i\i\\\\\\\\\\\\\\\\\\\\\\\\\\\\i\\\\\\\\\\\\\\\\\\\\i affix;
3129 0062
, University
"dram
This is to certify that the
thesis entitled
STATE FORMULATION OF LARGE-SCALE LINEAR
TIME-INVARIANT BOND GRAPH MODELS
presented by
Benjamin Moultrie
has been accepted towards fulfillment
of the requirements for
Ph . D . Jggee in Mechanical Engineering
firfi/ J?
Major professor
(f.
Date 5/3/77
0-7 639
ovmnur FINES ARE 25¢ PER DAY _
PER ITEM
to remove
Return to book drop
r record.
this checkout from you
‘ I
STATE FORMULATION OF LARGE-SCALE LINEAR
TIME-INVARIANT BOND GRAPH MODELS
BY
Benjamin Moultrie
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirement
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering
1979
ABSTRACT
STATE FORMULATION OF LARGE-SCALE LINEAR
TIME-INVARIANT BOND GRAPH MODELS
by
Benjamin Moultrie
In this dissertation, topology-based equations are
developed which give the effort-flow basis order for the
juncture structure transformation of an arbitrary weighted
junction structure. These equations 3e used to develop
three upper bounds for the numberof distinct sets of port
variables which can be used to specify weighted junction
structure input-output relations. Each successive bound is
shown to be numerically smaller and computationally more
expensive than its predecessor. Examples are given which
use the established bounds.
Also, a causal assignment procedure is specified which
simplifies the state model formulation process for linear
time-invariant bond graphs. This result is used to develop
an efficient computer implemented state model formulator for
linear time-invariant bond graphs. The key storage features
and novel matrix manipulation procedures of this state model
formulator are explored, and key computer subroutines are
given. The enhanced performance characteristics of this
formulator are validated by computer test results.
ACKNOWLEDGMENTS
With apologies to Buffon:
For in those few men whose head is steady, whose heart is
compassionate, and whose sense is exquisite - there is
substance, thought and reason; there is the art of speaking
to the mind.
Thanks Doc.
ii
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . vii
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . ix
Chapter
I. INTRODUCTION. 1
1.1 Background. . . . l
1.2 The Status Of Bond Graph Theory
And Practice. . . . . . 3
1.3 Research Highlights And Dissertation
Organization. . . . . . . . . . . . . . . . 5
II. BOND GRAPH JUNCTION STRUCTURES. . . . . . . . . 8
2.1 Junction Structure Terminology
And Notation. . . . . . . . . . . . . . . . 8
2.2 Power Orientation . . . . . . . . 10
2.3 Analytic Properties Of Junction
Structure Nodes . . . . . . . . . . . . . . 11
2.4 Basis Order Rules . . . . . . .‘. . . . . . 12
2.5 Causal Concepts . . . . . . . . . . . . . 15
2.5.1 Causal Assignment . . . . . . . . . . 16
2.5.2 Causal Completeness . . . . . . . . . 16
2.5.3 Causal Consistency. . . . . . . . . . 16
2.5.4 Causal Extension. . . . . . . . . . . 17
2.6 Causal Complexes. . . . . . . . . . . . . . 20
III. STATE MODEL FORMULATION FOR LINEAR
TIME- INVARIANT BOND GRAPHS. . . . . . . . . . . 28
3.1 Field Multiports. . . . . . . . 29
3.2 The Standard Sequential Causality
Assignment Procedure. . . . . . 31
3. 3 The State Model Formulation Algerithm . . . 32
iii
Chapter
IV.
V.
BIBLIOG
APPENDI
Appendi
A.
A COMPUTER IMPLEMENTED STATE MODEL
FORMULATOR OF INCREASED EFFICIENCY
4.1 Design Features. .
4.1.1 Data Structures.
4.1.2 Causal Assignment.
4.1.3 Determination of Junction
Structure Reducibility
4.1.4 Junction Matrix Formulation.
4.1.5 Matrix Inversion .
4.2 State Model Formulator Computer
Test Results . . . . . .
SUMMARY.
RAPHY.
CES.
x
DERIVATION OF THE BASIS ORDER RULES.
A.l Basis Order And Simple Junction
Structures . .
A. 1. 1 Basis Order For Proper Simple
Junction Structure Forests
A.1.2 Basis Order For Proper Simple
Junction Structures.
A.1.3 Basis Order For Standard Simple.
Junction Structures.
A.1.4 Port Basis Order For Standard
Simple Junction Structures
A.2 Basis Order And Weighted Junction
Structures . . .
A. 2.1 Basis Order For Standard
Weighted Junction Structures
A.2.2 Port Basis Order For Standard
Weighted Junction Structures
AN APPLICATION OF THE BASIS ORDER RULES.
B.l An Upper Bond For C As A Direct
Application Of The Basis Order Rules
8.2 A Refined Upper Bound For C.
THE RESOLUTION OF A CONFLICT RESULTING
FROM A PORT BOND CAUSAL ORIENTATION.
THE IMPACT OF THE STANDARD SEQUENTIAL
CAUSALITY ASSIGNMENT PROCEDURE (SSCAP)
ON THE REDUCED JUNCTION MATRIX .
iv
Page
40
41
41
42
42
43
44
46
53
55
S8
58
S8
58
66
69
71
72
73
74
8O
81
84
92
94
Appendix Page
E. PRUNING AND SOME ASPECTS OF JUNCTION
STRUCTURES. . . . . . . . . . . . . . . . . . . 101
F. COMPUTER SUBROUTINES. . . . . . . . . . . . . . 107
Table
10
LIST OF TABLES
Analytic Properties Of Junction
Structure Multiports
Analytic Properties Of Field Multiports
(with exactly one incident bond)
Junction Matrix Storage For Test
Examples
Processing Time For State Model
Formulation.
Subroutine For Causal Complex
Identification
Subroutine For The Determination Of
Causal Complex Solvability
Subroutine For The Computation Of The
Reduced Junction Matrix. . .
Subroutine For The Construction Of The
Junction Matrix.
Subroutine For The Determination Of
Junction Matrix Entry Position
Subroutine For Matrix Inversion.
vi
Page
25
35
52
52
107
110
115
125
129
130
Figure
2.
1
LIST OF FIGURES
Generic categories of bond
graph multiports
Junction structure examples.
Examples of power orientation.
A11 consistent causal forms for
junction structure multiports.
Examples of junction structure
multiports with inconsistent
causal forms
Example of a causal complex
(shown in solid lines)
Consistent causal forms for
field multiports
Diagram of the standard sequential
causal assignment procedure.
The reduced junction matrix equation
The junction matrix equation
Derivation of the reduced
junction matrix.
Example of a list pair with
the corresponding matrix
Test example 1
Test example 2
Test example 3
vii
’Page
22
23
24
26
26
27
36
37
38
39
39
49
50
50
51
Figure
A
A.
.1
2
A l-junction proper SJS.
A O-junction proper SJS.
Example of Lemma A.l
construction procedure
Transformation for converting a
standard SJS to a proper SJS
Transformation for converting a
WJS to a SJS
Example of a weighted junction
structure. . . . . . .
Causally augmented weighted
junction structure .
Unlabelled causal orientations
of a weighted junction structure
Symbolic bond graph representation
with fields identified
Example of subgraph generation
by pruning .
Symbolic bond graph representation
with source field pruned .
Junction structure with
labelled nodes
Example of the pruning procedure
viii
Page
76
76
77
79
79
89
90
91
98
99
100
105
106
Note
number
number
sum of
sum of
number
number
number
number
number
number
number
number
number
number
number
number
that
of
of
NOMENCLATURE
external 1-junctions
external O-junctions
the degrees of the l-junctions
the degrees of the O-junctions
of
of
of
of
of
of
of
of
of
of
of
of
A
N + Nf<(N +B -N
F 0 1 1 B f)max=“o 1 1’
since NfiNB° Therefore, Nf<0 implies that C is not an n-port.
Suppose NF=O where Nf=(Nf)max' Then
N =0 + O=NB-(NO+B
F -N
=NB-(J++NI)
1 l)
=(P1+PO)-J+=NP-J+ + J+=Np + J_=O
Therefore, NF=O implies that C is not an n-port (this con-
clusion represents an exclusion of a case where C acts like
a source of flow to its environment [26]). Similarly, N510
implies that C is not an n-port. Thus, in order for C to be
an n-port, it is necessary that 0\'1/<:
m
n+m:l l
I
Dissipation: gR/fi
n+m:l
3
Figure 3.1. Consistent causal forms for
field multiports.
ohsonOHQ ucoscuwmmw Hmmsmu Hmwucozcom ppmwcmpm ocu mo Empmmflm .N.m whamwm
‘ . 22.2%on __ 00‘
5 mm» .5325 I :5ng MSG
‘ mzom 322
fig... 3?... E. I as... j
- 7 amass n28
T , 021w
4 mg zoEESmE €832 -
. 53.28 .1
m>Z<>Ema mac . L1 _ gomEH mac - ,
- _ cz Ease 928 _
.M w muéofi gag
, . oz
? 1.85.28 Emazou mam . w #
- 02— , 3.558 :28
mm» 85% .522
LS
VGCtOT
VCCCOT
vector
VGCtOT
vector
VGCtOF
VGCtOT
VGCtOT
38
_ . - — 1 - u
—1 S11 S12 S13 S14 41
3d = 821 S22 S23 S24 Ed
Bin S31 832 83:5 334 —Dout
LE. .5418“ 543 544-191
of inputs to independent storage elements
of inputs to dependent storage elements
of inputs to the dissipation elements
of inputs to the source elements
of outputs from the independent storage elements
of outputs from the dependent storage elements
of outputs from the dissipation elements
of outputs from the source elements
Figure 3.3. The reduced junction
matrix equation.
39
gout S1 S2 2in
Kant 83 S4 Kant
Kin - vector of all inputs to the junction structure
Kant - vector of all junction structure internal variables
yout - vector of all outputs from the junction structure
Figure 3.4. The junction matrix equation
Xout = SIZ-in + SZX-int
Y-‘mt = 531m + S4y-int
(a)
v = (I - S )-IS v
—int 4 3—in
(b)
_ _ -1
Xout ' [31 + Sz(I 54) Sijin
(C)
Figure 3.5. Derivation of the reduced junction matrix. (a) The
expanded junction matrix equation. (b) Internal
variables in terms of inputs. (c) The unpartitioned
reduced junction matrix equation.
IV. A COMPUTER IMPLEMENTED STATE MODEL FORMULATOR
OF INCREASED EFFICIENCY
The ENPORT—4 program is a powerful tool for the modeling,
analysis, and simulation of multiport systems. When given a
bond graph description of a system, ENPORT-4 selects physi-
cally-meaningful state variables and derives the system
state model, eigenvalues, and time response. The many addi-
tional features, available options and outputs, and the struc-
ture of ENPORT-4 are discussed in the program's documenta-
tion [8].
Although ENPORT—4 provides the system analyst with a
broad array of system information, it has significant in-
adequacies which have a profound affect on program perfor-
mance. Most of these inadequacies are revealed in the graph
reduction and state model formulation procedures. In the
following sections, deficiencies are identified in the
ENPORT-4 graph reduction and state model formulation pro-
cedures, and modifications are discussed which increase
overall program efficiency. These modifications have been
implemented in the ENPORT-5 program which is currently under
development. Key ENPORT-S graph reduction and state model
formulation subroutines are listed in Appendix F.
40
41
4.1 Design Features
4.1.1 Data Structures
At various stages in the bond graph processing procedure
assorted graph parameter and structural information must be
retained or manipulated. In general, when interpreted in
matrix form, this information results in a sparse matrix
analogous to a graph incidence or adjacency matrix [27]. A
major deficiency of ENPORT-4 is its use of full storage
(storage which includes all matrix zero entries) in multi-
dimensional arrays for the retention and manipulation of
bond graph information.
A major improvement in efficiency is realized in ENPORT-5
by minimizing data storage requirements through the use of
a sparse-matrix-based storage format. This is achieved by
using push-down stacks and linked data structures.[28-30].
In particular, the ENPORT-5 graph reduction and state model
formulation procedures employ simple lists for the retention
and manipulation of data. These lists are grouped in pairs,
where the entries of each list are ordered. In each list
pair, one list contains the nonzero entries of an implied
matrix of known dimensions, and the other list contains the
coordinates of the matrix entries where each coordinate pair
is converted to a unique number. The conversion of a coor-
dinate pair is accomplished by representing the position of
a matrix entry as the entry's column coordinate added to the
product of the matrix column dimension and one less than the
42
entry's row coordinate. An example of a list pair is given
in Figure 4.1.
4.1.2 Causal Assignment
The assignment of causality is an important stage in
the processing of a bond graph. The ENPORT-4 program uses
a causal assignment scheme which is a modification of SSCAP
(the standard sequential causality assignment procedure) in
that the scheme gives priority to user specified causal orien-
tations. Although this feature provides the knowledgeable
user with a great deal of flexibility, the unwary user may
specify causal orientations which may violate system con-
straints (such as constraints imposed by sources), give a
false indication of system order, or create uncertainty in
the state model formulation procedure.
The causal assignment scheme employed by the ENPORT-5
program is a direct implementation of SSCAP in which the
user cannot specify causalorientatiOnSLHHjl.after all source
bonds and storage bonds have been causally oriented. This
scheme not only eliminates the difficulties discussed above,
but also guarantees that if a reduced junction matrix exists,
then it has the simplified form identified in section 3.2.
4.1.3 Determination Of Junction Structure Reducibility
The ENPORT-4 procedure for determining the reducibility
of the junction matrix represents an additional area of
inefficiency. The junction matrix equation was given in
43
Figure 3.4. As illustrated in Figure 3.5, the junction
matrix is reducible if the matrix (I-S4) is nonsingular.
In ENPORT-4, junction matrix reducibility only can be deter-
mined during the process of attempting to invert the matrix
(I-S4).
Based on previous work, the reducibility of the junc-
tion matrix depends on the solvability of causal complexes
[12-14,l9,20]. Stating the case more explicitly, the
junction matrix is reducible if and Only if each causal
complex is solvable. As an explicit step in the ENPORT-5
causal assignment procedure, causal complexes are identi-
fied and their graph locations are communicated to the user.
Prior to the formulation of the junction matrix, each causal
complex is tested for solvability in order to determine
junction matrix reducibility. If any causal complex is
determined to be unsolvable, then bond graph processing
aborts and the user is notified of all unsolvable causal
complexes.
4.1.4 Junction Matrix Formulation
As an intermediate step in the formulation of the
junction matrix in ENPORT-4, a matrix equation is explicitly
formed for each junction structure node. The entries of
each node matrix are then placed in the junction matrix in
accordance with bond classifications and orderings.
The ENPORT-5 program does not explicitly form a matrix
equation for each junction structure node. Instead, the
44
junction matrix is constructed directly by using node causal
forms, bond power orientations, and graph model parameters
to obtain the coefficients of the summation, identity, and
proportionality output equations for each junction struc-
ture node. Specifically, causal, power, and parameter in-
formation is used to identify the flow output variable and
the coefficients of the corresponding flow input variables
for each O-junction, the effort output variable and the
coefficients of the corresponding effort input variables
for each l-junction, the identity relations for each 0-
junction and l-junction, and the pr0portionality relation-
ships for each transformer and gyrator. Once determined,
each of the above coefficients (except zeros) is directly
stored in a compact junction matrix where each entry
position is determined by bond classifications and order-
ings, and the dimensions of the implied full storage junc-
tion matrix.
4.1.5 Matrix Inversion
As illustrated in section 3.3, several calculations in
the state model formulation algorithm require the computa-
tion of a matrix inverse. The matrix inversion routine
used by ENPORT-4 is a Gauss-Jordan procedure which selects
a matrix entry of greatest magnitude for the pivot at each
45
stage of the deflation process. In ENPORT-4, the selection
of a pivot requires a row and column scan of mostly zero
entries since each matrix is generally sparse and in a full
storage format.
A result of this research was the development of the
sparse matrix inversion subroutine which is employed by the
ENPORT-5 program. This subroutine is called "INPRD"
(INverse-PRoDuct). Its development was motivated by the
lack of a matrix inversion routine which can take advantage
of the special features of the equations in the state model
formulation algorithm.
A very important consideration in the development of
any sparse matrix inversion routine is the possible increase
in the storage requirements for the inverse of a sparse
matrix [31]. The INPRD subroutine effectively eliminates
the problem of storage growth by taking advantage of the
features of the matrix calculations in the graph reduction
and state model formulation procedures. In these procedures,
matrix inverses in calculations appear in the form A'lB
where the matrix product A-18 relates sets of junction
structure variables. INPRD is a Gauss-Jordan type pro-
cedure which controls storage requirements by accepting
the generally sparse matrices A and B as inputs and return-
ing the generally sparse matrix A-1B as output. Note that
A.1 is not explicitly computed unless B is the compatible
identity matrix. INPRD applies directly to B a set of
46
transformations which represent elementary row operations for
the reduction of A to the identity matrix. In order to mini-
mize round-off errors, a matrix entry of greatest magnitude
in A is selected as the pivot at each stage in the process
of deflating A. The benefits of INPRD are evidenced by the
performance characteristics of the ENPORT-5 program.
4.2 State Model Formulator Computer Test Results
In this section, some performance aspects of the ENPORT-
4 and ENPORT-5 state model formulators will be considered.
In particular, processing times and junction matrix storage
requirements will be assessed for three test examples inter-
actively processed on the CDC 6500 computer.
The processing time will be interpreted as the CP
(central processor) execution time consumed from the point
of parameter input tothe pointcfi?state model output.
Storage considerations are limited to the junction matrix,
since it is the largest system matrix in the formulation
process. For each example considered, the processing time
and junction matrix storage space requirements of ENPORT-4
will be used as benchmarks.
The first test example is a structural model of a lever
mechanism with inertia load (see Figure 4.2). The second
test example is a structural model of a beam-block trans-
ducer system (see Figure 4.3). The final test example is
a structural model of a radar pedestal position control
system (see Figure 4.4). Each of the above examples may be
47
found in the user's manual for the ENPORT-4 program, where
each is studied in detail [8].
For each test example, the computational results are
contained in Table 3 and Table 4 for storage requirements
and processing time respectively. From Table 3, it is
observed that the ENPORT-S formulator requires significantly
less storage (as typified by the junction matrix) than does
the ENPORT-4 formulator for a given bond graph model. .The
ENPORT-4 storage requirement for the junction matrix is
given by (Np+2NI)2. The ENPORT-5 storage requirement for
the junction matrix is given by 4(Np+2NI-1). Thus, the
difference between the bond graph model storage demands of
the ENPORT-4 and ENPORT-5 formulators becomes increasingly
dramatic as the number of bonds in a graph model increases.
From Table 4, it is observed that the ENPORT-5 formu-
lator provides a significant savings in processing time for
the given test examples. In general, the size (and sign)
of this savings is a function of several variables, e.g.,
the number of causal complexes, and the density and dimen-
sions of matrices to be manipulated. As an explicit case,
consider the multiplication of an (nxm) matrix by an (mxp)
matrix, neither of which contains a zero entry. In a full
storage format, this matrix multiplication requires nmp
scalar multiplications and n(m-l)p scalar additions. In
ENPORT-5, this matrix multiplication requires nmp scalar
multiplications, nmp scalar additions, and 2nm(p+l) element
48
comparisons. Note that full storage matrix multiplication
requires the same number of scalar operations irrespective
of the sparsity of either matrix factor. In general, the
above matrix multiplication in ENPORT-5 requires n(m-r)(p-s)
scalar multiplications, n(m-r)(p-s) scalar additions, and
2nm(p+l)-rn(p+2)-2ns(m-r/2) element comparisons where r is
the average number of zeros per row in the (nxm) matrix and
s is the average number of zeros per row in the (mxp) matrix.
Note that the worst case is given for the number of element
comparisons. It is seen that as r and 5 increase, matrix
multiplication in a full storage format rapidly becomes com-
putationally more demanding than matrix multiplication in
ENPORT-5. Thus, although it is possible for the processing
time required by ENPORT-S to exceed the processing time
required by ENPORT-4, this possibility is minimized by the
general sparsity of system matrices and the "inverse-multi-
plication" feature of the INPRD subroutine.
In conclusion, the combined results of Table 3 and
Table 4 suggest that the ENPORT-5 state model formulator not
only enhances the processing performance and capabilities
of the ENPORT-5 program, but also contributes to a reduced
dollar cost for the operation of a linear time-invariant
bond graph simulation program.
49
'2‘ [1‘
-l 3
5 6
3 8
:4. 1.12..
Entry list Position list
Figure 4.1. Example of a list pair with the
corresponding matrix.
50
SE C
1 4
1 6 TF 7 1 5
Z ‘ 3
I R
Figure 4.2. Test example 1
2 C 1
12 TE 11 O 10 1 9 GY
Figure 4.3.
Test example 2
51
I R
13
14
12 1 15
TF CY
11 16
0
0—-1-—C 1—17—1
9
~\4L\\\
18
/ 1 R
6 5
2 19
GY 4 1 1 GY
3 1
I SE
Figure 4.4.
Text example 3
52
Table 3. Junction Matrix Storage For Test Examples
Example (1)W4 (2)ws WS/W4
1 81 32 .395
2 289 64 .221
3 841 112 .133
(l) W4ENumber of storage words for junction matrix
in ENPORT-4.
(2) WSENumber of storage words for junction matrix
in ENPORT-5.
Table 4. Processing Time for State Model Formulation
Example I (1)PT4 (Z)PTS PTS/PT4
1 0.604 ’ 0.138 0.228
2 0.905 0.346 01382
3 1.420 0.927 0.653
(1) PT4EProcessing time (in seconds) for ENPORT-4.
(2) PTSEProcessing time (in seconds) for ENPORT-5.
V. SUMMARY
The basis order rules are among the most significant
results achieved in this investigation. For weighted junc-
tion structures, these topology-based formulations provide
the bond graph analyst with the composition of a basis for
the junction structure transformation. In addition, when
used with Theorem 3, the basis order rules provide a "good"
estimate of the number of distinct basis variable sets for
the junction structure transformation.
Herein, it was shown that the standard sequential
causality assignment procedure assures that the S S
23’ 32’
and 833 blocks of the reduced junction matrix are zero for
°a reducible junction structure. This resulted in'a major
simpfijicationijlRosenberg's state model formulation algo-
rithm which serves as a model for the ENPORT-S state model
formulator [1fl.
As indicated by the computer results in Chapter IV, the
ENPORT-5 state model formulator provides heretofore unreal-
ized efficiency and speed in the automated processing of
linear time-invariant bond graphs. The key features of
this formulator are (l) the use of sparse-matrix-based
storage, (2) the determination of junction structure reduci-
bility prior to the formation of the junction matrix, (3)
the direct construction of the junction matrix, and (4) the
54
versatile sparse-matrix-based INPRD subroutine. As a result
of the storage and processing efficiency of the ENPORT-5
state model formulator, the ENPORT-S program has a greatly
enhanced capacity for the processing of large bonds graphs.
Future advances in graph processing efficiency can be
achieved by the development of a general technique which
does not require matrix inversions for the determination of
junction structure reducibility. A step in this direction
can be made by the development of ”basis order rules" which
are applicable to any junction structure containing a gyrator.
Such formulations would offer necessary conditions for the
reducibility of an arbitrary junction structure, as well as
serve as aids for the determination of a basis for the junc-
tion structure transformation.
BIBLIOGRAPHY
[11
[2]
[31
['41
[51
[61
[71
[8]
[9]
BIBLIOGRAPHY
Chorafas, D.N., Systemsand.Simulation, Academic Press,
1965.
Malmberg, A.F., "NET-2 Network Analysis Program-User's
Manual Release 9", HDL-OSO-l, Braddock, Dunn, and
McDonald, Inc., El Paso, Texas, 1973.
Nagel, L. W. , and Pederson, D. O. "SPICE (Simulation Pro-
gram with Integrated Circuit Emphasis)", ERL- M382,
Electronics Research Laboratory, College of Engineering,
Univ. of California, Berkeley, Ca. 94720, April, 1973.
Chace, M. A. , and Angell, J. C. "Users Guide to DRAM
(Dynamic Response of Articulated Machinery)", Design
Engineering Computer Aids Laboratory, Department of
Mechanical Engineering, University of Michigan, Ann
Arbor, Mich., Feb., 1976.
Dix, R. C. "A Users Manual for MEDUSA (MEchanism-
Dynamics- -Universal System Analyzer)", Mechanics,
MechanicaT and Aerospace Engineering Department,
Illinois Institute of Technology, Chicago, ILL. , Jan.,
1975.
Pilkey, W., and Pilkey, B., eds, Shock and Vibration
Computer Programs - Reviews and Summaries, SVM-lO,
The Shock and Vibfation Information Center, Naval
Research Laboratory, Washington, D.C., 1975.
Bowers, J.C., O'Reilly, J.E.,znu1Shaw, G.A., "SUPER*
SCEPTRE - User's Manual", DAAA-Zl-73-C-0655, AMC,
Picatinny Arsenal, Dover, N.J., 1975.
Rosenberg, R.C., "A User' 5 Guide to ENPORT- 4", John
Wiley, Inc., 1974.
Paynter, H.M., Analysis and Design of Engineeripg
Systems, M.I.T. Press, 1960.
55
[101
[ll]
[12]
[13]
[14]
[15]
[161
[17]
[18]
[191
[20]
[211
56
Karnopp, D.C., and Rosenberg, R.C., Analysis and Simu-
lation of Multiport Systems - The Bond Graph Approach
to Physical System Dynamics, M.I.T. Press, 1968
(68A35012).
Gebben, V.D., "Bond Graph Bibliography for 1961-1976",
Trans. ASME, J. Dyn. Sys., Meas., and Control, 99(1977)
pp. 143-145.
Nobuhide, 8., "Bond Graphs: Structural Properties and
Augmentation Algorithm", Presented at the Conference
on System Control Theory and its Applications to Dynamic
Economic Models, Nagoya City University, Mar. 27-29,
1976.
Ort, J.R., and Marten, H.R., "The Properties of Bond
Graph Junction Structure Matrices", Trans. ASME, J.
Dyn. Sys., Meas., and Control, 95(1973) pp. 362-367.
Perelson, A.S., "Bond Graph Junction Structures", Trans.
ASME, J. Dyn. Sys., Meas., and Control, 97(1975)
pp. 189-195.
Rosenberg, R.C., "State-Space Formulation for Bond
Graph Models of Multiport Systems”, J. Dyn. Sys.,
Meas., and Control, Trans. ASME, 93(1971) pp. 35-40.
Dixhoorn, J.J. van, "Simulation of Bond Graphs on Mini-
computers", J. Dyn. Sys., Meas., and Control, Trans.
ASME, 99(1977) pp. 9-14.
Busacker, R.G., and Saaty, T.L., Finite Graphs and Net-
works - An Introduction with Applications, McGraw-Hill,
1965.
Harary, F., Graph Theory, Addison-Wesley, 1972.
Rosenberg, R.C., and Andry, A.N., Jr., "Solvability of
Bond Graph Junction Structures with Loops", IEEE Trans.
on Circuits and Systems, 26(1979) pp. 130-137.
Rosenberg, R.C., and Andry, A.N., Jr., "Solvability of
Certain Classes of Bond Graph Junction Structures", Pro-
ceedings of the Second International Symposium on Large
Engineering Systems, Univ. of Waterloo, Waterloo, Ontario,
Canada, May, 1978, pp. 537-541.
Ort, J.R., and Martens, H.R., "A Topological Procedure
for Converting a Bond Graph to a Linear Graph", Trans.
ASME, J. Dyn. Sys., Meas., and Control, 96(1974)
pp. 307-314.
\\I
I\)
. J
(Erapns'fi .I.
r—v
Di
Perglson, A.S., Discussion re [13], Trans. Asme, J.
' 'ys., Meas., and Control, 98(1976) pp. 209-210
Peix)!son, L S. :1nd (Ester. (3.F., "Bond.(3rarflus and.I.1nea1‘
I ranklin Ilh stit1te 302(1976)159-185
Karnopp, 9. C., and Rosenberg, R.C., System Dynamics
\ “Unilie u \pproach, John Wiley Inc., 1374.
Roscntm org, R.C.. ”Essential Gyrators and Reciprocity
in 111 ticn Structures”, (accepted for publication)
,.J
C
J. Franklin Institute.
Karnopp, D.C., ”Some Bond Graph Identities Involving
Junction Structures”, 7Trans. ASME, J. Dyn. Sys.,
Meas., and Control, - (197 5) pp. 439-440.
rste Computational Structures,
Korfhagc, R.R., )1
jcau mic Press. 19:
Berztiss, A.T., Data Structures - Theory and Practice,
Academic Press, 1971.
Knuth, I).E., The jnn:(1f (omzute 1f Progranmruig, Vol. I,
Chap. 3, \ddison—Wesley, I973.
Stone, H.S., Introduction to Computer Organization and
Data Structures, McGraw—hill, 1973
Iowarson, R.P., Sparse Matrices, Academic Press, 1973.
Riordan, J., Combinatorial Identities, Wiley, 1968.
APPENDIX A
DERIVATION OF THE BASIS ORDER RULES
A;l_ Basis Order and Simple Junction Structures
In this section we derive a pair of general compu-
tational rules for predicting the order and variable-type
composition of an N-port SJS basis. An N-port JS has exactly
N EN-nodes. It is common usage to refer to bonds (0,EN) or
(1,EN) as port bonds.
Motivation for these rules is derived by consider-
ing the number of free variables which remain following the
imposition of a set of independent constraint equations on a
set of system variables. Several types of proper SJS's are
studied first; then the results are extended to standard SJS's.
In passing, alternate forms of the order rules for proper
SJS's are given. The order rules are presented here as
Theorem 1.
Theorem 1: Every standard SJS satisfies the relations
(1)E = NB + NO - B0 - N1
and
(11)F = NB + N1 - B1 - N0.
A.1.l Basis Order for Proper Simple Junction Structure
Forests
Initially we establish Theorem 1 for an arbitrary
proper SJS tree G by demonstrating that G can be obtained
58
59
from a forest of separate 1-junctions and O-junctions by a
series of subgraph concatenations. It is observed that
junctions satisfy the order rules. Following the assumption
that G contains more than a single junction, a O-junction in
G is identified as a base node to which is added an appropriate
number of l-junctions and O-junctions, of specified degrees,
which yields a SJS equivalent to G. Additionally, it is noted
that the concatenation of a junction to a proper SJS tree
yields a proper SJS tree which satisfies the order rules, thus
yielding the results for G. Finally, by considering the order
rules for each component, the results are extended to an
arbitrary proper SJS forest.
Lemma A.1: Every proper SJS forest satisfies the relations
E = NB + N0 - B0 - N1 and F = NB + N1 - B1 - No.
Prior to proving Lemma A.l, two definitions are needed.
First we state that two distinct SJS's are conformable if one
contains a 0 and the other contains a 1.
We now define the graph concatenation operator C
where C(G,H) = K is a binary operation performed on two proper
SJS's (G and H) which are conformable to yield a third proper
SJS (K). Let G be a connected proper SJS and H be a connected
proper SJS where VG n VH = 0. Also, let u be a l-junction and
uEN be an EN-node 1n VG,
EN-node in VH’ where (u,uBN) EKG and (v,vEN) eXH. Then C[G(u),
and v be a O-junction and vEN be an
H(v)] will denote the connected proper SJS K where
60
VK [V u V { } and
G H] VEN’uEN
X
K [X
G u XH u {(v,u)}] - {(v,vEN),(u,uEN)}.
Note that if K = C[G(u), H(v)] and K' = C[G(u), H(v)]
(different EN-nodes are removed) then K and K' are isomorphic.
For C[G(u), H(v)], it will be said that G and H are "concaten-
ated". Note that C[H(v), G(u)] = C[G(u), H(v)]. We now pro-
ceed with the proof of Lemma A.l.
Proof: Let G (m) denote a connected proper SJS such that
1
VG (m) consists of exactly one l-junction and exactly
1
m EN-nodes, where m22. Thus C (m) has the form shown
1
in Figure A.l.
Observe that the definition of a l-junction applied to 61(m)
yields the results
E = N + N - B - N
B 0 0 1 U“) + (0) " (0) ' (1) In - l and
’1']
ll
2
+
Z
I
w
I
2
II
II
H
B 1 1 0 U“) + (1) " 0n) ‘ (0)
These results agree with Lemma A.l.
Let G0(n) denote a connected proper SJS such that
VG (n) consists of exactly one O-junction and exactly n EN-
0
nodes, where n22. Thus, G0(n) has the form shown in
Figure A.2.
G (n)
Observe that the definition of a O-junction applied to 0
yields the results
E B+NO-BO-N1 (n)+(1)-(n)-(O)
ll
2
l and
'11
ll
2
+
Z
I
w
I
2
II
II
:5
a
H
B 1 1 0 (n)+(0)-(01-(1)
61'
These results agree with Lemma A.1.
Note that Lemma A.1 applies to a forest with an
arbitrary number of components, each of which is a 0- or
l-junction together with a set of EN-nodes.
Now consider a concatenation involving G1(m) for mZZ.
Let G be an arbitrary connected pr0per SJS where VG contains at
least one 0-junction, say v, and let EG and FG be given.
Observe that K = C[G(v),Gl(m)], where m22, contains one less
effort variable and one less flow variable than GLJGl(m)
< V 11V (m), XGIJXG (m)>. since 0(X (m)) - l.
c; c K) = 00(19ch
1 l l
(”0" denotes "order of".) Also, K and GLJGl(m) yield the
same number of independent flow constraint equations, since
the number of junctions and junction degrees are unchanged.
Then, clearly
EK = EG + EG UN)‘ 1 and FK = FG + FG UB)- l.
1 l
(m): _ (m): ,
Let AE1 _ EK EG and AFl - FK FG.
Then
AE(m)=E(m)-l=m-2andAF(m)=F(m)-l=0.
1 G1 1 G1
where m22.
That is, as a result of a concatenation involving Gl(m) the
incremental changes in E and F are known.
Now consider a concatenation involving 60(n) for
n22. Let G be an arbitrary connected proper SJS where VG
contains at least one l-junction, say u, and let EG and FG
be given. Observe that K = C[G(u), 60(n)], where n22
62
contains one less effort variable and one less flow variable
than GlJGO(n). Also, K and GIJG (n) yield the same number
0
of independent effort constraint equations and the same num-
ber of independent flow constraint equations. Therefore,
EK = EG + EG(n) - l and F FG + FG(n) - l.
K
(n) , (n) -
Let AEO _ EK EG and F0 _ FK FG' Then
1E (n) E (n) — 1 = 0 and 1F (n) = F (n) - 1 = n - 2,
0 GO 0 G0
where n22.
We now establish lemma A.1 for an arbitrary proper SJS tree.
Let G be an arbitrary proper SJS tree. Suppose G contains
and N are
N l-junctions and N O-junctions; not both N
1 0 1 0
zero, since G is proper. If NO=0, G is a l-junction com-
ponent; if N1=O, G is a O-junction component. In either case,
we are done. Therefore, assume N1>0 and NO>0.
Enumerate the O-junctions in V by v ,v ,...,vJ ,
G l 2 N0
and the l-junctions 1n VG by vN +1,vN +2,...,vN +N . Let
0 0 0 l
(0) . - (1) .
VG {v1,v2,...,vNO} and VG {VN0+1’VNO+2"'”VNO+N1}'
Then VG(1)r1VG(O) = g and VG(1)lJVG(O)
contains all junctions in G. The junctions in G will now be
partitioned according to their distances from v1. Let S_l = 0
and S1 = {veVG(1)11VG(0)|d(v1,v) = i}, where i = 0,1,2,...
Note that Siswsj = 0 if i f j. The order of VG(1)LJVG(O) is
finite and G is a tree imply that there exists a smallest
63
positive kiN + N -1 such that d(v1,v):k for all veVG(1) VG(O).
l 0
Without loss of generality, assume k is odd. Then
k-l k-l
I—z ) (0) (T) (1)
.3 521 ‘ V0 and .3 S21-1 7 VG °
1-0 1-0
Relabel the elements in Si so that vi j is the jth element
in 81’ where Oiiik and l:j:o(Si). Let ai,j = deg(vi,j) for
in G. Also, let G = G (deg VI), and let G. . be the
0,0 0 1.)
proper SJS tree corresponding to vi j where Gi j is either
GICm) or G0(n)
V..
1,1
if i is odd or even, respectively, and m or
11:61. ..
1,3
Now we will reconstruct G from a proper SJS forest
of NO O-junction and N1 l-junctions by using the internal bonds
of G as a directory. Note that the concatenation of two con-
formable proper SJS trees yields a proper SJS tree.
Let G = 00,0, and let 01,1 = G1,0(S0)be the proper
1,0
SJS tree obtained from the series of concatenations of 01 0
’
1,j)€xG; i.e.,
133:0(81). Let GZ,O=Gl,o(SO)’ and let 62,1 be the proper SJS
with Gl,j at v0,1 for each 3 such that (v0’1,v
tree obtained from the series of concatenations of 02 0 with
9
G at v1,1 for each j such that (vl,l,v2 .)eXG. Let G
2,1 ,J 2,2
be the proper SJS tree obtained from the series of concatena-
tions of G2 with G 1,2 for each j such that (v1,2’
’1 2,j
V2,j)€xG‘
In general, let Gi n be the proper SJS tree obtained
from the series of concatenations of G, 1
1,n-
for each j such that (vi-lnfvi,j)€XG’ where Gi
with G1 . at v.
,0: i-1,0(S.-
liiikand l:n:o(Si_ Then G =0.
1)' k,o(Sk_1)
64
An example of the construction procedure is given
in Figure A.3.
The construction procedure involves NO-l concaten-
ations of proper SJS trees having the form 60(n), and N1
(m)
concatenations of proper SJS trees having the form G1
Therefore,
N II+N
EG = E6 = EG + _20 AE0(deg vi) + .20 1 AE1(deg Vi)
k,o(Sk_1) 0,0 i=2 1=N0+l
NO+N1 NO+Nl
= (l) + (0) + 2 (deg v.-2) = 2 deg v.-2N1+1 and
i=N +1 1 i=N +1 1
0 0
N LI+N
PG = FG = PC + .20 1P0(deg V1) + .20 1 AFl(deg Vi) =
k,o(Sk_1) 0,0 1=2 1=N0+1
N0 N0
(deg vl-l) + 2 (deg v.-2) + 0 = 2 deg vi-ZNO + 1.
i=2 1 i=1
N0
Observe that (i) B = Z deg v.,
0 . 1
1=l
N +N
(11) B = 20 1 deg v ,
l . 1
1=N0+l
(iii) N = N + N - l (Euler's Rule),
(iv) N = B + B - N
(v) P = B - N
(vi) P = B - N
65
Therefore, if G is a proper SJS tree, then
N0+N1
E r .8 deg vi -2N1 + l = NB + N0 - BO - N1
1—N +1
0
N0
F = 121 deg vi = 2N0 + 1 = NB + N1 - Bl - NO'
Note that deg via 2, Is isN implies that E2 1 and F2 1.
0,
We now extend the results to an arbitrary proper SJS forest
G. Let G1, G2,..., Gn be the components of G. Then each
component G1 is a proper SJS tree.
Therefore,
BG. 7 NBC. + N00. 7 N10. and
1 1
PG. = NBG. + NlG. - BIG. - NOG.’ where ls ISIL
1 1 l 1 1
Then
11 n
E = 2: E = z (N 1+ N - B - N )
G i=1 oi i=1 B0i 001 00 10i
n n n n
= z N + z N - z B G. - E N = N N B N ,
i=1 BGi i=1 00i i=1 0 1 1:1 10 BC 00 00 1G
and
n 11
Fe 7 .E F0. ' E (NBC. + N10. BIG. ‘ Nos.)
1-1 1 1-1 1 1 1 1
n n n
7 .E NBG. + E N10. ' .E ”00 7 NBG NlG B10 NOG’
1—1 1 1-1 1 1—1
where EG 2 n and F(3 2 11.
Hence, if G is a proper SJS forest, then G satisfies the rela-
tionsE=NB+NO-B -N andF=N +N -B -N
0 1 B 1 1 0'
66
Recall that B0 is the total number of (external
and internal) bonds incident to O-junctions. Observe that
in a proper SJS every internal bond is incident to exactly
one l-junction and exactly one O-junction; thus, in a
proper SJS, NI is included in BO. Therefore, for a proper
SJS, NB = P1 + BO' Similarly, for a proper SJS, NB = P0.+ B1.
These results and Lemma A.1 validate the following corollary.
Corollary 1.1: Every proper SJS satisfies the relations
(1) E N + P - N
0 l 1
(ii) F
N1 + PO - NO.
A.1.2 Basis Order for Proper Simple Junction Structures
Now Theorem 1 will be established for an arbitrary
connected proper SJS G. This will be accomplished by removing
cycle bonds from G until a spanning tree is obtained, and then
demonstrating that the previous relations remain valid when
these bonds are replaced. These results will then be extended
to an arbitrary proper SJS.
Lemma A.2: Every proper SJS satisfies the relations
NB + N0 - B0 - N1 and
[T1
II
F = NB + N1 - B1 - NO.
Prior to proving Lemma A.2 we define a transformation T
which removes cycles from a connected proper SJS, and define
a transformation S which creates a cycle in a connected
proper SJS.
67
To obtain T, let G be an arbitrary connected
proper SJS. Assume G contains at least one cycle, say C.
Let b be an arbitrary bond in C, and let vEN and uEN be
EN-nodes. Then there exists a unique l-junction veVG and a
unique O-junction ueVG such that b = (v, u). Let T [G(v, u)]
H 7 V0 U {VEN’UEN EN)’
(u,uEN)}]-{(v,u)}. Then, clearly, H is a connected proper SJS
denote the SJS H where V } and XH£E[XGu{(v,v
which contains one less cycle than G. Observe that N1H = NlG’
”on 7 ”00’ p111 7 p10 + 1' and Pon 7 P00 + 1'
To obtain S, let G be an arbitrary connected proper
SJS. Suppose there exist l-junction veVG and O-junction ueVG
such that (v,u)¢XG. Also, assume there exist EN-nodes vEN
and uEN in VG such that (v,vEN) and (u,uEN) are in X6. Then
} and
let S (G,v,u) denote the SJS H where VHEEV {v
G ' EN’ uEN
XH Etxcu{(V,u)}] - {(v, VEN),(u,uEN)}.
Then H is unique (to an isomorphism), and H is a
connected proper SJS which contains one more cycle than G.
Observe that S(G,v,u) contains One less flow variable and one
less effort variable than C, where S(G,v,u) and G yield the
same number of independent flow constraint equations and the
same number of independent effort constraint equations. There-
fore, if H = S(G,v,u), then
B - 1 and F = F - 1.
EH 7 G H G
Let ABS 5 EH - EG and AFS s FH - FG
Then AES = -l and.AFS = -1.
We now proceed with the proof of Lemma A.2.
68
Proof: Since for a proper SJS we have NB = P1 + B0 and NB =
P + B it is sufficient to show that every proper
0 1’
SJS satisfies the relations F = N1 + P0 - N0 and
E = NO + P1 - N1.
Consider an arbitrary proper SJS G. If G is a
forest, then, done, by Lemma A.1. Therefore, assume G con-
tains at least one cycle. Let G be connected and let GOEEG.
Also, let vi and ui denote l-junctions and O-junctions
respectively, for all 1. Then G contains some cycle C1.
0
Let b1 = (v1, ul) be an arbitrary bond in C1, and set G1 =
TIGO,(v1,u1)]. In general, 1f Gi-l contains some cycle Ci’
let bi=(vi,ui) be an arbitrary bond in 2., and set Gi=T[
G.
1 1-1’
Wi,ui)]. Then the order ofXG finite implies that G contains
a finite number of cycles, which implies that there exists a
smallest positive integer k such that Gk is a spanning tree.
Note that N = N 7 N16, P06 = p06 + k’ and PIG =
06k k k k
P16 + k. G is a proper SJS tree. Therefore,
N
0G’ 16
k
E = N + P - N ,and F = N + P - N
Gk 0Gk 1Gk le GK 1Gk
Observe that SIGi’Vi’ui] = Gi-l’ 1=k,k-l,...,l; i.e., k appli-
cations of S to Gk yields GO.
Therefore,
+ k (AE ) = (N + P - N ) + k(-l)
S 06k le le
B7 .. _ = ..
(”00 I pic I k N10) k N00 I P10 N10 and
F=F +k(AF)=(N +P -N )+k(-1)
G Gk S 1Gk 0Gk OGk
= (N16 I Poc I k ' N00) 7 k 7 N10 I poo 7 N00'
E = E
G Gk
69
Thus, if G is a connected proper SJS, then G satisfies
E = NO + P1 - N1 and F = N1 + P0 - NO. Assume G 15 not
connected. Let 61’ GZ"°°’Gn be the components of G. Then
each eomponent of G is a connected proper SJS which implies
that
BC. I NOG. I PlG. ' N16. and FG. I NlG. I POG. ' N0c; ’
1 1 1 1 1 1 1
where lsisn.
Therefore,
n n n n
E == 2 .F = 23 N + Z: P - Z N' "N P N ,
G i=1 Gi i=1 0Gi 1:1 161 1:1 1G 0G 16 16
n n n n
and F = 23 F E IN + Z I’ - 22 N = N P N .
G i=1 Gi 1:1 lGi 1:1 OGi i=1 0Gi 1G CG CG
Hence, if G is a proper SJS, then G satisfies the relations
E = NO + P1 - N1 and F = N1 + P0 - NO,
the relations E = NB + NO - B0 - N1 and F = NB + N1 - B1 - N0.
and thus, G satisfies
A.1.3 Basis Order for Standard Simple Junction Structures
In this section we prove Theorem A.1.
Theorem 1: Every standard SJS satisfies the relations
E=NB+NO-BO-Nl and
F = NB + N1 - Bl - NO.
The proof of Theorem 1 is preceeded by the definition of the
transformation T which reduces the number of bonds formed by
nodes of the same type in a standard SJS.
70
Let G be an arbitrary standard SJS. Assume G is
not proper, i.e., G contains a bond of the form (v1,v2)
where v1 and v2 are nodes of the same type.
Then for (vl,v2)eXG, where v1 and v2 are nodes of
the same type, let u be a junctionof a node-type distinct
v1 and v2. Without loss of generality, if v1 and v2 are EN—
nodes, then let u be a l-junction. Then T(G,v1,v2) will
denote the SJS H where 'HEVGU{u} and XHEEXG - {(v1,v2)}]u
{(vl,u),(u,v2)}. Observe that H is a standard SJS which con-
tains one less bond of the form (1,1), (0,0) or (EN,EN) than
G (see Figure A.4).
Note that H = T(G,v1,v2) contains one more effort variable
and yields one more independent effort constraint equation
than G. Therefore, EH=FG' Similarly, H contains one more
flow variable and yields one more independent flow constraint
equation than G. Thus, FH=FG. Observe that if v1 and v2 are
N
l-junctions, then N 1,,N N N N
BH= BGI ‘1H= lG’ 0H: 06
B0H=BOG+2. Also, if v1 and v2 are O-junctions or EN-nodes,
+1, BlH=BlG’ and
1H=N16I1’ 0H= OG’ 1H= icI 0H=
We are now prepared to establish the order rules for an
then NBH=NBG+1, N N N B B 2, and B B
06'
arbitrary standard SJS.
Proof: Let G be an arbitrary standard SJS. If G is a proper
SJS, then done, by Lemma A.2. Therefore, assume G is not
proper. Let k0,k1, and k2 be the number of bonds in G of the
forms (0,0),(1,l) and (EN,EN) respectively. Let k=kn+k1+k2
nil, mil
EN
1
EN
n>0, m>0
n>l, m>l
Transformation for converting a standard
SJS to a proper SJS
Transformation for converting a WJS
to a SJS.
EN -——————-D'
EN
Figure A.4.
Figure A.S.
APPENDIX B
AN APPLICATION OF THE BASIS ORDER RULES
Definition: For a WJS (weighted junction structure), a
causal form is feasible if it does not
violate any l-junction, O-junction, or TF
node constraint, and every port bond is
causally oriented.
The effort-flow variable composition of a basis can be
determined from the topological properties of a WJS.
The composition is given by the basis order rules, the
general froms of which are
NE = NB + NO - BO - Nl - NT
and (1)
NF = NB + N1 - B1 - NO - NT
Example 1
The WJS in Figure 3.1 has the following topological pro-
perties: NB = 12, N0 = 2, N1 = 3, N = 2, B0 = 5, B =
T 1
The basis order rules yield
NE=12+2-5-3-2=4
and
80
9.
NF=12+3-9-2-2=2
This input pattern is illustrated by the causally augmented
WJS in Figure 3.2.
Henceforth, all weighted junction structures will be con-
sidered as n-port structures.
In the process of obtaining a "good" upper bound for the
number of unlabelled feasible port bond causal orienta-
tions, three expressions for upper bounds will be derived
where each successive expression requires greater knowledge
of the junction structure and yields a smaller upper
bound.
B.1 An Upper Bound For C As A Direct Application Of
The Basis Order Rules.
A question of particular interest concerns the number of
distinct port-variable bases which an n-port possesses.
The basis order rules yield an upper bound for the number
of such bases.
From the fact that every port bond can accept exactly
two causal orientations it is easily seen that
(
c_% (m
where C is the total number of unlabelled feasible port
bond causa1 orientation and U0 = 2 NP.
82
However, an improvement over UO can be obtained by a
direct application of the basis order rules.
Given NP port bond with NE efforts as inputs, a (gen-
erally) smaller upper bound can be expressed as
CiUi (n
where
NP
U=< > M)
1 NE
and
m%mn fi 01mim
(9= (3
0 otherwise,
for integers n and m.
Notice that Ul represents a significant reduction from
the coarse upper bound U0.
Noting that
NP = NE + NF, (6)
one obtains the related form
N NP 1 N
P P
()= . =(). (7)
NE NEINF! NF
As would be expected, the results are symmetric with
respect to effort and flow variables.
83
The inequality U1 3 U0 can be demonstrated by expressing
ZNP as a binomial expansion.
I
Recalling the Binomial Expansion Theory,
n D
(a + b)r1 = z (bamkbk,
k=O I
let a = 1, b = 1, and n = NP. Then
N N NP NP
2P=(l+1)P=Z(k). (8)
N N
Then for O 5 NE E NP and 0 1 NF i NP, (NE) and (NP)
F
are merely symmetrical terms in (8). Thus U1 1 U0.
In particular, if NP 3 1, then U1 < UO .
Example 2
Referring to Figure B.1,
N = 6, N = 4, and N = 2.
P E P
Then
U0 = 26 = 64
and
U1I(2)IZ§IB_II15
Thus, C i 15 < 64.
84
8.2 A Refined Upper Bound for C
In general, U1 can be improved upon considering the
constraint equations associated with WJS elements. It
will be assumed that the WJS of interest contains no
external TF elements. This assumption results in no loss
of generality, due to the causal properties of TF element
[25].
Let e be the number of port effort inputs to external
O-junctions. Similarly, let f be the number of port
flow inputs to external 1-junctions.
For a given e, if "e" port efforts are inputs to the
A0 external O-junctions, then P0 - e port flows are
inputs to the remaining PO - e port bonds which are
incident to O-junctions. Thus, NF - (PO - e) port flows
are inputs to the Al external l-junctions, and NE - e
port efforts are inputs to the remaining Pl — (NF - PO + e)
port bonds which are incident to l-junctions.
Note that
- P + e (9)
and
85
Then
(10)
- ME A0
IJ2 I E ( e) (N
e-LE
A1
) (ll)
F-P0+e
for some ME and LE.
If NE i A0, then ME = NE‘ If NE > A0, then ME = A0.
Thus,
ME = min(NE,AO) . (12)
Suppose Pl 2 NE' Then there are at least NE - Pl port
effort inputs to the A0 external O-junctions. Then
L = N - Pl' Observe that f = 0 when e = NE - Pl’
E E
Suppose Pl > NE' Then NF > PO, and there can be a
minimum of zero port efforts inputs to the AO external
O-junctions. Then LE = 0. Observe that f = NF - PO > 0
when e = 0. Thus,
(13)
I"
ll
max (0, NE - P1)
Equation (11) is symmetric in e and f. Thus, U2 can
be formulated in terms of specified port flow inputs, if
desired.
86
By (9) we have e NE - Pl + f, since N. + N = P + P
E F 0 1'
Then, by (5),
ME A A NE A A Pi A A
Z (e0)(N -Pl+e) I Z (e0)(N -% +e) I Z (N -3 +f)(fl)
e=LE ‘F 0 e=LE F o f=LF E 1
$1 A1 ) A1 gr AO ) A1
= ( _ ( ) = ( _ ( )
szF NE Pl+f f fILF NE P1+f f
where M? = min (NF, A1) and LF = max (0, NF - PO).
It will now be shown that U2 i U1.
Suppose e port efforts are inputs to the A0 external
(A1)
NF-P0+e
consistent input assignments of NE - e port efforts to
O-junctions. Then there are at most causally
the Pl port bonds incident to l-junctions, given
causally consistent input assignments of NF - PO + e
port flows to the P1 port bonds incident to l-junctions.
P
l
)
Observe that (N -e is the number of unrestricted input
E
assignments of NE - e port efforts to the Pl port bonds
incident to l-junctions. Therefore,
Ai Pi
(NF-P0+e) i (NE-e) (14)
Therefore, by (5) and (14).
87
MI (A0>< I1 > NI (AO>< A1 >
egLE e NF-P0+e egO e NF-P0+e
N N .
EA P EP P P+P N
o 1 o l_Ol_P
< ego (e )(NeIe) : ego (e )(NEIe) - ( NE ) - (NE)
Therefore, U2 1 U1. In particular, whenever Al < P1
or A0 < PO, U2 is a significant improvement over U1.
Example 3
Again referring to Figure B.1, A0 = 0, A1 = 3, P0 = 0,
P1 = 6, NE = 4, and NF = 2. Therefore, M
and LE = max (0, 4-6) = 0. Thus,
E
_ 0 3 _
which is much lower than the upper bound of 15 obtained
in example 2. In addition, the junction structure in
Figure B.l has exactly three unlabelled causal orienta-
tions which are given in Figure 3.3, hence, C = U2 in
this example.
The significance of U2 is captured by the following
theorem.
Theorem 3: The number of distinct basis variable sets
for a WJS transformation is bounded above by
MEA A M? A A
u2 = z < 0 < 8 +f)
e=LE NE- 1
l
)( ) =
= min(4, 0) = O
88
where M = min(N = max(O, N
E E’ E EIPL)’
MF = min(NF, Al), and LF = max(O, NF-PO).
A0), L
89
EN.__.1___EN
EN EN
\
l—TF—O —TF—o—-1
EN \EN
Figure B.l. Example of a weighted junction
structure.
90
EN _m___EN
EN EN
\IP—Tn—O—ATF—jo—h/
.N/
N
Figure B.2. Causally augmented weighted
junction structure.
91
EN -Ill—EN
EIII/\il— TFl—_l_—| TF—-|O ——1i/\EN
M /\
EN
IN’N EN I—II—EN /EN
1|— TFl—— OF—TFF—OF-l
EX “\EN
EN }-—1|—— EN
EN EN
-l—-1TF ——|(|)—1TF——{o ——-|1/\
EN EN
Figure 3.3. Unlabelled causal orientations
of a weighted junction structure.
APPENDIX C
THE RESOLUTION OF A CONFLICT RESULTING FROM A PORT BOND
CAUSAL ORIENTATION
Property'ZJJ Let G be a junction structure with a port bond
b. Suppose a causal orientation (and subsequent
causal extension) of b results in a causal con-
flict, then the opposite causal orientation
(and subsequent causal extension) of b will not
yield a causal conflict.
Proof:
It will be assumed that each causal orientation of b is
followed by the causal extension process, and that G is acausal
(since the pruning process of Appendix E can be initially
applied to remove all causally oriented bonds and causally
completed nodes).
G contains a node of degree greater than two since a
causal orientation of b results in a causal conflict. From a
graph theoretic perspective, a causal orientation of b defines
a "walk” of G [18]. Let ENO be the field-node incident to b.
Then there exists a shortest path in G joining ENO and a
JS-node, V0, of degree greater than two such that (1) every
JS-node (exclusively) between ENO and V0 has degree two, and
(2) there exists no shorter path in G joining ENO and a JS-
node of degree greater than two.
92
93
Let d denote the path bond incident to V0. A causal
orientation of b resulting in a causal conflict implies that
the resulting causal orientation of d gives d strong causal
implication with tespect to V0. The reversal of the causal
orientation of b results in a reversal of d's causal orien-
tation, since ENO and V0 are joined by a sequence of JS-nodes
of degree two. This gives d weak causa1 implication with
respect to V0, which leaves VO causally incomplete; conse-
quently, no causal information propagates beyond V0 and no
causal conflict can result. Hence, if a causal orientation
of b yields a causal conflict, then the opposite causal
orientation of b will not yield a causal conflict.
APPENDIX D
THE IMPACT OF THE STANDARD SEQUENTIAL CAUSALITY ASSIGNMENT
PROCEDURE (SSCAP) ON THE REDUCED JUNCTION MATRIX
Theorem D.l: Let G be a bond graph with an algebraically
reducible junction structure. Assume that
G can be completely and consistently causally
oriented by SSCAP. Then dependent storage
field inputs are determined by source field
and independent storage field outputs. More-
over, no dissipation field input is determined
by a dependent storage field output.
Proof:
It will be assumed that (l) a causal assignment is
always followed by causal extension, (2) each field multiport
is a one—port, and (3) each field multiport is adjacent to a
O-junction or l-junction. It has been shown that a linear
time-invariant field multiport with n ports can be replaced
by n one-port field multiports [25]. Thus, for this reason
and due to the causal characteristics of junction structure
nodes, the above assumptions result in no loss of generality.
Consider the acasual representation of G = G given in
1
1 is the number of sources, m1 is the
number of storage multiports, and p1 is the number of dis-
Figure D.l where n
sipation multiports. Without loss of generality, assume nlzl.
94
95
In the following discussion only consistent causal orienta-
tions of source bonds will be considered.
Causally orient the bond incident to an arbitrary source
in G1. Let u1 denote the output from this source. Prune
(see Appendix E) from G1 all resulting causally oriented bonds
and causally completed nodes. (Note that no causal conflicts
can result from source bond orientations due to the theorem's
hypothesis). Let G2 be the bond graph obtained from G1 by
the pruning procedure (see Figure D.2). If any dependent
storage field bond or any dissipation field bond is pruned from
G then each corresponding dependent storage field input or
1’
dissipation field input is determined by 111 together with
prior information. If G2 # Q, then done. Assume G2 f B.
Then G has the acausal form given in Figure D.l where n1,
2
m1, and p1 are replaced by n2, m2, and p2 respectively.
Observe that if nzzl, then the source bond causal orienta-
tion process together with the bond graph pruning procedure
can be repeated (at most n1 times) until all source bonds
of G have been causally oriented. Then there exists a
smallest positive integer ksn1+l such that Gk contains no
sources. (If n1=0, then k=l). nlzl a. k22. Then each
dependent storage input and each dissipation input specified
in Gh is determined by u uz, . . ., uh where lshsk-l
1’
and uh is defined similar to ul.
96
Consider Gk' Gk has the acausal form given in Figure
D.3. If mk=0, then done. Assume mk>0. Assign integral
causality to an arbitrary storage bond, b, in Gk' If a
causal conflict results, then restore Gk to its acausal
form and select a different storage bond to causally orient.
In this case, property 2.2 of the causal extension process
indicates that sufficient input information was available
to determine the variables associated with bond b, i.e.,
the storage element incident to b is dependent and its in-
put is determined by ul, uz, . . ., uk_1. Suppose the inte-
gral causal orientation of b does not result in a causal
conflict in Gk' Let x1 be the resulting output of the
storage node incident to the oriented b. Let G be the
k+l
bond graph obtained from Gk by the pruning procedure. If
any dependent storage bond or any dissipation bond is pruned
from Gk’ then each corresponding dependent storage input
or dissipation input isdetermined by x1 in Gk’ and therefore,
is determined by ul, uz, . .I., uk_1, x1 in G.
Gk+1 has the acasual form given in Figure D.3 with k
replaced by k+l. If mk+l=0’ then done. If mk+1f0, then the
above storage bond integral orientation and graph pruning
process can be repeated (at most mk times) until the integral
orientation of any remaining storage bond yields a causal
conflict, i.e., there is a smallest integer 130 such that
either the integral causal orientation of each storage bond
in Gk+2 results in a causal inconsistency or mk+2=0 where
limk, As noted above, if the integral causa1 orientation of
97
a storage bond results in a causal conflict, then the bond's
associated variables are determined by previously specified
source and (independent) storage outputs. Then eacl depen—
dent storage input and each dissipation input specified in
Gh is determined by
u., . . ., 111 if l2
iHZOL-J HO
C-l—«ZI "V-H
'm¢UFCUW
'HDU mmz
C CHWXQM
a 12 yucca).
‘1. «Hmm tun
XA i U20 KO
' i—¢ OWD04 o
A 03" Coo—noucuu
D Z” *ZZPQOI£
0 2H '0 onuh
H 04 me» 2»
U 00 l um '02“.
x 00 COJ2H iZUP UOU
NJPQO OX i<~fl— U):
I» .AA.“<.V1 nmp-4P!-
X r-O—OOJZODVH—HJ Z
3. .1me 0 Oi cum: mun.
2 OZVHQMCUQ- ~u¢zzzeu PjHéu.
U \42U owl—oz: 2O
KH\ O\:‘.. O‘NL—CL oun—
uzxnamooa-«uamomm
(murmurs-u. IF‘IHUZ“
--o \m OGXUC£P XOLU
F'z '~a\ “new 0mm u-l o
DUZZWZZZOQWUJ OWN
UL‘JUUVUX 'C-"L.JZ'~LILK40
murxxxxr-r-a ¢HZO
Q U
2
U .4
U H
R _l ‘I
N d o
x U) a-
, D H
0'" (A.
3‘ U-u-«I
H H
3 15cm
' UNLAJ
A ’m
j VIXZ.
W v um.)
Lu U) zoz
X I-0 n. O\
Lul- H .J a: mom
.12 I m a :0
mm a- H Lu LAO-m:
I? o-IHv u-o 13¢
L2 04.. oo- . PNL‘J
UHt—n-oou c: .-
"V8230 O .4 GO XL:
.JHP'F-DQL'J ‘DCF‘UNO ‘NC
< ZZv-‘IND mmmomn sew
(DOG-.0. ZZlI—OIIHH 27-bi—
«HHHhr-v-Dacozomr-sz:
u u u HH‘F‘L‘PL‘FKL—Pk‘zzo
(Jamxxouuumommmuuuz
d
7-. uuuuo HUG
a: Nine-nun 000
EL HHHHH uuu
“N”
U
Of Causal Complex Solvability
l'
Fhe Determination
Subroutine For
Table 6.
IESIPLX
I o
X“
0 ha' 0
A (I)- n .
0 203 .H A
O IN .070) o 3
"‘ 0.! OULLUA :0
3:: Co O‘lqflm v
2 n2 mite-Q c:
H it 9 up . .gjv y—
(n ILA QUW‘7H O.
4 I: ongP P
O U“ Zuzacx H
A P!— DJUC): .-
O 22 (I) 0.13:: A
W O“. (A cue-1 o O
V «a: ~ma~~< ID!
& CH 0 mtm— All
( ufl. ‘~VOHL< ‘0<
0.: an qmwr-Q v—<
c4 rm mhmox oz
Nu HH Umhn< No
UN 2" XU'LQZ 1'"
c: O 0U mgr-om; o as
<~ GO (VJ Our-£0) 'CO‘
UGAOO QZH JXA~L 99.4.4 o- t— I-
O vanaCw
' Q!- AJAA (\AO‘)
moo anfiV u GHQ
02!: .1): O O xu-c I H
r-UAAVLPD- < I “IA
umouxucé :uxm
uJI—UQiZI o 0.4 CAN! 0
>CL ou031AA9r-r-IIU
"U4 I Ao-u-n-UUAu
Q'UVIIA—I-JJ “LI- 0
:1 OGAOUUQI-NDQ.
FZU~UUr—v-u-~2
.JVIIXXVU-IU'IKW"
UHJUUF¢HHHWHH
<
D
O.
.3
(continued)
Table 6.
113
U!
99 NI" 2
NM AH U
wag-I HF'I .I
o o +
CO 2 CO 2 A
I-I- hJ I-I- U I
00 .I GO .1 +
(3‘9 ‘9 I 00 9 02
AA ‘A AA A Z":
GIN qu- Nu) P- 0.1.]
o 0 4;.) o o :J .1.
00 4'0. GO 0. #A
mu; NH mm H HH
0 o O z- o o O V O :1
AA 1* om AA A (I) h- #N
«Ho-4 H zo. «Iv-1 H (L A z):
9 f LLJH ? 'I' H O W”
HH U .1" AH O NH N O .4"
.I-J I- . A .J-I l- :A A I- ! A
Us. 0 Al— UV Q 0!— O A)-
xx 9 HD xx o 23-0 0 a:
1:3“. A.1 l o. 2.2. A um. I— A I o.
UU <<2v o o .— it! u {— 2U
DD (\mm QC <_IAV) A 441.510)
Z‘ L) 040.. 44 a<—m. u: “‘40.
<< XH+H <4 X>IH 0 X>+~
00 (Iv-0A lo (Iv-4A U (INA
com IIIZO" 60‘ tuxm U ZlIIO‘
0 01-0 OAU o 0 OH oAv 0 01-4 ans-t o
GU§PPIIU UU’I—I—IIU A‘IN-PIIU
“UFO-JAM wa-UJAU Ar-UUAUJ
U o 0.: .mp— no“ a 0;.) «1,:— 0mg; or..v- cu
DmmthDQNDfl-QQPHJQDUQPHDQD
Z<-u.< all—.1032!
fiWI—ZX'IO< "D 0:!
6—32 O 0
ImU 0C
933%.“:
«u. 3x0.
CMOO. Z
CZI-Z OH
C3‘I
Z ((0‘3
mm z-I-¥ :UC
(IOU 12‘
a: ( F“.
012.] can.
<0 )I-CI C
NU<- I“
afia>=una
Unjuozu
Atiiiifiififii
Ififififififiiifiifittttiittiiitttiiiit
0
U
N
H c-
m c
0‘
<
I- O
< P
O o
W U
A
I— <
m I-
£ <
H D
< x
U <
4 A13:
9 0
Lu (III--
N 3U
H z c
U) "A
(DU)
m '37
a: .‘z
D A I
II- wx
U OLAJ
-' um
c: 62
I- 'v
U) 4A I
tau-0N
4 4%?
U ”-4
H HWH
I- ".103
U no.3:
20 M1
Stunt-04'
‘)V) as!
1(3le
22:35-04
U
m In
I
U
UUUUUUUUUUUUUUUJU
x
U
.1
D.
Z
0 Lu
U .1
a: o
.4 H' o
q U m
m D
D D O
< qu I-
0 C21 U
D '9
N I-V) A
.1 UH I:
m =4
H Am (III-0N
U 23¢ <1an
.Jx o: I-IZH
D.N.ut- 01-413
UCL‘U I-Z‘
JI- O.) tat-I?
aswuxz om-x
Huan- XZU
I-Kfl mum
a: u deD
OJHZ U22" II
LLUVO Paw—04°
club-c HLLmur—
XUHI‘ .JHdEJ—q
U U <
u 4 H
J: D 2
U ‘7 H
UU UUU
(continued)
Table 7.
(I)
p—
D
l
I- A
D I
U I'-
3
I- 0
Z LL
L‘ 14.!
IL A
IL on
U G AI-
:0 U
_I 0‘ (JO
4 ALL.
I 0 CM
0: r- L9H
U 0 AA
I- 6 no
x AG 0 0
DJ amor—
'ZUL‘J
IL L: o o 0
U u—IAA
116
A
p—
D
Q
A O
o O
I- v-I
.J
O 0
LL I-
LLJ 0 m
H 9 m0
.3 A 01°
2:: A In
”D F‘ O
3" V A PC
21.: (I) 7 CI-
x 0!: war-g U 00
H 2A 23H < “O
m “a A; I A At—Q QA
I- .1 o .JHI-A‘jot-untq 00
¢ Um A’OQH--UUU U.
a: v-z I «H ovmu4 U
H“ .I
IUJ '-
>I-J; AA
< I- Iv-I
aim KXF'IA
cum—o UUUIZ
(PUII 2 SDI-".1
UPI-U) UAZZIIJ
VJZOW—I—III II III-IUA
3U‘le °! N v . 3H404529AAAfl u 90404” m
‘F'N'H'IAIQ A HIOAOOI—Io—o-n-w-M: a: CHOOI( z
“'WV’9Z IIZZIIdGIIAAAIo-Ulvvvn 4 OIIAAAH a
Ont-INA? .JIggJHIIhJ'J‘J‘D'jv-ImuzzczczAum I “AH-«mama
NVIIIIIV‘IZ .IJ A O vvsz-o I-I-I-NZJD c-Iv-I o UUUNDA
U)A¥. ' II
In A
I— 2 H
D o «o
O. A x
I‘- ‘J U
D U a;
C I- 2
G! v
LL I-I _’
U 0 z
D A
L9 2 A
z < 1
H ‘H XXV
x ¢3~+ Iuwx:
U in 'H man:
0 #:UX? 22 0
I " "UII 0 QC!
0 AH 'QA HAM
II II AZ-4 II H O
QUHV‘Jme—OfiA
.lIl no" N: H
Udan—F‘OZVG’ V
"VSIUNJZKZHIDWJ
EUNGJHUHHNNX
' vgvg v
uzouudzooom
“HQUF‘HZUQQA
_
G
I- O t
In a) In
I N u:
I
U U
\J
U" Ao-ImII AAv-I
chnazzu 7'21:
2 HH" UVII
IOJJJOWWA
II I~ II II X‘JQLL '2
ch-INuNI-u-nw
Luv-IQ mm" an 1131
on HOHHHUu-NQH
ZXAQJJ)-h—IH2;H¥
dS-IRANSFCRHATION
0
GO
sm-
U
C
I {‘4 “I
I
U
(continued)
LL
I
H
II
A
XL”
MI '
33H
(II
’A
HI-I
vv
:4
2 xx
U AA
H CO
F- . O
< I-I-
LX _I_.l
«um 00
cm: AA
“2". AA
0) 0“ HA
‘g-IA UV
(II XLLZZ
GHUIXZ
F- m\4~v
I OZIII-I-
W0 9 AQJ}
jnF-IHHO-I
”Hwy
POI-“LAI-
ZOXKHA
H
a: o.
0. O
I n
I
U
121
IE 64509KHIKBEX*IK)QKHIIKIIIASUHIJI9J31’IADEX1)
A
H
S
C' A
H ’3
II Emu
J Ao-Iq
o Inno—
“ <
7 000
X” I-I—m
MI OQII
4 gx won
0 Z . AAo-g
H In IIA H A049
h O «"3 x x cam—I
< x 22-. m u Amp-I33
I: U A..I CDXX C3 D-JJH
m m xx A 2mm 2 LL 0 0.1
0 £ we 9 xomm H HFPI
LL. " CD 0 HxU—IZZH 0 '40A
mow-I (a Immlloocq—Iogo-Oj
can 00 HQ C‘NN"'I.JII-AAU
(uh-o I-N I." I Amman?" J‘J )0)“
mil Law II I v-I I-u-Ir-I A veLD
P ac. Autaddqmfiawmuz
I “NO—Ila ,< II IINII II IIU-v-IILQVH
(III-NIHZI- UHNMHNAFIZMHHII-
7H II VFW-ICES (12:32 2 vaz JH
:0 FILL. 3: II zHAumuv—utaommmo II a:
PJQfiszHJAUAJJQ
LIJo-IN (HO
xx r-I—II
xuuHIZUI—
HQDIIwC-KU
oczzr-IL 1!
P—HHHPA
< U
L U)
I In
w m
UU. 'J
UL)
O
¢o
U
a P
G' U
N NE!
I-XCI IXA
Duw- ma-
QQU OH
22c: ZI
AAA U-II-
..— OH
QHI-I A-
2x:o it. A
DUO U0 H
mauu OILI 1
20‘ 2. ~—
UII-IAU HA x
III-1U III-d 2.
ILo-I'JIL Afilum
0")" 393::
xZ—I x2
DULUIImz—III
ZIOUr-fl-IOUI-I-
U vr-U ~24
(DOLL-((CDLLOO
QH’HQHUH
O H
4 -0- Inc
A I'Iu Inq-
LL Q
U
(continued)
Table 8.
=2
1
ICUIIIIINE.I)ICIREC
A
X
U
D
O
c:
H
. Z
I— A
H In a:
O a- Z
A +
3 o x
O I- Lu
u. C Q
U U U
H A z
o x H
I- l...l II
M D m w
I- O 222:
< H OI-IA
Z V MEDIA
A X .440
D u OIU
x oxxxu
OHZUULJ 0U
OIIHQQQLLD
UXKOQOqz
U CKZPH
JQJHH—IZI—
OUUHVHV4
¢¢~zz| It—ZUMOJUHZDJUHUI—HI—uI-ZZI—HF-UO
.48" II V II II 4 II VII UJQVUUJOUVJVCVI‘vV-IVu GO (>14
00—003— O 4.363 0‘0 “ U‘Q
v-I IZH N 2 o x CO (4 X 00
ZX 0A zx OQ (Am 0 0 (HQ
“MI-Q ”UP-m XIII-0 605 III:—
XQQ I XDL'J CA. 04“: o 0—0 on-
mu '0 mo an vt—I- II actor—r- II
OCXLJ uuxwo-ozaa mun-o.»-
“KHU “H‘HUJJ 'QPU 0 .J .‘LI—
IIAIIQLLIIHIIQUQI—t—JDCLQCLI-HDO
XKHO¢XENOXHDVQZ<¢ G <>U U a:
,0 vuI-aa vuo-ma‘ 1: II F! U a: II H H a:
Q) goo a x33 0 OH OA-o CA -A- G La
3 mQ-u uO-cuvhhu AVPPH o
C omxu uuxuupwua.dpu:a go 0
,Hm «cam-dcmu-oaompuqnomhu Do a
H: IIHII QQIIHIIommuJ-HDDUmI-HDD Nth U
:2 ALLHOQ<¢tLX .
tht o:
¢~< m
I t>~~o
IASUM
it.iiii‘ififiiiititiifitiiitt.Iiitiitittifitittttfiifififitfitttttt
IF1.GT.IASUH.CR.MATDIH2.GT.IASUH)GOTO 910
03
C:
U
U
U
x
(0200:307‘40‘ cm a: 22 >uJ
gmogm¢ou OI< Q<< <
KAAILvnNJU CI— Auz I10;
gooxzwcox o mzz~o a:
HAmGKX'JWA: OILI-II-Imza; (H F. ""
OcvtOOGQo co ozmx o w AH H
mm: OAK) OOH- C «II-PU XI- 3 I'-
ngJcmA x CPPAL cc to 40¢.
H\~2 m a At" N~ZAnHmH XHZH
wilmfivI/h-IL‘O C II-r-‘Iu 1: a que- '4 ")va so woman"-
DOUGAKWKUP aou<~zc u a Hg“ oz OHHmHmOJouummHm
WUUHmthAH «mm: mm H o FAQ oxouzmnmnoxomxn~4u
++++++¢ om zutm I ' ~ 2
i‘wurhd I a 4 NH fl 00
gundpon I a H m
‘IIIIII I a I I
'IIIIII I a I I
uUUUUUUUUUUUUb H U
Q
....l I-
CC 2
I" <
I Z
IIU L: "4
CI- .l to:
AH .10 C CD
“OI- co I un-
®< .LA \ I-Iu
:3: A) AA UK
5: 2A 0H HH & on
0H “HOG. we 2 .
Havoc H c>co
g aunt-m r-I— HHg o
Oxraa << mmuc
U II< ' 'QILU’ ‘INIAJ
VHIOOA¢¢D>P 0
£4.19 "HID-nucl-
°°II (401.1"ngou
C—IGIIII z>I—II "[50
H JU~30HZ>P v
‘Oomm—Aaowuxm
gath-IHQc/IHQQGUH
U
9 0H4;
m a "U
E" I
I
uu
131
COLUMNS 10 bE IIHERCHANGED
.AND.IPIV.LE.N2)GCTO 20
\c :0
H H
H O O
a. r- I—
H O O
0 L3 A a A
I— A z A U
q 4 II U 3
O t 3 U :l U
4'" ' HHH HAZE: UAZQ
<3: 3 IZZHZB‘I’J z3~4
H—IO—n— ”—2 00.30 003:0
(no I a: «tor-o I umx: cmz:
rI—JHIzl-Iar->Iu—II II “HI! I!
O£QJJJmU383UUD
IH z-I-o-aw—oxcaooxczoz
>v-IIIUIIIc-ILLHIICHZU I :2"qu I H
LLZazv-ImZHZUI—GHZIIHQHZIII-
I—II.’ J|.UIIJV_JUUP~_’UUFZ
HnuuummflmmOAauuoztmo
zzzuz‘szHIxon—Ixxuu
Lu
Q I00 w ¢
g-g FIN 0-0 F‘
I
I
u
d
0T0 23
I
I
HO HA
l—A I-P'
(N dd!
H :2 t:
(I) I: 2 0 mm
2 H mI—H—IH
I O .chZAA
.J I-o-IH '- I I 30
.3 (ZEHAAOJ
UH XHij‘flzu
U); OOOI—UVF“
H HPPJAHI.
MO II (4" I-I-Ghs
(DP Httfi<UH7
A
l
\
A
r-I
I-v ;’
H A
'— CL
4 \
2:. III
LL' 3
C Csu D
a-u 1:3 \£J\13 ‘CDIIL
‘3' -J "’ (:3 ‘:’l-Q
In V Ch 0‘ 00 ORV
I r-I— a:
O D O U DO 03
I— m I— I- 023 Pm
<:, fill £_I \.a a- a- 1:)«¢::4-
d! +0) ID u: AN 6.5 III-n
aIIC‘] a-It‘jI-Ia-b III v-I.¢:;¢: dill-QI-fl
Cii: I-It-iil: O I::;--. r- o 1. “:p-¢.-q
0.42th 0 XH'J 32.12! xH<
q'IzzI—o 43.:- .: 0 O dl-a.
UHAJU 4
N 0 H 0
Z I— Q I-
. t9 \ 0
p—n O A I
Q A H A
O H up H >
A v N v H
H N I- N (Lo-I
w I- d I" \1:
N < 2. < AH
I- x I: a. 3.4
< A x C a: CI
N 2. H H A O H 01A
‘3 22 z x o .I c: u 0 =HH
G: H N H v .1 V c U‘ K U‘fi’v
H D 3: o I O I O IN
A u z a I 4 o o a car
PO < CNN 0 U) H U) I- I— H I-U)<
UH L hat a m z < c G t O4;
40 C < H H + H + :1) LB ‘3 H max
”‘90 ANHN 1200 NZ A .IHA uNA A ‘7 NJ AAH
'UQA ”It: ‘II-I- P. H OzI-I Hz. XA Q I—.. Xhfi' A
(Dex: IHHH AO< «Luv-Iv Luv-c1: 1H0 47;: «(u JUQ H
3&0 nooo not tJZN 44w PQO va :4 XNA a-»
c G JAPF 'ACN‘OIH «It 2A9 «20 40 QPO an:
'Hou o<<¢ A3ALUAA< «a: u'!HuJ cuLUAfi-D mm
‘: