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Файл:Физика лазеров. Часть 2. Основы теории лазеров. Учебное пособие
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ɂɬɚɤ, ɨɫɧɨɜɧɵɦ ɷɥɟɦɟɧɬɨɦ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɜ
Z
Z
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ɭɤɚɡɚɧɧɨɦ ɩɪɢɦɟɪɟ ɹɜɥɹɟɬɫɹ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɣ ɬɪɚɧɫɮɨɪɦɚɬɨɪ, ɤɨɧɰɵ
ɜɬɨɪɢɱɧɨɣ ɤɚɬɭɲɤɢ ɤɨɬɨɪɨɝɨ ɞɨɥɠɧɵ ɛɵɬɶ ɜɤɥɸɱɟɧɵ ɦɟɠɞɭ ɛɚɡɨɣ ɢ
ɷɦɢɬɬɟɪɨɦ ɬɪɚɧɡɢɫɬɨɪɚ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɬɨɤɚ
ɤɨɥɥɟɤɬɨɪɚ ɭɜɟɥɢɱɢɜɚɥɨɫɶ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɬɪɢɰɚɬɟɥɶɧɨɟ ɫɦɟɳɟɧɢɟ ɧɚ
ɛɚɡɟ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɞɨɥɠɟɧ ɭɜɟɥɢɱɢɜɚɬɶɫɹ ɤɨɥɥɟɤɬɨɪɧɵɣ ɬɨɤ.
ȼ ɫɥɭɱɚɟ ɥɚɡɟɪɚ ɪɨɥɶ ɷɥɟɦɟɧɬɚ ɩɨɥɨɠɢɬɟɥɶɧɨɣ
ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ
ɜɵɩɨɥɧɹɟɬ ɨɬɤɪɵɬɵɣ ɨɩɬɢɱɟɫɤɢɣ ɪɟɡɨɧɚɬɨɪ ɫ ɡɚɩɚɫɟɧɧɵɦ ɜ ɧɟɦ
ɫɜɟɬɨɜɵɦ ɩɨɥɟɦ. ɋɜɟɬɨɜɵɟ ɜɨɥɧɵ, ɢɧɞɭɰɢɪɨɜɚɧɨ ɢɡɥɭɱɟɧɧɵɟ ɜ
ɩɪɟɞɵɞɭɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɢ ɫɨɯɪɚɧɢɜɲɢɟɫɹ ɜ ɪɟɡɨɧɚɬɨɪɟ,
ɜɵɡɵɜɚɸɬ ɧɨɜɵɟ ɢɧɞɭɰɢɪɨɜɚɧɧɵɟ ɩɟɪɟɯɨɞɵ ɞɪɭɝɢɯ ɚɬɨɦɨɜ, ɩɪɢɱɟɦ
ɛɥɚɝɨɞɚɪɹ ɨɞɧɨɦɭ ɢɡ ɫɜɨɣɫɬɜ ɢɧɞɭɰɢɪɨɜɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɱɚɫɬɨɬɚ
ɢɧɞɭɰɢɪɨɜɚɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɬɨɱɧɨɫɬɢ ɪɚɜɧɚ ɱɚɫɬɨɬɟ ɢɧɞɭɰɢɪɭɸɳɢɯ
ɤɨɥɟɛɚɧɢɣ. Ɏɚɡɨɜɨɟ ɭɫɥɨɜɢɟ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɛɥɚɝɨɞɚɪɹ ɞɪɭɝɨɦɭ
ɫɜɨɣɫɬɜɭ ɢɧɞɭɰɢɪɨɜɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ – ɧɚɥɢɱɢɸ ɠɟɫɬɤɨɣ ɤɨɪɪɟɥɹɰɢɢ
ɦɟɠɞɭ ɮɚɡɚɦɢ ɢɧɞɭɰɢɪɨɜɚɧɧɵɯ ɢ ɢɧɞɭɰɢɪɭɸɳɢɯ ɜɨɥɧ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɰɟɩɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɜ ɥɚɡɟɪɟ
ɡɚɦɵɤɚɟɬɫɹ ɤɚɤ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬɚɤ ɢ ɜɨ ɜɪɟɦɟɧɢ, ɜ ɨɬɥɢɱɢɟ ɨɬ
ɬɪɚɧɡɢɫɬɨɪɧɨɝɨ ɝɟɧɟɪɚɬɨɪɚ, ɜ ɤɨɬɨɪɨɦ ɤɨɥɟɛɚɧɢɹ ɩɪɚɤɬɢɱɟɫɤɢ ɦɝɧɨɜɟɧɧɨ
ɩɟɪɟɞɚɸɬɫɹ ɢɡ ɰɟɩɢ ɤɨɥɥɟɤɬɨɪɚ ɜ ɰɟɩɶ ɛɚɡɵ.
Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɜ ɫɥɭɱɚɟ
ɥɚɡɟɪɚ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɭɫɢɥɟɧɢɹ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ ɨɬ ɱɚɫɬɨɬɵ ɨɩɢɫɵɜɚɟɬɫɹ ɮɨɪɦɨɣ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɥɢɧɢɢ ɥɚɡɟɪɧɨɝɨ ɩɟɪɟɯɨɞɚ g(
ɫɜɹɡɢ ɨɬ ɱɚɫɬɨɬɵ ɟɫɬɶ ɪɟɡɨɧɚɧɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ
ɪɟɡɨɧɚɬɨɪɚ (ɫɦ. ɪɢɫ. 7.9). ɍɱɢɬɵɜɚɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɲɢɪɢɧɚ ɫɩɟɤɬɪɚɥɶɧɨɣ
), ɚ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ ɨɛɪɚɬɧɨɣ
ɥɢɧɢɢ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ, ɤɚɤ ɩɪɚɜɢɥɨ, ɧɚɦɧɨɝɨ ɩɪɟɜɨɫɯɨɞɢɬ ɲɢɪɢɧɭ
ɪɟɡɨɧɚɧɫɧɨɣ ɤɪɢɜɨɣ ɦɨɞɵ, ɦɨɠɧɨ ɫɱɢɬɚɬɶ,
ɱɬɨ ɱɚɫɬɨɬɚ ɝɟɧɟɪɢɪɭɟɦɵɯ
ɤɨɥɟɛɚɧɢɣ ɩɪɚɤɬɢɱɟɫɤɢ ɜ ɬɨɱɧɨɫɬɢ ɫɨɜɩɚɞɚɟɬ ɫ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ
ɤɨɥɟɛɚɧɢɣ ɷɬɨɣ ɦɨɞɵ
ɪɚɫɫɬɪɨɣɤɚ ɱɚɫɬɨɬɵ ɝɟɧɟɪɢɪɭɟɦɵɯ ɤɨɥɟɛɚɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɜɟɥɢɱɢɧɭ ɦɧɨɝɨ ɦɟɧɶɲɟ ɲɢɪɢɧɵ ɪɟɡɨɧɚɧɫɧɨɣ ɤɪɢɜɨɣ ɦɨɞɵ) ɩɪɢɜɟɥɚ ɛɵ
ɤ ɜɨɡɧɢɤɧɨɜɟɧɢɸ ɪɚɡɧɨɫɬɢ ɮɚɡ ɦɟɠɞɭ ɢɧɞɭɰɢɪɭɸɳɢɦɢ ɢ
. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɞɚɠɟ ɫɪɚɜɧɢɬɟɥɶɧɨ ɧɟɛɨɥɶɲɚɹ
(ɧɚ
ɢɧɞɭɰɢɪɨɜɚɧɧɵɦɢ ɜɨɥɧɚɦɢ. ɗɬɨ ɨɡɧɚɱɚɥɨ ɛɵ ɡɧɚɱɢɬɟɥɶɧɨɟ ɭɦɟɧɶɲɟɧɢɟ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɢ, ɤɚɤ ɫɥɟɞɫɬɜɢɟ, – ɧɟɜɨɡɦɨɠɧɨɫɬɶ
ɝɟɧɟɪɚɰɢɢ.
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Ɋɢɫ. 7.9.
Ɂɚɜɢɫɢɦɨɫɬɢ ɨɬ ɱɚɫɬɨɬɵ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ ɢ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɜ ɥɚɡɟɪɟ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɚɫɬɨɬɚ ɝɟɧɟɪɢɪɭɟɦɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɥɚɡɟɪɟ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɱɚɫɬɨɬɨɣ ɬɨɣ ɦɨɞɵ, ɞɥɹ
ɤɨɬɨɪɨɣ ɩɟɪɟɜɵɩɨɥɧɟɧɨ ɭɫɥɨɜɢɟ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɭɫɥɨɜɢɟ ɝɟɧɟɪɚɰɢɢ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ
ɞɥɹ ɧɟɫɤɨɥɶɤɢɯ ɦɨɞ, ɛɭɞɟɬ ɜɨɡɦɨɠɧɚ ɝɟɧɟɪɚɰɢɹ ɥɚɡɟɪɚ
ɫɪɚɡɭ ɧɚ
ɧɟɫɤɨɥɶɤɢɯ ɱɚɫɬɨɬɚɯ. ɍɱɢɬɵɜɚɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɱɚɫɬɨɬɧɨɟ ɪɚɫɫɬɨɹɧɢɟ
ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɦɨɞɚɦɢ ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɧɚɦɧɨɝɨ ɦɟɧɶɲɟ ɲɢɪɢɧɵ
ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ, ɜ ɥɚɡɟɪɟ ɦɨɠɟɬ ɢɦɟɬɶ ɦɟɫɬɨ ɝɟɧɟɪɚɰɢɹ ɧɚ
ɧɟɫɤɨɥɶɤɢɯ ɢɥɢ ɞɚɠɟ ɧɚ ɦɧɨɝɢɯ ɱɚɫɬɨɬɚɯ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɟɫɥɢ ɧɟ
ɩɪɢɧɹɬɵ ɫɩɟɰɢɚɥɶɧɵɟ ɦɟɪɵ ɞɥɹ ɨɛɟɫɩɟɱɟɧɢɹ ɝɟɧɟɪɚɰɢɢ ɬɨɥɶɤɨ ɧɚ
ɨɞɧɨɣ ɱɚɫɬɨɬɟ.
Ʌɟɤɰɢɹ ʋ8
ɍɋɅɈȼɂȿ
ɋȺɆɈȼɈɁȻɍɀȾȿɇɂə ɂ ɋɄɈɊɈɋɌɇɕȿ ɍɊȺȼɇȿɇɂə ȾɅə
ɅȺɁȿɊȺ, ɆɈɓɇɈɋɌɖ ȽȿɇȿɊȺɐɂɂ ɅȺɁȿɊȺ
8.1. ɍɫɥɨɜɢɟ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ ɞɥɹ ɥɚɡɟɪɚ
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɜɵɪɚɠɟɧɢɣ (1.20) ɢ (1.21), ɩɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ
ɫɜɟɬɨɜɨɣ ɜɨɥɧɵ ɜ ɚɤɬɢɜɧɨɣ ɫɪɟɞɟ, ɩɨɦɟɳɟɧɧɨɣ ɦɟɠɞɭ ɞɜɭɦɹ ɫɬɪɨɝɨ
ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɨɬɪɚɠɚɬɟɥɹɦɢ, ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɜɨɥɧɵ ɩɨɫɥɟ ɩɪɨɛɟɝɚ
ɪɚɫɫɬɨɹɧɢɹ L ɦɟɠɞɭ ɡɟɪɤɚɥɚɦɢ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɪɟɡɨɧɚɬɨɪ
ɩɨɥɧɨɫɬɶɸ ɡɚɩɨɥɧɟɧ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ, ɛɭɞɟɬ
e
0
- ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ,
ɝɞɟ
– ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɜ ɧɚɱɚɥɟ ɩɪɨɛɟɝɚ.
0
L
,
32
ȿɫɥɢ ɭɱɢɬɵɜɚɬɶ ɩɨɬɟɪɢ ɷɧɟɪɝɢɢ ɫɜɟɬɚ ɬɨɥɶɤɨ ɩɪɢ ɨɬɪɚɠɟɧɢɢ ɨɬ
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ɡɟɪɤɚɥ, ɬɨ ɩɪɢ ɞɜɨɣɧɨɦ ɩɪɨɛɟɝɟ ɩɨ ɪɟɡɨɧɚɬɨɪɭ ɩɨɪɨɝɭ ɝɟɧɟɪɚɰɢɢ ɛɭɞɟɬ
ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ ɫɥɟɞɭɸɳɟɟ ɨɱɟɜɢɞɧɨɟ ɭɫɥɨɜɢɟ
2
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12
, R2 – ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɨɬɪɚɠɟɧɢɹ ɨɛɨɢɯ ɡɟɪɤɚɥ.
ɝɞɟ R
1
Ɉɞɧɚɤɨ, ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɟɥɢɱɢɧɚ
ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɜɨɥɧɵ ɢɡɦɟɧɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨ ɧɚ ɞɥɢɧɟ ɩɪɨɛɟɝɚ
ɪɚɫɫɬɨɹɧɢɹ L, ɩɨɪɨɝɨɜɨɟ ɭɫɥɨɜɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɪɚɜɟɧɫɬɜɚ
, (8.1)
L<<1 ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
ɚɛɫɨɥɸɬɧɵɯ ɡɧɚɱɟɧɢɣ ɭɞɟɥɶɧɨɣ ɦɨɳɧɨɫɬɢ ɢɧɞɭɰɢɪɨɜɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ
) ɢ ɭɞɟɥɶɧɨɣ ɦɨɳɧɨɫɬɢ ɩɨɬɟɪɶ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ (|Ɋv|
(|Ɋ
v|ɢɧɞ
ɩɨɞɯɨɞ ɩɨɡɜɨɥɹɟɬ ɜ ɜɟɥɢɱɢɧɟ |Ɋ
ɭɱɟɫɬɶ ɜɤɥɚɞ ɜɫɟɯ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜ
v|ɩɨɬ
). Ɍɚɤɨɣ
ɩɨɬ
ɥɟɤɰɢɢ ʋ6 ɮɚɤɬɨɪɨɜ ɩɨɬɟɪɶ ɜ ɪɟɡɨɧɚɬɨɪɟ.
ɂɬɚɤ, ɧɚɱɚɥɨ ɢ ɪɚɡɜɢɬɢɟ ɜɨ ɜɪɟɦɟɧɢ ɝɟɧɟɪɚɰɢɢ ɜ ɪɟɡɨɧɚɬɨɪɟ
ɥɚɡɟɪɚ ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɩɪɢ ɩɟɪɟɜɵɩɨɥɧɟɧɢɢ ɫɥɟɞɭɸɳɟɝɨ ɭɫɥɨɜɢɹ:
ɩɨɬ ɢɧɞ
vv
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɝɟɧɟɪɚɰɢɹ ɢɦɟɟɬ ɦɟɫɬɨ ɬɨɥɶɤɨ ɧɚ ɨɞɧɨɣ ɢɡ ɦɨɞ
PP
. (8.2)
ɪɟɡɨɧɚɬɨɪɚ. Ɍɨɝɞɚ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɯ ɩɨɬɟɪɶ ɜ
ɪɟɡɨɧɚɬɨɪɟ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, – ɦɚɥɨɝɨ ɭɫɢɥɟɧɢɹ ɜ ɚɤɬɢɜɧɨɣ ɫɪɟɞɟ
ɡɚɩɢɲɟɦ ɭɞɟɥɶɧɵɟ ɦɨɳɧɨɫɬɢ ɩɨɬɟɪɶ ɢ ɢɧɞɭɰɢɪɨɜɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɩɨɬ
ɢɧɞ
.
d
dt
ɩɨɬ
c
(8.3)
/
t
teP
z e dz dt
ɉɪɢɪɚɜɧɢɜɚɹ ɦɨɞɭɥɢ ɷɬɢɯ ɭɞɟɥɶɧɵɯ ɦɨɳɧɨɫɬɟɣ, ɩɨɥɭɱɢɦ
d
ɢɧɞ
P
v
dt n
,,
v
d
z
,,
,
dz n
c
ɢɧɞ
ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ ɜ
ɫɥɭɱɚɟ ɝɟɧɟɪɚɰɢɢ ɥɚɡɟɪɚ ɧɚ ɨɞɧɨɣ ɦɨɞɟ ɪɟɡɨɧɚɬɨɪɚ:
dd
dt dt n ɫ
ɩɨɬ ɢɧɞ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɪɨɝɨɜɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɨɛɪɚɬɧɨ
,, . (8.4)
ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɞɥɢɧɟ ɩɪɨɛɟɝɚ ɜɨɥɧɵ ɦɟɠɞɭ ɡɟɪɤɚɥɚɦɢ ɥɚɡɟɪɚ ɡɚ
ɜɪɟɦɹ
ɧɚɫɟɥɟɧɧɨɫɬɶ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ
.
ɍɱɬɟɦ ɫɜɹɡɶ
ɩɨɪ nk ɩɨɪ
()N
()
N
ɩɨɪ
cn
ɢ ɧɚɣɞɟɦ ɩɨɪɨɝɨɜɭɸ ɢɧɜɟɪɫɧɭɸ
n
ɫ
nk
()
ɩɨɪ
. (8.5)
33
ȼɵɞɟɥɢɦ ɢɡ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɫɟɱɟɧɢɹ ɩɟɪɟɯɨɞɚ ɜ ɹɜɧɨɦ ɜɢɞɟ ɮɨɪɦɭ
S Z
V Z V
S
V
Z
W
Z|Z
Q
Q Q
Q
Q
' Z ' Z Z
S Z Z
W V
Z
Q
Z
'Z=Z
Q
Z
Q
'
W V
Q
Z W 'Z
'
SW
ɪ
Z
Q
'
Z
'
ɥɢɧɢɢ (ɫɦ. ɥɟɤɰɢɸ ʋ2):
()
g
max
()
nk nk
Ɍɨɝɞɚ, ɭɱɢɬɵɜɚɹ ɬɨ, ɱɬɨ ɱɚɫɬɨɬɚ ɝɟɧɟɪɚɰɢɢ ɥɢɲɶ ɧɟɡɧɚɱɢɬɟɥɶɧɨ
L
, ɝɞɟ
T
2
max
nk
ɦɨɠɟɬ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɵ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ (
ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
T
2
()
g
Lnk
ɫ
n
ɦax
nk
() ( )
NNF
ɩɨɪ ɩɨɪ nk
ɝɞɟ F(
ɪɚɫɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ ɦɨɞɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɱɚɫɬɨɬɵ ɩɟɪɟɯɨɞɚ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɜɟɞɟɧɧɵɦ ɜɵɲɟ ɜɵɪɚɠɟɧɢɟɦ, ɝɪɚɮɢɤ
-
) – ɡɚɜɢɫɢɦɨɫɬɶ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɨɬ
nk
22
nk sp
ɦɢɧ
2
cT
n
2
. (8.6)
, (8.7)
),
ɡɚɜɢɫɢɦɨɫɬɢ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɨɬ ɱɚɫɬɨɬɧɨɣ
-
ɪɚɫɫɬɪɨɣɤɢ (
ɩɪɟɞɫɬɚɜɥɟɧɧɵɣ ɧɚ ɪɢɫ. 8.1.
) ɛɭɞɟɬ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɢɦɟɬɶ ɜɢɞ,
nk
ǻN
N
ɦɢɧ
ɩɨɪ
(
)
ɩɨ
0
Ɋɢɫ. 8.1.
Ʉɚɱɟɫɬɜɟɧɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɨɬ
ɪɚɫɫɬɪɨɣɤɢ ɱɚɫɬɨɬɵ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɱɚɫɬɨɬɵ
ɩɟɪɟɯɨɞɚ.
ȼ ɫɥɭɱɚɟ ɬɨɱɧɨɝɨ ɫɨɜɩɚɞɟɧɢɹ ɱɚɫɬɨɬɵ ɦɨɞɵ ɫ ɱɚɫɬɨɬɨɣ ɩɟɪɟɯɨɞɚ
ɜɵɪɚɠɟɧɢɟ (8.7) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɫ
n
.
max
nk
ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ
2
ɦɢɧ
N
ɩɨɪ
ȼɵɪɚɡɢɜ ɲɢɪɢɧɭ ɥɨɪɟɧɰɟɜɨɣ ɥɢɧɢɢ ɱɟɪɟɡ Ɍ
ɜɵɪɚɠɟɧɢɹɦɢ (8.5), (8.6), ɩɨɥɭɱɢɦ
23
n
sp
ɦɢɧ
N
ɩɨɪ
nk L
2
34
. (8.8)
3
c
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɪɨɝɨɜɚɹ ɢɧɜɟɪɫɧɚɹ ɧɚɫɟɥɟɧɧɨɫɬɶ ɬɟɦ ɦɟɧɶɲɟ,
SV '
* Z Z
Q Q
Z W
' Z
S Z Z Z Z
W V S W
Z Z
Q
Z W
'
Z
' Z
S W
Q Q Q Q Q
Q
U U U U V Z W
ª
º
'
«
»
W
¬
¼
Q Q Q
Q
U U V Z W
ª º
'
« »
W
¬ ¼
ɱɟɦ ɭɠɟ ɫɩɟɤɬɪɚɥɶɧɚɹ ɥɢɧɢɹ.
Ɂɚɩɢɲɟɦ ɬɟɩɟɪɶ ɩɨɪɨɝɨɜɨɟ ɭɫɥɨɜɢɟ ɜ ɫɥɭɱɚɟ ɥɚɡɟɪɨɜ ɫ
ɧɟɨɞɧɨɪɨɞɧɨ ɭɲɢɪɟɧɧɨɣ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ, ɤ ɤɨɬɨɪɵɦ ɨɬɧɨɫɹɬɫɹ, ɜ
ɨɫɧɨɜɧɨɦ, ɝɚɡɨɜɵɟ ɥɚɡɟɪɵ. ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ
ɝɚɡɚ ɛɵɥɨ ɩɨɥɭɱɟɧɨ ɜ ɥɟɤɰɢɢ ʋ 2 (ɫɦ. (2.40)) ɢ ɜɵɝɥɹɞɢɬ ɬɚɤ
max
N
.
ɉɨɞɫɬɚɜɢɜ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜ ɩɨɪɨɝɨɜɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ
nk
T
2
()
g
D
ɭɫɢɥɟɧɢɹ (8.4), ɩɨɥɭɱɢɦ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɜ
ɫɥɭɱɚɟ ɝɚɡɨɜɨɝɨ ɥɚɡɟɪɚ ɫ ɞɨɩɥɟɪɨɜɫɤɨɣ ɮɨɪɦɨɣ ɥɢɧɢɢ
23
N
()
ɩɨɪ
ɉɨɥɨɠɢɦ
ɫɜɹɡɶ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ ɫ ɲɢɪɢɧɨɣ ɞɨɩɥɟɪɨɜɫɤɨɣ
n
ɦax 2 3
ɫ c
nk
ɢ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɨɣ (2.19), ɩɨɥɭɱɢɦ
nk
T
2
gg
() ()
Dnk Dnk
nk
n
sp
1
ɥɢɧɢɢ
23
n
nk
sp
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢ ɜ ɫɥɭɱɚɟ ɞɨɩɥɟɪɨɜɫɤɨɝɨ ɭɲɢɪɟɧɢɹ ɩɨɪɨɝɨɜɚɹ
ɢɧɜɟɪɫɧɚɹ ɧɚɫɟɥɟɧɧɨɫɬɶ ɬɚɤɠɟ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɲɢɪɢɧɟ
N
()
ɩɨɪ nk
3/2 3
c
D
4ln2
. (8.9)
ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ.
8.3. ɋɤɨɪɨɫɬɧɵɟ ɭɪɚɜɧɟɧɢɹ ɥɚɡɟɪɚ
ȼ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɦɨɳɧɨɫɬɶ ɢɧɞɭɰɢɪɨɜɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ
ɩɪɟɜɨɫɯɨɞɢɬ ɦɨɳɧɨɫɬɶ ɩɨɬɟɪɶ, ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ
ɥɚɡɟɪɚ ɞɨɥɠɧɚ ɪɚɫɬɢ ɜɨ ɜɪɟɦɟɧɢ. ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɜɵɪɚɠɟɧɢɹɦɢ
(8.3), ɡɚɩɢɲɟɦ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜ ɪɟɡɨɧɚɬɨɪɧɨɣ
ɦɨɞɟ ɥɚɡɟɪɚ, ɫɱɢɬɚɹ, ɱɬɨ ɝɟɧɟɪɚɰɢɹ ɪɚɡɜɢɜɚɟɬɫɹ ɬɨɥɶɤɨ ɜ ɷɬɨɣ ɨɞɧɨɣ
ɦɨɞɟ:
dd d c
dt dt dt n
ɍɱɬɟɦ ɜɤɥɚɞ ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɜ ɝɟɧɟɪɢɪɭɸɳɭɸ ɦɨɞɭ
ɪɟɡɨɧɚɬɨɪɚ, ɤɨɬɨɪɨɟ ɨɫɬɚɟɬɫɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ, ɜɜɟɞɹ ɭɞɟɥɶɧɭɸ
ɢɧɞ ɩɨɬ
ɦɨɳɧɨɫɬɶ ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɧɚ ɱɚɫɬɨɬɟ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ (
nk
()
(8.10)
1
.
N
sp
P ).
v
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ
ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ:
dc
dt n
ɋɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɫɩɨɧɬɚɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɨɬɞɟɥɶɧɵɯ
ɚɬɨɦɨɜ ɢɦɟɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɲɢɪɨɤɢɣ ɭɝɥɨɜɨɣ ɢ ɱɚɫɬɨɬɧɵɣ ɫɩɟɤɬɪ ɢ,
nk
()
NP
35
sp
1
. (8.11)
v
ɩɨɷɬɨɦɭ, ɥɢɲɶ ɧɟɛɨɥɶɲɚɹ ɱɚɫɬɶ ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɦɨɠɟɬ ɨɫɬɚɬɶɫɹ
J
ZJ Z J
W
! !
Q Q
Q
ª º
U U
'
Z J
« »
W ' W
« »
¬ ¼
!
Q
' ' '
V U '
Z!
W
Q
'
Q
Q
U
' ' ' '
Z
W ' Z!
Q Q
Q
Q
Q
ª º
U U
'
Z J
« »
W ' Z W
« »
¬ ¼
U
' ' ' '
Z W ' Z
!
!
J
J J J
ɜ ɪɟɡɨɧɚɬɨɪɟ. ɋ ɬɟɦ, ɱɬɨɛɵ ɭɱɟɫɬɶ ɷɬɭ ɱɚɫɬɶ, ɜɜɟɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬ
ɤɨɬɨɪɵɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɨɬɧɨɲɟɧɢɟ ɬɨɣ ɞɨɥɢ ɦɨɳɧɨɫɬɢ ɫɩɨɧɬɚɧɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ, ɤɨɬɨɪɚɹ ɨɫɬɚɟɬɫɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ, ɤ ɩɨɥɧɨɣ ɦɨɳɧɨɫɬɢ
ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ. ȼ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɲɢɪɢɧɚ ɫɩɟɤɬɪɚɥɶɧɨɣ
ɥɢɧɢɢ ɥɸɦɢɧɟɫɰɟɧɰɢɢ ɧɚɦɧɨɝɨ ɦɟɧɶɲɟ ɱɚɫɬɨɬɵ ɢɡɥɭɱɟɧɢɹ, ɦɨɠɟɦ
ɩɪɢɛɥɢɠɟɧɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɫɟ ɫɩɨɧɬɚɧɧɨɟ ɢɡɥɭɱɟɧɢɟ, ɨɫɬɚɸɳɟɟɫɹ ɜ
ɪɟɡɨɧɚɬɨɪɟ, ɢɦɟɟɬ ɱɚɫɬɨɬɭ, ɪɚɜɧɭɸ ɱɚɫɬɨɬɟ ɩɟɪɟɯɨɞɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɫɩɨɧɬɚɧɧɵɣ ɱɥɟɧ
ɋ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ (8.5),
ɩɪɢɨɛɪɟɬɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
sp
x
PNA
vknk
N
k
nk
sp
. (8.12)
ɩɪɟɞɫɬɚɜɢɦ ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ (8.11) ɜ ɬɚɤɨɦ ɜɢɞɟ
d
dt N
ɉɪɟɨɛɪɚɡɭɟɦ ɬɟɩɟɪɶ ɜɬɨɪɨɟ ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ – ɭɪɚɜɧɟɧɢɟ
ɞɥɹ ɪɚɡɧɨɫɬɢ ɧɚɫɟɥɟɧɧɨɫɬɟɣ, ɩɨɥɭɱɟɧɧɨɟ ɜ ɥɟɤɰɢɢ ʋ3,
dN N N c
dt T n
ɍɦɧɨɠɢɦ ɢ ɩɨɞɟɥɢɦ ɩɨɫɥɟɞɧɢɣ ɱɥɟɧ ɭɪɚɜɧɟɧɢɹ ɧɚ
ɩɨɫɥɟɞɧɢɣ ɱɥɟɧ ɭɪɚɜɧɟɧɢɹ
dN N N N
dt T N
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɫɢɫɬɟɦɚ ɫɜɹɡɚɧɧɵɯ ɫɤɨɪɨɫɬɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢɦɟɬ
ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
d
dt N
dN N N N
dt T N
8.4. Ɉɰɟɧɤɚ ɜɤɥɚɞɚ ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɜ ɷɧɟɪɝɢɸ ɩɨɥɹ
N
e
1nk
N
ɩɨɪ sp
e
1nkɩɨɪ
1
ɩɨɪ sp
2
N
ɢɡ ɜɵɪɚɠɟɧɢɹ (8.5), ɩɨɥɭɱɢɦ
ɩɨɪ
e
1nkɩɨɪ
1
()
2
N
k
nk
N
k
()
,
. (8.13)
N
.
. (8.14)
.
()
(8.15)
. ȼɜɟɞɹ ɜ
nk
2
nk
ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ ɪɟɡɨɧɚɬɨɪɚ ɥɚɡɟɪɚ
ɉɪɨɜɟɞɟɦ ɨɰɟɧɤɭ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɨɥɢ ɷɧɟɪɝɢɢ ɫɩɨɧɬɚɧɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ ɧɚ ɱɚɫɬɨɬɟ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ, ɤɨɬɨɪɚɹ ɨɫɬɚɟɬɫɹ ɜ
ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ. Ʉɨɷɮɮɢɰɢɟɧɬ
ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɢ ɫɩɟɤɬɪɚɥɶɧɭɸ
q . (8.16)
ɝɟɨɦ ɫɩɟɤɬɪ
ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɞɜɟ ɱɚɫɬɢ –
36
,
ɉɪɢ ɨɰɟɧɤɟ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɮɚɤɬɨɪɚ ɨɝɪɚɧɢɱɢɦɫɹ ɭɱɟɬɨɦ ɬɨɥɶɤɨ
S
J
§ ·
S
¨ ¸
© ¹
Q Q
Q
Q
'Z Z
J
'Z W
Q
J
W
J
W
Q
ɬɟɯ ɱɚɫɬɢɰ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ, ɤɨɬɨɪɵɟ ɪɚɫɩɨɥɨɠɟɧɵ ɜ ɰɟɧɬɪɟ ɪɟɡɨɧɚɬɨɪɚ.
ɗɬɨ ɞɚɫɬ ɧɚɦ ɦɚɤɫɢɦɚɥɶɧɭɸ ɨɰɟɧɤɭ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɮɚɤɬɨɪɚ. ɋɱɢɬɚɹ,
ɱɬɨ ɫɩɨɧɬɚɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɨɬɞɟɥɶɧɵɯ ɚɬɨɦɨɜ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɨ
ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɞɥɹ ɱɚɫɬɢɰ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɜ ɰɟɧɬɪɟ ɪɟɡɨɧɚɬɨɪɚ,
ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɮɚɤɬɨɪ ɩɪɟɞɫɬɚɜɢɦ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɭɞɜɨɟɧɧɨɣ ɩɥɨɳɚɞɢ
ɡɟɪɤɚɥ ɤ ɩɥɨɳɚɞɢ ɫɮɟɪɵ
ɫ ɪɚɞɢɭɫɨɦ L/2 (ɫɦ ɪɢɫ. 8.2):
22
aa
2
4
22
L
2
ɝɟɨɦ
2
S
2a
S
L
/2
2
L
. (8.17)
*
L
2ɚ
Ɋɢɫ. 8.2.
Ⱦɢɚɝɪɚɦɦɚ ɢɡɥɭɱɟɧɢɹ ɚɬɨɦɚ, ɩɨɦɟɳɟɧɧɨɝɨ ɜ ɰɟɧɬɪ ɨɩɬɢɱɟɫɤɨɝɨ
ɨɬɤɪɵɬɨɝɨ ɪɟɡɨɧɚɬɨɪɚ ɢ ɞɨɥɹ ɟɝɨ ɢɡɥɭɱɟɧɢɹ, ɨɫɬɚɸɳɚɹɫɹ ɜ ɪɟɡɨɧɚɬɨɪɟ.
ɋɩɟɤɬɪɚɥɶɧɵɣ ɮɚɤɬɨɪ ɩɪɟɞɫɬɚɜɢɦ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɲɢɪɢɧɵ
ɪɟɡɨɧɚɧɫɧɨɣ ɤɪɢɜɨɣ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ ɤ ɲɢɪɢɧɟ ɫɩɟɤɬɪɚɥɶɧɨɣ ɥɢɧɢɢ.
ɋɱɢɬɚɹ ɭɲɢɪɟɧɢɟ ɨɞɧɨɪɨɞɧɵɦ, ɩɨɥɭɱɢɦ
T
2
(8.18)
-7
=10
ɫ,
ɫɩɟɤɬɪ
ɢ ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ J ɩɪɢɦɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ
ɋɞɟɥɚɟɦ ɨɰɟɧɤɭ ɜɟɥɢɱɢɧɵ
-10
Ɍ
=10
c, ɬɨɝɞɚ ɩɨɥɭɱɢɦ - J = 10
2
ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɜ ɷɧɟɪɝɢɸ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ ɨɤɚɡɵɜɚɟɬɫɹ
2/ 2
QT
L
aT
L
-8.
2
2
2
. (8.19)
2
. ɉɨɥɨɠɢɦ: L=20 ɫɦ, a=0,2 ɫɦ,
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɬɧɨɫɢɬɟɥɶɧɵɣ ɜɤɥɚɞ
ɜɟɫɶɦɚ ɦɚɥɵɦ ɢ ɟɝɨ ɜɥɢɹɧɢɟ ɦɨɠɟɬ ɩɪɨɹɜɢɬɶɫɹ ɬɨɥɶɤɨ ɧɚ ɩɨɪɨɝɟ
ɝɟɧɟɪɚɰɢɢ, ɤɨɝɞɚ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ ɬɚɤɠɟ
ɜɟɫɶɦɚ ɦɚɥɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɩɨɥɭɱɟɧɧɚɹ
ɮɨɪɦɭɥɚ (8.18) ɩɨɡɜɨɥɹɟɬ ɩɪɨɜɟɫɬɢ ɦɚɤɫɢɦɚɥɶɧɭɸ ɨɰɟɧɤɭ ɜɤɥɚɞɚ
ɫɩɨɧɬɚɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɜ ɝɟɧɟɪɢɪɭɸɳɭɸ
ɦɨɞɭ ɪɟɡɨɧɚɬɨɪɚ. ȼ ɪɟɚɥɶɧɨɫɬɢ
ɷɬɨɬ ɜɤɥɚɞ ɛɭɞɟɬ ɟɳɟ ɦɟɧɶɲɟ.
37
8.5. ȼɵɜɨɞ ɫɤɨɪɨɫɬɧɵɯ ɭɪɚɜɧɟɧɢɣ ɫ ɭɱɟɬɨɦ ɧɟɩɨɥɧɨɝɨ ɡɚɩɨɥɧɟɧɢɹ
H
H
H
H
U
U
U
U U
³ ³
Q Q
ª º
U U
'
Z J
« »
W ' Z W W
« »
¬ ¼
³ ³ ³
!
E
ǻ
ǻ
ɪɟɡɨɧɚɬɨɪɚ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ
ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɚɤɬɢɜɧɚɹ ɫɪɟɞɚ ɥɢɲɶ ɱɚɫɬɢɱɧɨ ɡɚɩɨɥɧɹɟɬ
ɨɩɬɢɱɟɫɤɢɣ ɪɟɡɨɧɚɬɨɪ. Ɉɫɨɛɟɧɧɨ ɷɬɨ ɯɚɪɚɤɬɟɪɧɨ ɞɥɹ ɬɜɟɪɞɨɬɟɥɶɧɵɯ
ɥɚɡɟɪɨɜ ɫ ɨɩɬɢɱɟɫɤɨɣ ɧɚɤɚɱɤɨɣ, ɩɨɤɚɡɚɬɟɥɶ ɩɪɟɥɨɦɥɟɧɢɹ ɚɤɬɢɜɧɵɯ ɫɪɟɞ
ɤɨɬɨɪɵɯ ɡɚɦɟɬɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ
1.
ȼɨɡɶɦɟɦ ɡɚ ɨɫɧɨɜɭ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (8.15). ɉɪɢ ɷɬɨɦ ɭɱɬɟɦ ɬɨɬ
ɮɚɤɬ, ɱɬɨ ɦɨɞɵ ɜ ɪɟɡɨɧɚɬɨɪɟ ɥɚɡɟɪɚ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɤɚɤ ɤɜɚɡɢ-ɌȿɆ.
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɢ ɦɚɝɧɢɬɧɚɹ ɤɨɦɩɨɧɟɧɬɵ ɩɨɥɹ ɩɪɢ
ɩɚɞɟɧɢɢ ɜɨɥɧɵ ɧɚ ɝɪɚɧɢɰɭ ɪɚɡɞɟɥɚ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ ɢ ɜɨɡɞɭɯɚ,
ɩɚɪɚɥɥɟɥɶɧɭɸ ɡɟɪɤɚɥɚɦ, ɬɚɤɠɟ ɩɪɚɤɬɢɱɟɫɤɢ ɩɚɪɚɥɥɟɥɶɧɵ ɡɟɪɤɚɥɚɦ.
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɚɦɩɥɢɬɭɞɭ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ ɜɧɭɬɪɢ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ
ɦɨɠɟɦ ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɩɨɥɨɠɢɬɶ ɪɚɜɧɨɣ ɚɦɩɥɢɬɭɞɟ
ɜɧɟ ɫɪɟɞɵ (ɫɦ. ɪɢɫ. 8.3).
k
N=0
>
L
ɚ
L
N=0
Ɋɢɫ. 8.3. ɋɯɟɦɚɬɢɱɟɫɤɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɨɩɬɢɱɟɫɤɨɝɨ ɪɟɡɨɧɚɬɨɪɚ, ɱɚɫɬɢɱɧɨ
ɡɚɩɨɥɧɟɧɧɨɝɨ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ.
Ɋɚɡɞɟɥɢɦ ɨɛɴɟɦ ɪɟɡɨɧɚɬɨɪɚ ɧɚ ɞɜɟ ɱɚɫɬɢ: ɨɛɴɟɦ, ɡɚɧɹɬɵɣ
ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ ɷɬɨɦ ɫɥɟɞɭɟɬ ɭɱɟɫɬɶ, ɱɬɨ ɜ ɚɤɬɢɜɧɨɣ ɫɪɟɞɟ ɢɦɟɟɦ
. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɬɢɦ, ɨɛɨɡɧɚɱɢɦ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɜɧɟ ɫɪɟɞɵ ɤɚɤ
0
, ɬɨɝɞɚ ɜ ɚɤɬɢɜɧɨɣ ɫɪɟɞɟ ɛɭɞɟɬ
0
ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɤɚɠɞɵɣ ɱɥɟɧ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ (8.15) ɩɨ
ɨɛɴɟɦɭ ɪɟɡɨɧɚɬɨɪɚ.
V
ɢ ɨɛɴɟɦ, ɧɟ ɡɚɧɹɬɵɣ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ – V0. ɉɪɢ
a
2
=n
. ɋ ɭɱɟɬɨɦ ɫɤɚɡɚɧɧɨɝɨ
ɚ
0
=
2
n
, ɚ ɜɧɟ ɟɟ -
0
Ɍɨɝɞɚ ɥɟɜɚɹ ɱɚɫɬɶ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɢɬɫɹ ɫɭɦɦɨɣ ɞɜɭɯ
ɢɧɬɟɝɪɚɥɨɜ:
dd
00
dv n dv
dt dt
VV
0a
ɉɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɢɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɫɭɦɦɨɣ
2
+
,. (8.19)
ɢɧɬɟɝɪɚɥɨɜ:
2
ndvdvdv
VV
a0a
N
00
N
ɩɨɪ sp
1
()
38
V
N
k
(8.20)
nk
.
=
ɍɱɬɟɦ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɷɧɟɪɝɢɹ ɩɨɥɹ ɜ ɪɟɡɨɧɚɬɨɪɟ ɭɫɪɟɞɧɟɧɚ ɩɨ
Q
ª º
U U
'
Z J
« »
W ' Z W
« »
¬ ¼
!
G
ª
º
G G
¬
¼
U
U
Q
ª º
U U
'
Z JG
« »
W ' W
¬ ¼
!
'
Q
' Z
W
V Z G
Q
Q
' Z
W V Z G S Z
W V G
Q Q Q
U G U U
' ' G '
u
Z W ' Z Z W Z W ' Z G
G ' Z G
! ! !
U
U
'
ɨɛɴɟɦɭ ɪɟɡɨɧɚɬɨɪɚ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɢɡɦɟɧɟɧɢɟɦ
ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜɧɭɬɪɢ ɨɛɴɟɦɨɜ V
ɢ Vo. Ɍɨɝɞɚ, ɩɨɥɚɝɚɹ ɩɥɨɳɚɞɶ
a
ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɪɚɜɧɨɣ ɩɥɨɳɚɞɢ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ
ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ, ɚ ɩɨɩɟɪɟɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɨɥɹ – ɨɞɧɨɪɨɞɧɵɦ,
ɦɨɠɟɦ ɡɚɦɟɧɢɬɶ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɭɦɧɨɠɟɧɢɟɦ ɩɨɞɵɧɬɟɝɪɚɥɶɧɵɯ
ɜɵɪɚɠɟɧɢɣ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɨɛɴɟɦ. ȼ ɪɟɡɭɥɶɬɚɬɟ, ɩɨɥɭɱɢɦ
ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫɥɟɞɭɸɳɟɝɨ ɜɢɞɚ:
d
VVn Vn VVn V
0a a 0a nk a
ȼɜɟɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɩɨɥɧɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ (
22 2
dt N
Ɂɚɦɟɧɢɦ V
ɧɚ (Vc-Va) ɢ ɩɨɞɟɥɢɦ ɜɫɟ ɱɥɟɧɵ ɭɪɚɜɧɟɧɢɹ ɧɚ Vc.
0
N
()
ɩɨɪ sp
ɢ ɤɜɚɞɪɚɬ ɷɮɮɟɤɬɢɜɧɨɝɨ ɩɨɤɚɡɚɬɟɥɹ ɩɪɟɥɨɦɥɟɧɢɹ ɚɤɬɢɜɧɨɣ ɫɪɟɞɵ (
V
V
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɭɫɪɟɞɧɟɧɧɨɟ ɩɨ ɨɛɴɟɦɭ
ɪɟɡɨɧɚɬɨɪɚ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɩɨɥɹ
ɤɜɚɞɪɚɬɧɨɣ ɫɤɨɛɤɢ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ
22
a
, 1 ( 1)
nn
eff
c
. (8.22)
2
=
n
ɫ
eff
n . ȼ ɪɟɡɭɥɶɬɚɬɟ,
N
k
. (8.21)
. ȼɵɧɟɫɟɦ ɡɚ ɡɧɚɤ
0
2
eff
=Va/Vc)
2
n ):
eff
ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
d
ɫɫ
dt N
ɝɞɟ
N0 ɟɫɬɶ ɩɨɪɨɝɨɜɚɹ ɪɚɡɧɨɫɬɶ ɧɚɫɟɥɟɧɧɨɫɬɟɣ ɫ ɭɱɟɬɨɦ ɧɟɩɨɥɧɨɝɨ
ɡɚɩɨɥɧɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɚɤɬɢɜɧɨɣ ɫɪɟɞɨɣ
Ɂɚɩɢɲɟɦ (8.24) ɚɧɚɥɨɝɢɱɧɨ ɜɵɪɚɠɟɧɢɸ (8.7) ɜɜɟɞɹ ɜ ɧɟɝɨ ɜ ɹɜɧɨɦ
ɜɢɞɟ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɪɚɫɫɬɪɨɣɤɢ ɝɟɧɟɪɢɪɭɸɳɟɣ ɦɨɞɵ ɨɬɧɨɫɢɬɟɥɶɧɨ
N
N
0
1+
0sp
()
n
c()
N
k
. (8.24)
n
, (8.23)
nk
2
eff
nk
ɱɚɫɬɨɬɵ ɩɟɪɟɯɨɞɚ
22
nn
()
N
0
ɉɪɟɨɛɪɚɡɭɟɦ ɬɟɩɟɪɶ ɩɪɚɜɭɸ ɱɚɫɬɶ ɜɬɨɪɨɝɨ ɫɤɨɪɨɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɜ (8.15) ɭɦɧɨɠɢɜ ɢ ɩɨɞɟɥɢɜ ɩɨɫɥɟɞɧɢɣ ɱɥɟɧ ɜ ɟɝɨ ɩɪɚɜɨɣ ɱɚɫɬɢ ɧɚ
22 2
nnn
222
0eff0eff
nk ɩɨɪ nk nk 0
ȼɜɟɞɹ ɡɚɬɟɦ
ɧɚɫɟɥɟɧɧɨɫɬɟɣ
NNnN
NN
() ()
ɜɦɟɫɬɨ
ɫ
N0, ɩɨɥɭɱɢɦ ɜɬɨɪɨɟ ɫɤɨɪɨɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ:
eff eff
c() ()
22
nNn
eff ɩɨɪ eff
ng
nk L
2
n
eff
max
c
nk
ɢ ɩɨɪɨɝɨɜɭɸ ɢɧɜɟɪɫɧɭɸ ɪɚɡɧɨɫɬɶ
0
n
()
T
2
. (8.25)
2
nG:
2
eff
c
.
39
dN N N N
Q
U
' ' ' '
Z
G ' Z W!
U
U
'
U
U
'
' '
U U !! U
' ' '
'
'
'
! '
dt T N
e
10
2
nk
()
c
. (8.26)
8.6. ɋɬɚɰɢɨɧɚɪɧɵɣ ɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɣ ɪɟɠɢɦɵ ɝɟɧɟɪɚɰɢɢ ɥɚɡɟɪɚ
ɉɨɥɭɱɟɧɧɚɹ ɜ ɩɪɟɞɵɞɭɳɢɯ ɩɭɧɤɬɚɯ ɫɢɫɬɟɦɚ ɫɤɨɪɨɫɬɧɵɯ ɭɪɚɜɧɟɧɢɣ
ɩɨɡɜɨɥɹɟɬ ɢɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɥɚɡɟɪɚ ɜɨ ɜɪɟɦɟɧɢ. Ɇɨɠɧɨ ɜɵɞɟɥɢɬɶ
ɞɜɚ ɨɫɧɨɜɧɵɯ ɪɟɠɢɦɚ ɪɚɛɨɬɵ ɥɚɡɟɪɚ – ɫɬɚɰɢɨɧɚɪɧɵɣ
ɧɟɫɬɚɰɢɨɧɚɪɧɵɣ. ɋɬɚɰɢɨɧɚɪɧɵɣ ɪɟɠɢɦ ɜɨɡɦɨɠɟɧ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ
ɩɚɪɚɦɟɬɪɵ ɥɚɡɟɪɚ ɧɟɢɡɦɟɧɧɵ, ɚ ɩɨɫɥɟ ɜɤɥɸɱɟɧɢɹ ɧɚɤɚɱɤɢ ɩɪɨɲɥɨ
ɧɚɫɬɨɥɶɤɨ ɦɧɨɝɨ ɜɪɟɦɟɧɢ, ɱɬɨ ɩɟɪɟɯɨɞɧɵɟ ɩɪɨɰɟɫɫɵ, ɫɜɹɡɚɧɧɵɟ ɫ
ɪɚɡɜɢɬɢɟɦ ɝɟɧɟɪɚɰɢɢ ɜ ɩɟɪɜɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɡɚɤɨɧɱɢɥɢɫɶ. ȼ
ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɥɭɱɚɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɜɪɟɦɟɧɢ ɨɬ
d
ɩɨɥɨɠɢɬɶ ɪɚɜɧɵɦɢ ɧɭɥɸ:
c
dt dt
Ɏɨɪɦɚɥɶɧɵɣ ɚɧɚɥɢɡ ɫɤɨɪɨɫɬɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ
dN
0 0
, .
ɢ 'N ɫɥɟɞɭɟɬ
ɫ
ɜɨɡɦɨɠɧɵ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɞɜɚ ɫɬɚɰɢɨɧɚɪɧɵɯ ɪɟɲɟɧɢɹ:
1. ɩɨɪɨɝ ɝɟɧɟɪɚɰɢɢ:
2. ɫɬɚɰɢɨɧɚɪɧɚɹ ɝɟɧɟɪɚɰɢɹ:
ɫ
ɇɚɣɞɟɦ ɡɧɚɱɟɧɢɟ ɢɧɜɟɪɫɧɨɣ ɪɚɡɧɨɫɬɢ ɧɚɫɟɥɟɧɧɨɫɬɢ ɜ ɫɥɭɱɚɟ
NN N
~
,
sp
cst sp
e
,
0
.
ɫɬɚɰɢɨɧɚɪɧɨɣ ɝɟɧɟɪɚɰɢɢ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ
(8.23), ɨɬɛɪɨɫɢɜ ɜ ɧɟɦ ɫɩɨɧɬɚɧɧɵɣ ɱɥɟɧ ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɢɜɟɞɟɧɧɨɝɨ
ɜɵɲɟ ɧɟɪɚɜɟɧɫɬɜɚ. ɉɨɥɨɠɢɜ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɪɚɜɧɨɣ
ɧɭɥɸ, ɧɚɣɞɟɦ, ɱɬɨ ɢɧɜɟɪɫɧɚɹ ɪɚɡɧɨɫɬɶ ɧɚɫɟɥɟɧɧɨɫɬɟɣ ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɣ
ɝɟɧɟɪɚɰɢɢ ɜɫɟɝɞɚ ɞɨɥɠɧɚ ɛɵɬɶ ɪɚɜɧɚ ɟɟ ɩɨɪɨɝɨɜɨɦɭ ɡɧɚɱɟɧɢɸ:
NN N
ɋɥɟɞɭɟɬ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɵɟ
ɡɧɚɱɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɣ ɪɚɡɧɨɫɬɢ ɧɚɫɟɥɟɧɧɨɫɬɟɣ (
ɞɜɭɯ ɫɥɭɱɚɹɯ ɞɨɥɠɧɵ ɨɬɥɢɱɚɬɶɫɹ - ɜ ɩɟɪɜɨɦ ɨɧɨ, ɪɚɡɭɦɟɟɬɫɹ, ɦɟɧɶɲɟ,
ɩɨɫɤɨɥɶɤɭ ɞɨɥɠɧɨ ɬɨɱɧɨ ɪɚɜɧɹɬɶɫɹ ɩɨɪɨɝɨɜɨɣ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ,
e
0
.
e
N
) ɜ ɪɚɫɫɦɨɬɪɟɧɧɵɯ
ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ, ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɣ ɝɟɧɟɪɚɰɢɢ ɜɟɥɢɱɢɧɚ ɪɚɜɧɨɜɟɫɧɨɝɨ
ɡɧɚɱɟɧɢɹ ɢɧɜɟɪɫɧɨɣ ɧɚɫɟɥɟɧɧɨɫɬɢ, ɡɚɜɢɫɹɳɚɹ ɨɬ ɦɨɳɧɨɫɬɢ ɧɚɤɚɱɤɢ,
ɞɨɥɠɧɚ ɜɫɟɝɞɚ ɛɵɬɶ ɛɨɥɶɲɟ ɜɟɥɢɱɢɧɵ
ȼ ɩɪɨɦɟɠɭɬɤɟ ɦɟɠɞɭ ɞɜɭɦɹ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɪɟɲɟɧɢɹɦɢ ɫɤɨɪɨɫɬɧɵɯ
N
e
:
NN
0
.
0
ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɦɟɫɬɨ ɩɟɪɟɯɨɞɧɨɣ ɩɪɨɰɟɫɫ ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ ɝɟɧɟɪɚɰɢɢ,
ɤɨɬɨɪɵɣ ɩɪɢ ɩɨɫɬɨɹɧɧɨɣ ɦɨɳɧɨɫɬɢ ɧɚɤɚɱɤɢ ɢ ɞɪɭɝɢɯ ɫɬɚɛɢɥɶɧɵɯ
ɩɚɪɚɦɟɬɪɚɯ ɪɚɧɨ ɢɥɢ ɩɨɡɞɧɨ ɞɨɥɠɟɧ ɩɪɢɜɟɫɬɢ ɤ ɭɫɬɚɧɨɜɥɟɧɢɸ
ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.
ɉɪɢ ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ ɠɟ ɧɚɤɚɱɤɟ ɢɥɢ ɞɪɭɝɢɯ ɧɟɫɬɚɛɢɥɶɧɵɯ
ɩɚɪɚɦɟɬɪɚɯ ɥɚɡɟɪɚ (ɧɚɩɪɢɦɟɪ – ɩɪɢ ɦɨɞɭɥɹɰɢɢ ɞɨɛɪɨɬɧɨɫɬɢ
ɪɟɡɨɧɚɬɨɪɚ) ɛɭɞɟɬ ɢɦɟɬɶ ɧɟɫɬɚɰɢɨɧɚɪɧɚɹ ɝɟɧɟɪɚɰɢɹ, ɩɪɢɱɟɦ ɨɧɚ ɦɨɠɟɬ
ɩɪɢɧɢɦɚɬɶ ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɮɨɪɦɵ ɜ ɜɢɞɟ ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɢɥɢ
ɢ
40
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