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Файл:Теория расчета пластин и оболочек. Учебное пособие
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Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɉɥɚɫɬɢɧɵɢɦɟɸɬ ɲɢɪɨɤɨɟɩɪɢɦɟɧɟɧɢɟɜɫɬɪɨɢɬɟɥɶɫɬɜɟɜɜɢɞɟ
ɧɚɫɬɢɥɨɜ ɩɚɧɟɥɟɣ ɩɥɢɬ ɩɟɪɟɤɪɵɬɢɹ ɢ ɬ ɞ ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ
ɱɬɨɩɪɢɫɭɳɢɟɬɨɧɤɨɫɬɟɧɧɵɦ ɤɨɧɫɬɪɭɤɰɢɹɦɥɟɝɤɨɫɬɶ ɢ ɪɚɰɢɨɧɚɥɶ
ɧɨɫɬɶɮɨɪɦɫɨɱɟɬɚɸɬɫɹɫɢɯɜɵɫɨɤɨɣ ɧɟɫɭɳɟɣɫɩɨɫɨɛɧɨɫɬɶɸɷɤɨ
ɧɨɦɢɱɧɨɫɬɶɸɢɯɨɪɨɲɟɣɬɟɯɧɨɥɨɝɢɱɧɨɫɬɶɸ
ɋɚɦɵɣɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɣ ɜɢɞ ɩɥɚɫɬɢɧ – ɷɬɨ ɬɚɤɧɚɡɵɜɚɟɦɵɟ
ɬɨɧɤɢɟɩɥɚɫɬɢɧɵɭɤɨɬɨɪɵɯɨɬɧɨɲɟɧɢɟɬɨɥɳɢɧɵɤɧɚɢɦɟɧɶɲɟɦɭ
ɯɚɪɚɤɬɟɪɧɨɦɭ ɪɚɡɦɟɪɭ /100 h/b 1/5. ɉɪɢ h/b >1/5ɩɥɚɫɬɢɧɚ
ɨɬɧɨɫɢɬɫɹ ɤ ɬɨɥɫɬɵɦ ɩɥɢɬɚɦ, ɤɨɬɨɪɵɟ ɞɨɥɠɧɵ ɪɚɫɫɱɢɬɵɜɚɬɶɫɹ
ɭɠɟ ɤɚɤ ɦɚɫɫɢɜɧɵɟ ɬɟɥɚ ɉɪɢ h/b < 1/10 ɩɥɚɫɬɢɧɚɩɪɟɜɪɚɳɚɟɬɫɹ
ɜɦɟɦɛɪɚɧɭɤɨɬɨɪɚɹɦɨɠɟɬɪɚɛɨɬɚɬɶɬɨɥɶɤɨɩɪɢɡɚɤɪɟɩɥɟɧɧɵɯɩɨ
ɤɨɧɬɭɪɭɤɪɚɹɯ. ȿɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚ ɢɡɝɢɛ ɨɤɚɡɵɜɚɟɬɫɹ ɧɢɱɬɨɠɧɨ
ɦɚɥɵɦ, ɚ ɨɫɧɨɜɧɭɸ ɪɨɥɶ ɜ ɜɨɫɩɪɢɹɬɢɢ ɧɚɝɪɭɡɤɢ ɢɝɪɚɸɬ ɭɫɢɥɢɹ
ɪɚɫɬɹɠɟɧɢɹɢɫɞɜɢɝɚɜɫɪɟɞɢɧɧɨɣɩɨɜɟɪɯɧɨɫɬɢ
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɨɬɧɨɲɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɩɪɨɝɢɛɚɤ 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
ɢɝɪɚɸɬɢɡɝɢɛɧɵɟɫɢɥɨɜɵɟɮɚɤɬɨɪɵɞɟɮɨɪɦɚɰɢɹɦɢɜɫɪɟɞɢɧɧɨɣ
ɩɨɜɟɪɯɧɨɫɬɢɢ 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
ɤɨɣɨɩɪɟɞɟɥɹɟɬɫɹɧɟɬɨɥɶɤɨɟɟɝɟɨɦɟɬɪɢɱɟɫɤɢɦɢɩɚɪɚɦɟɬɪɚɦɢ ȼɟ
ɥɢɱɢɧɵ ɭɩɪɭɝɢɯɞɟɮɨɪɦɚɰɢɣɬɚɤɠɟɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɹɬɨɬ ɦɟɯɚ
ɧɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜɦɚɬɟɪɢɚɥɚɭɫɥɨɜɢɣɡɚɤɪɟɩɥɟɧɢɹ ɩɥɚɫɬɢɧɵ ɜɢɞɚ
ɧɚɝɪɭɡɤɢ ɫɬɚɬɢɱɟɫɤɚɹɢɥɢɞɢɧɚɦɢɱɟɫɤɚɹ.
ȼɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦɪɚɫɫɦɚɬɪɢɜɚɬɶɬɨɥɶɤɨɬɨɧɤɢɟɠɟɫɬɤɢɟ
ɩɥɚɫɬɢɧɵɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹɪɚɫɱɟɬɧɵɦɢɫɯɟɦɚɦɢ ɩɥɢɬɩɟɪɟɤɪɵ
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
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 Ȗ
ɨɫɢ z ɬɚɤɠɟ ɨɬɫɭɬɫɬɜɭɟɬ İ
= 0, Ȗzx= 0), ɥɢɧɟɣɧɚɹɞɟɮɨɪɦɚɰɢɹɜ ɧɚɩɪɚɜɥɟɧɢɢ
yz
= 0) Ⱦɚɧɧɚɹ ɝɢɩɨɬɟɡɚɚɧɚɥɨɝɢɱɧɚ ɝɢ
z
ɩɨɬɟɡɟɩɥɨɫɤɢɯɫɟɱɟɧɢɣɜɬɟɨɪɢɢɢɡɝɢɛɚɛɚɥɨɤ
2. Ƚɢɩɨɬɟɡɚ ɨ ɧɟɞɟɮɨɪɦɢɪɭɟɦɨɫɬɢ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ:
ɜɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɨɬɫɭɬɫɬɜɭɸɬɞɟɮɨɪɦɚɰɢɢɪɚɫɬɹɠɟɧɢɹɫɠɚ
ɬɢɹ ɢ ɫɞɜɢɝɚ ɬ ɟ ɩɪɢ ɢɡɝɢɛɟ ɩɥɚɫɬɢɧɵ ɷɬɚ ɩɥɨɫɤɨɫɬɶ ɨɫɬɚɟɬɫɹ
ɧɟɣɬɪɚɥɶɧɨɣ ɢɟɟɩɟɪɟɦɟɳɟɧɢɹ u
0=v0
= 0.
3. Ƚɢɩɨɬɟɡɚ ɨ ɧɟɧɚɞɚɜɥɢɜɚɧɢɢ ɫɥɨɟɜ: ɞɚɜɥɟɧɢɟɦ ɫɥɨɟɜ ɩɚ
ɪɚɥɥɟɥɶɧɵɯ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z ɩɪɟɧɟɛɪɟ
ɝɚɸɬɬ ɟɩɨɥɚɝɚɟɦ ı
= 0).
z
Ƚɢɩɨɬɟɡɵ Ʉɢɪɯɝɨɮɚ ɹɜɥɹɸɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ɝɢɩɨɬɟɡɵ ɩɥɨɫ
ɤɢɯ ɫɟɱɟɧɢɣ ɩɪɢɧɹɬɨɣ ɜ ɫɨɩɪɨɬɢɜɥɟɧɢɢ ɦɚɬɟɪɢɚɥɨɜ Ʉɪɨɦɟ ɬɨɝɨ
ɩɪɢɢɫɩɨɥɶɡɨɜɚɧɢɢɬɟɯɧɢɱɟɫɤɨɣɬɟɨɪɢɢɢɡɝɢɛɚɦɚɬɟɪɢɚɥɩɥɚɫɬɢɧɵ
ɫɱɢɬɚɟɬɫɹ ɥɢɧɟɣɧɨ-ɭɩɪɭɝɢɦ ɢ ɞɟɣɫɬɜɭɸɬ ɭɤɚɡɚɧɧɵɟ ɜɵɲɟɨɝɪɚɧɢ
ɱɟɧɢɹɩɨɩɪɨɝɢɛɚɦw/h ɢ ɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵh/b
1/5). Ɉɞɧɚɤɨɜɫɜɹɡɢɫɬɟɦɱɬɨɪɚɫɱɟɬɬɨɥɫɬɵɯɩɥɢɬɫɭɳɟɫɬɜɟɧɧɨ
ɫɥɨɠɧɟɟɜ ɪɹɞɟ ɫɥɭɱɚɟɜɩɨ ɬɟɯɧɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɞɨɩɭɫɤɚɟɬɫɹ ɪɚɫ
ɫɱɢɬɵɜɚɬɶ ɩɥɚɫɬɢɧɵ ɫɨɬɧɨɲɟɧɢɟɦ h/b ɞɨ 1/3.
ɉɪɢɪɟɲɟɧɢɢ ɡɚɞɚɱɢɢɡɝɢɛɚɩɥɚɫɬɢɧɵ ɡɚɨɫɧɨɜɧɭɸ ɧɟɢɡɜɟɫɬ
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
ɉɟɪɟɦɟɳɟɧɢɹɢɞɟɮɨɪɦɚɰɢɢɜɩɥɚɫɬɢɧɟ
ɋɨɝɥɚɫɧɨ ɩɟɪɜɨɣ ɝɢɩɨɬɟɡɟ Ʉɢɪɯɝɨɮɚ ɥɢɧɟɣɧɚɹ ɞɟɮɨɪɦɚɰɢɹ
w
w
, ɨɬɫɸɞɚ ɫɥɟɞɭɟɬ ɱɬɨ ɩɪɨɝɢɛɵ ɩɥɚɫɬɢɧɵ w ɧɟ ɡɚɜɢɫɹɬ
H
z
0
w
z
ɨɬɤɨɨɪɞɢɧɚɬɵ z, ɬ ɟ w=w(x, y)ɗɬɨɨɡɧɚɱɚɟɬɱɬɨɩɟɪɟɦɟɳɟɧɢɹ
ɜɫɟɯɬɨɱɟɤɨɞɧɨɣɧɨɪɦɚɥɢɜɞɨɥɶɨɫɢ z ɨɞɢɧɚɤɨɜɵɢɫɨɨɬɜɟɬɫɬɜɭɸɬ
12
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
ɩɟɪɟɦɟɳɟɧɢɹɦɬɨɱɤɢɧɚɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ ɋɥɟɞɨɜɚɬɟɥɶɧɨɞɨ
ɫɬɚɬɨɱɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɝɢɛɵ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ
ɱɬɨɛɵɡɧɚɬɶɩɟɪɟɦɟɳɟɧɢɹɜɫɟɯɟɟɬɨɱɟɤ
w
w
u
w
ɉɨ ɬɨɣ ɠɟ ɝɢɩɨɬɟɡɟ
ɫɸɞɚ ɩɨɥɭɱɚɟɦ
w
z
w
x
w
uww
ɩɨ z ɢɢɫɩɨɥɶɡɭɹɭɫɥɨɜɢɹ u
;
v
J
yz
w
z
w
vww
w
z
= 0 ɞɥɹɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɫɨ
0=v0
ɝɥɚɫɧɨɜɬɨɪɨɣɝɢɩɨɬɟɡɟ, ɩɨɥɭɱɚɟɦ
w
w
;
0
w
y
w
. ɂɧɬɟɝɪɢɪɭɹ ɷɬɢ ɭɪɚɜɧɟɧɢɹ
y
w
w
zu
x
w
w
J
zx
x
w
w
;
w
zv
y
w
, ɨɬ
0
w
z
. ɉɨɞɫɬɚɜ
ɥɹɹɷɬɢɡɚɜɢɫɢɦɨɫɬɢɜɭɪɚɜɧɟɧɢɹɄɨɲɢɢɦɟɟɦ
2
u
w
H
x
w
w
w
z
2
x
w
v
w
H
y
w
2
w
w
z
2
y
w
J
xyyx
v
u
w
w
x
y
w
w
2
w
w
(1.12)
.2;;
z
yx
ww
Ʉɚɤɜɢɞɧɨɞɟɮɨɪɦɚɰɢɢɩɪɨɢɡɜɨɥɶɧɨɝɨɝɨɪɢɡɨɧɬɚɥɶɧɨɝɨɫɥɨɹ
ɩɥɚɫɬɢɧɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ z ɦɟɧɹɸɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ
ɢɡɚɜɢɫɹɬɨɬɬɪɟɯɯɚɪɚɤɬɟɪɧɵɯɜɟɥɢɱɢɧ
1
F
x
U
2
w
w
;
2
x
w
x
1
F
y
U
2
w
w
2
y
w
y
2
w
w
(1.13)
.;
F
yx
ww
ȼɟɥɢɱɢɧɵ
ɬɨɣ ɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ
ɞɚɧɧɨɝɨɷɥɟɦɟɧɬɚ
ɫɨɫɬɚɜɥɹɸɬ ɤɪɢɜɢɡɧɵ ɷɥɟɦɟɧɬɚ dx×dy ɢɡɨɝɧɭ
FF ,
yx
– ɤɪɢɜɢɡɧɚ ɤɪɭɱɟɧɢɹ
F
– ɪɚɞɢɭɫɵɤɪɢɜɢɡɧɵ ɷɥɟɦɟɧɬɚɫɪɟɞɢɧɧɨɣ
UU ,
yx
ɩɨɜɟɪɯɧɨɫɬɢɫɨɨɬɜɟɬɫɬɜɟɧɧɨɜɞɨɥɶɨɫɢ x ɥɢɛɨ y.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜɫɟ ɩɟɪɟɦɟɳɟɧɢɹ ɢ ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ
ɨɤɚɡɵɜɚɸɬɫɹɜɵɪɚɠɟɧɧɵɦɢ ɱɟɪɟɡɨɞɧɭ ɮɭɧɤɰɢɸ ɩɪɨɝɢɛɨɜ ɟɟɫɪɟ
ɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢ w (x, y).
1.2.3. ɇɚɩɪɹɠɟɧɢɹɢɭɫɢɥɢɹɜɩɥɚɫɬɢɧɟ
Ɂɚɩɢɲɟɦɮɨɪɦɭɥɵɡɚɤɨɧɚ Ƚɭɤɚ (1.4) ɞɥɹɥɢɧɟɣɧɵɯ ɞɟɮɨɪɦɚ
ɰɢɣɫɭɱɟɬɨɦı
= ɫɨɝɥɚɫɧɨ ɬɪɟɬɶɟɣɝɢɩɨɬɟɡɟ Ʉɢɪɯɝɨɮɚ):
z
1
;)(
13
1
EE
VQV HVQV H
.)(
xyyyxx
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
y
y
y
ɋɥɨɠɢɦɷɬɢɪɚɜɟɧɫɬɜɚɩɨɨɱɟɪɟɞɧɨɭɦɧɨɠɢɜɤɚɠɞɨɟɢɡɧɢɯɧɚɤɨ
ɷɮɮɢɰɢɟɧɬɉɭɚɫɫɨɧɚȞ:
22
,
EE
/)1(;/)1(
VQ HQHVQ HQH
yxyxyx
ɨɬɫɸɞɚ
ɝɞɟ
ȿȿ HQH VHQH V
11
),(;)(
xyyyxx
1
2
.)1/(
Q EE
ɋɭɱɟɬɨɦ ɡɚɜɢɫɢɦɨɫɬɟɣ ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɹ ɧɨɪɦɚɥɶ
ɧɵɯ ɧɚɩɪɹɠɟɧɢɣɱɟɪɟɡɮɭɧɤɰɢɸɩɪɨɝɢɛɨɜ w:
2
§
w
w
¨
V
ȿ
1
¨
w
x
©
2
§
w
w
¨
V
ȿ
1
¨
w
y
©
2
·
w
w
¸
Q
2
2
2
w
y
2
w
w
Q
2
w
x
1
¸
¹
·
¸
¸
¹
QFF
;)(
zȿz
yxx
(1.14)
QFF
.)(
zȿz
1
xyy
Ʉɚɫɚɬɟɥɶɧɨɟɧɚɩɪɹɠɟɧɢɟ ɩɨɥɭɱɢɦɢɡ ɡɚɤɨɧɚ Ƚɭɤɚɜ ɨɛɪɚɬɧɨɣ
ɮɨɪɦɟɫɭɱɟɬɨɦɢ
2
E
W W W
)1(2
Q
ȿ
xyyxxy
1
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ ɧɚɩɪɹɠɟɧɢɹ ı
w
w
Q J
ww
1
yx
FQ
(1.15)
.)1()1(
zȿz
ıy, IJ
x
ɢɡɦɟɧɹɸɬɫɹɩɨɥɢɧɟɣɧɨɦɭɡɚɤɨɧɭ ɨɛɪɚɳɚɹɫɶ ɜ ɧɭɥɶ ɜ ɬɨɱɤɚɯ ɫɪɟ
ɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɭɤɚɡɚɧɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɩɨ ɜɵ
ɫɨɬɟ ɷɥɟɦɟɧɬɚ ɩɥɚɫɬɢɧɵ ɫ ɪɚɡɦɟɪɚɦɢ dx
ɚ)
h/2
h/2
dz
y
dy
z
My·dx
dx
Mx·dy
x
ı
x
ı
z
×dy ɩɨɤɚɡɚɧɨɧɚ ɪɢɫ. 1.2.
ɛ)
dy
h/2
h/2
H·dx
y
dx
IJ
= IJ
= IJ
x
z
x
H·dy
x
Ɋɢɫ 1.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟɧɚɩɪɹɠɟɧɢɣɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵ
ɚ – ɧɨɪɦɚɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹ; ɛ – ɤɚɫɚɬɟɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹ
14
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
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 ı
ɩɪɢɜɨɞɹɬɫɹɤɢɡɝɢɛɚɸɳɟɦɭɦɨɦɟɧɬɭ M
2/
h
xx
2/
h
1
)()(
ɧɚ ɝɪɚɧɢ ɷɥɟɦɟɧɬɚ h×dy
x
Âdy:
x
2/
h
2
dzzdyȿzdydzdyɆ
³³
2/
h
1
12
3
hE
QFF QFF V
dy
yxyx
,)(
ɝɞɟ ɜɟɥɢɱɢɧɚ M
ɦɟɧɬɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɧɚɩɪɹɠɟɧɢɸ ı
ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ɢɡɝɢɛɚɸɳɟɝɨ ɦɨ
x
. ɉɨ ɪɚɡɦɟɪɧɨɫɬɢ ɷɬɨ
x
ɦɨɦɟɧɬ ɞɟɥɟɧɧɵɣ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɫɟɱɟɧɢɹ (ɩɨɝɨɧɧɵɣ ɢɡɝɢɛɚ
ɸɳɢɣ ɦɨɦɟɧɬ ɟɝɨ ɪɚɡɦɟɪɧɨɫɬɶ ɤɇ Â ɦ/ɦ ɬ ɟ M
ɜɵɪɚɠɚɟɬɫɹ
x
ɜɟɞɢɧɢɰɚɯɫɢɥɵȼ ɞɚɥɶɧɟɣɲɟɦ ɢɧɬɟɧɫɢɜɧɨɫɬɶɢɡɝɢɛɚɸɳɟɝɨ ɦɨ
ɦɟɧɬɚM
ɛɭɞɟɦɧɚɡɵɜɚɬɶɩɪɨɫɬɨɦɨɦɟɧɬɨɦ M
x
ɜɞɚɧɧɨɣɬɨɱɤɟɫɟ
x
ɱɟɧɢɹɩɥɚɫɬɢɧɵɌɨɠɟɨɬɧɨɫɢɬɫɹɢɤɞɪɭɝɢɦɜɧɭɬɪɟɧɧɢɦɭɫɢɥɢɹɦ.
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚɩɪɹɠɟɧɢɹ IJ ɧɚ ɩɥɨɳɚɞɤɟ h
×dy ɩɪɢɜɨɞɹɬɫɹ
ɤɤɪɭɬɹɳɟɦɭɦɨɦɟɧɬɭ HÂdy:
12
3
hE
Q
)1(
F
.
dy
2/
h
)1()(
2/
h
1
2/
h
2
³³
2/
h
1
FQ W
dzzdyȿzdydzdyH
ɁɞɟɫɶɜɟɥɢɱɢɧɚH ɹɜɥɹɟɬɫɹɢɧɬɟɧɫɢɜɧɨɫɬɶɸɤɪɭɬɹɳɟɝɨɦɨɦɟɧɬɚ.
Ⱥɧɚɥɨɝɢɱɧɨɧɚɯɨɞɢɦɦɨɦɟɧɬɵ M
Âdx ɢ H Âdx. ɉɨɞɫɬɚɜɢɜɜɵ
y
ɪɚɠɟɧɢɹ ɤɪɢɜɢɡɧ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ
ɦɨɦɟɧɬɨɜɜɫɟɱɟɧɢɹɯɩɥɚɫɬɢɧɵɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯɤɟɟɫɪɟɞɢɧɧɨɣ
ɩɥɨɫɤɨɫɬɢ
2
·
w
w
¸
Q
Q
2
2
w
.)1()1(
yx
ww
;)(
2
¸
y
w
¹
2
·
w
w
¸
(1.16)
;)(
2
¸
x
w
¹
ȼɟɥɢɱɢɧɚ
2
§
w
w
¨
DDɆ
FQF
yxx
FQF
xyy
DDH
33
1
D
12
hEhE
ɧɚɡɵɜɚɟɬɫɹɰɢɥɢɧɞɪɢɱɟɫɤɨɣɠɟɫɬ-
2
)1(12
Q
2
¨
x
w
©
2
§
w
w
¨
DDɆ
¨
y
w
©
w
Q FQ
ɤɨɫɬɶɸɩɥɚɫɬɢɧɵ. ɗɬɚɜɟɥɢɱɢɧɚɹɜɥɹɟɬɫɹɮɢɡɢɤɨ-ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ
ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣɩɥɚɫɬɢɧɵ ɩɪɢɟɟ ɢɡɝɢɛɟ ɢɢɝɪɚɟɬɬɭɠɟɪɨɥɶɱɬɨ
15
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
y
y
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)
dy
dx
ɛ)
dy
dx
Q
y
H
M
h/2
h/2
ɚ – ɢɡɝɢɛɚɸɳɢɟɢɤɪɭɬɹɳɢɟɦɨɦɟɧɬɵ; ɛ – ɩɨɩɟɪɟɱɧɵɟɫɢɥɵɢɧɚɩɪɹɠɟɧɢɹ
H
M
y
z
Ɋɢɫ ɍɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹɜɩɥɚɫɬɢɧɟ
ɂɡɝɢɛɚɸɳɢɟɦɨɦɟɧɬɵ M
ɬɚɩɥɚɫɬɢɧɵɫɤɪɢɜɢɡɧɚɦɢ
x
x
ɢ M
x
ɢ
F
F
x
Q
x
h/2
h/2
IJ
z
Qyz
y
ɫɨɡɞɚɸɬɢɫɤɪɢɜɥɟɧɢɟɷɥɟɦɟɧ
y
ɄɪɭɬɹɳɢɟɦɨɦɟɧɬɵH ɫɨɡɞɚɸɬ
y
Q
x
IJ
xz
ɞɟɮɨɪɦɚɰɢɸ ɫɞɜɢɝɚ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɫɥɨɟɜ ɷɥɟɦɟɧɬɚ ɢɡɦɟɧɹɸɳɭ
ɸɫɹɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵɩɨɥɢɧɟɣɧɨɦɭɡɚɤɨɧɭ
ȼɵɪɚɡɢɜ ɤɪɢɜɢɡɧɵ ɜ (1.16) ɱɟɪɟɡ ɩɨɝɨɧɧɵɟ ɦɨɦɟɧɬɵ ɢ ɩɨɞ
ɫɬɚɜɢɜɜɮɨɪɦɭɥɵɞɥɹɧɚɩɪɹɠɟɧɢɣ1.14), (1.15)ɩɨɥɭɱɢɦ
M
Ezȿ
)(
yxx
11
D
M
Ezȿ
)(
xyy
)1(
11
D
H
Ezȿ
11
12
z
FQ W
D
M
12
z
QFF V
z
H
3
h
xx
z
;
3
h
M
12
yy
(1.17)
z
QFF V
.
z
;
3
h
x
Ʉɚɤɜɢɞɧɨɮɨɪɦɭɥɵɞɥɹɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣɫɨɜɩɚɞɚɸɬ
ɫɮɨɪɦɭɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜɩɪɢ ɢɡɝɢɛɟɛɚɥɤɢ ɩɪɹɦɨ
ɭɝɨɥɶɧɨɝɨɫɟɱɟɧɢɹɜɵɫɨɬɨɣ h ɢɲɢɪɢɧɨɣ ɪɚɜɧɨɣɟɞɢɧɢɰɟ
16
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
Ɇɚɤɫɢɦɚɥɶɧɵɟɡɧɚɱɟɧɢɹɧɚɩɪɹɠɟɧɢɣɜɨɡɧɢɤɚɸɬɩɪɢ
6
6
M
x
;
x
h
y
M
h
6
y
;
maxmaxmax
H
.
W V V
222
h
(1.18)
:
2hz r
Ʉɪɨɦɟ ɦɨɦɟɧɬɨɜ, ɜ ɫɟɱɟɧɢɹɯ ɩɥɚɫɬɢɧɵ ɞɟɣɫɬɜɭɸɬ ɩɨɩɟɪɟɱ
ɧɵɟɫɢɥɵɢɧɬɟɧɫɢɜɧɨɫɬɢɤɨɬɨɪɵɯɨɛɨɡɧɚɱɢɦ Q
ɸɬ ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ IJ
xzɢIJyz
, ɜɟɥɢɱɢɧɵ ɤɨɬɨɪɵɯ ɦɚɥɵ ɩɨ
ɫɪɚɜɧɟɧɢɸɫɜɟɥɢɱɢɧɚɦɢɨɫɧɨɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ V
ɞɟɥɟɧɢɟɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣIJ
, IJyzɩɨɬɨɥɳɢɧɟɩɥɚɫɬɢɧɵɫɨ
xz
ɢ Q
ɂɦɨɬɜɟɱɚ
x
y
, Vy, Wxyɪɚɫɩɪɟ
x
ɨɬɜɟɬɫɬɜɭɟɬɡɚɤɨɧɭɤɜɚɞɪɚɬɧɨɣɩɚɪɚɛɨɥɵ – ɪɢɫ. 1.3, ɛ).
Ʉɚɤɭɠɟ ɭɩɨɦɢɧɚɥɨɫɶ ɦɨɦɟɧɬɵ M
Q
ɢ Qyɩɨɥɨɠɢɬɟɥɶɧɵɟɫɥɢɞɥɹɬɨɱɤɢɩɥɚɫɬɢɧɵɫɤɨɨɪɞɢɧɚɬɨɣ z >0
x
, MyɢH, ɚ ɬɚɤɠɟ ɭɫɢɥɢɹ
x
ɨɧɢɞɚɸɬɩɨɥɨɠɢɬɟɥɶɧɵɟɧɚɩɪɹɠɟɧɢɹɂɧɞɟɤɫɵɩɪɢɭɫɢɥɢɹɯɫɨɨɬ
ɜɟɬɫɬɜɭɸɬɧɨɪɦɚɥɢɤɫɟɱɟɧɢɸɜɤɨɬɨɪɨɦɞɟɣɫɬɜɭɸɬɷɬɢɭɫɢɥɢɹ
1.2.4. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟɭɪɚɜɧɟɧɢɟɢɡɝɢɛɚɩɥɚɫɬɢɧɵ
Ɍɚɤɢɦɨɛɪɚɡɨɦɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹɢɧɚɩɪɹɠɟɧɢɹɜɩɥɚɫɬɢɧɟ
ɨɞɧɨɡɧɚɱɧɨ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɩɪɨɝɢɛɵ ɟɟ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ
ɉɨɫɥɟɬɨɝɨ 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 Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɷɬɨɣɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ 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
ɝɨɣɝɪɚɧɢ ɨɬɫɬɨɹɳɟɣ ɧɚɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɟ ɪɚɫɫɬɨɹɧɢɟ dx ɢɥɢ dy,
ɜɧɭɬɪɟɧɧɢɟ ɭɫɢɥɢɹ ɩɨɥɭɱɚɸɬ ɬɚɤɠɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɩɪɢɪɚɳɟ
ɧɢɹ ɧɚ ɜɟɥɢɱɢɧɭ ɱɚɫɬɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɧɚɩɪɢɦɟɪ
M
w
x
. ɉɪɢɷɬɨɦ ɜɫɟɩɨɝɨɧɧɵɟ ɭɫɢɥɢɹɫɥɟɞɭɟɬ ɭɦɧɨɠɚɬɶ
M
w
x
dx
x
w
ɧɚɞɥɢɧɭɝɪɚɧɢɩɨɤɨɬɨɪɨɣɨɧɢɞɟɣɫɬɜɭɸɬ
17
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
H
dy
dx
w
H
dx
x
w
w
M
x
M
x
H
dx
w
x
0
H
w
dy
y
w
z
Q
y
q
Q
x
Q
y
dy
H
M
y
0
M
x
z
H
M
dy
Ɋɢɫ ɗɥɟɦɟɧɬɫɪɟɞɢɧɧɨɣɩɥɨɫɤɨɫɬɢɩɥɚɫɬɢɧɵ
x
q
M
w
y
y
w
y
y
ɋɩɪɨɟɰɢɪɭɟɦɜɫɟɫɢɥɵɩɪɢɥɨɠɟɧɧɵɟɤɷɥɟɦɟɧɬɭɧɚ ɨɫɶ z:
Q
Q
w
Q
x
dydxqdxQdxdy
x
x
w
w
QdyQdydx
y
yx
y
w
y
ɉɨɫɥɟɩɪɢɜɟɞɟɧɢɹɩɨɞɨɛɧɵɯɱɥɟɧɨɜɩɨɥɭɱɢɦ
Q
w
Q
w
x
w
y
x
y
w
(1.19)
.q
Q
w
x
Q
w
w
y
y
x
dx
Q
x
w
x
y
dy
dx
.0)()(
Ⱥɧɚɥɨɝɢɱɧɨ ɢɡ ɭɪɚɜɧɟɧɢɣ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɨɫɟɣ y ɢ x, ɩɪɟɧɟɛɪɟɝɚɹɜɟɥɢɱɢɧɚɦɢ ɛɨɥɟɟɜɵɫɨɤɨɝɨ ɩɨɪɹɞɤɚɦɚɥɨ
ɫɬɢ, ɩɨɥɭɱɚɟɦ
w
w
M
w
H
x
w
x
Q
y
w
M
w
H
y
x
x
y
w
w
(1.20)
.;
Q
y
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɭɪɚɜɧɟɧɢɹ (1.19), (1.20) ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟ
ɥɹɸɬɪɚɜɧɨɜɟɫɢɟɷɥɟɦɟɧɬɚɩɥɚɫɬɢɧɵɂɫɤɥɸɱɢɦɢɡɷɬɢɯɭɪɚɜɧɟɧɢɣ
ɩɨɩɟɪɟɱɧɵɟ ɫɢɥɵ Q
x
ɢ Q
: ɨɩɪɟɞɟɥɢɜ ɢɯ ɢɡ ɭɪɚɜɧɟɧɢɣ
y
ɢɩɨɞɫɬɚɜɢɜɜɭɪɚɜɧɟɧɢɟ ɛɭɞɟɦɢɦɟɬɶ
2
Ɇ
w
x
2
x
w
2
w
2
Ɇ
w
H
yx
ww
y
.2
q
w
2
y
ɉɨɞɫɬɚɜɢɜɜɷɬɨɭɪɚɜɧɟɧɢɟɜɵɪɚɠɟɧɢɹɦɨɦɟɧɬɨɜɱɟɪɟɡɮɭɧɤ
ɰɢɸɩɪɨɝɢɛɨɜɢɭɩɪɨɫɬɢɜɩɨɥɭɱɢɦ
18
Ƚɥɚɜɚ 1. Ɉɫɧɨɜɵɬɟɨɪɢɢɪɚɫɱɟɬɚɢɡɝɢɛɚɟɦɵɯɩɥɚɫɬɢɧ
4
§
w
w
¨
D
Ɂɞɟɫɶ
2
4
¨
x
w
©
224
ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ;
4
w
w
ww
w
ww
w
2
4
·
w
w
¸
ɢɥɢ
22
yx
w
2
2
w
2
x
2
w
w
x
w
q
4
¸
y
¹
2
w
2
w
w
w
2
w
)()(
2
y
2
w
– ɨɛɵɱɧɵɣ ɝɚɪɦɨɧɢɱɟ
2
y
w
q
4
w
. (1.21)
D
– ɛɢɝɚɪɦɨɧɢɱɟɫɤɢɣ
ɫɤɢɣɨɩɟɪɚɬɨɪɅɚɩɥɚɫɚɨɬɮɭɧɤɰɢɢɩɪɨɝɢɛɨɜ w (x, y) ɩɥɚɫɬɢɧɵ
ɍɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟɢɡɨɝɧɭɬɨɣɫɪɟɞɢɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢɩɥɚɫɬɢɧɵ ɟɝɨɧɚɡɵ
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
ɬɨɱɤɟɩɥɚɫɬɢɧɵ ȼɦɟɫɬɟ ɫ ɬɟɦ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜ
ɧɟɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɨɩɪɟ
ɞɟɥɹɟɦɵɯɢɡɭɫɥɨɜɢɣɧɚɤɨɧɬɭɪɟɩɥɚɫɬɢɧɵ– ɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ.
Ɏɨɪɦɭɥɢɪɨɜɤɚɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɯɚɪɚɤɬɟɪɚ ɡɚɤɪɟɩɥɟɧɢɹ ɤɪɚɟɜ ɧɚ ɤɨɧɬɭɪɟ
ɩɥɚɫɬɢɧɵɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɩɪɨɝɢɛɵ ɢ ɭɝɥɵɩɨɜɨɪɨɬɚɫɪɟɞɢɧɧɨɣ
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 (ɬɟɦɨɦɟɧɬɵ ɢɫɢɥɵ)ɧɚɡɵ
ɜɚɸɬɫɹɫɬɚɬɢɱɟɫɤɢɦɢȿɫɥɢ ɠɟ ɧɚɨɞɧɨɦ ɭɱɚɫɬɤɟɤɨɧɬɭɪɚ ɡɚɞɚɧɵ
ɨɞɧɨɜɪɟɦɟɧɧɨɢɩɟɪɟɦɟɳɟɧɢɹ, ɢɭɫɢɥɢɹɬɨɬɚɤɢɟɝɪɚɧɢɱɧɵɟɭɫɥɨ
ɜɢɹɧɚɡɵɜɚɸɬɫɹɫɦɟɲɚɧɧɵɦɢ.
ɍɪɚɜɧɟɧɢɟɋɨɮɢ ɀɟɪɦɟɧ ɹɜɥɹɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜ
ɧɟɧɢɟɦɱɟɬɜɟɪɬɨɝɨɩɨɪɹɞɤɚɉɨɫɤɨɥɶɤɭɩɪɨɝɢɛɵɹɜɥɹɸɬɫɹɮɭɧɤɰɢ
ɟɣ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ (x, y) ɩɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɹ
ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɩɨɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɭɱɟɬɚ
ɜɨɫɶɦɢɝɪɚɧɢɱɧɵɯɭɫɥɨɜɢɣ– ɩɨɞɜɚɭɫɥɨɜɢɹɧɚɤɚɠɞɨɦɤɪɚɸ
19
ȺȺɅɭɤɚɲɟɜɢɱɌɟɨɪɢɹɪɚɫɱɟɬɚɩɥɚɫɬɢɧɢɨɛɨɥɨɱɟɤ
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=aɲɚɪɧɢɪɧɨɨɩɟɪɬɵɚɤɪɚɣy=bɫɜɨɛɨɞɟɧɨɬɡɚɤɪɟɩɥɟɧɢɣ
0
a
y
Ɋɢɫ ɉɪɹɦɨɭɝɨɥɶɧɚɹɩɥɚɫɬɢɧɚɫɪɚɡɥɢɱɧɵɦɢɡɚɤɪɟɩɥɟɧɢɹɦɢɤɪɚɟɜ
x
b
Ɂɚɞɟɥɚɧɧɵɣ ɢɥɢ ɡɚɳɟɦɥɟɧɧɵɣ) ɤɪɚɣȼɷɬɨɦɫɥɭɱɚɟɧɚɫɨɨɬ
ɜɟɬɫɬɜɭɸɳɟɦ ɭɱɚɫɬɤɟ ɤɨɧɬɭɪɚ ɩɥɚɫɬɢɧɵ ɨɬɫɭɬɫɬɜɭɸɬ ɩɪɨɝɢɛɵ
ɢɧɟɜɨɡɦɨɠɟɧɩɨɜɨɪɨɬɤɪɚɟɜɨɝɨɫɟɱɟɧɢɹɜɧɚɩɪɚɜɥɟɧɢɢɩɟɪɩɟɧɞɢ
ɤɭɥɹɪɧɨɦɤɷɬɨɦɭɤɪɚɸ Ɍɚɤɢɦɨɛɪɚɡɨɦɞɥɹɤɪɚɹy=0 ɢɦɟɟɦ
w
w
w
T
y
w
y
(1.22)
.0;0
00
yy
ɗɬɢɭɫɥɨɜɢɹɚɧɚɥɨɝɢɱɧɵɭɫɥɨɜɢɹɦɡɚɞɟɥɤɢɢɡɝɢɛɚɟɦɨɣɛɚɥɤɢ
ɒɚɪɧɢɪɧɨ ɨɩɟɪɬɵɣ ɤɪɚɣ Ɂɞɟɫɶ ɨɬɫɭɬɫɬɜɭɸɬ ɩɪɨɝɢɛɵ ɢ ɢɡ
ɝɢɛɚɸɳɢɟ ɦɨɦɟɧɬɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɤ ɷɬɨɦɭ
ɤɪɚɸɬɟ ɩɪɢ x=0 ɢ x=aɢɦɟɟɦw =0ɢM
=0.ȼɵɪɚɡɢɜ ɦɨɦɟɧɬ M
x
ɱɟɪɟɡɩɪɨɝɢɛɢ ɭɱɢɬɵɜɚɹɱɬɨɜɞɨɥɶɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɤɪɚɟɜ
2
w
ɢɡɝɢɛɚɧɟɬɬ ɟ
w
w
w
w
w
y
w
, ɜɢɬɨɝɟ ɩɨɥɭɱɢɦ
0
2
y
2
w
w
2
w
x
(1.23)
.0;0
,,
00
axax
ɍɫɥɨɜɢɹ ɫɩɪɚɜɟɞɥɢɜɵ ɞɥɹ ɨɩɢɪɚɧɢɹ ɤɪɚɟɜ ɩɥɚɫɬɢɧɵ ɧɚ
ɠɟɫɬɤɢɟ ɲɚɪɧɢɪɧɵɟ ɨɩɨɪɵ Ɍɚɤɠɟ ɨɬɦɟɬɢɦ ɱɬɨ ɟɫɥɢ ɤ ɤɚɤɨɦɭɥɢɛɨ ɲɚɪɧɢɪɧɨ ɨɩɟɪɬɨɦɭ ɤɪɚɸ ɛɭɞɟɬ ɩɪɢɥɨɠɟɧ ɜɧɟɲɧɢɣ ɪɚɫɩɪɟ
20
x
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