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А. А. ЛУКАШЕВИЧ
ТЕОРИЯ РАСЧЕТА
ПЛАСТИН И ОБОЛОЧЕК
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɇɢɧɢɫɬɟɪɫɬɜɨɨɛɪɚɡɨɜɚɧɢɹɢɧɚɭɤɢ
ɊɨɫɫɢɣɫɤɨɣɎɟɞɟɪɚɰɢɢ
ɋɚɧɤɬ-ɉɟɬɟɪɛɭɪɝɫɤɢɣɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ
ɚɪɯɢɬɟɤɬɭɪɧɨ-ɫɬɪɨɢɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ
ȺȺɅɍɄȺɒȿȼɂɑ
ɌȿɈɊɂəɊȺɋɑȿɌȺ
ɉɅȺɋɌɂɇɂɈȻɈɅɈɑȿɄ
ɍɱɟɛɧɨɟɩɨɫɨɛɢɟ
ɋɚɧɤɬ-ɉɟɬɟɪɛɭɪɝ
2017
1
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɍȾɄ 624.04
Ɋɟɰɟɧɡɟɧɬɵ ɞ-ɪ ɬɟɯɧ ɧɚɭɤ ɩɪɨɮɟɫɫɨɪ Ⱥ. Ɇ. ɍɡɞɢɧ ɉɟɬɟɪɛɭɪɝɫɤɢɣ
ɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣɭɧɢɜɟɪɫɢɬɟɬɩɭɬɟɣɫɨɨɛɳɟɧɢɹɂɦɩɟɪɚɬɨɪɚȺɥɟɤɫɚɧɞɪɚI);
ɞ-ɪɬɟɯɧɧɚɭɤɩɪɨɮɟɫɫɨɪɅ. Ɇ Ʉɚɝɚɧ-Ɋɨɡɟɧɰɜɟɣɝ (ɋɉɛȽȺɋɍ)
ɅɭɤɚɲɟɜɢɱȺȺ
Ɍɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ: ɭɱɟɛɩɨɫɨɛɢɟ ȺȺɅɭ
ɤɚɲɟɜɢɱɋɉɛȽȺɋɍ. – ɋɉɛ7. – 131 ɫ
ISBN 978-5-9227-0779-4
ɂɡɥɨɠɟɧɵɨɫɧɨɜɵɬɟɨɪɢɢɢɡɝɢɛɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤɚɬɚɤɠɟɱɢɫɥɟɧ
ɧɨ-ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɢ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɢɯ ɪɚɫɱɟɬɚ ɉɪɢɜɨɞɹɬɫɹ ɨɫɧɨɜɧɵɟ
ɩɨɥɨɠɟɧɢɹ ɬɟɯɧɢɱɟɫɤɢɯ ɬɟɨɪɢɣ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɢ ɨɛɨɥɨɱɟɤ ɤɥɚɫɫɢɮɢɤɚɰɢɹ
ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɪɚɫɱɟɬɚ ɬɨɧɤɨɫɬɟɧɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ Ɋɚɫɫɦɚɬɪɢɜɚ
ɸɬɫɹ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɜ ɱɚɫɬɧɨɫɬɢ ɦɟɬɨɞ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɢ ɦɟɬɨɞ Ȼɭɛɧɨɜɚ –
Ƚɚɥɺɪɤɢɧɚ ɉɪɢɜɨɞɹɬɫɹɜɚɪɢɚɰɢɨɧɧɵɟ ɩɨɫɬɚɧɨɜɤɢɞɥɹɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧ
ɢɨɛɨɥɨɱɟɤ ɢ ɨɫɧɨɜɧɵɟɦɟɬɨɞɵɢɯɪɟɲɟɧɢɹ: ɜɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣɦɟɬɨɞ
ɦɟɬɨɞɊɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨɦɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ ɂɡɥɨɠɟɧɢɟɦɚɬɟɪɢɚ
ɥɚɫɨɩɪɨɜɨɠɞɚɟɬɫɹɩɪɢɦɟɪɚɦɢɪɚɫɱɟɬɨɜ
ɉɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɦɚɝɢɫɬɪɚɧɬɨɜ ɢ ɫɬɭɞɟɧɬɨɜ ɫɬɪɨɢɬɟɥɶɧɵɯ ɫɩɟɰɢ
ɚɥɶɧɨɫɬɟɣ
Ɍɚɛɥ4ɂɥ46Ȼɢɛɥɢɨɝɪ16 ɧɚɡɜ
Ɋɟɤɨɦɟɧɞɨɜɚɧɨɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦɫɨɜɟɬɨɦ ɋɉɛȽȺɋɍ ɜ ɤɚɱɟɫɬɜɟ
ɭɱɟɛɧɨɝɨɩɨɫɨɛɢɹ
ISBN 978-5-9227-0779-4
‹ȺȺɅɭɤɚɲɟɜɢɱ7
” ɋɚɧɤɬ-ɉɟɬɟɪɛɭɪɝɫɤɢɣɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣ
ɚɪɯɢɬɟɤɬɭɪɧɨ-ɫɬɪɨɢɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ7
2
ȼȼȿȾȿɇɂȿ
Ɉɞɧɢɦɢɡɜɚɠɧɵɯɷɥɟɦɟɧɬɨɜɡɞɚɧɢɣ ɢɫɨɨɪɭɠɟɧɢɣɹɜɥɹɸɬɫɹ
ɪɚɡɥɢɱɧɨɝɨɪɨɞɚɬɨɧɤɨɫɬɟɧɧɵɟɤɨɧɫɬɪɭɤɰɢɢɜɬɨɦɱɢɫɥɟɢɡɝɢɛɚɟ
ɦɵɟɩɥɚɫɬɢɧɵɢɨɛɨɥɨɱɤɢ ɉɪɢɫɭɳɢɟɷɬɢɦɤɨɧɫɬɪɭɤɰɢɹɦɥɟɝɤɨɫɬɶ
ɢɪɚɰɢɨɧɚɥɶɧɨɫɬɶɮɨɪɦɫɨɱɟɬɚɸɬɫɹɫɢɯɜɵɫɨɤɨɣɧɟɫɭɳɟɣɫɩɨɫɨɛ
ɧɨɫɬɶɸɷɤɨɧɨɦɢɱɧɨɫɬɶɸɢɯɨɪɨɲɟɣɬɟɯɧɨɥɨɝɢɱɧɨɫɬɶɸ
ȼ ɧɚɫɬɨɹɳɟɦ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ ɢɡɥɨɠɟɧɵ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɨɫ
ɧɨɜɵ ɡɚɞɚɱɢ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɢ ɨɛɨɥɨɱɟɤ ɚ ɬɚɤɠɟ ɩɪɚɤɬɢɱɟɫɤɢɟ
ɦɟɬɨɞɵɢɯɪɟɲɟɧɢɹ
ȼɩɟɪɜɨɣɝɥɚɜɟɩɪɢɜɟɞɟɧɵɨɫɧɨɜɧɵɟɭɪɚɜɧɟɧɢɹɬɟɨɪɢɢɭɩɪɭ
ɝɨɫɬɢɧɚɛɚɡɟɤɨɬɨɪɵɯɜɟɞɭɬɫɹɞɚɥɶɧɟɣɲɢɟɬɟɨɪɟɬɢɱɟɫɤɢɟɨɛɨɫɧɨ
ɜɚɧɢɹ ɦɟɬɨɞɨɜ ɪɚɫɱɟɬɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɬɨɧɤɨɫɬɟɧɧɵɯ ɤɨɧɫɬɪɭɤ
ɰɢɣ Ⱦɚɸɬɫɹ ɨɛɳɢɟ ɩɨɧɹɬɢɹ ɢ ɨɩɪɟɞɟɥɟɧɢɹ ɢɡɝɢɛɚɟɦɵɯ ɩɥɚɫɬɢɧ
ɤɚɤ ɜɢɞɚ ɤɨɧɫɬɪɭɤɰɢɣ ɩɪɢɜɟɞɟɧɵ ɨɫɧɨɜɧɵɟ ɭɪɚɜɧɟɧɢɹ ɬɟɯɧɢɱɟ
ɫɤɨɣɬɟɨɪɢɢɢɡɝɢɛɚɬɨɧɤɢɯɩɥɚɫɬɢɧɊɚɫɫɦɨɬɪɟɧɵɩɪɢɦɟɪɵɚɧɚɥɢ
ɬɢɱɟɫɤɨɝɨɪɟɲɟɧɢɹɧɟɤɨɬɨɪɵɯɱɚɫɬɧɵɯɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧ
ȼɨɜɬɨɪɨɣɝɥɚɜɟɞɚɧɚɤɥɚɫɫɢɮɢɤɚɰɢɹɩɪɢɛɥɢɠɟɧɧɵɯɦɟɬɨɞɨɜ
ɪɚɫɱɟɬɚɬɨɧɤɨɫɬɟɧɧɵɯɤɨɧɫɬɪɭɤɰɢɣɪɚɫɫɦɨɬɪɟɧɵɯɚɪɚɤɬɟɪɧɵɟɦɟ
ɬɨɞɵ ɪɟɲɟɧɢɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɜ ɱɚɫɬɧɨɫɬɢ ɦɟɬɨɞ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɢ ɦɟɬɨɞ
Ȼɭɛɧɨɜɚ– Ƚɚɥɺɪɤɢɧɚ.
ȼɬɪɟɬɶɟɣɝɥɚɜɟɫɮɨɪɦɭɥɢɪɨɜɚɧɚɜɚɪɢɚɰɢɨɧɧɚɹɡɚɞɚɱɚɢɡɝɢɛɚ
ɩɥɚɫɬɢɧɪɚɫɫɦɨɬɪɟɧɵɱɢɫɥɟɧɧɨ-ɚɧɚɥɢɬɢɱɟɫɤɢɟɢɱɢɫɥɟɧɧɵɟɦɟɬɨ
ɞɵɟɟɪɟɲɟɧɢɹɂɡɥɨɠɟɧɵɨɫɧɨɜɵɜɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɨɝɨɦɟɬɨ
ɞɚ ɦɟɬɨɞɚ Ɋɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨ ɚ ɬɚɤɠɟ ɪɚɫɫɦɨɬɪɟɧ ɧɚɢɛɨɥɟɟ ɷɮ
ɮɟɤɬɢɜɧɵɣɜɧɚɫɬɨɹɳɟɟɜɪɟɦɹɦɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ
ɑɟɬɜɟɪɬɚɹ ɝɥɚɜɚ ɩɨɫɜɹɳɟɧɚ ɪɚɫɱɟɬɭ ɬɨɧɤɢɯ ɨɛɨɥɨɱɟɤ Ƚɥɚɜɚ
ɫɨɞɟɪɠɢɬ ɨɛɳɢɟ ɩɨɥɨɠɟɧɢɹ ɬɟɨɪɢɢ ɬɨɧɤɢɯ ɨɛɨɥɨɱɟɤ ɚ ɬɚɤɠɟ ɨɫ
ɧɨɜɧɵɟɫɨɨɬɧɨɲɟɧɢɹɢɪɚɡɪɟɲɚɸɳɢɟɭɪɚɜɧɟɧɢɹɞɥɹɩɨɥɨɝɨɣɨɛɨ
ɥɨɱɤɢɊɚɫɫɦɨɬɪɟɧɵɱɢɫɥɟɧɧɵɟɦɟɬɨɞɵɪɚɫɱɟɬɚɨɛɨɥɨɱɟɤɜɱɚɫɬɧɨ
ɫɬɢɜɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣɦɟɬɨɞɢɦɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ
ɂɡɥɨɠɟɧɢɟ ɦɚɬɟɪɢɚɥɚ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɩɪɢɦɟɪɚɦɢ ɪɚɫɱɟɬɚ
ɢɥɥɸɫɬɪɢɪɭɸɳɢɦɢ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟ
ɧɢɹɩɪɢɤɥɚɞɧɵɯɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧ ɢɨɛɨɥɨɱɟɤ.
3
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ⱦɚɥɟɟɩɪɟɞɫɬɚɜɥɟɧɵɮɢɡɢɱɟɫɤɢɟɫɨɨɬɧɨɲɟɧɢɹɦɟɠɞɭɞɟɮɨɪ
ɦɚɰɢɹɦɢɢɧɚɩɪɹɠɟɧɢɹɦɢɨɛɨɛɳɟɧɧɵɣɡɚɤɨɧȽɭɤɚ).
Ƚɥɚɜɚ ɈɋɇɈȼɕɌȿɈɊɂɂɊȺɋɑȿɌȺ
ɂɁȽɂȻȺȿɆɕɏɉɅȺɋɌɂɇ
ɁɚɤɨɧȽɭɤɚɜɩɪɹɦɨɣɮɨɪɦɟ:
ɇɟɤɨɬɨɪɵɟɫɜɟɞɟɧɢɹɢɡɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ
Hx
Ɉɫɧɨɜɧɵɟɭɪɚɜɧɟɧɢɹɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ
Hy
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɷɥɟɦɟɧɬɚ ɫɩɥɨɲ
ɧɨɝɨɬɟɥɚ (ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ):
wV x wW xy wW xz
gx
wx
wy
wz
wW yx wV y wW yz
gy
wx
wy
wz
wW zx wW zy wV z
gz
wz
wx
wy
(1.1)
0.
Ɂɞɟɫɶ V x , V y , V z , W xy , W xz , W yx , W yz , W zx , W zy – ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ
ɧɚɩɪɹɠɟɧɢɣ g x , g y , g z – ɤɨɦɩɨɧɟɧɬɵɨɛɴɟɦɧɵɯɫɢɥ
Ɂɚɤɨɧɩɚɪɧɨɫɬɢɤɚɫɚɬɟɥɶɧɵɯɧɚɩɪɹɠɟɧɢɣ:
W xy
W yx , W yz
W zy , W zx
W xz .
J xy
wu
; Hy
wx
wu wv
; J yz
wy wx
wv
; Hz
wy
(1.2)
ww
;
wz
wv ww
; J zx
wz wy
ww wu
.
wx wz
(1.3)
Ɂɞɟɫɶ H x , H y , H z , J xy , J yz , J zx – ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ ɞɟɮɨɪɦɚɰɢɣ
u , v, w – ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɩɟɪɟɦɟɳɟɧɢɣ ɋɨɨɬɧɨɲɟɧɢɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɜɩɪɟɞɩɨɥɨɠɟɧɢɢɦɚɥɨɫɬɢɩɟɪɟɦɟɳɟɧɢɣɩɨɫɪɚɜɧɟ
ɧɢɸɫɯɚɪɚɤɬɟɪɧɵɦɢɪɚɡɦɟɪɚɦɢɬɟɥɚ
4
G
W yz
;
;
G
W zx
.
G
(1.4)
Vx
O T 2G H x ; W xy
G J xy ;
Vy
O T 2G H y ; W yz
G J yz ;
Vz
O T 2G H z ; W zx
G J zx .
(1.5)
Ɂɞɟɫɶ E – ɦɨɞɭɥɶɭɩɪɭɝɨɫɬɢ Ȟ – ɤɨɷɮɮɢɰɢɟɧɬɉɭɚɫɫɨɧɚ G – ɦɨ
ɞɭɥɶɫɞɜɢɝɚ Ȝ– ɤɨɷɮɮɢɰɢɟɧɬɅɚɦɟ ș– ɨɛɴɟɦɧɚɹɞɟɮɨɪɦɚɰɢɹ.
G
Ʌɢɧɟɣɧɵɟ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɞɟɮɨɪɦɚɰɢ
ɹɦɢɢɩɟɪɟɦɟɳɟɧɢɹɦɢ (ɭɪɚɜɧɟɧɢɹ Ʉɨɲɢ):
Hx
W xy
ɁɚɤɨɧȽɭɤɚɜɨɛɪɚɬɧɨɣɮɨɪɦɟ:
0;
0;
Hz
1
V x Q (V y V z ); J xy
E
1
V y Q (V z V x ); J yz
E
1
V z Q (V x V y ); J zx
E
E
; O
2 (1 Q)
QE
; T Hx H y Hz.
(1 Q) (1 2Q)
(1.6)
ɋɨɨɬɧɨɲɟɧɢɹ ɫɩɪɚɜɟɞɥɢɜɵ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɥɢɧɟɣɧɨɭɩɪɭɝɨɝɨɩɨɜɟɞɟɧɢɹɦɚɬɟɪɢɚɥɚ
ɉɪɢ ɪɟɲɟɧɢɢ ɤɨɧɤɪɟɬɧɵɯ ɡɚɞɚɱ ɭɪɚɜɧɟɧɢɹ (1.1)– ɞɨɩɨɥ
ɧɹɸɬɫɹ ɝɪɚɧɢɱɧɵɦɢɭɫɥɨɜɢɹɦɢ.
Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟɭɫɥɨɜɢɹ ɧɚɭɱɚɫɬɤɟɩɨɜɟɪɯɧɨɫɬɢ Sp:
u
us ; v
vs ; w
ws .
(1.7)
ɋɬɚɬɢɱɟɫɤɢɟɝɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹ (ɭɫɥɨɜɢɹɄɨɲɢ):
pnx
pny
pnz
V x cnx W xy cny W xz cnz ;
W yx ɫnx V y ɫny W yz ɫnz ;
W zx ɫnx W zy ɫny V z ɫnz .
5
(1.8)
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɂɞɟɫɶ pnx , pny , pnz – ɤɨɦɩɨɧɟɧɬɵ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɧɚɝɪɭɡɤɢ ɧɚ
ɭɱɚɫɬɤɟ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ Sp ɫ ɧɨɪɦɚɥɶɸ n; ɫnx , ɫny , ɫnz – ɧɚɩɪɚɜ
ɥɹɸɳɢɟɤɨɫɢɧɭɫɵɧɨɪɦɚɥɢɤɩɨɜɟɪɯɧɨɫɬɢɬɟɥɚ.
ɫnx
cos(n, x); ɫny
cos(n, y ); ɫnz
cos(n, z ).
Ɉɛɳɢɟɩɨɞɯɨɞɵɤɪɟɲɟɧɢɸɡɚɞɚɱɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ
ɉɟɪɟɱɢɫɥɟɧɧɵɟ ɜɵɲɟ ɭɪɚɜɧɟɧɢɹ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɫɨɞɟɪɠɚɬ
ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ ɲɟɫɬɶ ɤɨɦɩɨɧɟɧɬ ɧɚɩɪɹɠɟɧɢɣ V x , V y ,
V z , W xy , W yz , W zx ɲɟɫɬɶ ɤɨɦɩɨɧɟɧɬ ɞɟɮɨɪɦɚɰɢɣ H x , H y , H z , J xy ,
J yz , J zx ɢɬɪɢɤɨɦɩɨɧɟɧɬɵɩɟɪɟɦɟɳɟɧɢɣ u , v, w .
Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɷɬɢɯ ɮɭɧɤɰɢɣ ɪɚɫɩɨɥɚɝɚɟɦ ɭɪɚɜɧɟɧɢɹɦɢ
ɬɪɟɦɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢɭɪɚɜɧɟɧɢɹɦɢɪɚɜɧɨɜɟɫɢɹɲɟɫɬɶɸ
ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ ɢ ɲɟɫɬɶɸ ɮɢɡɢɱɟɫɤɢɦɢ ɭɪɚɜ
ɧɟɧɢɹɦɢ ɢɥɢ ɉɪɢ ɷɬɨɦ ɬɪɟɛɭɟɬɫɹ ɜɵɩɨɥɧɟɧɢɟ ɝɪɚɧɢɱ
ɧɵɯɭɫɥɨɜɢɣɧɚɩɨɜɟɪɯɧɨɫɬɢɬɟɥɚɌɚɤɢɦɨɛɪɚɡɨɦɫɦɚ
ɬɟɦɚɬɢɱɟɫɤɨɣɬɨɱɤɢɡɪɟɧɢɹɡɚɞɚɱɚɦɨɠɟɬɛɵɬɶɪɟɲɟɧɚɢɫɜɨɞɢɬɫɹ
ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɭɤɚɡɚɧɧɵɯ 15 ɭɪɚɜɧɟɧɢɣ ɩɪɢ ɭɞɨɜɥɟɬɜɨɪɟɧɢɢ
ɡɚɞɚɧɧɵɦɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ.
Ɋɚɡɥɢɱɚɸɬ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱ ɬɟɨɪɢɢ
ɭɩɪɭɝɨɫɬɢ ɢɫɨɨɬɜɟɬɫɬɜɟɧɧɨɫɩɨɫɨɛɵɢɯɪɟɲɟɧɢɹ:
1. ȼ ɩɟɪɟɦɟɳɟɧɢɹɯ – ɜ ɤɚɱɟɫɬɜɟ ɨɫɧɨɜɧɵɯ ɧɟɢɡɜɟɫɬɧɵɯ ɜɵ
ɫɬɭɩɚɸɬɩɟɪɟɦɟɳɟɧɢɹ u , v, w .
2. ȼ ɧɚɩɪɹɠɟɧɢɹɯ – ɜ ɤɚɱɟɫɬɜɟ ɨɫɧɨɜɧɵɯ ɧɟɢɡɜɟɫɬɧɵɯ ɩɪɢ
ɧɢɦɚɸɬɫɹɧɚɩɪɹɠɟɧɢɹ V x , V y , V z , W xy , W yz , W zx .
3. ȼ ɫɦɟɲɚɧɧɨɣ ɮɨɪɦɟ – ɜ ɤɚɱɟɫɬɜɟ ɨɫɧɨɜɧɵɯ ɧɟɢɡɜɟɫɬɧɵɯ
ɜɵɫɬɭɩɚɸɬɢɩɟɪɟɦɟɳɟɧɢɹ, ɢɧɚɩɪɹɠɟɧɢɹ
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɜ ɩɟɪɟɦɟɳɟɧɢɹɯ
ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ (1.1), ɡɚɦɟɧɢɜ
ɜ ɧɢɯ ɤɨɦɩɨɧɟɧɬɵ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɩɟɪɟɦɟɳɟɧɢɹ Ɍɚɤɚɹ ɡɚɦɟɧɚ
ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ ɜ ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɤɨɦɩɨɧɟɧɬ ɧɚɩɪɹ
ɠɟɧɢɣ ɜɵɪɚɠɟɧɧɵɯ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɚ Ƚɭɤɚ ɢ ɝɟɨɦɟɬɪɢɱɟ
ɫɤɢɯɫɨɨɬɧɨɲɟɧɢɣɄɨɲɢɱɟɪɟɡɩɟɪɟɦɟɳɟɧɢɹ u , v, w .
ȼɪɟɡɭɥɶɬɚɬɟɩɨɥɭɱɢɦɫɢɫɬɟɦɭɪɚɡɪɟɲɚɸɳɢɯɭɪɚɜɧɟɧɢɣɜɩɟ
ɪɟɦɟɳɟɧɢɹɯ
6
wT
G ’ 2u g x 0;
wx
wT
(1.9)
(O G )
G ’ 2 v g y 0;
wy
wT
(O G )
G ’ 2 w g z 0.
wz
ɍɪɚɜɧɟɧɢɹ ɧɚɡɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ Ʌɚɦɟ Ɂɞɟɫɶ
wu wv ww
– ɨɛɴɟɦɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɜɵɪɚɠɟɧɧɚɹ ɱɟɪɟɡ ɤɨɦ
T
wx wy wz
w2
w2
w2
ɩɨɧɟɧɬɵɩɟɪɟɦɟɳɟɧɢɣ ’ 2
– ɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚ.
wx 2 wy 2 wz 2
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɣ
ɨɛɴɟɞɢɧɹɸɬɫɬɚɬɢɱɟɫɤɢɟɝɟɨɦɟɬɪɢɱɟɫɤɢɟɢɮɢɡɢɱɟɫɤɢɟɫɨɨɬɧɨɲɟ
ɧɢɹ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɬɚɬɢɱɟɫɤɢɟ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹɬɚɤɠɟɞɨɥɠɧɵɛɵɬɶɜɵɪɚɠɟɧɵɱɟɪɟɡɩɟɪɟɦɟɳɟɧɢɹ
Ɋɟɲɚɹ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɭɞɨɜɥɟɬɜɨɪɟɧɢɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨ
ɜɢɹɦ ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɵɟ ɮɭɧɤɰɢɢ ɩɟɪɟɦɟɳɟɧɢɣ u ( x, y, z ) ,
v ( x, y, z ) , w ( x, y, z ) ɉɨɧɚɣɞɟɧɧɵɦɩɟɪɟɦɟɳɟɧɢɹɦɢɡɫɨɨɬɧɨɲɟɧɢɣ
Ʉɨɲɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɞɟɮɨɪɦɚɰɢɢ ɚ ɡɚɬɟɦ ɢɡ ɮɨɪɦɭɥ ɡɚɤɨɧɚ
Ƚɭɤɚɜɨɛɪɚɬɧɨɣɮɨɪɦɟ– ɧɚɩɪɹɠɟɧɢɹ
ɉɪɢɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜɧɚɩɪɹɠɟɧɢɹɯ ɨɞɧɢɯɬɨɥɶɤɨɭɪɚɜɧɟɧɢɣ
ɪɚɜɧɨɜɟɫɢɹ ɡɚɩɢɫɚɧɧɵɯ ɜ ɧɚɩɪɹɠɟɧɢɹɯ ɧɟɞɨɫɬɚɬɨɱɧɨ. Ⱦɨ
ɩɨɥɧɢɬɟɥɶɧɵɟɭɪɚɜɧɟɧɢɹɦɨɠɧɨɩɨɥɭɱɢɬɶɢɫɤɥɸɱɢɜɢɡɝɟɨɦɟɬɪɢ
ɱɟɫɤɢɯ ɫɨɨɬɧɨɲɟɧɢɣ Ʉɨɲɢ ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɩɟɪɟɦɟɳɟ
ɧɢɣ ɉɨɥɭɱɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɭɪɚɜɧɟɧɢɣ ɧɟɪɚɡɪɵɜ
ɧɨɫɬɢɞɟɮɨɪɦɚɰɢɣɭɪɚɜɧɟɧɢɹɋɟɧ-ȼɟɧɚɧɚ).
(O G )
2
w 2H x w H y
wy 2
wx 2
w 2 J xy
wxwy
;
w 2H y
wz 2
w 2H z
wy 2
w 2 J yz
wywz
w § wJ zx wJ xy wJ yz ·
¨
¸
wx ¨© wy
wz
wx ¸¹
w § wJ xy wJ yz wJ zx ·
¨
¸
wy ¸¹
wy ¨© wz
wx
2
w 2H y
wzwx
;
2
w 2H z w 2H x
2
wx 2
wz
w 2H x
;
wywz
w § wJ yz wJ zx wJ xy ·
¨
¸
wz ¸¹
wy
wz ¨© wx
;
7
w 2 J zx
;
wzwx
(1.10)
2
w 2H z
.
wxwy
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɭɪɚɜɧɟɧɢɣ ɋɟɧ-ȼɟɧɚɧɚ – ɭɫɥɨɜɢɹ ɫɩɥɨɲ
ɧɨɫɬɢɞɟɮɨɪɦɢɪɭɟɦɨɝɨɬɟɥɚ; ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɫɦɵɫɥɟɨɧɢɜɵɪɚ
ɠɚɸɬɭɫɥɨɜɢɹɢɧɬɟɝɪɢɪɭɟɦɨɫɬɢɫɨɨɬɧɨɲɟɧɢɣɄɨɲɢɩɪɢɞɚɧ
ɧɵɯɤɨɦɩɨɧɟɧɬɚɯɞɟɮɨɪɦɚɰɢɣ
ȿɫɥɢ ɜ ɭɪɚɜɧɟɧɢɹɯ ɞɟɮɨɪɦɚɰɢɢ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɧɚɩɪɹ
ɠɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɚ Ƚɭɤɚ ɫ ɭɱɟɬɨɦ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟ
ɫɢɹɬɨɜɪɟɡɭɥɶɬɚɬɟɩɨɥɭɱɢɦɲɟɫɬɶɭɪɚɜɧɟɧɢɣɫɜɹɡɵɜɚɸɳɢɯ
ɤɨɦɩɨɧɟɧɬɵɧɚɩɪɹɠɟɧɢɣɭɪɚɜɧɟɧɢɹȻɟɥɶɬɪɚɦɢ – Ɇɢɱɟɥɥɚ),
w2S
w 2 S1
(1 Q) ’ 2 V x 21 0; (1 Q) ’ 2 W xy 0;
wxwy
wx
( pnx , pny , pnz ) ɫɢɥ Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɭɩɪɭɝɢɟ ɩɟɪɟɦɟɳɟɧɢɹ
ɧɚɩɪɹɠɟɧɢɹɢɨɬɧɨɫɢɬɟɥɶɧɵɟɞɟɮɨɪɦɚɰɢɢɜɨɜɫɟɯɬɨɱɤɚɯɪɚɫɫɦɚɬ
ɪɢɜɚɟɦɨɝɨɬɟɥɚɬ ɟɟɝɨɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɟɫɨɫɬɨɹɧɢɟ.
ɉɪɹɦɚɹ ɡɚɞɚɱɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɧɚɢɛɨɥɶɲɢɣ ɩɪɚɤɬɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ
ɜɬɟɨɪɢɢɪɚɫɱɟɬɚɤɨɧɫɬɪɭɤɰɢɣɢɫɨɨɪɭɠɟɧɢɣ
ɉɪɢ ɪɟɲɟɧɢɢ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ ɡɚɞɚɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɩɟɪɟ
ɦɟɳɟɧɢɣ ɢɥɢ ɧɚɩɪɹɠɟɧɢɣ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɦɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶ
ɧɵɦɭɪɚɜɧɟɧɢɹɦɋɟɧ-ȼɟɧɚɧɚ) ɜɩɟɪɜɨɦɫɥɭɱɚɟɢɭɪɚɜɧɟɧɢɹɦ
Ȼɟɥɶɬɪɚɦɢ– Ɇɢɱɟɥɥɚ– ɜɨɜɬɨɪɨɦɌɪɟɛɭɟɬɫɹ ɧɚɣɬɢɨɫɬɚɥɶ
ɧɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɦɩɨɧɟɧɬɵ ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨ
ɫɬɨɹɧɢɹɜɬɨɦɱɢɫɥɟɨɛɴɟɦɧɵɟɫɢɥɵɢɭɫɥɨɜɢɹɧɚɩɨɜɟɪɯɧɨɫɬɢ
Ɋɟɲɟɧɢɟ ɨɛɪɚɬɧɨɣ ɡɚɞɚɱɢ ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɨɳɟ ɱɟɦ ɩɪɹɦɨɣ
ɬɚɤ ɤɚɤ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɫɢɫɬɟɦɭ ɞɢɮɮɟɪɟɧɰɢ
ɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣɜɱɚɫɬɧɵɯɩɪɨɢɡɜɨɞɧɵɯɜɫɟɫɜɨɞɢɬɫɹɤɧɟɫɥɨɠ
ɧɵɦɦɚɬɟɦɚɬɢɱɟɫɤɢɦɨɩɟɪɚɰɢɹɦɜɱɚɫɬɧɨɫɬɢɤɞɢɮɮɟɪɟɧɰɢɪɨɜɚ
ɧɢɸɢɥɢɢɧɬɟɝɪɢɪɨɜɚɧɢɸɮɭɧɤɰɢɣ).
Ɍɚɤ ɟɫɥɢ ɡɚɞɚɧɵ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɣ
u ( x, y, z ) , v ( x, y, z ) , w ( x, y, z ) ɬɨɩɪɨɫɬɵɦɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦɢɡ
ɭɪɚɜɧɟɧɢɣɄɨɲɢɨɩɪɟɞɟɥɹɸɬɫɹɞɟɮɨɪɦɚɰɢɢɚɡɚɬɟɦɩɪɢɩɨ
ɦɨɳɢ ɡɚɤɨɧɚ Ƚɭɤɚ ɜ ɨɛɪɚɬɧɨɣ ɮɨɪɦɟ – ɧɚɩɪɹɠɟɧɢɹ ɍɪɚɜɧɟ
ɧɢɹɫɨɜɦɟɫɬɧɨɫɬɢɞɟɮɨɪɦɚɰɢɣɩɪɢɷɬɨɦɜɫɟɝɞɚɜɵɩɨɥɧɹɸɬɫɹɈɛɴ
ɟɦɧɵɟ ɫɢɥɵ ɧɚɯɨɞɹɬ ɢɡ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɇɚɜɶɟ ɚɩɨɜɟɪɯɧɨɫɬɧɵɟ– ɢɡɭɪɚɜɧɟɧɢɣ
ȿɫɥɢ ɡɚɞɚɧɵ ɧɚɩɪɹɠɟɧɢɹ ɬɨ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɞɟɮɨɪɦɚɰɢɢ
ɨɩɪɟɞɟɥɹɸɬɫɹɩɪɢɩɨɦɨɳɢɭɪɚɜɧɟɧɢɣɚɩɟɪɟɦɟɳɟɧɢɹ– ɢɧɬɟ
ɝɪɢɪɨɜɚɧɢɟɦ ɭɪɚɜɧɟɧɢɣ Ʉɨɲɢ Ɉɛɴɟɦɧɵɟ ɢ ɩɨɜɟɪɯɧɨɫɬɧɵɟ
ɫɢɥɵɩɨɥɭɱɚɸɬɬɚɤɠɟɤɚɤɢɜɩɟɪɜɨɦɫɥɭɱɚɟ
Ⱦɥɹ ɪɟɲɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɧɟɪɟɞɤɨ
ɩɪɢɦɟɧɹɸɬ ɩɨɥɭɨɛɪɚɬɧɵɣ ɦɟɬɨɞ ɋɟɧ-ȼɟɧɚɧɚ ɋɭɬɶ ɟɝɨ ɫɨɫɬɨɢɬ
ɜ ɬɨɦ ɱɬɨ ɧɚ ɨɫɧɨɜɟ ɪɟɡɭɥɶɬɚɬɨɜ ɪɟɲɟɧɢɹ ɩɨɞɨɛɧɵɯ ɢɥɢ ɷɥɟɦɟɧ
ɬɚɪɧɵɯ ɡɚɞɚɱ ɢɥɢ ɧɚ ɨɫɧɨɜɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɡɚɞɚɸɬ
ɤɚɤɭɸ-ɬɨ ɱɚɫɬɶ ɢɫɤɨɦɵɯ ɮɭɧɤɰɢɣ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɨɫɧɨɜɧɵɦ
ɭɪɚɜɧɟɧɢɹɦ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɢ ɮɢɡɢɱɟɫɤɨɦɭ ɫɦɵɫɥɭ ɪɚɫɫɦɚɬɪɢ
ɜɚɟɦɨɣ ɡɚɞɚɱɢ ȿɫɥɢ ɷɬɢ ©ɭɝɚɞɚɧɧɵɟª ɮɭɧɤɰɢɢ ɩɨɞɫɬɚɜɢɬɶ ɜ ɪɚɡ
ɪɟɲɚɸɳɭɸɫɢɫɬɟɦɭɭɪɚɜɧɟɧɢɣɬɨɡɚɞɚɱɚɫɭɳɟɫɬɜɟɧɧɨɭɩɪɨɫɬɢɬɫɹ
ɡɚ ɫɱɟɬ ɭɦɟɧɶɲɟɧɢɹ ɱɢɫɥɚ ɢɫɤɨɦɵɯ ɧɟɢɡɜɟɫɬɧɵɯ Ɋɟɲɚɹ ɬɚɤɭɸ
ɭɩɪɨɳɟɧɧɭɸɡɚɞɚɱɭɞɨɨɩɪɟɞɟɥɹɸɬɧɟɢɡɜɟɫɬɧɭɸɱɚɫɬɶɪɟɲɟɧɢɹ
ȼ ɡɚɤɥɸɱɟɧɢɟ ɨɬɦɟɬɢɦ ɱɬɨ ɪɟɲɟɧɢɟ ɥɸɛɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ
ɭɩɪɭɝɨɫɬɢ ɩɪɟɞɩɨɥɚɝɚɟɬ ɭɞɨɜɥɟɬɜɨɪɟɧɢɟ ɭɪɚɜɧɟɧɢɣ ɪɚɜɧɨɜɟɫɢɹ
ɜɵɩɨɥɧɟɧɢɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɢ ɭɫɥɨɜɢɣ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɞɟɮɨɪ
ɦɚɰɢɣȼɩɥɚɧɟɨɩɪɟɞɟɥɟɧɢɹɤɨɦɩɨɧɟɧɬɧɚɩɪɹɠɟɧɢɣɢɞɟɮɨɪɦɚɰɢɣ
8
9
(1 Q) ’ 2 V y w 2 S1
wy 2
0;
(1 Q) ’ 2 W yz w 2 S1
wywz
0;
(1 Q) ’ 2 V z w 2 S1
wz 2
0;
(1 Q) ’ 2 W zx w 2 S1
wzwx
0.
(1.11)
Ɂɞɟɫɶ S1 V x V y V z – ɩɟɪɜɵɣɢɧɜɚɪɢɚɧɬɬɟɧɡɨɪɚɧɚɩɪɹɠɟɧɢɣ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ
ɜ ɧɚɩɪɹɠɟɧɢɹɯ ɩɪɢɯɨɞɢɬɫɹ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɞɟɜɹɬɶ ɭɪɚɜɧɟɧɢɣ (1.1)
ɢɩɪɢɷɬɨɦɞɨɥɠɧɵɭɞɨɜɥɟɬɜɨɪɹɬɶɫɹɫɬɚɬɢɱɟɫɤɢɟɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹɇɚɥɢɱɢɟɬɪɟɯ«ɥɢɲɧɢɯ» ɭɪɚɜɧɟɧɢɣɥɢɲɧɢɟíɩɨɬɨ
ɦɭɱɬɨɧɟɢɡɜɟɫɬɧɵɯɮɭɧɤɰɢɣɧɚɩɪɹɠɟɧɢɣɜɫɟɝɨɲɟɫɬɶ ɧɟɨɛɯɨɞɢ
ɦɨ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɨɞɧɨɡɧɚɱɧɨɝɨ ɪɟɲɟɧɢɹ ɉɨɥɭɱɟɧɧɵɟ ɧɚɩɪɹɠɟ
ɧɢɹɩɨɞɫɬɚɜɥɹɸɬɫɹɜɮɨɪɦɭɥɵɡɚɤɨɧɚ Ƚɭɤɚɢɧɚɯɨɞɹɬɫɹɤɨɦ
ɩɨɧɟɧɬɵ ɞɟɮɨɪɦɚɰɢɣ ɞɚɥɟɟ ɩɭɬɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɣ
Ʉɨɲɢɨɩɪɟɞɟɥɹɸɬɫɹɩɟɪɟɦɟɳɟɧɢɹ
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɜ ɫɦɟɲɚɧɧɨɣ ɮɨɪɦɟ
ɪɚɡɪɟɲɚɸɳɢɟɭɪɚɜɧɟɧɢɹɫɨɫɬɚɜɥɹɸɬɫɹɱɚɫɬɢɱɧɨɨɬɧɨɫɢɬɟɥɶɧɨɩɟ
ɪɟɦɟɳɟɧɢɣ ɚ ɱɚɫɬɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɩɪɹɠɟɧɢɣ Ɍɚɤɨɣ ɩɪɢɟɦ
ɧɚɩɪɢɦɟɪɢɫɩɨɥɶɡɭɟɬɫɹɩɪɢɪɟɲɟɧɢɢɡɚɞɚɱɪɚɫɱɟɬɚɨɛɨɥɨɱɟɤ>@
ɉɟɪɟɱɢɫɥɟɧɧɵɟ ɫɩɨɫɨɛɵ ɩɪɢɦɟɧɢɦɵ ɤ ɪɟɲɟɧɢɸ ɬɚɤ ɧɚɡɵɜɚ
ɟɦɨɣɩɪɹɦɨɣɡɚɞɚɱɢ ɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢɤɨɝɞɚɡɚɞɚɧɵɞɟɣɫɬɜɭɸɳɢɟ
ɧɚ ɬɟɥɨ ɧɚɝɪɭɡɤɢ ɜ ɜɢɞɟ ɨɛɴɟɦɧɵɯ ( g x , g y , g z ) ɢ ɩɨɜɟɪɯɧɨɫɬɧɵɯ
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɪɟɲɟɧɢɟɡɚɞɚɱɢɹɜɥɹɟɬɫɹɟɞɢɧɫɬɜɟɧɧɵɦ ɧɚɨɫɧɨɜɚɧɢɢɞɨɤɚɡɚɧɧɨɣ
ɜɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢɬɟɨɪɟɦɵɨɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɪɟɲɟɧɢɹɢɧɟɡɚɜɢ
ɫɢɬ ɨɬɬɨɝɨɤɚɤɢɦɫɩɨɫɨɛɨɦ ɨɧɨɩɨɥɭɱɟɧɨɉɪɢɷɬɨɦɤɨɦɩɨɧɟɧɬɵ
ɜɟɤɬɨɪɚ ɩɟɪɟɦɟɳɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɠɟɫɬɤɨɝɨ
ɫɦɟɳɟɧɢɹɭɩɪɭɝɨɝɨɬɟɥɚ
Ɋɚɡɪɟɲɚɸɳɢɟɭɪɚɜɧɟɧɢɹɢɡɝɢɛɚɬɨɧɤɢɯɩɥɚɫɬɢɧ
Ɉɛɳɢɟɩɨɥɨɠɟɧɢɹɢɤɥɚɫɫɢɮɢɤɚɰɢɹɩɥɚɫɬɢɧ
ɉɥɚɫɬɢɧɨɣ, ɢɥɢ ɩɥɢɬɨɣ, ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɡɦɚɬɢɱɟɫɤɨɟ ɬɟɥɨ
ɨɝɪɚɧɢɱɟɧɧɨɟ ɞɜɭɦɹ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɩɥɨɫɤɨɫɬɹɦɢ ɪɚɫɫɬɨɹɧɢɟ
ɦɟɠɞɭ ɤɨɬɨɪɵɦɢ ɬɨɥɳɢɧɚ K ɦɚɥɨ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɞɪɭɝɢɦɢ ɯɚ
ɪɚɤɬɟɪɧɵɦɢɪɚɡɦɟɪɚɦɢa, b).
ɉɥɨɫɤɨɫɬɶ ɪɚɜɧɨɭɞɚɥɟɧɧɚɹ ɨɬ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɩɨɜɟɪɯɧɨ
ɫɬɟɣɩɥɚɫɬɢɧɵɧɚɡɵɜɚɟɬɫɹɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɶɸ. ɋɨ ɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɶɸ ɫɨɜɩɚɞɚɟɬɨɞɧɚɢɡɤɨɨɪɞɢɧɚɬɧɵɯɩɥɨɫɤɨɫɬɟɣɞɟɤɚɪɬɨ
ɜɨɣɫɢɫɬɟɦɵɤɨɨɪɞɢɧɚɬx, yɨɫɶ z ɨɛɵɱɧɨɧɚɩɪɚɜɥɹɟɬɫɹɜɧɢɡɅɢ
ɧɢɸɨɝɪɚɧɢɱɢɜɚɸɳɭɸɫɪɟɞɢɧɧɭɸɩɥɨɫɤɨɫɬɶɩɥɚɫɬɢɧɵɧɚɡɵɜɚɸɬ
ɤɨɧɬɭɪɨɦ ɩɥɚɫɬɢɧɵ ɪɢɫ 1.1).
x
b
z
h
y
h/2
a
Ɋɢɫ 1.1. ɂɡɝɢɛɚɟɦɚɹɩɥɚɫɬɢɧɚ
ɂɡɝɢɛ ɩɥɚɫɬɢɧɵ ɜɵɡɵɜɚɟɬɫɹ ɧɚɝɪɭɡɤɨɣ ɩɪɢɥɨɠɟɧɧɨɣ ɩɟɪ
ɩɟɧɞɢɤɭɥɹɪɧɨ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ. ȼ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹ
ɧɢɢ ɫɪɟɞɢɧɧɭɸ ɩɥɨɫɤɨɫɬɶ ɧɚɡɵɜɚɸɬ ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ
ɢɡɨɝɧɭɬɨɣɩɥɚɫɬɢɧɵȼɬɟɨɪɢɢɢɡɝɢɛɚɩɥɚɫɬɢɧɫɪɟɞɢɧɧɚɹɩɨɜɟɪɯ
ɧɨɫɬɶɢɝɪɚɟɬɬɚɤɭɸɠɟɜɚɠɧɭɸɪɨɥɶɤɚɤɜɫɨɩɪɨɬɢɜɥɟɧɢɢɦɚɬɟɪɢ
ɚɥɨɜɧɟɣɬɪɚɥɶɧɵɣɫɥɨɣɩɪɢɩɨɩɟɪɟɱɧɨɦɢɡɝɢɛɟɛɚɥɨɤ. Ʉɨɦɩɨɧɟɧɬɵ
ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɟɤ ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z
ɧɚɡɵɜɚɸɬɫɹɩɪɨɝɢɛɚɦɢɩɥɚɫɬɢɧɵ ɢɨɛɨɡɧɚɱɚɸɬɫɹw.
10
ɉɥɚɫɬɢɧɵɢɦɟɸɬ ɲɢɪɨɤɨɟɩɪɢɦɟɧɟɧɢɟɜɫɬɪɨɢɬɟɥɶɫɬɜɟɜɜɢɞɟ
ɧɚɫɬɢɥɨɜ ɩɚɧɟɥɟɣ ɩɥɢɬ ɩɟɪɟɤɪɵɬɢɹ ɢ ɬ ɞ ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ
ɱɬɨɩɪɢɫɭɳɢɟɬɨɧɤɨɫɬɟɧɧɵɦɤɨɧɫɬɪɭɤɰɢɹɦɥɟɝɤɨɫɬɶɢɪɚɰɢɨɧɚɥɶ
ɧɨɫɬɶɮɨɪɦɫɨɱɟɬɚɸɬɫɹɫɢɯɜɵɫɨɤɨɣɧɟɫɭɳɟɣɫɩɨɫɨɛɧɨɫɬɶɸɷɤɨ
ɧɨɦɢɱɧɨɫɬɶɸɢɯɨɪɨɲɟɣɬɟɯɧɨɥɨɝɢɱɧɨɫɬɶɸ
ɋɚɦɵɣɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɣɜɢɞɩɥɚɫɬɢɧ– ɷɬɨɬɚɤɧɚɡɵɜɚɟɦɵɟ
ɬɨɧɤɢɟɩɥɚɫɬɢɧɵɭɤɨɬɨɪɵɯɨɬɧɨɲɟɧɢɟɬɨɥɳɢɧɵɤɧɚɢɦɟɧɶɲɟɦɭ
ɯɚɪɚɤɬɟɪɧɨɦɭ ɪɚɡɦɟɪɭ /100 ” h/b ” 1/5. ɉɪɢ h/b > 1/5 ɩɥɚɫɬɢɧɚ
ɨɬɧɨɫɢɬɫɹ ɤ ɬɨɥɫɬɵɦ ɩɥɢɬɚɦ, ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɪɚɫɫɱɢɬɵɜɚɬɶɫɹ
ɭɠɟ ɤɚɤ ɦɚɫɫɢɜɧɵɟ ɬɟɥɚ ɉɪɢ h/b < 1/10 ɩɥɚɫɬɢɧɚ ɩɪɟɜɪɚɳɚɟɬɫɹ
ɜɦɟɦɛɪɚɧɭɤɨɬɨɪɚɹɦɨɠɟɬɪɚɛɨɬɚɬɶɬɨɥɶɤɨɩɪɢɡɚɤɪɟɩɥɟɧɧɵɯɩɨ
ɤɨɧɬɭɪɭ ɤɪɚɹɯ. ȿɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚ ɢɡɝɢɛ ɨɤɚɡɵɜɚɟɬɫɹ ɧɢɱɬɨɠɧɨ
ɦɚɥɵɦ, ɚ ɨɫɧɨɜɧɭɸ ɪɨɥɶ ɜ ɜɨɫɩɪɢɹɬɢɢ ɧɚɝɪɭɡɤɢ ɢɝɪɚɸɬ ɭɫɢɥɢɹ
ɪɚɫɬɹɠɟɧɢɹɢɫɞɜɢɝɚɜɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢ
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɨɬɧɨɲɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɩɪɨɝɢɛɚ ɤ ɬɨɥ
ɳɢɧɟ ɩɥɚɫɬɢɧɵ (w/h) ɪɨɥɶ ɢɡɝɢɛɚɸɳɢɯ ɢ ɦɟɦɛɪɚɧɧɵɯ ɭɫɢɥɢɣ
ɜ ɬɨɧɤɨɣ ɩɥɚɫɬɢɧɟ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɢɱɧɨɣ ɉɨɷɬɨɦɭ ɬɨɧɤɢɟ ɩɥɚ
ɫɬɢɧɵɪɚɡɞɟɥɹɸɬɧɚɫɥɟɞɭɸɳɢɟɤɥɚɫɫɵ
1) ɠɟɫɬɤɢɟɩɥɚɫɬɢɧɵ (w/h ”1/4ɜɤɨɬɨɪɵɯɨɫɧɨɜɧɭɸɪɨɥɶ
ɢɝɪɚɸɬ ɢɡɝɢɛɧɵɟ ɫɢɥɨɜɵɟ ɮɚɤɬɨɪɵ ɞɟɮɨɪɦɚɰɢɹɦɢ ɜ ɫɪɟɞɢɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɦɟɦɛɪɚɧɧɵɦɢ ɭɫɢɥɢɹɦɢ ɡɞɟɫɶ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ,
ɚɡɚɜɢɫɢɦɨɫɬɶɦɟɠɞɭɩɪɨɝɢɛɚɦɢɢɧɚɝɪɭɡɤɨɣɥɢɧɟɣɧɚ;
2) ɝɢɛɤɢɟ ɩɥɚɫɬɢɧɵ (1/4 < w/h ” 4 ɜ ɤɨɬɨɪɵɯ ɧɟɨɛɯɨɞɢɦɨ
ɭɱɢɬɵɜɚɬɶɤɚɤɢɡɝɢɛɧɵɟɬɚɤɢɦɟɦɛɪɚɧɧɵɟɞɟɮɨɪɦɚɰɢɢ;
3) ɚɛɫɨɥɸɬɧɨɝɢɛɤɢɟɩɥɚɫɬɢɧɵ (w/h > 4), ɜɤɨɬɨɪɵɯɞɨɦɢɧɢ
ɪɭɸɬ ɦɟɦɛɪɚɧɧɵɟ ɞɟɮɨɪɦɚɰɢɢ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɩɪɨɝɢɛɚɦɢ
ɢɧɚɝɪɭɡɤɨɣɹɜɥɹɟɬɫɹɧɟɥɢɧɟɣɧɨɣ
Ⱦɟɥɟɧɢɟ ɩɥɚɫɬɢɧ ɧɚ ɠɟɫɬɤɢɟ ɝɢɛɤɢɟ ɢ ɚɛɫɨɥɸɬɧɨ ɝɢɛɤɢɟ
ɜɡɧɚɱɢɬɟɥɶɧɨɣɫɬɟɩɟɧɢɭɫɥɨɜɧɨɉɨɜɟɞɟɧɢɟɩɥɚɫɬɢɧɵɩɨɞɧɚɝɪɭɡ
ɤɨɣɨɩɪɟɞɟɥɹɟɬɫɹɧɟɬɨɥɶɤɨɟɟɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢɩɚɪɚɦɟɬɪɚɦɢȼɟ
ɥɢɱɢɧɵ ɭɩɪɭɝɢɯɞɟɮɨɪɦɚɰɢɣɬɚɤɠɟɫɭɳɟɫɬɜɟɧɧɨɡɚɜɢɫɹɬɨɬɦɟɯɚ
ɧɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜɦɚɬɟɪɢɚɥɚɭɫɥɨɜɢɣɡɚɤɪɟɩɥɟɧɢɹ ɩɥɚɫɬɢɧɵɜɢɞɚ
ɧɚɝɪɭɡɤɢ ɫɬɚɬɢɱɟɫɤɚɹɢɥɢɞɢɧɚɦɢɱɟɫɤɚɹ.
ȼɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦɪɚɫɫɦɚɬɪɢɜɚɬɶɬɨɥɶɤɨɬɨɧɤɢɟɠɟɫɬɤɢɟ
ɩɥɚɫɬɢɧɵɤɨɬɨɪɵɟɹɜɥɹɸɬɫɹɪɚɫɱɟɬɧɵɦɢɫɯɟɦɚɦɢɩɥɢɬɩɟɪɟɤɪɵ
ɬɢɹ ɞɨɪɨɠɧɵɯ ɢ ɦɨɫɬɨɜɵɯ ɩɥɢɬ ɩɥɢɬ ɞɥɹ ɷɫɬɚɤɚɞ ɢ ɫɤɥɚɞɱɚɬɵɯ
ɤɨɧɫɬɪɭɤɰɢɣɮɭɧɞɚɦɟɧɬɧɵɯɩɥɢɬɩɨɞɡɞɚɧɢɹɢɫɨɨɪɭɠɟɧɢɹɢɬɩ
ɉɪɢɪɚɫɱɟɬɟɬɨɧɤɢɯ ɠɟɫɬɤɢɯ ɩɥɚɫɬɢɧɨɛɵɱɧɨɢɫɩɨɥɶɡɭɸɬɬɚɤ
ɧɚɡɵɜɚɟɦɭɸ ɬɟɯɧɢɱɟɫɤɭɸ ɬɟɨɪɢɸ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ, ɜ ɨɫɧɨɜɟ ɤɨ
ɬɨɪɨɣɥɟɠɚɬɝɢɩɨɬɟɡɵɩɪɟɞɥɨɠɟɧɧɵɟȽɄɢɪɯɝɨɮɨɦ
11
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
1. Ƚɢɩɨɬɟɡɚɩɪɹɦɵɯɧɨɪɦɚɥɟɣ: ɨɬɪɟɡɨɤɧɨɪɦɚɥɢɤɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ ɨɫɬɚɟɬɫɹ ɩɪɹɦɵɦ ɢ ɧɨɪɦɚɥɶɧɵɦ ɤ ɢɡɨɝɧɭɬɨɣ
ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɢ ɷɬɨɦ ɞɥɢɧɚ ɟɝɨ ɧɟ ɦɟɧɹɟɬɫɹ Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɱɬɨ ɫɞɜɢɝɢ ɜ ɩɥɨɫɤɨɫɬɹɯ yz ɢ zx ɨɬɫɭɬ
ɫɬɜɭɸɬ ɬ ɟ Ȗyz = 0, Ȗzx = 0), ɥɢɧɟɣɧɚɹɞɟɮɨɪɦɚɰɢɹɜ ɧɚɩɪɚɜɥɟɧɢɢ
ɨɫɢ z ɬɚɤɠɟ ɨɬɫɭɬɫɬɜɭɟɬ İz = 0) Ⱦɚɧɧɚɹ ɝɢɩɨɬɟɡɚ ɚɧɚɥɨɝɢɱɧɚ ɝɢ
ɩɨɬɟɡɟɩɥɨɫɤɢɯɫɟɱɟɧɢɣɜɬɟɨɪɢɢɢɡɝɢɛɚɛɚɥɨɤ
2. Ƚɢɩɨɬɟɡɚ ɨ ɧɟɞɟɮɨɪɦɢɪɭɟɦɨɫɬɢ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ:
ɜɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɨɬɫɭɬɫɬɜɭɸɬɞɟɮɨɪɦɚɰɢɢɪɚɫɬɹɠɟɧɢɹɫɠɚ
ɬɢɹ ɢ ɫɞɜɢɝɚ ɬ ɟ ɩɪɢ ɢɡɝɢɛɟ ɩɥɚɫɬɢɧɵ ɷɬɚ ɩɥɨɫɤɨɫɬɶ ɨɫɬɚɟɬɫɹ
ɧɟɣɬɪɚɥɶɧɨɣ ɢɟɟɩɟɪɟɦɟɳɟɧɢɹ u0 = v0 = 0.
3. Ƚɢɩɨɬɟɡɚ ɨ ɧɟɧɚɞɚɜɥɢɜɚɧɢɢ ɫɥɨɟɜ: ɞɚɜɥɟɧɢɟɦ ɫɥɨɟɜ ɩɚ
ɪɚɥɥɟɥɶɧɵɯ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z ɩɪɟɧɟɛɪɟ
ɝɚɸɬɬ ɟɩɨɥɚɝɚɟɦ ız = 0).
Ƚɢɩɨɬɟɡɵ Ʉɢɪɯɝɨɮɚ ɹɜɥɹɸɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɝɢɩɨɬɟɡɵ ɩɥɨɫ
ɤɢɯ ɫɟɱɟɧɢɣ ɩɪɢɧɹɬɨɣ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ ɦɚɬɟɪɢɚɥɨɜ Ʉɪɨɦɟ ɬɨɝɨ
ɩɪɢɢɫɩɨɥɶɡɨɜɚɧɢɢɬɟɯɧɢɱɟɫɤɨɣɬɟɨɪɢɢɢɡɝɢɛɚɦɚɬɟɪɢɚɥɩɥɚɫɬɢɧɵ
ɫɱɢɬɚɟɬɫɹ ɥɢɧɟɣɧɨ-ɭɩɪɭɝɢɦ ɢ ɞɟɣɫɬɜɭɸɬ ɭɤɚɡɚɧɧɵɟ ɜɵɲɟ ɨɝɪɚɧɢ
ɱɟɧɢɹɩɨɩɪɨɝɢɛɚɦw/h ”ɢ ɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵ”h/b ”
”1/5). Ɉɞɧɚɤɨɜɫɜɹɡɢɫɬɟɦɱɬɨɪɚɫɱɟɬɬɨɥɫɬɵɯɩɥɢɬɫɭɳɟɫɬɜɟɧɧɨ
ɫɥɨɠɧɟɟ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ ɩɨ ɬɟɯɧɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɞɨɩɭɫɤɚɟɬɫɹ ɪɚɫ
ɫɱɢɬɵɜɚɬɶ ɩɥɚɫɬɢɧɵ ɫɨɬɧɨɲɟɧɢɟɦ h/b ɞɨ 1/3.
ɉɪɢɪɟɲɟɧɢɢ ɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵɡɚɨɫɧɨɜɧɭɸɧɟɢɡɜɟɫɬ
ɧɭɸɮɭɧɤɰɢɸɩɪɢɧɢɦɚɟɬɫɹɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ w = w (x, yȼɵɪɚɡɢɜ
ɱɟɪɟɡ ɩɪɨɝɢɛ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ ɩɨɥɭɱɢɦ ɪɚɡ
ɪɟɲɚɸɳɟɟ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɡɜɟɫɬɧɵɯ w ɉɨɫɥɟ ɟɝɨ
ɪɟɲɟɧɢɹ ɨɫɬɚɥɶɧɵɟ ɤɨɦɩɨɧɟɧɬɵ ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ
ɫɨɫɬɨɹɧɢɹ ɩɥɚɫɬɢɧɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
ɜɵɪɚɠɟɧɢɣ ɱɟɪɟɡ ɩɪɨɝɢɛɵ w Ɍɚɤɨɜ ɨɛɳɢɣ ɩɭɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ
ɢɡɝɢɛɚɩɥɚɫɬɢɧ
ȼɟɥɢɱɢɧɵ F x , F y ɫɨɫɬɚɜɥɹɸɬ ɤɪɢɜɢɡɧɵ ɷɥɟɦɟɧɬɚ dx × dy ɢɡɨɝɧɭ
ɬɨɣ ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ F – ɤɪɢɜɢɡɧɚ ɤɪɭɱɟɧɢɹ
ɞɚɧɧɨɝɨɷɥɟɦɟɧɬɚ U x , U y – ɪɚɞɢɭɫɵɤɪɢɜɢɡɧɵ ɷɥɟɦɟɧɬɚɫɪɟɞɢɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢɫɨɨɬɜɟɬɫɬɜɟɧɧɨɜɞɨɥɶɨɫɢ x ɥɢɛɨ y.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜɫɟ ɩɟɪɟɦɟɳɟɧɢɹ ɢ ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ
ɨɤɚɡɵɜɚɸɬɫɹɜɵɪɚɠɟɧɧɵɦɢ ɱɟɪɟɡɨɞɧɭɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜɟɟɫɪɟ
ɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ w (x, y).
ɉɟɪɟɦɟɳɟɧɢɹɢɞɟɮɨɪɦɚɰɢɢɜɩɥɚɫɬɢɧɟ
1.2.3. ɇɚɩɪɹɠɟɧɢɹɢɭɫɢɥɢɹɜɩɥɚɫɬɢɧɟ
ɋɨɝɥɚɫɧɨ ɩɟɪɜɨɣ ɝɢɩɨɬɟɡɟ Ʉɢɪɯɝɨɮɚ ɥɢɧɟɣɧɚɹ ɞɟɮɨɪɦɚɰɢɹ
ww
Hz
0 , ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɱɬɨ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ w ɧɟ ɡɚɜɢɫɹɬ
wz
ɨɬɤɨɨɪɞɢɧɚɬɵ z, ɬ ɟ w = w (x, y)ɗɬɨɨɡɧɚɱɚɟɬɱɬɨɩɟɪɟɦɟɳɟɧɢɹ
ɜɫɟɯɬɨɱɟɤɨɞɧɨɣɧɨɪɦɚɥɢɜɞɨɥɶɨɫɢ z ɨɞɢɧɚɤɨɜɵɢɫɨɨɬɜɟɬɫɬɜɭɸɬ
ɁɚɩɢɲɟɦɮɨɪɦɭɥɵɡɚɤɨɧɚȽɭɤɚ(1.4) ɞɥɹɥɢɧɟɣɧɵɯɞɟɮɨɪɦɚ
ɰɢɣɫɭɱɟɬɨɦız = ɫɨɝɥɚɫɧɨ ɬɪɟɬɶɟɣɝɢɩɨɬɟɡɟ Ʉɢɪɯɝɨɮɚ):
12
13
ɩɟɪɟɦɟɳɟɧɢɹɦɬɨɱɤɢɧɚɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ ɋɥɟɞɨɜɚɬɟɥɶɧɨɞɨ
ɫɬɚɬɨɱɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɝɢɛɵ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ
ɱɬɨɛɵɡɧɚɬɶɩɟɪɟɦɟɳɟɧɢɹɜɫɟɯɟɟɬɨɱɟɤ
ww wu
wv ww
ɉɨ ɬɨɣ ɠɟ ɝɢɩɨɬɟɡɟ J yz
0 , ɨɬ
0 ; J zx
wx wz
wz wy
wu
ww wv
ww
ɫɸɞɚ ɩɨɥɭɱɚɟɦ
;
. ɂɧɬɟɝɪɢɪɭɹ ɷɬɢ ɭɪɚɜɧɟɧɢɹ
wz
wx wz
wy
ɩɨ z ɢɢɫɩɨɥɶɡɭɹɭɫɥɨɜɢɹ u0 = v0 = 0 ɞɥɹɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɫɨ
ww
ww
; v z
. ɉɨɞɫɬɚɜ
ɝɥɚɫɧɨɜɬɨɪɨɣɝɢɩɨɬɟɡɟ, ɩɨɥɭɱɚɟɦ u z
wx
wy
ɥɹɹɷɬɢɡɚɜɢɫɢɦɨɫɬɢɜɭɪɚɜɧɟɧɢɹɄɨɲɢɢɦɟɟɦ
Hx
wu
wx
z
w2w
; Hy
wx 2
wv
wy
z
w2w
; J xy
wy 2
wu wv
wy wx
2 z
w2w
. (1.12)
wx wy
Ʉɚɤɜɢɞɧɨɞɟɮɨɪɦɚɰɢɢɩɪɨɢɡɜɨɥɶɧɨɝɨɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨɫɥɨɹ
ɩɥɚɫɬɢɧɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z ɦɟɧɹɸɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ
ɢɡɚɜɢɫɹɬɨɬɬɪɟɯɯɚɪɚɤɬɟɪɧɵɯɜɟɥɢɱɢɧ
Fx
1
Ux
Hx
w2w
; Fy
wx 2
1
Uy
1
(V x Q V y ) ; H y
E
w2w
; F
wy 2
w2w
.
wx wy
1
(V y Q V x ) .
E
(1.13)
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɋɥɨɠɢɦɷɬɢɪɚɜɟɧɫɬɜɚɩɨɨɱɟɪɟɞɧɨɭɦɧɨɠɢɜɤɚɠɞɨɟɢɡɧɢɯɧɚɤɨ
ɷɮɮɢɰɢɟɧɬɉɭɚɫɫɨɧɚȞ:
Hx Q H y
(1 Q 2 ) V x / E ;
H y Q H x (1 Q 2 ) V y / E ,
ɨɬɫɸɞɚ
Vx
ȿ1 (H x Q H y ) ; V y
ȿ1 (H y Q H x ), ɝɞɟ E1 E /(1 Q 2 ) .
ɋɭɱɟɬɨɦɡɚɜɢɫɢɦɨɫɬɟɣɩɨɥɭɱɚɟɦɜɵɪɚɠɟɧɢɹɧɨɪɦɚɥɶ
ɧɵɯ ɧɚɩɪɹɠɟɧɢɣɱɟɪɟɡɮɭɧɤɰɢɸɩɪɨɝɢɛɨɜ w:
Vx
§ w2w
w2w ·
ȿ1 ¨¨ 2 Q 2 ¸¸ z
wy ¹
© wx
ȿ1 (F x QF y ) z ;
Vy
§ w2w
w2w ·
ȿ1 ¨¨ 2 Q 2 ¸¸ z
wx ¹
© wy
ȿ1 (F y QF x ) z .
(1.14)
ɄɚɫɚɬɟɥɶɧɨɟɧɚɩɪɹɠɟɧɢɟɩɨɥɭɱɢɦɢɡɡɚɤɨɧɚȽɭɤɚɜɨɛɪɚɬɧɨɣ
ɮɨɪɦɟɫɭɱɟɬɨɦɢ
W W xy
E
J xy
2 (1 Q)
W yx
ȿ1 (1 Q)
w2w
z
wx wy
ȿ1 (1 Q) F z . (1.15)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ ɧɚɩɪɹɠɟɧɢɹ ıx ıy, IJ ɢɡɦɟɧɹɸɬɫɹɩɨɥɢɧɟɣɧɨɦɭɡɚɤɨɧɭ ɨɛɪɚɳɚɹɫɶɜɧɭɥɶɜɬɨɱɤɚɯɫɪɟ
ɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɭɤɚɡɚɧɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɩɨ ɜɵ
ɫɨɬɟ ɷɥɟɦɟɧɬɚ ɩɥɚɫɬɢɧɵ ɫ ɪɚɡɦɟɪɚɦɢ dx × dy ɩɨɤɚɡɚɧɨɧɚ ɪɢɫ. 1.2.
ɚ)
ɛ)
dx
dx
dy
dy
x
h/2 dz
z
h/2
³ (V x dz dy ) z
h / 2
h/2
H ˜ dy
h/2
h / 2
h / 2
2
³ (W dz dy ) z ȿ1 (1 Q) F ˜ dy ³ z dz
E1h 3
(F x QF y ) ˜ dy ,
12
E1 (1 Q)h 3
F ˜ dy .
12
ɁɞɟɫɶɜɟɥɢɱɢɧɚH ɹɜɥɹɟɬɫɹɢɧɬɟɧɫɢɜɧɨɫɬɶɸɤɪɭɬɹɳɟɝɨɦɨɦɟɧɬɚ.
Ⱥɧɚɥɨɝɢɱɧɨɧɚɯɨɞɢɦɦɨɦɟɧɬɵ My Âdx ɢ H Âdx. ɉɨɞɫɬɚɜɢɜɜɵ
ɪɚɠɟɧɢɹ ɤɪɢɜɢɡɧ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ
ɦɨɦɟɧɬɨɜɜɫɟɱɟɧɢɹɯɩɥɚɫɬɢɧɵɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯɤɟɟɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɢ
§ w2w
w2w ·
Ɇ x D (F x Q F y ) D ¨¨ 2 Q 2 ¸¸ ;
wy ¹
© wx
Ɇy
H
IJxy = IJyx = IJ
h / 2
h/2
§ w2w
w2w ·
D ¨¨ 2 Q 2 ¸¸ ;
wx ¹
© wy
w2w
D (1 Q)
.
wx w y
D (F y Q F x )
x
H·dx
h/2
ȿ1 (F x QF y ) ˜ dy ³ z 2 dz
ɝɞɟ ɜɟɥɢɱɢɧɚ Mx ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨ
ɦɟɧɬɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɧɚɩɪɹɠɟɧɢɸ ıx. ɉɨ ɪɚɡɦɟɪɧɨɫɬɢ ɷɬɨ
ɦɨɦɟɧɬ ɞɟɥɟɧɧɵɣ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɫɟɱɟɧɢɹ (ɩɨɝɨɧɧɵɣ ɢɡɝɢɛɚ
ɸɳɢɣ ɦɨɦɟɧɬ ɟɝɨ ɪɚɡɦɟɪɧɨɫɬɶ ɤɇ Â ɦ/ɦ ɬ ɟ Mx ɜɵɪɚɠɚɟɬɫɹ
ɜ ɟɞɢɧɢɰɚɯ ɫɢɥɵ ȼ ɞɚɥɶɧɟɣɲɟɦ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨ
ɦɟɧɬɚMx ɛɭɞɟɦɧɚɡɵɜɚɬɶɩɪɨɫɬɨɦɨɦɟɧɬɨɦ Mx ɜɞɚɧɧɨɣɬɨɱɤɟɫɟ
ɱɟɧɢɹɩɥɚɫɬɢɧɵɌɨɠɟɨɬɧɨɫɢɬɫɹɢɤɞɪɭɝɢɦɜɧɭɬɪɟɧɧɢɦɭɫɢɥɢɹɦ.
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚɩɪɹɠɟɧɢɹ IJ ɧɚɩɥɨɳɚɞɤɟ h×dy ɩɪɢɜɨɞɹɬɫɹ
ɤɤɪɭɬɹɳɟɦɭɦɨɦɟɧɬɭ H Âdy:
h/2
ıx
ıy
Ɇ x˜ dy
H·dy
Mx·dy
h/2
Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤɢɟ ɭɫɢɥɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɚɩɪɹɠɟɧɢɹɦ
(1.14), (1.15) ɜ ɫɟɱɟɧɢɹɯ ɩɥɚɫɬɢɧɵ ɧɨɪɦɚɥɶɧɵɯ ɤ ɟɟ ɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɢ (ɫɦ ɪɢɫ 1.2). Ɉɛɪɚɬɢɦɫɹ ɜɧɚɱɚɥɟ ɤ ɩɥɨɳɚɞɤɟ ɫ ɧɨɪɦɚ
ɥɶɸ ɩɚɪɚɥɥɟɥɶɧɨɣ ɨɫɢ x. ɇɚɩɪɹɠɟɧɢɹ ıx ɧɚ ɝɪɚɧɢ ɷɥɟɦɟɧɬɚ h×dy
ɩɪɢɜɨɞɹɬɫɹɤɢɡɝɢɛɚɸɳɟɦɭɦɨɦɟɧɬɭ Mx Âdy:
D (1 Q) F
(1.16)
Ɋɢɫ 1.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟɧɚɩɪɹɠɟɧɢɣɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵ
ɚ – ɧɨɪɦɚɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹ; ɛ – ɤɚɫɚɬɟɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹ
E1h 3
E h3
ɧɚɡɵɜɚɟɬɫɹɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬ12 12 (1 Q 2 )
ɤɨɫɬɶɸɩɥɚɫɬɢɧɵ. ɗɬɚɜɟɥɢɱɢɧɚɹɜɥɹɟɬɫɹɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣɩɥɚɫɬɢɧɵ ɩɪɢɟɟɢɡɝɢɛɟ ɢɢɝɪɚɟɬɬɭɠɟɪɨɥɶɱɬɨ
14
15
y My·dx
z
y
z
ȼɟɥɢɱɢɧɚ D
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɢ ɠɟɫɬɤɨɫɬɶ ɫɟɱɟɧɢɹ EI ɩɪɢ ɢɡɝɢɛɟ ɛɚɥɨɤ Ɉɬɦɟɬɢɦ ɬɚɤɠɟ ɱɬɨ
ɜɜɢɞɭ ɩɚɪɧɨɫɬɢ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɤɪɭɬɹɳɢɣ ɦɨɦɟɧɬ H ɧɚ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯɝɪɚɧɹɯɷɥɟɦɟɧɬɚɩɥɚɫɬɢɧɵɨɞɢɧɚɤɨɜ
ɇɚɪɢɫ 1.3, ɚ ɩɨɤɚɡɚɧɵɩɨɥɨɠɢɬɟɥɶɧɵɟɡɧɚɱɟɧɢɹɢɡɝɢɛɚɸɳɢɯ
ɢ ɤɪɭɬɹɳɢɯ ɦɨɦɟɧɬɨɜ ɩɪɢɱɟɦ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɭɫɢ
ɥɢɣ ɫɨɜɩɚɞɚɸɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɞɟɣɫɬɜɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɨɥɨ
ɠɢɬɟɥɶɧɵɯɫɨɫɬɚɜɥɹɸɳɢɯɧɚɩɪɹɠɟɧɢɣ
ɚ)
ɛ)
dx
dy
dx
Qy
dy
H
h/2
H
h/2
My
Qx
Mx
x
IJxz
h/2
h/2
x
IJyz
Qx
z
Qy z
y
Ɋɢɫ ɍɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹɜɩɥɚɫɬɢɧɟ
ɚ – ɢɡɝɢɛɚɸɳɢɟɢɤɪɭɬɹɳɢɟɦɨɦɟɧɬɵ; ɛ – ɩɨɩɟɪɟɱɧɵɟɫɢɥɵɢɧɚɩɪɹɠɟɧɢɹ
y
ɂɡɝɢɛɚɸɳɢɟɦɨɦɟɧɬɵ Mx ɢ My ɫɨɡɞɚɸɬɢɫɤɪɢɜɥɟɧɢɟɷɥɟɦɟɧ
ɬɚɩɥɚɫɬɢɧɵɫɤɪɢɜɢɡɧɚɦɢ F x ɢ F y ɄɪɭɬɹɳɢɟɦɨɦɟɧɬɵH ɫɨɡɞɚɸɬ
ɞɟɮɨɪɦɚɰɢɸ ɫɞɜɢɝɚ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɫɥɨɟɜ ɷɥɟɦɟɧɬɚ ɢɡɦɟɧɹɸɳɭ
ɸɫɹɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵɩɨɥɢɧɟɣɧɨɦɭɡɚɤɨɧɭ
ȼɵɪɚɡɢɜ ɤɪɢɜɢɡɧɵ ɜ (1.16) ɱɟɪɟɡ ɩɨɝɨɧɧɵɟ ɦɨɦɟɧɬɵ ɢ ɩɨɞ
ɫɬɚɜɢɜɜɮɨɪɦɭɥɵɞɥɹɧɚɩɪɹɠɟɧɢɣ1.14), (1.15)ɩɨɥɭɱɢɦ
Vx
ȿ1 (F x QF y ) z
E1
Vy
ȿ1 (F y QF x ) z
E1
W
ȿ1 (1 Q) F z
E1
H
z
D
Mx
z
D
12 M x
z;
h3
My
12 M y
D
z
h3
z;
(1.17)
12 H
z.
h3
Ʉɚɤɜɢɞɧɨɮɨɪɦɭɥɵɞɥɹɧɨɪɦɚɥɶɧɵɯɧɚɩɪɹɠɟɧɢɣɫɨɜɩɚɞɚɸɬ
ɫɮɨɪɦɭɥɚɦɢɫɨɩɪɨɬɢɜɥɟɧɢɹɦɚɬɟɪɢɚɥɨɜɩɪɢɢɡɝɢɛɟɛɚɥɤɢɩɪɹɦɨ
ɭɝɨɥɶɧɨɝɨɫɟɱɟɧɢɹɜɵɫɨɬɨɣ h ɢɲɢɪɢɧɨɣ ɪɚɜɧɨɣɟɞɢɧɢɰɟ
16
Ɇɚɤɫɢɦɚɥɶɧɵɟɡɧɚɱɟɧɢɹɧɚɩɪɹɠɟɧɢɣɜɨɡɧɢɤɚɸɬɩɪɢ z
rh 2:
6M y
6M x
6H
(1.18)
; V y max
; W max
.
2
2
h
h
h2
Ʉɪɨɦɟ ɦɨɦɟɧɬɨɜ, ɜ ɫɟɱɟɧɢɹɯ ɩɥɚɫɬɢɧɵ ɞɟɣɫɬɜɭɸɬ ɩɨɩɟɪɟɱ
ɧɵɟɫɢɥɵɢɧɬɟɧɫɢɜɧɨɫɬɢɤɨɬɨɪɵɯɨɛɨɡɧɚɱɢɦ Qx ɢ Qyɂɦɨɬɜɟɱɚ
ɸɬ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ IJxz ɢ IJyz, ɜɟɥɢɱɢɧɵ ɤɨɬɨɪɵɯ ɦɚɥɵ ɩɨ
ɫɪɚɜɧɟɧɢɸɫɜɟɥɢɱɢɧɚɦɢɨɫɧɨɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ Vx, Vy, Wxy ɪɚɫɩɪɟ
ɞɟɥɟɧɢɟɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣIJxz, IJyz ɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵɫɨ
ɨɬɜɟɬɫɬɜɭɟɬɡɚɤɨɧɭɤɜɚɞɪɚɬɧɨɣɩɚɪɚɛɨɥɵ – ɪɢɫ. 1.3, ɛ).
Ʉɚɤ ɭɠɟ ɭɩɨɦɢɧɚɥɨɫɶ ɦɨɦɟɧɬɵ Mx, My ɢ H, ɚ ɬɚɤɠɟ ɭɫɢɥɢɹ
Qx ɢ Qy ɩɨɥɨɠɢɬɟɥɶɧɵɟɫɥɢɞɥɹɬɨɱɤɢɩɥɚɫɬɢɧɵɫɤɨɨɪɞɢɧɚɬɨɣ z > 0
ɨɧɢɞɚɸɬɩɨɥɨɠɢɬɟɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹɂɧɞɟɤɫɵɩɪɢɭɫɢɥɢɹɯɫɨɨɬ
ɜɟɬɫɬɜɭɸɬɧɨɪɦɚɥɢɤɫɟɱɟɧɢɸɜɤɨɬɨɪɨɦɞɟɣɫɬɜɭɸɬɷɬɢɭɫɢɥɢɹ
V x max
1.2.4. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɥɚɫɬɢɧɵ
Ɍɚɤɢɦɨɛɪɚɡɨɦɜɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹɜɩɥɚɫɬɢɧɟ
ɨɞɧɨɡɧɚɱɧɨ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɩɪɨɝɢɛɵ ɟɟ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ
ɉɨɫɥɟ ɬɨɝɨ ɤɚɤ ɩɟɪɟɦɟɳɟɧɢɹ w ɛɭɞɭɬ ɨɩɪɟɞɟɥɟɧɵ ɜɫɟ ɨɫɬɚɥɶɧɵɟ
ɪɟɡɭɥɶɬɚɬɵ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɩɪɢ ɩɨɦɨɳɢ ɨɩɟɪɚɰɢɣ ɞɢɮɮɟ
ɪɟɧɰɢɪɨɜɚɧɢɹ Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ ɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ w (x, y) ɩɨɥ
ɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɟɬ ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɩɥɚ
ɫɬɢɧɵ Ⱦɥɹɨɬɵɫɤɚɧɢɹɷɬɨɣ ɧɟɢɡɜɟɫɬɧɨɣɮɭɧɤɰɢɢ ɧɟɨɛɯɨɞɢɦɨɫɨ
ɫɬɚɜɢɬɶɪɚɡɪɟɲɚɸɳɟɟɭɪɚɜɧɟɧɢɟɨɬɧɨɫɢɬɟɥɶɧɨɩɪɨɝɢɛɨɜ w.
Ɋɚɫɫɦɨɬɪɢɦɪɚɜɧɨɜɟɫɢɟɷɥɟɦɟɧɬɚɩɥɚɫɬɢɧɵɪɚɡɦɟɪɚɦɢ dx×dy
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɧɟɦɭ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɫɪɟɞɢɧ
ɧɨɣɩɥɨɫɤɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɧɚɝɪɭɡɤɢ q ɢɩɨɝɨɧɧɵɯ ɜɧɭɬɪɟɧɧɢɯ
ɭɫɢɥɢɣ ɧɚ ɝɪɚɧɢɰɚɯ ɷɥɟɦɟɧɬɚ ɪɢɫ 1.4).
ɍɱɢɬɵɜɚɟɦɱɬɨ ɩɪɢɩɟɪɟɯɨɞɟɨɬɨɞɧɨɣɝɪɚɧɢɷɥɟɦɟɧɬɚɤɞɪɭ
ɝɨɣ ɝɪɚɧɢ ɨɬɫɬɨɹɳɟɣ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɟ ɪɚɫɫɬɨɹɧɢɟ dx ɢɥɢ dy,
ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɩɨɥɭɱɚɸɬ ɬɚɤɠɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɩɪɢɪɚɳɟ
ɧɢɹ ɧɚ ɜɟɥɢɱɢɧɭ ɱɚɫɬɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɧɚɩɪɢɦɟɪ
wM x
wM x
˜ dx . ɉɪɢɷɬɨɦɜɫɟɩɨɝɨɧɧɵɟɭɫɢɥɢɹɫɥɟɞɭɟɬɭɦɧɨɠɚɬɶ
wx
ɧɚɞɥɢɧɭɝɪɚɧɢɩɨɤɨɬɨɪɨɣɨɧɢɞɟɣɫɬɜɭɸɬ
17
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
H
H
x
My
q
0
Mx
M y
z
H
wH
dx
wx
M x
wM y
wy
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
wM x
wx
Qy
wH
H
dy
wy
dy
wQx
wx
dx
dx
0
Qx
Qy wQ y
wy
dy
z
y
dy
dx
Ɋɢɫ ɗɥɟɦɟɧɬɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɩɥɚɫɬɢɧɵ
ɋɩɪɨɟɰɢɪɭɟɦɜɫɟɫɢɥɵɩɪɢɥɨɠɟɧɧɵɟɤɷɥɟɦɟɧɬɭɧɚ ɨɫɶ z:
(Qx wQ y
wQx
˜ dx) ˜ dy Qx ˜ dy (Q y ˜ dy ) ˜ dx Q y ˜ dx q ˜ dx ˜ dy
wx
wy
0.
ɉɨɫɥɟɩɪɢɜɟɞɟɧɢɹɩɨɞɨɛɧɵɯɱɥɟɧɨɜɩɨɥɭɱɢɦ
wQx wQ y
wx
wy
q.
(1.19)
q ɢɥɢ ’ 4 w
q
.
D
(1.21)
w2
w2
2
’
w
(
)
(’ 2 w) – ɛɢɝɚɪɦɨɧɢɱɟɫɤɢɣ
wx 2
wy 2
w2w w2w
ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ; ’ 2 w
2 – ɨɛɵɱɧɵɣ ɝɚɪɦɨɧɢɱɟ
wx 2
wy
ɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚɨɬɮɭɧɤɰɢɢɩɪɨɝɢɛɨɜ w (x, y) ɩɥɚɫɬɢɧɵ
ɍɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟɢɡɨɝɧɭɬɨɣɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵɟɝɨɧɚɡɵ
ɜɚɸɬ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɟɦ ɋɨɮɢ ɀɟɪɦɟɧ – Ʌɚɝɪɚɧɠɚ ɢɥɢ ɩɪɨɫɬɨ
ɭɪɚɜɧɟɧɢɟɦ ɋɨɮɢ ɀɟɪɦɟɧ Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɝɪɚɟɬ ɮɭɧɞɚɦɟɧ
ɬɚɥɶɧɭɸɪɨɥɶɜɬɟɨɪɢɢɢɡɝɢɛɚɩɥɚɫɬɢɧ
ɉɨɫɥɟɧɚɯɨɠɞɟɧɢɹɢɡɭɪɚɜɧɟɧɢɹɮɭɧɤɰɢɢɩɪɨɝɢɛɨɜw (x, y)
ɩɨ ɩɪɢɜɟɞɟɧɧɵɦ ɪɚɧɟɟ ɮɨɪɦɭɥɚɦ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ ɤɨɦɩɨ
ɧɟɧɬɵ ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ
ɬɨɱɤɟ ɩɥɚɫɬɢɧɵ ȼɦɟɫɬɟ ɫ ɬɟɦ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜ
ɧɟɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɨɩɪɟ
ɞɟɥɹɟɦɵɯɢɡɭɫɥɨɜɢɣɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ– ɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ.
Ɂɞɟɫɶ ’ 4 w
q
dy
y
Qx x
dx
§ w4w
w4w
w4w ·
D ¨¨ 4 2 2 2 4 ¸¸
wx w y
wy ¹
© wx
’ 2’ 2 w
Ɏɨɪɦɭɥɢɪɨɜɤɚɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
Ⱥɧɚɥɨɝɢɱɧɨ ɢɡ ɭɪɚɜɧɟɧɢɣ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɟɣ y ɢ x, ɩɪɟɧɟɛɪɟɝɚɹɜɟɥɢɱɢɧɚɦɢ ɛɨɥɟɟɜɵɫɨɤɨɝɨ ɩɨɪɹɞɤɚɦɚɥɨ
ɫɬɢ, ɩɨɥɭɱɚɟɦ
wM y wH
wM x wH
(1.20)
Qx ;
Qy .
wx
wy
wy
wx
ɉɨɞɫɬɚɜɢɜɜɷɬɨɭɪɚɜɧɟɧɢɟɜɵɪɚɠɟɧɢɹɦɨɦɟɧɬɨɜɱɟɪɟɡɮɭɧɤ
ɰɢɸɩɪɨɝɢɛɨɜɢɭɩɪɨɫɬɢɜɩɨɥɭɱɢɦ
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɯɚɪɚɤɬɟɪɚ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɟɜ ɧɚ ɤɨɧɬɭɪɟ
ɩɥɚɫɬɢɧɵɦɨɝɭɬɛɵɬɶɡɚɞɚɧɵɩɪɨɝɢɛɵɢɭɝɥɵɩɨɜɨɪɨɬɚɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɢɢɡɝɢɛɚɸɳɢɟɢɤɪɭɬɹɳɢɟɦɨɦɟɧɬɵɚɬɚɤɠɟ ɩɨɩɟɪɟɱɧɵɟ
ɫɢɥɵ Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɪɢ ɤɨɬɨɪɵɯ ɧɚ ɤɚɤɨɦ-ɥɢɛɨ ɭɱɚɫɬɤɟ
ɤɨɧɬɭɪɚ ɡɚɞɚɸɬɫɹ ɩɟɪɟɦɟɳɟɧɢɹ (ɬ ɟ ɩɪɨɝɢɛɵ ɢ ɭɝɥɵ ɩɨɜɨɪɨɬɚ),
ɧɚɡɵɜɚɸɬɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɪɢ ɤɨɬɨɪɵɯ
ɧɚɭɱɚɫɬɤɟɤɨɧɬɭɪɚɡɚɞɚɸɬɫɹɭɫɢɥɢɹ (ɬɟɦɨɦɟɧɬɵɢɫɢɥɵ)ɧɚɡɵ
ɜɚɸɬɫɹɫɬɚɬɢɱɟɫɤɢɦɢȿɫɥɢɠɟɧɚɨɞɧɨɦɭɱɚɫɬɤɟɤɨɧɬɭɪɚ ɡɚɞɚɧɵ
ɨɞɧɨɜɪɟɦɟɧɧɨɢɩɟɪɟɦɟɳɟɧɢɹ, ɢɭɫɢɥɢɹɬɨɬɚɤɢɟɝɪɚɧɢɱɧɵɟɭɫɥɨ
ɜɢɹɧɚɡɵɜɚɸɬɫɹɫɦɟɲɚɧɧɵɦɢ.
ɍɪɚɜɧɟɧɢɟɋɨɮɢ ɀɟɪɦɟɧɹɜɥɹɟɬɫɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɭɪɚɜ
ɧɟɧɢɟɦɱɟɬɜɟɪɬɨɝɨɩɨɪɹɞɤɚɉɨɫɤɨɥɶɤɭɩɪɨɝɢɛɵɹɜɥɹɸɬɫɹɮɭɧɤɰɢ
ɟɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ (x, y) ɩɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɩɨɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɭɱɟɬɚ
ɜɨɫɶɦɢɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ– ɩɨɞɜɚɭɫɥɨɜɢɹɧɚɤɚɠɞɨɦɤɪɚɸ
18
19
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɭɪɚɜɧɟɧɢɹ (1.19), (1.20) ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟ
ɥɹɸɬɪɚɜɧɨɜɟɫɢɟɷɥɟɦɟɧɬɚɩɥɚɫɬɢɧɵɂɫɤɥɸɱɢɦɢɡɷɬɢɯɭɪɚɜɧɟɧɢɣ
ɩɨɩɟɪɟɱɧɵɟ ɫɢɥɵ Qx ɢ Qy: ɨɩɪɟɞɟɥɢɜ ɢɯ ɢɡ ɭɪɚɜɧɟɧɢɣ ɢɩɨɞɫɬɚɜɢɜɜɭɪɚɜɧɟɧɢɟ ɛɭɞɟɦɢɦɟɬɶ
2
w2Ɇ x
w2H w Ɇ y
2
wx 2
wx w y
wy2
q.
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɋɮɨɪɦɭɥɢɪɭɟɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɫɥɭɱɚɟɜ
ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɟɜ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ. ɇɚ ɪɢɫ 1.5 ɢɡɨɛɪɚ
ɠɟɧɚ ɩɥɚɫɬɢɧɚ ɭ ɤɨɬɨɪɨɣ ɤɪɚɣ y = 0 ɠɟɫɬɤɨ ɡɚɞɟɥɚɧ ɤɪɚɹ x = 0
ɢx = a ɲɚɪɧɢɪɧɨɨɩɟɪɬɵɚɤɪɚɣy = b ɫɜɨɛɨɞɟɧɨɬɡɚɤɪɟɩɥɟɧɢɣ
0
x
b
ɞɟɥɟɧɧɵɣ ɦɨɦɟɧɬ mx ɬɨ ɜɬɨɪɨɟ ɭɫɥɨɜɢɟ ɢɡ (1.23) ɧɭɠɧɨ ɡɚɩɢɫɚɬɶ
m
w2w
ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ 2 x .
D
wx
ɋɜɨɛɨɞɧɵɣ ɤɪɚɣ. ȼ ɨɬɫɭɬɫɬɜɢɟ ɧɚ ɷɬɨɦ ɭɱɚɫɬɤɟ ɤɨɧɬɭɪɚ
ɜɧɟɲɧɢɯ ɫɢɥɨɜɵɯ ɮɚɤɬɨɪɨɜ ɩɨɝɨɧɧɵɯ ɧɚɝɪɭɡɨɤ ɜɫɟ ɩɨɝɨɧɧɵɟ
ɜɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɬɚɤɠɟɞɨɥɠɧɵɛɵɬɶɪɚɜɧɵɧɭɥɸɌɚɤɞɥɹɤɪɚɹ
y=b
My
a
y
Ɋɢɫ ɉɪɹɦɨɭɝɨɥɶɧɚɹɩɥɚɫɬɢɧɚɫɪɚɡɥɢɱɧɵɦɢɡɚɤɪɟɩɥɟɧɢɹɦɢɤɪɚɟɜ
Ɂɚɞɟɥɚɧɧɵɣ ɢɥɢɡɚɳɟɦɥɟɧɧɵɣ) ɤɪɚɣȼɷɬɨɦɫɥɭɱɚɟɧɚɫɨɨɬ
ɜɟɬɫɬɜɭɸɳɟɦ ɭɱɚɫɬɤɟ ɤɨɧɬɭɪɚ ɩɥɚɫɬɢɧɵ ɨɬɫɭɬɫɬɜɭɸɬ ɩɪɨɝɢɛɵ
ɢɧɟɜɨɡɦɨɠɟɧɩɨɜɨɪɨɬɤɪɚɟɜɨɝɨɫɟɱɟɧɢɹɜɧɚɩɪɚɜɥɟɧɢɢɩɟɪɩɟɧɞɢ
ɤɭɥɹɪɧɨɦɤɷɬɨɦɭɤɪɚɸ Ɍɚɤɢɦɨɛɪɚɡɨɦɞɥɹɤɪɚɹy = 0 ɢɦɟɟɦ
w 0
y 0
;
Ty
ww
wy
0
y 0
.
(1.22)
ɗɬɢɭɫɥɨɜɢɹɚɧɚɥɨɝɢɱɧɵɭɫɥɨɜɢɹɦɡɚɞɟɥɤɢɢɡɝɢɛɚɟɦɨɣɛɚɥɤɢ
ɒɚɪɧɢɪɧɨ ɨɩɟɪɬɵɣ ɤɪɚɣ Ɂɞɟɫɶ ɨɬɫɭɬɫɬɜɭɸɬ ɩɪɨɝɢɛɵ ɢ ɢɡ
ɝɢɛɚɸɳɢɟ ɦɨɦɟɧɬɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɤ ɷɬɨɦɭ
ɤɪɚɸɬɟ ɩɪɢ x = 0 ɢ x = a ɢɦɟɟɦw = 0 ɢMx = 0. ȼɵɪɚɡɢɜ ɦɨɦɟɧɬ Mx
ɱɟɪɟɡɩɪɨɝɢɛɢɭɱɢɬɵɜɚɹɱɬɨɜɞɨɥɶɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯɤɪɚɟɜ
ww w 2 w
ɢɡɝɢɛɚɧɟɬɬ ɟ
0 , ɜɢɬɨɝɟ ɩɨɥɭɱɢɦ
wy wy 2
w 0
x
;
0, a
w2w
wx 2
0
x 0, a
.
(1.23)
ɍɫɥɨɜɢɹ ɫɩɪɚɜɟɞɥɢɜɵ ɞɥɹ ɨɩɢɪɚɧɢɹ ɤɪɚɟɜ ɩɥɚɫɬɢɧɵ ɧɚ
ɠɟɫɬɤɢɟ ɲɚɪɧɢɪɧɵɟ ɨɩɨɪɵ Ɍɚɤɠɟ ɨɬɦɟɬɢɦ ɱɬɨ ɟɫɥɢ ɤ ɤɚɤɨɦɭɥɢɛɨ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɦɭ ɤɪɚɸ ɛɭɞɟɬ ɩɪɢɥɨɠɟɧ ɜɧɟɲɧɢɣ ɪɚɫɩɪɟ
20
0
y b
; Qy
0
y b
; H
0
y b
.
(1.24)
Ɍɚɤɢɦɨɛɪɚɡɨɦ, ɜɦɟɫɬɨɞɜɭɯɧɟɨɛɯɨɞɢɦɵɯɭɫɥɨɜɢɣɡɞɟɫɶɩɨ
ɹɜɥɹɸɬɫɹ ɬɪɢ ɭɫɥɨɜɢɹ Ɉɞɧɚɤɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɪɢɛɥɢɠɟɧɧɨɣ
ɬɟɨɪɢɢɄɢɪɯɝɨɮɚ ɜɨɛɳɟɦɫɥɭɱɚɟɧɟɥɶɡɹɨɞɧɨɜɪɟɦɟɧɧɨɭɞɨɜɥɟɬɜɨ
ɪɢɬɶɞɜɭɦɩɨɫɥɟɞɧɢɦɭɫɥɨɜɢɹɦ>1@ɗɬɨɩɪɨɬɢɜɨɪɟɱɢɟɦɨɠ
ɧɨɭɫɬɪɚɧɢɬɶɡɚɦɟɧɢɜɩɨɩɟɪɟɱɧɭɸɫɢɥɭ Qy ɢɤɪɭɬɹɳɢɣɦɨɦɟɧɬ H
ɨɞɧɨɣɜɟɪɬɢɤɚɥɶɧɨɣɨɛɨɛɳɟɧɧɨɣɫɢɥɨɣ ɫɬɚɬɢɱɟɫɤɢ ɢɦɷɤɜɢɜɚɥɟɧɬ
ɧɨɣɢɨɬɜɟɱɚɸɳɟɣɩɪɨɝɢɛɭwɤɚɤɨɛɨɛɳɟɧɧɨɦɭɩɟɪɟɦɟɳɟɧɢɸ
Vy
Qy 'Qy ,
(1.25)
ɝɞɟ ¨Qy – ɞɨɩɨɥɧɢɬɟɥɶɧɚɹ ɩɨɩɟɪɟɱɧɚɹ ɫɢɥɚ ɫɬɚɬɢɱɟɫɤɢ ɷɤɜɢɜɚ
ɥɟɧɬɧɚɹɤɪɭɬɹɳɟɦɭɦɨɦɟɧɬɭ H. Ɍɨɝɞɚɤɚɤɢɜɩɪɟɞɵɞɭɳɢɯɫɥɭɱɚ
ɹɯ ɨɩɢɪɚɧɢɹ ɞɥɹ ɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ ɜɨɡɦɨɠɧɨ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɧɟ
ɬɪɟɦ ɚ ɬɨɥɶɤɨ ɞɜɭɦ ɫɢɥɨɜɵɦ ɭɫɥɨɜɢɹɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɬɨɥɶɤɨ
ɞɜɭɦɧɟɡɚɜɢɫɢɦɵɦɩɟɪɟɦɟɳɟɧɢɹɦɧɚɤɪɨɦɤɟ ɩɥɚɫɬɢɧɵ
Ɋɚɫɫɦɨɬɪɢɦ ɞɟɣɫɬɜɢɟ ɤɪɭɬɹɳɢɯ ɦɨɦɟɧɬɨɜ, ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ
ɜɞɨɥɶɝɪɚɧɢȼɋ, ɧɚɫɜɨɛɨɞɧɨɦɤɪɚɸ y = b ɪɢɫ 1.6, ɚ). ɇɚɷɥɟɦɟɧ
ɬɚɪɧɨɦɭɱɚɫɬɤɟ dx ɞɟɣɫɬɜɭɟɬɤɪɭɬɹɳɢɣɦɨɦɟɧɬɪɚɜɧɵɣ H Âdx, ɤɨ
H ˜d x
ɬɨɪɵɣɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɜɢɞɟɩɚɪɵɫɢɥ H
ɫɩɥɟɱɨɦ dx.
dx
ɇɚ ɫɨɫɟɞɧɟɦ ɭɱɚɫɬɤɟ dx ɷɬɚ ɩɚɪɚ ɫɢɥ ɩɨɥɭɱɢɬ ɩɪɢɪɚɳɟɧɢɟ dH
wH
ɢɛɭɞɟɬɪɚɜɧɚ H dx ɫɬɚɤɢɦɠɟɩɥɟɱɨɦ dx ɪɢɫ 1.6, ɛ).
wx
ɉɪɢɬɚɤɨɦɩɪɟɞɫɬɚɜɥɟɧɢɢɞɟɣɫɬɜɢɟɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɦɨɠ
ɧɨɡɚɦɟɧɢɬɶɜɟɪɬɢɤɚɥɶɧɨɣɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɨɣɢɧɬɟɧɫɢɜɧɨ
wH
ɢ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢɫɢɥɚɦɢ Hɜɨɡɧɢɤɚɸɳɢɦɢ ɜɭɝɥɚɯ
ɫɬɶɸ
wx
B ɢ C ɪɢɫ 1.6, ɜ). ɋɭɦɦɢɪɭɹɷɬɭɧɚɝɪɭɡɤɭ ɫɨɝɥɚɫɧɨ ɫɩɨɩɟ
21
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɪɟɱɧɨɣɫɢɥɨɣ Qy, ɩɨɥɭɱɢɦɫɥɟɞɭɸɳɢɟɞɜɚɭɫɥɨɜɢɹɞɥɹɫɜɨɛɨɞɧɨɝɨ
ɤɪɚɹ y = b:
wH
(1.26)
M y 0 y b ; Qy 0 y b.
wx
HÂdx
a)
B
C
x
H H
wH
dx
wx
B
C
x
H
ɜ)
dx
H
wH
wx
H
dx
B
wH
dx
wx
H
C
x
Ɋɢɫ 1.6. ɉɪɟɞɫɬɚɜɥɟɧɢɟɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɧɚɝɪɚɧɢ ɩɥɚɫɬɢɧɵ
ȼɵɪɚɡɢɜ ɭɫɢɥɢɹ ɱɟɪɟɡ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ ɡɚɩɢɲɟɦ ɝɪɚɧɢɱ
ɧɵɟɭɫɥɨɜɢɹ (1.26) ɜɬɚɤɨɦ ɜɢɞɟ
w2w
w2w
Q
wy2
wx 2
0
y
;
b
w § w2w
w2w ·
¨¨ 2 (2 Q) 2 ¸¸ 0
wy © wy
wx ¹
y b
(1.27)
.
ȼɫɥɭɱɚɟ ɠɟɟɫɥɢɫɜɨɛɨɞɧɵɣɤɪɚɣɛɭɞɟɬɡɚɞɚɧɧɚɝɪɚɧɹɯɩɥɚ
ɫɬɢɧɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɨɫɢ x ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹɞɥɹ x = 0, x = a ɡɚɩɢɲɭɬɫɹɬɚɤ:
w2w
w2w
Q
wx 2
wy2
0
x
;
0, a
w § w2w
w2w ·
¨¨ 2 (2 Q) 2 ¸¸ 0
wx © wx
wy ¹
x 0, a
.
(1.28)
Ʉɨɧɟɱɧɨ ɩɪɢ ɷɬɨɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɫɜɨɛɨɞɧɨɣ ɝɪɚɧɢ
ɛɭɞɭɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶɫɹ ɩɪɢɛɥɢɠɟɧɧɨ ɇɨ ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɢɧɰɢɩɚ
ɋɟɧ-ȼɟɧɚɧɚɡɚɦɟɧɚ ɩɨɩɟɪɟɱɧɨɣɫɢɥɵ ɢɤɪɭɬɹɳɟɝɨɦɨɦɟɧɬɚɫɬɚɬɢ
ɱɟɫɤɢ ɢɦ ɷɤɜɢɜɚɥɟɧɬɧɨɣ ɨɛɨɛɳɟɧɧɨɣ ɩɨɩɟɪɟɱɧɨɣ ɫɢɥɨɣ ɜɵɡɨɜɟɬ
ɥɢɲɶ ɦɟɫɬɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜɛɥɢɡɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɜɨɛɨɞɧɨɝɨ
ɤɪɚɹɩɥɚɫɬɢɧɵ[1, 12].
22
ɗɥɟɦɟɧɬɚɪɧɵɟɫɥɭɱɚɢɢɡɝɢɛɚɩɥɚɫɬɢɧ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɪɚɫɱɟɬ ɬɨɧɤɢɯ ɭɩɪɭɝɢɯ ɩɥɚɫɬɢɧ ɫɜɨɞɢɬɫɹ
ɤɪɟɲɟɧɢɸɤɪɚɟɜɨɣɡɚɞɚɱɢɞɥɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ ɢɡɨ
ɝɧɭɬɨɣɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵ
w4w
w4w
w4w
2
wx 4
wx 2 wy 2 w y 4
dx
ɛ)
1.3. Ɋɟɲɟɧɢɟɡɚɞɚɱɢɡɝɢɛɚɩɪɹɦɨɭɝɨɥɶɧɵɯ ɩɥɚɫɬɢɧ
q
,
D
(1.29)
ɝɞɟ w = w (x, y) – ɮɭɧɤɰɢɹɩɪɨɝɢɛɨɜɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɩɥɚɫɬɢ
ɧɵ q = q (x, y) – ɪɚɫɩɪɟɞɟɥɟɧɧɚɹɩɨɜɟɪɯɧɨɫɬɧɚɹɧɚɝɪɭɡɤɚɩɟɪɩɟɧ
ɞɢɤɭɥɹɪɧɚɹɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ.
Ʉɪɚɟɜɨɣɡɚɞɚɱɟɣ ɧɚɡɵɜɚɟɬɫɹɡɚɞɚɱɚɧɚɯɨɠɞɟɧɢɹɬɚɤɨɝɨɪɟɲɟ
ɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɤɨɬɨɪɨɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɨɩɪɟ
ɞɟɥɟɧɧɵɦɭɫɥɨɜɢɹɦɧɚɝɪɚɧɢɰɚɯɤɪɚɹɯɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣɨɛɥɚɫɬɢ
Ɍɨɱɧɨɟɟɟɪɟɲɟɧɢɟɜɚɧɚɥɢɬɢɱɟɫɤɨɣɮɨɪɦɟɜɨɡɦɨɠɧɨɥɢɲɶɜɧɟɤɨ
ɬɨɪɵɯɱɚɫɬɧɵɯɫɥɭɱɚɹɯɝɟɨɦɟɬɪɢɢɩɥɚɫɬɢɧɵɟɟɧɚɝɪɭɠɟɧɢɹɢɭɫɥɨ
ɜɢɣɡɚɤɪɟɩɥɟɧɢɹȼɦɟɫɬɟɫɬɟɦɬɨɱɧɵɟɚɧɚɥɢɬɢɱɟɫɤɢɟɪɟɲɟɧɢɹɡɚɞɚɱ
ɹɜɥɹɸɬɫɹɫɜɨɟɨɛɪɚɡɧɵɦɷɬɚɥɨɧɨɦɫɤɨɬɨɪɵɦɦɨɠɧɨɫɪɚɜɧɢɜɚɬɶɪɟ
ɲɟɧɢɹɩɨɥɭɱɟɧɧɵɟɪɚɡɥɢɱɧɵɦɢɩɪɢɛɥɢɠɟɧɧɵɦɢɦɟɬɨɞɚɦɢ
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɩɪɨɫɬɵɟ ɫɥɭɱɚɢ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧ ɢɦɟ
ɸɳɢɟ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɩɨɧɢɦɚɧɢɹ ɨɫɨɛɟɧɧɨɫɬɟɣ ɢɯ ɪɚɛɨɬɵ
ɉɪɢɷɬɨɦɪɟɲɟɧɢɟɞɚɧɧɵɯɡɚɞɚɱɧɟɜɵɡɵɜɚɟɬɨɫɨɛɵɯɡɚɬɪɭɞɧɟɧɢɣ.
1. ɐɢɥɢɧɞɪɢɱɟɫɤɢɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
ɉɪɟɞɫɬɚɜɢɦ ɫɟɛɟ ɛɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɭɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ y
ɩɥɚɫɬɢɧɭ ɪɢɫ 1.7) ɧɚɝɪɭɡɤɚ q ɧɚ ɤɨɬɨɪɭɸ ɩɨɫɬɨɹɧɧɚ ɜɞɨɥɶ ɷɬɨɣ
ɠɟɨɫɢɢɢɡɦɟɧɹɟɬɫɹɬɨɥɶɤɨɩɨɨɫɢxɬ ɟ q = q (x). Ɉɱɟɜɢɞɧɨɱɬɨ
ɜɫɟɩɨɥɨɫɤɢɟɞɢɧɢɱɧɨɣɲɢɪɢɧɵɜɵɞɟɥɟɧɧɵɟɢɡɷɬɨɣɩɥɚɫɬɢɧɵɜɞɨɥɶ
ɨɫɢ xɛɭɞɭɬɢɡɝɢɛɚɬɶɫɹɨɞɢɧɚɤɨɜɨɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɰɟɥɨɦɩɥɚɫɬɢɧɚ
ɨɤɚɠɟɬɫɹɢɡɨɝɧɭɬɨɣɩɨɰɢɥɢɧɞɪɢɱɟɫɤɨɣɩɨɜɟɪɯɧɨɫɬɢ w = w (x).
ɉɨɫɤɨɥɶɤɭ ɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ y,
ɩɨɥɚɝɚɟɦ ɜ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ y ɪɚɜɧɵɦɢ ɧɭɥɸ ȼ ɪɟɡɭɥɶɬɚɬɟ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɪɢɦɟɬɫɥɟɞɭɸɳɢɣɜɢɞ
d 4 w ( x)
d x4
q ( x)
.
D
23
(1.30)
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɉɪɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɢɡɝɢɛɟ ɤɨɝɞɚ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ y ɪɚɜɧɵ
ɧɭɥɸɜɵɪɚɠɟɧɢɹ ɞɥɹɦɨɦɟɧɬɨɜɩɪɢɦɭɬɜɢɞ
w2w
w2w
(1.34)
Ɇ x D 2 ; Ɇ y D Q 2 QɆ x ; H 0 .
wx
wx
0
y
a
1
q0
x
ɋɭɱɟɬɨɦɢɦɟɟɦɫɥɟɞɭɸɳɭɸɡɚɜɢɫɢɦɨɫɬɶ ɞɥɹɦɨɦɟɧɬɚ:
Ɋɢɫ ɐɢɥɢɧɞɪɢɱɟɫɤɢɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
Ɂɞɟɫɶ ɢɫɩɨɥɶɡɭɟɬɫɹɨɛɵɤɧɨɜɟɧɧɚɹɚɧɟɱɚɫɬɧɚɹɩɪɨɢɡɜɨɞɧɚɹ
ɩɨɫɤɨɥɶɤɭ w ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɝɨ ɚɪɝɭɦɟɧɬɚ x ɍɪɚɜɧɟɧɢɟ
(1.30) ɨɩɢɫɵɜɚɸɳɟɟ ɰɢɥɢɧɞɪɢɱɟɫɤɢɣ ɢɡɝɢɛ ɩɥɚɫɬɢɧɵ ɫɨɜɩɚɞɚɟɬ
ɫɭɪɚɜɧɟɧɢɟɦɢɡɝɢɛɚɛɚɥɤɢɭɤɨɬɨɪɨɣɠɟɫɬɤɨɫɬɶɫɟɱɟɧɢɹɧɚɢɡɝɢɛ
EI ɪɚɜɧɚɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬɤɨɫɬɢɩɥɚɫɬɢɧɵ D (EI = D). Ɉɬɫɸɞɚ
ɜɟɥɢɱɢɧɚ D ɩɨɥɭɱɢɥɚɧɚɢɦɟɧɨɜɚɧɢɟɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬɤɨɫɬɢ.
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɰɢɥɢɧɞɪɢɱɟ
ɫɤɨɝɨɢɡɝɢɛɚɩɥɚɫɬɢɧɵɩɪɢɥɸɛɨɦɡɚɤɨɧɟɩɨɩɟɪɟɱɧɨɣɧɚɝɪɭɡɤɢq (x)
ɧɟɜɵɡɵɜɚɟɬɨɫɨɛɵɯɫɥɨɠɧɨɫɬɟɣɩɨɷɬɨɦɭɡɚɞɚɱɚɪɟɲɚɟɬɫɹɬɨɱɧɨ
ɉɭɫɬɶɧɚɩɪɢɦɟɪq = q0 x/a (ɫɦɪɢɫ 1.7), ɬɨɝɞɚ ɩɨɫɥɟɱɟɬɵɪɟɯ
ɤɪɚɬɧɨɝɨɢɧɬɟɝɪɢɪɨɜɚɧɢɹɭɪɚɜɧɟɧɢɹɩɨɥɭɱɢɦ
C1 C2 x C3 x 2 C4 x 3 w
q0 x
,
120 a D
(1.31)
x 0
;
ww
wx
w w
wx 2
2
0
x 0
; w 0
x a
;
0
x a
.
(1.32)
ɉɨɞɫɬɚɜɢɜɩɪɨɝɢɛɵ w (1.31) ɢɢɯɩɪɨɢɡɜɨɞɧɵɟɜɝɪɚɧɢɱɧɵɟɭɫɥɨ
ɜɢɹɢɪɟɲɢɜɫɢɫɬɟɦɭɢɡɱɟɬɵɪɟɯɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜɧɟɧɢɣ
9q a
7 q0 a 2
ɧɚɯɨɞɢɦɋ1 = ɋ2 = 0, C3
, C4 0 . ȼɪɟɡɭɥɶɬɚɬɟɩɨɥɭ
240 D
240 D
ɱɢɦɫɥɟɞɭɸɳɟɟɜɵɪɚɠɟɧɢɟɞɥɹɩɪɨɝɢɛɨɜ:
w ( x)
q0 a 4 § x 2
x3
x5 ·
¨¨ 7 2 9 3 2 5 ¸¸ .
a
a ¹
240 D © a
24
q0 a 2 §
x
x3 ·
¨¨ 7 27 20 3 ¸¸ .
a
a ¹
120 ©
2. ɑɢɫɬɵɣɢɡɝɢɛɩɥɚɫɬɢɧɵ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɩɥɚɫɬɢɧɭ ɫɜɨɛɨɞɧɭɸ ɨɬ ɡɚɤɪɟɩ
ɥɟɧɢɣ ɢ ɧɚɝɪɭɠɟɧɧɭɸ ɩɨ ɤɨɧɬɭɪɭ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ ɦɨɦɟɧɬɚɦɢ
ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ m1 = const ɢ m2 = const ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɪɢɫ 1.8, ɚ). ɇɚɱɚɥɨɤɨɨɪɞɢɧɚɬɩɨɦɟɫɬɢɦɜɰɟɧɬɪɟɩɥɚɫɬɢɧɵ
ɛ)
ɚ)
x
z
5
ɝɞɟɋ1, ɋ2, ɋ3, ɋ4 – ɱɟɬɵɪɟɩɨɫɬɨɹɧɧɵɟɢɧɬɟɝɪɢɪɨɜɚɧɢɹɨɩɪɟɞɟɥɹɟ
ɦɵɟ ɢɡ ɱɟɬɵɪɟɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɨ ɞɜɚ ɭɫɥɨɜɢɹ ɞɥɹ ɡɚɞɟɥɚɧ
ɧɨɝɨɤɪɚɹ x = 0 ɢɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɝɨɤɪɚɹ x = a):
w 0
M x ( x)
y
x
m1
z
y
m2
Ɋɢɫ 1.8. ɑɢɫɬɵɣ ɢɡɝɢɛɩɥɚɫɬɢɧɵ
ɉɨɫɤɨɥɶɤɭɩɨɩɟɪɟɱɧɚɹɧɚɝɪɭɡɤɚɨɬɫɭɬɫɬɜɭɟɬɢɦɟɟɦɨɞɧɨɪɨɞ
ɧɨɟɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ
w4w
w4w
w4w
2
wy4
wx 4
wx 2 wy 2
0,
(1.35)
ɪɟɲɟɧɢɟɤɨɬɨɪɨɝɨɛɭɞɟɦɢɫɤɚɬɶɜɜɢɞɟ
w
0,5 (C1 x 2 C2 y 2 ) .
(1.36)
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɋ1 ɢ ɋ2 ɧɚɣɞɟɦ ɢɡ ɝɪɚɧɢɱɧɵɯ
ɭɫɥɨɜɢɣMx = m1 ɢMy = m2ɉɨɞɫɬɚɜɢɜɜɩɨɥɭɱɢɦ
(1.33)
25
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ɇx
Ɇy
§ w2w
w2w ·
D ¨¨ 2 Q 2 ¸¸
wy ¹
© wx
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
D C1 Q C2 m1 ;
D (1 Q)
H
§ w2w
w2w ·
D ¨¨ 2 Q 2 ¸¸ D C2 Q C1 m2 ;
wx ¹
© wy
ɂɡɪɟɲɟɧɢɹɩɟɪɜɵɯɞɜɭɯɭɪɚɜɧɟɧɢɣɧɚɯɨɞɢɦ
C1
Q m2 m1
; C2
D (1 Q 2 )
w w
wx w y
w ( x, y )
2
0.
Q m1 m2
.
D (1 Q 2 )
1
2 D (1 Q 2 )
ª( Q m m ) x 2 ( Q m m ) y 2 º .
2
1
1
2
«¬
»¼
(1.37)
ȼɨ ɜɫɟɯ ɫɟɱɟɧɢɹɯ ɩɥɚɫɬɢɧɵ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɹɦ x ɢ y, ɞɟɣ
ɫɬɜɭɸɬɬɨɥɶɤɨɩɨɫɬɨɹɧɧɵɟɢɡɝɢɛɚɸɳɢɟɦɨɦɟɧɬɵMx = m1 ɢMy = m2.
Ⱦɪɭɝɢɟ ɜɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɨɬɫɭɬɫɬɜɭɸɬ: H = Qx = Qy = 0.
Ɋɚɫɫɦɨɬɪɢɦɧɟɫɤɨɥɶɤɨɱɚɫɬɧɵɯɫɥɭɱɚɟɜɉɭɫɬɶm1 = m2 = mɬɨɝɞɚ
w ( x, y )
m
x2 y2 .
2 D (1 Q)
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɨɢɞɚ ɜɪɚɳɟɧɢɹ ɂɡɨɝɧɭɬɚɹ ɩɨɜɟɪɯɧɨɫɬɶ
ɜɷɬɨɦɫɥɭɱɚɟɩɪɟɞɫɬɚɜɥɹɟɬɱɚɫɬɶɫɮɟɪɵɬɚɤɤɚɤɪɚɞɢɭɫɵɤɪɢɜɢɡ
ɧɵɨɞɢɧɚɤɨɜɵɜɨɜɫɟɬɨɱɤɚɯɩɥɚɫɬɢɧɵ
ȼɨɡɶɦɟɦɞɪɭɝɨɣ ɱɚɫɬɧɵɣɫɥɭɱɚɣ m1 = m; m2 ɍɪɚɜɧɟɧɢɟ
ɩɪɢɦɟɬɫɥɟɞɭɸɳɢɣɜɢɞ
m
w ( x, y )
x2 Q y2 .
2
2 D (1 Q )
ɇɟɬɪɭɞɧɨ ɩɨɤɚɡɚɬɶ ɱɬɨ ɩɪɢ ɬɚɤɨɦ ɧɚɝɪɭɠɟɧɢɢ ɜ ɤɨɫɵɯ ɫɟɱɟɧɢɹɯ
ɩɥɚɫɬɢɧɵɧɚɤɥɨɧɟɧɧɵɯɤɨɫɹɦx ɢy ɧɚɭɝɨɥĮ ƒɩɥɚɫɬɢɧɚɢɫ
ɩɵɬɵɜɚɟɬɞɟɮɨɪɦɚɰɢɸɱɢɫɬɨɝɨɤɪɭɱɟɧɢɹɬ ɟ HĮ = –m, MĮ = 0.
Ɋɟɲɟɧɢɟɩɪɹɦɨɣɢɨɛɪɚɬɧɨɣɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧ
Ɍɨɝɞɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɪɨɝɢɛɨɜ ɬ ɟ ɭɪɚɜɧɟɧɢɟ ɢɡɨɝɧɭɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵɪɢɫ 1.8, ɛ), ɡɚɩɢɲɟɬɫɹɜɜɢɞɟ
w ( x, y )
m
x 2 y 2 .
2 D (1 Q)
ɉɪɹɦɚɹ ɡɚɞɚɱɚ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸ
ɳɢɦ ɨɛɪɚɡɨɦ: ɡɚɞɚɧɵ ɧɚɝɪɭɡɤɚ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɛɭɟɬɫɹ
ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ ɢ ɞɟɣɫɬɜɭɸɳɢɟ ɜ ɧɟɣ ɜɧɭɬɪɟɧɧɢɟ
ɭɫɢɥɢɹɊɟɲɟɧɢɟɩɪɹɦɨɣɡɚɞɚɱɢɡɚɬɪɭɞɧɟɧɨɧɟɨɛɯɨɞɢɦɨɫɬɶɸɢɧɬɟ
ɝɪɢɪɨɜɚɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ ɢ ɝɥɚɜɧɨɟ
ɭɞɨɜɥɟɬɜɨɪɹɬɶɪɚɡɧɨɨɛɪɚɡɧɵɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦ
Ɋɚɫɫɦɨɬɪɢɦ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɭɸ ɩɨ ɤɨɧɬɭɪɭ ɩɪɹɦɨɭɝɨɥɶɧɭɸ
ɩɥɚɫɬɢɧɭ ɧɚɝɪɭɠɟɧɧɭɸ ɩɨɩɟɪɟɱɧɨɣ ɧɚɝɪɭɡɤɨɣ ɂɧɬɟɧɫɢɜɧɨɫɬɶ
ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ ɡɚɞɚɞɢɦ ɜ ɜɢɞɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ
Sx
Sy
ɮɭɧɤɰɢɢ q q0 sin sin
ɪɢɫ 1.9). Ɋɟɲɟɧɢɟɡɚɞɚɱɢ– ɮɭɧɤɰɢɸ
a
b
ɩɪɨɝɢɛɨɜɩɥɚɫɬɢɧɵwɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸɡɚɞɚɧɧɨɣɧɚɝɪɭɡɤɟɢɝɪɚ
ɧɢɱɧɵɦɭɫɥɨɜɢɹɦɛɭɞɟɦɢɫɤɚɬɶɜɚɧɚɥɨɝɢɱɧɨɦ ɜɢɞɟ
q0
b/2
x
b/2
a/2
a/2
y
Ɋɢɫ 1.9ɉɪɹɦɨɭɝɨɥɶɧɚɹɲɚɪɧɢɪɧɨɨɩɟɪɬɚɹɩɥɚɫɬɢɧɚ
ɉɨɜɟɪɯɧɨɫɬɶ ɨɩɢɫɵɜɚɟɦɚɹ ɷɬɢɦ ɭɪɚɜɧɟɧɢɟɦ ɢɦɟɟɬ ɫɟɞɥɨɨɛɪɚɡ
ɧɭɸ ɮɨɪɦɭ ɢ ɧɚɡɵɜɚɟɬɫɹ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦ ɩɚɪɚɛɨɥɨɢɞɨɦ ɝɨɪɢ
ɡɨɧɬɚɥɹɦɢɟɟɹɜɥɹɸɬɫɹɝɢɩɟɪɛɨɥɵ Ȼɥɚɝɨɞɚɪɹɜɥɢɹɧɢɸɤɨɷɮɮɢɰɢ
ɟɧɬɚ ɉɭɚɫɫɨɧɚ ɩɥɚɫɬɢɧɚ ɢɡɝɢɛɚɟɬɫɹ ɧɟ ɬɨɥɶɤɨ ɜ ɩɥɨɫɤɨɫɬɢ
ɞɟɣɫɬɜɢɹ ɦɨɦɟɧɬɚ Mx = m ɧɨ ɩɨɥɭɱɚɟɬ ɢ ɨɛɪɚɬɧɵɣ ɜɵɝɢɛ ɜ ɩɟɪ
ɩɟɧɞɢɤɭɥɹɪɧɨɣɩɥɨɫɤɨɫɬɢ
ɇɚɤɨɧɟɰɩɪɢɦɟɦm1 = m; m2 = –mȼɷɬɨɦɫɥɭɱɚɟɭɪɚɜɧɟɧɢɟ
ɩɪɨɝɢɛɨɜɬɚɤɠɟɢɦɟɟɬɜɢɞ ɝɢɩɟɪɛɨɥɢɱɟɫɤɨɝɨ ɩɚɪɚɛɨɥɨɢɞɚ:
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɡɚɩɢɲɟɬɫɹ
ɜɬɚɤɨɦɜɢɞɟ
Sx
Sy
w 4 w q0
w4w
w4w
(1.38)
2
sin sin .
4
2
2
4
D
a
b
wy
wx wy
wx
26
27
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ƚɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹɞɥɹɲɚɪɧɢɪɧɨɨɩɟɪɬɨɣɩɨɤɨɧɬɭɪɭɩɥɚɫɬɢɧɵ
w w
wx 2
2
w 0
x
;
0, a
w w
wy2
2
0
x
;
0, a
w 0
;
0, b
y
0
y
. (1.39)
0, b
Ɋɟɲɟɧɢɟɭɪɚɜɧɟɧɢɹɡɚɞɚɞɢɦɜɮɨɪɦɟɟɝɨɩɪɚɜɨɣɱɚɫɬɢ:
Sx
Sy
(1.40)
sin .
a
b
Ɏɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɡɚɩɢɫɚɧɧɚɹ ɜ ɜɢɞɟ (1.40) ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɯɚɪɚɤɬɟɪɭ ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ ɢ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ 1.39).
Ɉɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɭɸ C ɪɚɜɧɭɸ ɚɦɩɥɢɬɭɞɟ ɩɪɨɝɢɛɚ ɜ ɰɟɧɬɪɟ
ɩɥɚɫɬɢɧɵ. ɉɨɫɥɟɩɨɞɫɬɚɧɨɜɤɢɮɭɧɤɰɢɢɩɪɨɝɢɛɚ1.40) ɜɭɪɚɜɧɟɧɢɟ
(1.38ɢɫɨɤɪɚɳɟɧɢɹɧɚɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟɦɧɨɠɢɬɟɥɢɩɨɥɭɱɚɟɦ
w C sin
§ S2 S2 ·
ɋ ¨¨ 2 2 ¸¸
b ¹
©a
2
q0
, ɨɬɫɸɞɚ ɋ
D
w0
q0 a 4 b 4
.
D S 4 (a 2 b 2 ) 2
ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɪɚɜɧɟɧɢɟ ɢɡɨɝɧɭɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ
ɩɪɢɦɟɬɜɢɞ
q0 a 4 b 4
Sx
Sy
w
sin sin .
4
2
2 2
a
b
D S (a b )
q
ȿɫɥɢɢɧɬɟɧɫɢɜɧɨɫɬɶɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢɡɚɞɚɟɬɫɹɜɜɢɞɟ
2
2
m Sx
n Sy
n2 ·
4§ m
, ɬɨ ɋ q0 D S ¨ 2 2 ¸ ɢɮɭɧɤɰɢɹɩɪɨɝɢɛɚ
q0 sin
sin
b ¹
a
b
©a
w
q0
DS m a n b
4
2
2
2
2 2
sin
m Sx
n Sy
sin
.
a
b
(1.41)
ɂɦɟɹ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɚ, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ ɤɨɦɩɨɧɟɧɬɵ
ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɩɥɚɫɬɢ
ɧɵ
ɉɨɫɬɚɧɨɜɤɢ ɨɛɪɚɬɧɨɣɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ ɛɨɥɟɟɪɚɡɧɨ
ɨɛɪɚɡɧɵ Ɂɞɟɫɶ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɥɢɛɨ ɭɪɚɜɧɟɧɢɟ ɭɩɪɭɝɨɣ ɩɨ
ɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ ɥɢɛɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɧɚɩɪɹɠɟɧɢɣ ɢɥɢ ɞɟɮɨɪ
ɦɚɰɢɣ Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɦɩɨɧɟɧɬɵ
ɨɩɪɟɞɟɥɹɸɳɢɟɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɟɫɨɫɬɨɹɧɢɟɩɥɚɫɬɢɧɵ
28
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɹɦɨɭɝɨɥɶ
ɧɭɸ ɩɥɚɫɬɢɧɭ ɪɢɫ Ɂɚɞɚ
ɞɢɦɫɹ ɭɪɚɜɧɟɧɢɟɦ ɢɡɨɝɧɭɬɨɣ
ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢ
Sx
3Sy
ɧɵ ɜ ɜɢɞɟ w C ˜ sin sin
,
a
b
C = const; a = 2b; Ȟ 0
x
b
a
y
Ɋɢɫ ɉɪɹɦɨɭɝɨɥɶɧɚɹ
ɩɥɚɫɬɢɧɚ
Ɍɪɟɛɭɟɬɫɹ
1) ɭɫɬɚɧɨɜɢɬɶɤɚɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɭɞɨɜɥɟɬɜɨɪɹɟɬɡɚ
ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟɭɩɪɭɝɨɣɩɨɜɟɪɯɧɨɫɬɢw (x, y);
2) ɨɩɪɟɞɟɥɢɬɶɩɨɫɬɨɹɧɧɵɣɤɨɷɮɮɢɰɢɟɧɬɋɢɫɩɨɥɶɡɭɹɞɢɮɮɟ
ɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ;
3) ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ Mx, My
ɢɤɪɭɬɹɳɟɝɨɦɨɦɟɧɬɚH;
4) ɩɨɫɬɪɨɢɬɶɷɩɸɪɵɦɨɦɟɧɬɨɜɜɞɨɥɹɯɨɬ q0 ɢbɜɫɟɱɟɧɢɹɯ,
ɩɚɪɚɥɥɟɥɶɧɵɯɨɫɹɦx ɢy.
Ɋɟɲɟɧɢɟ
ɋɨɫɬɚɜɢɦɜɵɪɚɠɟɧɢɹ ɞɥɹɭɝɥɨɜɩɨɜɨɪɨɬɚ.
ȼɧɚɩɪɚɜɥɟɧɢɢɨɫɢ x:
ww
S
Sx
3Sy
.
C cos sin
Tx
a
a
b
wx
w2w
0 ɧɨ T x z 0 ɫɥɟɞɨɜɚɬɟɥɶɧɨ
wx 2
ɩɨɷɬɨɣɝɪɚɧɢɩɥɚɫɬɢɧɚɲɚɪɧɢɪɧɨɨɩɟɪɬɚɉɪɢx = a ɬɚɤɠɟɢɦɟɟɦ
w2w
w 0;
0 , T x z 0 ɬ ɟɢɷɬɚ ɝɪɚɧɶ ɩɥɚɫɬɢɧɵ ɲɚɪɧɢɪɧɨɨɩɟɪɬɚ
wx 2
ȼɧɚɩɪɚɜɥɟɧɢɢɨɫɢ y:
ww
3S
Sx
3Sy
.
Ty
sin cos
C
wy
b
a
b
Ɍɚɤɤɚɤɩɪɢx = 0 ɢɦɟɟɦ w 0;
w2w
0 , T y z 0 ɬɨɩɨɷɬɨɣɝɪɚɧɢɩɥɚɫɬɢ
wy2
ɧɚ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɚ ɉɪɢ y = b ɩɨ ɬɟɦ ɠɟ ɩɪɢɱɢɧɚɦ ɩɥɚɫɬɢɧɚ
ɬɚɤɠɟ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɚ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɜɫɟɦ ɫɜɨɢɦ ɝɪɚɧɹɦ
ɩɥɚɫɬɢɧɚɹɜɥɹɟɬɫɹɲɚɪɧɢɪɧɨɨɩɟɪɬɨɣ ɪɢɫ 1.11).
Ɍɚɤɤɚɤɩɪɢy = 0 w 0;
29
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
2. ɋɨɫɬɚɜɢɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɨɝɧɭɬɨɣɫɪɟɞɢɧ
ɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ ȼɵɱɢɫɥɢɦ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɮɭɧɤɰɢɢ
ɩɪɨɝɢɛɚw (x, y)Ⱦɥɹɫɨɤɪɚɳɟɧɢɹɡɚɩɢɫɢ ɜɜɟɞɟɦɨɛɨɡɧɚɱɟɧɢɹ:
X
Sx
; Y
a
3S y
; wcxc
b
w4w
; w IVy
wx 4
wxIV
w2w
; wcyc
wx 2
w4w
IV
; wxy
wy4
w2w
;
wy2
w4w
.
wx 2 wy 2
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ
wcxc C
wxIV C
S2
sin X sin Y ; wcyc
a2
C
9S 2
sin X sin Y ;
b2
81S 4
S4
IV
sin
X
sin
Y
w
C
sin X sin Y ;
;
y
a4
b4
9S 4
IV
wxy
C 2 2 sin X sin Y .
ab
18 81 ·
§1
C S sin X sin Y ¨ 4 2 2 4 ¸
©a a b b ¹
Ɉɬɫɸɞɚɩɨɞɫɬɚɜɢɜa = 2bɩɨɥɭɱɢɦ C
Ɇy
D wcxc Qwcyc D wcyc Qwcxc D (1 Q) C S 2
3S 2
cos X cosY
ab
q0b 2
cos X cosY .
803
ɉɨɤɚɠɟɦɢɡɦɟɧɟɧɢɟɢɧɬɟɧɫɢɜɧɨɫɬɢɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢq ɜɞɨɥɶɨɫɟɣɫɢɦɦɟɬɪɢɢɩɥɚɫɬɢɧɵɇɚɨɫɢɫɢɦɦɟɬɪɢɢ x = a/2 ɩɨɥɭ
ɱɢɦ q q0 sin (3Sy/b) ɡɧɚɱɟɧɢɹ q ɜɵɱɢɫɥɹɟɦ ɩɪɢ y = 0; b/4; b/2;
3b/4; bɇɚɨɫɢɫɢɦɦɟɬɪɢɢy = bɢɦɟɟɦ q q0 sin (Sx/a ) ɡɧɚɱɟɧɢɹ
q ɜɵɱɢɫɥɹɟɦɩɪɢ x = 0; a/4; a/2; 3a/4; aɏɚɪɚɤɬɟɪɢɡɦɟɧɟɧɢɹɩɨɩɟ
ɪɟɱɧɨɣɧɚɝɪɭɡɤɢɩɪɢɜɨɞɢɬɫɹɧɚɪɢɫ 1.11.
5. ɋɬɪɨɢɦɷɩɸɪɵɢɡɝɢɛɚɸɳɢɯɦɨɦɟɧɬɨɜɩɨɨɫɹɦɫɢɦɦɟɬɪɢɢ
(x = aɢy = bɷɩɸɪɵɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɩɨɝɪɚɧɹɦ x = ɢy = 0.
ɉɨɞɫɬɚɜɥɹɹɚɛɫɰɢɫɫɭ x = a/2 ɩɪɢɷɬɨɦVLQ X = 1) ɜɜɵɪɚɠɟɧɢɹ
ɞɥɹ Mx ɢMy, ɚ x = 0 (cos Y = 1) ɜɜɵɪɚɠɟɧɢɟɞɥɹH, ɩɨɥɭɱɢɦ
q0 b 2
q b2
q0 b 2
3Sy
3Sy
3Sy
sin
; Ɇ y 0 sin
; H
cos
.
286
93,5
803
b
b
b
Ɋɟɡɭɥɶɬɚɬɵɜɵɱɢɫɥɟɧɢɣɩɪɢɜɟɞɟɧɵɜɬɚɛɥɢ
Ɍɚɛɥɢɰɚ
ȼɵɱɢɫɥɟɧɢɟɨɪɞɢɧɚɬɷɩɸɪMx, My ɢH ɜɫɟɱɟɧɢɹɯɩɚɪɚɥɥɟɥɶɧɵɯɨɫɢy
VLQʌy/b)
cosʌy/b)
Mx (q0 b2)
My (q0 b2)
H (q0 b2)
0
0
1
0
0
1,25 Â10–3
q
0 sin X sin Y .
D
b/4
0,707
–0,707
–2,47 Â10–3
–7,56 Â10–3
–0,88 Â10–3
b/2
–1
0
3,5 Â10–3
10,7 Â10–3
0
q0 b 4
85,56 D S 4
3b/4
0,707
0,707
–2,47 Â10–3
–7,56 Â10–3
0,88 Â10–3
b
0
–1
0
0
–1,25 Â10–3
q0 b 4
.
8317 D
9·
§1
D C S 2 sin X sin Y ¨ 2 Q 2 ¸
b ¹
©a
q b2
0 sin X sin Y ;
286
1·
§9
D C S 2 sin X sin Y ¨ 2 Q 2 ¸
a ¹
©b
30
w2w
wx w y
y
ȼɵɪɚɠɟɧɢɹɞɥɹ ɦɨɦɟɧɬɨɜ Mx, My ɢ H ɛɭɞɭɬɫɥɟɞɭɸɳɢɦɢ:
Ɇx
D (1 Q)
Ɇx
ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɸ ɩɨɩɟɪɟɱɧɨɣ ɧɚɝɪɭɡɤɢ ɛɭɞɟɦ ɡɚɞɚɜɚɬɶ
ɜɚɧɚɥɨɝɢɱɧɨɦɜɢɞɟ:
3Sy
Sx
sin
.
q q0 sin X sin Y q0 sin
a
b
ɉɨɞɫɬɚɜɢɦ ɡɧɚɱɟɧɢɹɩɪɨɢɡɜɨɞɧɵɯɢɜɵɪɚɠɟɧɢɟɞɥɹɧɚɝɪɭɡɤɢ
ɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ (1.29):
4
H
q0 b 2
sin X sin Y ;
93,5
ɉɨɞɫɬɚɜɥɹɹ y = b/2 (sin Y = –1) ɜɜɵɪɚɠɟɧɢɹɞɥɹ Mx ɢMy, ɚ y = 0
(cos Y = 1) ɜɜɵɪɚɠɟɧɢɟɞɥɹ H, ɩɨɥɭɱɢɦ
Ɇx
q0 b 2
Sx
sin ; Ɇ y
286
a
q0 b 2
Sx
sin ; H
93,5
a
ɗɩɸɪɵ Mx, My ɢH ɩɪɢɜɟɞɟɧɵɧɚɪɢɫ 1.11.
31
q0 b 2
Sx
cos .
803
a
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɍɚɛɥɢɰɚ2
Hx
ȼɵɱɢɫɥɟɧɢɟɨɪɞɢɧɚɬɷɩɸɪMx, My ɢH ɜɫɟɱɟɧɢɹɯɩɚɪɚɥɥɟɥɶɧɵɯɨɫɢx
X
sin(ʌx/a)
cosʌx/a)
Mx (q0 b2)
My (q0 b2)
H (q0 b2)
0
0
1
0
0
1,25 Â10–3
a/4
0,707
0,707
2,47 Â10–3
7,56 Â10–3
0,88 Â10–3
a/2
1
0
3,5 Â10–3
10,7 Â10–3
0
3a/4
0,707
–0,707
2,47 Â10–3
7,56 Â10–3
–0,88 Â10–3
0
a
–1
0
q0
x
0,707q0
0,707q0
a/2
2,47
0,707q0
3,5
b/2
–1,25 Â10
7,56
0
2,47
7,56
1,25
0
0,88
2,47
b/2
0,707q0
y
0
0 0,88
3,5
0
0
–3
10,7
1,25
Mx (10 q0b ) My (10 q0b2) H (10–3q0b2)
a/2
2
–3
Mx (10–3q0b2)
2,47
My (10–3q0b2)
7,56
1,25
0,88
10,7
7,56
H (10–3q0b2)
0,88
Vy
Ey
Q yx
W xy
Vx
; J xy
Ex
Ex
,
ɝɞɟ Q xy E x Q yx E y ȿɫɥɢ ɩɨɜɬɨɪɢɬɶ ɜɫɟ ɪɚɫɫɭɠɞɟɧɢɹ ɩɪɢɜɟɞɟɧɧɵɟ
ɞɥɹ ɢɡɨɬɪɨɩɧɵɯ ɩɥɚɫɬɢɧ ɬɨ ɜɦɟɫɬɨ ɩɪɢɞɟɦ ɤ ɫɥɟɞɭɸɳɢɦ
ɫɨɨɬɧɨɲɟɧɢɹɦɞɥɹɦɨɦɟɧɬɨɜ>2, 16]:
Ɇx
–3
V
Vx
Q xy y ; H y
Ex
Ey
§ w2w
w2w ·
D1 ¨¨ 2 Q xy 2 ¸¸ ;
wy ¹
© wx
§ w2w
w2w ·
Ɇ y D2 ¨¨ 2 Q yx 2 ¸¸ ;
wx ¹
© wy
Ɂɞɟɫɶ ɜɜɟɞɟɧɵɨɛɨɡɧɚɱɟɧɢɹɠɟɫɬɤɨɫɬɟɣ
D1
Ex h3
; D2
12 (1 Q xy Q yx )
DF
H
E y h3
12 (1 Q xy Q yx )
w2w
.
wx w y
; DF
(1.42)
G h3
.
12
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɝɢɛɚ ɚɧɚɥɨɝɢɱɧɨɟ ɭɪɚɜɧɟ
ɧɢɸɛɭɞɟɬɢɦɟɬɶɜɢɞ
D1
w4w
w4w
w4w
2
D
D
3
2
wy4
wx 2 w y 2
wx 4
q ( x, y ) ,
(1.43)
ɝɞɟ D3 D1Q xy 2 DF D2 Q yx 2 DF .
Ⱦɥɹɪɟɲɟɧɢɹɭɪɚɜɧɟɧɢɹɩɪɢɦɟɧɢɦɵɬɟɠɟɫɩɨɫɨɛɵ, ɱɬɨ
ɢ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɡɝɢɛɚ ɢɡɨɬɪɨɩɧɵɯ ɩɥɚɫɬɢɧ Ɍɚɤ ɩɪɢ ɡɚɞɚ
ɧɢɢɩɨɜɟɪɯɧɨɫɬɢɩɪɨɝɢɛɨɜɜɮɨɪɦɟɩɨɫɬɨɹɧɧɚɹ
1,25
Ɋɢɫ Ɋɟɡɭɥɶɬɚɬɵɪɟɲɟɧɢɹɨɛɪɚɬɧɨɣɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ
1.3.3. Ɉɪɚɫɱɟɬɟ ɨɪɬɨɬɪɨɩɧɵɯɩɥɚɫɬɢɧɢɩɥɚɫɬɢɧ
ɧɚɭɩɪɭɝɨɦɨɫɧɨɜɚɧɢɢ
Ɉɪɬɨɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɨɣ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚɡɵɜɚɟɬɫɹ ɩɥɚ
ɫɬɢɧɚ ɠɟɫɬɤɨɫɬɧɵɟ ɫɜɨɣɫɬɜɚ ɤɨɬɨɪɨɣ ɧɟɨɞɢɧɚɤɨɜɵ ɩɨ ɧɚɩɪɚɜɥɟ
ɧɢɹɦɤɨɨɪɞɢɧɚɬɧɵɯɨɫɟɣ x ɢ y.
ȼ ɱɚɫɬɧɨɫɬɢ ɤ ɬɚɤɢɦ ɩɥɚɫɬɢɧɚɦ ɨɬɧɨɫɹɬɫɹ ɩɥɚɫɬɢɧɵ ɢɡɝɨ
ɬɨɜɥɟɧɧɵɟɢɡɨɪɬɨɬɪɨɩɧɨɝɨɦɚɬɟɪɢɚɥɚɩɪɢɷɬɨɦɡɚɤɨɧȽɭɤɚɜɵɪɚ
ɠɚɟɬɫɹɫɥɟɞɭɸɳɢɦɢɡɚɜɢɫɢɦɨɫɬɹɦɢ
32
ɋ
q0
§ m
m2n2
n4 ·
S ¨ D1 4 2 D3 2 2 D2 4 ¸
ab
b ¹
© a
4
4
.
Ɂɚɦɟɬɢɦɱɬɨɢɧɨɝɞɚɞɥɹɭɩɪɨɳɟɧɢɹɪɟɲɟɧɢɹɡɚɞɚɱɢɩɪɢɛɥɢ
ɠɟɧɧɨɩɪɢɧɢɦɚɸɬ D3
D1 D2 ȼɷɬɨɦɫɥɭɱɚɟɤɚɤɦɨɠɧɨɭɫɬɚɧɨ
ɜɢɬɶɭɪɚɜɧɟɧɢɟɩɪɢɜɨɞɢɬɫɹɤɨɛɵɱɧɨɦɭɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨ
ɦɭɭɪɚɜɧɟɧɢɸɞɥɹɢɡɨɬɪɨɩɧɨɣɩɥɚɫɬɢɧɵ
w4w
w4w
w4w
2
wx 4
wx 2 wy12 w y14
33
q
,
D1
(1.44)
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɝɞɟɩɪɢɜɟɞɟɧɧɚɹɩɟɪɟɦɟɧɧɚɹ y1 y 4 D1 D2 . Ɋɟɲɟɧɢɟɷɬɨɝɨɭɪɚɜɧɟ
ɧɢɹɩɪɨɢɡɜɨɞɢɬɫɹɬɚɤɠɟɤɚɤɞɥɹɭɫɥɨɜɧɨɣɢɡɨɬɪɨɩɧɨɣɩɥɚɫɬɢɧɵ
ɭɤɨɬɨɪɨɣɜɦɟɫɬɨɞɥɢɧɵɫɬɨɪɨɧɵ b ɩɪɢɧɹɬɪɚɡɦɟɪ b1 b 4 D1 D2 .
Ɋɚɫɫɦɨɬɪɢɦɬɟɩɟɪɶɤɨɧɫɬɪɭɤɬɢɜɧɨɨɪɬɨɬɪɨɩɧɭɸɩɥɚɫɬɢɧɭ.
Ɍɚɤ ɧɚɡɵɜɚɟɬɫɹ ɢɡɨɬɪɨɩɧɚɹ ɩɥɚɫɬɢɧɚ ɭɫɢɥɟɧɧɚɹ ɪɟɛɪɚɦɢ ɠɟɫɬɤɨ
ɫɬɢ ɤɚɤ ɜ ɨɞɧɨɦ ɬɚɤ ɢ ɜ ɞɪɭɝɨɦ ɧɚɩɪɚɜɥɟɧɢɹɯ ɪɢɫ Ɍɚɤɚɹ
ɪɟɛɪɢɫɬɚɹɩɪɹɦɨɭɝɨɥɶɧɚɹ ɩɥɚɫɬɢɧɚɪɟɛɪɚɤɨɬɨɪɨɣɩɚɪɚɥɥɟɥɶɧɵɟɟ
ɫɬɨɪɨɧɚɦ ɩɪɨɹɜɥɹɟɬ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɡɥɢɱɧɵɟ ɠɟɫɬɤɨɫɬɧɵɟ ɯɚ
ɪɚɤɬɟɪɢɫɬɢɤɢɜɧɚɩɪɚɜɥɟɧɢɹɯx ɢy. ȿɟɪɚɫɱɟɬɦɨɠɧɨɩɪɢɛɥɢɠɟɧɧɨ
ɜɵɩɨɥɧɢɬɶ ɤɚɤ ɪɚɫɱɟɬ ɭɫɥɨɜɧɨɣ ɨɪɬɨɬɪɨɩɧɨɣ ɩɥɚɫɬɢɧɵ ɫ ɠɟɫɬɤɨ
ɫɬɹɦɢ D1, D2 ɢ D3, ɜɯɨɞɹɳɢɦɢɜɭɪɚɜɧɟɧɢɟ
x
s1
s2
a
y
Ɋɢɫ Ʉɨɧɫɬɪɭɤɬɢɜɧɨɨɪɬɨɬɪɨɩɧɚɹɩɥɚɫɬɢɧɚ
ɉɭɫɬɶ ɞɥɹ ɪɟɛɟɪ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢ x ɠɟɫɬɤɨɫɬɶ ɧɚ ɢɡɝɢɛ –
EI1 ɧɚ ɤɪɭɱɟɧɢɟ – GIɤɪ1 ɚ ɲɚɝ ɪɚɫɫɬɚɧɨɜɤɢ ɷɬɢɯ ɪɟɛɟɪ – s1 ɋɨɨɬ
ɜɟɬɫɬɜɟɧɧɨɞɥɹɪɟɛɟɪɩɚɪɚɥɥɟɥɶɧɵɯɨɫɢyɷɬɨɛɭɞɭɬEI2, GIɤɪ2 ɢs2.
ȿɫɥɢ ɢɡɝɢɛɚɸɳɢɟ ɢ ɤɪɭɬɹɳɢɟ ɦɨɦɟɧɬɵ ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɫɟɱɟɧɢɹɯ
ɪɟɛɟɪ ɭɫɥɨɜɧɨ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɢɬɶ ɩɨ ɞɥɢɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸ
ɳɟɝɨ ɲɚɝɚ ɪɚɫɫɬɚɧɨɜɤɢ ɪɟɛɟɪ ɬɨ ɭɤɚɡɚɧɧɵɟ ɠɟɫɬɤɨɫɬɢ ɞɥɹ ɨɪɬɨ
ɬɪɨɩɧɨɣɩɥɚɫɬɢɧɵɛɭɞɭɬ
1 § G I ɤɪ 1 G I ɤɪ 2 ·¸
D ¨
,
2 ¨© s1
s2 ¸¹
ɝɞɟ D – ɰɢɥɢɧɞɪɢɱɟɫɤɚɹ ɠɟɫɬɤɨɫɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɚɦɨɣ ɢɡɨ
ɬɪɨɩɧɨɣɩɥɚɫɬɢɧɵɤɨɬɨɪɚɹɭɫɢɥɢɜɚɟɬɫɹɪɟɛɪɚɦɢ ɠɟɫɬɤɨɫɬɢ.
ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɪɟɛɟɪ ɭɪɚɜɧɟɧɢɟ ɛɭɞɟɬ ɷɤɜɢɜɚɥɟɧɬɧɨ
ɭɪɚɜɧɟɧɢɸɚɩɪɢɨɬɫɭɬɫɬɜɢɢɫɚɦɨɣɩɥɚɫɬɢɧɵD ɭɪɚɜ
ɧɟɧɢɟ ɛɭɞɟɬ ɩɪɢɛɥɢɠɟɧɧɨ ɨɩɢɫɵɜɚɬɶ ɢɡɝɢɛ ɫɢɫɬɟɦɵ ɩɟɪɟ
ɤɪɟɫɬɧɵɯɛɚɥɨɤɛɚɥɨɱɧɨɣɤɥɟɬɤɢ).
D1
D
E I1
; D2
s1
D
E I2
; D3
s2
34
x
s1
b
s1
s2
s2
a
y
Ɋɢɫ 1.13. ɉɥɚɫɬɢɧɚɫɧɟɫɢɦɦɟɬɪɢɱɧɵɦ ɪɚɫɩɨɥɨɠɟɧɢɟɦ ɪɟɛɟɪ
b
s1
s2
Ɂɚɦɟɬɢɦɱɬɨɟɫɥɢɪɟɛɪɚɠɟɫɬɤɨɫɬɢɫɬɨɹɬɧɟɫɢɦɦɟɬɪɢɱɧɨɨɬ
ɧɨɫɢɬɟɥɶɧɨ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ ɪɢɫ ɬɨ ɪɚɫɱɟɬ
ɬɚɤɨɝɨɪɨɞɚɪɟɛɪɢɫɬɨɣɩɥɚɫɬɢɧɵɭɫɥɨɠɧɹɟɬɫɹɬɚɤɤɚɤɜɫɪɟɞɢɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢɩɨɹɜɥɹɸɬɫɹɦɟɦɛɪɚɧɧɵɟɭɫɢɥɢɹɞɚɠɟɩɪɢɦɚɥɵɯɩɪɨ
ɝɢɛɚɯ ɇɨ, ɭɩɪɨɳɚɹ ɡɚɞɚɱɭ ɧɟɪɟɞɤɨ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɧɹɸɬ
ɢɩɪɢɧɟɫɢɦɦɟɬɪɢɱɧɨɦɪɚɫɩɨɥɨɠɟɧɢɢɪɟɛɟɪɠɟɫɬɤɨɫɬɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɥɚɫɬɢɧɭ ɥɟɠɚɳɭɸ ɧɚ ɫɩɥɨɲɧɨɦ ɞɟɮɨɪɦɢɪɭɟ
ɦɨɦɨɫɧɨɜɚɧɢɢɪɢɫ 1.14)ȼɷɬɨɦɫɥɭɱɚɟɩɪɢɡɚɩɢɫɢɞɢɮɮɟɪɟɧɰɢ
ɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɡɝɢɛɚ ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɧɭɸ
ɩɨ ɩɥɨɳɚɞɢɩɥɚɫɬɢɧɵ ɪɟɚɤɰɢɸ (ɨɬɩɨɪɨɫɧɨɜɚɧɢɹɢɧɬɟɧɫɢɜɧɨɫɬɶɸ
r = r (x, y).
0
x
q (x,y)
b
q (x,y)
r (x,y)
a
y
Ɋɢɫ 1.14. ɉɥɚɫɬɢɧɚɧɚɭɩɪɭɝɨɞɟɮɨɪɦɢɪɭɟɦɨɦɨɫɧɨɜɚɧɢɢ
Ɍɨɝɞɚɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɥɚɫɬɢɧɵɡɚɩɢɲɟɬɫɹɬɚɤ
qr
(1.45)
,
D
ɝɞɟq – ɢɧɬɟɧɫɢɜɧɨɫɬɶɜɧɟɲɧɟɣɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢ
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɜɨɣɫɬɜ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɨɫɧɨɜɚɧɢɹ ɫɜɹɡɶ
ɦɟɠɞɭɪɟɚɤɰɢɟɣɨɫɧɨɜɚɧɢɹr ɢɟɝɨɞɟɮɨɪɦɢɪɨɜɚɧɧɵɦɫɨɫɬɨɹɧɢɟɦɦɨ
ɠɟɬɛɵɬɶɪɚɡɥɢɱɧɨɣɇɚɩɪɚɤɬɢɤɟɨɱɟɧɶɱɚɫɬɨɢɫɩɨɥɶɡɭɸɬɢɡɜɟɫɬɧɭɸ
’4w
35
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɦɨɞɟɥɶȼɢɧɤɥɟɪɚ, ɫɨɝɥɚɫɧɨɤɨɬɨɪɨɣ r (x, y) = kɭ Â w (x, yɝɞɟ kɭ – ɤɨɷɮ
ɮɢɰɢɟɧɬɠɟɫɬɤɨɫɬɢɭɩɪɭɝɨɝɨɨɫɧɨɜɚɧɢɹɤɨɷɮɮɢɰɢɟɧɬɩɨɫɬɟɥɢ).
ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ r ɜ ɢ ɩɟɪɟɧɟɫɹ ɱɥɟɧ ɫɨɞɟɪɠɚɳɢɣ
ɧɟɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ w ɜ ɥɟɜɭɸ ɱɚɫɬɶ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɥɚɫɬɢɧɵɥɟɠɚɳɟɣɧɚɭɩɪɭ
ɝɨɦɜɢɧɤɥɟɪɨɜɫɤɨɦɨɫɧɨɜɚɧɢɢ,
kɭ
q
(1.46)
’4w w
.
D
D
Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɩɪɢɦɟɧɹɟɬɫɹ ɥɸɛɨɣ ɢɡ ɪɚɫ
ɫɦɨɬɪɟɧɧɵɯ ɪɚɧɟɟ ɫɩɨɫɨɛɨɜ ɇɚɩɪɢɦɟɪ ɞɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣ ɩɨ
m Sx
n Sy
ɤɨɧɬɭɪɭɩɥɚɫɬɢɧɵɩɪɢɧɚɝɪɭɡɤɟɜɜɢɞɟ q q0 sin
ɩɨɥɭɱɢɦ
sin
a
b
q0
m Sx
n Sy
sin
sin
.
w
4
2
2
2
2 2
a
b
DS m / a n /b k
Ʉɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵ
1. Ʉɚɤɢɟ ɝɪɭɩɩɵ ɭɪɚɜɧɟɧɢɣ ɫɨɫɬɚɜɥɹɸɬ ɫɢɫɬɟɦɭ ɨɫɧɨɜɧɵɯ
ɭɪɚɜɧɟɧɢɣɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ"
2. Ʉɚɤɢɦɢ ɭɫɥɨɜɢɹɦɢ ɞɨɩɨɥɧɹɸɬɫɹ ɭɪɚɜɧɟɧɢɹɬɟɨɪɢɢɭɩɪɭɝɨ
ɫɬɢɩɪɢɪɟɲɟɧɢɢɤɨɧɤɪɟɬɧɵɯɡɚɞɚɱ"
3. ɍɤɚɠɢɬɟɨɫɧɨɜɧɵɟɫɩɨɫɨɛɵɪɟɲɟɧɢɹɡɚɞɚɱɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ
4. Ʉɚɤ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ ɫɢɫɬɟɦɚ ɪɚɡɪɟɲɚɸɳɢɯ ɭɪɚɜɧɟ
ɧɢɣɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢɜɩɟɪɟɦɟɳɟɧɢɹɯ"
5. Ʉɚɤɢɟ ɭɪɚɜɧɟɧɢɹ ɬɪɟɛɭɟɬɫɹ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɩɪɢ ɪɟɲɟɧɢɢ
ɡɚɞɚɱɜɧɚɩɪɹɠɟɧɢɹɯ"
6. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ
ɜɫɦɟɲɚɧɧɨɣɮɨɪɦɟ"
7. Ʉɚɤɚɹ ɡɚɞɚɱɚ ɧɚɡɵɜɚɟɬɫɹ ɩɪɹɦɨɣ ɡɚɞɚɱɟɣ ɬɟɨɪɢɢ ɭɩɪɭɝɨ
ɫɬɢ"
8. Ʉɚɤɨɟɬɟɥɨɧɚɡɵɜɚɟɬɫɹɩɥɚɫɬɢɧɨɣɱɬɨɩɨɧɢɦɚɟɬɫɹɩɨɞɫɪɟ
ɞɢɧɧɨɣɩɥɨɫɤɨɫɬɶɸ?
9. ɑɬɨ ɧɚɡɵɜɚɟɬɫɹ ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢɡɨɝɧɭɬɨɣ ɩɥɚ
ɫɬɢɧɵɢɟɟɩɪɨɝɢɛɨɦ"
10. ɉɪɢɜɟɞɢɬɟ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɩɥɚɫɬɢɧ ɩɨ ɨɬɧɨɲɟɧɢɸ ɢɯ
ɬɨɥɳɢɧɵɤɯɚɪɚɤɬɟɪɧɨɦɭɪɚɡɦɟɪɭ
36
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
11. Ʉɚɤ ɪɚɡɞɟɥɹɸɬɫɹ ɬɨɧɤɢɟ ɩɥɚɫɬɢɧɵ ɩɨ ɨɬɧɨɲɟɧɢɸ ɦɚɤɫɢ
ɦɚɥɶɧɨɝɨɩɪɨɝɢɛɚɤɢɯ ɬɨɥɳɢɧɟ"
12. ɉɪɢɤɚɤɢɯɬɨɥɳɢɧɚɯɢɩɪɨɝɢɛɚɯɩɥɚɫɬɢɧɚɫɱɢɬɚɟɬɫɹɬɨɧ
ɤɨɣɢɠɟɫɬɤɨɣ"
13. ɉɟɪɟɱɢɫɥɢɬɟɝɢɩɨɬɟɡɵɄɢɪɯɝɨɮɚɢɫɩɨɥɶɡɭɟɦɵɟɜɬɟɨɪɢɢ
ɪɚɫɱɟɬɚɬɨɧɤɢɯɩɥɚɫɬɢɧ
14. ɂɡɥɨɠɢɬɟɝɢɩɨɬɟɡɭɩɪɹɦɨɣɧɨɪɦɚɥɢ. ɑɬɨɢɡɧɟɟɫɥɟɞɭɟɬ?
15. ɉɪɢɜɟɞɢɬɟɝɢɩɨɬɟɡɭɨɧɟɞɟɮɨɪɦɢɪɭɟɦɨɫɬɢɫɪɟɞɢɧɧɨɣɩɨ
ɜɟɪɯɧɨɫɬɢɜɟɟɩɥɨɫɤɨɫɬɢ
16. ɋɮɨɪɦɭɥɢɪɭɣɬɟɝɢɩɨɬɟɡɭɨɧɟɧɚɞɚɜɥɢɜɚɧɢɢɫɥɨɟɜ. ɑɬɨɢɡ
ɧɟɟɫɥɟɞɭɟɬ"
17. Ʉɚɤɢɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɫɥɨɜɢɹ ɩɪɢɧɹɬɵ ɜ ɬɟɯɧɢɱɟɫɤɨɣ
ɬɟɨɪɢɢɢɡɝɢɛɚɩɥɚɫɬɢɧ"
18. Ʉɚɤɢɟɩɟɪɟɦɟɳɟɧɢɹɢɞɟɮɨɪɦɚɰɢɢɭɱɢɬɵɜɚɸɬɫɹɩɪɢɪɚɫ
ɱɟɬɟɬɨɧɤɢɯɩɥɚɫɬɢɧ"
19. Ʉɚɤɢɟɜɟɥɢɱɢɧɵɢɤɚɤɨɣɡɚɜɢɫɢɦɨɫɬɶɸɫɜɹɡɵɜɚɸɬɭɪɚɜ
ɧɟɧɢɹɄɨɲɢɜɩɥɚɫɬɢɧɟ"
20. ɑɬɨɬɚɤɨɟɤɪɢɜɢɡɧɵɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵ"
21. Ʉɚɤɢɟɤɨɦɩɨɧɟɧɬɵɧɚɩɪɹɠɟɧɢɣɭɱɢɬɵɜɚɸɬɫɹɩɪɢɪɚɫɱɟɬɟ
ɬɨɧɤɢɯɩɥɚɫɬɢɧ"
22. Ʉɚɤɢɟ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɭɱɢɬɵɜɚɸɬɫɹ ɩɪɢ ɪɚɫɱɟɬɟ ɬɨɧ
ɤɢɯɩɥɚɫɬɢɧ? Ʉɚɤɨɟɭɧɢɯɩɪɚɜɢɥɨɡɧɚɤɨɜ"
23. Ɂɚɩɢɲɢɬɟɜɵɪɚɠɟɧɢɹɞɥɹɦɨɦɟɧɬɨɜɜɫɟɱɟɧɢɹɯɩɥɚɫɬɢɧɵ
ɱɟɪɟɡɮɭɧɤɰɢɸɩɪɨɝɢɛɨɜ
24. ɉɪɢɜɟɞɢɬɟ ɮɨɪɦɭɥɭ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɠɟɫɬɤɨɫɬɢ ɩɥɚɫɬɢ
ɧɵ. ɑɬɨɨɧɚɯɚɪɚɤɬɟɪɢɡɭɟɬ"
25. Ɂɚɩɢɲɢɬɟ ɮɨɪɦɭɥɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɜ ɫɟɱɟ
ɧɢɹɯɩɥɚɫɬɢɧɵ
26. ɂɡ ɤɚɤɢɯ ɭɫɥɨɜɢɣ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɭɪɚɜɧɟɧɢɟ ɋɨɮɢ
ɀɟɪɦɟɧ– Ʌɚɝɪɚɧɠɚ"
27. ɉɪɢɜɟɞɢɬɟɭɪɚɜɧɟɧɢɟɋɨɮɢɀɟɪɦɟɧ– Ʌɚɝɪɚɧɠɚ. Ʉɚɤɨɧɨ
ɟɳɟɧɚɡɵɜɚɟɬɫɹ"
28. Ʉɚɤɢɟ ɭɫɥɨɜɢɹ ɞɥɹ ɩɥɚɫɬɢɧɵ ɧɚɡɵɜɚɸɬɫɹ ɝɪɚɧɢɱɧɵɦɢ?
Ʉɚɤɨɝɨɬɢɩɚɨɧɢɛɵɜɚɸɬ"
29. ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɬɢɩɵ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɟɜ ɩɪɹɦɨ
ɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ
30. Ɂɚɩɢɲɢɬɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɡɚɞɟɥɚɧɧɨɝɨ ɤɪɚɹ ɩɥɚ
ɫɬɢɧɵ. Ʉɚɤɨɝɨɬɢɩɚɷɬɢɭɫɥɨɜɢɹ"
37
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
31. ɉɪɢɜɟɞɢɬɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɝɨ
ɤɪɚɹ. Ʉɚɤɨɝɨɬɢɩɚɷɬɢɭɫɥɨɜɢɹ"
32. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ
ɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ
33. Ʉɚɤɨɝɨɜɢɞɚɡɚɞɚɱɚɧɚɡɵɜɚɟɬɫɹɤɪɚɟɜɨɣɡɚɞɚɱɟɣ"
34. ɉɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɜɨɡɧɢɤɚɟɬ ɰɢɥɢɧɞɪɢɱɟɫɤɢɣ ɢɡɝɢɛ
ɩɥɚɫɬɢɧɵ"
35. ɉɪɢɤɚɤɢɯɭɫɥɨɜɢɹɯɜɨɡɧɢɤɚɟɬɱɢɫɬɵɣɢɡɝɢɛɩɪɹɦɨɭɝɨɥɶ
ɧɨɣɩɥɚɫɬɢɧɵ"
36. ɋɮɨɪɦɭɥɢɪɭɣɬɟɩɪɹɦɭɸɡɚɞɚɱɭɢɡɝɢɛɚɩɥɚɫɬɢɧɵ.
37. ɉɪɢɜɟɞɢɬɟɩɨɫɬɚɧɨɜɤɭɨɛɪɚɬɧɨɣɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ
38. Ʉɚɤɚɹ ɩɥɚɫɬɢɧɚ ɧɚɡɵɜɚɟɬɫɹ ɨɪɬɨɬɪɨɩɧɨɣ ɤɨɧɫɬɪɭɤɬɢɜɧɨ
ɨɪɬɨɬɪɨɩɧɨɣ"
39. Ɂɚɩɢɲɢɬɟɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɞɥɹɨɪɬɨ
ɬɪɨɩɧɨɣɩɥɚɫɬɢɧɵ
40. ɉɪɢɜɟɞɢɬɟɭɪɚɜɧɟɧɢɟɋɨɮɢ ɀɟɪɦɟɧɞɥɹɩɥɚɫɬɢɧɵɥɟɠɚ
ɳɟɣɧɚɭɩɪɭɝɨɦɨɫɧɨɜɚɧɢɢ
38
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ƚɥɚɜɚ 2. ɆȿɌɈȾɕɉɊɂȻɅɂɀȿɇɇɈȽɈ Ɋȿɒȿɇɂə
ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɈȽɈɍɊȺȼɇȿɇɂəɂɁȽɂȻȺ
Ʉɚɤ ɭɠɟ ɨɬɦɟɱɚɥɨɫɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ
(1.29) ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ ɜɨɡɦɨɠɧɨ ɥɢɲɶ ɜ ɧɟɤɨɬɨɪɵɯ ɱɚɫɬ
ɧɵɯ ɫɥɭɱɚɹɯ ɡɚɞɚɧɢɹ ɧɚɝɪɭɡɤɢ ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ Ɍɨɱɧɨɝɨ ɪɟ
ɲɟɧɢɹɞɥɹɛɨɥɶɲɢɧɫɬɜɚɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧɞɨɫɢɯ
ɩɨɪ ɧɟ ɩɨɥɭɱɟɧɨ ɩɨɫɤɨɥɶɤɭ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ ɤ ɤɨɬɨɪɵɦ ɨɧɢ ɩɪɢɜɨɞɹɬɫɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɛɨɥɶɲɢɟ ɦɚ
ɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ ɉɨɷɬɨɦɭ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɢɧɠɟɧɟɪ
ɧɨɣɩɪɚɤɬɢɤɢɢɦɟɸɬɩɪɢɛɥɢɠɟɧɧɵɟɧɨɞɨɫɬɚɬɨɱɧɨɨɛɳɢɟɦɟɬɨɞɵ
ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɩɪɢɤɥɚɞɧɨɣ ɦɟɯɚɧɢɤɢ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɬɟɥɚ
ȼ ɞɚɧɧɨɦ ɩɨɫɨɛɢɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɯɚ
ɪɚɤɬɟɪɧɵɟɞɥɹɡɚɞɚɱɢɡɝɢɛɚɩɥɚɫɬɢɧɧɨɫɬɚɤɢɦɠɟɭɫɩɟɯɨɦɩɪɢɦɟ
ɧɹɟɦɵɟɞɥɹɪɚɫɱɟɬɚɨɛɨɥɨɱɟɤɢɞɪɭɝɢɯɬɨɧɤɨɫɬɟɧɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ
ȼɡɚɜɢɫɢɦɨɫɬɢɨɬɩɨɞɯɨɞɨɜɤɪɟɲɟɧɢɸɡɚɞɚɱɦɨɠɧɨɜɵɞɟɥɢɬɶ
ɨɫɧɨɜɧɵɟɤɥɚɫɫɢɮɢɤɚɰɢɢɩɪɢɛɥɢɠɟɧɧɵɯɦɟɬɨɞɨɜ
ɉɨ ɬɢɩɭ ɪɚɡɪɟɲɚɸɳɢɯ ɭɪɚɜɧɟɧɢɣ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ
ɦɨɠɧɨɪɚɡɞɟɥɢɬɶɧɚɞɜɟɝɪɭɩɩɵɉɟɪɜɭɸɫɨɫɬɚɜɥɹɸɬɬɚɤɧɚɡɵɜɚɟ
ɦɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɤɪɚɟɜɵɯɡɚɞɚɱɞɥɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜ
ɧɟɧɢɣɂɡɱɢɫɥɚ ɷɬɢɯ ɦɟɬɨɞɨɜɡɞɟɫɶɛɭɞɭɬɪɚɫɫɦɨɬɪɟɧɵɦɟɬɨɞɤɨ
ɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɆɄɊ ɢ ɦɟɬɨɞ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɆȻȽ
ȼɬɨɪɭɸ ɝɪɭɩɩɭ ɫɨɫɬɚɜɥɹɸɬ ɩɪɹɦɵɟ ɜɚɪɢɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɨɫɧɨ
ɜɚɧɧɵɟ ɧɚ ɜɚɪɢɚɰɢɨɧɧɵɯ ɩɪɢɧɰɢɩɚɯ ɦɟɯɚɧɢɤɢ ɩɪɟɠɞɟ ɜɫɟɝɨ ɧɚ
ɩɪɢɧɰɢɩɟ ɦɢɧɢɦɭɦɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ Ʌɚɝɪɚɧɠɚ ɢ ɩɪɢɧ
ɰɢɩɟɦɢɧɢɦɭɦɚɞɨɩɨɥɧɢɬɟɥɶɧɨɣɪɚɛɨɬɵɄɚɫɬɢɥɶɹɧɨɄɷɬɢɦɦɟ
ɬɨɞɚɦ ɜ ɱɚɫɬɧɨɫɬɢ ɨɬɧɨɫɹɬɫɹ ɦɟɬɨɞ Ɋɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨ ɆɊɌ
ɜɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣ ɦɟɬɨɞ ȼɊɆ ɚ ɬɚɤɠɟ ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨ
ɫɬɪɚɧɟɧɧɵɣɜɧɚɫɬɨɹɳɟɟɜɪɟɦɹɦɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜɆɄɗ
ɉɨ ɫɯɟɦɟ ɩɨɫɬɪɨɟɧɢɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɪɚɡɥɢ
ɱɚɸɬ ɪɚɡɧɨɫɬɧɵɟ ɢ ɩɪɨɟɤɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ȼ ɪɚɡɧɨɫɬɧɵɯ ɦɟɬɨɞɚɯ
ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣ ɮɨɪɦɟ
ɆɄɊ ȼɊɆ ɚ ɜ ɩɪɨɟɤɰɢɨɧɧɵɯ ɡɚɦɟɧɹɟɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦ ɤɨɧɟɱ
ɧɨɦɟɪɧɵɦ ɭɪɚɜɧɟɧɢɟɦɪɟɲɟɧɢɟɤɨɬɨɪɨɝɨɢɳɟɬɫɹ ɩɪɢ ɩɨɦɨɳɢ ɬɚɤ
ɧɚɡɵɜɚɟɦɵɯɤɨɨɪɞɢɧɚɬɧɵɯ (ɛɚɡɢɫɧɵɯ) ɮɭɧɤɰɢɣ ɆȻȽɆɊɌɆɄɗ.
39
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɂ ɧɚɤɨɧɟɰ ɩɨ ɜɢɞɭ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ ɩɪɢ
ɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɪɚɡɞɟɥɹɸɬɫɹ ɧɚ ɱɢɫɥɟɧɧɨ-ɚɧɚɥɢɬɢɱɟɫɤɢɟ (ɩɨ
ɥɭɚɧɚɥɢɬɢɱɟɫɤɢɟɢɱɢɫɥɟɧɧɵɟ (ɫɟɬɨɱɧɵɟȼɩɟɪɜɵɯɢɡɧɢɯɪɟɲɟ
ɧɢɟ ɡɚɞɚɱɢ ɢɳɟɬɫɹ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ ɫ ɩɨɞɥɟɠɚɳɢɦɢ ɨɩɪɟ
ɞɟɥɟɧɢɸ ɱɢɫɥɨɜɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɆȻȽ ɆɊɌ ȼ ɫɟɬɨɱɧɵɯ ɦɟɬɨɞɚɯ ɆɄɊ ȼɊɆ ɆɄɗ ɪɟɲɟɧɢɟ
ɢɳɟɬɫɹ ɜ ɜɢɞɟ ɱɢɫɥɨɜɵɯ ɡɧɚɱɟɧɢɣ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜ ɧɟɤɨɬɨɪɨɣ
ɞɢɫɤɪɟɬɧɨɣɫɨɜɨɤɭɩɧɨɫɬɢɬɨɱɟɤɭɡɥɚɯɫɟɬɤɢ).
Ʉɪɨɦɟ ɬɨɝɨ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɦɨɝɭɬ ɪɟɚɥɢɡɨɜɵɜɚɬɶɫɹ
ɤɚɤ ɜ ɮɨɪɦɟ ɦɟɬɨɞɚ ɩɟɪɟɦɟɳɟɧɢɣ ɨɫɧɨɜɧɵɦɢ ɧɟɢɡɜɟɫɬɧɵɦɢ
ɡɞɟɫɶɹɜɥɹɸɬɫɹɩɟɪɟɦɟɳɟɧɢɹ – ɜ ɜɢɞɟɢɫɤɨɦɵɯɮɭɧɤɰɢɣɢɥɢɱɢɫ
ɥɨɜɵɯɡɧɚɱɟɧɢɹɜɭɡɥɚɯɫɟɬɤɢɬɚɤɢɜɮɨɪɦɟɦɟɬɨɞɚɫɢɥ ɨɫɧɨɜ
ɧɵɦɢ ɧɟɢɡɜɟɫɬɧɵɦɢ ɹɜɥɹɸɬɫɹ ɭɫɢɥɢɹ ɢɥɢ ɧɚɩɪɹɠɟɧɢɹ ɉɪɢ ɪɚɫ
ɱɟɬɟ ɫɥɨɠɧɵɯ ɬɨɧɤɨɫɬɟɧɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ ɜ ɱɚɫɬɧɨɫɬɢ ɨɛɨɥɨɱɟɤ
ɧɟɪɟɞɤɨ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɢɳɟɬɫɹ ɜ ɫɦɟɲɚɧɧɨɣ ɮɨɪɦɟ ɨɫ
ɧɨɜɧɵɦɢɧɟɢɡɜɟɫɬɧɵɦɢɡɞɟɫɶɹɜɥɹɸɬɫɹɢɩɟɪɟɦɟɳɟɧɢɹɢɭɫɢɥɢɹ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɩɪɢɜɵɩɨɥɧɟɧɢɢɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɫɜɨɞɢɬɫɹɤɪɟɲɟɧɢɸɫɢɫɬɟɦɵ
ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɡɥɨɜɵɯ ɡɧɚ
ɱɟɧɢɣɢɫɤɨɦɨɣɮɭɧɤɰɢɢɈɩɪɟɞɟɥɢɜɨɫɧɨɜɧɵɟɧɟɢɡɜɟɫɬɧɵɟɞɚɥɟɟ
ɱɟɪɟɡɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢɨɩɪɟɞɟɥɹɸɬɭɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹ
2Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɣɨɩɟɪɚɬɨɪɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹɢɡɝɢɛɚ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɬɪɨɟɧɢɟɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯ ɨɩɟɪɚɬɨɪɨɜ ɄɊɈ
ɞɥɹ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɨɬ ɮɭɧɤɰɢɢ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ w (x, y).
ɇɚɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸɨɛɥɚɫɬɶɧɚɥɨɠɢɦɫɟɬɤɭɭɡɥɨɜɫɪɚɜɧɵɦɢɢɧ
ɬɟɪɜɚɥɚɦɢ ¨x ɢ ¨y ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɨ ɨɫɹɦ x ɢ y ɪɢɫ Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɦɵ ɩɟɪɟɯɨɞɢɦ ɨɬ ɧɟɩɪɟ
ɪɵɜɧɨɣɨɛɥɚɫɬɢɤɞɢɫɤɪɟɬɧɨɣɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣɫɟɬɤɟ. ɇɚɷɬɨɣ
ɫɟɬɤɟ ɛɭɞɟɦ ɫɬɪɨɢɬɶ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɩɪɨɢɡɜɨɞ
ɧɵɯɢɞɪɭɝɢɯɱɥɟɧɨɜɜɯɨɞɹɳɢɯɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟ
¨x
2Ɋɚɫɱɟɬɩɥɚɫɬɢɧɦɟɬɨɞɨɦɤɨɧɟɱɧɵɯɪɚɡɧɨɫɬɟɣ
40
x
ww
'y
l
¨y
i
g
b
h
a
k
c
f
d
e
¨y
Ɇɟɬɨɞ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɢɥɢ, ɤɚɤ ɟɝɨ ɟɳɟ ɧɚɡɵɜɚɸɬ, ɦɟ
ɬɨɞ ɫɟɬɨɤ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɞɨɫɬɚɬɨɱɧɨ ɷɮɮɟɤɬɢɜɧɵɯ ɱɢɫɥɟɧ
ɧɵɯɦɟɬɨɞɨɜɪɟɲɟɧɢɹɤɪɚɟɜɵɯɡɚɞɚɱɞɥɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜ
ɧɟɧɢɣɋɨɝɥɚɫɧɨɷɬɨɦɭɦɟɬɨɞɭɜɫɹɨɛɥɚɫɬɶɪɟɲɟɧɢɹɤɪɚɟɜɨɣɡɚɞɚ
ɱɢɫɪɟɞɢɧɧɚɹɩɥɨɫɤɨɫɬɶɩɥɚɫɬɢɧɵɩɨɜɟɪɯɧɨɫɬɶɨɛɨɥɨɱɤɢɢɬɩ
ɩɨɤɪɵɜɚɟɬɫɹɫɟɬɤɨɣɥɢɧɢɣɬɨɱɤɢɩɟɪɟɫɟɱɟɧɢɹɤɨɬɨɪɵɯɧɚɡɵɜɚɸɬ
ɫɹɭɡɥɚɦɢɁɚ ɧɟɢɡɜɟɫɬɧɵɟɜɡɚɞɚɱɟɩɪɢɧɢɦɚɸɬɫɹɱɢɫɥɨɜɵɟɡɧɚɱɟ
ɧɢɹ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ ɜ ɭɡɥɚɯ ɫɟɬɤɢ ɭɡɥɨɜɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɉɪɢ
ɷɬɨɦ ɩɪɨɢɡɜɨɞɧɵɟ ɜɯɨɞɹɳɢɟ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɡɚ
ɦɟɧɹɸɬɫɹ ɩɪɢɛɥɢɠɟɧɧɵɦɢ ɜɵɪɚɠɟɧɢɹɦɢ ɜ ɜɢɞɟ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯ ɨɬɧɨɲɟɧɢɣ ɦɟɠɞɭ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɭɡɥɨɜɵɦɢ ɧɟɢɡɜɟɫɬɧɵ
ɦɢ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɩɪɨɢɡɜɨɞɧɵɯ ɉɨɞɫɬɚɜɢɜ
ɷɬɢ ɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɢɜ
ɬɟɦ ɫɚɦɵɦ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɣ ɨɩɟɪɚɬɨɪ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹɬɪɟɛɭɟɦɟɝɨɜɵɩɨɥɧɟɧɢɹ ɜɤɚɠɞɨɦɭɡɥɟɫɟɬɤɢ
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɬɚɤɠɟ ɡɚɩɢɫɵɜɚɸɬ ɱɟɪɟɡ
ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɨɩɟɪɚɬɨɪɵ, ɫɜɹɡɵɜɚɹ ɜɫɜɨɸɨɱɟɪɟɞɶɪɚɡɧɨɫɬ
ɧɵɦɢ ɨɬɧɨɲɟɧɢɹɦɢ ɧɟɢɡɜɟɫɬɧɵɟ ɜ ɩɪɢɝɪɚɧɢɱɧɵɯ ɭɡɥɚɯ ɫɟɬɤɢ
ȼ ɤɨɧɟɱɧɨɦ ɫɱɟɬɟ ɡɚɞɚɱɚ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
¨x
y
–1
m
w2w
' 2x
–1
1
'x
u
k
1
k
–2
1
1
1
u
1
u
1
' 2x
1
w2w
' x' y
1
'y
1
u
1
' 2y
–1
u
k
–1
1
Ɋɢɫ 2.1. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɚɹ ɫɟɬɤɚɢɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵ
41
k
–2
n
ww
'x
k
w2w
' 2y
1
4' x ' y
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ȼɩɪɟɞɩɨɥɨɠɟɧɢɢɦɚɥɨɫɬɢɪɚɡɦɟɪɨɜɲɚɝɨɜɫɟɬɤɢ ¨x ɢ¨y ɯɨɬɹ
ɷɬɨɧɟɫɨɜɫɟɦɬɚɤɡɚɩɢɲɟɦɩɟɪɜɵɟɩɪɨɢɡɜɨɞɧɵɟɨɬɮɭɧɤɰɢɢ w ɞɥɹ
ɭɡɥɚk ɜɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣɮɨɪɦɟɩɪɢ¨x = const, ¨y = const):
ww
wx
k
w wa
;
| c
2' x
ww
wy
k
w wb
.
| d
2' y
(2.1)
ɉɨɥɭɱɟɧɧɵɟ ɨɩɟɪɚɬɨɪɵ ɩɪɨɢɡɜɨɞɧɵɯ ɜ ɫɯɟɦɚɬɢɱɧɨɦ ɜɢɞɟ
ɢɡɨɛɪɚɠɟɧɵɧɚɪɢɫ ɩɨɞɫɟɬɤɨɣɭɡɥɨɜɢɫɛɨɤɭɈɧɢɫɢɦɦɟɬɪɢɱ
ɧɵɨɬɧɨɫɢɬɟɥɶɧɨɭɡɥɚkɩɨɷɬɨɦɭɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɢɦɜɵɪɚɠɟɧɢɹ
ɧɚɡɵɜɚɸɬɫɹɰɟɧɬɪɚɥɶɧɵɦɢɤɨɧɟɱɧɵɦɢɪɚɡɧɨɫɬɹɦɢ.
ɉɪɚɜɵɟ ɢɥɟɜɵɟɤɨɧɟɱɧɵɟ ɪɚɡɧɨɫɬɢ ɞɥɹɭɡɥɚk ɡɚɩɢɲɭɬɫɹɬɚɤ
ww
wx
ɩɪ
k
w wk ww
;
| c
'x
wy
ɩɪ
k
w wk ww
;
| d
'y
wx
ɥɟɜ
k
w wa ww
;
| k
'x
wy
ɥɟɜ
k
w wb
| k
.
'y
ȼɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɮɭɧɤɰɢɢ w ɜ ɭɡɥɟ k ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɪɢ
ɦɟɧɹɹɨɩɟɪɚɬɨɪɵɩɟɪɜɨɣɩɪɨɢɡɜɨɞɧɨɣɤɩɟɪɜɵɦɠɟɩɪɨɢɡɜɨɞɧɵɦ
w2w
wx 2
w2w
wy2
ɩɪ
k
1 §¨ ww
|
' x ¨ wx
©
ɩɪ
k
1 §¨ ww
|
' y ¨ wy
©
w2w
wx wy
|
k
k
k
ww
wx
ww
wy
ɥɟɜ ·
k
¸ | wa 2 wk wc ;
¸
'2x
¹
ɥɟɜ ·
¸ | wb 2 wk wd ;
¸
'2y
¹
k
1 §¨ ww
ww
¨
2' y © w x d w x
(2.2)
· we w f wg wh
¸|
.
¸
4' x ' y
b¹
k
w Pwb 2 (1 P) wk wc Pwd
w2w
w2w
| a
2
2
wx k wy k
'2x
ɢɥɢ
’2w
k
Kwa wb 2 (1 K) wk Kwc wd
w2w
w2w
, (2.4)
|
2
2
'2y
wx k wy k
42
’ w
’ ’ w
4
2
k
1
P
'2y
'2x
.
2
k
|
’ 2 w P’ 2 w 2(1 P)’ 2 w ’ 2 w P’ 2 w
a
b
k
c
.
d
'2x
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨɞɫɬɚɜɢɜ ɫɸɞɚ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɨɩɟɪɚɬɨɪɵ
’ 2 w , j a, b, k , c, d , ɜɵɪɚɠɟɧɧɵɟ ɫ ɩɨɦɨɳɶɸ (2.4), ɢ ɩɪɢɜɟɞɹ
j
ɩɨɞɨɛɧɵɟɱɥɟɧɵɩɨɥɭɱɢɦ
’4w
k
|
wk (8 6P 6K) 4 wa (1 K) 4 wb (1 P) 4 wc (1 K)
'2x '2y
4 wd (1 P) 2 we 2 w f 2 wg 2 wh Kwi Pwl Kwm Pwn
'2x '2y
(2.5)
.
ɊɚɡɧɨɫɬɧɚɹɫɯɟɦɚɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨɪɚɅɚɩɥɚɫɚɢɡɨɛɪɚ
ɠɟɧɚɧɚɪɢɫ 2.2, ɚȼɬɨɦɫɥɭɱɚɟɟɫɥɢɢɫɩɨɥɶɡɭɟɬɫɹɤɜɚɞɪɚɬɧɚɹɫɟɬɤɚ
¨x = ¨y = ¨ȝ = Ș = ɨɩɟɪɚɬɨɪɩɪɢɧɢɦɚɟɬɜɢɞɪɢɫ 2.2, ɛ)
’4w |
(2.3)
'2x
; K
'2y
Ȼɢɝɚɪɦɨɧɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ ’ 4 w ɞɥɹ ɭɡɥɚ k ɦɨɠɧɨ
ɩɨɥɭɱɢɬɶ ɞɜɚɠɞɵ ɩɪɢɦɟɧɢɜ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɨɩɟɪɚɬɨɪ ’ 2 w ɡɚɩɢ
ɫɚɧɧɵɣ ɜ ɜɢɞɟ ɢ ɋɧɚɱɚɥɚ ɢɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ ɩɪɟɞɫɬɚɜɢɦɨɩɟɪɚɬɨɪ ’ 4 w ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵɬɚɤɠɟɩɪɢɜɟɞɟɧɵɧɚɪɢɫ 2.1.
ɋɤɥɚɞɵɜɚɹ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɢɦ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɜɵɪɚɠɟɧɢɹɞɥɹɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨɪɚɅɚɩɥɚɫɚɜɭɡɥɟk:
’2w
ɝɞɟ P
20 wk 8 ( wa wb wc wd ) 2 ( we w f wg wh ) wi wl wm wn
'4
k
.
ɇɟɤɨɬɨɪɵɟɨɞɧɚɤɨɜɩɨɥɧɟɪɚɡɪɟɲɢɦɵɟ ɫɥɨɠɧɨɫɬɢɩɪɟɞɫɬɚɜ
ɥɹɟɬ ɩɨɫɬɪɨɟɧɢɟ ɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɩɟɪɚɬɨɪɚ ɜ ɫɥɭɱɚɟ ɧɟɪɚɜɧɨ
ɦɟɪɧɨɝɨɲɚɝɚɫɟɬɤɢ¨x cRQVW¨y FRQVWɚɬɚɤɠɟɞɥɹɤɨɫɨɭɝɨɥɶ
ɧɨɣɢɥɢɩɨɥɹɪɧɨɣɫɢɫɬɟɦɵɤɨɨɪɞɢɧɚɬ>4].
ȼɤɨɧɟɱɧɨɦɫɱɟɬɟɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɣɨɩɟɪɚɬɨɪɞɢɮɮɟɪɟɧ
ɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɭɡɥɚ k ɡɚ
ɩɢɲɟɬɫɹɜɜɢɞɟ
qk
(2.6)
’4w
, k 1, ... , N ,
k
D
43
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɝɞɟN – ɨɛɳɟɟɱɢɫɥɨɜɧɭɬɪɟɧɧɢɯɭɡɥɨɜɫɟɬɤɢqk – ɫɪɟɞɧɹɹɢɧɬɟɧ
ɫɢɜɧɨɫɬɶ ɧɚɝɪɭɡɤɢ ɩɪɢɯɨɞɹɳɟɣɫɹ ɧɚ ɩɥɨɳɚɞɤɭ ¨xרy ɩɪɢɦɵɤɚɸ
ɳɭɸɤɭɡɥɭ k. ȿɫɥɢɜɭɡɥɟ k ɩɪɢɥɨɠɟɧɚɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹɫɢɥɚ Fkɬɨ
ɨɧɚ ɭɱɢɬɵɜɚɟɬɫɹ ɜ ɜɢɞɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɧɚɝɪɭɡɤɢ qk Fk (' x ' y ) .
ɚ)
’
k
Ș
2
–ȝ
2
–4(1+Ș)
8+6(ȝȘ)
–4(1+Ș)
–ȝ
2
2
¨x
ȝ
ɛ)
Ș
u
1
' 2x ' 2y
wk
¨y
’4
k
1
–8
2
–8
20
–8
2
–8
2
1
¨
u
1
'4
¨
Ɋɢɫ 2.2. ȻɢɝɚɪɦɨɧɢɱɟɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚ
Ɉɩɟɪɚɬɨɪ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɩɥɚɫɬɢɧɵ ɥɟ
ɠɚɳɟɣɧɚɭɩɪɭɝɨɦɜɢɧɤɥɟɪɨɜɫɤɨɦɨɫɧɨɜɚɧɢɢ (ɩɪɢɞɟɣɫɬɜɢɢɪɚɫɩɪɟ
ɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢɢɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯɫɢɥ), ɡɚɩɢɲɟɬɫɹɬɚɤ
’4w k
ky
D
wk
Fk ·¸
1 §¨
qk , k 1, ... , N .
¨
' x ' y ¸¹
D©
w wa
ww
| c
wx k
2' x
0, ɨɬɫɸɞɚ wc
wa .
(2.8)
w 2 wk wc
w2w
| a
0 , ɨɬɫɸɞɚ wc wa . (2.9)
2
wx k
'2x
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɥɹ ɡɚɞɟɥɚɧɧɨɝɨ ɢ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɝɨ ɤɪɚɹ
ɩɥɚɫɬɢɧɵ ɭɡɥɨɜɵɟ ɧɟɢɡɜɟɫɬɧɵɟ ɧɚ ɤɨɧɬɭɪɟ ɜ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡɥɚɯ
ɦɨɠɧɨɢɫɤɥɸɱɢɬɶɫɩɨɦɨɳɶɸɪɚɜɟɧɫɬɜɢ
ɋɜɨɛɨɞɧɵɣɤɪɚɣ ɪɢɫ 2.3, ɜ). ɇɚɷɬɨɦɭɱɚɫɬɤɟɤɨɧɬɭɪɚɞɥɹɭɡ
ɥɚk ɜɫɨɨɬɜɟɬɫɬɜɢɢɫɢɦɟɟɦ
wk
1
0;
ɒɚɪɧɢɪɧɨɨɩɟɪɬɵɣɤɪɚɣ ɪɢɫɛȼɫɨɨɬɜɟɬɫɬɜɢɢɫ
ɢɞɥɹɭɡɥɚk ɢɦɟɟɦ
1
2
2.1.2. ɍɱɟɬɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
Ɋɚɫɫɦɨɬɪɢɦ ɯɚɪɚɤɬɟɪɧɵɟ ɫɥɭɱɚɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɧɚɩɪɢ
ɦɟɪɞɥɹɩɪɚɜɨɝɨɤɪɚɹɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵx = a.
Ɂɚɞɟɥɚɧɧɵɣ ɤɪɚɣ ɪɢɫ 2.3, ɚ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɞɥɹ ɭɡɥɚ k ɧɚ
ɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵɜɫɨɨɬɜɟɬɫɬɜɢɢɫɢɢɦɟɟɦ:
ȝ
4
ɧɟɧɢɣ ɧɟɨɛɯɨɞɢɦɨ ɭɱɟɫɬɶ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɤɪɚɹɯ ɩɥɚɫɬɢɧɵ
ɬɚɤɠɟɡɚɩɢɫɚɧɧɵɟɱɟɪɟɡɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢ
(2.7)
0;
w2w
w2w
Q 2
2
wx
wy
0;
k
w § w2w
w2w ·
¨¨ 2 (2 Q) 2 ¸¸
wx © wx
wy ¹
0.
(2.10)
k
ɂɫɩɨɥɶɡɭɹ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟ ɜɵɪɚɠɟɧɢɹ ɢ ɩɪɢɦɟɧɢɜ ɨɩɟ
ɪɚɬɨɪɩɟɪɜɨɣɩɪɨɢɡɜɨɞɧɨɣɤɨɜɬɨɪɵɦɩɪɨɢɡɜɨɞɧɵɦɩɨɫɥɟɩɪɢ
ɜɟɞɟɧɢɹɩɨɞɨɛɧɵɯɱɥɟɧɨɜɢɢɫɤɥɸɱɟɧɢɹɜɟɥɢɱɢɧ¨x ɢ¨y ɩɨɥɭɱɢɦ
2 (1 PQ) wk ( wa wc ) PQ ( wb wd ) 0 ;
2K (3 Q) ( wc wa ) (2 Q)( we w f wg wh ) Kwi KQwm 0 .
(2.11)
ɇɚɤɥɚɞɵɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɩɟɪɚɬɨɪ ɢɥɢ ɧɚ ɜɫɟ
ɭɡɥɵ ɫɟɬɤɢ ɜ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɪɨɝɢɛɵ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ
ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɡɥɨɜɵɯ ɧɟɢɡ
ɜɟɫɬɧɵɯ wk Ɉɞɧɚɤɨ ɩɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɭɡɥɨɜ ɥɟɠɚ
ɳɢɯɜɛɥɢɡɢɤɨɧɬɭɪɚɩɥɚɫɬɢɧɵɜɧɢɯɜɨɣɞɭɬɧɟɢɡɜɟɫɬɧɵɟɩɪɨɝɢɛɵ
ɢɜɡɚɤɨɧɬɭɪɧɵɯɭɡɥɚɯ ɫɟɬɤɢɉɨɷɬɨɦɭɞɥɹɪɟɲɟɧɢɹɫɢɫɬɟɦɵɭɪɚɜ
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɫɯɟɦɵ ɨɩɟɪɚɬɨɪɨɜ ɞɥɹ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
ɢɢɡɨɛɪɚɠɟɧɵɧɚɪɢɫ Ⱥɧɚɥɨɝɢɱɧɨɫɨɫɬɚɜɥɹɸɬɫɹ
ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɨɩɟɪɚɬɨɪɵɞɥɹɥɟɜɨɝɨɤɪɚɹɩɥɚɫɬɢɧɵx = ɜɷɬɨɦ
ɫɥɭɱɚɟɩɪɢɜɟɞɟɧɧɵɟɨɩɟɪɚɬɨɪɧɵɟɫɯɟɦɵɪɚɡɜɟɪɧɭɬɫɹɧɚɝɪɚɞɭɫɨɜ.
Ⱦɥɹɜɟɪɯɧɟɝɨɢɧɢɠɧɟɝɨɤɪɚɹɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɨɩɟɪɚɬɨɪɵɪɚɡɜɟɪ
44
45
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɧɭɬɫɹɧɚɝɪɚɞɭɫɨɜɤɪɨɦɟɬɨɝɨɜɨɩɟɪɚɬɨɪɚɯɞɥɹɫɜɨɛɨɞɧɨɝɨɤɪɚɹ
ɤɨɷɮɮɢɰɢɟɧɬɵȝɢȘɩɨɦɟɧɹɸɬɫɹɦɟɫɬɚɦɢɪɢɫɝ).
ɚ)
ɛ)
x=a
x=a
x
x
a
k
c
¨x
a
k
¨x
¨x
y
c
¨x
y
wa
1
1
ɜ)
wc
wa
wc
–1
1
2.1.3. ȼɵɱɢɫɥɟɧɢɟ ɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣɢɧɚɩɪɹɠɟɧɢɣ
x=a
x
g
i
h
a
k
c
ȝȞ
m
–1
e
¨y
f
¨x
ȝȞ
–1
ȝȞ
wa
ɝ)
wk
2 – Ȟ
Ș
Ș(Ȟ–6)
2 – Ȟ
ɉɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɨɝɢɛɨɜ ɜɨ ɜɫɟɯ ɭɡɥɚɯ ɫɟɬɤɢ ɧɚɥɨɠɟɧ
ɧɨɣ ɧɚ ɩɥɚɫɬɢɧɭ ɫ ɩɨɦɨɳɶɸ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯɨɩɟɪɚɬɨɪɨɜɜɵɱɢɫɥɹɟɦɡɧɚɱɟɧɢɹɢɡɝɢɛɚɸɳɢɯɢɤɪɭɬɹɳɟɝɨɦɨ
ɦɟɧɬɨɜɜɷɬɢɯɭɡɥɚɯɉɨɞɫɬɚɜɢɜɜɮɨɪɦɭɥɵɜɵɪɚɠɟɧɢɹɜɬɨɪɵɯ
ɩɪɨɢɡɜɨɞɧɵɯɢɩɪɢɜɟɞɹɩɨɞɨɛɧɵɟɱɥɟɧɵɩɨɥɭɱɢɦ
¨x
y
wi
Ʉɚɤ ɜɢɞɢɦ ɜ ɫɥɭɱɚɟ ɫɜɨɛɨɞɧɨɝɨ ɤɪɚɹ ɜ ɪɚɫɱɟɬ ɜɜɨɞɹɬɫɹ ɞɜɚ
ɫɥɨɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɯ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡɥɨɜ – ɫɬɨɥɶɤɨ ɠɟ ɫɥɨɟɜ
ɫɤɨɥɶɤɨ ɢɩɪɢɧɚɥɨɠɟɧɢɢɨɩɟɪɚɬɨɪɚɧɚɭɡɥɵɧɟɡɚɤɪɟɩɥɟɧɧɨɝɨ
ɤɨɧɬɭɪɚɩɥɚɫɬɢɧɵɉɪɢɷɬɨɦɤɚɠɞɨɦɭɬɚɤɨɦɭɭɡɥɭɧɚɤɨɧɬɭɪɟɨɬ
ɜɟɱɚɸɬ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ȼ ɪɟɡɭɥɶɬɚɬɟ ɨɛɳɟɟ ɱɢɫɥɨ ɞɨɩɨɥɧɢ
ɬɟɥɶɧɵɯɡɚɤɨɧɬɭɪɧɵɯɧɟɢɡɜɟɫɬɧɵɯɛɭɞɟɬɪɚɜɧɨɨɛɳɟɦɭɱɢɫɥɭɞɨ
ɩɨɥɧɢɬɟɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜɵɪɚɠɚɸɳɢɯ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ
ɜɫɟɯɭɡɥɨɜɧɚɫɜɨɛɨɞɧɨɦɤɪɚɟɩɥɚɫɬɢɧɵ
ɗɬɢ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɫɨɫɬɚɜɥɟɧɧɵɟ ɫ ɩɨɦɨɳɶɸ
ɪɚɜɟɧɫɬɜ ɞɨɛɚɜɥɹɸɬɫɹ ɤ ɭɪɚɜɧɟɧɢɹɦ ɜɵɪɚɠɚɸɳɢɦ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɥɚɫɬɢɧɵȼɪɟɡɭɥɶɬɚɬɟɢɦɟ
ɟɦ ɩɨɥɧɭɸ ɪɚɡɪɟɲɚɸɳɭɸ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜ
ɧɟɧɢɣ ɢɡ ɪɟɲɟɧɢɹ ɤɨɬɨɪɨɣ ɧɚɯɨɞɹɬɫɹ ɜɟɥɢɱɢɧɵ ɩɪɨɝɢɛɨɜ ɜɨ ɜɫɟɯ
ɭɡɥɚɯɫɟɬɤɢɜɤɥɸɱɚɹɡɚɤɨɧɬɭɪɧɵɟ. Ɂɧɚɱɟɧɢɹw ɜɡɚɤɨɧɬɭɪɧɵɯ ɭɡ
ɥɚɯɜɞɚɥɶɧɟɣɲɟɦɢɫɩɨɥɶɡɭɸɬɫɹɩɪɢɜɵɱɢɫɥɟɧɢɢɜɧɭɬɪɟɧɧɢɯɭɫɢ
ɥɢɣɜɭɡɥɨɜɵɯɬɨɱɤɚɯ ɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ
wc
k
My
k
ȝ
wm
Ȟ– 2
0
Mx
2 – Ȟ ȝ(Ȟ–6) 2 – Ȟ
0
Ș(6–Ȟ) –ȘȞ
Ȟ– 2
Ȟ– 2 ȝ(6–2Ȟ) Ȟ – 2
H
k
D
2 (1 PQ) wk wa wɫ PQ(wb wd );
'2x
D
2 (1 KQ) wk wb wd KQ( wa wc ) ;
'2y
(2.12)
D (1 Q)
we w f wg wh .
4 ' x' y
–ȝȞ
ɋɯɟɦɵɨɩɟɪɚɬɨɪɨɜɞɥɹɜɵɱɢɫɥɟɧɢɹɦɨɦɟɧɬɨɜɩɪɢɜɟɞɟɧɵɧɚɪɢɫ
Ɋɢɫ 2.3. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɟɨɩɟɪɚɬɨɪɵɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
46
47
–ȝȞ
My
–1
ɚ)
–1
k
2ȝȞ)
u
¨x
–ȝȞ
¨y
k
–1
–ȘȞ
D
' 2x
2(1+ȘȞ)
–1
H
D
' 2y
¨y
¨x
–1
P
O
P’
S
2
1
2
3Ǝ
T
3
4
3
D (1 Q)
u
4' x ' y
2Ǝ
6֤
2
1
2
r
6M y
h2
2
–11,11
2
0,563 –6,25 22,04 –6,25 0,563
2
–11,11
2
y
F
(2.13)
ɉɪɢɦɟɪɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɨɣɩɥɚɫɬɢɧɵɆɄɊ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɪɚɫɱɟɬɚ ɠɟɥɟɡɨɛɟɬɨɧɧɨɣ ɩɥɢɬɵ ɪɚɡɦɟɪɚ
ɦɢa = ɦb = ɦɢɬɨɥɳɢɧɨɣh = ɦɧɚɝɪɭɠɟɧɧɨɣɪɚɜɧɨɦɟɪ
ɧɨɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɨɣɢɧɬɟɧɫɢɜɧɨɫɬɶɸq = ɤɇɦ2 ɢɞɜɭ
ɦɹɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢɫɢɥɚɦɢF = ɤɇɄɪɚɹɩɥɢɬɵɩɚɪɚɥɥɟɥɶɧɵɟ
ɨɫɢx – ɲɚɪɧɢɪɧɨɨɩɟɪɬɵɩɚɪɚɥɥɟɥɶɧɵɟɨɫɢy – ɡɚɞɟɥɚɧɵɪɢɫɚ).
Ɍɪɟɛɭɟɬɫɹɨɩɪɟɞɟɥɢɬɶɩɪɨɝɢɛɵɜɭɡɥɚɯɧɚɥɨɠɟɧɧɨɣɧɚɩɥɢɬɭɫɟɬɤɢ
î ɩɨɫɬɪɨɢɬɶ ɷɩɸɪɵ ɩɪɨɝɢɛɨɜ w ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ Mx
ɢ My ɞɥɹ ɫɪɟɞɧɢɯ ɫɟɱɟɧɢɣ ɩɥɢɬɵ ɩɪɢ x = a ɢ y = b ɷɩɸɪɭ
ɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜH ɞɥɹɝɪɚɧɢɱɧɨɝɨɫɟɱɟɧɢɹy = 0.
ɍɱɢɬɵɜɚɹ ɫɢɦɦɟɬɪɢɸ ɩɥɚɫɬɢɧɵ ɫ ɡɚɤɪɟɩɥɟɧɢɹɦɢ ɢ ɩɪɢɥɨ
ɠɟɧɧɨɣɧɚɝɪɭɡɤɨɣɨɬɧɨɫɢɬɟɥɶɧɨɨɫɟɣx = aɢy = bɧɭɦɟɪɚɰɢɸ
ɭɡɥɨɜ ɫɟɬɤɢ ɜɵɩɨɥɧɢɦ ɬɚɤɠɟ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɢɯ
ɨɫɟɣȼɟɥɢɱɢɧɵɲɚɝɨɜɫɟɬɤɢ¨x = a/4 = ɦ¨y = b/4 = ɦɬɨɝɞɚ
ɤɨɷɮɮɢɰɢɟɧɬɵ P '2x '2y 1,778 ; K 1 P 0,553.
48
1,771
1,771
1
6H
; W r 2 .
h
x
Rƍ
a/2 = 4 ɦa/2 = 4 ɦ
¨x
ɇɚɢɛɨɥɶɲɢɟɡɧɚɱɟɧɢɹɧɚɩɪɹɠɟɧɢɣɞɟɣɫɬɜɭɸɳɢɯɧɚɧɢɠɧɟɣ
ɢɜɟɪɯɧɟɣɩɨɜɟɪɯɧɨɫɬɹɯ ɩɥɚɫɬɢɧɵɨɩɪɟɞɟɥɹɟɦɩɨɮɨɪɦɭɥɚɦ
Vx
ɛ)
2ƍ
2Ǝ
Ɋɢɫ 2.4. Ɉɩɟɪɚɬɨɪɵɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣ
6M
r 2 x ; Vy
h
1ƍ
–1
0
k
R
–ȘȞ
u
1
2ƍ
¨y
Mx
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
b/2 = 3 ɦb/2 = 3 ɦ
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
q
¨x
F
¨x
¨x
Ɋɢɫ 2.5. Ʉɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɚɹɫɟɬɤɚɞɥɹɪɚɫɱɟɬɚɩɥɢɬɵ
ɉɟɪɟɧɟɫɟɦɡɧɚɦɟɧɚɬɟɥɶ '2x '2y ɢɡɥɟɜɨɣɜ ɩɪɚɜɭɸɱɚɫɬɶ ɞɢɮ
ɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹɢɡɝɢɛɚɢɩɨɞɫɬɚɜɢɜɜɛɢɝɚɪɦɨɧɢ
ɱɟɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚɫɦɪɢɫ 2.2, ɚ) ɤɨɷɮɮɢɰɢɟɧɬɵ ȝ Ș ɢ Ȟ
ɡɚɩɢɲɟɦɷɬɨɬɨɩɟɪɚɬɨɪɜɱɢɫɥɨɜɨɦɜɢɞɟɪɢɫ 2.5, ɛ).
ɉɪɢɦɟɦ ɞɥɹ ɠɟɥɟɡɨɛɟɬɨɧɚ: E = 3,2 Â104 Ɇɉɚ = 3,2 Â107 ɤɇɦ2,
Ȟ = 0,15, ɬɨɝɞɚɰɢɥɢɧɞɪɢɱɟɫɤɚɹɠɟɫɬɤɨɫɬɶɩɥɢɬɵ:
D
E h3
12 (1 Q 2 )
3,2 ˜ 107 ˜ 0,23
12 (1 0,15 2 )
21,82 ˜ 103 ɤɇ ˜ ɦ .
ɉɨɞɫɱɢɬɚɟɦ ɜɟɥɢɱɢɧɵ ɜɯɨɞɹɳɢɟ ɬɟɩɟɪɶ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ
ɭɪɚɜɧɟɧɢɣ
q '2x '2y
D
30 ˜ 2 2 ˜1,52
21,82 ˜ 103
12,37 ˜103 ɦ ;
F '2x '2y
D ' x' y
120 ˜ 2 ˜1,5
21,82 ˜ 103
16,5 ˜103 ɦ .
Ɂɚɩɢɲɟɦ ɪɚɜɟɧɫɬɜɚ ɜɵɬɟɤɚɸɳɢɟ ɢɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɪɚɫ
ɫɦɚɬɪɢɜɚɟɦɨɣ ɩɥɢɬɵ ȼɨ ɜɫɟɯ ɭɡɥɚɯ ɧɚ ɤɨɧɬɭɪɟ ɩɥɢɬɵ ɡɧɚɱɟɧɢɹ
ɩɪɨɝɢɛɨɜ ɪɚɜɧɵ ɧɭɥɸ Ⱦɥɹ ɡɚɤɨɧɬɭɪɧɵɯ ɭɡɥɨɜ ɜɞɨɥɶ ɲɚɪɧɢɪɧɨ
49
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɨɩɟɪɬɵɯ ɫɬɨɪɨɧ w1ƍ = –w1; w2ƍ = –w2, ɜɞɨɥɶ ɡɚɳɟɦɥɟɧɧɵɯ ɫɬɨɪɨɧ
w2Ǝ = w2, w3Ǝ = w3.
ɂɫɩɨɥɶɡɭɹɨɩɟɪɚɬɨɪɪɢɫ 2.5, ɛɫɭɱɟɬɨɦɩɪɢɜɟɞɟɧɧɵɯɜɵɲɟ
ɪɚɜɟɧɫɬɜ ɫɨɫɬɚɜɥɹɟɦ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɜɧɭɬɪɟɧɧɢɯ ɭɡɥɨɜ ɫɟɬɤɢ
1, 2, 3, 4.
ɍɡɟɥ1: 22,04 Âw1 – 6,25 (w2 + w2) – 11,11 Âw4 + 2 (w3 + w3) +
+ 1,78 (w1 – w1) = 12,37 Â10–3.
ɍɡɟɥ2: 22,04 Âw2 – 6,25 Âw1 – 11,11 Âw3 + 2 Âw4 + 0,563 (w2 + w2)
+
+ 1,78 (w2 – w2) = 12,37 Â10–3.
­ w1 ½
°w °
° 2°
® ¾
° w3 °
°¯w4 °¿
­ 8 ,04 ½
° 5 ,79 °
°
° 3
¾ 10 ɦ
®
8
,
48
°
°
°¯11,37 °¿
­ 8 ,04 ½
° 5 ,79 °
°
°
®
¾ ɦɦ .
8
,
48
°
°
°¯11,37 °¿
ɑɢɫɥɨɜɨɟ ɩɨɥɟ ɩɪɨɝɢɛɨɜ w ɜ ɭɡɥɚɯ ɫɟɬɤɢ ɩɪɟɞɫɬɚɜɥɟɧɨ ɧɚ
ɪɢɫ 2.6, ɚɉɨɜɟɪɯɧɨɫɬɶɢɡɨɝɧɭɬɨɣɩɥɢɬɵɢɡɨɛɪɚɠɟɧɚɧɚɪɢɫ 2.6, ɛ.
ɚ)
ɛ)
x
ɍɡɟɥ3: 22,04 Âw3 – 6,25 Âw4 – 11,11 (w2 + w2) + 2 (w1 + w1) +
+ 0,563 (w3 + w3) = (12,37 Â10–3 + 16,5 Â10–3) = 28,87 Â10–3.
ɍɡɟɥ4: 22,04 Âw4 – 6,25 (w3 + w3) – 11,11 (w1 + w1) +
+ 2 (w2 + w2 + w2 + w2) = 12,37 Â10–3.
ɉɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɢɫɬɟɦɭ ɥɢɧɟɣ
ɧɵɯɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜɧɟɧɢɣ:
22,04 Âw1 – 12,5 Âw2 + 4 Âw3 – 11,11 Âw4 = 12,37 Â10–3 ;
0
0
0
0
0
0
5,79
8,04
5,79
0
0
8,48
11,37
8,48
0
0
5,79
8,04
5,79
0
0
0
0
0
0
– 22,22 Âw1 + 8 Âw2 – 12,5 Âw3 + 22,04 Âw4 = 12,37 Â10–3 .
ȼɦɚɬɪɢɱɧɨɣɮɨɪɦɟɫɢɫɬɟɦɚɭɪɚɜɧɟɧɢɣɢɦɟɟɬɜɢɞ
12 ,5
11,11º ­ w1 ½
4 ,0
ª 22 ,04
« 6 ,25
23 ,17 11,11 2 ,0 » °°w2 °°
«
»u® ¾
22 ,22 23 ,17 6 ,25 » ° w3 °
« 4 ,0
«
»
12 ,5 22 ,04 ¼ °¯w4 °¿
8 ,0
¬ 22 ,22
­12 ,37 ½
°12 ,37 °
°
° 3
®
¾ 10 .
°28 ,87 °
°¯12 ,37 °¿
5,8
8,0
5,8
8,5
8,5
5,8
8,0 11,4
w, ɦɦ
y
Ɋɢɫ 2.6. Ɂɧɚɱɟɧɢɹɩɪɨɝɢɛɨɜɜɭɡɥɚɯɫɟɬɤɢ
–6,25 Âw1 + 23,17 Âw2 – 11,11 Âw3 + 2 Âw4 = 12,37 Â10–3 ;
4 Âw1 – 22,22 Âw2 + 23,17 Âw3 – 6,25 Âw4 = 28,87 Â10–3 ;
5,8
ɋɩɨɦɨɳɶɸɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɯɨɩɟɪɚɬɨɪɨɜɜɧɭɬɪɟɧɧɢɯɭɫɢ
ɥɢɣ ɫɦ ɪɢɫ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡɝɢɛɚɸɳɢɟ ɢ ɤɪɭɬɹɳɢɟ
ɦɨɦɟɧɬɵɜɭɡɥɚɯɩɨɫɟɱɟɧɢɹɦɩɥɢɬɵ
ȼɵɱɢɫɥɢɦɡɧɚɱɟɧɢɹɦɧɨɠɢɬɟɥɟɣɜɷɬɢɯɨɩɟɪɚɬɨɪɚɯ
D
'2x
21,82 ˜ 103
22
D (1 Q)
4 ' x' y
5,46 ˜ 103 ɤɇɦ ;
D
'2y
21,82 ˜ 103
1,52
9,7 ˜ 103 ɤɇɦ ;
21,82 ˜ 103 (1 0,15)
1,55 ˜ 103 ɤɇɦ ,
4 ˜ 2 ˜ 1,5
Ɂɚɦɟɬɢɦ ɱɬɨ ɦɚɬɪɢɰɭ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɫɢɦ
ɦɟɬɪɢɱɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɟɫɥɢ ɜɬɨɪɨɟ ɭɪɚɜɧɟ
ɧɢɟɭɦɧɨɠɢɬɶɧɚɚɱɟɬɜɟɪɬɨɟ ɪɚɡɞɟɥɢɬɶ ɧɚ
Ɋɟɲɢɜ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɩɨɥɭɱɢɦ ɜɟɤɬɨɪ ɭɡɥɨɜɵɯ ɩɟɪɟɦɟ
ɳɟɧɢɣɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣɩɥɢɬɵ
ɢɡɚɩɢɲɟɦɨɩɟɪɚɬɨɪɵɫɦɪɢɫ ɜɱɢɫɥɨɜɨɦɜɢɞɟɪɢɫ 2.7).
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɧɚɤɥɚɞɵɜɚɟɦ ɷɬɢ ɨɩɟɪɚɬɨɪɵ ɧɚ ɭɡɥɵ ɫɟɬɤɢ
ɞɥɹ ɫɟɱɟɧɢɣ ɜ ɤɨɬɨɪɵɯ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ
ɉɪɢɜɟɞɟɦ ɜɵɱɢɫɥɟɧɢɟ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ Mx, My ɜ ɭɡɥɟ ɢɤɪɭɬɹɳɟɝɨ ɦɨɦɟɧɬɚ H ɜɭɡɥɚɯ O, P ɢPƍ ɫɦɪɢɫɚ).
50
51
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Mx
–0,267
My
k
–1
2,533
–1
–0,084
H
îÂ3
îÂ3
¨y
¨x
–1
My, ɤɇ Â ɦ/ɦ
0
0
0
0
–63,1
23,8
31,4
23,8
–92,5
38,3
41,3
–63,1
23,8
0
0
–0,084
–1
0
k
2,169
–1
1
Mx, ɤɇ Â ɦ/ɦ
–1
k
îÂ3
–0,267
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
0
0
0
0
0
–63,1
–9,5
32,9
49,3
32,9
–9,5
38,3
–92,5
–13,9
56,7
69,4
56,7
–13,9
31,4
23,8
–63,1
–9,5
32,9
49,3
32,9
–9,5
0
0
0
0
0
0
x
0
y
0
x
0
y
1
H, ɤɇ Â ɦ/ɦ
8,9
24,8
0
–24,8
–8,9
Ɋɢɫ 2.7. Ɉɩɟɪɚɬɨɪɵɞɥɹɜɵɱɢɫɥɟɧɢɹɦɨɦɟɧɬɨɜ
0
17,6
0
–17,6
0
0
17,6
0
–17,6
0
0
–17,6
17,6
0
–8,9
–24,8
24,8
8,9
x
3
Mx1 = 5,46 Â10 (2,533 Âw1 – 1 (w2 + w2) – 0,267 (wO + w4) =
= 5,46 Â103 (2,533 Â8,04 – 1 Â2 Â5,79 – 0,267 Â11,37) = ɤɇ Â ɦɦ
My1 = 9,7 Â103 (2,169 Âw1 – 0,084 (w2 + w2) – 1 (wO + w4) =
= Â3 (2,169 Â8,04 – 0,084 Â2 Â5,79 – 1 Â11,37) = ɤɇ Â ɦɦ
HO = 1,55 Â103 (w2 – w2 + w2ƍ – w2ƍ) = 0.
HP = 1,55 Â103 (w1 – wS – w1ƍ) = 1,55 (8,04 + 8,04) = ɤɇ Â ɦɦ
HPƍ = 1,55 Â103 (wSƍ – w1 + w1ƍ) = 1,55 Â2 (–8,04) = –ɤɇ Â ɦɦ
Ⱥɧɚɥɨɝɢɱɧɨ ɜɵɱɢɫɥɹɸɬɫɹ ɢɡɝɢɛɚɸɳɢɟ ɢ ɤɪɭɬɹɳɢɟ ɦɨɦɟɧɬɵ
ɜ ɨɫɬɚɥɶɧɵɯ ɭɡɥɚɯ ɫɟɬɤɢ ɑɢɫɥɨɜɵɟ ɩɨɥɹ ɜɧɭɬɪɟɧɧɢɯ ɭɫɢɥɢɣ Mx,
My, H ɩɪɟɞɫɬɚɜɥɟɧɵɧɚɪɢɫ 2.8.
Ɂɧɚɱɟɧɢɹ ɨɩɨɪɧɵɯ ɪɟɚɤɰɢɣ ɜ ɭɡɥɚɯ ɩɨ ɡɚɤɪɟɩɥɟɧɧɵɦ ɫɬɨɪɨ
ɧɚɦɩɥɢɬɵɦɨɠɧɨɧɚɣɬɢɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶɭɪɚɜɧɟɧɢɟɦɜɜɢ
ɞɟ ’ 4 w k (qk Vk / ' x ' y ) D , ɨɬɫɸɞɚ Vk ( D ’ 4 w k q ) ' x ' y . Ɍɨ
ɝɞɚɪɚɫɩɪɟɞɟɥɟɧɢɟɪɟɚɤɰɢɣɩɨɤɪɚɹɦɩɚɪɚɥɥɟɥɶɧɵɦɨɫɹɦx ɢy:
Vy
k
Vk ' x
( D ’ 4 w k q k ) ' y ; Vx
52
k
Vk ' y
( D ’ 4 w k qk ) ' x .
0
0
y
Ɋɢɫ 2.8. Ɂɧɚɱɟɧɢɹɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣɜɭɡɥɚɯɫɟɬɤɢ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɧɚɤɥɚɞɵɜɚɹɨɩɟɪɚɬɨɪɪɢɫɛɧɚɭɡɥɵO,
P, R ɜɵɱɢɫɥɹɟɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɩɨɪɧɨɣ ɪɟɚɤɰɢɢ Vy ɩɨ ɜɟɪɯɧɟɦɭ
ɲɚɪɧɢɪɧɨɨɩɟɪɬɨɦɭɤɪɚɸɩɥɢɬɵɋɭɱɟɬɨɦɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɧɚ
ɷɬɨɦɤɪɚɟw1ƍ = –w1; w2ƍ = –w2ɩɨɥɭɱɚɟɦ ’ 4 w O , P , R 0 ɬɨɝɞɚ
VyO = VyP = (0 – q/2) Â ¨y = –15 Â1,5 = –22,5 ɤɇ/ɦ;
VyR = (0 – q/4) ¨y = –7,5 Â1,5 = –11,25 ɤɇ/ɦ.
Ɂɧɚɤɦɢɧɭɫɝɨɜɨɪɢɬɨɬɨɦɱɬɨɷɬɚɪɟɚɤɰɢɹɤɚɤɜɧɟɲɧɹɹɞɥɹ
ɩɥɢɬɵɫɢɥɚɧɚɩɪɚɜɥɟɧɚɩɪɨɬɢɜɨɫɢzɬ ɟɜɜɟɪɯ
Ɋɟɡɭɥɶɬɚɬɵɪɚɫɱɟɬɚɚɢɦɟɧɧɨɷɩɸɪɵɩɪɨɝɢɛɨɜɢɢɡɝɢɛɚɸɳɢɯ
ɦɨɦɟɧɬɨɜMx, My ɞɥɹɫɪɟɞɧɢɯɫɟɱɟɧɢɣɩɥɢɬɵɷɩɸɪɚɤɪɭɬɹɳɢɯɦɨ
53
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɦɟɧɬɨɜH ɜɫɟɱɟɧɢɢɩɨɜɟɪɯɧɟɣɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɣɤɪɨɦɤɟɩɥɢɬɵ
ɚɬɚɤɠɟɪɚɫɩɪɟɞɟɥɟɧɢɟɨɩɨɪɧɵɯɪɟɚɤɰɢɣVy ɩɨɷɬɨɦɭɤɪɚɸ, ɩɪɢɜɨ
ɞɹɬɫɹɧɚɪɢɫ 2.9.
Ɉɛɪɚɬɢɦɜɧɢɦɚɧɢɟɧɚɬɨɱɬɨɟɫɥɢɷɩɸɪɵɩɪɨɝɢɛɨɜɪɟɚɤɰɢɣ
ɢɢɡɝɢɛɚɸɳɢɯɦɨɦɟɧɬɨɜɞɥɹɫɢɦɦɟɬɪɢɱɧɨ ɧɚɝɪɭɠɟɧɧɨɣɩɥɢɬɵɹɜ
ɥɹɸɬɫɹ ɫɢɦɦɟɬɪɢɱɧɵɦɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ ɫɢɦɦɟɬɪɢɢ ɬɨ ɷɩɸɪɵ
ɤɪɭɬɹɳɢɯɦɨɦɟɧɬɨɜɤɨɫɨɫɢɦɦɟɬɪɢɱɧɵ
R
F
S
O
2
1
3
T
Sƍ
P
2
b/2
3
2
a/2
y
x
Rƍ
2
F
4
1
Pƍ
a/2
8,0
0
8,5
92,5
8,5
11,4
49,3
41,3
31,4
11,4
w, ɦɦ
0
31,4
8,0
b/2
69,4
49,3
Mx, ɤɇ Â ɦ/ɦ My, ɤɇ Âɦɦ
41,3
13,9
8,5
My, ɤɇ Â ɦɦ
56,8
0
8,5
24,8
H, ɤɇ Â ɦɦ
Vy, ɤɇɦ
11,25
11,25
22,5
0.
ȿɫɥɢ ɨɞɧɚ ɢɡ ɮɭɧɤɰɢɣ ɧɚɩɪɢɦɟɪ f (x) ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɚ
ɧɭɥɸɬɨɨɧɚɛɭɞɟɬɨɪɬɨɝɨɧɚɥɶɧɚɤɥɸɛɨɣɮɭɧɤɰɢɢ Mi (x) ɇɚɩɪɢ
ɦɟɪ ɮɭɧɤɰɢɹ f ( x) EIwIV q, ɹɜɥɹɸɳɚɹɫɹ ɥɟɜɨɣ ɱɚɫɬɶɸ ɞɢɮɮɟ
ɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɡɨɝɧɭɬɨɣ ɨɫɢ ɛɚɥɤɢ ɞɥɢɧɨɣ l ɬɨɠɞɟ
ɫɬɜɟɧɧɨɪɚɜɧɚɧɭɥɸɩɪɢɜɫɟɯɡɧɚɱɟɧɢɹɯɯɩɨɷɬɨɦɭ
³ ( EIw
IV
q ) ˜ Mi ( x) dx
0.
l
38,3
69,4
24,8
³ f ( x) ˜ M ( x) dx
l1
92,5
13,9
56,8
l2
w, ɦɦ
Mx, ɤɇ Â ɦɦ
38,3
ȼ ɨɫɧɨɜɟ ɦɟɬɨɞɚ ɥɟɠɢɬ ɩɨɧɹɬɢɟ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɭɧɤɰɢɣ
Ⱦɜɟ ɮɭɧɤɰɢɢ: f (x) ɢ M (x) – ɧɚɡɵɜɚɸɬɫɹ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɜ ɢɧ
ɬɟɪɜɚɥɟ l1 d ɯd l2 (ɚɧɚɥɨɝɢɱɧɨɢɜɤɚɤɨɣ-ɥɢɛɨɦɧɨɝɨɦɟɪɧɨɣɨɛɥɚ
ɫɬɢɟɫɥɢɜɵɩɨɥɧɹɟɬɫɹɭɫɥɨɜɢɟ
22,5
ȿɫɥɢ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ w (x) ɩɪɢɛɥɢɠɟɧɧɨ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ wN
IV
i 1
ɭɠɟ ɧɟ ɛɭɞɟɬ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɨ ɧɭɥɸ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟ ɛɭɞɟɬ
ɨɪɬɨɝɨɧɚɥɶɧɨ ɥɸɛɨɣ ɮɭɧɤɰɢɢ Ɍɨɝɞɚ ɩɨɬɪɟɛɭɟɦ ɱɬɨɛɵ ɨɧɨ ɛɵɥɨ
ɨɪɬɨɝɨɧɚɥɶɧɨɩɨɤɪɚɣɧɟɣɦɟɪɟɤɤɚɠɞɨɣɢɡɮɭɧɤɰɢɣ Mi (x) ɫɨɫɬɚɜ
ɥɹɸɳɢɯ wN , ɬ ɟɱɬɨɛɵɜɵɩɨɥɧɹɥɢɫɶɭɫɥɨɜɢɹ
³ ( EIwN
IV
Ɋɢɫ Ɋɟɡɭɥɶɬɚɬɵɪɚɫɱɟɬɚɩɥɢɬɵ
N
¦ D i Mi ( x) , ɬɨ ɜɵɪɚɠɟɧɢɟ ( EIwN q)
q ) ˜ Mi ( x) dx
0, i
1, 2, ... , N .
l
Ɇɟɬɨɞ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɆȻȽ ɨɬɧɨɫɢɬɫɹ ɤ ɱɢɫɥɟɧɧɨɚɧɚɥɢɬɢɱɟɫɤɢɦ ɩɪɨɟɤɰɢɨɧɧɵɦ ɦɟɬɨɞɚɦ ɪɟɲɟɧɢɹ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɗɬɨɬ ɦɟɬɨɞ ɛɵɥ ɩɪɟɞɥɨɠɟɧ
ɜɝɂȽȻɭɛɧɨɜɵɦɢɧɟɡɚɜɢɫɢɦɨɨɬɧɟɝɨɜɝȻȽȽɚ
ɥɺɪɤɢɧɵɦ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ N ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ
ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ N ɤɨɷɮɮɢɰɢɟɧɬɨɜ D i ɜɯɨɞɹɳɢɯ
ɜɩɪɢɛɥɢɠɟɧɧɨɟɪɟɲɟɧɢɟ wN .
Ɏɭɧɤɰɢɢ Mi (x) ɧɚɡɵɜɚɸɬɫɹɛɚɡɢɫɧɵɦɢ, ɢɥɢ ɤɨɨɪɞɢɧɚɬɧɵɦɢ,
ɮɭɧɤɰɢɹɦɢ Ɉɧɢ ɞɨɥɠɧɵ ɛɵɬɶ ɡɚɞɚɧɵ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɱɬɨɛɵ ɭɞɨ
ɜɥɟɬɜɨɪɹɬɶɭɫɥɨɜɢɹɦɧɚɝɪɚɧɢɰɚɯɨɛɥɚɫɬɢɄɪɨɦɟɬɨɝɨ ɫɢɫɬɟɦɚɛɚ
ɡɢɫɧɵɯɮɭɧɤɰɢɣ Mi ( x), i 1, 2, ... , N , ɞɨɥɠɧɚɛɵɬɶɥɢɧɟɣɧɨɧɟɡɚɜɢ
ɫɢɦɨɣ ɬ ɟ ɧɢ ɨɞɧɭ ɢɡ ɮɭɧɤɰɢɣ Mi (x) ɧɟɥɶɡɹ ɜɵɪɚɡɢɬɶ ɥɢɧɟɣɧɨɣ
ɤɨɦɛɢɧɚɰɢɟɣɞɪɭɝɢɯɮɭɧɤɰɢɣɷɬɨɣɠɟɫɢɫɬɟɦɵ
54
55
ɆɟɬɨɞȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ
ɈɫɧɨɜɧɵɟɩɨɥɨɠɟɧɢɹɦɟɬɨɞɚȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɉɪɢɜɟɞɟɧɧɵɟ ɪɚɫɫɭɠɞɟɧɢɹ ɩɪɢɦɟɧɢɦɵ ɢ ɤ ɮɭɧɤɰɢɹɦ ɞɜɭɯ
ɢɛɨɥɟɟɩɟɪɟɦɟɧɧɵɯɍɫɥɨɜɢɟɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɭɧɤɰɢɣ f ( x, y )
ɢ M ( x, y ) ɜɞɜɭɦɟɪɧɨɣɨɛɥɚɫɬɢȍɡɚɩɢɲɟɬɫɹɬɚɤ
³³ f ( x, ɭ) ˜ M ( x, y) dx dy
Ʉɚɠɞɨɟ i-ɟɭɪɚɜɧɟɧɢɟɷɬɨɣɫɢɫɬɟɦɵɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɜɢɞɟ
N
¦
j 1
0.
Ai j D j
Bi ɢɥɢ Ai1 D1 Ai 2 D 2 ... Ai N D N
Bi ,
ɝɞɟɤɨɷɮɮɢɰɢɟɧɬɵɩɪɢɧɟɢɡɜɟɫɬɧɵɯ D j ɢɫɜɨɛɨɞɧɵɟɱɥɟɧɵ
:
ɉɪɟɞɫɬɚɜɢɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ
ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
q
(2.14)
L( w) 0.
D
ɁɞɟɫɶɫɢɦɜɨɥɨɦL ɨɛɨɡɧɚɱɟɧɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣɨɩɟɪɚɬɨɪɭɪɚɜɧɟ
ɧɢɹɢɡɝɢɛɚɬ ɟ L( w) ’ 4 w .
Ȼɭɞɟɦɢɫɤɚɬɶɩɪɢɛɥɢɠɟɧɧɨɟɪɟɲɟɧɢɟɭɪɚɜɧɟɧɢɹɜɜɢɞɟ
N
¦ D j M j ( x, y ) ,
ab
Ai j
³ ³ Mi ˜ L(M j ) dx dy ;
Bi
00
1
D
ab
³ ³ Mi ˜ q ( x, y) dx dy .
00
ȿɫɥɢɧɚɩɥɚɫɬɢɧɭɩɨɦɢɦɨɪɚɫɩɪɟɞɟɥɟɧɧɨɣɧɚɝɪɭɡɤɢ ɞɟɣɫɬɜɭ
ɟɬɫɢɫɬɟɦɚɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯɫɢɥ Fk , k 1, 2, ... , nF ɬɨ
Bi
Bi q Bi F
1
D
ab
³ ³ q ( x, y) ˜ Mi ( x, y) dx dy
00
1
D
nF
¦ Fk ˜ Mi ( xk , yk ) .
k 1
(2.15)
ȼɪɟɡɭɥɶɬɚɬɟɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵɭɪɚɜɧɟɧɢɣ ɧɚɯɨɞɢɦ ɡɧɚɱɟɧɢɹ
ɤɨɷɮɮɢɰɢɟɧɬɨɜ D j ɢ ɬɟɦ ɫɚɦɵɦ ɩɪɢɛɥɢɠɟɧɧɭɸ ɮɭɧɤɰɢɸ ɩɪɨɝɢ
ɝɞɟ D j í ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɞɥɟɠɚɳɢɟ ɨɩɪɟɞɟɥɟɧɢɸ
M j ( x, y ) í ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɟ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ ɜɵɛɪɚɧɧɵɟ
ɬɚɤɢɦɨɛɪɚɡɨɦɱɬɨɛɵɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶɩɪɟɞɩɨɥɚɝɚɟɦɵɟɩɟɪɟɦɟ
ɳɟɧɢɹɜɧɚɩɪɚɜɥɟɧɢɢɨɫɢ z ɢɭɞɨɜɥɟɬɜɨɪɹɬɶɜɫɟɦɤɢɧɟɦɚɬɢɱɟɫɤɢɦ
ɢɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ>1].
ɉɨɞɫɬɚɜɢɜɜɭɪɚɜɧɟɧɢɟɩɨɥɭɱɢɦ
ɛɨɜ wN ( x, y ) . ȿɫɥɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɥɢɧɟɣ
ɧɨɟ ɬɨ ɢ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ D j ɛɭɞɟɬ ɬɚɤɠɟ ɧɟɥɢ
ɧɟɣɧɨɣ
ȼɫɥɭɱɚɟ ɟɫɥɢɞɥɹɡɚɞɚɧɢɹɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹ ɢɫɩɨɥɶ
ɡɭɟɬɫɹɨɞɧɚɛɚɡɢɫɧɚɹ ɮɭɧɤɰɢɹɬ ɟ w D ˜ M ( x, y ) ɬɨɢɫɤɨɦɵɣɤɨ
ɷɮɮɢɰɢɟɧɬ D B A , ɝɞɟ
wN
j 1
N
¦ [ D j ˜ L (M j ) ] j 1
q
D
(2.16)
0.
ɉɨɬɪɟɛɭɟɦɱɬɨɛɵɥɟɜɚɹɱɚɫɬɶɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ
(2.16) ɛɵɥɚ ɨɪɬɨɝɨɧɚɥɶɧɚ ɤɚɠɞɨɣ ɢɡ ɛɚɡɢɫɧɵɯ ɮɭɧɤɰɢɣ ɪɹɞɚ
Mi ( x), i 1, 2, ... , N . ɗɬɢ ɭɫɥɨɜɢɹ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ
ɪɚɡɦɟɪɚɦɢ a×b (0 d ɯd a, 0 d y d b) ɩɪɢɜɨɞɹɬɤɫɥɟɞɭɸɳɟɣɫɢɫɬɟ
ɦɟ ɥɢɧɟɣɧɵɯ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɫɤɨɦɵɯ
ɧɟɢɡɜɟɫɬɧɵɯ D j :
N
ab
j 1
00
1
ab
¦[ D j ³ ³ L(M j ) Mi dx dy ] D ³ ³ q ( x, y ) Mi dx dy
00
i
1, 2, ... , N .
56
0,
(2.17)
ab
A
³ ³ M ˜ L(M) dx dy ;
00
B
1
D
ab
³ ³ q ( x, y) ˜ M dx dy 00
1
D
nF
¦ Fk ˜ M ( xk , yk ) .
k 1
ɏɨɬɹ ɨɩɢɫɚɧɢɟ ɦɟɬɨɞɚ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɩɪɢɜɟɞɟɧɨ ɞɥɹ
ɨɞɧɨ- ɢ ɞɜɭɦɟɪɧɨɣ ɨɛɥɚɫɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɨɧ ɟɫɬɟɫɬɜɟɧɧɨ ɩɪɢ
ɦɟɧɢɦ ɢ ɞɥɹ ɬɪɟɯɦɟɪɧɵɯ ɡɚɞɚɱ ɗɬɨɬ ɦɟɬɨɞ ɩɪɢɦɟɧɢɦ ɬɚɤɠɟ
ɢɤɫɢɫɬɟɦɚɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
ȼ ɷɬɨɣ ɫɜɹɡɢ ɫɥɟɞɭɟɬ ɭɩɨɦɹɧɭɬɶ ɨ ɦɟɬɨɞɚɯ ɪɨɞɫɬɜɟɧɧɵɯ
ɆȻȽɗɬɨɨɛɨɛɳɟɧɧɵɣɦɟɬɨɞȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚɦɟɬɨɞɉɟɬɪɨɜɚ –
Ƚɚɥɺɪɤɢɧɚ ɦɟɬɨɞ Ʉɚɧɬɨɪɨɜɢɱɚ – ȼɥɚɫɨɜɚ ɦɟɬɨɞɵ ɤɨɥɥɨɤɚɰɢɣ
ɜ ɪɚɡɥɢɱɧɵɯ ɜɢɞɚɯ ɜɧɭɬɪɟɧɧɟɣ ɝɪɚɧɢɱɧɨɣ ɤɨɦɛɢɧɢɪɨɜɚɧɧɨɣ
ɤɨɥɥɨɤɚɰɢɢɦɟɬɨɞɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜɢɬɞ
57
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Ɍɚɤ ɜ ɦɟɬɨɞɟ ɉɟɬɪɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ
ɡɚɞɚɟɬɫɹɱɟɪɟɡɨɞɧɭɫɢɫɬɟɦɭɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣɚɭɫɥɨɜɢɹɨɪɬɨɝɨ
ɧɚɥɶɧɨɫɬɢɡɚɩɢɫɵɜɚɸɬɫɹɩɨɨɬɧɨɲɟɧɢɸɤɞɪɭɝɨɣɛɚɡɢɫɧɨɣɫɢɫɬɟɦɟ
ȿɫɥɢɞɜɟɷɬɢɫɢɫɬɟɦɵɫɜɹɡɚɧɵɦɟɠɞɭɫɨɛɨɣɩɨɫɪɟɞɫɬɜɨɦɥɢɧɟɣɧɨɝɨ
ɨɩɟɪɚɬɨɪɚɬɨɞɚɧɧɵɣɦɟɬɨɞɧɚɡɵɜɚɟɬɫɹɦɟɬɨɞɨɦɦɨɦɟɧɬɨɜ.
ȼɦɟɬɨɞɚɯɤɨɥɥɨɤɚɰɢɣ ɧɟɢɡɜɟɫɬɧɵɟɤɨɷɮɮɢɰɢɟɧɬɵ D j ɜɪɚɡ
ɥɨɠɟɧɢɢɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɢɳɭɬɫɹɢɡɭɫɥɨɜɢɣɱɬɨɛɵ
ɷɬɨ ɩɪɢɛɥɢɠɟɧɢɟ ɭɞɨɜɥɟɬɜɨɪɹɥɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
ɜɡɚɞɚɧɧɵɯɬɨɱɤɚɯk (xk, ykɧɚɡɵɜɚɟɦɵɯɭɡɥɚɦɢɤɨɥɥɨɤɚɰɢɢ,
ɩɪɢɷɬɨɦɫɯɨɞɢɦɨɫɬɶɦɟɬɨɞɚɡɚɜɢɫɢɬɨɬɜɵɛɨɪɚɷɬɢɯɭɡɥɨɜ
ɋɭɬɶɦɟɬɨɞɚɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ ɡɚɤɥɸɱɚɟɬɫɹɜɧɚɯɨɠɞɟ
ɧɢɢ ɢɫɤɨɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ D j ɢɡ ɭɫɥɨɜɢɹ ɦɢɧɢɦɢɡɚɰɢɢ ɜ ɡɚ
ɞɚɧɧɵɯ ɬɨɱɤɚɯ k ɧɟɜɹɡɤɢ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɜ ɮɨɪɦɟ
ɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ
2
§N
q ·¸
o min ,
) (D ) ¦ ¨ ¦ [ D j ˜ L (M j ) ] ¨
D ¸¹
k 1©j 1
ɝɞɟnk – ɱɢɫɥɨɬɨɱɟɤɤɨɥɥɨɤɚɰɢɢnk • N).
ɆɟɬɨɞɄɚɧɬɨɪɨɜɢɱɚ – ȼɥɚɫɨɜɚ ɩɨɡɜɨɥɹɟɬɫɜɟɫɬɢɞɜɭɦɟɪɧɭɸ
ɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢ ɬɪɟɯɦɟɪɧɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɞɥɹ ɞɢɮɮɟɪɟɧ
ɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣɜɱɚɫɬɧɵɯɩɪɨɢɡɜɨɞɧɵɯɤɤɪɚɟɜɨɣɡɚɞɚɱɟɞɥɹ
ɫɢɫɬɟɦɵɨɛɵɤɧɨɜɟɧɧɵɯɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣȾɥɹɪɟɲɟ
ɧɢɹ ɬɚɤɢɯ ɫɢɫɬɟɦ ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɦɚɬɟɦɚɬɢɤɟ ɫɭ
ɳɟɫɬɜɭɟɬɞɨɫɬɚɬɨɱɧɨɷɮɮɟɤɬɢɜɧɵɯɦɟɬɨɞɨɜ
ɂɦɟɸɬɫɹ ɢ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɜ ɤɨɬɨɪɵɯ ɡɚɞɚɱɚ ɛɨɥɟɟ ɜɵɫɨɤɨɣ
ɪɚɡɦɟɪɧɨɫɬɢ ɫɜɨɞɢɬɫɹ ɤ ɡɚɞɚɱɟ ɦɟɧɶɲɟɣ ɪɚɡɦɟɪɧɨɫɬɢ ɩɪɚɜɞɚ ɤɚɤ
ɩɪɚɜɢɥɨɡɚɫɱɟɬɭɜɟɥɢɱɟɧɢɹɱɢɫɥɚɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣ
ɌɪɭɞɧɨɫɬɢɜɩɪɢɦɟɧɟɧɢɢɆȻȽɢɞɪɭɝɢɯɩɪɨɟɤɰɢɨɧɧɵɯɩɨɥɭ
ɚɧɚɥɢɬɢɱɟɫɤɢɯɦɟɬɨɞɨɜɜɨɫɧɨɜɧɨɦɫɜɹɡɚɧɵɫɩɨɞɛɨɪɨɦɛɚɡɢɫɧɵɯ
ɮɭɧɤɰɢɣ Mi ɬɚɤ ɤɚɤ ɩɨɫɥɟɞɧɢɟ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɧɟ ɬɨɥɶɤɨ
ɜɫɟɦɤɢɧɟɦɚɬɢɱɟɫɤɢɦɧɨɢɩɨɜɨɡɦɨɠɧɨɫɬɢ) ɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱ
ɧɵɦɭɫɥɨɜɢɹɦ
nk
2.2.2. Ȼɚɡɢɫɧɵɟɮɭɧɤɰɢɢ
Ȼɚɡɢɫɧɵɟɮɭɧɤɰɢɢɤɚɤɭɠɟɝɨɜɨɪɢɥɨɫɶɞɨɥɠɧɵɩɪɟɞɫɬɚɜɥɹɬɶ
ɫɨɛɨɣ ɩɪɟɞɩɨɥɚɝɚɟɦɵɟ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ ɢ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɤɪɚɟ
58
ɜɵɦɭɫɥɨɜɢɹɦɡɚɞɚɱɢȼɤɚɱɟɫɬɜɟɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣɨɛɵɱɧɨɩɪɢ
ɦɟɧɹɸɬɫɹɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɢɫɬɟɩɟɧɧɵɟ ɮɭɧɤɰɢɢ
Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɪɚɡɦɟɪɚɦɢ a×b ɜɵɛɨɪ ɛɚɡɢɫ
ɧɵɯ ɮɭɧɤɰɢɣ ɭɩɪɨɳɚɟɬɫɹ ɬɟɦ ɱɬɨ ɨɧɢ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɜ ɜɢɞɟ
ɩɪɨɢɡɜɟɞɟɧɢɹɞɜɭɯɮɭɧɤɰɢɣɨɬɨɞɧɨɝɨɚɪɝɭɦɟɧɬɚ:
M ( x, y ) M1 ( x) ˜ M2 ( y ) ,
ɝɞɟ M1 ( x) ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɧɚ ɤɪɚɹɯ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢy ɩɪɢɯ 0; a); M2 ( y ) í ɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɧɚ
ɤɪɚɹɯɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢɯ ɩɪɢy = 0; b).
Ɋɚɫɫɦɨɬɪɢɦɩɥɚɫɬɢɧɭɪɢɫ ɡɚɳɟɦɥɟɧɧɭɸ ɩɨɫɬɨɪɨɧɚɦ
ɩɚɪɚɥɥɟɥɶɧɵɦɨɫɢy (ɜɷɬɨɦɫɥɭɱɚɟ ɡɚɞɚɧɵɬɨɥɶɤɨɤɢɧɟɦɚɬɢɱɟɫɤɢɟ
ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɢ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɭɸ ɩɨ ɫɬɨɪɨɧɚɦ ɩɚɪɚɥ
ɥɟɥɶɧɵɦɨɫɢx ɫɦɟɲɚɧɧɵɟɝɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹɉɨɞɛɟɪɟɦɞɥɹɧɟɟ
ɛɚɡɢɫɧɵɟɮɭɧɤɰɢɢɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟɜɩɟɪɜɭɸɨɱɟɪɟɞɶɤɢɧɟɦɚɬɢ
ɱɟɫɤɢɦɢɩɨɜɨɡɦɨɠɧɨɫɬɢɫɬɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦ
w (0) = 0
wƎ (0) = 0
x
0
b
w (b) = 0
wƎ (b) = 0
a
y
y
x
w (a) = 0
wƍ (a) = 0
w (0) = 0
wƍ (0) = 0
Ɋɢɫ 2.10ɉɪɹɦɨɭɝɨɥɶɧɚɹɩɥɚɫɬɢɧɚɫɝɪɚɧɢɱɧɵɦɢɭɫɥɨɜɢɹɦɢ
ɉɨɞɛɟɪɟɦ ɫɧɚɱɚɥɚɮɭɧɤɰɢɸ M1 ( x) ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸɡɚɳɟɦ
ɥɟɧɢɸ ɩɨɤɪɚɹɦ ɯ 0, aɁɞɟɫɶɩɨɞɨɣɞɟɬɧɚɩɪɢɦɟɪɬɪɢɝɨɧɨɦɟɬɪɢ
ɱɟɫɤɚɹɮɭɧɤɰɢɹɜɢɞɚ
Sx ·
§
M1 ( x) ¨1 cos 2m ¸ ;
a ¹
©
M1c ( x)
59
2mS
Sx
˜ sin 2m ,
a
a
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɝɞɟ m – ɰɟɥɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɩɨɞɫɬɚɜɢɜ
ɜɷɬɭ ɮɭɧɤɰɢɸɢɟɟɩɪɨɢɡɜɨɞɧɭɸɡɧɚɱɟɧɢɹ ɯ= 0 ɢ ɯ= a, ɩɨɥɭɱɢɦ
M1 (0)
1 cos 0
M1 (a )
1 cos 2mS
2mS
˜ sin 0 0 ;
a
2mS
M1c (a )
˜ sin 2mS 0 .
a
M1c (0)
(1 1) 0 ;
(1 1) 0 ;
Ɍɟɩɟɪɶ ɩɨɞɛɟɪɟɦ ɮɭɧɤɰɢɸ M2 ( y ) ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ ɲɚɪ
ɧɢɪɧɨɦɭɨɩɢɪɚɧɢɸɩɨɤɪɚɹɦ y = 0; b. ȼ ɷɬɨɦɫɥɭɱɚɟ ɧɚɢɛɨɥɟɟɩɨɞ
ɯɨɞɹɳɟɣɹɜɥɹɟɬɫɹɮɭɧɤɰɢɹɫɢɧɭɫɚ
Sy
M2 ( y ) sin m ;
b
Sy
mS
2 sin m .
b
b
2
Mc2c ( y )
2
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɩɪɢ y = 0; b:
M2 (0) sin 0 0 ;
Mc2 (0)
M2 (b) sin mS 0 ; Mc2 (b)
mS
m 2 S2
cos 0 z 0 ; Mc2c (0) 2 sin 0 0 ;
b
b
mS
m 2 S2
cos mS z 0 ; Mc2c (b) 2 sin mS 0 .
b
b
ȼ ɢɬɨɝɟ ɛɚɡɢɫɧɚɹ ɮɭɧɤɰɢɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɝɨ ɜɢɞɚ ɞɥɹ
ɩɪɢɜɟɞɟɧɧɵɯɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɛɭɞɟɬɢɦɟɬɶ ɜɢɞ
Sy
Sx ·
§
M ( x, y ) ¨1 cos 2m ¸ ˜ sin m .
a
b
¹
©
ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ ɩɪɢɛɥɢɠɟɧɧɚɹ ɮɭɧɤɰɢɹ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ
w ( x, y ) ɩɪɢ ɤɨɧɟɱɧɨɦ ɱɢɫɥɟ ɱɥɟɧɨɜ ɜ ɪɚɡɥɨɠɟɧɢɢ ɦɨɠɟɬ
ɛɵɬɶɡɚɞɚɧɚɫɥɟɞɭɸɳɢɦɨɛɪɚɡɨɦ
w ( x, y )
N
N
ª
§
¦ D m Mm ( x, y) ¦ «D m ¨©1 cos 2m
m 1
m 1¬
Sx ·
Sy º
¸ sin m » .
a ¹
b¼
ɇɚɩɪɢɦɟɪɩɪɢɞɜɭɯɱɥɟɧɚɯɫɭɦɦɵɜɪɚɡɥɨɠɟɧɢɢ
Sx ·
Sy §
Sx ·
Sy
§
w ( x, y ) ¨1 cos 2 ¸ ˜ sin ¨1 cos 4 ¸ ˜ sin 2 .
a ¹
b ©
a ¹
b
©
60
Ⱥɧɚɥɨɝɢɱɧɨɩɨɞɛɟɪɟɦ ɛɚɡɢɫɧɭɸ ɮɭɧɤɰɢɸ ɫɬɟɩɟɧɧɨɝɨ (ɩɨɥɢ
ɧɨɦɢɚɥɶɧɨɝɨ) ɜɢɞɚ. Ⱦɥɹ M1 ( x) ɡɚɞɚɞɢɦɫɹɫɥɟɞɭɸɳɢɦɩɨɥɢɧɨɦɨɦ
M1 ( x)
x 2 (a x) 2 ;
M1c ( x)
2a 2 x 6ax 2 4 x 3 .
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣɩɪɢ ɯ 0; a:
M1 (0) 0 (a 0) 2
M1 (a )
0;
a (a a)
2
2
M1c (0) 0 0 0 0 ;
0 ; M1c (a )
2a 3 6a 3 4a 3
0.
ȼɤɚɱɟɫɬɜɟ ɮɭɧɤɰɢɢ M2 ( y ) ɩɪɢɦɟɦɩɨɥɢɧɨɦ
M2 ( y )
y (b 2 y 2 ) ;
Mc2c ( y )
6 y .
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ ɩɪɢ y = 0; b:
M2 (0) 0 (b 2 0 2 ) 0 ; Mc2 (0) b 2 z 0;
M2 (b) b (b b ) 0 ;
2
2
Mc2 (b)
Mc2c (0) 0 ;
2b z 0; Mc2c (b)
2
6b z 0 .
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɛɚɡɢɫɧɚɹ ɮɭɧɤɰɢɹ M ( x, y ) M1 ( x) ˜ M2 ( y ) ɛɭ
ɞɟɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɛɨɢɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ
ɢɨɞɧɨɦɭɢɡɞɜɭɯɫɬɚɬɢɱɟɫɤɨɦɭɗɬɨɞɨɩɭɫɬɢɦɨɬɚɤɤɚɤɩɪɢɪɚɫ
ɱɟɬɟ ɩɥɚɫɬɢɧɵ ɧɚ ɢɡɝɢɛ ɜɵɩɨɥɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɭɫɥɨɜɢɣ ɹɜ
ɥɹɟɬɫɹɨɛɹɡɚɬɟɥɶɧɵɦɚɫɬɚɬɢɱɟɫɤɢɯ– ɧɟɨɛɹɡɚɬɟɥɶɧɵɦɨɞɧɚɤɨɠɟ
ɥɚɬɟɥɶɧɵɦɞɥɹɩɨɥɭɱɟɧɢɹɛɨɥɟɟɬɨɱɧɵɯɪɟɡɭɥɶɬɚɬɨɜ
ɉɨɥɭɱɟɧɧɚɹɫɬɟɩɟɧɧɚɹɛɚɡɢɫɧɚɹɮɭɧɤɰɢɹ ɛɭɞɟɬɢɦɟɬɶɜɢɞ
M ( x, y )
x 2 (a x) 2 y (b 2 y 2 ) .
Ɉɱɟɜɢɞɧɨɱɬɨɩɪɢɜɡɚɢɦɧɨɣɡɚɦɟɧɟɭɫɥɨɜɢɣɡɚɤɪɟɩɥɟɧɢɹɩɨɤɪɚɹɦ
ɩɥɚɫɬɢɧɵɫɥɟɞɭɟɬɩɨɦɟɧɹɬɶɦɟɫɬɚɦɢɜɟɥɢɱɢɧɵɯ, aɢy, b).
Ɍɚɤɢɦ ɠɟ ɨɛɪɚɡɨɦ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ
ɢɞɥɹɞɪɭɝɢɯɫɥɭɱɚɟɜɡɚɤɪɟɩɥɟɧɢɹɤɪɚɟɜ ɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ
ɇɟɤɨɬɨɪɵɟ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɢ ɫɬɟɩɟɧɧɵɟ ɮɭɧɤɰɢɢ M1 ( x) ɞɥɹ
ɪɚɡɥɢɱɧɵɯ ɫɥɭɱɚɟɜ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɟɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢ y
(ɬ ɟ ɩɪɢɯ 0; a), ɩɪɢɜɟɞɟɧɵɜɬɚɛɥ. ȼɟɥɢɱɢɧɚm – ɰɟɥɨɟɩɨ
ɥɨɠɢɬɟɥɶɧɨɟɱɢɫɥɨm = 1«’
Ʉɚɤ ɦɨɠɧɨ ɜɢɞɟɬɶ ɩɪɢɜɟɞɟɧɧɵɟ ɜ ɬɚɛɥɢɰɟ ɮɭɧɤɰɢɢ ɭɞɨɜɥɟ
ɬɜɨɪɹɸɬ ɜɫɟɦ ɤɢɧɟɦɚɬɢɱɟɫɤɢɦ ɢ ɩɨ ɛɨɥɶɲɟɣ ɱɚɫɬɢ ɫɬɚɬɢɱɟɫɤɢɦ
ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ Ⱦɥɹ ɤɪɚɟɜ ɩɥɚɫɬɢɧɵ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɢ ɯ
ɩɪɢy = 0; b), ɜɦɟɫɬɨ M1 ( x) ɩɪɢɧɢɦɚɟɬɫɹ M2 ( y ) ɚɜɟɥɢɱɢɧɵɯ, a)
ɡɚɦɟɧɹɸɬɫɹɧɚy, b).
61
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Ɍɚɛɥɢɰɚ2.1
Ɍɢɩɡɚɤɪɟɩ
ɥɟɧɢɟɤɪɚɟɜ
Ɂɚɞɟɥɤɚ–
ɡɚɞɟɥɤɚ
Ɂ– Ɂ
Ɂɚɞɟɥɤɚ–
ɨɩɢɪɚɧɢɟ
Ɂ– Ɉ
Ɉɩɢɪɚɧɢɟ–
ɡɚɞɟɥɤɚ
Ɉ– Ɂ
Ɉɩɢɪɚɧɢɟ–
ɨɩɢɪɚɧɢɟ
Ɉ– Ɉ
Ɂɚɞɟɥɤɚ–
ɫɜɨɛɤɪɚɣ
Ɂ– ɋ
Ɉɩɢɪɚɧɢɟ–
ɫɜɨɛɤɪɚɣ
Ɉ– ɋ
ɋɯɟɦɚ
ɡɚɤɪɟɩɥɟɧɢɹ
Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟ
ɫɤɚɹɮɭɧɤɰɢɹ
0
x
0
x
0
x
1 cos 2m
1
ɋɬɟɩɟɧɧɚɹ
ɛɚɡɢɫɧɚɹɮɭɧɤɰɢɹ
Sx
a
x 2 (a x) 2
x
3Sx
cos
a
2a
x 2 (a x)
3Sx
x
2a
a
x (a x) 2
sin
b/2 = 3 ɦb/2 = 3 ɦ
Ɉɫɧɨɜɧɵɟɛɚɡɢɫɧɵɟɮɭɧɤɰɢɢɞɥɹɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ
a/2 = 4 ɦa/2 = 4 ɦ
y
F = 120 ɤɇ
3ɦ
0
0
x
x
sin m
Sx
a
Sx
1 cos
2a
q = 30 ɤɇɦ2
3ɦ
x
sin
Sx
2a
ɇɟɢɡɜɟɫɬɧɵɣɤɨɷɮɮɢɰɢɟɧɬ D
x ( 4a 2 x 2 )
ab
A
³ ³ M ˜ L(M) dx dy ; B
Bq BF
ȼɵɩɨɥɧɢɦ ɪɚɫɱɟɬ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɪɚɧɟɟ (ɜ ɩ ) ɩɪɹɦɨ
ɭɝɨɥɶɧɨɣɩɥɢɬɵ ɪɚɡɦɟɪɚɦɢîɦ ɬɨɥɳɢɧɨɣ h = 0,2 ɦ ɪɢɫ 2.11).
Ɏɭɧɤɰɢɸɩɪɨɝɢɛɨɜɩɥɢɬɵɡɚɞɚɞɢɦɜ ɬɚɤɨɦ ɜɢɞɟ
ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɛɚɡɢɫɧɨɣ ɮɭɧɤɰɢɢ ijx, y ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɞɜɭɯ
ɮɭɧɤɰɢɣɨɞɧɨɝɨɚɪɝɭɦɟɧɬɚɫɭɳɟɫɬɜɟɧɧɨɭɩɪɨɫɬɢɬɪɟɲɟɧɢɟɡɚɞɚɱɢ
62
ab
³ ³ q ˜ M dx dy 00
F
D
2
¦ M ( xk , y k ) .
k 1
§ w 4M
w 4M
w 4M ·
¨¨ 4 2 2 2 4 ¸¸ M1IV M 2 2M1cc Mc2c M1 MIV2
wx w y
wy ¹
© wx
4
4
4
2Sx
2Sx
Sy
2Sx ·
Sy 8S
Sy § S · §
§ 2S ·
sin 2 2 cos
sin ¨ ¸ ¨1 cos
¨ ¸ cos
¸ sin .
a
a
b
a
b
a
b
b
a
b
¹
© ¹ ©
© ¹
ɈɩɪɟɞɟɥɹɟɦɜɟɥɢɱɢɧɭA:
ɝɞɟɡɚ M1 ( x) , M2 ( y ) ɩɪɢɦɟɦɫɥɟɞɭɸɳɢɟɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟɮɭɧɤɰɢɢ:
Sy
M2 ( y ) sin .
b
1
D
’ 4M
w D ˜ M ( x, y ) D ˜ M1 ( x) ˜ M2 ( y ) ,
2Sx ·
§
M1 ( x) ¨1 cos
¸;
a ¹
©
B A , ɝɞɟ
ɇɚɯɨɞɢɦɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣɨɩɟɪɚɬɨɪL(ij):
L(M)
2.2.3. ɉɪɢɦɟɪɪɚɫɱɟɬɚɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ
3ɦ
Ɋɢɫ 2.11. ɋɯɟɦɚɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɢɬɵ
x (a 2 x 2 )
x2
F = 120 ɤɇ
3ɦ
00
0
x
ab
A
³³
00
ª § 2S · 4
2Sx §
2Sx · 2 Sy º
»
¨1 cos
¸ sin
« ¨ ¸ cos
a ©
a ¹
b »
« © a ¹
« 8S 4
2Sx §
2Sx · 2 Sy »
« 2 2 cos
» dx dy .
¨1 cos
¸ sin
a ©
a ¹
b
»
« ab
»
«
4
2
»
« §¨ S ·¸ §¨1 cos 2Sx ·¸ sin 2 Sy
»¼
«¬ © b ¹ ©
a ¹
b
63
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ɇɚɯɨɞɢɦɡɧɚɱɟɧɢɟɤɨɷɮɮɢɰɢɟɧɬɚ D :
ȼɵɱɢɫɥɹɟɦɢɧɬɟɝɪɚɥɵ:
a
³ cos
0
2Sx
dx
a
2Sx
dx
a
a
2
³ cos
0
b
³ sin
2
0
2Sx
a
sin
2S
a
Sy
dy
b
D
a
0;
4Sx ·
a § 2Sx 1
sin
¨
¸
2S © 2a 4
a ¹
b
0
a
a
;
2
0
b
.
2
4
4
8S 4 ab § S · §
a· b
§ 2S · ab
2 2
¨ ¸ ¨a ¸
¨ ¸
2¹ 2
ab 4
©b¹ ©
© a ¹ 4
8
3·
S 4 § 16
ab ¨ 4 2 2 4 ¸ .
4
ab
b ¹
©a
ɉɪɢɡɚɞɚɧɧɵɯɡɧɚɱɟɧɢɹɯ a = 8 ɦ b = 6 ɦ ɜɵɱɢɫɥɹɟɦA ɦ2).
Ɉɩɪɟɞɟɥɹɟɦɜɟɥɢɱɢɧɭ Bq :
Bq
q
D
ab
§
³ ³ ¨©1 cos
00
Sy
2Sx ·
¸ sin dx dy
a ¹
b
2Sx · b
Sy
q a§
¨1 cos
¸ dx ³ sin dy.
³
D 0©
a ¹ 0
b
ɂɧɬɟɝɪɢɪɭɹɧɚɯɨɞɢɦ:
2Sx ·
§
³ ¨©1 cos a ¸¹ dx
0
a
Sy
³ sin b dy
0
a §
a
2Sx ·
sin
¸
¨x
a ¹
2S ©
2S
b
Sy
cos
b
S
b
b
0
a
a;
0
ª §a b·
§ 3a b ·º
«M ¨© 4 , 2 ¸¹ M ¨© 4 , 2 ¸¹»
¼
¬
F ª§
3S ·
Sº
S §
S·
¨1 cos ¸ sin ¨1 cos ¸ sin »
D «¬©
2¹
2 ©
2¹
2¼
F
D
2F
.
D
ɉɪɢɡɚɞɚɧɧɵɯɡɧɚɱɟɧɢɹɯ a, b, F ɢD = 21,82 Â103 ɤɇ Â ɦɜɵɱɢɫɥɹɟɦ:
B
Bq B F
q 2b 2 F
a D S
D
64
Sx ·
Sy
§
4 ,678 ˜ 10 3 ¨1 cos 2 ¸ ˜ sin (ɦ) .
a ¹
b
©
ȼɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɜ ɩɥɚɫɬɢɧɟ ɧɚɯɨɞɢɦ ɩɨ ɢɡɜɟɫɬɧɵɦ ɡɚɜɢ
ɫɢɦɨɫɬɹɦɩɨɞɫɬɚɜɥɹɹɜɧɢɯɧɚɣɞɟɧɧɨɟɜɵɪɚɠɟɧɢɟ w (x, y):
§ w2w
w2w ·
D ¨¨ 2 Q 2 ¸¸
wy ¹
© wx
DD M1cc M2 Q M1 Mc2c § 4S 2
Sy
S2 §
Sy ·
2Sx
2Sx ·
D D ¨¨ 2 cos
Q 2 ¨1 cos
sin
¸ sin ¸¸ ;
a
b
a ¹
b ¹
b ©
© a
§ w2w
w2w ·
Ɇ y D ¨¨ 2 Q 2 ¸¸ DD M1 Mc2c Q M1cc M2 wx ¹
© wy
§ S2 §
Sy ·
2Sx ·
4S 2
2Sx
Sy
sin
cos
sin ¸¸ ;
D D ¨¨ 2 ¨1 cos
Q
¸
2
a ¹
b
a
b ¹
a
© b ©
2
2S
2Sx
Sy
sin
cos .
H D D (1 Q) M1c Mc2 DD (1 Q)
ab
a
b
ɉɨɞɫɬɚɜɢɜɡɧɚɱɟɧɢɹ D, D, Q 0,15 ɢɭɩɪɨɫɬɢɜɩɨɥɭɱɢɦ
Sx
Sy ·
§ Sy
4,197 ¨ sin
16 cos sin ¸ ;
6
4
6 ¹
©
Sx
Sy ·
§ Sy
Ɇ y 27,96 ¨ sin
1,325 cos sin ¸ ;
6
4
6 ¹
©
Sx
Sy
H 35,68 sin cos .
4
6
Ɂɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɹ ɩɪɨɝɢɛɨɜ ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɞɥɹ
ɫɟɱɟɧɢɣɩɪɨɯɨɞɹɳɢɯɱɟɪɟɡɰɟɧɬɪ ɩɥɚɫɬɢɧɵ.
Ɇx
2b
.
S
ɉɨɞɫɬɚɜɢɜɢɧɬɟɝɪɚɥɵɩɨɥɭɱɢɦ:
BF
4 ,678 ˜ 10 3 (ɦ) .
w ( x, y )
Ɇx
ɉɨɫɥɟɩɪɢɜɟɞɟɧɢɹɩɨɞɨɛɧɵɯɱɥɟɧɨɜɩɨɥɭɱɢɦ
A
0,053 11,33
Ɍɚɤɢɦɨɛɪɚɡɨɦɮɭɧɤɰɢɹɩɪɨɝɢɛɨɜɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢ
0
2Sy ·
b § Sy 1
¨ sin
¸
S © 2b 4
b ¹
B A
0,042 0,011 0,053 (1 / ɦ) .
Sx ·
§
4 ,68 ¨1 cos ¸ (ɦɦ ) ;
4¹
©
Sx ·
Sx ·
§
§
4,2 ¨1 16 cos ¸ ; Ɇ y 28,0 ¨1 1,325 cos ¸ (ɤɇ ˜ ɦɦ ).
4¹
4¹
©
©
ɉɪɢ y = b/2: w ( x)
Ɇx
65
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Sy
(ɦɦ ) ;
6
Sy
65,0 sin
(ɤɇ ˜ ɦɦ ).
6
Ʉɨɧɬɪɨɥɶɧɵɟɜɨɩɪɨɫɵ
ɉɪɢ x = a/2: w ( y ) 9 ,36 sin
Sy
; Ɇy
6
ɉɨɧɹɬɧɨɱɬɨɨɝɪɚɧɢɱɢɜɲɢɫɶɨɞɧɢɦɱɥɟɧɨɦɜɡɚɞɚɧɢɢ ɮɭɧɤ
ɰɢɢ ɩɪɨɝɢɛɨɜɦɵɧɟ ɩɨɥɭɱɢɦɪɟɲɟɧɢɟɯɨɪɨɲɟɣɬɨɱɧɨɫɬɢɄɪɨɦɟ
ɬɨɝɨɚɩɩɪɨɤɫɢɦɚɰɢɹɪɟɲɟɧɢɹɫɩɨɦɨɳɶɸɧɟɩɪɟɪɵɜɧɨɝɥɚɞɤɢɯɛɚ
ɡɢɫɧɵɯ ɮɭɧɤɰɢɣ ɧɟ ɦɨɠɟɬ ɜ ɞɨɥɠɧɨɣ ɦɟɪɟ ɭɱɢɬɵɜɚɬɶ ɫɨɫɪɟɞɨɬɨ
ɱɟɧɧɵɟɧɚɝɪɭɡɤɢɚɬɚɤɠɟɫɥɨɠɧɵɟɤɪɚɟɜɵɟɭɫɥɨɜɢɹ
ȼ ɷɬɨɣ ɫɜɹɡɢ ɦɟɬɨɞ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɩɨ ɛɨɥɶɲɟɣ ɱɚɫɬɢ
ɢɫɩɨɥɶɡɭɟɬɫɹɤɚɤɜɫɩɨɦɨɝɚɬɟɥɶɧɵɣɦɟɬɨɞɩɨɡɜɨɥɹɸɳɢɣɞɨɫɬɚɬɨɱ
ɧɨɩɪɨɫɬɨɩɨɥɭɱɢɬɶɩɪɢɛɥɢɠɟɧɧɵɟɚɧɚɥɢɬɢɱɟɫɤɢɟɜɵɪɚɠɟɧɢɹɞɥɹ
ɩɪɨɝɢɛɨɜɢɜɧɭɬɪɟɧɧɢɯɭɫɢɥɢɣɨɝɪɚɧɢɱɢɜɲɢɫɶɥɢɲɶɨɞɧɢɦ-ɞɜɭɦɹ
ɱɥɟɧɚɦɢ ɜ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ Ɍɚɤɨɟ ɪɟɲɟɧɢɟ
ɦɨɠɟɬɛɵɬɶɢɫɩɨɥɶɡɨɜɚɧɨɜɤɚɱɟɫɬɜɟɩɪɟɞɜɚɪɢɬɟɥɶɧɨɣɨɰɟɧɤɢɧɚɩɪɹ
ɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨɫɨɫɬɨɹɧɢɹɩɥɚɫɬɢɧɵɢɞɥɹɤɨɧɬɪɨɥɹɪɟɲɟ
ɧɢɹɩɨɥɭɱɟɧɧɨɝɨɛɨɥɟɟɬɨɱɧɵɦ (ɧɨɢɫɥɨɠɧɵɦ) ɦɟɬɨɞɨɦ
1. ɉɪɢɜɟɞɢɬɟ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɩɨ ɬɢ
ɩɭɪɚɡɪɟɲɚɸɳɢɯɭɪɚɜɧɟɧɢɣ.
2. ɇɚɡɨɜɢɬɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
ɞɥɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯɭɪɚɜɧɟɧɢɣ
3. Ʉɚɤɢɟ ɢɡ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɨɬɧɨɫɹɬɫɹ ɤ ɩɪɹɦɵɦ ɜɚ
ɪɢɚɰɢɨɧɧɵɦɦɟɬɨɞɚɦ"
4. Ⱦɚɣɬɟ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɦɟɬɨɞɨɜ ɩɨ ɫɯɟɦɟ ɩɨɫɬɪɨɟɧɢɹ ɩɪɢ
ɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɡɚɞɚɱɢ
5. Ʉɚɤɢɟ ɢɡ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɨɬɧɨɫɹɬɫɹ ɤ ɩɪɨɟɤɰɢɨɧ
ɧɵɦɦɟɬɨɞɚɦ"
6. Ⱦɚɣɬɟɤɥɚɫɫɢɮɢɤɚɰɢɸɩɪɢɛɥɢɠɟɧɧɵɯɦɟɬɨɞɨɜɩɨɜɢɞɭɢɫ
ɤɨɦɨɝɨɪɟɲɟɧɢɹ
7. Ʉɚɤɢɟ ɢɡ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɨɬɧɨɫɹɬɫɹ ɤ ɱɢɫɥɟɧɧɵɦ
ɦɟɬɨɞɚɦ"
8. ɉɪɢɜɟɞɢɬɟ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɩɨ ɬɢ
ɩɭɨɫɧɨɜɧɵɯɧɟɢɡɜɟɫɬɧɵɯ.
9. ȼɱɟɦɫɭɬɶɢɞɟɹɦɟɬɨɞɚɤɨɧɟɱɧɵɯɪɚɡɧɨɫɬɟɣɆɄɊ"
10. ɑɬɨ ɩɨɧɢɦɚɟɬɫɹ ɩɨɞ ɭɡɥɚɦɢ ɢ ɭɡɥɨɜɵɦɢ ɧɟɢɡɜɟɫɬɧɵɦɢ
ɜɆɄɊ"
11. ɑɬɨɧɚɡɵɜɚɟɬɫɹɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɵɦɨɩɟɪɚɬɨɪɨɦɜɆɄɊ"
12. ɉɪɢɜɟɞɢɬɟɪɚɡɧɨɫɬɧɭɸɫɯɟɦɭɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨ
ɪɚɅɚɩɥɚɫɚɞɥɹɤɜɚɞɪɚɬɧɨɣɫɟɬɤɢ
13. ɁɚɩɢɲɢɬɟɭɪɚɜɧɟɧɢɟɆɄɊɞɥɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɵɧɚɭɩɪɭ
ɝɨɦɨɫɧɨɜɚɧɢɢ.
14. Ʉɚɤɨɣ ɜɢɞ ɢɦɟɸɬ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɜ ɆɄɊ ɞɥɹ ɡɚɳɟɦ
ɥɟɧɧɨɝɨɢɨɩɟɪɬɨɝɨɤɪɚɟɜ?
15. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɆɄɊ ɢɫɤɥɸɱɚɸɬɫɹ ɡɚɤɨɧɬɭɪɧɵɟ ɭɡɥɵ
ɩɪɢɡɚɞɟɥɤɟɢɩɪɢɨɩɟɪɬɨɦɤɪɚɟ"
16. Ʉɚɤɨɣ ɜɢɞ ɢɦɟɸɬ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɜ ɆɄɊ ɞɥɹ ɫɜɨɛɨɞ
ɧɨɝɨɤɪɚɹɩɥɚɫɬɢɧɵ"
17. ɋɤɨɥɶɤɨɡɚɤɨɧɬɭɪɧɵɯɭɡɥɨɜɜɜɨɞɢɬɫɹɩɪɢɪɚɫɱɟɬɟɩɥɚɫɬɢ
ɧɵɫɨɫɜɨɛɨɞɧɵɦɤɪɚɟɦ"
18. ȼɱɟɦɫɭɬɶɩɪɨɟɤɰɢɨɧɧɵɯɦɟɬɨɞɨɜɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟ
ɧɢɹɡɚɞɚɱ"
19. ɄɚɤɨɟɭɫɥɨɜɢɟɥɟɠɢɬɜɨɫɧɨɜɟɦɟɬɨɞɚȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢ
ɧɚ ɆȻȽ"
66
67
Ɇx
71,4 sin
ɋɩɨɦɨɳɶɸɩɨɥɭɱɟɧɧɵɯɜɵɪɚɠɟɧɢɣɧɚɯɨɞɢɦɱɢɫɥɟɧɧɵɟɡɧɚ
ɱɟɧɢɹ ɢ ɫɬɪɨɢɦ ɷɩɸɪɵ ɩɪɨɝɢɛɨɜ ɢ ɢɡɝɢɛɚɸɳɢɯ ɦɨɦɟɧɬɨɜ ɞɥɹ
ɫɪɟɞɧɢɯ ɫɟɱɟɧɢɣ ɩɥɢɬɵ ɪɢɫ 2.12). ɇɚɩɪɹɠɟɧɢɹ ɜ ɩɥɢɬɟ ɦɨɠɧɨ
ɧɚɣɬɢɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶɮɨɪɦɭɥɚɦɢ
x
F
F
b/2
a/2
y
6,6
a/2
9,4
w, ɦɦ
0
0
4,7
4,7
9,4
63
4,2
46
50,5
6,6
b/2
71,4
50,5
65
46
Mx, ɤɇ Â ɦ/ɦ My, ɤɇ Â ɦɦ
w, ɦɦ
63
Mx, ɤɇ Â ɦ/ɦ
4,2
71,4
9,1
9,1
28
65
28
My, ɤɇ Â ɦ/ɦ
Ɋɢɫ 2.12. ɉɪɨɝɢɛɵɢɭɫɢɥɢɹɜɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɢɬɟ
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
20. ɑɬɨ ɬɚɤɨɟ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ ɤɚɤɢɟ ɬɪɟɛɨɜɚɧɢɹ ɤ ɧɢɦ
ɩɪɟɞɴɹɜɥɹɸɬɫɹ"
21. ȼɤɚɤɨɦɜɢɞɟɡɚɞɚɟɬɫɹɮɭɧɤɰɢɹɩɪɨɝɢɛɨɜɜɦɟɬɨɞɟȻɭɛɧɨ
ɜɚ – Ƚɚɥɺɪɤɢɧɚ?
22. ɇɚɡɨɜɢɬɟ ɦɟɬɨɞɵ ɪɨɞɫɬɜɟɧɧɵɟ ɦɟɬɨɞɭ Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪ
ɤɢɧɚ.
23. ɑɟɦ ɦɟɬɨɞ ɉɟɬɪɨɜɚ – Ƚɚɥɺɪɤɢɧɚ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɦɟɬɨɞɚ
Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ?
24. ȼɱɟɦɡɚɤɥɸɱɚɟɬɫɹɫɭɬɶɦɟɬɨɞɚɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ"
25. ɑɟɦ ɨɬɥɢɱɚɟɬɫɹ ɦɟɬɨɞ Ʉɚɧɬɨɪɨɜɢɱɚ – ȼɥɚɫɨɜɚ ɨɬ ɦɟɬɨɞɚ
Ȼɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ?
26. Ʉɚɤɨɝɨɜɢɞɚɮɭɧɤɰɢɢɨɛɵɱɧɨɩɪɢɦɟɧɹɸɬɫɹɜɤɚɱɟɫɬɜɟɛɚ
ɡɢɫɧɵɯɮɭɧɤɰɢɣ"
27. Ʉɚɤɢɦ ɭɫɥɨɜɢɹɦ ɢ ɤɚɤ ɫɬɪɨɝɨ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɛɚ
ɡɢɫɧɵɟɮɭɧɤɰɢɢɜɦɟɬɨɞɟȻɭɛɧɨɜɚ – Ƚɚɥɺɪɤɢɧɚ?
28. ɉɪɢɜɟɞɢɬɟ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ ɞɥɹ ɫɥɭɱɚɹ ɡɚɳɟɦɥɟɧɢɹ
ɨɛɨɢɯɤɪɚɟɜɩɥɚɫɬɢɧɵ
29. Ɂɚɩɢɲɢɬɟɛɚɡɢɫɧɵɟɮɭɧɤɰɢɢɞɥɹɫɥɭɱɚɹɲɚɪɧɢɪɧɨɝɨɨɩɢ
ɪɚɧɢɹɤɪɚɟɜɩɥɚɫɬɢɧɵ
30. ɉɪɢɜɟɞɢɬɟ ɛɚɡɢɫɧɵɟ ɮɭɧɤɰɢɢ ɞɥɹ ɫɥɭɱɚɹ ɡɚɳɟɦɥɟɧɢɹ
ɢɨɩɢɪɚɧɢɹɤɪɚɟɜɩɥɚɫɬɢɧɵ
68
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
Ƚɥɚɜɚ 3. ȼȺɊɂȺɐɂɈɇɇɕȿɆȿɌɈȾɕɊȿɒȿɇɂəɁȺȾȺɑ
Ȼɨɥɶɲɭɸ ɝɪɭɩɩɭ ɩɪɢɛɥɢɠɟɧɧɵɯ ɦɟɬɨɞɨɜ ɫɨɫɬɚɜɥɹɸɬ ɬɚɤ
ɧɚɡɵɜɚɟɦɵɟ ɩɪɹɦɵɟ ɜɚɪɢɚɰɢɨɧɧɵɟ ɦɟɬɨɞɵ (ɩɪɹɦɵɟ ɦɟɬɨɞɵ ɪɟ
ɲɟɧɢɹɜɚɪɢɚɰɢɨɧɧɵɯɡɚɞɚɱ). ɋɭɬɶ ɷɬɢɯɦɟɬɨɞɨɜɜɬɨɦɱɬɨ, ɦɢɧɭɹ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɚ ɨɫɧɨɜɟ ɜɚɪɢɚɰɢɨɧɧɵɯ ɩɪɢɧɰɢ
ɩɨɜ ɦɟɯɚɧɢɤɢ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ ɬɟɥɚ ɫɬɪɨɹɬ ɩɪɨɰɟɞɭɪɵ ɞɥɹ ɨɬɵɫ
ɤɚɧɢɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɗɬɢɦ ɦɟɬɨɞɚɦ ɩɨɫɜɹɳɟɧɨ
ɦɧɨɝɨ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɵ ɜ ɬɨɦ ɱɢɫɥɟ >4, 6, 7@ ȼ ɞɚɧɧɨɣ ɝɥɚɜɟ
ɪɚɫɫɦɨɬɪɟɧɵ ɧɚɢɛɨɥɟɟɯɚɪɚɤɬɟɪɧɵɟɦɟɬɨɞɵɷɬɨɣ ɝɪɭɩɩɵɜɚɪɢɚɰɢ
ɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣɦɟɬɨɞɦɟɬɨɞɊɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨɢɲɢɪɨɤɨɩɪɢ
ɦɟɧɹɟɦɵɣɜɧɚɫɬɨɹɳɟɟɜɪɟɦɹɦɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ
ȼɚɪɢɚɰɢɨɧɧɚɹɩɨɫɬɚɧɨɜɤɚɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ
Ɂɚɞɚɱɚɢɡɝɢɛɚɩɥɚɫɬɢɧɤɚɤɢɞɪɭɝɢɟɡɚɞɚɱɢɦɟɯɚɧɢɤɢɞɟɮɨɪ
ɦɢɪɭɟɦɨɝɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɞɨɩɭɫɤɚɟɬ ɤɚɤ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɭɸ ɬɚɤ
ɢɜɚɪɢɚɰɢɨɧɧɭɸɩɨɫɬɚɧɨɜɤɭɡɚɞɚɱɢȼɩɨɫɥɟɞɧɟɦɫɥɭɱɚɟɩɪɨɛɥɟɦɚ
ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɡɚɦɟɧɹɟɬɫɹ ɩɪɨɛɥɟɦɨɣ
ɨɩɪɟɞɟɥɟɧɢɹ ɮɭɧɤɰɢɣ ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɫɬɚɰɢɨɧɚɪɧɨɟ ɡɧɚɱɟɧɢɟ
ɦɢɧɢɦɭɦɢɥɢɦɚɤɫɢɦɭɦɧɟɤɨɬɨɪɨɝɨɮɭɧɤɰɢɨɧɚɥɚ ɗ ɤɨɝɞɚɜɚɪɢ
ɚɰɢɹįɗ ɍɫɥɨɜɢɹɫɬɚɰɢɨɧɚɪɧɨɫɬɢɜɡɚɞɚɱɚɯɦɟɯɚɧɢɤɢɞɟɮɨɪ
ɦɢɪɭɟɦɨɝɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜɵɪɚɠɚɸɬ ɨɛɳɢɟ ɫɜɨɣɫɬɜɚ ɭɩɪɭɝɢɯ ɫɢ
ɫɬɟɦɢɜɬɚɤɨɣɮɨɪɦɭɥɢɪɨɜɤɟɧɨɫɹɬɧɚɡɜɚɧɢɟɜɚɪɢɚɰɢɨɧɧɵɯɩɪɢɧ
ɰɢɩɨɜ.
Ɏɭɧɤɰɢɨɧɚɥ – ɷɬɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɩɨɧɹɬɢɟ ɞɥɹ ɨɛɨɡɧɚɱɟɧɢɹ
ɧɟɤɨɬɨɪɨɣ ɩɟɪɟɦɟɧɧɨɣɜɟɥɢɱɢɧɵ, ɡɚɞɚɧɧɨɣɧɚɦɧɨɠɟɫɬɜɟɮɭɧɤɰɢɣ
ɢ ɡɚɜɢɫɹɳɟɣ ɨɬ ɜɵɛɨɪɚ ɷɬɢɯ ɮɭɧɤɰɢɣ, ɦɟɠɞɭ ɬɟɦ ɤɚɤ ɮɭɧɤɰɢɹ –
ɷɬɨɩɟɪɟɦɟɧɧɚɹɡɚɜɢɫɹɳɚɹɨɬɜɟɥɢɱɢɧɵɚɪɝɭɦɟɧɬɨɜɷɬɨɣɮɭɧɤɰɢɢ
ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɢɡɦɟɧɹɹ ɜɢɞ ɮɭɧɤɰɢɣ ɜɚɪɶɢɪɭɹ ɢɯ ɢɡɦɟɧɹɟɦ
ɜɟɥɢɱɢɧɭ ɮɭɧɤɰɢɨɧɚɥɚ ɚ ɜɨ ɜɬɨɪɨɦ ɢɡɦɟɧɹɹ ɜɟɥɢɱɢɧɭ ɚɪɝɭɦɟɧ
ɬɨɜ ɨɩɪɟɞɟɥɹɟɦ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɨɩɟɪɚɰɢɹ ɜɚ
ɪɶɢɪɨɜɚɧɢɹ ɩɪɟɞɩɨɥɚɝɚɟɬɱɬɨɩɪɢɮɢɤɫɢɪɨɜɚɧɧɵɯɚɪɝɭɦɟɧɬɚɯɢɦɟɟɬ
ɦɟɫɬɨɩɟɪɟɯɨɞɨɬɨɞɧɨɣɮɭɧɤɰɢɢɤɞɪɭɝɨɣɬ ɟɦɟɧɹɟɬɫɹɟɟɜɢɞ
Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɜɚɪɢɚɰɢɨɧɧɨɣ ɡɚɞɚɱɢ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ ɜɨɫ
ɩɨɥɶɡɭɟɦɫɹ ɩɪɢɧɰɢɩɨɦ ɦɢɧɢɦɭɦɚ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
69
Ƚɥɚɜɚ 3. ȼɚɪɢɚɰɢɨɧɧɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɡɚɞɚɱ
Ƚɥɚɜɚ 2. Ɇɟɬɨɞɵɩɪɢɛɥɢɠɟɧɧɨɝɨɪɟɲɟɧɢɹɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨɭɪɚɜɧɟɧɢɹ…
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
ɫɢɫɬɟɦɵ (ɩɪɢɧɰɢɩɅɚɝɪɚɧɠɚɫɨɝɥɚɫɧɨɤɨɬɨɪɨɦɭɢɡɜɫɟɯɜɨɡɦɨɠ
ɧɵɯɩɟɪɟɦɟɳɟɧɢɣɫɢɫɬɟɦɵ ɢɫɬɢɧɧɵɦɢɹɜɥɹɸɬɫɹɬɚɤɢɟɩɟɪɟɦɟɳɟ
ɧɢɹɤɨɬɨɪɵɟɫɨɨɛɳɚɸɬɮɭɧɤɰɢɨɧɚɥɭɩɨɥɧɨɣɩɨɬɟɧɰɢɚɥɶɧɨɣɷɧɟɪ
ɝɢɢɦɢɧɢɦɚɥɶɧɨɟɡɧɚɱɟɧɢɟ( ɗ ( w) o min ).
ɉɨɥɧɚɹɩɨɬɟɧɰɢɚɥɶɧɚɹɷɧɟɪɝɢɹɞɟɮɨɪɦɚɰɢɢ ɭɩɪɭɝɨɣɫɢɫɬɟɦɵ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ U
ɢɩɨɬɟɧɰɢɚɥɚɜɧɟɲɧɢɯɫɢɥɉ:
ɗ
U ɉ.
1
Vij Hij dV ,
2 ³³³
V
1
(V x H x V y H y W xy J xy ) dV .
2 ³³³
V
ª§ w 2 w
w2w · w2w § w2w
w2w · w2w º
«¨¨ 2 Q 2 ¸¸ 2 ¨¨ 2 Q 2 ¸¸ 2 »
wy ¹ wx
wx
wx ¹ wy
ȿ
© wy
» dV .
z 2 «©
2 ³³³
2
2
«
»
2(1 Q ) V
w w w w
2 (1 Q)
«
»
wx wy wx wy
¬
¼
ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜɩɨz ɨɬ–hɞɨhɢɩɪɢɜɟɞɹɩɨɞɨɛɧɵɟɱɥɟ
ɧɵɩɨɥɭɱɢɦ
U
2
D ª§ w 2 w ·
«¨ 2 ¸ ¨ wx ¸
2 ³³
¹
: «
©
2
§ w2w ·
§ w2w ·
w2w w2w
¨ 2 ¸ 2Q 2
¨
¸
Q
2
(
1
)
¨ wy ¸
¨ wx wy ¸
wx wy 2
©
¹
©
¹
ab
³³
00
2
º
» d:
»
¼
ª w 2 w w 2 w § w 2 w ·2 º
¸ » dx dy
« 2
¨¨
wx wy ¸¹ »
« wx wy 2
©
¬
¼
U
D
2
2
§ w2w w2w ·
³ ³ ¨¨ wx 2 wy 2 ¸¸ dxdy.
¹
00 ©
ab
70
(3.5)
ɉɨɬɟɧɰɢɚɥ ɜɧɟɲɧɢɯ ɫɢɥ ɉ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɪɚɛɨɬɚ ɫɨ ɡɧɚ
ɤɨɦɦɢɧɭɫɜɧɟɲɧɢɯɫɢɥ qdxdy ɧɚɩɟɪɟɦɟɳɟɧɢɹɯ w ɩɪɢɩɟɪɟɯɨɞɟ
ɢɡɨɝɧɭɬɨɣɩɥɚɫɬɢɧɵɜɧɚɱɚɥɶɧɨɟɧɟɞɟɮɨɪɦɢɪɭɟɦɨɟɫɨɫɬɨɹɧɢɟ
ɉ
T
³³ q ( x, y ) w ( x, y ) d : .
(3.6)
:
Ⱦɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɧɚɝɪɭɠɟɧɧɨɣ ɤɪɨɦɟ ɪɚɫɩɪɟɞɟ
ɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ q (x, y) ɬɚɤɠɟ ɢ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢ ɫɢɥɚɦɢ Fk ,
ɜɜɵɪɚɠɟɧɢɟ(3.6ɞɨɥɠɧɚɛɵɬɶɞɨɛɚɜɥɟɧɚɪɚɛɨɬɚɷɬɢɯɫɢɥɧɚɫɨɨɬ
ɜɟɬɫɬɜɭɸɳɢɯɩɟɪɟɦɟɳɟɧɢɹɯw (xk, yk):
ab
³ ³ q ( x, y ) w ( x, y ) dxdy ɉ
00
nF
¦ Fk w ( xk , yk ) .
(3.7)
k 1
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜɵɪɚɠɟɧɢɟ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ɞɟɮɨɪɦɚɰɢɢɞɥɹɡɚɤɪɟɩɥɟɧɧɨɣɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵɢɦɟɟɬɜɢɞ
ɗ
ab
³³
Uɉ
D
2
2
§ w2w w2w ·
³ ³ ¨¨ wx 2 wy 2 ¸¸ dxdy ¹
00 ©
ab
q ( x, y ) w ( x, y ) dxdy nF
¦ Fk w ( xk , yk ) .
k 1
00
ɢɥɢ
0.
ȼɷɬɨɦɫɥɭɱɚɟɜɵɪɚɠɟɧɢɟɞɥɹɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɪɢɦɟɬɜɢɞ
(3.3)
ɉɨɞɫɬɚɜɢɦɜɮɨɪɦɭɥɭɜɵɪɚɠɟɧɢɹɞɟɮɨɪɦɚɰɢɣɢɧɚɩɪɹɠɟɧɢɣɱɟɪɟɡɩɪɨɝɢɛɵw (x, y):
U
ɝɞɟ ȍ– ɩɥɨɳɚɞɶɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵ
Ⱦɥɹɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵc ɡɚɤɪɟɩɥɟɧɧɵɦɢ(w = 0) ɤɪɚɹɦɢ
(3.2)
ɝɞɟ V – ɨɛɴɟɦ ɬɟɥɚ ıij İij – ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɨɜ ɧɚɩɪɹɠɟɧɢɣ
ɢɞɟɮɨɪɦɚɰɢɣ
ȾɥɹɬɨɧɤɨɣɩɥɚɫɬɢɧɵɫɭɱɟɬɨɦɝɢɩɨɬɟɡɄɢɪɯɝɨɮɚɩɪɢ ız = 0;
Ȗxz Ȗyz ɜɵɪɚɠɟɧɢɟɩɪɢɧɢɦɚɟɬɜɢɞ
U
2
§ w 2 w w 2 w § w 2 w · 2 ·º
D ª«§ w 2 w w 2 w ·
¨
¸ ¸» d : , (3.4)
¸
¨
¨
Q
2
(
1
)
¨ wx 2 wy 2 ¸
¨ wx 2 wy 2 ¨ wx wy ¸ ¸»
2 ³³
«
¹ ¹¼
©
¹
: ©
©
¬
(3.1)
Ɉɛɳɟɟ ɜɵɪɚɠɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ
ɧɚɤɨɩɥɟɧɧɨɣɜɭɩɪɭɝɨɦɬɟɥɟ
U
U
71
(3.8)
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 3. ȼɚɪɢɚɰɢɨɧɧɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɡɚɞɚɱ
ɉɪɢɜɟɞɟɦ ɜɵɪɚɠɟɧɢɟɩɨɥɧɨɣɷɧɟɪɝɢɢɜɦɚɬɪɢɱɧɨɣɮɨɪɦɟ
ɗ
1
ı T İ dV ³³³ pu dV
2 ³³³
V
V
1
İ T D İ dV ³³³ pu dV
2 ³³³
V
V
1
(3.9)
(A w)T D (A w) d : ³³ q ( x, y ) w d : .
2 ³³
:
:
Ɂɞɟɫɶu, p, ı, İ – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨɜɟɤɬɨɪɵ ɩɟɪɟɦɟɳɟɧɢɣɧɚɝɪɭɡ
ɤɢ ɧɚɩɪɹɠɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ D – ɦɚɬɪɢɰɚ ɭɩɪɭɝɢɯ ɤɨɧɫɬɚɧɬ
A – ɦɚɬɪɢɰɚ ɨɩɟɪɚɰɢɣ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ Ⱦɥɹ ɡɚɞɚɱɢ ɢɡɝɢɛɚ ɩɥɚ
ɫɬɢɧɵɭɤɚɡɚɧɧɵɟɜɟɤɬɨɪɵ ɢɦɚɬɪɢɰɵɢɦɟɸɬɫɥɟɞɭɸɳɟɟɫɨɞɟɪɠɚɧɢɟ
u
^w`;
p
^q`;
ı
­Vx ½
° °
® V y ¾; İ
°W °
¯ xy ¿
­ Hx ½
° °
® H y ¾;
°J °
¯ xy ¿
(3.10)
ª
º
2
2
ª
º
x
Q
w
w
1
0
«
»
«
»
D D « Q 1 0 » ; A « w 2 wy 2 » .
«
1 Q»
« 2 w 2 wxwy »
« 0 0
»
¬
¼
¬
2 ¼
ȼɵɪɚɠɟɧɢɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɜ ɜɢɞɟ ɹɜɥɹɟɬɫɹ
ɨɛɳɢɦ ɞɥɹ ɥɸɛɵɯ ɫɬɚɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɦɟɯɚɧɢɤɢ ɞɟɮɨɪɦɢɪɭɟɦɨɝɨ
ɬɜɟɪɞɨɝɨɬɟɥɚɜɞɚɧɧɨɦɫɥɭɱɚɟɞɜɭɦɟɪɧɵɯ
ȼɵɪɚɠɟɧɢɟ ɢɥɢ ɬɚɤɠɟɧɚɡɵɜɚɸɬɮɭɧɤɰɢɨɧɚɥɨɦɩɨɥ
ɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ (ɮɭɧɤɰɢɨɧɚɥɨɦ Ʌɚɝɪɚɧɠɚ ɨɧɨ ɹɜɥɹɟɬɫɹ
ɜɚɠɧɨɣɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨɫɨɫɬɨɹɧɢɹ
ɭɩɪɭɝɨɝɨɬɟɥɚȼɱɚɫɬɧɨɫɬɢɮɨɪɦɭɥɢɪɭɟɬɫɹɩɪɢɧɰɢɩɦɢɧɢɦɭɦɚɩɨ
ɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɪɢɧɰɢɩ Ʌɚɝɪɚɧɠɚ ɹɜɥɹɸɳɢɣɫɹ ɨɞɧɢɦ ɢɡ
ɨɫɧɨɜɧɵɯɜɚɪɢɚɰɢɨɧɧɵɯɩɪɢɧɰɢɩɨɜɦɟɯɚɧɢɤɢ>3, 7].
Ɍɚɤɢɦɨɛɪɚɡɨɦɜɚɪɢɚɰɢɨɧɧɚɹɩɨɫɬɚɧɨɜɤɚɡɚɞɚɱɢɢɡɝɢɛɚ ɩɥɚ
ɫɬɢɧɵɡɚɤɥɸɱɚɟɬɫɹɜɦɢɧɢɦɢɡɚɰɢɢɮɭɧɤɰɢɨɧɚɥɚɅɚɝɪɚɧɠɚɬ ɟ
Gɗ
Gɗ
G (ɗ ɉ ) 0,
lim 'ɗ, 'ɗ
'w o 0
(3.11)
ɗ ( w 'w) ɗ ( w).
Ɂɞɟɫɶ ¨w – ɩɪɢɪɚɳɟɧɢɟ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ į – ɫɢɦɜɨɥ ɜɚɪɢɚɰɢɢ
ɥɢɧɟɣɧɨɣɫɨɫɬɚɜɥɹɸɳɟɣɩɪɢɪɚɳɟɧɢɹɮɭɧɤɰɢɨɧɚɥɚ
72
ɉɪɢ ɩɪɢɛɥɢɠɟɧɧɨɦ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ
ɨɡɧɚɱɚɟɬ ɱɬɨ ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɚɰɢɢ ɜɵɱɢɫɥɟɧɧɚɹ ɧɚ
ɩɨɥɭɱɟɧɧɵɯɩɪɢɛɥɢɠɟɧɧɵɯɩɟɪɟɦɟɳɟɧɢɹɯɛɭɞɟɬɛɨɥɶɲɟɞɟɣɫɬɜɢ
ɬɟɥɶɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ ɜɵɱɢɫɥɟɧɧɨɣ ɞɥɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ
ɬ ɟɬɨɱɧɵɯɩɟɪɟɦɟɳɟɧɢɣ
ɂɫɩɨɥɶɡɨɜɚɧɢɟɜɚɪɢɚɰɢɨɧɧɨɝɨɭɪɚɜɧɟɧɢɹɅɚɝɪɚɧɠɚɧɟɬɪɟɛɭ
ɟɬ ɡɚɪɚɧɟɟ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɫɬɚɬɢɱɟɫɤɢɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɬɚɤ
ɤɚɤ ɨɧɢ ɜɵɩɨɥɧɹɸɬɫɹ ɚɜɬɨɦɚɬɢɱɟɫɤɢ ɬɚɤɢɟ ɭɫɥɨɜɢɹ ɧɚɡɵɜɚɸɬɫɹ
ɟɫɬɟɫɬɜɟɧɧɵɦɢ Ʉɢɧɟɦɚɬɢɱɟɫɤɢɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɤɨɬɨɪɵɟ
ɧɚɡɵɜɚɸɬɫɹ ɝɥɚɜɧɵɦɢ ɫɥɟɞɭɟɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɛɹɡɚɬɟɥɶɧɨ Ʉɪɨɦɟ
ɬɨɝɨɜɮɭɧɤɰɢɨɧɚɥɟɅɚɝɪɚɧɠɚɤɚɤɜɢɞɢɦɢɫɩɨɥɶɡɭɸɬɫɹɩɪɨɢɡ
ɜɨɞɧɵɟɤɭɞɚɛɨɥɟɟɧɢɡɤɨɝɨɩɨɪɹɞɤɚɱɟɦɜɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭɪɚɜ
ɧɟɧɢɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵɜɬɨɪɵɟɩɪɨɢɡɜɨɞɧɵɟɩɪɨɬɢɜɱɟɬɜɟɪɬɵɯ
Ɉɬɦɟɬɢɦ ɬɚɤɠɟ ɱɬɨ ɢɡ ɜɚɪɢɚɰɢɨɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɫɟ
ɝɞɚ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɪɚɟɜɨɣ
ɡɚɞɚɱɢɜɧɚɲɟɦɫɥɭɱɚɟɷɬɨɭɪɚɜɧɟɧɢɟɋ. ɀɟɪɦɟɧɤɨɬɨɪɨɟɜɷɬɨɦ
ɫɥɭɱɚɟɧɚɡɵɜɚɟɬɫɹɭɪɚɜɧɟɧɢɟɦɗɣɥɟɪɚɞɥɹɜɚɪɢɚɰɢɨɧɧɨɣɡɚɞɚɱɢ.
ȼɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣɦɟɬɨɞ
ȼɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɵɣ ɦɟɬɨɞ ȼɊɆ ɹɜɥɹɟɬɫɹ ɚɧɚɥɨɝɨɦ
ɦɟɬɨɞɚ ɤɨɧɟɱɧɵɯ ɪɚɡɧɨɫɬɟɣ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɟɲɟɧɢɸ ɜɚɪɢɚɰɢ
ɨɧɧɵɯ ɡɚɞɚɱ Ɉɫɧɨɜɧɚɹ ɢɞɟɹ ȼɊɆ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ
ɜɚɪɢɚɰɢɨɧɧɨɣ ɡɚɞɚɱɢ ɜ ɫɟɬɨɱɧɨɣ ɨɛɥɚɫɬɢ ɫ ɩɨɦɨɳɶɸ ɤɨɧɟɱɧɵɯ
ɪɚɡɧɨɫɬɟɣ Ɍɨ ɟɫɬɶ ɤɚɤ ɢ ɜ ɆɄɊ ɡɞɟɫɶ ɢɫɩɨɥɶɡɭɟɬɫɹ ɪɚɡɧɨɫɬɧɚɹ
ɫɟɬɤɚ ɫ ɧɟɢɡɜɟɫɬɧɵɦɢ ɩɟɪɟɦɟɳɟɧɢɹɦɢ ɜ ɟɟ ɭɡɥɚɯ Ɉɞɧɚɤɨ ɞɚɥɟɟ
ɢɫɩɨɥɶɡɭɟɬɫɹɧɟɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɤɚɤɜɦɟɬɨɞɟɫɟɬɨɤ
ɚ ɮɭɧɤɰɢɨɧɚɥ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɢɡɝɢɛɚ ɩɥɚɫɬɢɧɵ
ɉɪɨɢɡɜɨɞɧɵɟ ɜ ɜɵɪɚɠɟɧɢɢ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɞɟɮɨɪɦɚɰɢɢ ɡɚ
ɦɟɧɹɸɬɫɹɪɚɡɧɨɫɬɧɵɦɢɨɬɧɨɲɟɧɢɹɦɢɆɢɧɢɦɢɡɚɰɢɹɮɭɧɤɰɢɨɧɚɥɚ
ɩɪɢɜɨɞɢɬɤɫɢɫɬɟɦɟɚɥɝɟɛɪɚɢɱɟɫɤɢɯɭɪɚɜɧɟɧɢɣɨɬɧɨɫɢɬɟɥɶɧɨɭɡɥɨ
ɜɵɯɩɟɪɟɦɟɳɟɧɢɣȼɧɭɬɪɟɧɧɢɟɭɫɢɥɢɹɜɭɡɥɚɯɫɟɬɤɢɜɵɱɢɫɥɹɸɬɫɹ
ɤɚɤɢɜɆɄɊɱɟɪɟɡɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɪɚɡɧɨɫɬɧɵɟɨɬɧɨɲɟɧɢɹ
ɉɪɢɢɫɩɨɥɶɡɨɜɚɧɢɢȼɊɆɬɪɟɛɭɟɬɫɹɭɞɨɜɥɟɬɜɨɪɹɬɶɬɨɥɶɤɨɤɢ
ɧɟɦɚɬɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɫɬɚɬɢɱɟɫɤɢɟɝɪɚɧɢɱɧɵɟ ɭɫɥɨ
ɜɢɹ ɭɞɨɜɥɟɬɜɨɪɹɸɬɫɹ ɩɪɢ ɦɢɧɢɦɢɡɚɰɢɢɩɨɥɧɨɣɷɧɟɪɝɢɢɞɟɮɨɪɦɚ
ɰɢɢ ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɜ ɧɟɤɨɬɨɪɨɣ ɫɬɟɩɟɧɢ ɭɩɪɨɫɬɢɬɶ ɜɚɪɢɚɰɢɨɧɧɨɪɚɡɧɨɫɬɧɵɣɦɟɬɨɞ ɩɨɫɪɚɜɧɟɧɢɸɫɆɄɊ.
73
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 3. ȼɚɪɢɚɰɢɨɧɧɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɡɚɞɚɱ
D
2
ɗ
N
N
nF
k 1
k 1
k 1
¦ (’ k2 ) 2 ' x ' y ¦ qk wk ' x ' y ¦ Fk wk ,
Ɉɬɦɟɬɢɦɫɯɨɞɫɬɜɨɨɩɟɪɚɬɨɪɚɫɨɩɟɪɚɬɨɪɚɦɢɩɪɢ
ɦɟɧɹɟɦɵɦɢɩɪɢɪɚɫɱɟɬɟɩɥɚɫɬɢɧɆɄɊ
Ɍɚɤɢɦɠɟɨɛɪɚɡɨɦɤɚɤɛɵɥɨɩɨɥɭɱɟɧɨɪɚɡɧɨɫɬɧɨɟɜɵɪɚɠɟɧɢɟ
ɞɥɹɛɢɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨɪɚ ’ 4 wk ɫɦɪɚɡɞɟɥɦɨɠɧɨɩɨ
ɥɭɱɢɬɶɢɪɚɡɧɨɫɬɧɨɟɜɵɪɚɠɟɧɢɟɞɥɹɥɟɜɨɣɱɚɫɬɢɭɪɚɜɧɟɧɢɹ
ɡɚɩɢɫɚɧɧɨɟɱɟɪɟɡɡɧɚɱɟɧɢɹɩɟɪɟɦɟɳɟɧɢɣɜɭɡɥɚɯɫɟɬɤɢɪɢɫ 3.1, ɚ).
ɚ)
l
i
(3.12)
(’ k )
2
§ w2w
w2w
¨
¨ wx 2
wy 2
©
k
·
¸
¸
k¹
2
wɗ
¦ ww
k 1
k
N
D
2
2
2
2
§ 2
§ w2w ·
¨
¸ 2w w ˜ w w ¨w w
¨ wy 2
¨ wx 2 ¸
wx 2 k wy 2
©
k¹
k ©
(3.13)
n
N
w (’ 2 ) 2
¦ wwk ' x ' y ¦ qk ' x ' y ¦ Fk
k 1
k 1
k 1
k
N
F
2
·
¸.
¸
k¹
0.
Fk ·¸
1 §¨
qk , k 1, ... , N ,
¨
' x ' y ¸¹
D©
ɛ)
h
a
k
c
f
d
e
1 w (’ k2 )2
2 wwk
Ș
(3.15)
2
–ȝ
2
–4(1+Ș)
8+6(ȝȘ)
–4(1+Ș)
2
–ȝ
2
ȝ
Ș
¨x
u
1
' 2x ' 2y
¨y
Ɋɢɫ 3.1. Ɋɚɡɧɨɫɬɧɵɣɨɩɟɪɚɬɨɪɜɚɪɢɚɰɢɨɧɧɨɝɨɭɪɚɜɧɟɧɢɹ
ɉɪɨɞɟɦɨɧɫɬɪɢɪɭɟɦɜɵɜɨɞɪɚɡɧɨɫɬɧɨɝɨɨɩɟɪɚɬɨɪɚɧɚɩɪɢɦɟɪɟ
ɩɟɪɜɨɝɨɫɥɚɝɚɟɦɨɝɨɜɪɚɡɥɨɠɟɧɢɢ (’ k2 ) 2 (3.13).
w2w
ɬɨɝɞɚ ɜɡɹɜ ɩɪɨɢɡɜɨɞɧɭɸ ɨɬ ɫɥɨɠɧɨɣ
wx 2 k
ɮɭɧɤɰɢɢ ɢ ɩɪɢɦɟɧɢɜ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɜɬɨɪɨɣ
ɩɪɨɢɡɜɨɞɧɨɣɩɨɥɭɱɢɦ
Ɉɛɨɡɧɚɱɢɦ wkcc
ɝɞɟ N – ɨɛɳɟɟ ɱɢɫɥɨ ɭɡɥɨɜ ɫɟɬɤɢ qk – ɫɪɟɞɧɹɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɧɚɝɪɭɡɤɢɩɪɢɯɨɞɹɳɟɣɫɹɧɚɩɥɨɳɚɞɤɭ¨xרyɩɪɢɦɵɤɚɸɳɭɸɤɭɡɥɭk.
74
m
ȝ
(3.14)
Ɂɚɩɢɫɵɜɚɹɷɬɨɭɫɥɨɜɢɟɞɥɹɤɚɠɞɨɝɨɭɡɥɚɫɟɬɤɢɩɨɥɭɱɚɟɦɫɢ
ɫɬɟɦɭ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɢɡ ɪɟɲɟɧɢɹ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɹ
ɸɬɫɹɩɟɪɟɦɟɳɟɧɢɹɜɷɬɢɯɭɡɥɚɯ
ɉɪɟɞɫɬɚɜɢɦ ɪɚɡɧɨɫɬɧɵɣ ɨɩɟɪɚɬɨɪ ɜɚɪɢɚɰɢɨɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɞɥɹɩɪɨɢɡɜɨɥɶɧɨɝɨɭɡɥɚɫɟɬɤɢ k ɜɫɥɟɞɭɸɳɟɦɜɢɞɟ
1 w (’ k2 ) 2
2 wwk
b
¨x
y
ɍɫɥɨɜɢɟ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɨɧɚɥɚ ɛɭɞɟɬ ɢɦɟɬɶ ɫɥɟɞɭɸ
ɳɢɣɜɢɞ:
wɗ
ww
g
n
ɝɞɟɤɜɚɞɪɚɬɝɚɪɦɨɧɢɱɟɫɤɨɝɨɨɩɟɪɚɬɨɪɚ Ʌɚɩɥɚɫɚ
2
x
¨y
ɉɨɦɢɦɨ ɷɬɨɝɨ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦ ɭɪɚɜɧɟɧɢɢ ɢɡɝɢɛɚ ɩɥɚ
ɫɬɢɧɵɢɫɩɨɥɶɡɭɸɬɫɹɩɪɨɢɡɜɨɞɧɵɟɞɨɱɟɬɜɟɪɬɨɝɨɩɨɪɹɞɤɚ, ɞɥɹɪɚɡ
ɧɨɫɬɧɨɣɚɩɩɪɨɤɫɢɦɚɰɢɢɤɨɬɨɪɵɯɜɆɄɊɩɪɢɯɨɞɢɬɫɹɜɜɨɞɢɬɶ ɨɞɢɧɞɜɚ ɪɹɞɚ ɡɚɤɨɧɬɭɪɧɵɯ ɬɨɱɟɤ ȼ ɮɭɧɤɰɢɨɧɚɥɟ ɠɟ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ
ɞɟɮɨɪɦɚɰɢɢ ɩɪɨɢɡɜɨɞɧɵɟɧɟɩɪɟɜɵɲɚɸɬɜɬɨɪɨɝɨɩɨɪɹɞɤɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɚɥɢɡɚɰɢɸ ɜɚɪɢɚɰɢɨɧɧɨ-ɪɚɡɧɨɫɬɧɨɝɨ ɦɟɬɨɞɚ
ɩɪɢɪɟɲɟɧɢɢɡɚɞɚɱɢɢɡɝɢɛɚɩɪɹɦɨɭɝɨɥɶɧɵɯɩɥɚɫɬɢɧ
Ɍɚɤɤɚɤɢɧɬɟɝɪɢɪɨɜɚɧɢɟɜɩɨɪɚɡɧɨɫɬɧɨɣɫɯɟɦɟɜɵɩɨɥɧɹ
ɟɬɫɹɩɪɢɛɥɢɠɟɧɧɨɩɨɮɨɪɦɭɥɟɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜɬɨɩɪɢɞɢɫɤɪɟ
ɬɢɡɚɰɢɢ ɡɚɞɚɱɢ ɤɨɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣ ɫɟɬɤɨɣ (ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫ ɲɚ
ɝɨɦ¨x ɢ¨y) ɢɧɬɟɝɪɚɥɵɜɜɵɪɚɠɟɧɢɢɩɨɥɧɨɣɷɧɟɪɝɢɢɞɟɮɨɪɦɚɰɢɢ
ɡɚɦɟɧɹɟɦɫɭɦɦɢɪɨɜɚɧɢɟɦɩɨɜɫɟɦɭɡɥɚɦɫɟɬɤɢ
75
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
1 w ( wkcc ) 2
2 wwk
wwcc ·
wwcc
wwcc
2§
¨¨ wacc a wkcc k wccc c ¸¸
2©
wwk ¹
wwk
wwa
§ ( wi 2 wa wk ) w ( wi 2 wa wk ) ( wa 2 wk wc ) ·
u ¸
¨
'2x
'2x
wwk '2x
¸
¨
¨ w ( w 2w w ) ( w 2w w ) w ( w 2w w ) ¸
a
k
c
c
m
k
c
m
k
¨¨ u
¸¸
2
2
2
w
w
w
'
'
w
'
¹
©
k x
x
k x
1
( wi 2wa wk ) ˜1 ( wa 2wk wc ) ˜ (2) ( wk 2wc wm ) ˜1
'4x
1
wi 4wa 6wk 4wc wm .
'4x
Ⱥɧɚɥɨɝɢɱɧɨɜɵɜɨɞɹɬɫɹɢɨɫɬɚɥɶɧɵɟɪɚɡɧɨɫɬɧɵɟɫɨɨɬɧɨɲɟɧɢɹ
ɜɯɨɞɹɳɢɟ ɜ ɥɟɜɭɸ ɱɚɫɬɶ ɜɚɪɢɚɰɢɨɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ȼ ɪɟ
ɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɣ ɜɢɞ ɷɬɨɝɨ ɨɩɟɪɚɬɨɪɚ ɪɢɫ 3.1, ɛ).
Ʉɚɤɜɢɞɢɦɞɥɹɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵɫɪɚɜɧɨɦɟɪɧɨɣɫɟɬɤɨɣɨɧ
ɩɨɥɧɨɫɬɶɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɛɢɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɨɩɟɪɚɬɨɪɭ ’ 4 wk , ɢɫ
ɩɨɥɶɡɭɟɦɨɦɭ ɩɪɢɪɚɫɱɟɬɟɩɥɚɫɬɢɧɆɄɊ
Ɉɞɧɚɤɨɩɪɢɪɚɫɱɟɬɟɩɥɚɫɬɢɧɫɥɨɠɧɨɣɤɨɧɮɢɝɭɪɚɰɢɢɫɦɟɧɹ
ɸɳɢɦɢɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɧɟɪɚɜɧɨɦɟɪɧɨɣ
ɫɟɬɤɨɣɫɬɚɬɢɱɟɫɤɢɦɢɝɪɚɧɢɱɧɵɦɢɭɫɥɨɜɢɹɦɢɜɬɨɦɱɢɫɥɟɫɨɫɜɨ
ɛɨɞɧɵɦɢ ɢɥɢ ɧɚɝɪɭɠɟɧɧɵɦɢ ɤɪɚɹɦɢ ȼɊɆ ɢɦɟɟɬ ɩɪɟɢɦɭɳɟɫɬɜɚ
ɩɟɪɟɞɆɄɊɩɨɡɜɨɥɹɹɩɨɥɭɱɚɬɶɞɨɫɬɚɬɨɱɧɨɬɨɱɧɵɟ ɪɟɡɭɥɶɬɚɬɵ>5].
ȼɫɜɨɸɨɱɟɪɟɞɶ ɛɨɥɟɟɬɨɱɧɨɟ ɪɟɲɟɧɢɟɞɥɹɤɨɧɫɬɪɭɤɰɢɣɩɪɨ
ɫɬɨɣɝɟɨɦɟɬɪɢɢɦɨɠɟɬɛɵɬɶɩɨɥɭɱɟɧɨɟɫɥɢɩɪɢɦɟɧɹɬɶɞɥɹɢɯɪɚɫ
ɱɟɬɚɪɚɡɧɨɫɬɧɵɟɫɨɨɬɧɨɲɟɧɢɹɡɚɩɢɫɚɧɧɵɟɞɥɹɤɨɧɫɬɪɭɤɰɢɣɛɨɥɟɟ
ɫɥɨɠɧɨɣ ɝɟɨɦɟɬɪɢɢ ȼ ɱɚɫɬɧɨɫɬɢ ɬɨɱɧɨɫɬɶ ɪɟɡɭɥɶɬɚɬɨɜ ɞɥɹ ɩɥɚ
ɫɬɢɧɵ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɨɱɟɪɬɚɧɢɹ ɦɨɠɧɨ ɩɨɜɵɫɢɬɶ ɩɪɢɦɟɧɹɹ ɩɪɢ
ɟɟ ɪɚɫɱɟɬɟ ɫɨɨɬɧɨɲɟɧɢɹ ȼɊɆ ɜɵɜɟɞɟɧɧɵɟ ɞɥɹ ɩɥɚɫɬɢɧɵ ɩɪɨɢɡ
ɜɨɥɶɧɨɝɨ ɨɱɟɪɬɚɧɢɹ ɬ ɟ ɢɫɩɨɥɶɡɭɹ ɜɵɪɚɠɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢɞɟɮɨɪɦɚɰɢɢɜɜɢɞɟɚɧɟ
ɆɟɬɨɞɊɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨ
Ɇɟɬɨɞ Ɋɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨ ɆɊɌɹɜɥɹɟɬɫɹɩɪɢɛɥɢɠɟɧɧɵɦ
ɱɢɫɥɟɧɧɨ-ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɜɚɪɢɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ Ɋɚɡɪɚɛɨɬɚɧ
76
Ƚɥɚɜɚ 3. ȼɚɪɢɚɰɢɨɧɧɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɡɚɞɚɱ
ɲɜɟɣɰɚɪɫɤɢɦ ɮɢɡɢɤɨɦ ȼ Ɋɢɬɰɟɦ ɜ ɝ. ɧɟɡɚɜɢɫɢɦɨ ɢ ɩɪɚɤɬɢ
ɱɟɫɤɢɨɞɧɨɜɪɟɦɟɧɧɨɢɫɩɨɥɶɡɨɜɚɥɫɹɋɉɌɢɦɨɲɟɧɤɨɞɥɹɪɟɲɟɧɢɹ
ɡɚɞɚɱɢɡɝɢɛɚɢɭɫɬɨɣɱɢɜɨɫɬɢɤɨɧɫɬɪɭɤɰɢɣ
ȼ ɨɫɧɨɜɟ ɦɟɬɨɞɚ – ɜɚɪɢɚɰɢɨɧɧɵɣ ɩɪɢɧɰɢɩ Ʌɚɝɪɚɧɠɚ ɦɢɧɢ
ɦɭɦɚɩɨɥɧɨɣɩɨɬɟɧɰɢɚɥɶɧɨɣɷɧɟɪɝɢɢɫɢɫɬɟɦɵɈɞɧɚɤɨɜɦɟɫɬɨɤɨ
ɧɟɱɧɨ-ɪɚɡɧɨɫɬɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ ɤɚɤ ɜ ȼɊɆ
ɜɦɟɬɨɞɟɊɢɬɰɚ– Ɍɢɦɨɲɟɧɤɨɩɪɢɛɥɢɠɟɧɧɨɟɪɟɲɟɧɢɟɡɚɞɚɟɬɫɹ ɜɜɢɞɟ
ɥɢɧɟɣɧɨɣɤɨɦɛɢɧɚɰɢɢɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣɬɚɤɠɟɤɚɤɜɆȻȽ
ɉɨɞɫɬɚɜɢɦ ɜ ɮɭɧɤɰɢɨɧɚɥ Ʌɚɝɪɚɧɠɚ ɡɚɩɢɫɚɧɧɵɣ ɞɥɹ
ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ wN (x, y) ɡɚɞɚɧɧɭɸ
ɜɜɢɞɟ
wN
N
N
i 1
j 1
¦ Di Mi (x, y) ¦ Di M1i (x) M2i ( y) .
(3.16)
Ɂɞɟɫɶ D i í ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɞɥɟɠɚɳɢɟ ɨɩɪɟɞɟɥɟɧɢɸ M1i ( x) ,
M2i ( y ) í ɧɟɤɨɬɨɪɵɟɡɚɞɚɧɧɵɟɮɭɧɤɰɢɢɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟɝɟɨɦɟɬ
ɪɢɱɟɫɤɢɦɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɧɚɝɪɚɧɹɯɩɥɚɫɬɢɧɵ
ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɞɟ
ɮɨɪɦɚɰɢɢ ɩɪɹɦɨɭɝɨɥɶɧɨɣɩɥɚɫɬɢɧɵ ɗ (D i ) ɹɜɥɹɟɬɫɹ ɤɜɚɞɪɚɬɢɱɧɨɣ
ɮɭɧɤɰɢɟɣɩɚɪɚɦɟɬɪɨɜ D i :
ɗ (D i )
D
2
2
N
§N
·
³ ³ ¨¨ ¦ Di M1cci ( x) M2i ( y) ¦ Di M1i ( x) Mc2ci ( y) ¸¸ dxdy ¹
i 1
0 0 ©i 1
ab
ab
N ª
º N ª n
º
¦ «D i ³ ³ q ( x, y ) M1i ( x) M2i ( y ) dxdy » ¦ «D i ¦ Fk M1i ( xk ) M2i ( yk )» .
i 1 ¬
¼
¼ i1¬ k 1
00
F
Ɇɢɧɢɦɭɦɭ ɩɨɥɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɥɚɫɬɢɧɵ ɫɨɨɬɜɟɬ
ɫɬɜɭɟɬɭɫɥɨɜɢɟɪɚɜɟɧɫɬɜɚɧɭɥɸɟɟɩɪɨɢɡɜɨɞɧɵɯɩɨɩɚɪɚɦɟɬɪɚɦ D i
wɗ
wD i
0, i
1, 2, ... , N .
(3.17)
ɉɪɨɢɡɜɨɞɧɚɹ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɭɧɤɰɢɢ ɩɚɪɚɦɟɬɪɨɜ D i ɩɪɟɞ
ɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɥɢɧɟɣɧɭɸ ɮɭɧɤɰɢɸ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɬɚɤɢɦ ɨɛɪɚ
ɡɨɦɩɨɥɭɱɚɟɦɫɢɫɬɟɦɭN ɥɢɧɟɣɧɵɯɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣɨɬɧɨ
ɫɢɬɟɥɶɧɨ ɢɫɤɨɦɵɯ ɧɟɢɡɜɟɫɬɧɵɯ D i Ɂɚɦɟɬɢɦ ɱɬɨ ɭɪɚɜɧɟɧɢɹ 77
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
Ƚɥɚɜɚ 3. ȼɚɪɢɚɰɢɨɧɧɵɟɦɟɬɨɞɵɪɟɲɟɧɢɹɡɚɞɚɱ
ɜɵɪɚɠɚɸɬ ɤɚɤ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɬɚɤ ɢ ɫɬɚɬɢɱɟɫɤɢɟ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹ ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɩɪɢ ɡɚɞɚɧɢɢ ɮɭɧɤɰɢɣ M1i , M2i ɜɯɨɞɹɳɢɯ
ɜɜɵɪɚɠɟɧɢɟɩɪɨɝɢɛɨɜ3.16ɨɛɹɡɚɬɟɥɶɧɨɭɞɨɜɥɟɬɜɨɪɹɬɶɥɢɲɶɤɢ
ɧɟɦɚɬɢɱɟɫɤɢɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ.
ɉɪɢ ɭɞɚɱɧɨɦɜɵɛɨɪɟɛɚɡɢɫɧɵɯɮɭɧɤɰɢɣ M1i ( x) ɢ M2i ( y ) , ɩɪɢ
ɭɫɥɨɜɢɢɢɯɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɢɫɬɟɦɚɭɪɚɜɧɟɧɢɣɭɩɪɨɳɚɟɬɫɹ, ɢ ɞɥɹ
ɧɚɯɨɠɞɟɧɢɹɤɚɠɞɨɝɨɤɨɷɮɮɢɰɢɟɧɬɚ D i ɩɨɥɭɱɚɟɦ ɮɨɪɦɭɥɭ [2]:
Di
Bi
Ai
1
D
ab
³³
q ( x, y ) M1i M2i dxdy 00
³ ³ >M1cci M2i ab
2
nF
¦ Fk M1i M2i
k 1
@
.
(3.18)
2 M1i M1cci M2i Mc2ci M1i Mc2ci dxdy
2
00
Ƚɚɥɺɪɤɢɧɚ ɩɪɢɛɥɢɠɟɧɧɨ ɜɵɪɚɠɚɸɬ ɪɚɜɟɧɫɬɜɨ ɧɭɥɸ ɫɭɦɦɵ
ɪɚɛɨɬ ɜɫɟɯ ɜɧɟɲɧɢɯ ɢ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɩɥɚɫɬɢɧɟ ɧɚ ɜɨɡɦɨɠɧɵɯ
ɩɟɪɟɦɟɳɟɧɢɹɯ Mi .
Ⱦɚɧɧɨɟ ɩɨɥɨɠɟɧɢɟ ɨɬɜɟɱɚɟɬ ɜɚɪɢɚɰɢɨɧɧɨɦɭ ɩɪɢɧɰɢɩɭ ɜɨɡ
ɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɣ. ɋ ɷɬɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɦɟɬɨɞɵ Ȼɭɛɧɨɜɚ –
Ƚɚɥɺɪɤɢɧɚ ɢɊɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨɪɚɜɧɨɰɟɧɧɵɢɩɪɢɜɨɞɹɬɤɨɞɢɧɚ
ɤɨɜɵɦɪɟɡɭɥɶɬɚɬɚɦɟɫɥɢɮɭɧɤɰɢɢ Mi ɜɵɛɪɚɧɵɬɚɤɱɬɨɛɵɭɞɨɜɥɟ
ɬɜɨɪɹɬɶ ɜɫɟɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɇɨ ɩɪɢ ɷɬɨɦ ɡɚɦɟɬɢɦ ɱɬɨ
ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɜɵɩɨɥɧɟɧɢɟ ɫɬɚɬɢɱɟɫɤɢɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɧɟ
ɹɜɥɹɟɬɫɹɫɬɪɨɝɨɨɛɹɡɚɬɟɥɶɧɵɦ
3.4. Ɇɟɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ
3Ɉɫɧɨɜɧɵɟɩɨɥɨɠɟɧɢɹɦɟɬɨɞɚɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ
ɉɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɵɟ ɩɚɪɚɦɟɬɪɵ D i ɜ , ɩɨɥɭɱɢɦ ɢɫɤɨ
ɦɭɸ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ ɩɥɚɫɬɢɧɵ wN (x, y ) . ɉɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ
ɹɜɥɹɟɬɫɹɩɪɢɛɥɢɠɟɧɧɵɦɟɝɨɫɯɨɞɢɦɨɫɬɶɡɚɜɢɫɢɬ ɨɬɱɢɫɥɚɱɥɟɧɨɜ
ɜɡɚɞɚɧɢɢɮɭɧɤɰɢɢ wN .
ɉɪɢ ɷɬɨɦ ɨɬɦɟɬɢɦ ɱɬɨ ɯɨɬɹ ɭɞɨɜɥɟɬɜɨɪɟɧɢɟ ɫɬɚɬɢɱɟɫɤɢɦ
ɝɪɚɧɢɱɧɵɦɭɫɥɨɜɢɹɦɜɦɟɬɨɞɟɊɢɬɰɚ – Ɍɢɦɨɲɟɧɤɨɫɬɪɨɝɨɧɟɨɛɹ
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