STABILITY ANALYSIS OF A NONLINEAR COUPLED-CORE REACTOR CONTROL-SYSTEM

STABILITY ANALYSIS OF A NONLINEAR COUPLED-CORE REACTOR CONTROL SYSTEM

Tain-Sou Tsay

Institute of Electronics, National Chiao-Tung

University,

Taiwan,

R.O.C.

ABSTRACT-In this paper,

stability-equation

method is

applied

to the

analysis

of a

large

coupled-core reactor control system

having

multiple nonlinearities and

adjustable

para-meters. The characteristics of the limit-cycle and the

asymptotically

stable

regions

can be

easily

defined in a parameter

plane.

A numerical example is

given

and

comparisons

with other methods in current literature are made.

I. INTRODUCTION

In current

literature,

several methods have been applied to the

analysis

of

large

coupled-core reactor control

systems[l-3].

Raju and

Stone[l]

have derived an

analytical

model and

investigated

system

stability

using

the

describing

function

approach;

Raju

and Josselson[2] have obtained conditions of

stability

using

the Popov

criterion;

Tsouri and

Rootenberg[3]

have

applied

the

Tsypkin

locus method for limit cycle and

stability

analysis.

In the above mentioned

methods,

all the systems are considered

symmetrical,

and it is assumed that each system can be reduced into two

single-input,

single-output

systems [1,3], then the

single-output

systems are

analyzed. In this paper, a general method based upon the stability-equation method[4,5] is proposed. The considered systems need not be symmetrical and reduced. In

addition,

the systems may have both nonlinearities and adjustable parameters. The main approach of the proposed method is to analyze system

stability

and the existence of limit cycles by finding the simultaneous solutions of both the

stability-equations[4,5]

and the harmonic-balance

equations[6-13].

and Kuang-Wei Han

Chung-Shan

Institute and

adjunt

professor at National Chiao-Tung University, Taiwan, R.O.C.

b c= -A-n

Ac1

c b n c =

A~-

2-

Xc2

(1-c)

(l-d)

(l-e)

T =Kn

-aT1

T =Kn -a

aT2

_(1-f) where:

nl,n2

-deviations of power in core#l and core#2, respectively,and n is taken as proportional to neutron flux. cl,c2 -deviations in average concentration

of delayed neutrons in core#l and in core#2, respectively.

T1,T2 -deviations in temperature for core#l and core#2, respectively.

K -proportionality constant between power and temperature.

D -power coupling coefficient between cores.

x -effective delayed neutron decay-time constant.

b -fraction of neutrons delayed. A -prompt neutron generation time. r

-reactivity-temperature

coefficient.

a -heat removal coefficient. h -steady-state power level. p -reactivity.

Taking the Laplace transformation of

Eq.(l),

the block diagram of the model including the controller[1-3] is shown in Fig.l(a), where II. AN ANALYTICAL MODEL OF THE

COUPLED-CORE REACTOR

The

linearized

equations of the coupled-core reactor control system considered in this paper are as follows[1-3]:

D D b rh h

n1-

A

1nl

A,-

2-

A 11A

TA

lA -l G (5)=G2(S)= h AS(1+T S) m G

(5) =G22 (S)

_ _ _

-(S+a)(S+X

) (1-a) D D b rh h n₂ =--n-n -_n +-.-T + 1b A 2 lA1 A 2+2 A 2 A

(1-*Manuscript

is first received by IEEE namely May 28, 1987.

(2-a)

(S+-A ) (S+X) (S+a)+ A S(S+a) + A

(2b)

(2-b)

12( ) 21(S DA (2-c)

Tm is the time constant of the control-rod drive motor; N1 and N2 represent the on-off 0018-9499/87/1200-1827$01.00 © 1987IEEE

let

61

to be zero,then the harmonic-balance equations[6-13] of

loop-i

and loop-2 are

Fig.l(a). Block

diagram

of the control system for a large coupled-core reactor. relays. The equivalent system block

diagram

of Fig.l(a) is shown in

Fig.l(b),

where

W11

(S) =G1(S)

G11

(S)

/

A(S)

(3-a)

21(S)

=G1

(S)

G11

(S)

G21(S)

G22(S)

/ A(S) (3-b)

12(S) =G2

(S)

G11

(S)

G12

(S)

G22(S)

/

A(S)

(3-c)

(3-d)

W22

(S) =G2 (S) G22(S)

/ A(S)

and

A(S)=l-G11(S)G21(S)G21(S)G22(S).

Fig.l(b).

Equivalent

block

diagram

of the system shown in

Fig.l(a).

III. THE BASIC APPROACH

Consider the system shown in

Fig.l(b).

Assume that the

input

signals

to the nonli-nearities

N1

and

N2

are

(4-a)

a

C=A1exp[j(wt+01)]

and

a

2=A2exp[j(wt+ 2)]

respectively,

where A1 and

A2

are the

ampli-tudes; 01 and

02

are the

phase angles.

Con-sider

a1

as the reference

signal;

i.e.,

to

(4-b)

A N

1(al)w

(jw

)

+A2ej

%2(

a2)

w12(jc

)

=-1

(5-a) and

A1N1

(a1)W21

(j

w)

+A2eje92(a2)W22

(j

w)

=-A2ej62

(5-b) respectively, where e is the phase angle of the input signal to

tie

nonlinearity N2 with

a1

as the reference signal;

Nj(aj)

and N2(a2) are the describing functions(or equi-valent gains [14,15]) of the nonlinearities

Ni

and N2, respectively.

From Eq.(5-a), one has

e2-

A1[

1+N

1(a1

)

w11(iW)

I

e

A2N2(a2) W12()

Similarly,

Eq. (5-b)

gives

j02

N1(a1)W21(i

) e

A2[1+N2(a2)W22

(X)I

(6)

(7)

Equating

Eqs. (6) and (7) , one has F

(jw)

=l+N1(a1)W

11

(jw)

+N2(a2)

W22

(jw)

+N1(a1)N2(a2)

[wll(jc)w22(ij)

-W12

(jW)

W21

(jw)

I

=0

(8)

which is the characteristic

equation

of the considered system. Note that

Nj(aj)

and N2(a2) are considered as

varying

parameters.

Since N1 and N2 are two

single-valued

nonlinearities[l,2],

Eq.(8)

can be decom-posed into two

stability-equations[4,5]

as

F

(w)=B1(W)+N1(a1)C1(w)+N2(a2)

1(

+N1(a1)N2

(a2)E2

(to)

=0 and

F

(w)=B2(w)

+N1(a1)

C2

(w)

+N2(a2)

2(w)

+N1

(a1)

N2

(a2)

E2

(t)

=0 From Eqs.(9), one has

B1

(t)

+N1

(a1)

c1

()

N2(a

2)=-D1

(wi)

+N1

(al)E

1(

Similarly,

Eq.

(10)

gives

B2(wi)

+N1

(al)c

2(w

N2(a

2)=-D(o

)

+N1

(a1)

E2

(

(9)

(10) (11) (12)

Equating Eqs.(11) and (12), one has

[C1 (w)E2 (w)-C2 (w)E1

()]N1

(a)

2+[C

()D ()

+B2

()E

l(w)-CI

(w)D2(w)-B

(w)E

2(w)]

x

N1

(a1)

+[B2

(w)

01(w)-B1

()D

(w)

]

=0

(13)

For specified values of

frquency(w),

the values of

Nj(a1)

can be found

by

solving Eq.(13), then the

corresponding

values of N2(a2) can be found from Eq.(ll) or Eq.(12).

Fig.2. Root-loci of the stability-equations

for Case 1 with M=22.

For a number of suitable values of (A , the

real solutions(roots) of N1(a1) and N2(a2)

can be plotted in a

N1(al)

vs. N2(a2) plane.

The typical root loci for a latter case are

shown in Fig.2.

By use of Fig.2, the conditions of having

a limit cycle are explained as follows:

(i) Every point on the curves as shown in

Fig.2 represents a set of N1 (al), N2 (a2) and

w which can satisfy the condition that a

limit cycle may exist if the roots wei and wo0 of the even and odd stability-equations Fej

Iw)

and

Fo

w), respectively, are all real and alternaX ing in sequence. There is an

exception, however, when one root pair is equal to the other(i.e.,

Wei

=

Woj=0

[4,5].But

unfortunately for nonlinear multivariable systems, there are infinite number of

solu-tions which can satisfy this condition[13].

This is quite different from that of the single-input, single-output systems.

(ii) If the root-loci shown in Fig.2

sepa-rate the stable and unstable regions, then a

limit cycle may exist. The reason is that,

if the system becomes stable(unstable) when the amplitudes A1 and A2

increase(decrease),

a stable limit cycle may exist at the stabi-lity boundary[4,5,16].

(iii) A limit cycle may exist only if the corresponding values of N1(a1) and N2(a2) of the root-loci are less than the maximal gains

(Nlmax and N2max) of the nonlinearities. For

example,

in

Fig.2

only

the section between

points

Q2

and

Q3

can

give

a limit

cycle.

(iv)

A limit

cycle

may exist

only

if the roots

N1(a,)

and

N2(a2)

satisfy

both

Eqs.(5-a)

and

(5-b).

From

Eqs.(5-a)

and (5-b), the

possible

simultaneous solution can be found

by

equating

the real and

imaginary

parts of

Eqs.(6)

and

(7), respectively;

i.e.,

==0

(14)

where 026 and 027

represent

the

phase

angles found from

Eqs.(6)

and

(7),

respectively.

If the considered nonlinear

system

can

satisfy

all the above four

conditions,

a limit

cycle

may exist. The three

parameters

A1,

A2

and w of the limit

cycle

are defined

by

Eqs.(9),

(10)

and

(14).

Additional

expla-nations are given in the

following

section. IV. ANALYSIS OF THE CONTROL SYSTEM CASE 1: Assume that the numerical values of the

parameters

of the

system

considered

are at

b=.0064,A=0.l,X=.00lsec,K=l0

F/MW.sec,

a=10/sec, r=.001/

F,h-=30MW,D=.0l5,B=.25MW,

T

=0.07sec,and

V=MxlO

6k/k.sec[3].

For M=22 and for a number of

frequencies(w),the

simul-taneous solutions of

Eqs.(9)

and

(10)

are shown in

Fig.2

where the

stability

of each

region

has been checked. At every

point

on

the root

loci,

it has been checked that the roots

wei

and

woj

of the

stability-equations

Fei(w)

and

Fo

(w),

respectively,

are all real and

alternacing

in sequence

except

that one root

pair

is

equal

to the

other(i.e., wei=

woj=W).

By

inspecting

the root-loci shown in

Fig.2,

the section between

points Q2 and

Q3

can

satisfy

Conditions

(i)

to

(iii).

Solving

Eq.(14)

along

the section between points

Q2

and

Q3,

the

point

Q

(0.0163,0.0163) with

oscillating

frequency

w=22.77

rad/sec

and

amplitudes A1=A2=1.703

can

satisfy

condition

(iv).

Therefore,

a limit cycle may exist at

point Q1.

This fact is supported by

checking

the roots woei and

w0o

of the

stability-equations

in the

neighborhood

of

point

Q1[16]. Fig.3

shows the wei and

woj

loci for N1

(a1)

is fixed at

0.0163(i.e.,

Ai=1.703)

while

N2

(a2)

is

varying.

From Fig.3(a), one can see that if the value of N (a2 ) is less than 0.0163

(i.e.,

A2 = 1.703),

the

roots wei and

woj

are alternative in sequence, then the

corresponding

system is

stable[4,5,16].

If the value of N2 (a2) is larger than

0.0163,the

corresponding

system is unstable. A similar result can be obtained when N2 (a2) is fixed at 0.0163(i.e.,A2=1.703) and N1 (a1) is

varying.

Therefore, a stable

limit-cycle

will exist at the stability boundary where

N2

(a2)=0.0163;

i.e.,

A2 =1.703.

In

Fig.2,

for another branch of the

root-loci,

the corresponding values of

N,

(al) and N

(a )

are

larger

than the maximal gains of N,

(al)

and N2 (a2), respectively; therefore,

simulated result is quite obtained by calculation.

2

-

stable -4- unstable

o

W=

22

771tMdSej'

-21

WaW

A0_2(a2)

0.01

N,

(a)=.o163

0.02

Fig.3.

Root-loci of wei and

wo.

of the stability-equations with fixed

Nl(a1)

and

varying

N2(a2).

Note that the root-loci can also be plotted in the A1 vs. A plane by

solving

Eqs.(11), (12) and (13 which

Nl(a1)

and N2(a2) directly relate to the

describing

functions of the nonlinearities N1 and N2; i.e., N_i

(a

_i 4V=_{TrA 1 -}B2--_{A.2 )1/2}

/

i 1 i=1,2 (15) close to that

ai

W

22.63rad/Scc;

Cr=02

t-sec

Fig.5. The simulated limit-cycle of Case 1 for M=22.

CASE 2: For the system considered in Case 1, assume that the nonlinearities N1 and N2 are replaced by two double-valued nonlinea-rities[3] as shown in Fig.6. Then the

descri-I

V

-(a)

(b)

The result is

given

in

Fig.4

where

point

Q4 represents a stable limit

cycle.

In this

case condition

(iii)

is not necessary for analysis.

3-stable

2-Fig.6. Nonlinearities of Case 2.

bing functions

N1(a1)

and N2(a2) of Eqs.(5)-(8) are replaced by

Nk(ak)=Nkr(ak)+jNki

(ak) where

Nkr=

A

I[(1-

A 2 )/ (1-k = k N -- 2V

(P-B)

ki TrA 2 k=1,2 (16)

p2

1/2

Ak2

) I

Ak

> By

using

the same

approach

as in Case Eq.(8) is

decomposed

into

B.

1,

Fe

(()=B(w)+Nlr(a1)c1(u)-N

i(a

) 2(w)

+N2r

(a2)D1

(w)

-N2i

(a2)D2

(w)

+[Nlr(a)N2r

(a

2)-Ni(a

)N2i(

2)]

E1

(w)

-

[Nlr

(a1)

N2i

(a2)

+N1i

(a

1)

I--j

i

i

-

-i

2

Fig.4. Root-loci of the stability-equations for Case 1 with M=22.

By computer simulation, Fig.5 shows the limit cycle of the system for M=22. The

N2r

(a2) ]E2(wX)=0 (17)

and

F0

(w)=B2(w)+Nlr(al)C2(w)+N1i(a

1)

C1()

+N2 (a2)D2(w)+N2i (a2) D1(w) +[Nlr (a1)x

-GL,WJs

UfWe5

60[

30

F

W04

Zk2-~~~I .

0

l±

unstable

p

Anw-.. . I

lolp;;;;Illmllllkm

-&m

-t= I"-X

-W

N2r

(a2)-Ni(al)

N22i

(a2)

]E2(W)

+

[Nlr

(al)

N2i

(a2)

+N i

(al)

X

N2r(a2)]E1(()=0 (18) where

Bi(w), Ci(w),

Di(w)

and

Ei(w)

are the

same as those of

Eqs.(9)

and

(10).

For

M=22,

B=0.15MW and

P=0.25MW,

and for a number of

frequencies(w),

the root locus of the stabi-lity-equations is

plotted

as shown in

Fig.7.

It has been checked that every

point

on this

root-locus can

satisfy

conditions

(i)

and (ii).

Solving

Eq.(14)

along

this

root-locus,

the point Q

(1.752,1.752)

with an

oscilla-ting frequency w=22.526

rad/sec

represents

a

stable limit

cycle.

A2

V. CONSIDERATION OF PARAMETER ADJUSTMENT In this

section,

control systems with adjustable parameters are considered. Assume

that two

adjustable

parameters

K1 and

K2

are

cascaded by the nonlinearities N1 and

N2,

respectively,then

Eqs.(5-a)

and

(5-b)

become k

1A1N(aI)W11(jwO)+k1A2e02(a2)W12(jw)

=--A1

(19-a) and

k2A1N1

(a1)W21(jw)+k2A2e

02(a2)W22(jw)=-A2ejG2

(19-b)

respectively.

Eq.(19-a)

gives

WI

34+

eie2

=

i22.7

stable

2+

1*

0

eje2

=

A1

[l+k1N1

(a1)

W11

(iw)

k1

A2N2(a2)w2(i

(20)

Similarly,

Eq.(19-b) gives

k2N1(a1)W21

(iW)

A2

[l+k2N2

(a2)

W22

(iw)

(21)

Equating

Eqs.

(20)

and

(21),

one has

unstable

A,

41

1

F

(jw)

=l+k1N1

(a1)

W11

(jw)

+k2N2 (a2)

W22

(iw)

k1k2N1

(a1)

N2(a2)

[W11

(iW)W22 (i

)

-W12

(j

W)

W21

(iw)

] =0 (22) Fig.7. Root-loci of

stability-equations

for

Case 2 with

M=22,B=.15MW

and P=.25MW. By computer

simulation,

the limit

cycle

is shown in

Fig.8,

which is

quite

close to that obtained by calculation.

2

0

ka

1.765

r\-A

ft6

/A\

which is the characteristic

equation

of the system under consideration. The

stability-equations

are

Fe

(w)=B1(w)+k1[Nlr(a1)C1(w)

li(a1)

2(

+k2[N2r

(a2)D1(w)-N2i

(a2) 2( )]

+k1k2{[N

lr(a1)N2r

(a2)-Nli

(a1)x

N2

(a2)]E

(w)-[Nr

(a)N2

(a

2)

+N i (a

)N2r

(a2)

]

}E2

(w)=0

C

LU=

22.43

RdseC.)GI=a2

Fig.8. The simulated

limit-cycle

of Case 2 for M=22,B=0.15MW and P=0.25MW. Note that, although Eqs. (17) and

(18)

are more complex than Eqs.(9) and

(10),

the approach is

straightforward

and the compu-tations can be

easily

made on a computer.

(23)

and

F

(w)=B2(w)+k

[Nlr(a1)c2(w)+N

i(al)

1(

I

+k2 [N2r(a2)D2(w)

+N2i

(a2)

D1

(w)

+k1k2{

[Nlr

(a1)

N2r(a2)

-N1i

(a1)

N2i

(a2)]E2(w)+[N

lr(a1)N2i

(a2) +N

i(a )N2r

(a

2)]E

()=0

(24)

i

2

i

i

I

where

Bi

(w) ,

Ci

() , D (X) and E (w) are the same as those in

Eqs.(6)

and (103. The

con-dition defined in Eq.(14), now, becomes

ej6220

_ej6221

= 0 ₍₂₅₎

where 0220 and 0221 represent the phase angles found by Eqs.(20) and (21),

respec-tively. The desirable solutions are A1, A2, K1, K2, and w. Thus for specified values of

A1

and

w,

one can find the solutions A2, k1 and k2 by use of Eqs.(23)-(25). Now, for a

specfied value of

A1

and a number of values of w, a limit-cycle locus can be plotted in a K1 vs. K2 plane. For a number of

constant-A1

limit-cycle loci, the limit-cycle region and the asymptotically stable region can be found in the K1 vs. K2 plane[16]. Similarly, one can plot the

constant-A2

limit-cycle loci in the K1 vs.

K2

plane for

specified

values of A2 and a number of values of w. Case 3: Consider the system in Case 2. Assume that two adjustable parameters K1 and K2 are cascaded by nonlinearities

N1

and

N2

as shown in Fig.6, respectively. For M=22, B=0.15MW and P=0.25MW, following the above presented procedure the limit-cycle loci are

plotted as shown in Fig.9, where

Fig.9. Limit-cycle loci of Case 3 for M=22, B=0.15MW and P=0.25MW.

the solid lines and dash lines are the

con-stant-Al

and the constant-A limit-cycle loci, respectively; the shaded region shows

the asymptoically stable region[16]. For illustration, the limit cycle represented by point Q (0.5718,2.1683),with amplitudes A1=1,

A2=3.78E,

and with oscillating frequency

w=22.5rad/sec,

has been simulated; the result is shown in Fig.10.

CASE 4: Consider the system in Case 2.

Assume that the nonlinearities N1 and N2 as

shown in Fig.6 are followed by two adjustable parameters K1 and K2, respectively. This is

3.82

0a2

,,J AX ! /; ,;

~~~t-sec

U1=22.44

ucdIeC

Fig.10. The simulated limit-cycle of Case 3 with

K1=0.5718

and

K2=2.6183.

equivalent to the case that the amplitudes of the nonlinearities N1 and N2 are adjustable.

The harmonic-balance equations of the system

are found as

k1AN1 (a1)W1 (jw)

+k2A2ejeN2

(a2)W12(jw))=-A1

(26-a) and

k1 11(a)W21 (jw)

+k2A2ejN2

(a2)W22(jw)=-A ej2

(26-b) Following the same procedure as indicated by

Eqs.(22) to (25), the

constant-Al

limit-cycle loci are plotted as shown in Fig.ll, where the shaded region is the asymptotically stable region.

L

asymptotIcally

stable

3

k

W

A,

Fig.l1. Limit-cycle loci of Case 4 for M=22, B=0.15MW and P=0.25MW.

Table 1 shows the calculated and simulated

results of some points in Fig.ll. It can be

4

2

101

-2

-4

i

N--0

-....11 ,I - 1A

Table 1. Calculated and simulated results of Case 4.

seen that the simulated results are quite

close to those of calculated.

From Figs.9 and 11, one can see that the minimal values of K1 and K2 which give rise to a limit cycle are at K1=K2=0.188 for the symmetrical case[3]. Then the critical value of M for having a limit cycle is

MC=KlXM=

0.188x22=4.1, which is quite close to the result found by Tsouri and Rootenberg using the Tsypkin Locus Method[3].

Note that if the nonlinearities and the linear transfer fuctions are not symmetrical (such as K1=K2) the proposed method can be applied in the same way as for the symmetrical case. It is also worthwhile to point out that, by use of the asymptotically stable region, the limit cycle can be eliminated by adjusting the parameters in the system.

VI.CONCLUSIONS

In this paper, the stability-equation method has been applied for limit cycle analysis of a nonlinear coupled-core reactor control system.The proposed method is simpler than the other methods in current literature, and it has the potential to be applied to

very complicated, nonlinear, symmetrical and asymmetrical systems.

REFERENCES

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2. G. V. S. Raju and R. Josselson,"Stability of Reactor Control System in Coupled Core Reactors,"IEEE Trans. on Nuclear Science,

Vol.NS-18, No.1, pp.388-394, Feb. 1971. 3. N. Tsouri, J. Rootenberg and L. J.

Lidof-sky,"Stability Analysis of a Reactor Control System by the Tsypkin Locus Method,"IEEE Trans. on Nuclear Science,

Vol.NS-20, No.1, pp.649-660, Feb. 1973. 4. K. W. Kan and G. J. Thaler,"High Order

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Practical Method,"Academic Culture Company,

1977.

6. D. P. Atherton,"Nonlinear Control Engi-neering,"Van Nostrand Reinhold, London, 1975.

7. A. I. Mees, "Describing Functions, Circle Criteria and Multiloop Feedback Systems,"

PROC. IEE, Vol.120, No.1, ppl26-130, 1973. 8. N. Ramani and D. P. Atherton,"Frequency

Response Method for Nonlinear Multivari-able Systems,"Canadian Conference on Automatic Control, University of New Brunswick, Fredericton, 1973.

9. S. Shankar and D. P. Atherton,"Graphical Stability Analysis of Nonlinear Multiva-riable Control Systems,"Int. J. Control, Vol.25, pp.375-388, 1977.

10. A. K. El Shakkany and D. P. Atherton, "Computer Graphics Mehtod for Nonlinear Multivariable Systems,"IFAC Computer-Aided Design of Control Systems, pp.157-161, 1979.

11. J. 0. Gray and P. M. Taylor, "Frequency

Responses Method in the Design of Multivariable Nonlinear Feedback Systems," 4-th IFAC, Multivariable Technological Systems, pp.225-232, 1977.

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Aided Design of Multivariable Nonlinear Control Systems using Frequency Domain Techniques,"Automatica, Vol.15, pp.281-297, 1979.

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of Limit Cycle in Multivariable

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Nonlinear Sampled-data Proportional Na-vigation System having Adjustable Para-meters,"J. of Franklin Institute,Vol.321, No.4, pp.203-218, April 1986.

Parameters Calculated Simulated

k 1

k2

|

A1

2 |

A1

A2

0.3142 0.6555 22.4 0.75 0.9329 22.29 0.752 0.941 0.5729 0.1848 22.4 0.75 0.542 22.33 0.756 0.547 0.4958 0.7106 22.4 1.00 1.1147 22.32 1.008 1.124 0.6810 0.3715 22.4 1.00 0.8343 22.28 1.008 0.842 0.8364 1.3206 22.6 1.75 2.0051 22.46 1.768

2.017

1.6226 0.03463 23.4 1.75 0.934 23.27 1.771 0.972 2.0296 0.8124 22.8 2.75 2.115 22.77 2.777 2.143