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2. ɍɫɢɥɢɬɟɥɢ ɦɨɳɧɨɫɬɢ
ɬɟɨɪɢɢ ɧɟɥɢɧɟɣɧɵɯ ɭɫɢɥɢɬɟɥɟɣ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɧɟɣ, ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
ɢɦɩɭɥɶɫɵ ɬɨɤɚ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɢɞɟɚɥɶɧɵɟ ɨɬɪɟɡɤɢ ɤɨɫɢɧɭɫɨɢɞɵ, ɜɜɨɞɢɬɫɹ
ɩɨɧɹɬɢɟ ɭɝɥɚ ɨɬɫɟɱɤɢ ߠ (ɝɞɟ ߠ — ɩɨɥɨɜɢɧɚ ɞɥɢɬɟɥɶɧɨɫɬɢ ɢɦɩɭɥɶɫɚ ɜ
ɪɚɡɦɟɪɧɨɫɬɢ ɩɟɪɟɦɟɧɧɨɣ ߱ݐ), ɮɭɧɤɰɢɹ ݅
ɨɩɪɟɞɟɥɹɸɬɫɹ ɚɦɩɥɢɬɭɞɵ ɧɭɥɟɜɨɣ ܫ
(߱ݐ) — ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɪɹɞ Ɏɭɪɶɟ ɢ
ˍ
ɢ ɩɟɪɜɨɣ ܫ
ˍ
ɝɚɪɦɨɧɢɤ ɬɨɤɚ. ȼ ɪɚɛɨɬɟ
ˍଵ
[8] ɩɪɢɜɟɞɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɩɨɫɬɨɹɧɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɢ ɚɦɩɥɢɬɭɞɵ ɩɟɪɜɨɣ
ɝɚɪɦɨɧɢɤɢ ɬɨɤɚ ɨɬ ɭɝɥɚ ɨɬɫɟɱɤɢ ߠ — ɪɢɫ. 2.18. ɉɨ ɨɫɢ ɨɪɞɢɧɚɬ ɨɬɥɨɠɟɧɵ
ɧɨɪɦɢɪɨɜɚɧɧɵɟ ɚɦɩɥɢɬɭɞɵ ɬɨɤɚ
ܫ
ˍଵ
ߛ
ߛ
=
ଵ
ܫ
=
, (2.17)
ൗ
ܷܵ
˄
ˍ
. (2.18)
ൗ
ܷܵ
˄
ߛ
ଵǡߛ
ߠι
Ɋɢɫ. 2.18. Ɂɚɜɢɫɢɦɨɫɬɢ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɩɨɫɬɨɹɧɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɢ ɚɦɩɥɢɬɭɞɵ
ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɬɨɤɚ ɨɬ ɭɝɥɚ ɨɬɫɟɱɤɢ (ɜɟɪɯɧɹɹ ɤɪɢɜɚɹ — ߛ
ɧɢɠɧɹɹ — ߛሻ
ଵǡ
ɂɡ ɪɢɫ. 2.18 ɜɢɞɧɨ, ɱɬɨ ɫ ɪɨɫɬɨɦ ɞɥɢɬɟɥɶɧɨɫɬɢ ɢɦɩɭɥɶɫɚ ɜɨɡɪɚɫɬɚɟɬ ɢ ܫ
ܫ
, ɬɨ ɟɫɬɶ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ܲ
ˍ
ɦɨɳɧɨɫɬɶ, ɩɨɬɪɟɛɥɹɟɦɚɹ ɢɡ ɢɫɬɨɱɧɢɤɚ ɩɢɬɚɧɢɹ ܲ
ɭɜɟɥɢɱɟɧɢɢ ߠ ɨɬ 0 ɞɨ 180
0
ɨɬɧɨɲɟɧɢɟ
ܫ
ˍଵ
ɩɚɞɚɟɬ ɨɬ 2 ɞɨ 1.
ൗ
ܫ
ˍ
. ȼ ɬɨ ɠɟ ɜɪɟɦɹ, ɩɪɢ
ȼɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ
ܲ
= 0,5 ܫˍଵܷ
ଵ
, (2. 19)
ˍଵ
Ɇɨɳɧɨɫɬɶ, ɩɨɬɪɟɛɥɹɟɦɚɹ ɬɪɚɧɡɢɫɬɨɪɨɦ ɢɡ ɢɫɬɨɱɧɢɤɚ ɩɢɬɚɧɢɹ
ܲ
ɉɨɞɫɬɚɜɥɹɹ (2.19), (2.20) ɜ (2.16) ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ
= ܫˍ ܧ
. (2.20)
˒ˋ˕
ܷ
ˍଵ
ൗ § 1, ɩɨɥɭɱɢɦ
ܧ
˒ˋ˕
51
ˍଵ
ଵ
, ɢ
ɢ
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
ܫ
ˍଵ
ߟ
= 0,5
ଵ
(2.21)
ൗ
ܫ
ˍ
Ʌɟɝɤɨ ɡɚɦɟɬɢɬɶ ɢɡ (2.21), ɱɬɨ ɜ ɪɟɠɢɦɟ ɫ ɨɬɫɟɱɤɨɣ ɜɵɯɨɞɧɨɝɨ ɬɨɤɚ ɢ
ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɜɵɯɨɞɧɵɦ ɧɚɩɪɹɠɟɧɢɟɦ ɄɉȾ ɬɪɚɧɡɢɬɨɪɚ ɫɬɪɟɦɢɬɫɹ ɤ 100%
ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɭɝɥɚ ɨɬɫɟɱɤɢ, ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɢɬɟɥɶɧɨɫɬɢ ɢɦɩɭɥɶɫɚ
ɬɨɤɚ — ɪɢɫ. 2.19.
Ɋɢɫ. 2.19. Ɂɚɜɢɫɢɦɨɫɬɶ ɄɉȾ ɩɨ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɟ ɨɬ ɭɝɥɚ ɨɬɫɟɱɤɢ ɜɵɯɨɞɧɨɝɨ ɬɨɤɚ.
2.4.4. ȼɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɵɯɨɞɧɨɣ ɦɨɳɧɨɫɬɢ ɤɨɥɟɛɚɧɢɣ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ
ɜɵɪɚɠɟɧɢɟɦ (2.14), ɩɨɞɫɬɚɜɢɜ ɬɭɞɚ ɚɦɩɥɢɬɭɞɭ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɬɨɤɚ ɢ
ɢɫɩɨɥɶɡɨɜɚɜ ɞɥɹ ɚɦɩɥɢɬɭɞɵ ɜɵɯɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɫɨɨɬɧɨɲɟɧɢɟ
ܷ
ɝɞɟ ܴ
— ɜɯɨɞɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɵɯɨɞɧɨɣ ɫɨɝɥɚɫɭɸɳɟɣ ɰɟɩɢ ɧɚ ɱɚɫɬɨɬɟ
ˍ
= ܫˍଵܴ
ˍଵ
ɜɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ߱Ǥ ȼ ɪɟɡɭɥɶɬɚɬɟ, ɦɨɳɧɨɫɬɶ ɜɵɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɟɪɜɨɣ
ɝɚɪɦɨɧɢɤɢ
ܲ
= 0,5ܫˍଵܴ
ଵ
ଶ
ൌͲǡͷܵଶܷ
ˍ
Ʌɟɝɤɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɩɪɢ ɡɚɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ ɚɦɩɥɢɬɭɞɵ ɜɯɨɞɧɨɝɨ
ɧɚɩɪɹɠɟɧɢɹ ܷ
ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɜɨɡɪɚɫɬɚɧɢɟɦ ɭɝɥɚ
˄
ɨɬɫɟɱɤɢ ߠǤ ɇɚɢɛɨɥɶɲɚɹ ɦɨɳɧɨɫɬɶ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ߠ = 180
ɨɬɫɟɱɤɢ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɢ ɠɟɥɚɧɢɢ ɨɩɪɟɞɟɥɢɬɶ ɦɚɤɫɢɦɚɥɶɧɨ
ɞɨɫɬɢɠɢɦɭɸ ɜɵɯɨɞɧɭɸ ɦɨɳɧɨɫɬɶ, ɨɬɞɚɜɚɟɦɭɸ ɬɪɚɧɡɢɫɬɨɪɨɦ, ɟɫɬɶ
ɜɨɡɦɨɠɧɨɫɬɶ ɭɫɬɚɧɨɜɢɬɶ ɨɩɬɢɦɚɥɶɧɭɸ ɚɦɩɥɢɬɭɞɭ ɜɯɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹܷ
ɷɬɨɦ ɫɥɭɱɚɟ, ɦɚɤɫɢɦɚɥɶɧɚɹ ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɤɨɥɟɛɚɧɢɣ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ
ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ ɭɝɥɟ ɨɬɫɟɱɤɢ ߠ = 120
Ɉɞɢɧ ɢɡ ɜɚɠɧɟɣɲɢɯ ɩɚɪɚɦɟɬɪɨɜ ɭɫɢɥɢɬɟɥɹ — ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɦɨɳɧɨɫɬɢ
0
ܭ
=
, (2.22)
ˍ
ଶ
ଶ
ߛ
(ߠሻܴ
(2.23)
ˍ
ଵ
˄
0
, ɬɨ ɟɫɬɶ ɪɚɛɨɬɟ ɛɟɡ
[8].
ܲ
ଵ
ൗ
ܲ
˅˘
˄
. ȼ
— ɜɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ ɬɪɚɧɡɢɫɬɨɪɚ:
ɝɞɟ ܲ
˅˘
52
2. ɍɫɢɥɢɬɟɥɢ ɦɨɳɧɨɫɬɢ
ଶ
= 0,5 ܫ
ܲ
˅˘
ܫ
— ɦɨɞɭɥɶ ɚɦɩɥɢɬɭɞɵ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɬɨɤɚ ɛɚɡɵ, ܴ
˄ଵ
˄ଵ
ܴ
,
˅˘
— ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ
˅˘
ɱɚɫɬɶ ɜɯɨɞɧɨɝɨ ɢɦɩɟɞɚɧɫɚ ɬɪɚɧɡɢɫɬɨɪɚ, ɨɩɪɟɞɟɥɹɟɦɚɹ ɩɨɬɟɪɹɦɢ ɜ ɬɪɚɧɡɢɫɬɨɪɟ
ɢ ɷɮɮɟɤɬɨɦ ɩɪɹɦɨɝɨ ɩɪɨɯɨɠɞɟɧɢɹ ɜɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɧɚɝɪɭɡɤɭ ɭɫɢɥɢɬɟɥɹ.
Ⱥɧɚɥɢɡ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɜɟɥɢɱɢɧɚ ܴ
ɩɪɨɯɨɠɞɟɧɢɹ ɜɯɨɞɧɨɣ ɦɨɳɧɨɫɬɢ ɜ ɧɚɝɪɭɡɤɭ ɱɟɪɟɡ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ܮ
ɡɚɜɢɫɢɬ, ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ, ɨɬ ɩɪɹɦɨɝɨ
˅˘
ɜɵɜɨɞɚ
ˠ
ɬɪɚɧɡɢɫɬɨɪɚ, ɨɛɳɟɝɨ ɞɥɹ ɜɯɨɞɧɨɣ ɢ ɜɵɯɨɞɧɨɣ ɰɟɩɟɣ ɭɫɢɥɢɬɟɥɹ:
ܮ
ˠ
ൌ
ൗ
,
߬
௧
ܴ
˅˘
— ɫɪɟɞɧɟɟ ɜɪɟɦɹ ɩɪɨɥɟɬɚ ɷɥɟɤɬɪɨɧɨɜ ɱɟɪɟɡ ɛɚɡɭ ɬɪɚɧɡɢɫɬɨɪɚ.
ɝɞɟ߬
௧
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɭɫɢɥɟɧɢɹ ɦɨɳɧɨɫɬɢ, ɫɩɪɚɜɟɞɥɢɜɨɟ ɞɥɹ
ɱɚɫɬɨɬ ɤɨɥɟɛɚɧɢɣ, ɛɥɢɡɤɢɯ ɤ ɝɪɚɧɢɱɧɨɣ ݂
ܭ
ɝɞɟ ݒ
— ɫɪɟɞɧɹɹ ɞɪɟɣɮɨɜɚɹ ɫɤɨɪɨɫɬɶ ɷɥɟɤɬɪɨɧɨɜ, l — ɞɥɢɧɚ ɩɪɨɥɟɬɧɨɝɨ
ˇ
ǡɢɦɟɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ [8]
௧
ఊ
భሺഇሻೃˍೡ
=
ˇ
, (2.24)
ఠమˠ
ɩɪɨɫɬɪɚɧɫɬɜɚ.
Ɇɚɤɫɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɧɚ ɡɚɞɚɧɧɨɣ ɱɚɫɬɨɬɟ ɤɨɥɟɛɚɧɢɣ
ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧ ɩɪɢ ɭɫɥɨɜɢɹɯ:
í ɭɝɨɥ ɨɬɫɟɱɤɢ ɤɨɥɥɟɤɬɨɪɧɨɝɨ ɬɨɤɚ ߠ= 180
í ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚɝɪɭɡɤɢ ɬɪɚɧɡɢɫɬɨɪɚ ܴ
0,
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɝɪɚɧɢɱɧɨɦɭ
ˍ
ɪɟɠɢɦɭ ɪɚɛɨɬɵ,
í ɦɢɧɢɦɚɥɶɧɚɹ ɞɥɢɧɚ ɛɚɡɵ,
í ɦɢɧɢɦɚɥɶɧɚɹ ɢɧɞɭɤɬɢɜɧɨɫɬɶ ɜɵɜɨɞɚ ɬɪɚɧɡɢɫɬɨɪɚ, ɨɛɳɟɝɨ ɞɥɹ ɜɯɨɞɧɨɣ
ɢ ɜɵɯɨɞɧɨɣ ɰɟɩɢ.
ɇɚ ɭɤɚɡɚɧɧɵɯ ɱɚɫɬɨɬɚɯ ɜɯɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɪɨɢɡɜɟɞɟɧɢɟ
ଶ
ܭ
݂
§ const,
ɬɨ ɟɫɬɶ ɫ ɪɨɫɬɨɦ ɱɚɫɬɨɬɵ ɦɚɤɫɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɩɚɞɚɟɬ ɫɨ
ɫɤɨɪɨɫɬɶɸ 6ɞȻ ɧɚ ɨɤɬɚɜɭ.
2.4.5. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɪɟɠɢɦɨɜ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ ɜ
ɭɫɢɥɢɬɟɥɟ
ȼ ɫɥɭɱɚɟ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ ɜ ɧɟɥɢɧɟɣɧɨɦ ɪɟɠɢɦɟ ɩɨɹɜɥɹɸɬɫɹ ɝɚɪɦɨɧɢɤɢ
ɤɨɥɟɛɚɧɢɣ ɜɯɨɞɧɨɣ ɱɚɫɬɨɬɵ. ɉɨɥɟɡɧɵɦ ɷɮɮɟɤɬɨɦ ɭɫɢɥɢɬɟɥɹ ɹɜɥɹɟɬɫɹ
ɭɜɟɥɢɱɟɧɢɟ ɦɨɳɧɨɫɬɢ ɤɨɥɟɛɚɧɢɣ ɜɯɨɞɧɨɣ ɱɚɫɬɨɬɵ, ɬɨ ɟɫɬɶ ɧɚɥɢɱɢɟ ɜɵɯɨɞɧɨɣ
ɦɨɳɧɨɫɬɢ ܲ
ܷ
. Ⱥɦɩɥɢɬɭɞɵ ɝɚɪɦɨɧɢɤ ɜɵɯɨɞɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ
ଵ
= |ܼˍ (݊߱)ȁܫ
ˍ
(2. 25)
ˍ
53
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɧɟ ɛɵɥɨ ɦɨɳɧɨɫɬɢ ɜɵɫɲɢɯ ɝɚɪɦɨɧɢɤ ɧɚ ɜɵɯɨɞɟ ɭɫɢɥɢɬɟɥɹ,
ɜɵɯɨɞɧɚɹ ɫɨɝɥɚɫɭɸɳɚɹ ɰɟɩɶ ɢɡɛɢɪɚɬɟɥɶɧɚ ɢ ɜɵɯɨɞɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɢɦɟɟɬ
ɝɚɪɦɨɧɢɱɟɫɤɭɸ ɮɨɪɦɭ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ
(߱ሻ| ب |ܼˍ (݊߱)ȁ, n = 2, 3, 4,…
ȁܼ
ˍ
Ɋɟɠɢɦɵ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ ɫ ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɜɵɯɨɞɧɵɦ ɧɚɩɪɹɠɟɧɢɟɦ
ɪɚɡɥɢɱɚɸɬɫɹ ɞɥɢɬɟɥɶɧɨɫɬɶɸ ɢɦɩɭɥɶɫɚ ɜɵɯɨɞɧɨɝɨ ɬɨɤɚ ɢɥɢ ɭɝɥɨɦ ɟɝɨ ɨɬɫɟɱɤɢ.
ȼ ɬɚɛɥɢɰɟ 2.1 ɩɪɢɜɟɞɟɧɚ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɪɟɠɢɦɨɜ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ ɫ
ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɜɵɯɨɞɧɵɦ ɧɚɩɪɹɠɟɧɢɟɦ.
Ɍɚɛɥɢɰɚ 2.1. Ɋɟɠɢɦɵ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ
ɍɝɨɥ ɨɬɫɟɱɤɢ ߠ
ɇɚɢɦɟɧɨɜɚɧɢɟ
180
A
0
0
90
< ߠ <180
0
AB B C
Ɉɬɦɟɬɢɦ ɨɫɨɛɟɧɧɨɫɬɢ ɪɟɠɢɦɨɜ ɪɚɛɨɬɵ ɬɪɚɧɡɢɫɬɨɪɚ ɜ ɭɫɢɥɢɬɟɥɟ.
Ʉɥɚɫɫ A. Ʌɢɧɟɣɧɵɣ, ɦɚɤɫɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ, ɛɥɢɡɤɚɹ ɤ
ɦɚɤɫɢɦɚɥɶɧɨɣ ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ, ɧɢɡɤɢɣ ɄɉȾ.
Ʉɥɚɫɫ AB. ɋɥɚɛɨ ɧɟɥɢɧɟɣɧɵɣ, ɦɚɤɫɢɦɚɥɶɧɚɹ ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ, ɛɥɢɡɤɢɣ
ɤ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ.
Ʉɥɚɫɫ B. ɇɟɥɢɧɟɣɧɵɣ, ɭɞɨɛɧɵɣ ɞɥɹ ɭɜɟɥɢɱɟɧɢɹ ɜɵɯɨɞɧɨɣ ɦɨɳɧɨɫɬɢ ɫ
ɩɨɦɨɳɶɸ ɞɜɭɯɬɚɤɬɧɵɯ ɫɯɟɦ ɭɫɢɥɟɧɢɹ.
Ʉɥɚɫɫ ɋ. ɋɢɥɶɧɨ ɧɟɥɢɧɟɣɧɵɣ, ɧɚɢɛɨɥɶɲɢɣ ɄɉȾ .
ɋɨɜɪɟɦɟɧɧɵɟ ɭɫɢɥɢɬɟɥɢ ɦɨɳɧɨɫɬɢ ɜɵɩɨɥɧɹɸɬ ɜ ɜɢɞɟ ɦɨɧɨɥɢɬɧɵɯ
ɦɢɤɪɨɜɨɥɧɨɜɵɯ ɢɧɬɟɝɪɚɥɶɧɵɯ ɫɯɟɦ, ɪɚɛɨɬɚɸɳɢɯ ɧɚ ɱɚɫɬɨɬɚɯ ɫɚɧɬɢɦɟɬɪɨɜɨɝɨ
ɢ ɦɢɥɥɢɦɟɬɪɨɜɨɝɨ ɞɢɚɩɚɡɨɧɨɜ ɞɥɢɧ ɜɨɥɧ. ɇɚ ɪɢɫ. 2.20 ɩɨɤɚɡɚɧɚ ɬɨɩɨɥɨɝɢɹ
ɦɢɤɪɨɫɯɟɦɵ ɭɫɢɥɢɬɟɥɹ
ɦɨɳɧɨɫɬɢ ɞɢɚɩɚɡɨɧɚ ɱɚɫɬɨɬ 70 — 80 ȽȽɰ.
90
0
< 90
0
Ɋɢɫ. 2.20. Ɍɨɩɨɥɨɝɢɹ ɭɫɢɥɢɬɟɥɹ ɦɨɳɧɨɫɬɢ ɦɢɥɥɢɦɟɬɪɨɜɨɝɨ ɞɢɚɩɚɡɨɧɚ
Ɉɛɲɢɪɧɚɹ ɢɧɮɨɪɦɚɰɢɹ ɨ ɫɨɜɪɟɦɟɧɧɵɯ ɭɫɢɥɢɬɟɥɹɯ ɦɨɳɧɨɫɬɢ ɋȼɑ
ɫɨɞɟɪɠɢɬɫɹ ɜ ɪɚɛɨɬɟ [9].
54
3. Ⱥɜɬɨɝɟɧɟɪɚɬɨɪɵ
Ⱦɥɹ ɩɟɪɟɞɚɱɢ ɢɧɮɨɪɦɚɰɢɢ ɩɨ ɪɚɞɢɨ ɧɟɨɛɯɨɞɢɦ ɢɫɬɨɱɧɢɤ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ
ɤɨɥɟɛɚɧɢɣ. ɉɪɨɫɬɟɣɲɢɣ ɢɫɬɨɱɧɢɤ — ɚɜɬɨɝɟɧɟɪɚɬɨɪ, ɬɨ ɟɫɬɶ ɪɚɞɢɨɬɟɯɧɢɱɟɫɤɨɟ
ɭɫɬɪɨɣɫɬɜɨ, ɜ ɤɨɬɨɪɨɦ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ ɜɨɡɧɢɤɚɸɬ
ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨ. ɑɚɫɬɨɬɚ ɜɨɡɧɢɤɲɢɯ ɤɨɥɟɛɚɧɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɟɡɨɧɚɬɨɪɨɦ,
ɷɧɟɪɝɢɹ ɢɯ ɱɟɪɩɚɟɬɫɹ ɢɡ ɢɫɬɨɱɧɢɤɚ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, ɞɥɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
ɷɧɟɪɝɢɢ ɩɨɫɬɨɹɧɧɨɝɨ ɩɨɥɹ ɜ ɷɧɟɪɝɢɸ ɤɨɥɟɛɚɧɢɣ ɧɟɨɛɯɨɞɢɦ ɚɤɬɢɜɧɵɣ ɷɥɟɦɟɧɬ.
ɋɬɪɭɤɬɭɪɧɚɹ ɫɯɟɦɚ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɩɪɢɜɟɞɟɧɚ ɧɚ ɪɢɫ. 3. 1.
Ɋɢɫ. 3. 1. ɋɬɪɭɤɬɭɪɚ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ
ȼ ɤɚɱɟɫɬɜɟ ɪɟɡɨɧɚɬɨɪɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɦɟɧɟɧ ɤɨɥɟɛɚɬɟɥɶɧɵɣ ɤɨɧɬɭɪ,
ɨɬɪɟɡɨɤ ɩɟɪɟɞɚɸɳɟɣ ɥɢɧɢɢ, ɨɛɴɟɦɧɵɣ ɷɥɟɦɟɧɬ ɢ ɞɪ. ɂɫɬɨɱɧɢɤ ɩɢɬɚɧɢɹ — ɷɬɨ
ɜɵɩɪɹɦɢɬɟɥɶ, ɚɤɤɭɦɭɥɹɬɨɪ, ɝɚɥɶɜɚɧɢɱɟɫɤɢɣ ɷɥɟɦɟɧɬ ɢ ɩɪɨɱɟɟ. Ⱥɤɬɢɜɧɵɦ
ɷɥɟɦɟɧɬɨɦ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɱɚɳɟ ɜɫɟɝɨ ɹɜɥɹɟɬɫɹ ɬɪɚɧɡɢɫɬɨɪ, ɦɨɝɭɬ ɛɵɬɶ
ɝɟɧɟɪɚɬɨɪɧɵɟ ɞɢɨɞɵ, ɚ ɬɚɤɠɟ ɜɚɤɭɭɦɧɵɟ ɩɪɢɛɨɪɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɜ ɞɢɚɩɚɡɨɧɟ
ɋȼɑ — ɦɚɝɧɟɬɪɨɧɵ, ɥɚɦɩɵ ɛɟɝɭɳɟɣ ɜɨɥɧɵ, ɤɥɢɫɬɪɨɧɵ.
3.1. Ɇɟɯɚɧɢɡɦ ɪɚɛɨɬɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ
Ʉɨɥɟɛɚɧɢɹ ɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ ɜɨɡɧɢɤɚɸɬ, ɜɫɥɟɞɫɬɜɢɟ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ
ɯɚɪɚɤɬɟɪɚ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɪɟɡɨɧɚɬɨɪɟ ɩɪɢ ɜɤɥɸɱɟɧɢɢ ɧɚɩɪɹɠɟɧɢɹ
ɢɫɬɨɱɧɢɤɚ ɩɢɬɚɧɢɹ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɤ ɤɨɧɞɟɧɫɚɬɨɪɭ, ɡɚɪɹɠɟɧɧɨɦɭ ɞɨ
ɧɚɩɪɹɠɟɧɢɹ ܷ
ɛɭɞɟɬ ɢɡɦɟɧɹɬɶɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ɝɞɟ ߱
— ɪɟɡɨɧɚɧɫɧɚɹ ɱɚɫɬɨɬɚ, ߙ — ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ. ɋ ɩɨɦɨɳɶɸ
˓ˈˊ
ɰɟɩɢ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɱɚɫɬɶ ɧɚɩɪɹɠɟɧɢɹ ɪɟɡɨɧɚɬɨɪɚ ɩɨɫɬɭɩɚɟɬ ɧɚ ɜɯɨɞ
ɚɤɬɢɜɧɨɝɨ ɷɥɟɦɟɧɬɚ, ɤɨɬɨɪɵɣ, ɡɚɛɢɪɚɹ ɷɧɟɪɝɢɸ ɢɡ ɢɫɬɨɱɧɢɤɚ ɩɢɬɚɧɢɹ ɢ
ɩɪɟɨɛɪɚɡɭɹ ɟɟ ɜ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɭɸ ɷɧɟɪɝɢɸ, ɞɨɛɚɜɥɹɟɬ ɷɧɟɪɝɢɸ ɜ ɪɟɡɨɧɚɬɨɪ,
ɫɩɨɫɨɛɫɬɜɭɹ ɬɨɦɭ, ɱɬɨɛɵ ɤɨɥɟɛɚɧɢɹ ɫɬɚɥɢ ɧɟɡɚɬɭɯɚɸɳɢɦɢ.
, ɩɨɞɤɥɸɱɢɬɶ ɤɚɬɭɲɤɭ ɢɧɞɭɤɬɢɜɧɨɫɬɢ, ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɧɟɦ
ݑሺݐሻ = ܷ
ିఈ௧
݁
߱
˓ˈˊ
t,
55
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
Ɍɪɚɧɡɢɫɬɨɪɧɵɣ ɚɜɬɨɝɟɧɟɪɚɬɨɪ ɧɚ LC — ɤɨɧɬɭɪɟ.
ɇɚ ɪɢɫ. 3.2 ɩɪɢɜɟɞɟɧɚ ɫɯɟɦɚ ɩɪɨɫɬɟɣɲɟɝɨ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɧɚ ɛɢɩɨɥɹɪɧɨɦ
ɬɪɚɧɡɢɫɬɨɪɟ — ɫɯɟɦɚ Ʉɨɥɩɢɬɰɚ, «ɢɥɢ ɟɦɤɨɫɬɧɚɹ ɬɪɟɯɬɨɱɤɚ».
Ɋɢɫ. 3.2. Ɍɪɚɧɡɢɬɨɪɧɵɣ ɚɜɬɨɝɟɧɟɪɚɬɨɪ
Ɋɟɡɨɧɚɬɨɪ ɨɛɪɚɡɨɜɚɧ ɷɥɟɦɟɧɬɚɦɢ L, ܥଵ ǡܥ
ɩɢɬɚɧɢɹ ܧ
ɪɚɡɞɟɥɢɬɟɥɶɧɚɹ ɟɦɤɨɫɬɶ ʠ
ɟɦɤɨɫɬɶ ɫɜɹɡɢ ɫ ɧɚɝɪɭɡɤɨɣʠ
, ɛɥɨɤɢɪɨɜɨɱɧɵɟ ɟɦɤɨɫɬɶ ʠ˄ˎ ɢ ɢɧɞɭɤɬɢɜɧɨɫɬɶܮ
˒ˋ˕
, ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɦɟɳɟɧɢɹ ɧɚ ɛɚɡɭ ܴ˔ˏǡɚ ɬɚɤɠɟ
˓
. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɫɦɟɳɟɧɢɹ ɨɞɧɨɜɪɟɦɟɧɧɨ
˔˅
. ȼ ɫɯɟɦɟ ɢɦɟɟɬɫɹ ɢɫɬɨɱɧɢɤ
ଶ
˄ˎ
ɹɜɥɹɟɬɫɹ ɛɥɨɤɢɪɨɜɨɱɧɵɦ. ȼ ɷɬɨɣ ɫɯɟɦɟ ɩɪɢɦɟɧɟɧɚ ɟɦɤɨɫɬɧɚɹ ɨɛɪɚɬɧɚɹ ɫɜɹɡɶ,
ɨɛɪɚɡɨɜɚɧɧɚɹ ɟɦɤɨɫɬɹɦɢ ʠ
ɢ ʠ
. Ⱦɥɹ ɜɨɡɛɭɠɞɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɤ ɤɨɥɥɟɤɬɨɪɭ
ଵ
ଶ
ɬɪɚɧɡɢɫɬɨɪɚ ɫɥɟɞɭɟɬ ɩɨɞɜɟɫɬɢ ɧɚɩɪɹɠɟɧɢɟ ɩɢɬɚɧɢɹ ɢ ɩɨɞɨɛɪɚɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɚɜɬɨɫɦɟɳɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɬɪɚɧɡɢɫɬɨɪ ɨɬɤɪɵɜɚɟɬɫɹ.
Ⱥɜɬɨɝɟɧɟɪɚɬɨɪ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɪɟɯ ɤɨɦɩɨɧɟɧɬɨɜ,
ɡɚɦɤɧɭɬɵɯ ɜ ɤɨɥɶɰɨ: ɪɟɡɨɧɚɬɨɪɚ, ɷɥɟɦɟɧɬɨɜ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɢ ɬɪɚɧɡɢɫɬɨɪɚ. ɋ
ɰɟɥɶɸ ɩɨɥɭɱɟɧɢɹ ɭɫɥɨɜɢɣ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɧɟɡɚɬɭɯɚɸɳɢɯ
ɤɨɥɟɛɚɧɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɩɚɪɚɦɟɬɪɚ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ:
1) S — ɤɪɭɬɢɡɧɚ ɥɢɧɟɣɧɨɣ ɱɚɫɬɢ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɪɚɧɡɢɫɬɨɪɚ
݀݅
ˍ
ɝɞɟ ݅
— ɤɨɥɥɟɤɬɨɪɧɵɣ ɬɨɤ, ݑ
ˍ
S =
— ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɛɚɡɟ (ɦɟɠɞɭ ɛɚɡɨɣ ɢ
˄
,
ൗ
݀ݑ
˄
ɷɦɢɬɬɟɪɨɦ);
ఝ
2) ࡿ
= ܵଵ݁
ೞ
െɫɪɟɞɧɹɹ ɤɪɭɬɢɡɧɚ ɬɪɚɧɡɢɫɬɨɪɚ
ࡵ
ˍ
=
ࡿ
,
ൗ
ࢁ
˄ଵ
,
56
ɝɞɟ ࡵ
ࢁ
— ɤɨɦɩɥɟɤɫɧɚɹ ɚɦɩɥɢɬɭɞɚ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɤɨɥɥɟɤɬɨɪɧɨɝɨ ɬɨɤɚ,
ˍ
— ɤɨɦɩɥɟɤɫɧɚɹ ɚɦɩɥɢɬɭɞɚ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɛɚɡɟ;
˄
3)
= ˑ˔
ˑ˔
୨
ˑ˔
െɤɨɷɮɮɢɰɢɟɧɬ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɩɨ ɧɚɩɪɹɠɟɧɢɸ
ࢁ
ࡷ
˄
=
ˑ˔
൘
,
ࢁ
˓
3. Ⱥɜɬɨɝɟɧɟɪɚɬɨɪɵ
ɝɞɟ ࢁ
— ɤɨɦɩɥɟɤɫɧɚɹ ɚɦɩɥɢɬɭɞɚ ɩɟɪɜɨɣ ɝɚɪɦɨɧɢɤɢ ɧɚɩɪɹɠɟɧɢɹ ɧɚ
˓
ɪɟɡɨɧɚɬɨɪɟ;
ఝ
1) ࢆ
= ܼ˓݁
˓
˓
— ɢɦɩɟɞɚɧɫ ɪɟɡɨɧɚɬɨɪɚ ɜ ɬɨɱɤɚɯ ɤɨɥɥɟɤɬɨɪ —
ɷɦɢɬɬɟɪ.
ʑ ɩɪɢɜɟɞɟɧɧɵɯ ɜɵɪɚɠɟɧɢɹɯ ߮
(t), ߮ˑ˔ — ɦɟɠɞɭ ݑˍ(t) ɢ ݑ˄ሺݐሻ, ߮˓ — ɦɟɠɞɭ ݅ˍ(t) ɢ ݑ
݅
ˍ
— ɪɚɡɧɢɰɚ ɮɚɡ ɦɟɠɞɭ ɤɨɥɟɛɚɧɢɹɦɢݑ
௦
(t).
ˍ
(t) ɢ
˄
3.2. ɍɫɥɨɜɢɹ ɜɨɡɛɭɠɞɟɧɢɹ ɢ ɭɫɬɨɣɱɢɜɨɝɨ
ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ ɤɨɥɟɛɚɧɢɣ
ɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ
ɋɬɚɰɢɨɧɚɪɧɵɦ ɧɚɡɵɜɚɟɬɫɹ ɪɟɠɢɦ ɩɨɫɬɨɹɧɧɨɣ ɱɚɫɬɨɬɵ ɢ ɚɦɩɥɢɬɭɞɵ
ɤɨɥɟɛɚɧɢɣ. Ⱦɥɹ ɟɝɨ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɬɪɟɛɭɟɬɫɹ ɜɵɩɨɥɧɟɧɢɟ ɫɥɟɞɭɸɳɢɯ ɭɫɥɨɜɢɣ.
1. ɍɫɥɨɜɢɟ ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɹ
Sܭˑ˔ܼ
Ⱦɥɹ ɧɚɞɟɠɧɨɝɨ ɜɨɡɛɭɠɞɟɧɢɹ ɤɨɥɟɛɚɧɢɣ ɜɜɨɞɹɬ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɩɚɫɚ ɩɨ
ɫɚɦɨɜɨɡɛɭɠɞɟɧɢɸ
߯ = Sܭ
2. ɍɫɥɨɜɢɟ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ ɤɨɥɟɛɚɧɢɣ
ࡿࡷˑ˔ࢆ˓ൌͳ (3.2),
ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ ɜɢɞɟ:
> 1 (3.1)
˓
= 2…5
ˑ˔ܼ˓
ܵ
߮
(n = 0,1,2, …).
௦
߮ˑ˔+ ߮˓ =2ߨ݊ (3.2ɛ),
ൌͳ (3.2ɚ)
ଵܭˑ˔ܼ˓
ɋɨɨɬɧɨɲɟɧɢɟ (3.2ɚ) ɧɚɡɵɜɚɸɬ ɛɚɥɚɧɫɨɦ ɚɦɩɥɢɬɭɞ, ɚ (3.2ɛ) — ɛɚɥɚɧɫɨɦ
ɮɚɡ.
3. ɍɫɥɨɜɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɚɥɵɦ
ɨɬɤɥɨɧɟɧɢɹɦ ɚɦɩɥɢɬɭɞɵ ɢ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ
57
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
ܷ݀
˄ଵ
݀ܤ
˓
ൗ
*
< 0 (3.3),
݀߱
ͳ
=
˓
.
ൗ
ࢆ
˓
݀ܵ
ɝɞɟ ܤ
— ɦɧɢɦɚɹ ɱɚɫɬɶ ɩɨɥɧɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɪɟɡɨɧɚɬɨɪɚ ࢅ
˓
ଵ
ൗ
ɋɨɨɬɧɨɲɟɧɢɟ (3.3) ɦɨɠɟɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɨ ɜ ɜɢɞɟ ɞɜɭɯ ɜɚɪɢɚɧɬɨɜ
݀ܵ
ଵ
ൗ
ܷ݀
൏Ͳ ɢ
˄ଵ
݀ܤ
˓
ൗ
> 0 (3.3,ɚ),
݀߱
ɥɢɛɨ
݀ܵ
ଵ
ൗ
ܷ݀
Ͳ ɢ
˄ଵ
݀ܤ
˓
ൗ
< 0 (3.3,ɛ).
݀߱
ʑ˃ɪɢɚɧɬ (3.3ɚ) ɪɟɚɥɢɡɭɟɬɫɹ ɩɪɢ ɩɚɪɚɥɥɟɥɶɧɨɦ ɪɟɡɨɧɚɧɫɟ ɤɨɥɟɛɚɬɟɥɶɧɨɣ
ɫɢɫɬɟɦɵ ɜ ɬɨɱɤɚɯ ɤɨɥɥɟɤɬɨɪ — ɷɦɢɬɬɟɪ ɬɪɚɧɡɢɫɬɨɪɚ. ɉɪɢ ɷɬɨɦ, ɨɝɪɚɧɢɱɟɧɢɟ
ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɩɪɢ ɢɯ ɜɨɡɛɭɠɞɟɧɢɢ ɢ ɧɚɪɚɫɬɚɧɢɢ ɩɪɨɢɫɯɨɞɢɬ ɡɚ ɫɱɟɬ
ɨɝɪɚɧɢɱɟɧɢɹ ɤɨɥɥɟɤɬɨɪɧɨɝɨ ɬɨɤɚ. ȼɚɪɢɚɧɬ (3.3ɛ) ɜɨɡɦɨɠɟɧ ɩɪɢ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɦ ɪɟɡɨɧɚɧɫɟ ɢ ɨɝɪɚɧɢɱɟɧɢɢ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ
ɧɚɩɪɹɠɟɧɢɟɦ ɧɚ ɛɚɡɟ ɬɪɚɧɡɢɫɬɨɪɚ.
ɋɜɹɡɶ ɱɚɫɬɨɬɵ ɢ ɚɦɩɥɢɬɭɞɵ ɝɟɧɟɪɢɪɭɟɦɵɯ ɤɨɥɟɛɚɧɢɣ ɫ ɩɚɪɚɦɟɬɪɚɦɢ
ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ.
Ʉɚɤ ɩɪɚɜɢɥɨ, ɤɨɥɟɛɚɧɢɹ ɜ ɬɪɚɧɡɢɫɬɨɪɧɨɦ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ ɩɪɨɢɫɯɨɞɹɬ ɜɛɥɢɡɢ
ɱɚɫɬɨɬɵ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɪɟɡɨɧɚɧɫɚ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ. ȼ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ
ɪɢɫ. 3.2 ɪɟɡɨɧɚɧɫɧɚɹ ɱɚɫɬɨɬɚ
߱
˓ˈˊ
=
ͳ
൘
, ( 3.4)
ܮܥ
ඥ
σ
ܥ
ɝɞɟ ܥ
ଵܥଶ
=
σ
ൗ
ሺܥଵܥଶሻ
Ȅ
ɫɭɦɦɚɪɧɚɹ ɟɦɤɨɫɬɶ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɤɨɧɬɭɪɚ.
ɇɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɧɟɫɤɨɥɶɤɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɪɟɡɨɧɚɧɫɧɨɣ
ɱɚɫɬɨɬɵ ɡɚ ɫɱɟɬ ɜɥɢɹɧɢɹ ɮɚɡɨɜɵɯ ɡɚɞɟɪɠɟɤ ɜ ɬɪɚɧɡɢɫɬɨɪɟ ߮
ɫɜɹɡɢ ߮
. ɍɬɨɱɧɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɱɚɫɬɨɬɵ ɝɟɧɟɪɚɰɢɢ ߱ɦɨɠɧɨ ɧɚɣɬɢ,
ˑ˔
ɢ ɰɟɩɢ ɨɛɪɚɬɧɨɣ
௦
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɭɫɥɨɜɢɟɦ ɛɚɥɚɧɫɚ ɮɚɡ (3.2,ɛ).
ɂɡ ɬɪɟɯ ɫɥɚɝɚɟɦɵɯ, ɜɯɨɞɹɳɢɯ ɜ (3.2,ɛ), ɨɞɧɨ — ߮
ɫɢɥɶɧɨ ɡɚɜɢɫɢɬ ɨɬ
˓
ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ, ɜ ɬɨ ɜɪɟɦɹ, ɤɚɤ ɨɫɬɚɥɶɧɵɟ ɫɥɚɛɨ. ɉɨɥɚɝɚɹ n = 0, ɱɚɫɬɨɬɭ
ɝɟɧɟɪɚɰɢɢ ߱
߮
ɦɨɠɧɨ ɧɚɣɬɢ, ɪɟɲɢɜ ɭɪɚɜɧɟɧɢɟ
ሺ߱ሻ = — (߮ˑ˔߮௦ሻ (3.5)
˓
Ⱥɦɩɥɢɬɭɞɭ ɝɟɧɟɪɢɪɭɟɦɵɯ ɤɨɥɟɛɚɧɢɣ ɧɟɫɥɨɠɧɨ ɧɚɣɬɢ ɢɡ ɭɫɥɨɜɢɹ ɛɚɥɚɧɫɚ
ɚɦɩɥɢɬɭɞ (3.2, ɚ). ɉɨɫɤɨɥɶɤɭ ɪɟɡɨɧɚɬɨɪ ɢ ɰɟɩɶ ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ ɨɛɪɚɡɨɜɚɧɵ
ɥɢɧɟɣɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ, ɫɨɦɧɨɠɢɬɟɥɢ ܭ
ˋܼ
ɨɬ ɚɦɩɥɢɬɭɞɵ ɧɟ ɡɚɜɢɫɹɬ, ɢ ɞɥɹ
ˑ˔
˓
58
3. Ⱥɜɬɨɝɟɧɟɪɚɬɨɪɵ
ɧɚɯɨɠɞɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɪɟɠɢɦɟ ɫɥɟɞɭɟɬ ɪɟɲɢɬɶ
ɭɪɚɜɧɟɧɢɟ
ͳ
ܵ
ଵሺܷ˓
) =
(3.6)
ൗ
ܭ
ˑ˔ܼ˓
Ɂɞɟɫɶ ܷ
- ɦɨɞɭɥɶ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɪɟɡɨɧɚɬɨɪɟ.
˓
ɇɚ ɪɢɫ. 3.3 ɩɨɤɚɡɚɧ ɝɪɚɮɢɱɟɫɤɢɣ ɫɩɨɫɨɛ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ (3.5) ɢ (3.6).
߮
, ɝɪɚɞ
˓
߱
˓ˈˊ
߱
ɚ)
ɛ)
Ɋɢɫ. 3.3. Ɉɩɪɟɞɟɥɟɧɢɟ ɱɚɫɬɨɬɵ ߱
ሺ˃ሻɢ ɚɦɩɥɢɬɭɞɵ ܷ
ሺ˄ሻ
ɤɨɥɟɛɚɧɢɣ ɜ
˔˕
ɫɬɚɰɢɨɧɚɪɧɨɦ ɪɟɠɢɦɟ ɪɚɛɨɬɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ.
Ɉɫɧɨɜɧɵɟ ɩɚɪɚɦɟɬɪɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ
Ⱦɥɹ ɚɜɬɨɝɟɧɟɪɚɬɨɪɨɜ ɢɫɩɨɥɶɡɭɸɬ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɩɚɪɚɦɟɬɪɵ:
1) ɝɟɧɟɪɢɪɭɟɦɚɹ ɱɚɫɬɨɬɚ,
59
Ɋɨɦɚɧɸɤ ȼ.Ⱥ. Ⱥɧɚɥɨɝɨɜɵɟ ɭɫɬɪɨɣɫɬɜɚ ɩɪɢɟɦɨɩɟɪɟɞɚɬɱɢɤɨɜ
2) ɞɨɥɝɨɜɪɟɦɟɧɧɚɹ ɫɬɚɛɢɥɶɧɨɫɬɶ ɱɚɫɬɨɬɵ,
3) ɭɪɨɜɟɧɶ ɲɭɦɚ,
4) ɜɵɯɨɞɧɚɹ ɦɨɳɧɨɫɬɶ.
3.3. Ⱦɨɥɝɨɜɪɟɦɟɧɧɚɹ ɫɬɚɛɢɥɶɧɨɫɬɶ ɱɚɫɬɨɬɵ
ɤɨɥɟɛɚɧɢɣ
ɑɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɟɡɨɧɚɧɫɧɨɣ ɱɚɫɬɨɬɨɣ
ɪɟɡɨɧɚɬɨɪɚ (3.4), ɨɧɚ ɦɨɠɟɬ ɢɡɦɟɧɹɬɶɫɹ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɢɧɞɭɤɬɢɜɧɨɫɬɢ ɢ
ɟɦɤɨɫɬɢ ɤɨɧɬɭɪɚ ɩɨɞ ɜɥɢɹɧɢɟɦ ɜɧɟɲɧɢɯ ɤɥɢɦɚɬɢɱɟɫɤɢɯ ɢ ɦɟɯɚɧɢɱɟɫɤɢɯ
ɜɨɡɞɟɣɫɬɜɢɣ, ɚ ɬɚɤɠɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɬɚɪɟɧɢɹ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɨɫɤɨɥɶɤɭ ɪɟɡɨɧɚɬɨɪ
ɜɤɥɸɱɟɧ ɜ ɰɟɩɶ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ, ɧɚ ɪɟɡɨɧɚɧɫɧɭɸ ɱɚɫɬɨɬɭ ɦɨɝɭɬ ɨɤɚɡɚɬɶ
ɜɥɢɹɧɢɟ ɢɡɦɟɧɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ, ɧɚɩɪɢɦɟɪ,
ɟɦɤɨɫɬɟɣ ɬɪɚɧɡɢɫɬɨɪɚ .
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɧɚ ɱɚɫɬɨɬɟ ɝɟɧɟɪɚɰɢɢ ɞɨɥɠɧɨ ɛɵɬɶ ɜɵɩɨɥɧɟɧɨ
ɭɫɥɨɜɢɟ ɛɚɥɚɧɫɚ ɮɚɡ (3.2,ɛ) ɢ, ɟɫɥɢ ɩɪɢ ɷɬɨɦ ɨɤɚɠɟɬɫɹ, ɱɬɨ ɞɥɹ ɟɝɨ ɜɵɩɨɥɧɟɧɢɹ
ɮɚɡɨɜɵɣ ɫɞɜɢɝ ɦɟɠɞɭ ɤɨɥɟɛɚɧɢɹɦɢ ɬɨɤɚ ɢ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɪɟɡɨɧɚɬɨɪɟ ߮
ɱɚɫɬɨɬɚ ɝɟɧɟɪɚɰɢɢ ɛɭɞɟɬ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɪɟɡɨɧɚɧɫɧɨɣ. Ɋɚɫɫɦɨɬɪɢɦ ɬɪɟɛɨɜɚɧɢɹ,
ɤɨɬɨɪɵɦ ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɪɟɡɨɧɚɬɨɪ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɞɥɹ ɩɨɥɭɱɟɧɢɹ
ɜɵɫɨɤɨɫɬɚɛɢɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ.
ɉɨɫɬɨɹɧɫɬɜɨ ɪɟɡɨɧɚɧɫɧɨɣ ɱɚɫɬɨɬɵ ɪɟɡɨɧɚɬɨɪɚ.
Ⱦɥɹ ɜɵɩɨɥɧɟɧɢɹ ɷɬɨɝɨ ɬɪɟɛɨɜɚɧɢɹ ɪɟɡɨɧɚɬɨɪ ɞɨɥɠɟɧ ɛɵɬɶ ɢɡɝɨɬɨɜɥɟɧ ɢɡ
ɫɬɚɛɢɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ ɢ ɦɚɬɟɪɢɚɥɨɜ, ɫɥɚɛɨ ɩɨɞɜɟɪɠɟɧɧɵɯ ɜɥɢɹɧɢɸ
ɬɟɦɩɟɪɚɬɭɪɵ, ɜɥɚɠɧɨɫɬɢ, ɜɢɛɪɚɰɢɣ, ɭɞɚɪɨɜ, ɞɪɭɝɢɯ ɤɥɢɦɚɬɢɱɟɫɤɢɯ ɢ
ɦɟɯɚɧɢɱɟɫɤɢɯ ɜɨɡɞɟɣɫɬɜɢɣ.
˓
0,
ɋɥɚɛɨɟ ɜɥɢɹɧɢɟ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɧɚ ɱɚɫɬɨɬɭ
ɪɟɡɨɧɚɬɨɪɚ.
ɑɚɫɬɶɸ ɩɨɥɧɨɣ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɚɜɬɨɝɟɧɟɪɚɬɨɪɚ ɹɜɥɹɸɬɫɹ ɟɦɤɨɫɬɢ
ɬɪɚɧɡɢɫɬɨɪɚ, ɤɨɬɨɪɵɟ ɨɤɚɡɵɜɚɸɬɫɹ ɩɨɞɤɥɸɱɟɧɧɵɦɢ ɩɚɪɚɥɥɟɥɶɧɨ ɪɟɡɨɧɚɬɨɪɭ.
ɗɬɢ ɟɦɤɨɫɬɢ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɟɟ ɫɬɚɛɢɥɶɧɵ, ɱɟɦ ɷɥɟɦɟɧɬɵ ɪɟɡɨɧɚɬɨɪɚ, ɢɯ
ɜɥɢɹɧɢɟ ɠɟɥɚɬɟɥɶɧɨ ɨɫɥɚɛɢɬɶ. ȿɫɥɢ ɪɟɡɨɧɚɬɨɪɨɦ ɹɜɥɹɟɬɫɹ LC-ɤɨɧɬɭɪ ɢ ɱɚɫɬɨɬɚ
ɝɟɧɟɪɚɰɢɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɟɝɨ ɩɚɪɚɥɥɟɥɶɧɨɦɭ ɪɟɡɨɧɚɧɫɭ, ɬɨ, ɤɚɤ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ
[1], ɞɥɹ ɭɦɟɧɶɲɟɧɢɹ ɜɥɢɹɧɢɹ ɜɧɟɲɧɢɯ ɟɦɤɨɫɬɟɣ ɠɟɥɚɬɟɥɶɧɨ ɬɚɤ ɜɵɛɪɚɬɶ
ɷɥɟɦɟɧɬɵ L ɢ C, ɱɬɨɛɵ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɤɨɧɬɭɪɚ ߩൌ
ξሺܮȀ
ܥሻ ɛɵɥɨ ɧɟɛɨɥɶɲɢɦ.
ȿɫɥɢ ɠɟ ɜ ɚɜɬɨɝɟɧɟɪɚɬɨɪɟ ɩɪɢɦɟɧɟɧ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɣ ɪɟɡɨɧɚɧɫ, ɬɨ ɞɥɹ
ɨɫɥɚɛɥɟɧɢɹ ɜɥɢɹɧɢɹ ɜɧɟɲɧɢɯ ɟɦɤɨɫɬɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ߩ
ɫɥɟɞɭɟɬ ɭɜɟɥɢɱɢɜɚɬɶ.
60
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