��������������������� �������������� �������������������� !"!#$

%&'

()*+,-./0*123+,)-42*2 5+4+62 +5728,)9+5),++:.;<:+= >7/)60,)3 .052 */2.26,+?@A@BCDEFGHED?GEIJK@BGAHDE?BHELMNOP QR STUVWT XY RZ[T\\TZ[V RX R\YTXYQ]_ abcdaefgdhcij ckijgldamljnodpq

rstuvvvwxyz{|}}}}~�����������

������������������������������ ¡¢£ ¤¢¥ �¦§©¢ �¦§¦§ª«¬­®«¬ ¥°±²³±µ³¶±°·¹º³ °··»¼½¾¿¼½À¶ÁÂÃÄÅÂÆÄÇÂÁÈÉÊËÄ ÁÈÉÈÉÌÍÎÏÐÑÎÒÇ

ÓÔÕÖ×ÔØÕÖÙÚÛÜÝÚÖÞÕÖßÖàáÛâãäåæç⋅èÔÕáé⋅Ûâ

êëìíîëï

ðñòóôõö÷øùúûüúýþÿýþÿ�ù�þÿýü���

���� ������ ��� �� � ���� �����������������������

�������� !�" �!��#��$ %� ��#" �!#�&�&� ��'( )&��*'! �� ��&�+�� '�&", -!�"� �(���&�� '!�.��#�("#" �!��&#/0���1�$ %#!' ��&�$ .#��#�������!� '! �� ��&��� '&"�,

23

45

67

89

89

89

:;

<=>?

@

A

B

CD

CE

CF

GH

GH

GH

BH

IJKL

M

A

B

u1

u2

u3

NOPQOPRSPOTUVWXSVQYSVZY[TYSR\V]YSPR\Y _UVQ]TSY [PSR\UX[aYb cVbVde

fgh

ijhjklmlnopqrnmqslrtuvsqtupwoxq

yz

{|

}~

����

u1

��

��

��

ϕ�

ϕ�

α

����

v12

u2

α

��

��

��

���

���

����

v33u'

u3

α

����������������� ¡�¢¢��£���� ¤��£�¥����£�¦�§�¦�§� ���£��©

α

α

=

=

= ⋅

arctan .

cos

5

35903624347

12 2

o

v u

ª«

ª¬­«¬

2 90⋅ −α

α−90

v33u3'

u3

α

αα

α ª®

ª«

­®¬

­«¬α

°±²°±³±°µ¶·¹·²º ·»º¼µº ³½·¾º¿ ±³½º À¶·²¾µºÁ¼±³½¶¹¼ÂºÃ ķ÷ÅÆ

ÇÈÉ

( ) ( ) ( )

( ) ( ) ( )α⋅⋅=α⋅α⋅⋅α

=α⋅α⋅⋅=α⋅=

α⋅⋅=α⋅⋅α

=α⋅⋅=−α⋅⋅=

α=

α=→=α

α⋅=

2sinusin2sinsinu

sin2sin'usinvv

cosu22sinsinu

2sin'u902cos'uv

sinu

'u

tanu

vvu

tan

cosuv

33

2313

33

3323

33

333

33

3

212

ÊËÌÍÎËÏÌÐÍÌÍÑÒËÓÐÓÍÔ( )

( )( ) ( )

( )( )

( ) ( )

( )( ) ( )

( ) ( ) 323

32

22

1123

323

332

323213

13222

12

223212

111

111

2

332

33

232

222

2321

211

2333

233

2232

222

213121

211

utanM

Mcos4M2sinM2sincosMuuT

dtd

utanM

uMcosu4M2sin2sinucosuMuT

2sincosMuuMcosMuT

dtd

uMcos2sinucosuMuT

uMuT

dtd

uMuT

2tanu

M

2uM

2cosu2M

2uM

22sinucosuM

2uM

2vM

2uM

2vM

2uM

2vvM

2uM

T

ÕÕÕÕÕ

ÕÕÕÕÕÕ

ÕÕÕÕÕ

ÕÕÕÕ

ÕÕÕÕÕ

ÕÕ

ÕÕÕÕÕ

ÕÕÕÕÕÕÕ

⋅

α++α⋅⋅+α⋅⋅+α⋅⋅α⋅⋅−=

∂∂

⋅α

+⋅+α⋅⋅⋅+α⋅⋅α⋅⋅−α⋅⋅−=∂∂

α⋅⋅α⋅⋅−⋅+α⋅=

∂∂

⋅+α⋅α⋅⋅−α⋅⋅=∂∂

⋅=

∂∂→⋅=

∂∂

α⋅

+⋅+

α⋅⋅⋅+⋅+α⋅⋅−α⋅⋅+⋅=

⋅+⋅+⋅+⋅+−⋅+⋅=

Ö×ØÙ×Ú×ÛÜÝÞ×ßàÛ×Þáâ

ãäåæäåçèåäéêëìíèëæîèëïîðéîèçñëòîóèåçñî ôêëæòéèîõðåèçñêíðöî÷ øë÷ëùú

ûüý

[ ] ( )

( ) ( )M

M

M M M

M M M MM

= ⋅ + − ⋅ ⋅ ⋅

− ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ + +

= −−

1

12

2 1

1 12

22

33

2

0 0

0 2

0 2 2 4

0 0

0

0

cos cos sin

cos sin sin costan

α α α

α α α αα

5000.

11323.52941176471 2269.834215121441

2269.834215121441 34880.96885813149

þÿ�ÿ����������������

�

�

�

x1=1

�

�

�

����

34

10

8574929257

⋅ =sin

.

α

x2=1

�

�

[M2]

x3=1

�

����

�

������� ��!"#$% #�& #'&(!& �)#*&+ ��)& ,"#�*! &-(� �)"%(.&/ 0#/#12

343

EI Nm

EI Nm

EI Nm

dEI EI EI

dEI EI

dEI

dEI

1

39 8 2

2

39 9 2

3

39 9 2

111 2 3

7

122 3

7

133

8

222

04 06

1230 10 216 10

06 09

1230 10 10

08 12

1230 10 10

10

10

10

=⋅

⋅ ⋅ = ⋅

=⋅

⋅ ⋅ = ⋅

=⋅

⋅ ⋅ = ⋅

= = ⋅

= = ⋅

= = ⋅

=

−

−

−

. ..

. .

. .

1.0935

3.456

41.666666666+

340.1388605+

923.234057.710959159848536

141.6666666666666+

531.6666666666662.833921563023926

194.36506325.623989097222223

66.08412147999999+

306.1821407

EI

dEI

dEI

3

7

233

8

333

8

10

10

10

= ⋅

= = ⋅

= = ⋅

−

−

−

1.490279510158749

111.66666666659993.231095679010415

48.59126571.405997271412037

[ ]d =

⋅ −

77.10959159848536 28.33921563023926 5.623989097222223

28.33921563023926 14.90279510158749 3.231095679010415

5.623989097222223 3.231095679010415 1.405997271412037

10 8

567689:;<9=:;<;>?9@;

[ ] [ ]DM = ⋅ =d M

⋅⋅

−810

14120371.4059972790104153.2310956772222235.62398909

90104153.23109567015874914.9027951302392628.3392156

72222235.62398909302392628.3392156984853677.1095915

5000.

11323.52941176471 2269.834215121441

2269.834215121441 34880.96885813149

=

38.55479579924268 30.81344188170113 13.18448672914579

14.16960781511963 16.14181871257802 7.887687353467791

2.811994548611111 3.339602624048184 4.170849551419221

0 0

0

0

10 4

−−

=

⋅ −

ABCBDEFGHIJKILMIHNIOHPEFGHO MILGQKRO

STUVTUWXUTYZ[\]X[VX[_Y XWa[bcXUWa dZ[VbYXe UXWaZ]fg h[g[ij

kll

ω ν1 1= → =13.82197953644657 2.199836366540495rad

sHz

mΦ1 =

0.01194568732866988

0.004901655340973729

0.001037127799391879

ω ν2 2= → =46.79483383686124 7.447629116300349rad

sHz

mΦ2 = −

−

0.007043883555290691

0.006329899077313233

0.003364718463554524

ω ν3 3= → =71.47165113723052 11.37506656942972rad

sHz

nΦ3 = −

0.002772049549550144

0.005038355133697751

0.00408054778243177

opqrstuvrwxuyz{q|}~|u�w����s���|x��|}�usuyz|v|�|ux�~��qy w|wxr~y x���wxuyz�rxqy}��wr�|puyz|v|�|ux�~��qr��t���x|~rxy� �rw��~�q|� t��qr�v�|u|�qrstuw}|����|vtt���x|~rvy �rvy�|}�x �wu�u|�|{�q�r�yvu|������������������� ����������������� ������������������� ���  �¡¢¢£¤¥¦§ ¥©ª¥«¬­® ­ª°®±²¬³ª² ¥¬¥²ª ²¥¬©µ¶ ¶®³ª·²¬³ª µ¬°®¹¥¬©±®²³«°©²¬³ªº»®«¬ ­°®·µ²®² ¶®³ª² ¼°« ²½ ³¬¾ννν

1

2

3

===

2.20213641831129Hz

7.44825445926024Hz

11.3742989380368 Hz

[ ]¿Φ = − −−

0.01194568732866988 0.007043883555290691 0.002772049549550144

0.004901655340973729 0.006329899077313233 0.005038355133697751

0.001037127799391879 0.003364718463554524 0.00408054778243177

ÀÁÂÃÄÅÆÇÈÉ ÊÆËÅÌÍ

ÎÆÏÌÐÉÏÆÑÒÅÓÔÆ

ÕÖ×ØÖ×ÙÚ×ÖÛÜÝÞßÚÝØàÚÝáàâÛàÚÙãÝäàåÚ×Ùãà æÜÝØäÛÚàçâ×ÚÙãÜßâèàé êÝéÝëì

íîï

( ) ( ) ( ) ( )( )

( ){ } ( )( )

W F t v v F t u u

P t

W

uW

uW

u

F t

= ⋅ − = ⋅ ⋅ ⋅ − ⋅

=

= ⋅ −⋅

13 12 3 2

1

2

3

2

0

2

sin cos

cos

sin

α α

∂∂∂∂∂∂

αα

ðñïòóôõö÷øùøúõùõóøúûüõúøýþõÿûþøú÷ø

�õóøúõõúøýþõÿûþøú÷ø�õÿöø�û÷��[ ] { } [ ] { } ( ){ }M u K u P t⋅ + ⋅ =��

�ôõö÷õ����øú�����øû÷�{ } [ ] { }u y= ⋅

ΦΦ �� �����

[ ] [ ] { } [ ] [ ] { } ( ){ }M y K y P t⋅ ⋅ + ⋅ ⋅ =� �� �ΦΦ ΦΦ

��������������������� [ ]�ΦΦ T �� ������

[ ] [ ] [ ] { } [ ] [ ] [ ] { } [ ] ( ){ }[ ] { } [ ] { } [ ] ( ){ }

� � �� � � ��� �

ΦΦ ΦΦ ΦΦ ΦΦ ΦΦ

ΦΦ

T T T

T

M y K y P t

I y y P t

⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ = ⋅

⋅ + ⋅ = ⋅ω 2

��� ��

[ ]ωω

ωω

21

2

22

32

0 0

0 0

0 0

=

!� �"������" ��� �� ���� �" ��� ������ ��#������$ ������� �������� �����"����������%�����$ �� ��% �&���������������'(�� ���������� ��� ) ������

[ ] ( ){ } ( )�

sinΦT

P t t⋅ =

⋅ ⋅-19.28121723996411

3.454043700169982

74.31234801676533

5

��

*+,-+,./,+01234/2-5/265705/.8295:/,.85 ;12-90/5<7,/.8147=5> ?2>2@A

BCD

1 0 0

0 1 0

0 0 1

1

2

3

⋅

EEEEEE

y

y

y

+

⋅

191.0471183059477 0 0

0 2189.756473819454 0

0 0 5108.196916281987

y

y

y

1

2

3

( )=

⋅ ⋅-19.28121723996411

3.454043700169982

74.31234801676533

sin 5 t

FGHIJKHJLMN OMPGQJKRSRI TJUJOMTJ LGVJWJTXGMYTJ JTMZ[J\ SG NG] WJQGKRUHMSRPRHJ[JN

( )( )( )

EEsin

EEsin

EEsin

y y t

y y t

y y t

1 1

2 2

3 3

5

5

5

+ ⋅ = − ⋅ ⋅+ ⋅ = ⋅ ⋅+ ⋅ = ⋅ ⋅

191.0471183059477 19.28121723996411

2189.756473819454 3.454043700169982

5108.196916281987 74.31234801676533

_ abcdefghcibjcklcdkgmnoepcfjq l lerfs pcfghct( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

y t t t

y t t t

y t t t

1

2

3

= ⋅ ⋅ ⋅ − ⋅ ⋅= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅

4.20051866773408610 13.82197525681478 0.11611893018242 5

1.5955764149374528510 5 1.7048624809925796110 46.79487151387426

1.461919656909033810 5 1.02272662823765910 71.471672710242343

-2

-3 -4

-2 -3

sin sin

sin sin

sin sinupognnvjenlgwxccdpglemewq t

{ } [ ] { }

( ) ( )

( )

u y

t t

t

= ⋅

=− ⋅− ⋅− ⋅

⋅ ⋅ +⋅⋅⋅

⋅ ⋅

+− ⋅

⋅⋅

⋅ ⋅

+− ⋅

⋅− ⋅

y

sin sin

sin

Φ

1.33535624116429610

6.52931516077913710

6.61445058102472210

5.01780826229923710

2.05894947625586310

4.35647468217156410

13.82197525681478

1.20088527938957210

1.07916074453808710

5.73638226761710610

46.79487151387426

2.83504888911914110

5.15285995775060210

4.17328487488910210

-3

-4

-5

-4

-4

-5

-6

-6

-7

-6

-6

-

5

( )6

71.47167271024234

⋅ ⋅sin t

zeplqg mdgq godeog bdgbepeqcjeimlhalgwelqas xg be i xhejagopakg{ _ mepjcpogneoelhedghcs ime|gcledlg cpognoebgkhcxcmt

}~��~����~���������������������������� �������������������� ������

���

{ } ( )u tst =− ⋅− ⋅− ⋅

⋅ ⋅1.33535624116429610

6.52931516077913710

6.61445058102472210

-3

-4

-5

sin 5

�������� ¡¢£�¤¥¦£�§¢©�¡ª«©¦¬¢©�£«£¢­�

®�¦��� ¡¢£�¤¥¦£�§¢©�¡ª«©¦¬¢©�£«£¢­�¡ª«­°�£«¥�£«¥­�±¦«¢��¢¦ ²§«²�¦°��³�²«¬¦£«¥¦§ ´ ¥¤�­�¦£««¬�£¢¦£«¤«¢¥­�¡¤±¦µ« ª«£¢«¶¦£«­«©«¥­¢«£�­�¦¤«¥¥­�±¦«¢��¢¦£«²�¦°«£·

[ ] { } ( )k u tst⋅ =−−

⋅ ⋅166.9209397896666

6355.025912081007

10567.59357296104

sin 5

�� ²¦�©��£«°£«£¢­«° §�¤«¥¤§ª�£« ²�¬«£�¤¥¦£�§¢¦ £«£¢­ ¢�¥­«ª¦§ ¢�£¥­¡°ª­µ�¥ª«²¢©� § £¢­��¹«£¢­¬«­�£·

( ) ( ) ( )( ) ( )

( ) ( )( )t5sin63851643325.13915

t5sin729610410567.59355

t5sin20810076355.02591574929257.8

t5sin7896666166.92093915tM spodaj

⋅⋅−=⋅⋅⋅+

⋅⋅−⋅+

⋅⋅−⋅=

º� ©� �� ¢«¥­�°¢« ��£«¢¦ ¢«»¡¢¤±¦µ« µ µ�¥¢« ²�¬«£�¤¥¦£�§¢¦ £«£¢­¤�� ¢�¤·

( )M tspodaj = 3325.139156385164Nm

¼½¾¿�À�½Á½Â�ÃÄÅ¿ÀÁ¹Æ��³«§« ¦­°¦¢­�ª«§�±¦µ¥¤¦ »¡¢¤±¦µ

Ã�°�¦¢­�ª«§�±¦µ¥¤�»¡¢¤±¦µ�

ÇÈÉÊÈÉËÌÉÈÍÎÏÐÑÌÏÊÒÌÏÓÒÔÍÒÌËÕÏÖÒ×ÌÉËÕÒ ØÎÏÊÖÍÌÒÙÔÉÌËÕÎÑÔÚÒÛ ÜÏÛÏÝÞ

ßàá

âãäåæçãèãéæ êãë

ìííîáïðñòóôõö÷øùóóúúûüýþüÿ�ÿ������û��üýþü��ÿ

���ûü�üýþü��ÿ���û 34

ýþü��ÿ

���ûü�üýþü��ÿ���û 34

ýþü��ÿ���úû 34

ýþü ���

����� �� �������úû��üýþ���úûü�üý ������⋅���úúû��üýþ���⋅���úúûü�üýÿ�ÿ���

���úû 34ýþ���úûü�üý ���

���⋅���úúû 34ýþ���⋅���úúûü�üýÿ�ÿ���

���ÿ������������������������ �� ���� �����ÿ��� ������������������������������������������� ( )v x' ' ' ' = 0 ��

( )v xC x C x

C x C111

32

2

3 46 2=

⋅+

⋅+ ⋅ + ! ( )v x

C x C xC x C12

53

62

7 86 2=

⋅+

⋅+ ⋅ +

�

( )v xC x C x

C x C139

310

2

11 126 2=

⋅+

⋅+ ⋅ +

"�������������������������������� �� ���� ���ÿ�

( )( )

( )

v x = -3.37890873641057310 u x + 0.2844727184102642u x - u

v x = 8.03602546929065110 u x -1.00115814412165110 u x

3.10545631794712410 u x

v x = -1.09766958349501410 u x +1.28009170755886110 u x

- 3.73207658388304610 u x

11-3

13

1 1

12-4

13 -2

12

-21

13-4

13 -3

12

-31

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅+ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

#����������������������

$%%

$%&

$%'

u1

(

(

(

)*+,*+-.+*/0123.1,4.1546/4.-71849.+-74 :01,8/.4;6+.-7036<4= >1=1?@

ABC

DEFGH IEJEKH LEM

NOPQRSTRIEUHVNOPWWQRSTRUEUXGYNOPQAZRST[\O⋅LHGα

IEUHV

NOOQRZRST[\OIEUHVNOOQ 34

STRIEUHV

NO]QRZRSTRIEUHVNO]Q 34

STRIEUHVNO]WQ 34

STR _L\V

EJEKH NXGELYHNOPWQAZRSTNOOWQRZRS _L\VabP⋅NOPWWQAZRS TabO⋅NOOWWQRZRSUEUXGY

NOOWQ 34STNO]WQRZRS _L\V

abO⋅NOOWWQ 34STab]⋅NO]WWQRZRSUEUXGY

cIEUEdKEeEFGHfIEJEKXNHGIEJEKXN NXGELYH IEHgdXUEGXG_GXVEGLY_GYXZcIhEgGXeXgHYNXiHjXeXGkH_hGXXG_dFX ( )v x' ' ' ' = 0

LEM

( )v xC x C x

C x C211

32

2

3 46 2=

⋅+

⋅+ ⋅ + l ( )v x

C x C xC x C22

53

62

7 86 2=

⋅+

⋅+ ⋅ +

HG

( )v xC x C x

C x C239

310

2

11 126 2=

⋅+

⋅+ ⋅ +

mF\IEgYXN_GK\eEFGHfIEJEKXNHGIEJEKXN NXGELYH iEFHUEM ( )( )

( )

v x = 6.96562823192877310 u x - 0.3456392909407282u x

v x = -3.6949607477435810 u x + 2.06388984649741410 u x

+ 0.1767828264539298u x - u

v x = 1.19369642375900510 u x -1.39207728479752510 u x +

4.05856784078061410 u x

21-3

23

2

22-3

23 -2

22

2 2

23-3

23 -2

22

-22

⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

neXYK_HGYXeIEh_kHKLV_j\GVkHK_

opq

opp

opr

u2 ⋅cosα

u2

s

s

s

tuvwuvxyvuz{|}~y|w�y|���z�yx�|���yvx�� �{|w�zy���vyx�{~���� �|�|��

���

����������� ���

���������� �¡���¢¢����� � £�¤�����¥���¦�§�⋅ ��α�¦�⋅©�⋅ ��§α �� �¡

��§��¥������ �¡

��§� 34��

α

π−α⋅

⋅−sin

22sin

u3

�� �¡

�����¥���¦©�ª«α

−sin

u3 ¬­®°

±²²³ 34 «µ¬­®°±²²¶³ 34 «µ·¹º°

»­¼­½ ·±¾·¿­¹À±²Á¶³Âõ«±²Ä¶³µÃµ ·¹º°ÅÆÁ⋅±²Á¶¶³Âõ «ÅÆÄ⋅±²Ä¶¶³µÃµ ®­®¾¿À

±²Ä¶³ 34 «±²²¶³µÃµ ·¹º°ÅÆÄ⋅±²Ä¶¶³ 34´«ÅƲ⋅±²²¶¶³µÃµ ®­®¾¿À

Ǭ­®­È½­É­Ê¿Ë¬­¼­½¾± ¿¬­¼­½¾±·±¾·¿­¹À ¬­ÌȾ®­¿¾·¿¿¾°­¿¹À¿À¾ÃǬͭ̿¾É¾ÌÀ±¾ÎϾɾ¿Ð Í¿¾¾¿Èʾ ( )v x' ' ' ' = 0 ¹­Ñ

( ) 43

22

31

31 CxC2

xC

6

xCxv +⋅+

⋅+

⋅= Ò ( ) 87

26

35

32 CxC2

xC

6

xCxv +⋅+

⋅+

⋅= ¿

( ) 1211

210

39

33 CxC2

xC

6

xCxv +⋅+

⋅+

⋅=

Óʺ¬­ÌÀ¾±¿½ºÉ­Ê¿Ë¬­¼­½¾± ¿¬­¼­½¾±·±¾·¿­¹À έʮ­Ñ

( )( )

( )33

23

2-33

3-33

23

23

3-33

3-32

33

3-3

31

u8969061.16619037 -x u82540980.22165912

xu1026505882.68706973xu1057232413-5.2453197=xv

u-xu455007660.23047951-

xu1027496387.38353550-xu1035784895.27690905=xv

xu213672640.04358377-xu1032178002-2.4919432=xv

⋅⋅⋅+⋅⋅⋅+⋅⋅⋅

⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅

ÔÕÖ

ÔÕ×

ÔÕÕ

( )u3 2⋅ ⋅sin α

2 3⋅ ⋅u cosα

u3

u3

tanα

Ø

Ø

Øu3’

ÙÚÛÜÚÛÝÞÛÚßàáâãÞáÜäÞáåäæßäÞÝçáèäéÞÛÝçä êàáÜèßÞäëæÛÞÝçàãæìäí îáíáïð

ñòó

ôõö÷øöùúûüõýûõþ

ÿ��õö÷øöùöúûüõýûõþ �õ�õùú� �öüöû�ú�öøö�þõ�üö�þö�üöúûüöûõþö���ø�õ�öþü�ù�ú� ��ú��üõý�þ�ù�ü�õý� �öø�ú���öùú�øõ õü��÷øõøöü��

k P v v dx P v v dx P v v dxi j ukl i jx

ukl i jx

ukl i jx

, ' ' sin ' ' sin ' '= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅= = =∫ ∫ ∫1 1

0

5

2 20

34

3 30

34

α α

ö�ú�ú�þ��üö÷�úù��øõ�û�ö�ú÷úû�üöùö�øõ�ö���

[ ]k Pg ukl= ⋅− −

− −− −

0.2292426884155164 0.2293241410138043 0.09033850063374726

0.2293241410138043 0.4281903869509456 0.06266107825572284

0.09033850063374726 0.06266107825572284 0.4375201513028297

�þ�ü� �ú÷úû�ü� ùö�øõ�� û��ö� õ�øö�üöùú û �úùú��ú õü��ø�õ�� �� �üöü��ú�ö�üúû�ü�ùö�øõ��öþõ�öû�úùú��úõ�øö�üöüõýõü��ø�úþö�õ�û�õý�ü��õ� �ú��

k EI v v dx EI v v dx EI v v dxi j i jx

i jx

i jx

, " " " " " "= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅= = =∫ ∫ ∫1 1 1

0

5

2 2 20

34

3 3 30

34

ö�úõ�øö�üöüö�ú÷úû�üöùö�øõ�ö���

[ ]k =⋅ − ⋅ ⋅

− ⋅ ⋅ − ⋅⋅ − ⋅ ⋅

4.37906572238810410 9.02745418857969110 3.22955842890269210

9.02745418857969110 3.19836155698468210 3.73911238064878810

3.22955842890269210 3.73911238064878810 1.44133492232684610

6 6 6

6 7 7

6 7 8

õü��û����ö�üö�öùö�øõ�õ� �õ �ú�ú�õùú�õü��ø�õ�ú�ú�ö�üúû�ü�ùö�øõ����øú�þ�ù�úþú�õ����øõ�õ�ü��þúüû��ûõþ�û��ø������ú�þõ�ú�

[ ] [ ]( )k k

u

u

ug− ⋅

=

1

2

3

0

0

0

�������⋅�

��

���� ! ⋅�

P

�

�

"#$�

%&'(&')*'&+,-./*-(0*-102+0*)3-405*')30 6,-(4+*072'*)3,/2809 :-9-;<

=>?

@ABCBDEFGHIFEJHFJBIFKFLBMFBNOEPDLOQ

[ ] [ ]k k

P P

P

g

ukl ukl

ukl

− =

− ⋅ + ⋅ ⋅

− ⋅ ⋅ + ⋅ =

0

10 0

3 2

21

0.01294691731254715 6.12370624703525910

4.27122712834835710 4.165258821968367

6

14

@ABCBDEOGHIFEJHOJBIOJFQRSTU VWXYXZ[\]Z]Z\Z_\_Xabcdeefghicjklm ndofpgqrspppttsorhuepbcdopfgqicjklm vdrfqpgstsqrhqrqetsepwxyz{|}~��x

��������������������������������������������� �y��~z��|�}���|z{�xy��}~��x

������� ������������������������ ��}¡¡¢ £¤¥¦§ ©ª¤§«¤¬­®¬­® °­®¬«±²

³³ µ¶·¶¹º»¼ ½º»¼¹·¾ ¹¿ ·»À» Á¹¿¾Â¹¿¹ÃÄŶƺ¶Æº¿¹Ãƺ¶Ä»¿Ç

ȹƺ»Âº¼¶Å¹Ç¶ ºÅ¹ Â¾Æ»É Ê¹ ¹Â¾Ç¶ ¼ ž¼¿¹¿¹ ¼ÆÄ·¶½¿»ÂË ÄŶƺ¶Æº¿¹Ã ƺ¶Ä»¿Ç ÌŹǻ »·»Â»¿º¹ÄÅ»Áƺ¾¼·Ç¾Ç¶ºÅ¹ »¿¾Í»¼»Î¹É ʾÅĶ»¿¹É Á¾¹Â¾Æ¹Æº»ÂºÅ¹ ÄŶƺ¶Æº¿»Æº¶Ä¿Ç»

³Ïµ¶·¶¹º»¼ ·»¿¶¼Â¾Æ¿»Â¾ºÅ¹Ê»

ÐÑ

ÐÒ

ÐÓ

ÔÕ

ÔÕ

ÔÕ

ÖÕ

×ØÙÚ

Û

Ü

Ö

u1

u2

u3

ÝÞßàÞßáâßÞãäåæçâåàèâåéèêãèâáëåìèíâßáëè îäåàìãâèïêßâáëäçêðèñ òåñåóô

õöõ

÷ø

÷ù

÷ú

ûüýþ

u1

÷ø

÷ù

÷ú

ϕø

ϕø

α

ûüýþ

v12

u2

α

÷ø

÷ù

÷ú

ÿøú

ÿùú

ûüýþ

u3

α

��������������� ����������� ������������������� �������α

α

=

=

= ⋅

arctan .

cos

5

35903624347

12 2

o

v u

÷ù

÷úÿùú

2 90⋅ −αu3

α

αα

α ÷ø

÷ù

ÿøú

ÿùúα

( ) ( )( ) α⋅α⋅⋅=α⋅=

α⋅⋅=−α⋅⋅=sin2sinusinvv

2sinu902cosuv

32313

3323

��������������������

����������� !"#�!�$�!%$&�$��'!($)���'$ * !�(��$+&���' #&,$- .!-!/0

123

( )

( )( )

( )( )

( )( )

( ) ( )

( )( ) ( ) ( )

( ) ( ) ( )( ) 332

222

1123

332

323213

13222

12

223212

111

111

233

232

222

2321

211

233

2232

222

213121

211

uM2sinM2sinsinM2sincossinMuuT

dtd

uM2sinuM2sinsin2sinsinucosuMuT

2sincossinMuuMcosMuT

dtd

uMcos2sinsinucosuMuT

uMuT

dtd

uMuT

2uM

22sinuM

2uM

22sinsinucosuM

2uM

2uM

2vM

2uM

2vvM

2uM

T

44444

44444

44444

4444

44444

44

4444

444444

⋅+α⋅⋅+α⋅⋅α⋅+α⋅⋅α⋅α⋅⋅−=

∂∂

⋅+α⋅⋅⋅+α⋅⋅α⋅α⋅⋅α⋅−α⋅⋅−=∂∂

α⋅⋅α⋅α⋅⋅−⋅+α⋅=

∂∂

⋅+α⋅α⋅⋅α⋅−α⋅⋅=∂∂

⋅=

∂∂→⋅=

∂∂

⋅+α⋅⋅⋅+

⋅+α⋅⋅α⋅−α⋅⋅+⋅=

⋅+⋅+⋅+−⋅+⋅=

[ ] ( )( ) ( ) ( )

−−=

+α⋅⋅+α⋅⋅α⋅α⋅⋅α⋅α⋅−α⋅⋅α⋅α⋅−+α⋅=

192143425647.77122006921946.366780

2006921946.36678117647111323.52940

005000.

M2sinM2sinsinM2sincossinM0

2sincossinMMcosM0

00M

M

32

222

11

122

1

1

567689:;<=9>?=;@;?ABC;DEAF@F=B >F>;@9?AF?<B :B;GA;@B=E9>HF:BI=9JG9H< ?B>?B IHF=BK CB B@;<9B=:FC>76

LMNOMNPQNMRSTUVQTOWQTXWYRWQPZT[W\QNPZW ]STO[RQW YNQPZSVY_W aTTbc

dee

34

x3=1

f

ghij

8

333

8

323

8

313

1028240741.91215629EI

866.0841214d

1038657413.76807269EI

3130.224592d

1033796296.55864197EI

6226.666666 d

−

−

−

⋅==

⋅==

⋅==

klmnoplqrpnsnrtuvnowrnvlx

[ ] 810

28240741.9121562938657413.7680726933796296.55864197

38657413.76807269015874914.9027951302392628.3392156

33796296.55864197302392628.3392156984853677.1095915

d −⋅

=

yz{z|upnsu}pnsnrtuvn

[ ] [ ]DM = ⋅ =d M

810

28240741.9121562938657413.7680726933796296.55864197

38657413.76807269015874914.9027951302392628.3392156

33796296.55864197302392628.3392156984853677.1095915−⋅

−−⋅

192143425647.77122006921946.366780

2006921946.36678117647111323.52940

005000.

~������������������������������������� �������������������� ������

���

410

14192214.1708495584297913.8946124466898153.27932098

44851826.76363610125780216.1418187151196314.1696078

911148111.3056040817011330.8134418992426838.5547957

= −⋅

��������� ¡¢ £¤ �¥ ¦�§����¦ ¤ £� ¢©¦

Hz65568272.19983636s

rad365491813.8219795 11 =ν→=ω

=Φ84611461610.00120948

53410195910.00490165

7328724730.01194568ˆ

1

Hz27911727.44762911s

rad148124346.7948338 22 =ν→=ω

−−=Φ

23097964450.00392390

90672249740.00632989

35494109670.00704388ˆ

2

Hz514833911.3750665s

rad244704371.4716510 33 =ν→=ω

−=Φ

55568980030.00475869

151463274310.00503835

95642543860.00277204ˆ

3

ª��¢¦£�§����¦«¤ £�¢© ¤

[ ]

−−−=Φ

55568980030.0047586923097964450.0039239084611461610.00120948

51463274310.0050383590672249740.0063298953410195910.00490165

95642543860.0027720435494109670.007043887328724730.01194568ˆ

��¬­®� �¦ ¤ £� ¢° § �¦§ ±²�³© 

µ¶·µ¶¹¶µº»¼½¾¹¼·¿¹¼À¿Áº¿¹ ¼ÿĹ¶Â¿ Å»¼·Ãº¹¿ÆÁ¶¹Â»¾ÁÇ¿È É¼È¼ÊË

ÌÍÎ

( ) ( ) ( ) ( )( )

( ){ } ( )( )

W F t v v F t u u

P t

W

uW

uW

u

F t

= ⋅ − = ⋅ ⋅ ⋅ − ⋅

=

= ⋅ −⋅

13 12 3 2

1

2

3

2

0

2

sin cos

cos

sin

α α

∂∂∂∂∂∂

αα

ÏÐÍÑÒÓÔÕÖ×Ø×ÙÔØÔÒ×ÙÚÛÔÙ×ÜÝÔÞÚÝ×ÙÖ×ßàÝá×ØÙ×Ø×ÙÔâÓáÚâÔáãäàÝÚâàå

[ ] ( ){ } ( )t5sin

230366374.3123480

01828173.45404355

426623919.2812172-

tPˆ T⋅⋅

=⋅Φ

ÚÙæ×çàèÕÔäÚå

1 0 0

0 1 0

0 0 1

1

2

3

⋅

ééééééy

y

y

⋅

+

3

2

1

y

y

y

01636875108.1969000

017559132189.756470

003087843191.047118

( )t5sin

230366374.3123480

01828173.45404355

426623919.2812172-

⋅⋅

=

êÚèæÔâèÔä×Ö Ò×ÓÚëÔâàçàæ ÙÔØÔÒ×ÙÔ äÚìÔáÔÙíÚ×ÕÙÔ ÔÙ×ÜÝÔî çÚ ÖÚÛ áÔëÚâàØè×çàÓàèÔÝÔÖå

( )( )( )t5sin230366374.3123480y01636875108.19690y

t5sin01828173.45404355y17559132189.75647y

t5sin426623919.2812172y3087843191.047118y

33

22

11

⋅⋅=⋅+⋅⋅=⋅+⋅⋅−=⋅+

éééééé

ïàÝÕÖÔÙÔÔÙ×ÜÝÔèÔÓá×çæÚÜÙàÓàÓàÕÙàâ×ãÖÔâ×ÖàÒÔÙ×ÜÝ×âÚî äàÝÕÖÔÙÚâÚ ØÓáØÔâÓáÚèæàÓãÐðÔèÔÒ×ÙÖÔãÓàá×ÝÚÖàèÓÕàëÙÔáÔëÚæØÔî èÕÔäÚå

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )t1024234371.4716727sin1082376591.02272662t5sin10690903381.46191965ty

t138742646.7948715sin100992579611.70486248t5sin104937452851.59557641ty

t5sin42611893018214.6990.11t568147813.8219752sin10773408614.20051866ty

3-2-3

4-3-2

-21

⋅⋅⋅−⋅⋅⋅=

⋅⋅⋅−⋅⋅⋅=

⋅⋅−⋅⋅⋅=

ñòóôòóõöóò÷øùúûöùôüöùýüþ÷üöõÿù�ü�öóõÿü �øùô�÷öü�þóöõÿøûþ�ü� �ù�ù��

�

� ��������������� ���������

{ } [ ] { }

( ) ( )

( )

( )t102423471.4716727sin

1017158264.86684466

1077506025.15285995

1091191412.83504888

t138742646.7948715sin

1070520836.68971382

1045380871.07916074

1093895721.20088527

t568147813.8219752sin

1094534235.08047885

1062558632.05894947

1022992375.01780826

t5sin

109755267.71370863

1007791376.52931516

1011642961.33535624

yˆu

6-

6-

6-

7-

6-

6-

5-

4-

4-

5-

4-

3-

⋅⋅

⋅−⋅⋅−

+

⋅⋅

⋅⋅⋅−

+

⋅⋅

⋅⋅⋅

+⋅⋅

⋅−⋅−⋅−

=

⋅Φ=

�� ��� ����������� ����� ��������� !������!" �� ����� ��!�� !#�$ %��� ��� �������� ��� �" ���&������� � �������#� �����

{ } ( )t5sin

109755267.71370863

1007791376.52931516

1011642961.33535624

u5-

4-

-3

st ⋅⋅

⋅−⋅−⋅−

=

'()*(+*,(-)./012(345 67485496:;<9=>65?@AB>74:A@6989=6789>45B7<9?B7C45DAD7>AE9F D9589C7=98AG65 H<9?B7 B94 @=5I @AB>74 @=A C4ADG69BA 69=C47>6A 7D >5=>A?F B5 =769@ADA65 @C45B9G96A85G49J;6;F B94 :AC49@ >9B7B74AB>A64AG;<>9>K LE<A:87=5=AD9:F B9: =A =C4A8A65F JA 589 6C4K >4A>:5 <9=>65 674854965 @AB>74 69=C47>6AC4ADG69BAM

−

−=Φ

55568980030.00475869

151463274310.00503835

95642543860.00277204ˆ

3

N9>45B9<9=>65?@AB>74:A@:A=AD9:

[ ]

−−−

−=Φ

55568980030.0047586923097964450.0039239084611461610.00120948

51463274310.0050383590672249740.0063298953410195910.00490165

95642543860.0027720435494109670.007043887328724730.01194568ˆ

OPQRPQSTQPUVWXYTWRZTW[Z\UZTS]WZ_TQS]Z VWRUTZa\QTS]VY\bZc dWcWef

ghi

jklkmn opqkrpmpstrnspuvowxprylpzpxn nxt{op|xn }p~otl�pytxt}xtpxk~�

( ){ } ( )( )

α⋅α−⋅=

2sin

cos

0

tFtP

�ln n�yps�k}n xp}p�kxn�pxk�{pqn{kx�krpmk� mt{nzt�

[ ] ( ){ } ( )t5sin

230366374.3123480

01828173.45404355

426623919.2812172

tPˆ T⋅⋅

−

−=⋅Φ

nx

1 0 0

0 1 0

0 0 1

1

2

3

⋅

������y

y

y

⋅

+

3

2

1

y

y

y

01636875108.1969000

017559132189.756470

003087843191.047118

( )t5sin

230366374.3123480

01828173.45404355

426623919.2812172-

⋅⋅

−=

�nropzrpmk� �kyn�pzt~to xp}p�kxp mn�plpx�nksxp pxk�{p� ~n �n� lp�nzt}rk~tytrp{p��

( )( )( )t5sin230366374.3123480y01636875108.19690y

t5sin01828173.45404355y17559132189.75647y

t5sin426623919.2812172y3087843191.047118y

33

22

11

⋅⋅−=⋅+⋅⋅=⋅+⋅⋅−=⋅+

������

�p�kmt{s�pxppxk�{p�ytlk{nztryst�xtlp�nop}� mt{nzt�

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )t1024234371.4716727sin1082376591.02272662t5sin10690903381.46191965ty

t138742646.7948715sin100992579611.70486248t5sin104937452851.59557641ty

t5sin42611893018214.6990.11t568147813.8219752sin10773408614.20051866ty

3-2-3

4-3-2

-21

⋅⋅⋅+⋅⋅⋅−=

⋅⋅⋅−⋅⋅⋅=

⋅⋅−⋅⋅⋅=

�m�n}}qsk}xn�~ttlmnxkok��pytxt}xtpxk~�

�������������������������������������� �������� ��������¡�¢ £�¢�¤¥

¦§

{ } [ ] { }

( ) ( )

( )

( )t102423471.4716727sin

1017158264.86684466

1077506025.15285995

1091191412.83504888

t138742646.7948715sin

1070520836.68971382

1045380871.07916074

1093895721.20088527

t568147813.8219752sin

1094534235.08047885

1062558632.05894947

1022992375.01780826

t5sin

109755267.71370863

1007791376.52931516

1011642961.33535624

yˆu

6-

6-

6-

7-

6-

6-

5-

4-

4-

5-

4-

3-

⋅⋅

⋅−⋅⋅−

+

⋅⋅

⋅⋅⋅−

+

⋅⋅

⋅⋅⋅

+⋅⋅

⋅−⋅−⋅−

=

⋅Φ=

©ª«¬­® °®­±®²°ª²® ³°®³ª«ª­ µª¶¬±· ¬®¹ª¬­ º »® ³ª¶±»·ªµ ®²«¼®½ ¾± ª«±µ«²®¿ª²ª¬±·ª°®·º ¶ªÀ® ¬ª°¬® «²®¿²ª³®¼±· » Á

{ } ( )t5sin

109755267.71370863

1007791376.52931516

1011642961.33535624

u5-

4-

-3

st ⋅⋅

⋅−⋅−⋅−

=

°±³°®Ãªµ® ¶· ¶±º «ª¬ª¶³° ¬® ³°±«²¬ª»³°® ¬°ª·®°ª¬­ µª¶¬®¹¿±» °­±¿¬±®Ä°ª¿µ ı³°® ®²°ªÃ¬ ®«²®¿ª¿Äµª¿¬®¹» °«®¬ªª¹½

ÅÆÇÈÉÊÉÆËÆÌÉÂÍÎÈÊËÏ

½Ðѵ ®¬±°³µªÀ®­¶»®¹Ò¬»À®­Â°¿ª ®¬±°³µªÀ®­¶»ª Ò¬»À®­ª ®·ª ¶®À±° ¬±»µ®» «°ÄªÃ±Óµ »®°ª¬ Ô½ ³° ¶ ¶¬¶ ³¬­ º ¿±¬«ª° ¶ °Ó¬® ³Ä­® Õ®¬³¶µ±«®Ã¬ °±¼®¿±Ö±¬ª»®½

×ØÙÚØÙÛÜÙØÝÞßàáÜßÚâÜßãâäÝâÜÛåßæâçÜÙÛåâ èÞßÚæÝÜâéäÙÜÛåÞáäêâë ìßëßíî

ïðñ

òóôõö÷óøóùö úóûüýýþÿ����ý÷ó�ö�üýý��þÿ��ÿ�ó��õ�üýýþïÿ��ÿ÷ó�ö�

üýþÿÿ��ÿ÷ó�ö�üýþ 34

��ÿ÷ó�ö�

üý�þÿÿ��ÿ÷ó�ö�üý�þ 34

��ÿ÷ó�ö�üý��þ 34

��ÿ� ú���óøóùö �ü��õóú�öüýý�þïÿ��üý�þÿÿ�� ú����ý⋅üýý��þïÿ����⋅üý��þÿÿ��ó��õ�

üý�þ 34��üý��þÿÿ�� ú��

��⋅üý��þ 34�����⋅üý���þÿÿ��ó��õ�

�÷ó�ó�ùó�óôõö�÷óøóù�üöõ÷óøóù�ü�ü��õóú�ö ÷óö����óõ��õ õ��óõú� õ��öõóô�÷ó���ü õù��óôõö�÷óøóù�üöõ÷óøóù�ü�ü��õóú�ö�óôö�óû

( )( )

( )

v x = -3.37890873641057310 u x + 0.2844727184102642u x - u

v x = 8.03602546929065110 u x -1.00115814412165110 u x

3.10545631794712410 u x

v x = -1.09766958349501410 u x +1.28009170755886110 u x

- 3.73207658388304610 u x

11-3

13

1 1

12-4

13 -2

12

-21

13-4

13 -3

12

-31

⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅

+ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

���ø öõ���÷ó� �öùú� ��õ��öù ���ö ��� ù ù�÷��÷����õö ÷ó�ö�õ��ó�ö�ó���ø ��õ� ü�õ� � úó �óôõö ÷óøóùö öõ���ö�ü�÷óõóüõóö��õ�ö�õö

���

���

��

u1

!

!

!

"#$%#$&'$#()*+,'*%-'*.-/(-'&0*1-2'$&0- 3)*%1('-4/$'&0),/5-6 7*6*89

:;<

=>?@AB>C>DA E>F

GHIJKLMKB>NAOGHIPPJKLMKN>NQ@RGHIJ:SK

LMTUH⋅EA@α

B>NAO

GHHJKSKLMTUHB>NAO

GHHJ 34LMKB>NAO

GHVJKSKLMKB>NAO

GHVJ 34LMKB>NAO

GHVPJ 34LMKWXEUO

Y>C>DA WGQW@>ERAGHIPJ:SK

LMGHHPJKSKLWXEUO

Z[I⋅GHIPPJ:SK

LMZ[H⋅GHHPPJKSK

LN>NQ@R

GHHPJ 34LMGHVPJKSK

LWXEUOZ[H⋅GHH

PPJ34

LMZ[V⋅GHVPPJKSK

LN>NQ@R

\B>N>]D> >?@A_B>C>DQGA@B>C>DQGWGQW@>ERA B>A]QN>@QW@X@QO>@ERX@RQA@>?UB>RQGX@DU >?@A_B>C>DQGA@B>C>DQGWGQW@>ERA a>?AN>F

( )( )

( )

v x = 6.96562823192877310 u x - 0.3456392909407282u x

v x = -3.6949607477435810 u x + 2.06388984649741410 u x

+ 0.1767828264539298u x - u

v x = 1.19369642375900510 u x -1.39207728479752510 u x +

4.05856784078061410 u x

21-3

23

2

22-3

23 -2

22

2 2

23-3

23 -2

22

-22

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

bQRDXA@RQB>cXdADEOXeU@OdADX

fgh

fgg

fgi

u2 ⋅cosα

u2

j

j

j

klmnlmopmlqrstupsnvpswvxqvpoyszv{pmoyv |rsnzqpv}xmpoyrux~v� �s�s��

���

����������� ���

������������������������������������

�� ( ) α⋅α⋅⋅− cos2sinu3

�����

���������������

���� 34��

π−α⋅⋅−

22sinu3

�����

�����������������

���� 34��������

����� 34���� ���

¡����� �����������������

������������ ���

¢£�⋅���������

��¢£�⋅���������

�������

����� 34����������

�� ���¢£�⋅���

���34

��¢£�⋅���������

�������

¤����¥��¦����§������������������������� ���¥������� ������� ����

¤�©���¦�����ª�«�¦��¬� ©���� ¥�� ( )v x' ' ' ' = 0���

( ) 43

22

31

31 CxC2

xC

6

xCxv +⋅+

⋅+

⋅= ­ ( ) 87

26

35

32 CxC2

xC

6

xCxv +⋅+

⋅+

⋅=

��

( ) 1211

210

39

33 CxC2

xC

6

xCxv +⋅+

⋅+

⋅=

®������� ���¦����§������������������������� ª������ ( )( )

( )33

23

2-33

3-33

23

23

3-33

3-32

33

3-3

31

u-x u439749860.19007113

xu1066441762.30414328xu1084927034-4.4978245=xv

u-xu32483520.19763455+

xu1003552276.33132946-xu1030720294.52491218=xv

xu6283110330.03737277-xu1092869889-2.1368236=xv

⋅⋅+⋅⋅⋅+⋅⋅⋅

⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅

� �¦ �����§��©

°±²

°±³

°±±

( )α⋅⋅α⋅ 2sinsinu3

( )α⋅⋅ 2sinu3

u3

´

´

´

µ¶· ¶·¹º·¶»¼½¾¿º½ Àº½ÁÀ»Àº¹Ã½ÄÀź·¹ÃÀ Ƽ½Ä»ºÀÇ·º¹Ã¼¿ÂÈÀÉ Ê½É½ËÌ

ÍÎÏ

ÐÑÒÓÔÕÖÔ×ÔØÙÚÓÛÙÓÜ ÝÓÒÓ×ØÞ ÒÔÚÔÙßØàÔÖÔÑÜÓáÚÔßÜÔáÚÔØÙÚÔÙÓÜÔÝÝâÖßÓãÔÜÚâ×ãØß ÝàØäâÝÚÓÛâÜâ×âÚßÓÛÞ ãÔÖåàØäßâÝÔ×ØàÖÓ ÓÚßâÕÖÓÖÔÚæåç

k P v v dx P v v dx P v v dxi j ukl i jx

ukl i jx

ukl i jx

, ' ' sin ' ' sin ' '= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅= = =∫ ∫ ∫1 1

0

5

2 20

34

3 30

34

α α

èÔãØÒØéÜæâÚÔÕâØ×âßÖÓæÙãÔßØÕØÙßÚÔ×ÔßÖÓãÔæâç

[ ]

−−−−−−

⋅=360502180.321705991321802560.053731435212916430.07746462

1321802560.05373143695094560.42819038101380430.22932414

5212916430.07746462101380430.22932414841551640.22924268

Pk uklg

êÜâÚâ ßØÕØÙßÚâ ×ÔßÖÓãâ ÙâÒÔæ ÓÑÖÔáåÚÔ×Ø Ù àØ×ØáæØ ÓÚÝâÖÑÓæâ ëâ ÑÚÔÚâàØÒÔæÚØÙßÚâ×ÔßÖÓãâÔÜÓ àÔÙàØ×ØáæØÓÑÖÔáåÚÔÚÓÛÓÚßâÖàØÜÔìÓæÙãÓÛíåÚãìÓæ ãØßç

k EI v v dx EI v v dx EI v v dxi j i jx

i jx

i jx

, " " " " " "= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅= = =∫ ∫ ∫1 1 1

0

5

2 2 20

34

3 3 30

34

èÔãØÓÑÖÔáåÚÔÚÔßØÕØÙßÚÔ×ÔßÖÓãÔæâç

[ ]

⋅⋅−⋅⋅−⋅⋅−

⋅⋅−⋅=

876

776

666

1099462111.059805081048505273.206262411059593822.76932350

1048505273.206262411069846823.198361551085796919.02745418

1059593822.769323501085796919.027454181023881044.37906572

k

ÓÚæâÙâÝâÒÔâÚÔãÔ×ÔßÖÓãÓÞ ãÓ æØÒØéÓ×ØÑÓÚÝâÖÑÓæØàØÒÔæÚØÙßÚâ×ÔßÖÓãâî

ïÖØéÜâ×ÒØÜØáÓßÝâãÖÓßÓáÚâåãÜØÚÙãâÙÓÜâÙâàÖâÝâÒâÝØéÜÓãØç

[ ] [ ]( )k k

u

u

ug− ⋅

=

1

2

3

0

0

0

ðÖÓßÓáÚØåãÜØÚÙãØÙÓÜØÒØéÓ×ØÓÑâÚÔáéâç

ñòóôõöõ⋅÷

ñ÷

ñòóôõöõ⋅÷

P

�

�

øùú�

ûüýþüýÿ�ýü�������þ���������ÿ����ýÿ� ���þ��� �ý�ÿ������ ������

���

[ ] [ ]

01002708583.06269031P1026090853.14060818

P1016435724.50272518P21415787830.00951979

0kk

21ukl

14

2ukl

63ukl

g

=⋅+⋅⋅−

⋅⋅+⋅−

=−

����������� �!��!���! "#$%& '(

)*)+,-�+�+-+.�-�+

)/01())*

+�21#$%& 3(�*

/+-4,///..,�4�) )/01(�/*

+-21#$%& 5(4*

-/+,.,-4�-4-).� )/61(4-/*

+.21