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Файл:Оптимизация режимов резания при токарной обработке металлов. Монография
.pdf
Ɇ. ȼ. ɋɨɤɨɥɨɜ
ɈɉɌɂɆɂɁȺɐɂə ɊȿɀɂɆɈȼ ɊȿɁȺɇɂə
ɉɊɂ ɌɈɄȺɊɇɈɃ ɈȻɊȺȻɈɌɄȿ ɆȿɌȺɅɅɈȼ
Ɇɨɧɨɝɪɚɮɢɹ
Ɇɨɫɤɜɚ ȼɨɥɨɝɞɚ
«ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ»
2024
ɍȾɄ 621.91.01
ȻȻɄ 34.5
ɋ59
ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɯ ɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ»
Ɋɟɤɨɦɟɧɞɨɜɚɧɨ ɇɚɭɱɧɨ-ɬɟɯɧɢɱɟɫɤɢɦ
ɫɨɜɟɬɨɦ ɎȽȻɈɍ ȼɈ «Ɍɚɦɛɨɜɫɤɢɣ
Ɋɟɰɟɧɡɟɧɬɵ:
ɞɨɤɬɨɪ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɡɚɜ. ɤɚɮɟɞɪɨɣ «Ɇɚɬɟɪɢɚɥɵ ɢ ɬɟɯɧɨɥɨɝɢɹ»
Ɍɚɦɛɨɜɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ
Ɇɨɪɞɚɫɨɜ Ⱦɟɧɢɫ Ɇɢɯɚɣɥɨɜɢɱ;
ɡɚɦ. ɧɚɱɚɥɶɧɢɤɚ ɩɪɨɢɡɜɨɞɫɬɜɚ ȺɈ «Ɍɚɦɛɨɜɫɤɢɣ ɡɚɜɨɞ “ɊȿȼɌɊɍȾ”»
Ⱦɨɥɨɬɨɜ ȼɢɤɬɨɪ ɂɜɚɧɨɜɢɱ
ɋɨɤɨɥɨɜ, Ɇ. ȼ.
ɋ59 Ɉɩɬɢɦɢɡɚɰɢɹ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ ɩɪɢ ɬɨɤɚɪɧɨɣ ɨɛɪɚɛɨɬɤɟ ɦɟɬɚɥɥɨɜ : ɦɨ-
ɧɨɝɪɚɮɢɹ / Ɇ. ȼ. ɋɨɤɨɥɨɜ
. –
Ɇɨɫɤɜɚ
; ȼɨɥɨɝɞɚ : ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ, 2024. –
96 ɫ. : ɢɥ., ɬɚɛɥ.
ISBN 978-5-9729-1907-9
Ɉɩɢɫɚɧɚ ɤɨɧɰɟɩɰɢɹ ɫɨɡɞɚɧɢɹ ɩɪɨɝɪɚɦɦɵ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɢ ɨɩɢɫɚɧɚ ɟɝɨ ɪɚɡɪɚɛɨɬɤɚ. Ⱦɚɧɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɩɪɨɝɪɚɦɦɧɵɯ
ɫɪɟɞɫɬɜ, ɢɫɩɨɥɶɡɭɟɦɵɯ ɞɥɹ ɫɨɡɞɚɧɢɹ ɩɪɨɝɪɚɦɦɵ. ɉɪɨɜɟɞɟɧɵ ɱɢɫɥɟɧɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɵ
ɩɨ ɩɪɨɜɟɪɤɟ ɚɞɟɤɜɚɬɧɨɫɬɢ ɪɚɛɨɬɵ ɩɪɨɝɪɚɦɦɵ ɞɥɹ ɗȼɆ. ɉɪɟɞɫɬɚɜɥɟɧɵ ɦɟɬɨɞɢɤɢ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɢ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ, ɤɨɬɨɪɵɟ ɩɨɤɚɡɵɜɚɸɬ ɨɝɪɨɦɧɵɣ ɩɨɬɟɧɰɢɚɥ CAD-ɫɢɫɬɟɦ ɜ ɢɫɫɥɟɞɨɜɚɧɢɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ.
Ⱦɥɹ ɫɩɟɰɢɚɥɢɫɬɨɜ ɜ ɨɛɥɚɫɬɢ ɦɚɲɢɧɨɫɬɪɨɟɧɢɹ, ɚ ɬɚɤɠɟ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɫɬɚɪɲɢɯ ɤɭɪɫɨɜ ɢ ɦɚɝɢɫɬɪɚɧɬɨɜ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ ɛɚɤɚɥɚɜɪɢɚɬɚ 15.03.05, 15.03.01 ɢ ɦɚɝɢɫɬɪɚɬɭɪɵ
15.04.05, 15.04.01 «Ʉɨɧɫɬɪɭɤɬɨɪɫɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɟ ɨɛɟɫɩɟɱɟɧɢɟ ɦɚɲɢɧɨɫɬɪɨɢɬɟɥɶɧɵɯ ɩɪɨɢɡɜɨɞɫɬɜ» ɩɪɨɮɢɥɶ «Ɍɟɯɧɨɥɨɝɢɹ ɦɚɲɢɧɨɫɬɪɨɟɧɢɹ» ɢ «Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ» ɩɪɨɮɢɥɶ «ɐɢɮɪɨɜɨɟ ɦɚɲɢɧɨɫɬɪɨɟɧɢɟ», ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ɍȾɄ 621.91.01
ȻȻɄ 34.5
ISBN 978-5-9729-1907-9 ɋɨɤɨɥɨɜ Ɇ. ȼ., 2024
ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024
Ɉɮɨɪɦɥɟɧɢɟ. ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024
ɈȽɅȺȼɅȿɇɂȿ
ȼȼȿȾȿɇɂȿ .................................................................................................................. 4
1. Ʌɢɬɟɪɚɬɭɪɧɨ-ɩɚɬɟɧɬɧɵɣ ɨɛɡɨɪ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ ....... 7
1.1. Ɇɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ ...................................................... 7
1.2. Ɉɫɧɨɜɧɵɟ ɩɪɢɧɰɢɩɵ ɜɵɛɨɪɚ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ........................... 18
ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱ ɢɫɫɥɟɞɨɜɚɧɢɹ .......................................................................... 19
2. Ɋɚɡɪɚɛɨɬɤɚ ɩɪɨɝɪɚɦɦɵ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɧɵɯ
ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ................................................................................... 21
2.1. ɏɚɪɚɤɬɟɪɢɫɬɢɤɚ ɫɪɟɞɵ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ Delphi 7 .................................. 21
2.2. Ɋɚɡɪɚɛɨɬɤɚ ɛɥɨɤ-ɫɯɟɦɵ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ .................... 27
2.3. Ɉɩɢɫɚɧɢɟ ɪɚɛɨɬɵ ɫɨɡɞɚɧɧɨɝɨ ɩɪɢɥɨɠɟɧɢɹ .................................................. 30
ȼɵɜɨɞɵ ɤ ɝɥɚɜɟ 2 ................................................................................................... 33
3. ɉɪɨɜɟɞɟɧɢɟ
ɱɢɫɥɟɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ .............................................................. 34
ȼɵɜɨɞɵ ɤ ɝɥɚɜɟ 3 ................................................................................................... 39
4. ɉɪɢɦɟɧɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɢɫɫɥɟɞɨɜɚɧɢɣ ɜ ɦɨɞɟɪɧɢɡɚɰɢɢ ɨɛɪɚɡɨɜɚɬɟɥɶɧɨɝɨ
ɩɪɨɰɟɫɫɚ ɩɪɢ ɨɛɭɱɟɧɢɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ ɦɚɲɢɧɨɫɬɪɨɢɬɟɥɶɧɨɝɨ ɩɪɨɮɢɥɹ
15.00.00 Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ ........................................................................................ 40
4.1. ɋɢɫɬɟɦɚ ɚɜɬɨɦɚɬɢɡɢɪɨɜɚɧɧɨɝɨ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ
ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ ................................................................................................. 40
4.2. ȼɢɡɭɚɥɢɡɚɰɢɹ ɦɟɯɚɧɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɨɛɪɚɛɨɬɤɢ
ɤɚɤ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɮɚɤɬɨɪ ɜɥɢɹɧɢɹ ɧɚ ɨɛɪɚɡɨɜɚɧɢɟ ...................................... 42
ȼɵɜɨɞɵ ɤ ɝɥɚɜɟ 4 ................................................................................................... 45
Ɂɚɤɥɸɱɟɧɢɟ ................................................................................................................. 46
ɋɩɢɫɨɤ ɢɫɩɨɥɶɡɨɜɚɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ...................................................................... 47
ɉɪɢɥɨɠɟɧɢɟ Ⱥ. ɉɪɨɝɪɚɦɦɚ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɧɵɯ
ɩɚɪɚɦɟɬɪɨɜ
ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ....................................................................................................... 53
3
ȼȼȿȾȿɇɂȿ
ȼɨ ɜɫɟ ɜɪɟɦɟɧɚ ɫɨɤɪɚɳɟɧɢɟ ɫɪɨɤɨɜ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɢ ɩɨɞɛɨɪ ɨɩɬɢɦɚɥɶ-
ɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ – ɷɬɨ ɜɚɠɧɟɣɲɢɟ ɬɪɟɛɨɜɚɧɢɹ, ɩɪɟɞɴɹɜɥɹɟɦɵɟ
ɤ ɪɚɡɪɚɛɨɬɤɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ. Ɉɩɪɟɞɟɥɟɧɢɟ ɨɩɬɢɦɚɥɶɧɵɯ ɤɨɧɫɬɪɭɤ-
ɬɢɜɧɵɯ ɢ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɹɜɥɹɟɬɫɹ ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɪɟ-
ɲɚɟɦɨɣ ɡɚɞɚɱɟɣ ɩɪɢ ɩɪɨɟɤɬɢɪɨɜɚɧɢɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɦɚɲɢɧɨ-
ɫɬɪɨɟɧɢɢ.
ɉɨɞ ɨɩɬɢɦɚɥɶɧɵɦɢ ɤɨɧɫɬɪɭɤɬɢɜɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ
ɫɥɟɞɭɟɬ ɩɨɧɢɦɚɬɶ ɧɚɯɨɠɞɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɢ ɝɟɨɦɟɬɪɢɢ ɪɟɠɭɳɟ-
ɝɨ ɢɧɫɬɪɭɦɟɧɬɚ. Ɉɩɪɟɞɟɥɟɧɢɸ ɨɩɬɢɦɚɥɶɧɨɣ ɝɟɨɦɟɬɪɢɢ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ
ɩɨɫɜɹɳɟɧɵ ɢɫɫɥɟɞɨɜɚɧɢɹ Ɏ. Ɍɟɣɥɨɪɚ, Ɇ. ɇ. Ʌɚɪɢɧɚ, Ⱥ. ɂ. Ȼɟɬɚɧɟɥɢ, Ƚ. Ʌ. ɏɚɟ-
ɬɚ, ɋ. ɂ. ɉɟɬɪɭɲɢɧɚ, Ⱥ. ɋ. ȼɟɪɟɳɚɤɚ, ɘ. Ƚ. Ʉɚɛɚɥɞɢɧɚ ɢ ɦɧɨɝɢɯ ɞɪɭɝɢɯ.
ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɧɟɬ ɟɞɢɧɨɝɨ ɩɨɞɯɨɞɚ ɤ ɜɵɛɨɪɭ ɧɚɢɜɵɝɨɞɧɟɣɲɢɯ
ɝɟɨɦɟɬɪɢ-
ɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɪɟɠɭɳɟɣ ɱɚɫɬɢ ɧɚ ɫɬɚɞɢɢ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɢɧɫɬɪɭɦɟɧɬɨɜ ɢ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɚ ɢɦɟɸɳɢɟɫɹ ɜ ɫɩɪɚɜɨɱɧɢɤɚɯ ɢ ɧɨɪɦɚɬɢɜɚɯ ɪɟɤɨ-
ɦɟɧɞɚɰɢɢ ɧɨɫɹɬ ɨɛɳɢɣ ɯɚɪɚɤɬɟɪ. ɗɬɨ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ ɧɚ ɩɪɚɤɬɢɤɟ ɝɟɨɦɟɬ-
ɪɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɩɨɞɜɟɪɝɚɸɬɫɹ ɤɨɪɪɟɤɬɢɪɨɜɤɟ ɜ ɩɪɨɰɟɫɫɟ ɷɤɫɩɥɭɚɬɚɰɢɢ ɢɧ-
ɫɬɪɭɦɟɧɬɚ, ɱɬɨ ɜɥɟɱɟɬ ɡɚ ɫɨɛɨɣ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɢ ɮɢɧɚɧɫɨɜɵɟ
ɩɨɬɟɪɢ.
ȼɨɩɪɨɫɚɦɢ ɚɜɬɨɦɚɬɢɡɚɰɢɢ
ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɟɧɢɹ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢ-
ɦɨɜ ɪɟɡɚɧɢɹ ɩɨɫɜɹɳɟɧɵ ɪɚɛɨɬɵ Ƚ. Ʉ. Ƚɨɪɚɧɫɤɨɝɨ, Ƚ. ɘ. əɤɨɛɫɚ, ȿ. ɋɢɤɨɪɚ,
ȿ. ɉ. ɑɭɪɚɤɨɜɚ ɢ ɞɪ. Ɉɞɧɚɤɨ ɜ ɪɚɛɨɬɚɯ ɷɬɢɯ ɚɜɬɨɪɨɜ ɭɞɟɥɟɧɨ ɦɚɥɨ ɜɧɢɦɚɧɢɹ
ɜɥɢɹɧɢɹ ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɧɚ ɪɟɠɢɦɵ ɪɟɡɚɧɢɹ.
Ɉɞɧɨɜɪɟɦɟɧɧɨɟ ɪɟɲɟɧɢɟ ɤɨɦɩɥɟɤɫɚ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ ɤɨɧɫɬɪɭɤɬɢɜɧɵɯ
ɢ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɜɟɫɶɦɚ ɬɪɭɞɨɟɦɤɨ.
Ɉɬɫɸɞɚ ɜɨɡɧɢɤɚɟɬ
ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɜ ɪɚɡɪɚɛɨɬɤɟ ɫɩɟɰɢɚɥɶɧɨɝɨ ɦɟɬɨɞɢɱɟɫɤɨɝɨ ɢ ɩɪɨɝɪɚɦɦɧɨɝɨ
ɨɛɟɫɩɟɱɟɧɢɹ, ɨɫɭɳɟɫɬɜɥɹɸɳɟɝɨ ɨɩɬɢɦɢɡɚɰɢɸ ɤɨɧɫɬɪɭɤɬɢɜɧɵɯ ɢ ɪɟɠɢɦɧɵɯ ɩɚ-
ɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ, ɢ ɟɝɨ ɪɟɚɥɢɡɚɰɢɹ ɜ ɜɢɞɟ ɢɧɮɨɪɦɚɰɢɨɧɧɨɣ ɫɢɫɬɟɦɵ
ɩɨɞɞɟɪɠɤɢ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ ɫ ɷɥɟɦɟɧɬɚɦɢ ɢɫɤɭɫɫɬɜɟɧɧɨɝɨ ɢɧɬɟɥɥɟɤɬɚ.
4
Ⱥɤɬɭɚɥɶɧɨɫɬɶ ɩɪɨɛɥɟɦɵ ɫɨɫɬɨɢɬ ɜ ɫɨɤɪɚɳɟɧɢɢ ɫɪɨɤɨɜ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɢ
ɩɨɞɛɨɪ ɨɩɬɢɦɚɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ – ɷɬɨ ɜɚɠɧɟɣɲɢɟ ɬɪɟɛɨɜɚ-
ɧɢɹ, ɩɪɟɞɴɹɜɥɹɟɦɵɟ ɤ ɪɚɡɪɚɛɨɬɤɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ. Ɉɩɪɟɞɟɥɟɧɢɟ ɨɩ-
ɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɞɧɭ ɢɡ ɱɚɫɬɵɯ ɡɚɞɚɱ ɜ ɩɪɨɰɟɫɫɟ
ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ.
ɉɨɷɬɨɦɭ ɨɫɨɛɨ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢɨɛɪɟɬɚɟɬ ɩɪɚɜɢɥɶɧɵɣ ɜɵɛɨɪ ɞɟɣɫɬɜɢ-
ɬɟɥɶɧɨ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ
ɪɟɡɚɧɢɹ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɧɚɢɛɨɥɶɲɢɣ ɷɤɨɧɨɦɢ-
ɱɟɫɤɢɣ ɷɮɮɟɤɬ ɫ ɭɱɟɬɨɦ, ɩɨ ɜɨɡɦɨɠɧɨɫɬɢ, ɜɫɟɯ ɮɚɤɬɨɪɨɜ, ɜɥɢɹɸɳɢɯ ɧɚ ɩɪɨɢɡ-
ɜɨɞɢɬɟɥɶɧɨɫɬɶ, ɬɟɯɧɢɱɟɫɤɢɟ ɤɚɱɟɫɬɜɚ ɢ ɷɤɨɧɨɦɢɤɭ ɩɪɨɢɡɜɨɞɫɬɜɚ.
ɐɟɥɶɸ ɪɚɛɨɬɵ ɹɜɥɹɟɬɫɹ ɫɨɡɞɚɧɢɟ ɚɥɝɨɪɢɬɦɚ ɢ ɩɪɨɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ
ɞɥɹ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɧɚ ɩɪɢɦɟɪɟ ɬɨɤɚɪɧɨɣ
ɨɛɪɚɛɨɬɤɢ (ɪɚɡɪɚɛɨɬɤɚ ɩɨɪɹɞɤɚ ɪɚɫɱɺɬɨɜ, ɩɪɨɢɡɜɨɞɢɦɵɯ ɩɪɨɝɪɚɦɦɨɣ, ɪɚɡɪɚɛɨɬ-
ɤɚ ɚɥɝɨɪɢɬɦɚ ɪɚɛɨɬɵ ɩɪɨɝɪɚɦɦɵ).
Ⱦɥɹ ɞɨɫɬɢɠɟɧɢɹ
ɰɟɥɢ ɛɵɥɢ ɩɨɫɬɚɜɥɟɧɵ ɫɥɟɞɭɸɳɢɟ ɡɚɞɚɱɢ:
1) ɪɚɡɪɚɛɨɬɚɬɶ ɩɨɪɹɞɨɤ ɪɚɫɱɺɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚ-
ɧɢɹ;
2) ɪɚɡɪɚɛɨɬɚɬɶ ɚɥɝɨɪɢɬɦ ɪɚɛɨɬɵ ɩɪɨɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ;
3) ɩɪɨɜɟɫɬɢ ɱɢɫɥɟɧɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɵ ɩɨ ɩɪɨɜɟɪɤɟ ɚɞɟɤɜɚɬɧɨɫɬɢ ɪɚɛɨɬɵ
ɫɨɡɞɚɧɧɨɝɨ ɛɥɨɤɚ.
Ɉɛɴɟɤɬɨɦ ɢɫɫɥɟɞɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɨɩɬɢɦɢɡɚɰɢɹ ɩɪɨɰɟɫɫɨɜ ɥɟɡɜɢɣɧɨɣ ɨɛ-
ɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ.
ɉɪɟɞɦɟɬɨɦ ɢɫɫɥɟɞɨɜɚɧɢɹ ɹɜɥɹɟɬɫɹ ɩɪɨɝɪɚɦɦɚ, ɨɫɭɳɟɫɬɜɥɹɸɳɚɹ ɩɨɥɧɭɸ
ɨɩɬɢɦɢɡɚɰɢɸ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɨɜ ɦɟɯɚɧɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟ
-
ɪɢɚɥɨɜ.
ɇɚɭɱɧɨɣ ɧɨɜɢɡɧɨɣ ɹɜɥɹɟɬɫɹ ɪɚɡɪɚɛɨɬɚɧɧɚɹ ɤɨɧɰɟɩɰɢɹ ɫɨɡɞɚɧɢɹ ɩɪɨ-
ɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ɞɥɹ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨ-
ɰɟɫɫɚ ɪɟɡɚɧɢɹ, ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɛɥɨɤɨɦ ɨɩɬɢɦɢɡɚɰɢɢ ɋȺɉɊ ɩɪɨɰɟɫɫɨɜ ɦɟɯɚɧɢ-
ɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɧɚ ɩɪɢɦɟɪɟ ɬɨɤɚɪɧɨɣ ɨɛɪɚɛɨɬɤɢ.
5
ɉɪɚɤɬɢɱɟɫɤɨɣ ɡɧɚɱɢɦɨɫɬɶɸ ɹɜɥɹɟɬɫɹ ɫɨɡɞɚɧɢɟ ɚɥɝɨɪɢɬɦɚ ɢ ɩɪɨɝɪɚɦɦɧɨɝɨ
ɨɛɟɫɩɟɱɟɧɢɹ ɞɥɹ ɪɚɫɱɟɬɚ ɨɩɬɢɦɚɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɧɚ ɩɪɢɦɟɪɟ
ɬɨɤɚɪɧɨɣ ɨɛɪɚɛɨɬɤɢ.
Ɋɟɚɥɢɡɚɰɢɹ ɪɚɛɨɬɵ. Ɉɫɧɨɜɧɵɟ ɢɞɟɢ ɢ ɪɟɡɭɥɶɬɚɬɵ ɪɚɛɨɬɵ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ
ɭɱɟɛɧɨɦ ɩɪɨɰɟɫɫɟ ɩɪɢ ɩɨɞɝɨɬɨɜɤɟ ɫɬɭɞɟɧɬɨɜ ɩɨ ɧɚɩɪɚɜɥɟɧɢɹɦ 15.03.05,
15.04.05 «Ʉɨɧɫɬɪɭɤɬɨɪɫɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɟ ɨɛɟɫɩɟɱɟɧɢɟ ɦɚɲɢɧɨɫɬɪɨɢɬɟɥɶɧɵɯ
ɩɪɨɢɡɜɨɞɫɬɜ» ɢ ɧɚɩɪɚɜɥɟɧɢɹɦ 15.03.01, 15.04.01 «Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ».
Ɉɩɪɟɞɟɥɟɧɢɟ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɞɧɭ ɢɡ ɱɚɫɬɵɯ
ɡɚɞɚɱ ɜ ɩɪɨɰɟɫɫɟ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ.
Ɋɟɠɢɦɵ ɪɟɡɚɧɢɹ ɫɥɭɠɚɬ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɨɫɧɨɜɨɣ ɞɥɹ ɪɚɡɪɚɛɨɬɤɢ
ɤɢɧɟɦɚɬɢɤɢ ɫɬɚɧɤɚ – ɱɢɫɟɥ ɨɛɨɪɨɬɨɜ, ɱɢɫɟɥ ɯɨɞɨɜ, ɜɟɥɢɱɢɧɵ ɩɨɞɚɱ; ɞɢɧɚɦɢɤɢ
ɫɬɚɧɤɚ – ɦɨɳɧɨɫɬɢ ɷɥɟɤɬɪɨɦɨɬɨɪɚ, ɭɫɢɥɢɣ, ɜɨɡɧɢɤɚɸɳɢɯ ɩɪɢ ɪɟɡɚɧɢɢ, ɜɟɥɢɱɢɧ
ɤɪɭɬɹɳɢɯ ɦɨɦɟɧɬɨɜ ɧɚ ɲɩɢɧɞɟɥɹɯ ɢ ɜɚɥɚɯ ɫɬɚɧɤɚ, ɩɪɨɱɧɨɫɬɢ ɢ ɠɟɫɬɤɨɫɬɢ ɨɬ-
ɞɟɥɶɧɵɯ ɞɟɬɚɥɟɣ ɢ ɭɡɥɨɜ ɫɬɚɧɤɚ
ɢ ɬ. ɞ.
ɇɚ ɨɫɧɨɜɟ ɜɵɛɪɚɧɧɵɯ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɩɟɪɢɨɞɢɱɧɨɫɬɶ ɢ
ɩɨɪɹɞɨɤ ɫɦɟɧɵ ɪɟɠɭɳɢɯ ɢɧɫɬɪɭɦɟɧɬɨɜ, ɪɚɫɯɨɞ ɢɧɫɬɪɭɦɟɧɬɨɜ, ɨɩɪɟɞɟɥɹɟɬɫɹ
ɱɢɫɥɨ ɪɚɛɨɬɚɸɳɢɯ, ɮɨɧɞɵ ɡɚɪɩɥɚɬɵ, ɱɢɫɥɨ ɫɬɚɧɤɨɜ, ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɜɵɩɨɥ-
ɧɟɧɢɹ ɡɚɞɚɧɧɨɣ ɩɪɨɝɪɚɦɦɵ, ɩɥɨɳɚɞɢ ɰɟɯɨɜ ɢ ɭɱɚɫɬɤɨɜ, ɧɟɨɛɯɨɞɢɦɵɟ ɤɚɩɢɬɚ-
ɥɨɜɥɨɠɟɧɢɹ ɢ ɦɧɨɝɢɟ ɞɪɭɝɢɟ ɷɥɟɦɟɧɬɵ ɨɪɝɚɧɢɡɚɰɢɢ ɢ ɷɤɨɧɨɦɢɤɢ ɩɪɨɢɡɜɨɞɫɬɜɚ.
ɉɨɷɬɨɦɭ ɨɫɨɛɨ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢɨɛɪɟɬɚɟɬ ɩɪɚɜɢɥɶɧɵɣ
ɬɟɥɶɧɨ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ
ɜɵɛɨɪ ɞɟɣɫɬɜɢ-
ɧɚɢɛɨɥɶɲɢɣ ɷɤɨɧɨɦɢ-
ɱɟɫɤɢɣ ɷɮɮɟɤɬ ɫ ɭɱɟɬɨɦ, ɩɨ ɜɨɡɦɨɠɧɨɫɬɢ, ɜɫɟɯ ɮɚɤɬɨɪɨɜ, ɜɥɢɹɸɳɢɯ ɧɚ ɩɪɨɢɡ-
ɜɨɞɢɬɟɥɶɧɨɫɬɶ, ɬɟɯɧɢɱɟɫɤɢɟ ɤɚɱɟɫɬɜɚ ɢ ɷɤɨɧɨɦɢɤɭ ɩɪɨɢɡɜɨɞɫɬɜɚ [1].
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɪɚɡɪɚɛɨɬɤɚ ɩɪɨɝɪɚɦɦɵ ɪɚɫɱɟɬɚ ɨɩɬɢ-
ɦɚɥɶɧɵɯ ɪɟɠɢɦɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɪɟɡɚɧɢɹ ɧɚ ɩɪɢɦɟɪɟ ɬɨɤɚɪɧɨɣ ɨɛɪɚ-
ɛɨɬɤɢ.
6
1. ɅɂɌȿɊȺɌɍɊɇɈ-ɉȺɌȿɇɌɇɕɃ ɈȻɁɈɊ ɆȿɌɈȾɈȼ
ɈɉɌɂɆɂɁȺɐɂɂ ɉɊɈɐȿɋɋɈȼ ɊȿɁȺɇɂə
Ⱦɥɹ ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱɢ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɢɬɶ ɨɛɡɨɪ ɫɭɳɟ-
ɫɬɜɭɸɳɢɯ ɦɟɬɨɞɨɜ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ. Ɋɚɫɫɦɨɬɪɢɦ ɜɵɲɟɭɩɨɦɹ-
ɧɭɬɨɟ ɩɨɷɬɚɩɧɨ.
1.1. Ɇɟɬɨɞɵ ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ
Ɋɚɫɱɟɬ ɨɩɬɢɦɚɥɶɧɵɯ ɩɚɪɚɦɟɬɪɨɜ (ɪɟɠɢɦɨɜ ɪɟɡɚɧɢɹ, ɩɚɪɚɦɟɬɪɨɜ ɤɚɱɟɫɬɜɚ
ɢ ɞɪ.) ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɢɥɢ ɨɩɟɪɚɰɢɢ ɩɪɢ ɡɚɞɚɧɧɨɣ ɫɬɪɭɤɬɭɪɟ ɫ ɩɨɡɢ-
ɰɢɢ ɧɟɤɨɬɨɪɨɝɨ ɤɪɢɬɟɪɢɹ
ɧɚɡɵɜɚɸɬ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɨɩɬɢɦɢɡɚɰɢɟɣ, ɤɨɬɨɪɚɹ
ɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬ ɨɩɪɟɞɟɥɟɧɢɟ ɬɚɤɢɯ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ ɯ, ɩɪɢ ɤɨɬɨɪɵɯ ɧɟɤɨ-
ɬɨɪɚɹ ɮɭɧɤɰɢɹ F(x), ɧɚɡɵɜɚɟɦɚɹ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɟɣ, ɢɥɢ ɮɭɧɤɰɢɟɣ ɷɮɮɟɤɬɢɜɧɨ-
ɫɬɢ (ɧɚɩɪɢɦɟɪ, ɩɪɢɜɟɞɟɧɧɵɟ ɡɚɬɪɚɬɵ, ɬɟɯɧɨɥɨɝɢɱɟɫɤɚɹ ɫɟɛɟɫɬɨɢɦɨɫɬɶ, ɲɬɭɱɧɨɟ
ɜɪɟɦɹ, ɲɬɭɱɧɚɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ, ɬɟɯɧɨɥɨɝɢɱɟɫɤɚɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ,
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɟ ɜɪɟɦɹ ɢ ɞɪ.), ɩɪɢɧɢɦɚɟɬ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɦ ɩɪɨɟɤɬɢɪɨɜɚɧɢɢ ɢɫ-
ɩɨɥɶɡɭɸɬ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɢ ɬɚɤɢɟ ɦɟɬɨɞɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢ-
ɪɨɜɚɧɢɹ, ɤɚɤ ɥɢɧɟɣɧɨɟ, ɰɟɥɨɱɢɫɥɟɧɧɨɟ, ɞɢɧɚɦɢɱɟɫɤɨɟ, ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɢ ɞɪ.
ȼ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɦ ɩɪɨɟɤɬɢɪɨɜɚɧɢɢ ɨɩɟɪɚɰɢɨɧɧɵɟ ɦɨɞɟɥɢ, ɨɩɢɫɚɧɧɵɟ ɦɟ-
ɬɨɞɚɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɡɚɩɢɫɵɜɚɸɬ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ [2]:
ɝɞɟ ɜɫɟ ɭɩɪɚɜɥɹɟɦɵɟ ɯ
ɦɨɝɭɬ ɩɪɢɧɢɦɚɬɶ ɡɧɚɱɟɧɢɹ ɢɡ ɦɧɨɠɟɫɬɜɚ [ɚ1i, a2i]
i
xxxF
o
min(max);...,
n
,21
,21
jnj
niaxa
dd
,1,
iii
21
½
°
mjbxxxg
d
;,1,...,
(1.1)
¾
°
¿
ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ ɱɢɫɟɥ;
F(x) ɢ g
b
(ɯ) – ɫɤɚɥɹɪɧɵɟ ɮɭɧɤɰɢɢ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ;
j
– ɡɚɞɚɧɧɵɟ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɮɭɧɤɰɢɢ.
j
7
Ɂɚɞɚɱɢ ɩɨɞɨɛɧɨɝɨ ɬɢɩɚ ɜ ɬɟɯɧɨɥɨɝɢɢ ɦɚɲɢɧɨɫɬɪɨɟɧɢɹ ɜɨɡɧɢɤɚɸɬ ɩɪɢ
ɨɩɪɟɞɟɥɟɧɢɢ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ ɨɛɪɚɛɨɬɤɢ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɝɭɬ ɛɵɬɶ ɢɫ-
ɩɨɥɶɡɨɜɚɧɵ ɦɟɬɨɞɵ ɥɢɧɟɣɧɨɝɨ ɢ ɧɟɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ.
ɉɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɜɵɡɵɜɚɟɬ ɬɪɭɞɧɨɫɬɢ,
ɫɜɹɡɚɧɧɵɟ ɫ ɥɢɧɟɣɧɨɫɬɶɸ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɢ ɨɝɪɚɧɢɱɟɧɢɣ. ɇɚɩɪɢɦɟɪ,
ɩɪɢ ɧɚɡɧɚɱɟɧɢɢ ɩɥɚɧɚ ɱɟɪɧɨɜɨɣ ɨɛɪɚɛɨɬɤɢ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɝɨɬɨɜɤɢ ɞɨɥɠɧɵ ɛɵɬɶ
ɭɱɬɟɧɵ ɨɝɪɚɧɢɱɟɧɢɹ, ɫɜɹɡɚɧɧɵɟ
ɫ ɬɟɯɧɢɱɟɫɤɢɦɢ ɞɚɧɧɵɦɢ ɨɛɨɪɭɞɨɜɚɧɢɹ, ɯɚɪɚɤ-
ɬɟɪɢɫɬɢɤɚɦɢ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ, ɪɚɡɦɟɪɚɦɢ ɞɟɬɚɥɢ ɢ ɞɪ. ɗɬɢ ɨɝɪɚɧɢɱɟɧɢɹ
ɜɵɪɚɠɚɸɬɫɹ ɱɟɪɟɡ ɩɚɪɚɦɟɬɪɵ ɩɟɪɟɯɨɞɨɜ (ɪɚɛɨɱɢɯ ɯɨɞɨɜ) – ɪɟɠɢɦɵ ɪɟɡɚɧɢɹ ɢ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɟɥɢɱɢɧɵ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɭɫɥɨɜɢɹ ɨɛɪɚɛɨɬɤɢ (ɦɨɳɧɨɫɬɶ
ɩɪɢɜɨɞɚ ɨɛɨɪɭɞɨɜɚɧɢɹ; ɞɨɩɭɫɬɢɦɚɹ ɫɢɥɚ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɦɟɯɚɧɢɡɦ ɩɨɞɚɱɢ
ɫɬɚɧɤɚ; ɩɪɨɱɧɨɫɬɶ ɢ ɫɬɨɣɤɨɫɬɶ ɪɟɠɭɳɟɝɨ ɢɧɫɬɪɭɦɟɧɬɚ; ɞɨɩɭɫɬɢɦɨɟ ɩɟɪɟɦɟɳɟ-
ɧɢɟ ɡɚɝɨɬɨɜɤɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥ ɪɟɡɚɧɢɹ
).
Ⱦɥɹ ɫɨɝɥɚɫɨɜɚɧɢɹ ɡɧɚɱɟɧɢɣ ɩɨɞɚɱɢ s ɢ ɱɚɫɬɨɬ ɜɪɚɳɟɧɢɹ ɲɩɢɧɞɟɥɹ n
ɫ ɩɚɫɩɨɪɬɧɵɦɢ ɞɚɧɧɵɦɢ ɨɛɨɪɭɞɨɜɚɧɢɹ ɢɫɩɨɥɶɡɭɸɬ ɤɨɷɮɮɢɰɢɟɧɬɵ ɝɟɨɦɟɬɪɢɱɟ-
ɫɤɢɯ ɪɹɞɨɜ ɩɨɞɚɱ (ij
) ɢ ɱɚɫɬɨɬ ɜɪɚɳɟɧɢɹ ɲɩɢɧɞɟɥɹ (ijɩ):
s
1
z
s
½
ss
nn
;
M
1
1
°
s
(1.2)
¾
1
z
n
.
M
°
n
¿
Ʌɭɱɲɟɦɭ ɜɚɪɢɚɧɬɭ ɩɥɚɧɚ ɨɛɪɚɛɨɬɤɢ ɛɭɞɭɬ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ ɦɢɧɢɦɚɥɶɧɵɟ
ɡɚɬɪɚɬɵ:
p
CC
,
¦
i
1
i
ɝɞɟ ɋi – ɡɚɬɪɚɬɵ ɧɚ ɜɵɩɨɥɧɟɧɢɟ ɩɟɪɟɯɨɞɚ (ɪɚɛɨɱɟɝɨ ɯɨɞɚ);
ɪ – ɱɢɫɥɨ ɩɟɪɟɯɨɞɨɜ (ɪɚɛɨɱɢɯ ɯɨɞɨɜ).
Ɂɚɬɟɦ ɩɭɬɟɦ ɥɨɝɚɪɢɮɦɢɪɨɜɚɧɢɹ ɨɝɪɚɧɢɱɟɧɢɣ ɢ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ, ɫɜɹɡɚɧ-
ɧɵɯ ɫ ɨɞɧɢɦ ɩɟɪɟɯɨɞɨɦ (ɪɚɛɨɱɢɦ ɯɨɞɨɦ), ɩɨɥɭɱɚɸɬ ɥɢɧɟɣɧɭɸ ɡɚɞɚɱɭ:
bxa d
; i = 1, n,
ijjij
ɝɞɟ
Z = k
+ k
1x1
+ k
2x2
0
x
= lnt; x2 = zs; x3 = zn;
1
+ k
ĺ min ɩɪɢ ɨɝɪɚɧɢɱɟɧɢɹɯ
3ɯ3
¦
31
8
t, s – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɝɥɭɛɢɧɚ ɪɟɡɚɧɢɹ ɢ ɩɨɞɚɱɚ ɩɪɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɟ-
ɪɟɯɨɞɟ;
a
– ɤɨɷɮɮɢɰɢɟɧɬɵ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɩɨɤɚɡɚɬɟɥɟɣ ɫɬɟɩɟɧɢ ɩɪɢ ɝɥɭɛɢɧɟ ɪɟɡɚ-
ij
ɧɢɹ, ɩɨɞɚɱɢ ɢ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ ɜ ɮɨɪɦɭɥɚɯ ɫɢɥ ɢ ɫɤɨɪɨɫɬɢ ɪɟɡɚɧɢɹ,
ɚ ɬɚɤɠɟ ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɪɹɞɨɜ ɩɨɞɚɱ ɢ ɱɚɫɬɨɬ ɜɪɚɳɟɧɢɹ
ɲɩɢɧɞɟɥɹ;
k
, ..., k3, bi – ɤɨɷɮɮɢɰɢɟɧɬɵ ɢ ɜɟɥɢɱɢɧɵ ɨɝɪɚɧɢɱɟɧɢɣ, ɡɚɜɢɫɹɳɢɟ
0
ɨɬ ɤɨɧɤɪɟɬɧɵɯ ɭɫɥɨɜɢɣ ɨɛɪɚɛɨɬɤɢ.
Ɉɞɧɢɦ ɢɡ ɜɨɡɦɨɠɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɧɟɥɢɧɟɣɧɵɯ ɡɚɞɚɱ
ɹɜɥɹɟɬɫɹ ɦɟɬɨɞ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɣ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɦɚɬ-
ɪɢɜɚɬɶ ɡɚɞɚɱɢ ɫ ɭɱɟɬɨɦ ɨɫɨɛɟɧɧɨɫɬɟɣ ɢɯ ɢɧɠɟɧɟɪɧɨɣ ɩɨɫɬɚɧɨɜɤɢ. Ɉɫɧɨɜɧɨɟ ɬɪɟɛɨ-
ɜɚɧɢɟ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ ɜɫɟ ɬɟɯɧɢɱɟɫɤɢɟ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɛɵɥɢ ɜɵɪɚɠɟɧɵ ɜ ɜɢɞɟ ɩɨɥɨɠɢɬɟɥɶɧɵɯ
ɩɨɥɢɧɨɦɨɜ (ɩɨɡɢɧɨɦɵ) ɨɬ
ɪɟɝɭɥɢɪɭɟɦɵɯ ɩɚɪɚɦɟɬɪɨɜ. ȼɨ ɦɧɨɝɢɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɡɚɞɚɱɚɯ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠ-
ɞɭ ɩɚɪɚɦɟɬɪɚɦɢ ɩɪɢɜɨɞɹɬ ɤ ɮɭɧɤɰɢɹɦ ɬɢɩɚ ɩɨɡɢɧɨɦɨɜ.
Ⱦɥɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧ ɦɟɬɨɞ ɞɢɧɚ-
ɦɢɱɟɫɤɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɣ ɫɜɨɞɢɬɫɹ ɤ ɪɟɤɭɪɪɟɧɬɧɵɦ ɫɨɨɬɧɨɲɟɧɢ-
ɹɦ (ɧɚɩɪɢɦɟɪ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɪɢɩɭɫɤɚ ɩɨ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦ ɩɟɪɟɯɨɞɚɦ). Ⱦɢɧɚ-
ɦɢɱɟɫɤɨɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɟ ɹɜɥɹɟɬɫɹ ɜɵɱɢɫɥɢɬɟɥɶɧɵɦ ɦɟɬɨɞɨɦ, ɩɪɢɜɨɞɹɳɢɦ
ɤ ɝɥɨɛɚɥɶɧɨɦɭ ɨɩɬɢɦɭɦɭ.
ɂɫɩɨɥɶɡɭɸɬ
ɬɚɤɠɟ ɪɚɡɥɢɱɧɵɟ ɦɟɬɨɞɵ ɩɨɢɫɤɚ, ɢɫɤɥɸɱɚɸɳɢɟ ɩɨɥɧɵɣ ɩɟ-
ɪɟɛɨɪ (ɧɚɩɪɢɦɟɪ, ɪɟɝɭɥɹɪɧɨɝɨ ɩɨɢɫɤɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɩɬɢɦɚɥɶɧɵɯ ɪɟɠɢɦɨɜ
ɪɟɡɚɧɢɹ ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɫɬɭɩɟɧɱɚɬɵɯ ɜɚɥɨɜ ɧɚ ɬɨɤɚɪɧɨɦ ɝɢɞɪɨɤɨɩɢɪɨɜɚɥɶɧɨɦ
ɩɨɥɭɚɜɬɨɦɚɬɟ). Ɂɚɞɚɸɬ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ (ɪɚɡɦɟɪɵ ɢ ɦɚɬɟɪɢɚɥ ɞɟɬɚɥɢ, ɪɟɠɭɳɢɣ
ɢɧɫɬɪɭɦɟɧɬ, ɝɥɭɛɢɧɭ ɪɟɡɚɧɢɹ, ɠɟɫɬɤɨɫɬɶ ɭɡɥɨɜ ɫɬɚɧɤɚ, ɰɢɤɥɨɜɵɟ ɢ ɜɧɟɰɢɤɥɨɜɵɟ
ɩɨɬɟɪɢ ɜɪɟɦɟɧɢ ɪɚɛɨɬɵ ɨɛɨɪɭɞɨɜɚɧɢɹ). Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɟɠɢɦ ɨɛɪɚɛɨɬɤɢ, ɭɞɨ-
ɜɥɟɬɜɨɪɹɸɳɢɣ ɭɫɥɨɜɢɹɦ ɩɨ ɬɨɱɧɨɫɬɢ ɨɛɪɚɛɨɬɤɢ, ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɢ,
ɦɨɳɧɨɫɬɢ, ɪɚɫɯɨɞɭɟɦɨɣ ɧɚ ɪɟɡɚɧɢɟ, ɤɢɧɟɦɚɬɢɤɟ ɫɬɚɧɤɚ ɢ ɩɪɢɜɨɞɹɳɢɣ ɰɟɥɟɜɭɸ
ɮɭɧɤɰɢɸ ɤ ɦɚɤɫɢɦɭɦɭ.
9
ɉɪɢ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɨɰɟɧɢɜɚɸɬ
ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɢɝɨɞɧɨɫɬɢ ɢɯ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɞɥɹ ɪɟɲɟɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ
ɡɚɞɚɱ ɜ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɭɫɥɨɜɢɹɯ. ɂɯ ɨɰɟɧɢɜɚɸɬ ɫ ɩɨɦɨɳɶɸ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ
ɚɧɚɥɢɡɚ ɩɭɬɟɦ:
1) ɫɪɚɜɧɟɧɢɹ ɞɜɭɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɤɨɧɤɪɟɬɧɨɣ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɡɚɞɚɱɢ –
ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɧɨɪɦɚɬɢɜɧɵɯ ɞɚɧɧɵɯ;
ɩɪɢ ɷɬɨɦ ɩɪɨɜɟɪɹɸɬ ɝɢɩɨɬɟɡɭ ɫɨɨɬɜɟɬɫɬɜɢɹ ɡɧɚɱɟɧɢɣ ɞɜɭɯ ɜɵɛɨɪɨɤ;
2) ɩɪɨɜɟɪɤɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ
ɦɨɞɟɥɢ ɧɚ ɱɭɜɫɬɜɢɬɟɥɶɧɨɫɬɶ ɜɥɢɹɧɢɹ ɫɥɭ-
ɱɚɣɧɵɯ ɮɚɤɬɨɪɨɜ;
3) ɩɪɨɜɟɪɤɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɧɚ ɟɟ ɚɞɟɤɜɚɬɧɨɫɬɶ ɪɟɚɥɶɧɨɦɭ ɬɟɯɧɨ-
ɥɨɝɢɱɟɫɤɨɦɭ ɩɪɨɰɟɫɫɭ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɟ ɬɟɯɧɢɱɟɫɤɢɟ ɨɝɪɚɧɢɱɟɧɢɹ, ɜɡɹɬɵɟ ɢɡ ɪɚ-
ɛɨɬɵ [1]. ȼɫɟ ɨɝɪɚɧɢɱɟɧɢɹ ɩɪɢɜɨɞɹɬɫɹ ɤ ɧɟɪɚɜɟɧɫɬɜɚɦ, ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɤɨɬɨɪɵɯ
ɡɚɮɢɤɫɢɪɨɜɚɧɵ ɩɚɪɚɦɟɬɪɵ ɪɟɠɢɦɚ ɪɟɡɚɧɢɹ, ɚ ɜ ɩɪɚɜɨɣ – ɩɚɪɚɦɟɬɪɵ, ɯɚɪɚɤɬɟɪɢ-
ɡɭɸɳɢɟ ɷɥɟɦɟɧɬɵ ɋɉɂȾ:
Ɉɝɪɚɧɢɱɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɪɟɡɚɧɢɹ, ɨɛɭɫɥɨɜ-
1)
ɥɟɧɧɨɣ ɩɪɢɧɹɬɨɣ ɫɬɨɣɤɨɫɬɶɸ ɢɧɫɬɪɭɦɟɧɬɚ, ɦɚɬɟɪɢɚɥɨɦ ɪɟɠɭɳɟɣ ɱɚɫɬɢ ɢɧɫɬɪɭ-
ɦɟɧɬɚ, ɟɝɨ ɝɟɨɦɟɬɪɢɟɣ, ɝɥɭɛɢɧɨɣ ɪɟɡɚɧɢɹ, ɩɨɞɚɱɟɣ, ɦɟɯɚɧɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ
ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɫ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɢ ɫɤɨɪɨɫɬɶɸ ɪɟɡɚɧɢɹ, ɨɩɪɟɞɟɥɹ-
ɟɦɨɣ ɤɢɧɟɦɚɬɢɤɨɣ ɫɬɚɧɤɚ, ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ:
1000
CK
y
X
ns
ɝɞɟ
ɋX – ɩɨɫɬɨɹɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɧɨɪɦɚɬɢɜɧɵɟ ɭɫɥɨɜɢɹ
d
XX
x
m
S
Tt d
, (1.3)
X
ɨɛɪɚɛɨɬɤɢ;
Ʉ
– ɨɛɳɢɣ ɩɨɩɪɚɜɨɱɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɧɚ ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ, ɭɱɢɬɵɜɚɸ-
X
ɳɢɣ ɢɡɦɟɧɟɧɧɵɟ ɭɫɥɨɜɢɹ ɨɛɪɚɛɨɬɤɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɧɨɪɦɚɬɢɜɧɵɦɢ;
x
, yX – ɩɨɤɚɡɚɬɟɥɢ ɫɬɟɩɟɧɟɣ; ɫɤɨɪɨɫɬɶ ɪɟɡɚɧɢɹ
X
X
= S ā d ā n/1000;
d – ɞɢɚɦɟɬɪ ɡɚɝɨɬɨɜɤɢ.
10
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