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4. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ
ɋɢɧɬɟɡɚɬɨɪ ɱɚɫɬɨɬ — ɷɬɨ ɪɚɞɢɨɬɟɯɧɢɱɟɫɤɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɝɟɧɟɪɢɪɭɸɳɟɟ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ ɜɵɫɨɤɨɣ ɫɬɚɛɢɥɶɧɨɫɬɢ ɜ ɡɚɞɚɧɧɨɦ ɞɢɚɩɚɡɨɧɟ
ɱɚɫɬɨɬ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɲɚɝɨɦ ɞɢɫɤɪɟɬɧɨɣ ɩɟɪɟɫɬɪɨɣɤɢ ο݂Ǥ ɋɢɧɬɟɡɚɬɨɪɵ
ɱɚɫɬɨɬ ɮɨɪɦɢɪɭɸɬ ɧɟɫɭɳɭɸ ɱɚɫɬɨɬɭ ɤɨɥɟɛɚɧɢɣ ɪɚɞɢɨɫɢɫɬɟɦɵ ɜ ɩɟɪɟɞɚɬɱɢɤɚɯ
ɢ ɹɜɥɹɸɬɫɹ ɝɟɬɟɪɨɞɢɧɚɦɢ ɜ ɩɪɟɨɛɪɚɡɨɜɚɬɟɥɹɯ ɱɚɫɬɨɬɵ. ɉɪɢɧɰɢɩɵ ɪɚɛɨɬɵ
ɫɢɧɬɟɡɚɬɨɪɨɜ ɱɚɫɬɨɬ ɪɚɡɥɢɱɧɵ, ɧɨ ɜ ɞɢɚɩɚɡɨɧɟ ɋȼɑ ɩɪɢɦɟɧɹɸɬɫɹ, ɜ ɨɫɧɨɜɧɨɦ,
ɫɢɧɬɟɡɚɬɨɪɵ ɞɜɭɯ ɬɢɩɨɜ:
- ɫɢɧɬɟɡɚɬɨɪɵ ɧɚ ɨɫɧɨɜɟ ɭɩɪɚɜɥɹɟɦɵɯ ɩɨ ɱɚɫɬɨɬɟ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ,
ɨɯɜɚɱɟɧɧɵɯ ɰɟɩɶɸ ɮɚɡɨɜɨɣ ɚɜɬɨɩɨɞɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ (ɎȺɉɑ),
- ɰɢɮɪɨɜɵɟ ɫɢɧɬɟɡɚɬɨɪɵ, ɜ ɤɨɬɨɪɵɯ ɤɨɥɟɛɚɧɢɹ ɮɨɪɦɢɪɭɸɬɫɹ ɜ ɰɢɮɪɨɜɨɦ
ɜɢɞɟ ɢ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɚɧɚɥɨɝɨɜɭɸ ɮɨɪɦɭ ɫ ɩɨɦɨɳɶɸ ɰɢɮɪɨ-ɚɧɚɥɨɝɨɜɵɯ
ɩɪɟɨɛɪɚɡɨɜɚɬɟɥɟɣ (ɐȺɉ).
Ɉɫɧɨɜɧɵɟ ɩɚɪɚɦɟɬɪɵ ɫɢɧɬɟɡɚɬɨɪɨɜ ɱɚɫɬɨɬ.
1) Ⱦɢɚɩɚɡɨɧ ɝɟɧɟɪɢɪɭɟɦɵɯ
2) ɒɚɝ ɫɟɬɤɢ ɱɚɫɬɨɬ ο݂Ǥ
3) Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɧɟɫɬɚɛɢɥɶɧɨɫɬɶ ɝɟɧɟɪɢɪɭɟɦɨɣ ɱɚɫɬɨɬɵ
4) ɍɪɨɜɟɧɶ ɮɚɡɨɜɨɝɨ ɲɭɦɚ ɩɪɢ ɡɚɞɚɧɧɵɯ ɨɬɫɬɪɨɣɤɚɯ ɨɬ ɱɚɫɬɨɬɵ
ɝɟɧɟɪɚɰɢɢ.
5) Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɦɨɳɧɨɫɬɶ ɩɨɛɨɱɧɵɯ ɤɨɥɟɛɚɧɢɣ.
6) ȼɪɟɦɹ ɩɟɪɟɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ.
Ⱦɥɹ ɫɨɡɞɚɧɢɹ ɧɚɢɛɨɥɟɟ ɜɵɫɨɤɢɯ ɱɚɫɬɨɬ ɩɪɢɦɟɧɹɸɬɫɹ ɫɢɧɬɟɡɚɬɨɪɵ ɧɚ ɛɚɡɟ
ɝɟɧɟɪɚɬɨɪɨɜ, ɭɩɪɚɜɥɹɟɦɵɯ ɧɚɩɪɹɠɟɧɢɟɦ (Ƚɍɇ), ɜɤɥɸɱɟɧɧɵɯ ɜ ɰɟɩɶ ɎȺɉɑ.
ɱɚɫɬɨɬ ݂
…݂௫
Ǥ
ߜ݂
ൗ .
݂
4.1. ɋɢɫɬɟɦɚ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ.
Ɏɭɧɤɰɢɨɧɚɥɶɧɚɹ ɫɯɟɦɚ ɫɢɫɬɟɦɵ ɮɚɡɨɜɨɣ ɚɜɬɨɦɚɬɢɱɟɫɤɨɣ ɩɨɞɫɬɪɨɣɤɢ
ɱɚɫɬɨɬɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ (ɎȺɉɑ ȺȽ) ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 4.1.
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Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
Ɋɢɫ. 4.1. ɋɯɟɦɚ ɮɚɡɨɜɨɣ ɚɜɬɨɦɚɬɢɱɟɫɤɨɣ ɩɨɞɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ.
Ɉɫɧɨɜɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɫɢɫɬɟɦɵ ɎȺɉɑ ɹɜɥɹɸɬɫɹ: ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɣ
ɝɟɧɟɪɚɬɨɪ, ɭɩɪɚɜɥɹɟɦɵɣ ɧɚɩɪɹɠɟɧɢɟɦ, ɧɢɡɤɨɱɚɫɬɨɬɧɵɣ ɷɬɚɥɨɧɧɵɣ
ɚɜɬɨɝɟɧɟɪɚɬɨɪ (ɤɚɤ ɩɪɚɜɢɥɨ, ɫɬɚɛɢɥɢɡɢɪɨɜɚɧɧɵɣ ɤɜɚɪɰɟɜɵɦ ɪɟɡɨɧɚɬɨɪɨɦ),
ɮɚɡɨɜɵɣ ɞɟɬɟɤɬɨɪ, ɞɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ ɢ ɮɢɥɶɬɪ ɧɢɠɧɢɯ ɱɚɫɬɨɬ. ɇɚɡɧɚɱɟɧɢɟ
Ƚɍɇɚ — ɝɟɧɟɪɚɰɢɹ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɯ ɤɨɥɟɛɚɧɢɣ ɬɪɟɛɭɟɦɨɣ ɱɚɫɬɨɬɵ ݂
ˆ˖ː
ɧɚɩɪɹɠɟɧɢɟɦ
(t) = ܷ
ݑ
ˆ˖ː
ȼɵɫɨɤɨɫɬɚɛɢɥɶɧɵɣ ɷɬɚɥɨɧɧɵɣ ɚɜɬɨɝɟɧɟɪɚɬɨɪ ɱɚɫɬɨɬɨɣ ݂
ˆ˖ː
ܿݏ߱
ˆ˖ː
t.
ɩɪɟɞɧɚɡɧɚɱɟɧ ɞɥɹ
ˠ
ɫɬɚɛɢɥɢɡɚɰɢɢ ɱɚɫɬɨɬɵ Ƚɍɇɚ, ɟɝɨ ɧɚɩɪɹɠɟɧɢɟ
ݑ
(t) = ܷˠ߱
ˠ
t
ˠ
Ⱦɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ ɧɚ N ɭɦɟɧɶɲɚɟɬ ɱɚɫɬɨɬɭ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ Ƚɍɇɚ ɞɨ
ɜɟɥɢɱɢɧɵ, ɪɚɜɧɨɣ ɱɚɫɬɨɬɟ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ,
ɩɪɢ ɷɬɨɦ,
߱
ˆ˖ː
ൌܰ߱
, (4.1)
ˠ
ɝɞɟ N — ɰɟɥɨɟ ɢɥɢ ɞɪɨɛɧɨɟ ɱɢɫɥɨ, ɛɨɥɶɲɟɟ 1.
ɇɚɩɪɹɠɟɧɢɟ ɧɚ ɜɵɯɨɞɟ ɞɟɥɢɬɟɥɹ
ݑ
ɝɞɟ ߱
(t) = ܷˑ˔ܿݏ߱
ˑ˔
߱
ˆ˖ː
=
ˑ˔
ൗ
— ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ.
ܰ
t,
ˑ˔
Ɏɚɡɨɜɵɣ ɞɟɬɟɤɬɨɪ (ɎȾ) ɧɭɠɟɧ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɫɢɝɧɚɥɚ ɨɲɢɛɤɢ ɜ ɫɥɭɱɚɟ,
ɤɨɝɞɚ ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɪɟɦɟɧɢ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ
ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ߔ
ሺݐሻ
ߔ
ˑ˔
t ɧɟ ɫɨɜɩɚɞɚɸɬ. Ɏɢɥɶɬɪ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɩɪɟɞɧɚɡɧɚɱɟɧ ɞɥɹ
= ߱
ˑ˔
ሺݐሻൌ߱
ˠ
t ɢ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ
ˠ
ɩɪɨɩɭɫɤɚɧɢɹ ɤɨɥɟɛɚɧɢɣ ɫɢɝɧɚɥɚ ɨɲɢɛɤɢ ɢ ɭɫɬɪɚɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɨɬɵ
ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ ɢ ɟɝɨ ɝɚɪɦɨɧɢɤ.
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4. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ
ɋɢɫɬɟɦɚ ɎȺɉɑ ȺȽ ɪɚɛɨɬɚɟɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɉɪɢ ɪɚɜɟɧɫɬɜɟ
ɮɚɡߔ
ሺݐሻɢ ߔ
ˠ
ሺݐሻ
ɪɚɜɧɵ ɱɚɫɬɨɬɵ ߱
ˑ˔
ɢ ߱
ˠ
, ɧɚ ɜɵɯɨɞɟ ɮɚɡɨɜɨɝɨ ɞɟɬɟɤɬɨɪɚ
ˑ˔
ɨɬɫɭɬɫɬɜɭɟɬ ɫɢɝɧɚɥ ɨɲɢɛɤɢ ɢ ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ (4.1) . ɗɬɨ, ɬɚɤ
߱
ˆ˖ː
ɧɚɡɵɜɚɟɦɵɣ, ɫɢɧɯɪɨɧɧɵɣ ɪɟɠɢɦ ɪɚɛɨɬɵ. ȿɫɥɢ ɠɟ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ
߱
, ɩɨɞɜɨɞɢɦɵɟ ɤ ɜɯɨɞɚɦ ɮɚɡɨɜɨɝɨ ɞɟɬɟɤɬɨɪɚ, ɧɟ ɪɚɜɧɵ, ɬɨ ɧɚ ɜɵɯɨɞɟ ɎȾ
ˠ
ൗ
ɢ
ܰ
ɩɨɹɜɥɹɟɬɫɹ ɧɚɩɪɹɠɟɧɢɟ ɨɲɢɛɤɢ, ɤɨɬɨɪɨɟ ɩɪɨɣɞɹ ɱɟɪɟɡ Ɏɇɑ, ɩɨɫɬɭɩɚɟɬ ɧɚ
ɜɚɪɢɤɚɩ Ƚɍɇɚ, ɢɡɦɟɧɹɹ ߱
ɞɨ ɜɵɩɨɥɧɟɧɢɹ ɪɚɜɟɧɫɬɜɚ (4.1). ȼ ɫɢɧɯɪɨɧɧɨɦ
ˆ˖ː
ɪɟɠɢɦɟ ɪɚɛɨɬɵ ɧɚ ɜɵɯɨɞɟ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɢɦɟɸɬɫɹ ɤɨɥɟɛɚɧɢɹ, ɱɚɫɬɨɬɚ
ɤɨɬɨɪɵɯ ɪɚɜɧɚ ɱɚɫɬɨɬɟ Ƚɍɇɚ, ɚ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɬɚɛɢɥɶɧɨɫɬɶ ɱɚɫɬɨɬɵ —
ɫɬɚɛɢɥɶɧɨɫɬɢ ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɝɟɧɟɪɢɪɭɸɬɫɹ ɫɬɚɛɢɥɶɧɵɟ
ɤɨɥɟɛɚɧɢɹ ɜɵɫɨɤɢɯ ɱɚɫɬɨɬ, ɞɥɹ ɤɨɬɨɪɵɯ ɨɬɫɭɬɫɬɜɭɸɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɞɨɪɨɝɢɟ
ɜɵɫɨɤɨɞɨɛɪɨɬɧɵɟ ɪɟɡɨɧɚɬɨɪɵ.
4.2. ɋɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɫɢɫɬɟɦɵ ɎȺɉɑ
ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
ɋɢɫɬɟɦɚ ɎȺɉɑ — ɷɬɨ ɫɥɟɞɹɳɚɹ ɫɢɫɬɟɦɚ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ,
ɡɚɞɚɱɚ ɤɨɬɨɪɨɣ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɤɨɥɟɛɚɧɢɹ ɦɝɧɨɜɟɧɧɨɣ ɮɚɡɵ ɧɚɩɪɹɠɟɧɢɹ
ሺݐሻ
ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ Ƚɍɇɚ ߔ
ɦɝɧɨɜɟɧɧɨɣ ɮɚɡɵ ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ߔ
ɜɯɨɞɧɵɯ ɢ ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɢɫɬɟɦɵ ɎȺɉɑ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ
ɫɢɦɜɨɥɢɱɟɫɤɨɦ ɜɢɞɟ
ߔ
ɝɞɟ ܭ
(p) — ɨɩɟɪɚɬɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ ɎȺɉɑ, ൌ
˗˃˒˚
ˆ˖ː
ሺݐሻ
ɨɩɟɪɚɬɨɪ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ.
ɋ ɰɟɥɶɸ ɩɨɥɭɱɟɧɢɹ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ܭ
ɜɯɨɞɧɵɦɢ ɢ ɜɵɯɨɞɧɵɦɢ ɮɢɡɢɱɟɫɤɢɦɢ ɜɟɥɢɱɢɧɚɦɢ ɤɚɠɞɨɝɨ ɷɥɟɦɟɧɬɚ ɫɢɫɬɟɦɵ
ɎȺɉɑ ɢ ɫɨɫɬɚɜɢɬɶ ɟɟ ɫɬɪɭɤɬɭɪɧɭɸ ɫɯɟɦɭ.
ɨɬɫɥɟɠɢɜɚɥɢ ɢɡɦɟɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ
ˆ˖ː
(t). ɋɜɹɡɶ ɮɚɡ
ˠ
(p)ߔ
= ܭ
˗˃˒˚
˗˃˒˚
(t), (4.2)
ˠ
(p) ɧɭɠɧɨ ɭɫɬɚɧɨɜɢɬɶ ɫɜɹɡɶ ɦɟɠɞɭ
ௗ
—
ௗ௧
Ƚɟɧɟɪɚɬɨɪ, ɭɩɪɚɜɥɹɟɦɵɣ ɧɚɩɪɹɠɟɧɢɟɦ
Ɉɫɧɨɜɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ Ƚɍɇɚ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɢɪɨɜɨɱɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ,
ɬɨ ɟɫɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɱɚɫɬɨɬɵ ɝɟɧɟɪɚɰɢɢ ݂
ɦɚɥɵɯ ɧɚɩɪɹɠɟɧɢɹɯ ɨɧɚ ɥɢɧɟɣɧɚ ɢ ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧɚ ɤɪɭɬɢɡɧɨɣ
οఠ
ˆ˖ː
=
݇
ˆ˖ː
— ɪɢɫ. 4.2.
ο௨
˅
ɨɬ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɜɚɪɢɤɚɩɟ ݑ
ˆ˖ː
73
. ɉɪɢ
˅
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
Ɋɢɫ. 4.2. Ɏɪɚɝɦɟɧɬ ɪɟɝɭɥɢɪɨɜɨɱɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ Ƚɍɇɚ
Ɏɚɡɨɜɵɣ ɞɟɬɟɤɬɨɪ
ȼ ɫɢɫɬɟɦɟ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɜ ɮɚɡɨɜɨɦ ɞɟɬɟɤɬɨɪɟ
ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɫɜɹɡɶ ɦɟɠɞɭ ɪɚɡɧɨɫɬɶɸ ɮɚɡ οߔൌߔ
ɨɲɢɛɤɢ ݑ
ɝɞɟ ݇
ˇ
ɫ ɩɨɦɨɳɶɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɟɬɟɤɬɨɪɚ (ɪɢɫ. 4.3):
ˑ˛
= ݇ˇ¨ߔ,
ݑ
ˑ˛
— ɤɪɭɬɢɡɧɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ.
Ɋɢɫ. 4.3. ɏɚɪɚɤɬɟɪɢɫɬɢɤɚ ɮɚɡɨɜɨɝɨ ɞɟɬɟɤɬɨɪɚ
Ȅߔ
ˠ
ɢ ɧɚɩɪɹɠɟɧɢɟɦ
ˑ˔
ɇɚɝɪɭɡɤɨɣ ɮɚɡɨɜɨɝɨ ɞɟɬɟɤɬɨɪɚ ɜ ɫɢɫɬɟɦɟ ɎȺɉɑ ɹɜɥɹɟɬɫɹ ɮɢɥɶɬɪ ɧɢɠɧɢɯ
ɱɚɫɬɨɬ, ɩɨɷɬɨɦɭ ݑ
— ɷɬɨ ɜɯɨɞɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɮɢɥɶɬɪɚ.
ˑ˛
Ɏɢɥɶɬɪ ɫɢɫɬɟɦɵ ɎȺɉɑ
ɋɜɹɡɶ ɜɯɨɞɧɨɝɨ ݑ
(t) ɢ ɜɵɯɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ݑ
ˑ˛
ሺݐሻ
ɮɢɥɶɬɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ
˅
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ
ሺݐሻ
ɝɞɟ ܭ
(p) — ɨɩɟɪɚɬɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɮɢɥɶɬɪɚ. ɇɚɩɪɹɠɟɧɢɟ ɫ
˗
ɜɵɯɨɞɚ ɮɢɥɶɬɪɚ ݑ
ݑ
˅
ሺݐሻ
ɩɨɫɬɭɩɚɟɬ ɧɚ ɜɚɪɢɤɚɩ Ƚɍɇɚ.
˅
ൌܭ
˗
(p) ݑ
ˑ˛
(t),
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɯɟɦɚ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ ɫɨɞɟɪɠɢɬ ɫɥɟɞɭɸɳɢɟ ɡɜɟɧɶɹ:
1) ɷɥɟɦɟɧɬ ɫɪɚɜɧɟɧɢɹ ɮɚɡ ɧɚɩɪɹɠɟɧɢɣ ɷɬɚɥɨɧɧɨɝɨ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ߔ
ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ߔ
ሺݐሻ, ɚ ɬɚɤɠɟ, ɨɩɪɟɞɟɥɟɧɢɹ ɪɚɡɧɢɰɵ ɮɚɡ οߔሺݐሻǡ
ˑ˔
2) ɩɪɟɨɛɪɚɡɨɜɚɬɟɥɶ ɪɚɡɧɢɰɵ ɮɚɡ οߔሺݐሻ ɜ ɧɚɩɪɹɠɟɧɢɟ ɫɢɝɧɚɥɚ
ɨɲɢɛɤɢݑ
ˑ˛
(t),
74
ሺݐሻ ɢ
ˠ
4. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ
3) ɩɪɟɨɛɪɚɡɨɜɚɬɟɥɶ ɧɚɩɪɹɠɟɧɢɹ ɨɲɢɛɤɢ ݑ
ሺݐሻ
ݑ
ǡɩɨɞɜɨɞɢɦɨɟ ɤ ɜɚɪɢɤɚɩɭ Ƚɍɇɚ,
˅
4) ɡɜɟɧɨ, ɫɜɹɡɵɜɚɸɳɟɟ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ݑ
Ƚɍɇɚ ο߱
5) ɫɭɦɦɚɬɨɪ ɱɚɫɬɨɬ — ɫɜɨɛɨɞɧɨɝɨ Ƚɍɇɚ ߱
ʒʢʜ
ሺݐሻ,
ሺሻ
ɢ ɞɨɛɚɜɤɢ ¨߱
ʒʢʜ
(t) ɜ ɧɚɩɪɹɠɟɧɢɟ
ˑ˛
ሺݐሻ
ɫ ɢɡɦɟɧɟɧɢɟɦ ɱɚɫɬɨɬɵ
˅
ሺݐሻ,
ʒʢʜ
6) ɩɪɟɨɛɪɚɡɨɜɚɬɟɥɶ ɱɚɫɬɨɬɵ ɜ ɮɚɡɭ: ߔሺݐሻൌ߱ሺݐሻ݀ݐ, ɢɥɢ ɜ ɫɢɦɜɨɥɢɱɟɫɤɨɣ
( p =
ௗ
),
ௗ௧
ͳ
ൗ .
ܰ
ɮɨɪɦɟ ߔൌ
߱
ൗ
7) ɞɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ Ƚɍɇɚ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɩɟɪɟɞɚɱɢ
ȼɯɨɞɧɵɦ ɜɨɡɞɟɣɫɬɜɢɟɦ ɧɚ ɫɢɫɬɟɦɭ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ ɹɜɥɹɟɬɫɹ ɮɚɡɚ
ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ߔ
Ƚɍɇɚ ߔ
ɪɢɫ. 4.4. ߔ
(t). ɋɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɫɢɫɬɟɦɵ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ ɩɨɤɚɡɚɧɚ ɧɚ
ˆ˖ː
(t)
ˆ˖ː
(t), ɜɵɯɨɞɨɦ — ɮɚɡɚ ɧɚɩɪɹɠɟɧɢɹ
ˠ
Ɋɢɫ. 4.4. ɋɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɫɢɫɬɟɦɵ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
4.3. Ɉɩɟɪɚɬɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ
ɎȺɉɑ
ɇɚ ɨɫɧɨɜɟ ɫɬɪɭɤɬɭɪɧɨɣ ɫɯɟɦɵ ɪɢɫ. 4.4 ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ
ɨɩɟɪɚɬɨɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ ɎȺɉɑܭ
ɫɢɫɬɟɦɟ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɨɩɟɪɚɬɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
˒˓ሺሻ
ܭ(p) =
ɝɞɟ ʙ
ɫɢɫɬɟɦɵ ɎȺɉɑ, ܭ
(p) — ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɩɪɹɦɨɣ ɜɟɬɜɢ ɨɬ ɜɯɨɞɚ ɞɨ ɜɵɯɨɞɚ
˒˓
(p) — ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɪɚɡɨɦɤɧɭɬɨɣ ɩɟɬɥɢ ɎȺɉɑ
˓˃ˊ
ଵା
, (4.3)
˓˃ˊሺሻ
ɨɬ ɩɟɪɜɨɝɨ ɞɨ ɜɬɨɪɨɝɨ ɜɯɨɞɚ ɩɟɪɜɨɝɨ ɫɭɦɦɚɬɨɪɚ. Ʉɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɰɟɩɢ
ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɤɚɤ ɩɪɨɢɡɜɟɞɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɟɪɟɞɚɱɢ ɨɬɞɟɥɶɧɵɯ
ɡɜɟɧɶɟɜ, ɩɪɢ ɷɬɨɦ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɟɪɟɞɚɱɢ ɫɭɦɦɚɬɨɪɨɜ ɪɚɜɧɵ 1.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɫɯɟɦɨɣ ɪɢɫ. 4.4,
75
(p). ȼ ɡɚɦɤɧɭɬɨɣ
˗˃˒˚
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
ܭ
ܭ
(p) = ݇ˇܭ˗(p)݇
˒˓
(p) = ݇ˇܭ˗(p)݇
˓˃ˊ
/p, (4.4)
ˆ˖ː
/pN (4.5)
ˆ˖ː
ɉɨɞɫɬɚɜɢɜ (4.4) ɢ (4.5) ɜ (4.6), ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɟɪɚɬɨɪɧɨɝɨ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
ܭ
ɝɞɟ ߬ =
ܰ
ൗ
- ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ ɰɟɩɢ ɎȺɉɑ.
݇
ˆ˖ː݇ˇ
˗˃˒˚
(p) =
ଵା
ఛ
ே
, (4.6)
ൗ
ሺ୮ሻ
˗
ɉɨɪɹɞɨɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɫɜɹɡɵɜɚɸɳɟɝɨ ɮɥɭɤɬɭɚɰɢɢ ɮɚɡɵ
ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ ߔ
(t) ɢ ɮɚɡɵ ɧɚɩɪɹɠɟɧɢɹ Ƚɍɇɚ ߔ
ˠ
ɪɚɜɟɧ ɫɬɟɩɟɧɢ ɤɨɷɮɮɢɰɢɟɧɬɚ p ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ܭ
ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɮɢɥɶɬɪɚ (ܭ
(p) = 1) ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ ɎȺɉɑ
˗
(p). ȼ ɤɪɚɣɧɟɦ ɫɥɭɱɚɟ,
˗˃˒˚
ˆ˖ː
ሺݐሻ
ɩɨɞɨɛɟɧ ɩɪɨɫɬɟɣɲɟɦɭ RC — ɮɢɥɶɬɪɭ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɢ ɮɚɡɵ ɫɜɹɡɚɧɵ
ɭɪɚɜɧɟɧɢɟɦ 1-ɝɨ ɩɨɪɹɞɤɚ.
ȿɫɥɢ ɞɨɩɭɫɬɢɬɶ, ɱɬɨ ɤɨɥɟɛɚɧɢɹ ߔ
(t) ɢ ߔ
ˠ
ሺݐሻ
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ
ˆ˖ː
ɫɨɜɨɤɭɩɧɨɫɬɶ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɨɬ ߗൌʹߨܨ c ɤɨɦɩɥɟɤɫɧɵɦɢ
ɚɦɩɥɢɬɭɞɚɦɢ Ɏ, ɬɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɟɪɚɬɨɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ
ɦɨɠɧɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɜ ɮɨɪɦɭɥɭ ɤɨɦɩɥɟɤɫɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ. Ⱦɥɹ
ɷɬɨɝɨ ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹܭ
(p) ɫɥɟɞɭɟɬ ɩɨɥɨɠɢɬɶ p = jߗ. Ʉɨɦɩɥɟɤɫɧɵɣ
˗˃˒˚
ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɫɢɫɬɟɦɵ ɎȺɉɑ
ࡷ
˗˃˒˚
(jȳሻ =
ଵା
୨ఆఛ
ே
(4.7)
൘
ሺ୨ఆሻ
˗
,
ɝɞɟ ࡷ
˗˃˒˚
(jȳሻ= ܭ
˗˃˒˚
(ȳሻ݁
ఝሺఆሻ
.
4.4. Ⱥɦɩɥɢɬɭɞɧɨ-ɱɚɫɬɨɬɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɫɢɫɬɟɦɵ
ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
Ɂɚɜɢɫɢɦɨɫɬɶ ɦɨɞɭɥɹ ɤɨɦɩɥɟɤɫɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ ܭ
ɱɚɫɬɨɬɵ ܨ ɮɥɭɤɬɭɚɰɢɣ ɮɚɡɵ ߔ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ (Ⱥɑɏ) ɫɢɫɬɟɦɵ ɎȺɉɑ, (ܨൌ
ሺݐሻ
— ɷɬɨ ɚɦɩɥɢɬɭɞɧɨ-ɱɚɫɬɨɬɧɚɹ
ˆ˖ː
ߗ
). ȼɢɞ Ⱥɑɏ ɨɩɪɟɞɟɥɹɟɬɫɹ
ൗ
ʹߨ
ɮɢɥɶɬɪɨɦ ɧɢɠɧɢɯ ɱɚɫɬɨɬ, ɜɚɪɢɚɧɬɵ ɤɨɬɨɪɵɯ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 4.5.
76
˗˃˒˚
(F) ɨɬ
4. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ
Ɋɢɫ. 4.5. Ɏɢɥɶɬɪɵ ɫɢɫɬɟɦɵ ɎȺɉɑ
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɢɫɬɟɦɵ ɎȺɉɑ ɫ ɮɢɥɶɬɪɨɦ ɪɢɫ. 4.5,ɚ — ɷɬɨ
ɭɪɚɜɧɟɧɢɟ 2-ɝɨ ɩɨɪɹɞɤɚ, ɫ ɮɢɥɶɬɪɨɦ ɪɢɫ. 4.5, ɛ — ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ, ɫ
ɮɢɥɶɬɪɨɦ ɪɢɫ.4.5, ɜ — ɱɟɬɜɟɪɬɨɝɨ ɩɨɪɹɞɤɚ. Ⱥɑɏ ɫɢɫɬɟɦ ɎȺɉɑ ɪɚɡɧɵɯ
ɩɨɪɹɞɤɨɜ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 4.6.
Ɋɢɫ. 4.6. Ⱥɑɏ ɫɢɫɬɟɦ ɎȺɉɑ ɪɚɡɧɵɯ ɩɨɪɹɞɤɨɜ: 1-ɝɨ, 2-ɝɨ ɢ 3-ɝɨ.
(ɫɢɫɬɟɦɚ 1-ɝɨ ɩɨɪɹɞɤɚ ɛɟɡ ɮɢɥɶɬɪɚ, 2-ɝɨ ɩɨɪɹɞɤɚ ɫ ɮɢɥɶɬɪɨɦ ɪɢɫ. 4.6,ɚ, 3-ɝɨ ɩɨɪɹɞɤɚ
ɫ ɮɢɥɶɬɨɦ ɪɢɫ. 4.6. ɛ; ܨ
– ɱɚɫɬɨɬɚ ɫɪɟɡɚ ɫɢɫɬɟɦɵ 1-ɝɨ ɩɨɪɹɞɤɚ).
˔˓ˊଵ
ɉɨɥɨɫɚ ɩɪɨɩɭɫɤɚɧɢɹ ɫɢɫɬɟɦɵ ɎȺɉɑ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɪɢɫ. 4.6, ɫɢɫɬɟɦɚ ɎȺɉɑ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɤɨɥɟɛɚɧɢɹɦ ɮɚɡɵ
ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ ȺȽ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɮɢɥɶɬɪ ɧɢɠɧɢɯ
ɱɚɫɬɨɬ. ɉɨɥɨɫɚ ɩɪɨɩɭɫɤɚɧɢɹ ɫɢɫɬɟɦɵ ɎȺɉɑ ¨ܨ
ɮɥɭɤɬɭɚɰɢɣ ɮɚɡɵ ɨɬ 0 ɞɨ ɱɚɫɬɨɬɵ ɫɪɟɡɚ Ⱥɑɏ ܨ
ɪɚɜɧɚ ɩɪɨɦɟɠɭɬɤɭ ɱɚɫɬɨɬ
˗˃˒˚
. ɑɚɫɬɨɬɚ ɫɪɟɡɚ ɪɚɡɞɟɥɹɟɬ Ⱥɑɏ
˔˓ˊ
ɧɚ ɞɜɟ ɨɛɥɚɫɬɢ:
1) F < ܨ
2) F > ܨ
— ɨɛɥɚɫɬɶ ɩɪɨɩɭɫɤɚɧɢɹ ɢ
˔˓ˊ
— ɨɛɥɚɫɬɶ, ɝɞɟ ɤɨɥɟɛɚɧɢɹ ɮɚɡɵ ɧɚɩɪɹɠɟɧɢɹ Ƚɍɇɚ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ
˔˓ˊ
ɨɬɫɥɟɠɢɜɚɸɬ ɤɨɥɟɛɚɧɢɹ ɮɚɡɵ ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚǤ
ɑɚɫɬɨɬɚ ɫɪɟɡɚ — ɭɫɥɨɜɧɚɹ ɜɟɥɢɱɢɧɚ. Ⱦɥɹ ɫɢɫɬɟɦ ɎȺɉɑ ɪɚɡɧɵɯ ɩɨɪɹɞɤɨɜ
ɩɪɢɧɹɬɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɱɚɫɬɨɬɚ ɫɪɟɡɚ ܨ
— ɷɬɨ ɱɚɫɬɨɬɚ ɮɚɡɨɜɵɯ
˔˓ˊ
77
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
ɮɥɭɤɬɭɚɰɢɣ, ɩɪɢ ɤɨɬɨɪɨɣ ɦɨɞɭɥɶ ɤɨɦɩɥɟɤɫɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢ
ɪɚɡɨɦɤɧɭɬɨɣ ɩɟɬɥɢ ܭ
ሺܨሻ ɪɚɜɟɧ 1.
˓˃ˊ
Ʉɚɤ ɜɢɞɧɨ ɢɡ ɪɢɫ.4.6, ɜ ɫɢɫɬɟɦɟ 1-ɝɨ ɩɨɪɹɞɤɚ ɦɚɤɫɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ
ɩɟɪɟɞɚɱɢ ɪɚɜɟɧ N, ɚ ɱɚɫɬɨɬɚ ɫɪɟɡɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɧɚɱɟɧɢɸ ܭ
˗˃˒˚
ൌ
ܰ
. ɋ
ൗ
ʹ
ξ
ɪɨɫɬɨɦ ɩɨɪɹɞɤɚ ɫɢɫɬɟɦɵ ɭɦɟɧɶɲɚɟɬɫɹ ɩɨɥɨɫɚ ɩɪɨɩɭɫɤɚɧɢɹ ɢ ɧɟɫɤɨɥɶɤɨ
ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɜɛɥɢɡɢ ɧɢɡɤɢɯ ɱɚɫɬɨɬ.
4.5. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ ɧɚ ɨɫɧɨɜɟ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ,
ɨɯɜɚɱɟɧɧɵɯ ɰɟɩɶɸ ɎȺɉɑ
ɋɢɧɬɟɡɚɬɨɪ ɱɚɫɬɨɬ ɧɚ ɛɚɡɟ ɫɢɫɬɟɦɵ ɎȺɉɑ Ƚɍɇɚ ɜɵɩɨɥɧɹɸɬ ɩɨ ɫɯɟɦɟ ɪɢɫ.
4.1 ɫ ɧɟɤɨɬɨɪɨɣ ɦɨɞɢɮɢɤɚɰɢɟɣ. Ⱦɥɹ ɜɨɡɦɨɠɧɨɫɬɢ ɱɚɫɬɨɬɧɨɣ ɩɟɪɟɫɬɪɨɣɤɢ
ɞɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ ɞɟɥɚɸɬ ɭɩɪɚɜɥɹɟɦɵɦ, ɩɪɢ ɷɬɨɦ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ
ɢɡɦɟɧɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɟɥɟɧɢɹ N. Ɍɚɤ, ɞɥɹ ɭɜɟɥɢɱɟɧɢɹ ɱɚɫɬɨɬɵ ɧɚɩɪɹɠɟɧɢɹ
= N݂
Ƚɍɇɚ ݂
ˆ˖ː
ɇɨɜɚɹ ɱɚɫɬɨɬɚ Ƚɍɇ
ɩɪɢ ɷɬɨɦ, ɲɚɝ ɩɟɪɟɫɬɪɨɣɤɢ ¨f = ݂
ɋ ɰɟɥɶɸ ɭɦɟɧɶɲɟɧɢɹ ɲɚɝɚ ɩɟɪɟɫɬɪɨɣɤɢ ɜɤɥɸɱɚɸɬ ɜɬɨɪɨɣ ɞɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ
ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɞɟɥɟɧɢɹ n ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫ ɷɬɚɥɨɧɧɵɦ ɝɟɧɟɪɚɬɨɪɨɦ. ȼ
ɮɚɡɨɜɨɦ ɞɟɬɟɤɬɨɪɟ ɫɪɚɜɧɢɜɚɸɬɫɹ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɞɜɭɯ ɱɚɫɬɨɬ: ɱɚɫɬɨɬɵ
ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ݂
ɧɚɩɪɹɠɟɧɢɹ ɫɪɚɜɧɟɧɢɹ ݂
ɪɚɜɟɧɫɬɜɨ
ɞɨɫɬɚɬɨɱɧɨ ɭɫɬɚɧɨɜɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɞɟɥɟɧɢɹ, ɪɚɜɧɵɦ N+1.
ˠ
ᇱ
݂
= (N+1)݂ˠ =݂
ˆ˖ː
݂
ˠ
ൗ
=
˔˓
Ǥ
ˠ
ȼ ɫɢɧɯɪɨɧɧɨɦ ɪɟɠɢɦɟ ɪɚɛɨɬɵ ɜɵɩɨɥɧɹɟɬɫɹ
݊Ǥ
݂
ˆ˖ː
ൗ
ൌ
ܰ
+ ¨f,
ˆ˖ː
݂
ˆ˖ː
ൗ
=
ˑ˔
݂
ˠ
ൗ
(4.8)
݊
ɢ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ
ܰ
ɩɪɢ ɷɬɨɦ, ݂
ˆ˖ː
ൌ
ே
݂
ɢ ɩɪɢ ɢɡɦɟɧɟɧɢɢ N ɲɚɝ ɩɟɪɟɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ ɪɚɜɟɧ
ˠ
݂
ˠ
ൗ
Ǥ
݊
ȿɳɟ ɨɞɧɚ ɦɨɞɟɪɧɢɡɚɰɢɹ ɫɯɟɦɵ ɪɢɫ.4.1 ɫɨɫɬɨɢɬ ɜ ɩɪɢɦɟɧɟɧɢɢ ɱɚɫɬɨɬɧɨɮɚɡɨɜɨɝɨ ɞɟɬɟɤɬɨɪɚ (ɑɎȾ) ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɛɵɥɚ ɜɨɡɦɨɠɧɨɫɬɶ ɧɟ ɬɨɥɶɤɨ
ɫɪɚɜɧɟɧɢɹ ɮɚɡ ɞɜɭɯ ɤɨɥɟɛɚɧɢɣ, ɧɨ ɢ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ ɩɨɞɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ ɩɪɢ
ɛɨɥɶɲɨɣ ɪɚɡɧɢɰɟ
݂
ˆ˖ː
݂
ˠ
ൗ
ൗ
ˋ
ܰ
.
݊
Ɏɭɧɤɰɢɨɧɚɥɶɧɚɹ ɫɯɟɦɚ ɫɢɧɬɟɡɚɬɨɪɚ ɱɚɫɬɨɬ, ɨɫɧɨɜɚɧɧɨɝɨ ɧɚ ɎȺɉɑ Ƚɍɇɚ,
ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ.4.7.
78
4. ɋɢɧɬɟɡɚɬɨɪɵ ɱɚɫɬɨɬ
Ɋɢɫ. 4.7. Ɏɭɧɤɰɢɨɧɚɥɶɧɚɹ ɫɯɟɦɚ ɫɢɧɬɟɡɚɬɨɪɚ ɱɚɫɬɨɬ ɧɚ ɛɚɡɟ ɫɢɫɬɟɦɵ ɎȺɉɑ
ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ
4.6. Ɏɚɡɨɜɵɣ ɲɭɦ ɫɢɧɬɟɡɚɬɨɪɨɜ ɱɚɫɬɨɬ
Ɏɚɡɨɜɵɣ ɲɭɦ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ ɨɰɟɧɢɜɚɸɬ ɧɨɪɦɢɪɨɜɚɧɧɨɣ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɩɥɨɬɧɨɫɬɶɸ ɦɨɳɧɨɫɬɢ ɮɚɡɨɜɵɯ ɮɥɭɤɬɭɚɰɢɣ, ɡɚɜɢɫɹɳɟɣ ɨɬ ɱɚɫɬɨɬɵ
ɮɥɭɤɬɭɚɰɢɣ F
ᇱ
ሺܨሻ = 10
ܵ
ɝɞɟ ܵሺܨ
ሻ
— ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɮɚɡɨɜɨɝɨ ɲɭɦɚ, ɢɡɦɟɪɟɧɧɚɹ ɩɪɢ
ɨɬɫɬɪɨɣɤɟ ɧɚ ɱɚɫɬɨɬɭ F ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɟɧɟɪɢɪɭɟɦɨɣ ɱɚɫɬɨɬɵ, ܲ
ɝɟɧɟɪɚɬɨɪɚ.
Ɏɚɡɨɜɵɣ ɲɭɦ ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɢɧɬɟɡɚɬɨɪɚ ɱɚɫɬɨɬ ɡɚɜɢɫɢɬ ɨɬ ɜɫɟɯ
ɤɨɦɩɨɧɟɧɬɨɜ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɫɯɟɦɵ ɪɢɫ. 4.7, ɧɨ, ɜ ɨɫɧɨɜɧɨɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ
ɲɭɦɚɦɢ ɝɟɧɟɪɚɬɨɪɨɜ: ɨɩɨɪɧɨɝɨ ɢ Ƚɍɇɚ. Ɏɥɭɤɬɭɚɰɢɢ ɮɚɡɵ ɨɩɨɪɧɨɝɨ
ɝɟɧɟɪɚɬɨɪɚ ɩɟɪɟɞɚɸɬɫɹ ɧɚ ɜɵɯɨɞ ɫɢɧɬɟɡɚɬɨɪɚ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɩɟɪɟɞɚɱɢ
ܭ
(ܨሻ. Ɇɨɳɧɨɫɬɶ ɮɚɡɨɜɨɝɨ ɲɭɦɚ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɮɚɡɨɜɵɯ
˗˃˒˚
ɮɥɭɤɬɭɚɰɢɣ, ɩɨɷɬɨɦɭ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɮɚɡɨɜɨɝɨ ɲɭɦɚ
ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɢɧɬɟɡɚɬɨɪɚ
ܵ
ሺܨሻ = ܭ
ˆ˖ː
ɝɞɟܵ
(F) — ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɮɚɡɨɜɨɝɨ ɲɭɦɚ ɷɬɚɥɨɧɧɨɝɨ
ˠ
ɝɟɧɟɪɚɬɨɪɚ.
ɉɪɢ ɨɰɟɧɤɟ ɮɚɡɨɜɨɝɨ ɲɭɦɚ ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɢɧɬɟɡɚɬɨɪɚ ɩɪɢɧɹɬɨ
ɢɞɟɚɥɢɡɢɪɨɜɚɬɶ Ⱥɑɏ ɫɢɫɬɟɦɵ ɎȺɉɑ: ɜ ɩɨɥɨɫɟ ɩɪɨɩɭɫɤɚɧɢɹ ɫɢɫɬɟɦɵ ɫɱɢɬɚɸɬ
ܭ
= N, ɚ ɜɧɟ ɩɨɥɨɫɵ ܭ
˗˃˒˚
= 0. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ
˗˃˒˚
ɦɨɳɧɨɫɬɢ ɮɚɡɨɜɨɝɨ ɲɭɦɚ Ƚɍɇɚ, ɨɯɜɚɱɟɧɧɨɝɨ ɰɟɩɶɸ ɎȺɉɑ, ɜ ɩɨɥɨɫɟ
ɩɪɨɩɭɫɤɚɧɢɹ ¨ܨ
˗˃˒˚
ܵሺܨሻ
ଶ
(ሻܵˠሺܨሻ,
˗˃˒˚
ൗ
, ɞȻ/Ƚɰ,
ܲ
ˆ
— ɦɨɳɧɨɫɬɶ
ˆ
79
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
ଶ
ሺܨሻ
ܵ
ˆ˖ː
§ ܰ
ܵˠሺܨሻ
ɢɥɢ
ܵ
ˆ˖ːǡˇʐ
(F) § ܵ
(F) + 20 ܰ.
ˠǡˇʐ
ȿɫɥɢ ɜ ɫɢɫɬɟɦɟ ɎȺɉɑ ȺȽ ɩɪɢɦɟɧɟɧ ɜɬɨɪɨɣ ɞɟɥɢɬɟɥɶ ɱɚɫɬɨɬɵ, ɬɨ ɩɨɞɫɬɚɜɢɜ
ͳ
ɜ ɫɬɪɭɤɬɭɪɧɭɸ ɫɯɟɦɭ ɪɢɫ.4.4 ɡɜɟɧɨ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɩɟɪɟɞɚɱɢ
ൗ
— ɪɢɫ. 4.8,
݊
ɩɨɥɭɱɢɦ
ே
ൗ
ࡷ
˗˃˒˚
(jȳሻ =
ଵା
୨ఆఛ
൘
ሺ୨ఆሻ
˗
ɢ ɮɚɡɨɜɵɣ ɲɭɦ ɫɢɧɬɟɡɚɬɨɪɚ ɜ ɩɨɥɨɫɟ οܨ
ܵ
ˆ˖ːǡˇʐ
(F) § ܵ
ˠǡˇʐ
˗˃˒˚
(F) + 20
ܰ
ൗ
. (4.9)
݊
Ɂɞɟɫɶ N — ɤɨɷɮɮɢɰɢɟɧɬ ɞɟɥɟɧɢɹ ɱɚɫɬɨɬɵ ɧɚɩɪɹɠɟɧɢɹ ȽɍɇȺ, n —
ɤɨɷɮɮɢɰɢɟɧɬ ɞɟɥɟɧɢɹ ɱɚɫɬɨɬɵ ɧɚɩɪɹɠɟɧɢɹ ɷɬɚɥɨɧɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ.
Ɋɢɫ. 4.8. Ɏɪɚɝɦɟɧɬ ɫɬɪɭɤɬɭɪɧɨɣ ɫɯɟɦɵ ɫɢɫɬɟɦɵ ɎȺɉɑ ȺȽ ɫ ɜɬɨɪɵɦ ɞɟɥɢɬɟɥɟɦ
ɱɚɫɬɨɬɵ.
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɨɰɟɧɢɬɶ ɮɚɡɨɜɵɣ ɲɭɦ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɫɢɧɬɟɡɚɬɨɪɚ
ɜɧɟ ɩɨɥɨɫɵ ɩɪɨɩɭɫɤɚɧɢɹ, ɫɨɫɬɚɜɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɟɪɚɬɨɪɧɨɝɨ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɟɪɟɞɚɱɢܭ
(p), ɫɜɹɡɵɜɚɸɳɟɝɨ ɮɚɡɨɜɵɟ ɮɥɭɤɬɭɚɰɢɢ Ƚɍɇɚ
ˆ˖ː
ɩɪɢ ɪɚɡɨɦɤɧɭɬɨɣ ɢ ɡɚɦɤɧɭɬɨɣ ɰɟɩɢ ɎȺɉɑ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɤɨɷɮɮɢɰɢɟɧɬ
ɩɟɪɟɞɚɱɢ ɩɪɹɦɨɣ ɜɟɬɜɢ
˒˓
(p) = 1,
ܭ
ˆ˖ː
ɤɨɷɮɮɢɰɢɟɧɬ ɩɟɪɟɞɚɱɢ ɪɚɡɨɦɤɧɭɬɨɣ ɰɟɩɢ
˓˃ˊ
ܭ
(p) = ݇ˇܭ˗(p)݇
ˆ˖ː
ˆ˖ː
/pN
ɢ
୨ఆఛ
൘
ሺ୨ఆሻ
˗
ࡷ
ʒʢʜ
(jȳሻ =
ଵା
୨ఆఛ
(4.10)
൘
ሺ୨ఆሻ
˗
80
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