Федоров М.П. и др. Информационно-компьютерные технологии в строительстве. Использование программы SCAD для расчета сооружений из плоских элементов
Подождите немного. Документ загружается.
10
ɦɚɬɟɪɢɚɥɨɜ ɢ ɩɪɢ ɪɚɡɪɚɛɨɬɤɟ ɦɟɬɨɞɨɜ ɪɚɫɱɟɬɚ ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦ ɜ
ɫɬɪɨɢɬɟɥɶɧɨɣ ɦɟɯɚɧɢɤɟ ɩɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɪɚɫɱɟɬɧɵɯ ɫɯɟɦ ɜ
ɛɨɥɶɲɢɧɫɬɜɟ ɡɚɞɚɱ ɩɪɢɧɢɦɚɥɢɫɶ ɫɥɟɞɭɸɳɢɟ ɨɫɧɨɜɧɵɟ ɞɨɩɭɳɟɧɢɹ.
1. Ɇɚɬɟɪɢɚɥ, ɢɡ ɤɨɬɨɪɨɝɨ ɜɵɩɨɥɧɟɧ ɤɚɠɞɵɣ ɨɬɞɟɥɶɧɵɣ ɫɬɟɪɠɟɧɶ,
ɨɛɥɚɞɚɟɬ ɫɜɨɣɫɬɜɨɦ ɫɩɥɨɲɧɨɫɬɢ (ɨɧ ɧɟɩɪɟɪɵɜɟɧ ɩɨ ɜɫɟɦɭ ɨɛɴɟɦɭ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɷɥɟɦɟɧɬɚ ɫɨɨɪɭɠɟɧɢɹ). Ɇɨɥɟɤɭɥɹɪɧɨɟ ɫɬɪɨɟɧɢɟ ɦɚɬɟɪɢɚɥɚ
ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɟɬɨɞɢɤɢ ɪɚɫɱɟɬɚ ɇȾɋ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ.
ɗɬɨ ɞɨɩɭɳɟɧɢɟ ɩɨɡɜɨɥɹɟɬ ɩɪɟɞɫɬɚɜɥɹɬɶ ɩɟɪɟɦɟɳɟɧɢɹ ɬɟɥɚ ɤɚɤ
ɧɟɩɪɟɪɵɜɧɵɟ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɟ ɮɭɧɤɰɢɢ ɤɨɨɪɞɢɧɚɬ ɢ ɩɪɢɦɟɧɹɬɶ ɞɥɹ
ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɚɩɩɚɪɚɬ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɢ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ.
2. Ɇɚɬɟɪɢɚɥ ɹɜɥɹɟɬɫɹ ɢɞɟɚɥɶɧɨ ɭɩɪɭɝɢɦ ɢ ɩɨɞɱɢɧɹɸɳɢɦɫɹ ɡɚɤɨɧɭ Ƚɭɤɚ,
ɤɨɬɨɪɵɣ, ɧɚɩɪɢɦɟɪ, ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ (ɫɠɚɬɢɢ) ɫɬɟɪɠɧɹ ɫ ɩɥɨɳɚɞɶɸ
ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ F ɢ ɩɪɨɞɨɥɶɧɨɣ ɫɢɥɨɣ N ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
E
F
lN
l '
, ɢɥɢ
H
V
E
.
(1.2)
Ɂɞɟɫɶ
l
l'
H
– ɭɩɪɭɝɚɹ ɞɟɮɨɪɦɚɰɢɹ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ l, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ
ɟɝɨ ɭɞɥɢɧɟɧɢɸ (ɭɤɨɪɨɱɟɧɢɸ) ɨɬ ɫɢɥɵ N ɧɚ ɜɟɥɢɱɢɧɭ
l
' ; F
N
/
V
– ɫɪɟɞɧɟɟ
ɧɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɜ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ ɫɬɟɪɠɧɹ; E – ɦɨɞɭɥɶ
ɩɪɨɞɨɥɶɧɨɣ ɭɩɪɭɝɨɫɬɢ ɦɚɬɟɪɢɚɥɚ ɫɬɟɪɠɧɹ (ɦɨɞɭɥɶ ɘɧɝɚ).
ɂɡ ɩɟɪɜɨɝɨ ɜɵɪɚɠɟɧɢɹ (1.2) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɨɞɨɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɫɬɟɪɠɧɹ
l
l
/
'
H
ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɪɨɞɨɥɶɧɨɣ ɫɢɥɟ N ɢ ɨɛɪɚɬɧɨ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɪɨɞɨɥɶɧɨɣ ɠɟɫɬɤɨɫɬɢ ɫɬɟɪɠɧɹ
E
F . ɂɡ ɜɬɨɪɨɝɨ ɜɵɪɚɠɟɧɢɹ
(1.2) ɜɢɞɧɨ, ɱɬɨ ɧɨɪɦɚɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ F
N
/
V
ɜ ɫɬɟɪɠɧɟ ɩɪɹɦɨ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɩɪɨɞɨɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɢ.
ɉɨɫɥɟ ɭɞɚɥɟɧɢɹ ɫɢɥɵ
N
ɜɫɟ ɱɚɫɬɢɰɵ ɫɬɟɪɠɧɹ ɜɨɡɜɪɚɬɹɬɫɹ ɜ ɟɝɨ
ɩɟɪɜɨɧɚɱɚɥɶɧɨɟ (ɞɨ ɩɪɢɥɨɠɟɧɢɹ ɫɢɥɵ) ɩɨɥɨɠɟɧɢɟ, ɬ.ɟ. ɞɟɮɨɪɦɚɰɢɹ ɫɬɟɪɠɧɹ ɢ
ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɧɟɦ ɩɨɥɧɨɫɬɶɸ ɢɫɱɟɡɧɭɬ.
3. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɦɚɬɟɪɢɚɥ ɨɛɥɚɞɚɟɬ ɫɜɨɣɫɬɜɨɦ ɨɞɧɨɪɨɞɧɨɫɬɢ.
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɬɟɥɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɨɞɢɧɚɤɨɜɵɯ ɧɚɩɪɹɠɟɧɢɣ
ɩɪɨɢɫɯɨɞɹɬ ɨɞɢɧɚɤɨɜɵɟ ɞɟɮɨɪɦɚɰɢɢ.
4. Ɇɚɬɟɪɢɚɥ ɫɱɢɬɚɟɬɫɹ ɢɡɨɬɪɨɩɧɵɦ, ɬ.ɟ. ɭɩɪɭɝɢɟ ɫɜɨɣɫɬɜɚ ɦɚɬɟɪɢɚɥɚ
ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦɢ ɜɨ ɜɫɟɯ ɧɚɩɪɚɜɥɟɧɢɹɯ.
5. ɉɟɪɟɦɟɳɟɧɢɹ ɬɨɱɟɤ ɢ ɞɟɮɨɪɦɚɰɢɢ ɭɩɪɭɝɨɝɨ ɫɬɟɪɠɧɹ ɢɥɢ ɫɬɟɪɠɧɟɜɨɣ
ɫɢɫɬɟɦɵ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɪɚɡɦɟɪɚɦɢ ɫɬɟɪɠɧɹ ɢɥɢ ɫɬɟɪɠɧɟɜɨɣ ɫɢɫɬɟɦɵ.
ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɜ ɪɚɫɱɟɬɚɯ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɨɥɶɤɨ ɪɚɫɱɟɬɧɭɸ ɫɯɟɦɭ
ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɷɥɟɦɟɧɬɚ ɫɨɨɪɭɠɟɧɢɹ ɢɥɢ ɜɫɟɝɨ ɫɨɨɪɭɠɟɧɢɹ ɢ
11
ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ (ɩɪɢɧɰɢɩ ɧɚɥɨɠɟɧɢɹ ɢ ɩɪɢɧɰɢɩ
ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɞɟɣɫɬɜɢɹ ɫɢɥ).
ȼ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɜ ɨɫɧɨɜɧɨɦ ɢɫɩɨɥɶɡɭɸɬɫɹ ɷɬɢ ɠɟ ɞɨɩɭɳɟɧɢɹ.
ȼ ɤɭɪɫɚɯ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɢ ɫɬɪɨɢɬɟɥɶɧɨɣ ɦɟɯɚɧɢɤɢ
ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦ ɤɪɨɦɟ ɭɤɚɡɚɧɧɵɯ ɞɨɩɭɳɟɧɢɣ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ ɪɚɡɥɢɱɧɨɝɨ
ɪɨɞɚ ɪɚɛɨɱɢɟ ɝɢɩɨɬɟɡɵ, ɤɨɬɨɪɵɟ ɭɩɪɨɳɚɸɬ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɧɨ ɡɚɜɟɞɨɦɨ
ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɜɥɢɹɸɬ ɧɚ ɪɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ.
ɉɪɢɦɟɪɚɦɢ ɬɚɤɢɯ ɝɢɩɨɬɟɡ ɹɜɥɹɸɬɫɹ: ɝɢɩɨɬɟɡɚ ɩɥɨɫɤɢɯ ɫɟɱɟɧɢɣ ɩɪɢ
ɨɩɪɟɞɟɥɟɧɢɢ ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɜ ɬɨɧɤɢɯ ɫɬɟɪɠɧɹɯ ɩɪɢ ɩɨɩɟɪɟɱɧɨɦ
ɢɡɝɢɛɟ; ɩɪɟɧɟɛɪɟɠɟɧɢɟ ɩɟɪɟɦɟɳɟɧɢɹɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ ɭɡɥɨɜ
ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɦɟɬɨɞɚ ɩɟɪɟɦɟɳɟɧɢɣ ɜ
ɫɬɪɨɢɬɟɥɶɧɨɣ ɦɟɯɚɧɢɤɟ ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦ ɢ ɬ.ɞ.
ȼ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɩɨɞɨɛɧɨɝɨ ɪɨɞɚ ɪɚɛɨɱɢɟ ɝɢɩɨɬɟɡɵ ɬɚɤɠɟ
ɢɫɩɨɥɶɡɭɸɬɫɹ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɟɬɨɞɢɤɢ ɪɚɫɱɟɬɚ ɬɨɧɤɢɯ ɩɥɢɬ
ɢɫɩɨɥɶɡɭɟɬɫɹ (ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɝɢɩɨɬɟɡɨɣ ɩɥɨɫɤɢɯ ɫɟɱɟɧɢɣ ɞɥɹ ɬɨɧɤɢɯ ɫɬɟɪɠɧɟɣ)
ɝɢɩɨɬɟɡɚ ɩɪɹɦɵɯ ɧɨɪɦɚɥɟɣ.
ɉɪɢ ɪɟɲɟɧɢɢ ɪɚɡɥɢɱɧɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ
ɩɪɢɧɰɢɩ ɋɚɧ-ȼɟɧɚɧɚ, ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ ɜ ɬɨɱɤɚɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɬɟɥɚ,
ɞɨɫɬɚɬɨɱɧɨ ɭɞɚɥɟɧɧɵɯ ɨɬ ɦɟɫɬɚ ɩɪɢɥɨɠɟɧɢɹ ɜɧɟɲɧɟɣ ɧɚɝɪɭɡɤɢ, ɧɚɩɪɹɠɟɧɢɹ
ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɫɩɨɫɨɛɚ ɩɪɢɥɨɠɟɧɢɹ ɷɬɨɣ ɧɚɝɪɭɡɤɢ. ɗɬɨ ɩɨɡɜɨɥɹɟɬ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ ɪɚɫɱɟɬɚɯ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɫɬɚɬɢɱɟɫɤɢ ɷɤɜɢɜɚɥɟɧɬɧɵɟ
ɧɚɝɪɭɡɤɢ.
ɉɪɢɧɰɢɩ ɋɚɧ-ȼɟɧɚɧɚ ɦɨɠɧɨ ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɬɶ ɧɚ ɩɪɢɦɟɪɟ ɩɥɚɫɬɢɧɵ,
ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ.1.3.
ɋɨɝɥɚɫɧɨ ɷɬɨɦɭ ɩɪɢɧɰɢɩɭ ɤɚɪɬɢɧɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɜ ɫɪɟɞɧɟɦ
ɨɫɥɚɛɥɟɧɧɨɦ ɫɟɱɟɧɢɢ ɢ ɡɧɚɱɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɢɡɦɟɧɹɬɫɹ, ɟɫɥɢ
ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɟ ɪɚɫɬɹɝɢɜɚɸɳɢɟ ɫɢɥɵ ɡɚɦɟɧɢɬɶ ɥɸɛɨɣ ɧɚɝɪɭɡɤɨɣ (ɧɚɩɪɢɦɟɪ,
ɪɚɜɧɨɦɟɪɧɨ-ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ), ɞɥɹ ɤɨɬɨɪɨɣ ɫɢɥɚ P ɛɭɞɟɬ ɹɜɥɹɬɶɫɹ
ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣ. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɫɢɥɚ ɢ ɪɚɜɧɨɦɟɪɧɨ
ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ ɹɜɥɹɸɬɫɹ ɫɬɚɬɢɱɟɫɤɢ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ.
1.3. ɉɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɫɜɟɞɟɧɢɹ ɨ ɩɨɫɬɚɧɨɜɤɟ ɢ ɦɟɬɨɞɚɯ ɪɟɲɟɧɢɹ ɨɫɧɨɜɧɨɣ
ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ
ɍɱɚɳɟɦɭɫɹ ɢɡ ɤɭɪɫɨɜ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɢ ɫɬɪɨɢɬɟɥɶɧɨɣ
ɦɟɯɚɧɢɤɢ ɫɬɟɪɠɧɟɜɵɯ ɫɢɫɬɟɦ ɭɠɟ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɢɫɫɥɟɞɨɜɚɧɢɟ ɇȾɋ ɫɨɨɪɭɠɟɧɢɣ
ɢɥɢ ɢɯ ɷɥɟɦɟɧɬɨɜ ɫɜɨɞɢɬɫɹ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɬɪɟɯ ɬɢɩɨɜ ɧɟɢɡɜɟɫɬɧɵɯ ɜɟɥɢɱɢɧ:
ɧɚɩɪɹɠɟɧɢɣ (ɢɥɢ ɭɫɢɥɢɣ, ɩɨ ɤɨɬɨɪɵɦ ɡɚɬɟɦ ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɩɪɹɠɟɧɢɹ),
12
ɩɟɪɟɦɟɳɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ.
ɉɪɢ ɷɬɨɦ ɦɟɬɨɞɢɤɚ ɪɚɫɱɟɬɨɜ ɨɛɵɱɧɨ ɫɬɪɨɢɬɫɹ ɬɚɤ, ɱɬɨ ɫɧɚɱɚɥɚ ɧɚɯɨɞɹɬɫɹ
ɫɨɫɬɚɜɥɹɸɳɢɟ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɧɟɤɨɬɨɪɵɦ ɤɨɨɪɞɢɧɚɬɧɵɦ
ɨɫɹɦ, ɩɨ ɤɨɬɨɪɵɦ ɡɚɬɟɦ ɜɵɱɢɫɥɹɸɬɫɹ ɞɪɭɝɢɟ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ ɩɨ ɥɸɛɵɦ
ɧɚɩɪɚɜɥɟɧɢɹɦ.
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ, ɩɟɪɟɦɟɳɟɧɢɣ ɢ
ɞɟɮɨɪɦɚɰɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɬɪɢ ɝɪɭɩɩɵ ɭɪɚɜɧɟɧɢɣ:
1. ɍɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɫɬɚɬɢɤɢ), ɨɬɪɚɠɚɸɳɢɟ ɪɚɜɧɨɜɟɫɢɟ ɥɸɛɨɝɨ
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɷɥɟɦɟɧɬɚ ɬɟɥɚ;
2. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ, ɫɜɹɡɵɜɚɸɳɢɟ ɦɟɠɞɭ ɫɨɛɨɣ ɩɟɪɟɦɟɳɟɧɢɹ ɢ
ɞɟɮɨɪɦɚɰɢɢ;
3. Ɏɢɡɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ ɫɜɹɡɢ ɧɚɩɪɹɠɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ (ɜ ɥɢɧɟɣɧɨ-
ɞɟɮɨɪɦɢɪɭɟɦɨɦ ɬɟɥɟ – ɭɪɚɜɧɟɧɢɹ ɡɚɤɨɧɚ Ƚɭɤɚ).
Ʉɪɨɦɟ ɬɨɝɨ, ɧɚ ɝɪɚɧɢɰɟ ɬɟɥɚ ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɩɨɥɧɟɧɵ ɡɚɞɚɧɧɵɟ
ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ.
ɇɚ ɨɫɧɨɜɟ ɫɞɟɥɚɧɧɵɯ ɜɵɲɟ ɞɨɩɭɳɟɧɢɣ ɨ ɫɜɨɣɫɬɜɚɯ ɦɚɬɟɪɢɚɥɚ ɞɥɹ
ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ ɫɨɨɪɭɠɟɧɢɹ ɢɥɢ ɟɝɨ ɷɥɟɦɟɧɬɚ ɜɫɟ
ɭɪɚɜɧɟɧɢɹ ɹɜɥɹɸɬɫɹ
ɥɢɧɟɣɧɵɦɢ, ɬɨ ɟɫɬɶ ɧɟɢɡɜɟɫɬɧɵɟ ɜɟɥɢɱɢɧɵ, ɢ ɨɩɟɪɚɰɢɢ ɧɚɞ ɧɢɦɢ ɜɯɨɞɹɬ ɬɨɥɶɤɨ
ɜ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ. ɗɬɨ ɞɟɥɚɟɬ ɫɩɪɚɜɟɞɥɢɜɵɦ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɩɪɢɧɰɢɩɚ
ɫɭɩɟɪɩɨɡɢɰɢɢ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɇȾɋ.
ȼ ɞɚɧɧɨɦ ɍɆɄ ɩɨɥɭɱɟɧɢɟ ɩɨɥɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɪɟɲɟɧɢɹ
ɨɫɧɨɜɧɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɩɨɤɚɡɚɧɨ ɧɚ ɩɪɢɦɟɪɟ ɩɥɨɫɤɨɣ ɡɚɞɚɱɢ.
Ɋɟɲɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢ ɩɨɫɬɚɜɥɟɧɧɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ
ɜɵɩɨɥɧɹɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɢɥɢ ɱɢɫɥɟɧɧɵɦɢ ɦɟɬɨɞɚɦɢ [6, 7]. ȼ ɍɆɄ ɪɚɫɱɟɬ
ɇȾɋ ɞɥɹ ɤɨɧɤɪɟɬɧɵɯ ɫɨɨɪɭɠɟɧɢɣ ɢɥɢ ɢɯ ɷɥɟɦɟɧɬɨɜ ɜɵɩɨɥɧɹɟɬɫɹ ɧɚ ɉɄ ɫ
ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɱɢɫɥɟɧɧɨɝɨ ɦɟɬɨɞɚ ɜ ɮɨɪɦɟ ɆɄɗ, ɤɨɬɨɪɵɣ ɪɟɚɥɢɡɭɟɬɫɹ ɫ
ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɦɵ SCAD.
ɉɟɪɟɣɞɟɦ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ ɨɛɨɡɧɚɱɟɧɢɣ ɧɟɢɡɜɟɫɬɧɵɯ ɜɟɥɢɱɢɧ,
ɢɫɩɨɥɶɡɭɟɦɵɯ ɜ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ (ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɟɪɟɦɟɳɟɧɢɣ, ɧɚɩɪɹɠɟɧɢɣ ɢ
ɞɟɮɨɪɦɚɰɢɣ ɩɨ ɤɨɨɪɞɢɧɚɬɧɵɦ ɨɫɹɦ).
1.4. Ɉɛɨɡɧɚɱɟɧɢɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ
ȼ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɫɭɳɟɫɬɜɭɸɬ ɪɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ ɨɛɨɡɧɚɱɟɧɢɹ
ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ. Ɂɞɟɫɶ ɩɪɢɜɟɞɟɦ ɜɚɪɢɚɧɬ ɨɛɨɡɧɚɱɟɧɢɣ, ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ
ɢɫɩɨɥɶɡɭɸɳɢɣɫɹ ɜ ɭɱɟɛɧɢɤɚɯ ɩɨ ɫɨɩɪɨɬɢɜɥɟɧɢɸ ɦɚɬɟɪɢɚɥɨɜ, ɬɟɨɪɢɢ
ɭɩɪɭɝɨɫɬɢ, ɫɩɟɰɢɚɥɶɧɵɯ ɞɢɫɰɢɩɥɢɧɚɯ ɢ ɜ ɢɧɠɟɧɟɪɧɵɯ ɢ ɧɚɭɱɧɵɯ ɪɚɛɨɬɚɯ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɢɡɜɨɥɶɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɟ ɬɟɥɨ (Ɋɢɫ. 1.5), ɧɚɯɨɞɹɳɟɟɫɹ
13
ɜ ɪɚɜɧɨɜɟɫɢɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɢɯ ɢ ɜɧɭɬɪɟɧɧɢɯ (ɧɚɩɪɢɦɟɪ, ɫɨɛɫɬɜɟɧɧɵɣ
ɜɟɫ ɬɟɥɚ) ɫɢɥ ɢ ɨɬɧɟɫɟɧɧɨɟ ɤ ɩɪɚɜɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ X,
Y, Z. Ʉɨɦɩɨɧɟɧɬɵ ɩɟɪɟɦɟɳɟɧɢɣ ɤɚɤɨɣ-ɥɢɛɨ ɬɨɱɤɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x, y, z ɩɨ
ɧɚɩɪɚɜɥɟɧɢɸ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɜ ɷɬɨɦ ɜɚɪɢɚɧɬɟ ɨɛɨɡɧɚɱɚɸɬɫɹ
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜ ɜɢɞɟ (ɫɦ. ɪɢɫ. 1.5):
),,(),,,(),,,( zy
x
wwzy
x
v
v
zy
x
uu
(1.3)
ɉɨɥɨɠɢɬɟɥɶɧɵɦɢ ɫɱɢɬɚɸɬɫɹ ɩɟɪɟɦɟɳɟɧɢɹ, ɤɨɬɨɪɵɟ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ
ɫɨɜɩɚɞɚɸɬ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ.
Ɉɛɨɡɧɚɱɟɧɢɹ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɤɨɦɩɨɧɟɧɬ ɧɚɩɪɹɠɟɧɢɣ ɜ ɷɬɨɣ ɬɨɱɤɟ,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɫɹɦ ɨɛɳɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɜɚɪɢɚɧɬɟ,
ɢɡɨɛɪɚɠɟɧɨ ɧɚ ɪɢɫ. 1.6, ɝɞɟ ɞɥɹ ɭɞɨɛɫɬɜɚ ɩɨɤɚɡɚ ɩɥɨɫɤɨɫɬɟɣ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ
ɧɟɤɨɬɨɪɭɸ ɬɨɱɤɭ ɩɚɪɚɥɥɟɥɶɧɨ ɤɨɨɪɞɢɧɚɬɧɵɦ ɩɥɨɫɤɨɫɬɹɦ, ɨɤɨɥɨ ɬɨɱɤɢ ɜɵɞɟɥɟɧ
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ ɬɚɤ, ɱɬɨɛɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɬɨɱɤɚ ɨɤɚɡɚɥɚɫɶ
ɜ ɟɝɨ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ.
O
P
6
Y
P
5
Ɋɢɫ.1.5
u
v
P
4
X
Z
y
x
z
P
2
P
3
w
P
1
x
V
X
n /
/
Y
yx
W
y
V
x
V
xy
W
X
Z
y
V
zy
W
yz
W
xz
W
zx
W
z
V
Ɋɢɫ.1.6
14
ɉɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ ɜ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ ɬɟɥɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɧɟɲɧɟɣ ɧɨɪɦɚɥɢ n ɤ
ɩɥɨɳɚɞɤɟ, ɩɪɨɜɟɞɟɧɧɨɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ (ɫɦ. ɧɚ ɪɢɫ. 1.6 ɧɨɪɦɚɥɶ
X
n /
/
).
ȿɫɥɢ ɜɧɟɲɧɹɹ ɧɨɪɦɚɥɶ ɤ ɩɥɨɳɚɞɤɟ – ɝɪɚɧɢɰɟ ɜɵɞɟɥɟɧɧɨɝɨ ɷɥɟɦɟɧɬɚ
ɧɚɩɪɚɜɥɟɧɚ ɜ ɬɭ ɠɟ ɫɬɨɪɨɧɭ, ɱɬɨ ɢ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ, ɤɨɬɨɪɨɣ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɳɚɞɤɚ, ɬɨ ɢ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɨɫɬɚɜɥɹɸɳɢɯ
ɧɚɩɪɹɠɟɧɢɣ ɫɨɜɩɚɞɚɸɬ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɧɚɩɪɚɜɥɟɧɢɹɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ
ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ. ɉɪɢ ɨɛɪɚɬɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɜɧɟɲɧɟɣ ɧɨɪɦɚɥɢ ɤ ɩɥɨɳɚɞɤɟ
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɩɥɨɳɚɞɤɟ ɢɦɟɸɬ
ɬɚɤɠɟ ɨɛɪɚɬɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɹɦ
ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ.
Ʉɪɨɦɟ ɭɤɚɡɚɧɧɵɯ ɜɟɥɢɱɢɧ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ ɬɚɤɠɟ ɨɩɪɟɞɟɥɹɸɬɫɹ:
ɬɪɢ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɩɪɨɞɨɥɶɧɵɯ ɞɟɮɨɪɦɚɰɢɢ
),,(ˢˢ),,,(ˢˢ),,,(ˢˢ
zzyyxx
zyxzyxzyx
(1.4)
ɜɵɞɟɥɟɧɧɨɝɨ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɫɹɦ ɤɨɨɪɞɢɧɚɬɢ
ɟɝɨ ɬɪɢ ɫɞɜɢɝɨɜɵɯ ɞɟɮɨɪɦɚɰɢɢ
),,(),,,(),,,(
zxzxyzyzxyxy
zyxzyxzyx
J
J
J
J
J
J
.
(1.5)
ɏɚɪɚɤɬɟɪ ɩɪɨɞɨɥɶɧɵɯ ɢ ɫɞɜɢɝɨɜɵɯ ɞɟɮɨɪɦɚɰɢɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɢɯ
ɨɛɨɡɧɚɱɟɧɢɟɦ ɩɨɹɫɧɟɧ ɧɚ ɪɢɫ.1.7,
ɚ,ɛ, ɝɞɟ ɨɬɪɚɠɟɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɩɪɨɞɨɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ
x
H
ɷɥɟɦɟɧɬɚ ɜ ɧɚɩɪɚɜɥɟɧɢɢ X ɢ
ɩɨɥɨɠɢɬɟɥɶɧɚɹ ɫɞɜɢɝɨɜɚɹ ɞɟɮɨɪɦɚɰɢɹ
yxxy
J
J
.
Ɋɢɫ. 1.7
dx
Y
X
Z
ɚ)
yxxy
J
J
dx
x
H
ɛ)
15
ɋɞɜɢɝɨɜɚɹ ɞɟɮɨɪɦɚɰɢɹ
yxxy
J
J
ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɪɹɦɨɣ ɭɝɨɥ ɦɟɠɞɭ
ɝɪɚɧɹɦɢ ɷɥɟɦɟɧɬɚ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ YOZ ɢ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ (ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɜɧɟɲɧɟɣ ɧɨɪɦɚɥɢ) ɝɪɚɧɹɦɢ ɷɥɟɦɟɧɬɚ
ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ XOZ ɭɦɟɧɶɲɚɟɬɫɹ ɧɚ ɭɝɨɥ
yxxy
J
J
.
ȼ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɢ ɞɪɭɝɢɟ ɨɛɨɡɧɚɱɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ
ɜɟɥɢɱɢɧ. Ɉɞɧɚɤɨ ɤɚɤɢɟ ɛɵ ɨɛɨɡɧɚɱɟɧɢɹ ɧɟ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ, ɨɧɢ ɧɟ ɦɟɧɹɸɬ ɫɭɬɶ
ɩɨɫɬɚɧɨɜɨɤ ɢ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɡɚɞɚɱ.
1.5. Ⱦɜɚ ɜɚɪɢɚɧɬɚ ɩɥɨɫɤɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ
ȼɫɟ ɬɟɥɚ ɢɦɟɸɬ ɬɪɢ ɢɡɦɟɪɟɧɢɹ ɢ ɹɜɥɹɸɬɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦɢ ɬɟɥɚɦɢ.
ɉɨɷɬɨɦɭ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɢɯ ɪɚɫɱɟɬ ɫ ɰɟɥɶɸ ɩɨɥɭɱɟɧɢɹ ɇȾɋ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɡɚɞɚɱɢ.
Ɉɞɧɚɤɨ ɟɫɬɶ ɞɜɚ ɜɢɞɚ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɬɟɥ ɩɪɢ ɨɩɪɟɞɟɥɟɧɧɨɦ ɯɚɪɚɤɬɟɪɟ
ɢɯ ɡɚɝɪɭɠɟɧɢɣ, ɤɨɝɞɚ ɪɟɲɟɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɞɥɹ
ɷɬɢɯ ɬɟɥ ɦɨɠɧɨ ɡɚɦɟɧɢɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɪɟɲɟɧɢɹɦɢ ɩɥɨɫɤɨɣ ɡɚɞɚɱɢ.
ɉɥɨɫɤɨɟ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɬɟɥɚ
ɉɪɢɦɟɪɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɬɟɥɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɭɫɥɨɜɢɹɯ ɩɥɨɫɤɨɝɨ
ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɹɜɥɹɟɬɫɹ ɫɬɟɧɚ (ɩɥɚɫɬɢɧɚ) ɞɥɢɧɨɣ l ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ
ɨɫɢ X, ɜɵɫɨɬɨɣ h ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɨɫɢ Y (ɪɢɫ.1.8) ɢ ɬɨɥɳɢɧɨɣ b<<l, b<<h ɜ
ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Z, ɡɚɝɪɭɠɟɧɧɚɹ ɩɨ ɬɨɪɰɟɜɵɦ ɫɬɨɪɨɧɚɦ (ɫɬɨɪɨɧɵ, ɤɨɬɨɪɵɟ
ɢɦɟɸɬ ɬɨɥɳɢɧɭ ɩɥɚɫɬɢɧɵ b) ɬɚɤ, ɱɬɨ ɧɚɝɪɭɡɤɭ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɤ ɧɚɝɪɭɡɤɟ,
ɥɟɠɚɳɟɣ ɜ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ.
ɋɨɫɪɟɞɨɬɨɱɟɧɧɚɹ ɧɚɝɪɭɡɤɚ P ɫɱɢɬɚɟɬɫɹ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ
ɬɨɥɳɢɧɟ b ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ b
P
p
/
.
ɉɪɢ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɟ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɧɚɝɪɭɡɤɢ, ɧɚɩɪɢɦɟɪ ɧɚ
f
q
ɩɨ 1 - 1
X
Z
p
1
1
P
q(x)
q
b
(x)
Y
Y
f
b
Ɋɢɫ. 1.8
A
16
ɜɟɪɯɧɸɸ ɝɪɚɧɶ ɩɥɚɫɬɢɧɵ, ɫɨɫɬɚɜɥɹɟɬ bff
b
, ɝɞɟ
b
f í ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ ɩɨ ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ Ȼɨɤɨɜɵɟ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ
(ɫ ɪɚɡɦɟɪɚɦɢ
l ɢ h) ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɫɥɭɱɚɟ ɫɜɨɛɨɞɧɵ ɨɬ ɧɚɝɪɭɡɤɢ.
Ⱦɥɹ ɬɨɧɤɨɣ ɩɥɚɫɬɢɧɵ ɜɫɟ ɢɫɤɨɦɵɟ ɤɨɦɩɨɧɟɧɬɵ ɧɚɩɪɹɠɟɧɢɣ (ɫɦ. ɪɢɫ. 1.8)
ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ ɩɨ ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ
(ɜ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɧɟ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɤɨɨɪɞɢɧɚɬɵ
z).
ɉɪɢ ɷɬɨɦ, ɬɚɤ ɤɚɤ ɧɚ ɛɨɤɨɜɵɯ ɝɪɚɧɹɯ ɧɟɬ ɧɚɝɪɭɡɤɢ, ɫɱɢɬɚɟɬɫɹ, ɱɬɨ
ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɣ
zyzxz
W
W
V
,, ɪɚɜɧɵ ɧɭɥɸ ɧɟ ɬɨɥɶɤɨ ɧɚ ɛɨɤɨɜɵɯ ɝɪɚɧɹɯ
ɩɥɚɫɬɢɧɵ, ɧɨ ɢ ɩɨ ɟɟ ɬɨɥɳɢɧɟ.
ɋ ɭɱɟɬɨɦ ɢɡɜɟɫɬɧɨɝɨ ɟɳɟ ɢɡ ɤɭɪɫɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɫɜɨɣɫɬɜɚ
ɜɡɚɢɦɧɨɫɬɢ (ɩɚɪɧɨɫɬɢ) ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ
ɩɥɨɳɚɞɤɚɯ (ɫɦ. ɟɝɨ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɚɤɠɟ ɜ ɩɨɞɪɚɡɞɟɥɟ 2.1) ɛɭɞɭɬ ɪɚɜɧɵ ɧɭɥɸ ɢ
ɤɚɫɚɬɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ
xzzx
WW
ɢ
yzzy
W
W
(ɫɦ. ɪɢɫ. 1.6).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ-ɡɚ ɨɫɨɛɟɧɧɨɫɬɢ ɝɟɨɦɟɬɪɢɢ ɬɟɥɚ ɢ ɯɚɪɚɤɬɟɪɚ ɟɝɨ
ɡɚɝɪɭɠɟɧɢɹ, ɡɚɞɚɱɚ ɪɚɫɱɟɬɚ ɇȾɋ ɜ ɧɟɦ ɫɜɟɥɚɫɶ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɬɨɥɶɤɨ ɬɪɟɯ
ɤɨɦɩɨɧɟɧɬɨɜ ɧɚɩɪɹɠɟɧɢɣ, ɤɨɬɨɪɵɟ ɨɬɧɨɫɹɬɫɹ ɤ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɬɟɥɚ.
ɉɪɢ ɜɵɛɪɚɧɧɨɣ ɧɚ ɪɢɫ. 1.8 ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɷɬɢɦɢ ɤɨɦɩɨɧɟɧɬɚɦɢ
ɹɜɥɹɸɬɫɹ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ
y
ɢ
V
V
x
ɢ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ
yxxy
W
W
.
Ɉɩɢɫɚɧɧɨɟ ɧɚɩɪɹɠɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɩɥɨɫɤɢɦ ɧɚɩɪɹɠɟɧɧɵɦ
ɫɨɫɬɨɹɧɢɟɦ.
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ
ɩɪɢ ɞɟɣɫɬɜɢɢ ɧɚ ɩɥɚɫɬɢɧɭ ɧɚɝɪɭɡɤɢ,
ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɛɵɬɶ ɫɜɟɞɟɧɚ ɤ ɧɚɝɪɭɡɤɟ ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ
ɩɥɚɫɬɢɧɵ (ɫɦ. ɪɢɫ. 1.8), ɩɪɨɢɫɯɨɞɹɬ ɩɨɩɟɪɟɱɧɵɟ (ɫɢɦɦɟɬɪɢɱɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ) ɞɟɮɨɪɦɚɰɢɢ ɩɥɚɫɬɢɧɵ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɩɥɨɫɤɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɯɚɪɚɤɬɟɪɧɨ
ɨɬɫɭɬɫɬɜɢɟ ɫɨɫɬɚɜɥɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ
ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ, ɧɨ ɧɚɥɢɱɢɟ ɞɟɮɨɪɦɚɰɢɣ ɩɥɚɫɬɢɧɵ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ.
ɉɥɨɫɤɚɹ ɞɟɮɨɪɦɚɰɢɹ
ɉɪɢɦɟɪɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɬɟɥɚ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ
ɧɚɯɨɞɹɳɟɟɫɹ ɜ ɭɫɥɨɜɢɹɯ ɩɥɨɫɤɨɣ ɞɟɮɨɪɦɚɰɢɢ, ɹɜɥɹɟɬɫɹ ɛɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɚɹ
ɩɨɞɩɨɪɧɚɹ ɫɬɟɧɤɚ ɫ ɨɞɢɧɚɤɨɜɵɦ ɩɨ ɜɫɟɣ ɞɥɢɧɟ ɩɨɩɟɪɟɱɧɵɦ ɫɟɱɟɧɢɟɦ. Ⱦɥɹ
ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɧɚ ɪɢɫ. 1.9 ɢɡɨɛɪɚɠɟɧɨ ɧɟɤɨɬɨɪɨɟ ɩɨɩɟɪɟɱɧɨɟ ɫɟɱɟɧɢɟ ɬɚɤɨɣ
ɫɬɟɧɤɢ, ɩɚɪɚɥɥɟɥɶɧɨɟ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɥɨɫɤɨɫɬɢ
XOY.
ȼ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ
Z ɫɬɟɧɤɚ ɹɜɥɹɟɬɫɹ ɛɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɨɣ ( f
2
l ).
17
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɬɟɧɤɚ ɨɩɢɪɚɟɬɫɹ ɩɨ ɧɢɠɧɟɣ ɫɬɨɪɨɧɟ («ɩɨɞɨɲɜɟ») ɧɚ
ɧɟɤɨɬɨɪɨɟ ɫɨɜɟɪɲɟɧɧɨ ɨɞɢɧɚɤɨɜɨɟ ɩɨ ɜɫɟɣ ɞɥɢɧɟ
f
2
l ɫɬɟɧɤɢ ɨɫɧɨɜɚɧɢɟ (ɨɧɨ
ɭɫɥɨɜɧɨ ɩɨɤɚɡɚɧɨ ɲɬɪɢɯɨɜɤɨɣ).
ɇɚɝɪɭɡɤɚ ɧɚ ɫɬɟɧɤɭ ɬɚɤɨɜɚ, ɱɬɨ ɟɟ ɜɢɞ ɨɞɢɧɚɤɨɜ ɞɥɹ ɤɚɠɞɨɝɨ ɩɨɩɟɪɟɱɧɨɝɨ
ɫɟɱɟɧɢɹ (ɧɚɝɪɭɡɤɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɨɫɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ
ɩɨɩɟɪɟɱɧɨɦɭ ɫɟɱɟɧɢɸ ɫɬɟɧɤɢ).
ȼɵɞɟɥɢɦ ɢɡ ɫɬɟɧɤɢ ɫɥɨɣ ɞɜɭɦɹ ɫɟɱɟɧɢɹɦɢ, ɨɬɫɬɨɹɳɢɦɢ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ ɨɞɢɧ ɦɟɬɪ. ɉɨɥɭɱɢɦ ɩɥɚɫɬɢɧɭ ɟɞɢɧɢɱɧɨɣ ɬɨɥɳɢɧɵ.
ȼ ɨɬɥɢɱɢɟ ɨɬ ɫɥɭɱɚɹ ɩɥɨɫɤɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɬɚɤɚɹ ɩɥɚɫɬɢɧɚ ɧɟ
ɢɦɟɟɬ ɜɨɡɦɨɠɧɨɫɬɢ ɞɟɮɨɪɦɢɪɨɜɚɬɶɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Z, ɩɨɬɨɦɭ ɱɬɨ
ɫɨɫɟɞɧɢɟ ɫ ɧɟɣ ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ ɫɥɨɢ ɫɬɟɧɤɢ ɬɚɤɨɣ ɠɟ ɟɞɢɧɢɱɧɨɣ ɬɨɥɳɢɧɵ ɩɪɢ
f
2
l ɧɚɯɨɞɹɬɫɹ ɬɨɱɧɨ ɜ ɬɚɤɢɯ ɠɟ ɭɫɥɨɜɢɹɯ ɢ ɧɟ ɞɚɞɭɬ ɞɟɮɨɪɦɢɪɨɜɚɬɶɫɹ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦɭ ɫɥɨɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ
Z.
ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɩɪɨɢɫɯɨɞɢɬ ɬɨɥɶɤɨ
ɩɥɨɫɤɚɹ ɞɟɮɨɪɦɚɰɢɹ ɩɨɞɩɨɪɧɨɣ
ɫɬɟɧɤɢ.
ȼɫɥɟɞɫɬɜɢɟ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɫɥɨɟɜ ɩɨɥɭɱɢɦ
0
z
H
, ɩɨ ɢɯ
ɛɨɤɨɜɵɦ ɝɪɚɧɹɦ ɜɨɡɧɢɤɧɭɬ ɧɨɪɦɚɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ
z
V
.
ȼ ɪɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɩɨɞɩɨɪɧɚɹ ɫɬɟɧɤɚ ɧɟ ɹɜɥɹɟɬɫɹ ɛɟɫɤɨɧɟɱɧɨ ɞɥɢɧɧɨɣ.
Ɍɨɝɞɚ ɭɫɥɨɜɢɹ, ɯɚɪɚɤɬɟɪɧɵɟ ɞɥɹ ɩɥɨɫɤɨɣ ɞɟɮɨɪɦɚɰɢɢ, ɩɪɢ ɟɟ ɪɚɫɱɟɬɟ ɦɨɝɭɬ
ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɬɨɥɶɤɨ ɩɪɢɛɥɢɠɟɧɧɨ (ɞɥɹ ɫɟɱɟɧɢɣ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɧɚ
ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɤɨɧɰɨɜ ɫɬɟɧɤɢ).
ɑɚɫɬɨ ɬɚɤɨɣ ɩɨɞɯɨɞ ɜɩɨɥɧɟ ɞɨɩɭɫɬɢɦ ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ.
O
l
1
Ɋɢɫ.1.9
X
h(y)
q(y)
Y
18
2. ɉɅɈɋɄȺə ɁȺȾȺɑȺ ɌȿɈɊɂɂ ɍɉɊɍȽɈɋɌɂ
2.1. ɋɬɚɬɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ
ɇɚ ɪɢɫ. 1.6 ɫ ɩɨɦɨɳɶɸ ɝɪɚɧɟɣ ɢɡɨɛɪɚɠɟɧɧɨɝɨ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɛɵɥɢ
ɩɨɤɚɡɚɧɵ ɩɥɨɳɚɞɤɢ, ɩɪɨɜɟɞɟɧɧɵɟ ɱɟɪɟɡ ɧɟɤɨɬɨɪɭɸ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɬɨɱɤɭ
ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɬɟɥɚ (ɫɦ. ɪɢɫ. 1.5).
ȼɵɞɟɥɟɧɧɵɣ ɷɥɟɦɟɧɬ ɢɦɟɟɬ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɪɚɡɦɟɪɵ
dx, dy, dz (ɫɦ.
ɪɢɫ. 1.6).
Ɉɞɧɚɤɨ ɩɪɢ ɢɡɨɛɪɚɠɟɧɢɢ ɜɟɤɬɨɪɨɜ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɝɪɚɧɹɯ
ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ, ɱɬɨ ɷɬɢ ɪɚɡɦɟɪɵ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ, ɩɨɫɤɨɥɶɤɭ
ɩɥɨɳɚɞɤɢ ɫ ɩɨɤɚɡɚɧɧɵɦɢ ɧɚ ɪɢɫ. 1.6 ɫɨɫɬɚɜɥɹɸɳɢɦɢ ɧɚɩɪɹɠɟɧɢɣ ɩɪɨɯɨɞɹɬ
ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɱɟɪɟɡ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɬɨɱɤɭ. ɉɨɷɬɨɦɭ ɨɞɧɨɢɦɟɧɧɵɟ
ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɥɟɜɨɣ ɢ ɩɪɚɜɨɣ ɝɪɚɧɹɯ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ
(ɚɧɚɥɨɝɢɱɧɨ ɧɚ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɝɪɚɧɹɯ) ɢɡɨɛɪɚɠɟɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵɦɢ
ɞɪɭɝ ɞɪɭɝɭ.
Ⱦɥɹ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɫ ɭɤɚɡɚɧɧɵɦɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɦɢ ɪɚɡɦɟɪɚɦɢ
dx, dy, dz ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɥɨɳɚɞɤɚɯ ɛɭɞɭɬ
ɨɬɥɢɱɚɬɶɫɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɭɸ ɜɟɥɢɱɢɧɭ. ɗɬɨ ɩɨɤɚɡɚɧɨ ɧɚ ɩɪɢɦɟɪɟ
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɷɥɟɦɟɧɬɚ ɩɥɚɫɬɢɧɵ, ɜɵɞɟɥɟɧɧɨɝɨ ɭ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ
A ɧɚ
ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ (ɪɢɫ. 2.1).
dx
dy
dx
y
V
X
Y
O
dy
y
yx
yx
w
w
W
W
yx
W
x
V
dx
x
x
x
w
w
V
V
dx
x
xy
xy
w
w
W
W
Ɋɢɫ. 2.1
A
X
Y
dy
y
y
y
w
w
V
V
xy
W
19
Ɍɚɤ ɤɚɤ ɫɨɫɬɚɜɥɹɸɳɢɟ ɧɚɩɪɹɠɟɧɢɣ ɧɟ ɢɡɦɟɧɹɸɬɫɹ ɩɨ ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ,
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ ɪɚɡɦɟɪ
dz ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɡɚɦɟɧɢɦ ɤɨɧɟɱɧɵɦ ɪɚɡɦɟɪɨɦ b.
ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɞɥɹ ɩɥɨɫɤɨɣ ɡɚɞɚɱɢ ɜɦɟɫɬɨ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɜɨ ɜɫɟɯ
ɧɚɩɪɚɜɥɟɧɢɹɯ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɚ ɦɨɠɧɨ ɢɡɨɛɪɚɡɢɬɶ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ
ɩɪɹɦɨɭɝɨɥɶɧɢɤ, ɜɵɞɟɥɟɧɧɵɣ ɢɡ ɫɪɟɞɢɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ ɬɚɤ, ɱɬɨɛɵ
ɬɨɱɤɚ A ɧɚɯɨɞɢɥɚɫɶ ɜ ɰɟɧɬɪɟ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ, ɚ ɟɝɨ ɬɨɥɳɢɧɭ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ
ɬɨɥɳɢɧɟ ɩɥɚɫɬɢɧɵ
b (ɞɥɹ ɜɚɪɢɚɧɬɚ «ɉɥɨɫɤɚɹ ɞɟɮɨɪɦɚɰɢɹ» 1 b ).
Ʉɪɨɦɟ ɫɨɫɬɚɜɥɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ, ɩɪɢɯɨɞɹɳɢɯɫɹ ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ
ɝɪɚɧɟɣ ɷɥɟɦɟɧɬɚ, ɧɚ ɪɢɫ. 2.1 ɜ ɬɨɱɤɟ
A ɩɨɤɚɡɚɧɵ ɫɨɫɬɚɜɥɹɸɳɢɟ X ɢ Y ɨɛɴɟɦɧɵɯ
ɫɢɥ ɷɥɟɦɟɧɬɚ, ɩɪɢɯɨɞɹɳɢɟɫɹ ɧɚ ɟɞɢɧɢɰɭ ɨɛɴɟɦɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɜɧɨɜɟɫɢɟ ɜɵɞɟɥɟɧɧɨɝɨ ɷɥɟɦɟɧɬɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɬɚɬɢɱɟɫɤɢ
ɩɪɢɥɨɠɟɧɧɵɯ ɧɚ ɧɟɝɨ ɜɧɟɲɧɢɯ ɢ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ.
Ɍɚɤ ɤɚɤ ɷɥɟɦɟɧɬ ɧɚɯɨɞɢɬɫɹ ɜ ɭɫɥɨɜɢɹɯ ɩɥɨɫɤɨɣ ɡɚɞɚɱɢ, ɬɨ ɞɟɣɫɬɜɭɸɳɢɟ
ɫɢɥɵ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɬɪɟɦ ɭɪɚɜɧɟɧɢɹɦ ɪɚɜɧɨɜɟɫɢɹ:
¦¦¦
.0;0;0 YXM
A
(2.1)
ɋɧɚɱɚɥɚ ɫɨɫɬɚɜɢɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ (ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɫɟɯ
ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ
A ɪɚɜɧɚ ɧɭɥɸ), ɬɚɤ ɤɚɤ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɨɡɜɨɥɹɟɬ
ɞɨɤɚɡɚɬɶ ɫɜɨɣɫɬɜɨ ɩɚɪɧɨɫɬɢ (ɜɡɚɢɦɧɨɫɬɢ) ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɜɡɚɢɦɧɨ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɩɥɨɳɚɞɤɚɯ ɷɥɟɦɟɧɬɚ:
0
2
))((
2
)(
2
))((
2
)(
w
w
w
w
dy
bdxdy
y
dy
bdx
dx
bdydx
x
dx
bdy
yx
yxɭx
xy
xyxy
W
WW
W
WW
Ɇɨɦɟɧɬɵ ɨɬ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɢɯ ɧɨɪɦɚɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ ɩɨ ɝɪɚɧɹɦ
ɷɥɟɦɟɧɬɚ ɢ ɨɬ ɫɨɫɬɚɜɥɹɸɳɢɯ ɨɛɴɟɦɧɵɯ ɫɢɥ ɷɥɟɦɟɧɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ
A
ɪɚɜɧɵ ɧɭɥɸ, ɩɨɷɬɨɦɭ ɨɧɢ ɧɟ ɜɨɲɥɢ ɜ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ.
ɉɨɫɥɟ ɩɪɢɜɟɞɟɧɢɹ ɩɨɞɨɛɧɵɯ ɜɟɥɢɱɢɧ, ɫɨɤɪɚɳɟɧɢɹ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ
ɜɟɥɢɱɢɧ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɦɢ
ɜɟɥɢɱɢɧɚɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢ ɞɟɥɟɧɢɹ ɧɚ ɜɟɥɢɱɢɧɭ
bdxdy ɩɨɥɭɱɢɦ
ɭɩɨɦɹɧɭɬɨɟ ɫɜɨɣɫɬɜɨ
ɜɡɚɢɦɧɨɫɬɢ ɤɚɫɚɬɟɥɶɧɵɯ ɧɚɩɪɹɠɟɧɢɣ:
ɭxxy
W
W
.
(2.2)
Ⱦɜɚ ɞɪɭɝɢɯ ɭɪɚɜɧɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɷɥɟɦɟɧɬɚ ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ
ɩɪɢɧɢɦɚɸɬ ɜɢɞ (2.3).
.0Y
;0X
w
w
w
w
w
w
w
w
yx
yx
yxy
xy
x
VW
W
V
(2.3)