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Файл:Методология расчётов гидродинамических параметров шахтных автоматизированных стационарных установок с центробежными нагнетателями. Монография
.pdf
Z
Q
2Qɇt
1
, (3.92)
max
ɝɞɟ Q – ɪɚɫɯɨɞ ɠɢɞɤɨɫɬɢ ɱɟɪɟɡ ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɨɬɜɟɪɫɬɢɟ ɞɢɚɦɟɬɪɨɦ d0;
ȝ – ɤɨɷɮɮɢɰɢɟɧɬ ɪɚɫɯɨɞɚ ɨɬɜɟɪɫɬɢɹ (ȝ = 0,615);
– ɩɥɨɳɚɞɶ ɩɪɨɯɨɞɧɨɝɨ ɫɟɱɟɧɢɹ ɨɬɜɟɪɫɬɢɹ (Ȧ0 = 0,785ād
Ȧ
0
Ȧ – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɟɡɟɪɜɭɚɪɚ (Ȧ = 0,784 d
2
);
0
2
);
d – ɜɧɭɬɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɬɪɭɛɵ;
ɇ0 – ɩɪɢɜɟɞɟɧɧɵɣ ɧɚɩɨɪ ɩɟɪɟɞ ɨɬɜɟɪɫɬɢɟɦ, ɤɨɬɨɪɵɣ ɜ ɫɥɭɱɚɟ ɢɫɬɟɱɟɧɢɹ
ɠɢɞɤɨɫɬɢ ɜ ɚɬɦɨɫɮɟɪɭ ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ ɫɬɨɥɛɭ ɠɢɞɤɨɫɬɢ ɦɟɠɞɭ ɬɨɱɤɚɦɢ ȼ
ɢ ɋ;
– ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɜɵɫɨɬɚ ɢɫɫɥɟɞɭɟɦɨɝɨ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɭɱɚɫɬɤɚ ɬɪɭɛɨ-
ɇ
1
ɩɪɨɜɨɞɚ, ɤɨɬɨɪɚɹ ɞɥɹ ɬɢɩɨɜɵɯ ɝɢɞɪɚɜɥɢɱɟɫɤɢɯ ɫɯɟɦ ɝɥɚɜɧɨɝɨ ɜɨɞɨɨɬɥɢɜɚ
ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɚ ɩɨ ɮɨɪɦɭɥɟ:
1
ɇɇɇ , (3.93)
5,6935,0
ɜɫɇ
ɝɞɟ ɇɇ – ɧɨɦɢɧɚɥɶɧɵɣ ɧɚɩɨɪ ɧɚɫɨɫɚ (ɩɚɫɩɨɪɬɧɚɹ ɜɟɥɢɱɢɧɚ);
– ɜɚɤɭɭɦɦɟɬɪɢɱɟɫɤɚɹ ɜɵɫɨɬɚ ɜɫɚɫɵɜɚɧɢɹ ɧɚɫɨɫɚ (ɩɚɫɩɨɪɬɧɚɹ ɜɟɥɢɱɢ-
ɇ
ɜɫ
ɧɚ);
ɇ2 – ɨɬɦɟɬɤɚ, ɞɨ ɤɨɬɨɪɨɣ ɩɚɞɚɟɬ ɭɪɨɜɟɧɶ ɩɪɢ ɧɟɩɨɥɧɨɦ ɨɩɨɪɨɠɧɟɧɢɢ
ɢɫɫɥɟɞɭɟɦɨɝɨ ɭɱɚɫɬɤɚ ɬɪɭɛɨɩɪɨɜɨɞɚ;
t0 – ɜɪɟɦɹ ɧɟɩɨɥɧɨɝɨ ɨɩɨɪɨɠɧɟɧɢɹ;
t – ɜɪɟɦɹ ɩɨɥɧɨɝɨ ɨɩɨɪɨɠɧɟɧɢɹ (ɩɪɢ ɇ
– ɦɚɤɫɢɦɚɥɶɧɵɯ ɪɚɫɯɨɞ ɠɢɞɤɨɫɬɢ ɱɟɪɟɡ ɨɬɜɟɪɫɬɢɟ ɜ ɧɚɱɚɥɟ ɢɫɬɟɱɟɧɢɹ
Q
max
Q
= qÂQH.
max
= 0);
2
Ɏɨɪɦɭɥɵ (3.90), (3.91) ɢ (3.92) ɫɩɪɚɜɟɞɥɢɜɵ ɬɨɥɶɤɨ ɥɢɲɶ ɩɪɢ ɯɨɪɨɲɨ ɪɚɡɜɢɬɨɣ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ ɢɫɬɟɱɟɧɢɹ, ɤɨɝɞɚ:
5
Re
10Re t ; (3.94)
dV
00
, (3.95)
ɝɞɟ Re – ɱɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ ɤɚɤ ɦɟɪɚ ɬɭɪɛɭɥɟɧɬɧɨɫɬɢ;
– ɫɤɨɪɨɫɬɶ ɢɫɬɟɱɟɧɢɹ ɢɡ ɨɬɜɟɪɫɬɢɹ;
V
0
Ȟ – ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ ɜɨɞɵ (Ȟ = 10
V
0
Z
0
-6
ɦ/ɫ2).
QQ
785,0 d
.
2
0
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜɟɥɢɱɢɧɵ V0 ɜ ɮɨɪɦɭɥɭ (3.95) ɢ ɷɥɟɦɟɧɬɚɪɧɵɯ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɢɦɟɟɦ:
101
n
G
G
|
Q
274,1Re
. (3.96)
d
Q
0
ɋ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɮɨɪɦɭɥɵ (3.96) ɜɵɱɢɫɥɹɟɬɫɹ ɩɪɚɜɚɹ ɝɪɚɧɢɰɚ ɞɢɚɩɚɡɨɧɚ ɨɩɪɟɞɟɥɟɧɢɹ ɞɢɚɦɟɬɪɚ ɨɬɜɟɪɫɬɢɹ, ɡɚ ɩɪɟɞɟɥɚɦɢ ɤɨɬɨɪɨɣ ɮɨɪɦɭɥɵ
(3.90), (3.91) ɢ (3.92) ɧɟ ɧɚɞɟɠɧɵ. Ɇɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɵɣ ɞɢɚɦɟɬɪ ɨɬɜɟɪɫɬɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ (3.96) ɫ ɭɱɺɬɨɦ ɭɫɥɨɜɢɹ (3.94) ɩɪɢ
ɦɢɧɢɦɚɥɶɧɨɦ ɪɚɫɯɨɞɟ Q = Q
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɧɚɯɨɞɢɦ:
ɝɞɟ q
– ɦɢɧɢɦɚɥɶɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɭɬɟɱɤɚ, ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɛɵɬɶ
min
= q
min
74,12 , (3.97)
. ɉɨɫɥɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
mināQH
Qqd d
ɇ
minmax0
ɨɩɪɟɞɟɥɟɧɚ ɩɨ ɮɨɪɦɭɥɟ (3.89) ɫ ɭɱɺɬɨɦ ɩɨɝɪɟɲɧɨɫɬɟɣ ɨɩɪɟɞɟɥɟɧɢɹ
ɜɯɨɞɹɳɢɯ ɜ ɧɟɺ ɜɟɥɢɱɢɧ Q1 ɢ Q2. ɉɪɢɧɹɜ ɩɨɝɪɟɲɧɨɫɬɶ ɪɚɫɯɨɞɨɦɟɪɚ ɬɢɩɚ
ɂɊ-51, ɪɚɜɧɨɣ 1,5 % (įQ = 0,015), ɨɩɪɟɞɟɥɢɦ ɩɨɝɪɟɲɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɹ
ɭɬɟɱɤɢ ɩɪɢ ɩɨɦɨɳɢ ɭɤɚɡɚɧɧɨɣ ɮɨɪɦɭɥɵ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɪɚɜɢɥ
ɨɛɪɚɛɨɬɤɢ ɩɪɢɛɥɢɠɟɧɧɵɯ ɜɵɱɢɫɥɟɧɢɣ:
03,0015,022
Qqq
mi
.
ɍɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɭɬɟɱɤɚ ɫɨɫɬɚɜɥɹɟɬ
0,15, ɛɨɥɟɟ ɤɨɬɨɪɨɣ ɩɨɜɪɟɠɞɟɧɢɟ ɤɜɚɥɢɮɢɰɢɪɭɟɬɫɹ ɤɚɤ ɪɚɡɪɵɜ
q
max
ɬɪɭɛɨɩɪɨɜɨɞɚ. Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɭɬɟɱɤɚ ɥɟɠɢɬ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0,03 ɞɨ 0,15 ɢ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɚɪɢɮɦɟɬɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ ɫ ɲɚɝɨɦ įq = 0,03.
15,0;12,0;09,0;06,0;03,0 q . (3.98)
Ʌɟɜɚɹ ɝɪɚɧɢɰɚ ɨɩɪɟɞɟɥɟɧɢɹ ɞɢɚɦɟɬɪɚ ɨɬɜɟɪɫɬɢɹ (d
ɧɚɣɞɟɧɚ ɢɡ ɮɨɪɦɭɥɵ (3.90) ɩɪɢ q = q
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɢɦɟɟɦ:
684,0
0
684,0
, H0 = H1:
min
5,0
minmin0
5,05,0
ɇQqd
ɇ
ɇ
25,0
. (3.99)
0
5,0
25,0
ɇQqd
1
) ɦɨɠɟɬ ɛɵɬɶ
0 min
. (3.100)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɢɚɩɚɡɨɧ ɞɢɚɦɟɬɪɨɜ ɨɬɜɟɪɫɬɢɣ, ɧɚɞɟɠɧɨ ɨɩɪɟɞɟɥɹɟɦɵɣ ɩɨ ɮɨɪɦɭɥɟ (3.99), ɧɚɯɨɞɢɬɫɹ ɜ ɩɪɟɞɟɥɚɯ ɨɬ d
0 min
ɞɨ d
. Ƚɪɚɧɢɰɵ
0 max
ɷɬɨɝɨ ɞɢɚɩɚɡɨɧɚ ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (3.97) ɢ (3.100) ɞɥɹ ɤɨɧɤɪɟɬɧɨɣ ɜɨɞɨɨɬɥɢɜɧɨɣ ɭɫɬɚɧɨɜɤɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɩɪɢɦɟɧɹɟɦɨɝɨ ɧɚɫɨɫɚ,
ɞɢɚɦɟɬɪɚ ɧɚɩɨɪɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ ɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɫɯɟɦɵ ɜɨɞɨɨɬɥɢɜɚ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɤɭɳɢɯ ɡɧɚɱɟɧɢɣ d0 ɩɨ ɮɨɪɦɭɥɟ (3.99) ɧɟɨɛɯɨɞɢɦɨ
ɪɚɫɩɨɥɚɝɚɬɶ ɬɟɤɭɳɢɦɢ ɡɧɚɱɟɧɢɹɦɢ ɇ
. ɋ ɷɬɨɣ ɰɟɥɶɸ ɧɚɣɞɺɦ ɡɚɜɢɫɢɦɨɫɬɶ
0
102
ɇ0 = f(t). ɂɡ ɮɨɪɦɭɥɵ (3.32) ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɢ ɪɟɲɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶ-
ɧɨ ɩɟɪɟɦɟɧɧɨɣ ɇ ɧɚɯɨɞɢɦ:
tQq
ɇ
637,0
0
ɇ
, (3.101)
2
d
ɝɞɟ t – ɜɪɟɦɹ, ɮɢɤɫɢɪɭɟɦɨɟ ɬɚɣɦɟɪɨɦ ɩɨ ɩɪɢɜɟɞɟɧɧɨɣ ɜɵɲɟ ɦɟɬɨɞɢɤɟ.
ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ ɇ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ, ɩɨɥɭɱɢɦ:
0
ɢɡ (3.101) ɜ (3.99) ɩɨɫɥɟ ɷɥɟɦɟɧɬɚɪɧɵɯ
0
25,025,025,025,0
7656,0
ɇ
. (3.102)
tdQqd
ɉɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ (3.101) ɢ (3.102) ɹɜɥɹɸɬɫɹ ɪɚɛɨɱɢɦɢ ɞɥɹ
ɨɩɪɟɞɟɥɟɧɢɹ ɦɟɫɬɨɩɨɥɨɠɟɧɢɹ ɢ ɪɚɡɦɟɪɚ ɩɨɜɪɟɠɞɟɧɢɹ ɬɪɭɛɨɩɪɨɜɨɞɚ ɢ
ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ ɞɢɚɩɚɡɨɧɟ ɢɡɦɟɧɟɧɢɣ ɬɚɤɢɯ
ɩɚɪɚɦɟɬɪɨɜ ɤɚɤ: q; d
, H0 ɢ t. Ⱦɥɹ ɞɜɭɯ ɩɚɪɚɦɟɬɪɨɜ (d0 ɢ q) ɬɚɤɢɟ ɞɢɚɩɚɡɨɧɵ
0
ɨɩɪɟɞɟɥɟɧɵ ɜɵɲɟ, ɞɥɹ ɨɫɬɚɥɶɧɵɯ ɧɟɨɛɯɨɞɢɦɵ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɢɫɫɥɟɞɨɜɚɧɢɹ, ɩɪɢɜɟɞɟɧɧɵɟ ɧɢɠɟ.
ȼɟɥɢɱɢɧɚ ɇ
(ɬɨɱɤɚ ȼ, ɪɢɫ. 3.9) ɦɨɠɟɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɞɢɚɩɚɡɨɧɟ ɨɬ ɇ
Ɇɢɧɢɦɚɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ ɇ
ɹɜɥɹɟɬɫɹ ɪɚɫɫɬɨɹɧɢɟɦ ɞɨ ɦɟɫɬɚ ɭɬɟɱɤɢ ɨɬ ɭɫɬɶɹ ɫɬɜɨɥɚ
0
0 ɞɨ ɇ0 = ɇ1.
0 min
(ɬɨɱɤɚ D, ɪɢɫ. 3.9) ɨɛɭɫɥɨɜɥɟɧɨ
0 min
ɩɨɝɪɟɲɧɨɫɬɶɸ ɦɚɧɨɦɟɬɪɚ Ɋ (ɬɨɱɤɚ Ⱥ, ɪɢɫ. 3.9), ɩɨɤɚɡɚɧɢɹ ɤɨɬɨɪɨɝɨ ɨɞɧɨɡɧɚɱɧɨ ɫɜɹɡɚɧɨ ɫ ɜɟɥɢɱɢɧɨɣ ɇ0 ɤɚɤ ɞɨɩɨɥɧɟɧɢɟ ɞɨ ɇ1. ɉɪɢɧɹɜ ɩɨɝɪɟɲɧɨɫɬɶ ɦɚɧɨɦɟɬɪɚ ɬɢɩɚ Ɇɋ-ɗ1 (ɗ2), ɪɚɜɧɨɣ 1 % (įP = 0,01), ɜɟɥɢɱɢɧɚ ɇ
ɫɨɫɬɚɜɢɬ:
01,0 ɇɇ .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɢɚɩɚɡɨɧ ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɫɫɬɨɹɧɢɹ ɇ
ɩɪɟɞɟɥɚɯ ɨɬ ɇ
ɮɨɪɦɭɥɵ (3.91), ɩɨɥɨɠɢɜ H
ɞɨ ɇ1. Ɇɢɧɢɦɚɥɶɧɨɟ ɜɪɟɦɹ ɢɫɬɟɱɟɧɢɹ t
0 min
= H
2
t
min0
= H1 – H
2 min
2
max
ɇd
Qq
1min0
ɧɚɯɨɞɢɬɫɹ ɜ
0
ɧɚɣɞɟɦ ɢɡ
0 min
= 0,99āH1:
0 min
2
ɇd
1
0523,000785,0
1
. (3.103)
Q
ɇɇ
0 min
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɨɜɟɞɟɧɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɪɚɡɪɚɛɨɬɚɧɚ ɦɟɬɨɞɢɤɚ
ɨɩɪɟɞɟɥɟɧɢɹ ɨɫɧɨɜɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɭɬɟɱɤɢ, ɩɨɥɭɱɟɧɵ ɪɚɛɨɱɢɟ ɮɨɪɦɭɥɵ
ɞɥɹ ɜɵɱɢɫɥɟɧɢɣ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ, ɭɫɬɚɧɨɜɥɟɧɵ ɞɢɚɩɚɡɨɧɵ ɢɯ ɢɡɦɟɧɟɧɢɣ,
ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɨɡɞɚɧɵ ɩɪɟɞɩɨɫɵɥɤɢ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɩɪɨɰɟɫɫɨɜ
ɢɫɬɟɱɟɧɢɹ ɠɢɞɤɨɫɬɢ ɢɡ ɩɨɜɪɟɠɞɟɧɧɵɯ ɧɚɩɨɪɧɵɯ ɬɪɭɛɨɩɪɨɜɨɞɨɜ ɲɚɯɬɧɵɯ
ɜɨɞɨɨɬɥɢɜɧɵɯ ɭɫɬɚɧɨɜɨɤ.
103
3.11. ɂɫɫɥɟɞɨɜɚɧɢɹ ɞɢɧɚɦɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɬɚɧɨɜɨɤ
3.11.1. Ɉɛɳɢɟ ɫɜɟɞɟɧɢɹ ɨ ɫɩɟɰɢɮɢɤɟ ɪɚɛɨɬɵ ɲɚɯɬɧɵɯ
ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɬɚɧɨɜɨɤ
Ɉɫɨɛɟɧɧɨɫɬɶɸ ɷɤɫɩɥɭɚɬɚɰɢɢ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɯ ɭɫɬɚɧɨɜɨɤ ɹɜɥɹɟɬɫɹ
ɡɧɚɱɢɬɟɥɶɧɵɟ ɧɟɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɟ ɡɚɬɪɚɬɵ ɜɨɞɵ ɧɚ ɫɨɛɫɬɜɟɧɧɵɟ ɧɭɠɞɵ,
ɫɨɫɬɚɜɥɹɸɳɢɟ ɨɤɨɥɨ 25 % ɨɛɳɢɯ ɡɚɬɪɚɬ ɧɚ ɬɪɚɧɫɩɨɪɬɢɪɨɜɚɧɢɟ. ȼ ɛɨɥɶɲɟɣ ɦɟɪɟ ɫɨɤɪɚɳɟɧɢɸ ɷɬɢɯ ɡɚɬɪɚɬ ɫɩɨɫɨɛɫɬɜɭɟɬ ɩɪɢɧɹɬɵɣ ɫɩɨɫɨɛ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ.
Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɜ ɩɨɞɪɚɡɞɟɥɟ
2.2, ɟɝɨ ɫɭɬɶ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɣ ɫɦɟɧɟ ɬɪɟɯ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɯ ɪɟɠɢɦɨɜ, ɢɦɟɸɳɢɯ ɦɟɫɬɨ ɜ
ɬɪɚɧɫɩɨɪɬɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ ɩɪɢ ɪɟɝɭɥɢɪɨɜɚɧɢɢ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɧɟɫɭɳɟɣ
ɠɢɞɤɨɫɬɢ [6]. Ɋɚɛɨɱɢɟ ɪɟɠɢɦɵ, ɜɨɡɧɢɤɚɸɳɢɟ ɩɪɢ ɷɬɨɦ, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɝɪɚɮɢɱɟɫɤɢ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.10. ɍɫɥɨɜɧɵɦɢ ɝɪɚɧɢɰɚɦɢ ɫɦɟɧɵ
ɪɟɠɢɦɨɜ ɹɜɥɹɸɬɫɹ ɤɪɢɬɢɱɟɫɤɚɹ ɫɤɨɪɨɫɬɶ V
ɢ ɫɤɨɪɨɫɬɶ ɬɪɨɝɚɧɢɹ Vɬɪ
ɤɪ
ɬɜɺɪɞɵɯ ɱɚɫɬɢɰ ɜ ɱɚɫɬɢɱɧɨ ɡɚɢɥɟɧɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ.
Ɋɟɠɢɦ «Ⱥ». ɏɚɪɚɤɬɟɪɢɡɭɟɬ ɧɨɪɦɚɥɶɧɭɸ ɪɚɛɨɬɭ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ
ɭɫɬɚɧɨɜɤɢ ɫ ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ ɩɨɬɨɤɚ ɢ ɧɨɦɢɧɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɶɸ
ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ. ɗɬɨɬ ɪɟɠɢɦ ɧɨɫɢɬ ɭɫɬɨɣɱɢɜɵɣ, ɞɥɢɬɟɥɶɧɵɣ ɯɚɪɚɤɬɟɪ ɢ
ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɦ ɩɟɪɢɨɞɨɦ ɜ ɷɤɫɩɥɭɚɬɚɰɢɢ ɭɫɬɚɧɨɜɤɢ. Ɉɧ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɛɨɥɟɟ ɪɚɜɧɨɦɟɪɧɨɣ ɫɬɪɭɤɬɭɪɨɣ ɩɨɬɨɤɚ ɢ ɯɨɪɨɲɨ
ɢɡɭɱɟɧɧɨɣ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɣ ɨɛɫɬɚɧɨɜɤɨɣ ɜɧɭɬɪɢ ɬɪɭɛɨɩɪɨɜɨɞɚ
. Ⱦɥɹ
ɷɬɨɝɨ ɪɟɠɢɦɚ ɪɚɡɪɚɛɨɬɚɧɵ ɮɨɪɦɭɥɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɫɧɨɜɧɵɯ ɢɧɬɟɝɪɚɥɶɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɬɪɚɧɫɩɨɪɬɢɪɨɜɚɧɢɹ (ɩɨɬɟɪɶ ɞɚɜɥɟɧɢɹ ɩɨ ɞɥɢɧɟ ɢ
ɤɪɢɬɢɱɟɫɤɨɣ ɫɤɨɪɨɫɬɢ), ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɭɫɩɟɲɧɨ ɩɪɨɟɤɬɢɪɨɜɚɬɶ ɢ ɷɤɫɩɥɭɚɬɢɪɨɜɚɬɶ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɵɟ ɭɫɬɚɧɨɜɤɢ ɜ ɷɬɨɦ ɪɟɠɢɦɟ.
Ɋɟɠɢɦ «ȼ» – ɩɟɪɟɯɨɞɧɵɣ, ɧɟɭɫɬɚɧɨɜɢɜɲɢɣɫɹ ɪɟɠɢɦ, ɜɨɡɧɢɤɚɸɳɢɣ
ɩɪɢ ɫɧɢɠɟɧɢɢ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɧɢɠɟ ɤɪɢɬɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ. ɏɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɧɟɪɚɜɧɨɦɟɪɧɨɣ ɫɬɪɭɤɬɭɪɨɣ ɩɨɬɨɤɚ ɢ ɨɛɪɚɡɨɜɚɧɢɟɦ ɩɨɞɜɢɠɧɨɝɨ
ɫɥɨɹ ɨɫɚɠɞɟɧɢɹ, ɹɜɥɹɸɳɟɝɨɫɹ ɨɫɧɨɜɧɨɣ ɩɪɢɱɢɧɨɣ ɡɚɤɭɩɨɪɤɢ ɬɪɭɛɨɩɪɨɜɨɞɨɜ [6].
Ɋɟɠɢɦ «ɋ» – ɭɫɬɨɣɱɢɜɵɣ ɪɟɠɢɦ, ɜɨɡɧɢɤɚɸɳɢɣ ɩɪɢ ɞɚɥɶɧɟɣɲɟɦ
ɫɧɢɠɟɧɢɢ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ ɧɢɠɟ, ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ, ɫɤɨɪɨɫɬɢ ɬɪɨɝɚɧɢɹ
ɬɜɟɪɞɵɯ ɱɚɫɬɢɰ ɜ ɬɪɭɛɨɩɪɨɜɨɞɟ. ɏɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɨɥɧɵɦ ɪɚɫɫɥɨɟɧɢɟɦ
ɩɨɬɨɤɚ ɢ ɨɛɪɚɡɨɜɚɧɢɟɦ ɧɟɩɨɞɜɢɠɧɨɝɨ ɫɥɨɹ ɨɬɥɨɠɟɧɢɹ, ɧɚɞ ɤɨɬɨɪɵɦ ɦɨɠɟɬ ɬɪɚɧɫɩɨɪɬɢɪɨɜɚɬɶɫɹ ɜɨɞɚ ɜ ɧɟɡɧɚɱɢɬɟɥɶɧɨɦ
ɤɨɥɢɱɟɫɬɜɟ ɞɥɹ ɩɨɞɞɟɪɠɚɧɢɹ ɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ ɜ ɪɚɛɨɬɚɸɳɟɦ ɫɨɫɬɨɹɧɢɢ ɜ ɬɟɱɟɧɢɟ ɫɤɨɥɶɭɝɨɞɧɨ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɝɨ ɩɟɪɢɨɞɚ ɜɪɟɦɟɧɢ, ɨɛɭɫɥɨɜɥɟɧɧɨɝɨ ɱɢɫɥɨɦ ɢ
104
ɞɥɢɬɟɥɶɧɨɫɬɶɸ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɧɚ ɞɚɧɧɨɦ ɜɢɞɟ ɬɪɚɧɫɩɨɪɬɟ.
m
a
x
Ƚ
Ƚ
U
U
0
U
U
Ƚ
U
U
U
U
ɗɬɨɬ ɪɟɠɢɦ ɪɚɛɨɬɵ ɩɨɡɜɨɥɹɟɬ ɫɨɤɪɚɬɢɬɶ ɞɨ 50 % ɜɫɟɯ ɧɟɩɪɨɢɡɜɨɞɢɬɟɥɶɧɵɯ ɡɚɬɪɚɬ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɜɨɞɵ ɡɚ ɫɱɟɬ ɢɫɤɥɸɱɟɧɢɹ ɨɩɟɪɚɰɢɣ ɩɨɥɧɨɣ
ɩɪɨɦɵɜɤɢ ɬɪɚɧɫɩɨɪɬɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ ɩɟɪɟɞ ɟɝɨ ɨɫɬɚɧɨɜɤɨɣ ɢ ɡɚɩɨɥɧɟɧɢɹ ɜɨɞɨɣ ɬɪɭɛɨɩɪɨɜɨɞɧɨɣ ɫɟɬɢ ɩɟɪɟɞ ɟɺ ɡɚɩɭɫɤɨɦ.
maxȽȽ
0
Ƚ
Ɋɢɫɭɧɨɤ 3.10. Ɋɚɛɨɱɢɟ ɪɟɠɢɦɵ ɭɝɥɟɫɨɫɧɨɣ ɭɫɬɚɧɨɜɤɢ
ɩɪɢ ɪɟɝɭɥɢɪɨɜɚɧɢɢ
105
3.11.2. Ƚɢɞɪɨɬɪɚɧɫɩɨɪɬɧɚɹ ɭɫɬɚɧɨɜɤɚ ɤɚɤ ɨɛɴɟɤɬ ɪɟɝɭɥɢɪɨɜɚɧɢɹ
Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɜɵɲɟ, ɪɟɝɭɥɢɪɨɜɚɧɢɟ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ
ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɨɫɭɳɟɫɬɜɢɬɶ ɩɭɬɺɦ ɞɪɨɫɫɟɥɢɪɨɜɚɧɢɹ ɧɚɩɨɪɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ ɫ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɦ ɩɟɪɟɜɨɞɨɦ ɭɝɥɟɫɨɫɚ ɧɚ ɪɚɛɨɬɭ ɩɨ ɜɨɞɟ ɢ ɱɚɫɬɢɱɧɨɣ ɩɪɨɦɵɜɤɨɣ ɬɪɭɛɨɩɪɨɜɨɞɚ (ɩɪɢ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɟ) ɢɥɢ
ɩɨɥɧɨɣ ɩɪɨɦɵɜɤɨɣ ɬɪɭɛɨɩɪɨɜɨɞɚ (ɩɪɢ ɜɟɪɬɢɤɚɥɶɧɨɦ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɟ).
ȼ ɬɚɤɢɯ ɫɥɭɱɚɹɯ ɭɝɥɟɫɨɫɧɵɟ ɭɫɬɚɧɨɜɤɢ ɧɢɱɟɦ ɧɟ ɨɬɥɢɱɚɸɬɫɹ ɨɬ
ɧɚɫɨɫɧɵɯ
ɭɫɬɚɧɨɜɨɤ ɢ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɩɨɫɥɟɞɧɢɟ, ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɞɢɧɚɦɢɱɟɫɤɢɣ
ɨɛɴɟɤɬ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɜɯɨɞɧɵɦɢ ɢ ɜɵɯɨɞɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ, ɫɜɹɡɚɧɧɵɦɢ
ɦɟɠɞɭ ɫɨɛɨɣ ɪɚɡɥɢɱɧɵɦɢ ɤɚɧɚɥɚɦɢ, ɞɢɧɚɦɢɤɚ ɤɨɬɨɪɵɯ ɨɩɢɫɵɜɚɟɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ.
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɪɟɚɥɶɧɵɣ ɨɛɴɟɤɬ ɨɬɧɨɫɢɬɫɹ ɤ ɩɪɨɦɵɲɥɟɧɧɵɦ
ɨɛɴɟɤɬɚɦ, ɨɛɥɚɞɚɸɳɢɦ ɫɜɨɣɫɬɜɨɦ ɫɚɦɨɜɵɪɚɜɧɢɜɚɧɢɹ ɢ, ɤɚɤ ɨɬɦɟɱɟɧɨ ɜ
ɪɚɛɨɬɟ [12], ɬɚɤɢɟ ɨɛɴɟɤɬɵ ɨɩɢɫɵɜɚɸɬɫɹ ɩɪɢ ɩɨɦɨɳɢ ɩɟɪɟɞɚɬɨɱɧɵɯ
ɮɭɧɤɰɢɣ, ɚɧɚɥɨɝɢɱɧɵɯ (3.50). ȼɢɞ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɣ ɮɭɧɤɰɢɢ, ɚ ɬɚɤɠɟ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ, ɜɯɨɞɹɳɢɯ ɜ ɟɺ ɫɨɫɬɚɜ, ɨɩɪɟɞɟɥɹɸɬɫɹ ɞɥɹ
ɤɨɧɤɪɟɬɧɨɝɨ ɫɥɭɱɚɹ ɫ ɭɱɺɬɨɦ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɨɫɨɛɟɧɧɨɫɬɟɣ ɢ ɭɫɥɨɜɢɣ
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɨɛɴɟɤɬɚ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɭɱɚɫɬɤɨɜɭɸ ɭɝɥɟɫɨɫɧɭɸ ɫɬɚɧɰɢɸ ɤɚɤ ɨɛɴɟɤɬ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɜ
ɭɫɥɨɜɢɹɯ ɫɩɟɰɢɮɢɤɢ ɪɚɛɨɬɵ ɲɚɯɬɧɨɝɨ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɚ ɫ ɭɱɺɬɨɦ ɪɚɧɟɟ
ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɯ
ɡɚɞɚɱ ɭɩɪɚɜɥɟɧɢɹ. ɇɚ ɪɢɫ. 3.11, ɚ ɩɪɢɜɟɞɟɧɚ ɭɩɪɨɳɟɧɧɚɹ ɫɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɨɛɴɟɤɬɚ, ɜ ɤɨɬɨɪɨɣ ɨɫɧɨɜɧɵɦɢ ɜɯɨɞɧɵɦɢ (ɪɟɝɭɥɢɪɭɸɳɢɦɢ) ɢ ɜɨɡɦɭɳɚɸɳɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɩɪɢɧɹɬɵ:
– ɜɟɥɢɱɢɧɚ ɩɪɢɬɨɤɚ ɝɢɞɪɨɫɦɟɫɢ Q
(t), ɩɨɫɬɭɩɚɸɳɟɣ ɨɬ ɡɚɛɨɟɜ ɜ
ɉ
ɩɪɢɺɦɧɭɸ ɺɦɤɨɫɬɶ ɫɬɚɧɰɢɢ ɢ ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɚɹɫɹ ɩɨ ɤɚɧɚɥɭ I – I;
– ɝɢɞɪɚɜɥɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɪɚɛɨɱɟɝɨ ɨɪɝɚɧɚ ȟ(t) ɞɪɨɫɫɟɥɢɪɭɸ-
ɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ, ɡɚɜɢɫɹɳɟɟ ɨɬ ɥɢɧɟɣɧɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɲɬɨɤɚ ɢɫɩɨɥɧɢɬɟɥɶɧɨɝɨ ɦɟɯɚɧɢɡɦɚ ɡɚɞɜɢɠɤɢ ɢ ɢɡɦɟɧɹɸɳɟɟ ɨɛɳɟɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɧɚɩɨɪɧɨɣ ɫɟɬɢ. ɗɬɨ ɜɨɡɞɟɣɫɬɜɢɟ ɩɟɪɟɞɚɟɬɫɹ ɩɨ ɤɚɧɚɥɚɦ 2 – I ɢ 2 – I';
– ɝɢɞɪɚɜɥɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚɩɨɪɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ Ȝ(t), ɜɟɥɢɱɢɧɚ ɩɟɪɟɦɟɧɧɚɹ, ɡɚɜɢɫɹɳɚɹ, ɧɚɩɪɢɦɟɪ, ɨɬ ɤɨɥɢɱɟɫɬɜɚ ɬɜɟɪɞɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɜɵɩɚɜɲɟɝɨ ɧɚ ɞɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ ɢ ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɟɟɫɹ ɩɨ ɤɚɧɚɥɚɦ 3 – I' ɢ 3 – II, ɢ ɨɤɚɡɵɜɚɸɳɟɟ ɜɥɢɹɧɢɟ ɧɚ ɨɛɳɟɟ
ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɪɭɛɨɩɪɨɜɨɞɚ ɬɨɱɧɨ ɬɚɤ ɠɟ, ɤɚɤ ɦɟɫɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɡɚɞɜɢɠɤɢ;
– ɤɨɥɢɱɟɫɬɜɨ ɜɨɡɞɭɯɚ Qȼ(t), ɩɨɫɬɭɩɚɸɳɟɝɨ ɜɨ ɜɫɚɫɵɜɚɸɳɭɸ ɥɢɧɢɸ
ɭɝɥɟɫɨɫɚ ɢ ɨɤɚɡɵɜɚɸɳɟɟ ɜɥɢɹɧɢɟ ɧɚ ɩɚɪɚɦɟɬɪɵ ɨɛɴɟɤɬɚ ɩɨ ɤɚɧɚɥɭ 4 – I;
– ɩɥɨɬɧɨɫɬɶ ɝɢɞɪɨɫɦɟɫɢ ȡȽ(t), ɩɨɫɬɭɩɚɸɳɟɣ ɜ ɫɢɫɬɟɦɭ ɱɟɪɟɡ ɜɫɚɫɵɜɚɸɳɢɣ ɧɚɤɨɧɟɱɧɢɤ ɞɨɡɢɪɭɸɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ ɩɨ ɤɚɧɚɥɚɦ 5 – I, 5 – I'
ɢ
106
5 – II, ɢ ɜɥɢɹɸɳɚɹ ɧɚ ɧɚɩɨɪɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɤɚɤ ɭɝɥɟɫɨɫɚ, ɬɚɤ ɢ
[
O
Ƚ
U
[
O
O
Ƚ
U
[
ɬɪɭɛɨɩɪɨɜɨɞɧɨɣ ɫɟɬɢ;
– ɱɚɫɬɨɬɚ ɜɪɚɳɟɧɢɹ ɝɥɚɜɧɨɝɨ ɜɚɥɚ ɭɝɥɟɫɨɫɚ n(t), ɨɤɚɡɵɜɚɸɳɚɹ ɜɥɢɹɧɢɟ ɧɚ ɢɧɞɢɜɢɞɭɚɥɶɧɭɸ ɧɚɩɨɪɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɭɝɥɟɫɨɫɚ ɩɨ ɤɚɧɚɥɚɦ
6 – I ɢ 6 – I'.
Qɉ(t)n(t)
6
)(t
2
3
)(t
4
(t)
Q
ȼ
5
)(t
ɚ)
1
II’Q
(t)
ɍ
Ɋ
(t)
ɍ
Iȱ
(t)
Ɋ
Ɍ
Q
ɉ
Q
n(t)
3
(t)
ȼ
4
6
2
)(t
1
(t)
Q
ɍ
I
Iȱ
(t)
Ɋ
ɛ) ɜ)
Ɍ
Q
(t)
ȼ
4
6IQ
n(t)
)(t
5
Q
ɉ
3
2
1
(t)
ɍ
Iȱ
(t)
Ɋ
Ɍ
Ɋɢɫɭɧɨɤ 3.11. ɋɬɪɭɤɬɭɪɧɵɟ ɫɯɟɦɵ ɨɛɴɟɤɬɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ
ȼɵɯɨɞɧɵɦɢ (ɪɟɝɭɥɢɪɭɟɦɵɦɢ) ɜɟɥɢɱɢɧɚɦɢ ɩɪɢ ɷɬɨɦ ɩɪɢɧɹɬɵ:
ɩɨɞɚɱɚ ɭɝɥɟɫɨɫɚ Q
ɭɝɥɟɫɨɫɚ Ɋ
(t) (ɜɵɯɨɞ I') ɢ ɞɚɜɥɟɧɢɟ, ɫɨɡɞɚɜɚɟɦɨɟ ɞɜɢɠɭɳɟɣɫɹ ɫɪɟɞɨɣ ɜ
ɍ
(t) (ɜɵɯɨɞ I), ɞɚɜɥɟɧɢɟ ɜ ɧɚɝɧɟɬɚɬɟɥɶɧɨɦ ɩɚɬɪɭɛɤɟ
ɍ
107
ɧɚɩɨɪɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ ɩɪɢ ɧɟɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ (ɜɵɯɨɞ II).
Ɋɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɜɵɯɨɞɧɵɟ ɤɚɧɚɥɵ ɜɡɚɢɦɨɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ. Ɍɚɤ,
ɤɨɧɬɪɨɥɢɪɭɟɦɵɟ ɜɟɥɢɱɢɧɵ Q
(t) ɢ Ɋɍ(t) ɨɞɧɨɡɧɚɱɧɨ ɜɵɪɚɠɚɸɬɫɹ ɞɪɭɝ ɱɟ-
ɍ
ɪɟɡ ɞɪɭɝɚ ɢɡ ɭɪɚɜɧɟɧɢɹ ɧɚɩɨɪɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɚɲɢɧɵ (ɭɝɥɟɫɨɫɚ). ɇɚ
ɩɪɚɤɬɢɤɟ ɞɥɹ ɰɟɥɟɣ ɭɩɪɚɜɥɟɧɢɹ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɨɤɚɡɵɜɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɵɦ ɤɨɧɬɪɨɥɶ ɨɞɧɨɣ ɢɡ ɭɤɚɡɚɧɧɵɯ ɜɟɥɢɱɢɧ (ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɦɢɧɢɦɚɥɶɧɚɹ ɡɚɳɢɬɚ ɩɨ ɪɚɫɯɨɞɭ, ɫɦ. ɩ. 3.9), ɹɜɥɹɸɳɟɣɫɹ ɨɛɵɱɧɨ ɝɪɚɧɢɱɧɵɦ
ɭɫɥɨɜɢɟɦ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɞɪɭɝɨɣ ɭɩɪɚɜɥɹɟɦɨɣ ɜɟɥɢɱɢɧɵ – ɞɚɜɥɟɧɢɹ ɫɪɟɞɵ ɜ
ɬɪɚɧɫɩɨɪɬɧɨɦ ɬɪɭɛɨɩɪɨɜɨɞɟ.
ȼ ɤɨɧɬɟɤɫɬɟ ɷɬɨɝɨ ɩɚɪɚɦɟɬɪ Q
(t) ɢɥɢ Ɋɍ(t) ɜɵɫɬɭɩɚɟɬ ɤɚɤ ɜɯɨɞɧɚɹ
ɍ
ɤɨɨɪɞɢɧɚɬɚ ɞɥɹ ɤɚɧɚɥɚ I – II (I' – II), ɩɪɟɞɫɬɚɜɥɹɸɳɟɝɨ ɬɪɚɧɫɩɨɪɬɧɵɣ
ɬɪɭɛɨɩɪɨɜɨɞ, ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɹɬ ɨɫɧɨɜɧɵɟ ɞɢɧɚɦɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɜ
ɦɨɦɟɧɬɵ ɧɨɪɦɚɥɶɧɨɣ ɷɤɫɩɥɭɚɬɚɰɢɢ ɭɫɬɚɧɨɜɤɢ ɢ ɜ ɦɨɦɟɧɬɵ ɪɟɝɭɥɢɪɨɜɚɧɢɹ
ɟɺ ɩɨ ɩɨɞɚɱɟ. Ʉɪɨɦɟ ɬɨɝɨ, ɜɟɥɢɱɢɧɚ Q
(t) ɩɪɢ ɨɩɪɟɞɟɥɟɧɧɵɯ ɭɫɥɨɜɢɹɯ
ɭ
ɨɤɚɡɵɜɚɟɬ ɜɥɢɹɧɢɟ ɧɚ ɢɡɦɟɧɟɧɢɟ ɜɯɨɞɧɨɣ ɜɟɥɢɱɢɧɵ Ȝ(t), ɫɩɨɫɨɛɫɬɜɭɹ ɬɟɦ
ɫɚɦɵɦ ɩɨɹɜɥɟɧɢɸ ɧɟɠɟɥɚɬɟɥɶɧɨɝɨ ɜɧɭɬɪɟɧɧɟɝɨ ɤɨɧɬɭɪɚ ɫ ɩɨɥɨɠɢɬɟɥɶɧɨɣ
ɨɛɪɚɬɧɨɣ ɫɜɹɡɶɸ, ɪɟɡɤɨ ɭɯɭɞɲɚɸɳɟɝɨ ɭɫɬɨɣɱɢɜɨɫɬɶ ɨɛɴɟɤɬɚ.
ɉɨ ɯɚɪɚɤɬɟɪɭ ɮɨɪɦɢɪɨɜɚɧɢɹ, ɜɯɨɞɧɵɟ ɜɟɥɢɱɢɧɵ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɪɟɝɭɥɢɪɭɸɳɢɦ ȟ(t), n(t), ȡ
(t), Qȼ(t) ɢ ɜɨɡɦɭɳɚɸɳɢɦ ȟ, Qɉ, Ȝ, Qȼ
Ƚ
ɜɨɡɞɟɣɫɬɜɢɹɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɪɚɡɥɢɱɧɵɦ ɬɨɱɤɚɦ ɨɛɴɟɤɬɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ.
ɉɨ ɯɚɪɚɤɬɟɪɭ ɩɪɨɬɟɤɚɧɢɹ ɜɨ ɜɪɟɦɟɧɢ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɜɯɨɞɧɵɟ
ɤɨɨɪɞɢɧɚɬɵ, ɤɪɨɦɟ ȟ(t) ɢ ɤɚɧɚɥ I – II (I' – II), ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɹɜɥɹɸɬɫɹ ɧɟ-
ɫɬɚɰɢɨɧɚɪɧɵɦɢ, ɩɪɨɢɡɜɨɥɶɧɨ ɢɡɦɟɧɹɸɳɢɦɢɫɹ ɩɪɢ ɦɢɧɢɦɚɥɶɧɨ ɢɦɟɸɳɟɣɫɹ ɚɩɪɢɨɪɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɢɯ ɤɨɨɪɞɢɧɚɬ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɨɡɦɨɠɧɚ ɨɛɳɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
: ɪɚɡɪɚɛɨɬɚɬɶ ɫɬɪɭɤɬɭɪɭ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ, ɢɧɜɚɪɢɚɧɬɧɭɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɧɟɲɧɢɯ ɜɨɡɦɭɳɟɧɢɣ, ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɹ ɧɚ ɷɬɨɦ ɷɬɚɩɟ ɢɦɟɸɳɟɟ ɦɟɫɬɨ ɢɡɦɟɧɟɧɢɟ ɫɨɛɫɬɜɟɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɫɢɫɬɟɦɵ [10]. Ɂɚɞɚɱɚ, ɞɚɠɟ ɜ ɬɚɤɨɦ ɭɩɪɨɳɟɧɧɨɦ ɜɚɪɢɚɧɬɟ ɜɵɡɵɜɚɟɬ ɡɧɚɱɢɬɟɥɶɧɵɟ ɡɚɬɪɭɞɧɟɧɢɹ ɜ ɟɺ ɪɟɲɟɧɢɢ, ɜ
ɫɜɹɡɢ ɫ ɧɚɥɢɱɢɟɦ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɜɧɟɲɧɢɯ ɜɨɡɦɭɳɟɧɢɣ, ɢɡɦɟɧɹɸɳɢɯɫɹ
ɜɨ ɜɪɟɦɟɧɢ, ɚ ɬɚɤɠɟ ɜ ɫɜɹɡɢ ɢ ɢɦɟɸɳɢɦɢ ɦɟɫɬɨ ɜɧɭɬɪɟɧɧɢɦɢ ɜɡɚɢɦɨɫɜɹɡɹɦɢ ɦɟɠɞɭ ɜɯɨɞɧɵɦɢ ɢ ɜɵɯɨɞɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɤɨɧɤɪɟɬɢɡɢɪɭɟɦ ɟɺ ɩɨɫɬɚɧɨɜɤɭ ɫ
ɭɱɺɬɨɦ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɨɫɨɛɟɧɧɨɫɬɟɣ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɨɛɴɟɤɬɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ. ɉɪɢ ɷɬɨɦ ɨɤɚɡɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɵɦ ɩɪɟɞɫɬɚɜɢɬɶ ɨɛɴɟɤɬ ɫ ɨɝɪɚɧɢɱɟɧɧɵɦ ɱɢɫɥɨɦ ɜɧɟɲɧɢɯ ɜɨɡɦɭɳɟɧɢɣ, ɢɡɦɟɧɹɸɳɢɯɫɹ ɜɨ ɜɪɟɦɟɧɢ, ɚ
ɨɫɬɚɥɶɧɵɟ ɜɨɡɦɭɳɟɧɢɹ ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɦɨɝɭɬ ɛɵɬɶ ɩɪɢɧɹɬɵ ɜɟɥɢɱɢɧɚɦɢ ɩɨɫɬɨɹɧɧɵɦɢ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɜ ɬɟɱɟɧɢɟ ɜɫɟɝɨ ɰɢɤɥɚ
108
ɪɟɝɭɥɢɪɨɜɚɧɢɹ. ȼɨɡɦɨɠɧɨɫɬɶ ɬɚɤɨɝɨ ɩɨɞɯɨɞɚ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ ɭɩɪɚɜɥɟɧɢɹ ɲɚɯɬɧɨɣ ɭɱɚɫɬɤɨɜɨɣ ɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɨɣ ɨɛɭɫɥɨɜɥɟɧɚ ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɛɪɚɠɟɧɢɹɦɢ:
1) ɩɥɨɬɧɨɫɬɶ ɝɢɞɪɨɫɦɟɫɢ ɜ ɨɫɧɨɜɧɨɦ ɨɩɪɟɞɟɥɹɟɬ ɜɚɠɧɟɣɲɢɣ ɩɚɪɚɦɟɬɪ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɷɤɨɧɨɦɢɱɧɨɫɬɶ ɪɚɛɨɬɵ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ – ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɫɬɚɧɰɢɢ ɩɨ ɬɜɺɪɞɨɦɭ. ɉɨɷɬɨɦɭ ɜ ɧɨɦɢɧɚɥɶɧɨɦ ɬɪɚɧɫɩɨɪɬɧɨɦ ɪɟɠɢɦɟ ɩɥɨɬɧɨɫɬɶ ɝɢɞɪɨɫɦɟɫɢ ɨɛɵɱɧɨ ɫɬɚɛɢɥɢɡɢɪɭɸɬ
ɧɚ ɡɚɞɚɧɧɨɦ ɭɪɨɜɧɟ (ɟɫɥɢ ɷɬɚ
ɜɟɥɢɱɢɧɚ ɧɟ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɢɪɭɸɳɢɦ ɜɨɡɞɟɣɫɬɜɢɟɦ). Ɍɚɤɚɹ ɫɬɚɛɢɥɢɡɚɰɢɹ ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢɦɟɧɟɧɢɟɦ ɫɩɟɰɢɚɥɶɧɨɝɨ
ɞɨɡɢɪɭɸɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ ɧɚ ɜɫɚɫɵɜɚɸɳɟɣ ɥɢɧɢɢ ɭɝɥɟɫɨɫɚ, ɚ ɩɪɢ ɧɟɨɛɯɨɞɢɦɨɫɬɢ – ɩɪɢɦɟɧɟɧɢɟɦ ɨɬɞɟɥɶɧɨɣ ɋȺɊ, ɨɫɧɨɜɚɧɧɨɣ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ
ɬɚɤɨɝɨ ɭɫɬɪɨɣɫɬɜɚ. ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɞɥɹ ɲɚɯɬɧɵɯ ɭɱɚɫɬɤɨɜɵɯ ɭɫɬɚɧɨɜɨɤ ɨɤɚɡɵɜɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɵɦ ɩɪɢɦɟɧɟɧɢɹ ɞɥɹ ɷɬɢɯ ɰɟɥɟɣ ɥɢɲɶ ɞɨɡɢɪɭɸɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ, ɧɚɩɪɢɦɟɪ ɬɢɩɚ ɍȼ, ɪɚɡɪɚɛɨɬɚɧɧɨɝɨ
ɜ ȾɨɧɇɌɍ
(Ⱦɉɂ), ɩɨɞɞɟɪɠɢɜɚɸɳɟɝɨ ɩɨɫɬɨɹɧɧɭɸ ɩɥɨɬɧɨɫɬɶ ɝɢɞɪɨɫɦɟɫɢ ɜ ɡɚɞɚɧɧɵɯ
ɩɪɟɞɟɥɚɯ ɡɚ ɫɱɺɬ ɭɫɬɚɧɨɜɥɟɧɢɹ ɨɩɪɟɞɟɥɺɧɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɩɨɞɩɢɬɨɱɧɨɝɨ
ɢ ɮɢɥɶɬɪɚɰɢɨɧɧɨɝɨ ɪɚɫɯɨɞɨɜ ɜɨ ɜɫɚɫɵɜɚɸɳɟɦ ɭɡɥɟ, ɱɬɨ ɞɨɫɬɢɝɚɟɬɫɹ
ɤɨɧɫɬɪɭɤɬɢɜɧɵɦɢ ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɭɫɬɪɨɣɫɬɜɚ [11]. ȿɫɥɢ ɠɟ ɜɟɥɢɱɢɧɚ ȡȽ(t)
ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɢɪɭɸɳɟɣ (ɪɢɫ. 3.11, ɛ), ɬɨ ɫɬɚɛɢɥɢɡɚɰɢɸ ɩɚɪɚɦɟɬɪɚ ȡ
(t) ɧɟ
Ƚ
ɩɪɨɜɨɞɹɬ, ɢ ɞɨɡɢɪɭɸɳɟɟ ɭɫɬɪɨɣɫɬɜɨ ɍȼ ɜ ɬɚɤɨɦ ɫɥɭɱɚɟ ɜɵɩɨɥɧɹɟɬ ɪɨɥɶ
ɪɟɝɭɥɹɬɨɪɚ ɩɥɨɬɧɨɫɬɢ ɝɢɞɪɨɫɦɟɫɢ, ɚ ɩɨɫɪɟɞɫɬɜɨɦ ɟɺ – ɩɨɞɚɱɢ ɭɝɥɟɫɨɫɚ.
Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɢ ɪɟɝɭɥɢɪɨɜɚɧɢɢ ɭɝɥɟɫɨɫɚ ɩɨ ɩɨɞɚɱɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɩɟɪɟɜɨɞɚ ɭɫɬɚɧɨɜɤɢ ɧɚ ɪɚɛɨɬɭ ɩɨ ɜɨɞɟ, ɜ ɦɨɦɟɧɬ
ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɫɬɚɧɨɜɢɬɶɫɹ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɨɞɵ [6]
= ȡ0 = const;
ɬ. ɟ. ȡ
Ƚ
2) ɜɟɥɢɱɢɧɚ Q
(t) ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɨɫɬɭɩɥɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɜɨɡɞɭɯɚ ɜɨ
ȼ
ɜɫɚɫɵɜɚɸɳɭɸ ɥɢɧɢɸ ɭɝɥɟɫɨɫɚ ɡɚ ɫɱɺɬ ɩɨɞɫɨɫɨɜ ɱɟɪɟɡ ɢɡɧɨɲɟɧɧɵɟ
ɫɚɥɶɧɢɤɨɜɵɟ ɭɩɥɨɬɧɟɧɢɹ ɦɚɲɢɧɵ ɢ ɜɵɞɟɥɟɧɢɹ ɜɨɡɞɭɯɚ ɢɡ ɩɨɪ ɬɜɺɪɞɨɝɨ
ɦɚɬɟɪɢɚɥɚ, ɫɛɪɚɫɵɜɚɟɦɨɝɨ ɜ ɩɪɢɺɦɧɭɸ ɺɦɤɨɫɬɶ ɭɝɥɟɫɨɫɧɨɣ ɫɬɚɧɰɢɢ ɩɨɫɥɟ
ɩɪɨɯɨɠɞɟɧɢɹ ɟɝɨ ɩɨ ɛɟɡɧɚɩɨɪɧɨɦɭ ɬɪɚɧɫɩɨɪɬɭ. ɗɬɨɬ ɩɚɪɚɦɟɬɪ ɬɚɤɠɟ
ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɧɹɬ ɜɟɥɢɱɢɧɨɣ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Q
= const, ɬɚɤ ɤɚɤ ɨɛɴɺɦ
ȼ
ɜɨɡɞɭɯɚ, ɩɨɞɫɚɫɵɜɚɟɦɵɣ ɱɟɪɟɡ ɫɚɥɶɧɢɤɨɜɵɟ ɭɩɥɨɬɧɟɧɢɹ, ɪɟɝɥɚɦɟɧɬɢɪɭɟɬɫɹ ɭɫɥɨɜɢɹɦɢ ɬɟɯɧɢɱɟɫɤɨɣ ɷɤɫɩɥɭɚɬɚɰɢɢ ɭɝɥɟɫɨɫɨɜ ɢ ɩɟɪɢɨɞɢɱɟɫɤɢ ɩɪɢɜɨɞɢɬɫɹ ɤ ɧɨɪɦɟ ɩɭɬɺɦ ɡɚɦɟɧɵ ɢɡɧɨɲɟɧɧɵɯ ɭɩɥɨɬɧɟɧɢɣ ɜɨ ɜɪɟɦɹ ɪɟɦɨɧɬɧɨ-ɩɪɨɮɢɥɚɤɬɢɱɟɫɤɢɯ ɪɚɛɨɬ. Ɉɛɴɺɦ ɠɟ ɜɨɡɞɭɯɚ, ɜɵɞɟɥɹɸɳɟɝɨɫɹ ɢɡ ɩɨɪ
ɬɪɚɧɫɩɨɪɬɢɪɭɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɢɡɦɟɧɹɟɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɜɨ ɜɪɟɦɟɧɢ, ɬɚɤ
ɤɚɤ ɡɚɜɢɫɢɬ ɜ ɨɫɧɨɜɧɨɦ ɨɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɜɺɪɞɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɨɬ ɟɝɨ ɤɨɥɢɱɟɫɬɜɚ ɢ ɫɩɨɫɨɛɚ ɞɨɛɵɱɢ, ɞɥɢɧɵ ɢ ɜɢɞɚ ɬɪɚɧɫɩɨɪɬɢɪɨɜɚɧɢɹ, ɚ ɬɚɤɠɟ ɨɬ
ɚɬɦɨɫɮɟɪɧɵɯ ɭɫɥɨɜɢɣ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ. ȼɫɟ ɩɟɪɟɱɢɫɥɟɧɧɵɟ ɮɚɤɬɨɪɵ
ɧɚ ɪɟɚɥɶɧɵɯ ɨɛɴɟɤɬɚɯ ɞɚɧɧɨɝɨ ɤɥɚɫɫɚ ɨɫɬɚɸɬɫɹ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɟ ɜɪɟɦɹ
ɩɨɫɬɨɹɧɧɵɦɢ. ȼɦɟɫɬɟ ɫ ɬɟɦ, ɜɟɥɢɱɢɧɚ Qȼ(t) ɦɨɠɟɬ ɜɵɫɬɭɩɚɬɶ ɜ ɪɨɥɢ
109
ɪɟɝɭɥɢɪɭɸɳɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɩɨɞɚɱɢ ɭɝɥɟɫɨɫɚ, ɤɚɤ ɷɬɨ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 3.11, ɛ ɢ ɪɢɫ. 3.11, ɜ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɤɨɥɢɱɟɫɬɜɨ ɜɨɡɞɭɯɚ,
ɧɚɩɪɚɜɥɹɟɦɨɟ ɜɨ ɜɫɚɫɵɜɚɸɳɭɸ ɥɢɧɢɸ ɭɝɥɟɫɨɫɚ ɪɟɝɥɚɦɟɧɬɢɪɭɟɬɫɹ ɪɚɫɱɺɬɨɦ, ɩɪɨɜɟɞɟɧɧɨɦ ɩɨ ɫɩɟɰɢɚɥɶɧɨɣ ɦɟɬɨɞɢɤɟ, ɤɨɬɨɪɚɹ ɫɨɞɟɪɠɢɬɫɹ ɜ ɩɨɞɪɚɡɞɟɥɟ 2.2.
ɉɪɢɧɹɬɵɟ ɞɨɩɭɳɟɧɢɹ ɡɧɚɱɢɬɟɥɶɧɨ ɭɩɪɨɳɚɸɬ ɡɚɞɚɱɭ ɭɩɪɚɜɥɟɧɢɹ ɢ
ɩɨɡɜɨɥɹɸɬ ɧɚ ɢɯ ɨɫɧɨɜɟ ɪɚɡɪɚɛɨɬɚɬɶ ɜɨɡɦɨɠɧɵɟ
ɜɚɪɢɚɧɬɵ ɫɬɪɭɤɬɭɪɧɵɯ
ɫɯɟɦ ɨɛɴɟɤɬɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ, ɩɨɥɭɱɢɬɶ ɩɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɩɨ ɨɬɞɟɥɶɧɵɦ ɡɜɟɧɶɹɦ ɢ, ɜ ɤɨɧɟɱɧɨɦ ɢɬɨɝɟ, ɧɚɣɬɢ
ɩɪɢɟɦɥɟɦɭɸ, ɜ ɩɥɚɧɟ ɬɟɯɧɢɱɟɫɤɨɣ ɪɟɚɥɢɡɚɰɢɢ, ɨɛɳɭɸ ɫɬɪɭɤɬɭɪɭ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɩɨɞɚɱɢ ɝɢɞɪɨɬɪɚɧɫɩɨɪɬɧɨɣ ɭɫɬɚɧɨɜɤɢ.
ȼɚɪɢɚɧɬɵ ɫɬɪɭɤɬɭɪɧɵɯ ɫɯɟɦ ɨɛɴɟɤɬɨɜ ɪɟɝɭɥɢɪɨɜɚɧɢɹ, ɩɨɥɭɱɟɧɧɵɟ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɧɹɬɵɯ ɞɨɩɭɳɟɧɢɣ, ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ. 3.11, ɛ ɢ ɪɢɫ. 3.11, ɜ.
ɋɯɟɦɚ ɪɢɫ. 3.11, ɛ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɫɥɨɜɢɸ ɪɚɛɨɬɵ ɭɝɥɟɫɨɫɧɨɣ ɭɫɬɚɧɨɜɤɢ ɜ
ɩɟɪɟɯɨɞɧɨɦ ɪɟɠɢɦɟ ɫ ɡɚɢɥɟɧɧɵɦ ɧɚɩɨɪɧɵɦ ɬɪɭɛɨɩɪɨɜɨɞɨɦ, ɚ ɫɯɟɦɚ
ɪɢɫ. 3.11, ɜ – ɫ ɧɟɡɚɢɥɟɧɧɵɦ ɬɪɭɛɨɩɪɨɜɨɞɨɦ ɢ ɩɪɢ ɪɚɛɨɬɟ ɭɫɬɚɧɨɜɤɢ ɧɚ
ɱɢɫɬɨɣ ɜɨɞɟ. ȼ ɨɛɟɢɯ ɫɯɟɦɚɯ ɜ ɤɚɱɟɫɬɜɟ ɜɵɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɢɧɹɬɵ
ɨɞɧɢ ɢ ɬɟ ɠɟ ɜɟɥɢɱɢɧɵ Qɍ(t) ɢ Ɋɍ(t). ɉɪɟɞɥɨɠɟɧɧɵɟ ɫɯɟɦɵ ɤɪɨɦɟ
ɢɞɟɧɬɢɱɧɵɯ ɜɵɯɨɞɨɜ I – II ɫɨɞɟɪɠɚɬ ɬɚɤɠɟ ɢɞɟɧɬɢɱɧɵɟ ɜɯɨɞɧɵɟ
ɤɨɨɪɞɢɧɚɬɵ Qɉ(t) ɢ ȟ(t), ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɢɟɫɹ ɩɨ ɨɞɢɧɚɤɨɜɵɦ ɞɥɹ ɨɛɟɢɯ
ɫɯɟɦ ɨɞɧɨɢɦɟɧɧɵɦ ɤɚɧɚɥɚɦ I – I ɢ 2 – I. ɋɭɳɟɫɬɜɟɧɧɵɦ ɨɬɥɢɱɢɟɦ
ɭɤɚɡɚɧɧɵɯ ɫɯɟɦ ɹɜɥɹɟɬɫɹ ɧɚɥɢɱɢɟ ɜ ɫɨɫɬɚɜɟ ɨɞɧɨɣ ɢɡ ɧɢɯ (ɪɢɫ. 3.11, ɛ)
ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɜɯɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɵ Ȝ(t), ɨɤɚɡɵɜɚɸɳɟɣ ɜɥɢɹɧɢɟ ɧɚ
ɜɵɯɨɞɧɵɟ ɩɚɪɚɦɟɬɪɵ ɨɛɴɟɤɬɚ ɩɨ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦ ɤɚɧɚɥɚɦ 3 – I ɢ 3 – II.
ɉɪɢ ɪɚɡɪɚɛɨɬɤɟ ɫɯɟɦɵ ɪɢɫ. 3.11, ɛ ɭɱɢɬɵɜɚɥɨɫɶ, ɱɬɨ
ɧɚ ɦɨɦɟɧɬ
ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɩɨɞɚɱɢ ɭɝɥɟɫɨɫɚ ɩɨɫɥɟɞɧɢɣ ɩɟɪɟɜɟɞɟɧ ɧɚ ɪɚɛɨɬɭ ɩɨ ɜɨɞɟ ɢ
ɫ ɷɬɨɝɨ ɦɨɦɟɧɬɚ ɟɝɨ ɧɚɩɨɪɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ.
ȼɥɢɹɧɢɟ ɤɨɨɪɞɢɧɚɬɵ Ȝ(t) ɧɚ ɜɵɯɨɞɧɭɸ ɜɟɥɢɱɢɧɭ Q
(t) ɜ ɷɬɨɦ ɫɥɭɱɚɟ
ɍ
ɚɧɚɥɨɝɢɱɧɨ ɜɥɢɹɧɢɸ ɩɚɪɚɦɟɬɪɚ ȟ(t), ɢ ɞɢɧɚɦɢɤɚ ɤɚɧɚɥɚ 3 – III ɨɩɢɫɵɜɚɟɬɫɹ
ɬɟɦɢ ɠɟ ɭɪɚɜɧɟɧɢɹɦɢ, ɱɬɨ ɢ ɤɚɧɚɥɚ 2 – I. Ɉɞɧɚɤɨ ɧɚɥɢɱɢɟ ɬɜɺɪɞɨɝɨ
ɦɚɬɟɪɢɚɥɚ ɜ ɩɨɬɨɤɟ ɞɨ ɦɨɦɟɧɬɚ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɜɨ ɜɪɟɦɟɧɢ ɧɟ ɨɫɬɚɸɬɫɹ ɩɨɫɬɨɹɧɧɵɦɢ, ɤɚɤ ɷɬɨ ɛɭɞɟɬ
ɩɨɤɚɡɚɧɨ ɜ ɫɥɭɱɚɟ ɪɢɫ. 3.11, ɜ, ɚ ɩɪɟɬɟɪɩɟɜɚɟɬ ɢɡɦɟɧɟɧɢɟ
ɫ ɢɡɦɟɧɟɧɢɟɦ
ɜɵɯɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɵ, ɬɨɠɟ ɩɪɨɢɫɯɨɞɢɬ ɢ ɫ ɜɟɥɢɱɢɧɨɣ Ȝ(t). ɉɨɷɬɨɦɭ, ɜ
ɨɬɥɢɱɢɟ ɨɬ ɤɚɧɚɥɚ 2 – I, ɤɚɧɚɥ 3 – I, ɬɚɤɠɟ ɤɚɤ ɢ ɤɚɧɚɥ 3 – II, ɨɤɚɡɵɜɚɟɬɫɹ
ɧɟɥɢɧɟɣɧɵɦ, ɱɬɨ ɭɫɥɨɠɧɹɟɬ ɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ. ȼ ɩɨɞɨɛɧɵɯ
ɫɥɭɱɚɹɯ, ɤɚɤ ɢɡɜɟɫɬɧɨ, ɩɪɢɛɟɝɚɸɬ ɤ ɪɚɡɥɢɱɧɵɦ ɩɪɢɟɦɚɦ ɥɢɧɟɚɪɢɡɚɰɢɢ
ɭɪɚɜɧɟɧɢɣ ɞɢɧɚɦɢɤɢ, ɥɢɛɨ ɩɟɪɟɞɚɬɨɱɧɭɸ ɮɭɧɤɰɢɸ ɬɚɤɢɯ ɡɜɟɧɶɟɜ ɧɚɯɨɞɹɬ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɩɭɬɟɦ, ɢɫɩɨɥɶɡɭɹ ɮɢɡɢɱɟɫɤɢɟ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ
ɦɨɞɟɥɢ.
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