Ի Ի ՈՒԹ ՈՒ Ի Ի Ի Ի Ղ Ի
65, №3, 2012
539.374
Ы
K Ш
Ц
и . .
ы : , , ,
.
Keyword: complex modules,complex compliant, dissipative forces, linear und exponential approximation.
. .
-- ,
, - :
,
ox
-, :
և :Խ ,
և ,
:
Vermishyan G.B.
Distribution of temperature in of visco-elastic plate with a washer imbedded under the effect of vibration
The distribution of temperature in an infinite plate of visco-elastic material with a circular hole,into which is embedded visko-elastic circular disc from another viaco-elastic material is examined. Applied load is a tensile force acting at infinity in the direction of the
ox
axis, which varies harmonically with constant amplitude. The case of omnidirectional tension of the plate is considered as well.The problem is solved under the condition that the dividing line of two-body temperature and heat flux are equal , and on the lateral surface of the plate and the washer is loose heat to the environment.
Calculation is made for the case of omnidirectional tension.
,
.
,
ox
, ..
, ,
.
.
,
.
. , .
,
ox
,,
p
0.[1],
. , ё
,
,
, . ,
,
, . . o
oz
.(
0)
T
T
T
z
∂
= α
−
∂
,α
– ., .
1. и .
R
,.
,
p
=
p
0cos
ω
t
,
p
0. ,
J
/ /(
T
,
ω
)
J
/(
T
,
ω
)
.ё
. 1
, , – 2.
[2]
( ) ( )
0
cos
,
j j
t
ρ ρ
σ = σ
ω
σ = σ
( )ϑj ( )0θjcos
ω
t
,
( )j ( )0jcos
t
,
(
r
,
j
1, 2
)
R
ρϑ ρθ
τ = τ
ω ρ =
=
(1.1)( ) ( ) ( ) 0
,
0,
0j j j
ρ ϑ ρϑ
σ ϑ τ
– .,
. .
ё [3 ]
( ) ( )
∫
(
)
( )( )
( )∫
(
)
( )( )
∞ − ∞
−
−
−
−
−
+
=
t jj t
j j j
j j
j j
d
t
T
K
E
d
t
T
K
E
,
ν
,
τ
σ
τ
τ
,
νσ
τ
τ
σ
τ
σ
ε
ϑ ϑρ ρ
ρ
(1.2)
( ) ( )
∫
(
)
( )( )
( )∫
(
)
( )( )
∞ − ∞
−
−
−
−
−
+
=
t jj t
j j j
j j
j j
d
t
T
K
E
d
t
T
K
E
,
ν
,
τ
σ
τ
τ
,
νσ
τ
τ
σ
τ
σ
ε
ρ ρϑ ϑ
ϑ
( )
(
)
( )(
)
∫
(
)
( )( ) (
)
∞ −
=
−
+
+
+
=
t jj j
j j
j
d
t
T
K
E
2
1
,
,
1
,
2
.
1
2
ν
τ
ν
τ
τ
τ
τ
γ
ρϑ ρϑ ρϑ( )j
,
( ) ( )j,
jρ ϑ ρϑ
σ σ
τ
(1.1) (1.2),ς = − τ
t
(
)
( )
(
)
(
)
(
)
( )(
)
( )(
)
* / / /
0
0 0
,
,
,
,
,
cos
,
sin
1, 2 ,
i
j j j j j j
j j
j j
J
T
K T
d
J
T
iJ
T
J T
iJ T
j
∞ − ως
ω =
ς
ς =
ω −
ω =
=
ω
φ −
ω
φ
=
∫
( )
(
( ) ( ))
1
/( )
,
cos
//( )
,
sin
,
00
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
=
J
T
t
J
T
t
E
j j jj j
j
σ
νσ
ω
ω
ω
ω
ε
ρ ρ ϑ
( )
(
( ) ( ))
1
/( )
,
cos
//( )
,
sin
,
0 0
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
−
=
J
T
t
J
T
t
E
j j jj j
j
σ
νσ
ω
ω
ω
ω
ε
ϑ ϑ ρ(1.4)
( )
(
)
( )( )
( )
,
sin
,
cos
,
1
1
2
0 / //⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
+
=
J
T
t
J
T
t
E
j j jj
j
ν
τ
ω
ω
ω
ω
γ
ρϑ ρϑ
, ,
( )j ( )j ( )j ( )j ( )j ( )j
(
1, 2
)
jd
d
d
W
dt
dt
dt j
dt
dt
dt
π π π
ω ω ω
ρ ϑ ρϑ
ρ ϑ ρϑ
π −π −π
− ω ω ω
ε
ε
γ
=
∫
σ
+
∫
σ
+
∫
τ
=
(1.5)(1.1) (1.4) (1.5),
(
)
( )
( ) 2 ( ) ( )( )
( ) 2( )
( )
( ) 2(
)
/ /0 0 0 0 0
,
j2
j j j2 1
j1, 2 .
j j j
W
= π
J
T
ω
⎢
⎡
σ
ρ− νσ σ + σ
ρ ϑ ϑ+
+ν τ
ρϑ⎤
⎥
j
=
⎣
⎦
(1.6), ,
(
)
*
1, 2 ,
2
j
j j
k
q
=
λω
W
j
=
π
(1.7)– , ,
λ
– ,, .
λ
λ
=
1
. [1],(
)
(
)
*
*
1
,
,
J
T
E T
ω =
ω
,(
)
(
)
(
)
(
)
/ /
2 2
/ / /
,
,
,
,
,
E T
J
T
E T
E
T
ω
ω =
⎡
ω
⎤
+
⎡
ω
⎤
⎣
⎦
⎣
⎦
(
)
(
)
(
)
(
)
/ / / /
2 2
/ / /
,
,
,
,
E
T
J
T
E T
E
T
ω
ω =
⎡
ω
⎤
+
⎡
ω
⎤
⎣
⎦
⎣
⎦
. (1.8)
(
)
/
,
j
E T
ω
(
)
/ /
,
j
E
T
ω
, ,(
)
/
,
j
E T
ω
(
)
(
)
(
)
/ / /
,
,
,
1, 2
j j j j j j j j
E T
ω =
A E
T
ω =
B
+
C T
j
=
. (1.9),
E
/ /(
T
j,
ω
)
<<(
)
/
,
j
E T
ω
,E
/ /(
T
j,
ω
)
(
)
/
,
j
E T
ω
(
)
(
) (
)
/ / /
2
1
,
,
,
J j jj j j j
j j
B
C T
J
T
J
T
A
A
+
ω =
ω =
. (1.10)
( )
( ) / /(
)
(
)
(
)
0
,
00
1, 2
j
j j j j j j j
L T
≡ Δ + μ
T
J
T
ω
f
− α
T
−
T
=
j
=
, (1.11)( ) * 02
0
2
j j
j j
k p
a c
λω
μ =
,(
)
( )
( ) 2 ( ) ( )( )
( ) 2(
)
( )
( ) 2(
)
0 0 0 0
,
j2
j j j2 1
j1, 2
j j
f
ρ ϑ = σ
ρ− νσ σ
ϑ+ σ
ϑ+
+ ν τ
ρϑj
=
(1.12)1 2 *
2 2
2 1
2
,
,
,
,
,
,
T
T
T r
R
dT
dT
k
k
r
R
dr
dr
T
r
=
=
=
=
=
< ∞
→ ∞
(1.13)
(
T
*– ), . (1.10) ,
(1.12)
(
)
(
)
,
j j j j1, 2 ,
j
B
C B
r
u
j
R
A
+
= ρ
=
=
(1.14)( )
( )
( ) ( )
( )(
)
1 1 1
,
1,
1,
1,
0
1, 0
2
,
L u
≡ Δ
u
ρ ϑ = −
F
ρ ϑ
u
ρ ϑ − Φ ρ ϑ
≤ ρ <
≤ ϑ ≤ π
(1.15)( )
(
)
(
)
(
,
) (
,
)
(
,
)(
1
,
0
2
)
,
,
,
2 2
2
2 2 2
2 2
π
ϑ
ρ
ϑ
ρ
ϑ
ρ
ϑ
ρ
ϑ
ρ
ε
ϑ
ρ
≤
≤
∞
<
<
Φ
−
−
=
−
Δ
≡
u
F
u
u
u
L
(1.16)
,
1
1
2 2
2 2
2
ϑ
ρ
ρ
ρ
ρ
∂
∂
+
∂
∂
+
∂
∂
=
Δ
(
)
2(
1 1 0)
( )11 1
1
,
R
B
C T
,
A
α
+
Φ ρ ϑ =
= α
(
)
(
)
( )2
2
2 2 0
2 1
2
,
R
B
C T
,
A
α
+
Φ ρ ϑ =
= α
(1.17)
( )2 2 2 ( )02 2 2
(
)
2 ( )022 2
0
2 2
2 1
,
C
R C
R
A
A
+ ν
μ
μ
α =
ε = α
−
(
)
1
,
F
ρ ϑ
F
2(
ρ ϑ
,
)
– ,T
0–. (1.14) (1.13)
:
(
)
1 * 1 1 1
1
,
A
T
C
B
u
ρ
ϑ
ρ==
+
,
(
,
)
,
2 * 2 2 1 2
A
T
C
B
u
ρ
ϑ
ρ==
+
.
,
,
2
1 1
1 1 1 1 2
2 2 2
∞
→
∞
<
=
==
ρ
ρ
ρ
ρ ρu
d
du
C
A
k
d
du
C
A
k
(1.15) (1.16) (1.18)
(
ρ
ϑ
)
=
( )
ϑ
+
2∫∫
π(
ρ
ϑ
ρ
ϑ
) (
μ
ρ
ϑ
)
ρ
ρ
ϑ
0 1
0
* 1 0
0 1 0
* 1 0 0
1
,
u
G
,
;
,
,
d
d
,
u
(1.19)
(
0
≤
ρ
0<
1
,
0
≤
ϑ
0≤
2
π
)
,
(
)
=
( )
+
∫∫
π∞(
ρ
ϑ
ρ
ϑ
) (
μ
ρ
ϑ
)
ρ
ρ
ϑ
ρ
ϑ
ϑ
ρ
20 1
* 2 0
0 2 2
0 0 * 2 0 0
2
,
G
,
;
,
,
d
d
,
u
u
(1.20)
(
1
<
ρ
0<
∞
,
0
≤
ϑ
0≤
2
π
)
1
G
– ,G
2–
(
)
(
(
)
)
,
cos
2
cos
2
1
ln
4
1
,
;
,
20 0 0
2
2 0 2 0 0
0 0
1
ρ
ρρ
ϑ
ϑ
ρ
ρ
ρ
ϑ
ϑ
ρρ
π
ϑ
ρ
ϑ
ρ
+
−
−
+
−
−
=
G
(1.21)
(
)
(
)
(
) (
)
2 0 0 0 0 0 0
1
1
,
; ,
,
2
,
cos
,
2
n nG
W
W
∞
=
⎡
⎤
ρ ϑ ρ ϑ =
⎢
ρ ρ +
ρ ρ
ϑ − ϑ
⎥
π ⎣
∑
⎦
(
)
( ) ( )
( ) ( )
( )
( )
( ) ( )
( ) ( )
( )
( )
0 0 0
0
0
0
, 1
,
,
, 1
,
n
n n n n
n n
n
n n n n
n
K
I
K
K
I
K
W
K
I
K
K
I
K
ερ
⎧
ερ
ε −
ερ
ε
≤ ρ < ρ
⎡
⎤
⎪⎣
⎦
ε
⎪
ρ ρ = ⎨
ερ
⎪
⎡
ερ
ε −
ερ
ε
⎤
≤ ρ < ρ
⎣
⎦
⎪
ε
⎩
(1.22)
( )
x
I
n – ,K
n( )
x
– .(
)
1
,
u
ρ ϑ
(1.19) (1.15),(
)
2 1(
) (
)
(
)
* * * *
1 0 0 1 0 0 1 1 0 0
0 0
,
K
,
; ,
,
d d
,
,
π
μ ρ ϑ +
∫ ∫
ρ ϑ ρ ϑ μ ρ ϑ ρ ρ ϑ = Φ ρ ϑ
(1.23)(
0,
0;
,
)
1(
,
) (
1 0,
0;
,
)
,
*
1
ρ
ϑ
ρ
ϑ
F
ρ
ϑ
G
ρ
ϑ
ρ
ϑ
K
=
(1.24)
(
)
[
(
)
]
( )
( )
( )11 0 // * 1 0 * 1 0
0 1 2 0 0 0 *
1
ρ
,
ϑ
=
ρ
ρ
,
ϑ
+
4
ϑ
+
ϑ
+
α
Φ
F
u
u
(1.25) (1.20),
u
2(
ρ ϑ
,
)
(1.16),(
)
2(
) (
)
(
)
* * * *
2 0 0 2 0 0 2 2 0 0
0 1
,
K
,
; ,
,
d d
,
,
π ∞
μ ρ ϑ +
∫ ∫
ρ ϑ ρ ϑ μ ρ ϑ ρ ρ ϑ = Φ ρ ϑ
(1.26)(
)
(
) (
)
*
2 0
,
0; ,
2,
2 0,
0; ,
,
(
)
( )2(
)
2( )
( )
( )2* 0 * */ /
2 0 0 2 2 0 0 4 2 2 0 4 2 0 1
0 0 0 0
4
1
,
⎡
α
,
⎛
ε
⎞
⎤
u
u
.
Φ ρ ϑ =
⎢
ψ ρ ϑ −
⎜
−
⎟
⎥
ϑ +
ϑ +α
ρ
⎝
ρ
ρ
⎠
ρ
⎣
⎦
(1.28)2. (1.23) ,
( )1
(
)
2
(
1
)
,
kcos
,
k
k
k
+
ϕ ρ ϑ =
ρ
ϑ
π
( )1
(
)
(
)
(
)
0 0 0 0
2
1
,
mcos
,
1, 2,...
m
m
m
k m
+
ϕ ρ ϑ =
ρ
ϑ
=
π
(2.1)(
)
( ) ( )1 1(
)
( )1(
)
*
1 0 0 0 0
1 1
,
; ,
km m,
k,
k m
K
A
∞ ∞
= =
⎧
⎫
ρ ϑ ρ ϑ =
⎨
ϕ ρ ϑ
⎬
ϕ ρ ϑ
⎩
⎭
∑ ∑
,( )
∫∫ ∫∫
(
)
( )(
)
( )(
)
=
⎭
⎬
⎫
⎩
⎨
⎧
=
2π πρ
ϑ
ρ
ϑ
ϕ
ρ
ϑ
ρ
ρ
ϑ
ϕ
ρ
ϑ
ρ
ρ
ϑ
0 1
0
1 2
0 1
0
0 0 0 0 0 1 0
0 * 1
1
K
,
;
,
,
d
d
,
d
d
A
km m k
(
)
( )(
)
(
)
( )( )
2 1 2 1
1 1
1 0 0 0 0 0 0 0 1
0 0 0 0
,
; ,
m,
,
k,
.
G
d
d
F
d d
π
⎧
π⎫
=
⎨
ρ ϑ ρ ϑ ϕ ρ ϑ ρ ρ ϑ
⎬
ρ ϑ ϕ ρ ϑ ρ ρ ϑ
⎩
⎭
∫ ∫ ∫ ∫
(2.2)1
G
(1.21) (2.2),( )1 0( )1
1
{
( ) ( )0 1 ( ) ( )2 2 ( ) ( )4 3}
,
2
1
km km km km km km km
l
m
A
J
J
J
k
+
=
ε
+ ε
+ ε
π
+
(2.3)( )
(
)
( )
( )
(
1
)
(
1
,
2
,
3
)
,
,
4
,
2
,
0
cos
cos
cos
1
0
1 2 2
0
∫
∫
=
−
=
=
=
+ +
p
dx
x
x
x
b
J
p
mxdx
kx
px
m k p
p km p km
π
ε
* 1
K
(2.1), (1.23),(
)
( ) ( )1 1(
)
2 1 ( )1(
) (
)
(
)
* * *
1 0 0 0 0 1 1 0 0
, 1 0 0
,
km m,
k,
,
,
,
k m
A
d d
π ∞
=
μ ρ ϑ +
∑
ϕ ρ ϑ
∫ ∫
ϕ ρ ϑ μ ρ ϑ ρ ρ ϑ = −Φ ρ ϑ
(2.4)
(
)
(
) ( )
( )
( )
( )1* 2 * */ / *
1 0
,
0 0F
1 0,
0u
1 0u
1 04
u
1 0 1Φ ρ ϑ = ρ
ρ ϑ
ϑ +
ϑ +
ϑ + α
–. ,
( )1 ( ) ( )1 1 *
(
)
1 1
1, 2,...
N
m km k m
k
X
A X
m
N
=
( )
( )( )
(
)
2 11
* *
1 1
0 0
,
,
1, 2,...
m
x y
mx y xdxdy m
N
π
Φ = −
∫ ∫
Φ
ϕ
=
. (2.6)( )1 km
А
ё (2.3), (2.4)(
,
)
(
,
)
( ) ( )(
,
)(
0
1
)
,
0 1
0 0 1 1 0
0 * 1 0 0 *
1
=
Φ
+
∑
≤
<
=
ρ
ϑ
ρ
ϕ
ϑ
ρ
ϑ
ρ
μ
Nk
m m
Y
(2.7)
( ) ( ) ( )
(
)
N
m
X
A
Y
N
k
k km
m
1
,
2
,...
1 1 1 1
*
=
∑
=
=
( )1 k
X
(2.5).(1.26), ,
( )
(
)
(
)
,
cos
1
,
3 00 0 0
0
2
ϑ
ρ
ρ
ϑ
ρ
ϕ
m=
c
m m+−
m
( )(
,
)
(
31
)
cos
,
2
ϑ
ρ
ρ
ϑ
ρ
ϕ
=
k k+−
kc
(2.8)
(
2
+
1
)(
2
+
2
)(
2
+
3
)
/
2
π
=
k
k
k
c
k
(
)
(
)
( ) ( )(
)(
)
,
1
,
,
,
01
0 0 2 2 0
0 * 2 0 0 *
2
=
Φ
+
∑
<
<
∞
=
ρ
ϑ
ρ
ϕ
ϑ
ρ
ϑ
ρ
μ
Nk
m m
Y
(2.9)
( )
A
( ) ( )X
(
m
N
)
Y
N
k
k km
m
1
,
2
,...
1
2 2 2
*
=
∑
=
=
,
( )2 k
X
( )2 ( ) ( )2 2 *
(
)
2 1
1, 2,...
N
m km k m
k
X
A X
m
N
=
+
∑
= Φ
=
, (2.10)
( )2 ( )2
{
( ) ( )1 0 ( ) ( )2 2 ( ) ( )3 4}
02
,
km k m km km km km km km
А
= πα
c c
H
ε +
H
ε +
H
ε
(2.11)( ) ( )
( ) ( )
(
)
,
3
,
2
,
1
1
1
2 2
∫
∞
+
=
−
=
dx
p
x
x
x
a
x
Q
H
kmp mm p k
( )2
( )
( )
2 1
1
,
,
mm m k
y
Q
x
W
x y
dy
y
∞
+
−
=
∫
(2.12)( )
x
a
p (1.23),W
m( )
x
,
y
– (1.28) ,ε
( )kmp(
p
=
0, 2, 4
)
– (2.5)( )
( )( )
( )
2 2
2
* * *
2 2 2 2
0 1 0 1
1
,
,
,
cos
.
m m m m
x
x y
x y xdxdy
c
x y
my
dxdy
x
π ∞ π ∞
+
−
Φ =
∫ ∫
Φ
ϕ
=
∫ ∫
Φ
(2.13)3. (1.18),
( )
(
) (
)
⎥
−
⎦
⎤
⎢
⎣
⎡
=
= ∞∫∫
1 2 0 1 * 2 0 0 2 0 2 2 * 2 0 * 0,
,
;
,
ρ πϑ
ρ
ρ
ϑ
ρ
μ
ϑ
ρ
ϑ
ρ
ρ
λ
ϑ
G
d
d
d
d
C
A
T
(3.1)(
) (
)
( )00 1 2 0 1 0 * 1 0 0 1 0 2 1 * 1 0
,
,
;
,
ϑ
ρ
ϑ
μ
ρ
ϑ
ρ
ρ
ϑ
β
ρ
ρ
λ
ρ π+
⎥
⎦
⎤
⎢
⎣
⎡
−
=∫∫
G
d
d
d
d
C
A
(
)
* 1 1 1 2,
2
k
k
k
λ =
+
(
)
* 2 2 1 2,
2
k
k
k
λ =
+
( )0
(
1* 1 2 *2 2 1)
01 2
2
B C
B C
C C
λ
+ λ
β = −
. (3.2)(
)
*
1
,
μ ρ ϑ
*(
)
2
,
μ ρ ϑ
(2.7)-(2.9) (3.1),
( )
( ) ( )
/ /( )
*
0
*2
,
*0
*2
.
T
=
T
π
T
=
T
π
(3.3)( )
( )
(
) ( )
( )
( )
( )
2 2
/ 0
* * * 0 0 *
1
0 0
2
0 0 * 0
* 1
1
1
ctg
,
2
2
1
2
0 cos 2
,
T
d
T
H
T
d
T
π
ϑ − ϑ
⎧
π⎫
ϑ ϑ =
⎨
−
ϑ +
ϑ ϑ
ϑ ϑ +
⎬
π
λ ⎩
⎭
+
Ψ
ϑ − π
ϑ
λ
∫
∫
(3.4)(
)
( )(
)
( )(
)
( )(
)
( ),
,
,
,
,
1 1 * 1 0 3 0 * 2 0 1 0 * 1 0 2 0 * 2 0ϑ
ψ
λ
ϑ
ϑ
λ
ϑ
ϑ
λ
ϑ
ϑ
λ
ϑ
ϑ
∂
∂
−
+
−
=
H
H
H
H
(3.5)
( )
(
,
)
[
(
( )(
,
)
)
4
]
(
,
;
,
)
,
1 1 0 0 0 1 2 1 1 0 2 0 0 1 0 0=
⎭
⎬
⎫
⎩
⎨
⎧
+
−
=
∫
ρρ
ρ
ϑ
ρ
ϑ
ρ
α
ϑ
ρ
ρ
ρ
ϑ
ϑ
l
f
R
G
d
d
d
H
( )(
)
( )(
)
( )
(
)
,
,
;
,
4
,
,
1 1 0 0 2 4 2 2 2 2 0 0 0 2 0 0= ∞⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
∫
ρρ
ρ
ϑ
ρ
ϑ
ρ
ρ
ερ
ϑ
ρ
ψ
ρ
α
ρ
ϑ
ϑ
G
d
d
d
H
(3.6) ( )(
)
,
,
1 1 3 2 2 2 0 0 3 0 0= ∞⎭
⎬
⎫
⎩
⎨
⎧
∂
∂
=
∫
ρρ
ρ
ϑ
ρ
ϑ
ϑ
G
d
d
d
H
( )
(
)
[
(
)
]
(
)
(
)
(
)
ln
2
[
1
cos
(
)
]
,
2
sin
sin
2
cos
0 0 0 0 0 0 1 1ϑ
ϑ
ϑ
ϑ
ϑ
ϑ
ϑ
ϑ
ϑ
ϑ
π
ϑ
ϑ
ψ
−
−
−
+
+
−
+
−
−
−
=
−
(3.7) ( )( )
( )(
)
( )(
)
( )( )
( ) ( )2 * *
2 2 2 2 1 1 1
0 0 0 0 0 0 0
1
2 1
0
2
* * 1
0
2 1 2 1 1 1 1
,
,
cos
.
N
m m
B
B
H
H
d
m
C
C
A K
A
π
=
⎧
λ
λ
⎫
Ψ
ϑ =
⎨
ϑ ϑ −
ϑ ϑ
⎬
ϑ+
Ψ
ϑ +
⎩
⎭
λ α
ε
λ α
+
−
+ β
∑
∫
[4,5 ], (3.4)
( )
( )
( ) ( )
( )( )
( )
2 2
2
/ 0
* 0 * * * * 0
1 0 0 1
*
* 0 1
1
1
ctg
,
2
2
2
0 cos 2
.
T
T
H t
T t dt
d
T
c
π
ϑ−ϑ
⎡
π⎤
ϑ =
⎢
ϑ −
ϑ
− Ψ
ϑ
⎥
ϑ +
πλ
⎣
λ
⎦
+ π
ϑ +
∫
∫
(3.9)(3.4) , ,
( )
(
) ( )
( )( )
2 2
2
* 0 0 * * 0 0 0
1
0 0
1
,
0.
T
H t
T t dt
d
π
⎡
π⎤
ϑ −
ϑ
−
Ψ
ϑ
ϑ =
⎢
λ
⎥
⎣
⎦
∫
∫
(3.10)0
ϑ
(3.9),( )
( )
( )
( )
( )
.
2
sin
2
ln
1
2
sin
2
ln
,
2
sin
2
ln
1
* 2 0 * 1 2
0
0 2
0 * 1
2
0
* 2
0
0 0
* 1 0 *
c
c
dy
y
y
dt
t
T
dy
y
y
t
H
t
T
+
+
−
Ψ
−
−
⎭
⎬
⎫
⎩
⎨
⎧
−
−
−
=
∫
∫
∫
ϑ
ϑ
λ
ϑ
ϑ
πλ
ϑ
π
π π
(3.11)
T
*( )
0
=
T
*( ) ( )
2
π
,
T
*/0
=
T
*/( )
2
π
,c
1*=
0
,(3.10)-(3.11)
( )
( )
22 *
2 * 0
1 0
1
.
2
c
tdt
π
=
Ψ
πλ
∫
(3.12)(3.11),
( )
2 *(
) ( )
*( )
* 0 0 * 0
0
,
,
T
K
t
T t dt
f
π
ϑ +
∫
ϑ
=
ϑ
(3.13)( )
( )
,
2
sin
2
ln
1
2
sin
2
ln
,
1
,
0* 1 2
0
0 *
1 0
*
ϑ
πλ
ϑ
πλ
ϑ
=
∫
πH
t
y
y
−
dy
−
t
−
t
K
(3.14)
( )
( )( )
,
2
sin
2
ln
1
*2 2
0
0 2
0 * 1 0
*
y
y
dy
c
f
=
−
∫
Ψ
−
+
π
ϑ
λ
ϑ
– , ,
2,
( )
2
( )
( )cos
,
1
0 0
0 *
2 2 * 2 0
*
∑
=
+
+
=
Nn
n
n
Y
f
C
B
T
ϑ
πλ
ϑ
ϑ
(3.15)
( ) ( ) ( )
(
1
,
2
,...
)
,
10 0 0
N
n
X
A
Y
N
m
m mn
n
=
−
∑
=
=
( )0 m
X
( ) ( ) ( ) ( )
(
)
,
,...
2
,
1
1
0 0 0 0
N
m
F
X
A
X
N
m
m n mn
m
+
∑
=
=
( )
( )
( )
2 20 *
0 0
2 2 2
*
1 0 0 0
,
cos
cos
1
,
ln 2 sin
cos
cos
2
mn
A
K
t x
mt
nxdtdx
y
x
H t y
dy
mt
nxdtdx
π π
π π π
=
=
⎧
−
⎫
=
⎨
⎬
−
πλ
⎩
⎭
∫ ∫
∫ ∫ ∫
2 2
* 1 0 0
1
ln 2 sin
cos
cos
.
2
t
x
mt
nxdtdx
π π
−
−
πλ
∫ ∫
(3.16)( )0 2 *
( )
0
1
cos
2
m
F
f
x
mxdx
π
=
π
∫
., ,
.
4. и . ,
. ,
,
R
+ ε
,R
– р.
ε
,. ,
,
.
. [2],
(
)
1 2(
)
1 0
1 2
4
,
2 1
2 1 2
E E
p
p
R E
E
ε
=
=
+ ν
⎡
⎣
+
− ν
⎤
⎦
(4.1)
(
)
1(
)
20 1 2
2
1
2 1 2
.
4
R
E
E
p
E E
+ ε
⎡
⎣
+
− ν
⎤
⎦
ε =
(4.2),
, , ё
(
1
)
r
=
R
ρ =
. .,
r
( )
,
(
1
,
2
)
.
2
// 2
0
=
=
k
p
J
T
j
q
jλω
j j jω
(1.11)
(
)
( )(
)
2
2 2
0 0
2 2
1
2
0
1, 2
j j j j j j
j
j
d T
dT
B
C T
R T
T
R
j
d
d
A
+
+
− α
−
+
μ
=
=
ρ
ρ ρ
(4.3)(1.18). (4.3) (1.14).
( ) 2
1 2
1 1
1 1 1
2
1
, 0
1,
d u
du
u
d
ρ
+
ρ ρ
d
− ε
= −α
≤ ρ <
(4.4) ( )2
2 2
2
1
2, 1
,
d u
du
u
+
− ε
= −α
< ρ < ∞
( )
( ) 2
(
0)
(
)
2 2 0
2 1
2
,
1, 2 .
j
j j j
j j
j j
R
B
C T
C
R
j
A
A
⎛
μ
⎞
α
+
ε =
⎜
⎜
α −
⎟
⎟
α =
=
⎝
⎠
(4.6)(4.4) (4.5)
( )
0 0( )
0 0( )
(
1, 2 ,
)
j j j
u
ρ =
A I
ε ρ +
B K
ε ρ
j
=
(4.7)( )
x
I
0 – ,K
0( )
x
– .0
A
B
0– , . . .,
B
0=
0
( )
( )
1 0 0 1
, 0
1.
u
ρ =
A I
ε ρ
≤ ρ <
(4.8)ρ → ∞
, ,A
0=
0
,( )
( )
2 0 0 2
,1
.
u
ρ =
B K
ε ρ
< ρ < ∞
(4.9) (4.4) (4.5)( )
( )
( )2,
0
1
,
1 1 1 1 0 0
1
=
+
ε
≤
ρ
<
α
ρ
ε
ρ
A
I
u
(4.10)
( )
( )
( )2,
1
.
2 2 1 2
0 0
2
=
+
ε
<
ρ
<
∞
α
ρ
ε
ρ
B
K
u
(1.18)
A
0,B
0. (4.10)( )
1 1 * ( )11 0( )
( )
1 1( )11 2 2
1 1 0 1 1
, 0
1
I
B
C T
u
A
I
⎛
+
α
⎞
ε ρ
α
ρ =
⎜
⎜
−
⎟
⎟
+
≤ ρ <
ε
ε
ε
⎝
⎠
(4.11)( )
2 2 * 1( )2 0( )
( )
2 1( )22 2 2
2 2 0 2 2
,1
.
K
B
C T
u
A
K
⎛
+
α
⎞
ε ρ
α
ρ =
⎜
⎜
−
⎟
⎟
+
< ρ < ∞
ε
ε
ε
⎝
⎠
(4.12)(1.18)
*
Т
,( )
( ) ( )
( )( ) ( )
( ) ( )
( ) ( )
[
]
,
2 1 1 0 2 2 2 0 1 1 1 1 2 1
2 0 1 1 1 0 2 1 2 1 1 0 2 0 1 2
*
δ
δ
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
K
I
k
K
I
k
K
I
M
k
K
I
M
k
T
+
+
=
(4.13)
( )
2
( ),
2
( ),
2 2(
1
,
2
)
.
2 0 2
0 0
0
=
+
=
−
=
R
j
=
A
C
A
B
T
M
j jj j j
j j
j j
j
α
μ
δ
α
μ
ε
δ
R
p
0. .[1,3,6]:
100
,
0
,
00234
,
20
,
0
,
71
.
,
.
.
28
,
0
.,
.
87
,
3
10
.
87
,
3
,
10
.
4
,
3
,
10
.
5
,
2
2 0
0 2
1
2 2 1 1 2
2 1
2 2
2 1 2 4
2 2 4
1
−
=
=
=
=
=
=
=
=
=
=
=
=
=
C
T
а
k
k
а
а
а
c
a
c
a
а
C
C
B
B
А
A
α
ω
0 * *
T
T
V
j=
,
(
1
,
2
)
0
=
=
j
T
T
V
jj
p
p
p
0*
=
, 2
3
10
p
=
,
R
r
=
ρ
, (4.14)
*
Т
– ,Т
o–,
p
– .. 1–3
V
j*,R
,p
*R
=10 1*
p
1 2 3 4 5 6 7* 1
V
1.248 1.324 1.450 1.628 1.863 2.171 2.592R
=12 2*
p
1 2 3 4 5 6 7* 2
V
1.119 1.189 1.305 1.471 1.694 1.994 2.415R
=14 3*
p
1 2 3 4 5 6 7* 3
V
1.029 1.094 1.204 1.362 1.578 1.875 2.3. 1–3 , , .
. 4–5
V
j ,ρ
,R
=10 ,p
*=34
ρ 0 0.2 0.4 0.6 0.8 0.7 0.9 1
1
V
1.007 1.007 1.011 1.026 1.049 1.11 1.211 1.45
Т 5
ρ
1 1.2 1.4 1.6 1.8 2 2.2 2.42
V
1.451 1.085 1.022 1.011 1.01 1.009 1.009 1.009. 4–5 ,
,
. ,
1 2 3 4 5 6 7 1.5
2.0 2.5
. 1
. 1
V
j* Rp
*R =10,12,14 .
0.5 1.0 1.5 2.0 r
1.1 1.2 1.3 1.4 VHrL
. 2.
. 2.
V
jρ
.
1. . . .: . , 1963. 535
.
2. . .
. .: , 1966. 201 .
3. . .
. // . . .1983. N3. .86–93.
4. . . . . .: , 1963. 62 .
5. . . . .: , 1968.
117 .
6. . . . // .
. . 1965. N6.
и :
и и х и – . - . , .
.: (25-06-71), .: (095)45 62 65
