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operated by

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

UNION CARBIDE CORPORATION m
for the
U.S5. ATOMIC ENERGY COMMISSION
OAK RIDGS NATIONAL LABORATORY L ORNL- TM- 1189
NN ERRENN s 37
3 4456 D54909k 5 DATE - June 24, 1965

A TECHNIQUE FOR CAICULATING FREQUENCY RESPONSE AND
ITS SENSITIVITY TO PARAMETER CHANGES FOR
MULTI-VARIABLE SYSTEMS

T. W. Kerlin and J. L. Lucius*

ABSTRACT

A general method for calculating the frequency response
of a dynamic system and the sensitivity of this frequency
response to changes in system parameters is described. The
development is carried out using the matrix differential
equation (or state variable) approach. SFR-1, a computer
code prepared to carry out the computations, is described.
Two sample problems serve to illustrate the method and the
use of the code.

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*
Oak Ridge Gaseous Diffusion Plant.

NOTICE This document contains information of a preliminary nature
and waos prepared primarily for internal use ot the Ook Ridge MNational
Loboratory. It is subject to revision or correction and therefore does
not represent o final report.

 
 

 

 

LEGAL NOTICE

This report was prepared as on account of Government sponsored work, Meither the United States,

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or his employment with such contractor.

 

 

 

 
 

 

Contenfs.
Page
INtroduction sve.iieeeereenennses Cerbeeaees Ceeerraaraas e 5
Frequency RESPONSE cueeseeseaees ceeean ceeeenn .;;...;.. ....... 5
Freguency Response Senéitivity‘ :.....; ..... Weesesanans ceeens 9
The Computer Program ..eeccescesesssseasss Chececenccaranenas 19
Input ..eeeecocrrececesnssosnnccns ceecos ceessrssenesens 13
Output  seiveevencnacnas ...........;...}..a... ...... voee 17
‘Sample Problems ...... ceeeneen Ceeeseatenssecenannneeos . 18
PLOBLEM L erenrenieieninnaninnnn e et 18
Problem.Q .......;...;...;;.........,.... ........... 19
Conclusions seeevesos. ceseecisens ,.‘............;.;... ...... 23
References ..... .............f........}j..;...;..f.......... 30
‘ AL LABORATORY LIBRARIES

il

|
' 3 445k 054909k &

I
'
L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

Intfodnetion“

Techniques for determining the frequency response of multi-variable
dynamic¢ systems are well knewfi; and'seVefél COmputef.COdes-have been
prepared which are useful for ealeu1atiné nucleaf'pofier reactor frequency
:res'pcbnse;lt"2 The frequency .reésponse is usnaily determined for the system
at the design condition and at several of f-désign conditions to determine
the sensitivity of the results to &henéés ih'syétefi pafemeters: This
sensitivity 1nformat10n can be useful 1n re- de51gn of dynamlcally unsatls-
factory systems and in determlnatlon of necessary tolerances in des1gn
specifications to 1nsure sultable dynamlc behav1or at lowest cost
Sensitivity 1nformatlon can also prev1de a -deéper understanding of system
dynamic characteristiCS to the system anaiyst and can help in matching

This report presents a technlque for determlnlng the frequency
response of multi-yariable systens., In addition; the sensitivities to
system parameters can be‘determined directly. A domputer code for
carrying out the calculation is described and numerical results are
shown for sample problems.: o | |

‘Frequency Response

 

The system equation for a linear; autonomous, lumped parameter

system may be written:

dz . . - |
— I 7 Bl .
W
where

7z = the response vector,

t = time,

A = the system matrix (the elements are the- usual coefficients in

the differential equatlons),: | |
- f = the disturbance vector

kq. (l) is usually called the state varlable representatlon of the system.
In Eq. (1), it is assumed that - the dependent varlables are wrltten as

perturbatlons_around an equlllbrlum point. This implies that all the
initial conditions are zero when the'equation’is‘Laplace transformed.

The Laplace transform of (l) is then given by the following equation:

A-sI]Z=-T , | f (2)
where T o
I = unit diagonal matrix, |
s = Laplace transform'fiéfameter,-
Z = Laplace transform of Zz, ”
f = Leplace transform of f.

Cramer's rule can be used to write the formal solution of (2):

B

_ P : : .
z, = (3)
|A - sI] K
where |
Ei = ith éomponent‘of E,
Bi'=,determinant of [A - sI] with the ith column replaced by -f.

In general, a transfer function expresses the relationship between
an independent variable and some dependent variable., The independent
variable appegrs as a factor in the disturbance vector, f, on the right
hand side of the system equation. Thus, T may be written as follows:

f = pg ‘ ‘ (L)
fihere
o = Laplace transform of the selected independent variable - a scalar,
g = a vector of coefficients. |

Use Eq. (4) with (3) to give

Z C

 

G == = L (5)
. P |A - sI| '
where |
G = transfer function between the independent variable, p, and the
dependent variable, z,, , i
C, = determinant of [A - sI] with the 1™ o6lumn repiaced by -g.

Q

 
 

For nuclear reactor applications, the selected independent variable is
most often reactivity, and the selected dependent variable is most often
the neutron flux or a temperature at some point in the system.u

The transfer function in Eq. (5) may be used to give the frequency
response. For this, the Laplace transform variable, s, is replaced by
Jjw, where Jj =\/:T“ and- w = the frequency of the perturbation. Thus, the

transfer function becomes a complex quantity:
G=a+ jB . ‘ | (6)

The appearance of G in the complex plane is shown in Fig. 1. It is common

to characterize G by a magnitude, M, and a phase angle, 8. These are given

by: o :
M =" + 52 s (7)
o = ten™ £ (8)

The variation of M and @ as a function of. the frequency, w, is called the
frequency response of the system. |

A number of approaches are possible for solving Eq. (5). The most
obvious is to form the numerator and denominator determinants and to
numerically evaluate these determinants in complex arithmetic. Another
approach is to transform the determinants into polynomials in s.l’2 This
has the advantage that evaluation for numerous frequencies (w = — js)
does not require re-evaluation of the determinants. The choice, then, is
whether to perform the bulk of the computation in finding the polynomial
or in evaluating the determinants. The preference seems to have been for
the polynomial method in most previous calculations. This was done because
the polynomial methods were sufficiently faster than direct determinant
solutions to offset twé difficulties characteristic of polynomial methods:
the accurate calculation of the coefficients of the polynomial is a dif-
ficult numerical problem, and the complex relation between the basic system
parameters and the coefficients of the polynomials complicates calculation

of the effect of changes in the parameters.

 
ORNL DWG. 65-7607

Im(G)

 

 

Re(G)

 

Fig. 1. Appearance of G in the Complex Plane

 

Ca

c
 

o

[i]

7

In this study the desire to determine the effect of changes in
system parameters on the fregquency response dictated the use of direct .
calculation of the determinants. In cofitrast'to the polynomial methods,
it is easy to keep track of the system parameters and to determine their
effect on the frequency response. It was also found thét direct calculation
of the determinants for calcuiating the frequency response alone is
inexpensive on large digital computers unless the matrix;is'quite large.

5

The running time for a FORTRAN IV Gaussian elimination scheme” on the

IBM 7090 has been'found to be given by:

T = 0,028 nl'9 ,
Where
T = running time (seconds/ffequehcy calculated),
n = order of the matrix. | |

If it is assumed that about 25 pointé are needed to define the
frequency response, then the running time is given approximately in

Table 1.

Table 1. Approximate IBM 7090 Running Time for
Direct Frequency Response Calculation

 

 

Ordef of : | Running Time
‘Matrix - o . (min)-

20 ’ 3.4

50 Te3

Lo | ‘10,5

50 | ' : 19.5

60 . L : 27.9.

 

Frequency Response Sensitivity

 

It is frequently valuable to know what changes in the frequency
response will occur if certain of the system paraméters should change.

It would be desirable to get this information without recalculating.the

 
10

whole frequency response repeatedly. A technique for accomplishing this
is given in this section. |

First, rewrite Eq. (5) as shown below:

N N : . :
G=Tr =] C (9)

Now differentiate BEq. (9) with respect to an element, aij’ of the.system

matrix, A,

aD
e . (10)

iy 1J . 1J

 

 

il
ol
1
o=

Equation (10) gives the sensitivity of the frequency response to changes
in the elements of the system matrix. The derivatives on the right sidé
of Eq. (10) are easily calculated. It can be-shownLL that the derivative
of a determinant of a matrix with respect to one of its elements is the

cofactor of that element in the matrix. Thus we get:

{

oG 1
e =D My m @y5) (11)
ig
where
nij = cofactor of aij"in the numerator matrix, N,
7ij = cofactor of aij in the denominator matrix, D.

It is alsoc necessary tc convert the G sensitivity into magnitude
sensitivity and phase sensitivity. PFirst, since a, is real, we can write:
oG _ ou . 0B
da.. oa,.  Pa_. - (12)
1J 1J 1d
Thus the real and imaginary parts of the'solufiion given by Eq. (11) are

 

actually ao/aaij and as/aaij. From the definitions of M and 6, it is

clear that the following relations exist:

 

,d o +B§g_

oM oa. . Bai.

9a. = _ 'lJ_ q. P | - (13)
L Jao + B ’ '

 

 

 

{4
11

 

 

o g@ - oo
a, ., oa. . .
- 08 _ ij ij : (14)
‘fa., ., - 2 .2 ’
13 a” + B

Equations (11) through (1) are adequate for finding the sensitivities

to matrix element changes. However, these matrix elements are made up

of algebraic combinations of basic éystem.parameters. The same system
parameters frequently occur in several matrix elements. It is desirable

to find the sensitivities to system parameter changes as well as the sensi-
tivities to matrix element changeé. This can easily be done using Egs. (15)

and (16):

 

 

 

 

 

' ' oa. . .

oM _ oM ij

ox - 23 oa. ox (15)
L iJ. ij b

T ‘aa » ,

06 o 06 ij ,

e S L BT s o (16)
£ ij ij £

where
th '
xz = the /4 system parameter.

The quantities 8a /ax are known since the algebraic relations between
matrix elements and system parameters must be known from the analytlcal
description of the system.

A special feature of the numerator determinant, N, should be noted.
The column whose elements consist of the disturbance vector, g, clearly
do not depend on the matrix elements, aij' Thus 6N/aaij does not contain
a contribution from the column replaced by g. . However, g can depend on

the system parameters. Thus 8N/9dx, may have a non-zero contribution from

£
the column of the matrix whose components are the components of g. Thus

the complete equations are:

 

 

. da 0
M\ oM 1j M G
=) e m L (17)

g i3 Piy 9% K 98 9%
12 .

 

 

96 90 9%i; 90 98
Ll B motl B (18)
g i3 Ty T k T8 9%y
The procedures for finding BM/ng and ae/agk are similar to those for
finding BM/Baij and BQ/aij. Slnce = a?pears only in N,
oG 1 ON :
ng D g, ’ o (19)
where
_ gg— = negative of the cofactor of the-element in.N containing g
k
From the definitions of M and 8, it is clear that
M By .. 98 (20)
ral :
“k Vo2 + g2
0
96 S K (21)
g, - '
| &y V& + 52
where
gg; = real part of 8G/8gk,'
8
g%— = imaginary part of-BG/agk.' o
k .

The Computer Program

 

A computer program called SFR-1 (Sensitivity of the Frequency
Response) was prepared for the IBM 7090. The cbmputer code is provided
with the system matrix, A (59 x 59 or-smaller), and the disturbance
vector, g. For specified values of w, the code calculates the frequency

response using Eq. (5) and s = jw. Equations (7) and (8) are used to

 

|L'.
135

give the magnitude afid phase. The determinarits in Eq. (5) are calculated
in complex arithmetic using a Gaussian elimination scheme with partial
pivofiing3 (obtained from:R. E._Funderlic of Oak Ridge Gaéeous Diffusion
Plant). The code also can calculate the sensitivities to matrix element
changes using Egs. (11), (13), and (14). The sensitivities to the system
parameters are calculated using Egs. (17) and (18). The method for
providihg the algebraic‘relationships between the matrix elements and

the system parameters are given below -in the section on input.

Input

The input to SFR-1 is short and simple. The only section requiring
extensive explanation is the algebra table. The algebra table serves ‘to
establish the relationship between matrix elements and system parameters
and . the relationship between elements of the disturbance vector and
system parameters; In-general, each matrix element or disturbance vector
element is made up of a sum of terms, each of which 1is an algebraic -

combination of various system parameters:

a = 7 x%'jb £%1+Z x%';% x%i+
ij -1 "1 T2 """ Tm 271 Tp "t Tp

-OXr.
I
_ am - _
oy = ) 7 gz x (22)

where

N
1

a constant,
qu = exponenfi of the qth factor in the mth term,

= the number of the term,
= index on the systém parameter,

the number of terms,

H =2 o B
i

= the number ofifactors in term m. -
1k -

For instance if

 

 

 

_ 2 ..8 -1 -2 3 1.8
38,9 =2x % X"+ L2 X % XF.
we could express a8 9 in tabular form as:
2
Coefficient of
i J m zZ_ 1 2 3 L
8 9 1 . 2.0 2.0 0.8 -1.0
8 9 2 .2 - | -2.0 3.0 1.8

 

 

A table of this type appears in the SFR infiut. The information in tfiis
table is also used by SFR-1 to calculate the derivatives shown in Egs.
(15-18). The general rule for differentiating terms of this type leads

 

 

 

 

to
M I qu
BaiJ 23 II %4 q
= Z P b (23)
axfl 2om Zm g=1 X, Z
where
83‘= 1if X, appears in the mth term
O if_xfl does not appear in the n*? term
The detailed description of the input is given below:
Type 1: ‘
Title card.
Type 2:
Column 1-5 6-10 11-15 16-20 21-25 26-30
Format 15 15 15 15 15 15 -
Input N NOW NCTS NOXI KIPD NOFV

 

 

 

 

 

 

 

 

 

 

 
 

where

N = order of the system matrix,

NOW = number of frequencies to be calculated,'

NCTS = number of different columns to be replaced by the disturbance

 

 

 

 

 

 

 

 

 

vector,

NOXI = number of system parameters being considered,

KIPD = derivative option. If KIPD is positive, SFR calculates the
frequency response - only. If KIPD is zero or negative, SFR
calculates the frequency response and the sensitivities,

NOFV = row number of the last non-zero entry in the disturbance
vector if the disturbance vector is specified in Type 3% input.
If the disturbance vector is specified in the algebra table
(Type 5 input), NOFV is omitted.

Type  3:
Column 1-10
Format TELO. 4 Repeat, 7.per card
Input Ci

where

Ci = components of

the disturbance vector

Note: Type % cards are omitted if all components of the disturbance

vector are calculated from the algebra table (Type 5 input).

 

 

 

Type k4:
Column 1-10
Format TE10.4
Input xfl

 

 

 

Repeat, 7 per card

 

 

 
16

where @

X, = value of the system parameter, values are listed sequentially

starting with Xq -

Note: Omit Type 4 cards if NOXI = O.

Type 5:

 

Column| 1-2|3-4|5-6]7-16 17-25!2&430 31-37138-44] 45-51] 52-58(59-65| 66 -72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Format || I2 | I2| I2{El0.5 8F7.2
Input I J m |2 P
where

I = row number of matrix element if I = 59.  If I = 60, a component
of the disturbance vector .is being specified,
J = column number of matrix element if I £ 59, .If I = 60, J is
the row of the component of the disturbance vector being
specified,
m = number of the term,
Z = constant multiplier 6f this term,
P = eprnent of the system parameter. »
Note: ZEnd Type 5 cards with a blank card. Omit Type 5 cards if NOXT = O.
No blank card is used to end Type 5 input if NOXI = O.

 

 

 

 

 

 

 

 

 

Type 6:
g
Column 1-2 |
| Format ’ I2 Repeat
Tnput | CR
where

CR = column number to be replaced by the disturbance vector, NCTS

entries should be made.

 
 

17

 

 

 

 

 

 

 

 

 

 

 

Type 7T:
Column 1-5 6-10 | 11-20 |
- Format I5: I5. F10.4 Repeat, three per card
Input I J a. .
1J
where
= rOW number,
J = column number
a = value of element, aij’ cf the system matrix,
1j .

Note: End Type 7 cards with a blank card.

 

If a matrix element is specified on a Type 7 card and also is
calculated from the algebra table, the value from the algebra
table will be used. '

 

 

 

 

 

 

 

 

 

Type 8:
Column 1-10
Format 7E10.4 | Repeat, seven per card
Input W

where f

w = fregquency for calculation. ©Specify NOW values.

The FORTRAN listing of the SFR code is available from J. L. Lucius.

Output

The output of SFR is clearly labeled in notation consistent with
the notation in previcus sections of this report. The first page is a
review of input data. It consists of a print-out of the following:

1. Title
Input system parameters (x)
Algebra table

Order of matrix

N W o

Number of frequencies (w's)

 
18 .

Columns to be replaced
Frequencies to be calculated

System matrix non-zero elements

O 0 9 Oh

Disturbance vector

The input summary page is followed by the resu;ts of the calculation.
The results for each specified frequency are. shown, one frequency to a

page. The print-out is as follows:

 

1. Frequency _
2. Non-zero elements of 8D/8aij (see Eg. 10)
3., Magnitude ratio (M) and phase angle (THETA) -
4. Column replaced by f vector
5. Values of a, B, D, and N (see Egs. 6 and 9)
oG oG oM
6. Values of 8N/aaij, 8D/8aij, Re 5o Im,EETT e and
. i3 i T
a8
da. | .
1J

7. Values of 8M/8xfl and ae/axfl (see Eqs. 17 and 18).

Sample Problems

 

Problem 1. The first illustrative problem is a calculation for a

second order system.

“

dxl ‘ '
at - ()
dx2

T = - % - 0.12x, - k (25)

Rewrite these in matrix notation:

dx _ B
EE'—AX-}-f ‘ (26)

 
 

i~
~J

19
The transfer .function, Ei/E, is given by

1 :
= 5 . : (27)
s+ 0.12s + 1 :

 

W||_,:><|

Mhis is the form of a quadratic lag with a damping ratio of 0.06. This
familiar problem was analyzed with SFR-1. The frequency response and the
sensitivity of the frequency response to changes in the damping ratio

(% a,, in the systenm matrix) were calculated. The sensitivities were used
to predict the frequency response when the’ damping ratio changes from

0.06 to 0.05. Table 2 shows the predicted results and a comparison with
exact values, It is clear that the sensitivities provided very reliable
information about the effect of chenges in the damfiing'ratid in this

problem. A copy of the SFR-1 input required for this problem is shown
in Table 3.

Problem 2. The second problem is the analysis of a reactor with one
group of delayed neutrons and two temperature feedbacks, one prompt and

one delayed. The linearized equations are:

a

<
i

 

 

n o T! n o T! n &k
dn' _ B, cL 2 Moo | BoTex )
5z - gt N 3 =3 7 (28)
1 1 .
Lo By o e PPAT | PP | TP e (29)
dt £ £ £
1
i ..y (50)
dt (MC_) (MC_), 1 (MC_) 2
o p'l STp'l p’l
{ | | .
T2 omA g DA g (51)
at (M.cpi2 1 (Mcp52 2
where
n! = deviation in neutron population from the initial condition,
C!' = deviation in the precursor concentration from the initial

condition,

 
Table 2, Results for Problem 1

 

 

 

02

Predicted Actual
Frequency Amplitude Amplitude Amplitude - Amplitude Percent
(radians/sec) (damping ratio Sensitivity (damping ratio (damping ratio Error
= 0.06) : = 0.05) = 0.05) -

0.1 1.01003 0.00123646 1.01005 1.01005 0
0.2 ‘1.0413kL 0.00542026 1.04145 1.0414L 0.001
0.3% 1.09804 0.0142982 1.09833 ‘ 1.09830 0.003
0.4 1.18854 '0.0322%359 1.18919 1.18913 0.005 .
0.5 1.%2909 0.070433%9 1.3%050 1.%3038 0.009
0.6 1.55271 0.161715 1.55594 1.55568 0.017.
0.7 1.93672 0.425824 -+ 1.9432) 1.94k257 0.034
0.8 2.68339 1.48492 2.71310 2.71163 0.054
0.9 b 57562 9.31140 4,76185 b 75651 0.112
0.95 6.66633 22,0841 7. 30801 T . 3459) 0.516
1.0 8.33333 69.4uLy 3.72219 10.00000 2.778
1.2 2.15999 1.741k1 2.19482 2.19265 0.099
1.4 1.02607 0.254081 1.03115 1.03076 0.038
1.6 0.636225 0.0791140 0.637807 0.637680 0.020
1.8 0.44L367 0.0341156 0. 445049 0.Lhkooh 0.012
2.0 0.332272 0.0176085 0.33%262k 0.3%2595 0.009
3.0 - 0.12487k 0.00210298 0.124916 o 0.124912 0.003
5.0 0.0416537 0.000216811 - 0.0416580 - 0.0416576 0.001
10.0 0.0101003 0.0000123646 0.010050 0.010050 0

 

 
 

 

Type
Type
Type

Type
Type

 

KEYPUNCHING INSTRUCTIONS:

Punch only those cords containing data.

UCMN-8302

13

12-03)

TEN COLUMN INPUT
SFR-1 SAMPLE
T. W. Kerlin

Table 3. Sample Input for Problem 1

REFERENCE

e
22

deviation in fuel temperature from the initial condition,

deviation in moderator temperature from the initial

condition,

delayed neutron fraction = 0.006L,
neutron lifetime = 0.5 X lO-u,
precursor decay constant = 0.125,

initial neutron population = 10.0,

fuel temperature coefficient of reactivity = -0O.
moderator temperature cdefficient of reactivity
external perturbation in keff’
heat capacity of fuel = 1.5,

(heat transfer coefficient) X (fuel area) = 3.0,

heat capacity of moderator = 2.0

The system parameters are identified with specific X, as shown below for

this problem:

mN I—JN

™

b
£ Ul

0.0064
0.00005
0.125

= 10.0
~0.0005
'—0.00005
1.5

1l
EER RS ¥ T
1l

i

H

(@
~—
1\

p’l
= 500

= 2.0

il
&
3
~
no

Substitution of these values into Egs. (28) through (31) yields the

following matrix equation:

where

dz
-128.0 0.125 =100.0
A = 128.0 —0.125 =0.64
0.667 0 —2.0
0 0 1.5

 

\.)‘

5 x 1072,

= —0.5 X 10'”,

(32)

-10.0
—~0.064

2.0
-1.5
 

23

2.0 X 105‘
1.28 x 10°
0
0

The frequency response for problem 2 is shown in Fig. 2. Figure 3
shows the sensitivity of the magnitude of the}frequency'responée to
changes in the fuel £emperature coefficient of reactivity (Qi) and the
moderator coefficient of reactivity (Qé).' Several:observations about the
behavior of the system are immediately obtained from the sensitivity plot.
At frequencies below 0.1 radians/sec, the effect df'changes in the fuel
temperature coefficient and the moderator temperature coefficient are the
same. However, since the fuel temperature coefficient (Qi) has a magnitude
which is 10 times as large as the magnitude of the moderator coefficient
(ag), it is clear that the effects of fractional changes in oy are 10 times
as large as equal fractional changes in Qé. It is also clear that the
frequency response is very sensitive to changes in the temperature coeffi-
¢ients in the frequency ranhge, 0.1 to 0.5 radians/sec. Above 0.5 radians/sec
the graphite effect is much smaller than the fuel effect until they both
diminish to small values at frequencies above 10 radians/sec. These
illustrative results are typical of the results obtained in sensitivity
analysis of reactor systems. The sensitivity data furnish useful infor-
mation about the system which can aid in obtaining the essential under-
standing of the dynamic structure of the systém that is needed in analysis,
design, and experiment planniné.

A copy of the SFR-1 input sheet for this problem‘is shown in Table
L.

-Conclusions

SFR-1 represents a preliminary attempf to obtain frequency response
sensitivities along with the usual frequency response data. The use of
Cramer's rule was expedient in developing the SFR method and certainly
does not represent the most efficient procedure. Nevertheless, the cdlcu-

lations on SFR-1 have proved useful in practical prOblems5 and the cost of

 
2l

the calculation has not been excessive. The only numerical difficulty
observed has occurred at high frequency where inaccurate sensitivities
have been obtained in some problems.

The success with SFR-1 has led to the development of a new computer
code which performs the SFR calculation more efficiently. This code has
been prepared and is now being tested. Thé oFR method is also being used
to furnish sensitifitigs to a foutine for automatically adjusting the
parameters in a theoretical model to fit experimental frequency response
dafia, The method being used is similgr‘to the "learning model" approach

used by Margolis and Leondes6 in adaptive control applications.

 
 

 

10,000

1000

Mégnitude Ratio (&n/&k)

10 .
. 0.001

0.01

Fig. 2a.

0.1 1
Frequency (radians/sec)

Frequency Response for Problem 2 — Magnitude Ratios

 

10

ORNL DWG. 65-7608

 

100 .

4
 

: 7

Lo

Phase Shift (degrees)

90

60

50

3

3

-
O

o

1
-
O

o
n
o

' Frequency (radians/sec)

:Fig. €b. Frequency Response for Problem 2 — Phase Shift

'

 

ORNL DWG. 65-7609

 

100

9%
Sensitivity x 1076

0.001

0.01

fuel temperature coefficient of reactivity

moderator temperatufé coefficient of reactivity

 

0.1 . 1l

10 " 100
Frequency (radians/sec) ;

' Fig. 3. Selected Sensitivities from Problem 2 Results..

LC
 

Type 1

Type

Type L

Type

 

KEYPUNCHING INSTRUCTIONS:

Punch only those cards eontoining dota.

TEN‘ COLUMN INPUT

REFERENCE

8¢

UCN-5393 . Teble 4. Sample Input for Problem 2

(3 12-83)
 

 

 

KEYPUNCHING INSTRUCTIONS:

Punch only those cards containing dota.

UCN-5393
{3 tl-lli)

TEN COLUMN INPUT

-}
T.W. -\

Table L. (continued);

REFERENCE

6C
20

References

®

S. M. Katz and D. S. St. John, "Lass — An IBM TO4 Program for Calcu- .
lating System Stability," USAEC Report DP-894, Savannah River

Laboratory, July, 1964.

C. B. Guppy, "Computer Programme for the Derivation of Transfer
Functions for Multivariable Systems," United Kingdom Atomic Energy
Authority Report AEEW-R189, Winfrith, March 192.

 

V. N. Faddeeva, Computational Methods of Linear Algebra, Dover
Publications, Inc., New York (1959). |

F. E. Hohn, Elementary Matrix Algebra, The MacMillan Co., New York
(1958).

S. J. Ball and T. W. Kerlin, "MSRE Stability Analysis," USAEC Report
ORNL-TM-1070, Oak Ridge National Laboratory (in preparation).

 

M. Margolis and C. T. Leondes, "On the Theory of Adaptive Control
Systems; The Learning Model Approach,” pp. 556-563 of Automatic and

Remote Control, Proceedings of the First International Congress of

 

&

the International Federation of Automatic Control, Moscow, 1960
edited by J. F. Coales, J. R. Ragazzini and A. T. Fuller, Butterworths,
London (1961).

 

 
 

O O—1 O =\ o H

e -

31

. Internal Distribution

 

R. K. Adams 28. A. M. Perry

C. L. Allen, K-25 29. B. E. Prince

5. J. Ball _ 30. C. W. Ricker

S. E. Beall 31, M. W, Rosenthal

A, A. Brooks, K-25 32, D. P. Roux

0. W. Burke ‘ 3%, H. W, Savage

F. L. Culler 34, A, W. Savolainen

A, P. Fraas %5. L. Spiewak

D. N. Fry 36, R. S..Stone

S. H. Hanauer 37. D. B. Trauger

P. N. Haubenreich : 38, G. E. Whitesides, K-25

T. W. Kerlin 39, G. D. Whitman

B. R. Lawrence 40. ORNL Patent Office

J. L. Lucius. 41-42, Central Research Library

M. I. Lundin 4%-4l, Y-12 Document Reference Section
R. N. Lyon 45-47, Laboratory Records Department
H. C. McCurdy 48, Laboratory Records Department,
A, J. Miller - LRD-RC

External Distribution

 

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