Готман В.И. Короткие замыкания и несимметричные режимы в электроэнергетических системах
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ɇɚ ɪɢɫ. 4.5, ɚ ɩɪɢɜɟ
ɞ
ɧɵɯ ɨɛɦɨɬɨɤ) ɜ ɩɪɨɞɨɥɶɧɨ
ɣ
ɨɛɦɨɬɤɢ – ɜɨɡɛɭɠɞɟɧɢɹ ɢ
ɫ
ɧɚ ɗȾɋ
qf
E
, ɨɬɜɟɱɚɸɳɚɹ
Ɂɚɦɟɧɹɹ ɦɚɝɧɢɬɧɭɸ ɫɜ
ɹ
ɡɶ
ɱɟɫɤɨɣ, ɩɨɥɭɱɚɟɦ ɫɯɟɦɭ ɡɚ
ɦ
ɝɢɱɧɚ ɫɯɟɦɟ ɡɚɦɟɳɟɧɢɹ ɞ
ɜ
ɧɵ ɚɤɬɢɜɧɨɣ ɜɟɬɜɢ ɫ
f
x
V
,
ɩɨɥɭɱɚɟɦ ɫɯɟɦɭ (
ɪ
ɢɫ. 4.5
,
ɩɪɟɞɫɬɚɜɥɟɧɚ ɫɜɨɢɦɢ ɩɟɪɟ
ɯ
Ɋɢɫ. 4.5. Ʉ ɨɩɪɟɞɟɥɟɧɢ
ɸ
ɜ ɩɪɨɞɨɥɶɧɨɣ ɨ
ɫ
ɛ, ɜ – ɞɜɭɯɤɨɧɬɭɪɧɵɟ ɫɯɟɦ
ɫɯɟɦɚ ɡɚɦɟɳɟ
ɧ
ȼ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ
q
ɫɬɚɬɨɪɧɚɹ ɨɛɦɨɬɤɚ (ɫɦ. ɪɢ
ɫ
ɲɟɧɢɹ ɪɟɠɢɦɚ ɨɧɚ ɯɚɪɚɤɬ
ɟ
ɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ:
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ
ɫɬɶɸ
ɜɧ
x
ɧɚɱɚɥɶɧɨɟ ɡɧɚɱɟ
ɧ
ɝɨ ɪɟɠɢɦɚ (ɧɚɱɚɥɶɧɵɣ ɩɟɪ
ɟ
51
ɞ
ɟɧɚ ɩɪɢɧɰɢɩɢɚɥɶɧɚɹ ɫɯɟɦɚ ɋɆ (ɛɟɡ
ɞ
ɣ
ɨɫɢ
d
, ɫɨɞɟɪɠɚɳɚɹ ɞɜɟ ɦɚɝɧɢɬɧɨ ɫ
ɜ
ɫ
ɬɚɬɨɪɚ. ȼ ɤɨɧɬɭɪ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧ
ɢ
ɟɟ
ɪ
ɟɡɭɥɶɬɢɪɭɸɳɟɦɭ ɩɨɬɨɤɨɫɰɟɩɥɟɧ
ɢ
ɭɤɚɡɚɧɧɵɯ ɤɨɧɬɭɪɨɜ ɷɤɜɢɜɚɥɟɧɬɧɨɣ
ɷ
ɦ
ɟɳɟɧɢɹ ɦɚɲɢɧɵ (ɪɢɫ. 4.5, ɛ), ɤɨɬɨɪ
ɚ
ɜ
ɭɯɨɛɦɨɬɨɱɧɨɝɨ ɬɪɚɧɫɮɨɪɦɚɬɨɪɚ. ɉɨ
ɫ
qf
E
ɢ ɩɚɫɫɢɜɧɨɣ ɫ
ad
x
ɨɞɧɨɣ ɷɤɜɢɜ
ɚ
,
ɜ), ɝɞɟ, ɤɚɤ ɢ ɫɥɟɞɨɜɚɥɨ ɨɠɢɞɚɬɶ,
ɯ
ɨɞɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ
q
E
c
ɢ
d
x
c
.
ɸ
ɩɟɪɟɯɨɞɧɨɣ ɗȾɋ ɢ ɪɟɚɤɬɢɜɧɨɫɬɢ ɦɚɲ
ɢ
ɫ
ɢ:
ɚ – ɢɫɯɨɞɧɚɹ ɩɪɢɧɰɢɩɢɚɥɶɧɚɹ ɫɯɟɦɚ;
ɵ ɡɚɦɟɳɟɧɢɹ; ɝ – ɷɤɜɢɜɚɥɟɧɬɧɚɹ ɨɞɧɨɤɨɧɬ
ɭɪ
ɧ
ɢɹ ɞɥɹ ɪɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ
q
ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ ɪɚɫɩɨɥɨɠɟɧ
ɚ
ɫ
. 4.1, ɜ), ɩɨɷɬɨɦɭ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟ
ɧ
ɟ
ɪɢɡɭɟɬɫɹ ɬɚɤɢɦɢ ɠɟ ɩɚɪɚɦɟɬɪɚɦɢ, ɱɬ
ɨ
0
d
E
c
,
qq
xx
c
.
ɄɁ ɜ ɫɬɚɬɨɪɧɨɣ ɰɟɩɢ ɡɚ ɜɧɟɲɧɟɣ ɪɟ
ɚ
ɧ
ɢɟ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɥɚɝɚɟɦɨɣ ɬɨɤɚ ɩɟ
ɪ
ɟ
ɯɨɞɧɵɣ ɬɨɤ) ɛɭɞɟɬ ɬɨɥɶɤɨ ɩɪɨɞɨɥɶɧ
ɵ
ɞ
ɟɦɩɮɟɪ-
ɜ
ɹɡɚɧɧɵɟ
ɢ
ɹ ɜɜɟɞɟ-
ɢ
ɸ
f 6
Ȍ
.
ɷ
ɥɟɤɬɪɢ-
ɚ
ɹ ɚɧɚɥɨ-
ɫ
ɥɟ ɡɚɦɟ-
ɚ
ɥɟɧɬɧɨɣ
ɦɚɲɢɧɚ
ɢ
ɧɵ
ɭɪ
ɧɚɹ
ɚ
ɬɨɥɶɤɨ
ɧ
ɬ ɧɚɪɭ-
ɨ
ɢ ɜ
ɭ
ɫ-
(4.11)
ɚ
ɤɬɢɜɧɨ-
ɪ
ɟɯɨɞɧɨ-
ɵ
ɦ:
d
I
ɉɟɪɟɯɨɞɧɚɹ
ɪ
ɟɚɤɬɢɜ
ɧ
ɯɪɨɧɧɨɣ (
d
x
) ɢ ɫɨɫɬɚɜɥɹ
ɟ
ɲɢɧ ɢ ɞɥɹ ɹɜɧɨɩɨɥɸɫɧɵɯ
x
ɉɟɪɟɯɨɞɧɚɹ ɗȾɋ, ɪ
ɚ
ɧɟɪɚɬɨɪɚ, ɜ ɨ. ɟ. ɜ ɫɪɟɞɧɟ
ɦ
ɫɢɧɯɪɨɧɧɚɹ –
2,15
q
E
.
ɇɚ ɪɢɫ. 4.6 ɩɪɟɞɫɬɚɜ
ɥ
ɩɚɪɚɦɟɬɪɨɜ (ɫɢɧɯɪɨɧɧɨɣ ɢ
ɧɨɣ ɰɟɩɢ) ɫɢɧɯɪɨɧɧɨɝɨ ɝɟ
ɧ
ɜɨɡɛɭɠɞɟɧɢɹ ɜ ɩɟɪɟɯɨɞɧɨ
ɦ
ɞɨɥɠɟɧɢɟɦ ɩɪɚɜɨɝɨ ɩɨ ɜɟ
ɪ
ɧɵɯ ɩɚɪɚɦɟɬɪɨɜ, ɢɫɯɨɞɹ ɢɡ
Ɋɢɫ. 4.6. ɂɡɦɟɧɟɧɢɟ ɩɚɪɚɦɟ
ɬ
ɚ – ɫɢɧɯɪɨɧɧɨɣ ɗȾɋ
q
E
; ɛ – ɩ
ɟ
ɝ – ɩɟɪ
ɢ
ɋɢɧɯɪɨɧɧɚɹ ɗȾɋ (ɪ
ɢ
ɠɢɦɟ ɬɨɤɨɦ ɜɨɡɛɭɠɞɟɧɢɹ
ɰɟɩɢ ɫɬɚɬɨɪɚ ɜɧɟɲɧɟɟ ɫɨɩ
ɪ
ɪɢɨɞɢɱɟɫɤɢɣ ɬɨɤ
0d
I
(ɪɢɫ.
0ddf
II
. ɗɬɨɬ ɫɤɚɱɨɤ
52
0
0 ɉ 0
ɜɧ
q
d
d
E
I
xx
c
c
.
ɧ
ɨɫɬɶ (
d
x
c
) ɜ ɫɪɟɞɧɟɦ ɧɚ ɩɨɪɹɞɨɤ ɦɟɧ
ɶ
ɟ
ɬ
0,15...0,37
d
x
c
ɞɥɹ ɧɟɹɜɧɨɩɨɥɸ
ɫ
0,2...0,5
d
x
c
.
ɚ
ɫɫɱɢɬɚɧɧɚɹ ɩɨ ɧɨɦɢɧɚɥɶɧɵɦ ɩɚ
ɪ
ɚɦɟ
ɬ
ɦ
ɫɨɫɬɚɜɥɹɟɬ:
1, 0 9
q
E
c
|
; ɫɨɨɬɜɟɬɫɬɜɭ
ɸ
ɥ
ɟɧɵ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɢɡɦɟɧɟɧɢɹ ɯɚɪ
ɚ
ɩɟɪɟɯɨɞɧɨɣ ɗȾɋ, ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤ
ɚ
ɧ
ɟɪɚɬɨɪɚ ɛɟɡ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥ
ɢ
ɦ
ɪɟɠɢɦɟ (ɥɟɜɚɹ ɱɚɫɬɶ ɪɢɫɭɧɤɚ ɹɜɥɹɟ
ɪ
ɬɢɤɚɥɶɧɨɣ ɨɫɢ). ɉɨɹɫɧɢɦ ɩɨɜɟɞɟɧɢ
ɟ
ɮɢɡɢɤɢ ɹɜɥɟɧɢɣ.
ɬ
ɪɨɜ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɋɆ ɜ ɩɟɪɟɯɨɞɧɨɦ
ɪ
ɟ
ɪɟɯɨɞɧɨɣ ɗȾɋ
q
E
c
; ɜ – ɧɚɩɪɹɠɟɧɢɹ ɝɟɧɟɪɚ
ɬ
ɢ
ɨɞɢɱɟɫɤɨɝɨ ɬɨɤɚ ɫɬɚɬɨɪɚ
d
I
ɢ
ɫ. 4.6, ɚ), ɨɛɭɫɥɨɜɥɟɧɧɚɹ ɜ ɧɨɪɦɚɥ
ɶ
0f
I
, ɪɚɜɧɚ
0q
E
. ɉɪɢ ɤɨɪɨɬɤɨɦ ɡɚɦ
ɵ
ɪ
ɨɬɢɜɥɟɧɢɟ ɭɦɟɧɶɲɚɟɬɫɹ, ɚ ɜɵɧɭɠɞɟ
ɧ
4.6, ɝ) ɫɤɚɱɤɨɨɛɪɚɡɧɨ ɜɨɡɪɚɫɬɚɟɬ ɧɚ
ɜ
ɬɨɤɚ ɢɧɞɭɰɢɪɭɟɬ ɜ ɨɛɦɨɬɤɟ ɜɨɡɛ
ɭ
ɶ
ɲɟ ɫɢɧ-
ɫ
ɧɵɯ ɦɚ-
ɬɪɚɦ ɝɟ-
ɸ
ɳɚɹ ɟɣ
ɚ
ɤɬɟɪɧɵɯ
ɚ
ɫɬɚɬɨɪ-
ɢ
ɪɨɜɚɧɢɹ
ɟ
ɬɫɹ ɩɪɨ-
ɟ
ɭɤɚɡɚɧ-
ɪ
ɟɠɢɦɟ:
ɬ
ɨɪɚ
q
U
;
ɶ
ɧɨɦ ɪɟ-
ɵ
ɤɚɧɢɢ ɜ
ɧ
ɧɵɣ ɩɟ-
ɜ
ɟɥɢɱɢɧɭ
ɭ
ɠɞɟɧɢɹ
53
ɩɨɫɬɨɹɧɧɵɣ ɬɨɤ
0
f
'I
, ɫɨɜɩɚɞɚɸɳɢɣ ɩɨ ɡɧɚɤɭ ɫ ɨɫɧɨɜɧɵɦ ɬɨɤɨɦ ɜɨɡ-
ɛɭɠɞɟɧɢɹ
0
f
I
. ɋɥɟɞɫɬɜɢɟɦ ɷɬɨɝɨ ɹɜɥɹɟɬɫɹ ɫɤɚɱɤɨɨɛɪɚɡɧɨɟ ɜɨɡɪɚɫɬɚɧɢɟ
ɫɢɧɯɪɨɧɧɨɣ ɗȾɋ ɧɚ ɜɟɥɢɱɢɧɭ
(0)q
'E
:
(0) 0 (0)qqq
EEE '
,
ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 4.6, ɚ.
ɋ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɣ ɬɨɤ ɰɟɩɢ ɜɨɡɛɭɠɞɟɧɢɹ
()
f
'I
ɡɚɬɭɯɚɟɬ ɩɨ ɡɚɤɨɧɭ ɷɤɫɩɨɧɟɧɬɵ, ɜɵɡɵɜɚɹ ɚɧɚɥɨɝɢɱɧɨɟ ɢɡɦɟɧɟɧɢɟ ɩɪɢɪɚ-
ɳɟɧɢɹ ɫɢɧɯɪɨɧɧɨɣ ɷɥɟɤɬɪɨɞɜɢɠɭɳɟɣ ɫɢɥɵ (
q
'E
). ȼ ɧɨɜɨɦ ɭɫɬɚɧɨɜɢɜ-
ɲɟɦɫɹ ɪɟɠɢɦɟ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ ɬɨɤ
f
'I
ɪɚɜɟɧ ɧɭɥɸ, ɚ ɫɢɧɯɪɨɧɧɚɹ
ɗȾɋ – ɫɜɨɟɦɭ ɩɪɟɞɲɟɫɬɜɭɸɳɟɦɭ ɡɧɚɱɟɧɢɸ ɜ ɧɨɪɦɚɥɶɧɨɦ ɪɟɠɢɦɟ:
0qqf
EE
.
ɋɨɫɬɚɜɥɹɸɳɚɹ ɧɚɩɪɹɠɟɧɢɹ ɝɟɧɟɪɚɬɨɪɚ ɩɨ ɨɫɢ q (
0q
U
) ɜ ɦɨɦɟɧɬ ɧɚ-
ɪɭɲɟɧɢɹ ɪɟɠɢɦɚ ɩɪɟɬɟɪɩɟɜɚɟɬ ɫɤɚɱɤɨɨɛɪɚɡɧɨɟ ɫɧɢɠɟɧɢɟ (ɫɦ. ɪɢɫ. 4.6, ɜ).
ȼɟɥɢɱɢɧɚ ɫɤɚɱɤɚ ɬɟɦ ɛɨɥɶɲɟ, ɱɟɦ ɛɥɢɠɟ ɦɟɫɬɨ ɩɨɜɪɟɠɞɟɧɢɹ ɤ ɝɟɧɟɪɚ-
ɬɨɪɭ. ȼ ɩɪɟɞɟɥɟ, ɤɨɝɞɚ ɧɚ ɤɥɟɦɦɚɯ ɝɟɧɟɪɚɬɨɪɚ ɩɪɨɢɫɯɨɞɢɬ ɬɪɟɯɮɚɡɧɨɟ
ɤɨɪɨɬɤɨɟ ɡɚɦɵɤɚɧɢɟ, ɩɟɪɜɨɧɚɱɚɥɶɧɨɟ ɫɧɢɠɟɧɢɟ ɱɢɫɥɟɧɧɨ ɪɚɜɧɨ ɧɚɩɪɹ-
ɠɟɧɢɸ ɜ ɩɪɟɞɲɟɫɬɜɭɸɳɟɦ ɪɟɠɢɦɟ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚɩɪɹɠɟɧɢɟ ɜ
ɷɬɨɦ ɫɥɭɱɚɟ ɪɚɜɧɨ ɧɭɥɸ. ɉɪɢ
ɫɪɟɞɧɟɣ ɭɞɚɥɟɧɧɨɫɬɢ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚ-
ɧɢɹ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ
q
U
ɜ ɩɟɪɟɯɨɞɧɨɦ ɪɟɠɢɦɟ ɛɭɞɟɬ ɬɚɤɢɦ, ɤɚɤ
ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 4.6, ɜ.
ɉɟɪɢɨɞɢɱɟɫɤɢɣ ɬɨɤ ɰɟɩɢ ɫɬɚɬɨɪɚ ɩɨɫɥɟ ɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ ɜɨɡɪɚɫɬɚ-
ɧɢɹ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɡɚɬɭɯɚɟɬ ɞɨ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɡɧɚɱɟɧɢɹ
df
I
.
Ɍɚɤɨɣ ɯɚɪɚɤɬɟɪ ɢɡɦɟɧɟɧɢɹ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɡɚɬɭɯɚɧɢɹ ɚɩɟɪɢɨɞɢɱɟ-
ɫɤɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɬɨɤɚ
f
'I
ɰɟɩɢ ɜɨɡɛɭɠɞɟɧɢɹ.
ɂɬɚɤ, ɫɢɧɯɪɨɧɧɚɹ ɗȾɋ, ɪɚɜɧɨ ɤɚɤ ɢ ɧɚɩɪɹɠɟɧɢɟ ɝɟɧɟɪɚɬɨɪɚ, ɜ ɦɨ-
ɦɟɧɬ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ ɫɤɚɱɤɨɨɛɪɚɡɧɨ ɢɡɦɟɧɹɸɬɫɹ. ȼɟɥɢɱɢɧɵ ɷɬɢɯ
ɫɤɚɱɤɨɜ ɡɚɜɢɫɹɬ ɨɬ ɢɫɤɨɦɨɝɨ ɬɨɤɚ
(0)d
I
ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ, ɢ ɩɨɷɬɨɦɭ
ɧɢ ɫɢɧɯɪɨɧɧɚɹ ɗȾɋ (
0q
E
), ɧɢ ɧɚɩɪɹɠɟɧɢɟ ɝɟɧɟɪɚɬɨɪɚ ɧɨɪɦɚɥɶɧɨɝɨ ɪɟ-
ɠɢɦɚ (
0q
U
) ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɞɥɹ ɪɚɫɱɟɬɚ ɬɨɤɚ
(0)d
I
.
ɉɟɪɟɯɨɞɧɚɹ ɗȾɋ
(0)q
c
E
ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɧɟɡɚɩɧɨɝɨ ɧɚɪɭɲɟ-
ɧɢɹ ɪɟɠɢɦɚ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ ɢ ɪɚɜɧɨɣ ɫɜɨɟɦɭ ɡɧɚɱɟɧɢɸ ɜ ɩɪɟɞɲɟ-
ɫɬɜɭɸɳɟɦ ɪɟɠɢɦɟ
0q
c
E
. ɗɬɨ, ɤɚɤ ɨɬɦɟɱɚɥɨɫɶ ɪɚɧɟɟ, ɩɨɡɜɨɥɹɟɬ ɢɫɩɨɥɶɡɨ-
ɜɚɬɶ ɟɟ ɞɥɹ ɪɚɫɱɟɬɚ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫɥɚɝɚɟɦɨɣ ɬɨɤɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ.
54
4.4. Ɍɝɠɫɰɪɠɫɠɰɩɟɨɶɠ ɘȿɌ ɣ ɫɠɛɥɭɣɝɨɩɬɭɣ ɬɣɨɰɫɩɨɨɩɤ ɧɛɳɣɨɶ
ɍɫɬɚɧɨɜɢɦ ɩɚɪɚɦɟɬɪɵ, ɤɨɬɨɪɵɦɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɦɚɲɢɧɵ ɫ
ɞɟɦɩɮɟɪɧɵɦɢ ɨɛɦɨɬɤɚɦɢ ɜ ɦɨɦɟɧɬ ɜɧɟɡɚɩɧɨɝɨ ɧɚɪɭɲɟɧɢɹ ɪɟɠɢɦɚ.
ɉɨɥɚɝɚɟɦ, ɱɬɨ ɩɨɦɢɦɨ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ ɧɚ ɪɨɬɨɪɟ ɢɦɟɟɬɫɹ ɩɨ
ɨɞɧɨɣ ɞɟɦɩɮɟɪɧɨɣ ɨɛɦɨɬɤɟ ɜ ɩɪɨɞɨɥɶɧɨɣ ɢ ɩɨɩɟɪɟɱɧɨɣ ɨɫɹɯ. ɇɚɥɢɱɢɟ
ɞɟɦɩɮɟɪɧɵɯ ɨɛɦɨɬɨɤ ɧɟ ɨɛɟɫɩɟɱɢɜɚɟɬ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɫɢɦɦɟɬɪɢɢ
ɦɚɲɢɧɵ, ɱɬɨ ɜɵɧɭɠɞɚɟɬ ɨɩɪɟɞɟɥɹɬɶ ɟɟ ɩɚɪɚɦɟɬɪɵ ɨɬɞɟɥɶɧɨ
ɜ ɩɪɨɞɨɥɶ-
ɧɨɣ ɢ ɩɨɩɟɪɟɱɧɨɣ ɨɫɹɯ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ, ɤɚɤ ɢ ɪɚɧɟɟ, ɨɩɢɪɚɟɦɫɹ ɧɚ
ɩɪɢɧɰɢɩ Ʌɟɧɰɚ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɦɚɝɧɢɬɧɨ-ɫɜɹɡɚɧɧɵɦ ɤɨɧɬɭɪɚɦ.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɜɵɤɥɚɞɨɤ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɜɨɡɦɨɠɧɨɫɬɶɸ ɡɚɦɟɧɵ ɬɚɤɢɯ
ɰɟɩɟɣ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɫɯɟɦɨɣ, ɚɧɚɥɨɝɢɱɧɨɣ ɫɯɟɦɟ ɬɪɚɧɫ-
ɮɨɪɦɚɬɨɪɚ.
ȼ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ
d ɫɢɧɯɪɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɬɪɢ
ɨɛɦɨɬɤɢ (ɫɦ. ɪɢɫ. 4.7, a): ɫɬɚɬɨɪɚ, ɤɨɧɬɭɪɚ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɩɪɨɞɨɥɶɧɚɹ
ɞɟɦɩɮɟɪɧɚɹ. ɋ ɰɟɥɶɸ ɭɩɪɨɳɟɧɢɹ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɫɱɢɬɚ-
ɟɦ, ɱɬɨ ɨɛɦɨɬɤɚ ɫɬɚɬɨɪɚ ɢ ɨɛɟ ɨɛɦɨɬɤɢ ɪɨɬɨɪɚ ɜ ɟɝɨ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɫɜɹ-
ɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɨɛɳɢɦ ɩɨɬɨɤɨɦ ɜɡɚɢɦɨɢɧɞɭɤɰɢɢ
ad
Ɏ , ɤɨɬɨɪɨɦɭ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɡɚɢɦɧɵɣ ɪɟɚɤɬɚɧɫ
ad
x
. ȼ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ
ɞɟɦɩɮɟɪɧɵɟ ɨɛɦɨɬɤɢ ɫɟɛɹ ɧɟ ɩɪɨɹɜɥɹɸɬ, ɩɨɫɤɨɥɶɤɭ ɧɟ ɫɨɞɟɪɠɚɬ ɫɨɛɫɬ-
ɜɟɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɗȾɋ. ɉɨ ɷɬɨɣ ɩɪɢɱɢɧɟ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ ɩɨɬɨɤɨɫɰɟ-
ɩɥɟɧɢɟ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ ɜ ɧɨɪɦɚɥɶɧɨɦ ɪɟɠɢɦɟ ɛɭɞɟɬ ɬɟɦ ɠɟ, ɱɬɨ ɢ
ɭ ɦɚɲɢɧɵ ɛɟɡ ɞɟɦɩɮɟɪɧɵɯ ɨɛɦɨɬɨɤ, ɬ. ɟ.
f
6
Ȍ
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɪɚ-
ɠɟɧɢɟɦ (4.7). Ɋɟɡɭɥɶɬɢɪɭɸɳɟɟ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɟ ɩɪɨɞɨɥɶɧɨɣ ɞɟɦɩɮɟɪ-
ɧɨɣ ɨɛɦɨɬɤɢ
1d6
Ȍ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɨɣ ɩɨɥɟɡɧɨɝɨ ɩɨɬɨɤɚ
dfad
x Ɏ I
ɢ ɨɬɜɟɬɧɨɣ ɪɟɚɤɰɢɟɣ ɫɬɚɬɨɪɚ
ad d ad
x
Ɏ I , ɱɬɨ ɜ ɫɤɚɥɹɪ-
ɧɨɦ ɜɵɪɚɠɟɧɢɢ ɡɚɩɢɲɟɬɫɹ ɬɚɤ:
1
ȌɎɎ
ddad6
. (4.12)
ɇɚ ɪɢɫ. 4.7, ɚ ɩɪɢɜɟɞɟɧɚ ɩɪɢɧɰɢɩɢɚɥɶɧɚɹ ɫɯɟɦɚ ɦɚɲɢɧɵ ɫ ɬɪɟɦɹ
ɦɚɝɧɢɬɧɨ ɫɜɹɡɚɧɧɵɦɢ ɤɨɧɬɭɪɚɦɢ. ȼ ɤɨɧɬɭɪ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ ɜɜɟ-
ɞɟɧɚ ɗȾɋ
qf
E
, ɨɬɜɟɱɚɸɳɚɹ ɟɟ ɪɟɡɭɥɶɬɢɪɭɸɳɟɦɭ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɸ
f
6
Ȍ
, ɚ ɜ ɤɨɧɬɭɪ ɩɪɨɞɨɥɶɧɨɣ ɞɟɦɩɮɟɪɧɨɣ ɨɛɦɨɬɤɢ –
1qd
E
, ɫɨɨɬɜɟɬɫɬ-
ɜɭɸɳɚɹ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɸ
1d6
Ȍ . ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɫɯɟɦɚ ɡɚɦɟɳɟɧɢɹ ɫɢɧ-
ɯɪɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ (ɫɦ. ɪɢɫ. 4.7, ɛ) ɚɧɚɥɨɝɢɱɧɚ ɫɯɟɦɟ ɡɚɦɟɳɟɧɢɹ ɬɪɟɯ-
ɨɛɦɨɬɨɱɧɨɝɨ ɬɪɚɧɫɮɨɪɦɚɬɨɪɚ. Ɉɧɚ ɫɨɞɟɪɠɢɬ ɬɪɢ ɩɚɪɚɥɥɟɥɶɧɵɟ ɜɟɬɜɢ,
ɞɜɟ ɢɡ ɤɨɬɨɪɵɯ – ɷɬɨ ɨɛɦɨɬɤɚ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɩɪɨɞɨɥɶɧɚɹ ɞɟɦɩɮɟɪɧɚɹ
ɨɛɦɨɬɤɚ – ɹɜɥɹɸɬɫɹ ɚɤɬɢɜɧɵɦɢ ɫ ɩɚɪɚɦɟɬɪɚɦɢ
qf
E
,
f
x
V
ɢ
1qd
E
,
1d
x
V
,
ɚ ɨɞɧɚ – ɩɚɫɫɢɜɧɨɣ – ɫ ɪɟ
ɚ
ɧɢɟ (ɪɢɫ. 4.7, ɜ) ɩɪɢɜɨɞɢɬ
ɤ
ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡ
ɫ
E
ɝɞɟ
q
E
cc
– ɫɜɟɪɯɩɟɪɟɯɨɞɧɚɹ
ɗ
d
x
cc
d
x
cc
– ɫɜɟɪɯɩɟɪɟɯɨɞɧɚɹ ɪɟɚ
ɤ
ɪ
ɚɫɫɟɹɧɢɹ ɩɪɨɞɨɥɶɧɨɣ ɞɟɦ
ɩ
ɋɜɟɪɯɩɟɪɟɯɨɞɧɚɹ ɗ
Ⱦ
ɠɢɦɚ ɫɨɯɪɚɧɹɟɬ ɫɜɨɟ ɡɧɚɱ
ɟ
ɬɢɱɟɫɤɭɸ ɰɟɧɧɨɫɬɶ. Ɉɧɚ ɹ
ɜ
ɩɚɪɚɦɟɬɪɚɦɢ ɧɨɪɦɚɥɶɧɨɝɨ
Ɋɢɫ. 4.7. Ʉ ɨɩɪɟɞɟɥɟɧɢ
ɸ
ɚ – ɢɫɯɨɞɧɚɹ ɩɪɢɧɰɢɩɢɚɥɶɧ
ɚ
ɨɞɧɨɤɨɧɬɭɪɧɚɹ ɫɯɟɦɚ ɡɚɦɟɳɟ
ɧ
55
ɚ
ɤɬɚɧɫɨɦ
ad
x
. ȿɟ ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɩɪɟɨ
ɤ
q
E
cc
ɢ
d
x
cc
.
ɫ
ɯɟɦɵ (ɪɢɫ. 4.7, ɜ) ɢɦɟɟɦ
00
0
qdd
q
E
UIx
cc cc
,
ɗ
Ⱦɋ ɜ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ;
1
// //
fdad
xx x x
VV V
,
ɤ
ɬɢɜɧɨɫɬɶ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ;
1d
x
V
–
ɪ
ɩ
ɮɟɪɧɨɣ ɨɛɦɨɬɤɢ.
Ⱦ
ɋ
q
E
cc
ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɟɪɟɯɨɞ
ɧ
ɟ
ɧɢɟ ɧɟɢɡɦɟɧɧɵɦ, ɱɬɨ ɢ ɨɩɪɟɞɟɥɹɟɬ
ɜ
ɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ ɪɚɫɱɟɬɧɨɣ ɢ ɨɩɪɟ
ɞ
ɪɟɠɢɦɚ (
0q
U
,
0d
I
).
ɸ
ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɋɆ ɜ ɨɫ
ɢ
ɚ
ɹ ɫɯɟɦɚ; ɛ, ɜ – ɫɯɟɦɵ ɡɚɦɟɳɟɧɢɹ; ɝ – ɷɤɜɢɜɚɥɟɧ
ɬ
ɧ
ɢɹ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɞɥɹ ɪɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪ
ɟ
ɛɪɚɡɨɜɚ-
(4.13)
(4.14)
ɪ
ɟɚɤɬɚɧɫ
ɧɨɝɨ ɪɟ-
ɟɟ ɩɪɚɤ-
ɞ
ɟɥɹɟɬɫɹ
ɢ
d:
ɬ
ɧɚɹ
ɟ
ɠɢɦɚ
ȼ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ
q
ɞɜɚ ɤɨɧɬɭɪɚ: ɨɛɦɨɬɤɚ ɫ
ɬ
(ɪɢɫ. 4.8, ɚ). ɂɯ ɜɡɚɢɦɨɞɟɣ
aq
Ɏ
, ɤɨɬɨɪɨɦɭ ɫɨɨɬɜɟɬɫɬ
ɜ
ɬɨɤɨɫɰɟɩɥɟɧɢɟ ɩɨɩɟɪɟɱɧɨ
ɣ
ɨɬɜɟɬɧɨɣ ɪɟɚɤɰɢɟɣ ɫɬɚɬɨɪɚ
Ȍ
ɗɥɟɤɬɪɢɱɟɫɤɢɟ ɫɯɟɦ
ɞɜɭɯɨɛɦɨɬɨɱɧɨɝɨ ɬɪɚɧɫɮɨ
ɪ
ɳɟɧɢɹ (ɪɢɫ. 4.8, ɜ) ɫɯɟɦɚ
ɞ
ɪ
ɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɵɪɚɠɟ
ɧ
E
x
ɝɞɟ
d
E
cc
– ɫɜɟɪɯɩɟɪɟɯɨɞɧɚɹ
ɪ
ɚɦɟɬɪɚɦɢ ɧɨɪɦɚɥɶɧɨɝɨ ɪ
ɟ
ɚɤɬɢɜɧɨɫɬɶ ɜ ɩɨɩɟɪɟɱɧɨɣ
ɪ
ɚɫɫɟɹɧɢɹ ɩɨɩɟɪɟɱɧɨɣ ɞɟɦ
ɩ
Ɋɢɫ. 4.8. Ʉ ɨɩɪɟɞɟɥɟɧɢ
ɸ
ɚ – ɢɫɯɨɞɧɚɹ ɩɪɢɧ
ɰ
ɝ – ɷɤɜɢɜɚɥɟɧɬɧɚɹ ɨɞɧ
ɨ
ɞɥɹ
ɪ
56
ɪ
ɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɦɚɲɢɧɵ ɜɡɚɢɦɨɞɟ
ɬ
ɚɬɨɪɚ ɢ ɩɨɩɟɪɟɱɧɚɹ ɞɟɦɩɮɟɪɧɚɹ
ɫɬɜɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɬɨɤɨɦ ɜɡɚɢɦɨɢ
ɧ
ɜ
ɭɟɬ ɪɟɚɤɬɢɜɧɨɫɬɶ
aq
x
. Ɋɟɡɭɥɶɬɢɪɭ
ɸ
ɣ
ɞɟɦɩɮɟɪɧɨɣ ɨɛɦɨɬɤɢ ɨɩɪɟɞɟɥɹɟɬɫ
ɹ
, ɬ. ɟ.
1qaqqaq
x
6
Ȍ
Ɏ I
.
ɵ ɡɚɦɟɳɟɧɢɹ ɦɚɲɢɧɵ ɜ ɩɨɩɟɪɟɱɧɨ
ɣ
ɪ
ɦɚɬɨɪɚ ɨɞɢɧɚɤɨɜɵ (ɪɢɫ. 4.8, ɛ). ɉɨɫ
ɥ
ɞ
ɚɟɬ ɢɧɬɟɪɟɫɭɟɦɵɟ ɩɚɪɚɦɟɬɪɵ
d
E
cc
ɢ
x
ɧ
ɢɹɦɢ:
00
0
dqq
d
E
UIx
cc cc
;
1
//
qqaq
x
xx x
VV
cc
,
ɗȾɋ ɜ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ, ɨɩɪɟɞɟɥɹɟɸ
ɳ
ɟ
ɠɢɦɚ (
0d
U
,
0q
I
);
q
x
cc
– ɫɜɟɪɯɩɟɪɟɯɨ
ɞ
ɨɫɢ (ɩɚɫɩɨɪɬɧɚɹ ɜɟɥɢɱɢɧɚ);
1q
x
V
–
ɪ
ɩ
ɮɟɪɧɨɣ ɨɛɦɨɬɤɢ.
ɸ
ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɋɆ ɜ ɨɫ
ɢ
ɰ
ɢɩɢɚɥɶɧɚɹ ɫɯɟɦɚ; ɛ, ɜ – ɫɯɟɦɵ ɡɚɦɟɳɟɧɢɹ;
ɨ
ɤɨɧɬɭɪɧɚɹ ɫɯɟɦɚ ɡɚɦɟɳɟɧɢɹ ɜ ɩɨɩɟɪɟɱɧɨɣ ɨɫɢ
ɪ
ɚɫɱɟɬɚ ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ
ɟ
ɣɫɬɜɭɸɬ
ɨɛɦɨɬɤɚ
ɧ
ɞɭɤɰɢɢ
ɸ
ɳɟɟ ɩɨ-
ɹ
ɬɨɥɶɤɨ
ɣ
ɨɫɢ ɢ
ɥ
ɟ ɭɩɪɨ-
q
x
cc
, ɤɨɬɨ-
(4.15)
(4.16)
ɳ
ɚɹɫɹ ɩɚ-
ɞ
ɧɚɹ ɪɟ-
ɪ
ɟɚɤɬɚɧɫ
ɢ
q:
57
ɗȾɋ –
q
E
cc
ɢ
d
E
cc
– ɫɨɯɪɚɧɹɸɬ ɫɜɨɢ ɡɧɚɱɟɧɢɹ ɧɟɢɡɦɟɧɧɵɦɢ ɜ ɧɚ-
ɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɧɚɪɭɲɟɧɢɹ ɪɟɠɢɦɚ, ɚ ɜ ɫɯɟɦɟ ɡɚɦɟɳɟɧɢɹ ɞɥɹ ɪɚɫɱɟɬɚ
ɩɟɪɟɯɨɞɧɨɝɨ ɪɟɠɢɦɚ ɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
d
x
cc
ɢ
q
x
cc
. ɉɪɢ-
ɫɬɚɜɤɚ «ɫɜɟɪɯ» ɜ ɬɟɪɦɢɧɟ «ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɟ» ɭɤɚɡɵɜɚɟɬ ɧɚ ɬɨ, ɱɬɨ ɞɚɧ-
ɧɵɟ ɩɚɪɚɦɟɬɪɵ ɋɆ ɭɱɢɬɵɜɚɸɬ ɜɥɢɹɧɢɟ ɞɟɦɩɮɟɪɧɵɯ ɤɨɧɬɭɪɨɜ.
Ⱦɥɹ ɹɜɧɨɩɨɥɸɫɧɨɣ ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ ɫ ɞɟɦɩɮɟɪɧɵɦɢ ɨɛɦɨɬ-
ɤɚɦɢ, ɪɚɛɨɬɚɸɳɟɣ ɫ ɨɬɫɬɚɸɳɢɦ ɬɨɤɨɦ, ɧɚ ɪɢɫ. 4.9 ɩɪɢɜɟɞɟɧɚ ɜɟɤɬɨɪɧɚɹ
ɞɢɚɝɪɚɦɦɚ. ɇɚ ɧɟɣ ɜ ɤɨɨɪɞɢɧɚɬɚɯ
d
, q ɩɨɤɚɡɚɧɵ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹ-
ɠɟɧɢɹ
q
U
ɢ
d
U
, ɩɚɞɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɯ ɪɟɚɤɬɢɜɧɨ-
ɫɬɹɯ
0dd
Ix
cc
ɢ
0qq
Ix
cc
, ɩɨɩɟɪɟɱɧɚɹ ɢ ɩɪɨɞɨɥɶɧɚɹ ɫɜɟɪɯɩɟɪɟɯɨɞɧɵɟ ɷɥɟɤ-
ɬɪɨɞɜɢɠɭɳɢɟ ɫɢɥɵ
q
E
cc
,
d
E
cc
ɢ ɩɨɥɧɚɹ ɜɟɥɢɱɢɧɚ ɷɬɨɣ ɗȾɋ.
ɉɪɢ ɱɢɫɬɨ ɢɧɞɭɤɬɢɜɧɨɣ ɰɟɩɢ ɫɬɚɬɨɪɚ ɩɪɨɞɨɥɶɧɚɹ ɢ ɩɨɩɟɪɟɱɧɚɹ
ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɫɜɟɪɯɩɟɪɟɯɨɞɧɨɝɨ ɬɨɤɚ ɨɩɪɟɞɟɥɹ-
ɸɬɫɹ ɜɵɪɚɠɟɧɢɹɦɢ:
ɜɧ
/
dqd
IExx
cc cc cc
;
ɜɧ
/
qdq
IExx
cc cc cc
ɢ ɟɝɨ ɩɨɥɧɚɹ ɜɟɥɢɱɢɧɚ
2
2
dq
II I
cc cc cc
.
ɋɨɩɪɨɬɢɜɥɟɧɢɹ
d
x
cc
ɢ
q
x
cc
ɹɜɥɹɸɬɫɹ ɯɚɪɚɤ-
ɬɟɪɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɦɚɲɢɧɵ ɢ ɧɚɪɹɞɭ ɫ ɞɪɭɝɢ-
ɦɢ ɩɪɢɜɨɞɹɬɫɹ ɜ ɟɟ ɩɚɫɩɨɪɬɧɵɯ ɞɚɧɧɵɯ.
Ⱦɥɹ ɫɨɜɪɟɦɟɧɧɵɯ ɝɟɧɟɪɚɬɨɪɨɜ ɫɜɟɪɯɩɟɪɟɯɨɞɧɚɹ
ɪɟɚɤɬɢɜɧɨɫɬɶ
0,14...0,27
d
x
cc
|
, ɢ ɧɟɫɤɨɥɶɤɨ
ɦɟɧɶɲɟ
q
x
cc
ɢ
d
x
c
. ɋɬɪɨɝɨ ɝɨɜɨɪɹ, ɪɚɡɥɢɱɢɟ ɪɟɚɤ-
ɬɢɜɧɨɫɬɟɣ
d
x
cc
ɢ
q
x
cc
ɬɪɟɛɭɟɬ ɪɚɡɞɟɥɶɧɨɝɨ ɧɚɯɨɠ-
ɞɟɧɢɹ ɫɨɫɬɚɜɥɹɸɳɢɯ
d
I
cc
,
q
I
cc
ɢ ɩɨɥɧɨɝɨ ɫɜɟɪɯɩɟ-
ɪɟɯɨɞɧɨɝɨ ɬɨɤɚ
I
cc
. ȼ ɩɪɚɤɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɬɨ-
ɤɨɜ ɤɨɪɨɬɤɨɝɨ ɡɚɦɵɤɚɧɢɹ ɫɱɢɬɚɸɬ
dq
xx
cc cc
, ɱɬɨ
ɩɨɡɜɨɥɹɟɬ ɨɬɤɚɡɚɬɶɫɹ ɨɬ ɪɚɡɥɨɠɟɧɢɹ
E
cc
,
I
cc
ɩɨ
ɨɫɹɦ
d
ɢ q ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ
ɢ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɟɬ ɪɚɫɱɟɬ. ɗɬɨ ɞɨɩɭɳɟ-
ɧɢɟ ɧɟ ɩɪɢɜɨɞɢɬ ɤ ɫɤɨɥɶɤɨ-ɥɢɛɨ ɨɳɭɬɢɦɨɣ ɩɨ-
ɝɪɟɲɧɨɫɬɢ.
Ɋɢɫ. 4.9. ȼɟɤɬɨɪɧɚɹ
ɞɢɚɝɪɚɦɦɚ ɋɆ
ɫ ɞɟɦɩɮɟɪɧɵɦɢ
ɨɛɦɨɬɤɚɦɢ
58
4.5. Ɏɫɛɝɨɠɨɣɺ ɪɠɫɠɰɩɟɨɩɞɩ ɪɫɩɱɠɬɬɛ ɬɣɨɰɫɩɨɨɩɤ ɧɛɳɣɨɶ
ɉɟɪɟɯɨɞɧɵɣ ɩɪɨɰɟɫɫ ɜ ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧ
ɫɢɫɬɟɦɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. ɉɪɢ ɜɵɜɨɞɟ ɷɬɢɯ ɭɪɚɜɧɟɧɢɣ
ɩɪɢɯɨɞɢɬɫɹ ɩɪɢɧɢɦɚɬɶ ɪɹɞ ɞɨɩɭɳɟɧɢɣ ɜ ɨɬɧɨɲɟɧɢɢ ɫɥɨɠɧɨ ɭɱɢɬɵɜɚɟ-
ɦɵɯ ɮɚɤɬɨɪɨɜ:
x ɨɬɫɭɬɫɬɜɢɟ ɩɨɬɟɪɶ ɜ ɦɚɝɧɢɬɧɨɣ ɫɢɫɬɟɦɟ;
x ɦɚɝɧɢɬɧɚɹ ɫɢɫɬɟɦɚ ɧɟɧɚɫɵɳɟɧɧɚ, ɜ ɫɢɥɭ ɷɬɨɝɨ ɢɧɞɭɤɬɢɜɧɨɫɬɢ
ɦɚɲɢɧɵ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɬɨɤɨɜ;
x
ɜ ɜɨɡɞɭɲɧɨɦ ɡɚɡɨɪɟ ɞɟɣɫɬɜɭɸɬ ɬɨɥɶɤɨ ɩɟɪɜɵɟ ɝɚɪɦɨɧɢɤɢ ɧɚ-
ɦɚɝɧɢɱɢɜɚɸɳɟɣ ɫɢɥɵ ɢ ɢɧɞɭɤɰɢɢ, ɩɪɢ ɷɬɨɦ ɗȾɋ ɫɬɚɬɨɪɚ ɹɜɥɹɸɬɫɹ ɫɢ-
ɧɭɫɨɢɞɚɦɢ ɨɫɧɨɜɧɨɣ ɱɚɫɬɨɬɵ;
x ɤɨɧɫɬɪɭɤɬɢɜɧɚɹ ɫɢɦɦɟɬɪɢɹ ɫɬɚɬɨɪɧɵɯ ɨɛɦɨɬɨɤ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɮɚɡɧɵɯ ɨɫɟɣ
A
,
B
,
C
ɢ ɪɨɬɨɪɚ – ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɞɨɥɶɧɨɣ d ɢ ɩɨɩɟ-
ɪɟɱɧɨɣ
q ɨɫɟɣ.
ɇɚ ɪɢɫ. 4.10 ɩɪɟɞɫɬɚɜɥɟɧɚ ɩɪɢɧɰɢɩɢɚɥɶɧɚɹ ɫɯɟɦɚ ɫɢɧɯɪɨɧɧɨɣ ɹɜ-
ɧɨɩɨɥɸɫɧɨɣ ɦɚɲɢɧɵ ɛɟɡ ɞɟɦɩɮɟɪɧɵɯ ɨɛɦɨɬɨɤ, ɝɞɟ ɫɬɪɟɥɤɚɦɢ ɭɤɚɡɚɧɵ
ɩɪɢɧɹɬɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɬɨɤɨɜ ɢ ɧɚɩɪɹɠɟɧɢɣ, ɚ ɬɚɤɠɟ ɤɨ-
ɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɫɬɚɬɨɪɚ (
A
,
B
,
C
) ɢ ɪɨɬɨɪɚ (d , q ).
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɦɚɲɢɧɚ ɫɨɞɟɪɠɢɬ ɱɟɬɵɪɟ ɦɚɝɧɢɬɧɨ ɫɜɹɡɚɧɧɵɯ
ɤɨɧɬɭɪɚ: ɬɪɢ ɤɨɧɬɭɪɚ ɮɚɡ ɫɬɚɬɨɪɚ ɢ ɤɨɧɬɭɪ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ. Ɂɚɩɢ-
ɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɗȾɋ ɢ ɩɚɞɟɧɢɣ ɧɚɩɪɹ-
ɠɟɧɢɣ. ɍɪɚɜɧɟɧɢɹ ɫɬɚɬɨɪɧɵɯ ɤɨɧɬɭɪɨɜ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɧɟɩɨɞɜɢɠɧɵɯ
ɤɨɨɪɞɢɧɚɬɚɯ ɮɚɡ
A
,
B
,
C
, ɢɦɟɸɳɢɯ ɫɞɜɢɝ ɜ 120
D
, ɚ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟ-
ɧɢɹ – ɜ ɩɨɞɜɢɠɧɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɪɨɬɨɪɚ
d , q:
A
AA
d
Uri
dt
<
,
B
B
B
d
Uri
dt
<
,
C
CC
d
Uri
dt
<
; (4.17)
f
f
ff
d
Uri
dt
<
, (4.18)
ɝɞɟ
A
U ,
B
U ,
C
U – ɦɝɧɨɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɡɧɵɯ ɧɚɩɪɹɠɟɧɢɣ;
A
< , ,
B
<
C
< ,
f
<
– ɪɟɡɭɥɶɬɢɪɭɸɳɢɟ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɹ ɨɛɦɨɬɨɤ ɫɬɚ-
ɬɨɪɚ ɢ ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ;
r,
f
r
– ɚɤɬɢɜɧɵɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɨɛɦɨɬɨɤ
ɫɬɚɬɨɪɚ ɢ ɰɟɩɢ ɜɨɡɛɭɠɞɟɧɢɹ.
Ɂɧɚɤ «
» ɩɟɪɟɞ ɩɪɨɢɡɜɨɞɧɨɣ ɪɟɡɭɥɶɬɢɪɭɸɳɢɯ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɣ
ɮɚɡ ɫɬɚɬɨɪɚ Ȍ ɩɨɫɬɚɜɥɟɧ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɚɜɢɥɨɦ Ʌɟɧɰɚ.
Ɋɢɫ. 4.10. ɋɢɫɬ
ɟ
ɚ –
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧ
ɢ
ɛ – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟ
ɧ
Ɋɚɫɤɪɨɟɦ ɜɵɪɚɠɟɧɢ
ɹ
ɬɵɯ ɞɨɩɭɳɟɧɢɹɯ ɩɪɟɞɫɬɚ
ɜ
ɪ
ɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɤɨɧɬɭɪɚ
Ʉɨɷɮɮɢɰɢɟɧɬɚɦɢ ɩɪɨɩɨɪɰ
ɢɧɞɭɤɬɢɜɧɨɫɬɶ
L
ɞɚɧɧɨɝɨ
ɞɪɭɝɢɦɢ ɤɨɧɬɭɪɚɦɢ. ȼɜɟɞ
ɹ
ɡɚɩɢɫɚɬɶ, ɱɬɨ
AAA
B
BA
CC
A
f
fA
Li
M
M
Mi
<
<
<
<
Ɂɞɟɫɶ
AB BA
MM
,
M
ɋɢɫɬɟɦɵ (4.17), (4.1
ɭɪɚɜɧɟɧɢɣ ɫ ɩɟɪɟɦɟɧɧɵɦ
ɢ
59
ɟ
ɦɚ ɤɨɨɪɞɢɧɚɬ ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ:
ɢ
ɹ ɦɚɝɧɢɬɧɵɯ ɨɫɟɣ ɪɨɬɨɪɚ ɢ ɮɚɡɧɵɯ ɨɛɦɨɬɨɤ ɫ
ɬ
ɧ
ɢɹ ɬɨɤɨɜ ɢ ɧɚɩɪɹɠɟɧɢɣ ɤɨɧɬɭɪɨɜ ɫɢɧɯɪɨɧɧɨɣ
ɦ
ɹ
ɞɥɹ ɩɨɬɨɤɨɫɰɟɩɥɟɧɢɣ, ɤɨɬɨɪɵɟ ɩɪ
ɢ
ɜ
ɥɹɸɬ ɫɨɛɨɣ ɥɢɧɟɣɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ
ɢ ɬɨɤɨɜ ɦɚɝɧɢɬɧɨ ɫɜɹɡɚɧɧɵɯ ɫ ɧɢɦ ɤ
ɨ
ɢɨɧɚɥɶɧɨɫɬɢ ɩɪɢ ɷɬɨɦ ɹɜɥɹɸɬɫɹ ɫɨɛ
ɫ
ɤɨɧɬɭɪɚ ɢ ɟɝɨ ɜɡɚɢɦɧɵɟ ɢɧɞɭɤɬɢɜɧɨ
ɫ
ɹ
ɭ
L
ɢ
M
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɢɧɞɟɤɫ
ɵ
;
;
;
.
AB B AC C Af f
ABB BCC Bff
A
ACBBCCCff
AfBBfCCff
Mi Mi Mi
iLiMiMi
iMiLiMi
i
Mi Mi Li
½
°
°
¾
°
°
¿
fB Bf
M
M
ɢ ɬ. ɞ.
8) ɫɨɫɬɨɹɬ ɢɡ ɥɢɧɟɣɧɵɯ ɞɢɮɮɟɪɟɧɰ
ɢ
ɢ
ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɫ
ɨ
ɬ
ɚɬɨɪɚ;
ɦ
ɚɲɢɧɵ
ɢ
ɩɪɢɧɹ-
ɨɬ ɬɨɤɚ
ɨ
ɧɬɭɪɨɜ.
ɫ
ɬɜɟɧɧɚɹ
ɫ
ɬɢ
M
ɫ
ɵ
, ɦɨɠɧɨ
(4.19)
ɢ
ɚɥɶɧɵɯ
ɨ
ɛɫɬɜɟɧ-
60
ɧɨɣ ɢɧɞɭɤɬɢɜɧɨɫɬɢ
f
L
ɨɛɦɨɬɤɢ ɜɨɡɛɭɠɞɟɧɢɹ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɩɨɫɬɨ-
ɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. Ɉɧɢ ɡɚɜɢɫɹɬ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɨɬɨɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɛ-
ɦɨɬɨɤ ɫɬɚɬɨɪɚ, ɱɬɨ ɫɨɡɞɚɟɬ ɩɪɢɧɰɢɩɢɚɥɶɧɭɸ ɫɥɨɠɧɨɫɬɶ ɜ ɪɟɲɟɧɢɢ ɢɫ-
ɯɨɞɧɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ. ȼɵɹɫɧɢɦ ɡɚɤɨɧɨɦɟɪɧɨ-
ɫɬɢ ɢɡɦɟɧɟɧɢɹ ɢɧɞɭɤɬɢɜɧɨɫɬɟɣ ɫɢɧɯɪɨɧɧɨɣ ɦɚɲɢɧɵ, ɫɱɢɬɚɹ, ɱɬɨ ɨɬɧɨ-
ɫɢɬɟɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ
ɦɚɝɧɢɬɧɨɣ ɨɫɢ ɮɚɡɵ
A
ɢ ɩɪɨɞɨɥɶɧɨɣ ɨɫɢ ɪɨɬɨɪɚ d ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ
ɭɝɥɨɦ
J
(ɫɦ. ɪɢɫ. 4.12).
ȼɡɚɢɦɧɵɟ ɢɧɞɭɤɬɢɜɧɨɫɬɢ
Af
M
,
B
f
M
,
Cf
M
ɡɚɜɢɫɹɬ ɨɬ ɩɨɥɨ-
ɠɟɧɢɹ ɪɨɬɨɪɚ (
J
). ɉɪɢɧɹɬɨɟ ɞɨɩɭɳɟɧɢɟ ɨ ɫɢɧɭɫɨɢɞɚɥɶɧɨɫɬɢ ɧɚɜɨɞɢɦɵɯ
ɜ ɫɬɚɬɨɪɟ ɗȾɋ ɭɤɚɡɵɜɚɟɬ ɧɚ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɢɯ ɢɡɦɟɧɟɧɢɹ. ȼɡɚɢɦɧɚɹ
ɢɧɞɭɤɬɢɜɧɨɫɬɶ
Af
M
ɩɨɥɭɱɚɟɬɫɹ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɪɚɜɧɨɣ
d
M
ɩɪɢ 0
J
.
Ɉɧɚ ɜɵɪɚɠɚɟɬɫɹ ɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɫ ɩɟɪɢɨɞɨɦ
2
S
cos
Af fA d
MMM
J
;
ɞɥɹ ɮɚɡ
B
ɢ C ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɧɢɦɚɬɶ ɚɪɝɭɦɟɧɬɵ ( 120
J
D
) ɢ ( 120
J
D
).
ɋɨɛɫɬɜɟɧɧɚɹ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ɮɚɡ
A
L ,
B
L ,
C
L ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɨ-
ɠɟɧɢɹ ɪɨɬɨɪɚ (
J
). Ɉɧɚ ɩɨɥɭɱɚɟɬɫɹ ɧɚɢɛɨɥɶɲɟɣ ɞɥɹ ɥɸɛɨɣ ɢɡ ɮɚɡ, ɤɨɝɞɚ
ɩɪɨɞɨɥɶɧɚɹ ɨɫɶ ɪɨɬɨɪɚ ɫɨɜɩɚɞɚɟɬ ɫ ɦɚɝɧɢɬɧɨɣ ɨɫɶɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ
ɮɚɡɵ ɫɬɚɬɨɪɚ, ɢ ɧɚɢɦɟɧɶɲɟɣ, ɤɨɝɞɚ ɷɬɢ ɨɫɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ (
90
J
D
).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɛɫɬɜɟɧɧɚɹ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ɢɡɦɟɧɹɟɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢ ɫ
ɩɟɪɢɨɞɨɦ
S
, ɬ. ɟ. ɫ ɞɜɨɣɧɨɣ ɱɚɫɬɨɬɨɣ, ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɜɪɚ-
ɳɟɧɢɹ ɪɨɬɨɪɚ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɮɚɡɵ
A
ɢɦɟɟɦ
02
cos2
A
Lll
J
. (4.20)
Ɏɚɡɚɦ
B
ɢ C ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɚɪɝɭɦɟɧɬɵ (
2120
J
D
) ɢ (
2 120
J
D
).
ȼɡɚɢɦɧɚɹ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ɮɚɡ ɫɬɚɬɨɪɚ
AB
M ,
AC
M ,
B
C
M ɹɜɥɹ-
ɟɬɫɹ ɜɟɥɢɱɢɧɨɣ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɢ ɞɨɫɬɢɝɚɟɬ ɧɚɢɦɟɧɶɲɟɝɨ ɚɛɫɨɥɸɬɧɨɝɨ
ɡɧɚɱɟɧɢɹ, ɤɨɝɞɚ ɨɫɶ
d ɫɨɜɩɚɞɚɟɬ ɫ ɛɢɫɫɟɤɬɪɢɫɨɣ ɭɝɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ
ɨɫɹɦɢ ɮɚɡ ɫɬɚɬɨɪɚ. Ɉɧɚ ɢɡɦɟɧɹɟɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢ ɫ ɩɟɪɢɨɞɨɦ
ʌ
ɢ ɦɟɠ-
ɞɭ ɮɚɡɚɦɢ
A
ɢ
B
ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
02
cos 2 120
AB
Mmm
J
D
. (4.21)
ȼɡɚɢɦɧɨɣ ɢɧɞɭɤɬɢɜɧɨɫɬɢ ɮɚɡ
B
ɢ C ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɚɪɝɭɦɟɧɬ 2
J
,
ɚ ɮɚɡ
A
ɢ C – ɚɪɝɭɦɟɧɬ (
2120
J
D
).
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɜ (4.20), (4.21) ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɢɧɞɭɤɬɢɜɧɨ-
ɫɬɢ, ɤɨɬɨɪɵɦɢ ɨɛɵɱɧɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɢɧɯɪɨɧɧɚɹ ɦɚɲɢɧɚ [11]. ȼ ɨɬ-
ɧɨɫɢɬɟɥɶɧɵɯ ɟɞɢɧɢɰɚɯ ɷɬɚ ɜɡɚɢɦɨɫɜɹɡɶ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ: