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Навигационная задача Цермело аналитическое решение для вихревого поля.

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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.17-22
ɇɚɜɢɝɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ ɐɟɪɦɟɥɨ: ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɞɥɹ
ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ
ɋ.ȼ. ɉɚɲɟɧɰɟɜ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɫɭɞɨɜɨɠɞɟɧɢɹ
Ⱥɧɧɨɬɚɰɢɹ. ɉɪɢɜɨɞɢɬɫɹ ɪɟɲɟɧɢɟ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɡɚɞɚɱɢ ɐɟɪɦɟɥɨ ɞɥɹ ɩɥɨɫɤɨɝɨ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ
ɫɤɨɪɨɫɬɟɣ, ɩɨɥɭɱɟɧɧɨɟ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɮɨɪɦɟ. ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɨɜ ɞɚɸɬɫɹ ɪɟɡɭɥɶɬɚɬɵ ɤɨɧɤɪɟɬɧɵɯ
ɪɟɲɟɧɢɣ ɡɚɞɚɱɢ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɫɨɨɬɧɨɲɟɧɢɹɯ ɫɤɨɪɨɫɬɟɣ ɜɢɯɪɹ ɢ ɞɜɢɠɭɳɟɝɨɫɹ ɨɛɴɟɤɬɚ. ɗɬɢ ɪɟɲɟɧɢɹ
ɩɨɥɭɱɟɧɵ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɩɚɤɟɬɚ MathCad.
Abstract. The solution of the Zermelo navigational task for the flat rotational velocities field has been obtained
in an analytical form. The specific solutions of the task under various relations of velocities of a curl and a
moving vehicle are given as examples. These solutions have been obtained with the help of the MathCad
application.
1. ȼɜɟɞɟɧɢɟ
Ɂɚɞɚɱɚ ɐɟɪɦɟɥɨ ɩɨ ɨɩɬɢɦɚɥɶɧɨɦɭ ɭɩɪɚɜɥɟɧɢɸ ɨɛɴɟɤɬɨɦ, ɩɟɪɟɦɟɳɚɸɳɢɦɫɹ ɜ ɡɚɞɚɧɧɨɦ
ɩɪɨɢɡɜɨɥɶɧɨɦ ɩɨɥɟ ɫɤɨɪɨɫɬɟɣ, ɹɜɥɹɟɬɫɹ ɤɥɚɫɫɢɱɟɫɤɨɣ. Ɉɛɵɱɧɨ ɟɟ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɱɢɫɥɟɧɧɨ, ɬ.ɤ.
ɩɨɥɭɱɟɧɢɟ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ ɜɜɢɞɭ ɧɟɥɢɧɟɣɧɨɫɬɢ ɛɨɥɶɲɢɧɫɬɜɚ ɡɚɞɚɱ ɭɩɪɚɜɥɟɧɢɹ ɩɪɚɤɬɢɱɟɫɤɢ
ɧɟɜɨɡɦɨɠɧɨ. ɗɬɚ ɧɟɥɢɧɟɣɧɨɫɬɶ ɡɚɞɚɱɢ ɭɩɪɚɜɥɟɧɢɹ ɢɦɟɟɬ ɦɟɫɬɨ, ɞɚɠɟ ɟɫɥɢ ɢɫɯɨɞɧɚɹ ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ
ɡɚɞɚɱɚ ɥɢɧɟɣɧɚ.
ɉɨɷɬɨɦɭ ɜɨɡɦɨɠɧɵɟ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɪɟɲɟɧɢɹ ɬɚɤɢɯ ɡɚɞɚɱ ɜɫɟɝɞɚ ɩɪɟɞɫɬɚɜɥɹɸɬ ɛɨɥɶɲɨɣ ɢɧɬɟɪɟɫ.
Ⱥɜɬɨɪɭ ɭɞɚɥɨɫɶ ɩɨɫɬɪɨɢɬɶ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɨɞɧɨɣ ɡɚɞɚɱɢ ɩɨɞɨɛɧɨɝɨ ɬɢɩɚ ɫ ɭɩɪɚɜɥɟɧɢɟɦ ɩɨ
ɛɵɫɬɪɨɞɟɣɫɬɜɢɸ. ɗɬɨ ɡɚɞɚɱɚ ɧɚɢɫɤɨɪɟɣɲɟɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɨɛɴɟɤɬɚ ɜ ɜɢɯɪɟɜɨɦ ɩɨɥɟ ɫɤɨɪɨɫɬɟɣ. ɉɨɥɭɱɟɧɨ
ɞɨɫɬɚɬɨɱɧɨ ɝɪɨɦɨɡɞɤɨɟ (ɧɨ ɚɧɚɥɢɬɢɱɟɫɤɨɟ!) ɪɟɲɟɧɢɟ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɜɪɟɦɹ ɩɟɪɟɦɟɳɟɧɢɹ ɩɪɢ ɡɚɦɵɤɚɧɢɢ
ɡɚɞɚɱɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ. ȿɝɨ, ɟɫɬɟɫɬɜɟɧɧɨ, ɩɪɢɯɨɞɢɬɫɹ ɪɟɲɚɬɶ ɱɢɫɥɟɧɧɨ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɩɪɨɜɟɪɟɧɨ ɞɥɹ ɱɚɫɬɧɵɯ ɫɥɭɱɚɟɜ, ɪɟɲɟɧɢɹ ɞɥɹ ɤɨɬɨɪɵɯ ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɵ,
ɢ ɩɨɥɭɱɟɧɵ ɢɯ ɩɨɥɧɵɟ ɫɨɜɩɚɞɟɧɢɹ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɞɥɹ ɪɹɞɚ ɫɥɭɱɚɟɜ ɩɪɨɜɟɞɟɧɨ ɱɢɫɥɟɧɧɨ ɫ ɩɨɦɨɳɶɸ
ɩɚɤɟɬɚ MathCad7.0.
2. Ɉɫɧɨɜɧɵɟ ɭɪɚɜɧɟɧɢɹ ɡɚɞɚɱɢ
ɉɭɫɬɶ ɨɛɴɟɤɬ ɞɜɢɠɟɬɫɹ ɜ ɩɥɨɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ x1, x2 ɫ ɩɨɫɬɨɹɧɧɨɣ ɩɨ ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɶɸ
V, ɢ ɭɩɪɚɜɥɟɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɬɨɥɶɤɨ ɜɵɛɨɪɨɦ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ – ɭɝɥɚ u ɦɟɠɞɭ ɜɟɤɬɨɪɨɦ ɫɤɨɪɨɫɬɢ
ɢ ɨɫɶɸ ɚɛɫɰɢɫɫ x1. Ɍɨɝɞɚ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɜɵɝɥɹɞɹɬ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ:
dx1/dt = v1(x1, x2) + V cos u
(1)
dx2/dt = v2(x1, x2) + V sin u,
ɝɞɟ v1(x1, x2) ɢ v2(x1, x2) – ɫɨɫɬɚɜɥɹɸɳɢɟ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ, ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɞɜɢɠɟɧɢɟ ɨɛɴɟɤɬɚ. ȿɫɥɢ
ɪɟɱɶ ɢɞɟɬ ɨ ɞɜɢɠɟɧɢɢ ɜ ɩɨɥɟ ɜɢɯɪɹ, ɬɨ ɷɬɢ ɫɨɫɬɚɜɥɹɸɳɢɟ ɬɚɤɨɜɵ:
v1(x1, x2) = Z x2
v2(x1, x2) = +Z x1,
(2)
ɝɞɟ Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɜɢɯɪɹ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
ɉɭɫɬɶ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɢɡ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɢ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x10, x20 ɜ ɤɨɧɟɱɧɭɸ ɬɨɱɤɭ ɫ
ɤɨɨɪɞɢɧɚɬɚɦɢ x1f, x2f. ȼɪɟɦɹ ɞɜɢɠɟɧɢɹ tf ɧɟ ɨɩɪɟɞɟɥɟɧɨ, ɧɨ ɞɨɥɠɧɨ ɛɵɬɶ ɦɢɧɢɦɚɥɶɧɵɦ.
ɂɫɩɨɥɶɡɭɟɦ ɞɥɹ ɪɟɲɟɧɢɹ ɦɟɬɨɞ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɚɤɚɞ. Ʌ.ɋ. ɉɨɧɬɪɹɝɢɧɚ, ɞɥɹ ɱɟɝɨ
ɫɨɫɬɚɜɢɦ Ƚɚɦɢɥɶɬɨɧɢɚɧ H ɡɚɞɚɱɢ ɜ ɜɢɞɟ ɥɢɧɟɣɧɨɣ ɮɨɪɦɵ (Ȼɨɥɬɹɧɫɤɢɣ, 1969):
H = p1(v1 + V cos u) + p2 (v2 + V sin u) 1,
ɜ ɤɨɬɨɪɨɦ ɩɨɹɜɢɥɢɫɶ ɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ p1 ɢ p2, ɧɚɡɵɜɚɟɦɵɟ ɫɨɩɪɹɠɟɧɧɵɦɢ.
17
(3)
ɉɚɲɟɧɰɟɜ ɋ.ȼ., ɇɚɜɢɝɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ ɐɟɪɦɟɥɨ: ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɞɥɹ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ
Ɉɩɬɢɦɚɥɶɧɨɟ ɜ ɫɦɵɫɥɟ ɛɵɫɬɪɨɞɟɣɫɬɜɢɹ ɭɩɪɚɜɥɟɧɢɟ ɪɟɚɥɢɡɭɟɬɫɹ ɩɪɢ ɦɚɤɫɢɦɭɦɟ Ƚɚɦɢɥɶɬɨɧɢɚɧɚ
ɜ ɨɛɥɚɫɬɢ ɢɡɦɟɧɟɧɢɹ ɩɚɪɚɦɟɬɪɚ ɭɩɪɚɜɥɟɧɢɹ u ɢ ɩɨɬɨɦɭ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɨ ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ
ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ Ƚɚɦɢɥɶɬɨɧɢɚɧɚ ɩɨ ɩɚɪɚɦɟɬɪɭ ɭɩɪɚɜɥɟɧɢɹ:
wH/wu = p1 V sin u + p2 V cos u = 0.
(4)
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɩɪɨɫɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬ ɭɩɪɚɜɥɟɧɢɟ ɜ ɜɢɞɟ tg u = p2/p1, ɢɥɢ ɜ ɜɢɞɟ ɨɬɞɟɥɶɧɵɯ
ɜɵɪɚɠɟɧɢɣ ɞɥɹ ɫɢɧɭɫɚ ɢ ɤɨɫɢɧɭɫɚ ɭɩɪɚɜɥɟɧɢɹ
sin u = p2/(p12 + p22)1/2
cos u = p1/(p12 + p22)1/2.
(5)
Ⱦɨɫɬɢɝɚɟɦɨɟ ɩɪɢ ɷɬɨɦ ɭɩɪɚɜɥɟɧɢɢ ɡɧɚɱɟɧɢɟ ɦɚɤɫɢɦɭɦɚ Ƚɚɦɢɥɶɬɨɧɢɚɧɚ ɪɚɜɧɨ ɧɭɥɸ ɜ ɬɟɱɟɧɢɟ ɜɫɟɝɨ
ɞɜɢɠɟɧɢɹ, ɢ ɫ ɭɱɟɬɨɦ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ (2) ɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
M = maxu{H} = p1v1 + p2v2 + V (p12 + p22)1/2 1 =
(6)
= p1(Z x2) + p2(+Z x1) + V (p12 + p22)1/2 1 = 0.
ɗɬɨ ɫɨɨɬɧɨɲɟɧɢɟ, ɡɚɩɢɫɚɧɧɨɟ ɞɥɹ ɤɨɧɤɪɟɬɧɨɝɨ ɦɨɦɟɧɬɚ (ɦɨɦɟɧɬɨɜ) ɜɪɟɦɟɧɢ, ɩɨɡɜɨɥɢɬ ɩɨɥɭɱɢɬɶ
ɧɟɞɨɫɬɚɸɳɟɟ ɭɫɥɨɜɢɟ (ɭɫɥɨɜɢɹ) ɞɥɹ ɡɚɦɵɤɚɧɢɹ ɡɚɞɚɱɢ.
3. ɉɨɫɬɪɨɟɧɢɟ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ
ɉɨɥɭɱɢɦ ɫɨɩɪɹɠɟɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɜɜɟɞɟɧɧɵɯ ɧɟɢɡɜɟɫɬɧɵɯ p1 ɢ p2, ɞɢɮɮɟɪɟɧɰɢɪɭɹ
Ƚɚɦɢɥɶɬɨɧɢɚɧ ɱɚɫɬɧɵɦ ɨɛɪɚɡɨɦ ɩɨ ɚɪɝɭɦɟɧɬɚɦ x1 ɢ x2:
dp1/dt = wH/wx1 = Z p2 dp2/dt = wH/wx2 = +Z p1 .
(7)
ɍɦɧɨɠɚɹ ɩɟɪɜɨɟ ɢɡ ɷɬɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɧɚ p1, ɚ ɜɬɨɪɨɟ ɧɚ p2 ɢ ɫɤɥɚɞɵɜɚɹ, ɩɨɥɭɱɢɦ
ɩɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɫɢɫɬɟɦɵ ɜ ɜɢɞɟ p12 + p22 = const. ɉɨɥɭɱɟɧɧɚɹ ɫɢɫɬɟɦɚ (7) ɥɢɧɟɣɧɚ, ɨɞɧɨɪɨɞɧɚ ɢ ɢɦɟɟɬ
ɩɪɨɫɬɨɟ ɨɛɳɟɟ ɪɟɲɟɧɢɟ:
p1 = A cos(Z t) + B sin(Z t)
p2 = B cos(Z t) + A sin(Z t),
(8)
ɝɞɟ Ⱥ ɢ ȼ – ɩɨɤɚ ɧɟɢɡɜɟɫɬɧɵɟ ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ p12 + p22 = Ⱥ2 + ȼ2 ɢ, ɬɟɦ
ɫɚɦɵɦ, ɩɨɫɬɨɹɧɧɨ, ɤɚɤ ɷɬɨ ɫɥɟɞɭɟɬ ɢɡ ɩɟɪɜɨɝɨ ɢɧɬɟɝɪɚɥɚ ɞɜɢɠɟɧɢɹ. ȿɫɥɢ ɜ ɡɚɞɚɱɟ ɡɚɞɚɧɵ ɤɨɨɪɞɢɧɚɬɵ
ɧɚɱɚɥɶɧɨɣ ɢ ɤɨɧɟɱɧɨɣ ɬɨɱɤɢ, ɬɨ ɭɫɥɨɜɢɹ ɬɪɚɧɫɜɟɪɫɚɥɶɧɨɫɬɢ ɫɜɨɞɹɬɫɹ ɤ ɡɚɞɚɧɢɸ ɧɚɱɚɥɶɧɵɯ ɢ ɤɨɧɟɱɧɵɯ
ɡɧɚɱɟɧɢɣ ɩɟɪɟɦɟɧɧɵɯ p1 ɢ p2:
p1(0)
p2(0)
p1(tf)
p2(tf).
(9)
ȿɫɥɢ ɤɨɧɟɱɧɨɟ ɭɫɥɨɜɢɟ ɫɨɫɬɨɢɬ, ɧɚɩɪɢɦɟɪ, ɜ ɩɨɩɚɞɚɧɢɢ ɧɚ ɨɤɪɭɠɧɨɫɬɶ ɡɚɞɚɧɧɨɝɨ ɪɚɞɢɭɫɚ R, ɬ.ɟ. ɩɪɢ t = tf
ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ x12 + x22 = R2, ɬɨ ɭɫɥɨɜɢɹ ɬɪɚɧɫɜɟɪɫɚɥɶɧɨɫɬɢ ɧɚ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ ɩɪɢɦɭɬ ɢɧɭɸ
ɮɨɪɦɭ ɫ ɧɨɜɨɣ ɤɨɧɫɬɚɧɬɨɣ /:
p1(tf) + / 2x1f = 0,
p2(tf) + / 2x2f = 0.
(10)
ɉɪɢ ɧɚɣɞɟɧɧɨɦ ɨɩɬɢɦɚɥɶɧɨɦ ɭɩɪɚɜɥɟɧɢɢ (5) ɢ ɩɨɥɭɱɟɧɧɨɦ ɪɟɲɟɧɢɢ ɫɨɩɪɹɠɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ (8),
ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ (1) ɩɪɢɨɛɪɟɬɚɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
dx1/dt = Z x2 + V (A cos(Z t) + B sin(Z t)) / (Ⱥ2 + ȼ2)1/2
(11)
dx2/dt = +Z x1 + V (B cos(Z t) + A sin(Z t)) / (Ⱥ2 +ȼ2)1/2.
ɋɢɫɬɟɦɚ ɥɢɧɟɣɧɚ, ɢɦɟɟɬ ɧɟɧɭɥɟɜɭɸ ɩɪɚɜɭɸ ɱɚɫɬɶ, ɢ ɟɟ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɬɪɨɢɬɫɹ ɤɚɤ ɫɭɦɦɚ
ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɫ ɩɪɚɜɵɦɢ ɱɚɫɬɹɦɢ:
x1(t) = C cos(Z t) + D sin(Z t) + t V (A cos(Z t) + B sin(Z t)) / (Ⱥ2 + ȼ2)1/2
x1(t) = C cos(Z t) + D sin(Z t) + t V (A cos(Z t) + B sin(Z t)) / (Ⱥ2 + ȼ2)1/2
18
(12)
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.17-22
ɫ ɞɜɭɦɹ ɧɨɜɵɦɢ ɩɪɨɢɡɜɨɥɶɧɵɦɢ ɤɨɧɫɬɚɧɬɚɦɢ C ɢ D.
ȼɫɟ ɪɟɲɟɧɢɹ ɩɨɥɭɱɟɧɵ, ɧɨ ɨɧɢ ɫɨɞɟɪɠɚɬ ɩɹɬɶ ɧɟɢɡɜɟɫɬɧɵɯ – ɱɟɬɵɪɟ ɩɪɨɢɡɜɨɥɶɧɵɯ ɤɨɧɫɬɚɧɬɵ A,
B, C, D ɢ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɞɨ ɤɨɧɟɱɧɨɣ ɬɨɱɤɢ tf. Ⱦɥɹ ɢɯ ɧɚɯɨɠɞɟɧɢɹ ɭ ɧɚɫ ɢɦɟɟɬɫɹ ɪɨɜɧɨ ɩɹɬɶ ɭɫɥɨɜɢɣ –
ɱɟɬɵɪɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹ ɡɚɞɚɧɧɵɯ ɡɧɚɱɟɧɢɣ ɤɨɨɪɞɢɧɚɬ ɢ ɭɫɥɨɜɢɟ (6) ɦɚɤɫɢɦɭɦɚ Ƚɚɦɢɥɶɬɨɧɢɚɧɚ ɜ
ɤɨɧɟɱɧɨɣ ɬɨɱɤɟ. ɂɦɟɧɧɨ ɩɨɷɬɨɦɭ ɡɚɞɚɱɚ ɡɚɦɵɤɚɟɬɫɹ, ɢ ɪɟɲɟɧɢɟ ɟɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɧɨɫɬɶɸ ɩɨɫɬɪɨɟɧɨ.
ɍɞɨɜɥɟɬɜɨɪɹɟɦ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ:
t=0
x1(0) = x10
x2(0) = x20,
ɱɬɨ ɞɚɟɬ ɢɡ (12) ɡɧɚɱɟɧɢɹ ɞɜɭɯ ɤɨɧɫɬɚɧɬ ɋ = x10, D = x20. ɍɞɨɜɥɟɬɜɨɪɹɟɦ ɤɨɧɟɱɧɵɦ ɭɫɥɨɜɢɹɦ:
t = tf
x1(tf) = x1f
x2(tf) = x2f.
ɗɬɨ ɩɪɢɜɨɞɢɬ ɤ ɞɜɭɦ ɪɚɜɟɧɫɬɜɚɦ, ɜ ɤɨɬɨɪɵɯ ɤɨɧɫɬɚɧɬɵ ɋ ɢ D ɡɚɦɟɧɟɧɵ ɢɯ ɡɧɚɱɟɧɢɹɦɢ, ɩɨɥɭɱɟɧɧɵɦɢ ɢɡ
ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ:
x1f = x10 cos(Z tf) x20 sin(Z tf) + tf V (A1 cos(Z tf) + B1 sin(Z tf)),
(13)
x2f = x20 cos(Z tf) + x10 sin(Z t) + tf V (B1 cos(Z tf) + A1 sin(Z tf)),
ɝɞɟ ɜɦɟɫɬɨ ɤɨɧɫɬɚɧɬ Ⱥ ɢ ȼ ɜɜɟɞɟɧɵ ɤɨɧɫɬɚɧɬɵ
A1 = A/(Ⱥ2 + ȼ2)1/2,
B1 = B/(Ⱥ2 + ȼ2)1/2,
ɬɚɤɢɟ, ɱɬɨ A12 + B12 = 1.
Ɋɚɡɪɟɲɚɹ ɫɢɫɬɟɦɭ (13) ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɢɯ ɤɨɧɫɬɚɧɬ, ɩɨɥɭɱɢɦ:
A1 = {x10 – (x1f cos(Z tf) + x2f sin(Z tf))} /V tf,
B1 = {x20 + (x2f cos(Z tf) x1f sin(Z tf))} /V tf.
ɇɚɣɞɟɧɧɵɟ ɤɨɧɫɬɚɧɬɵ ɩɨɡɜɨɥɹɸɬ ɹɜɧɨ ɜɵɩɢɫɚɬɶ ɪɟɲɟɧɢɟ – ɩɨɜɟɞɟɧɢɟ ɨɛɴɟɤɬɚ ɜɨ ɜɪɟɦɟɧɢ:
x1(t) = (x10 cos(Z t) x20 sin(Z t))(1t/tf ) + (x1f cos(Z (tft)) + x2f sin(Z (tft))) t / tf,
x2(t) = (x20 cos(Z t) + x10 sin(Z t))(1t/tf ) + (x2f cos(Z (tft)) x2f sin(Z (tft))) t / tf.
(14)
ɉɪɢ ɷɬɨɦ ɨɩɬɢɦɚɥɶɧɨɟ ɜ ɫɦɵɫɥɟ ɛɵɫɬɪɨɞɟɣɫɬɜɢɹ ɭɩɪɚɜɥɟɧɢɟ (5) ɬɚɤɠɟ ɜɵɩɢɫɵɜɚɟɬɫɹ ɹɜɧɨ ɫ ɩɨɦɨɳɶɸ
ɪɟɲɟɧɢɹ (8) ɢ ɩɨɥɭɱɟɧɧɵɯ ɤɨɧɫɬɚɧɬ A1 ɢ B1:
V sin u = {(x20 cos(Z t) + x10 sin(Z t)) + (x2f cos(Z (tft)) – x1f sin(Z (tft)))} / tf,
V cos u = {(x10 cos(Z t) x20 sin(Z t)) + (x1f cos(Z (tf-t)) + x2f sin(Z (tft)))} / tf.
(15)
ɉɨɥɭɱɟɧɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ (14)-(15) ɫɨɞɟɪɠɢɬ ɧɟɢɡɜɟɫɬɧɵɣ ɩɚɪɚɦɟɬɪ: ɜɪɟɦɹ
ɨɩɬɢɦɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ tf. ȿɝɨ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ, ɢɫɩɨɥɶɡɭɹ ɡɧɚɱɟɧɢɟ ɦɚɤɫɢɦɭɦɚ (6) Ƚɚɦɢɥɶɬɨɧɢɚɧɚ ɜ
ɤɨɧɟɱɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tf:
M = p1f(Z x2f) + p2f(+Z x1f) + V (p1f2 + p2f2)1/2 1 = 0.
ȼɜɨɞɹ ɜ ɷɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɧɚɣɞɟɧɧɵɟ ɪɟɲɟɧɢɹ ɞɥɹ p1,2 ɢ x1,2 ɜ ɤɨɧɟɱɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɢɦ
ɫɨɨɬɧɨɲɟɧɢɟ, ɜ ɤɨɬɨɪɨɟ ɜɯɨɞɢɬ ɧɟɢɡɜɟɫɬɧɚɹ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ ɞɜɢɠɟɧɢɹ tf:
{cos(Z tf)(x10 x2f x20 x1f) sin(Z tf)(x10 x1f + x20 x2f)} Z (Ⱥ2 + ȼ2)1/2/(V tf) + V (Ⱥ2 + ȼ2)1/2 1 = 0.
Ɉɞɧɚɤɨ ɜ ɷɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɜɨɲɥɚ ɬɚɤɠɟ ɤɨɦɛɢɧɚɰɢɹ ɢɡ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ ɤɨɧɫɬɚɧɬ Ⱥ ɢ ȼ.
ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ ɫɜɹɡɚɧɧɵɯ ɫ ɧɢɦɢ ɤɨɧɫɬɚɧɬ A12 + B12 = 1. ɗɬɨ ɞɚɫɬ
ɠɟɥɚɟɦɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɢ ɞɜɢɠɟɧɢɹ:
x102 + x202 + x1f2 + x2f2 2 sin(Z tf)(x10 x2f x20 x1f) 2 cos(Z tf)(x10 x1f + x20 x2f) = (V tf)2.
19
(16)
ɉɚɲɟɧɰɟɜ ɋ.ȼ., ɇɚɜɢɝɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ ɐɟɪɦɟɥɨ: ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɞɥɹ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ
4. ɑɢɫɥɟɧɧɵɟ ɪɟɲɟɧɢɹ
Ʉ ɫɨɠɚɥɟɧɢɸ, ɭɪɚɜɧɟɧɢɟ (16) ɞɥɹ tf ɧɨɫɢɬ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɣ ɯɚɪɚɤɬɟɪ ɢ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɨ
ɬɨɥɶɤɨ ɱɢɫɥɟɧɧɨ. Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɣ ɱɢɫɥɟɧɧɵɟ ɪɟɲɟɧɢɹ ɩɨɥɭɱɟɧɵ ɫ ɩɨɦɨɳɶɸ ɩɚɤɟɬɚ MathCad7, ɜ ɫɪɟɞɟ
ɤɨɬɨɪɨɝɨ ɜɨɡɦɨɠɧɨ ɨɩɪɟɞɟɥɹɬɶ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ ɞɜɢɠɟɧɢɹ ɢ ɡɚɬɟɦ ɫɬɪɨɢɬɶ ɩɟɪɟɦɟɳɟɧɢɟ ɨɛɴɟɤɬɚ ɢ
ɭɩɪɚɜɥɟɧɢɟ ɢɦ ɜ ɯɨɞɟ ɞɜɢɠɟɧɢɹ ɫ ɢɯ ɝɪɚɮɢɱɟɫɤɨɣ ɢɥɥɸɫɬɪɚɰɢɟɣ. Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ,
ɤɨɬɨɪɵɣ ɧɟɦɟɞɥɟɧɧɨ ɫɥɟɞɭɟɬ ɢɡ (16) ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ, ɤɨɝɞɚ Z = 0:
tf = {(x10 x20)2 + (x1f x2f)2}1/2 /V,
ɱɬɨ ɜɩɨɥɧɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɹɦɨɥɢɧɟɣɧɨɦɭ ɞɜɢɠɟɧɢɸ ɜ ɷɬɨɦ ɫɥɭɱɚɟ. Ⱦɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ ɞɜɚ ɪɟɲɟɧɢɹ,
ɥɟɠɚɳɢɯ ɜ ɢɡɜɟɫɬɧɨɣ ɦɟɪɟ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɝɪɚɧɢɰɚɯ ɨɛɥɚɫɬɢ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ ɩɨ ɫɨɨɬɧɨɲɟɧɢɸ
ɫɤɨɪɨɫɬɟɣ ɞɜɢɠɟɧɢɹ ɨɛɴɟɤɬɚ ɢ ɜɢɯɪɹ. ɉɪɢ ɷɬɨɦ ɧɚɱɚɥɶɧɚɹ ɢ ɤɨɧɟɱɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɧɢɹ ɜɵɛɪɚɧɵ
ɨɞɢɧɚɤɨɜɵɦɢ ɫɨ ɫɥɟɞɭɸɳɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ (ɞɜɢɠɟɧɢɟ ɧɚɱɢɧɚɟɬɫɹ ɫ ɩɟɪɢɮɟɪɢɢ ɜɢɯɪɹ):
x10 = 20
x20 = 0
x1f = 4
x2f = 0.
ɚ) ɋɥɭɱɚɣ ɨɩɬɢɦɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɩɪɢ ɦɚɥɨɣ ɫɨɛɫɬɜɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɨɛɴɟɤɬɚ V ɢ ɛɨɥɶɲɨɣ
ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɢɯɪɹ Z. ȼɵɛɪɚɧɵ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɨɛɴɟɤɬɚ V = 2 ɢ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɢɯɪɹ Z = 3.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɥɢɧɟɣɧɚɹ ɫɤɨɪɨɫɬɶ ɜ ɧɚɱɚɥɶɧɨɣ ɬɨɱɤɟ ɞɜɢɠɟɧɢɹ ɧɚ ɪɚɞɢɭɫɟ 20 ɪɚɜɧɚ 60,
ɱɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɫɨɛɫɬɜɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɛɴɟɤɬɚ 2.
Ɋɟɲɟɧɢɟ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (16) ɫ ɩɨɦɨɳɶɸ ɩɚɤɟɬɚ MathCad ɞɚɟɬ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ
ɞɜɢɠɟɧɢɹ tf = 8.623. ɗɬɨ ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɧɨ ɝɪɚɮɢɤɚɦɢ ɧɚ ɪɢɫ. 1ɚ. ɇɚ ɧɟɦ ɩɪɢɜɟɞɟɧɵ ɝɪɚɮɢɱɟɫɤɢ ɩɪɚɜɚɹ
ɢ ɥɟɜɚɹ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɢ ɩɨɤɚɡɚɧɵ ɬɨɱɤɢ ɢɯ ɩɟɪɟɫɟɱɟɧɢɹ. ȿɫɬɟɫɬɜɟɧɧɨ, ɜ ɤɚɱɟɫɬɜɟ ɪɟɲɟɧɢɹ ɜɵɛɢɪɚɟɬɫɹ
ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ tf – ɜɟɞɶ ɪɟɱɶ ɢɞɟɬ ɨɛ ɨɩɬɢɦɚɥɶɧɨɫɬɢ ɩɨ ɛɵɫɬɪɨɞɟɣɫɬɜɢɸ.
20
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.17-22
Ɋɢɫ. 1. Ƚɪɚɮɢɱɟɫɤɚɹ ɢɥɥɸɫɬɪɚɰɢɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɢ ɞɜɢɠɟɧɢɹ.
Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɧɚ ɪɢɫɭɧɤɚɯ:
ɚ) ɦɚɥɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɫɤɨɪɨɫɬɶ ɨɛɴɟɤɬɚ ɢ ɛɨɥɶɲɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɢɯɪɹ;
ɛ) ɛɨɥɶɲɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɫɤɨɪɨɫɬɶ ɨɛɴɟɤɬɚ ɢ ɦɚɥɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɢɯɪɹ
Ⱦɚɥɟɟ ɩɪɢɜɟɞɟɧɵ ɬɪɢ ɪɢɫɭɧɤɚ ɫ ɝɪɚɮɢɤɚɦɢ ɬɪɚɟɤɬɨɪɢɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɪɢɫ. 2ɚ) ɜ ɨɫɹɯ
ɯ1 ɢ ɯ2, ɡɚɜɢɫɢɦɨɫɬɢ ɭɩɪɚɜɥɹɸɳɟɝɨ ɩɚɪɚɦɟɬɪɚ ɜ ɝɪɚɞɭɫɚɯ ɨɬ ɤɨɨɪɞɢɧɚɬɵ ɯ1 (ɪɢɫ. 3ɚ) ɢ ɡɚɜɢɫɢɦɨɫɬɢ ɟɝɨ ɠɟ
ɨɬ ɜɪɟɦɟɧɢ (ɪɢɫ. 4ɚ).
Ɋɢɫ. 2. Ɍɪɚɟɤɬɨɪɢɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɛɴɟɤɬɚ
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ɉɚɲɟɧɰɟɜ ɋ.ȼ., ɇɚɜɢɝɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ ɐɟɪɦɟɥɨ: ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɞɥɹ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ
Ɋɢɫ. 3. Ɂɚɜɢɫɢɦɨɫɬɶ ɭɩɪɚɜɥɹɸɳɟɝɨ ɩɚɪɚɦɟɬɪɚ ɨɬ ɤɨɨɪɞɢɧɚɬɵ
Ɋɢɫ. 4. Ɂɚɜɢɫɢɦɨɫɬɶ ɭɩɪɚɜɥɹɸɳɟɝɨ ɩɚɪɚɦɟɬɪɚ ɨɬ ɜɪɟɦɟɧɢ
ȼ ɫɢɥɭ ɦɚɥɨɣ ɫɤɨɪɨɫɬɢ ɨɛɴɟɤɬɚ ɞɨɫɬɢɠɟɧɢɟ ɤɨɧɟɱɧɨɣ ɬɨɱɤɢ ɩɪɨɢɫɯɨɞɢɬ ɬɨɥɶɤɨ ɩɪɢ
ɦɧɨɝɨɤɪɚɬɧɵɯ (ɡɞɟɫɶ 4-ɤɪɚɬɧɵɯ) ɨɛɨɪɨɬɚɯ ɜɨɤɪɭɝ ɰɟɧɬɪɚ ɜɢɯɪɹ (ɪɢɫ. 2ɚ), ɩɪɢ ɷɬɨɦ ɫɬɨɥɶɤɨ ɠɟ ɪɚɡ
ɩɪɨɢɫɯɨɞɢɬ ɰɢɤɥ ɩɟɪɟɤɥɚɞɤɢ ɪɭɥɹ ɢ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɚɜɥɟɧɢɹ ɞɜɢɠɟɧɢɹ. ȼɟɪɬɢɤɚɥɶɧɵɟ ɥɢɧɢɢ ɧɚ ɪɢɫ. 3ɚ ɢ
4ɚ ɧɟ ɹɜɥɹɸɬɫɹ ɫɤɚɱɤɚɦɢ ɜ ɡɚɤɨɧɟ ɭɩɪɚɜɥɟɧɢɹ. Ɉɧɢ ɫɜɹɡɚɧɵ ɬɨɥɶɤɨ ɫɨ ɫɩɟɰɢɮɢɤɨɣ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɭɝɥɨɜ,
ɝɞɟ ɨɬɫɱɟɬɵ 0 ɢ 360 ɝɪɚɞɭɫɨɜ ɩɪɟɞɫɬɚɜɥɹɸɬ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɭɝɨɥ.
ɛ) ɋɥɭɱɚɣ ɨɩɬɢɦɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɩɪɢ ɛɨɥɶɲɨɣ ɫɨɛɫɬɜɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɨɛɴɟɤɬɚ V ɢ ɦɚɥɨɣ
ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɢɯɪɹ Z. Ɋɢɫɭɧɤɢ ɞɥɹ ɷɬɢɯ ɫɥɭɱɚɟɜ ɩɪɢɜɨɞɹɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɢ ɞɥɹ ɨɬɥɢɱɢɹ
ɢɧɞɟɤɫɢɪɭɸɬɫɹ ɛɭɤɜɚɦɢ "ɛ". Ɍɚɤ ɢɯ ɥɟɝɤɨ ɫɪɚɜɧɢɜɚɬɶ.
ɉɨɫɤɨɥɶɤɭ ɭɝɥɨɜɚɹ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɥɢɧɟɣɧɵɟ ɫɤɨɪɨɫɬɢ ɜ ɜɢɯɪɟ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɶɲɟ, ɱɟɦ ɜ
ɡɚɞɚɱɟ ɚ), ɬɨ ɢ ɪɟɲɟɧɢɟ ɦɟɧɹɟɬ ɫɜɨɣ ɯɚɪɚɤɬɟɪ. Ʉɨɧɟɱɧɚɹ ɬɨɱɤɚ ɞɨɫɬɢɝɚɟɬɫɹ ɛɟɡ ɨɛɨɪɨɬɨɜ ɜɨɤɪɭɝ ɰɟɧɬɪɚ
ɜɢɯɪɹ, ɚ ɭɩɪɚɜɥɟɧɢɟ ɢɡɦɟɧɹɟɬɫɹ ɝɥɚɞɤɨ, ɧɟ ɩɪɟɬɟɪɩɟɜɚɹ ɫɤɚɱɤɨɜ. ɗɬɨ ɢ ɞɟɦɨɧɫɬɪɢɪɭɸɬ ɩɪɢɜɟɞɟɧɧɵɟ
ɪɢɫɭɧɤɢ 2ɛ-4ɛ.
5. Ɂɚɤɥɸɱɟɧɢɟ
ɉɨɥɭɱɟɧɨ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɐɟɪɦɟɥɨ ɞɥɹ ɩɥɨɫɤɨɝɨ ɜɢɯɪɟɜɨɝɨ ɩɨɥɹ (14)-(15).
Ɉɞɧɚɤɨ, ɡɚɦɵɤɚɧɢɟ ɡɚɞɚɱɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ ɞɜɢɠɟɧɢɹ ɨɤɚɡɵɜɚɟɬɫɹ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɦ
ɭɪɚɜɧɟɧɢɟɦ, ɢ ɧɚ ɷɬɨɦ ɩɨɫɥɟɞɧɟɦ ɷɬɚɩɟ ɡɚɞɚɱɚ ɜɵɧɭɠɞɟɧɧɨ ɪɟɲɚɟɬɫɹ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɨɦ. ɇɨ ɷɬɨ
ɧɟɢɡɦɟɪɢɦɨ ɩɪɨɳɟ, ɱɟɦ ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɩɨɥɧɨɣ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ, ɢ ɦɨɠɟɬ
ɞɨɫɬɢɝɚɬɶɫɹ ɜ ɪɚɦɤɚɯ ɬɚɤɢɯ ɩɚɤɟɬɨɜ, ɤɚɤ MathCad, ɛɟɡ ɫɩɟɰɢɚɥɶɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ. ɇɚɛɨɪ
ɱɢɫɥɟɧɧɵɯ ɪɟɲɟɧɢɣ ɞɥɹ ɩɪɟɞɟɥɶɧɵɯ ɫɥɭɱɚɟɜ ɜɩɨɥɧɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɭɳɟɫɬɜɟɧɧɵɟ ɱɟɪɬɵ ɜɫɟɯ
ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ. ɉɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɩɪɢ ɨɰɟɧɨɱɧɵɯ ɪɚɫɱɟɬɚɯ
ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɭɬɢ ɫɭɞɧɚ ɜ ɪɚɣɨɧɚɯ ɫ ɢɧɬɟɧɫɢɜɧɨɣ ɰɢɤɥɨɧɚɥɶɧɨɣ ɞɟɹɬɟɥɶɧɨɫɬɶɸ.
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ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 3, ʋ1, 2000 ɝ.
ɫɬɪ.17-22
Ʌɢɬɟɪɚɬɭɪɚ
Ȼɨɥɬɹɧɫɤɢɣ ȼ.Ƚ. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ. Ɇ., ɇɚɭɤɚ, 378 ɫ., 1969.
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