oscilacije elasticnih tela
DESCRIPTION
mehanikaTRANSCRIPT
-
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
==+=
+========
&&
&&