 3. The Storage Field Reduction Technique

3.1 Identifying a Storage Field

The causal path provides a means for the elimination of derivative
causality in a bond graph model. During the second phase of SCAP
integral causality may be assigned to either a capacitance or
inertial element. This random application may result in the
assignment of derivative causality to another I or C element during
the extend portion of this phase. This particular problem can be
corrected by following the path created during the causal extension

process.

The identification of a storage field is the initial step in the
reduction technique. Once derivative causality has been identified
on a linear, 1-port inertial or capacitance element, the causal path
trace can begin. Consider a derivative inertial element and its
potential causal path. The inertial causal trace follows a flow path.
If the flow path enters a zero junction with more than two ports,
the exiting flow path is not unique and the trace cannot continue.
The dependent energy storing variable is a function of more than one

energy and/or source variable if this split path situation occurs.

The elements through which the causal path can travel include any
number of one junctions, zero junctions, and linear, constant
coefficient transformers. The inertial causal path trace leads to
either an independent energy storing variable or to a source variable.

If the path leads to a linear, independent, 1-port inertial element,
9

10

the storage field has been identified. However, if the causal path
leads to a source variable the technique terminates because the
dependent energy variable is not solely a function of an independent
energy variable and cannot be reduced by our algorithm. Once the

field has been identified, the reduction can commence.

Similarly, if a capacitance derivative element exists in the model
the causal trace follows an effort path. if the effort causal path
enters a one junction with more than two ports this trace
terminates due to the non-unique exiting path. The dependent
energy variable is a function of more than one energy and/or source
variable and cannot be reduced using this technique when this split
path occurs. The elements eligible to be on the causal path are the
same as for the inertial element's trace. If the causal path leads to
a linear, independent, 1-port capacitance element, the storage field

has been identified and reduction can commence.

The reduction of an inertial element into a capacitance element, or
vice versa, can occur only if the causal path between these two
elements contains an odd number of linear, constant coefficient
gyrators. The unique relationships of the gyrator make this
combination possible by reversing the causal path trace variable (9
or f) through the element and allowing for the combination of mass

and stiffness parameters.

3.2 The Reduction Technique

The storage field reduction derives from consideration of the
system equations. The coefficients of the elements along the causal
path will represent these equations in a reduced form. The
development of the relations will utilize inertia elements for the
energy variables as an illustration. Equivalent relationships for the
capacitance elements will be discussed after the initial

development. Consider the details shown in Figure 5.

causal path

|———l-——I——

l1 l2
v1=(1/m1)p1 pz=(m2)vz

Figure 5. Inertial Storage Field

Following the causal path entails traversing through the different
types of elements mentioned in section 3.1. The coefficients of
these elements are recorded for this procedure in order for the
reduction to account for all the system relationships. These
coefficients are represented on either an effort gain, ge, or a flow
gain, gf. Thus, by noting these coefficients we can reduce the
dependent energy variable's effects into the independent energy

variable. For the example in Figure 5, the total expression for e1

yields equation (5) on the following page.

11

12
e1=612+e1r (5)

The effort, e12, is the contribution due to l2 and an is the

contribution of the rest of the system. We can represent 912 as

shown in equation (6).
912 =(g,)i>2 (6)

The effort gain, ge, represents the coefficients obtained from the
effort path from l1 to l2. The integration of equation (6) yields

equation (6a).

912 - (geipz (6a)
Substituting the constitutive relationship for l2, equation (7) into
equation (6a) we find (8).

92 - Mafia (7)

p12 - (gem2)f2 (3)
Equation (9) represents the principle upon which the identification
of unique causal path is based. Notice that the dependent energy
variable, f2, is a function of only one independent energy variable,
f1. The causal flow path does not branch at any point in the trace.

The flow gain, gf, represents the coefficients obtained on the flow

path from l2 to l1.

f2 = (crib (9)

Substitution of equation (9) into equation (8) yields (10).

13
me = (gem291)f2 (10)

Equations (11-13) detail the integration and substitution steps
necessary for the computation of the equivalent parameter. This
computation requires a choice of element for reduction, and for the
sake of example l2 will be reduced. The integration of equation (5),
disregarding the effort associated with the rest of the model

because it plays no role in the reduction process, yields (11).

Pt=I(eu+eu)dt=Ieudt+Pu (11)

Substitution of equation (10) into equation (11) yields (12).
p1=JIekdt+(gengf)fl (12)

Substituting the constitutive relationship for l1 into equation (12)
yields (13).

mlfi—(9.ngJfi=Ia.dt (13)

Rearranging equation (13) yields the equivalent relationship (14).
(ml-gengf)fl=leudt (14)

Since the effort and flow gains can differ by a sign at most, the
recording of one of these values is sufficient for storage field
reduction. If the signs of the gains are opposite, the reduction
equations will introduce a negative sign to ensure that the resulting
equivalent coefficient is always positive, as in (14). Thus, the

equivalent inertial parameter is represented in (15).

14

mmfl=Ieudt (15)

The relationships for the causal path between capacitance elements
are similar. In this case, the effort relationship comparable to (9)
is the defining equation. If the dependent capacitance variable is a
function of only one other independent capacitance energy variable,
the reduction can take place. This reduction will yield an equivalent
stiffness parameter which can be inserted into the selected

capacitance energy variable.

The relationships for the causal path between an inertial element
and a capacitance element are similar. The odd number of gyrator
elements required for this combination of energy variables ensures
that the equivalent coefficient will have the correct form. Since
either energy variable may remain in the model, the element
reduction choice will determine the form of the equivalent

parameter, inertial or capacitance.

The effort and flow gains warrant a brief explanation. When the
causal path travels through a power conserving zero or one junction,
the power directions must be examined. Through a one junction, the
flow gain will have an identity relationship, while the effort gain
may have a sign change. Similarly, through a zero junction the flow
gain may have a sign change, while the effort gain will have an
identity relationship. Thus, the source of possible negative signs in
the equivalence calculation has been identified. Similarly, if a

linear, constant coefficient transformer is entered on the path, the

15

coefficient recorded is the transformer modulus. When the path
enters a linear, constant coefficient gyrator the recorded

coefficient is the gyrator modulus.

The storage field reduction algorithm, shown in the following

sequence, presents the ideas detailed in a systematic manner.

1. Allow causality to be assigned by SCAP. General non-linear

models are permissible at this point.

2 Identify the existence of derivative causality. Allow
simulation software to formulate implicit state equations for

models with derivative causality.

3. Trace the specific structure of each storage field, with the
result being complete (causal path is unique) or incomplete

(causal path is not unique).

4. For complete, linear storage fields:

calculate the causal path gains.
5. Select the location for the equivalent node.

6. Express the constitutive equation in terms of a stiffness or

mass parameter.

7. Form the modified model with a single storage element and

proper equivalence parameter.

8. Eliminate unnecessary elements from the model.

16

The elimination of unnecessary elements in the reduced bond graph
is a critical final step in the process. This elimination technique
uses the causal path to determine if there are elements in the bond
graph that serve no purpose. These elements include: a one junction,
zero junction, transformer or gyrator with only one bond. If found,
the process eliminates these elements, thus reducing unnecessary
steps in the equation formulation procedure. Once this elimination
is complete the process can repeat in order to reduce other sources
of derivative causality. If all storage fields containing derivative
causality can be reduced, then explicit state equations can be

formed for the entire system.

3.3 Example Problems

This technique can be applied to a wide range of physical system
models, including non-linear models which contain linear storage
fields. Perhaps the simplest case is that of two masses travelling
at the same velocity. Consider the portion of a bond graph model
shown in Figure 6, which contains an element (l2) with derivative

causality.

SE——>|1|—A

3?? > I”

(ma) (mb)

 

 

Figure 6. Bond Graph Model with Derivative Causality

17
The flow path from l1 to l2 yields (13).

The effort path between l1 and l2 yields (14).
99 = -1.0 (14)

If we select l2 for reduction, the equivalent parameter is calculated

as shown in equation (15).
meq a ma - (1.0)(mb)(-1.0) (15)

Thus, the equivalent parameter is: ma + mb. The model can now be

represented as shown in Figure 7.

se—a\I1I——\

l1
(ma+mb)

Figure 7. Reduced Bond Graph Model.

This simple example provides an illustration of the storage field
reduction technique. Consider the portion of a bond graph shown in
Figure 8. This example shows a derivative C element whose initial
effort path leads directly into a one junction that has two exiting

effort paths.

Figure 8. Bond Graph Model with Derivative Causality.

The trace would end at this point due to the ambiguity of the exiting
effort path. The solution to this type of problem would entail a
matrix reduction operation and is discussed in the recommendations

section in Chapter 5.

Chapter 4. Algorithm Implementation

4.1 Design of Implementation

This methodology has been implemented into the ENPORT software
simulation package. The form of the algorithm used to develop the
FORTRAN code is compact and takes advantage of the processes
discussed previously. These processes reduce the required storage
space and increase the speed of calculation of the equivalent
coefficient. The algorithm checks on the linearity of each element
in the trace by examining the form of each coefficient it encounters.
The computer implemented algorithm incorporates the identification
process into the actual reduction by noting the coefficients along
the initial examination of the path. The presentation of this process

is shown in Figure 9 on the following page.

The causal path trace begins at the derivative element and continues
through to the independent energy storing element. The computer
implemented algorithm requires that the linear coefficients
associated with each element passed during the trace be noted.
Several options are employed by the algorithm inserted in the
software. The process allows the user to select the element that is
to be reduced. Default is the derivative element originally found by
the algorithm. Additionally, the user can provide the form of the
equation for the modified element. Default is the original form of
the equation for the unaltered element. The software provides a
listing of the elements in the causal path strictly for user

information. If a model has multiple elements with derivative
19

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21

causality, the process is repeated until all eligible forms of

derivative causality have been reduced.

Upon encountering an ineligible storage field with derivative
causality the software determines under what class of problem this
particular storage field falls and provides the user with a message
to that effect. This identification tool was installed in the hopes
that future software development will utilize the identification
process already encoded in this portion of the program. The topic of

further research will be discussed in Chapter 5, section 5.2.

4.2 Examples of Use

The following physical system model can be described in bond graph
form. As the illustration shows, the system consists of two point
masses attached at either ends of a massless bar which is pivoted in
the middle.

a b

l<—>l<——>l

m1I Im2

74>

 

Figure 10. Lever Arm System

The bond graph of this system is shown in Figure 11. Derivative

causality results on the l2 element as a direct result of the

22

assignment of integral causality on the l1 element. This model is

eligible for the storage field reduction technique.

SE —l—>|;A |-3—\ TF I—‘I—X 13 (3)4000

‘L (b/a) EL

I1 l2
(m1) (m2)

Figure 11. Bond Graph Model of Lever Arm System

The flow path from l2 to 11 yields equation (16).

f6=(b/a)f2 (16)
The effort path from l1 to I2 yields equation (17)

e2=-(b/a)e,3 (17)
The equivalent parameter is shown in equation (18).

m..=m.+(b/a)’m. (18)
The text on the following page shows the exact format used by the

ENPORT software package when reducing a storage field in a bond
graph model.

23

*** The number of dependent C and I ports is:

L: List the current equations

M: Modify the equations

S: Show function types available

D: list nodes with Default equations
T: list/modify named_parameTers
C: Causality display or assignment
A: Alleviate derivative causality
P: Proceed to the solver menu

R: Return to the main menu

H: Help

Enter option (P):A

The path contains the following elements:

I2 13 TF 1A l1

Which element would you like to reduce?
l2 or l1
Enter selection (l2 ):
Hit <return> to continue...
Select equation type for l1
(GAIN ):

Trace is complete

1.

As the text shows, the algorithm alleviates the derivative causality

in a user interactive manner. The reduced bond graph model is shown

in Figure 12. Note the absence of derivative causality in the model;

l2 has been reduced.

24

SE 1—>|;A |i> TF |-3’—\ 1B (340(k)

—L (b/a)

I1
[m.+(b/a)’m.]

Figure 12. Reduced Form of Bond Graph Model

Equation development can now proceed to yield a system of explicit

state equations because no derivative causality exists in the model.

An example bond graph is shown in Figure 13. Derivative causality
results on the C1 element as a direct result of the assignment of

integral causality on the It element.

SE —1—>|1A (53—3 or A 03
If I:

Figure 13. Bond Graph of Field with GY Element

The algorithm eliminates the derivative causality in a user
interactive manner. The reduced bond graph model is shown in
Figure 14. Note the absence of derivative causality in the model.

The linear spring, C1, has been reduced.

25
SE —-1—>|1A2

I

I1
[m +(r2 /k)]

Figure 14. Reduced Form of Bond Graph

The unnecessary gyrator and O junction have been eliminated from
the bond graph model as they now serve no purpose and equation
development will now yield a system of explicit state equations.
These examples provide insight into the basic principles involved in

the reduction of linear storage fields with derivative causality.

4.3 Description of Software

The FORTRAN algorithm included in Appendix C contains six
subroutines: TBACK, INIT, TRACE, REDUCE, ELIM, and REASSIGN. The
controlling subroutine, TBACK, calls INIT, TRACE, and REDUCE. INIT
and TRACE make no subroutine calls, while REDUCE calls ELIM and
REASSIGN. When the controlling subroutine TBACK is called, it
identifies derivative elements and calls INIT. lNlT simply
initializes the variables for each causal path trace and returns to
TBACK. The controlling routine, TBACK, then calls TRACE. This
subroutine, which follows the causal path trace, records the

corresponding coefficients and returns to TBACK.

TBACK calls REDUCE for the reduction portion of the algorithm.
REDUCE asks the user for the element to be reduced and the form of
the equivalent element equation. The REDUCE subroutine then calls
ELIM. The elimination routine, ELIM, removes all unnecessary
elements from the bond graph and returns to REDUCE. The storage
field reduction routine, REDUCE, then calls REASSIGN to align
equation numbers after the removal of elements from the bond graph.
This reorganization is necessary for the proper storage of equations
and their coefficients. REASSIGN returns to REDUCE when this
process is complete. REDUCE then returns to TBACK and the method

is repeated for other linear storage fields.

Figure 15, shown on the following page, presents a flow chart of the

subroutines.

26

27

W

TBACK

— INIT
— TRACE
_— REDUCE

I: W
REASSIGN

Figure 15. Flow Chart of Subroutines

 

Chapter 5. Conclusions

5.1 Summary

The process detailed in Chapters 3 and 4 alleviates derivative
causality in certain linear storage fields by tracing a causal path
from the dependent variable to the independent energy storing
variable that forced the original application of the derivative
causality. Through the use of a straightforward algorithm, the
methodology records the coefficients of the elements in the path in
order to calculate an equivalence parameter. This equivalence
parameter is applied to the user selected element and the reduced
element can then be removed from the bond graph. Thus, integral
causality has been restored to the bond graph while keeping the
effects of the derivative element encoded in the equations for the
model. Equation formulation will thus yield a system of explicit

state equations.

The methodology was derived with the intent of assisting engineers
with their derivative causality problems. The causal path technique,
although presented in a different manner, has been detailed
previously in [1]. The form of the presentation of this technique and
its implementation into ENPORT represented the main efforts in this
project. The process enables the program to determine which bond
graph modes are eligible for a direct explicit computational solution
and which models need to be reduced before any explicit computation

can take place.

28

29

The reduction takes place in a user interactive manner, with the
user choosing the variable to be reduced along with the form of the
equation for the equivalent energy variable. This interactive
process serves not only as a means of alleviating derivative
causality in a linear storage field, but also as an enhancement of the
instructional powers of the ENPORT software package. This
enhancement will serve as a rapid equivalence calculation technique.
Thus, ENPORT users can rely on the software to calculate their

linear equivalence parameters quickly and accurately.

5.2 Recommendations for Further Development

This project began with the intent of alleviating derivative
causality in all its forms, including those that arise from non-linear
elements. The range of the project was narrowed to its existing
form when the details of dynamic augmentation were examined.
Although dynamic augmentation yielded insight, it encompassed a
range of problems much larger than originally anticipated. It is
recommended that this procedure be examined in greater detail. The
development of a set of standard rules for the insertion of

capacitance elements would then be in order.

The extension of the storage field reduction technique to include the
Split path problems is recommended. This development should yield
a matrix reduction technique which will enable the software to
eliminate those dependent energy variables which are functions of
more than one independent energy variable. It is hoped that this new

process will take advantage of the enclosed method of identification

30

and the model reduction algorithm's ability to correct those forms

of derivative causality which are already eligible for consideration.

Along the same line, the zero eigenvalue case may be eligible for a
storage field reduction. This class of problems involves models
with one stationary real mode. This stationary energy variable could
be reduced into another energy variable and the equations for the
model would be of lower order. This type of relationship arises for
two capacitance elements on a one junction. As long as the initial
displacements of the springs are equal, the effects of one element
may be incorporated into the other without losing any dynamic

information.

The nonlinear storage field may best be studied through a
comparison of the dynamic augmentation method with a direct
numerical solution based on implicit integration. It is not known
which of these two solutions would yield the most favorable results
generally. The final judgement should be made by the system

modeler.

APPENDICES

APPENDIX A

DYNAMIC AUGMENTATION OF ROLLER COASTER MODEL

The roller coaster system model was based on a simulation study
begun by Scott Emery, an MSU graduate student in Mechanical
Engineering. [3] This system introduces derivative causality to the
bond graph when attempts are made to force the coaster to follow
the track. Integral causality was restored to the system model
through dynamic augmentation; the insertion of a 0 junction with a
stiff spring between the vertical velocity 1 junction and the
transformer that represented the dy/dx connection point. The
modified system model was then examined using ENPORT and the
results compare favorably with theory, i.e., the coaster followed the
track and the velocity and acceleration profiles were as expected.

The physical model follows in Figure A-1.

31

32

 

 

Figure A-1. Roller Coaster Model

The bond graph is shown in Figure 2 section 2.1. The derivative
causality is reduced by the insertion of a stiff spring element at the
connection point between the point mass and the path. The

augmented bond graph is shown in Figure 4, section 2.2.

APPENDIX B

DYNAMIC AUGMENTATION OF ROBOT ARM MODEL

The robot arm system was based on the work presented by Tarokh in
[4]. Figure 8-1 shows the robot arm free to move in the vertical

plane.

//

 

 

Figure 8-1. Robot Arm in Vertical Plane
The bond graph of this model is shown in Figure 32 on the following

page. The model yielded derivative causality when the pinned body

and the unconstrained body were attached.

33

34

 

12
SE4
1 V
C1 MTFS MTF8
MTF2 _L
@[Mrra \ 1 MTF6 1
13

MTF7 {—3 1
MTF1 '1 )\/\I I4

I 853°»?

SE2

 

Figure B-2. Bond Graph Model of Robot Arm

The causality was corrected by dynamic augmentation; the insertion
of stiff spring elements, used to represent the connection point of
the two arms. One spring was inserted for the vertical velocity
component and one stiff spring for the horizontal velocity. The

augmented bond graph is shown in Figure B—3 on the following page.

 

 

 

C1
MTF2
1 P6 MTF3:
SE1
MTF1 I1
I )“o y
352 SE3 R2

Figure 18. Augmented Bond Graph Model of Robot Arm

This dynamic augmentation methodology was discovered in Zeid's
work [5]. The substitution served to alleviate the derivative
causality, as Figure 8-3 shows, and facilitate the formulation of
state equations. The modified system model was then examined
using ENPORT and the simulation results compared favorably with

those available in the literature.

APPENDIX C

THE FORTRAN ALGORITHM

CTBACK>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Last Change: 03/09/91

C
C

SUBROUTINE TBACK

C
C

C****t'hititiii'iiiiiii*********************************

C

C VARIABLE LIST:

0000000000OOOOOOOOOOOOOOO
5

CURRENT CONNECTOR NUMBER IN TRACE
CURRENT ELEMENT IN TRACE

CHECKS FOR MULTIPLE PATHS

CHECKS ID OF PATH

NUMBER OF TRACE PATHS IN MODEL

PATH NUMBER PASSED TO REDUCE

DERIVATIVE ELEMENT

EXIT SIGNAL (TRACE FAILED IF FLAG-TRUE)
CHARACTER NAME OF FUNCTION ASSOCIATED WITH IEQ
POINTER FOR CONNECTOR LIST

POINTER FOR CONNECTOR LIST

PATH IDENTIFICATION

ARRAY STORING NUMBER OF ELEMENTS IN EACH PATH
EQUATION NUMBER ASSOCIATED WITH CURRENT
E.EMENT

INTEGER FUNCTION TYPE OF IEO

mDE TYPE

PREVIOUS ELEMENT IN TRACE

ARRAY CONTAINING ELEMENT NUMBERS IN TRACE
PRIMARY COEFFICIENT

PRIMARY PATH TRACE

SECONDARY COEFFICIENT

SECONDARY PATH TRACE

36

37

Ci'itiiiii'*****i'iiitfiiiii'iiii'*tttiii’iifiit**************

C

C
INTEGER CB,DE,CE,OE,IEO,IFI'P,CT,ID2(5)
INTEGER 11 ,l2,CNT,CH1 ,CH2,ID(5),PATH(5,20)
DOUBLE PRECISION PC(5),SC(5)
CHARACTER PPT*1,SPT*1,NTYPE*4,FNAM*4
LOGICAL FLAG

00

INCLUDE 'SIZEBK.CBK'
INCLUDE 'GREDBK.CBK‘
INCLUDE 'FUNCBK.CBK'
INCLUDE 'UTILBK.CBK'

EXTERNAL BLNKLN,PROMPT,WRTSTR,NEWL

*iTBACKiiitiiiiii******i*****************ifiifitiiiiti

...IN|TIALIZE COUNTER FOR NUMBER OF PATHS

"OOOOOO 00

0 CONTINUE
CNT a 1

O

C...USE CAUSAL PATH SUBROUTINES TO IDENTIFY THE DERIVATIVE

C CAUSALITY AND, WHERE LINEAR ELEMENTS EXIST, CALCULATE

c COEFFICIENTS AND ELIMINATE DERIVATIVE CAUSALITY FROM

MODEL

0

C
DO 2000 1-1,1NELS

c

C...IDENTIFY ELEMENT TYPE

C
NTYPE .. NBXTP(I)(2:2)

c

C...1 ELEMENT

C
IF (NTYPE.EO.'I') THEN

C

C

...OBTAIN THIS ELEMENTS CONNECTOR(S) AND ASSIGN CB

090

090

090 090

0090

C

38

I1 - IBDPTRU)
I2 . IBDPTR(I+1)-1

..IF MULTIPORT ELEMENT FOUND SKIP THIS PATH

IF (l1.NE.|2) THEN
CALL BLNKLN
CALL WRTSTR(' A multiport l element was found')
CALL BLNKLN
GOTO 1900
ENDIF
CB - IBDSEL(I1)

..CHECK CAUSALI'IY FOR I ELEMENT

IF (BNDCAU(CB).NE.I) THEN
DE - I

..FIND EQUATION NUMBER

IEO - INEQP(I)

..FIND EQUATION TYPE

IH’P - IFCNTP(IEQ)
FNAM - FNNAM1(IFTP)

..IF EQUATION TYPE IS NOT ATTI'ENUATE OR GAIN SKIP THIS PATH

IF EQUATION IS OK, START TRACE

IF ((FNAM.EQ'GAIN').OR.(FNAM.EQ.'A1T)) THEN
FLAG - .FALSE.
ELSE
CALL BLNKLN
CALL WRTSTR(' A nonlinear I element was found')
CALL BLNKLN
GOTO 1900
ENDIF

.INITIALIZE FOR TRACE

CALL INIT(DE,PPT,SPT,PC,SC,CNT)

090

090

090

090 090

090 090 no

39

..RUN TRACE AND CHECK FOR PUNTS

OE - DE

CALL TRACE(CB,PPT,SPT,PC,SC,OE,FLAG,PATH,CNT,ID,IDZ)
IF (FLAG.EQ..TRUE.) GOTO 1900

CNT - CNT + 1

ENDIF

..C ELEMENT

ELSE IF (NTYPE.EQ.'C') THEN

..OBTAIN THIS ELEMENTS CONNECTOR(S)

I1 - IBDPTRII)
l2 - IBDPTR(I+1)-1

..IF MULTIPORT ELEMENT FOUND SKIP THIS PATH

IF (l1.NE.|2) THEN
CALL BLNKLN
CALL WRTSTR(' A multiport C element was found')
CALL BLNKLN
GOTO 1900
ENDIF
CB - IBDSEL(I1)

..CHECK CAUSALITY FOR C ELEMENT

IF (BNDCAU(CB).EQ.I) THEN
DE - I

..FIND EQUATION NUMBER

IEO - INEQP(I)

..FIND EQUATION TYPE

IFTP - IFCNTP(IEQ)
FNAM - FNNAMI(IFrP)

40

...IF EQUATION IS NOT ATTENUATE OR GAIN -- PUNT!
IF EQUATION IS OK, START TRACE

0000

IF ((FNAM.EQ.'GAIN').OR.(FNAM.EQ.‘ATT')) THEN
FLAG - .FALSE.
ELSE
CALL BLNKLN
CALL WRTSTFIC A nonlinear C element was found')
CALL BLNKLN
GOTO 1900
ENDIF

...INITIALIZE FOR TRACE
CALL IN|T(DE,PPT,SPT,PC,SC,CNT)

...RUN TRACE AND CHECK FOR PUNTS

000 000

OE = DE

COUNT s 1

CALL TRACE(CB,PPT,SPT,PC,SC,OE,FLAG,PATH,CNT,ID,IDZ)
IF (FLAG.EQ..TRUE.) GOTO 1900

CNT - CNT + 1 ‘

ENDIF

ENDIF

0000

—L

900 CONTINUE

NO

000 CONTINUE

...EXIT IF MODEL HAS NOT BEEN CHANGED

000

IF (CNT.EQ.1) THEN
CALL BLNKLN
CALL WRTSTRC No derivative causality was found')
CALL BLNKLN
RETURN
ENDIF

41

C...CHECK FOR REDUCTION POSSIBILITIES
C
CH1 - 1
DO 2100 I-1,CNT-1
CH2 - ID(I)
IF ((CH1.EO.1).AND.(CH2.EQ.1)) THEN
CT - I
CALL REDUCE(PATH,PC,SC,CT,IDZ,FLAG)
CH1 - 2
ELSE IF (CH2.EQ.2) THEN
CALL BLNKLN
CALL WRTSTRC Split path found...')
CALL BLNKLN
ELSE IF (CH2.EO.3) THEN
CALL BLNKLN
CALL WRTSTRC Nonlinear element in path...')
CALL BLNKLN
ENDIF
2100 CONTINUE
IF ((CH1.EO.2).AND.(CNT.GT.2)) GOTO 10

C
C...DONE
C
CALL BLNKLN
CALL WRTSTRC Trace is complete ')
CALL BLNKLN
CALL PROMPTC Press <return> key to continue...')
CALL NEWL
CALL BLNKLN
RETURN
ED
C
C>>>>>
C
C
C

CINIT>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Last Change: 03/09/91

SUBROUTINE INIT(DE,PPT,SPT,PC,SC,CNT)

00 00

42

C*********tittiti************tifiitiiiiititit*iitittttt

C
C VARIABLE LIST:
C
C CNT: NUMBER OF PATHS IN MODEL
C DE: DERIVATIVE ELEMENT
C N'IYPE: ELEMENT TYPE
C PC: PRIMARY COEFFICIENT
C PPT: PRIMARY PATH TRACE
C SC: SECONDARY COEFFICIENT
C SPT: SECONDARY PATH TRACE
C
C'tiiitifiiiiiiiiti‘iiiifififiiiii********i**i*******ttttiit
C
C
INTEGER DE,CNT
CHARACTER NTYPE*4,PPT*1,SPT*1
DOUBLE PRECISION PC(5),SC(5)
C
INCLUDE 'SIZEBK.CBK'
INCLUDE 'GREDBK.CBK'
INCLUDE 'FUNCBK.CBK'
INCLUDE 'UTILBK.CBK'
C
C**INIT*******i‘i'i'ktiiti'ifitttittttfii‘titfii‘tttt***t******
C
C

C...ASSIGN INITIAL VALUES DEPENDING ON FORM OF DE
C
NTYPE =- NBXTP(DE)(2:2)
IF (NTYPE.EO.'C') THEN
PPT - 'E'
SPT - 'F'

00

ELSE
PPT - 'F'
SPT - 'E'

ENDIF

0000

43

C...INIT|ALIZE PC, AND SC

C

PC(CNT) - 1.000
SC(CNT) -1.0D0

CTRACE>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Last Change: 03/09/91

C
C

C
C

SUBROUTINE TRACE(CB,PPT,SPT,PC,SC,OE,FLAG,PATH,CNT,ID,IDZ)

Cififiiiiiitiii*tifiiittifit*iiiitiitiiiitifiiiifiiflttttitiii

C

C VARIABLE LIST:

OOOOOOOOOOOOOOOOOOOO

AC:

CB.
CE:
CNT:

CONT:

ENAM:
FLAG:
FNAM:
l1 :

I2:

ID:
ID2:
IEQ:

IFI' P:
IVGY:
MTF:
NTYPE:

BNDTYP:

ACTUAL CONNECTOR (IDENITIFICATION TOOL)
CONNECTOR TYPE

CURRENT CONNECTOR NUMBER IN TRACE
CURRENT ELEMENT IN TRACE

NUMBER OF TRACE PATHS IN MODEL

NUMBER OF ELEMENTS IN TRACE

ELEMENT NAME IN SYSTEM GRAPH

EXIT SIGNAL (TRACE FAILED IF FLAG-.TRUE.)
NAME OF FUNCTION ASSOCIATED WITH IEQ
CONNECTOR LIST POINTER

CONNECTOR LIST POINTER

PATH IDENTIFICATION

ARRAY STORING NUMBER OF ELEMENTS IN PATH
EQUATION NUMBER ASSOCIATED WITH CURRENT
ELEMENT

INTEGER FUNCTION TYPE OF IEQ

GYRATOR MODULUS FOUND IN TRACE
TRANSFORMER MODULUS FOUND IN TRACE
ELEMENT TYPE

44

PREVIOUS ELEMENT IN TRACE

ARRAY CONTAINING ELEMENT NUMBERS IN TRACE
PRIMARY COEFFICIENT

DIRECTION OF CONNECTOR BETWEEN OE AND CE
DIRECTTON OF CONNECTOR BETWEEN CE AND NE
PRIMARY PATH TRACE

SECONDARY COEFFICIENT

NUM. OF PRIMARY PATH'S EXITING A 0 OR 1 ELEMENT
(NOTE: IF SCOUNT > 1 THE MODEL CANNOT BE
ADJUSTED WITH THIS METHOD)

SPT: SECONDARY PATH TRACE

STRK1 : ELEMENT NUMBER AT CAUSAL STROKE END OF CB
STRK2: ELEMENT NUMBER AT CAUSAL STROKE END OF NB
TEST: COM PLETTON CHECK (COMPLETE IF TEST=.TRUE.)

*iiiitttiiii'*********i'ttfiiifittitiitttiti‘tti**t******

000000000000000000

INTEGER CB,CE,OE,I1 ,I2,AC,STRK1,STRK2,SCOUNT,IEO,IFTP
INTEGER COUNT,CNT,ID(5),PATH(5,20),ID2(5)

CHARACTER PPT‘1 ,SPT*1,NTYPE‘4,FNAM*8,BNDTYP*4
DOUBLE PRECISION PC(5),SC(5),POWER1,POWER2,MGY,MTF
LOGICAL TEST,FLAG

00

INCLUDE 'SIZEBK.CBK'
INCLUDE 'GREDBK.CBK'
INCLUDE 'FUNCBK.CBK'
INCLUDE 'UTILBK.CBK'

EXTERNAL BLNKLN,WRTSTR,NEWL,PROMPT

*OTRACE************fitiiiiiifiiiitii‘ttfiiii*tfifiiiiiittti

...|NITIALIZE TEST, FLAG, SCOUNT, COUNT, ID AND PATH

000000 00

FLAG - .FALSE.
ID(CNT) - 1
TEST - .FALSE.
SCOUNT - 0
COUNT - 1

45

PATH(CNT,COUNT) - OE
C
C
10 CONTINUE
C

C...IDENTIFY CURRENT ELEMENT, CE, FROM CURRENT CONNECTOR, CB,
C AND OLD ELEMENT, OE. SIMILIARLY, IDENTIFY THE POWER
C...D|RECTION WITH +1 INDICATING THAT POWER IS CONSERVED

C (CB GOES FROM OE TO CE).

C
C
CE-IELSBD(CB,1)
POWER1 - -1.0D0
IF (CE.E0.0E) THEN
CE-IELSBD(CB,2)
POWER1-1.0DO
ENDIF
STRK1 - BNDCAU(CB)
C
C...IDENTIFY ELEMENT TYPE OF CE
C
NTYPE-NBXTP(CE)(2:2)
C
C
Cittttiiti'ktt*fifittitifliiifitiiiiiitfitfitifitti'iifitittttii
Cififiiiiifiiitttiii*ttitfitflittiiitttfiiifitiifiittittititti
C
C
IF (NTYPE.EO.'0') THEN
C

C...IDENTIFY THE CONNECTOR LIST FOR THIS ELEMEMT
C
I1 - IBDPTR(CE)
I2 - IBDPTR(CE+1)-1
C
C...IF SIGNAL EXISTS IN CONNECTOR LIST -- PUNT!
C
DO 20 J-I1,l2
AC . IBDSEL(J)
BNDTYP - BDTP(AC)(1 :1)
|F(BNDTYP.EQ.'SIGNAL') THEN
ID(CNT) .. 3
GOTO 10100

46

ENDIF
20 CONTINUE
C
C...EXAMINE EACH COIWECTOR ON THE ELEMENT
C

DO 100 l-l1,l2

AC . IBDSEL(I)

..FIND POWER DIRECTTON OF AC

090

IF (IELSBD(AC,1).EQ.CE) THEN
POWER2 - 1.0D0
ELSE
POWER2 = -1 .0D0
ENDIF
..IDENTIFY THE CAUSAL STROKE END CF AC
STRK2 - BNDCAU(AC)

.. .IF THE PRIMARY PATH IS FLOW, CHECK FOR SPLIT PATH,
ASSIGN NEW CONNECTOR NUMBER, AND ADJUST PC & SC

IF ((PPT.EO.'F').AND.(STRK2.NE.CE).AND.(AC.NE.CB)) THEN

SCOUNT - SCOUNT + 1

..USE CAUSAL INFO. ALONG WITH POWER DIRECTIONS OF
AC AND CB TO DETERMINE ADJUSTMENTS TO SC AND PC

0000 00 00:00 000

IF (POWER1.EO.POWER2) THEN
PC(CNT) . PC(CNT)
ELSE
PC(CNT) - -1.0DO*PC(CNT)
ENDIF
SC(CNT) - SC(CNT)
C
C...ASSIGN NEXT CONNECTOR
C
NB - AC
C
C...IF PPT IS EFFORT: ASSIGN NEw CON. NUMBER AND ADJUST SC

47

C
ELSE IF ((PPT.EQ.'E‘).AND.(STRK2.EO.CE)) THEN
C
C...USE CAUSAL INFO. ALONG WITH POWER DIRECTIONS OF
C AC AND CB TO DETERMINE ADJUSTMENTS TO SC AND PC
C
IF (POWER1.EO.POWER2) THEN
SC(CNT) - SC(CNT)
ELSE
SC(CNT) . -1.0DO*SC(CNT)
ENDIF
PC(CNT) - PC(CNT)
C
C..ASSIGN NEXT CONNECTOR
C
NB =- AC
C
ENDIF
100 CONTINUE
C
C...ASSIGN VALUES FOR NEXT STEP IN TRACE
C
COUNT I COUNT + T
PATT-I(CNT,COUNT) - CE
OE =- CE
CB - NB
TEST - .FALSE.
C
C...NOTE SPLIT PATH
C
IF (SCOUNT.GT.1) THEN
CALL BLNKLN
CALL WRTSTR(' Split path at 0 junction ')
CALL WRTSTR(' Trace failed... ')
CALL BLNKLN
ID(CNT) - 2
GOTO 10000
ENDIF
C
C

CiiiiiiifiiiiiiiittiifiI'iiii’iiiititiiiitittiifififittttiiit

C*****.*§*§**tiiiiitiiii***************ti*************

C

48

C
ELSE IF (NTYPE.EO.'1') THEN
c
C...IDENTIFY THE CONNECTOR UST FOR THIS ELEMENT
C
l1 . IBDPTR(CE)
I2 - lBDPTR(CE+1)-1
C

C...IF SIGNAL EXISTS IN CONNECTOR LIST -- PUNT!
C
DO 150 J-I1,l2
AC - IBDSEL(J)
BNDTYP .- BDTP(AC)(1 :1)
IF(BNDTYP.EQ.'SIGNAL') THEN

ID(CNT) - 3

GOTO 10100

ENDIF

150 CONTINUE
C
C...EXAMINE EACH CONNECTOR ON THE ELEMENT
C

DO 200 I=|1,I2

AC - IBDSEL(I)
c
C...FIND POWER DIRECTTON OF AC
C
IF (IELSBD(AC,1).EO.CE) THEN
POWER2 - 1.0D0
ELSE
POWER2 . -1 .0D0
ENDIF
...IDENTII-v THE CAUSAL STROKE END OF AC
STRK2 . BNDCAU(AC)

...IF THE PRIMARY PATH IS EFFORT, CHECK FOR SPLIT PATH,
ASSIGN NEW CONNECTOR NUMBER, AND ADJUST PC & SC

0000 000

IF ((PPT.EO.'E').AND.(STRK2.EO.CE).AND.(AC.NE.CB)) THEN

00

49

SCOUNT - SCOUNT + 1
C
C...USE CAUSAL INFO. ALONG WITH POWER DIRECTIONS OF
C AC AND CB TO DETERMINE ADJUSTMENTS TO SC AND PC
C
IF (POWER1.EQ.POWER2) THEN
PC(CNT) - PC(CNT)
ELSE
PC(CNT) . -1 .ODO*PC(CNT)
ENDIF
SC(CNT) - SC(CNT)
C
C..ASSIGN NEXT CONNECTOR
C
NB = AC
C
C...IF PPT IS FLOW: ASSIGN NEW CONNECTOR NUMBER AND ADJUST SC
C
ELSE IF ((PPT.EQ.'F').AND.(STRK2.NE.CE)) THEN
C
C...USE CAUSAL INFO. ALONG WITH POWER DIRECTIONS OF
C AC AND CB TO DETERMINE ADJUSTMENTS TO SC AND PC
C
IF (POWER1.EO.POWER2) THEN
SC(CNT) - SC(CNT)
ELSE
SC(CNT) .. -1.0DO*SC(CNT)
ENDIF
PC(CNT) - PC(CNT)
C
C..ASSIGN NEXT CONNECTOR
0
NB . AC
C
C
ENDIF
200 CONTINUE
C
C...ASSIGN VALUES FOR NEXT STEP IN TRACE
C

COUNT - COUNT + 1
PATH(CNT,COUNT) - CE

5O

OE - CE
CB - NB
TEST - .FALSE.
C
C...NOTE SPLIT PATH
C
IF (SCOUNT.GT.1) THEN
CALL BLNKLN
CALL WRTSTR(' Split path found at 1 junction ')
CALL WRTSTR(' Trace failed... ')
CALL BLNKLN
ID(CNT) . 2
GOTO 10000
ENDIF
C
C
Ctiittitttittttfitttttit.tttitit.tittttttttitfitttttttii'
Cititititttitittititttittttt*tttfitttitttfittttttfitttitt
C
C
ELSE IF (NTYPE.EQ.'T') THEN
C
C...IDENTIFY MOD.TF FROM CE
C
C...FIND EQUATION NUMBER FOR CE
C
IEO - INEOP(CE)
C
C...FIND EQUATION TYPE OF CE
C
IFT P - IFCNTP(IEO)
FNAM - FNNAM1(IFTP)
C
C...IF MTF IS NOT A CONSTANT -- PUNT!
C
IF (FNAM.EO.'CON') THEN
MTF - EOPAR(IEOPTR(IEO))
ELSE
ID(CNT) - 3
GOTO 10200
ENDIF
C

C...FIND CONNECTOR LIST FOR THIS ELEMENT

51

I1 - IBDPTR(CE)
I2 - IBDPTR(CE+1)-1
C
C...IF SIGNAL EXISTS IN CONNECTOR LIST -- PUNTI
C
00 250 J-I1,I2
AC - IBDSEL(J)
BNDTYP . BDTP(AC)(1 :1)
IF(BNDTYP.EO.'SIGNAL') THEN

ID(CNT) - 3
GOTO 10100
ENDIF
250 CONTINUE
C
C...FIND NEXT CONNECTOR
C
DO 300 I - I1,I2
Ac - IBDSEL(I)
IF (AC.NE.CB) THEN
NB . AC
C
C...CHECK CAUSAL DIRECTION
C
IF (BNDCAU(NB).EO.CE) THEN
C
C...ADJUST PC AND SO
C
PC(CNT) - MTF'PC(CNT)
SC(CNT) - (1.000/MTF)*SC(CNT)
ELSE
PC(CNT) - (1 .ODOIMTF)*PC(CNT)
SC(CNT) - MTF*SC(CNT)
ENDIF
ENDIF
300 CONTINUE
C

C...ASSIGN VALUES FOR NEXT STEP IN TRACE
C
COUNT - COUNT + 1
PATH(CNT,COUNT) .- CE
OE - CE
CB - NB

5 2
TEST - .FALSE.C

C
Ciiii*‘Iiiiiti*ifiiiiii*iii'iiiiifiiitiiifiti*tttiiii******
C******.***.*****titiiiiiiittiti*************titfliiiti
C
C
ELSE IF (NTYPE.EQ.'G') THEN
C
C IDENTIFY MGY FROM CE
C
C...FIND EOUATTON NUMBER OF CE
C
IEO - INEOP(CE)
C
C...FIND EQUATION TYPE OF CE
C
IFTP - IFCNTP(IEO)
FNAM .. FNNAM1(IFTP)
C
C...IF MGY IS NOT A CONSTANT - PUNT!
C
IF (FNAM.EO.'CON') THEN
MGY - EOPAR(IEOPTR(IEO))
ELSE
ID(CNT) - 3
GOTO 10200
ENDIF
C
C...FIND CONNECTOR LIST FOR THIS ELEMENT
C
I1 = IBDPTR(CE)
I2 - IBDPTR(CE+1)-1
C
C...IF SIGNAL EXISTS IN CONNECTOR LIST - PUNTI
C

DO 350 J-I1,l2
AC - IBDSEL(J)
BNDTYP . BDTP(AC)(1 :1)
IF(BNDTYP.EQ.'SIGNAL') THEN

ID(CNT) - 3
GOTO 10100
ENDIF
350 CONTINUE

53

C
C...FIND NEXT CONNECTOR
C
DO 400 I - I1 ,12
AC - IBDSEL(I)
IF (AC.NE.CB) THEN
NB - AC
...CHECK CAUSAL DIRECTION
IF (BNDCAU(NB).EO.CE) THEN

...SWITCH PATHS (ENTERED GY ON PRIMARY FLOW)

000 000

PPT - 'E'
SPT - 'F'

...ADJUST PC AND SC

000

PC(CNT) .. (1.000/MGY)*PC(CNT)
SC(CNT) - MGY*SC(CN1)

ELSE

...SWITCH PATHS (ENTERED GY ON PRIMARY EFFORT)

000 0

PPT - 'F'
SPT - 'E'

...ADJUST PC AND SC

000

PC(CNT) - MGY*PC(CN1)
SC(CNT) - (1 .000/MGY)*SC(CNT)
ENDIF
ENDIF
400 CONTINUE
C
C...ASSIGN VALUES FOR NEXT STEP IN TRACE
C
COUNT - COUNT + 1
PATH(CNT,COUNT) .. CE
OE . CE
CB . NB

5 4
TEST - .FALSE.

C
C
Ciiifliiiititiiiit*tiitttiittiiifiitttifiit*t*t****tttttt
Ciiiiifiittfiitititiiiiiiiiiiifittt*tiiiittitttittt*ititfl
C
C
ELSE IF (NTYPE.EQ.'I') THEN
C
C...FIND EQUATION NUMBER
C
IEQ - INEOP(CE)
C
C...FIND EQUATION TYPE
C
IFTP - IFCNTP(IEO)
FNAM . FNNAM1(IFTP)
C
C...IF EQUATION TYPE IS NOT ATr OR GAIN -- PUNTI
C
IF ((FNAM.EO.'A‘I'I").OR.(FNAM.EO.‘GAIN')) THEN
FLAG - .FALSE.
ELSE
ID(CNT) .. 3
GOTO 10200
ENDIF
C
C...FIND CONNECTOR LIST FOR THIS ELEMENT
C
I1 - IBDPTR(CE)
12 - IBDPTR(CE+1)-1
C

C...IF SIGNAL EXISTS IN CONNECTOR LIST -- PUNTI

0

DO 450 J-I1,I2
AC - IBDSEL(J)
BNDTYP - BDTP(AC)(1 :1)
IF(BNDTYP.EQ.'SIGNAL') THEN
ID(CNT) - 3
GOTO 10100
ENDIF
450 CONTINUE
C

5 5
C...IDENTIFY THIS ELEMENT IN PATH

c
ID(CNT) . 1
COUNT - COUNT + 1
PATH(CNT,COUNT) - CE
I02(CNT) - COUNT
TEST . .TRUE.
C
C
Ctfiiiiiiiflit.if...itititiifliititifiiiiiifi******i*******
Ciil‘fiiiii*ttttiiiittfiittittiittt*********iiiiiitttti'i't
C
C
ELSE IF (NTYPE.EQ.'C') THEN
C
C...FIND EQUATION NUMBER
C
IEQ - INEOP(CE)
C
C...FIND EQUATION TYPE
C
IFI'P - IFCNTP(IEO)
FNAM - FNNAM1(IFTP)
C
C...IF FUNCTION IS NOT ATT OR GAIN - PUNTI
C
IF ((FNAM.EO.'ATI").OR.(FNAM.EQ.’GAIN')) THEN
FLAG - .FALSE.
ELSE
ID(CNT)-3
GOTO 10200
ENDIF
C
C...FIND CONNECTOR LIST FOR THIS ELEMENT
C
I1 - IBDPTR(CE)
I2 - lBDPTR(CE+1)-1
C
C...IF SIGNAL EXISTS IN CONNECTOR LIST - PUNTI
0

DO 550 J-I1 ,12
AC . IBDSEL(J)
BNDTYP - BDTP(AC)(1:1)

5 6
IF(BNDTYP.EQ.'SIGNAL') THEN

ID(CNT)-3
GOTO10100
ENDIF
550 CONTINUE
C
C...IDENTIFY THIS ELEMENT IN PATH
C
ID(CNT)-1
COUNT - COUNT + 1
PATH(CNT,COUNT) - CE
ID2(CNT) - COUNT
TEST-.TRUE.
C
C
Ctitttiii*tttttttttttttitititittittttttttttitittttitit
Ctitittiitttitttfitttttttittttit.*ttttttttttt*ttiitttit
C
C
ELSE IF (NTYPE.EQ.'E') THEN
C
C...PRINT ERROR MESSAGE AND RETURN
C
CALLBLNKLN
CALL WRTSTR(' Trace found <Se> as forcing element')
CALL WRTSTR(' Trace failed... ')
CALLBLNKLN
ID(CNT)-=3
FLAG=.TRUE.
RETURN
C
C
Ciitttttt*fltittttttttttttitititi*tttttttttitttittttttt
C***i********tt***titttttitt*itiittittttiitttitt*ttitt
C
C
ELSE IF (NTYPE.EQ.'F') THEN
C
C...PRINT ERROR MESSAGE AND RETURN
C

CALL BLNKLN
CALL WRTSTR(' Trace found <Sf> as forcing element')
CALL WRTSTR(' Trace failed... ')

57

CALLBLNKLN
ID(CNT)-3
FLAG-.TRUE.
RETURN
C
C
Ctttiitiiititttttttttttttitttitt*ttttttttttttit.titttt
C*****ttt*tittiitttttiitiitt*ttittti'titfiiti'tttfittttiti
C
C
ELSE IF (NTYPE.EQ.'R') THEN
C
C...PRINT ERROR MESSAGE AND RETURN
C
CALLBLNKLN
CALL WRTSTR(' Trace found <R> as forcing element')
CALL WRTSTR(' Trace failed... ')
CALLBLNKLN
ID(CNT)-3
FLAG-.TRUE.
RETURN
C
C
Cttifiiiiiittt*titttttttitiititti*ttiittttttiititititit
Ctiitifitittittttiittfitfittiiiiittiitittttttifittttittttt
C
C
ELSE
C
C...TRACE HAS FOUND AN ILLEGAL ELEMENT TYPE
C
C
CALLBLNKLN
CALL WRTSTR(' Trace has found an illegal element')
CALL WRTSTR(' Trace failed... ')
CALLBLNKLN
ID(CNT)-3
FLAG-.TRUE.
RETURN
C
C...SPLIT PATH
C

10000 CONTINUE

58

C
FLAG - .TRUE.
RETURN
C
C...SIGNAL FOUND
C
10100 CONTINUE
C
CALL BLNKLN
CALL WRTSTR(' A signal was found in the path)
CALL WRTSTR(' Trace failed... ')
CALL BLNKLN
FLAG - .TRUE.
RETURN
C
C...ILLEGAL FUNCTION TYPE
C
10200 CONTINUE
C
CALL BLNKLN
CALL WRTSTR(' Illegal function type found)
CALL WRTSTR(' Trace failed... ')
CALL BLNKLN
FLAG - .TRUE.
RETUFN
C
C
Citttttttttitttitttttttttttttitttttt*ttttttttttttttttt
Cttttttiitiittittttttitttittittitttttttttittttittttttt
C
ENDIF
C
C
IF (T EST.EQ..FALSE.) GOTO 10
C
C
RETURN
3D
C
C

C>>>>>

59

C
C
C
CREDUCE>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Last Change: 04/23/91
C
C
SUBROUTINE REDUCE(PATH,PC,SC,CNT,IDZ,FLAG)
C
C
C**.****t*******************************fi*************
C
C VARIABLE LIST:
C
C ANS: ANSWER
C CNT: PATH NUMBER
C CONT: ELEMENT NUVIBER
C II: DERIVATIVE ELEMENT
C DFLT: DEFAULT ANSWER
C DL: ELEMENT NLMBER
C EC: EQUIVALENT COEFFICIENT
C EL: ELEMENT NUMBER
C ENAM1 : ELEMENT NAME
C ENAM : ELEMENT NAME
C FE: FORCING ELEMENT
C FLAG: ERROR INDICATOR
C l1 : POINTER
C IB: POINTER
C ID2: ARRAY STORING NUMBER OF ELEMENTS IN PATH
C IEQ: EQJATION NUMBER
C K: STIFFNESS PARAMETER
C M: INERTIAL PARAMETER
C N: ELEMENT NAME
C NAM: ELEMENT NAME
C NAME: ELEMENT NAMES
C (D: ORIGINAL COEFFICIENT
C PATH: MATRIX CONTAINING ELEMENTS IN TRACE
C PC: PRIMARY COEFFICIENT
C SC: SECONDARY COEFFICIENT
C Z: POINTER
C
C*****1...fi****.*.*************i****§*fi****.§***fi*fifitfi‘I'
C

INTEGER PATH(5,20),CNT,IDZ(5),DE,FE,IEQ,EL,DL,I1,IB,Z

60

INTEGER COUNT

DOUBLE PRECISION PC(5),SC(5),OC,EC,K,M

CHARACTER NTYPE*4,ENAM1*3,ENAM2*B,ANS*3,STRING*40
CHARACTER NAM*8,NAME(20)*8,DFLT*8,N*8

LOGICAL FLAG
C
EXTERNAL WRTSTR,PROMPT,GETANS,BLNKLN,NEWL,MENSET,GOON
C
INCLUDE 'SIZEBK.CBK'
INCLUDE 'GREDBK.CBK'
INCLUDE 'FUNCBK.CBK‘
INCLUDE 'UTILBK.CBK‘
C
CtiREDUCEttiitiiitittiititttttitttttt*itttttitfifititttt
C
COUNT =- ID2(CNT)
CALL MENSET(.TRUE.)
FULL - MMFULL
C
C...INFORM USER OF PATH CONTENTS
C
DO 4 I=1,COUNT
Z - PATH(CNT,I)
NAME(I) - ELNAM(Z)
4 CONTINUE
C
CALL BLNKLN
CALL WRTSTR(' The path contains the following elements:')
CALL BLNKLN

WRITE(STRING,5)(NAME(I),I=1 ,COUNT)
5 FORMAT(4X,6(1X,A3))

CALL WRTSTR(STRING)
CALL BLNKLN
C
C...OBTAIN REDUCTION CHOICES
C
DE - PATH(CNT,1)
FE - PATH(CNT,COUNT)
ENAM1 - ELNAM(DE)
ENAM2 - ELNAM(FE)
c

C...PROMPT USER FOR REDUC‘ITON CHOICE (DEFAULT = DE)
C

100

C

61

CONTTNUE

CALL BLNKLN

CALL WRTSTR(' Which element would you like to reduce?)
CALL BLNKLN

WRITE(STRING,100)ENAM1 ,ENAM2

FORMAT(5X,A8,'or ',5X,A8)

CALL WRTSTR(STRING)

CALL BLNKLN

STRING-' Enter selection ('l/ENAM1/l'):'

CALL PROMPT(STRING)

C...READ AND INTERPRET ANSWER

C

C

ANS - '11'

CALL GETANS(ANS)

IF ((ANS.EO.'#').OR.(ANS.EO.' )) THEN
ANS - ENAM1

ENDIF

C...ASSIGN VALUES FOR REDUCTION

C

C

IF (ANS.EO.ENAM1) THEN

EL - FE

DL - DE
ELSE IF (ANS.EO.ENAMZ) THEN
EL - DE

DL - FE
ELSE

CALL BLNKLN

CALL WRTSTR(' Bad entry...Please try again‘)
CALL BLNKLN

GOTO 10
ENDIF

C...FIND CC

C

NTYPE - NBXTP(DL)(2:2)
IEQ - INEOP(DL)

IF (NTYPE.EQ.'I') THEN

OC - EQPAR(IEQPTR(IEQ))
IF (OC.NE.0.0DO) THEN
OC - 1.0D0/OC

090 090

090

090 090 OO

on 090 on

62

ENDIF

ELSE

OC - EQPAR(IEQPTR(IEQ))
ENDIF

..OBTAIN CORRECT COEFFICIENTS

NTYPE - NBXTP(EL)(2:2)

..IDENTIFY CONNECTOR LIST

I1 - IBDPTR(EL)
IB - IBDSEL(I1)

IF (NTYPE.EQ.'I') THEN

..FIND EQUATION NUMBER

IEQ :- INEOP(EL)

..FIND COEFFICIENT

M - EQPAR(IEQPTR(IEQ))
IF (M.NE.0.0D0) THEN

M - 1.0D0/M
ENDIF

..CALCULATE EQUIVALENT COEFFICIENT

EC - M-PC(CNT)*SC(CNT)*OC
IF (EC.NE.0.0D0) THEN

EC - 1.0D0/EC
ENDIF

ELSE IF (NTYPE.EQ.'C') THEN

..FIND EQUATION NUMBER

IEQ - INEOP(EL)

..FIND COEFFICENT

63

K - EQPAR(IEQPTR(IEQ))
...CALCULATE EQUIVALENT COEFFICENT

EC - K - PC(CNT)"’SC(CNT)*OC

ELSE

...ERROR IN UST

000 00 000 0

CALL BLNKLN

CALL WRTSTR(' An error in the procedure has occurred')
CALL BLNKLN

FLAG - .TRUE.

RETUM

ENDIF

C...ELIMINATION

0

CALL ELIM(PATH,DL,CNT,ID2)
DO 200 1:25
200 E7CFLG(I) - .FALSE.
CALL REASSIGN(.TRUE.)
C
C...CORRECT THE EQUATION
C
DO 1000 I-1,INELS
NAM - ELNAM(I)
IF (ENAM1.EO.NAM) THEN
DE - I
IEQ . INEOP(DE)
N - NAM
ELSE IF (ENAM2.EQ.NAM) THEN
FE - I
IEQ . INEOP(FE)
N . NAM
ENDIF
1000 CONTTNUE
C
C...ASSIGN CORRECT COEFFICIENT
C

64

EQPAR(IEQPTR(IEQ)) - EC
C
C...PROMPT USER FOR EOUATTON TYPE
C
1400 CONTINUE
CALL BLNKLN
IF (NTYPE.EQ.'I') THEN
DFLT - 'GAIN'
ELSE IF (NTYPE.EQ.'C') THEN
DFLT - 'ATT'
ENDIF
WRITE(STRING,1450)N
1450 FORMAT(5X,'SeIect equation type for ',A8)

CALL WRTSTR(STRING)
CALL BLNKLN
STRING-' ('//DFLT//'):'
CALL PROMPT(STRING)
C
C...READ AND INTERPRET ANSWER
C
ANS - '#'
CALL GETANS(ANS)
IF ((ANS.EO.'#').OR.(ANS.EQ.' ')) THEN
ANS - DFLT
ENDIF
IF ((ANS.NE.'ATT‘).OR.(ANS.NE.'GAIN')) THEN
IFT P - IFCNTP(IEO)
FNNAM1 (IFT P) =- ANS
ELSE
CALL BLNKLN
CALL WRTSTR(' Bad entry please try again...')
GOTO 1400
ENDIF
C
C
RETUFN
30
C
C
C>>>>>
C
C

65

CELIM>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>Last Change: 04/23/91

C

C

SUBROUTINE ELIM(PATH,DE,CNT,ID2)
C
C
Ci*i*****i*fi***i**i*tiii********************tiiiiiitii
C
C VARIABLE LIST:
C
C A: COUNTER
C B: COUNTER
C Q COLNTER
C CF.- CURRENT ELEMENT
C CNT: NUMBER OF TRACE PATHS IN MODEL
C COUNT ELEMENT NUMBER IN PATH
C DE: ELEMENT SELECTED FOR REDUCTION
C I1 : CONNECTOR LIST POINTER
C I2: CONNECTOR LIST POINTER
C ID2: ARRAY STORING NUMBER OF ELEMENTS IN TRACE
C NTYPE: NODE TYPE
C PATH: ARRAY CONTAINING ELEMENT NUMBERS IN TRACE
C
C***********t*tttiititiiifiittiii**i**************ittti
C

INTEGER PATH(5,20),DE,CNT,ID2(5),l1,I2,CE,A,B,C

INTEGER COUNT

CHARACTER NTYPE*4

INCLUDE 'SIZEBK.CBK'

INCLUDE 'GREDBK.CBK'

INCLUDE 'FUNCBK.CBK'

INCLUDE 'UTILBK.CBK'
C
CtiELlMitfiiiiiiitiiit*iitttfifi*tttifittittiittttitiit...
C
C

C...BEGIN ELIMINATION BY DELETING THE SELECTED ELEMENT
C
CALL DELND(DE)
COUNT . ID2(CNT)
IF (COUNT.LE.3) GOTO 200
C
C...CONTINUE ELIMINATION

000 00 000 00

000 00 000

000

66

IF (DE.EQ.PATH(CNT,1)) THEN

A - 2

B - COUNT-2

C - 1

ELSE IF (DE.EQ.PATH(CNT,COUN1)) THEN
A - COUNT-1

B .. 3

C - -1

ENDIF

DO 100 I - A,B,C
...FIND TYPE OF NEXT ELEMENT IN TRACE
CE .. PATH(CNT,I)
NTYPE - NBXTP(CE)(2:2)
IF ((NTYPE.EQ.'T).OR.(NTYPE.EQ.'G)) THEN
...CHECK NUMBER OF CONNECTORS
I1 - IBDPTR(CE)
12 - lBDPTR(CE+1)-1
IF (l1.EQ.I2) THEN
...IF ONLY ONE CONNECTOR EXISTS, DELETE CE AND ITS CONNECTOR
CALL DELND(CE)
ENDIF
ELSE IF ((NTYPE.EQ.'O').OR.(NTYPE.EQ.'1')) THEN
...IDENTIFY NUMBER OF CONNECTORS

I1 .. IBDPTR(CE)
I2 :- IBDPTR(CE+1)-1

...IF ONLY ONE CONNECTOR EXISTS, DELETE CE AND ITS CONNECTOR

67
IF (11.EQ.12) THEN

CALL DELND(CE)

ENDIF
C
C

ENDIF
C
C
1 00 CONTTNUE
C
C
200 CONTTNUE
C
C

RETURN

ED
C
C
C
C>>>>>
C
C
CREASSIGN >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 02/09/91 JM
C

SUBROUTTNE REASSIGN(PROCFG)
C
C—- PURPOSE: REASSIGNING EQUATIONS AND CAUSALITY TO MODEL
C
C--- OUTPUT: PROCFG, =.T. if user wishes to proceed, =.F. else
C

INCLUDE 'SIZEBK.CBK‘

INCLUDE 'GREDBK.CBK‘

INCLUDE 'UTILBK.CBK'
C
C--- VARIABLES: E7SFLG(2), was causality ever set? (yes-.true.)
E7SFLG(3), were equations ever set?
E7CFLG(2), does causality match graph data?
E7CFLG(3), do equations match causal data?

0000

CHARACTER STRING‘72, ANS‘1
INTEGERI
LOGICAL PROCFG, FULL, BGPART, AUTO, CHANGE, OKAY

C

68

EXTERNAL GRPROC, MENSET, BLNKLN, WRTSTR, PROMPT, GETANS
EXTERNAL LSTLB1, LSTEQS, SETFCN, RHFILE, INVOPT, DEFEQS
EXTERNAL DMPFLG, REDOEQ, GOON, LSTDQS, PARDRV

CtiiREASSIGNfiifiiiiiiii*i'*titiiiittfitt*****t**tttti*itt

C

000

000

C

CALL MENSET(.TRUE.)
FULL. MMFULL

...Check on causality status first...

BGPART- NEL.GT.0
IF (BGPART) THEN
IF ((.NOT.E7SFLG(2)).OR.(.NOT.E7CFLG(2))) THEN

...Assign auto causality to the entire graph...

DO 20 I- 3,5
E7CFLG(l)-.FALSE.

AUTO-.TRUE.
CALL GRPROC( AUT0,0KAY,CHANGE )
E7CFLG(2)- OKAY
IF (OKAY) THEN

E7SFLG(2)-.TRUE.
ELSE

C...Let the user try now...

C

CALL BLNKLN
CALL WRTSTR(' Automatic causality assignment produced')
CALL WRTSTR(' a system graph with derivative causality.')
AUTO - .FALSE.
CALL GRPROC( AUT0,0KAY,CHANGE )
E7CFLG(2)- OKAY

IF (OKAY) THEN

E7SFLG(2)-.TRUE.

ENDIF

ENDIF
ENDIF
ELSE
E7SFLG(2)-.TRUE.
E7CFLG(2)-.TRUE.
ENDIF

000 0000

0

6 9
IF (DBGFLG) CALL DMPFLG('FUNCD2')

...Check on equation status next...

IF (.NOT.E7SFLG(3)) THEN

...Set all node equations to their default status...

E7CFLG(4)-.FALSE.
E7CFLG(5)-.FALSE.

I- 1
CALL DEFEQS( I,OKAY)
E7SFLG(3)- OKAY
CALL DEFEQS( I,OKAY)
E7SFLG(3)- OKAY
E7CFLG(3)- OKAY

ELSE

IF (.NOT.E7CFLG(3)) THEN

C...Recover the equations to the maximum extent possible...

E7CFLG(4)-.FALSE.
E7CFLG(5)-.FALSE.
CALL REDOEQ( OKAY)
E7CFLG(3)- OKAY
ENDIF

ENDIF

LIST OF REFERENCES

_L
O

N

UST OF REFERENCES

Rosenberg. R. and KarnOPP D. ALIDILoduotionJLEhxsigaLsttem
Wigs, 1st edition, New York, New York, MCGraw-Hill, 1983.

ROSENCODE Associates Inc. IhLENEQBLBafeLenoiManual.
ROSENCODE Associates, Inc., Lansing, MI, 1990, Vol. 7.3.1.

Emery, S. ”Roller Coaster Analysis.” Michigan State University,
Department of Mechanical Engineering, ME 851 Final Project,
Dec.1989.

Tarokh, M. ”Simulation analysis of dynamics and
decentralized control of robot manipulators.” Technical
Article, Simulation, October 1989, pp 169-176.

Zeid, A. and Chang D. ”A Modular Computer Model for the
Design of Vehicle Dynamics Control Systems.” Vehicle
System Dynamics, 18 (1989), pp. 201-221.

Zeid, A. ”Bond Graph Modeling of Planar Mechanisms With
Realistic Joint Effects.” Journal of Dynamic Systems,
Measurement, and Control, March 1989, Vol. 111, pp. 15-23.

Karnopp. D. and Margolis, D. "Analysis and Simulation of Planar
Mechanism Systems Using Bond Graphs.” Journal of Mechanical
Design, April 1979, Vol. 101, pp. 187-191.

70