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thesis entitled

Modal Analysis of Damped Systems
Using a Bond-Graph Approach

presented by

Thomas L. Bush

has been accepted towards fulfillment
of the requirements for

_Mas_ter_'_s_ degree in Manual Engineering

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4r; fi—————-i ~-—————

MODAL ANALYSIS OF DAMPED SYSTEMS
USING A BOND-GRAPH APPROACH

BY

Thomas Lee Bush

A THESIS

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1989

GOO ISOCO

ABSTRACT

MODAL ANALYSIS OF DAMPED SYSTEMS

USING A BOND-GRAPH APPROACH

BY
Thomas Lee Bush

Modal analysis is one method used in the analysis of
linear dynamic systems. This method cannot be applied to
damped systems unless the damping is of the proper modal
form. To overcome this problem designers sometimes approxi-
mate the actual damping with modally-distributed damping. A
study into the errors resulting from these approximations
was conducted using bond-graphs. A bond-graph model was de-
veloped that represents the system in modal coordinates and
included an element to account for the coupling effect
caused by nonmodally-distributed damping. Simulations were
conducted using this bond—graph to try to correlate power
action in the coupling element to resulting output error.
Computer software was written to allow computation of dif-
ferences from successive ENPORT simulations. The results of
these simulations did show some of the expected trends.

However the desired general correlation was not obtained.

ACKNOWLEDGMENTS

I would like to thank my Major Professor, Dr. Ronald
Rosenberg for his knowledge and guidance during my graduate
studies.

A special thanks also goes to my wife Linda, for all

her love, support and understanding.

iii

TABLE OF CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . .1
The Modal Bond-Graph Model . . . . . . . . . . . . . .4
2.1 Systems with Modally Distributed Damping . . . . 5
2.2 Modifications for Non-modal Damping . . . . . . 17
2.3 The Need for This Type of Model . . . . . . . . 24
Automated Error Analysis Using ENPORT . . . . . . . .27
3.1 The Need for Error Analysis . . . . . . . . . . 27
3.2 Program Development . . . . . . . . . . . . . . 28
3.3 An Example of the Analysis Feature . . . . . . .32
3.4 Program Implementation and Future

Enhancement Suggestions . . . . . . . . . . .40
Approximations to Non-modal Systems . . . . . . . . .43
4.1 The Objectives of the Study . . . . . . . . . . 43
4.2 The Study Implementation . . . . . . . . . . . .44
4.3 The Results . . . . . . . . . . . . . . . . . . 53
4.4 Problems . . . . . . . . . . . . . . . . . . . .56
4.5 Suggestions for Future Research . . . . . . . . 58
Summary . . . . . . . . . . . . . . . . . . . . . . .60
List of References . . . . . . . . . . . . . . . . . 64
Appendix A ENPORT Enhancement Code Listing . . . . . 65
Appendix B An ENPORT File Listing of the System

Used in the study . . . . . . . . . . . . . . . .85

iv

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure

Figure

Figure

Figure

10

11

12

13

LIST OF FIGURES

The 2 D.O.F. System Example . . . .
Bond-Graph for a 2 D.O.F. System
with Modal Damping . . . . . . .
The Modal Bond-Graph of Figure 2
with Causality . . . . . . . . .
Modally Damped Bond-Graph with
Specific values . . . . . . . . .
The 2-Port R Element . . . . . . . .
A Coupled-Mode Bond-Graph for
Nonmodally Damped Systems . . . .
The Coupled-Mode Bond-Graph
with Causality . . . . . . . . .
The Reference Velocity . . . . . . .
The Modified Post-Processor Menu . .
The Run-Data Menu in ENPORT . . . .
The Response with Assumed Modal
Damping . . . . . . . . . . . . .
The Run-Data Analysis Menu in ENPORT
Change in Velocity for Modally

Damped Assumption . . . . . . . .

20

21

.22

33

34

.34

35

37

38

Figure

Figure

Figure
Figure

Figure

Figure
Figure

Figure

14

15

16

17

18

19

20

21

Tabular Listing of the Change

in Damper Energy Dissipated . . . .
A Coupled-Mode Bond-Graph for

a Non-modally Damped System . . . .
A Typical 2-port R Coupling Element . .
The Base 2 D.O.F. System. . . . . . . .
A Coupled-Mode Bond-Graph for

the System in Figure 17 . . . . . .
A Typical POwer Versus Error Plot . . .
Modal Velocity Response Curves . . . .
The ENPORT Bond-Graph For

The System Studied . . . . . . . . .

vi

Chapter 1

Introduction

Modal analysis is commonly used in the design process
for predicting the response of linear dynamic systems with
multiple degrees of freedom. This analysis technique allows
the designer to decompose the coupled, second-order, linear
differential equations into a set of decoupled, second-
order, linear, equations in a new set of coordinates. These
uncoupled equations can then be solved independently, yield-
ing the modal responses which can then be transformed back
to the original coordinate frame to obtain the overall sys-
tem response.

The problem with this analysis technique is that for
many damped systems it is not possible to obtain a transfor-
mation that will decouple the original differential equa-
tions. If the standard decoupling technique is employed the
modal equations may remain coupled by their damping terms.
Such a system is said to be non-modally damped. It is gen-
erally accepted that the modal analysis technique can only
be truly applied to systems with Rayleigh damping, that is,
systems with a damping matrix proportional to the mass and
stiffness matrices. However, a more general condition which
is both necessary and sufficient for determining the possi-
bility for modal decomposition is presented by I. Fawzy (1).
In either case, accurate representation of a system response

using this method is limited to a modal form of damping.

1

2

The alternative to the modal decomposition procedure is
to convert the problem into a state-space formulation. This
formulation converts n second-order differential equations
into 2*n first-order differential equations. Because this
approach requires the solution of twice as many equations it
is generally used for numerical computations but avoided
when generalized solutions are desired.

One approximate method which has commonly been used to
deal with systems which are non-modally damped is to elimi-
nate the damping from the original equations, perform the
transformation, then add damping to the various modes based
on experience. This method is generally employed success-
fully for systems with light damping. However, there still
exists a question as to how much error is incurred when
using this approach. Additionally, the question arises as
to when the damping is small enough so that this method can
be employed without incurring substantial errors.

This thesis presents a study into non-modally damped
systems using bond-graphs. A bond-graph model is developed
which allows a modal representation of the system augmented
by intermode coupling elements. This structure allows one
to study the coupling activity and to determinate how
strongly the modes are coupled for a variety of inputs.

As part of this research, computer subroutines were
developed for the ENPORT software (4) to assist in the
evaluation of the errors produced by the elimination of the

intermode coupling. This software allows the user to

compare the results of a series of simulation runs made with
varied parameters. From these comparisons predictions can
be made as to the sensitivity of the original model to a
given parameter.

The development of the modal bond—graph is described in
Chapter 2. Chapter 3 then presents and describes the soft-
ware developed for use in comparing simulation results from
modal and non-modal systems. The actual study is presented

in Chapter 4 and Chapter 5 gives a summary of the findings.

Chapter 2

The Modal Bond-Graph Model

The purpose of this section is to develop a modal bond-
graph model for use in studying the intermode coupling ef-
fects. Margolis and Young (2) present a method for convert-
ing large bond-graph models to a modal bond-graph represen-
tation. However, their method falls somewhat short in that
they take the conventional approach and require that the
actual damping be replaced by modally distributed damping in
the modal model. The model developed here will allow the
transformed equations to remain coupled and include this
effect in the new bond graph as a coupling element between
the modes. Through this element potentially valuable infor-
mation can be obtained as to the effect of this coupling.

In this form the model will be an exact representation of
the original system. Therefore simulation results are
valid.

Section 2.1 presents the development of a modal bond-
graph for a system in which the damping is modally distrib-
uted. No intermode coupling element is needed since the
modal equations completely decouple. This model is modified
in section 2.2 to allow for coupling elements when the damp-

ing is not modally distributed.

5

2.; Systems with Modally Distributed Damping

The best way to present the development of the damped
modal model is by an example. This example begins with the
2 Degree of Freedom (D.O.F.) spring-mass-damper system shown

in Figure 1.

f__|_ X's. F2 X32.
J_% I /\ 2K [— ASK
2M _ M
+
02

I E 3‘6 0 O cs

 

 

 

 

 

 

 

FT
’LL

 

 

 

 

Figure 1

The 2 D.O.F. System Example

The differential equation of motion can be obtained in
the X coordinates shown using a Lagrangian or similar ap-
proach. Equation 2.1 give the representation of the two
coupled, linear, ordinary differential equations for this

system in terms of X1 and X2.

2M 0 x1 01+cz -cz x1 3K -2K x1 F1 (2.1)

o M x2 + -c2 C2+C3 x2 + -2K 5K x2 = F2

The superdot denotes a time derivative throughout this

study.

Equation (2.2) is the matrix form of eq. (2.1), namely,

Mx+cx+xx=r (2.2)

To ensure that the system is modally damped the damping can

be selected such that
c=o(M+flK, (2.3)
where (X and K are real constants.
Proceeding with the condition that the damping is mo-
dally distributed, the set of equations (2.1) can be de-
coupled using the transformation,

x= dn (2.4)

where ¢ denotes the eigenvector matrix for the undamped

equation

Mx+Kx=o (2-5)

and rldenotes the modal coordinate vector.
(Note that the eigenvector matrix has not been normalized.)

Substituting equation (2.3) into (2.2) and premultiplying by

7

¢ yields a new equation in modal coordinates, namely,

d'M¢fl+¢'C¢h+¢‘Kpn=¢'F. (2.5)
Letting
M' = d'M d (2.7a)
c' = d'c d (2.7b)
and K' = ¢'K d (2.7c)

equation (2.6) becomes
M1t1+ c'h + K'fl(= d'F. (2.8)

Since the damping matrix in the original model was
assumed to be structured such that the damping was of the
modal form, this new equation will contain only diagonal
matrices (except for ¢'). That is, the original coupled
equations have been decomposed into uncoupled equations in

the new coordinatefl.

Expressing the diagonal matrices M' , C' , and K' for

our example as

M' = MA 0] (2.9a)

c' = CA 0 (2.9b)
0 cs

RA 0 (2-90)

0 KB

and the eigenvalue matrix as
d = V11 v12 (2.10)
V21 V22

the bond-graph shown in Figure 2 can be constructed.

KI

 

 

(MA) (MB)
(CA) IA 18 (C8)
RA RB
(KA) \ / (KB)
CA*¢-—~— IA IB —-w~7 CB
(VII) (v22)
TFIA TFZB
(VIZ) VZI)
l TFIB TFZA r
/’ ..
SEI 6i 7 0' V\~02 laigL-SEZ
XI x2
Figure 2

Bond-Graph for a 2 D.O.F. System with Modal Damping.

This bond-graph consists of 2 damped oscillators which
are connected to the external forcing through a transformer
structure. These oscillators are the 2 modes for the system
and each 1 junction (1A, 13) represents a common velocity
junction for each of the modes. The velocities are R1, and
A2: respectively. The values for the I, C, and K modal
elements come from the transformed independent modal equa-
tions. In short, they are the diagonal terms from the M',
C', and K' matrices. Applying causality to the bond-graph,
as shown in Figure 3, allows further insights into the equa-

tion structure.

(MA) (MB)
(CA) IA (CB)
RA T
(KA) \/’RB (KB)
(:A\AL___—~‘ IA. IBIl*-*"“17 CI3
(VII) (v22)
TFIA TFZB
-(V|2) v2I) -—
‘T TFIB TFZA f
XI x2
Figure 3

The Modal Bond-Graph of Figure 2 with Causality.

10

The transformer junction structure performs the trans—
formation of the forcing terms into the modal space. Spe-
cific values for transformer (TF) elements are obtained from
the d matrix. In addition to transforming the forcing terms
into the modal space, the transformer junction structure
also transforms the velocity terms from the modal coordi-
nates back to the original X coordinate base. This becomes
apparent by viewing the model as a 'black box' the inputs of
which are the forcing terms in the original coordinate
frame. The outputs, as required by causality conditions,
must be the resulting velocities, which are also in the
original coordinate frame.

For this particular example, modal analysis leads to
the following values for M', C', K', and ¢. (Recall that the
damping was assumed proportional to the mass and stiffness

matrices (eq. 2.3)).

2.25 M d] (2.11a)
o 2.25 M

EzsmMo +310 0 3 (2.115)

180tM + 99,8

K' = 2. 25 K (2.11c)
o 99 K

¢ = 1 1 (2.12)
.50 -4.0

MI

CI

11

For these particular values the bond-graph of Figure 2
takes on specific form as shown in Figure 4. Note that ¢
has not been normalized in the standard fashion at this

point: the leading element of each column vector has been

set to unity.

(2 25M) (2.25M)
IB (I8«M+995K)

2. 25( (OKMHSK) T T
(.2 25M) A\l /R ()99K

CA‘L———4 IB h—"—7-CB
(|.O) (~4.0)
TFIA TFZB

T TF l B TFZAR T

Fl //A(IAD) (.50) F2
SEI—~ :4‘3' 02 Prggf‘SEZ

 

Figure 4

Modally Damped Bond-Graph with Specific Values.

Using this specific bond-graph as an example, four

state-space equations can be written in the (modal) state

variables WA, RA, 7T5, RB:
7g= -(2.25(o(M +flK)/2.25 M) ’TT'A-2.25K RA+ (F1+.5 F) (2.13a)

RA = (1/2.25 Man (2.1316)

12

’frB = -((180¢M+99,£K)/18 M) 7% -99K RB +(F1-4.0 F2) (2.13c)

m. = (1/ 18 mm; (2.13d)

where ’H3_and 'Wg are modal momenta associated with IA and

IB, respectively. Additionally, 2 output equations can be

written across the junction structure, converting the state
variables in modal coordinates to X coordinate velocity

values. That is,

)2]. = “A + '13 (2.148)

x2 .50 RA - 4.0 RB. (2.14b)

Taking the Laplace transform of equations (2.13) with

zero initial conditions and solving for 7T'(S) yields

F1(S) + .50F2(S)
7Q(S) = -------------------- (2.15a)

s + (oIM+,6K)/M + K/MS

F1(S) - 4.0F2(S)
(S) = -------------------- (2.155)

s + (13 0(M + 99 310/1814 + 99K/18MS.

Setting the velocities in the output equations (2.14a)

and (2.14b) equal to the derivative of the output

13

displacements and Laplacing these equations yields

SXl(S) (1/2.25M) 7Q(S) + (1/18M)’rfB(S) (2.16a)

SX2(S) (.50/2.25M) 713(5) — (4.0/18M)?fB(S). (2.16b)
Substituting equations (2.15a) and (2.15b) into (2.16a) and
(2.16b) yields the Laplace solution for the displacements in

the original X coordinates (equations (2.17a) and (2.17b).

F1(S)+.50F2(S) F1(S)-4.0f2(S)
x1(3)= ---------------- + -------------------- (2.17a)
2 . 25 (M32+ (0t M+fl K) S+K) 18 (M32+ (18a M+99 flx) s+991<)
.50(F1(S)+.50F2(s) -4.0(F1(S)-4.0F2(S))
X2(S)= ------------------ + -------------------- (2.175)

2 . 25 (MSZ+(0(M+ 3K) S+K) 18 (MSZ+ (18 oIM+99£ K) S+99K)
This solution for X(S) obtained from the state equations of
the modal model can be compared to the solution obtained
using the Laplace of equations (2.4) and (2.8).
X(S) = (IRIS) (2.18)

M'82 “(S)+ c'sll(S)+ K'n (S) =)0T F(S) (2.19)

solving the matrix equation (2.19) for “1(8) and "2(8)

14

Fl(S) + .50F2(S)
“1(5): ------------------ (2.20a)

s + (otM+ [IQ/M + K/MS

Fl(S) + .50F2(S)
"1(5): ------------------ (2.20b)

s + (18 0m + 99flx)/18M + 99K/18MS
and substitute into equation (2.18) and solve for (98)

F1(S)+.50F2(S) Fl(S)-4.0F2(S)
X1(S) = --------------- + --------------- (2.213)

2 . 25 (M82+(otM+ ,S’K)s+r<) 18 (M82+ (180IM+99£K)S+99K

.50(F1(S)+.50F2(S)) -4.0(Fl(S)-4.0F2(S))
x2(5) = ---------------- + ---------------- (2.21b)

2 . 25 (M32+ ( otM-I- 3K) s+1<) 18 (1452+ (180tM+99£ K) S+99K

As expected, the Laplace solution obtained from the
bond-graph model is identical to that obtained directly
from the transformed differential equations.

This analysis has demonstrated the modal model's
ability to replicate the original differential equations.
Insights are also gained into the transformation junction

structure's job in linking the original physical

15

coordinates to the derived modal coordinates. That is, the
input forcing terms in the original coordinates are trans-
formed to modal coordinates and act on the individual modal
oscillators. The resulting modal response is then trans-
formed back to the original coordinate space through the
same structure, yielding the output response. This can most
easily be seen by studying the causality conditions shown in

Figure 3.
A remark on normalization.

Some form of normalization of the eigen matrix is com-
mon in modal analysis. The classic method is to set the
transformed mass matrix equal to the identity matrix; that

is,
I
M' = d M d = I (2.22)
This results in a modal stiffness matrix of the form

K' = "12 o (2.23)

2
0 w2 e

The Wi are interpreted as the undamped natural frequen-
cies. However, for the purposes of this study a particular
normalization is not required since any form of the eigen

matrix will transfer the original mass, stiffness, and

16

damping matrices to modal coordinates. However, it should
be noted that once the transformed modal mass, damping, and
stiffness matrices are obtained, further normalization (i.e.
dividing each equation through by its mass term) should be
undertaken with some care. Since these modal matrices are

obtained using equations of the form

M' = (I'M 95 (2.24)

a squared normalization term appears in each of the modal
M', C', and K' matrices where only a linear term appears in
the matrix. Dividing the entire equations through by a
linear term will result in an incorrect modal bond-graph.
For this reason, if some normalization scheme is desired, it
should be performed on the eigenvector matrix, before the
bond-graph parameters are specified.

This section has outlined the development of a modal
bond-graph for a system with modally distributed damping.
Each mode was represented in modal coordinates with a simple
spring-mass-damper oscillator. These modal oscillators were
then coupled to the original coordinate space by a transfor-
mation junction structure. This modal bond graph structure
can be used to provide insights to the mode activity which
may not be available in a conventional bond-graph model.

Separating the various modes allows the designer to de-
termine the relative importance of a particular mode to the

overall response for a given input forcing combination. In

17

some instances this may allow model simplifications if a
particular mode or range of modes is determined to be inac-
tive. This is especially useful if the system under study
is a subsystem of a larger model. However, this modeling
technique is still limited to systems with modally distrib-
uted damping. The next section will expand on this modal

bond-graph to include non-modal damping.

2.2. Modifications for Non-modal Damping

The last section presented a bond-graph model which was
based on the assumption that the damping was modally dis-
tributed. In general this will not be the case and the
eigenvector transformation will not decouple the equations
of motion. This section presents a modified version of the
previous model which includes an intermode coupling element
to allow for the coupling of the transformed equations. As
before, this model will be an exact representation of the
original system equations.

Return to the system shown in Figure 1 and assume that
the damping is non—modal. The differetial equations are of

the same form.

2M 0 X1 + c1+c2 -c2 x1 + 3K -2K x1 = Fl (2.25)

0 M x2 -c2 c2+c3 x2 -2K 5K x2 F2

Since the eigenvector matrix is based strictly on the

18

undamped equations it also remains unchanged.
v11 v12 (2.26)
¢ = v21 V22

Proceeding as if the eigenvalue matrix could uncouple
the differential equations, as before, the following matrix
representation for the differential equations in the modal

variable fl are obtained.
M' fi+ on + K'll = d'F (2.27)

M' and K' are the same diagonal matrices obtained previ-

ously,

M' = d' M d = MA 0 (2.28a)
0 MB

K' = ¢' K ¢ = KA 0 (2.28b)
0 KB

but C' is no longer a diagonal matrix. It has the form

c' = d' c d = CA CN (2.29)
CN CB

Notice, however, that C' is a symmetric matrix and can
be represented with 3 parameters: CA, CB, and CN. The rea-

son that C' is symmetric stems from the transformation

19

¢IC d and the fact that the eigenvector matrix is composed
of orthogonal vectors. It can be shown that this transfor-
mation will always result in a symmetric matrix.

Generally this procedure would be considered of little
value since the transformed equations remain coupled by
their damping terms, especially if a generalized solution
equation is sought.

Since the bond-graph model will generally be solved
using numerical techniques, this intermode coupling is of
little consequence and the modal bond-graph of section 2.1
can therefore be modified to include the coupling effect
without disrupting the solution procedure.

Let the diagonal modal damping matrix be represented as
the sum of a diagonal matrix and a matrix containing the off

diagonal terms, namely,
C. = CA 0 + 0 CN (2.30)
0 CB CN 0
Equation (2.29) then takes the form

MIMI: :12; E31] WI). :: 21:12”

Notice that with the exception of the off-diagonal
damping matrix, this is exactly the same as equation (2.8)

used in the development of the modal bond-graph from section

20

2.1. Notice also that the effect of this off-diagonal damp-
ing matrix is simply to add forcing terms to each mode that
are proportional to the velocity in the other mode. That
is, a force is applied to mode A which is proportional to
the velocity in mode B ((13), and vice-versa. In bond-graph
terms an element which reproduces a force proportional to a
velocity is an R element or a gyrator. Since in this par-
ticular case the forcing effect is cross-coupled, a special
form of the R element, called a 2-port R element, can be
incorporated. This element takes the form shown in Figure

5.

0 (3|
CI? (3
el
FTI—mv R D 4—————I
Figure 5

The 2-Port R Element
This 2-port R element uses a matrix type of equation struc-

ture of the form

e1 0 c1 f2 (2.32)
e2 = c2 0 f1
Where f1 and f2 are the input flows, and el and e2 are the

resulting efforts. (Note that this matrix may be inverted

depending on the causal demands of the system.) This

21

element equation is of the exact form of the off—diagonal
damping term in equation 2.31.

Using this element, the bond-graph developed in section
2.1 can now be modified as shown in Figure 6 to include the

effects of non-modally distributed damping.

 

 

(MA) (MB)
(CA) IA 18 (CB)
RA RB
(KA) \ (CN) / (KB)
CA‘¢~~- IA , RD .111_mm- IB -~m7 CB
(VII) (v22)
TFIA TFZB
(VI2) v2I)
f TFIB TFZA r
// \
Fl Ol 02 F2 c~
—'.—-"7 e E2
SEI XI x2 .3
Figure 6

A Coupled-Mode Bond-Graph for Non-modally Damped Systems

Except for the addition of the 2-port R element, this
bond-graph is identical to the graph shown in Figure 2. It
contains the same 2 damped modal oscillators connected to
the external forcing through the same transformer structure.
Adding causality as shown in Figure 7 will further illus-

trate the interactions across the R element.

(MA) 22 (MB)

 

 

(CA) IA (CB)
RA if
(KA) \\\v (CN) «//4% (KB)
CA‘L*—-fl IA I , RD A? I IB B———7 CB
(VII) (vzz)
TFIA TFZB
_. (VI2) v2I)
f \TFIB TFZA ‘f
XI x2
Figure 7

The Coupled-Mode Bond-Graph with Causality Added

From this bond-graph, the state space equations can be

written in the (modal) state variables’TfA,n ”VB, "3:
’rfA=-(CA/MA)’TfA - KA RA - (CN/MB) WE + VllFl + V21F2(2.38a)

(1/MA)7fA (2.38b)

;.
3’
II

’ffB=-(CB/MB)’TTB - KB n3 - (CN/MA) 7g + VlZFl + V22F2(2.380)

(l/MB)’rrB (2.38d)

:
m
II

Differentiating equations (2.38b) and (2.38d) yields

23

’frA = MA “A (2.393)

MB ('13 (2.39b)

7TB

Now substituting equations (2.39a), (2.39b), (2.38b), and

(2.38d) into equations (2.38a) and (2.38c) yields

MA RA + CA RA + CN RB + KA RB = VllFl + V21F2 (2.40a)

MB RB + CB RB + CN RA + KB m V12F1 + V22F2 (2.4013)

or in matrix form,

[:MA 0] (V [CA 0] ml? Chilmlfm (flu: d' [p1] (2.41)
0 MB 0 CB CN 0 0 KB F2

Notice that these equations are exactly the transformed
equations (2.36) from which the model in Figure 7 was devel-
oped. Since the transformed equations are simply the origi-
nal equations taken into a different coordinate frame, this
shows that the modified modal model exactly represents the

original system equations.

24

2.3. The Need for This Type of Model

At this point the question arises as to why go to the
trouble of putting the system into this coupled-mode bond-
graph form. It seems hardly worth the extra trouble since a
more conventional bond-graph for the system can easily be
developed and a computer solution based on that model easily
obtained. It would appear that this representation offers
little advantage. The answer lies in the potential for
model simplification. This modal structure allows a view
into the modal response of the system not generally avail-
able in other bond-graph structures. If, for the expected
type of input, a particular mode is found to be inactive, it
may be possible to eliminate that part of the model, thereby
reducing the model complexity and saving computer time.

This would be of particular interest if the model were part
of a large system on which repeated simulations were in-
tended. This situation frequently occurs in the design of
control systems.

One type of model reduction of particular interest is
the modal damping assumption. Earlier it was stated that
one way in which designers avoid the problems with non-modal
damping is to replace the damping in the original system
equations with modally distributed damping in the trans-
formed equations. If the damping is light this procedure is
assumed to introduce only minimal error. The questions here

are:

25

1. When can the damping be considered light?
2. How much error is introduced by assuming damping is

light?

Chapter 4 presents a study into these questions using
the newly developed modal bond-graph model. The objectives
of this study were to determine if any conclusions could be

drawn in regards to:

1. The error introduced by eliminating the coupling element
by looking at some measure of the force or power

transmitted through the 2-port R element.

2. If some indication of the strength of the coupling could
be obtained by comparing the coupling coefficient with

some grouping of the other system parameters.

If some "rules of thumb" could be established in this
area it would give some insights as to when a modeler could
safely make the modal damping assumption.

Before proceeding with this study it was necessary to
develop some way to quantify the difference between the re-
sponses obtained for various levels of modal coupling. At
the time of this study, bond-graph analysis packages such as
ENPORT did not allow numerical comparisons between consecu-
tive simulations. It was therefore, desirable to enhance
the ENPORT package to allow this type of comparison. Chap—

ter 3 presents the development of this feature. It should

26

be pointed out that this is also a very useful feature for
general system modeling and design. It could be very help-
ful in determining the sensitivity of a system response to

changes in the design variables.

Chapter 3

Automated Error Analysis Using ENPORT.

3.1. he Need for Error Ana sis.

Very often in the design modeling process it is desir-
able to evaluate the difference in the output response as
the model parameters are changed. This is especially impor-
tant when the goal is model reduction. For this case these
differences are the output errors resulting from the pro-
posed reduction. The most straightforward way for determin-
ing these response differences is to compare the results
from comparable simulations of the two models.

In the past the problem has been that ENPORT, a bond-
graph simulation software, did not facilitate comparisons of
separate simulations. Designers were forced to "overlay"
response curves from consecutive simulations in order to
visually perform these types of comparisons. The informa-
tion obtained from these comparisons is generally qualita-
tive in nature and lacks the numerical basis for making
accurate comparisons. Since a substantial portion of the
study presented in Chapter 4 consists of making these types
of comparisons, it was desirable to enhance ENPORT to allow
the numerical evaluation of output response comparisons

obtained from comparative simulation runs.

27

28

3.2. Program Development

Before explaining the specifics regarding the new
analysis feature, a brief explanation is in order on how
ENPORT operates. ENPORT is a dynamic analysis package which
provides numerical solutions for dynamic systems which have
been represented in bond-graph/block-diagram form.

This study focuses on models represented in bond-graph
form. In developing the differential system equations from

a bond-graph, four types of variables are used: P, Q, E, F.

(1) P is the momentum variable associated with energy

storage element I.

(2) Q is the displacement variable associated with

energy storage element C.

(3) E is the effort (force) variable associated with

each bond.

(4) F is the flow (velocity) variable associated with

each bond.

Using these variables, ENPORT sorts the system equa-

tions and arrives at a formulation equivalent to

>2 = f(X,U,t) (3.1)

where X is the state vector,

29

U is the input vector,

and t is the time (the independent variable)

A linearized form of this equation can be generated by
a numerical approximation at any given state X and time t.

The result is

x = A*X + B*U (3.2)

where A is generated by ( dfi/ de)

and B is generated by ( dfi/ de)

For a linear system the A and B matrices should be independ-
ent of the X and t chosen.

Once ENPORT has obtained the equations in this form, it
is simply a matter of numerical integration to obtain solu-
tions for the state (energy) variables. These values are
then stored in an internal array for subsequent display. In
addition to these state variables, the efforts and flows
associated with each bond and the input values are also
saved. These can be used later to derive other quantities
of interest such as power and energy transferred through the
bond structure. With these solutions store in internal
memory, output plots of selected quantities can easily be
obtained, although some quantities, such as power or energy,
are derived and need to be computed prior to display.

These stored values can represent a substantial portion

30

of memory. Considering a simple model might contain 10
storage elements, 30 bonds, and 3 or 4 inputs, saving 300
time points for each variable would result in a storage
array containing 22,200 real numbers. For this reason
ENPORT reuses the same internal memory for each new solu-
tion and this is the reason that numerical comparisons of
subsequent simulations could not be made.

The first step in incorporating the analysis feature
was to provide ENPORT with the capability of storing and
retrieving the result values. Since these result files may
be used over a period of several ENPORT runs, it was deter-
mined that they should be saved in disk files and not simply
in internal memory. This also avoids excessive demands on
the required system memory. In order to keep the result
file size to a minimum, the original system model and equa-
tions are not included. Besides the results themselves,
these files contain only a header including the original
model filename, the results filename, i.e. a 1 line descrip-
tion, and a list of the saved variables. It is assumed that
the original model has been reloaded into ENPORT prior to
reloading a saved results file, using the standard method.
This assumption is verified by comparing the model name with
the model name saved in the results file. Although this is
obviously not a failsafe check, it does provide some safe-
guard. Although not yet incorporated, a better check would
be to compare the results variable list with the model vari-

able list. Also in an effort to make the results files

31

smaller, they are saved in an unformatted form.

Once the ability to save and retrieve the computed
results was developed, incorporating a difference analysis
was begun. It was simply a matter or retrieving the results
values into a 3 dimensional array, with the third dimension
indicating a particular set of results. Difference values
can then be computed for any set of selected variables.
These values can be listed or plotted just like an original
set of solutions. From this information quantitative deci-
sions can be made regarding the errors associated with the
proposed changes.

Another procedure along the same lines as error analy-
sis is sensitivity analysis. To assist in this type of
analysis it would be useful to be able to plot some measure
of the change in the output variables as a function of the
variation in system parameters. The problem in incorporat-
ing this type of feature for arbitrarily selected result
variables is that it would potentially require huge amounts
of internal memory. Considering that each variation would
require a full set of results and that several variations
would need to be compared in order to draw accurate conclu-
sions for a particular parameter. Even for a simple model
this would require the storage of 300,000 real numbers.

In order to avoid using large amounts of internal mem-
ory to store loaded results files, the analysis package is
currently limited to a comparison of only 2 files at a time.

This serves the purpose of this study since only error

32

values caused by changing the intermode coupling strength
are of interest. The effect as this parameter is varied
over a range of values can be studied by comparison of the
results two at a time. This is an area for potential future

improvement which will be discussed in a later section.

3.3. An Example of the Analysis Feature

As stated earlier, it is often desirable to obtain
quantitative comparisons of the output response for a system
as the model parameters are varied. ENPORT is now capable
of these types of comparisons. The following example is
provided as a demonstration of this feature.

Using the bond-graph for the non-modally damped system
developed in section 2.2 as an example, the effects of as-
suming modal damping on output response can be studied.

This is the actual application in this study for which the
software was developed. Figure 1 shows the standard 2 de-
gree of freedom spring-mass-damper system from section 2.2.
Using the methods described in that section the bond-graph
in Figure 2 can be obtained for a given set of parameter
values.

This model and its associated parameter values can be
input into ENPORT in the usual way. (An ENPORT file listing
is included in the appendix.) A simulation can then be run
to yield the system response to a given set of inputs and

initial conditions. Selecting the velocities of the masses

33

as the variables of interest, the plot shown in Figure 8 can
be obtained of these velocities as a function of time using
the standard ENPORT feature. In the modal bond-graph of
Figure 2 the velocities of the masses are the flows on bonds
1 and 2: f.1 and f.2. Since no assumptions have been in-
troduced into the model, these resulting velocities are the

true results and therefore are used as the baseline for

future comparisons.

 

 

 

 

 

 

3&9 .
TAIL
a» :17. - i 3 I-
‘J- . .' l
I 1:3 Li II a
0.3 "
(I ' .
3 I GG H II
III 'I '
'|' '.‘_J°\-.. .
L I
l! II~ 3; ‘ ‘-
1 I DU r .5 "_..--—---~-—..._ “a
[B I. - “H‘s; Aw. I -
'1 f if ah" ' ”if”
' 1 l a“ Iii“..- __r"‘- ‘1'.
gr“ y
Ix. -
-::I.IIII , - - -
“it!“ B4+fl ka er§ [IEU £186
TIHE 116'; 1

LEEEHD) F.Sl-——-F.52-.m

RBFHEMEEETWEE
Figure 8

The Reference Velocity

At this point the results data should be filed using

the new analysis feature.

34

Post-processor options

G: Graph (High resolution)
P: Plot (low resolution)
T: Table

8: System graph display

L: List processed data

D: Operations on run Data

R: Return to the main menu

Figure 9

The Modified Post-processor Menu

The new option, D: Operation on Run_Data, can be called
from the modified ENPORT post-processor menu (Figure 9) to

obtain the analysis options menu as shown in Figure 10.

Run_data processing options

W: Write run_data to file

L: Load run_data from file
S: Status of run_data
C: Clear the run_data

A: Analyze the run_data

R: Return to the post-processor

Figure 10

The Run_Data Menu in ENPORT

35

From this options menu, W: Write Run_Data to File, can
be selected. This choice will call a subroutine that writes
all the computed result values to an unformatted binary file
under a user selected name. Call it "ref.dat" for this ex-
ample. Additionally, the user is allowed a one line file
description which is included in the file. This is helpful
for future identification of the correct file.

Now that the results from the baseline simulation have
been saved, a second simulation can be run with the assump-
tion of a modally damped system. This change can be imple-
mented by setting the parameter of the coupling element, RN,

to zero. The output response from this run is shown in

 

 

 

 

 

 

 

Figure 12.
H! I [‘33 .
mu: ,5 ".
1 1. iii ' l -‘
. .555 (‘1 (I
1.3:... ‘
3 I “G kl ‘| .‘I
,II I
u 'u
,3“ '1" .r" " "‘2.
i I an F, 3"? .I: - '-
i i 5’ ”ha. Cgééz—
K x; ET*~“fiT”
' 1 a E Ii} 'I".""‘.._ ._,,.".‘:I'.‘
1. "‘ :3
{3:953 "* ‘1 fl _’_ ‘ -
T IHE 3 1)}; 1

I£Bmm FBI—“FEE."
ASSIi-E {I HIIIZaf-L EI-‘Ié—IFII-EI
Figure 11

The Response with Assumed Modal Damping.

36

One important point to note is that simulations which
are to be analyzed together must currently be run under the
same time conditions. That is, the initial and final times,
as well as the number of time stages, must match in order to
obtain accurate response comparisons. Also, simulations
which are to be compared must originate from the same topo-
logical model. Only numerical values of the model parame-
ters may be changed. For example it would not be acceptable
to invoke modal damping by eliminating the coupling element.

Now that this second simulation has been run, its re-
sults can be written to a separate file. Call it
"model.dat", using the same method as for the reference
case.

With the simulation results for the two cases now
stored in the respective files, comparisons can be made
between these sets of results for selected variables. These
comparisons are obtained by again selecting option D: op-
erations on run_Data from the ENPORT post processor menu
which brings up the Run_Data menu as before. Selecting L:
Load a set of run_data will allow the first file to be
loaded into the storage buffer. This first file is consid-
ered to contain the reference data set or the set containing
the values to be subtracted from the second file loaded. In
the case of this example, this would be the data set saved
under "ref.dat", from the first simulation. After retriev-
ing this reference data set, the second data file can be

loaded using the same method as for the reference set.

37

This data set should contain the results to be compared to
the reference case. For example the file "moda1.dat" would
be loaded.

Now, with the 2 data sets loaded and stored in the
internal memory buffer, the analysis can proceed by select
option A: Analyze run_data. This option presents the Analy-

sis Options Menu as shown below.

Run_data analysis options

G: Graph difference values
T: Table of select difference values

R: Return

Figure 12

Run_Data Analysis Menu in ENPORT

Currently, the options allow the user to obtain tabular
listings and hi-res graphic plots of actual difference val-
ues for user selected variables. In the future the option
could easily be modified to display normalized values such
as actual average, RMS average or absolute average. Select-
ing option '6: Graph', from this menu produces a hi-res
plot of the difference in value of the user selected vari-
ables. The program automatically computes difference values
as the plots are constructed so there are no other steps

involved from the user perspective. The graph routine used

 

38

to obtain difference plots is virtually identical to the

standard ENPORT graph routine and all the standard features

have been retained. Figure 13 is a plot of the displacement

difference values versus time for the two files loaded

above.

 

BBB
am:

.7
313.: ‘z

:1 ' [5.5
mm 3' __f_._
‘LBB V“C

..

~ 3 I G 3 [Di

 

 

 

 

-5189 <
H.132! Edi) 13.9% LEI] U313 2.30
TIME ilGE 1
LEGEND: F.31—— FUSE - --
DIFFEIE'IEE (BLUES
Figure 13

Change in Velocity for Modal Damping Assumption.

Plots could also be obtained for the difference in any
other system variables. The tabular data can also be dis-
played by selecting 'T: Table'. For instance, if the in-
formation regarding the change in net energy into the modal
dampers was desired at each time step, option, T: Table,
could be selected. Choosing the display variable as the
energy (T) on the bonds connected to the modal damping ele-

ments, a tabular listing of the change in net energy into

the damper elements will be produced, Figure 14. Since

39

energy is an integral quantity of the state variables, the
system will ask for the initial energy value. This value is
taken to be the offset in the initial energy for the two

cases and would generally be zero.

DIFFERENCE VALUES
Time T.DA T.DB T.NA

Q.QQEE+QQ E.QEBEE+00 Q.@053E+GQ $.009@E+EQ
5.QQ@E+@$ -7.1824E-33 -7.198EE-@3 7.5281E-g3
1.9@®E+01 -7.9916E-03 -7.4B4éE-03 7.58fi9E-Q3
1.5%QE+Q1 -7.7132E-@3 -7.5135E-03 7.5389E-E3

Figure 14

Tabular Listing of the Change in Damper Energy Dissipated

After completing the desired plots and tables from the
loaded data sets options R: Return can be selected which
returns to the run_data processing options menu. The other

options which may be selected from this menu include:

C: Clear the run_data
S: Status of the run_data

D: Debug the run_data

Option C clears the internal memory to allow room for new
sets of data to be loaded. Option 8 displays the names of
the Data sets which are currently loaded. Option D allows
the user to display a list containing the variable vector
and the numerical values of each variable at each time step

for a loaded results file, generally used as an aid to de-

bugging the program.

40

3.4 Ppoggam Implementation and Future Enhancement
Suggestions.

The Fortran 77 subroutines which perform this new

analysis feature fall into one of 3 categories:

1) New subroutines
2) Modified ENPORT subroutines

3) Unmodified standard ENPORT subroutines

The new subroutines and the modified ENPORT subroutines have
been collected under 2 files: DRUNI and DRULOC. The file
DRULOC contains the subroutines which must be modified for a
particular operating system. These are the subroutines that
perform file manipulations. The rest of the subroutines are
collected under the file heading DRUNI. Appendix A contains
a listing of each of these files for reference purposes.

The added capability of storing and retrieving ENPORT
results data opens the door to a multitude of features that
would further enhance ENPORT's usefulness in design. Some
of these have already been mentioned such as the computation
of normalized measures of the difference values (RMS, abso-
lute average, actual average, etc.). Additionally, the
program could be modified to allow comparisons of results
data that were not computed with the same time steps. This
could be accomplished by simple interpolation and will be a
necessary change if results are calculated with a variable

timestep integrator.

41

A somewhat more complicated enhancement would be the
ability to compare specific results from several simulations
simultaneously. Such a feature might be used to plot some
measure of error (Such as RMS) versus percent change in a
system variable, in essence showing the sensitivity of the
system to changes in that variable. Earlier it was stated,
however, that this type of analysis would currently require
a large amount of system memory. This is because under the
current design all system variables are saved and retrieved
from the results files, which, to compare several result
data sets, would require a prohibitive amount of memory.

The solution to this problem would be to allow the user to
reload only those system variables required to produce the
desired plots. This would require much less system memory
which would make it feasible to produce sensitivity plots.

The discussion above points to another potential oppor-
tunity for improving the program which would reduce the
result data file size. This opportunity arises because the
current design saves values for all the system variables.
For a given model, many of these variables can be derived
from a set of other system variables through algebraic equa-
tions. If the system could recognize these independent
variables, the size of the result data sets could be reduced
by including only these variables and the necessary alge-
braic equations. From these independent variables the post-
processor could compute any requested variable value using

the algebraic equations. This data handling scheme would

42

result in a minimal data set file size but would require a

substantial change in the design of the post-processor.

Chapter 4

Approximations to Non-modal Systems

The model developed in section 2.2 was shown to be an
exact "coupled-mode" bond-graph representation for a non-
modally damped system. Solving this bond-graph would yield
the exact response for the system under study. For larger,
multiple degree of freedom systems this solution can be very
time consuming. This is especially true when the model will
be solved repeatedly, under a variety of input or initial
conditions. It is therefore desirable to reduce the model
complexity wherever possible. This can most easily be ac-
complished by eliminating elements and bonds which can be
shown to have minimal effect on the output response. This
procedure is routinely used in the modal analysis of con-
tinuous beam structures when only the first few modes are
considered in the analysis. It is this same reasoning that
allows designers to model lightly damped non-modal systems
with modal damping assumptions. In the case of the non-
modal bond-graph developed in Chapter 2 this assumption
eliminates the non-modal 2-port R damping element resulting

in a simplified representation.

4.; The Objectives of The Study

The problem with the modal damping assumption is that

it is usually unclear as to when the assumption can

43

44

accurately be applied. Using the modal bond-graph from
section 2.2 to provide a unique perspective, this study
attempts to quantify this decisions. In essence some "rule
of thumb" was sought that would indicate the amount of error
which would be incurred if the non-modal damping element
were eliminated, reducing the system to the modally damped
case.

To provide this rule of thumb would require some type
of indicator which could be correlated to the error incurred
in the modal damping assumption. Mathematical analysis did
not reveal the desired indicator. An in-depth study was
therefore undertaken comparing simulation results as system
parameters were varied. It was believed that some correla-
tion could be found by comparing the output response error
between the non-modal model and the corresponding reduced
modal model to easilty measure system quantities. Since
there was no way to accurately compare results from succes-
sive simulations the first task of this study was the ENPORT
enhancement detailed in Chapter 3. Once this new tool was

developed the experimental investigation could proceed.

4.2. The Study Implementation

Figure 15 shows the modal bond-graph of the non-modally
demped system developed in section 2.2. The structure of
this bond-graph can basically be broken down into 2 inde-

pendent modal oscillators that are coupled by a 2-port

45

damping element.

 

(MA) (MB)

(CA) IA IB (CB)
RA RB
(KA) \ (CN) / (KB)
CA‘w—l IA I—-——- / RD A ——————— ——~--I\ IB (~—-—-°-/ CB
(VII) (V22)

TFIA TFZB
(VIZ) V2I)

r \TFIB TFZA T
SE1 Jil— I 0' 02 |p_f_2_ SE2
Figure 15

A Coupled-Mode Bond-Graph for a Non-modally Damped System.

As explained in section 2.2 this coupling element ex-
ists to provide for the damped coupling of the transformed
modal equations (see eq. 2.36). The parameter value for
this element is determined by the off diagonal terms in the
transformed damping matrix. Since the transformed damping
matrix is symmetric, a single parameter value is associated
with this 2-port R element. Because of the symmetry, this
element provides a unique power interaction between the
modes which can be illustrated a typical element as shown in

Figure 16.

46

 

 

el (Cti) <32
7.); ‘1
a r—f I f2? 1
Figure 16

A Typical 2-port R Coupling Element

Assigning the power direction and causality the same as
used in the non-modal bond-graph of Figure 15 implies that
the inputs are the flows fa and fb. The algebraic nature of
the R element then allows the direct computation of the

output efforts ea and eb as follows.

(cn) * fb (4.1a)

ea

eb (cn) * fa (4.1b)

The power on bonds a and b can then be computed.

Power a ea * fa (4.2a)

eb * fb (4.2b)

Power b

substituting equations 4.1a and 4.1b into 4.2a and 4.2b

yields

Power a (CN) * fb * fa (4.3a)

Power b (CN) * fa * fb (4.3b)

47

01'

Power a = Power b (4.4)

This result is interesting since it shows that the
nature of this coupling element is to act alternately as a
power source or a power sink to both elements simultane-
ously, depending on the relative signs of the input flows.
Additionally, an equal amount of power is provided to, or
dissipated from, both modes.

Since the addition of this coupling element is the only
factor that differentiates a non-modally damped system from
a system with modal damping, it would seem that the strength
of its interaction could be used to determine the potential
for model reduction. One measure of this interaction typi-
cal in bond-graph studies is the power transferred. Using
the power interaction of the coupling element as a basic
measurement to quantify the potential for a modal assumption
was therefore the first path explored as part of this study.

To implement this portion of the investigation a simple

2 degree of freedom was chosen as shown in Figure 17.

F I X '3 X29.

..“.9_ I:
”"2 I’ ””3 F—
-___(3) _-_-.-_-___7 2) .
‘ W (I) - V (2)
:3“ ’"flm‘

CI 50:0“ 3:0“: cs

Figure 17

 

 

 

 

 

 

 

 

The Base 2 D.O.F. System

Using the methods developed in Chapter 2 the modal
mass, stiffness and damping parameter matrices were com-

puted, as well as the modal transformation matrix.

 

_
M' = 10.90 0

0 1.10 (4.5a)
K. = 6000 0

0 6.00 (4.5b)

CI = c1+1.5002+4.95c3 c1-1.50c2-.50c3
c1-1.50c2-.50c3 c1+1.50c2+.051c3 (4.5c)
and

¢ = 1.00 1.00

2.225 -.225 (4.6)

From these parameter matrices a coupled-mode bond-graph can

49

be obtained as in Figure 18.

(IO.9) (I. I)

(CA) IA IB
RA T
(6.0) \ (CN) 8/3235
CA (.1 IA I—————-——---/ RD z—~-—-—~—~-——-~—~I ——~—— 2- bCB
( I .0) 22)
TFIA TFZB
.. ( I .0) 2.22) —-
( TF I B TFZA/ f
\
SEI -—-[:-l-~7I 0' 02 Fig“ SE2
XI x2

Figure 18

A Coupled-Mode Bond-Graph For the System in Figure 17

It should be noted at this point that although a non-
modal coupling element has been included, this system may or
may not be modally damped. This does not present any prob-
lems since if the system were modally damped, the parameter
of the non-modal coupling element would simply be zero.

The initial focus of this investigation was to corre-
late the modal damping assumption error to some measure of
the power activity of the 2-port R element. It was believed

that larger assumption errors would result from increased

50

intermode coupling activity, which could be measured as
larger power interaction. If a generalized correlation
between power and error could be found then a decision could
be made as to the potential for model reduction based on
power measurements. The idea was to run a simulation with
the full system under the expected forcing conditions and
measure the power interactions. From these power measure-
ments quantitative decisions could be made about the poten-
tial for model reduction.

An experiment to test this idea and to reveal the cor-
relation between power and error was then conducted. Using
the above system, groups of simulations were performed while
varying the damping parameter values. During each group of
simulations, the values of the modal damping parameters, Ra
and Rb, remained fixed. Several simulations were then run
while the value of the intermode coupling element (Rn) was
varied. For each simulation a measure of power transferred
through the intermode-coupling bonds was recorded. Addi-
tionally, the percentage error that would occur if the sys-
tem were assumed to be modally damped was recorded. Plots
of output error vs power were then created for each group of
simulations.

For the purpose of this experiment, error was defined
as the difference in output velocities, f.1 and f.2, between
a non-modally damped system and the same system under a
modal damping assumption. Recall that these are the actual

velocities of the masses in the original system. The RMS

51

average over some time period was selected as a means to
quantify the amount of error. To avoid scaling problems
these raw quantities were normalized by dividing the RMS
error by an RMS average of the true velocity values. This
normalization provided a measure of the percentage error
associated with making a modal damping assumption.

The second piece of information necessary to obtain the
desired plots was a quantitative measure of the power activ-
ity of the intermode coupling element. The RMS average was
selected for this measurement also. This measurement is
also affected by scaling and must be normalized. The normal
procedure in ENPORT is to normalize these power measurements
by the maximum power average of the elements selected,
yielding power as a percentage of maximum. Since the effect
of the non-modal element is to exchange power with each
modal oscillator, it would seem that normalizing this power
by some measure of the power activity within a mode,namely
the maximum power bond, would be appropriate.

This method was used in early portions of the study.

It was noticed during the course of the early investigation
that the element which exchanged the maximum amount of power
would vary. It was felt that this inconsistency might
hamper the process of developing the desired correlation.

It was believed that a better scheme would be to compare the
power activity in the damping bonds. Because the origin of
intermode coupling is in the original system damping, this

normalization would seem to provide a better insight into

52

the relative strength of the intermode coupling element. It
is this relative strength which is the core of the issue
since the modal damping assumption simply eliminates the
intermode element and therefore its effect. If the effect
is relatively weak then the elimination of the effect will
not cause much variation. For this reason only the average
power in the R element bonds was computed. The highest
average was used as a baseline to normalize the other damp-
ing bonds. This proved to be fairly consistent over the
course of the experiment since the first mode always seemed
to contain the highest power level.

With these insights in mind, the experiment continued
in an attempt to find a generic correlation between the
normalized intermode power activity and the percentage of
error resulting from a proposed modal assumption. Many
groups of simulations were conducted for varied modal damp-
ing ratios. To avoid influencing the results with the input
frequency, unit step forces were simultaneously applied at
both inputs. This type of input forced a transient analysis
as opposed to a steady state analysis which would be pos-
sible under a periodic form of input. Still it was felt
that the correlations would be more easily obtained if
forcing frequency was not a factor. The transient analysis
did present a problem however, which was over what time

period to compute the averages. Three times the longer

53

modal time constant was selected for this study. Potential
problems resulting from this decision will be discussed in a

later section.

4.3. e Results.

Each group of simulations consisted of holding all
parameters fixed except the intermode coupling. This pa-
rameter was then varied over an allowable range as simula-
tions were run for each setting. The allowable range for
this parameter was determined using the transformed damping
matrix. Since real physical systems were of interest in
this study, the original damping parameters were selected to
be positive. This restricted the range over which the off-
diagonal term in equation 4.3c for a give set of diagonal
terms. Recall that the off-diagonal term is the intermode
coupling element parameter, while the diagonal terms are the
parameters for the modal damping elements. For each of
these groups of simulations, curves representing the average
damping power activity versus percentage error were pro-
duced. A typical plot is shown in Figure 19. (Appendix 8
contains an ENPORT listing of the mode with the parameter

values)

54

)

 

 

<

(I
232 20—"- /} ‘/p
(I D

m I ,/ //
2m I /

0 I5—- /
Em / / 0 - NEGATIVE RN
g, I / / 0 I - POSITIVE RN
QC) H}._. 1’
mu) 4;,” —-- CA=I CB=I
922'. / — — CA=2 CB=2
Z<
UJE
OII
CEO
LUZ I l

25 30

 

PERCENTAGE OUTPUT ERROR

Figure 19

A Typical Power Versus Error Plot

The curves represented in this plot are the power dis-
sipated in the coupling element (RD) for various levels of
modal damping. The RMS values have been normalized by the
average power in the first mode damping element (Ra). The
power scale is therefore represented as percentage of power.
Note that each data point is for a specific value of cou-
pling parameter.

Each curve developed during the experiment exhibited
the same expected trend of increased error as the modal

coupling power activity increased. Notice however that each

55

power vs error curve is represented by two lines. This
split occurs as the intermode element parameter is varied
from its maximum negative value, through zero, and to its
maximum positive value. This causes the split in the error
curve since error increases as the magnitude of the inter—
mode parameter is increased. At first it may seem strange
that positive damping parameters in the original system
could produce a negative damping parameter in the trans-
formed equations. However, the analysis shows this to be
perfectly acceptable. What does seem strange is that the
power vs error curves are not the same for a negative cou-
pling element parameter as for a positive parameter. In
general the effect of the sign of intermode parameter on
system behavior is not understood.

Although individually each group of simulations produce
the expected results, the overall trends were not apparent.
The slopes of the power vs error curves varied between
groups of simulations as the modal damping parameters were
changed. One set of simulations would show large errors
corresponding to very small power percentages, while the
next set would be the opposite. In the final analysis no
trends could be found which could be used to provide the
desired rule of thumb for making a modal damping assumption.

Some progress was made in observing trends in symmetric
systems. These are systems in which the damping ratios are
the same for both modes. The general trend observed in

these systems was that as the damping ratio was increased

56

the sensitivity of output error to power percentage was also
increased. That is, that as the amount of modal damping is
increased, a relatively smaller amount of power activity in
the coupling element causes a larger output error. This
seems to go along with the common wisdom that for small
damping the modal damping assumption can be made with little
consequential error. This trend was only an observation and

was not thoroughly verified.

4.4. Problems

Although the objectives of this study were not
achieved, a great deal of information was obtained. Among
this information are some of the potential reasons for not
finding the desired correlations. It is possible that the
desired correlation does not exist. However this author
still believes that such a correlation can be found.

One of the important factors effecting this study was
the selection of step forcing inputs. As stated earlier,
this decision forced a transient form of analysis. Due to
the nature of the transient response, simulation time has
large effect on the averaged values used in this study.
This point can be illustrated using the response curves

shown in Figure 20.

£57

 

 

I" "
l ‘51 .n
r” r:-
- r‘I- (5:1
-1
I

 

 

 

 

 

 

2.3.0 ’
“+186
£1.80 8.60 1.3.1 LBJ BAG 3.3a)
TIHE iIUE 1
LEGEND) F .ELI ..——— FIB - --
KBIFETBEE
Figure 20

Modal Velocity Response Curves.

The curves shown in Figure 20 are the modal velocities
plotted as a function of time. Notice that they are each
typical underdamped responses of differing time constants.
Using the methods from the experiment, these curves can be
averaged over a time period of three times the longer time
constant. If there is significant difference in time con-
stants between the longer and the shorter, significant er-
rors are induced in the average of the short time constant
curve. These errors occur because for a substantial portion
of the average period no activity is present. This tends to
artificially reduce the averages, leading to power percent-

ages that vary depending on the relationship between the

58

modal time constants. It is believed that this is part of
the reason that some progress was made in the symmetric
systems, since in these systems both modes posses the same
time constant.

Another factor that strongly effected the analysis was
the value of the transformer parameters. These parameters
attenuate the inputs to and outputs from each of the modes,
scaling the effect of each mode in the output response.
Since the transformed output responses were used in the
error percentage computations, large error in one mode may
not result in a large error in the analysis, the second mode
had far less influence in the output than the first. There-
fore, large errors in the second mode produced very little
change in the output response. Somehow, in order to find
the desired general rule of thumb, this effect must be ac-

counted for in the analysis.

4.5 Suggestions for Future Research.

During the course of this research a great deal of time
was devoted to the development and initial analysis of the
model and to the development of tools and techniques neces-
sary to perform the analysis. Although the resulting re-
search did not produce the desired results, the information
gathered and documented in this report provides a good
starting point for a second phase of research. With this

information available at the start, it is believed that

59

further research into the behavior of the modal model will
reveal the desired correlations.

One area of interest is to consider the effect of the
two port coupling element as a modulated effort source since
its effect is to provide an additional forcing input to each
mode. Comparing the force level in the coupling elements to
the other force inputs to the mode could provide some useful
insights.

At this point it is also believed that the step inputs
to the system should be replaced by periodic forcing. This
will provide a periodic steady state response and avoid the
problems associated with averaging the transients as de-
scribed above. Elimination of this problem should make the
trends more apparent although forcing frequency effects will
now need to be accounted for.

Based on the results that were obtained in this experi-
ment it is believed that the proposed trends are present and
that some measure of coupling activity can be correlated to
the amount of error associated with a modal assumption. It
is envisioned that a second phase of research that combines
the information presented in this report with some further

creative insights will produce these correlations.

Chapter 5

Summary

This research began as an investigation into linear
non-modally damped systems using bond-graphs. It was hoped
that a bond-graph approach might lead to new insights into
the effects of damped coupling in the transformed equations.
The ultimate goal was to develop some guidelines indicating
the amount of output error potentially incurred by making a
modal damping assumption on a given system. When possible,
this assumption allows a simplified analysis, as well as
reduces the model complexity.

The first step in the process was to develop a modal
model that accurately represents non-modally damped systems.
Using a set of coupled transformed equations, a standard
modal bond-graph was modified to include the coupling ef-
fect. It was found that a standard 2-port R element, con-
nected between the modes would accurately represent the
coupling effect. Although this element is commonly governed
by two parameters, only one was necessary in this case due
to symmetry of the transformed damping matrix. This fact
lead to the interesting property that the element acts a1-
ternately as a power source or sink to both modes simultane-
ously, providing or dissipating equal amounts of power to

both modes.

60

61

After developing and verifying the model the next step
was to begin to use the model to study the effect of inter-
mode coupling strength on output response. This study re-
quired the comparison of output responses of a series of
simulations. The problem here was that ENPORT software did
not facilitate these types of direct comparisons. Because
these types of comparisons are often desired when using
ENPORT in the design process, a decision was made to enhance
the package by developing software that would include this
feature.

With all the tools in place the study proceeded in an
attempt to find a correlation between intermode coupling ac-
tivity and output response error. Net power activity was
used as the basis to quantify the amount of coupling effect.
Output error was defined as the difference between the out-
put velocities with intermode coupling and the same veloci-
ties with coupling set to zero, as if assuming modal damp-
ing. RMS average was used to quantify both effects. To
avoid a reflection of forcing frequency in the modal re-
sponse, a step input was used in the study. This resulted
in a transient form for the corresponding responses. Three
times the longest time constant was selected as the standard
time base for computing the RMS averages.

Groups of simulations were run for varied damping para-
meters. In each group, the value of the intermode coupling
parameter was varied. Plots of net average power versus

output response error were constructed for each group of

62

simulations. To include the effect of scaling between
groups or simulations, the power average was normalized by
the highest average power transferred in the modal damping
elements. The average velocity errors were normalized by
there respective average velocities. These normalizations
provided measures that were interpreted as percent power and
percent error and it was from these values that plots were
constructed.

As predicted each plot showed the expected trend of in-
crease? error percentage a coupling power percentage in—
creased. The problem came in attempting to draw any corre-
lations from one group to the next. The slope of each plot
seemed to vary depending on the overall amount of damping.
That is to say that for one plot 10% power would correspond
to 3% error while in the next plot 10% power would corre-
spond to 15% error. Because of this plot to plot variation,
no universal correlations could be found. However, it was
observed that for smaller damping higher power percentages
seemed to correspond to smaller errors.

although the desired 'rule of thumb' for assuming modal
damping was not found, much valuable information was gained.
Tools and techniques were developed that might be valuable
to future research in this area or possibly some other re-
lated areas. Additionally, one avenue was explored and the
possible road blocks pointed out which provide insights and
direction to future research in this area. I hope some

future research will reveal the desired correlations.

63

This information includes tools and techniques that can
be used as a starting point for a second phase of investiga-

tion that might reveal the desired correlations.

LIST OF REFERENCES

1/

2/

3/

4/

64

LIST OF REFERNECES

I. Fawzy. "A Theorm on the Free Vibration of Damped

Systems." Journal of Applied Mechanics, Trans.
ASME, March 1977, 132-134.

D.L. Margolis: and G.E. Young, "Reduction of Models
of Large Scale Lumped Structures Using Normal Modes
and Bond Graphs." Journal of the Franklin Institute.
Vol. 304, No. 1, July 1977, 65-79.

R.C. Rosenberg; and D.C. Karnopp, Introduction to
Physical System Dynamics, New York: McGraw-Hill,
1983.

ENPORT Reference Manual, Rosencode Associates, Inc.
Lansing, Michigan, 1989.

APPENDIX A

65
APPENDIX A

ENPORT Enhancement Code Listing

CFlLE:DRUNli --------------- DRUNll ------------------------------------
EPILEzDRUNll --------------- DRUNll ------------------------------------
C--- Purpose: Subroutines in this file support storing and retreving
C of simulation results data and the caICulation and
C displau of difference values between sets of retrieved
2 data.
g--- Description: A main module called from the post processor.
C--- Contents:
C RUNBL block data unit for initializing common
C DATRUN controls module flow with menu
C DATPUT put the run_data in file
C DATCET et the run_data from file (dummu)
C DBCRUN isplaz retrieved data for debugging _
C ANLDVR presen s menu for selecting ana usis and displau routines
C DIFPLT driver for hi-res graphics
C DFCURV draws actual hi-res curves
C DlFTBL driver for tabular data displa:
C DIFDEL gets displa: increments for ta le printing
C DlFLST oes the ac ual table grintin
C DIFVRH gets the displag varia le lis from user
C DlFFIL ills the derived variables if necessarg
C lNTDlF inte rates displa variables if necessarq
C DFRSLT fcn o evaluate t e difference values
C DFTLNT sets the displag time limit parameters
g DFYLHT finds the min/mas value for each display vbl
C
C--- Index:
C ANLDVR
C DATCET
C DATPUT
C DATRUN
C DBCRUN
C DFCURV
C DFRSLT (fcn)
C DFTLN
C DFYLHT
C DlFDEL
C DlFFIL
C DlFLST
C DlFPLT
C DIFTBL
C DlFVRH
C lNTDlF
C RUNBLK
C
C
g--- Programmer: Tom Bush Harch l5. 1989
E--- Last revision: flag 2. l989 TLB
CEOFHzDRUNll - ----------------------------------------------------------
C>>>>>
g .
ERUNBLK >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 12/21/88 RR
C BLOCK DATA RUNBLK
E--- PURPOSE: Initialize common vbls in DATZBK.CBK
lNCLUDE"ENPORT>ROSENBERC>SH1P72>SlZEBK.CBK’
INCLUDE 'ENPORT>ROSENBERC>SHIP72>SOLNBK.CBK‘
INCLUDE 'RDATBK.CBK'
END
C>>>>>
g
EDATRUN >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 01/09/89 TB
C SUBROUTINE DATRUN

C--- PURPOSE:

C

Err- DESCRIPTION:
INCLUDE
INCLUDE

INCLUDE

Presents main menu for simulation results data handling.
Hain menu called from post processor.

'ENPORT)ROSENBERG>SHXP72>SlZEBK.CBK‘
'ENPORT)ROSENBERC>SHIP72>SOLNBK.CBK'
'RDATBK.CBK'

(56

CFlLE:DRUNll --------------- DRUNll --------------------------

INCLUDE 'ENPORT)ROSENBERC)SHIP72)UTILBK.CBK‘

C
INTEGER I
LOGICAL FULL. OKAY
C CHARACTER TOKENDI. STRING'BO

EXTERNAL HENSET. BLNKLN. URTSTR. PROMPT. CETANS. COON.
EXTERNAL DATPUT. DATCET. DBCRUN. ANLDVR

INVOPT

C
CeeeDATRUNeeaeeeeeeeeeeeaeeeeaeeaeeeaaaaeaeeaaesseaeaeaeaeeeeeeeeaaeaeee

CALL NENSET(.TRUE.)
FULL - HHFULL
NDSET' O

C---- Present the run _data menu
10 CONTINUE
IF (FULL) THEN
CALL BLNKLN
CALL BLNKLN
CALL HRTSTR(' Run _data processing options‘)
CALL URTSTR(: ---------- )

 

CALL HRTSTR( Hrite run data to file’)
CALL HRTSTR( Load run Kata from file')
CALL URTSTRI’ Status 0? run _data’)
CALL HRTSTR(’ Clear the run data‘)
CALL URTSTR(’ Debug print 0? the data’)
CALL HRTSTRI' Analyze the run _data')
CALL HRTSTR(’ . Return to the main menu ')
CA ALL HRTSTR(' --------------------------- )
STRINCI’ Enter option (R): '

ELSE
CALL BLNKLN
CALL BLNKLN
CALL HRTSTRi' Run _data processing options')
CALL URTSTR( -------------------------- ’)
STRINc- ' Hrite. Load. Status. Clear. Debug. Analyze.

”9????9’9713

lifull): '
ENDIF
C-r- NON READ THE COHHAND TOKEN
CALL PROHPT(STRINC)
TOKEN
C CALL CETANSiTOKEN)
C--- NON INTERPRET THE TOKEN

IF (TOKEN.EO.’O’) THEN

IF (TOKEN.EO.'H') THEN
CALL DATPUTIOKAY)
COON

ELSEIF (TOKEN.EO.’L') THEN
CALL DATCET(OKAY)

ALL 0
ELSEIF (TOKEN. E0. ’5' ) THEN
CALL URTSTRI‘ Names of data sets loaded ')
IF (NDSET.CT.O) THEN
DO 50 l- l.NDSET
50 CALL HRTSTR(' 'l/DSNAH(I))
CALL GOON
ELSE
EN g?%L HRTSTR!‘ No run data has been loaded.‘)
ELSEIF (TOKEN.EO.'C’) THEN

ME 0
ELSEIF (TOKEN.EO.‘D') THEN

ELSEEF (TOKEN.EO.'A') THEN
ELSEIF (TOKEN.EO.'R') THEN
URN

CALL INVOPT

Return?

(57

CFlLE:DRUNll --------------- DRUNI) ------------------------------------

C
END
E))))>

C
CDATPUT >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>§>>>>>> Last Change: 03/09/89 TLB

C

C SUDROUTINE DATPUTIOKAY)

E--- PURPOSE: Store the results data in a binary file.

g--- INPUTS: Generated by dialog.

E--- OUTPUTS: Data to file.
INCLUDE 'ENPORT)ROSENBERC>SHIP72)SIZEBK.CBK'
INCLUDE 'ENPORT)ROSENBERC>SHIP72)TITLBK.CBK'
INCLUDE 'ENPORT)ROSENBERC>SHIP72)SOLN8K.CBK'
INCLUDE ’ENPORT>ROSENBERC>SHIP72)POSTBK.CBK'
INCLUDE ’RDATBK.CBK'
INCLUDE 'ENPORT>ROSENBERC>SHlP72>UTlLBK.CBK‘
INCLUDE 'ENPORT)ROSENBERC>SHIP72)INPTBK.CBK'

C INCLUDE ’ENPORT>ROSENBERC>SHlP72>CRFLBK.CBK’
CHARACTER FlLNAHO32. STRlNCe72. TIHOS. DATOB
INTEGER UNIT. 1. J

C LOGICAL OKAY. lSYES

EXTERNAL BLNKLN. HRTSTR. PROHPT. YDRN. OUTBUF
EXTERNAL CETUFL. CETTD

CaeeDATPUYeeeeeeeeeeeaeeeeeeeeeeeeeeeeeeaeeeeaeeeeeeaeeeeeeeaeeeaeeeeaea
C

g---- Checl if original model .5. filed
UNlT-PRUNlT
0PEN¢UNlT.FlLE=NAHE.STATus-‘OLD‘.ERR-950.
l FORfla’FORHATTED’.ACCESS=‘SEOUENTIAL’)

E---- File exists. model has been filed

C CLOSE¢UNlT)

C---- Get the run label

10 CONTINUE

CALL BLNKLN
CALL URTSTRt' T
CALL HRTSTRi' ’

CALL BLNKLN
CALL PROHPT(' Do you want to change it? (N):‘)
ISYES'.FALSE.
CALL YORN(ISYES)
IF (ISYES) THEN
CALL BLNKLN
CALL URTSTR(' Enter the new run label on one line ‘)
READ(e.lOSS) HRTLBL
l055 FORHAT(A)
HRITE(O.'(1X.A)‘) HRTLBL
CALL OUTBUFiZ)
COTO lO
ENDIF

he current run label line isz')
//HRTLBL)

C
C---- Get the file name
FILNAHI’DUHHY.NAH'
CALL CETUFL(UNlT.FlLNAH)
C lF (UNIT.E0.0) RETURN
C---- Hrite the heading line. results file name. original model title.
C and run label
REUIND(UNIT)
CALL CETTDIT!H.DAT)
HRITE(UNIT.ERR-900) PROCID.VRSION.TIH.DAT
URlTEIUNlT.ERRt900) FlLNAH
HRlTEIUNlT.ERR-900) NAME
HRITElUNlT.ERRs900) HRTLBL
C---- Hrite the number of stages saved. the initial and final times.

e vector
C

68

CFlLE;DRUNll --------------- DRUNll ------------------------------------
uRlTE(UNIT.ERR-900) NS.TlN.T2$OLV
HRITE<UNIT.ERR-900) NOVV
C HRITE(UNIT.ERR-900) (OVNL(I).l-l.NOVV)
C---- Write the run data
DO 100 III l.N
£00 CONTINUE
C---- Close the file and return
C CLOSE(UNlT)
CALL BLNKLN
STRING- ‘ Run data has been written to file; '//FlLNAH
CALL URTSTR(STRlNG)
OKAY-.TRUE.
C RETURN
C---- Uh-oh return

900 OKAY-.FALSE.
CALL BLNKLN
CALL HRTSTRI’ 0'9 Error while writing to file '//FlLNAH)
RETURN

C---- No model filed

950 OKAY-.FALSE.
CALL BLNKLN
CALL URTSTR(' 000 The model has not been filed.’)
CALL URTSTR(‘ Please file origional model before
i writin the results to a file.‘)
CLOSE<U IT)
RETURN

C
END

C>>>)>

g ,.

EDATGET >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 03/09/89 TLB‘

C SUBROUTINE DATGETIOKAY)

E--- PURPOSE: Read the results data from a bin file

E--- INPUTS: Run data common vbls in file.

E--- OUTPUTS: Data in DATZBK.CBK filled in.

INCLUDE 'ENPORT>ROSENBERG>SHIP72>SlZEBK.CBK'
INCLUDE 'ENPORT>ROSENBERG>SHIP72>SOLNBK.CBK’
INCLUDE 'RDATBK.CBK'

INCLUDE 'ENPORT>ROSENBERG>SHIP72)UTlLBK.CBK’
INCLUDE 'ENPORT>ROSENBERG>SHIP72)lNPTBK.CBK'

C lNCLUDE 'ENPORT>ROSENBERG>SHIP72)GRFLBK.CBK'
CHARACTER DSNAHT032. STRING972. NAHTHPOBZ. FlLNAHe32
CHARACTER TPRGlDOi. TlHeB. DATOS
INTEGER UNIT. 1. J. TVRSN

C LOGICAL OKAY. E7YORN

C EXTERNAL BLNKLN. HRTSTR. FNDUFL. E7YORN

CeaeDATCETeeeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeeeeeeeaeeaeeaeeaaeaaeaaeeaeaoe

C

IF (NDSET.GE.HXNDST) THEN
CALL BLNKLN
CALL URTSTRI' see The run data base is full.')
uRlTEISTRlNG.lO20) HXNDST

lO2O FORHAT(4X.12.' is the masimum.')
CALL HRTSTR(STRING)
OKAY-.FALSE.
RETURN

C ENDIF

lO CONTINUE

Crrrr Get the file name
FILNAH-‘eeee'
CALL FNDUFLIUNIT.FILNAH)
IF (UNIT.E0.0) THEN
OKAY-.FALSE.

69

CFILEzoRUNIl --------------- DRUNI! ------------------------------------

C---- Set the data set name temporarilu
DSNAHT- FILNAH
DO 20 I- l.NDSE
IF (DSNAHT E0. DSNAHiI)) THEN
CALL BLNKLN
CALL URTSTR(‘ .09 Duplication of data set already loaded.’)

CALL HRTSTR(’ Please try again.’)
CDT 0 10
ENDIF
go CONTINUE
NDSET- NDSET +1
C DSNAH¢NDSETlI DSNAHT

C---- Read the heading line. results File name. original model title.
C and run label
REUIND(UNIT)
READ(UNIT.ERR-900.END-950) TPRCID.TVRSN.TIH.DAT
READ(UNIT.ERR-900.END-950) FILNAH
READ(UNIT.ERR-900.END-950) NAHTHP
READ(UNIT.ERR-900.END-950) RUNLBL(NDSET)

C---- Check to see if correct model is loaded
‘ IF (NAHTHP .NE. NAHE) THEN
CALL URTSTR(' Incorrect model loaded’)
CALL HRTSTR(' The model 'l/NAHTHP)
CALL HRTSTR(' Hust be loaded to proceed')
NDSET-NDSET-l

RETURN
C NDIF
C---- Check i9 file is correct
IF (FILNAH.NE.DSNAHT) THEN
EALL BLNKLN

ALL HRTSTR(' File name mismatch.‘)
ENgIFL HRTSTRC’ File ‘l/DSNAHT/l' was requested’)
HRITE(STRING.10IO) TPRCID.TVRSN
1010 FORHAT(' written b ENPORT-‘.A1.’.’.I2)
CALL HRTSTR(STRING .
STRINCI’ Date written: ’l/DAT
CALL HRTSTR(STRIN¢)
ST RINGt' Time written ’l/TIH
CALL HRTSTR<STRINGI
CALL BLNKLN
CALL HRTSTR(‘ The run labela')
CALL NRTSTR(RUNLBL(NDSET))
C---- Check if it is oh ,
CALL BLNKLN
IF (.NOT. E7YORN(' Is this the data set gou want?'..TRUE.)) THEN
NDSET-NDSET-l

RETURN
C ENDIF
C---- Read the number and list of run vbl nam
READ(UNIT.ERR-900.END-950) RNSCNDSET). RTIN(NDSET). RTZSLV¢NDSETI
READ¢UNIT.ERR-900.END-950) RNOVV(NDSET)
C READ<UNIT.ERR-900.END-950) (ROVNL(I.NDSET).I-1.RNOVV(NDSET))
C---- Check 1! time data match re'erence set.
STR ING-' 090 Fil no t com atible with reference set'
IF (RTIN(NDSET). NE. RTIN(l)) HEN
CALL BLNKLN
CALL HRTSTRCSTRING)
CALL URTSTR(’ Hismatch in the initial times.')
NDSET-NDSET-l
RETURN
ENDIF
IF (RT2SLV(NDSET). NE. RT2$LV(i)l THEN
CALL BLM
CALL URTSTR(STRINC)
CALL URTST Hismatch in the Final times.’)
NDSET-NDSET-l
RETURN
ENDIF
IF (RNS(NDSET). NE. RNS(l)) THEN
CALL BLNKLN
CALL HRTSTR(STRING)
CALL HRT ST Hismatch in the number of stages saved ’)
NDSET-NDSET- l
RETURN

ENDIF

‘70

CFILE:DRUNII --------------- DRUNIl -------------- ' ----------------------

C
C---~ Re ad the run data
DO 100 I- l.RNS(NDSET)
READ (U lT.ERR-900.END-950) (DSET(I.J.NDSET).J-l.RNOVV¢NDSET)l
éOO CONTINUE
C---- Close the File and return
C CLOSEt UNIT
CALL BLNKLN _
STRING- ' Run data has been read From File: ‘IIFILNAH
CALL HRTSTR<STRINC)
OKAY-.TRUE.
C RETURN
C---- Uh-oh return

900 OKAY-.FALSE.
NDSET- NDSET -l
CALL BLNKLN .
CALL HRTSTR(' GOO Error while reading File ’//FILNAH)
RETURN

C---- Another uh-oh return
950 OKAY-.FALSE.

CALL HRTSTR(' .00 Unexpected end-oF-File encountered.')

CALL HRTSTR(’ Data read are not complete or reliable.’)
CALL HRTSTR(' This File has been ignored.’)
C RETURN
END
C>>)>>
E
EDBCRUN ))>>)>)))>))>)>>>)>>>>>>Z>>>>>>>>)>>>>> Last Change: 01/09/89 TB
C SUBROUTINE DBCRUN
g--- PURPOSE: Displag data stored in memoru buFFers For debugging.
C--- INPUTS; RUNLBL
E DSNAH
E--- OUTPUTS:
INCLUDE 'ENPORT)ROSENBERC)SHIP72)SIZEBK.CBK'
C INCLUDE 'RDATBK.CBK'
CHARACTER STRINGOBO
INTEGER I J
C LOGICAL ISYES
C EXTERNAL BLNKLN. HRTSTR. YORN
Ceeepacnuueeeeeeeeaeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
C

CALL BLNKLN
CALL HRTSTRt‘ 0'0 This ogtion only disp
CALL HRTSTR(' From t e last File lo

IF (NDSET CT. NO) THEN

CALL URTSTR!’ Data From File: ‘//DSNAH(NDSET))
CALL HRTSTR(' The run label is;')
CALL URTSTR<RUNLBL(NDSET))

CALL BLNKLN

STRINC-' Hou ou like to see the saved vbls? (Yl'
CALL NRTSTRCSTR NC)

ISYESI. TRUE.

CALL YORN¢ISYESl

IF (.NOT. ISYES) COTO 100

CALL BLNKLN
50 III. RNOVV(NDSET)
BRITE‘STRINC.l020) ROVNLiI.NDSET)
I020 FORMATIIX.AI2)
30 CALL HRTSTR¢STRINOl

IOO CALL HRTSTRC' Do gou want a data listing? (Y)')
ISYESI. TRUE

71

CFILEzDRUNIl --------------- DRUNIl ------------------------------------

CALL YORN(ISYES)
IF (.NOT.ISYES) COTO 200

CALL BLNKLN

0073 I-I.RNS(NDSET)
0085 J-I.RNOVV(NDSET)
HRITE(STRING.IO40) DSET(I.J.NDSET)
I040 FORHAT(IX.(IP.EI5.4))
CALL URTSTR¢5TRINCl
85 CONTINUE
75 CONTINUE
c200 CONTINUE

LSE
CALL BLNKLN
CALL URTSTR(’ No data recovered.’)

C ENDIF
RETURN
END

E)))>)

EANLDVR >>>>>>>>)>))))))))))>>>>>>>>>>>>>D>>>>>D Last Change: 5/2/89 TLB
SUBROUTINE ANLDVR

g-—- PURPOSE: Controls Flow For run data options.

g--- INPUTS: Henu Driven.

C

C--- VARIABLES:

g FULL Determines iF Full or abreviated menu is displayed.
C

INCLUDE 'ENPORT)ROSENBERC>SHIP72)SIZEBK.CBK’
INCLUDE ’ENPORT>ROSENBERG>SHIP72)SOLNBK.CBK‘
INCLUDE 'RDATBK.CBK’

no

CHARACTER STRING’BO. TOKENll
LOGICAL FULL

EXTERNAL URTSTR. BLNKLN. COON. PROHPT. CETANS
EXTERNAL INVOPT. DIFPLT. DIFTBL

CeeeANLDVReeeeeeeeeeeeeeeeeeeeeeeeeeeeeaeaeeeeeeeeeeeeeeeeaeeeeeeeeeeeee

C
g--- First check iF suFFicient data sets have been loaded
IF (NDSET LT. N2) THEN
CALL BLNKL

CALL NRTSTR(’ GOO InsuFFicient data OOO’)
CALL URTSTR(’ At least 2 data sets must be loaded to analyze’)

RETURN
C ENDIF
E--- Present the options menu
10 CONTINUE

IF (FULL) THEN
CALL BLNKLN
CALL BLNKLN
CALL HRTSTR(’ Run data analysis options‘)

CALL HRTSTR(’ --- --’)

CALL HRTSTR(’ G: Gra h diFFerence values ’)

CALL HRTSTR(: T: Tab e oF select diFFerence values ’)

 

CALL HRTSTR( L: List processed diFFerence values.’)
CALL HRTSTR( Return’)
CALL HRTSTR(
STRING-’ Enter option (R): ’

ELSE
CALL BLNKLN
CALL BLNKLN
CALL HRTSTR(’ Run data analysis options’)
STRING-’ Graph. Table. List. Return? (Full)’

 

................ .)

ENDIF
C--- Now read the command
CALL PPONPT<STRING)

72

CFILE DRUNII --------------- DRUNI: ------------------------------------

TOKENI’O’
CALL GETANSCTOKEN)

C--- Now inter et the command token
IF (TOKE E0. THEN
IF (FULL) THEN
TOK EN ’R’

ELSE
FULL'.TRUE.
GOTO 10

IF (TOKEN.EO.’G’) THEN
CALL DIFPLT
ELSEIF (TOKEN.EO.’T’) THEN
CALL DIFTBL
CALL GOON
ELSEIF (TOKEN.EO.’L’) THEN
CALL URTSTR(’ List option not yet available’)
ELSEIFRéTOKEN.EO.’R’) HEN

SE

CALL INVOPT
DIF

GOTO IO

END

DIFPLT )>)>>)))))))>))>>>>>>>>>>>>>>>>>>>>> Last Change; 4/9/89 TLB
SUBROUTINE DIFPLT

--- Purpose: Provide driver For hi-res graphicsT plots
diFFerence in valhes store
Screen scale Factors: 1 line a 22 points
I column I 14 paints

(WOCHDOCH) Offlfifi (H1

INCLUDE 'ENPORT)ROSENBERG>SHIP72>SIZEBK.
INCLUDE 'ENPORT)ROSENBERG>SHIP72>SOLNBK.
INCLUDE 'ENPORT)ROSENBERG)SHIP72>POSTBK.
INCLUDE 'ENPORT)ROSENBERG>SHIP72>UTILBK.
INCLUDE ’ENPORT>ROSENBERG>SHIP72>GRCHBK.
INCLUDE ’RDATBK. CB

OCWNWO
mcnnmai
XXJUFX

on

INTEGER SCLVARiD). NUHSCL. JTIH
LOGICAL CONT. TIH
REAL YHX(0:5). YHN(O:5). SCLHIN(O:5). SCLNAX(O:5)

LOGICAL PROCFG. E7YORN
INTEGER CURTER. CURDEV. VC2IS

EXTERNAL NEHDEV. NENL. SETDEV. E7YORN. DALINE. TGRTYPE
EXTERNAL HRTSTR. BLNKLN. GETLAB. DIFVRH. DFTL

EXTERNAL DFYLHT. SCALY. CONTUE. GRIDLE. DFCURV. ANHODE
EXTERNAL DAVIS. COLOR. DAFULL.VC2IS

eeeo1FPLTeeeeeeeeeeeeeeeeeeeeaeeeeeeeeeeeeeeeeeeaeeeeeeeeeeeeeeeee

OCHH1

IF (ITERH.E0.0) THEN

CALL BLNKL

SETLM HRTSTR(’ COO This is not a graphics terminal ’)
NDIF

C
C--- Set deFu

It t increments to maximum density
JTIH-VC2éS: ’)
l

NUHSCL-l
CALL GETLAB¢LABEL.DATIH)

CALL DIFVRH(.TRUE.)
IF (NREOD.E0.0) RETURN

CALL DFTLHT

73.

CFILfizDRUNIl --------------- DRUNIl ------------------------------------

CALL DFYLHTIYHX.YHN)
CALL SCALY(SCLHIN.SCLHAX.YHN.YHX.SCLVAR.NUHSCL)

CALL GRTYPE

C
CALL CONTUE(CONT)
IF (.NOT.CONT) RETURN
CALL GRIDLE(SCLHIN.SCLHAX.DATIH.SCLVAR.NUHSCL)
C CALL DFCURV(SCLHIN.SCLHAX.SCLVAR)
C--- Hait For next input to return (Ieeps screen clear)
CALL NEHL
C CALL PAGE
CALL ANHODE
C CALL DAVIS(.TRUE.)
CALL DALINE(2)
CURDIA-.FALSE.
CALL BLNKLN
CALL BLNKLN
CALL BLNKLN
IF (E7YORN(’ Hould ou like this image stored in a File? ’.
l .FALSE.) THEN
CURTER I ITERH
C CURDEV - DEVICE
CALL NEHDEV(PROCFG)
IF (.NOT. PROCFG) THEN
RETUR
C ENDIF
CALL GRIDLE(SCLHIN.SCLHAX.DATIH.SCLVAR.NUHSCL)
C CALL DFCURV(SCLHIN.SCLHAX.SCLVAR)
g--- And we put the pen away on HPGL hard copy plotters
C CALL COLOR(-l) ’
ITERH-CURTER
CALL SETDEV(CURDEV)
C ENDIF
CALL DAFULL
CALL PAGE
CALL ANHODE
RETURN
C
C
END
C>)>>>

EDFCURV >>>))>)))>)>>>>)>>>)>)>>))))>>>>>>))> Last Change' 04/l5/89 TLB

000000000

n

00 000

SUBROUTINE DFCURV(SCLHIN.SCLHAX.SCLVAR)

DFCURV DRAHS THE Y VERSUS T CURVES
FOR PROCESSED DIFFERENCE VALUES

INPUTS: Tl.II.I2.NREGD.JDX.RES
I- STAGE (X) INDEX
J- VARIABLES (CURVE) INDEX
L. TYPE OF LINE DRAWN

INCLUDE 'ENPORT)ROSENBERG>SHIP72)UTILBK.CDK’
INCLUDE 'ENPORT)ROSENBERG>SHIP72>SIZEBK.CBK'
INCLUDE 'ENPORT>ROSENBERG)SHIP72>SOLNBK.CBK’
INCLUDE 'ENPORT>ROSENBERG)SHIP72>POSTDK.CBK’
INCLUDE 'ENPORT>ROSENBERG>SHIP72>GRCHBK.CBK’

INCLUDE 'RDATBK.CBK'

REAL SCLHIN‘O:5). SCLHAX(0:5). DFRSLT. TIH
INTEGER SCLVAR(3). J. L. IVAR. INIT. JTIH. VCZIS

EXTERNAL RECOVR. DUINDO. COLOR. NOVEA. DFRSLT. DASHA. HOME. BELL
EXTERNAL ANHODE. VC2IS

eeeDFCURVeeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

CALL RECOVR

74

CFILEtoRUNIl --------------- DRUNI) ----—---------‘ ---------------------

C Find the First point in time we are interested in
C

C JTIH I E7KVBG(‘TIHE') + l
JTIHI VC2ISI’TIHE‘)
INIT I 0

ll CONTINUE

INIT I INIT * 1
TIM I DSET(INIT.JTIH.I)
IF(TIH.LT.TI) GOTD II

C
DELTAIDSET(2.JTIH.I)-DSET(I.JTIH.I)
DO IOO JII.NREOD
CALL DUINDO(SCLHIN(O).SCLHAX(O).
I SCLHIN(SCLVAR(J7).SCLHAX(SCLVAR(J)))
IF(DEVICE.NE.2 .AND. DEVICE.NE.5) CALL COLOR(J)
LIJ-l
CALL HOVEA(DFRSLT(INIT.O).DFRSLT(INIT.J))
C IVAR I INIT

IO IVAR I IVAR O I
IF(IVAR .GT. RNS(I)) GOTO IOO
TIH I DSET(IVAR.JTIH.I)
IF(TIH.GT.T2) GOTO 100
CALL DASHA(DFRSLT(IVAR.O).DFRSLT(IVAR.J).L)
C GOTO 10
éOO CONTINUE
CALL HOHE
CALL DELL
CALL ANHODE
CLOSE (18)
RETURN
END
))))) x
DIFTBL ))))>))>>>>>>>>>>>)>>>>>>>2>>>>>>>> Last Change: 419/89 TLB
SUBROUTINE DIFTBL

This is a driving routine For listing run data diFFerence values
in tabular Form.

INCLUDE 'ENPORT)ROSENBERG>SHIP72>SIZEBK.CD

INCLUDE 'ENPORT>ROSENBERG)SHIP72)SOLNBK.CB

INCLUDE 'ENPORT)ROSENBERG>SHIP72)POSTDK.CB

LOGICAL CONT. DATIH

EXTERNAL GETLAD. DIFVRH. DFTLHT. CONTUE. DIFLST.DIFDEL
eeeotFTBLeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

CALL GETLAB(LABEL.DATIH)

CALL DIFVRN(.FALSE.)
IF (NREOD.E0.0) RETURN

0 CONTINUE
CALL DFTLHT

CALL DIFDEL

CALL CONTUE(CONT)

IF (.NOT.CONT) GOTO l0
CALL DIFLST(DATIH)
RETURN

END
>>)))

0000 00000 0 0

O
O
O

0 000 0 0
XXX

*0

0

DIFDEL >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 05/l6/89 TLB
SUBROUTINE DIFDEL
DIFDEL: SETS DISPLAY INCREHENT USED IN TABLE PRINTING.

000 00000 0 0 0

7S

CFILE DRUNIl --------------- DRUNIl ------------- d; ---------------------
C INPUTS: DELTA.TI.T2
E OUTPUTS: IDEL

INCLUDE 'SIZEBK.CBK’
INCLUDE 'SOLNBK.CBK’
INCLUDE ’POSTBK.CBK’

C

REAL DTHI. TINC

LOGICAL NEHLIN.ENDLIN
C CHARACTERI70 STRING
C EXTERNAL BLNKLN. HRTSTR. PROHPT. GETRL
CeeeDIFDELeeeeeeeeeeeeaeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeeeeeeeeeeeeeee
C

CALL BLNKLN
CALL HRTSTR(’ Set time increment For display ’)
CALL BLNKLN

TINCIDELTA
uRlTE<STRING.l0l0) DELTA
lOlO FORHAT(’ Enter display increment (’.iPEll.4.’): ’)
CALL PROHPT(STRING
NEHLINI.TRUE.
DTHl-T2-Tl
CALL GETRL(DELTA.TINC.DTHI.NEHLIN.ENDLIN)

C

C IDELI (DELTA+.OOIOTINC)/TINC
RETURN

C>>>)>

E

SDIFLST )>>>))>)>>)>>)>>>>>>>>)>>)))>))>)>>) Last Change: 4/9/89 TLB

C SUBROUTINE DIFLST(DATIH) x

C Purgose:

E Lis the diFFerence values in the loaded run data.
INCLUDE ’ENPORT>ROSENBERG>SHIP72>SIZEBK.CBK’
INCLUDE ’ENPORT>ROSENBERG>SHIP72>SOLNBK.CBK’
INCLUDE ’ENPORT>ROSENBERG>SHIP72>POSTBK.CDK’

C INCLUDE ’RDATBK.CBK’

LOGICAL DATIH

INTEGER I. J. N. JTIH. VCQIS

REAL TIN. DFRSLT

CHARACTER DATEOS. TIHECHOS. STRINGOBO

EXTERNAL GETTD. DLNKLN. HRTSTR. VC2IS. DFRSLT

C
C
CeeeoIFLSTeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
c
C

--- Put up list heading
CALL BLNKLN
HRITE(STRING.lOIO) NDSET. DSNAH(NDSET)
IOIO FORHAT(lX.’Data set ’.I2." ’.(A))
CALL NRTSTR(STRING)
CALL URTSTR(’ Hinus’
HRITE(STRING.IO20) DSNAH(l)
1020 FORHATth.’ReFerence data set: ’.(A))
CALL URTSTR(STRING)
CALL BLNKLN

C--- Put u the date and time
IF ( TIH) THEN
CALL GETTDéTlHECH.DATE)

IF (DATE TH HEN
HRITE(STRING.1005) DATE. TIHECH
IOOS FORMAT(IX.’Date: ’.A8.’ Time: ’.A8)
CALL HRTSTR(STRING)
ENDI
C ENDIF
C--- Put up title and column headings
CALL BLNKLN

STRINGI’ 'l/LABEL(I:7I)

CALL NRTSTR(STRING)

CALL BLNKLN

HRITE(STRING.IOOO) (CLBL(J).JII.NREOD)

76

CFILELDRUNII

FORHAT(' Ti '.5X.5(IX.AIO.1X))
CALL HRTSTR(STRING)
C CALL BLNMLN

C--- Now com
JTIHIV

1000

ute a the table oF data

TIH .
1100 DEI 2.

100

CALL NR
CONTINUE

RETURN
END

(3 0
V
J
V
V
V

DIFVRH ))))))>)>>)>)>>>>>>)>>>>>>))>>>>>>>>>>> Last Change:

SUBROUTINE DIFVRH<HORZFG)
--- PURPOSE: Get
Note:

the display vbl list

This routine onl
that it calls DI

HORZFG. I.T.

JDX.
CLBL.

FIL not

INPUTS:
--- OUTPUTS:

get horiz vbl

index oF di

play vbls
names oF di

play vbls

(H50FH50CH50CKT 0CH10
'0

INCLUDE
INCLUDE
INCLUDE
INCLUDE

CHARACTER ANSOIZ. BLANKOIZ.
INTEGER KVBLX. I. J. NCHARS.
LOGICAL NEHLIN. ENDLIN.

EXTERNAL BLNKLN. NRTSTR.
EXTERNAL NCHARS. GETUD.
INTRINSIC HIN

DATA BLANK/'OOOOOOQOOOOO’I

STRINGOBO.
E7YORN.

E7YORN.
GETANS.

HORZFG.

DIFFIL

(70C) 0

CALL DLNKLN

CALL HRTSTR(’ Choose the display variables...

C
C---- Present the current list
10 CONTINUE
C---- Set a deFault list iF necessary
IF (NREGD. E0. 0) THEN
C ------ Set the x axis deFault to ’TIHE’..

CLDL(0)I
IF (NXI.GT.O) THEN

NREODI HIN(5.NXI)

DO II II l.NREOD
CLBL(I)I OVNL(XIX(I))
JDX(I)I XIX‘I)

ELSEIF (NY.GT.O) THEN

NREODI HIN(5.NY)

I2 II l.NREOD
CLBLCIlI OVNL(YX(I))
JDX(I)I YX(I)

(NU.GT.0) THEN

NREODI HIN(5.NU)

DO I3 II l.NREOD
CLBL(I)I OVNL(UX(I))
JDXCIII UX(I)

CALL BLNKLN
CALL HRTSTR(’
TURN

II

l2

I3

’ENPORT)ROSENBERG)SHIP72)SIZEBK.
'ENPORT)ROSENBERG>SHIP72>SOLNBK.
’ENPORT)ROSENBERG>SHIP72>POSTBK.
’ENPORT)ROSENBERG>SHIP72)UTILBK.

DVNAH(HAXNRO)012
RVBLX. NR ED

PROMPT.

eee Nothing available For display,

(DF RSLT(I.N).NII.NREOD)
4)

04/15/89 TLB

diFFers From DVAROH in

FILRSD.

CBK‘
CBK’
CBK’
CBK'

SPEAKF
RVBLX

eeeDIFVRHeeeeeeeeeeeeeeeeeeeoeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

‘)

')

CFILE:

c--_-
I5

c----

1020
C

c-—_-

'77

onunxi --------------- DRUNll ---------—-—--¥ ---------------------

Do the horizontal axis here (one variable only)

rixontal axis? (’//
I).I)

CALL GETANS(ANS)

SPEAKFI.TRUE.

KVBLXI RVBLX(ANS.SPEAKF)

IF (KVBLX.E0.0) THEN
GOTO 13

ELS SE
JDX(
CLBL

ENDIF

ENDIF

And now the vertical axis variables.

CALL BLNKLN

CALL HRTSTR (’ The Current display list ’)
CALL BLNKLN

HRITE(STRING.1020) (CLBL(J).JI1.NREOD)
CALL HRTSTR(STRING)
FORHAT(5(1X.AI2))

Check iF the list is okay
IF (E7YORN(’ NDo you want to change the list?’. .FALSE.)) THEN
CALL BLNKL
CALL HRTSTR(’ Enter a new list on one line (or HELP):’)
CALL BLNKLN
CALL PROHPT(’ )')
NEHLIN-.TRUE.
NREDI 0
- Commence to read the line
ANSIBL
CALL GETHD (ANS.NEHLIN.ENDLIN)
IF (ANS. NE. BLANK) THEN
NREDI NRED+1
DVNAH(NRED)I ANS
IF (NRED.EO.HAXNRO) GOTO 203
NEHLIN-.FALSE. °
GOTO 200
ENDIF

I KVBLX

0)
(OII ANS

C------ Commence to identiFy the entries

203

c .....

I

9010

0000

SPEAKFI. TRUE.
NREODI 0
DO 220 II l.NRED
KVBLXI RVBLX(DVNAH(I). SPEAKF)
--- Implicitly ignore bad entries
IF (KVBLX GT 0) THEN
NREODI NREOD+1
JDX(NREOD)- KVBLX
CLBL(NREOD)- DVNAH(1)
ENDIF
CONTINUE
- This list has been cracked
IF (NREOD E0 0) GOTO 10
IF (E7YORN(’ Do you want to see the list again?’..FALSE.))
GOTO 10
ENDIF

Set the JDX entries For derived vbls to proper indices
NODVI 0
IF (JDX(0).GT NOVV) THEN
NODVI NODV+1
JDX(0)I NOVV+NODV
ENDIF
DO 250 II l.NREOD
IF (JDX(I).GT.NOVV) THEN
NODVI NODV+1
JDX(I)- NOVV+NODV
ENDIF
CONTINUE

IF (DDGFLG) THEN

CALL BLNKLN

CALL HRTSTR(' 009 Debu data For d

CALL HRTSTR(' DEBUG HO E NOT ACTIV

HRITE(STRING.9010) NREOD. NODV. NOVV
FORMAT(3X.’NREOD. NODV. NOVV: '.3

CALL HRTSTR¢STRINGl

HRITE¢STRING.9020) (JDX(I).I=O.5)

ved vbls ’)

'78

CFILE onunx: --------------- DRUNll -------------- 1 ---------------------
C9020 FORHAT(5X.’JDX: ’.bl5)
C CALL HRTSTR(STRING)
C HRITE(STRING.9030) (CLBL(I).I-0.2)
C9030 FORHAT(SX.’CLBL: ’.3(A12.2X))
C CALL HRTSTR(STRING)
C HRITE(STRING.9030) (CLBL(I).II3.5)
C CALL HRTSTR(STRING)
C ENDIF
C---- Fill in the derived results iF necessary
CALL DIFFIL
C IF (DBGFLG) THEN
C CALL BLNKLN
C DO 940 JI i.5
C HRITE(STRING.9040) (RSLTD(J.I).II1.NODV)
C9040 FORHAT(3X.1P.6E12.4)
C CALL HRTSTR(STRING)
C940 CONTINUE
E ENDIF
RETURN
END
C>)>>>
E
EDIFFIL >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change: 03/)1/89 RR
C SUBROUTINE DIFFIL
E--- PURPOSE: Fills derived vbl buFFer with the proper diFFerence values.
C--- INPUTS: CLBL. names oF all display vbls
g JDX. indices oF all disp ay vbls
g--- OUTPUTS: DRVVAL values oF derived vbls
INCLUDE ’ENPORT)ROSENBERG>SHIP72>SIZEBK.CBK’
INCLUDE 'ENPORT>ROSENBERG>SHIP72)GREDBK.CBK’
INCLUDE ’ENPORT)ROSENBERG>SHIP72>SOLNBK.CBK’
INCLUDE ’ENPORT>ROSENBERG>SHIP72>POSTBK.CBK'
INCLUDE ’ENPORT)ROSENBERG>SHIP72)UTILBK.CBK’
C INCLUDE ’RDATBK.CBK’
INTEGER I. JRD. N. VC2IS. IXE. IXF. NCON. IBIAS. J
CHARACTER CH202. RESTOIO
LOGICAL SPEAKF
C REAL TEHPI. TEHP2
C EXTERNAL INTGR. VC2IS
ceeeDIFF1Leeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
C

IF (NREOD.E0.0) RETURN
SPEAKFI.TRUE.

C---- Get DRVVAL column index For this vbl (JRD)
DO 100 II O.NREOD
IF (JDX(I).LE.NOVV) GOTO 100
JRDI JDX(I) -NOVV

rrrrrr Interpret the implied operation
is derived momentum
0 is derived displacement
H is power
T is energy
N is tota nodal energy

CH2I CLBL(I)(1:2)
RESTI CLBL(I)(3:12)

IF (CH2.E0.’P.’) THEN
IXEI VC2IS('E.'//REST)
DO 20 NI l.RNS(l)
20 DRVVAL(N.JRD)I DSET(N.IXE.NDSET)-DSET(N.IXE.I)
CALL INTGR(CLDL(I).JRD.SPEAKF)

ELSEIF (CH2.EO.’0.’) THEN
IXF- VC2IS(’F.’//REST)
DO 30 u- 1.RNS(1)
30 onvvac¢~.unoi- DSET(N.IXF.NDSET)-DSET(N.IXF.1)
CALL INTGR(CLBL(I).JRD.SPEAKF)

0CWNDOFWN5

0

'79

CFILE'DRUNII --------------- DRUNll ------------------------------------

ELSEIF (CH2
IXEI VC2IS
IXFI VC2IS
DO 40 NI 1

TEHPIIDS

40 DRVVAL(N.J
IF (CH2 E0. '
CALL INTGR

ENDIF

ELSEIF (CH2.E0.’N.’) THEN

DO 60 N- 1.RNS(l)
60 DRVVAL(N.JRD)I 0.
C -------- IdentiFy this node

DO 62 JI .INELS

IF (REST E0 ELNAH(J)) GOTO 63

ONTINUE
ETURN
or each BOND on this node
BIASI IBDPTR(J) -1
CON- IBDPTR(J*1) -IBDPTR(J)
0 b9 JI l.NCON

IF (BDTP(I8DSEL(IBIAS+J)).NE.‘8’) GOTO 69

REST- BDNAH(IBDSEL(IBIAS¢J))

IXE- VC2IS(’E.’//REST)

IXFI VC2IS('F.’//REST)

DO 65 NI 1.RNS(1)
TEHPl-DSET(N.IXE.1)¢DSET(N.IXF.1)
TEHP2IDSET(N.IXE.NDSET)ODSET(N.IXF.NDSET)

b5 DRVVAL(N.JRD)IDRVVAL(N.JRD)+(TEHP2-TEHP1)
CALL INTGR<CL8L(I).JRD.SPEAKF)
b9 CONTINUE
ENDIF
00 CONTINUE

RETURN x
END

.OR CH2.E0 ‘T ') THEN
)

flP-IAM

I)ODSET(N.IXF.I
NDSET)’DSET(N.I XF. NDSET)
HP2- TEHPI

N
).JRD.SPEAKF)

62

c
R
c-------- r
63 I
N
0

0H

V
V
V
V
V

INTGR )>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Last Change 04/30/89 TLB
SUBROUTINE INTGR(NAH.JVBL.SPEAKF)
--- PURPOSE: Integrates the display vbl in INTVEC
INPUTS: NAH. dis lay vbl name
JVBL. DRV AL vbl (column) inder

SPEAKF. a T call For initial value
DRVVAL(O.JVBL) contains the vector to be integrated

--- OUTPUTS: DRVVAL(O.JVBL) contains integrated vector

INCLUDE 'ENPORT)ROSENBERG>SHIP72>SIZEBK_CBK‘
INCLUDE 'ENPORT>ROSENBERG>SHIP72>SOLNBK.CBK'
INCLUDE 'ENPORTDROSENBERGDSHIP72}POSTBK CBK’
INCLUDE ‘ENPORT>ROSENBERG)SHIP72)UTILDK CBK'
INCLUDE 'RDATBK CBK’

CHARACTER NAHOI2

INTEGER JTIH. LNAH. NCHARS. N. VC2IS. JVBL
LOGICAL SPEAKF. NEHLIN. ENDLIN

REAL VAL(HAXRES). TINSTP

EXTERNAL BLNKLN. PROHPT. NCHARS. CETRL.VC2IS

eeeINTCReeeeeoooeoeoeoeeeeeeeaeeoeeeoeeeoeeeaeeaeeeeeeeeeeeaeeeeeeeeeee

0("50fH50FH50 (H50FH1

0

C
C
C
C---- Set initial value
VAL(l)I 0.
IF (SPEAKF) THEN
CALL BLNKLN
LNAHI NCHARSINAH)
“ CsLL PROHPT(’ Enter initial oFFset value For '//NAH(1.LNAH)//
NENLINI TRUE.
EN DIFL GETRL(VAL(I).-10. £25.10. E25.NEHLIN.ENDLIN)

C
C--—- Now integrate
DO 2.RNS(i)

8()

CFILE:DRUNI1 --------------- DRUNI] -----------——-4 ---------------------

C--- First Find the time step
JTIH-VCZIS(‘TIHE')
TIHSTP-DSET(N.JTIH.l)-DSET(N-l.JTlH.l)

C
20 VAL(N)- VAL(N-l) +DRVVAL(N-I.JVBL)OTIHSTP
0 40 N' I.RNS(I)
DRVVAL(N.JVBL) ‘ VALIN)
RETURN

END
g>)>))

C

C
EDFRSLT >>))>)))>)))>>>)>>D>>>>>>>>>>>>>>>>>>>> Last Change: 08/29/88 RR
REAL FUNCTION DFRSLTIISTACE.N)

C

gr-- PURPOSE: Retrieve data From the storage buffer DSET or DRVVAL.
C--- INPUTS: ISTAGE. the stage 0? storage

E N. the N- displau variable

C--- OUTPUTS: DFRSLT. value of the result requested
INCLUDE ’ENPORT>ROSENBERG>SHIP72>SIZEBK.CBK'
INCLUDE ’ENPORT>ROSENBERC>SHIP72>SOLNBK.CBK'
INCLUDE 'ENPORT)ROSENBERC>SHIP72>POSTBK.CBK'
INCLUDE ’RDATBK.CBK'

INTEGER ISTACE. N. J. JTIH. VC2IS

EXTERNAL VCZIS

OOODFRSLT 6996.00.00.999909900909999ooaeeaeeeeeeeeeeeeeeeeeeeeeeeee

O

J'JDXIN) x

Check if re e9
JTIH-VC2IS(
IF (JTIH .EO.
DFRSLTIDSE
RETURN
ENDIF

Of} OFHWO (T O

1?; a time value
TH EN
Ac

U0
TI
T E.J.l)

osti
IHE’
J)
(IST

C---- Set359(simple value if result alreadu stored
IF (J. CT. NOVV) THEN
C ------ Derived vbl
J- J-NOW
DFRSLT- DRVVAL(ISTACE.J)
ELSEIF (Jn CT bO) HEN
C ------ Soluti
LDERSLT-DSETIISTACE. JDX(N). NDSET)-DSET(ISTACE JDX(N).i)

C ------ Hho knows

DFRSLTI O.

C ENDIF

RETURN

END
C>>)>>
CEOF
E
gDFTLHT)>>>>))>>>>>>§>D>>>§>DFTLHT>>>>>>>>>>> Last Change 04/16/89 TLB
C SUBROUTINE DFTLHT
E--- Sets the time limit parameters For RUN DATA
C INPUTS: DSET(0.0.l)
g DELTA
C OUTPUTS: Tl.T2
E II.I2

INCLUDE 'ENPORT)ROSENBERC>SHIP72)SIZEBK.CBK‘
INCLUDE ’ENPORT>ROSEN8ERC)SHIP72)SOLNBK.CBK'
INCLUDE ’ENPORT>ROSENBERC)SHIP72>POSTBK.CBK‘
INCLUDE 'RDATBK.CBK'

81

CFILE DRUNI! --------------- DRUNI! ------------------------------------

00"} n

n

lOIO

n

(DOC) 0

REAL TFINAL

INTEGER JTIH. VCZIS. TZERO
LOGICAL NEHLIN.ENDLIN
CHARACTERO7O STRING

EXTERNAL BLNKLN. HRTSTR. PROHPT. GETRL. VC2IS

eevDFTLHTeeeeeeeeoeeeeeeeeeoeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeesee

CALL BLNKLN
CALL HRTSTRI' Set time limits For displau ’)

JTIH-VC2ISI'TINE’)
TZERO'DSETII.JTIH.I)
TI=TZERO
TFINAL‘DSETIRNSII).JTIH.l)
T2= TFIN NA
DELTA-DSETI2.JTIH.Il-TZERO

CALL BLNKLN

HRITEISTRINC.IOIO) Tl

FORHATI’ Enter initial time ('.lPEll.4.‘)' ')
CALL PROHPTISTRINC)

NEHLIN‘ TRUE

CALL GETRLITI. TZERO.TFINAL NEHLIN. ENDLIN)

HRITEISTRINC.IO20) T2

FORHATI’ Enter Final time I’.IPEll.4.‘). ’)
CALL PROHPTISTRINC)

NEHLINI. TRUE

CALL CETRLIT2. Tl. TFINAL. NEHLIN. ENDLIN)

IFIDELTA.LT.1.0E-25) THEN
CALL NRTSTRI' 099 Bad time control resolution.’)
CALL HRTSTRI' Too small time steps ’)
RETURN ,

ENDIF

Il-ITI*.OOIODELTA-TZ
IZ-IT2*.OOIODELTA-TZ

RETURN
N

ERO)/DELTA*I
EROl/DELTA‘I

DFYLHT)))>>)))>))))>>}>>>>>>§DFYLHT>>>>>§>D§>>§> Last Change O4/lb/89 TLB

SUBROUTINE DFYLHTIYHAX.YHIN)

FINDS HINIHUH AND HAXIHUH VALUES FOR EACH DISPLAY VARIABLE
INCLUDE 'ENPORT)ROSENBERG>SHIP72>SIZEBK,CBK’

INCLUDE 'ENPORT>ROSENBERG>SHIP72>SOLNBK.CBK'

INCLUDE 'ENPORT)ROSENBERC)SHIP72>POSTBK.CBK'

INCLUDE 'RDATBK.C8K'

INTEGER I J
REAL YHAXIO: 0). YHINIOZO). TRY. DFRSLT

EXTERNAL DFRSLT

eeeDFYLHTeooeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeaeeeeeeeeeeeeeeeeeeeeee

DOIO I'O.NREGD
YHAXIIII-IO.E3O
YHINII)‘ IO.E30

éO CONTINUE
DOZO I-II.I2
DOBO J'O.NREOD
TRY-DFRSLTII.J)
IF (TRY.GT.YHAXI J) I Y" AX IJ) l=TRY
IF ITRY.LT.YHINI J) ) YH IN (J) 'TRY
30 CONTINUE
go CONTINUE
RETURN
END
C>)>)))

CEOF.

82

CFILETDRULOC ---------------- DRULOC----—------------r ------------------
EFILE.DRULOC ---------------- DRULOC ------------------------------------
C--- PURPOSE: Local utilities File. For UNFormatted File mani ulation.
g Hag need modiFication For a given operating sus em.
E--- DESCRIPTION: Used For compact storage oF results Files
C--- CONTENTS:
CETUFL opens and checks a File For unFormated output
FNDUFL Finds and opens unFormated File For reading
--- INDEX:
CETUFL
FNDUFL
--- Programmer: Tom Bush
--- Last HodiFication: April 8. I989
EOFH DRULOC -----------------------------------------------------------
>>>>

CETUFL )))>>>>>)>>>>)>>>>>>>>>)>>>>>>>>>>>>>>> Last Change. 4/8/89 TLB
SUBROUTINE CETUFLIUNIT. FNAHE)
Purpose: SpeciFicallu For UNFORHATTED influt/output

Cet File name and open it on FT unit.
--- Input: FNAHE The deFault name For File (can be ‘OUIT’)
--- Outputs: UNIT The FORTRAN unit number on which File was opened.
Set to :ero iF File was not opened.

FNAHE The name oF the File (can be ‘OUIT')
INCLUDE ‘ENPORT>ROSENBERC>SHIP72>INPTBK.CBK‘
INTEGER NCHARS. UNIT

LOGICAL E7YORN '
EXTERNAL GETHD. INOUIR. PROHPT. BLNKLN. HRTSTR. E7YORN. NCHARS

f) OCKNWOFKMWOCIIWO (“finsyNWOFNNWOCWNWO

C
CHARACTER FNAHEIIO). DEFALTe32. TESTLNGBO. LFNAHEOBZ
INTEGER JCDFLT. JCFILE
LOGICAL NEHLIN. ENDLIN. EXISTS

ceeeCETUFLeaeeeeeeeeeeeeeooeeeeooeeaeeeeeeoeeeeeeeeeeeeeeeoeeaeeeoeeeeee

UNIT I PRUNIT
DEFALT I FNAHE
LFNAHEI FNAHE
IO CONTINUE
CALL BLNKLN
JCDFLT I NCHARSIDEFALT)
CALL PROHPTI' Please enter the File name I'l/
DEFALTIlzJCDFLTIII
0).!)

NEHLIN I .TRUE.
LFNAHE I DEFALT
CALL CETHDILFNAHE.NEHLIN.ENDLIN)
IF ILFNAHE.E0.'OUIT‘.OR.LFNAHE.EO.‘/‘) THEN
CALL BLNKLN
CALL HRTSTRI' 000 No File will be created ')
UNIT I 0
GO TO 20
ENDIF
JCFILE I NCHARS(LFNAHE)
C---- PRIHOS stgle (deFault)
OPENIUNIT.FILEILFNAHE.STATUSI‘OLD‘.ERRI8010.
l FORHI’UNFORHATTED'.ACCESSI‘SEOUENTIAL‘)

READIUNIT.ENDI8020.ERR-8010) TESTLN

2

--- The File already exists

CALL BLNKLN

:CALL HRTSTRI' The File ”’l/LFNAHEIlzJCFILElll’“ alreadq'll
' exists ')

IF (E7YORNI' Shall I overwrite it?'.LFNAHE.EO.DEFALT)) THEN

ELSEO T0 16

CFILE:

c--—

8010
801!

C
C---
C
8020

c-—_-

C

12

C
c---

C
8030

83

DRULOC ---------------- DRULOC ----------------- ‘ -------------------

CLOSEIUNIT)
GO TO 18
ENDIF

The File most probablg does not exist or it is not a sequential.
unFormatted one.

CONTINUE
CLOSEIUNIT.ERRISOII)
CONTINUE
CALL INOUIR(LFNAHE.’UNFORHATTED’.EXISTS)
IF (EXISTS) THEN
CALL BLNKLN
CALL HRTSTRI' GIG The File ”'l/LFNAHEII:JCFILE)//'" alreadg’l/
i ’ exists and')
CALL URTSTRI: has incompatible Form and/or access mode.’)

CALL HRTSTRI Please speciFu another File name.‘)

GO TO 18
ELSE
--- PRIHOS tule (deFault)
OPENIUNIT.FIL EILFNAHE.STATUSI'NEU'.ERR=8040.
l FORHI’UN FORMATTED’.ACCESSI'SEOUENTIAL')
GO TO l6
ENDIF
The File exists. but it seems to be empty.
CONTINUE
CLOSEIUNIT)
PRIHOS stu le
OPENIUNIT. FILEILFNAHE. STATUSI’OLD’ .ERRI8010.
I FORHI'FORHATTED'.ACCESSI’SEOUENTIAL’l
REHINDIUNIT. ERR-8010)
READIUNIT.'IA)’.ENDII2.ERRI8030) TESTLN
GO TO 8010
CONTINUE
GO TO 16
The File exists. however. it is inaccessible.
CONTINUE
CLOSEIUNIT)

CALL BLNKLN
xCALL URTSTRI

CALL HRTSTRI
CALL HRTSTRI
GO TO I8

The new File cannot be opened.

CONTINUE

CLOSEIUNIT.ERRI8041)

CONTINUE

CALL BLNKLN

lCALL URTSTRI' eea The File ”'l/LFNAHEIlzJCFILElll'" cannot'l/
' o ened.‘

CALL HRTSTRI‘ Please specin another File.')

00 T018

909 The File ”’l/LFNAHEII:JCFILE)//‘“ alreadg’l/
exists.’)

Horeover. it is inaccessible.')

Please speciFu another File.')

\\\\

TEST For writing privileges and/or disk space availability

CONTINUE

REUINDI UNIT
HRITE(UNIT.ERRI8050) TESTLN
REHINDI UNIT)

GO TO 20

Unable to write to the File.
CONTINUE

CLOSE(UNIT)

CALL BLNKLN

CALL URTSTRI' GOG Unable to write to the File.’)
CALL HRTSTRI' Please speciFg another Filename ‘)

CHANCE deFault name iF necessary.
CONTINUE

84

crst DRULOC ---------------- DRULoc-------——-------4 -------------------

IF ILFNAHE.EO.DEFALT) THEN
DEFALT I ’GUIT'

ENDIF
GO TO IO
go CONTINUE
FNAHE I LFNAHE
RETURN
END
E>)>>)
EFNDUFL )>)>>>>)>))>)>>>>>>>>>>>>>>>)>>> Last Change: 4/8/89 TLB
C SUBROUTINE FNDUFLIUNIT.FNAHE)
C Purpose: Get Filename and open it For UNFORHATED I/O
g Hust be modiFied For VHS use.
E INPUT: FNAHE The deFaIt name For File (can be 'OUIT')
C OUTPUT: UNIT The FORTRAN unit number on which the File was opened
C Set to zero iF the File was not a ened
E FNAHE The name oF the File (can be 'GUI ')
C INCLUDE ’ENPORT>ROSENBERG>SHIP72)INPTBK.CBK‘
CHARACTER FNAHEOII). DEFALT032. TESTLNOSO. LFNAHEOJZ
INTEGER UNIT. JCDFLT. JCFILE. NCHARS
C LOGICAL NEHLIN. ENDLIN
C EXTERNAL GETUD. PROHPT. BLNKLN. URTSTR. NCHARS
Ceea FINDUFL eeoeeoaeaaeeeeeeeaaeeeeeaeoeeeeeeeeeeeaoea
C

UNIT I PRUNIT
DEFALT I FNAHE
LFNAHE I FNAflE
IO CONTINUE
CALL BLNKLN
JCDFLT I NCHARS(DEFALT) .
CALL PROHPTI’ Please enter the File name ('/l
5 DEFALTIlzJCDFLTlll
NEHLIN I .TRUE.
LFNAHE I DEFALT
CALL GETHDILFNAHE.NEHLIN.ENDLIN)
IF (LFNAHE.E0.'0UIT’.0R.LFNAHE.E0.’/') THEN
CALL BLNKLN
CALL HRTSTRI' 000 No File will be sought'eoo')

UNIT I 0
RETURN
C ENDIF
C--- Crunch on this File name
JCFILE I NCHARSILFNAHE)
C--- PRIHOS style IdeFalt)
0PENIUNIT.FILEILFNAHE.STATUSI’OLD'.ERRI8010.
C l FORHI’UNFORHATTED’.ACCESSI’SEOUENTIAL’)
C READIUNIT.ENDIBOIO.ERR-8010) TESTLN
C--- The File is open and readable
CALL BLNKLN
CALL URTSTRI' File “'l/LFNAHEIllJCFILElll‘” Found ’)
FNAHE I LFNAHE
C RETURN
C--- The File most probably does not exist or it is not a sequential

C UNFormated one
SOiO CONTINUE
CLOSEIUNIT)
CALL BLNKLN
CALL HRTSTRI' 05* Unable to Find or read 'l/LFNAHE)

C CALL HRTSTRI' Please speciFy another File name.')
C--- Chan e deFalt name iF necessary.
IF I FNAHE.E0.DEFALT) DEFALT I 'OUIT'
C GO TO IO
END
C)>)>>
CEOF ---------------------------------------------------------

APPENDIX B

85
APPENDIX B

An ENPORT File Listing of
The System Used

HEADING
In The Study
HEADING
FILE 13:51:54 11/01/88 ENPORT-7.2
RESRCH
TITLE

RESEARCH INTO NODAL DECOHPOSITION H/ DAHPING

SYSTEH GRAPH DESCRIPTION

NUDE TYPE XLOC YIUC
IA MIC 300. 600.
I3 "IQ U00. (‘00.
IA MIG 300. 900.
CA ”CO ICC. 600.
OI "CC 300, 300.
TFIA HTC 300. 2:00.
02 "CC 800 300.
TFZA ”TC 300. 400.
TFQB H10 800 300,
ID MIC “00. (“)0
(:I3 f1(:(3 I ()(Jf). £i()()
552 "EC U00. 200.
SEI "EC 300. 200.
TFIB "TC 400. 400.
RA "RC 100. 800.
R8 NRC IOOO. 000.
FIT“ P1IIC; (‘()(I (i(’(‘.

CONNECTOR TYPE FROM TO VERTICFS
SI DC 51' I OI

2 BC SF# 0?

PA BC IA If

0“ [4G IA LI‘.

PB BC )0 I“

08 BC I“ (N

I “C U) I I- ll‘

2 RC TFIA IA

3 DC OI II I"

‘3 [1(3 1 5' l I‘ l I.

5 BC 0? Ti H"

6 BC TFIWI I“

7 BC ()9 I l-a'l‘.

8 BC TFTA IA

DA Dc IA Hfl '
OB BC IU RU 800. 600.
NA BC IA RN

NB “C In RN

GRAPHICAL ENVIRONMENT

Node size: 1.0000EFOO
Connector size. 6.?500E-01
Scale Factor; 9.0000F-Ol
Horizontal window minimum: -8.8894E+OI
Horizontal window maximum: 1.1889E+03
Vertical window minimum: 1.0000EF02
Vertical window maximum: 9.0000E+02
Gridding enabled: ON
Griddino visible: ON
Giid size: l.0000€+02
DeFault node regime enabled: ON
DeFault connector regime enabled: OFF
DeFault node regime: G
DeFault connectr regime: G
Postprocessor grid: ON
Postprocessor box: ON
Color I 0 30 0 Dark Gray (background)
Color 2 O )00 0 white (deFault line and label color)
Color 3 )2 50 100 Red (attention)
Color 4 240 30 100 Green (Mechanical)
Color 3 0 BO 100 Blue (Rotational)
Color 6 300 50 )00 Cyan (Electrical)
~Color 7 60 50 100 Ha enta (Hadraulic)
Color 8 ISO 50 IOO low (T ermal

'HEADING

NODE EQUATIONS

Number oF outputs:

14

Node. IA Connectors:
Equation: Y I GAIN .
Y_list X list
F.PA PTPA
Node: CA Connectors:
Equation: Y I GAIN I X.
Y_list X_list
E.0A 0.0A
Node: TFIA Connectors:
F.) I HOD.TF)A
.2 I HOD.TF)A
Equation: Y I CON I X.
Y list X_list
MOD.TFIA
Node: TF2A Connectors:
F.7 a HOD.TFPA
E.8 I HOD.TFPA
Equation: Y = CON I X.
Y list X_list
HUD.TF2A
Node: TF2B Connectors:
F.5 I NOD.TP?B
E.b I HOD.TFPB
Equation: Y = CON I X.
Y list Xfilist
HOD.TF28
Node: IB Connectors:
Equation: Y = GAIN I X.
Y_list X-list
F.PB P.PB
Node: CB Connectors:
Equation: Y I AIN I X.
Y_list X_list
E.GB 0.08
Node: SE2 Connectors:
Equation: Y I STEP I X.
Y_list X list
E.52 TIME
Node: SE) Connectors:
Equation: Y I STEP I X.
Ynlist X list
E.Sl TIME
Node. TFIB Connectors:
.. I HOD.TFJB
E.4 I HOD.TFIB
Equation: Y I CON I X.
Y list X_list
HOD.TFIB
Node: RA Connectors:
Equation: Y I GAIN I X.
Y list X list
ETDA FFDA
Node: RB Connectors;
Equation: Y I GAIN I X.
Y_list X_list
E.DB F.DB
Node: RN ' Connectors:
Equation: Y I GAIN _ I X.
Y list X list
ETNA FENB
. Equation: Y I GAIN I X.
Y-list X list
ECNB FENA

86

PA
) l I 1
Parameters
9.1743E-O2

GA

P ) I l 1
Parameters
6.0000EFOO

i 2

e F.2
O E.)

P ) l O 1
Parameters
I.OOOOE+OO

7 8

G F.8
D F.7

P ) I 0 1
Parameters
2.2250E+00

5 b

e P.b
I 5.5

P ) l O 1

Parameters
-P.2500E-Oi

PB

P ) I I 1
Parameters
9.0827E-01

GB

P ) i l 1
Parameters
6.0000E+OO

S2

P ) l i 2
Parameters:
0.0000EF00
1.00006900

S)

P ) i I 2
Parameters
0.0000E+OO
i.OOOOEF00

3 4

G F.4
O E.

P ) l 0 1
Parameters
l.0000E+OO

DA

P ) l l I
Parameters
l.0000E+OO

DB

P ) l l l
Parameters
I.0000E+00

NA NB

P ) I I 1
Parameters
o.ooooe+oo

) I I I
Parameters

0.0000EFOO

87

umflosum amummm may
now nmmuouvcom emomzm one

am musmwm

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