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Ʉɲ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
j
y
4.2. dzȌșȕȋȢ ȗȇȘȞȌșȇ ȟȏȜșȢ
Ʉ
= (100 í yɲ) / 100,
ɲ
– ɭɝɚɪ ɜɫɟɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɲɢɯɬɵ ɜ ɩɪɨɰɟɫɫɟ ɩɪɢɝɨɬɨɜɥɟɧɢɹ ɫɩɥɚɜɚ.
ɝɞɟ y
ɲ
n
ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ (4.5).
ɝɞɟ y
j
=
y
ɲ
¦
j
,
1
ȼ ɩɪɢɛɥɢɠɟɧɧɵɯ ɪɚɫɱɟɬɚɯ ɡɧɚɱɟɧɢɟ ɭɝɚɪɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɧɹɬɨ ɞɥɹ
ɷɥɟɤɬɪɨɞɭɝɨɜɨɣ ɩɟɱɢ: y
= 1,5…5,0ௗ%; ɢɧɞɭɤɰɢɨɧɧɨɣ yɲ = 1…2ௗ%.
ɲ
1. ɉɪɢ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɦɟɬɨɞɟ ɡɚɞɚɱɚ ɪɚɫɱɟɬɚ ɲɢɯɬɵ ɮɨɪɦɢɪɭɟɬɫɹ
ɜ ɜɢɞɟ ɪɚɜɟɧɫɬɜ. ȿɫɥɢ ɭɝɚɪ ɪɚɫɱɟɬɧɵɯ ɷɥɟɦɟɧɬɨɜ ɢɡ ɨɬɞɟɥɶɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɲɢɯɬɵ ɭɱɢɬɵɜɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɧɨ, ɬɨ ɪɚɫɱɟɬɧɵɟ ɭɪɚɜɧɟɧɢɹ (4.2), (4.7), (4.8) ɫ ɭɱɟɬɨɦ (4.3) ɢɦɟɸɬ ɜɢɞ:
n
ɗɄ Ʉ ɗ
¦
ij ɷij j j i
ɲɲ ɠ
j
1
ɏ ;
ɫɪ
(4.9)
ɏjɲ Ʉj = h; (4.10)
n
¦
j
ɲ
j
1
.ɏ 1
Ʉ
(4.11)
j
ɉɪɢ ɪɚɫɱɟɬɟ ɲɢɯɬɵ ɩɨ ɫɪɟɞɧɟɦɭ ɡɧɚɱɟɧɢɸ ɭɝɚɪɚ ɷɥɟɦɟɧɬɨɜ ɪɚɫ-
ɱɟɬɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ:
n
ɗɄ
ɏ
¦
j
¦
j
1
n
ɷɲ
ɲ
ij ij j
= h / Ʉj; (4.13)
jɲ
ɲ
j
1
21
;ɏ
(4.12)
.ɏ 1/
Ʉ
(4.14)
ɲ
Ɋɚɫɱɟɬɧɵɟ ɭɪɚɜɧɟɧɢɹ (4.9) – (4.11) ɢɥɢ (4.12) – (4.14) ɩɪɟɞɫɬɚɜɥɹɸɬ
ɫɨɛɨɣ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ, ɞɥɹ ɪɟɲɟɧɢɹ ɤɨɬɨɪɨɣ ɪɟɤɨɦɟɧɞɭɸɬ ɦɟɬɨɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɢɫɤɥɸɱɟɧɢɹ Ƚɚɭɫɫɚ ɢɥɢ
ɮɨɪɦɭɥɭ Ʉɪɚɦɟɪɚ. ɉɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɨɣ Ʉɪɚɦɟɪɚ ɢ ɦɟɬɨɞɨɦ Ƚɚɭɫɫɚ
ɩɪɢ ɱɢɫɥɟ ɧɟɢɡɜɟɫɬɧɵɯ ɛɨɥɟɟ ɬɪɟɯ ɫɱɢɬɚɟɬɫɹ ɧɟɰɟɥɟɫɨɨɛɪɚɡɧɵɦ [7] ɢɡ-ɡɚ
ɩɨɜɵɲɟɧɧɨɣ ɫɥɨɠɧɨɫɬɢ ɢ ɝɪɨɦɨɡɞɤɨɫɬɢ ɜɵɱɢɫɥɟɧɢɣ.
2. Ƚɪɚɮɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɪɚɫɱɟɬɚ ɲɢɯɬɵ ɨɫɧɨɜɚɧɵ ɧɚ ɪɟɲɟɧɢɢ ɫɢ-
ɫɬɟɦɵ
ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɢɞɚ (4.9) – (4.11) ɢɥɢ (4.12) – (4.14) ɩɭɬɟɦ
ɝɪɚɮɢɱɟɫɤɢɯ ɩɨɫɬɪɨɟɧɢɣ. ɗɬɢ ɦɟɬɨɞɵ ɪɚɫɱɟɬɚ ɨɬɥɢɱɚɸɬɫɹ ɛɨɥɶɲɨɣ ɩɪɨɫɬɨɬɨɣ. Ʉ ɢɯ ɧɟɞɨɫɬɚɬɤɚɦ ɫɥɟɞɭɟɬ ɨɬɧɟɫɬɢ: ɧɟɜɨɡɦɨɠɧɨɫɬɶ ɢɥɢ ɫɥɨɠɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɫɨɫɬɚɜɚ ɲɢɯɬɵ ɩɪɢ ɛɨɥɶɲɨɦ ɱɢɫɥɟ ɪɚɫɱɟɬɧɵɯ ɷɥɟɦɟɧɬɨɜ ɢ ɫɨɫɬɚɜɥɹɸɳɢɯ, ɫɪɚɜɧɢɬɟɥɶɧɨ ɧɢɡɤɭɸ ɬɨɱɧɨɫɬɶ ɜɵɱɢɫɥɟɧɢɣ,
ɨɛɭɫɥɨɜɥɟɧɧɭɸ ɢɡɦɟɪɟɧɢɟɦ ɨɬɪɟɡɤɨɜ.
3. Ɋɚɫɱɟɬ ɲɢɯɬɵ ɦɟɬɨɞɨɦ ɩɨɞɛɨɪɚ ɨɫɧɨɜɚɧ ɧɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɦ ɜɵ-
ɛɨɪɟ
ɟɟ ɛɚɡɨɜɨɝɨ ɫɨɫɬɚɜɚ, ɩɪɨɜɟɪɤɟ ɦɚɬɟɪɢɚɥɶɧɨɝɨ ɛɚɥɚɧɫɚ ɩɨ ɪɚɫɱɟɬɧɵɦ
ɷɥɟɦɟɧɬɚɦ ɢ ɧɚ ɩɨɫɥɟɞɭɸɳɟɣ ɤɨɦɩɟɧɫɚɰɢɢ ɧɟɞɨɫɬɚɸɳɢɯ ɷɥɟɦɟɧɬɨɜ ɞɨɛɚɜɤɚɦɢ ɮɟɪɪɨɫɩɥɚɜɨɜ, ɱɢɫɬɵɯ ɦɟɬɚɥɥɨɜ, ɧɟ ɜɯɨɞɹɳɢɯ ɜ ɛɚɡɨɜɭɸ ɲɢɯɬɭ.
Ʉɚɤɢɟ-ɥɢɛɨ ɱɟɬɤɨ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɟ ɩɪɢɧɰɢɩɵ ɜɵɛɨɪɚ ɛɚɡɨɜɨɝɨ
ɫɨɫɬɚɜɚ ɲɢɯɬɵ ɨɬɫɭɬɫɬɜɭɸɬ. ȿɝɨ ɜɵɛɢɪɚɸɬ ɷɦɩɢɪɢɱɟɫɤɢ, ɢɫɯɨɞɹ ɢɡ
ɧɚɤɨɩɥɟɧɧɨɝɨ ɨɩɵɬɚ ɫɨɫɬɚɜɥɟɧɢɹ ɲɢɯɬɵ. Ⱦɥɹ ɷɬɨɝɨ ɢɫɩɨɥɶɡɭɸɬ ɢɧɮɨɪɦɚɰɢɸ ɨ ɩɪɢɦɟɪɧɨɦ
ɫɨɫɬɚɜɟ ɲɢɯɬɵ ɩɪɢ ɜɵɩɥɚɜɤɟ ɫɩɥɚɜɨɜ, ɛɥɢɡɤɢɯ ɩɨ
ɫɨɫɬɚɜɭ ɤ ɡɚɞɚɧɧɨɦɭ, ɨɩɪɟɞɟɥɹɸɬ ɩɟɪɟɱɟɧɶ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ ɞɥɹ
ɛɚɡɨɜɨɣ ɲɢɯɬɵ ɢ ɡɚɞɚɸɬɫɹ ɪɚɫɯɨɞɨɦ ɨɬɞɟɥɶɧɵɯ ɤɨɦɩɨɧɟɧɬɨɜ (ɜɨɡɜɪɚɬɚ,
ɫɬɪɭɠɤɢ ɢ ɬɚɤ ɞɚɥɟɟ). ɗɬɚ ɡɚɞɚɱɚ ɭɩɪɨɳɚɟɬɫɹ, ɟɫɥɢ ɛɚɡɨɜɚɹ ɲɢɯɬɚ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɩɨ ɨɞɧɨɦɭ-ɞɜɭɦ ɨɫɧɨɜɧɵɦ ɷɥɟɦɟɧɬɚɦ ɫɩɥɚɜɚ, ɨɛɵɱɧɨ ɩɨ
ɭɝɥɟɪɨɞɭ ɢɥɢ ɭɝɥɟɪɨɞɭ ɢ ɨɫɧɨɜɧɨɦɭ ɥɟɝɢɪɭɸɳɟɦɭ ɷɥɟɦɟɧɬɭ,
ɫ ɩɨɦɨɳɶɸ
ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɬɢɩɚ (4.9) – (4.11) ɢɥɢ (4.12) – (4.14). ɉɨɫɥɟ
ɜɵɛɨɪɚ ɧɚɱɚɥɶɧɨɝɨ ɜɚɪɢɚɧɬɚ ɛɚɡɨɜɨɝɨ ɫɨɫɬɚɜɚ ɲɢɯɬɵ ɩɪɨɜɟɪɹɸɬ ɦɚɬɟɪɢɚɥɶɧɵɣ ɛɚɥɚɧɫ ɩɨ ɜɫɟɦ ɪɚɫɱɟɬɧɵɦ ɷɥɟɦɟɧɬɚɦ ɫ ɭɱɟɬɨɦ ɬɟɯɧɢɱɟɫɤɨɣ
ɫɯɟɦɵ ɩɥɚɜɤɢ ɫɬɚɥɢ. ȿɫɥɢ ɫɨɞɟɪɠɚɧɢɟ ɪɚɫɱɟɬɧɵɯ ɷɥɟɦɟɧɬɨɜ ɦɟɧɶɲɟ ɞɨɩɭɫɬɢɦɨɝɨ, ɬɨ ɨɩɪɟɞɟɥɹɸɬ ɤɨɥɢɱɟɫɬɜɨ ɞɨɛɚɜɨɤ ɩɨ ɮɨɪɦɭɥɟ
ɂ
ɞɨɛ
= ((Ɇɲ Ʉɲ) / Ʉ
ɲ
ɞɨɛ
) ((ɗ
ɪɚɫɱ
iɠ
í ɗ
ɢɫɯ
iɠ
) / (ɗ
ɞɨɛ
Ʉɷi)), (4.15)
iɲ
ɝɞɟ Ɇ
Ʉ
Ʉ
– ɪɚɫɱɟɬɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɲɢɯɬɵ, ɤɝ;
ɲ
– ɤɨɷɮɮɢɰɢɟɧɬ ɜɵɯɨɞɚ ɠɢɞɤɨɝɨ ɦɟɬɚɥɥɚ ɢɡ ɜɫɟɣ ɲɢɯɬɵ;
ɲ
– ɤɨɷɮɮɢɰɢɟɧɬ ɜɵɯɨɞɚ ɠɢɞɤɨɝɨ ɦɟɬɚɥɥɚ ɢɡ ɞɨɛɚɜɤɢ.
ɞɨɛ
22
ɉɪɢ ɭɩɪɨɳɟɧɧɵɯ ɪɚɫɱɟɬɚɯ ɩɪɢɧɢɦɚɸɬ Ʉ
ɞɨɛ
ɧɢɟ ɪɚɫɱɟɬɧɨɝɨ ɷɥɟɦɟɧɬɚ ɜ ɝɨɬɨɜɨɦ ɫɩɥɚɜɟ,ௗ%; ɗ
= 1; ɗ
ɢɫɯ
iɠ
ɪɚɫɱ
– ɫɨɞɟɪɠɚ-
iɠ
– ɫɨɞɟɪɠɚɧɢɟ ɪɚɫ-
ɱɟɬɧɨɝɨ ɷɥɟɦɟɧɬɚ ɜ ɫɩɥɚɜɟ ɞɨ ɜɜɟɞɟɧɢɹ ɞɨɛɚɜɨɤ, ɩɪɢɜɟɞɟɧɧɨɟ ɤ ɦɚɫɫɟ ɝɨɬɨɜɨɝɨ ɫɩɥɚɜɚ,ௗ%; ɗ
ɢɡ ɞɨɛɚɜɨɤ,ௗ%; Ʉ
ɞɨɛ
– ɫɨɞɟɪɠɚɧɢɟ ɪɚɫɱɟɬɧɨɝɨ ɷɥɟɦɟɧɬɚ, ɭɫɜɚɢɜɚɟɦɨɟ
iɲ
– ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɜɨɟɧɢɹ ɪɚɫɱɟɬɧɨɝɨ ɷɥɟɦɟɧɬɚ ɢɡ ɞɨ-
ɷi
ɛɚɜɤɢ.
ȿɫɥɢ ɠɟ ɫɨɞɟɪɠɚɧɢɟ ɯɨɬɹ ɛɵ ɨɞɧɨɝɨ ɪɚɫɱɟɬɧɨɝɨ ɷɥɟɦɟɧɬɚ ɛɭɞɟɬ
ɛɨɥɶɲɟ ɞɨɩɭɫɬɢɦɨɝɨ ɜ ɝɨɬɨɜɨɦ ɫɩɥɚɜɟ, ɬɨ ɜɵɛɢɪɚɸɬ ɧɨɜɵɣ ɜɚɪɢɚɧɬ ɛɚɡɨɜɨɝɨ ɫɨɫɬɚɜɚ ɲɢɯɬɵ ɢɡ ɧɨɜɵɯ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ. Ɂɚɬɟɦ ɜɧɨɜɶ ɩɪɨɜɟɪɹɸɬ ɦɚɬɟɪɢɚɥɶɧɵɣ ɛɚɥɚɧɫ ɲɢɯɬɵ ɫ ɪɚɫɱɟɬɨɦ ɞɨɛɚɜɨɤ.
4. Ɋɚɫɱɟɬ ɲɢɯɬɵ ɤɚɤ ɡɚɞɚɱɚ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. Ɂɚɞɚɱɚ
ɜ ɤɨɬɨɪɨɣ ɨɬɵɫɤɢɜɚɟɬɫɹ ɷɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɣ ɜɨ ɦɧɨɠɟɫɬɜɟ, ɨɬɧɨɫɢɬɫɹ
ɤ ɡɚɞɚɱɚɦ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ [8].
ȼ ɨɛɳɟɣ ɡɚɞɚɱɟ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɨɩɪɟɞɟɥɹɸɬ ɡɧɚɱɟ-
ɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɏ
, ɏ2, …, ɏn, ɤɨɬɨɪɵɟ ɨɛɟɫɩɟɱɢɜɚɸɬ ɦɢɧɢɦɭɦ (ɢɥɢ
1
ɦɚɤɫɢɦɭɦ) ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ
,
F
= O1ɏ1 + O2ɏ2 + … + Ojɏj + Onɏn (4.16)
min
ɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɭɫɥɨɜɢɹɦ-ɨɝɪɚɧɢɱɟɧɢɹɦ:
n
ɚɜ
ɝɞɟ O
, ɚij, ɜj – ɢɡɜɟɫɬɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, j = 1,2…n; m t n.
j
t
¦
ij
jj
1
j
j
,ɏ ; ɏ 0
(4.17)
ȼ ɨɛɳɟɣ ɡɚɞɚɱɟ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɭɫɥɨɜɢɹ (4.17) ɩɨɥɧɨɫɬɶɸ ɫɨɜɩɚɞɚɸɬ ɫ ɫɢɫɬɟɦɨɣ ɛɚɥɚɧɫɨɜɵɯ ɭɪɚɜɧɟɧɢɣ, (4.1), (4.6) – (4.8),
ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɶɸ ɡɚɞɚɱɢ ɪɚɫɱɟɬɚ ɲɢɯɬɵ. ȿɫɥɢ
ɫɢɫɬɟɦɚ, ɫɨɫɬɚɜɥɟɧɧɚɹ ɢɡ ɭɫɥɨɜɢɣ (4.1), (4.6) – (4.8), ɞɚɟɬ ɥɢɲɶ ɞɨɩɭɫɬɢɦɵɟ ɪɟɲɟɧɢɹ, ɬɨ ɫɢɫɬɟɦɚ, ɫɨɫɬɚɜɥɟɧɧɚɹ ɢɡ ɭɪɚɜɧɟɧɢɣ (4.16), (4.17), ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɚɫɱɟɬ
ɨɩɬɢɦɚɥɶɧɨɝɨ ɫɨɫɬɚɜɚ ɲɢɯɬɵ ɪɚɜɧɨɫɢɥɟɧ
ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɜ ɤɨɬɨɪɨɣ ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɨɩɬɢɦɢɡɚɰɢɢ.
O
ɉɪɢɞɚɜɚɹ ɩɨɫɬɨɹɧɧɵɦ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ
ɪɚɡɥɢɱɧɵɣ ɫɦɵɫɥ, ɦɨɠ-
j
ɧɨ ɩɨɥɭɱɢɬɶ ɪɟɲɟɧɢɹ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɬɟɦ ɢɥɢ ɢɧɵɦ ɡɚɞɚɱɚɦ. ɇɚɢɛɨɥɟɟ ɱɚɫɬɨ ɰɟɥɟɜɭɸ ɮɭɧɤɰɢɸ ɢɫɩɨɥɶɡɭɸɬ ɞɥɹ ɦɢɧɢɦɢɡɚɰɢɢ ɡɚɬɪɚɬ ɧɚ ɦɚɬɟɪɢɚɥɵ [9; 10].
23
ɉɪɢ ɪɚɫɱɟɬɟ ɲɢɯɬɵ ɞɜɭɫɬɨɪɨɧɧɢɟ ɧɟɪɚɜɟɧɫɬɜɚ ɩɪɟɨɛɪɚɡɭɸɬ ɤ ɞɜɭɦ
j
ɨɞɧɨɫɬɨɪɨɧɧɢɦ ɢ ɩɪɢɜɨɞɹɬ ɤ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɟ ɜɜɟɞɟɧɢɟɦ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɏ
. Ɂɧɚɤ ɩɨɫɥɟɞɧɢɯ ɜɵɛɢɪɚɟɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɤ
ɯɚɪɚɤɬɟɪɚ ɧɟɪɚɜɟɧɫɬɜɚ.
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɧɟɪɚɜɟɧɫɬɜɚ (4.1) ɩɨɥɭɱɚɟɦ:
n
ɗɗ
¦
ij j
ɠɠɤ
1
j
ɜ
;ɏɏ
(4.18)
ɠ
i
n
ɏɏ .
ɗɗ
¦
ij j
ɠɠɤ1 ɠ
1
j
ɧ
i
ɉɪɢ ɷɬɨɦ Ʉ > n, ɝɞɟ n – ɱɢɫɥɨ ɨɫɧɨɜɧɵɯ ɩɟɪɟɦɟɧɧɵɯ.
Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɜɯɨɞɹɬ ɜ ɰɟɥɟɜɭɸ ɮɭɧɤɰɢɸ ɫ ɧɭɥɟɜɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɸɬ ɭɧɢɜɟɪɫɚɥɶɧɵɣ
ɫɢɦɩɥɟɤɫɧɵɣ ɦɟɬɨɞ. ɉɨ ɷɬɨɦɭ ɦɟɬɨɞɭ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɨɩɪɟɞɟɥɹɸɬ ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɦɧɨɝɨɲɚɝɨɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ.
ɋɧɚɱɚɥɚ ɧɚɯɨɞɹɬ ɨɞɧɨ ɢɡ ɪɟɲɟɧɢɣ, ɤɨɬɨɪɨɟ ɩɪɢɧɢɦɚɸɬ ɡɚ ɛɚɡɢɫɧɨɟ. Ɂɚɬɟɦ ɩɪɢ ɩɨɦɨɳɢ
ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɭɫɬɚɧɚɜɥɢɜɚɸɬ, ɩɨɥɭɱɟɧɨ ɥɢ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ. ɉɪɢ ɷɬɨɦ ɩɟɪɟɯɨɞ ɨɬ ɨɞɧɨɝɨ ɛɚɡɢɫɧɨɝɨ
ɪɟɲɟɧɢɹ ɤ ɞɪɭɝɨɦɭ ɫɨɜɟɪɲɚɟɬɫɹ ɧɟ ɩɪɨɢɡɜɨɥɶɧɨ, ɚ ɬɚɤ, ɱɬɨ ɩɪɢ ɤɚɠɞɨɦ
ɢɡ ɧɢɯ ɡɧɚɱɟɧɢɟ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɭɦɟɧɶɲɚɟɬɫɹ. ɗɬɢ ɲɚɝɢ ɩɨɜɬɨɪɹɸɬ ɞɨ
ɬɟɯ ɩɨɪ, ɩɨɤɚ ɧɟ ɛɭɞɟɬ ɧɚɣɞɟɧɨ ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ.
ɑɬɨɛɵ ɭɩɪɨɫɬɢɬɶ ɧɚɯɨɠɞɟɧɢɟ ɩɟɪɜɨɝɨ ɪɟɲɟɧɢɹ ɩɪɢɛɟɝɚɸɬ ɤ ɫɨ
ɡɞɚɧɢɸ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɧɟɢɡɜɟɫɬɧɵɯ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɵ ɪɚɡɦɟɪɧɨɫɬɶɸ m m, ɝɞɟ m ɱɢɫɥɨ ɨɝɪɚɧɢɱɟɧɢɣ (4.17).
ɉɪɟɠɞɟ ɜɫɟɝɨ ɨɩɪɟɞɟɥɹɸɬ, ɫɨɞɟɪɠɢɬ ɥɢ ɦɚɬɪɢɰɚ, ɫɨɫɬɚɜɥɟɧɧɚɹ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɚ
ɭɫɥɨɜɢɣ ɨɝɪɚɧɢɱɟɧɢɣ, ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ (ɨɩɪɟɞɟɥɢɬɟɥɶ
ij
ɤɨɬɨɪɨɣ ɪɚɜɟɧ ɟɞɢɧɢɰɵ), ɬɨ ɟɫɬɶ ɦɨɠɧɨ ɥɢ ɩɟɪɟɫɬɚɧɨɜɤɨɣ ɫɬɪɨɤ ɢ ɫɬɨɥɛɰɨɜ
ɦɚɬɪɢɰɵ ɭɫɥɨɜɢɣ ɩɨɥɭɱɢɬɶ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ ɧɟɨɛɯɨɞɢɦɨɣ ɪɚɡɦɟɪɧɨɫɬɢ.
ȿɫɥɢ ɢɫɯɨɞɧɚɹ ɦɚɬɪɢɰɚ ɧɟ ɫɨɞɟɪɠɢɬ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ, ɬɨ ɫɨɫɬɚɜɥɹɸɬ ɢɫɤɭɫɫɬɜɟɧɧɵɣ ɛɚɡɢɫ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɢɞɚ ɨɝɪɚɧɢɱɟɧɢɣ ɞɥɹ
ɫɨɫɬɚɜɥɟɧɢɹ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɵ ɩɪɢɦɟɧɹɸɬ ɧɟɫɤɨɥɶɤɨ ɩɪɢɟɦɨɜ. ȿɫɥɢ
ɨɝɪɚɧɢɱɟɧɢɹ ɢɦɟɸɬ
ɜɢɞ
n
ɚɜ
¦
ij j
1ɏj
,
-
24
ɬɨ ɩɪɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ ɧɟɪɚɜɟɧɫɬɜ ɜ ɪɚɜɟɧɫɬɜɚ, ɨɫɭɳɟɫɬɜɥɹɟɦɨɦ ɜɜɟɞɟ-
j
ɧɢɟɦ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɏ
, ɨɛɪɚɡɭɟɬɫɹ ɧɨɜɚɹ ɦɚɬɪɢɰɚ, ɫɨ-
ɤ
ɞɟɪɠɚɳɚɹ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ +1 ɩɪɢ ɧɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ. ɉɪɢ ɷɬɨɦ ɩɟɪɜɨɟ ɛɚɡɢɫɧɨɟ ɪɟɲɟɧɢɟ ɩɨɥɭɱɚɟɬɫɹ ɩɪɢ ɏ
ɬɨ ɟɫɬɶ ɩɪɢ ɡɧɚɱɟɧɢɹɯ ɏ
= ɜi.
ɤ
d h = 0,
j
ȼ ɫɥɭɱɚɟ, ɟɫɥɢ ɧɟɤɨɬɨɪɵɟ ɢɥɢ ɜɫɟ ɫɬɪɨɤɢ ɧɟ ɫɨɞɟɪɠɚɬ ɤɨɷɮɮɢɰɢɟɧɬ +1, ɧɟɨɛɯɨɞɢɦɵɣ ɞɥɹ ɨɛɪɚɡɨɜɚɧɢɹ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɵ, ɬɨ ɩɪɢɛɟɝɚɸɬ ɤ ɦɟɬɨɞɭ ɢɫɤɭɫɫɬɜɟɧɧɨɝɨ ɛɚɡɢɫɚ (Ɇ-ɦɟɬɨɞɭ).
ɉɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ ɜɢɞɚ
n
ɚɜ
!
¦
ij j j
1
j
ɜɜɨɞɹɬ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɏ
ɏ
ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɪɚɜɟɧɫɬɜɚ
ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ í1, ɤɨɬɨɪɵɟ
ɤ
ɧɟ ɦɨɝɭɬ ɨɛɪɚɡɨɜɵɜɚɬɶ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ (ɬɚɤ ɠɟ, ɤɚɤ ɢ
n
ɚɜ
ɩɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ ɜɢɞɚ
¦
j
1
!
ij j j
ɛɚɜɥɹɸɬ ɢɫɤɭɫɫɬɜɟɧɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɏ
,ɏ
ɤ ɩɨɥɭɱɟɧɧɵɦ ɪɚɜɟɧɫɬɜɚɦ ɩɪɢ-
(t > Ʉ), ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɟɞɢɧɢɱ-
t
ɧɭɸ ɦɚɬɪɢɰɭ. Ɍɨɝɞɚ ɰɟɥɟɜɚɹ ɮɭɧɤɰɢɹ ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
n
ɏɏɏ,0
O
1
j
¦¦¦
jt
Ɇ
ɤ
ɝɞɟ ɡɧɚɱɟɧɢɟ Ɇ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɦ ɩɨɥɨɠɢɬɟɥɶɧɵɦ
ɱɢɫɥɨɦ (Ɇ t O
).
j
ɉɪɢɦɟɧɟɧɢɟ ɫɢɦɩɥɟɤɫɧɨɝɨ ɦɟɬɨɞɚ ɤ ɪɚɫɲɢɪɟɧɧɨɣ ɡɚɞɚɱɟ (ɫ ɢɫɤɭɫɫɬɜɟɧɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ) ɨɛɟɫɩɟɱɢɜɚɟɬ ɬɚɤɨɟ ɪɟɲɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ ɤɚɠɞɨɟ ɢɡ ɢɫɤɭɫɫɬɜɟɧɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɏ
ɪɚɜɧɨ ɧɭɥɸ. ȼ ɩɪɨɰɟɫɫɟ ɪɟɲɟɧɢɹ
t
ɡɚɞɚɱɢ ɩɟɪɟɦɟɧɧɵɟ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ Ɇ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɜɨɞɹɬɫɹ ɢɡ
ɛɚɡɢɫɚ, ɜ ɨɩɬɢɦɚɥɶɧɨɦ ɪɟɲɟɧɢɢ ɨɧɢ ɧɟ ɫɨɞɟɪɠɚɬɫɹ.
ȿɫɥɢ ɢɦɟɸɬɫɹ ɨɝɪɚɧɢɱɟɧɢɹ ɫɦɟɲɚɧɧɨɝɨ ɬɢɩɚ, ɬɨ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɟɪɜɨɟ ɛɚɡɢɫɧɨɟ ɪɟɲɟɧɢɟ ɩɨɥɭɱɚɸɬ ɩɪɢɟɦɚɦɢ,
ɨɩɢɫɚɧɧɵɦɢ ɜɵɲɟ.
ɉɨɫɥɟ ɩɪɢɜɟɞɟɧɢɹ ɡɚɞɚɱɢ ɤ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɟ ɢ ɩɨɥɭɱɟɧɢɹ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɵ ɫɨɫɬɚɜɥɹɸɬ
ɫɢɦɩɥɟɤɫɧɭɸ ɬɚɛɥɢɰɭ ɢ ɩɪɢɫɬɭɩɚɸɬ ɤ ɪɟ-
ɲɟɧɢɸ.
ɉɪɨɰɟɞɭɪɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɫɢɦɩɥɟɤɫɧɵɦ ɦɟɬɨɞɨɦ ɫɥɨɠɧɚ ɞɥɹ ɪɟɲɟɧɢɹ ɜɪɭɱɧɭɸ, ɬɚɤ ɤɚɤ ɧɟɨɛɯɨɞɢɦɨ
ɜɵɩɨɥɧɢɬɶ ɛɨɥɶɲɨɣ ɨɛɴɟɦ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ ɢ ɥɨɝɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɩɪɢ
25
ɧɚɥɢɱɢɢ ɞɨ 30 ɩɟɪɟɦɟɧɧɵɯ ɢ ɞɨ 20 ɨɝɪɚɧɢɱɟɧɢɣ. ɇɨ ɚɥɝɨɪɢɬɦ ɫɢɦ-
4.3. ǵȘȕȈȌȔȔȕȘșȏ ȘȕȘșȇȉȒȌȔȏȦ ȟȏȜșȢ ȋȒȦ ȖȒȇȉȑȏ ȘșȇȒȏ
ɩɥɟɤɫɧɨɝɨ ɦɟɬɨɞɚ ɭɞɨɛɟɧ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɧɚ ɗȼɆ, ɩɨɫɤɨɥɶɤɭ ɫɨɫɬɨɢɬ,
ɩɨ ɫɭɳɟɫɬɜɭ, ɢɡ ɨɞɧɨɝɨ ɰɢɤɥɚ. ɉɪɢ ɷɬɨɦ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɫɨɫɬɚɜɥɹɬɶ
ɞɥɹ ɷɬɨɝɨ ɩɪɨɝɪɚɦɦɭ, ɬɚɤ ɤɚɤ ɜ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɰɟɧɬɪɚɯ ɢɦɟɸɬɫɹ ɧɚɛɨɪɵ ɫɬɚɧɞɚɪɬɧɵɯ ɩɪɨɝɪɚɦɦ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɫɬɚɧɞɚɪɬɧɵɟ ɩɪɨɝɪɚɦɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɥɢɧɟɣɧɨɝɨ
ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ.
Ɉɫɨɛɟɧɧɨɫɬɶ ɪɚɫɱɟɬɚ ɲɢɯɬɵ ɞɥɹ ɩɥɚɜɤɢ ɫɬɚɥɢ ɫɜɹɡɚɧɚ ɫ ɛɨɥɶɲɨɣ
ɭɧɢɜɟɪɫɚɥɶɧɨɫɬɶɸ ɫɬɚɥɟɩɥɚɜɢɥɶɧɵɯ ɚɝɪɟɝɚɬɨɜ, ɦɧɨɝɨɨɛɪɚɡɢɟɦ ɦɟɬɨɞɨɜ
ɩɥɚɜɤɢ ɢ ɲɢɪɨɤɢɦ ɞɢɚɩɚɡɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜɨ ɜɪɟɦɹ ɩɥɚɜɤɢ. ɉɨɞɚɜɥɹɸɳɟɟ ɛɨɥɶɲɢɧɫɬɜɨ ɷɥɟɦɟɧɬɨɜ, ɜɯɨɞɹɳɢɯ
ɜ ɫɨɫɬɚɜ ɫɬɚɥɢ, ɧɟɨɞɢɧɚɤɨɜɨ ɨɤɢɫɥɹɸɬɫɹ ɩɨ ɩɟɪɢɨɞɚɦ ɩɥɚɜɤɢ. ȼ ɩɟɪɢɨɞɵ
ɩɥɚɜɥɟɧɢɹ ɢ ɨɤɢɫɥɟɧɢɹ ɷɬɢ ɷɥɟɦɟɧɬɵ ɜɵɝɨɪɚɸɬ ɫɢɥɶɧɟɟ, ɱɟɦ ɜ ɩɪɨɰɟɫɫɟ
ɞɨɜɨɞɤɢ
ɟɬ ɜ ɨɤɢɫɥɢɬɟɥɶɧɵɣ ɩɟɪɢɨɞ ɩɪɢ ɜɜɟɞɟɧɢɢ ɜ ɜɚɧɧɭ ɩɟɱɢ ɤɢɫɥɨɪɨɞɚ ɢɥɢ
ɠɟɥɟɡɧɨɣ ɪɭɞɵ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ ɨɬɞɟɥɶɧɵɯ ɩɟɪɢɨɞɨɜ
ɩɥɚɜɤɢ, ɨɫɨɛɟɧɧɨ ɧɚɱɚɥɶɧɵɯ, ɦɨɠɟɬ ɡɧɚɱɢɬɟɥɶɧɨ ɤɨɥɟɛɚɬɶɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɤɚɱɟɫɬɜɚ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ ɢ ɨɬ ɬɟɯɧɨɥɨɝɢɢ ɩɥɚɜɤɢ. Ȼɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ ɢ ɚɛɫɨɥɸɬɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɷɥɟɦɟɧɬɨɜ
ɨɛɪɚɡɨɦ, ɞɚɠɟ ɩɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɜ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɩɟɱɢ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɜɚɪɢɚɧɬɵ, ɢ ɭɝɚɪ ɷɥɟɦɟɧɬɨɜ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ
ɜ ɲɢɪɨɤɢɯ ɩɪɟɞɟɥɚɯ. ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɭɝɚɪɚ ɷɥɟɦɟɧɬɨɜ
ɹɜɥɹɟɬɫɹ ɬɪɭɞɧɨɭɫɬɚɧɨɜɢɦɨɣ ɢ ɩɪɢɛɥɢɠɟɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɩɨɜɟɞɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɜ ɩɪɨɰɟɫɫɟ ɩɥɚɜɤɢ. Ɋɚɫɫɱɢɬɵɜɚɬɶ ɟɟ ɩɨ ɤɨɷɮɮɢɰɢɟɧɬɭ
ɭɫɜɨɟɧɢɹ
ɤɚɤ ɧɟ ɢɡɜɟɫɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɤɨɥɢɱɟɫɬɜɚ ɫɨɫɬɚɜɥɹɸɳɢɯ ɲɢɯɬɵ (ɷɬɨ
ɹɜɥɹɟɬɫɹ ɰɟɥɶɸ ɪɚɫɱɟɬɚ).
ɩɟɪɢɨɞɵ ɩɥɚɜɤɢ, ɢ ɷɥɟɦɟɧɬɵ ɜ ɧɢɯ ɭɫɜɚɢɜɚɸɬɫɹ ɧɟɨɞɢɧɚɤɨɜɨ. ɉɨɷɬɨɦɭ
ɩɪɢ ɪɚɫɱɟɬɟ ɲɢɯɬɵ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɭɝɚɪ ɷɥɟɦɟɧɬɨɜ ɢɡ ɨɬɞɟɥɶɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɲɢɯɬɵ, ɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɯ
ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ ɩɟɪɟɫɱɢɬɵɜɚɟɬɫɹ ɧɚ ɭɫɜɚɢɜɚɟɦɨɟ ɠɢɞɤɨɣ ɫɬɚɥɶɸ
ɫɨɞɟɪɠɚɧɢɟ ɷɥɟɦɟɧɬɨɜ. ɉɨɞɛɨɪɨɦ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ ɫ ɩɪɢɜɟɞɟɧɧɵɦ
ɢ ɪɚɫɤɢɫɥɟɧɢɹ. ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɭɝɚɪɚ ɷɥɟɦɟɧɬɨɜ ɪɟɡɤɨ ɜɨɡɪɚɫɬɚ-
ɜ ɫɬɚɥɢ. Ɍɚɤɢɦ
ɷɥɟɦɟɧɬɨɜ ɢɡ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɫɬɪɨɝɨ ɝɨɜɨɪɹ, ɧɟɥɶɡɹ, ɬɚɤ
ɉɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɲɢɯɬɨɜɵɟ ɦɚɬɟɪɢɚɥɵ ɡɚɝɪɭɠɚɸɬɫɹ ɜ ɪɚɡɥɢɱɧɵɟ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ. ɋɨɫɬɚɜ
26
ɫɨɞɟɪɠɚɧɢɟɦ ɷɥɟɦɟɧɬɨɜ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɡɚɞɚɧɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɷɥɟɦɟɧɬɨɜ
ɜ ɠɢɞɤɨɣ ɫɬɚɥɢ.
ɉɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ, ɤɚɤ ɩɪɚɜɢɥɨ, ɜ ɡɚɜɚɥɤɭ ɞɚɸɬ ɥɢɲɶ ɜɬɨɪɢɱɧɵɟ
ɦɟɬɚɥɥɵ ɢ ɤɚɪɛɸɪɢɡɚɬɨɪɵ (ɢɧɨɝɞɚ ɦɟɞɶ, ɧɢɤɟɥɶ).
ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɮɟɪɪɨɫɩɥɚɜɵ ɢ ɥɟɝɢɪɭɸɳɢɟ ɩɪɢɫɚɞɤɢ ɜɜɨɞɹɬ ɜ ɠɢɞɤɢɣ ɦɟɬɚɥɥ ɜ ɤɨɧɰɟ ɩɥɚɜɤɢ. ɉɨɷɬɨɦɭ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦ
ɜɫɟ ɲɢɯɬɨɜɵɟ ɦɚɬɟɪɢɚɥɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɩɥɚɜɤɢ ɫɬɚɥɢ, ɭɫɥɨɜɧɨ ɪɚɡ
ɞɟɥɢɬɶ ɧɚ ɞɜɟ ɝɪɭɩɩɵ.
ɉɟɪɜɚɹ ɝɪɭɩɩɚ – ɨɬɯɨɞɵ, ɤ ɤɨɬɨɪɵɦ ɨɬɧɨɫɹɬɫɹ ɜɫɟ ɦɚɬɟɪɢɚɥɵ,
ɜɜɨɞɢɦɵɟ ɜ ɡɚɜɚɥɤɭ. ɗɬɨ ɨɛɵɱɧɨ ɜɨɡɜɪɚɬ ɫɨɛɫɬɜɟɧɧɨɝɨ ɩɪɨɢɡɜɨɞɫɬɜɚ,
ɥɨɦ ɭɝɥɟɪɨɞɢɫɬɨɣ ɢ ɥɟɝɢɪɨɜɚɧɧɨɣ ɫɬɚɥɟɣ, ɤɚɪɛɸɪɢɡɚɬɨɪɵ. Ɉɧɢ ɩɨɞɜɟɪɝɚɸɬɫɹ ɛɨɥɶɲɨɦɭ ɨɤɢɫɥɟɧɢɸ, ɢ ɩɨɷɬɨɦɭ ɷɥɟɦɟɧɬɵ ɢɡ ɨɬɯɨɞɨɜ ɭɫɜɚɢɜɚɸɬɫɹ ɜ ɦɟɧɶɲɟɣ ɫɬɟɩɟɧɢ.
ȼɬɨɪɚɹ ɝɪɭɩɩɚ – ɮɟɪɪɨɫɩɥɚɜɵ, ɤ ɤɨɬɨɪɵɦ ɨɬɧɨɫɹɬɫɹ ɫɨɛɫɬɜɟɧɧɨ
ɮɟɪɪɨɫɩɥɚɜɵ ɢ
ɞɪɭɝɢɟ ɩɪɢɫɚɞɤɢ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɤɨɪɪɟɤɬɢɪɨɜɤɢ ɯɢɦɢɱɟɫɤɨɝɨ ɫɨɫɬɚɜɚ, ɥɟɝɢɪɨɜɚɧɢɹ, ɪɚɫɤɢɫɥɟɧɢɹ. ɂɯ ɜɜɨɞɹɬ ɜ ɩɟɱɶ ɜ ɤɨɧɰɟ
ɩɥɚɜɤɢ, ɢ ɨɧɢ ɭɫɜɚɢɜɚɸɬɫɹ ɡɧɚɱɢɬɟɥɶɧɨ ɥɭɱɲɟ.
ɉɪɚɤɬɢɤɨɣ ɭɫɬɚɧɨɜɥɟɧɵ ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɡ
ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɢɯ ɚɛɫɨɥɸɬɧɨɝɨ ɫɨɞɟɪɠɚɧɢɹ
ɜ ɫɬɚɥɢ, ɬɢɩɚ ɩɥɚɜɢɥɶɧɨɝɨ ɚɝɪɟɝɚɬɚ ɢ ɬɟɯɧɨɥɨɝɢɢ ɩɥɚɜɤɢ (ɬɚɛɥ. 4.1, 4.2).
Ɍɚɛɥɢɰɚ 4.1
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɡ
ɨɬɯɨɞɨɜ (ɥɨɦɚ)
ɢ ɮɟɪɪɨɫɩɥɚɜɨɜ ɩɪɢ ɜɵɩɥɚɜɤɟ ɫɬɚɥɢ ɜ ɷɥɟɤɬɪɨɞɭɝɨɜɵɯ ɩɟɱɚɯ
Ʌɟɝɢɪɭ-
ɸɳɢɣ
ɷɥɟɦɟɧɬ
1 2 3 4 5 6 7
ɋ – */0,9 – */0,9 – 1,00
Si – 0/0,6 – 0/0,6 3 0,90
Mn 5 0,3/0,8*
S – 0,9 – 0,9 – 1,00
Ɋ – 0,3-0,5 – 0,3-0,5 – 0,80
ɋr 3 0,8-0,85 3 0,8-0,85 3 0,95
Ni 10 0,97 10 0,95 – 0,97
ɋɨɞɟɪ-
ɠɚɧɢɟ ɧɟ
ɛɨɥɟɟ,ௗ%
Ɉɬɯɨɞɵ (ɥɨɦ) Ɏɟɪɪɨɫɩɥɚɜɵ
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
ɋɨɞɟɪ-
ɠɚɧɢɟ
ɛɨɥɟɟ,ௗ%
5 0,7/0,9 5 0,95
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
ɋɨɞɟɪ-
ɠɚɧɢɟ
ɛɨɥɟɟ,ௗ%
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
-
27
Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ 4.1
Ʌɟɝɢɪɭ-
ɸɳɢɣ
ɷɥɟɦɟɧɬ
1 2 3 4 5 6 7
Cu – 0,95 – 0,95 – 0,97
Al – 0 – 0 – 0,75
Ti – 0/0,10 – 0/0,10 1 0,50
W 3 0,90 3 0,90 – 0,95
V 0,7 0,20/0,50 0,7 0,5/0,85 – 0,90
Mo – 0,95 – 0,95 – 0,97
Co – 0,95 – 0,95 – 0,97
* ȼ ɱɢɫɥɢɬɟɥɟ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɜɨɟɧɢɹ ɥɟɝɢɪɭɸɳɢɯ ɷɥɟɦɟɧɬɨɜ ɩɪɢ ɜɵɩɥɚɜɤɟ
ɫɬɚɥɢ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɤɢɫɥɨɪɨɞɚ (ɫ ɩɨɥɧɵɦ ɨɤɢɫɥɟɧɢɟɦ), ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɛɟɡ
ɨɤɢɫɥɟɧɢɹ (ɩɟɪɟɩɥɚɜɨɦ)
ɋɨɞɟɪ-
ɠɚɧɢɟ ɧɟ
ɛɨɥɟɟ,ௗ%
Ɉɬɯɨɞɵ (ɥɨɦ) Ɏɟɪɪɨɫɩɥɚɜɵ
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
ɋɨɞɟɪ-
ɠɚɧɢɟ
ɛɨɥɟɟ,ௗ%
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
ɋɨɞɟɪ-
ɠɚɧɢɟ
ɛɨɥɟɟ,ௗ%
Ʉɨɷɮɮɢ-
ɰɢɟɧɬ
ɭɫɜɨɟɧɢɹ
Ɍɚɛɥɢɰɚ 4.2
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɡ ɨɬɯɨɞɨɜ ɢ ɮɟɪɪɨɫɩɥɚɜɨɜ
ɩɪɢ ɜɵɩɥɚɜɤɟ ɫɬɚɥɢ ɜ ɢɧɞɭɤɰɢɨɧɧɵɯ ɩɟɱɚɯ
Ʌɟɝɢɪɭ-
ɸɳɢɣ
ɷɥɟɦɟɧɬ
ɋ 0,90 1,00 Al 0 0,75
Si 0,80 0,95 Ti 0,50 0,90
Mn 0,80 0,90 W 0,95 0,95
S 1,00 1,00 V 0,85 0,95
Ɋ 1,00 1,00 Mo 0,97 0,99
Cr 0,85 0,95 Nb 0,95 0,95
Ni 0,97 0,99 Zr 0 0,50
Cu 0,97 0,99 Co 0,95 0,95
Ʉɨɷɮɮɢɰɢɟɧɬ ɭɫɜɨɟɧɢɹ
ɢɡ ɨɬɯɨɞɨɜ,
ɥɨɦɚ
ɢɡ ɮɟɪɪɨ-
ɫɩɥɚɜɨɜ
Ʌɟɝɢɪɭ-
ɸɳɢɣ
ɷɥɟɦɟɧɬ
Ʉɨɷɮɮɢɰɢɟɧɬ ɭɫɜɨɟɧɢɹ
ɢɡ ɨɬɯɨɞɨɜ,
ɥɨɦɚ
ɢɡ ɮɟɪɪɨ-
ɫɩɥɚɜɨɜ
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɬɚɛɥ. 4.1 ɢ 4.2 ɧɟɨɛɯɨɞɢɦɨ ɢɦɟɬɶ ɜ ɜɢɞɭ ɧɟɤɨɬɨɪɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɪɚɫɱɟɬɚ ɲɢɯɬɵ ɩɨ ɫɨɞɟɪɠɚɧɢɸ ɨɬɞɟɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ
ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɮɭɬɟɪɨɜɤɢ ɢ ɜɢɞɚ ɩɥɚɜɢɥɶɧɨɝɨ ɚɝɪɟɝɚɬɚ.
28
ȼ ɬɚɛɥ. 4.1 ɭɤɚɡɚɧɵ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɭɝɥɟɪɨɞɚ ɩɪɢ ɜɵɩɥɚɜɤɟ ɜ ɷɥɟɤɬɪɨɞɭɝɨɜɵɯ ɩɟɱɚɯ. ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɭɝɚɪɨɦ ɭɝɥɟɪɨɞɚ ɜ ɩɟɪɢɨɞ
ɤɢɩɟɧɢɹ ɢ ɧɚɭɝɥɟɪɨɠɢɜɚɧɢɟɦ ɫɬɚɥɢ ɨɬ ɷɥɟɤɬɪɨɞɨɜ ɜɨ ɜɪɟɦɹ ɩɥɚɜɤɢ.
ɉɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɫ ɨɤɢɫɥɟɧɢɟɦ ɜ ɷɥɟɤɬɪɨɞɭɝɨɜɨɣ ɩɟɱɢ ɫ ɨɫɧɨɜɧɨɣ
ɮɭɬɟɪɨɜɤɨɣ ɜ ɩɪɨɰɟɫɫɟ ɤɢɩɟɧɢɹ ɨɤɢɫɥɹɸɬ ɞɨ 0,25…0,35ௗ% ɭɝɥɟɪɨɞɚ [4; 5].
Ɍɨɝɞɚ ɪɚɫɱɟɬɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ
ɜ ɠɢɞɤɨɣ ɫɬɚɥɢ ɩɨɫɥɟ ɪɚɫ-
ɩɥɚɜɥɟɧɢɹ, ɛɭɞɟɬ:
ɋ
= ɋɠ + (0,25 í 0,35).
ɫɬ
ɉɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɫ ɨɤɢɫɥɟɧɢɟɦ ɜ ɤɢɫɥɨɣ ɷɥɟɤɬɪɨɞɭɝɨɜɨɣ ɩɟɱɢ
ɜ ɩɪɨɰɟɫɫɟ ɤɢɩɟɧɢɹ ɨɤɢɫɥɹɟɬɫɹ ɞɨ 0,15…0,20ௗ% ɭɝɥɟɪɨɞɚ. ɉɨɷɬɨɦɭ ɪɚɫɱɟɬɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɠɢɞɤɨɣ ɫɬɚɥɢ ɩɪɢ ɩɥɚɜɤɟ ɜ ɤɢɫɥɨɣ ɷɥɟɤɬɪɨɞɭɝɨɜɨɣ ɩɟɱɢ ɛɭɞɟɬ:
ɋ
= ɋɠ + (0,10 í 0,20).
ɫɬ
ɉɪɢ ɩɥɚɜɤɟ ɛɟɡ ɨɤɢɫɥɟɧɢɹ ɜɫɥɟɞɫɬɜɢɟ ɧɚɭɝɥɟɪɨɠɢɜɚɧɢɹ ɫɬɚɥɢ ɨɬ
ɷɥɟɤɬɪɨɞɨɜ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɲɢɯɬɟ ɞɨɥɠɧɨ ɛɵɬɶ ɧɢɠɟ, ɱɟɦ ɜ ɝɨɬɨɜɨɣ ɫɬɚɥɢ [4; 5]:
ɧ
ɋ
= ɋ
í (0,1 í 0,2),
ɠ
ɝɞɟ ɋ
ɲ
ɧ
– ɧɢɠɧɟɟ ɞɨɩɭɫɬɢɦɨɟ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɝɨɬɨɜɨɣ ɫɬɚɥɢ,ௗ%.
ɠ
ɉɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɜ ɢɧɞɭɤɰɢɨɧɧɨɣ ɩɟɱɢ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ
ɜ ɲɢɯɬɟ ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ
ɧ
ɋ
= ɋ
ɲ
+ (0,05 í 0,1).
ɠ
Ʉɪɟɦɧɢɣ ɩɪɢ ɩɥɚɜɤɟ ɜ ɨɫɧɨɜɧɵɯ ɷɥɟɤɬɪɨɞɭɝɨɜɵɯ ɩɟɱɚɯ ɢɡ ɨɬɯɨɞɨɜ
ɜɵɝɨɪɚɟɬ ɩɨɥɧɨɫɬɶɸ, ɩɪɢ ɩɥɚɜɤɟ ɩɟɪɟɩɥɚɜɨɦ ɭɫɜɚɢɜɚɟɬɫɹ ɧɚ 60ௗ%.
ɉɪɢ ɩɥɚɜɤɟ ɜ ɤɢɫɥɵɯ ɩɟɱɚɯ ɫɨɫɬɚɜ ɫɬɚɥɢ ɩɨ ɤɪɟɦɧɢɸ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɧɟ ɫɨɞɟɪɠɚɧɢɟɦ ɟɝɨ ɜ ɲɢɯɬɟ, ɚ ɬɟɯɧɨɥɨɝɢɟɣ ɩɥɚɜɤɢ (ɤɪɟɦɧɟɜɨɫɫɬɚɧɨɜɢɬɟɥɶɧɵɦ ɚɤɬɢɜɧɵɦ ɢɥɢ ɩɚɫɫɢɜɧɵɦ ɩɪɨɰɟɫɫɨɦ). ɋɨɞɟɪɠɚɧɢɟ ɤɪɟɦɧɢɹ ɪɟɝɭɥɢɪɭɸɬ ɢɡɦɟɧɟɧɢɟɦ ɪɟɠɢɦɚ ɩɥɚɜɤɢ, ɢ
ɪɚɫɱɟɬ ɜɟɞɭɬ ɞɥɹ ɤɨɧɤɪɟɬɧɨɣ
ɬɟɯɧɨɥɨɝɢɢ.
Ɏɨɫɮɨɪ ɢ ɫɟɪɚ ɩɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɜ ɨɫɧɨɜɧɵɯ ɷɥɟɤɬɪɨɞɭɝɨɜɵɯ ɩɟɱɚɯ
ɦɨɝɭɬ ɛɵɬɶ ɭɞɚɥɟɧɵ ɧɚ 50…70ௗ% ɨɬ ɫɨɞɟɪɠɚɧɢɹ ɢɯ ɜ ɲɢɯɬɟ. ȼ ɤɢɫɥɵɯ
ɷɥɟɤɬɪɨɞɭɝɨɜɵɯ ɩɟɱɚɯ ɫɨɞɟɪɠɚɧɢɟ ɫɟɪɵ ɢ ɮɨɫɮɨɪɚ ɧɟ ɦɟɧɹɟɬɫɹ.
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5. ǶǷǯdzǬǷȂ ǷǧǸǾǬǹǧ ǿǯǼǹȂ ǫDzȆ ǶDzǧǩDZǯ ǸǹǧDzǯ
5.1. ǷȇȘȞȌș ȟȏȜșȢ ȋȒȦ ȖȒȇȉȑȏ ȘșȇȒȏ ȓȇȗȑȏ 110Ǫ13Dz
ȖȌȗȌȖȒȇȉ
ȕȓ, ȖȗȏȈȒȏȍȌȔȔȢȓ ȘȖȕȘȕȈȕȓ
1. ɂɫɯɨɞɹ ɢɡ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɬɪɟɛɨɜɚɧɢɣ ɤ ɨɬɥɢɜɤɟ, ɜɵɛɢɪɚɟɦ
ɫɬɚɥɶ ɦɚɪɤɢ 110Ƚ13Ʌ. Cɬɚɥɶ 110Ƚ13Ʌ ɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɯɢɦɢɱɟɫɤɢɣ ɫɨɫɬɚɜ [3] ɜ ɩɪɨɰɟɧɬɚɯ:
ɋ Si Mn S P Cr Ni
0,9…1,3 0,4…1,0 11,5…14,5 ɞɨ 0,05 ɞɨ 0,1 0,5 0,5
2. ȼɵɛɢɪɚɟɦ ɦɟɬɨɞ ɩɥɚɜɤɢ, ɬɢɩ ɩɥɚɜɢɥɶɧɨɝɨ ɚɝɪɟɝɚɬɚ /3/, ɨɩɪɟɞɟɥɹɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ (ɬɚɛɥ. 5.1). ɉɥɚɜɤɭ ɫɬɚɥɢ
110Ƚ13Ʌ ɫɥɟɞɭɟɬ ɜɟɫɬɢ ɜ ɷɥɟɤɬɪɨɞɭɝɨɜɨɣ ɩɟɱɢ ɫ ɨɫɧɨɜɧɨɣ ɮɭɬɟɪɨɜɤɨɣ.
ɋɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɲɢɯɬɟ ɩɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ ɩɟɪɟɩɥɚɜɨɦ ɢɡ-ɡɚ ɧɚɭɝɥɟɪɨɠɢɜɚɧɢɹ ɨɬ ɷɥɟɤɬɪɨɞɨɜ ɞɨɥɠɧɨ ɛɵɬɶ ɧɢɠɟ, ɱɟɦ ɜ ɝɨɬɨɜɨɣ ɫɬɚɥɢ ɧɚ
0,1…0,2ௗ% (ɫɦ. ɪɚɡɞɟɥ 4.3).
ɋɪɟɞɧɟɟ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɲɢɯɬɟ ɪɚɜɧɨ:
ɋ
= ((0,9 + 1,3) / 2) í (0,1 í 0,2) = 1,1 í 0,2 = 0,9ௗ%.
ɲ
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɡ ɲɢɯɬɵ ɩɪɢ ɩɥɚɜɤɟ ɫɬɚɥɢ
110Ƚ13Ʌ ɩɟɪɟɩɥɚɜɨɦ ɞɚɧɵ ɜ ɬɚɛɥ. 5.1.
Ɍɚɛɥɢɰɚ 5.1
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɭɫɜɨɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɢɡ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ
Ƚɪɭɩɩɚ
ɦɚɬɟɪɢɚɥɚ
Ɉɬɯɨɞɵ, ɥɨɦ 0,9 0,6 0,80 0,5 0,5 0,85 0,97
Ɏɟɪɪɨɫɩɥɚɜɵ 1,0 0,9 0,95 1,0 0,8 0,95 0,97
Ʉ
Ʉ
ɋ
Ʉɨɷɮɮɢɰɢɟɧɬ ɭɫɜɨɟɧɢɹ
Ʉ
Si
Mn
Ʉ
Ʉ
S
Ʉ
P
Ʉ
Cr
Ni
ȼɵɛɢɪɚɟɦ ɦɚɪɤɢ ɢ ɫɨɫɬɚɜɵ ɢɫɯɨɞɧɵɯ ɲɢɯɬɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ [3]
ɢ ɫɜɨɞɢɦ ɢɯ ɜ ɬɚɛɥ. 5.2.
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