Куликов А.А. (сост.) Смешанные задачи для уравнения теплопроводности и уравнения колебаний
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51
Ɍɚɤ ɠɟ, ɤɚɤ ɢ ɪɚɧɶɲɟ, ɦɵ ɩɨɥɭɱɚɟɦ ɡɚɞɚɱɭ Ʉɨɲɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɮɭɧɤɰɢɣ
)t(T
n
:
0
2
cc
)t(Ta)t(T
nnn
O
, (8.20)
nn
)(T
M
0 ,
nn
)(T
\
c
0 , !,,, 210
n . (8.21)
ȼ ɫɥɭɱɚɟ
0
n ɭɪɚɜɧɟɧɢɟ (8.20) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
0
0
c
c
)t(T . (8.22)
Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (8.22) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
tBA)t(T
000
,
ɝɞɟ
0
A ,
0
B – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
(8.21), ɧɚɯɨɞɢɦ, ɱɬɨ
00
M
A ,
00
\
B .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɫɥɭɱɚɟ
0
n
ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.20), (8.21) ɢɦɟɟɬ ɜɢɞ
t)t(T
000
\
M
.
ȼ ɫɥɭɱɚɟ
1
t
n
ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
n
O
ɩɨɥɨɠɢɬɟɥɶɧɵ, ɢ ɪɟɲɟɧɢɟ
ɡɚɞɚɱɢ (8.20), (8.21) ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ (8.18):
ta
a
ta)t(T
n
n
n
nnn
O
O
\
OM
sincos
l
tan
an
l
l
tan
n
n
S
S
\
S
M
sincos
, !,, 21
n .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ
1
21
D
D
, 0
21
E
E
ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (8.1) –
(8.5) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
¦¦
f
f
1
0
0 n
nn
n
nn
)x(X)t(T)t(T)x(X)t(T)tx,(u
l
xn
l
tan
an
l
l
tan
t
n
n
n
SS
S
\
S
M\M
cossincos
¦
f
¸
¸
¹
·
¨
¨
©
§
1
00
.
(8.23)
2. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
ɫɬɪɭɧɵ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ ɮɨɪ-
ɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ
)
t
x,(uu
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨ-
ɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ
52
)tx,(fuau
xxtt
2
, l
x
0 , 0
!
t , (8.24)
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
00
)
x,(u , lx
d
d
0 , (8.25)
00
),(xu
t
, l
x
d
d
0 (8.26)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
000
11
)t,(u)t,(u
x
E
D
, 0
!
t , (8.27)
0
22
)tl,(u)tl,(u
x
E
D
, 0
!
t . (8.28)
Ɂɞɟɫɶ
)
t
x,(
f
– ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ
2
a ,
1
D
,
1
E
,
2
D
,
2
E
ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ ɡɚɞɚɱɟ (8.1) – (8.5).
ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɸɚɦɟɥɹ (ɫɦ. §5) ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.24) – (8.28)
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
³
t
d)IJIJ;tx,(v)tx,(u
0
W
, (8.29)
ɝɞɟ
)
IJ
t;x,(vv
– ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
vav
2
, l
x
0 , 0
!
t , (8.30)
00
)
IJ;x,(
v
, lx
d
d
0 , (8.31)
)IJx,(f)IJ;x,(v
t
0 ,
lx
d
d
0
, (8.32)
000
11
);t,(v);t,(v
x
W
E
W
D
, 0
!
t , (8.33)
0
22
)t;l,(v)t;l,(v
x
W
E
W
D
, 0
!
t . (8.34)
ȼ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ ɮɢɤɫɢɪɨɜɚɧɧɨɦ
0
!
W
ɮɭɧɤɰɢɹ
)
,
x
(
f
W
ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɚ ɜ ɪɹɞ Ɏɭɪɶɟ ɩɨ ɫɢɫɬɟɦɟ
ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ:
¦
f
1n
nn
)x(X)IJ(f)x,(f
W
,
ɝɞɟ
xd)x(X)x,(f
X
)IJ(f
n
l
n
n
³
0
2
1
W
, !,, 21
n .
ɉɭɫɬɶ
0
2
2
2
1
!
E
E
. Ɍɨɝɞɚ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɡɚɞɚɱɚ (8.30) – (8.34)
ɢɦɟɟɬ ɪɟɲɟɧɢɟ
53
¦
f
1n
nn
n
n
)x(XtȜa
a
)IJ(f
);tx,(v sin
O
W
. (8.35)
ɉɨɞɫɬɚɜɥɹɹ (8.35) ɜ (8.29) ɢ ɢɡɦɟɧɹɹ ɦɟɫɬɚɦɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝ-
ɪɢɪɨɜɚɧɢɟ, ɧɚɯɨɞɢɦ, ɱɬɨ
¦
f
1n
nn
)x(X)t(T)tx,(u ,
ɝɞɟ
> @
³
t
nn
n
n
d)IJ(f)tIJ(a
a
)t(T
0
1
WO
O
sin
, !,, 21
n .
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ, ɱɬɨ
1
21
D
D
, 0
21
E
E
. ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ Ⱦɸɚ-
ɦɟɥɹ ɪɟɲɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (8.29), ɝɞɟ
)
IJt;x,(
v
v
– ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
vav
2
, lx
0 , 0
!
t , (8.36)
00
)
IJ
;x,(v , l
x
d
d
0 , (8.37)
)IJx,(f)IJ;x,(v
t
0 , l
x
d
d
0 , (8.38)
00
);t,(v
x
W
, 0
!
t
, (8.39)
0
)t;l,(v
x
W
, 0
!
t . (8.40)
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.36) – (8.40) ɢɦɟɟɬ ɜɢɞ
¦
f
xtȜa
a
)IJ(f
t)IJ(f)IJt;x,(v
n
n
n
n
n
O
O
cossin
1
0
l
xn
l
tan
n
f
a
l
t)IJ(f
n
n
SS
W
S
cossin
)(
¦
f
1
0
, (8.41)
ɝɞɟ
³
l
xd)IJx,(f
l
)IJ(f
0
0
1
,
³
l
n
xd
l
xn
)IJx,(f
l
)IJ(f
0
2
S
cos
, !,, 21
n .
ɉɨɞɫɬɚɜɥɹɹ (8.41) ɜ (8.29) ɢ ɢɡɦɟɧɹɹ ɦɟɫɬɚɦɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝ-
ɪɢɪɨɜɚɧɢɟ, ɧɚɯɨɞɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ
1
21
D
D
, 0
21
E
E
ɪɟɲɟɧɢɟɦ ɡɚ-
ɞɚɱɢ (8.24) – (8.28) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
54
¦
f
1
0
n
n
l
xn
tT)t(T)tx,(u
S
cos)( , (8.42)
ɝɞɟ
W
d)IJt()IJ(f)t(T
t
³
0
00
,
W
S
S
d
l
)IJt(an
)IJ(f
an
l
)t(T
t
nn
³
sin
0
, !,, 21
n .
Ɋɚɫɫɦɨɬɪɢɦ ɬɚɤɠɟ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ:
)tx,(fuau
xxt
2
, l
x
0 , 0
!
t , (8.43)
)
x(
)
x,(u
M
0 , lx
d
d
0 , (8.44)
)x()x,(u
t
\
0
, l
x
d
d
0 , (8.45)
000
11
)t,(u)t,(u
x
E
D
, 0
!
t , (8.46)
0
22
)tl,(u)tl,(u
x
E
D
, 0
!
t
. (8.47)
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.30) – (7.33) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)tx,(u)tx,(u)tx,(u
21
,
ɝɞɟ
1
u ɢ
2
u – ɪɟɲɟɧɢɹ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɪɚɧɟɟ ɡɚɞɚɱ:
xxt
uau
1
2
1
, lx
0 , 0
!
t ,
)x()x,(u
M
0
1
, l
x
d
d
0 ,
)x()x,(u
t
\
0
1
, lx
d
d
0 ,
000
1111
)t,(u)t,(u
x
E
D
, 0
!
t ,
0
1212
)tl,(u)tl,(u
x
E
D
, 0
!
t ,
ɢ
)tx,(fuau
xxt
2
2
2
, lx
0 , 0
!
t ,
00
2
)x,(u , l
x
d
d
0 ,
00
2
)x,(u
t
, lx
d
d
0 ,
000
2121
)t,(u)t,(u
x
E
D
, 0
!
t ,
0
2222
)tl,(u)tl,(u
x
E
D
, 0
!
t .
55
3. ɋɦɟɲɚɧɧɚɹ ɡɚɞɚɱɚ ɞɥɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
ɫɬɪɭɧɵ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ. ɍɤɚɡɚɧɧɚɹ ɡɚɞɚɱɚ
ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɮɭɧɤɰɢɸ
)
tx,(uu
, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɧɟɨɞɧɨ-
ɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɤɨɥɟɛɚɧɢɣ ɫɬɪɭɧɵ
)tx,(fuau
xx
t
t
2
, l
x
0 , 0
!
t , (8.48)
ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ
)
x(
)
x,(u
M
0
,
lx
d
d
0
, (8.49)
)x()x,(u
t
\
0
, l
x
d
d
0 (8.50)
ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
)t()t,(u)t,(u
x 111
00
J
E
D
, 0
!
t , (8.51)
)t()tl,(u)tl,(u
x 222
J
E
D
, 0
!
t . (8.52)
Ɂɞɟɫɶ
)
tx,(
f
,
)
x
(
M
,
)
x
(
\
, )t(
1
J
)t(
2
J
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ, ɚɤɨ-
ɷɮɮɢɰɢɟɧɬɵ
2
a ,
1
D
,
1
E
,
2
D
,
2
E
ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɬɟɦ ɠɟ ɭɫɥɨɜɢɹɦ, ɱɬɨ ɢ ɜ
ɡɚɞɚɱɟ (8.1) – (8.5).
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.48) – (8.52) ɜ ɜɢɞɟ
)
t
x,(w
)
t
x,(
v
)
t
x,(u
,
ɝɞɟ
)
tx,(v – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ
)
tx,(w – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤ-
ɰɢɹ, ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɩɨ
x
ɢɩɨt ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ ɝɪɚɧɢɱ-
ɧɵɦ ɭɫɥɨɜɢɹɦ
)t()t,(w)t,(w
x 111
00
J
E
D
, 0
!
t , (8.53)
)t()tl,(w)tl,(w
x 222
J
E
D
, 0
!
t . (8.54)
ɇɚɯɨɠɞɟɧɢɟ ɮɭɧɤɰɢɢ
)
t
x,(w
ɪɚɫɫɦɨɬɪɟɧɨ ɜ §7, ɩ. 3. ȼ ɞɚɥɶɧɟɣɲɟɦ
ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ, ɱɬɨ ɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɢɡɜɟɫɬɧɚ.
ɉɨɞɫɬɚɜɥɹɹ ɮɭɧɤɰɢɸ
w
v
u
ɜ (8.48) – (8.52) ɢ ɭɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ
(8.53), (8.54), ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɨɜɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ
)
tx,(v :
)tx,(fvav
xxtt
~
2
,
lx
0
, 0
!
t , (8.55)
)
x(
)
x,(
v
M
~
0
,
lx
d
d
0
, (8.56)
)x()x,(v
t
\
~
0
, l
x
d
d
0 , (8.57)
000
11
)t,(v)t,(v
x
E
D
, 0
!
t , (8.58)
56
0
22
)tl,(v)tl,(v
x
E
D
,
0
!
t
, (8.59)
ɝɞɟ
)tx,(w)tx,(wa)tx,(f)tx,(f
txx
2
~
,
)
x,(w
)
x
(
)
x
( 0
M
M
~
, )x,(w)x()x(
t
0
\
\
~
.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜɢɞɚ (8.55) – (8.59) ɪɚɫɫɦɚɬɪɢɜɚɥɨɫɶ ɜ §8, ɩ. 2.
ɉɪɢɦɟɪ 1.
Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
2
32
2
x
txuau
xxtt
S
sincos
, 10
x
, 0
!
t , (8.60)
1
2
3
20
x
)x,(u
S
sin , (8.61)
2
5
0
x
)x,(u
t
S
sin
, (8.62)
t
)
t
,u( cos
0
,
2
1 ttu
x
),( . (8.63)
Ɋɟɲɟɧɢɟ
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.60) – (8.63) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)
tx,(w
)
tx,(v
)
tx,(u
,
ɝɞɟ
)
tx,(v – ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ, ɚ ɮɭɧɤɰɢɹ
)
tx,(w ɭɞɨɜɥɟɬɜɨɪɹ-
ɟɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
t
)
t
,(w cos
0 ,
2
1 t)t,(w
x
. (8.64)
Ɏɭɧɤɰɢɸ
)
tx,(w ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
)
t
(dx
)
t
(c
)
t
x,(w
.
ɂɡ (8.64) ɫɥɟɞɭɟɬ, ɱɬɨ
t
)
t(d cos
,
2
t)t(c
, ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ttx)tx,(w cos
2
.
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɧɨɜɚɹ ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ
)
tx,(v ɹɜɥɹɟɬɫɹ ɪɟ-
ɲɟɧɢɟɦ ɡɚɞɚɱɢ
)tx,(fvav
xxt
~
2
, 10
x
, 0
!
t , (8.65)
)
x(
)
x,(
v
M
~
0 , (8.66)
)x()x,(v
t
\
~
0 , (8.67)
00
)
t,(v ,
01
)t,(v
x
, (8.68)
57
ɝɞɟ
2
3
2
x
)tx,(w)tx,(wa)tx,(f)tx,(f
ttxx
S
sin
~
,
2
3
20
x
)x,(w)x()x(
S
MM
sin
~
,
2
5
0
x
)x,(w)x()x(
t
S
\
\
sin
~
.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.65) – (8.68) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
)tx,(v)tx,(v)tx,(v
21
,
ɝɞɟ
1
v – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
vav
1
2
1
, (8.69)
2
3
20
1
x
)x,(v
S
sin
, (8.70)
2
5
0
1
x
)x,(v
t
S
sin
, (8.71)
00
1
)t,(v , 01
1
)t,(v
x
, (8.72)
ɚ
2
v – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
2
3
2
2
2
x
vav
xxt
S
sin
, (8.73)
00
2
)x,(v , (8.74)
00
2
)x,(v
t
, (8.75)
00
2
)t,(v , 01
2
)t,(v
x
. (8.76)
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (8.69) – (8.72). Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚ-
ɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-
Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
4
12
22
S
O
)n(
n
,
2
12
x)n(
)x(X
n
S
sin , !,, 21
n .
Ɍɚɤ ɤɚɤ
)x(X
x
)x(
2
2
2
3
2
S
M
sin
~
,
ɢ
58
)x(X
x
)x(
3
2
5
S
\
sin
~
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
2
2
M
~
, 0
n
M
~
ɞɥɹ 2
z
n ,
1
3
\
~
, 0
n
\
~
ɞɥɹ 3
z
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
)tx,(v
1
)x(Xta
a
)x(Xta
33
3
3
222
O
O
\
OM
sincos
2
5
2
5
5
2
2
3
2
3
2
xta
a
xta
S
S
S
S
S
sinsinsincos
.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɡɚɞɚɱɭ (8.73) – (8.76).
Ɍɚɤ ɤɚɤ
)x(X
x
)tx,(f
1
3
2
3
S
sin
~
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)tx,(f
~
ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
3
1
f
~
, 0
n
f
~
ɞɥɹ 1
z
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
2
1112
x
)t(T)x(X)t(T)tx,(v
S
sin
,
ɝɞɟ
> @
³
t
df)tIJ(a
a
)t(T
0
11
1
1
1
WO
O
sin
¸
¹
·
¨
©
§
2
1
12
1
3
22
1
1
2
ta
a
ta
a
S
S
O
O
coscos .
ɂɬɚɤ,
22
1
12
22
2
xta
a
)tx,(v
S
S
S
sincos
¸
¹
·
¨
©
§
,
ɢ
59
)tx,(w)tx,(v)tx,(v),tx(u
21
2
5
2
5
5
2
2
3
2
3
2
xta
a
xta
S
S
S
S
S
sinsinsincos
ttx
xta
a
cossincos
¸
¹
·
¨
©
§
2
22
22
1
12
S
S
S
.
ɉɪɢɦɟɪ 2.
Ɋɟɲɢɬɶ ɫɦɟɲɚɧɧɭɸ ɡɚɞɚɱɭ
432
2
xuau
xxt
S
cos ,
10
x
, 0
!
t , (8.77)
220
x
)
x,(u
S
cos , (8.78)
30
)x,(u
t
, (8.79)
00
)t,(u
x
,
01
)t,(u
x
. (8.80)
Ɋɟɲɟɧɢɟ
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.77) – (8.80) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ (ɫɦ. §8, ɩ. 2)
)tx,(u)tx,(u)tx,(u
21
ɝɞɟ
1
u – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
xxt
uau
1
2
1
, 10
x
, 0
!
t
, (8.81)
)x()x,(u
M
0
1
, (8.82)
)x()x,(u
t
\
0
1
, (8.83)
00
1
)t,(u
x
,
0
1
)tl,(u
x
, (8.84)
22
x
)
x
(
S
M
cos , 3
)
x
(
\
,
ɚ
2
u – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
)tx,(fuau
xxt
2
2
2
, 10
x
, 0
!
t , (8.85)
00
2
)x,(u
, (8.86)
00
2
)x,(u
t
, (8.87)
00
2
)t,(u
x
,
0
2
)tl,(u
x
, (8.88)
432
x
)
tx,(
f
S
cos .
ɋɨɛɫɬɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɢ ɫɨɛɫɬɜɟɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɨɨɬɜɟɬɫɬ-
ɜɭɸɳɟɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɹɜɥɹɸɬɫɹ
60
2
)ʌn(
n
O
, xn)x(X
n
S
cos
, !,,, 210
n .
Ɍɚɤ ɤɚɤ
)x(X)x(Xx)x(
20
222
S
M
cos ,
ɢ
)x(X)x(
0
33
\
,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
2
0
M
, 1
2
M
, 0
n
M
ɞɥɹ
20,
z
n
,
3
0
\
, 0
n
\
ɞɥɹ 1
t
n .
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 1 ɫɥɟɞɭɟɬ, ɱɬɨ
)tx,(u
1
¸
¸
¹
·
¨
¨
©
§
¦
f
xntan
an
tant
n
n
n
SS
S
\
SM\M
cossincos
1
00
x
tat
S
S
2232 coscos
.
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ (8.85) – (8.88).
Ɍɚɤ ɤɚɤ
)x(X)x(Xx)tx,(f
30
24432
S
cos ,
ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
)
tx,(
f
ɩɨ ɫɢɫɬɟɦɟ ɫɨɛɫɬɜɟɧɧɵɯ ɮɭɧɤ-
ɰɢɣ ɡɚɞɚɱɢ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɪɚɜɧɵ
4
0
f , 2
3
f , 0
n
f ɞɥɹ
30,
z
n
.
Ɉɬɫɸɞɚ ɢ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ §8, ɩ. 2 ɫɥɟɞɭɟɬ, ɱɬɨ
)x(X)t(T)t(T)x(X)t(T)tx,(u
n
nn
330
0
2
¦
f
,
ɝɞɟ
2
0
00
2td)IJt(f)t(T
t
³
W
,
> @
³
WS
S
dIJ)(taf
a
)t(T
t
3
3
1
0
33
sin
ta
a
S
S
31
9
2
22
cos
.