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100
Ⱦɥɹ
j
– ɨɣ ɋɎɄ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɚɧɚɥɨɝɢɱɧɨɟ (2.28).
ɂɫɩɨɥɶɡɭɟɦ ɞɥɹ ɷɬɢɯ ɞɜɭɯ ɩɪɨɢɡɜɨɥɶɧɵɯ ɮɨɪɦ ɤɨɥɟɛɚɧɢɣ ɬɟɨɪɟɦɭ ɨ
ɜɡɚɢɦɧɨɫɬɢ ɪɚɛɨɬ [21]:
jiij
ujuj
ɬɬ
)()( .
(2.31)
ɍɱɢɬɵɜɚɹ, ɱɬɨ
MuuMj
ɬ2ɬ2ɬ
)()()(
jjjjj
Z Z ɢ MuuMj
ɬ2ɬ2ɬ
)()()(
iiiii
Z Z ,
ɩɨɥɭɱɢɦ
jiiijj
uMuuMu
ɬ2ɬ2
)()( Z Z .
ɇɨ
ijji
uMuuMu
ɬɬ
)()(
,
(2.32)
ɩɨɷɬɨɦɭ
0))((
ɬ22
ZZ
ijij
uMu .
Ɍɚɤ ɤɚɤ ɩɨ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɢɦɟɸɬɫɹ ɞɜɟ ɪɚɡɧɵɟ ɮɨɪɦɵ ɤɨɥɟɛɚɧɢɣ, ɭ ɤɨɬɨɪɵɯ
0)(
22
zZZ
ij
, ɬɨ ɩɨɥɭɱɢɦ ɪɚɜɟɧɫɬɜɨ (2.29), ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.
ɋɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ ɦɨɠɟɬ ɛɵɬɶ ɬɚɤɠɟ ɩɪɟɞɫɬɚɜɥɟɧɨ ɜ ɜɢɞɟ
ɩɪɨɢɡɜɟɞɟɧɢɹ (2.33), ɜ ɤɨɬɨɪɨɦ ɜɦɟɫɬɨ ɦɚɬɪɢɰɵ ɦɚɫɫ ɜ ɩɪɢɜɟɞɟɧɧɨɦ ɜɵɲɟ
ɜɵɪɚɠɟɧɢɢ (2.29) ɢɫɩɨɥɶɡɨɜɚɧɚ ɦɚɬɪɢɰɚ ɠɟɫɬɤɨɫɬɢ
K
:
0)(
ɬ
ij
uKu .
(2.33)
Ⱦɥɹ ɩɨɹɫɧɟɧɢɹ ɦɚɬɪɢɱɧɨɟ ɭɪɚɜɧɟɧɢɟ (2.15), ɡɚɩɢɫɚɧɧɨɟ ɞɥɹ ɤɚɤɨɣ-ɬɨ
i -ɨɣ
ɋɎɄ, ɭɦɧɨɠɢɦ ɫɥɟɜɚ ɧɚ ɦɚɬɪɢɰɭ ɫɬɪɨɤɭ (
j
u )
ɬ
. Ɍɨɝɞɚ ɩɨɥɭɱɢɦ:
0)()(
ɬɬ
M i
jiij
uMuuKu .
(2.34)
ɉɨɫɤɨɥɶɤɭ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɭɪɚɜɧɟɧɢɢ (2.34) ɪɚɜɧɨ ɧɭɥɸ ɜɫɥɟɞɫɬɜɢɟ
ɫɜɨɣɫɬɜɚ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ (2.29), ɬɨ ɪɚɜɧɨ ɧɭɥɸ ɢ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ:
ɋɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ ɹɜɥɹɟɬɫɹ ɜɚɠɧɵɦ ɫɜɨɣɫɬɜɨɦ, ɤɨɬɨɪɨɟ
ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɢ ɪɟɲɟɧɢɢ ɞɢɧɚɦɢɱɟɫɤɢɯ ɡɚɞɚɱ.
ɉɪɢɦɟɱɚɧɢɟ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɩɪɨɜɟɪɤɟ ɫɜɨɣɫɬɜɚ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ ɜ ɜɢɞɟ
(2.29) ɦɨɠɧɨ ɜɦɟɫɬɨ ɜɟɤɬɨɪɨɜ
j
u ɢ
i
u ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɟɤɬɨɪɵ
j
y ɢ
i
y (ɫɦ.
ɜɵɪɚɠɟɧɢɟ (2.15), ɚ ɜɦɟɫɬɨ ɦɚɬɪɢɰɵ ɦɚɫɫ
M ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɚɬɪɢɰɭ ɜɟɫɨɜ ɦɚɫɫ g MG .
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɪɚɜɟɧɫɬɜɨ (2.29) ɧɟ ɢɡɦɟɧɢɬɫɹ ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɟɝɨ ɧɚ ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ
j
u ,
i
u ɢ
g
.
ɉɪɢɦɟɪ 1. ȼɵɩɨɥɧɢɦ ɜɪɭɱɧɭɸ ɩɪɨɜɟɪɤɭ ɫɜɨɣɫɬɜɚ (2.29) ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɞɥɹ ɩɟɪɜɨɣ
ɢ ɜɬɨɪɨɣ ɢ ɞɥɹ ɩɟɪɜɨɣ ɢ ɬɪɟɬɶɟɣ ɋɎɄ ɫɢɫɬɟɦɵ ɫ ɬɪɟɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜ
ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ. ɋ ɭɱɟɬɨɦ ɫɞɟɥɚɧɧɨɝɨ ɩɪɢɦɟɱɚɧɢɹ ɜɦɟɫɬɨ (2.29) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:
101
;0
1
0
1
839.07.2839.0
1
0
1
2.1
7.2
2.1
699.01699.0
)()(
2
ɬ
1
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¸
¹
·
¨
¨
¨
©
§
»
»
»
¼
º
«
«
«
¬
ª
yGy
(2.35)
.0839.0678.1839.0
1
6216.0
1
2.1
7.2
2.1
839.07.2839.0
)()(
3
ɬ
2
¸
¸
¸
¹
·
¨
¨
¨
©
§
»
»
»
¼
º
«
«
«
¬
ª
yGy
(2.36)
ɉɪɢɦɟɪ 2. ɂɫɩɨɥɶɡɭɟɦ ɫɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ (2.29) ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ
ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜ ɜɵɪɚɠɟɧɢɢ (2.17) ɫ ɩɨɦɨɳɶɸ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɞɜɢɠɟɧɢɹ
ɦɚɫɫ ɜ ɜɢɞɟ (2.7). Ⱦɥɹ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨɞɫɬɚɜɢɦ (2.17) ɜ (2.10). Ɍɨɝɞɚ
ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɹ (2.37) ɢ (2.38):
)sincos(
21
1
tAtA
iiii
n
i
i
ZZ
¦
uu
;
(2.37)
)cossin(
21
1
tAtA
iiiiii
n
i
i
ZZZZ
¦
uu
.
(2.38)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɫɸɞɚ
0 t
, ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɡɚɩɢɲɭɬɫɹ ɜ ɜɢɞɟ
o
1
1
uu
¦
i
n
i
i
A
;
oo
2
1
)( uu Z
¦
ii
n
i
i
A
.
(2.39)
ɍɦɧɨɠɢɜ ɷɬɢ ɜɵɪɚɠɟɧɢɹ ɧɚ ɦɚɬɪɢɰɭ-ɫɬɪɨɤɭ
Mu
j
ɬ
)( , ɫ ɭɱɟɬɨɦ (2.29), ɧɚɣɞɟɦ:
iii
i
i
ii
i
i
AA
uMu
uMu
uMu
uMu
ɬ
ooɬ
2
ɬ
oɬ
1
)(
)(
;
)(
)(
Z
.
(2.40)
2.4. ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɢ ɟɟ ɪɟɲɟɧɢɟ
ɦɟɬɨɞɨɦ ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɢɫɤɨɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɩɨ ɜɟɤɬɨɪɚɦ
ɩɟɪɟɦɟɳɟɧɢɣ ɜ ɋɎɄ
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɫɱɟɬɧɭɸ ɫɯɟɦɭ ɫɨɨɪɭɠɟɧɢɹ ɜ ɜɢɞɟ ɧɟɜɟɫɨɦɨɝɨ ɭɩɪɭɝɨɝɨ
ɥɢɧɟɣɧɨ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɫɬɟɪɠɧɹ, ɧɚ ɤɨɬɨɪɨɦ ɪɚɫɩɨɥɚɝɚɟɬɫɹ
n ɬɨɱɟɱɧɵɯ ɦɚɫɫ
(ɫɦ. ɪɢɫ. 2.1).
Ɍɚɤɠɟ ɤɚɤ ɩɪɢ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫɬɟɪɠɧɹ ɩɪɢ ɭɱɟɬɟ ɬɨɥɶɤɨ
ɩɨɩɟɪɟɱɧɵɯ ɤ ɨɫɢ ɫɬɟɪɠɧɹ ɩɟɪɟɦɟɳɟɧɢɹɯ ɬɨɱɟɱɧɵɯ ɦɚɫɫ ɛɭɞɟɦ ɢɦɟɬɶ ɫɢɫɬɟɦɭ
ɫ
n ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ.
ɋɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɟɟ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ, ɤɚɤ ɢ ɩɪɢ
ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ (ɫɦ. ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (2.1) ɢ (2.2)), ɡɚɩɢɲɟɦ ɜ
ɦɚɬɪɢɱɧɨɦ ɜɢɞɟ ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɭɪɚɜɧɟɧɢɹɦɢ ɞɥɹ ɫɯɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ
ɫɜɨɛɨɞɵ (ɫɦ. ɭɪɚɜɧɟɧɢɹ (1.60) ɢ (1.61)):
0KuuCuM
.
(2.41)
102
)( uCuMDu
(2.42)
ȼ ɨɬɥɢɱɢɟ ɨɬ ɫɢɫɬɟɦ ɭɪɚɜɧɟɧɢɣ (2.1)ɢ (2.2) ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɹ ɞɥɹ
ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɫɨɞɟɪɠɚɬ ɟɳɟ ɨɞɧɨ ɫɥɚɝɚɟɦɨɟ, ɤɨɬɨɪɨɟ ɭɱɢɬɵɜɚɟɬ ɫɢɥɵ
ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɚ. Ⱦɥɹ ɦɚɬɪɢɱɧɵɯ ɭɪɚɜɧɟɧɢɣ ɷɬɨ ɨɡɧɚɱɚɟɬ
ɞɨɛɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɫɢɥ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɤɨɬɨɪɵɣ ɜ
ɫɤɨɪɪɟɤɬɢɪɨɜɚɧɧɨɦ ɦɟɬɨɞɟ Ɏɨɣɝɬɚ [4] ɢɦɟɟɬ ɜɢɞ:
uCr
ɧ
.
(2.43)
ȼɟɤɬɨɪ
u
ɜ (2.43) ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɨɦ ɫɤɨɪɨɫɬɟɣ ɦɚɫɫ ɢ ɢɦɟɟɬ ɜɢɞ
ɚɧɚɥɨɝɢɱɧɵɣ ɜɢɞɭ ɜɟɤɬɨɪɨɜ ɩɟɪɟɦɟɳɟɧɢɣ
u ɢ ɭɫɤɨɪɟɧɢɣ u
ɦɚɫɫ
(ɫɦ. (2.3)); ɦɚɬɪɢɰɚ
C ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɚɬɪɢɰɭ ɞɟɦɩɮɢɪɨɜɚɧɢɹ.
Ɋɟɲɟɧɢɟ ɭɤɚɡɚɧɧɵɯ ɫɢɫɬɟɦ ɭɪɚɜɧɟɧɢɣ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɚɡɥɨɠɟɧɢɹ
(2.10) ɩɪɢ ɭɱɟɬɟ ɫɢɥ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜ ɜɢɞɟ ɜɟɤɬɨɪɚ (2.43)
ɨɤɚɡɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɵɦ, ɟɫɥɢ ɧɚɪɹɞɭ ɫɨ ɫɜɨɣɫɬɜɨɦ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ ɜ
ɜɢɞɟ (2.29) ɢ (2.33) ɩɪɟɞɩɨɥɨɠɢɬɶ ɧɚɥɢɱɢɟ ɫɜɨɣɫɬɜɚ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɜ ɜɢɞɟ
0)(
ɬ
ij
uCu , ij z .
(2.44)
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɦɚɫɫ ɜ ɩɟɪɜɨɣ (2.42) ɢ
ɜɬɨɪɨɣ (2.43) ɮɨɪɦɚɯ ɡɚɩɢɫɢ ɪɚɡɞɟɥɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ɫɢɫɬɟɦɵ
ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ (2.15), (2.16) ɢ ɭɪɚɜɧɟɧɢɟ (2.45) ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
ɮɭɧɤɰɢɢ ɜɪɟɦɟɧɢ
)(˓ t
i
ɞɥɹ ɤɚɠɞɨɣ ɨɣi ɋɎɄ:
0)(˓)(˓2)(˓
2
ZH ttt
iiiii
, ni ...,3,2,1 ,
(2.45)
ɝɞɟ ɜɜɟɞɟɧɨ ɨɛɨɡɧɚɱɟɧɢɟ
ii
ii
i
uMu
uCu
ɬ
ɬ
)(
)(
2 H .
(2.46)
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2.45) ɢɡɜɟɫɬɧɨ (ɫɦ. ɩɨɞɪɚɡɞɟɥ 1.2):
)sincos()(˓
21
tAtAet
iDiiDi
t
i
i
ZZ
H
, ni ...,3,2,1
.
(2.47)
ɗɬɚ ɮɭɧɤɰɢɹ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɤɚɠɞɨɣ ɣ
i ɋɎɄ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɤɨɬɨɪɨɟ
ɝɚɪɦɨɧɢɱɟɫɤɨɟ ɡɚɬɭɯɚɸɳɟɟ ɤɨɥɟɛɚɧɢɟ.
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɤɨɷɮɮɢɰɢɟɧɬ
i
H ,
ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧ ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɚɦ, ɢɡɜɟɫɬɧɵɦ ɢɡ ɩɨɞɪɚɡɞɟɥɚ
1.2 (ɫɦ. (1.76):
DiDiDii
Z
J
Z
S
\
Z
S
G
H
242
ɧɥ
,
(2.48)
ɝɞɟ
iD
Z – ɱɚɫɬɨɬɚ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɨɣi ɋɎɄ ɢ
ɨɩɪɟɞɟɥɹɟɦɚɹ ɞɥɹ ɤɚɠɞɨɣ ɋɎɄ ɢɡ ɜɵɪɚɠɟɧɢɣ ɚɧɚɥɨɝɢɱɧɵɯ (1.77).
103
ȼ ɫɜɹɡɢ ɫ ɬɟɦ, ɱɬɨ ɜ ɮɨɪɦɭɥɚɯ (1.77) ɞɥɹ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ
1)()
2
()(
222
ɧ
S
G
S
\
J
ɥ
, ɩɨɥɭɱɢɦ
iiD
Z|Z ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
iiD
TT | .
ɉɨɷɬɨɦɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ
i
H
ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɚɤɠɟ ɜɵɪɚɠɟɧɢɟ (1.80) ɫ
ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ (1.81).
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜ (2.45) ɨɩɪɟɞɟɥɹɸɬɫɹ ɬɚɤɠɟ ɤɚɤ ɞɥɹ
ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɢ, ɩɨɫɥɟ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɫɜɨɣɫɬɜɚ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ
ɋɎɄ (2.29), ɩɨɥɭɱɚɸɬɫɹ ɜ ɜɢɞɟ
i
ɧ
iii
i
i
ii
i
i
AAA
1
ɬ
ooɬ
2
ɬ
oɬ
1
2
)(
)(
;
)(
)( J
Z
uMu
uMu
uMu
uMu
.
(2.49)
2.5. ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɢ ɟɟ ɪɟɲɟɧɢɟ ɦɟɬɨɞɨɦ
ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɢɫɤɨɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɢ ɜɟɤɬɨɪɚ ɡɚɞɚɧɧɵɯ ɧɚɝɪɭɡɨɤ
ɩɨ ɜɟɤɬɨɪɚɦ ɩɟɪɟɦɟɳɟɧɢɣ ɜ ɋɎɄ
Ɋɚɫɫɦɨɬɪɢɦ ɯɨɞ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɧɚ ɩɪɢɦɟɪɟ ɭɩɪɭɝɨɣ ɥɢɧɟɣɧɨ
ɞɟɮɨɪɦɢɪɭɟɦɨɣ ɫɢɫɬɟɦɵ, ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ. 2.1. Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɤ ɦɚɫɫɚɦ ɜ
ɧɚɩɪɚɜɥɟɧɢɢ ɩɨɩɟɪɟɤ ɨɫɢ ɤɨɧɫɨɥɢ ɩɪɢɥɨɠɟɧ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɞɢɧɚɦɢɱɟɫɤɢɯ
ɫɢɥ, ɢɡɦɟɧɹɸɳɢɯ ɫɜɨɟ ɡɧɚɱɟɧɢɟ ɜɨ ɜɪɟɦɟɧɢ:
ɬ
21
)(...)()( tPtPtP
n
p .
(2.50)
ȼɜɟɞɹ ɷɬɨɬ ɜɟɤɬɨɪ ɜ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ,
ɡɚɩɢɫɚɧɧɭɸ ɜ ɮɨɪɦɟ ɦɟɬɨɞɚ ɩɟɪɟɦɟɳɟɧɢɣ (2.42) ɢɥɢ ɜ ɮɨɪɦɟ ɦɟɬɨɞɚ ɫɢɥ (2.43),
ɩɨɥɭɱɢɦ ɞɜɟ ɚɧɚɥɨɝɢɱɧɵɯ ɮɨɪɦɵ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ:
pKuuCuM
;
(2.51)
)(
puCuMDu
.
(2.52)
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɩɪɢ 0
t
ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɧɭɥɟɜɵɦɢ.
ɏɨɞ ɪɟɲɟɧɢɹ ɪɚɫɫɦɨɬɪɢɦ ɧɚ ɩɪɢɦɟɪɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (2.51).
ȼɵɩɨɥɧɢɦ ɫɥɟɞɭɸɳɢɟ ɞɟɣɫɬɜɢɹ.
1. ɉɨɞɫɬɚɜɢɦ ɜ (2.51) ɜɟɤɬɨɪ ɢɫɤɨɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ ɦɚɫɫ ɩɪɢ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɋɎɄ (2.10) ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ
ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɨɜ ɩɪɨɢɡɜɨɞɧɵɯ.
2. ɉɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɭɦɧɨɠɢɦ ɫɥɟɜɚ ɧɚ ɦɚɬɪɢɰɭ-ɫɬɪɨɤɭ
ɬ
)(
j
u ɢ ɢɫɩɨɥɶɡɭɟɦ ɫɜɨɣɫɬɜɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ ɜ ɜɢɞɟ (2.29), (2.33) ɢ
(2.44). ȼ ɪɟɡɭɥɶɬɚɬɟ ɜɦɟɫɬɨ ɫɢɫɬɟɦɵ (2.51) n ɭɪɚɜɧɟɧɢɣ ɫ n ɧɟɢɡɜɟɫɬɧɵɦɢ
ɩɨɥɭɱɢɦ n ɧɟɡɚɜɢɫɢɦɵɯ ɭɪɚɜɧɟɧɢɣ ( ni ...,,2,1
):
104
^
`
^
`
^
`
puuKuuCuuMu
ɬɬɬɬ
)()()(˓)()(˓)()(˓
iiiiiiiiii
ttt
,.
(2.53)
ɉɨɫɤɨɥɶɤɭ ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɦɚɬɪɢɰɵ-ɫɬɪɨɤɢ
ɬ
)(
j
u ɧɚ ɦɚɬɪɢɰɵ
K
CM ,,
ɩɨɥɭɱɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɦɚɬɪɢɰɵ-ɫɬɪɨɤɢ, ɬɨ ɪɟɡɭɥɶɬɚɬɨɦ ɭɦɧɨɠɟɧɢɹ ɦɚɬɪɢɰ
ɜ ɮɢɝɭɪɧɵɯ ɫɤɨɛɤɚɯ ɛɭɞɭɬ ɫɤɚɥɹɪɧɵɟ ɜɟɥɢɱɢɧɵ.
3. Ɋɚɡɞɟɥɢɦ ɭɪɚɜɧɟɧɢɟ (2.53) ɧɚ ɫɤɚɥɹɪɧɭɸ ɜɟɥɢɱɢɧɭ
ii
uMu
ɬ
)(. ɉɪɢ
ɷɬɨɦ ɜɨ ɜɬɨɪɨɦ ɫɥɚɝɚɟɦɨɦ ɭɪɚɜɧɟɧɢɹ ɭɱɬɟɦ ɭɠɟ ɢɡɜɟɫɬɧɨɟ ɨɛɨɡɧɚɱɟɧɢɟ (2.46).
ȼ ɬɪɟɬɶɟɦ ɫɥɚɝɚɟɦɨɦ ɭɱɬɟɦ, ɱɬɨ ɢɡ ɭɪɚɜɧɟɧɢɹ (2.15) ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɟɝɨ ɧɚ
ɦɚɬɪɢɰɭ-ɫɬɪɨɤɭ
ɬ
)(
i
u ɩɨɥɭɱɚɟɬɫɹ
2
ɬ
ɬ
)(
)(
i
ii
ii
Z
uMu
uKu
, ni ...,,2,1 .
Ⱦɥɹ ɫɥɚɝɚɟɦɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɩɟɪɟɧɟɫɟɧɧɨɝɨ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ, ɩɨɫɥɟ ɟɝɨ
ɞɟɥɟɧɢɹ ɧɚ ɫɤɚɥɹɪɧɭɸ ɜɟɥɢɱɢɧɭ
ii
uMu
ɬ
)( ɜɜɟɞɟɦ ɧɨɜɨɟ ɨɛɨɡɧɚɱɟɧɢɟ
)(
)(
)(
ɬ
ɬ
tH
i
ii
i
uMu
pu
, ni ...,,2,1 .
(2.54)
Ɏɭɧɤɰɢɹ
)(tH
i
ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ ɡɚɞɚɧɧɨɣ
ɧɚɝɪɭɡɤɢ p ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɨɪɦɚɦ ɤɨɥɟɛɚɧɢɣ ɜ ɜɢɞɟ:
)(
1
tH
ii
n
i
uMp
¦
.
(2.55)
ɍɦɧɨɠɢɜ (2.55) ɧɚ ɦɚɬɪɢɰɭ-ɫɬɪɨɤɭ
ɬ
)(
j
u (ɫɦ. (2.53)), ɫ ɭɱɟɬɨɦ ɫɜɨɣɫɬɜɚ
ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɋɎɄ (2.29) ɩɨɥɭɱɢɦ (2.54).
ɍɪɚɜɧɟɧɢɟ (2.53) ɩɪɢɦɟɬ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ (1.82)
)()(
˓)(˓2)(˓
2
tHttt
iiiiii
ZH
, ni ...,,2,1 ,
(2.56)
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɩɪɢ ɧɭɥɟɜɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɟɬ ɜɢɞ ɢɧɬɟɝɪɚɥɚ
Ⱦɸɚɦɟɥɹ (1.84):
³
WWZ
WH
W
Z
t
i
i
i
i
i
dt
t
eHt
0
)(sin
)(
)(
1
)(˓ .
(2.57)
ȼ ɨɛɳɟɦ ɜɢɞɟ ɜɟɤɬɨɪ p (2.50) ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɵɦ. Ɉɞɧɚɤɨ, ɞɥɹ
ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɧɚɩɪɢɦɟɪ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɢ ɩɪɢ
ɪɚɫɱɟɬɟ ɫɨɨɪɭɠɟɧɢɣ ɧɚ ɫɟɣɫɦɢɱɟɫɤɨɟ ɜɨɡɞɟɣɫɬɜɢɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɱɚɫɬɧɵɟ
ɫɥɭɱɚɢ, ɤɨɝɞɚ ɜɟɤɬɨɪ p ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
)(
o
tfpp .
(2.58)
105
ɝɞɟ
o
p – ɡɚɞɚɧɧɵɣ ɜɟɤɬɨɪ ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ ɫɢɥ; )(
t
f
– ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ,
ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɢɡɦɟɧɟɧɢɟ ɫɢɥ ɜɨ ɜɪɟɦɟɧɢ. Ɍɨɝɞɚ ɢɡ (2.54) ɩɨɥɭɱɢɦ
)()( tftH
ii
K .
(2.59)
Ɂɞɟɫɶ ɩɨɫɬɨɹɧɧɚɹ ɞɥɹ ɨɣi ɋɎɄ ɜɟɥɢɱɢɧɚ
i
K
ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɮɨɪɦɭɥɵ:
ii
i
i
uMu
pu
ɬ
o
ɬ
)(
)(
K
, ni ...,,2,1 .
(2.60)
ɉɨɞɫɬɚɜɢɦ (2.59) ɜ ɢɧɬɟɝɪɚɥ Ⱦɸɚɦɟɥɹ (2.57) ɢ ɩɪɟɞɫɬɚɜɢɦ ɟɝɨ ɜ ɜɢɞɟ
(2.61), ɝɞɟ (ɫ ɭɱɟɬɨɦ
iiD
Z|Z ) )(t
i
E ɨɩɪɟɞɟɥɢɬɫɹ ɢɡ ɢɧɬɟɝɪɚɥɚ (2.62).
)()(˓
2
tt
i
i
i
E
Z
K
, ni ...,,2,1 .
(2.61)
³
WWZ
WH
WZ E
t
ii
dt
t
eft
i
i
0
)(sin
)(
)()( .
(2.62)
ɍɱɢɬɵɜɚɹ (2.61), ɢɫɤɨɦɵɣ ɜɟɤɬɨɪ (2.10) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
)()(˓
2
111
tt
ii
n
i
ii
n
i
i
n
i
i
EZ
¦¦¦
ˤ
uuu ,
(2.63)
ɝɞɟ
iii
u
ˤ
K
(2.64)
ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɨɦ ɥɨɤɚɥɶɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɨɣ
i ɋɎɄ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
ɤɚɠɞɨɣ ɦɚɫɫɟ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɟɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ.
Ʉɚɤ ɜɢɞɧɨ ɢɡ (2.30), ɜɟɤɬɨɪɭ ɩɟɪɟɦɟɳɟɧɢɣ
i
u ɜɨɣi ɋɎɄ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɜɟɤɬɨɪ ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ ɫɢɥ ɢɧɟɪɰɢɢ
iii
uMj
2
Z
(2.28).
ȼɵɩɨɥɧɢɜ ɫɬɚɬɢɱɟɫɤɢɣ ɪɚɫɱɟɬ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɤɨɧɫɨɥɢ ɨɬ ɷɬɢɯ ɫɢɥ,
ɩɨɥɭɱɢɦ ɢɡɝɢɛɚɸɳɢɟ ɦɨɦɟɧɬɵ
ik
M
,
ɢ ɩɨɩɟɪɟɱɧɵɟ ɫɢɥɵ
ik
Q
,
ɜ ɟɟ ɫɟɱɟɧɢɹɯ k
ɣi
ɋɎɄ, ɚ ɩɨ ɧɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɧɨɪɦɚɥɶɧɵɟ ɢ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ.
Ɉɛɨɡɧɚɱɢɦ ɥɸɛɭɸ ɬɚɤɭɸ ɜɟɥɢɱɢɧɭ
ik
v
,
.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɚɡɥɨɠɟɧɢɟɦ (2.63) ɨɤɚɡɵɜɚɸɬɫɹ ɪɚɡɥɨɠɟɧɧɵɦɢ ɩɨ ɋɎɄ
ɢ ɜɫɟ ɧɚɝɪɭɡɤɢ, ɢ ɜɫɟ ɭɫɢɥɢɹ, ɢ ɜɫɟ ɧɚɩɪɹɠɟɧɢɹ.
Ɋɚɡɥɨɠɟɧɢɟ ɧɚɝɪɭɡɨɤ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ:
¦¦¦¦
E Z
n
i
ii
n
i
iii
n
i
ii
n
i
i
ttt
11
2
11
)()(˓)(˓ˤ
ˤ
MuMjss ,
(2.65)
ɝɞɟ
ɧ
rjps ɹɜɥɹɟɬɫɹ ɫɭɦɦɚɪɧɵɦ ɜɟɤɬɨɪɨɦ ɜɫɟɯ ɧɚɝɪɭɡɨɤ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɫɢɫɬɟɦɭ;
)()(˓ tt
iiiii
E
ˤ
Mjs
(2.66)
106
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɤɬɨɪ ɧɚɝɪɭɡɨɤ, ɩɪɢɯɨɞɹɳɢɯɫɹ ɧɚ ɫɢɫɬɟɦɭ ɜ i - ɣɋɎɄ.
Ɋɚɡɥɨɠɟɧɢɟ ɜɟɥɢɱɢɧɵ
)(tvv
kk
(ɭɫɢɥɢɹ ɢɥɢ ɧɚɩɪɹɠɟɧɢɹ) ɩɨ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɧɟ ɡɚɜɢɫɹɳɢɦ ɨɬ ɜɪɟɦɟɧɢ ɜɟɥɢɱɢɧɚɦ
ki
v ɜ ɋɎɄ ɢɦɟɟɬ ɜɢɞ:
¦¦¦
EZ
n
i
iiiki
n
i
iki
n
i
kik
t
ˤ
vtvvtv
1
2
11
)()(˓)( ,
(2.67)
Ⱥɧɚɥɨɝɢɱɧɵɣ ɪɟɡɭɥɶɬɚɬ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɟɬɨɞɚ ɪɚɡɥɨɠɟɧɢɹ ɜɟɤɬɨɪɚ
ɢɫɤɨɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ (2.10) ɢ ɜɟɤɬɨɪɚ ɡɚɞɚɧɧɵɯ ɧɚɝɪɭɡɨɤ (2.55) ɩɨ
ɫɨɛɫɬɜɟɧɧɵɦ ɜɟɤɬɨɪɚɦ
i
u ɩɨɥɭɱɢɥɫɹ ɛɵ ɩɪɢ ɪɟɲɟɧɢɢ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɮɨɪɦɟ (2.52).
ɉɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɩɨɡɜɨɥɹɸɬ ɧɚɦɟɬɢɬɶ ɫɥɟɞɭɸɳɭɸ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɦɟɬɨɞɨɦ ɪɚɡɥɨɠɟɧɢɹ ɜɟɥɢɱɢɧ, ɜɯɨɞɹɳɢɯ ɜ
ɭɪɚɜɧɟɧɢɹ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɩɨ ɋɎɄ.
1. Ɉɩɪɟɞɟɥɹɟɬɫɹ ɫɩɟɤɬɪ ɫɨɛɫɬɜɟɧɧɵɯ ɱɚɫɬɨɬ (2.23) ɧɟɡɚɬɭɯɚɸɳɢɯ
ɤɨɥɟɛɚɧɢɣ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ ɫɨɨɪɭɠɟɧɢɹ ɢ ɞɥɹ ɤɚɠɞɨɣ ɣ
i
ɋɎɄ ɧɚɯɨɞɹɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɟ
i
Z
ɜɟɤɬɨɪɚ
ɩɟɪɟɦɟɳɟɧɢɣ ɦɚɫɫ
i
u ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɪɢɧɹɬɵɯ ɞɥɹ ɪɚɫɱɟɬɚ ɢɯ ɫɬɟɩɟɧɟɣ
ɫɜɨɛɨɞɵ,
ɗɬɚ ɱɚɫɬɶ ɪɚɫɱɟɬɚ ɫɨɨɪɭɠɟɧɢɹ ɧɚ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɪɢ
ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɪɨɝɪɚɦɦɵ SCAD ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ «Ɇɨɞɚɥɶɧɵɣ
ɚɧɚɥɢɡ», ɞɚɠɟ, ɟɫɥɢ ɨɧɚ ɛɵɥɚ ɜɵɩɨɥɧɟɧɚ ɨɬɞɟɥɶɧɨ.
2. ɉɪɢ ɡɚɞɚɧɢɢ ɜɟɤɬɨɪɚ ɜɨɡɦɭɳɚɸɳɟɣ ɧɚɝɪɭɡɤɢ ɜ ɜɢɞɟ (2.58)
ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨɫɬɨɹɧɧɵɟ ɞɥɹ ɤɚɠɞɨɣ ɋɎɄ ɤɨɷɮɮɢɰɢɟɧɬɵ
i
K
,
ni ...,,2,1
,
(2.60) ɢ ɜɟɤɬɨɪɵ ɥɨɤɚɥɶɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɋɎɄ
iii
u
ˤ
K (2.64).
3. ɉɨ ɮɨɪɦɭɥɟ (2.28) ɨɩɪɟɞɟɥɹɸɬɫɹ ɚɦɩɥɢɬɭɞɧɵɟ ɡɧɚɱɟɧɢɹ ɢɧɟɪɰɢɨɧɧɵɯ
ɫɢɥ
iii
uMj
2
Z ɢ ɜɵɩɨɥɧɹɟɬɫɹ ɫɬɚɬɢɱɟɫɤɢɣ ɪɚɫɱɟɬ ɫ ɰɟɥɶɸ ɨɩɪɟɞɟɥɟɧɢɹ
ɥɸɛɵɯ
ɵɯk
ɭɫɢɥɢɣ ɢɥɢ ɧɚɩɪɹɠɟɧɢɣ ɜ ɣi ɋɎɄ (
ni ...,,2,1
).
4. Ⱦɥɹ ɤɚɠɞɨɣ ɋɎɄ ɞɨɥɠɟɧ ɜɵɱɢɫɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥ (2.62).
5. Ⱦɚɥɟɟ ɜɵɩɨɥɧɹɸɬɫɹ ɫɭɦɦɢɪɨɜɚɧɢɹ (2.63), (2.65) ɢ (2.67) ɞɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɜ ɢɧɬɟɪɟɫɭɸɳɢɟ ɪɚɫɱɟɬɱɢɤɚ
ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ.
Ɉɞɧɚɤɨ ɱɟɬɜɟɪɬɵɣ ɢ ɩɹɬɵɣ ɩɭɧɤɬɵ ɹɜɥɹɸɬɫɹ ɜɟɫɶɦɚ ɬɪɭɞɨɟɦɤɢɦɢ.
ɉɨɷɬɨɦɭ, ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɞɥɹ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ
ɨɛɵɱɧɨ ɜ ɩ.4 ɨɝɪɚɧɢɱɢɜɚɸɬɫɹ ɨɩɪɟɞɟɥɟɧɢɟɦ ɜ ɤɚɠɞɨɣ ɋɎɄ ɬɨɥɶɤɨ
ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɢɧɬɟɝɪɚɥɚ:
107
)(max
t
t
ii
E E .
(2.68)
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ ɜ ɋɎɄ ɫɬɚɧɨɜɢɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦ ɨɬ
ɜɪɟɦɟɧɢ (ɫɦ. ɫɥɚɝɚɟɦɵɟ ɜ ɮɨɪɦɭɥɚɯ (2.63), (2.65) ɢ (2.67):
iiiii
t EZ
2
t
o
)(max
ˤ
ua ;
(2.69)
gt
iiiiii
/)(max
t
o
E E G
ˤˤ
Mss ;
(2.70)
iiiiii
vtvv EZK
2
t
o
)(max
(2.71)
ȼ ɢɧɠɟɧɟɪɧɨɣ ɩɪɚɤɬɢɤɟ, ɞɥɹ ɫɭɦɦɢɪɨɜɚɧɢɹ ɷɬɢɯ ɧɟɡɚɜɢɫɢɦɵɯ ɨɬ ɜɪɟɦɟɧɢ
ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ, ɩɨɥɭɱɟɧɧɵɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɚɡɥɨɠɟɧɢɹ ɢɫɤɨɦɵɯ
ɜɟɥɢɱɢɧ ɩɨ ɋɎɄ, ɢ ɨɩɪɟɞɟɥɟɧɢɹ ɢɯ ɪɚɫɱɟɬɧɵɯ ɡɧɚɱɟɧɢɣ ɩɪɢɦɟɧɹɸɬ
ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ, ɤɨɬɨɪɵɟ ɪɚɫɫɦɨɬɪɢɦ ɞɚɥɟɟ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ
ɤɨɧɤɪɟɬɧɵɦ ɡɚɞɚɱɚɦ.
2.6. Ɇɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɩɨ ɋɎɄ ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ
ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɚ ɫɢɫɬɟɦɭ ɫ
n ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ ɡɚɞɚɧ ɜɟɤɬɨɪ
ɜɨɡɦɭɳɚɸɳɟɣ ɧɚɝɪɭɡɤɢ ɜ ɜɢɞɟ (2.58). Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɟɪɟɦɟɳɟɧɢɹ,
ɭɫɢɥɢɹ ɢ ɧɚɩɪɹɠɟɧɢɹ ɩɪɢ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫɢɫɬɟɦɵ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɢɫɩɨɥɶɡɭɟɦ ɦɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɩɨ ɋɎɄ ɢ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɮɨɪɦɭɥɵ, ɩɨɥɭɱɟɧɧɵɟ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɨɞɪɚɡɞɟɥɟ.
Ʉɨɧɤɪɟɬɢɡɢɪɭɟɦ ɜɢɞ ɮɨɪɦɭɥ ɞɥɹ ɥɸɛɨɣ ɣ
i ɋɎɄ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦɭ ɜɚɪɢɚɧɬɭ, ɤɨɝɞɚ
t
t
f
T sin)(.
ɉɨɞɫɬɚɜɢɜ ɷɬɭ ɮɭɧɤɰɢɸ ɜ ɢɧɬɟɝɪɚɥ (2.62), ɡɚɩɢɲɟɦ ɞɥɹ ɧɟɝɨ ɪɟɲɟɧɢɟ,
ɢɡɜɟɫɬɧɨɟ ɢɡ ɩɨɞɪɚɡɞɟɥɚ (1.4), ɜ ɜɢɞɟ:
)sin()(
iii
tt DTE E
(2.72)
ɝɞɟ ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɜɵɪɚɠɟɧɢɹɦɢ (1.95) ɢ (1.99)
2
2
2
ɧ
2
2
2
2
2
22
2
2
4
2
2
2
2
2
)1(
1
4)1(
1
4)1(
1
iiiii
i
i
i
Z
T
J
Z
T
Z
T
]
Z
T
Z
T
H
Z
T
E
.
(2.73)
2
2
ɧ
2
2
2
2
2
1
tgarc
1
2
tgarc
1
2
tgarc
i
i
i
i
i
i
i
i
Z
T
Z
T
J
Z
T
Z
T
]
Z
T
Z
T
H
D
.
(2.74)
108
Ɍɨɝɞɚ ɜɟɤɬɨɪɵ ɢɫɤɨɦɵɯ ɩɟɪɟɦɟɳɟɧɢɣ (2.63) ɢ ɫɭɦɦɚɪɧɵɯ ɧɚɝɪɭɡɨɤ (2.65)
ɡɚɩɢɲɭɬɫɹ ɜ ɜɢɞɟ:
)sin()()(˓
1
o
2
111
i
n
i
iii
n
i
ii
n
i
i
n
i
i
ttt DT EZ
¦¦¦¦
a
ˤ
uuu
;
(2.75)
¦¦¦¦
DT Z
n
i
ii
n
i
iii
n
i
ii
n
i
i
ttt
1
o
1
2
11
)sin()(˓)(˓ suMjss
,
(2.76)
ɝɞɟ
ii oo
ɢ sa
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɜɟɤɬɨɪɵ ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɜɟɤɬɨɪɨɜ, ɨɩɪɟɞɟɥɹɟɦɵɯ ɩɨ ɮɨɪɦɭɥɚɦ (2.69), (2.70) ɞɥɹ
ɤɚɠɞɨɣ ɋɎɄ.
ɂɫɩɨɥɶɡɨɜɚɜ ɢɡɜɟɫɬɧɭɸ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɭɸ ɮɨɪɦɭɥɭ
iii
ttt DTrDT DrT sincoscossin)sin(
,
(2.77)
ɩɪɟɞɫɬɚɜɢɦ ɪɟɲɟɧɢɹ (2.76) ɢ (2.77) ɜ ɜɢɞɟ:
ttt
i
n
i
i
T
cc
T
c
DT
¦
cossin)sin(
1
o
aaau
;
(2.78)
ttt
n
i
ii
T
cc
T
c
DT
¦
cossin)sin(
1
o
ssss
,
(2.79)
ɉɪɢɦɟɱɚɧɢɟ ɤ ɮɨɪɦɭɥɚɦ (2.78) ɢ (2.79). ȼ ɞɚɧɧɨɦ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ ɩɪɨɰɟɫɫ ɩɨɥɭɱɟɧɢɹ
ɮɨɪɦɭɥ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɚɦɩɥɢɬɭɞɧɵɯ ɜɟɤɬɨɪɨɜ
ssaa
cccccc
,,,
ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ. ɋ ɭɱɟɬɨɦ
ɩɪɢɧɹɬɵɯ ɪɚɧɟɟ ɨɛɨɡɧɚɱɟɧɢɣ (ɜ ɬɨɦ ɱɢɫɥɟ ɜ ɩɨɞɪɚɡɞɟɥɟ 1.4) ɩɪɢɜɟɞɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ
ɮɨɪɦɭɥɵ ɜ ɨɤɨɧɱɚɬɟɥɶɧɨɦ ɜɢɞɟ:
¦
c
c
n
i
i
1
aa
, ɝɞɟ
iiii
a ua
2
Z
c
,
iiii
a FEK
2
;
¦
cc
cc
n
i
i
1
aa
, ɝɞɟ
iiii
b ua
2
Z
cc
,
ɧ
2
JEK
iii
b ;
¦
c
c
n
i
i
1
ss
, ɝɞɟ
iiiiiii
a
ˤ
MuMaMs
22
EF
c
Z
;
¦
cc
cc
n
i
i
1
ss
, ɝɞɟ
iiiiii
b
ˤ
MuMaMs
2
ɧ
2
EJ
cc
Z
cc
.
ɂɡ ɧɢɯ, ɤɚɤ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ, ɩɨɥɭɱɚɸɬɫɹ ɮɨɪɦɭɥɵ (1.97)* ɢ (1.111), ɩɨɥɭɱɟɧɧɵɟ ɞɥɹ
ɚɦɩɥɢɬɭɞɧɵɯ ɡɧɚɱɟɧɢɣ A
cc
ɢ A
cc
ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɚɦɩɥɢɬɭɞɧɵɯ
ɡɧɚɱɟɧɢɣ S
c
ɢ S
cc
ɢɧɟɪɰɢɨɧɧɨɣ ɫɢɥɵ ɜ ɫɢɫɬɟɦɟ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ. ɉɪɢ ɷɬɨɦ
ɫɤɚɥɹɪɧɭɸ ɜɟɥɢɱɢɧɭ ɜ ɱɢɫɥɢɬɟɥɟ ɜɵɪɚɠɟɧɢɹ (2.60) ɞɥɹ ɫɤɚɥɹɪɧɨɣ ɜɟɥɢɱɢɧɵ
i
K ɦɨɠɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɨɛɨɛɳɟɧɧɭɸ ɫɢɥɭ ɜ ɋɎɄ ɫ ɧɨɦɟɪɨɦ i :
o
ɬ
o
)( pu
ii
P , ɚ ɫɤɚɥɹɪɧɭɸ
ɜɟɥɢɱɢɧɭ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜɵɪɚɠɟɧɢɹ (2.60) – ɤɚɤ ɨɛɨɛɳɟɧɧɭɸ ɦɚɫɫɭ
iii
m uMu
ɬ
)( ɜ ɷɬɨɣ ɠɟ
ɋɎɄ.
109
ɉɪɢɜɟɞɟɧɧɵɟ ɮɨɪɦɭɥɵ ɚɧɚɥɨɝɢɱɧɵ ɢɫɩɨɥɶɡɭɟɦɵɦ ɜ ɩɪɨɝɪɚɦɦɟ SCAD [23] ɢ
ɩɨɥɭɱɟɧɧɵɦ ɜ ɭɱɟɛɧɢɤɟ [11] (ɫ ɭɱɟɬɨɦ ɩɪɢɧɹɬɵɯ ɜ ɭɤɚɡɚɧɧɵɯ ɪɚɛɨɬɚɯ ɨɛɨɡɧɚɱɟɧɢɣ ɢ
ɧɟɫɤɨɥɶɤɨ ɨɬɥɢɱɚɸɳɟɣɫɹ ɮɨɪɦɵ ɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɹ).
ɉɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɤɬɨɪɨɜ ɧɚɝɪɭɡɨɤ
(2.79), ss
cc
c
ɜɵɩɨɥɧɹɸɬɫɹ ɞɜɚ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɫɬɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚ ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɫɢɥɢɹ, ɧɚɩɪɢɦɟɪ,
ɢ
kk
MM
ccc
ɞɥɹ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɜ ɫɟɱɟɧɢɢ
k
. Ɋɚɫɱɟɬɧɵɟ ɭɫɢɥɢɹ,
ɧɚɩɪɢɦɟɪ,
ɪk,
M
ɞɥɹ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨɦɟɧɬɚ ɜ ɫɟɱɟɧɢɢ
k
, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ
ɮɨɪɦɭɥɟ
22
p,
)()(
kkk
MMM
cc
c
.
(2.80)
ɉɪɢɦɟɪ 1. ȼ ɩɪɢɦɟɪɟ 1 ɩɨɞɪɚɡɞɟɥɚ 1.4 ɛɵɥɨ ɪɚɫɫɦɨɬɪɟɧɨ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɠɟɥɟɡɨɛɟɬɨɧɧɨɣ ɛɚɥɤɢ ɫ ɷɥɟɤɬɪɨɞɜɢɝɚɬɟɥɟɦ, ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɩɨ
ɫɟɪɟɞɢɧɟ ɟɟ ɩɪɨɥɟɬɚ. Ɋɚɫɱɟɬ ɜɵɩɨɥɧɹɥɫɹ ɨɬ ɞɜɭɯ ɫɬɚɬɢɱɟɫɤɢɯ ɡɚɝɪɭɠɟɧɢɣ (ɡɚɝɪɭɠɟɧɢɟ 1 ɫ
ɢɦɟɧɟɦ «ɫɜ-ɛ» ɨɬ ɫɨɛɫɬɜɟɧɧɨɝɨ ɜɟɫɚ ɛɚɥɤɢ ɢ ɡɚɝɪɭɠɟɧɢɟ 2 ɫ ɢɦɟɧɟɦ «ɫɜ-ɞ» ɨɬ ɫɨɛɫɬɜɟɧɧɨɝɨ
ɜɟɫɚ ɞɜɢɝɚɬɟɥɹ) ɚ ɬɚɤɠɟ ɨɬ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɡɚɝɪɭɠɟɧɢɹ ɜɟɪɬɢɤɚɥɶɧɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ
ɧɚɝɪɭɡɤɨɣ
tPtP
o
T sin)(
(ɡɚɝɪɭɠɟɧɢɟ 3 ɫ ɢɦɟɧɟɦ «ɝɚɪɦ-1») ɩɨ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɟ ɛɚɥɤɢ ɜ
ɜɢɞɟ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ (ɫɦ. ɪɢɫ. 1.24, ɛ).
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɲɟɧɢɟ ɷɬɨɣ ɠɟ ɡɚɞɚɱɢ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ ɆɄɗ, ɤɨɝɞɚ
ɛɚɥɤɚ ɩɨ ɞɥɢɧɟ ɩɪɨɥɟɬɚ ɪɚɡɞɟɥɟɧɚ ɧɚ ɱɟɬɵɪɟ ɪɚɜɧɵɯ ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɚ ɬɢɩɚ 2
(ɫɦ. ɪɢɫ. 2.2, ɚ, ɛ). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢ ɡɚɝɪɭɠɟɧɢɢ ɭɡɥɚ 3 ɞɢɧɚɦɢɱɟɫɤɨɣ ɧɚɝɪɭɡɤɨɣ ɜ ɜɢɞɟ
tPtP T sin)(
o
ɩɨɥɭɱɢɦ ɪɚɫɱɟɬɧɭɸ ɫɯɟɦɭ ɜ ɜɢɞɟ ɫɢɫɬɟɦɵ ɫ ɬɪɟɦɹ ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ,
ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ, ɩɪɢɜɟɞɟɧɧɨɦ ɧɚ ɪɢɫ. 2.6.
ɋɧɚɱɚɥɚ ɫɨɡɞɚɞɢɦ ɫɬɚɬɢɱɟɫɤɢɟ ɡɚɝɪɭɠɟɧɢɹ 1 ɢ 2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɬ ɫɨɛɫɬɜɟɧɧɨɝɨ ɜɟɫɚ
ɛɚɥɤɢ ɢ ɞɜɢɝɚɬɟɥɹ. Ɂɚɬɟɦ ɩɟɪɟɣɞɟɦ ɤ ɞɢɧɚɦɢɱɟɫɤɨɦɭ ɡɚɝɪɭɠɟɧɢɸ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɨɛ
ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫɢɫɬɟɦɵ, ɜɵɡɜɚɧɧɵɯ ɜɪɚɳɟɧɢɟɦ ɪɨɬɨɪɚ
ɷɥɟɤɬɪɨɞɜɢɝɚɬɟɥɹ ɫ ɬɟɦɢ ɠɟ ɩɚɪɚɦɟɬɪɚɦɢ, ɤɨɬɨɪɵɟ ɪɚɫɫɦɚɬɪɢɜɚɥɢɫɶ ɩɪɢ ɪɟɲɟɧɢɢ
ɚɧɚɥɨɝɢɱɧɨɣ ɡɚɞɚɱɢ ɞɥɹ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ (ɫɦ. ɩɪɢɦɟɪ 1 ɜ ɩɨɞɪɚɡɞɟɥɟ 1.4).
Ɋɢɫ. 2.6
ɗɬɚɩ 3. ɋɨɡɞɚɧɢɟ ɡɚɝɪɭɠɟɧɢɹ 3 «Ƚɚɪɦ-3» ɞɥɹ ɞɢɧɚɦɢɱɟɫɤɨɣ ɡɚɞɚɱɢ «Ƚɚɪɦɨɧɢɱɟɫɤɢɟ
ɤɨɥɟɛɚɧɢɹ». Ⱦɥɹ ɷɬɨɝɨ ɧɚɠɦɟɦ ɤɧɨɩɤɭ
ɧɚ ɢɧɫɬɪɭɦɟɧɬɚɥɶɧɨɣ ɩɚɧɟɥɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ
ɪɚɡɞɟɥɭ Ɂɚɝɪɭɠɟɧɢɟ ɢ ɨɬɤɪɨɟɦ ɩɟɪɜɭɸ ɫɬɪɚɧɢɰɭ «ȼɜɨɞ ɩɚɪɚɦɟɬɪɨɜ ɞɢɧɚɦɢɱɟɫɤɨɣ ɧɚɝɪɭɡɤɢ»
2
ɞ
G
3
2
ɛ
q
2
1
5
2
2
4
)(tP