A method for filtering out the free response from roll records

I

Rninc

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE

TECHNISCHE HOGESCHOOL DELFT

LABORATORIUM VOOR SCHEEPSHVDROMECHANICA

A METHOD FOR FILTERING OUT

THE FREE RESPONSE FROM ROLL RECORDS

Jacek S. Pawlowski

Report no.559 August 1982

Deift University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD DELFT

The Netherlands Phone 015 -786882

1 4 6 7 8 11 16 26 2. 3. 4. 5. 6. 7. 8. 9.

The Outline of The Method.

An Example of Application.

Conclusion.

References.

Appendix 1. A Summary of Discrete Fourier Analysis.

Appendix 2. Fourier Analysis of Response Records.

Appendix 3. Short Description of the Computing Programs and Listings.

Introduction.

When the roll motion is excited by a harmonic external moment the measured values of amplification factors often show con-siderable scatter. Besides, the phenomenon known as beating,

JiJ

, can be observed frequently on the corresponding roll records.

Taking into account that the motion in the roll mode is usually very weakly damped it is plausible to suppose that both the

scatter and the beating are due to the presence of significant free response components in the records. In the present report a method for filtering out the free response components from roll records is presented and illustrated by an example. The method involves the use of three computer programs the listings of which are presented in the Appendix 3.

The Outline of The Method.

It is assumed that the relation between the harmonic exciting moment and the roll response is adequately modeled by the governing differential equation of the weakly damped linear

system of one degree of freedom:

M L +

+ t)c

(4]

&',

with the initial conditions

= .xo

(-=

0 ,

LCov-

'7'

The notation in the above equations is as follows: - displacement

-time

ti - generalised mass

S

with: and: =

('-- x2jd,

-

+

- A

+ ts

tr

U3')

- the amplitude of the generalised exciting force

It is well known, see e.g. _[27 , that the response X(-&) of

the system (1), (2), takes the form:

where and denote respectively so called free

and steady response of the system. In the equation (4):

(5)

and the condition of weak damping is expressed by:

77fl

_C)

The purpose of the analysis of response records of duration

1,

i.e. for:

is to determine the amplitudes A and for several values of C-S) _{which are spread in a region about the frequency (...J,1}

The records for = are considered to be

known as well as the values of and

The analysis is based on the fact that Fourier coefficients are linear functionals on the set of functions fulfilling Dirichlet conditions jnaclosed interval, see e.g. _C31 , and that for a harmonic function of the frequency which is a natural harmonic of the fundamental frequency all out of tune coefficients are equal zero.

Hence by choosing:

T

S

a

(oj

kQ4tz. 'VYi1

_J

represent the free response components of the record for

kk

, and for

kk'

-t- adllcJ

b =

bi.i

with the subscript 1 denoting the free response contribution. It follows from the linearity of the Fourier coefficients that:

(c11

Ck

(A1

and it follows that:

p

-

-for

kk

, with the matrix oi. the left-hand side

being a function of and . The components of the matrix

can be determined from the equations (4) and for a given record j the set of equations (11) with

k2.I..)kkJ

is obtained which can be solved +or _A1 in the sense

of least squares, see e.g. E4J

Although estimates of r and are often known form a

separate experiment another approach is recommended here, since, as it is shown in the example, the estimates may be not accurate enough especially as far as L31 is concerned. If for

simpli-city _Es" is assumed to be equal zero it is

found1

[1] , that:

For

j

'I1z1.

(YL. the equations (13) constitute a set of equations

which again can be solved by the least squares method for

P

and

LQ

, and next can be found from (5)

The method of analysis is characterized as an iterative process

for L=

S

the Fourier coefficients 2-ov

are determined numerically for the records

') _c

the i-th approximations of the amplitudes fk,j are ob-tained from (10) with:

(o (o

=

_o

for j=/112.'b1..-;

fl(t)

/

LL)

the set of equations (13) is solved for I and is found from (5);

are determined from the equations (11)

and to are estimated for each

ikJ

the process can be continued by moving to point b) until sufficient convergence is attained.

3. An Example of Application.

Seventeen runs of a model have been carried out at ± O.2SQ

During each run a harmonic roll exciting moment was applied of

,=1.5I'iwt

and frequency (J

,'fl-

varying between

runs. The values of and

_=QA9Lo

-with f generally defined as:

have been found from a separate experiment.

For every run the values of the record of roll displacement have been discretized at 51 points evenly spaced in the

interval determined according to the relation (9) and

the useful duration of the record. From the discretized values the Fourier coefficients have been computed by means of the program

FR

, see Appendix I

_,

and are shown in Table 1. In Table 2 these results are summarized by giving the values of

oJ.j

in degrees and radians. The primes indicate

that for the analysis of the records the time origin was chosen such that in general

E" 0

in (1). Hence with the exciting force of the form:

for the j-th run, tX (4) can be expressed as:

S

S

tCs+&

and by comparison with (4) it follows that:

c= Vco(E-*&1

i3=-Y

E'+j'

and

y

(4)

(Q

The values of Y are shown in Table 2 and Figure 1. Clearly the relations of the kind (18), (19) apply also to the

coeffi-cients and b

The values of E- have been measured from the records for

cp'

and from them the values of A1 and

are derived according to the formulae (18) by putting E! 0. These values are shown in Table 3. The last step can be avoided if the time origin for the analysis of each record is chosen so

that E,/*

₀

for (YL

(o

The derived values and

have been used for estimating the values of 1' _and from

the set of equations (13) and equation (5) To this end the program AN2. has been applied and the output is shown in

Table 4a. Taking into account the formula. (5) it is found that:

p (o

It should be noticed that this values differ considerably from those estimated independently. The difference in the value of is of particular importance for the filtering out of freee response components.

The values

r

and the coefficients

°k

j2'ç

for

each run supply sufficient data for the evaluation of the free response components. The necessary computations have been car-ries out by means of the program AN'I. The input data is shown in Table 5, compare with Appendix 2, and in Table 6 . The re-suits are presented in Table6 , which in particular contains

I / (4

the values of 0... and _A1 * as the free response Fourier

4k

coefficients for k=k1. These values are listed in Table 1 ,

A / (t /(4 'I (4

shown in Figure

1,

against the values of y(o

At this stage the process of deriving not-primed values is repeated with

a

values smoothed according to a drawing.

% (-1\

The calculated values of and are listed in

Table3

rl(1) (4)

These values have been used for computing I and L)1 the results are presented in Table 4b. It is found that:

L. _Cz4

r

11

O.5Zo

In comparison with the values in (20), shows sharp

difference with P , another iteration should be carried out in order to check if the convergence of the

P

sequence has been attained.

It should be pointed out that the application of the method has been hindered in this example due to the inappropriate choice of time origins in the analysis of the records, which

was made by mistake and resulted in

0

for most of the

records. This led to the necessity of employing

E

values estimated from the records. If the proper choice of time

origins is made the process in consecutive iterations gives

L)

approximations of both and values.

4. Conclusion.

OThe

example of the analysis of forced roll records presented

above shows that filtering out of free response components may have important influence upon the form of the observed curve of amplification factors, see Figure 4

The scatter of the points on the curve can be eliminated and the character of the curve, especially in the vicinity of the resonance frequency, can be revealed.

It seems that the method applied can produce reliable esti-mates of the resonance frequency and the damping coefficient.

A refinement of the method is possible by considering the damping coefficient as dependent upon the frequency of exci-tation.

The application of the method may have favourable influence upon the effctiveness of the comparison of measured roll characteristics with their theoretically predicted values. Hence the application of the method is recommended, with

S

References.

[i] Frank S. Crawford, Jr.,

"Waves. Berkeley physics course-volume 3, McGraw-Hill

1965.

2] _{Walter C. Hurty, Moshe F. Rubinstein,}

"Dynamics of Structures", Prentice Hall, 1964.

O[3]

Cornelius Lanczos,

"Applied Analysis", Prentice Hall, Englewood Cliffs N.J., 1956.

[41

Cornelius Lanczos,

"Linear Differential Operators", D. van Nostrand Company, London 1961.

Appendix 1.

ASunhrnary of Discrete Fourier Analysis.

We assume a function -

(x)

fulfilling the Dirichiet condi-tions,[fl ,

to be given on the interval Ke<-flfl>

of

the independent variable. This function is decomposed into its even and odd parts respectively by:

.i)

[f( -

(-1

on

S.

It is known that under the above mentioned conditions the

func-tions and can be represented by the series:

Q,0

t

a,1co,)c -

-L

±

with:

cn--Lc

oL

For further derivations the variable )( is normalized by

the relation

Xtt)

which gives:

t<11?

fo

and the functions 1L)(

,h C

are considered

as functions of

_b

without changing their names.

+

(;';

[f(i

= Q + -1- Q

c,Z1Ti

..

k (

₌

L/YL'1R t bpwL 2t +..

1kk

. N

ak=

-

---rn-N N for

k

Q,'L1

-. NJ with the2 indicating that the

first and the last term of the sum should be taken with the coefficient

_4.

The formulae (1O.4)can be considered as providing approximations of the formulae

_(8.'fland

it can be shown

,CIJ

, that the finite series:

(=

4a0-t- O1C,'o( +

1-constitute the least squares approximations of

) ('lOX)

9(

and

hJ±)

Let be given on equidistant points:

Approximated in the sense of the trapezoidal rule the inte-grals (8.1) take the form:

respectively on the discrete set of points O

Fourier Analysis of Response Records.

In order to apply the relations presented in the Appendix 1 to the analysis of response records, the time variable of the records is normalized according to the relation:

for the j-th record. The ecuation (9) gives:

and similarly the definition:

(L +Lj

is introduced, with:

6 (o

(3.2:

Hence, in terms of the normalized time variable and the notation introduced by (9) and (3) the j-th response can

be expressed as:

Xj(

x1(')

_L)(Z(') I

(.2.)

with:

vç

okT11 +

I

where:

Defining and as the even and oid parts

respectively of the steady response , it is found

imme-diately that: 4

S

S

-Hence, following (3.1.) 'I kj 0

bkJ

0

and it is found that:

o

jø

kk

k=k1

o

kk

1

f-

k=k.

It should be emphasized that the same result follows from the discrete formulae (10.1), since it can be proved,[31, that for

'

A(kc

= 0.

N

çç

N

The even and odd parts of the free oscillation component take the form:

f(-

G

[A1(L

_(1.2)

LYtTt] *

C1b L1

kL

S

it is found that:

kji

=

(4Lk

&j

[

(o.z5)

+(L1*kALj

+ A t

_--I!

LO.25jZ

_(L-k+AL

(-o.z5

t

TrUJ +kAL

-1-S

-

+

[

irCLç-k AL)

1-(05&IL -k+Aj)

(-o.a5

±

-t _J(

O25G-1)]

'1tLL

-t

4 (4j+k

_{-Q25G L}

(osj)

_z(LkL

'1 Tr(t1

-Fk+AL-J)

(o.z

fl

-+

(L tkL

iT (L-k-ALJ

-

(o.zsj] o4LI

)

Hence form the formulae (8.1.) and:

S

°-'

--'<

and:

(0.z5

-k

Gj +

m(LI ±k+L)

P(0

*

r(L -k+Lfl

O.Z5 G--

i(112(

L-k+&Lj

-

-t

'1 ± L1 +k

-0.25

j

_L (o.zs

t

_1

-kLj

( -o.5

-r

-

(0.Q5

1 COT

bk

Al

t(4)Li+k{

L + (45.z) '1

r(L -k+&L

(L -k*

ALJ1

-F

S

S

From the formulae (9.2.), (14.2.), (15.2.) and the relations:

a1

=

bk

Appendix 3.

Short Description of the Computing Programs and Listings.

Program FR2.

Computs Fourier coefficients for a discretized recrod and is based on the formulae (5.1.), (6.1.), (9.1.) and (10.1.) of the Appendix 1.

Input:

First line (card) : NRUN, IC, N, M, in format 13, 3X, Ii, 2X, 12, 2X, 12. NRUN - the ordinal number of the run

IC=Ø- control parameter

N - 2N+1 is the number of evenly spaced reference points on

tle time axis of the record, 2N+11'1

M - the number of harmonics for which the Fourier coefficients are to be computed,

following ZN +1 lines (cards) : SFNT(I) in format F8.3 , in each line,

SFNT(I) - values of the recorded function at the points, from left to right (increasing time) with the order

pre-served, iLz,tJi

Output:

Apart form checking output which is selfexplanatory, M lines,

each containing the value of k1ct of the formulae

(10.1.) for

ko1,

.-. ti , in the format:

12, 2X, F8.3, 2X, F8.3. Program AN1.

Computes free response amplitudes and Fourier coefficients on the basis of the formulae (14.2.), (13.2.) and (11).

Input:

first line (card) : Li, M, DEL, G, in format 12, 12, F9.5, F9.5 Li = lj of the formulae (2.2)

M - the number of harmonics used for the computation

DEL=lj

of the formulae (2.2.)

G = Gj of the formulae (6.2.)

S

_{KN(I) - the ordinal number of the harmonic, according to}in each line,

the output of FR2,

₀

<T k1

r cu

L odc

B(I) = 4,

T

the values of the corresponding Fourier coefficients, according to the output of FR2, I

= 1,2,..,

M

Output:

Apart from the checking output which is selfexplanatory, first line: Al, Bl in format F9,5, F9.5,

Pi=

3'l=

following 26 lines, in each line k'1 A1(KL

in format 12, 2X, F9.5, 2X, F9.5 for

kQ111.-K = k the number of the harmonic

M(v=

_-ki

1(v _kj1

3) Program AN2

r-1 (

\-Computes the values of I and according to the

formulae (13) Input:

First line (card) : M in format I2_

M - the number of runs used for the computation following M lines, in each line:

F(I), A(I), B(I) in format F6.3, F9.5, F9.5, for

F(I) according to the formulae (9),

in,1,

A(I) = A., according to the formula (lO)(Ej=O)

B(I) = B., according to the formula (10) in

Output:

Apart from checking output which is selfexplanatory the program

prints out the values of

r

1in and ,

in[zl,

.t1)S'?I' ) i:'

NJ. A A

(AH)Vw'1

ucc

Li., r . i, ,, .j I )

ca

IL' Sl)A.:i)1;i. VWJJS 1VO1J..dO

J 00&)

3 09&00

( . 31INLLNU:i Li::,

0V00

cr .t i>

000

.1 + N : t) 1 00

0000

06.o0

1 4 -UU'

jj,

I 3 J.. JHj, ) UdvOL 3

t00

'S-.4-! . (_ H.1) H$ I IL) J i41) .dH.1 H I ii V.I'1U

' /

')iY vU' o

o>too

: .;:<.

9) u_i 4ft

((1 )11d:)4

1((

o:oc

IilN.i

hh./UV.3i

N I0'c'Oa

0)00

( ..L

9) ::llxlm

08:C0

V.t.t'il Nu::1:)Nil.

j ooe

3 (9O0

L I I " 1. 1 1V4 4U

-'i..L '

IW 1 )

f'iII Hi

.

I L''tI

') (.ICJ

f)

i.cI i 4 -J.., -) 1 j .jti

i 4"ilIi

H. i ' .I. '14 43_i 3 I, ' ._ f. il) ' 11) }i I' 2 / L5Ui) d l.'il

i_-(-3 I-3 I13 u

3'3H-Lu di

i ''Irl H....

ii ''Ui

Li U'i

'l

3I 13 ,' ,, 3,2i ,.',. .d1 I'1

(0) 3.:E

0E300

( I) ' 1

1 3);

l 3J 1 IV . :JflN:f Or dU.LS

00C

o.00

J LL& I' -1

';.i

Ifl i i. ' ) '

''

ii',.) dU.LS

00c)

081.00 u ) I'. -1 I )

u'c

> -, I 3.) .di u JflNX .LNu; (H 09 1:00 dO_IS

O00

i ) u. x 0t 3 1)J

ji 1')

-j ..'. j.(jl

000

01 E00 ,

,I1N' I

-3 I I ),H U,' ) 3 J11 I i1.'1111V-I 1l IJ13 I

J 'JÔJL'

tjl.ü..

, 5c' /

9 cYa 3 , j

j3( (

.1. -1-( ') -IJ 3 ( -1

'''

p41)3 I I -"i -1 -f ,fl,OU U ( L' 1L 3. 4 Iv v I,' IJ')J 1J) I 3 J ,'3,itl1 ç9a1.i1j ( I ( f ) JJ_, .4_ ' 3 ' 3 .. -' 11,11/1 1 ( 3';

J1t',WIU

LL1L' 1 I , Ij3'V i1 I'II ) 2 . 3 J . ('' Li 113:1 N

.uo.s/ 0000

1)3 1 J 14 / (; 3. LiL' I JI 3 .1 I I'til) 3f1' -' 'i -3.3)3 .. -I I

-/

lj )iji_i. 33.)

'.

ii:iu:

ooc' C

C' I'OhF T f. uHF TF b flF F IC' 11- tTc C 1 C HFF r -tt N

0Oc''0 C

y) C

if 'tTFC 't1tF 1

f. FF C' TFHT 00A60

0:)80

9-Tj c_Ufl

r , 007C'O ) (B :t: ) j

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11

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/'4(J 1 r s(\1 1 , .,

00750

5TOP

00760

00770 C

oo:; c: CL1F

r HF

T( h" ('1 f. 1: H r 008:10

T=FLflT(N)

00820

:uox 2

i1'

0080

E=Fi..UT < ).-1 1' 00850 81C':::1 . 693::BTN C TFT;) /TETA u :r ' : :

00820

Vfl::VT.ST.G

) -'i

1 tiNi ' 008.90 V C 1 ) :0 0 Ut ' tl ' 1 0091.0 RETURN

00920

0090 C

009Y) C

UPF OLITTI'F CflFF -U ' I-U Hi , , r 1 11 ( 1 r tt0HB -L ( ' .) ' -(L T

00980

T:tt)ft.E(F!.0i(N)

00990

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t'

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'fl0i (H

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0100

t1020J=2 'N 01.040

tiBiFCFi(cT(J1 ) )

01050

TETF'I.SW/T

' I )6') (

'

1 4 1! 11 >' flt t-' I TTA / (T

-(fl+V'i)<ii'ITr;

01. 010 20 C'U:1NT :':NUE 01090 ( T ..( I 011.00 0111 0 10 (.:ONTINHE 01. 1.20 (:1) s1o* (1) 11 0 . fl. 01:140 RETURN 01150 END 01 1. o i11;c' 01120 415 0 25 26 0S) ./*. <)840 7/ EN.:i o :iTc

(XN0i.X ilv)'Dii)':

iic

O"0O

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003O

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t'/'7

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'I'

0:. ic/

' ::: o

000

(9)..LIM

0.00

131J'kc I

6',j

3 fl )NS ::93 kJ'zQ( '

iJu i

lJ

9).dIIdM

3

(cO

V..VU

IN1d

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Oi0()

0 o8.O0

0 , I Li ..L Ul t

j

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I'!

3 U(c Ucj .''(.4J tdIc/ 1) i1Ii) 1.1.1 ri. I'c IIV-c 1 -I-i 1 III IH-'

d..1J

) '

3 t'

''.i.V

i'. SaV3

J OJOO

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C) I0O

; c) '4cc V.L'"u c ii..L H1 c il ic/ H 1 -t

-,)

c ) '4h <U V r411 '-c I -'4 -{

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i c ,'4)61

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r{ 6 6()4 /

lt)c'1lJhI!i

1'

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NJ1N.W1u

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jjJiJ tI133It

)

h''(J')

C) Ott.?OO 3 U9c/OO

)I j-J) I

4I'fl I UHv 3

Ic((iI,j

::i.1l 11 IdHV

d''4

:'4:'4.'l S1i1dWU:)

VJUid

0 Ot'OOO * ii

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010

0d>.I

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1' 1 i .f' J

fPi) IN

-(.i "(1)1 ii J H J 1)

S.NUd!4f):)

H.i. 1LRdW0C)

osoo

Q 10 1.j3 ' c Ic ci < 1: : 14

01 000 0 1 0 1 0 01 c 20 01030 0 1 040 01 050 01060 (', C)7C) 0108 c 01090 C) 1 1 00 01110 011 20 01 1 30 01 :1 40 0 11 o 011 60 011 70 011 50 011 90 0120 0 WF: I TE. (6 ...19) WRXTE(6:z :1.29) DO 9C'

i.::I.26

:' .-ALl.. ocm iC A:l.. :1. 'f'.L :

EI

Ii (1 , 1 7 1

+t

r )' 1 CALL r'Et'FN (A 1.. 1 ORi. I ) ' " ) ' U ' 1 1 ; ?A 1 1 +jt I ( / " i

' 2'

IF(T+NF:1) (ft

Iii 80

2 : :i: ) ./FL.OA1 C 2 80

C.: U NT I N U E

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8B7 C .1) SNUI. C

C I ) (If. I A, 1 49 C & ' '-J-9 0 (.: 0 NT I N LIE 1 1

f'flI'fri, TIli ")HEE

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1-uhf. I '9 F flf. Pi,'i ç 1 1, ) iH I t1 I I-flI-N(IT(1

t2,2/L"

' ' L9 WR:iIE(6, 169) S TO P N Li C0EFF :c C:1. E:ii'1s,' ) 00 600 IF(R.NF4(s) L:(i 10 60 006 :i. 0 c.t.. J:t cR 00620 co 1 C.i

:o

00)

60 C. ON 1 x u E 006 1 0 CALL. ):'EVEN((I. : ( 00 A :30 CONT X N(JE 00660 WF:TE(6, :1.69) 7() WF:TTE(6. 19) 006R0 ti 0 :1. 00 I 1 fri 0 C) A 90

I fl

I -1 ' ) A ' ( I 2 00700 :1. ("0 CONT :NUE 00 ? 2 '9 159 F if t'lc1 1 3I'') I, Ii '1, ,

FORfriAI( 1>$Hri irF::t>:/

007)

-9

1- 0" Miii (j I 00740 c.

007) c:.

00760 C

CONFUTES FREE RESPONSE ,mF'I...

00770 C

00780 DO 70 :1 .1 M 00790 OR C .0 1 ) ( t: i. 00800 OF: C I 2....c' C ,( 2) 0081 0 ;'o CONT:(HLIF:: 0 OS 20 CALL

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36 THF F1JNr.T rON LA1.(1FS i 31.. '. 2t. 2é rFIE FUNCTION Vf1. UF S

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Tab2.

S

(0) Nrun scale f. k. a . b a . b . . a b . y. j j

kj

k.j

kj

kJJ

kj

j 1/sec mm m deg deg rad rad deg 229 0.1 .700 7 -21.489 - .942 -2.149 - .0942 - .0375 - .0016 2.150 230 0.1 .700 5 -20.194 7.498 -2.019 .7498 - .0352 .0131 2.154 231 0.2 .602 6 -21.524 -4.726 -4.305 - .9452 - .0751 - .0165 4.408 232 0.2 .597 6 -21.588 -4.986 -4.318 - .9972 - .0754 - .0174 4.432 233 0.2 .552 6 -61.827 -4.964 -12 .365 - .9928 - .2158 - .0173 12.405 234 0.4 .541 6 -33.790 -6.726 -13.516 -2.6904 - .2359 - .0470 13.781 308 0.4 .525 5 49.896 -1.242 19.958 - .4968 .3483 - .0087 19.964 309 0.4 .527 5 52.210 1.006 20.884 .4024 .3645 .0070 20.888 310 0.4 .498 5 43.616 4.171 17.446 1.6684 .3045 .0291 17.526 311 0.4 .500 5 45.085 1.460 18.034 .5840 .3148 .0102 18.043 312 0.4 .472 5 30.860 -0.521 12.344 - .2084 .2154 - .0036 12.345 313 0.4 .473 5 32.036 4.813 12.814 1.9252 .2237 .0336 12.958 315 0.2 .449 5 45.995 0.345 9.199 .0690 .1606 .0012 9.199 316 0.2 .399 4 52.512 2.511 5.251 .2511 .0917 .0044 5.257 317 0.1 .399 4 54.068 8.443 5.407 .8443 .0937 .0147 5.472 318 0.1 .300 3 30.233 0.123 3.023 .0123 .0528 .0002 3.023 319 0.1 .300 3 30.921 0.527 3.092 .0527 .0540 .0009 3.093

a

Table 3.

Nrun

f.

J

1/sec

E. J

deg

rad

J

rad

J

rad

J

229

.700

176.4

.03753

-.03746

.00240

230

.700

158.8

.03756

-.03502

.01358

231

.602

164.7

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-.07416

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232

.597

174.1

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-.07697

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233

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157.0

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-.19929

.08459

234

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134.4

.24050

-.16827

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308

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96.4

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-.03884

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309

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85.4

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310

.497

53.8

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.18067

.24685

311

.500

45.0

.3150

.22274

.22274

312

.472

23.8

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.19708

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313

.473

18.7

.2262

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.07252

315

.449

22.6

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.14827

.06172

316

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11.6

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.08995

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317

.399

8.6

.09551

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318

.300

0.0

.05277

.05277

.00000

319

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6.5

.05398

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S

S

a

a

Table 4 a.

.C)A Tn fri F .: :1. C) .. 'C)'.)):+O.) I .) ' tI1 ft( L

)tjfl(

3 0 .. éOYJi+00

) ryj '

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9/:'I-00

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C) :9 44 i-01

i"i1+(i

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0 .5630).t-01

' 1 i (hf. C pELt

Table 4 b.

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Table 5.

Nrun

f.

J

1/sec

k.

J T.J

sec

G. J

1.

J

Al.

J M.J 229

.700

7

10.00

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5

.217

10 230

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5

7.14

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3

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16 231

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6

9.97

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5

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12 232

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6

10.50

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5

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16 233

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6

10.87

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5

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18 234

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6

11.10

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5

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8 308

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5

9.52

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4

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16 309

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5

9.49

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4

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12 310

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5

10.04

.6734

5

.238

14 311

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5

10.00

.6708

5

.217

14 312

.472

5

10.59

.7103

5

.525

12 313

.473

5

10.57

.7090

5

.514

18 315

.449

5

11.13

.7465

5

.807

14 316

.399

4

10.03

.6728

5

.233

14 317

.399

4

10.03

.6728

5

.233

16 318

.300

3

10.00

.6708

5

.217

20 319

.300

3

10.00

.6708

5

.217

16

p

p

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0.

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