Download pdf - Oscilacije Elasticnih Tela

7.

, , , . . , , , . , , . . . . , , . , , , . . , . , , () , . , . 7.1

, , , . . , , . . , . , , , . A i B tako da u svakom preseku postoji normalni napon =z . , , (.7.1), .

dz dmy

7.1

F

r F r

zd+= . ( ) , , . , :

zy

) (7.1.1) ,( tzyy = . dm F

r F r ( )FFF == rr :

FFdmarrr +=

Oy :

sin)sin(22

FdFdmty

z += . (7.1.2)

zd , : dz dz

zdda z )(sin)(sinsin)sin(

==+ . ,

y , :

zy= tgsin , dz

zydz 2

2)(sin

= . :

AFAdzdVdm === , , : , A , , (7.1.1) :

==

2

2

22

2

2 0 c

zyc

ty , (7.1.3)

. , .

c

(7.1.3) .

(7.1.1) , . (7.1.3) : ) , (7.1.4) ()( tTzZy = :

TZdt

TdZtyTZT

dzZd

zy &&==

==

2

2

2

2

2

2

2

2 , .

(7.1.3) : 0 (7.1.5) 2 = TZcTZ &&, :

hTc

TZZ == 2

&&, (7.1.6)

, ,

h

z t . (7.1.5) : (7.1.7) .0 ,0 2 == ThcThZZ && , : h

(7.1.8)

).sin()cos( : 0 3)

),sinh()cosh( : 0 2)

, : 0 1)

2

2

kzDkzCZkh

kzDkzCZkh

DCzZh

+==

+==

(7.1.4) ) , : , 0( lzz == 0)( 0),( ,0)0( 0),0( ==== lZtlyZty . (7.1.9) (7.1.8) 3) , (7.1.7) :

h

(7.1.10) ),( 0 ,0 22222 ckTTZkZ ==+=+ && : ) (7.1.11) sin()cos( kzDkzCZ += )sin()cos( tBtAT += . (7.1.12) (7.1.11) (7.1.9), : (7.1.13) ,0 0)0( == CZ , )0( D , :

,0)sin( =kl (7.1.14) :

l

nkn= ),,2,1( = Kn . (7.1.15)

, (7.1.11) (17.1.2), :

n

(7.1.16) )sin( zkDZ nnn =

==+=

lcncktBtAT nnnnnn

n )sin()cos( , (7.1.17) , (7.1.4), [ )sin()sin()cos( zktBtAy nnnnnn ] += , (7.1.18) : nnnnnn DBDAA == B , . (7.1.3) (7.1.18), . :

, (7.1.19) [ )sin()sin()cos(1

zktBtAy nn

nnnn

=+= ]

:

) , (7.1.20) cos()sin(1

nnnn

n tzkRy =

= :

n

nnnn A

BBAR =+= arctg , n22 . (7.1.21)

nn BA , (7.1.19) , nnR , (7.1.20), . ) n- (.7.2) .

sin( zkR nn

. )0( 0 =t , : )( ),()0,( ,0

00 zt

yzfzytt

=

==

=, (7.1.22)

(20) :

)(sin ,)(sin11

zzl

nBzfzl

nAn

nnn

n =

=

=

= . (7.1.23)

nnA B , :

==

l

nmnm

ldzz

lmz

ln

0 21 0

sinsin ),...,2,1,( =nm . (7.1.24)

(7.1.23)

zl

msin [ ]l,0 , (7.1.24), :

=

= l lnn dzzlnz

cndzz

lnzf

lA

0 0sin)(2B ,sin)(2

. (7.1.25) (7.1.19) (7.1.20) . . , n , . 7.2 .

7.2

7.1.1 : b, , , . 7.3. .

7.3

: , , :

.0)()0,(

2),(

501

20,

50)()0,( ==

== zzy

bzbzb

bzz

zfzy & (1)

(7.1.10) :

bnc

bn

n == . (2) :

0,2

sin25

2sin50

2sin50

2

2/22

2/

0==+= n

b

b

b

n Bn

nbdz

bznzb

bdz

bznz

bA

, (3) :

=

=1

22 cossin2sin1

252),(

nt

bn

bznn

nbtzy

. (4)

7.1.2 l :

lzvzzy

lzyzfzy 5sin)()0,(,2sin)()0,( 00 ==== & . (1)

, A F . : (7.1.24) :

====

===

5,0

5,5sin5sin2

2,02,

sin2sin2

0

5

0

00

0

00

n

nFAlvv

dzlzn

lzv

lB

nny

dzlzn

lzy

lA

l

nn

l

n

(2)

:

tA

Fll

zFAlvt

AF

llzytzy

5sin5sin5

2cos2sin),( 00 += , (3) , , . 7.2 ()

( ) , . . , :

l

) , , . , ) , ) , , . , , :

Oz

z w

. (7.2.1) lztzww = 0 ),( z z :

zw

z = ,

zwEE zz == , (7.2.2)

E . (7.2.1) , . , (.7.4). dz dm

7.4 F

r F

r

.

ktwa

rr2

2

= Oz , , :

FFdma

rrr += . (7.2.3) :

)( d22

FFFFddmtw

zz == , (7.2.4)

, , :

Fdzz

dzzwAEdz

zAEdz

zAdz

zFFdAdzdm zzz 2

2 ,

==

=== , (7.2.5)

: , . (7.2.4) :

A

Ec

zwc

tw ==

2

2

22

2

2 , 0 (7.2.6)

. . .

c

(7.2.6) (7.1.3) , . , , ) , 7.2.7) ()( tTzZw = , (7.2.6), :

===+=+

EkckTTZkZ 222222 0 ,0 && , (7.2.8)

: )sin()cos( ),sin()cos( tBtATkzDkzCZ +=+= . (7.2.9) (7.2.9), (7.2.7).

k

( )

. , , , . . 1. . .7.5 , .

7.5 : ,0),( ,0),0( == tlwtw (7.2.10) , (7.2.7) (7.2.9) : .0)sin(0)( ,00)0( ==== klDlZCZ (7.2.11) 0D , (7.2.11) :

),,2,1( 0)sin( === Knl

nkkl n . (7.2.13)

, . , :

,sincos ,sin

+

=

= ctl

nBctl

nATzl

nDZ nnnnn (7.2.14)

+

= =

zl

nctl

nBctl

nAwn

nn sinsincos

1, (7.2.15)

: nnnnnn DBDAA == B , . 2. . (.7.6) .

7.6 , , , , , :

,0 ,00

=

=

== lzz zw

zw (7.2.16)

, (7.2.7) (7.2.9) : 0)sin(0)( ,00)0( ==== klClZDZ . (7.2.17) )0( C :

),,2,1( 0)sin( === Knl

nkkl n . (7.2.18)

(7.2.9):

+

=

= ctl

nBctl

nATzl

nCZ nnnnn sincos ,cos , (7.2.19)

:

=

+

=1

cossincosn

nn zlnct

lnBct

lnAw . (7.2.20)

3. . .7.7 .

7.7 :

,0 ,0),0( =

=

=lzzwtw (7.2.21)

: ,0)cos(0)( ,00)0( ==== klDlZCZ (7.2.22) :

),,2,1( 2

)12( 0)cos( ,0 === Knl

nkklD n , (7.2.23)

:

=

+

=1 2

)12(sin2

)12(sin2

)12(cosn

nn zlnct

lnBct

lnAw .(7.2.24)

4. . A (.7.8) , , .

1m

7.8

A , (7.2.7) (7.2.8),

)()()()( 222

tTlZtTlZtwa

lzA ==

=

=&& , (7.2.25)

, , A: (7.2.26) ).()(211 tTlZmamF A == , , , , , , , :

).()(),( 211 lZmlZAEamzwAEFtlA A

lzz ==

=

= (7.2.27)

, , (7.2.9) (7.2.27), , (7.2.28) )sin()cos( ,0 ,0 21 klkcmklAEDC ==, : AlmEc == ,

2 , :

klmmkl 1)(ctg = . (7.2.29)

xy ctg= xmmy 1= . ),,2,1( = Knxn

, lxk nn =

.

7.9

.7.9 ,

, : n

),5,4 ( K=n

lnkn

)1( . (7.2.30) 5. . (.7.10) , , , , .

n

7.10

(7.2.6),

),,,2,1( 022

22

2ni

zwc

tw

i

ii

i K==

(7.2.31) , .7.10 . , n2

) ,0( 1 nn lzz == , , 22 n . , :

),1,,2,1( ),0(),( ),,0(),( 11 === ++ nitFtlFtwtlw iiiiii K (7.2.32) :

),0()( ),0()( 1111 ++++ == iiiiiiiiii ZAElZAEZlZ )1,,2,1( = ni K (7.2.33) , , , , : 11 ++= iiii kckc . (7.2.34) 7.2.1: c l . A ( E).

Slika 7.11

: . 7.2 , , (7.2.24):

( ) ( )[ ] ( )l

nkEczkctkBctkAw nn

nnnnn 2)12(,,sinsincos

1

==+==

. (1)

:

,0)0,()( == zwz & (2) :

.0=nB (3) , :

,1000

lclcF == (4)

:

,100

)0,( lczzwEA =

(5) :

),0,0(100

)0,( wzEAlczw += (6)

, (): .0)0,0( =w (7)

:

,100

)0,()( zEAlczwzf == (8)

:

).cos(sin50

sin100

22

0lklklk

EAkcdzzkz

EAlc

lA nnn

n

l

nn == (9) (7.2.23) :

)cos(sinsin150

),(1

2 ctkzklkkEActzw nn

nn

n=

= . (10) 7.2.2: , m, l, A E, , F . . F .

7.12

: (7.2.15):

( ) ( )[ ] ( )l

nkm

EAlEczkctkBctkAw nn

nnnnn

===+=

=,,sinsincos

1. (1)

:

.0=nB (2) :

7.12

:

,2,0

12,)1(2

sin2

sin1

=

===

snsnnlk sn (10)

(16) :

.)12(cos)12(sin)12(

)1(2),(1

2

1

2 mEAl

lts

lzs

sEAFltzw

s

s

= =

(11)

7.2.3: AB m, l, A E, OA BD . OA , R, BD R. , AB.

xEI

: . 7.13. (. 7.13). [XXX] :

7.13

,3

31

1xEI

RFf = (1) ( Alberto Castigliao, 1847-1888) [XX]:

f2

,2

2 FAf d= (2)

7.13

:

=2/

0

2 .2

1 Rf

xd dsMEI

A (3)

(. 7.13) : .sin2 RFM f = (4)

(4) (3) (2) :

.4

])sin(2

1[3

22/

0

22

22

xx EIRFRdRF

EIFf

== (5) , :

.4,3 32

223

1

11 R

EIfFc

REI

fFc xx ==== (6)

A B. AB - :

),(),(),,0(),0( 21 tlwcztlwEAtwc

ztwEA ==

, (7)

7.13

:: ),()(),0()0( 21 lZclZEAZcZEA == (8)

:

.0)cossin()cossin(,0)()(

22

1

=+++=+

DklEAkklcCklcklEAkDEAkCc

(9)

(10) :

,0cossincossin 22

1 =++

klEAkklcklcklEAkEAkc

(10)

:

,)(

)()(tg

2221

22

21

AEcclkl

klEA

cclkl

+= (11)

(6). 1c 2c 7.2.4: m, l A, . 7.14. 2 1 3 E, .

7.14

: 1 3 , :

),()sincos()()(),(),()sincos()()(),(

2222222222

1111111111

tTzkDzkCtTzZtzwtTzkDzkCtTzZtzw

+==+==

(1)

: .sincos)( tBtAtT += (2)

1 3 :

,21 mEAlEccc ==== (3)

: 1k 2k

.21 ckkk === (4)

, O : ,0),0(1 =tw (5)

C: 0),(2 = tlw , (6)

A B : ).,0(),( 21 twtlw = (7)

, 2, , :

),,(),0(),0( 122 tlwEAtwEAtwm =&& (8) . 7.14 2.

7.14

(1) (2) (5), (6), (7) (8) :

),()0()0()(),0()(

,0)(,0)0(

1222

21

21

lZEAZEAZkcmZlZ

lZZ

====

(9)

:

),cossin(

,sincos,0cossin

,0

112222

211

22

1

klkDklkCEAEAkDCcmk

CklDklCklkDklkC

C

+==+

=+=

(10)

:

,0cos

01sincossin0

2=

EAmkcklEAkl

klkl (11)

: .0)cossin(cossin 22 =+ klEAklmkcklklEA (12)

(12) (3) :

klkl 22tg = . (13)

7.14

(13) . 7.14 . :

,...90594.4,40701.3,96758.1,63230.0 4321 ==== lklklklk (14)

7.2.5: 1, 2A, 2, 4 A, m. l

E. c . 7.15. , .

7.15

: :

.0

2cos)2(2sin)3(

22sin)2(2cos)3(00

2sin2)2()2cos1(2cos2sin201sincos002

21

221

2=

++

klkEAklcmkc

cklkEAklcmkc

klckEAklcklEAkklEAkklkl

EAkc

(11)

7.2.6 .7.16 (1-5) .. .

7.16

:1.

),(),(2]6[),(2),0(2),0(]5[

),0(),(]4[),0(2),(]3[),0(),(]2[0),0(]1[,,

33233

3221

211321

tlwctlwAEtlwAEtwAEtwm

twtlwtwAEtlwEAtwtlwtwEkkkkk

====

======

&&

2.

0),(]6[)),0(),0((2),2(),0(),0(]5[

),0(),2(]4[)),0(),0((2),2(),0(),0(]3[

),0(),2(]2[0),0(]1[,,2,

323233

3223122

211321

===+=

======

tlwEAtwtwctlwEAtwEAtwm

twtlwtwtwctlwEAtwEAtwm

twtlwtwEkkkkkk

&&&&

3.

),(2),(),(]6[

),(2),0(2),0(]5[),0(),(]4[),0(2),(]3[

),0(),(]2[0),0(]1[,,

333

233

3221

211321

tlwAEtlwctlwm

tlwAEtwAEtwmtwtlwtwAEtlwEA

twtlwtwEkkkkk

====

======

&&&&

4.

0),(]6[),(),0(),0(]5[),0(),(]4[)),(),0((),0(]3[)),(),0((),(]2[

0),0(]1[,,,,2,

323332

122121

1231

============

tlwtlwEAtwEAtwmtwtlwtlwtwctwEAtlwtwctlwEA

twAlmlEAcEkkkkkk

&&

5.

0),2(]6[),0(),2(]5[),0(),2(]4[),0(),(]3[

),0(),(]2[0),0(]1[,)2(,2,

3

323221

211231

====

======

tlwtwEAtlwEAtwtlwtwEAtlwEA

twtlwtwEkkkkkk

7.3 () , , () . , (.7.17) ( , ), ),( tz = .

7.17 () :

z

GIM oz = , (7.3.1)

: G , oI . (.7.17),

dz zM zM .

zzz MMtdJ =

2

2 , (7.3.2)

. :

zdJ Oz

, , 22

dzz

GIdzz

MMdMMdzIdJ ozzzzoz =

=== (7.3.3) , (7.3.2) : . 0 22

22

2

2

==

Gcz

ct

(7.3.4)

(7.1.3) (7.2.6) , , . : )()( tTzZ= (7.3.5) (7.3.4) : (7.3.6) ),( 0 ,0 22 kcTTZkZ ==+=+ && :

)sin()cos( ),sin()cos( tBtATkzDkzCZ +=+= . (7.3.7) () , 7.2 . 7.3.1: 2l D d l. G. .

7.18

: :

)2,1(,22

22

2== i

zc

t iii

, (1)

:

)2,1(),()(),( == itTzZtz iiii , (2) :

.sincos)()2,1(,sincos)(

tBtAtTizkDzkCzZ iiiiiiii

+==+=

(3)

:

Gc = , (4)

: kkk == 21 . (5)

O : 0),0(1 =t , (6)

B ( ): 0),(22 = tlGIO , (7)

A : ),0(),( 21 ttl = , (8)

: ),0(),( 2211 tGItlGI OO = , (9)

:

32)(,

32

44

2

4

1 dDIDI OO == . (10)

:

,32

)()cossin(32

,sincos,0cossin

,0

2

44

11

4211

22

1

kDdDGklkDklkCDG

CklDklCklkDklkC

C

=+=+

=+=

(11)

:

,0)(0cos

01sincossin0

444=

dDklDkl

klkl (12)

:

44

42tg

dDDkl = . (13)

7.3.2: 3l - . 1 3 0I

G, 2 . , .

G

: , , :

lIGc 0

= . (1)

Slka 7.19

:

),sin(

),sin(cos

,0sincos

122

121

22

klDCGGlkD

klDCGDklGlk

klDklC

==

=+

(2)

: 0cos)sin2cos( =+ klklGklGkl . (3)

7.3.3: m, l, D G .

, . .

0M

7.20

: :

2

22

2

2 ),(),(z

tzct

tz

= . (1)

:

mlGDGc

4

2 == . (2)

, O : 0),0( =t , (3)

A : 0M

00 ),( MtlGI = , (4) :

32

4

0DI = . (5)

:

zGIMtztz

0

0),(),( += , (6) (1) :

2

22

2

2 ),(),(z

tzct

tz

= , (7)

(3) (4) :

.0),(,0),0(

==

tlt

(8)

),( tz = . :

.0)0,()0,()(

,)0,()0,()(0

0

0

0

======

zzz

zGIMz

GIMzzzf

&&

(9)

(7) (8) (9) :

...,2,1,2

12 == nl

nkn (10) :

tckzkk

lklGI

Mtz nnn n

n cossinsin2),(1

20

0 =

= . (11) (11) (6) :

)cossinsin2(),(1

20

0 tckzklk

lkzGIMtz nn

n n

n=

= , (18) c, (2), (5) (10). 0I nk 7.3.4 .7.21 (1-5) . . .

7.21

: 1.

0),(]6[),(),0(),0(2]5[

),0(),(]4[),(),0(),0(]3[

),0(),(]2[0),0(]1[,,

320303

3210202

211321

====

======

tltlGItGItJ

ttltlGItGItJ

ttltGkkkkk

&&&&

2.

),2(),2(]4[

),(2),0(),(4),0(3]3[

),0(2),(]2[0),0(]1[,,

202

102012

21121

tlGItlJ

tlGItGItlJtJ

ttltGkkkk

=+=

=====

&&&&&&

3.

),(),(]6[),0(),(2]5[

),0(),(]4[),(2),0(2),0(]3[

),0(),(]2[0),0(]1[,,

31303020

3210202

211321

tlctlGItGItlGI

ttltlGItGItJ

ttltGkkkkk

==========

&&

4.

0),2(]6[),0(),(]5[),0(),(]4[),0(),(]3[),0(),(]2[

0),0(]1[15,16,,

303303202

3220210121

101030102321

=====

=======

tlGItGItlGIttltGItlGIttl

tIIIIGkkkkk

5.

0),(]6[),0(2),(]5[),0(),(]4[),0(),2(2]3[

),0(),2(]2[0),0(]1[,,2,

33020

322010

211231

====

======

tltGItlGIttltGItlGI

ttltGkkkkkk

7.4

(.7.22), .

yOzOz

7.22

: ) ,

v Ozl

) , ) . , , : dz AdzdVdm == , (7.4.1) : , A . TF

r TF

r

(.7.22), , , :

fMr

fM r

22

, zvEIM

zM

F xff

T =

= , (7.4.2) : E , xI Ox . Oy

YYdmtv =

2

2, (7.4.3)

cos)cos( TT FdFYY += . 1)cos(cos + d , , (7.4.2),

dzzvEIdz

zFFdFFYY xTTzTT 4

4

=

=== . (7.4.4) (7.4.1) (7.4.4) (7.4.3) :

==+

A

EIczvc

tv x

2

4

42

2

2 0 . (7.4.5)

, , , : ) (7.4.6) ,( tzvv = , . : ) , (7.4.7) ()( tTzZv = :

)()( ),()( 44

2

2tTzZ

zvtTzZ

tv IV=

= && .

(7.4.5), , :

hTc

TZ

Z IV == 2&&

,

, , , . , (7.4.5) :

h4kh =

(7.4.8) ).( 0

,022

4

ckTT

ZkZ IV

==+=

&&

zaeZ = , : 0 ,0 0 222244 ==+= kkk : kik == 4,32,1 , . (7.4.9) , (7.4.8) , . : )sinh()cosh()sin()cos( 4321 kzCkzCkzCkzCZ +++= , (7.4.10) : . (7.4.11) )( )sin()cos( 2cktBtAT =+= . (7.4.5) z, . .

k

1. (.7.23) :

.0),(00),(

,0),(0),(

,0),0(00),0(

,0)0(0),0(

2

2

02

2

==

===

==

===

=

=

tlZzvtlM

tlZtlv

tZzvtM

Ztv

lzf

zf

(7.4.12)

7.23

, (7.4.10), :

(7.4.13)

,0)sinh()cosh()sin()cos(0)sinh()cosh()sin()cos(

00

4321

4321

31

31

=++=+++

=+=+

klCklCklCklCklCklCklCklC

CCCC

: 0)sin( ,0 ,0 ,0 ,0 2431 ==== klCCCC , (7.4.14)

:

),,2,1( , 2

=

== Kncl

nl

nk nn . (7.4.15)

(7.4.10) (7.4.11) :

+

=

= ctl

nBctl

nATzl

nCZ nnnnn sincos ,sin2 , (7.4.16)

(7.4.7) :

+

= zl

nctl

nBctl

nAv nnn sinsincos , (7.4.17)

: . (7.4.18) nnnnnn CBCAA 22 B , == ,

=

+

=1

sinsincosn

nn zlnct

lnBct

lnAv , (7.4.19)

= = nn n

ctl

nzl

nRv cossin1

, (7.4.20)

:

n

nnnnn A

BBAR =+= arctg , 22 . (7.4.21)

:

)( ),()0,(0

ztvzfzv

t=

=

=, (7.4.22)

, :

=

= l lnn dzzlnz

cnBdzz

lnzf

lA

0 0

sin)(2 ,cos)(2 . (7.4.23)

2. (.7.24), :

l

.0)(0),(

,0)(0),(

,0)0(0

,0)0(0),0(

0

======

==

=

lZtlF

lZtlM

Zzv

Ztv

T

f

z (7.4.24)

7.24

, (7.4.10), :

.0)cosh()sinh()cos()sin( 0)sinh()cosh()sin()cos(

0 0

=++=++

=+=+

klDklCklBklAklDklCklBklA

DBCA

(7.4.25)

, , . :

0

)cosh()sinh()cos()sin()sinh()cosh()sin()cos(

10100101

=

klklklklklklklkl

, (7.4.26)

:

1coscosh =klkl (7.4.27)

7.24

, , . .7.24 . 7.4.1: AB E, l, m, A, F. F. ( 3.10.8).

xI

7.25

: :

00)( 0

===

=n

tBz

tv (1)

: )()0,( zfzv = (2)

. :

)( zfEIM xf = (3) :

=,

2),

2(

2

,2

0,2

lzllzFzF

lzzF

M f (4)

: ,0)(,0)0( == lff (5)

:

++

+=

.2

,])2

(61

483

12[

,2

0),48

312

()(

323

23

lzllzzlz

EIF

lzzlzEIF

zf

x

x (6)

(7.4.23) :

++=

l

ln

l

nx

n dzzklzzzldzzkzzl

lEIFA

2/

3322/

0

32sin])

2(

61

1216[sin)

1216(2 , (7)

:

2sin2

2sin2 44

3

4

n

EInFllk

klEIFA

x

n

nxn == . (8)

:

=

=

12

22

44

3cossin

2sin12),(

n

x

xt

mlEI

ln

lznn

nEIFltz , (9)

, :

=

=

12

22

4

1

4

3 )12(cos)12(sin)12(

)1(2),(p

xp

xt

mlEI

lp

lzp

pEIFltz . (10)

( 3.10.8) :

333231 14)21116(332,

2332,

14)21116(332

mlEI

mlEI

mlEI xxx +=== . (11)

(7.4.15):

,9,4, 32

232

232

1 mlEI

mlEI

mlEI xxx === (12)

:

1 2 3 ( ) / i i i 100% -0.03% -0.73% -6.75%

. , . . 3.23a. (9) :

)3,2,1(),/sin( =nlzn , (13) :

z n 1 2 3 l 4 2 2 1 2 2 l 2 1 0 -1

3l 4 2 2 -1 2 2 . 7.4.2: AB, m, l, E , B . .

Ixc

0=c =c .

Slka 7.26

Reeje: : )()sinhcoshsincos(),( 4321 tTkzCkzCkzCkzCtz +++= . (1)

, A : 0),0(,0),0( == tt , (2)

B : 0),( = tlEIx , (3)

: ),(),( tlctlEIx = . (4)

(1) (2), (3) (4) :

),sinhcoshsincos()coshsinhcossin(

,0sinhcoshsincos,0,0

4321

43213

4321

42

31

klCklCklCklCcklCklCklCklCkEI

klCklCklCklCCCCC

x

+++==++

=++=+=+

(5)

: 4321 ,,, CCCC

0

sinhhcos

coshhsin

sincos

cossin

sinhcoshsincos10100101

3333=

klcklkEI

klcklkEI

klcklkEI

klcklkEI

klklklklxxxx

, (6)

: .0)cossinhsin(cosh)coscosh1(3 =++ klklklklcklklkEIx (7)

(. 7.26) (7):

0=c

.1coscosh =klkl (8)

7.26 7.26 c B (. 7.26). (7) :

=

0lim3= c

kEI xc

, (9)

: klkl tgtgh = . (10)

7.4.3: , l, A, xEI , m R. . Reeje: :

)()sinhcoshsincos(),( 4321 tTkzCkzCkzCkzCtz +++= , (1) :

AEIkcktBtAtT x

22,sincos)( ==+= , (2) O :

0),0(,0),0( == tt . (3)

7.27

B :

.21,

,

2mRJRFMJ

Fm

tBfB

tBT

=+==

&&

&& (4)

(. 7.27):

).,(),,(tlEIM

tlEIF

xfB

xtB

==

(5)

7.27a

T (. 7.27) :

),,(),,(),(

tltlRtlT

=+=

(6)

:

).,(

)],,(),([2

2

tl

tlRtlT

=

+=&&&&

(7)

7.27

(1), (2), (5) (7) (3) (4) :

.0

cosh2sinh2cosh

sinh2cosh2

sinh

cos2sin2cos

sin2cos2sin

coshcosh

sinh

sinhsinh

cosh

coscos

sin

sinsin

cos10100101

32323232

2222

=

++

++

+

+

+

klARkklA

klkmR

klARkklAklkmR

klARkklA

klkmR

klARkklAklkmR

klAklRmk

klmk

klAklRmk

klmk

klAklRmk

klmk

klAklRmk

klmk

(8)

7.4.4: m, 2l, E, A , R, . .

Ix 1m

7.28

: AM NB, :

),()sinhcoshsincos(),(),()sinhcoshsincos(),(

2423222122

1413121111

tTkzDkzDkzDkzDtztTkzCkzCkzCkzCtz

+++=+++=

(1)

:

mlEIkcktBtAtT x2,sincos)( 22 ==+= . (2)

A B :

.0),(,0),0(,0),(,0),0(

21

21

====

tlEItEItlt

xx

(3)

, M N :

).,0(),(),,0(),(

21

21

ttlttl

=

= (4)

, , - M N, (. 7.28) :

.41,)],0([

,),0(

2122

21

RmJMMtt

J

FFtm

fNfM

tMtN

===

&&

(5)

7.28

:

).,0(

),,(),,0(),,(

2

1

2

1

tEIM

tlEIMtEIFtlEIF

xfN

xfM

xtN

xtM

==

==

(6)

:

),,0()],0([

),,0(),0(

22

22

22

22

ttt

tt

=

=&& (7)

(1), (2) (7) (3), (4) (5), : C C1 3 0= = . (8)

:

0

22sinhsin22coshcos

1010coshcos0101sinhsin

sinhcoshsincos00sinhcoshsincos00

3311

=

lJkmlJkmklmklmmlkmmklmklmklm

klklklkl

klklklklklklklkl

(9)

7.4.5 .7.29 ( 1-10 ) , . . .

7.29 : 1.

0),2(]8[0),2(]7[))),(()),0((),0(]6[),0(),(]5[),0(),(]4[),0(),(]3[

0),0(]2[0),0(]1[,)(,2,

22122

212121

1124

21

======

=====

tltltlEItEItmtEItlEIttlttl

ttAEIkkkkk

xx

xx

x

&&

2.

),2(),2(16

]8[

)),2((),2(]7[),0()),2((),0(]6[

),0(),2(]5[),0(),2(]4[),0(),2(]3[0),0(]2[0),0(]1[,)(,

22

222212

212121

112

21

tlEItlmD

tlEItlmtEItlEItc

tEItlEIttlttlttAEIkkkk

x

xxx

xx

x

===+

========

&&

&&

3.

0),(]8[0),(]7[

)3

3),0(6

3),(),(),0((),0(12

]6[

)),(()),0(()),0(3

3),0((]5[

),0(),0(2

3),(]4[),0(),(]3[

0),0(]2[0),0(]1[,)(,

22

21122

2

1222

22121

112

21

==+++=

=

=+======

tltl

atatltltEItma

tlEItEItatm

ttatlttl

tEItAEIkkkk

x

xx

xx

&&

&&&&

4.

),(2

),(),(12

5]8[

)),(()),(2

),((]7[),0(),0(),(]6[

),0(),(]5[),0(),(]4[),0(),(]3[0),0(]2[0),0(]1[,)(,

222

2

222221

2121

21112

21

tlEIatlEItlma

tlEItlatlmtEItctlEI

tEItlEIttlttlttAEIkkkk

xx

xxx

xx

x

=

=+=+==

======

&&

&&&&

5.

),2(),0()),0(),0((]5[),0(2),2(),0(]4[),0(),2(]3[

0),0(]2[0),0(]1[,)(,

1222

21221

112

21

tREItEItRtmtRtRtttR

tEItAEIkkkk

xx

xx

+=+==

=====

&&&&

0),2(]8[0),2(]7[

)),0(),2(),0(),2((),0(2

]6[

22

21212

2

==++=

tRtR

tRtRRttREItmR x

&&

6.

),(),(]8[0),(]7[),0(),2(),0(]6[

),0(),2(]5[),0(),2(]4[),0(),2(]3[0),0(]2[0),0(]1[,)2(,2,

222

212

212121

1124

21

tlEItlmtlEItctlEItEI

tEItlEIttlttltEItAEIkkkkk

xx

xx

xx

xx

==+=

========

&&

7.

0),2(]8[0),2(]7[),0(),3(),0(]6[

),0(),3(]5[),0(),3(]4[),0(),3(]3[),0(),0(]2[0),0(]1[,)(,

22212

212121

11112

21

==+=========

tltltctlEItEItEItlEIttlttl

tctEItAEIkkkk

xx

xx

xx

8.

0),2(2]8[0),2(]7[),0(2),3(]6[),0(2),3(]5[),0(),3(]4[),0(),3(]3[

0),0(]2[0),0(]1[,)(,2,

2221

212121

1124

21

===========

tlEItltEItlEItEItlEIttlttl

tEItAEIkkkkk

xxx

xx

xx

9.

0),(]8[0),(]7[),0(),(),0(]6[),0(),(]5[0),0(]4[

0),(]3[0),0(]2[0),0(]1[,)(,

22

2112212

1112

21

===========

tlEItltctlEItEIttlt

tlttAEIkkkk

x

xx

x

10.

),(),(]8[0),(]7[

),0(),(),0(12

)2(]6[

),(),0(),0(]5[),0(),(]4[),0(),(]3[0),0(]2[0),0(]1[,)(,

222

212

21222121

112

21

tlctlEItlEI

tEItlEItam

tlEItEItmttlttlttAEIkkkk

xx

xx

xx

x

==+=

+========

&&

&&