Imperfections measurements of a perfect shell with specially designed equipment (UNIVIMP)

'__

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r

-,

I~ .

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!

Imperfections Measurements of a

Perfect Shell with Speciallv Designed

Equipment (UNIVIMP)

Bibliotheek TU Delft

" "1111111

C 3021910

Series 05: Aerospace Structures

Imperfections Measurements of a

Perfect Shell with Specially

Designed Equipment (UNIVIMP)

L.

Gunawan

Published and distributed by:

Delft University Press

Mekelweg 4

2628

CD Delft

The Netherlands

Telephone

+31 (0)152783254

Fax

+31 (0)15278 1661

e-mail: DUP@DUP.TUDelft.NL

by order of:

Faculty of Aerospace Engineering

Delft University of Technology

Kluyverweg

1

P.O

.

Box

5058

2600 GB

Delft

The Netherlands

Telephone

+31 (0)15278 1455

Fax

+31 (0)152781822

e-mail: Secretariaat@LR

.

TUDelft.NL

website: http://www.lr.tudelft.nl

Cover:

Aerospace Design Studio,

66.5 x 45.5

cm

,

by:

Fer Hakkaart, Dullenbakkersteeg

3

,

2312

HP Le

i

den

,

The Netherlands

Tel.

+ 31 (0)71 512 67 25

90-407-1586-6

Copyright

©

1998

by Faculty of Aerospace Engineering

All rights reserved

.

No part of the material protected by this copyright notice may be

reproduced or utilized in any form or by any means, electron ic or

mechanical

,

including photocopying, recording or by any information storage

and retrieval system, without written permission from the publisher: Delft

University Press

.

Contents

1

Introduction

1

2

The Shells

1

3

The Imperfections Models

1

4 The Imperfections Measurements

2

5

The Data Processing and Reduction

4

6

The Measurement Results and Discussions

6

7 Conclusions

7

References

7

List of Figures

1

The Sectional View

of the Shell Assembly

8

2

The Measurement set

ups

. . .

9

3

Sketch of the test

set

up

. . . .

. . .

.

9

4

Flowchart of Data Reduction Process

10

5

The parameters of the

measurements

11

6

3d plot of

liupf01c.

12

7

3d

plot of

liupf03c

.

12

8

3d

plot of

liupf05c

.

13

9

3d

plot of

liupf07c.

13

10

3d plot of

liupf09c .

14

11

3d plot

of

liupf11c .

14

13

3d

plot

Of

l

liUPfOif

.

16

14

3d

plot

of

liupf03f

.

16

1

5

3d plot

of

l

liupf05f

.

17

16

3d

plot

of

!

liUPf07f

.

17

17

3d plot of

liupf09f

.

18

18

3d p

l

ot of

l

liupflif

.

18

19

The eosin

d

Fourier series

eoeffie

ient

s

of

liupfOl

vs

k

31

20

The

eosin

~

Fouri

er

series

eoeffie

ients

of

li

upfO

1

vs

I .

31

21

The

s

in

e

f\ourier

se

ries

eoeffieients of

liupfOl

vs

k

..

32

22

The sine

1

urier

series

eoefficients of

liupfOl

vs

I ..

32

23

The eosine

l

Fourier

se

ries

eoeffie

ients

of

liupf03

vs

k

33

24

The eosine

l

Fourier series eoeffieients of

liupf03

vs

I .

33

25

The sine Fburier

ser

ies

eoeffieients of

li

upf0

3 vs

k ..

34

26

The sine F

b

urier

ser

i

es eoeffieients of

li

upf03

vs I

. .

34

27

The eosine

i

Fourier series eoeffieients of

liupf05

vs

k

35

28

The eosine

l

Fourier

series eoeffieients of

liupf05

vs

I .

35

29

The sine F<Durier

ser

ies

eoeffieients of

liupf05

vs

k .

.

36

I

30

The sine Fljlurier

series

eoeffieients of

liupf05

vs

I

36

31

The eosine

\

Fouri

er

series

eoeffieients of

liupf07

vs

k

37

32

The

eosine

Fouri

er

series coeffieients of

liupf07

vs

I

.

37

33

Th, ,in, Fri"

,,,i~

,oeffi,i,nt,

of

hupf07"

k .

38

34

The

sine F

urier

series eoeffieients of

liupf07

vs

I

..

38

35

The eosine

r

ourier series eoeffieients of

liupf09

vs

k

39

36

The eOSine{Ourier series eoeffieients of

liupf09

vs

I

.

39

:37

The

sine F

urier

se

ries

eoeffieients of

liupf09

vs

k

..

40

38

The sine F

d

urier

ser

ies

eoeffieients of

liupf0

9 vs

I

.

40

I

39

The eosine lFourier series

eoeffie

ients

of

li

upf

11

vs

k

41

I

40

The eosine fourier series eoeffieients of

li upf 11

vs

I .

4

1

41

The sine F

d

urier

series eoeffieients of

11 upf 11

vs

k

.

42

42

Th, ,in,

F

~

mi"

",i" eo,ffi,i,nt,

of

hupf11

"

1

.

42

43

3d

plot

of li

upfOlk

.

. . . .

.

. . . ..

43

44

3d plot of

1

iupf03k.

43

45

3d

plot

of

1

iupf05k

.

44

46

3d plot

of

1

iupf07k

.

44

47

3d p

l

ot of

l

l

iupf09k

.

45

48

3d

plot

of

l

t

UPfllk

. .

.

. .

.

. . . . .

.

. .

. . .

. . . .

.

45

49

The

eompa~isons

of

the

axisymmetrie imperfeetion

mode

s.

46

50

The

compa

rison

s of

i

mpe

rfecti

on modes

with

1=

1

46

51

The

compa

rison

s

of

imperfection m

odes wit

h

1=

2

47

52

The

co

mpari

sons

of imperfection modes with

1=3

4

7

53

Th

e comp

arison

s

of imperfection modes with

1=

4

48

54

Th

e compa

rison

s of

imp

e

rfe

ct

ion

moeles

with

1=

5

48

List of Tables

1

Wal! thickness eli

str

ibution

of shel!liupfOl

8

2

Ak.l

a

nd

Bk.!

of

liupfOl

19

3

C

k•l

anel

Dk.l

of

liupfOl

2

0

4

Ak.l

anel

Bk.l

of

liupf03

21

5

Ck .l

anel

Dk,l

of liupf03

22

6

,4k.l

a

nel

Bk,l

of

liupf05

23

7

C

k •l

anel

Dk.l

of

liupf05

24

8

Ak.l

a

nel

Bk,l

of

liupf07

25

9

C

k .l

anel

Dk,l

of

liupf07

26

10

Ak.l

a

nel

Bk.l

of

liupf09

2

7

11

C

k .!

and

Dk.l

of

liupf09

28

1

2

Ak.l

an

el

Bk.!

of

liupf11

29

Acknow ledgement

The

work

presented in

this

report was

carried out

by

the author

as

a part of

the nonlinear

vi-bration

tests of cylindrical shells. The supports

of lndonesian Aircraft lndustry

(PT.lPTN),

Bandung lnstitute

of Technology

(ITB), and Faculty

of

Aerospace Engineering

of

the Delft

University of

Technology are

gratefully

acknowledged.

The author

wishes

to express

his

thanks to Mr.A.W.H

.

Klompé for his assistance

and

advice

during the

measurements and the preparation

of th is report, to Mr.J.H

.

Weerheim

and

Mr.H.v.d.Hoek for

their

help during

the

preparation

of the

test

setup.

The

author

also

wishes to thank Prof.R.J

.

Zwaan for his careful reading

and

comments

on

the final version

of

this

report

.

1

Introduction

In

conjunction with the validation of the

nonlinear

vibration analysis of imperfect cylindrical

shells carried out

by Liu

[4], a vibration experiment

is

being performed. For this purpose

four cylindrical shells as the test

objects

were

made,

of which

two

shells were

designed

with

known

initial axisymmetric

imperfections

and the ot

h

er

two with small

imperfections

.

Initial imperfections of each shell, which wil! be correlated to

its

vibration

cha

racteristics,

were verified with

measur

ements.

This report presents the imperfection measurements of the

first

perfect shell. There

we re

six

measurements

performed

on

thi

s

shell for different

con

ditions

to check

the

repro-duceability of the

results.

The

results

are

ca

lled liupfOl

,

liupf03

,

liupf05, liupf07,

liupf09

,

and

liupfl1.

2

The Shells

The shells were made by machining seamless aluminium tubes to the specified dimensions.

Each shel! was manufactured on a mandrel with the aid of an accurately machined steel

mould.

In

the experiments the shell ends we re encased in aluminium end rings to enable the

application of the axial compressive

forces.

The rings we re attached to the shell

by

placing

the

shell

end into

the

circular channel of the

ring

and filling the channel with melting

ce

rrobend

which solidifies when cooling down. A

firm

attachement was obtained since the

cerrobend

expands

slightly during solidification.

Figur

e

1

shows

the sectional view of this

assembly. The

she

ll,

af ter

it

s

imperfection was

verified

with measurements, was

ready

for

the vibration tests.

The

above procedure will be

applied

to the other shelJs one

af

ter another. This

is

because only one set

of

two

e

nd

rings

i

s avai

lable

and once the shell has been

assembied

and

its

imperfections

has

been measured, it win be

k

ept

in the same

condit

ions for

the

vibration tests.

3

The Imperfections Models

The

imperfections

of

the

shells considered are the deviations

of

the

radius

with

respect

to the

ideal

one

.

The

imperfections

can be

represented

as double Fourier series

in

c

ircumferential

direction

(y

or ()

=

y /

R)

with a

half-wav

e

cosine shape

in

the axial

direction (x)

M

i7rX

M N

k7rx

(

ly

ly)

w

=

t ~

AiO

cos

L

+

t

E

~

cos L

Aki

cos

R

+

Bkl

sin

R

(1)

or with a half-wave

sine

shape in the axial direction

M .

i7rx

M N .

k7rx (

ly

.

ly)

with:

t

R

L

k

I

A,B

,

C,D

shell wall thickness

shell

radius

shell

length

axial half-wave number of the

axisymmetric

modes

(l

=

0)

axial half-wave number of the asymmetrie modes

(I

=I-

0

circumferential

full-wave number

nondimensionalized double Fourier

series

coefficients

The coefficients

A, B,

C,

and

D

can be

calculated

from the imperfection distributions

Jij

by using Fourier analysis

as shown

in [2].

4

The Imperfections Measurements

The Measurements procedure

The imperfection measurement was done

by

using the

UNIVIMP

,

a

machine specially

de-signed

to measure the initial imperfections

and

the deflections of thin-walled cylindrical

shells

under compressive loads. Figure 2.a

shows

the UNIVIMP

configured

for the

mea-surement of the initial imperfections of the shell. This instrument mainly consists of a base,

a rotating platform on which the shell is mounted, and

a vertical

column. An LVDT, which

is used as

a shape

transducer, is attached

on a

carriage

that

can

travel along

the

vertical

column

.

In the

operation

the

stylus

of

the

LVDT touches

the shell sUl·face.

The output of the

LVDT represents the variation in distance between

shell outer

radius at the point touched

by the stylus

and

a reference point. The instrument measures the distribution of this

distance

over the

shell's

surface

by controlling the

axial

position of the

carriage,

rotating

the

shell, and

recording the output of the LVDT

at several

points. The

scanning

starts at

the lowest

axial

position

on the shell

surface. The measurements

were carried

out

at

200

points for one shell revolution. Then

the carriage

with the shape transducer was

moved

upward 2 mm

and

at this new axial position

another series

of 200 points were measured.

The procedure was repeated until the upper position on

the shell

was reached.

The shell

was

measured for six different

conditions

by using two set ups. The first

setup,

shown in figure 2.a, was the typical UNIVIMP

configuration

for imperfections measurement.

The second setup was designed such that it can easily be modified for the vibration tests.

For this purpose a capacitive displacement transducer and its cable were installed inside

the shell.

It

was mounted to a rod that

was

positioned

along

the longitudinal axis of the

shell,

fixed to the lower

end

ring via an

aluminium

plate,

and constrained

to move in radial

direction by the upper

end ring via another aluminium

plate. Figure

2.b

shows

the shell

with

the

transducer installed inside without the plate

at

the upper

end

ring.

A hexagonal ring platform was attached on the

UNIVIMP.

To provide room for the

platform, the vertical

column

had to be

shifted

to its most

outer

position on the base

and

consequently

longer rods

were

needed

to

hold the LVDT's

such

that

they could

operate in

their ideal working range. The shell was placed on the UNIVIMP with a long filler ring in

between. Top parts which consist of a short filler ring, a top bearing mechanism, and some

wooden discs could be installed on top of the shell to enable the application ofaxial loads

on the shel!. Figure 2.c shows this setup with the shell top parts installed. For the vibration

tests this setup can be used by removing all the LVDT's, moving the vertical column from

the base to the hexagonal platform, and installing another dis placement transducer on the

carriage of the vertical column. Hence the shell can be left untouched and consistency of

the measured imperfections can be maintained.

The sequence of the measurements is shown in figure 3

.

The first measurement which

is referred to as liupfOl was the only one carried out by using the first setup. The second

measurement was performed by using the second setup without the shell top parts is referred

to as liupf03. The third measurement liupf05 was carried out with the top parts installed.

Although in the latter case the measured data contained the e1astic deformations of the

shell due to the top parts, the measured data was treated as the initial imperfections of the

shel!.

Since the cable of the capacitive transducer inside the shell was forgotten to be installed,

the shell had to be removed from the UNIVIMP. This opportunity was used to gain some

information about the reproduceability of the measurements. First the top parts we re

removed from the shell and the fourth measurement

,

liupf07, was performed. Then the

shell was removed from the UNIVIMP, the cable of the capacitive transducer was installed,

the shell was placed again on the UNIVIMP, and the fifth measurement, liupf09, was

performed

.

Af ter installing the top parts the sixth measurement

,

liupfll

,

was carried out.

During the installation of the capacitive transducer inside the shell

,

the shell was handled

by gripping the upper and lower end rings. After the installation was completed, the part

of the vertical rod that prot ru des above the aluminium plate of the upper end ring was

used as grip instead of the upper end ring. In this way the shell would be subjected to less

forces and thus the imperfections could be kept from any undesireable change

.

The Data Corrections

The above measurement principle would give error-free data if same ideal conditions we re

fulfilled. They were: (i) the carriage moves following a straight line which is parallel to the

rotation axis; (ii) the cylindrical shell axis is aligned with the rotation axis of the platform;

and (iii) the rotation axis of the platform is fixed in space.

Although this instrument was manufactured with high precision, there were still some

small deviations to the conditions mentioned above. Ta overcome the errors introduced by

these deviations, the following corrections were applied during the measurements:

Column shape correction: The data was corrected for the initial shape of the column

and the unparallelism of the column with respect to the rotation axis of the rotating

platform. The shape and unparallelism of the column were measured before the

imperfection measurement took place (and they we re used during the imperfection

1111) 1 . "==-=' .---L.._--<'.l

-Rigid body motions correction:

The

data was

corrected for

the

rigid

body mot ion

of

the shel! which

is introduced by the misalignment between the

shel! axis and the

rotation

axis.

This rigid body motion

of

the

shell was

measured

at the

beginning

of

the imperfection measurement and

then

used for

correct

i on during the imperfection

measurement.

Periodicity correction:

During

the

rotation

of the

platform

there was also a

small motion

of

the

rotation

axis which causes a smal!

dis

cont

inuity between the first

and the

last data during

scanning

of

the

points

along

eac

h

circle

.

The

larg

est

discontinuity

measured

was

in the order of

2Jlm.

This error was

corrected

by distributing the

discontinuity linearly

to all

data from

that

circle.

The

techni

ca

l details and

the

operation manual of

the UNIVIMP can

be found in

[3].

Another

co

ndition

assumed

in the

measurement was

that the

shell

wal! thickness

IS

constant and

hence the

variation

of the

outer

radius

is equa

l

to that of the mid-radius.

Since the measurement data was not corre

c

ted for the

error

due to wal! thickness variation,

the reliability of the measur

e

d data depends

on

the actual wall thickness distribution of the

shel!.

From the wall thickn

ess

data measured

at

40 points

shown

in

table

1

,

the

maximum

variations is

11

J.lm

and the standard deviations is

3.7

Jlm

,

or

respecti

ve

l

y

0.045

and

0.015

times

wall

thickness.

5

The Data Processing and Reduction

To obtain

the

Fourier

coefficients

of the measured imperfections,

a

data reduction program

was used, which was

capable

to process

the

measured imperfection data from al! the

im-perfection

survey

instruments of the Structures and Mat

e

rials

Lab

oratory

including the

UN

IVIMP

.

This program

consists

of

several steps

which perform specific tasks

and creates

an information file about the process

and (a)

data fil

ets)

for use in

(before)

the next step.

The flowchart of the program is

shown

in figur

e

4.

with

the notation

[shellcodeJ

according to

[2]

is to be replaced with th

e

identification

of

the shel!

being

process

e

d

.

All

the created files which

correspond

to a

certain

shell are named

[shellcodeJ

followed by

one

or

a

few

characters.

Since

all

the measurements presented in this report

corresponded

to one

she

ll,

the [shellcodeJ was replaced

with

the identification of the measurement. Hence in

this

r

eport

the

[shellcodeJ

shou

ld

be replaced by the identification

of

the measurements,

i.e.

liupf01

, liupf03

,

liupf05

,

liupf07

,

liupf09, and

liupf1!.

Befor

e

using this program

,

the output file of each measurement process was split into the

numeric data file and the information file which contains all the measurement parameters.

For each measurement

,

they

are

respectively files

[shellcodeJ

and

[shellcodeJ

a

.

The

measurement parameters needed for

the

reduction process

were

filled in a

file

[shellcodeJ var!.

Figures

5 shows

those parameters and their

values

are presented in the

corresponding

tabie.

The first

step

was processing the file

[shellcodeJ

with program xxOO. This program,

af ter asking with which instrument

the

measurement was

carried

out

,

reformats the data

into

standard

forms,

averages

the data

,

and calcu

lat

es

the

imperf

ections with

respect

to

the averaged value. Two

files

are created in this step

,

[shelleode] b-Dut which

contains

the

information

about the

process

and

[shelleode]b

which contains the

processed data

in

the standard

format for data reduction

program.

The

second

step is processing

the file

[shelleode]b with

program

xx01. This program

is

capable

to

correct

the data for

some errors

due to the measurement instrument

inaccu-racies and the shell

structure

such as missing scans,

hatches,

rivets, and extra

plates. This

program can also carry out the rigid body motion correction if it was not done

during

the

measurement. Since the UNIVIMP data

has been

corrected for th

e

instrument

inaccuracies

and no further corrections are needed

,

this step

has been

performed without any correction.

The only act ion taken

has been

removing the

rigid

body

information from

the

data file

.

The outputs are an

information

file

[shelleode] cout

and data

file [shelleode] e.

The third step is processing the

file [shelleode] e

with program

xx02.

This

program

calculates a perfect shell which best-fit the imperfection data by

using the least-squares

method and recalculates the imp

e

rfections with

respect

to this best-fit shel!.

The

output

files

of this

process are the information

file

[shelleodeJf -Dut

and data

file [shelleodeJf

which is ready

for the Fourier

analysis. Figure

12 illustrates

the output variables

of the

best-fit process and the

best-fit results

of all measurements are presented

in

the corresponding

tabie.

Th

e

fourth step

is

done on the file

[shelleode]

f

with the program

xx03

which

de-composes the imperfections

into

the Fourier coefficients A

k

.{, B

k.l'

Ck,t,

and

Dk.

1

of the

equation 1 and 2. The output files are the

information file [shelleode] h_out,

the file

[shelleode] h_e

which contains the half-wave cosine coefficients

Ak.

1

and

Bk

.

/,

and the file

[shelleode]

h~

which contains the half-wave sine coefficients

C

k •1

and

D

k •l •

Program xx04 can

be used

to reconstruct the

imperfections

from the Fourier coefficients

[shellcodeJ he or [shelleodeJ hs. The

output

file which

contains the

reconstructed

im-perfections is [shelleodeJ k.

Files [shelleode] b, [shelleode] e, [shelleodeJf

and

[shelleode] k

can

be

viewed

with a pseudo 3-D graphic program

graph3d. This

program

is

very

useful

to

check

and

justify the

reduction process.

The coefficients of the

double

Fourier series

in

the

files [shelleode] he

and

[shelleode] hs

can be presented as tables b

y

using program

mapprint

and plotted with

respect

to

k

or

1

with program

hargraph.

The axisymmetric coefficients can be plotted separately with

program

axigraph.

The same procedure were performed on data

from the

other measurements.

Reference

[2]

-

-

- - -

---~

6

The Measurement Results and Discussions

Figures

6

,

ï

,

8

,

9,

10,

and 11 show

the

3-D graphs of the

imperfeetions

af ter

proeessed

with

programs xxOO and xx01. These imperfections are

referred

to

the averaged

value

of

the

measured data. Note that there is a dent in the middle

axial and

eireumferential positions

of

the

shell

with

a

depth

of about

0.5

wall thiekness.

The results of

the

best-fit proeess

(xx02) are shown

in figure 12.

It

ean

be

seen that

the

best fit

shell

dimensions for all

eases are very similar, exeept

for

measurement

liupf01.

This shows that the

various

proeesses

applied

to

the shell

during th

e

setup change

from

liupfOl to liupf03 introduce some

smal!

deformations to the shel!. This

ean

be minimized

by keeping the shell

conditions

unehanged

as are

indieated by the

smal!

differences

among

results

of

liupf03

to

liupfl1. Figures 13

,

14

,

15

,

16

,

17

,

and

18

show the 3-D graphs

of

the imperfeetions with

respect

to the best-fit

shell.

The results of the harmonie analysis

of the

imperfeetions (xx03)

are given

in tab les 2

to 13. The Fourier eoeffieients Ak,/

and

Bk,/

refer to the representation

of

the imperfections

with axial half-wave

eosine terms

and

C

k,/

and

Dk,/

refer

to

the axial half-wave sine

terms

representations.

The

amplitudes

of

the

imperfection

modes

,

JAt/

+ Bf

,

/

and

JCf

,

/

+

Df,1

are

plotted

against

k

and

I

in

figures

19

to 42.

The maximum amplitudes of the imperfection modes

oeeur

when the short filler

ring

and the top bearing meehanism are installed

on

the top

of the

shell, liupf05. For axial

half-wave eosine

representations

it

is 0.08

wall thiekness for

k=O and 1=2

.

For axial

half-wave

sine

representations it

is

0.11

wall thiekness

for k=1

and

1=2

.

Figures

43 to 48

show

the

3-D graphs

of the

imperfeetions reconstrueted from the

eal-eulated Fourier

eosine series eoeffieients

with program xx04.

By

eomparing

the 3-D figures

of imperfections af ter best-fit and the

reeonstructed

imperfections for

eaeh

measurement

(i.e.

by

eomparing

figure 43

with

figure

6

,

ete.), it ean be seen

that the

axial

half-waves

and eireumferential

ful!-waves numbers

in the

Fourier

analysis are suffieient to represent

the imperfections of the

shell

.

The dent in the middle

axial

and circumferencial positions can be reeonstructed again

by using the Fourier coeffieients. This means that in the harmonie analysis the influenee of

the dent was spread to all imperfection modes.

Figures 49 to

54 compare the

coefficients of

the eosine axial

half-wave imperfections from

all measurements

.

It

can

be

seen

that

only the

axisymmetric modes

are

mostly influenced

by the

ehange

of measurement configurations. The differences were

significant

between

liupfOl and liupf03

and small

among liupf03

to

liupf11.

Sinee the

vibration tests

will be performed on

vibration

modes with lowest natural

frequencies, i

.

e

.

the

vibration

modes

with axial

half-wave number m=1

and circumferential

wave number n=·5 to 13,

not all

of the imperfection modes

are of

interest. Aecording

to

the theoretical results

of

Liu [4], the imperfection modes of interest

are

then the

eosine

axial

half-wave axisymmetric imperfection modes

with i=2

and the

sine axial

half-wave

asymmetrie imperfeetion modes with

the same

pattern

as the vibration

modes

,

i.e. k=1

and

1=5

to

13. Figure

49

indicates

that th

e

amplitud

e

s of the axis

y

mmetrie imperfection

modes

with

k(

=

i)

=

2

is less

than

0.05

wal! thickness. Figures 22, 26

,

30, 34

,

38,

and 42

indicate

that the amplitudes of the sine axial

half-wave imperfection

modes with k

=

1

and

I

=

5 to 13 are

less

than

0.08

wal! thickness.

Hence this

shell can practical!y

be considered

as a perfect one.

7

Conclusions

An

imperfections

survey on shell

liupfOl

has

been carried out successfully. The

imperfec-tion modes of

interest

,

i.e those with

I

>

2 which

may have

significant

influences

on the

vibration characteristics of the shell

,

have amplitudes less

than

0.04

wal! thickness.

Hence

this

shell

may practically be

considered as a

perfect one.

The reproduceability of the measurements

was shown

by performing

the

measurements

at

different

conditions. As

long

as

the

changes

of shell

conditions were

minor, such

as

adding or removing

sm all masses

on top of the

shel!, the

measurements gave

reproduceable

results

with

slight variations. The

installation

of the

capacitive

transducer inside the shell

which involved

more complex

activity such as

disassembling the

shell from

the UNIVIMP,

carrying the shell,

installing

the transducer, and

reassembling

th

e

shell

to

the

UNIVIMP,

gave some significant changes

in

the

imperfections

characteristic of the shel!.

References

[1]

J.

Arbocz. Shell

buckling

research at

Delft

(1976-1992). Memorandum M-596, Faculty

of

Aerospace Engineering, Delft

University

of Technology, The

Netherlands,

1993.

[2]

A.W.H.Klompé

Ch.Cartalas,

H.J.C.Van

der Hoeven.

Guide to

the data reduction

of

imperfection

surveys

on

circular shells.

Memorandum

M-622, Faculty of

Aerospace

Engineering, Delft

University of Technology,

The

Netherlands

,

1990.

[3]

A.W.H. Klompé. UNIVIMP

,

a UNIVersal

instrument for

the survey of

initial

IMPer-fections

of thin-walled shells. Report

LR-570

,

Faculty

of Aerospace Engineering

,

Delft

University of

Technology,

The Netherlands

,

December 1988.

[4] D.K. Liu.

Nonlinear Vibration Characteristics

of Imperfect

Shells

under

Compressive

Axial

Laad.

Ph.D.thesis,

Faculty of

Aerospace Engineering, Delft

University

of

Tech-nology, The Netherlands, 1988.

Figures and Tables

Detail

of A

_

cerrobend

L

_t

_R

_shell

L2

A

R---J

1

Ro

R

125 rnrn

L2

30 rnrn

L

240

rnrn

Ic

Îrnrn

t

0.25 rnrn

he

4rnrn

Ro

129 rnrn

E

7000 kg/rnrn

2

R·

1

84

rnrn

J-l

0.3

Figure 1: The

Seetional View of the Shell

Assernbly

Shell wallthickness distributions (in

J-lm)

Axial positions

Circurnferential positions in deg.

relative

to lower end

(mm)

0

36

72

108

144 180

216 252

288

324

40

255 247

242

247 247

254

257

254

252

254

80

252 250

248

246

249

255

254

256

253

252

120

260

256

251

252

249

251

257 258

258

261

160

256

255

252

251

249 251

254

255

256

257

200

256

257

256

251

252

250

255

258 250

260

A veraged wallthickness

: 253

J-lrn

Table 1: Wall thickness distribution

of

shellliupf01

(a)

(b)

(c)

Figure 2:

The

UNIVIMP

for imperfection measurements

.

(a) For measurement

liupfOl,

(b) the shell with the capacitive transducer installed

inside,

and

(

c) the setup for

measure-ments

liupf03

to liupfll (c)

Remarks

11

10

1

UNIVIMP base

=

? ?

2

Vertical column

5

5

9

3

Shell

-2

₆

3

-

4

-2

6

3

-

4

2

-

5

4

Lower RBM

LVDT

5

Upper RBM LVDT

6

Shape LVDT

7

Hexagonal platform

8

I

T

l

8

11

71

I

~

₃

-

4

I

1

I

r

l

8

Short filler ring

9

Long

filler

ring

10

Top bearing

mech.

11

Wooden dis cs

(a)

(b)

(c)

Figure 3: The sketch of the test setup

.

(a) For liupfOl

, (

b) for liupf03, liupf07,

JO

Measured

Imperfeetion Data

Create Formatted

Data File

Create Corrected

Data File

Calculate Imperfection Data

w.r.t Best-fit Shell

Ca!Culate Fourier

Harmonie Coefficients

mapprint

[shellcode]hc~+-

__

~

[shellcodeJhs

Reealculate

Imperfection Data

Pseudo 3-D Plot

Tables of Coefficients

Ak,I,Bk,I,Ck,l,

and

Dk,l

Plots of Harmonie

Coefficients

Plots ofAxi-symmetric

Coefficients

Date

D

dA

dB

For harmonie analysis

NAXIAL

of measurement

Designation

(mm) (mm) (mm)

LHA

NCIRCxNAXIAL

15-05-1995

liupf01

285

25

2

118

234

201x118

20-06-1995

liupf03

286

27

4

117

232

201x117

20-06-1995

liupf05

286

27

4

117

232

201x117

22-06-1995

liupf07

286

27

4

117

232

201 x117

23-06-1995

liupf09

286

27

4

117

232

20lx117

23-06-1995

liupfl1

286

27

4

117

232

201x117

For all measurements:

1.

The axial step

during

the measurement is 2 mmo

2.

The dA and

dB

of the measurements liupf03 to liupf11 were aetually the same

as those of liupf01. However, sinee the data of the first axial positions for those

measurements were not used in the harmonie analysis, the eorresponding parameters

presented above are the modified on es for the harmonie analysis.

Figure 6: Imperfections of sheli liupfOl af ter formatting and corrections step

1 i

up-f03c

3

Figure 7: Imperfections of sheli liupf03 aft er formatting and corrections step

Figure 8: Imperfections of shellliupf05 aft er formatting and corrections step

Figure 10: Imperfections of shell liupf09 aft er formatting and corrections step

Figure 11: Imperfections of shellliupfli af ter formatting and corrections step

z

_z'

i" measurod point

~+-1--~+-.,L---I~ Y

yl

Xl X

liupf01

liupf03

liupf05

liupf07

liupf09

liupfl1

Xl

(

in mm) 0

.

021

0

.

000

0

.

001

0

.

000

0.004

0.003

Y

I

(in mm) 0.066

0.060

0

.

057

0.055

0

.

045

0

.

050

El

(in deg)

90.002

89.996

89.997

89.99

8

89.998

89

.

998

E2

(in deg)

90.022

90.019

90

.

178

90.017

90.017

90

.

016

E3

(in deg)

-.010

0.022

0

.

023

0.016

0.012

0

.

027

R

2

(

in mm) 124.980 125.044 12

5

.047

125.032 125.024 125

.

0

5

5

Figure 13: Imperfections of shellliupf01

af

ter the best-fit step

Figure 14: Imperfections of shellliupf03 af ter the best-fit step

E

E

C

13

Figure 15: Imperfections of

liupf05

af ter the best-fit step

1 ; up-f07-f

Figure 17: Imperfections of

liupf09

af

ter the best-fit step

Figure 18: Imperfections of

liupfll

af

ter the best-fit step

-Ak,1

eomponents

-

cos

kI

x

cos

~

\ L= 0 3 4 6 7 8 9 10 11 12 13 14 I" 0 .0001 -.0002 .0509 .0051 -.0381 .0094 -.0162 .0038 -.0390 .0048 -.0033 .0039 .0041 -.0078 -.0140 1= 1 .0034 -.0003 .0129 .0025 -.0474 -.0002 -.0014 -.0173 -.0001 .0062 .0058 -.0183 .0044 .0122 .0038 I" 2 -.0415 .0038 -.0103 .0026 .0065 -.0052 .0141 -.0074 .0145 -.0010 -.0026 .0063 -.0043 .0090 .0218 I- 3 -.0276 -.0016.0 .0009 .0016 .0011 -.0008 -.0013 -.0033 -.0017 -.0074 .0082 .0047 -.0022 -.0003 Iz 4 -.0065 -.0014 -.0023 -.0012 .0029 -.0004 .0008 .0031 .0056 -.0037 .0005 -.0060 -.0026 -.0025 -.0082 (- 5 .0 .0013 .0017 -.0005 .0021 .0006 -.0007 .0 -.0014 .0007 .0029 .0037 .0024 .0004 .0008 (= 6 -.0028 .0002 -.0012 .0004 .0014 .0 .0011 .0013 .0026 .0014 -.0002 -.0023 -.0022 .0038 .0016 (- 7 .0041 -.0007 .0013 .0009 -.0005 .0009 -.0002 -.0008 -.0016 .0017 -.0012 .0021 .0003 -.0012 -.0025 I- 8 .0024 -.0002 -.0007 -.0007 .0014 .0 .0005 -.0006 .0014 -.0005 -.0002 -.0009 .0005 -.0009 .0007 1= 9 -.0008 .0002.0 .0009 .0010 -.0005 .0003 -.0011 -.0006 -.0007 -.0003 .0015 .0010 .0 -.0020 1=10 -.0038 -.0005.0 -.0002 .0009 .0007 .0002 .0011 .0016 -.0005 -.0005 -.0010 -.0017 .0009 .0015 1=11 .0007 .0001 .0003 .0006 -.0004 -.0003 -.0007 -.0004 -.0004 -.0005 -.0001 .0017 .0004 -.0005 .0008 1=12 -.0005 -.0007.0 -.0003 .0003 .0 .0010 .0006 .0006 .0 .0005 -.0012 -.0003 .0005 -.0001 1"13-.0023.0 .0 .0003-.0003-.0005 1=14 .0002 -.0007 -.0006.0 .0007 .0003 .0 -.0009 -.0010 .0 -.0002 .0010 .0 .0005 -.0005 .0010 .0002 .0001 -.0010 .0 -.0001 -.0002 .0002 .0002 1"15 -.0031 .0007 .0006 -.0002 -.0003 .0002 .0 -.0007 .0 .0002 -.0004 . 0006 .0002 -.0005 .0002 1=16 .0015 .0003 -.0007 -.0004 1-17 -.0003 .0005 .0002 -.0004 1-18 -.0014 -.0006 -.0007 -.0004 Iz19 .0026.0 .0002 -.0003 1-20 .0016 -.0001 -.0006 .0004 \ L" 0 2 3 .0007 -.0001 .0 .0 .0007 -.0005 .0004 -. 0006 .0 .0 -.0006 .0003 -.0003 .0001 .0002 .0007 .0 .0003 .0003 . 0 .0002. 0003 -.0006 -.0004 .0004 .0001 .0 -.0005 -.0002 -.0006 -.0001 .0 .0 .0 .0001 -.0007 .0 -.0004 .0004 -.0005

Bk,/

eomponents -

COS

k~x sin ~

4 6 7 8 9 10 11 .0002 -.0003 -.0002 .0 -.0002.0 .0 .0006 .0003 .0002 -.0005 -.0002 .0003 - .0002 .0 12 13 14 I- 0 -.0002 -.0298 -.0501 -.0133 .0499 .0362 .0231 -.0134 .0159 .0361 .0059 -.0019 .0051 -.0028 (- 1 ••• -.0014 -.0036 -.0055 .0044 .0008 .0044 -.0063 .0135 -.0094 .0002 -.0077 -.0043 -.0022 .0059 I-2 ••• .0004 .0044 .0036 -.0041 -.0167 -.0173 -.0072 .0030 -.0012 -.0237 .0016 -.0010 -.0113 .0065 I- 3 ••• -.0006 -.0017 -.0009 .0019 -.0009 -.0004 .0012 -.0012 .0020 .0024 .0067 .0071 -.0044 -.0028 I- 4 ••• -.0037 -.0017 .0015 -.0011 -.0036 -.0029 -.0049 .0084 .0018 .0008 -.0014 -.0008 .0003 -.0033 1= 5 . . . .0019 -.0003 -.0016 .0006.0 .0007 -.0004 -.0024 .0037 -.0016 .0016 -.0013 -.0002 .0001 I" 6 1= 7 ••• -.0003 -.0022 .0013 .0027 -.0013 -.0016 -.0009 ••• .0022 -.0011 .0009 .0013 -.0009 .0003 -.0008 .0040 .0 -.0041 .0004 .0010 -.0003 .0003 .0001 .0008 .0004 .0006 -.0014 -.0011 .0006 I" 8 ••• -.0012 -.0008 .0002 -.0001 -.0007 -.0014 -.0016 .0011 -.0014 -.0002 .0002 .0015 .0011 -.0018 (" 9 ••• .0021 .0004 .0009 .0007 -.0013 -.0006 -.0005 .0003 .0012 -.0003 .0008 .0009 -.0007 .0 (=10 ••• -.0004 -.0010 .0 .0 -.0010 -.0012 -.0005 .0016 .0002 -.0002 .0 .0 -.0017 .0003 1=11 ••• .0025 -.0005 -.0002 .0004 .0 .0004 -.0005 -.0010 .0007 -.0003 -.0007 -.0014 -.0001 .0004 ••• -.0006 -.0004 -.0005 ••• .0035 - .0004 .0005 ••• -.0011 -.0006 .0009 .0009 .0007 .0003 -.0005 .0007 -.0008 .0 -.0001 .0001 -.0003 -.0006 .0 .0009.0 .0006 .0002 -.0001.0 -.0005 .0009 -.0002 .0 .0006 .0003.0002.0004 .0003 -.0007 -.0002 -.0006 .0001 -.0010 .0003 1=15 ••• .0016 .0004 .0005 -.0001 -.0004 .0 .0003 -.0005 .0006 .0002 .0001 -.0003 .0004 -.0008 1"'16 ••• -.0007 .0004 -.0004 .0001 .0005 -.0007 -.0004 .0007 -.0001 .0 .0 .0002 -.0001 .0008 1"'17 ••• .0039 .0 -.0002 -.0002.0 .0 .0 .0 .0002 -.0003 -.0001 .0001 -.0002 -.0003 1"18 ••• -.0008 -.0009 .0005 .0005 -.0004 -.0005 -.0005 .0006 -.0003 -.0003 .0003 .0001 -.0001 .0003 1"19 ••• .0030 -.0002 -.0003 .0001 -.0001.0 -.0002 .0003 -.0003.0 .0006 .0002.0 -.0004 1=20 ••• .0003.0002 .0005 .0001 .0 -. 0004 .0 .0003 -.0003 .0003 -.0007 .0002 .0003 .0004

C

k ,/

components - sin

k~x cos ~

\ L= 0 3 4 5 6 7 8 9 10 11 12 13 14 I- 1 .0184 -.0018 .0694 .0055 -.0517 .0142 -.0258 .0077 -.0564 .0069 -.0031 .0028 .0074 -.0137 -.0265 I- 2 .0168 .0004 .0106 .0016 -.0413 -.0008 -.0007 -.0139 .0019 .0060 .0084 -.0204 .0010 .0116 .0035 I- 3 -.0278 .0036 .0153 .0048 -.0132 .0002 .0032 -.0059 -.0091 .0032 -.0036 .0097 .0002 .0044 .0148 I- 4 -.0193 -.0019 .0033 .0016 -.0161 .0003 -.0006 -.0065 -.0013 .0003 -.0048 -.0029 .0034 .0025 .0011 I- 5 -.0157 I- 6 -.0091 It-7 -.0127 .0002 .0007 .0008 I- 8 -.0010 -.0006 I- 9 - .0042 .0008 1-10 -.0027 -.0003 1-11 -.0081 .0005 1-12 .0001 -.0002 1-13 -.0065 .0 .0091 .0012 -.0069 .0004 -.0001 .0002 -.0036 -.0024 -.0012 .0010 -.0008 -.0032 -.0039 .0032 -.0002 -.0083 .0003 -.0009 -.0035 -.0008 .0063 .0019 -.0050 .0005 .0004 .0011 -.0021 .0005 .0020 -. 0009 .0036 .0032 .0011 - .0010 .0 - .0023 .0026 .0030 .0 . 0034 .0004 -. 0077 .0048 . 0008 -.0034 .0020 .0008 -.0047 .0043 .0008 -.0025 .0019 .0010 -.0045 .0041 .0005 -.0026 .0013 -.0007 -.0027 -.0015 .0026 -.0002 -.0004 .0018 .0010 .0002 .0 .0002 -.0008 -.0020 .0002 -.0008 .0004 .0003 -.0005 -. 0002 .0003 .0002 -.0029 -.0011 .0010 -.0001 -.0004 .0021 .0007 -.0004 .0006 -.0009 -.0003 -.0013 .0003 -.0016 .0003 -.0005 -.0020 -.0005 .0003.0 .0004 .0019 .0002 .0005 .0011 -.0013 .0 -.0005 -.0002 -.0011 .0016 - .0018 .0005 .0014 .0008 .0002 .0007 .0004 1-14 -.0016 -.0005 .0013 .0011 -.0040 -.0004 -.0002 -.0022 -.0013 .0005.0 .0005 .0012 .0011 -.0002 1"15 -.0057 -.0006 .0034 .0006 -.0020 .0006 .0006 .0 -.0005 .0004 -.0006 -.0005 -.0010 .0006 .0006 1-16 -.0044 .0001 .0017 .0008 -.0039 .0001 .0 -.0025 -.0006 .0007 -.0004 .0002 .0013 .0006 .0 1-17 -.0026 .0004 .0029 .0003 -.0013 .0 '-18 -.0043 .0005 .0014 .0003 -.0034 .0 .0002 .0004 -.0003 -.0004 .0002 -.0004 - .0005 -.0002 .0 - . 0006 -.0013 -.0006 .0010 .0001 .0007 .0010 .0008 .0002 '-19 -.0051 -.0004 .0025 -.0002 -.0016 .0004 .0004 .0007 -.0003 .0004 -.0009 -.0002 -.0008 .0006 .0004 '=20 -.0005 .0 .0015 .0001 -.0027 .0003 -.0003 -.0018 -.0004 .0002 .0001 .0001 .0013 .0003 .0

Dk,/

components - sin

klx

sin

~

\ L= 0 2 3 4 6 7 8 9 10 11 12 13 14 1= 1 ••• .0 -.0395 -.0655 -.0152 .0709 .0638 .0329 -.0192 .0206 .0561 .0070 -.0020 .0113 -.0060 I- 2 ••• -.0016 -.0021 -.0042 .0025 .0013 .0038 -.0058 .0124 -.0096 -.0009 -.0102 -.0070 .0005 .0064 , - 3 1= 4 I" 5 (- 6 I- 7 (- 8 I- 9 1-10 1-11 ••• .0025 -.0079 -.0195 -.0086 .0107 .0042 .0073 -.0088 .0049 -.0026 ••• -.0031 -.0021 -.0019 .0022 -.0001 .0008 -.0009 .0050 -.0041 .0027 ••• -.0020 -.0059 -.0115 -.0071 .0060 .0032 .0011 .0007 .0052 .0050 ••• -.0017 -.0009 -.0033 .0010 .0008 .0012 -.0007 .0009 .0003 -.0005 ••• -.0002 -.0057 -.0072 -.0017 .0044 .0024 .0019 .0021 .0040 -.0007 ••• -.0012 -.0018 -.0015 .0016 .0003 .0014 -.0010 .0013 -.0005 .0005 ••• -.0009 -.0041 -.0059 -.0024 .0037 .0016 .0006 .0005 .0015 .0006 ••• -.0012 -.0004 -.0004 .0014 -.0009 .0 -.0010 .0019 .0003 .0001 ••• -.0004 -.0039 -.0048 -.0023 .0020 .0005 .0007 .0011 .0018 .0003 .0045 -.0014 -.0066 .0056 .0012 .0047 -.0036 -.0002 .0007 - .0022 -.0017 - .0010 .0008 .0009 -.0011 .0003 .0012 -.0010 -.0017 .0011 .0004 - .0010 -.0015 .0009 .0011 .0007 .0009 -.0013 .0012 .0007 .0013 - .0017 .0 -.0015 .0003 .0 '-12 ••• -.0011 -.0008 -.0012 .0010 .0001 .0006 -.0012 .0 .0006 -.0002 -.0005 -.0010 -.0014 .0007 1-13 -.0002 -.0031 -.0051 -.0011 .0030 .0017 .0003 .0011 .0020 .0012 .0012 -.0003 -.0002 -.0009 '-14 ••• .0006 -.0012 -.0006 .0015 -.0006 .0006 -.0011 .0016 -.0003 .0 .0004 .0001 -.0010 .0013 '-15 ••• -.0008 -.0032 -.0031 -.0013 .0019 .0012 .0008 .0004 .0010 .0007 .0002 -.0003 -.0012 -.0007 1-16 ••• -.0014 -.0004.0 .0010 -.0006 .0004 -.0004 .0005 .0003 .0003 .0003 -.0003 -.0003 .0002 1-17 ••• -.0007 -.0017 -.0038 -.0014 .0026 .0008 .0005 .0007 .0011 .0008 .0003 -.0001 -.0009 .0001 1-18 ••• .0008 -.0003 -.0005 .0006 -.0003 .0003 -.0004 .0005 .0005 -.0003 -.0002 -.0001 -.0006 .0003 1=19 ••• -.0013 -.0027 -.0029 -.0008 .0018 .0006 .0 .0008 .0009 .0002 .0010 -.0002 -.0009 .0 1=20 ••• .0013 -.0005 -.0012 .0009 -.0003 .0003 -.0007 .0009 -.0001 -.0003 .0008.0 -.0003 .0

Table 3

:

Fourier coefficient of the half-wave sine imperfection representation of

hupfOl

20

Ak,/

eomponents -

COS

k~x

COS

!Jt

\ L= 0 3 4 6 7 8 9 10 11 12 13 14 Jt~ 0 .0002 .0009 .0064 -.0251 -.0219 .0378 .0393 .0018 .0022 .0124 .0336 .0020 -.0067 .0079 .0098 1= 1 .0102 -.0004 -.0078 .0026 -.0280 .0020 .0099 -.0095 .0134 -.0102 -.0050 .0046 -.0080 -.0117 -.0022 1= 2 -.0150 -.0009 -.0055 .0105 .0020 -.0076 -.0145 .0044 -.0043 .0022 -.0205 -.0013 .0026 -.0133 -.0151 I~ 3 -.0639 -.0021 -.0018 -.0013 .0027 -.0027 -.0008 .0011.0 .0044 .0054 .0005 .0025 -.0005 -.0006 I- 4 -.0014 -.0043 -.0031 -.0004 .0003 -.0001 -.0051 -.0010 .0060 .0016.0 -.0004 .0003 .0032 .0051 1= 5 -.0324 -.0003 .0008 -.0024 .0019 -.0008 .0003 .0004 -.0028 .0034 -.0023 -.0007 -.0020 -.0022 -.0002 1= 6 .0039 -.0011 -.0016 .0027 1= 7 -.0045 -.0013 .0012 .0003 1= 8 .0092 .0008 -.0010 .0006 1= 9 .0020 .0008.0 .0002 1=10 -.0050 -.0016 .0002 -.0005 1=11 .0028 .0014 -.0002 .0007 1=12 .0076 -.0004 .0003 - .0004 1=13 .0020 .0011 -.0006 .0013 .0019 -.0012 -.0021 -.0006 .0024 -.0005 -.0038 .0012 .0018 -.0040 -.0009 .0021 .0004 - .0005 - .0008 .0002 .0003 .0015 .0001 -.0005 .0005 .0019 .0009 -.0010 -.0020 -.0003 .0007 .0002 .0001.0 .0005 .0017 -.0011 .0009 -.0011 -.0007 -.0003 .0010 .0005 .0001 -.0005 -.0004 -.0017 .0019 .0014 .0002 -.0006 -.0002 .0004 .0006 -.0001 .0009 .0013 -.0010 -.0007 .0 .0 -.0004 -.0006 -.0008 .0005 -.0003 -.0016 -.0006 .0005 -.0006 .0008 .0005.0 .0001 .0005 -.0007 -.0006 .0 .0002 .0003 .0005 .0012 .0006 - .0004 .0007 -.0004 - .0001 -.0002.0 .0 .0004 .0005 1=14 .0041 -.0015 -.0005 .0004 .0005 .0001 -.0008 .0004 -.0004 .0 -.0006 .0 .0002 -.0004 .0 1=15 - .0102 1=16 .0050 1=17 -.0017 .0002 .0006 . 0002 .0018 .0003 -.0010 .0 -.0009.0 .0004 -.0001 -.0005 -.0004 -.0003 .0008 .0004 -.0003 -.0002 .0002 .0006 -.0008 .0 .0003 -.0003 -.0010 .0 -.0002 - .0001 .0 .0005 .0002 .0002 .0001 -.0005 -.0003 .0003 . 0003 -.0002 .0003 .0 .0003 .0 1=18 -.0012 -.0005 -.0008 -.0003 .0010.0 -.0006 -.0003 .0008 -.0002.0 .0006 .0 -.0003 -.0003 1=19 .0007.0 .0001 -.0004 -.0003 -.0002 -.0003 -.0003 .0003 -.0002 .0003.0 -.0002 .0007 .0001 1=20 .0030 -.0010 -.0002 .0010 .0 -.0003.0 .0004 -.0002 -.0001 .0006 -.0003 .0005 -.0001 -.0001 \ L- 0 1= 0 1= 1

I" 2

1= 3 1= 4 1= 5 1= 6 1= 7 1= 8 1= 9 1=10 1=11 1-12 1=13 1=14 1=15 1=16 1"17 1=18 1=19 1=20

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Bk,/

eomponents -

COS

k~x sin

!Jt

3 4 5 6 7 8 9 10 11 12 13 14 -.0005 -.0574 -.0380 .0142 .0149 .0234 .0061 .0343 -.0171 -.0126 -.0050 -.0032 -.0021 .0071 -.0015 -.0051 -.0055 .0355 -.0002 .0023 .0099 .0005 -.0028 -.0046 .0153 -.0003 -.0043 -.0082 -.0009 .0037 .0041 -.0103 -.0052 -.0157 .0040 -.0077 .0047 .0102 -.0083 .0033 .0058 -.0137 .0039 -.0028 .0010 .0006 -.0020 -.0002 .0022 .0025 .0008 .0058 -.0075 -.0088 .0057 .0026 -.0011 -.0015 -.0011 -.0051 -.0012 -.0010 -.0040 -.0060 .0032 -.0007 .0045 .0024 .0002 .0 .0049 .0005 .0026 -.0003 .0004 -.0010 .0 .0011 .0007 .0016 -.0006 -.0023 -.0036 -.0011 -.0002 -.0011 -.0002 .0012 -.0022 -.0016 -.0012 -.0019 -.0010 .0029 .0006 .0010 -.0010 -.0004 .0029 -.0005 .0008 .0003 -.0013 .0 .0 .0009 -.0013 .0009 -.0014 .0003 .0025 .0 .0027 -.0003 .0018 -.0020 .0001 -.0001 .0009 -.0006 .0014 .0004.0 -.0012 -.0010 .0013 -.0009 -.0002 -.0009 -.0003 .0002 -.0005 .0010.0 -.0003 -.0002 -.0013 -.0011 .0002 .0005 .0014 -.0003 .0006 -.0006 -.0008 -.0008 -.0008 -.0018 .0011 .0013 .0007 .0010 .0013 -.0007 .0022 -.0010 -.0003 .0012.0 .0002 -.0004 .0004 .0003 -.0006 -.0010 .0006.0 -.0006 .0005 -.0004 -.0002 -.0005 .0004 -.0004 .0001 -.0006 -.0001 -.0002 .0012 -.0001 -.0001 .0006 .0025 .0002 .0007 .0002 -.0003 .0 .0008.0006.0003.0002 -.0010 -.0005 .0 - .0004 .0 - .0001 .0006 .0001 .0009 .0005 .0028 .0 . 0005 -.0007 . 0026 -.0003 .0014 .0010 .0009 -.0004 -.0004 -.0003 .0004 -.0013 .0004 -.0002 .0013 -.0001 .0006 .0001 .0 .0 -.0002.0 .0007.0 -.0002 -.0002 -.0005 -.0005 -.0002 .0 .0003 -.0007 -.0004 -.0003 -.0002 -.0002 .0008 -.0002 .0 .0 .0002 -.0003 .0003 .0007 -.0005 .0005.0 .0 -.0003 .0009 -.0005 .0 .0007 -.0004 -. 0002 -.0003 -. 0003 -. 0005 -.0002 .0006 .0002 .0 .0005 -.0003 -.0005 -.0003 .0011 .0005 .0006 -.0002 -.0005 -.0001 .0005 -.0003 .0001 -.0004 .0003 .0 .0006 -.0006 .0006 .0 .0 .0006 .0002 -. 0002 .0 .0001 -.0001 - .0003