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Построение зоны навигационной безопасности объекта и его кинематических характеристик на основе обсервации двух разнесенных точек объекта.

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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.13-16
ɉɨɫɬɪɨɟɧɢɟ ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ ɨɛɴɟɤɬɚ ɢ ɟɝɨ
ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɧɚ ɨɫɧɨɜɟ ɨɛɫɟɪɜɚɰɢɢ ɞɜɭɯ
ɪɚɡɧɟɫɟɧɧɵɯ ɬɨɱɟɤ ɨɛɴɟɤɬɚ
ɋ.ȼ. ɉɚɲɟɧɰɟɜ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɫɭɞɨɜɨɠɞɟɧɢɹ
Ⱥɧɧɨɬɚɰɢɹ. ȼ ɪɚɛɨɬɟ ɪɚɫɫɦɨɬɪɟɧɚ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɫɬɪɨɟɧɢɹ ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ
ɩɨɞɜɢɠɧɨɝɨ ɨɛɴɟɤɬɚ ɢ ɟɝɨ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɩɪɢ ɨɛɫɟɪɜɚɰɢɢ ɞɜɭɯ ɟɝɨ ɪɚɡɧɟɫɟɧɧɵɯ ɬɨɱɟɤ.
ɉɨɫɬɪɨɟɧɵ ɬɨɱɧɨɫɬɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɥɸɛɨɣ ɬɨɱɤɢ ɨɛɴɟɤɬɚ ɩɪɢ ɢɡɜɟɫɬɧɵɯ ɷɥɥɢɩɫɚɯ ɪɚɫɫɟɢɜɚɧɢɹ
ɨɛɫɟɪɜɨɜɚɧɧɵɯ ɬɨɱɟɤ. ɇɚɜɢɝɚɰɢɹ, ɤɨɬɨɪɚɹ ɦɨɠɟɬ ɛɵɬɶ ɨɫɧɨɜɚɧɚ ɧɚ ɩɨɞɨɛɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɛ ɨɛɴɟɤɬɟ,
ɧɚɡɜɚɧɚ ɚɜɬɨɪɨɦ ɞɜɭɯɬɨɱɟɱɧɨɣ.
Abstract. The possibility of the construction of a navigational safety zone of a mobile vehicle and kinematic
characteristics under observation of its two distant points has been considered in the work. The method of
observing of coordinates errors of any point of the vehicle, when dispersion ellipses of observation points are
known, has been worked out in the paper. Navigation which can be based on this kind of information
concerning the vehicle has been called by the author as a two-point navigation.
1. ȼɜɟɞɟɧɢɟ
ɉɪɢ ɡɧɚɱɢɬɟɥɶɧɵɯ ɪɚɡɦɟɪɚɯ ɩɨɞɜɢɠɧɵɯ ɦɨɪɫɤɢɯ ɨɛɴɟɤɬɨɜ, ɫɪɚɜɧɢɦɵɯ ɫ ɪɚɡɦɟɪɚɦɢ ɫɚɦɢɯ
ɚɤɜɚɬɨɪɢɣ, ɜ ɤɨɬɨɪɵɯ ɩɪɨɢɫɯɨɞɢɬ ɟɝɨ ɦɚɧɟɜɪɢɪɨɜɚɧɢɟ, ɜ ɰɟɥɹɯ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɟɩɥɚɜɚɧɢɹ ɧɟɨɛɯɨɞɢɦɨ
ɡɧɚɬɶ ɩɨɥɨɠɟɧɢɟ ɜɫɟɯ (ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ) ɬɨɱɟɤ ɨɛɴɟɤɬɚ, ɚ ɧɟ ɬɨɥɶɤɨ ɟɝɨ ɰɟɧɬɪɚ ɦɚɫɫ.
Ⱦɥɹ ɠɟɫɬɤɨɝɨ ɬɟɥɚ ɞɨɫɬɚɬɨɱɧɨ ɡɧɚɧɢɟ ɩɨɥɨɠɟɧɢɹ ɜɫɟɝɨ ɞɜɭɯ ɬɨɱɟɤ, ɱɬɨɛɵ ɡɚɬɟɦ ɨɩɪɟɞɟɥɹɬɶ ɩɨɥɨɠɟɧɢɟ
ɥɸɛɨɣ ɟɝɨ ɬɨɱɤɢ. ȿɫɥɢ ɤ ɬɨɦɭ ɠɟ ɩɨɥɭɱɟɧɚ ɨɰɟɧɤɚ ɬɨɱɧɨɫɬɢ ɨɛɫɟɪɜɨɜɚɧɧɵɯ ɬɨɱɟɤ, ɬɨ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ
ɩɨɫɬɪɨɟɧɢɹ ɚɫɫɨɰɢɢɪɨɜɚɧɧɨɣ ɫ ɨɛɴɟɤɬɨɦ ɮɢɝɭɪɵ: ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ (ɁɇȻ).
Ʉɪɨɦɟ ɬɨɝɨ, ɡɧɚɧɢɟ ɜɟɤɬɨɪɨɜ ɫɤɨɪɨɫɬɟɣ ɷɬɢɯ ɠɟ ɛɚɡɨɜɵɯ ɬɨɱɟɤ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɥɧɨɫɬɶɸ
ɪɚɫɫɱɢɬɚɬɶ ɤɢɧɟɦɚɬɢɤɭ ɨɛɴɟɤɬɚ ɢ ɨɩɪɟɞɟɥɢɬɶ ɬɚɤɢɟ ɜɚɠɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɞɜɢɠɟɧɢɹ, ɤɚɤ ɜɟɤɬɨɪ ɩɭɬɟɜɨɣ
ɫɤɨɪɨɫɬɢ, ɭɝɨɥ ɞɪɟɣɮɚ, ɦɝɧɨɜɟɧɧɵɣ ɰɟɧɬɪ ɜɪɚɳɟɧɢɹ ɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɩɨɜɨɪɨɬɚ ɜɨɤɪɭɝ ɧɟɝɨ.
ɉɪɢ ɫɨɜɪɟɦɟɧɧɵɯ ɫɪɟɞɫɬɜɚɯ ɨɩɪɟɞɟɥɟɧɢɹ ɦɟɫɬɚ ɫ ɢɯ ɜɵɫɨɤɨɣ ɬɨɱɧɨɫɬɶɸ ɞɨɫɬɚɬɨɱɧɨ ɢɦɟɬɶ
ɨɛɫɟɪɜɚɰɢɢ ɷɬɢɯ ɞɜɭɯ ɬɨɱɟɤ ɫ ɥɢɧɟɣɧɨɣ ɩɨɝɪɟɲɧɨɫɬɶɸ, ɦɟɧɶɲɟɣ ɧɚ ɩɨɪɹɞɨɤ ɪɚɡɦɟɪɚ ɛɚɡɵ, ɬ.ɟ.
ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɜɭɦɹ ɨɛɫɟɪɜɨɜɚɧɧɵɦɢ ɬɨɱɤɚɦɢ ɨɛɴɟɤɬɚ. ɇɚɜɢɝɚɰɢɸ, ɨɫɧɨɜɚɧɧɭɸ ɧɚ ɨɩɢɫɚɧɧɨɣ ɜɵɲɟ
ɢɧɮɨɪɦɚɰɢɢ, ɦɨɠɧɨ ɧɚɡɜɚɬɶ "ɞɜɭɯɬɨɱɟɱɧɨɣ".
2. ɉɨɫɬɪɨɟɧɢɟ ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɞɜɢɠɧɵɣ ɨɛɴɟɤɬ ɜ ɧɟɤɨɬɨɪɨɣ ɚɛɫɨɥɸɬɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ X, Y (ɪɢɫ. 1). ɉɭɫɬɶ
ɤɨɨɪɞɢɧɚɬɵ ɧɨɫɨɜɨɣ ɢ ɤɨɪɦɨɜɨɣ ɬɨɱɟɤ ɨɛɴɟɤɬɚ ɛɭɞɭɬ (Xɇ, Yɇ) ɢ (XɄ, YɄ) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ,
ɱɬɨ ɨɛɴɟɤɬ ɫɧɚɛɠɟɧ ɫɪɟɞɫɬɜɚɦɢ ɨɛɫɟɪɜɚɰɢɢ, ɩɨɡɜɨɥɹɸɳɢɦɢ ɨɩɪɟɞɟɥɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɷɬɢɯ ɞɜɭɯ ɛɚɡɨɜɵɯ
ɬɨɱɟɤ ɢ ɨɰɟɧɢɬɶ ɩɨɝɪɟɲɧɨɫɬɶ ɬɚɤɨɣ ɨɛɫɟɪɜɚɰɢɢ ɫ ɩɨɦɨɳɶɸ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ ɤɚɠɞɨɣ ɢɡ ɬɨɱɟɤ.
ȼɜɟɞɟɦ ɞɚɥɟɟ ɥɨɤɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɭɸ ɫ ɨɛɴɟɤɬɨɦ. Ʉɨɨɪɞɢɧɚɬɧɚɹ ɨɫɶ X1
ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ Ⱦɉ ɨɛɴɟɤɬɚ ɨɬ ɤɨɪɦɨɜɨɣ ɬɨɱɤɢ ɤ ɧɨɫɨɜɨɣ, ɨɫɶ Y1 ɟɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ. ɉɨɩɪɨɛɭɟɦ
ɜɵɪɚɡɢɬɶ ɤɨɨɪɞɢɧɚɬɵ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ ɛɨɪɬɚ Ⱦ ɱɟɪɟɡ
ɤɨɨɪɞɢɧɚɬɵ ɧɨɫɨɜɨɣ ɢ ɤɨɪɦɨɜɨɣ ɬɨɱɟɤ. ɉɭɫɬɶ ɬɨɱɤɚ ȼ –
ɩɪɨɟɤɰɢɹ Ⱦ ɧɚ Ⱦɉ – ɭɞɚɥɟɧɚ ɨɬ ɤɨɪɦɵ ɢ ɧɨɫɚ ɧɚ ɪɚɫɫɬɨɹɧɢɹ
LɄ ɢ Lɇ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɬɚɤ ɱɬɨ ɞɥɢɧɚ ɨɛɴɟɤɬɚ ɜɞɨɥɶ Ⱦɉ
L = LɄ + Lɇ. ȿɫɥɢ ɜɜɟɫɬɢ ɨɬɧɨɲɟɧɢɹ a = LɄ/L ɢ b = Lɇ/L, ɬɨ
ɥɟɝɤɨ ɧɚɣɬɢ ɤɨɨɪɞɢɧɚɬɵ ȼ, ɤɚɤ ɬɨɱɤɢ, ɞɟɥɹɳɟɣ ɨɬɪɟɡɨɤ ɇɄ
ɜ ɨɬɧɨɲɟɧɢɢ O = a/b:
Xȼ = (Xɇ O + XɄ)/(1+O),
Yȼ = (Yɇ O + YɄ)/(1+O).
(1)
ɉɪɢ ɷɬɨɦ ɤɨɪɦɨɜɨɣ ɬɨɱɤɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɧɚɱɟɧɢɟ O = 0, ɚ
ɧɨɫɨɜɨɣ ɬɨɱɤɟ – ɡɧɚɱɟɧɢɟ O = f. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ
ɩɪɹɦɨɣ, ɤɨɬɨɪɚɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ Ⱦɉ ɜ ɬɨɱɤɟ ȼ:
Ɋɢɫ. 1. ɋɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɨɛɴɟɤɬɚ
Y – Yȼ = (X – Xȼ) / KȾɉ,
13
(2)
ɉɚɲɟɧɰɟɜ ɋ.ȼ., ɉɨɫɬɪɨɟɧɢɹ ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ ɨɛɴɟɤɬɚ...
ɭɱɢɬɵɜɚɹ ɢɡɜɟɫɬɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɭɝɥɨɜɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɩɪɹɦɵɯ k1 = 1/k2. Ɂɞɟɫɶ
KȾɉ = (Yɇ – YɄ)/(Xɇ XɄ) – ɭɝɥɨɜɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɹɦɨɣ ɥɢɧɢɢ Ⱦɉ. ɇɚ ɷɬɨɣ ɩɪɹɦɨɣ, ɜ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ
ɬɨɱɤɢ Ⱦ, ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ:
YȾ – Yȼ = (XȾ – Xȼ)/KȾɉ.
(3)
Ʉɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ d ɜɞɨɥɶ ɷɬɨɝɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɚ (1) ɨɬ ɬɨɱɤɢ ȼ ɞɨ ɬɨɱɤɢ Ⱦ ɧɚ ɨɛɜɨɞɟ ɫɭɞɧɚ ɦɨɠɧɨ
ɧɚɣɬɢ ɨɛɵɱɧɵɦ ɨɛɪɚɡɨɦ:
d2 = (Xȼ XȾ)2 + (Yȼ – YȾ)2,
ɢ ɫ ɭɱɟɬɨɦ (3):
d2 (Xȼ XȾ)2 = (Xȼ XȾ)2/KȾɉ2,
ɢɥɢ
Xȼ – XȾ = rd (1 + 1/KȾɉ2)1/2,
ɱɬɨ ɩɪɢɜɨɞɢɬ, ɧɚɤɨɧɟɰ, ɤ ɜɵɪɚɠɟɧɢɹɦ ɞɥɹ ɤɨɨɪɞɢɧɚɬ ɛɨɪɬɨɜɨɣ ɬɨɱɤɢ Ⱦ ɨɛɴɟɤɬɚ:
XȾ = (XɄ + Xɇ O)/(1 + O) r d/L (Yɇ – YɄ),
YȾ = (YɄ + Yɇ O)/(1 + O) r d/L (Xɇ – XɄ)
(4)
(ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ L2 = (Xɇ XɄ)2 + (Yɇ YɄ)2, ɚ 1 + 1/KȾɉ2 = L/(Yɇ YɄ)). ɉɪɢ ɷɬɨɦ ɞɜɨɣɧɨɣ ɡɧɚɤ ɜɬɨɪɨɝɨ
ɫɥɚɝɚɟɦɨɝɨ ɨɩɪɟɞɟɥɹɟɬ ɤɨɨɪɞɢɧɚɬɭ ɥɟɜɨɝɨ (ɩɪɚɜɨɝɨ) ɛɨɪɬɨɜ.
ȼɵɜɟɞɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɢɫɤɨɦɵɯ ɤɨɨɪɞɢɧɚɬ ɥɢɧɟɣɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɭɱɟɧɧɵɯ ɩɪɢ
ɨɛɫɟɪɜɚɰɢɢ ɤɨɨɪɞɢɧɚɬ ɧɨɫɨɜɨɣ ɢ ɤɨɪɦɨɜɨɣ ɬɨɱɟɤ ɨɛɴɟɤɬɚ, ɢ ɷɬɚ ɥɢɧɟɣɧɨɫɬɶ ɩɨɡɜɨɥɢɬ ɥɟɝɤɨ ɧɚɣɬɢ
ɞɢɫɩɟɪɫɢɢ ɷɬɢɯ ɬɨɱɟɤ ɩɪɢ ɢɡɜɟɫɬɧɵɯ ɞɢɫɩɟɪɫɢɨɧɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɬɨɱɟɤ Ʉ ɢ ɇ. ɉɪɚɜɞɚ, ɩɪɢɦɟɧɢɬɶ
ɨɩɟɪɚɬɨɪ ɞɢɫɩɟɪɫɢɢ ɤ ɨɛɟɢɦ ɱɚɫɬɹɦ ɪɚɜɟɧɫɬɜ (4) ɫɥɨɠɧɨ, ɬɚɤ ɤɚɤ ɨɬɞɟɥɶɧɵɟ ɫɥɚɝɚɟɦɵɟ ɩɪɚɜɵɯ ɱɚɫɬɟɣ
ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɡɚɜɢɫɢɦɵ ɞɪɭɝ ɨɬ ɞɪɭɝɚ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, d/L ɟɫɬɶ ɮɭɧɤɰɢɹ O, ɡɚɜɢɫɹɳɚɹ ɨɬ ɮɨɪɦɵ
ɨɛɜɨɞɨɜ ɫɭɞɧɚ. ɉɪɢ ɜɡɹɬɢɢ ɞɢɫɩɟɪɫɢɢ ɨɬ ɩɪɚɜɨɣ ɱɚɫɬɢ ɩɪɢɲɥɨɫɶ ɛɵ ɭɱɢɬɵɜɚɬɶ ɷɬɭ ɫɜɹɡɶ, ɚ ɧɟ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɨɫɬɨɟ ɩɪɚɜɢɥɨ – "ɞɢɫɩɟɪɫɢɹ ɫɭɦɦɵ ɪɚɜɧɚ ɫɭɦɦɟ ɞɢɫɩɟɪɫɢɣ". ɉɨɷɬɨɦɭ ɩɨɣɞɟɦ ɞɪɭɝɢɦ
ɩɭɬɟɦ. ȼɨɡɶɦɟɦ ɨɬ ɨɛɟɢɯ ɱɚɫɬɟɣ ɪɚɜɟɧɫɬɜ (4) ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɠɢɞɚɧɢɹ, ɚ ɡɚɬɟɦ ɫ ɢɯ ɩɨɦɨɳɶɸ ɧɚɣɞɟɦ
ɨɬɤɥɨɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɨɬ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɨɠɢɞɚɧɢɣ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɠɢɞɚɧɢɹ ɤɨɨɪɞɢɧɚɬ XȾ ɢ YȾ:
M(XȾ) = (M(XɄ) + M(Xɇ) O)/(1 + O) + d/L (M(Yɇ) – M(YɄ)),
M(YȾ) = (M(YɄ) + M(Yɇ) O)/(1 + O) + d/L (M(Xɇ) – M(XɄ)).
(5)
Ɉɬɤɥɨɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬ XȾ ɢ YȾ ɨɬ ɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɨɠɢɞɚɧɢɣ:
VɏȾ = XȾ Ɇ(XȾ) = (Vɏɇ + VɏɄ O)/(1 + O) + G (VYɇ – VYɄ),
VYȾ = YȾ Ɇ(YȾ) = (VYɇ + VYɄ O)/(1 + O) + G (Vɏɇ – VɏɄ),
(6)
ɝɞɟ ɞɥɹ ɩɪɨɫɬɨɬɵ ɩɨɫɥɟɞɭɸɳɢɯ ɡɚɩɢɫɟɣ ɨɛɨɡɧɚɱɟɧɨ G = d/L. Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɫɬɪɭɤɬɭɪɚ
ɨɬɤɥɨɧɟɧɢɣ ɩɨɥɧɨɫɬɶɸ ɚɧɚɥɨɝɢɱɧɚ ɫɬɪɭɤɬɭɪɟ ɫɚɦɢɯ ɤɨɨɪɞɢɧɚɬ, ɜ ɫɢɥɭ ɢɯ ɥɢɧɟɣɧɨɫɬɢ, ɨ ɤɨɬɨɪɨɣ ɛɵɥɨ
ɫɤɚɡɚɧɨ ɜɵɲɟ. ȼ ɫɨɨɬɧɨɲɟɧɢɹɯ (5) ɢ (6) ɢ ɞɚɥɟɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɧɢɠɧɢɟ ɢɧɞɟɤɫɵ Ⱦ, ɇ ɢɥɢ Ʉ ɨɡɧɚɱɚɸɬ
ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɨɱɤɟ Ⱦ, ɇ ɢɥɢ Ʉ.
Ɂɧɚɹ ɨɬɤɥɨɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬ, ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ ɦɟɠɞɭ ɧɢɦɢ, ɤɨɬɨɪɨɝɨ
ɧɚɦ ɤɚɤ ɪɚɡ ɢ ɧɟ ɯɜɚɬɚɥɨ ɞɥɹ ɪɚɫɱɟɬɚ ɞɢɫɩɟɪɫɢɣ ɤɨɨɪɞɢɧɚɬ. ɂɫɯɨɞɢɦ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ
ɦɨɦɟɧɬɚ ɤɚɤ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɨɬ ɩɪɨɢɡɜɟɞɟɧɢɹ ɨɬɤɥɨɧɟɧɢɣ:
MXYȾ = M{[VɏɄ VYɄ + OVɏɇ VYɄ + OVɏɇ VYɇ + O2Vɏɇ VYɇ]/(1+O)2 +
+ G2 [VYɇ Vɏɇ – VYɄ Vɏɇ – VYɇ Vɏɇ + VYɄ Vɏɇ] +
+ G/(1+O) [VɏɄ Vɏɇ + OVɏɇVɏɇ – Vɏɇ VɏɄ OVɏɇ VɏɄ +
+ VYɄ VYɇ + O VYɇ VYɇ – VYɄ VYɄ O VYɇ VYɄ] } =
= (MXYɄ + O2MXYɇ)/(1+O)2 + G2 (MXYɄ + MXYɇ) + G/(1+O)(ODXɇ DXɄ + ODYɇ DYɄ).
(7)
ɇɚɣɞɟɦ ɬɚɤɠɟ ɤɜɚɞɪɚɬɵ ɨɬɤɥɨɧɟɧɢɣ (6):
VɏȾ 2 = (Vɏɇ + Vɏɇ O)2/(1+O)2 + G2(VYɇ – VYɄ)2+ 2G/(1+O) (VɏɄ + Vɏɇ O)(VYɇ – VYɄ) =
= (VɏɄ 2 + O2Vɏɇ2 + 2OVɏɄ Vɏɇ)/ (1+O)2 + G2(VYɇ 2 + VYɄ 2 2VYɄ VYɇ) +
+ 2G/(1+O)(VɏɄ VYɇ + OVɏɇ VYɇ VɏɄ VYɄ OVɏɇ VYɄ).
(8)
ɉɪɢɦɟɧɢɜ ɨɩɟɪɚɰɢɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɤ ɨɛɟɢɦ ɱɚɫɬɹɦ ɪɚɜɟɧɫɬɜɚ (8), ɩɨɥɭɱɚɟɦ
ɞɢɫɩɟɪɫɢɢ ɤɨɨɪɞɢɧɚɬ:
14
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.13-16
D(XȾ) = [D(XɄ)+O2D(Xɇ)]/(1+O)2+G2[D(YɄ)+D(Yɇ)] + 2G/(1+O)[OMXYɇ MXYɄ],
(9)
D(YȾ) = [D(YɄ)+O2D(Yɇ)]/(1+O)2+G2[D(XɄ)+D(Xɇ)] + 2G/(1+O)[OMXYɇ MXYɄ].
ɉɨɥɭɱɟɧɵ ɜɫɟ ɧɟɨɛɯɨɞɢɦɵɟ ɮɨɪɦɭɥɵ ɞɥɹ ɮɨɪɦɢɪɨɜɚɧɢɹ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ ɬɨɱɤɢ Ⱦ,
ɬ.ɟ. ɨɩɪɟɞɟɥɟɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɟɟ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ. Ⱦɥɹ ɷɬɨɝɨ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɢɡɜɟɫɬɧɵɦɢ ɮɨɪɦɭɥɚɦɢ
ɨɛɴɟɞɢɧɟɧɢɹ ɞɜɭɯ ɜɟɤɬɨɪɢɚɥɶɧɵɯ ɩɨɝɪɟɲɧɨɫɬɟɣ V1 ɢ V2 ɜ ɷɥɥɢɩɫ ɩɨɝɪɟɲɧɨɫɬɟɣ (Ⱥɛɟɡɝɚɭɡ, 1970). Ɉɧɢ
ɨɩɪɟɞɟɥɹɸɬ ɨɪɢɟɧɬɚɰɢɸ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɷɥɥɢɩɫɚ – ɭɝɨɥ D ɦɟɠɞɭ ɛɨɥɶɲɨɣ ɨɫɶɸ ɷɥɥɢɩɫɚ ɢ ɛɨɥɶɲɟɣ
ɜɟɤɬɨɪɢɚɥɶɧɨɣ ɩɨɝɪɟɲɧɨɫɬɶɸ:
tg 2D = (V22 sin2T + 2kV1 V2 sinT)/(V12 + V22 cos2T + 2kV1 V2 cosT).
(10)
Ⱦɜɟ ɞɪɭɝɢɟ ɮɨɪɦɭɥɵ ɜɵɱɢɫɥɹɸɬ ɤɜɚɞɪɚɬɵ ɩɨɥɭɨɫɟɣ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ a ɢ b:
a2(b2) = 0.5{(V12 + V22 + 2kV1 V2 cosT) r
[ (V1 + V2 cos2T + 2kV1 V2 cosT)2 + (V22 sin2T + 2kV1 V2 sinT)2 ]1/2 }.
2
2
(11)
Ɂɞɟɫɶ k ɢ T ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɢ ɭɝɨɥ ɦɟɠɞɭ ɜɟɤɬɨɪɢɚɥɶɧɵɦɢ ɩɨɝɪɟɲɧɨɫɬɹɦɢ V1 ɢ V2.
ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɮɨɪɦɭɥɵ (10), (11) ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ, ɬɚɤ ɤɚɤ ɜɟɤɬɨɪɢɚɥɶɧɵɟ ɩɨɝɪɟɲɧɨɫɬɢ
ɨɪɬɨɝɨɧɚɥɶɧɵ, ɢ ɭɝɨɥ T = 0. Ɍɨɝɞɚ
tg 2D = 2kV1 V2 / (V12 V22)
(12)
2 2
a (b ) = 0.5{(V12 + V22) + [(V12 V22)2 + (2kV1 V2)2 ]1/2 }.
(13)
ɇɚɣɞɟɧɧɵɟ ɧɚɦɢ ɜɵɲɟ ɞɢɫɩɟɪɫɢɢ (9) ɤɨɨɪɞɢɧɚɬ ɬɨɱɤɢ Ⱦ ɢ ɟɫɬɶ ɤɜɚɞɪɚɬɵ ɨɛɴɟɞɢɧɹɟɦɵɯ
ɜɟɤɬɨɪɢɚɥɶɧɵɯ ɩɨɝɪɟɲɧɨɫɬɟɣ, ɚ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ (7) ɟɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɤɨɪɪɟɥɹɰɢɢ ɢ ɜɟɤɬɨɪɢɚɥɶɧɵɯ ɩɨɝɪɟɲɧɨɫɬɟɣ:
MXYȾ = k V1 V2,
D(XȾ) = V12,
D(YȾ) = V22.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɵɟ ɩɚɪɚɦɟɬɪɵ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ (12), (13) ɬɨɱɤɢ Ⱦ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ
ɬɟɪɦɢɧɚɯ ɞɢɫɩɟɪɫɢɣ ɢ ɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ ɦɨɦɟɧɬɚ ɜ ɷɬɨɣ ɬɨɱɤɟ:
tg 2D = 2MXYȾ / (D(XȾ D(YȾ))
a2(b2) = 0.5{(D(XȾ) + D(YȾ)) + [(D(XȾ) D(YȾ))2 + (2 MXYȾ)2]1/2 }.
(14)
ɉɨɫɤɨɥɶɤɭ ɩɪɢɟɦɨɢɧɞɢɤɚɬɨɪɵ ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɵɯ ɫɩɭɬɧɢɤɨɜɵɯ ɫɢɫɬɟɦ ɧɚɜɢɝɚɰɢɢ ɨɩɪɟɞɟɥɹɸɬ,
ɩɨɦɢɦɨ ɫɚɦɢɯ ɤɨɨɪɞɢɧɚɬ, ɩɚɪɚɦɟɬɪɵ ɫɬɚɧɞɚɪɬɧɨɝɨ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ ɨɛɫɟɪɜɨɜɚɧɧɨɣ ɬɨɱɤɢ, ɬɨ
ɫɥɟɞɭɟɬ ɜɵɪɚɡɢɬɶ ɢɦɟɧɧɨ ɱɟɪɟɡ ɧɢɯ ɩɚɪɚɦɟɬɪɵ (14) ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɢ Ⱦ ɨɛɜɨɞɚ
ɤɨɪɩɭɫɚ ɨɛɴɟɤɬɚ. Ɉɛɨɡɧɚɱɢɦ ɷɬɢ ɩɚɪɚɦɟɬɪɵ ɨɛɫɟɪɜɨɜɚɧɧɵɯ ɬɨɱɟɤ ɤɚɤ aɤ, bɤ, Dɤ ɢ aɧ, bɧ, Dɧ ɩɨɥɭɨɫɢ
ɷɥɥɢɩɫɚ ɢ ɭɝɨɥ ɧɚɤɥɨɧɚ ɟɝɨ ɛɨɥɶɲɨɣ ɨɫɢ ɞɥɹ ɬɨɱɟɤ Ʉ ɢ ɇ.
Ⱦɥɹ ɤɚɠɞɨɣ ɢɡ ɷɬɢɯ ɬɨɱɟɤ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɧɚɛɨɪ ɫɨɨɬɧɨɲɟɧɢɣ, ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɵɯ ɢɡ ɬɟɨɪɢɢ
ɷɥɥɢɩɬɢɱɟɫɤɢɯ ɩɨɝɪɟɲɧɨɫɬɟɣ.
MXYɄ = 0.5 tg2Dɤ [D(XɄ) – D(YɄ)] = 0.5
[D(XɄ) – D(YɄ)] = cos2Dɤ (aɤ2 bɤ2)
[D(XɄ) + D(YɄ)] = (aɤ2 + bɤ2).
(15)
Ⱥɧɚɥɨɝɢɱɧɵ ɢ ɮɨɪɦɭɥɵ ɞɥɹ ɧɨɫɨɜɨɣ ɬɨɱɤɢ ɇ, ɞɨɫɬɚɬɨɱɧɨ ɥɢɲɶ ɫɦɟɧɢɬɶ ɜ ɧɢɯ ɜɫɸɞɭ "Ʉ" ɧɚ "ɇ".
ɉɨɞɫɬɚɜɢɜ ɫɨɨɬɧɨɲɟɧɢɹ (15) ɜ ɜɵɪɚɠɟɧɢɟ (7) ɞɥɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ ɦɨɦɟɧɬɚ ɜ ɬɨɱɤɟ Ⱦ, ɩɨɥɭɱɢɦ
ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ ɪɚɫɱɟɬɚ ɤɨɦɛɢɧɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ:
2MXYȾ = (aɤ2 bɤ2) sin2Dɤ[1/(1 + O)2 + G2] + (aɧ2 – bɧ2) sin2Dɧ[O2/(1 + O)2 + G2] +
+ 2G/(1 + O)[O (aɧ2 + bɧ2) (aɤ2 + bɤ2)],
D(XȾ) + D(YȾ) = [O2 (aɧ2 + bɧ2) + (aɤ2 + bɤ2)]/ (1 + O)2 + G2[(aɧ2 + bɧ2) + (aɤ2 + bɤ2)] +
+ 2G/(1 + O)[O (aɧ2 bɧ2) sin2Dɧ (aɤ2 bɤ2) sin2Dɤ],
D(XȾ) D(YȾ) = [1/(1 + O)2 + G2] (aɤ2 bɤ2) cos2Dɤ + [O2/(1 + O)2 + G2] (aɧ2 – bɧ2)sin2Dɧ.
(16)
ɗɬɢɯ ɬɪɟɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ, ɩɨɞɫɬɚɜɢɜ ɢɯ ɜ ɮɨɪɦɭɥɵ (14), ɨɩɪɟɞɟɥɢɬɶ
ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɵ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ, ɫɜɹɡɚɧɧɨɝɨ ɫ ɬɨɱɤɨɣ Ⱦ.
3. ɉɪɢɦɟɪ ɱɢɫɥɟɧɧɨɝɨ ɪɚɫɱɟɬɚ
ɉɪɢɜɟɞɟɦ ɩɪɢɦɟɪ ɪɚɫɱɟɬɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɷɥɥɢɩɫɚ ɩɨɝɪɟɲɧɨɫɬɟɣ ɞɥɹ ɫɥɟɞɭɸɳɟɝɨ ɫɥɭɱɚɹ
(ɩɨɝɪɟɲɧɨɫɬɢ ɜ ɦɟɬɪɚɯ, ɭɝɥɵ ɜ ɝɪɚɞɭɫɚɯ):
ɉɚɪɚɦɟɬɪɵ ɷɥɥɢɩɫɚ ɤɨɪɦɨɜɨɣ ɬɨɱɤɢ aɤ = 10, bɤ = 8, Dɤ = 20.
15
ɉɚɲɟɧɰɟɜ ɋ.ȼ., ɉɨɫɬɪɨɟɧɢɹ ɡɨɧɵ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɛɟɡɨɩɚɫɧɨɫɬɢ ɨɛɴɟɤɬɚ...
ɉɚɪɚɦɟɬɪɵ ɷɥɥɢɩɫɚ ɧɨɫɨɜɨɣ ɬɨɱɤɢ aɧ = 12, bɧ = 10, Dɧ = 30.
Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ ɩɚɪɚɦɟɬɪɨɜ ɷɥɥɢɩɫɚ ɞɥɹ ɬɨɱɤɢ Ⱦ ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥɢɰɟ ɞɥɹ ɧɟɫɤɨɥɶɤɢɯ ɬɨɱɟɤ
ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɨɛɜɨɞɚ ɫɭɞɧɚ ɩɪɢ G = 3/20.
ȼɵɱɢɫɥɟɧɢɟɦ ɩɪɨɜɟɪɟɧɵ ɜɫɟ ɮɨɪɦɭɥɵ, ɤɨɬɨɪɵɟ ɜ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ ɬɨɱɟɤ Ʉ(O = 0, G = 0) ɢ ɇ(O =
100, G = 0) ɞɚɸɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɷɥɥɢɩɫɨɜ ɩɨɝɪɟɲɧɨɫɬɟɣ ɢɦɟɧɧɨ ɷɬɢɯ ɬɨɱɟɤ.
OȾ
DȾ
0.5
13
8
7
aȾ
bȾ
1.0
32
99
6
2.0
36
9
6
4.0
38
10
7
8.0
38
11
7
16.0
39
11
7
0
20
10
8
f
30
12
10
4. Ɉɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɨɛɴɟɤɬɚ
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɬɚɤɠɟ, ɱɬɨ ɫɩɭɬɧɢɤɨɜɵɟ ɩɪɢɟɦɨɢɧɞɢɤɚɬɨɪɵ, ɤɨɬɨɪɵɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɧɚ ɨɛɴɟɤɬɟ,
ɩɨɡɜɨɥɹɸɬ ɨɩɪɟɞɟɥɹɬɶ ɧɟ ɬɨɥɶɤɨ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ, ɧɨ ɢ ɜɟɤɬɨɪ ɟɟ ɫɤɨɪɨɫɬɢ ɫ ɩɨɦɨɳɶɸ ɞɨɩɩɥɟɪɨɜɫɤɨɝɨ
ɷɮɮɟɤɬɚ (ɚ ɧɟ ɩɭɬɟɦ ɜɵɱɢɫɥɟɧɢɹ ɩɪɢɪɚɳɟɧɢɣ ɤɨɨɪɞɢɧɚɬ).
ɂɬɚɤ, ɩɭɫɬɶ ɢɡɜɟɫɬɧɵ ɜɟɤɬɨɪɵ ɫɤɨɪɨɫɬɟɣ ɧɨɫɨɜɨɣ Vɇ ɢ
ɤɨɪɦɨɜɨɣ VɄ ɬɨɱɟɤ ɨɛɴɟɤɬɚ (ɪɢɫ. 2). ȼɵɛɟɪɟɦ ɦɝɧɨɜɟɧɧɵɣ
ɰɟɧɬɪ ɜɪɚɳɟɧɢɹ ɨɛɴɟɤɬɚ ɧɚ ɟɝɨ Ⱦɉ ɜ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ Ɉ*.
ȼɟɤɬɨɪ ɩɨɫɬɭɩɚɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɨɛɴɟɤɬɚ ɧɚɩɪɚɜɥɟɧ ɜɞɨɥɶ
ɧɟɤɨɬɨɪɨɣ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɷɬɭ ɬɨɱɤɭ Ɉ*.
ɉɪɹɦɚɹ ɫɨɫɬɚɜɥɹɟɬ ɭɝɨɥ E ɫ Ⱦɉ ɨɛɴɟɤɬɚ. Ɉɩɪɟɞɟɥɢɦ
ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ Ɉ*, ɧɚɩɪɚɜɥɟɧɢɟ ɢ ɜɟɥɢɱɢɧɭ
ɩɨɫɬɭɩɚɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɩɨɜɨɪɨɬɚ
ɨɛɴɟɤɬɚ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
ɦɝɧɨɜɟɧɧɵɣ ɰɟɧɬɪ Ɉ*. ȼɫɟ ɷɬɢ ɜɟɥɢɱɢɧɵ ɥɟɝɤɨ ɧɚɣɬɢ, ɡɧɚɹ
ɥɢɲɶ ɜɟɤɬɨɪɵ ɫɤɨɪɨɫɬɟɣ ɞɜɭɯ ɬɨɱɟɤ ɨɛɴɟɤɬɚ.
ɋɩɪɨɟɰɢɪɭɟɦ ɨɛɚ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɟɣ VɄ ɢ Vɇ ɧɚ
ɜɵɛɪɚɧɧɭɸ ɩɪɹɦɭɸ ɢ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ, ɤ ɧɟɣ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɟ. ȿɫɥɢ ɭɤɚɡɚɧɧɚɹ ɩɪɹɦɚɹ ɫɨɫɬɚɜɥɹɟɬ ɫ Ⱦɉ
Ɋɢɫ. 2. Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ
ɭɝɨɥ E, ɬɨ ɩɪɨɟɤɰɢɢ ɛɭɞɭɬ ɫɥɟɞɭɸɳɢɟ:
ɨɛɴɟɤɬɚ
ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɩɪɹɦɨɣ:
Vɉ = VɏɄ cosE + VYɄ sinE,
Vɉ = Vɏɇ cosE + VYɇ sinE,
(17)
ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɟ ɩɪɹɦɨɣ:
VȼɊɄ = VɏɄ sinE + VYɄ cosE,
VȼɊɇ = Vɏɇ sinE + VYɇ cosE.
(18)
Ʌɟɜɵɟ ɱɚɫɬɢ ɮɨɪɦɭɥ (17) ɨɞɢɧɚɤɨɜɵ. ɗɬɨ ɜɵɪɚɠɚɟɬ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɜɞɨɥɶ ɢɫɤɨɦɨɣ ɩɪɹɦɨɣ
ɧɚɩɪɚɜɥɟɧɚ ɟɞɢɧɚɹ ɞɥɹ ɨɛɴɟɤɬɚ ɟɝɨ ɩɨɫɬɭɩɚɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ Vɉ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
VɏɄ cosE + VYɄ sinE = Vɏɇ cosE + VYɇ sinE,
ɨɬɤɭɞɚ ɧɟɦɟɞɥɟɧɧɨ ɫɥɟɞɭɟɬ ɡɧɚɱɟɧɢɟ ɭɝɥɚ ɞɪɟɣɮɚ E
tgE = (Vɏɇ – VɏɄ)/(VYɇ – VYɄ).
(19)
ȼ ɮɨɪɦɭɥɚɯ (18) ɮɢɝɭɪɢɪɭɸɬ ɥɢɧɟɣɧɵɟ ɫɤɨɪɨɫɬɢ ɬɨɱɟɤ Ʉ ɢ ɇ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɜɪɚɳɟɧɢɹ ɨɛɴɟɤɬɚ
ɜɨɤɪɭɝ ɢɫɤɨɦɨɣ ɬɨɱɤɢ Ɉ*. ɗɬɢ ɫɤɨɪɨɫɬɢ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɪɚɫɫɬɨɹɧɢɹɦ LɄ ɢ Lɇ ɨɬ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ ɢ
ɩɨɷɬɨɦɭ ɜɟɪɧɚ ɩɪɨɩɨɪɰɢɹ:
LɄ / Lɇ = VȼɊɄ / VȼɊɇ = (Vɏɇ sinE + VYɄ cosE)/(Vɏɇ sinE + VYɇ cosE).
Ɍɚɤ ɤɚɤ ɭɝɨɥ E ɭɠɟ ɨɩɪɟɞɟɥɟɧ ɫɨɨɬɧɨɲɟɧɢɟɦ (19), ɬɨ ɜɨɡɦɨɠɧɨ ɧɚɣɬɢ ɢ ɨɬɧɨɲɟɧɢɟ LɄ / Lɇ, ɬ.ɟ.
ɩɨɥɨɠɟɧɢɟ ɰɟɧɬɪɚ ɜɪɚɳɟɧɢɹ ɧɚ Ⱦɉ, ɚ ɡɚɬɟɦ ɨɩɪɟɞɟɥɢɬɶ ɢ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɨɛɴɟɤɬɚ ɜɨɤɪɭɝ
ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɚɣɞɟɧɧɭɸ ɬɨɱɤɭ Ɉ*.
Z = VȼɊɄ / LɄ = (Vɏɇ sinE + VYɄ cosE) / LɄ = (Vɏɇ sinE + VYɇ cosE) / Lɇ = VȼɊɇ / Lɇ,
ɝɞɟ ɭɝɨɥ E ɢɡɜɟɫɬɟɧ ɢɡ (19).
5. Ɂɚɤɥɸɱɟɧɢɟ
16
(20)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.13-16
ɉɪɟɞɥɨɠɟɧɧɚɹ ɜɵɲɟ ɢɞɟɹ ɢ ɩɨɞɤɪɟɩɥɹɸɳɢɟ ɟɟ ɮɨɪɦɭɥɵ ɟɫɬɶ ɨɫɧɨɜɚ ɞɥɹ ɜɜɟɞɟɧɢɹ ɧɨɜɨɝɨ
ɩɨɧɹɬɢɹ – ɞɜɭɯɬɨɱɟɱɧɨɣ ɧɚɜɢɝɚɰɢɢ. Ɉɧɚ ɩɨɡɜɨɥɢɬ ɜɨɫɩɪɢɧɢɦɚɬɶ ɫɭɞɧɨ ɧɟ ɤɚɤ ɬɨɱɤɭ, ɚ ɤɚɤ ɨɛɴɟɤɬ
ɤɨɧɟɱɧɵɯ ɪɚɡɦɟɪɨɜ, ɢ ɭɩɪɚɜɥɹɬɶ ɷɬɢɦ ɨɛɴɟɤɬɨɦ, ɢɫɯɨɞɹ ɢɡ ɡɚɞɚɱɢ ɛɟɡɨɩɚɫɧɨɫɬɢ ɦɨɪɟɩɥɚɜɚɧɢɹ. ɗɬɨ ɬɚɤɠɟ
ɨɬɤɪɵɜɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɜ ɛɭɞɭɳɟɦ ɢɞɟɧɬɢɮɢɰɢɪɨɜɚɬɶ ɩɚɪɚɦɟɬɪɵ ɦɨɞɟɥɢ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɧɚ, ɧɟ
ɩɪɢɦɟɧɹɹ ɧɢɤɚɤɢɯ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɫɭɞɨɜɵɯ ɞɚɬɱɢɤɨɜ.
Ʌɢɬɟɪɚɬɭɪɚ
Ⱥɛɟɡɝɚɭɡ Ƚ.Ƚ. ɋɩɪɚɜɨɱɧɢɤ ɩɨ ɜɟɪɨɹɬɧɨɫɬɧɵɦ ɪɚɫɱɟɬɚɦ. Ɇ., ȼɨɟɧɢɡɞɚɬ, ɫ. 374, 1970.
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