close

Вход

Забыли?

вход по аккаунту

?

Еще раз об уравнении управляемости Номото.

код для вставкиСкачать
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 6, ʋ1, 2003 ɝ.
ɫɬɪ.69-74
ȿɳɟ ɪɚɡ ɨɛ ɭɪɚɜɧɟɧɢɢ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɇɨɦɨɬɨ
Ɋ.Ƚ. ɋɬɟɩɚɯɧɨ
ɋɭɞɨɜɨɞɢɬɟɥɶɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɭɩɪɚɜɥɟɧɢɹ ɫɭɞɧɨɦ
ɢ ɩɪɨɦɵɫɥɨɜɨɝɨ ɪɵɛɨɥɨɜɫɬɜɚ
Ⱥɧɧɨɬɚɰɢɹ. ɉɪɟɞɥɚɝɚɟɬɫɹ ɧɨɜɵɣ ɩɨɞɯɨɞ ɤ ɨɛɪɚɛɨɬɤɟ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɬɭɪɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɨ ɫɬɚɧɞɚɪɬɧɨɦɭ
ɦɚɧɟɜɪɭ "Ɂɢɝɡɚɝ". Ɇɨɞɟɥɶ, ɤɨɬɨɪɚɹ ɩɪɢ ɷɬɨɦ ɢɞɟɧɬɢɮɢɰɢɪɭɟɬɫɹ, ɟɫɬɶ ɩɪɨɫɬɟɣɲɚɹ ɥɢɧɟɣɧɚɹ ɦɨɞɟɥɶ, ɧɨɫɹɳɚɹ
ɧɚɡɜɚɧɢɟ ɦɨɞɟɥɢ ɇɨɦɨɬɨ (ɢɥɢ ɇɨɦɨɬɨ-ɇɨɪɪɛɢɧɚ). Ɉɧɚ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɡɚɪɭɛɟɠɧɨɣ ɩɪɚɤɬɢɤɟ
ɨɛɪɚɛɨɬɤɢ, ɯɨɬɹ ɜ ɨɬɟɱɟɫɬɜɟɧɧɨɦ ɫɭɞɨɫɬɪɨɟɧɢɢ ɢ ɷɤɫɩɥɭɚɬɚɰɢɢ ɮɥɨɬɚ ɨɬɧɨɲɟɧɢɟ ɤ ɧɟɣ ɧɟɨɞɧɨɡɧɚɱɧɨɟ. ȼ
ɡɧɚɱɢɬɟɥɶɧɨɣ ɦɟɪɟ ɷɬɨ ɫɜɹɡɚɧɨ ɫ ɧɟɭɞɨɛɫɬɜɨɦ ɬɪɚɞɢɰɢɨɧɧɵɯ ɩɪɢɟɦɨɜ ɨɛɪɚɛɨɬɤɢ ɬɚɤɢɯ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ
ɞɚɧɧɵɯ. ɂɦɟɧɧɨ ɷɬɨɬ ɜɨɩɪɨɫ ɨɛɫɭɠɞɚɟɬɫɹ ɜ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ, ɝɞɟ ɞɟɦɨɧɫɬɪɢɪɭɟɬɫɹ ɢɧɨɣ ɩɨɞɯɨɞ ɤɚɤ ɤ
ɩɨɥɭɱɟɧɢɸ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ, ɬɚɤ ɢ ɤ ɢɯ ɨɛɪɚɛɨɬɤɟ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɟɟ ɭɩɪɨɳɟɧɢɹ.
Abstract. In the paper the new approach to processing of the outcomes of full-scale experiments on the standard
manoeuvre "Zigzag" has been considered. The model which thus has been identified is an elementary linear
model called the Nɨɦɨɬɨ model (or Nɨɦɨɬɨ-Nɨrrbin). It is widely used in the processing practice abroad,
though in domestic shipbuilding and maintenance of fleet the relation to it is ambiguous. Largely it is connected
to inconvenience of conventional techniques of these experimental data processing. Just this problem has been
considered in the work, the author has demonstrated another approach both to obtaining of experimental data
and to their processing trying to simplify it.
1. ȼɜɟɞɟɧɢɟ
ɉɨɫɬɚɧɨɜɤɚ ɧɚɬɭɪɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɨ ɦɚɧɟɜɪɢɪɨɜɚɧɢɸ ɦɨɪɫɤɢɯ ɫɭɞɨɜ ɢ ɨɛɪɚɛɨɬɤɚ ɢɯ
ɪɟɡɭɥɶɬɚɬɨɜ ɩɨɞɱɢɧɟɧɵ ɨɫɧɨɜɧɨɣ ɡɚɞɚɱɟ ɨɰɟɧɤɟ ɩɚɪɚɦɟɬɪɨɜ ɜɵɛɪɚɧɧɨɣ ɦɨɞɟɥɢ ɫɭɞɧɚ (Ȼɚɫɢɧ, 1977).
ɗɬɨ ɫɥɨɠɧɚɹ ɡɚɞɚɱɚ, ɨɫɨɛɟɧɧɨ ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ ɫɚɦɚ ɦɨɞɟɥɶ ɨɛɵɱɧɨ ɧɟɥɢɧɟɣɧɚ. ɉɨɷɬɨɦɭ
ɢɫɫɥɟɞɨɜɚɬɟɥɢ ɫɬɪɟɦɹɬɫɹ ɭɩɪɨɫɬɢɬɶ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɜɵɛɢɪɚɹ ɦɨɞɟɥɢ ɧɚɫɬɨɥɶɤɨ ɩɪɨɫɬɵɟ, ɧɚɫɤɨɥɶɤɨ ɷɬɨ
ɜɨɡɦɨɠɧɨ ɛɟɡ ɩɨɬɟɪɢ ɜ ɨɬɪɚɠɟɧɢɢ ɯɚɪɚɤɬɟɪɧɵɯ ɱɟɪɬ ɩɨɜɟɞɟɧɢɹ ɪɟɚɥɶɧɨɝɨ ɨɛɴɟɤɬɚ. ȼ 30-ɟ ɝɨɞɵ XX ɜ. ɷɬɨ
ɨɫɨɛɟɧɧɨ ɬɟɫɧɨ ɫɜɹɡɵɜɚɥɨɫɶ ɫ ɩɨɩɵɬɤɨɣ ɫɨɡɞɚɬɶ ɫɬɚɧɞɚɪɬɧɵɟ ɦɟɬɨɞɵ ɢɫɩɵɬɚɧɢɣ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɨɜ ɢ
ɜɜɟɫɬɢ ɫɪɚɜɧɢɬɟɥɶɧɵɟ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɤɪɢɬɟɪɢɢ ɷɬɨɝɨ ɤɚɱɟɫɬɜɚ (Kempf, 1932).
ɇɨɜɭɸ ɠɢɡɧɶ ɷɬɨɣ ɢɞɟɟ ɞɚɥ ɇɨɦɨɬɨ (Nomoto et al., 1957; Nomoto and Nɨrrbin, 1969), ɤɨɬɨɪɵɣ
ɩɪɟɞɥɨɠɢɥ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɨɫɬɟɣɲɭɸ ɦɨɞɟɥɶ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɧɚ, ɫɨɞɟɪɠɚɳɭɸ ɥɢɲɶ ɞɜɚ ɩɚɪɚɦɟɬɪɚ. ɗɬɢ
ɩɚɪɚɦɟɬɪɵ ɨɧ ɧɚɯɨɞɢɥ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɫɩɟɰɢɚɥɶɧɨɝɨ ɢɫɩɵɬɚɧɢɹ ɫɭɞɧɚ ɦɚɧɟɜɪɚ "Ɂɢɝɡɚɝ". Ⱦɥɹ ɢɯ ɩɨɥɭɱɟɧɢɹ
ɇɨɦɨɬɨ ɢɫɩɨɥɶɡɨɜɚɥ ɦɟɬɨɞɢɤɭ, ɤɨɬɨɪɚɹ ɡɚɬɟɦ ɩɪɢɡɧɚɜɚɥɚɫɶ ɫɥɢɲɤɨɦ ɫɥɨɠɧɨɣ, ɝɪɨɦɨɡɞɤɨɣ (Ʉɚɰɦɚɧ ɢ ɞɪ., 1970).
ȼɩɪɨɱɟɦ, ɩɪɢ ɫɭɳɟɫɬɜɨɜɚɜɲɟɦ ɜ ɬɟ ɝɨɞɵ ɭɪɨɜɧɟ ɪɚɡɜɢɬɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ ɬɚɤɚɹ ɨɰɟɧɤɚ ɜɩɨɥɧɟ
ɩɪɚɜɨɦɨɱɧɚ. ɏɨɬɹ ɩɨɫɥɟɞɭɸɳɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɞɪɭɝɢɯ ɚɜɬɨɪɨɜ (Ⱥɫɢɧɨɜɫɤɢɣ, 1969) ɩɪɟɞɥɚɝɚɥɢ ɧɟɤɨɬɨɪɵɟ
ɭɩɪɨɳɟɧɢɹ ɜ ɨɛɪɚɛɨɬɤɟ ɪɟɡɭɥɶɬɚɬɨɜ ɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ, ɷɬɢ ɦɟɬɨɞɵ ɧɟ ɩɨɥɭɱɢɥɢ ɞɨɥɠɧɨɝɨ
ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜ ɨɬɟɱɟɫɬɜɟɧɧɨɣ ɫɭɞɨɫɬɪɨɢɬɟɥɶɧɨɣ ɢ ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɨɣ ɩɪɚɤɬɢɤɟ. ɉɨɞɜɟɪɝɚɥɚɫɶ ɤɪɢɬɢɤɟ
(Ⱥɫɢɧɨɜɫɤɢɣ, Ƚɨɮɦɚɧ, 1967) ɢ ɫɚɦɚ ɭɩɪɨɳɟɧɧɚɹ ɦɨɞɟɥɶ ɇɨɦɨɬɨ, ɩɚɪɚɦɟɬɪɵ ɤɨɬɨɪɨɣ ɡɚɜɢɫɹɬ ɧɟ ɬɨɥɶɤɨ ɨɬ ɫɚɦɨɝɨ
ɫɭɞɧɚ, ɧɨ ɢ ɨɬ ɜɢɞɚ ɦɚɧɟɜɪɟɧɧɨɝɨ ɢɫɩɵɬɚɧɢɹ. Ɍɟɦ ɧɟ ɦɟɧɟɟ, ɢɧɬɟɪɟɫ ɤ ɦɨɞɟɥɢ ɇɨɦɨɬɨ ɢ ɢɫɩɵɬɚɧɢɸ "Ɂɢɝɡɚɝ" ɜ
ɦɢɪɟ ɫɨɯɪɚɧɢɥɫɹ, ɨɫɨɛɟɧɧɨ ɟɫɥɢ ɭɱɟɫɬɶ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɫɜɹɡɚɧɧɨɝɨ ɫ ɧɢɦɢ.
ɂɦɟɧɧɨ ɜ ɫɢɥɭ ɷɬɨɝɨ ɢɧɬɟɪɟɫɚ ɪɚɫɫɦɨɬɪɢɦ ɞɪɭɝɨɣ ɦɟɬɨɞ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɢɞɟɧɬɢɮɢɤɚɰɢɢ
ɦɨɞɟɥɢ ɇɨɦɨɬɨ, ɢɫɩɨɥɶɡɭɹ ɬɨɬ ɠɟ ɬɢɩɨɜɨɣ ɦɚɧɟɜɪ.
2. Ⱥɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɇɨɦɨɬɨ
Ɇɨɞɟɥɶ ɇɨɦɨɬɨ ɢɦɟɟɬ ɩɪɨɫɬɟɣɲɢɣ ɜɢɞ ɥɢɧɟɣɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ 1-ɝɨ ɩɨɪɹɞɤɚ,
T1dZ /dt + Z = KG,
(1)
ɢɦɟɸɳɟɝɨ ɞɜɚ ɩɚɪɚɦɟɬɪɚ T1 ɢ K. ɉɚɪɚɦɟɬɪ T1 ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɭɞɧɚ ɧɚ ɤɭɪɫɟ, ɩɚɪɚɦɟɬɪ Ʉ ɟɝɨ
ɩɨɜɨɪɨɬɥɢɜɨɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɭɯɨɞɚ ɫɭɞɧɚ ɫ ɩɨɫɬɨɹɧɧɨɝɨ ɤɭɪɫɚ. ɋɚɦɨ ɭɪɚɜɧɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ Z ɩɨɜɨɪɨɬɚ ɫɭɞɧɚ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ ɨɬ ɤɥɚɞɤɢ
ɪɭɥɹ G. Ⱦɥɹ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɚɧɟɜɪ "Ɂɢɝɡɚɝ", ɤɨɬɨɪɵɣ ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ.
ɇɚ ɫɭɞɧɟ, ɢɞɭɳɟɦ ɧɚ ɩɪɹɦɨɦ ɤɭɪɫɟ, ɪɭɥɶ ɤɥɚɞɭɬ, ɫɤɚɠɟɦ, ɧɚ 10° ɧɚ ɩɪɚɜɵɣ ɛɨɪɬ. ɋɭɞɧɨ ɧɚɱɢɧɚɟɬ
ɩɨɜɨɪɚɱɢɜɚɬɶ ɜɩɪɚɜɨ ɢ ɤɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɤɭɪɫɚ (ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɤɭɪɫ ɭɜɟɥɢɱɢɜɚɟɬɫɹ) ɞɨɫɬɢɝɧɟɬ ɬɨɝɨ ɠɟ
ɡɧɚɱɟɧɢɹ ɜ 10°, ɪɭɥɶ ɩɟɪɟɤɥɚɞɵɜɚɸɬ ɧɚ 10° ɜɥɟɜɨ. ɋɭɞɧɨ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ɩɪɨɞɨɥɠɚɟɬ ɩɨɜɨɪɨɬ ɧɚɩɪɚɜɨ,
69
ɋɬɟɩɚɯɧɨ Ɋ.Ƚ. ȿɳɟ ɪɚɡ ɨɛ ɭɪɚɜɧɟɧɢɢ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɇɨɦɨɬɨ
ɡɚɦɟɞɥɹɟɬ ɟɝɨ, ɡɚɬɟɦ ɧɚɱɢɧɚɟɬ ɩɨɜɨɪɨɬ ɧɚɥɟɜɨ. Ʉɨɝɞɚ ɢɡɦɟɧɟɧɢɟ ɤɭɪɫɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɚɱɚɥɶɧɨɦɭ (ɬɟɩɟɪɶ
ɤɭɪɫ ɭɦɟɧɶɲɚɟɬɫɹ) ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ 10°, ɪɭɥɶ ɩɟɪɟɤɥɚɞɵɜɚɸɬ ɜɩɪɚɜɨ ɢ ɬ.ɞ. ɋ ɧɟɤɨɬɨɪɨɝɨ ɦɨɦɟɧɬɚ ɷɬɨɬ
ɩɪɨɰɟɫɫ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɪɢɛɥɢɠɟɧɧɨ ɩɟɪɢɨɞɢɱɟɫɤɢɦ, ɟɫɥɢ ɨɬɜɥɟɱɶɫɹ ɨɬ ɞɟɣɫɬɜɢɹ ɜɧɟɲɧɢɯ ɭɫɥɨɜɢɣ ɩɥɚɜɚɧɢɹ
ɜɟɬɪɚ ɢ ɬɟɱɟɧɢɹ. ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɬɚɤɨɝɨ ɦɚɧɟɜɪɚ ɫɥɟɞɭɟɬ ɜɵɛɪɚɬɶ "ɜɪɟɦɹ ɢ ɦɟɫɬɨ".
Ɇɚɧɟɜɪ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɝɪɚɮɢɱɟɫɤɢ (ɪɢɫ. 1). ɇɚ ɧɟɦ ɯɨɪɨɲɨ ɜɢɞɧɨ, ɤɚɤ ɢɡɦɟɧɹɟɬɫɹ ɤɥɚɞɤɚ
ɪɭɥɹ ɜ ɬɟɱɟɧɢɟ ɦɚɧɟɜɪɚ ɢ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɚ ɨɬ ɫɜɨɟɝɨ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ. Ɍɪɟɛɨɜɚɧɢɟ ɩɪɢ
ɜɵɩɨɥɧɟɧɢɢ ɦɚɧɟɜɪɚ ɪɚɜɟɧɫɬɜɚ ɢɡɦɟɧɟɧɢɹ ɤɭɪɫɚ ɢ ɤɥɚɞɤɢ ɪɭɥɹ ɜɵɪɚɠɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɤɪɢɜɚɹ ɢɯ
ɝɪɚɮɢɱɟɫɤɨɝɨ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɜɨ ɜɪɟɦɟɧɢ ɢɦɟɸɬ ɨɛɳɢɟ ɬɨɱɤɢ, ɧɚɩɪɢɦɟɪ, Ⱥ ɢ ȼ.
ɉɪɢ
ɫɭɳɟɫɬɜɭɸɳɢɯ
ɦɟɬɨɞɢɤɚɯ
ɨɛɪɚɛɨɬɤɢ
ɪɟɡɭɥɶɬɚɬɨɜ ɬɪɟɛɭɟɬɫɹ ɡɚɩɢɫɶ ɤɭɪɫɚ ɫɭɞɧɚ ɱɟɪɟɡ ɤɚɠɞɵɟ 510 ɫɟɤ. Ɋɚɡɭɦɟɟɬɫɹ, ɷɬɢ ɞɚɧɧɵɟ ɜɫɟɝɞɚ ɩɨɥɟɡɧɵ, ɨɫɨɛɟɧɧɨ
ɟɫɥɢ ɨɧɢ ɩɨɥɭɱɟɧɵ ɫ ɫɚɦɨɩɢɫɰɚ. ȿɳɟ ɥɭɱɲɟ, ɟɫɥɢ ɬɚɤɚɹ
ɡɚɩɢɫɶ ɜɟɞɟɬɫɹ ɢ ɞɥɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɩɨɜɨɪɨɬɚ. ɇɨ ɞɥɹ
ɩɪɟɥɚɝɚɟɦɨɣ ɦɟɬɨɞɢɤɢ ɞɨɫɬɚɬɨɱɧɨ ɮɢɤɫɢɪɨɜɚɬɶ ɜ
ɷɤɫɩɟɪɢɦɟɧɬɟ ɬɨɥɶɤɨ ɞɜɚ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ: ɜɪɟɦɹ
ɭɪɚɜɧɢɜɚɧɢɹ ɢɡɦɟɧɟɧɢɹ ɤɭɪɫɚ ɢ ɤɥɚɞɤɢ ɪɭɥɹ (ɚ ɷɬɨ
ɫɭɳɟɫɬɜɟɧɧɨ ɧɟɨɛɯɨɞɢɦɨ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɫɚɦɨɝɨ
ɢɫɩɵɬɚɧɢɹ) ɢ ɜɪɟɦɹ, ɤɨɝɞɚ ɫɭɞɧɨ ɜɨɡɜɪɚɳɚɟɬɫɹ ɧɚ
ɩɟɪɜɨɧɚɱɚɥɶɧɵɣ ɤɭɪɫ. ȿɫɥɢ ɛɵ ɩɪɨɰɟɫɫ ɛɵɥ ɢɞɟɚɥɶɧɨ
Ɋɢɫ. 1. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜɨ
ɩɟɪɢɨɞɢɱɟɫɤɢɦ, ɞɨɫɬɚɬɨɱɧɨ ɧɚɛɥɸɞɚɬɶ ɷɬɢ ɦɨɦɟɧɬɵ ɥɢɲɶ
ɜɪɟɦɟɧɢ ɤɥɚɞɤɢ ɪɭɥɹ, ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɨɜɨɝɨ ɨɞɢɧ ɪɚɡ. ɇɨ ɥɭɱɲɟ ɫɞɟɥɚɬɶ ɷɬɨ ɧɟɫɤɨɥɶɤɨ ɪɚɡ, ɬ.ɟ.
ɭɝɥɚ ɢ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɦɚɧɟɜɪɚ "ɁɂȽɁȺȽ" ɜɵɩɨɥɧɢɬɶ ɩɪɨɰɟɞɭɪɭ ɩɟɪɟɤɥɚɞɤɢ ɪɭɥɹ, ɫɤɚɠɟɦ, 6-ɤɪɚɬɧɨ.
(T = 75 c, t1 = 10 c, t3 = 45 c, T1 = 7.55 c)
Ɍɨɝɞɚ ɦɨɠɧɨ ɨɰɟɧɢɬɶ ɫɬɟɩɟɧɶ ɧɟɩɟɪɢɨɞɢɱɧɨɫɬɢ ɩɪɨɰɟɫɫɚ ɢ
ɜɡɹɬɶ ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɧɚɛɥɸɞɚɟɦɵɯ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ.
ɑɬɨɛɵ ɫɜɨɛɨɞɧɨ ɨɩɟɪɢɪɨɜɚɬɶ ɞɚɥɟɟ ɤɥɚɞɤɨɣ ɪɭɥɹ ɜ ɦɨɞɟɥɢ (1), ɪɚɡɥɨɠɢɦ ɟɟ ɤɚɤ ɮɭɧɤɰɢɸ
ɜɪɟɦɟɧɢ ɜ ɪɹɞ Ɏɭɪɶɟ. ɉɪɢ ɷɬɨɦ ɹɫɧɨ, ɱɬɨ ɷɬɨ ɛɭɞɟɬ ɪɚɡɥɨɠɟɧɢɟ ɬɨɥɶɤɨ ɩɨ ɫɢɧɭɫɚɦ, ɬ.ɤ. ɮɭɧɤɰɢɹ G(t)
ɧɟɱɟɬɧɚɹ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ ɪɹɞɟ ɛɭɞɭɬ ɩɪɢɫɭɬɫɬɜɨɜɚɬɶ ɬɨɥɶɤɨ ɱɥɟɧɵ ɫ ɧɟɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ, ɩɨɫɤɨɥɶɤɭ ɧɚ
ɜɬɨɪɨɦ ɩɨɥɭɩɟɪɢɨɞɟ (Ɍ, 2Ɍ) ɡɧɚɱɟɧɢɹ ɤɥɚɞɤɢ ɩɨɥɭɱɟɧɵ ɡɟɪɤɚɥɶɧɵɦ ɨɬɪɚɠɟɧɢɟɦ ɜ ɨɫɢ ɜɪɟɦɟɧɢ ɤɥɚɞɤɢ
ɩɟɪɜɨɝɨ ɩɨɥɭɩɟɪɢɨɞɚ (0, Ɍ). ɂɬɚɤ, ɟɫɥɢ ɩɨɥɭɩɟɪɢɨɞ ɮɭɧɤɰɢɢ ɟɫɬɶ Ɍ, ɬɨ ɪɚɡɥɨɠɟɧɢɟ ɤɥɚɞɤɢ ɪɭɥɹ ɬɚɤɨɜɨ:
G(t) = 6k=1,3,5bksin(Sk˜t/T), k = 1, 3, 5, …
T
bk = 2 [ ³ G(t) sin(Sk˜t/T)dt]/T.
(2)
0
ȿɫɥɢ ɬɟɩɟɪɶ ɩɪɟɞɫɬɚɜɢɬɶ ɤɥɚɞɤɭ ɪɭɥɹ ɤɚɤ ɤɭɫɨɱɧɨ-ɡɚɞɚɧɧɭɸ ɮɭɧɤɰɢɸ ɧɚ ɩɟɪɜɨɦ ɩɨɥɭɩɟɪɢɨɞɟ ɜ ɮɨɪɦɟ
G(t) =
´G0 t/t1, t < t1
®G0, t1 < t < Tt1
¯G0 (Tt)/t1, Tt1 < t < T,
ɬɨ, ɜɡɹɜ ɨɬ ɷɬɨɣ ɮɭɧɤɰɢɢ ɢɧɬɟɝɪɚɥ (2), ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɹ Ɏɭɪɶɟ ɤɥɚɞɤɢ
ɪɭɥɹ
(3)
bk = 4G0sin(Sk˜t1/T) / [(Sk/T)2T˜t1].
ɉɨɥɭɱɟɧɧɵɣ ɪɹɞ Ɏɭɪɶɟ ɧɟɩɥɨɯɨ ɫɯɨɞɢɬɫɹ (ɫɨ ɫɤɨɪɨɫɬɶɸ, ɨɛɪɚɬɧɨɣ ɤɜɚɞɪɚɬɭ ɧɨɦɟɪɚ k); ɜ
ɩɨɞɬɜɟɪɠɞɟɧɢɟ ɷɬɨɝɨ ɩɪɢɜɟɞɟɦ ɜ ɬɚɛɥ. 1 ɪɟɡɭɥɶɬɚɬɵ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɡɧɚɱɟɧɢɣ ɤɥɚɞɤɢ ɪɭɥɹ ɞɥɹ 3, 6 ɢ 9
ɱɥɟɧɨɜ ɜ ɫɭɦɦɟ ɟɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɹɞɨɦ (2). ȼ ɬɚɛɥ. 1 ɜɵɛɪɚɧɵ ɞɥɹ ɪɚɫɱɟɬɚ Ɍ = 75 ɫ, t1 = 10 ɫ, G0 = 10°.
ɉɪɟɞɫɬɚɜɥɟɧɚ ɬɨɥɶɤɨ ɥɟɜɚɹ ɩɨɥɨɜɢɧɚ ɩɨɥɭɩɟɪɢɨɞɚ (0, 35), ɬ.ɤ. ɩɪɚɜɚɹ ɩɨɥɨɜɢɧɚ ɩɨɥɧɨɫɬɶɸ ɫɢɦɦɟɬɪɢɱɧɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɪɟɞɧɟɣ ɬɨɱɤɢ Ɍ/2 = 37.5 ɫ ɢɧɬɟɪɜɚɥɚ (0, T).
ɏɨɪɨɲɨ ɜɢɞɧɨ, ɤɚɤ ɭɥɭɱɲɚɟɬɫɹ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɤ ɬɨɱɧɨɦɭ ɡɧɚɱɟɧɢɸ ɤɥɚɞɤɢ, ɩɪɢɜɟɞɟɧɧɨɦɭ ɜ ɧɢɠɧɟɦ
ɪɹɞɭ ɩɟɪɜɨɣ ɫɬɪɨɤɢ ɬɚɛɥ. 1 ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɱɥɟɧɨɜ ɪɹɞɚ. Ɋɚɡɭɦɟɟɬɫɹ, ɨɬ ɹɜɥɟɧɢɣ ɤɨɥɟɛɚɧɢɣ Ƚɢɛɛɫɚ,
ɨɫɨɛɟɧɧɨ ɜɛɥɢɡɢ ɬɨɱɟɤ t1 ɢ T-t1 ɪɚɡɪɵɜɚ ɩɪɨɢɡɜɨɞɧɨɣ ɮɭɧɤɰɢɢ ɧɟ ɭɞɚɟɬɫɹ ɢɡɛɚɜɢɬɶɫɹ ɩɨɥɧɨɫɬɶɸ. ɇɨ ɩɨɞɨɛɧɚɹ
ɬɨɱɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɧɚɫ ɜɩɨɥɧɟ ɭɫɬɪɚɢɜɚɟɬ, ɨɫɨɛɟɧɧɨ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɟɲɚɟɦɵɦ ɞɚɥɟɟ ɡɚɞɚɱɚɦ.
Ɍɚɛɥɢɰɚ 1
t, c
N /G, ɝɪɚɞ
3
6
9
0
0
0
0
0
5
5
5.37
5.10
5.02
10
10
8.99
9.40
9.56
15
10
10.32
10.18
9.98
70
20
10
10.16
9.85
10.06
25
10
9.80
10.12
9.94
30
10
9.87
9.92
10.04
35
10
10.15
10.03
9.98
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 6, ʋ1, 2003 ɝ.
ɫɬɪ.69-74
ɋɱɢɬɚɹ ɢɡɜɟɫɬɧɵɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ (2), ɛɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɜ
ɜɢɞɟ ɛɨɥɟɟ ɨɛɳɟɝɨ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɹɞɚ
Z(t) = 6k[Akcos(Sk˜t/T) + Bksin(Sk˜t/T)],
(6)
ɚ ɫ ɟɝɨ ɩɨɦɨɳɶɸ ɜɵɪɚɡɢɬɶ ɢ ɢɡɦɟɧɟɧɢɟ ɤɭɪɫɨɜɨɝɨ ɭɝɥɚ M ɧɚ ɨɫɧɨɜɚɧɢɢ ɢɡɜɟɫɬɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ Z = dM/dt:
M(t) = 6k[Aksin(Sk˜t/T) Bkcos(Sk˜t/T)]/(Sk/T).
(7)
ɉɨɞɫɬɚɜɥɹɹ ɩɪɟɞɩɨɥɚɝɚɟɦɨɟ ɪɟɲɟɧɢɟ ɢ ɟɝɨ ɩɪɨɢɡɜɨɞɧɭɸ ɜ ɭɪɚɜɧɟɧɢɟ (1) ɜɦɟɫɬɟ ɫ ɪɹɞɨɦ,
ɩɪɟɞɫɬɚɜɥɹɸɳɢɦ ɟɝɨ ɩɪɚɜɭɸ ɱɚɫɬɶ, ɢ ɭɪɚɜɧɢɜɚɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɫɢɧɭɫɚɯ ɢ ɤɨɫɢɧɭɫɚɯ ɨɞɧɨɣ ɱɚɫɬɨɬɵ,
ɧɚɣɞɟɦ ɜ ɢɬɨɝɟ ɷɬɨɣ ɨɛɵɱɧɨɣ ɩɪɨɰɟɞɭɪɵ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɟɲɟɧɢɹ Ak ɢ Bk:
Ak = K bk T1 (Sk/T) / [1 + (SkɌ1/T)2]
Bk = K bk / [1 + (SkɌ1/T)2].
(8)
ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɡɚɩɢɫɚɬɶ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɜ ɮɨɪɦɟ ɟɞɢɧɨɣ ɮɭɧɤɰɢɢ, ɚ ɧɟ ɩɨɥɶɡɨɜɚɬɶɫɹ
ɤɭɫɨɱɧɵɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɪɟɲɟɧɢɹ, ɤɚɤ ɷɬɨ ɞɟɥɚɥɨɫɶ ɨɛɵɱɧɨ ɞɥɹ ɞɚɧɧɨɣ ɦɨɞɟɥɢ. ɂɦɟɧɧɨ ɷɬɨ
ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɩɨɡɜɨɥɹɟɬ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ Ɍ1 ɢ Ʉ ɦɨɞɟɥɢ (1).
3. ɂɞɟɧɬɢɮɢɤɚɰɢɹ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ
ɉɨ ɫɚɦɨɦɭ ɫɭɳɟɫɬɜɭ ɦɚɧɟɜɪɟɧɧɵɯ ɢɫɩɵɬɚɧɢɣ ɜɢɞɚ "Ɂɢɝɡɚɝ" ɦɵ ɢɦɟɟɦ ɞɜɚ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹ ɪɚɜɟɧɫɬɜɨ ɢɡɦɟɧɟɧɢɹ ɤɭɪɫɨɜɨɝɨ ɭɝɥɚ ɢ ɭɝɥɚ ɤɥɚɞɤɢ ɪɭɥɹ (ɫ ɭɱɟɬɨɦ ɡɧɚɤɨɜ) ɜ ɡɚɮɢɤɫɢɪɨɜɚɧɧɵɟ ɩɪɢ
ɷɤɫɩɟɪɢɦɟɧɬɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ t1 ɢ (T-t1). ɉɪɚɜɞɚ, ɜ ɫɢɥɭ ɰɢɤɥɢɱɧɨɫɬɢ ɩɪɨɰɟɫɫɚ, ɨɛɚ ɷɬɢ ɝɪɚɧɢɱɧɵɯ
ɭɫɥɨɜɢɹ ɩɪɢɜɨɞɹɬ ɤ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɭɪɚɜɧɟɧɢɸ, ɜ ɤɨɬɨɪɨɟ ɜɯɨɞɹɬ ɞɜɚ ɢɞɟɧɬɢɮɢɰɢɪɭɟɦɵɯ ɩɚɪɚɦɟɬɪɚ Ɍ1
ɢ Ʉ (ɜ ɷɬɨɦ ɭɪɚɜɧɟɧɢɢ ɩɨɞɫɬɚɜɥɟɧɵ ɜɫɟ ɩɪɨɦɟɠɭɬɨɱɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ):
M(t1) = Ʉ˜C˜6k{sin(Sk˜t1/T)/[(Sk/T)2(1 + (SkɌ1/T)2)]u
u[cos(Sk˜t/T) / (Sk/T) T1 sin(Sk˜t/T)] / (Sk/T)} = G0,
(9)
ɝɞɟ C = 4G0/(T˜t1).
ɉɨɫɤɨɥɶɤɭ ɢɞɟɧɬɢɮɢɰɢɪɭɸɬɫɹ ɞɜɚ ɩɚɪɚɦɟɬɪɚ, ɬɨ ɫɥɟɞɭɟɬ ɜɵɛɪɚɬɶ ɟɳɟ ɨɞɧɨ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ,
ɱɬɨɛɵ ɢɦɟɬɶ ɫɢɫɬɟɦɭ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɢɯ ɨɩɪɟɞɟɥɟɧɢɹ. Ɍɚɤɢɦ ɭɫɥɨɜɢɟɦ ɦɨɠɟɬ ɛɵɬɶ ɜɪɟɦɹ t3
ɜɨɡɜɪɚɳɟɧɢɹ ɫɭɞɧɚ ɧɚ ɢɫɯɨɞɧɵɣ ɤɭɪɫ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢɡɦɟɧɟɧɢɟ ɤɭɪɫɚ
ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ: M(t3) = 0. ɉɪɢ ɷɬɨɦ, ɫ ɰɟɥɶɸ ɭɩɪɨɳɟɧɢɹ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɢɫɯɨɞɧɵɣ ɤɭɪɫ ɪɚɜɟɧ
ɧɭɥɸ. ɗɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɮɢɤɫɢɪɭɟɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɢɫɩɵɬɚɧɢɣ ɛɟɡ ɜɫɹɤɨɝɨ ɬɪɭɞɚ. ȼɜɟɞɟɧɧɨɟ
ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ ɩɪɢɜɨɞɢɬ ɤ ɫɥɟɞɭɸɳɟɦɭ ɫɨɨɬɧɨɲɟɧɢɸ:
Ʉ˜ 6k bk [cos(Sk˜t3/T) / (Sk/T) + T1˜ sin(Sk˜t3/T)] / [1 + (SkɌ1/T)2] = 0.
(10)
Ɉɧɨ ɹɜɥɹɟɬɫɹ ɬɪɚɧɫɰɟɧɞɟɧɬɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɬɨɥɶɤɨ ɞɥɹ ɨɞɧɨɝɨ ɢɡ ɢɞɟɧɬɢɮɢɰɢɪɭɟɦɵɯ
ɩɚɪɚɦɟɬɪɨɜ Ɍ1. ȼɬɨɪɨɣ ɩɚɪɚɦɟɬɪ Ʉ ɜɯɨɞɢɬ ɜ ɧɟɝɨ ɦɭɥɶɬɢɩɥɢɤɚɬɢɜɧɨ ɢ ɩɪɨɫɬɨ ɫɨɤɪɚɳɚɟɬɫɹ.
ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɪɟɲɚɬɶ ɟɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɚ Ɍ1 ɦɨɠɧɨ ɬɨɥɶɤɨ ɱɢɫɥɟɧɧɨ, ɞɥɹ ɱɟɝɨ ɭɪɚɜɧɟɧɢɟ
ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ, ɜɵɪɚɡɢɜ ɹɜɧɨ ɨɞɧɨ ɢɡ ɜɯɨɞɹɳɢɯ ɜ ɧɟɝɨ Ɍ1. Ⱦɥɹ ɷɬɨɝɨ ɢɡ ɫɭɦɦɵ ɜɵɞɟɥɢɦ
ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɢɡ ɤɨɬɨɪɨɝɨ ɜɵɪɚɡɢɦ ɹɜɧɨ Ɍ1 ɜ ɜɢɞɟ:
T1 = [(6k*)˜(S/T)2(1 + (SɌ1/T)2) / sin(St1/T) + cos(St3/T)/(S/T)] / sin(St3/T),
(11)
(6k*)
ɝɞɟ ɩɨɞ ɫɢɦɜɨɥɨɦ
ɤɪɨɟɬɫɹ ɫɭɦɦɚ ɢɡ ɬɟɯ ɠɟ ɱɥɟɧɨɜ, ɱɬɨ ɢ ɜ ɜɵɪɚɠɟɧɢɢ (10), ɧɨ ɫɭɦɦɢɪɨɜɚɧɢɟ ɧɚɱɢɧɚɟɬɫɹ
ɫ ɢɧɞɟɤɫɚ k, ɪɚɜɧɨɝɨ 3. Ɍɚɤɚɹ ɮɨɪɦɚ ɢɫɯɨɞɧɨɝɨ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɨɩɭɫɤɚɟɬ ɩɪɨɫɬɨɟ ɱɢɫɥɟɧɧɨɟ
ɪɟɲɟɧɢɟ ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ ɇɶɸɬɨɧɚ. ɉɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɩɚɪɚɦɟɬɪɚ Ɍ1 ɥɟɝɤɨ ɧɚɣɬɢ
ɩɚɪɚɦɟɬɪ Ʉ ɢɡ ɭɪɚɜɧɟɧɢɹ (9) ɩɪɨɫɬɵɦ ɞɟɥɟɧɢɟɦ G0 ɧɚ ɦɧɨɠɢɬɟɥɶ ɩɪɢ Ʉ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ.
ɑɢɫɥɟɧɧɚɹ ɨɛɪɚɛɨɬɤɚ ɦɨɞɟɥɶɧɵɯ ɞɚɧɧɵɯ
ɉɪɨɜɟɞɟɦ ɦɨɞɟɥɶɧɵɟ ɪɚɫɱɟɬɵ ɩɨ ɩɪɟɞɥɨɠɟɧɧɵɦ ɮɨɪɦɭɥɚɦ ɞɥɹ ɯɚɪɚɤɬɟɪɧɵɯ ɫɥɭɱɚɟɜ
ɦɚɧɟɜɪɢɪɨɜɚɧɢɹ ɡɢɝɡɚɝɨɦ. Ʉɚɤ ɢ ɩɪɢ ɪɚɫɱɟɬɚɯ, ɩɪɢɜɟɞɟɧɧɵɯ ɜ ɬɚɛɥ. 1, ɜɨɡɶɦɟɦ Ɍ = 75 ɫ, ɨɬɤɥɨɧɟɧɢɟ ɪɭɥɹ
ɧɚ ɤɚɠɞɵɣ ɛɨɪɬ G0 = 10° ɢ ɜɪɟɦɹ ɩɟɪɟɜɨɞɚ ɪɭɥɹ ɜ ɷɬɨ ɩɨɥɨɠɟɧɢɟ t1 = 10 ɫ. ɗɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɚɪɚɦɟɬɪɚɦ
ɦɚɧɟɜɪɢɪɨɜɚɧɢɹ ɡɢɝɡɚɝɨɦ ɫɭɞɧɚ ɫɪɟɞɧɟɝɨ ɬɨɧɧɚɠɚ.
ɍ ɧɚɫ ɨɫɬɚɥɫɹ ɜ ɡɚɩɚɫɟ ɩɚɪɚɦɟɬɪ t3 ɜɪɟɦɹ ɜɨɡɜɪɚɳɟɧɢɹ ɫɭɞɧɚ ɧɚ ɩɟɪɜɨɧɚɱɚɥɶɧɵɣ ɤɭɪɫ.
ɉɪɢɞɚɜɚɹ ɟɦɭ ɪɚɡɥɢɱɧɵɟ ɡɧɚɱɟɧɢɹ, ɦɵ ɛɭɞɟɦ ɩɨɥɭɱɚɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɡɧɚɱɟɧɢɹ ɢɞɟɧɬɢɮɢɰɢɪɭɟɦɵɯ
ɩɚɪɚɦɟɬɪɨɜ T1 ɢ K. ɉɪɟɞɟɥɵ ɜɨɡɦɨɠɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɠɟɫɬɤɢ. ɋ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, t3 ɧɟ ɦɨɠɟɬ
71
ɋɬɟɩɚɯɧɨ Ɋ.Ƚ. ȿɳɟ ɪɚɡ ɨɛ ɭɪɚɜɧɟɧɢɢ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɇɨɦɨɬɨ
ɛɵɬɶ ɛɨɥɶɲɟ ɱɟɦ (T - t1), ɬ.ɤ. ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɨɬ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɤɭɪɫɚ ɞɨɥɠɟɧ ɫɬɚɬɶ
ɪɚɜɧɵɦ G0. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɦɟɧɶɲɟ t1, ɬ.ɤ. ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɚ ɨɬ
ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɪɚɜɟɧ -G0. ɉɪɨɝɪɚɦɦɚ ɧɚ ɥɸɛɨɦ ɹɡɵɤɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ Ɍ1
ɩɢɲɟɬɫɹ ɷɥɟɦɟɧɬɚɪɧɨ. ɉɪɢɜɟɞɟɦ ɮɪɚɝɦɟɧɬ ɬɚɤɨɣ ɩɪɨɝɪɚɦɦɵ ɧɚ ɹɡɵɤɟ VB5, ɜ ɤɨɬɨɪɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɩɚɪɚɦɟɬɪɌ1 ɧɚ ɨɫɧɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɹ (11) ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ. Ɏɪɚɝɦɟɧɬ ɨɮɨɪɦɥɟɧ
ɜ ɜɢɞɟ ɮɭɧɤɰɢɢ count_t1(), ɤɨɬɨɪɚɹ ɜɨɡɜɪɚɳɚɟɬ ɱɟɪɟɡ ɫɩɟɰɢɚɥɶɧɭɸ ɩɨɥɶɡɨɜɚɬɟɥɶɫɤɭɸ ɫɬɪɭɤɬɭɪɭ
ExpTimes ɢɫɤɨɦɨɟ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ Ɍ1 ɫ ɡɚɞɚɧɧɨɣ ɫɬɟɩɟɧɶɸ ɩɪɢɛɥɢɠɟɧɢɹ. ȼ ɮɭɧɤɰɢɸ ɩɟɪɟɞɚɸɬɫɹ
ɫɥɟɞɭɸɳɢɟ ɚɪɝɭɦɟɧɬɵ: ɭɠɟ ɭɩɨɦɹɧɭɬɚɹ ɫɬɪɭɤɬɭɪɚ, ɩɨɥɹ ɤɨɬɨɪɨɣ ɫɜɹɡɚɧɵ ɫ ɜɪɟɦɟɧɧɵɦɢ ɞɚɧɧɵɦɢ ɦɨɦɟɧɬɚɦɢ t1, t3, ɩɟɪɢɨɞɨɦ T ɢ ɡɧɚɱɟɧɢɟɦ ɢɞɟɧɬɢɮɢɰɢɪɭɟɦɨɝɨ ɩɚɪɚɦɟɬɪɚ, ɡɚɞɚɧɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɟɝɨ
ɨɩɪɟɞɟɥɟɧɢɹ Eps0, ɧɨɦɟɪ ɩɨɫɥɟɞɧɟɝɨ ɭɱɢɬɵɜɚɟɦɨɝɨ ɫɥɚɝɚɟɦɨɝɨ Nobr ɜ ɫɭɦɦɟ (6k*) ɜɵɪɚɠɟɧɢɹ (11);
ɤɪɨɦɟ ɬɨɝɨ, ɫɚɦɚ ɮɭɧɤɰɢɹ ɜɨɡɜɪɚɳɚɟɬ ɤɨɥɢɱɟɫɬɜɨ ɢɬɟɪɚɰɢɣ NumbIter, ɪɟɚɥɢɡɨɜɚɧɧɵɯ ɜɧɭɬɪɢ ɮɭɧɤɰɢɢ,
ɞɥɹ ɞɨɫɬɢɠɟɧɢɹ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɪɟɡɭɥɶɬɚɬɚ.
‘ɨɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ
Public function count_t1(ET as ExpTimes, _
Eps0 as single, Nobr as integer ) as integer
Dim sums, pikt, eps, Y, tt0, tt1, nmb
nmb=0
tt0=ET.tt0
’ ɧɚɱɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ
DO WHILE eps > Eps0
sums = 0
with EN
FOR k = 3 TO Nobr
IF k MOD 2 = 0 THEN GOTO nxt1
pikt = pi * k/.T
Y = SIN(pikt*.t1)/pikt^2/(1 + (pikt*tt0)^2)
Y = Y*(COS(pikt*.t3)/pikt + tt0*SIN(pikt*.t3))
sums = sums + Y
nxt1:
NEXT k
nmb = nmb + 1
Y=(pi/.T)^2*(1+ (pi*tt0/.T)^2)/SIN(pi*.t1/.T)
Y=(Y + .T/pi*COS(pi*.t3/.T))/SIN(pi*.t3/.T)
tt1 = (-sums) * Y
eps = ABS(tt1 - tt0)
tt0 = tt1
LOOP
ET.tt0=tt0
‘ ɜɨɡɜɪɚɳɚɟɦ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ
count_t1=nmb
‘ ɜɨɡɜɪɚɳɚɟɦ ɱɢɫɥɨ ɢɬɟɪɚɰɢɣ
End function
Dim tt0,eps0 as single
Dim nobr, NumbIter as integer
Public Const pi=3.141592
‘ɨɩɪɟɞɟɥɹɟɦ ɩɨɥɶɡɨɜɚɬɟɥɶɫɤɭɸ ɫɬɪɭɤɬɭɪɭ
Def type ExpTimes
t1 as single
t3 as single
T as single
tt0 as single
End type
Dim ET as ExpTimes
With ET
.tt0 = 0
.t1=10
.t3=50
.T=75
end with
eps0 = 0.001nobr = 9
‘ɨɛɪɚɳɟɧɢɟ ɤ ɮɭɧɤɰɢɢ count_t1
NumbIter = count_t1(ET, eps, nobr)
END
Ⱦɥɹ ɪɚɫɱɟɬɨɜ ɛɵɥɢ ɜɵɛɪɚɧɵ ɫɥɟɞɭɸɳɢɟ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ: ɩɨɥɭɩɟɪɢɨɞ ɩɪɨɰɟɫɫɚ Ɍ = 75 ɫ, ɱɬɨ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɭɞɧɭ ɫɪɟɞɧɟɝɨ ɬɨɧɧɚɠɚ, 10° ɚɦɩɥɢɬɭɞɚ ɨɬɤɥɨɧɟɧɢɣ ɪɭɥɹ G0, ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t3
ɢɡɦɟɧɹɥɫɹ ɨɬ 10 ɫ ɞɨ 65 ɫ. Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɨɜ ɩɪɢɜɨɞɹɬɫɹ ɜ ɬɚɛɥ. 2 ɩɨ ɤɨɥɨɧɤɚɦ: ɦɨɦɟɧɬ, ɩɚɪɚɦɟɬɪɵ
ɦɨɞɟɥɢ T1 ɢ K, ɨɬɧɨɲɟɧɢɟ K/T1, ɧɚɡɵɜɚɟɦɨɟ "ɩɨɫɥɭɲɥɢɜɨɫɬɶɸ" ɪɭɥɸ, ɚɦɩɥɢɬɭɞɚ ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɚ ɨɬ
ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ Mmax, ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tmax, ɤɨɝɞɚ ɷɬɨ ɡɧɚɱɟɧɢɟ ɞɨɫɬɢɝɚɟɬɫɹ. ɉɨɫɥɟɞɧɢɣ ɟɫɬɶ,
ɮɚɤɬɢɱɟɫɤɢ, ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɜ ɤɨɬɨɪɵɣ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɩɨɜɨɪɨɬɚ ɪɚɜɧɚ ɧɭɥɸ t4.
ɉɪɨɰɟɫɫ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (11) ɢɬɟɪɚɰɢɨɧɧɵɣ, ɩɨɬɨɦɭ ɜ ɩɪɟɞɫɬɚɜɥɟɧɧɨɦ ɨɬɪɟɡɤɟ
ɩɪɨɝɪɚɦɦɧɨɝɨ ɤɨɞɚ ɮɢɤɫɢɪɭɟɬɫɹ ɤɨɥɢɱɟɫɬɜɨ ɢɬɟɪɚɰɢɣ ɞɥɹ ɞɨɫɬɢɠɟɧɢɹ ɭɤɚɡɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɜ 0.001 ɫ.
ɉɪɨɰɟɫɫ ɷɬɨɬ ɨɱɟɧɶ ɛɵɫɬɪɨ ɫɯɨɞɢɬɫɹ ɢ ɧɢɝɞɟ ɧɟ ɩɨɧɚɞɨɛɢɥɨɫɶ ɛɨɥɟɟ 4-ɯ ɢɬɟɪɚɰɢɣ.
Ɍɚɛɥɢɰɚ 2
t3, c
10
15
19
25
30
35
37
38
40
45
50
55
60
T1, c
-49.30
-30.50
-21.91
-13.14
-7.55
-2.50
-0.50
0.50
2.50
7.55
13.14
20.20
30.50
K
0.0973
0.0596
0.0463
0.0368
0.0336
0.0341
0.0358
0.0370
0.0400
0.0501
0.0687
0.1117
0.2638
100 K/T1
-0.197
-0.195
-0.211
-0.280
-0.445
-1.364
-7.160
7.400
1.600
0.664
0.523
0.553
0.865
72
Mmax , ɝɪɚɞ
-12.7
-12.7
-12.7
-12.7
-12.7
-12.7
-12.7
-12.7
-12.9
-15.3
-19.2
-27.7
-57.7
tmax, ɫ
0
0
0
0
0
0
0
0
3
7
11
15
20
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 6, ʋ1, 2003 ɝ.
ɫɬɪ.69-74
ɏɨɪɨɲɨ ɜɢɞɧɨ, ɱɬɨ ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɭɞɧɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɦɚɧɟɜɪɢɪɨɜɚɧɢɹ ɧɚɱɢɧɚɟɬɫɹ ɫ
ɡɧɚɱɟɧɢɹ t3 = 37, 38 ɫ, ɬ.ɤ. ɨɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ Ɍ1 > 0. ɉɨɷɬɨɦɭ ɜɫɟ ɜɵɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɦɟɧɶɲɢɦ ɜɟɥɢɱɢɧɚɦ t3, ɦɵ ɨɛɫɭɠɞɚɬɶ ɧɟ ɛɭɞɟɦ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɥɢɧɟɣɧɵɣ ɚɧɚɥɢɡ
ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɩɪɨɫɬɨ ɧɟ ɢɦɟɟɬ ɫɦɵɫɥɚ. ɉɪɢ ɪɨɫɬɟ t3 ɪɚɫɬɟɬ ɢ ɜɟɥɢɱɢɧɚ Ɍ1, ɬ.ɟ. ɭɜɟɥɢɱɢɜɚɟɬɫɹ
ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɭɞɧɚ ɧɚ ɤɭɪɫɟ. Ɋɭɥɶ ɭɠɟ ɩɨɥɨɠɟɧ ɧɚ ɞɪɭɝɨɣ ɛɨɪɬ, ɚ ɨɬɤɥɨɧɟɧɢɟ ɨɬ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɤɭɪɫɚ
ɜɫɟ ɟɳɟ ɪɚɫɬɟɬ, ɢɦɟɧɧɨ ɩɨɷɬɨɦɭ tmax > t1 = 10 (6 ɤɨɥɨɧɤɚ ɬɚɛɥ. 2). Ɋɚɫɬɟɬ ɩɪɢ ɷɬɨɦ ɢ ɫɚɦɚ ɚɦɩɥɢɬɭɞɚ
ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɚ (5 ɤɨɥɨɧɤɚ). ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɪɚɛɨɬɟ ɩɨ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɧɚ (ɋɨɛɨɥɟɜ, 1976)
ɩɪɟɞɥɨɠɟɧɚ ɩɪɨɫɬɚɹ ɮɨɪɦɭɥɚ, ɫɜɹɡɵɜɚɸɳɚɹ Ɍ1 ɢ tmax ɜ ɧɚɲɟɦ ɨɛɨɡɧɚɱɟɧɢɢ: Ɍ1 | 1.45 tmax. ɇɚɲɢ
ɪɟɡɭɥɶɬɚɬɵ ɜ ɩɪɢɧɰɢɩɟ ɩɨɯɨɠɢ ɧɚ ɷɬɨ ɫɨɨɬɧɨɲɟɧɢɟ, ɧɨ ɩɨɥɧɨɝɨ ɫɨɜɩɚɞɟɧɢɹ ɧɟɬ ɩɨ ɬɨɣ ɩɪɢɱɢɧɟ, ɱɬɨ
ɮɨɪɦɭɥɚ Ƚ.ȼ. ɋɨɛɨɥɟɜɚ ɩɨɥɭɱɟɧɚ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɦɝɧɨɜɟɧɧɨɣ ɩɟɪɟɤɥɚɞɤɢ ɪɭɥɹ (t1 = 0), ɚ ɭ ɧɚɫ ɨɧɚ
ɡɚɧɢɦɚɟɬ 10 ɫ. Ɉɞɢɧ ɢɡ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɫɱɟɬɚ (ɩɪɢ Ɍ1 = 7.55 ɫ) ɛɵɥ ɩɪɟɞɫɬɚɜɥɟɧ ɜɵɲɟ ɧɚ ɪɢɫ. 1.
ȼ ɬɚɛɥ. 2 ɩɪɟɞɫɬɚɜɥɟɧɵ ɪɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ ɞɥɹ ɤɨɧɤɪɟɬɧɨɝɨ ɡɧɚɱɟɧɢɹ ɦɨɦɟɧɬɚ
ɜɪɟɦɟɧɢ t1 = 10 c ɢ ɜɟɥɢɱɢɧɵ ɩɨɥɭɩɟɪɢɨɞɚ Ɍ = 75 ɫ. Ⱦɥɹ ɞɪɭɝɢɯ ɡɧɚɱɟɧɢɣ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɜɟɫɶ ɪɚɫɱɟɬ
ɩɪɢɞɟɬɫɹ ɩɨɜɬɨɪɢɬɶ. ɇɨ ɦɨɠɧɨ ɭɤɚɡɚɬɶ ɞɪɭɝɨɣ ɫɩɨɫɨɛ, ɤɨɝɞɚ ɦɵ ɛɭɞɟɦ ɨɩɟɪɢɪɨɜɚɬɶ ɧɟ ɚɛɫɨɥɸɬɧɵɦɢ
ɡɧɚɱɟɧɢɹɦɢ ɜɪɟɦɟɧɢ, ɚ ɢɯ ɨɬɧɨɫɢɬɟɥɶɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ, ɨɬɧɟɫɹ ɢɯ ɤ ɡɧɚɱɟɧɢɸ ɩɨɥɭɩɟɪɢɨɞɚ Ɍ. Ɍɨɝɞɚ ɜ
ɪɚɫɱɟɬɚɯ ɜɦɟɫɬɨ Ɍ ɛɭɞɟɬ ɮɢɝɭɪɢɪɨɜɚɬɶ ɜɫɟɝɞɚ 1, a ɜɦɟɫɬɨ t1 ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɜɟɥɢɱɢɧɚ W1 = t1/T. Ɍɚɤ ɤɚɤ
ɡɧɚɱɟɧɢɟ t1 ɧɨɪɦɢɪɨɜɚɧɨ ɫɤɨɪɨɫɬɶɸ ɩɨɜɨɪɨɬɚ ɪɭɥɹ ɩɪɢɦɟɪɧɨ 1 ɝɪɚɞ/ɫ, ɬɨ ɦɵ ɜ ɪɚɫɱɟɬɚɯ ɜɨɡɶɦɟɦ ɬɚɤɨɣ ɪɹɞ
ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ: ɨɬ 0.05 ɞɨ 0.20 ɫ ɲɚɝɨɦ ɜ 0.01. ɉɪɢ ɷɬɨɦ ɫɚɦɨ t1 ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
ɧɨɪɦɚɬɢɜɨɦ ɪɚɜɧɵɦ 10 ɫ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɟɞɥɨɠɟɧɧɵɣ ɪɹɞ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ W1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɫɥɟɞɭɸɳɟɦɭ ɪɹɞɭ ɡɧɚɱɟɧɢɣ ɩɨɥɭɩɟɪɢɨɞɚ Ɍ, ɫ: ɨɬ 200 ɞɨ 50 ɫ ɫ ɩɟɪɟɦɟɧɧɵɦ ɲɚɝɨɦ. ɇɨ ɫɚɦɨɟ ɡɚɦɟɱɚɬɟɥɶɧɨɟ
ɩɪɢ ɷɬɨɦ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ ɦɨɞɟɥɢ ‚ = Ɍ1/T ɨɤɚɡɵɜɚɟɬɫɹ ɡɚɜɢɫɹɳɢɦ ɧɟ ɨɬ W1, ɚ ɬɨɥɶɤɨ
ɨɬ W3 = t3/T. ɉɨɤɚɠɟɦ ɜɵɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɬɚɛɥ. 3.
Ɍɚɛɥɢɰɚ 3
W3 = t3/T
‚ = Ɍ1/T
0.60
0.1005
0.65
0.1547
0.70
0.2173
0.75
0.2953
0.80
0.4011
0.85
0.5634
0.90
0.9665
Ɇɨɠɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ ɨɛɪɚɛɨɬɤɢ ɞɚɧɧɵɯ ɦɚɧɟɜɪɢɪɨɜɚɧɢɹ ɬɚɛɥ. 3, ɧɨ ɦɨɠɧɨ ɭɤɚɡɚɬɶ ɩɪɨɫɬɭɸ
ɚɩɩɪɨɤɫɢɦɚɰɢɸ ɷɬɢɯ ɪɟɡɭɥɶɬɚɬɨɜ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɨɜ 2-ɨɣ ɢɥɢ 3-ɟɣ ɫɬɟɩɟɧɟɣ. Ɉɧɢ ɩɨɥɭɱɟɧɵ ɦɟɬɨɞɨɦ
ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɫ ɩɨɦɨɳɶɸ ɩɚɤɟɬɚ MathCad7 ɢ ɢɬɨɝɢ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɬɚɤɨɜɵ:
‚ = 1.632 5.598W3 + 5.095(W3)2
‚ = 6.018 + 26.579W3 39.646(W3)2 + 20.57(W3)3
ɫ ɋɄɉ V = 9˜10-3
c CɄɉ V = 2˜10-3.
(14)
ɋ ɩɨɦɨɳɶɸ ɬɚɛɥ. 3 ɢɥɢ ɨɞɧɨɝɨ ɢɡ ɭɪɚɜɧɟɧɢɣ (14) ɩɚɪɚɦɟɬɪ ɦɨɞɟɥɢ ɇɨɦɨɬɨ Ɍ1 ɧɚɯɨɞɢɬɫɹ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ. ȼ ɯɨɞɟ ɢɫɩɵɬɚɧɢɣ ɮɢɤɫɢɪɭɟɬɫɹ ɩɨɥɭɩɟɪɢɨɞ Ɍ ɢ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t3, ɤɨɝɞɚ ɨɬɤɥɨɧɟɧɢɟ ɤɭɪɫɚ
ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɧɭɥɸ. ɉɨ ɢɯ ɨɬɧɨɲɟɧɢɸ W3 ɢɡ ɬɚɛɥ. 3 ɫ ɢɧɬɟɪɩɨɥɹɰɢɟɣ ɢɥɢ ɩɨ ɨɞɧɨɣ ɢɡ ɮɨɪɦɭɥ (14)
ɧɚɣɞɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ ɦɨɞɟɥɢ ‚. ɍɦɧɨɠɚɹ ɟɝɨ ɧɚ ɩɨɥɭɩɟɪɢɨɞ Ɍ, ɩɨɥɭɱɚɟɦ ɡɧɚɱɟɧɢɟ
ɩɚɪɚɦɟɬɪɚ Ɍ1 ɦɨɞɟɥɢ. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɬɨɪɨɝɨ ɩɚɪɚɦɟɬɪɚ Ʉ ɩɪɢɞɟɬɫɹ ɜɫɟ ɠɟ ɢɫɩɨɥɶɡɨɜɚɬɶ ɮɨɪɦɭɥɭ (9).
5. ȼɨɡɦɨɠɧɵɟ ɪɚɫɲɢɪɟɧɢɹ ɫɮɟɪɵ ɩɪɢɦɟɧɢɦɨɫɬɢ
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɟɳɟ ɨɞɢɧ ɧɚɛɥɸɞɚɟɦɵɣ ɩɚɪɚɦɟɬɪ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɦɚɧɟɜɪɚ ɦɨɦɟɧɬ,
ɤɨɝɞɚ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɩɨɜɨɪɨɬɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ɗɬɨ ɜɨɡɦɨɠɧɨ ɫɞɟɥɚɬɶ, ɟɫɥɢ ɢɫɩɨɥɶɡɭɟɬɫɹ
ɫɚɦɨɩɢɫɟɰ ɤɭɪɫɚ ɫɭɞɧɚ ɢɥɢ ɚɤɫɢɨɦɟɬɪ. ɇɚɡɨɜɟɦ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t4 ɢ ɧɚɩɨɦɧɢɦ, ɱɬɨ ɜ ɬɚɛɥ. 2 ɨɧ ɧɟ
ɧɚɛɥɸɞɚɥɫɹ, ɚ ɜɵɱɢɫɥɹɥɫɹ. ȼɨɡɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬ ɬɚɤɨɝɨ ɧɚɛɥɸɞɟɧɢɹ ɢ ɧɚɣɬɢ ɩɚɪɚɦɟɬɪ
ɦɨɞɟɥɢ Ɍ1 ɢɡ ɭɫɥɨɜɢɹ Z(t4) = 0. Ɉɧɨ ɩɪɢɜɟɞɟɬ ɤ ɬɪɚɧɫɰɟɧɞɟɧɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ, ɩɨɞɨɛɧɨɦɭ (11):
T1 = [(6k**)˜(S/T)2(1 + (SɌ1/T)2) / sin(St1/T) + sin(St4/T)] / [(S/T) cos(St4/T)],
(12)
ɝɞɟ 6k** ɨɡɧɚɱɚɟɬ ɫɭɦɦɭ, ɤɨɬɨɪɚɹ ɭɱɚɫɬɜɭɟɬ ɜ ɜɵɪɚɠɟɧɢɢ (6) ɞɥɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ, ɧɨ ɫɭɦɦɢɪɨɜɚɧɢɟ
ɧɚɱɢɧɚɟɬɫɹ ɫ k = 3. Ɍɚɤɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ Ɍ1 ɞɚɫɬ ɛɨɥɟɟ ɦɟɞɥɟɧɧɭɸ ɫɯɨɞɢɦɨɫɬɶ ɢɬɟɪɚɰɢɣ, ɬ.ɤ. ɭɝɥɨɜɚɹ
ɫɤɨɪɨɫɬɶ ɟɫɬɶ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɭɝɥɚ ɨɬɤɥɨɧɟɧɢɹ. Ɂɚ ɫɱɟɬ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɤɚɠɞɵɣ k-ɵɣ
ɱɥɟɧ ɪɹɞɚ ɜɨɡɪɚɫɬɚɟɬ ɜ (Sk/T) ɪɚɡ, ɱɬɨ ɢ ɡɚɦɟɞɥɹɟɬ ɫɯɨɞɢɦɨɫɬɶ. Ɍɟɦ ɧɟ ɦɟɧɟɟ, ɛɵɥɢ ɩɪɨɜɟɞɟɧɵ ɪɚɫɱɟɬɵ ɫ
ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (12). Ɍɟɩɟɪɶ ɱɢɫɥɨ ɢɬɟɪɚɰɢɣ ɜɨɡɪɨɫɥɨ ɞɨ 7-8, ɱɬɨ ɧɟ ɢɝɪɚɟɬ ɨɫɨɛɨɣ ɪɨɥɢ ɩɪɢ
ɫɨɜɪɟɦɟɧɧɨɦ ɛɵɫɬɪɨɞɟɣɫɬɜɢɢ ȼɌ. ɉɪɢ ɷɬɨɦ ɪɟɡɭɥɶɬɚɬɵ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ ɩɪɚɤɬɢɱɟɫɤɢ
ɧɟ ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɪɟɡɭɥɶɬɚɬɨɜ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ (11).
ɇɨ ɡɚɱɟɦ ɝɨɜɨɪɢɬɶ ɨ ɩɪɢɟɦɟ, ɤɨɬɨɪɵɣ ɩɨ ɫɯɨɞɢɦɨɫɬɢ ɹɜɧɨ ɭɫɬɭɩɚɟɬ? Ⱦɟɥɨ ɜ ɬɨɦ, ɱɬɨ ɷɬɨ ɭɞɨɛɧɨɟ
ɤɚɤ ɩɨ ɫɩɨɫɨɛɭ ɧɚɛɥɸɞɟɧɢɹ, ɬɚɤ ɢ ɩɨ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɩɪɨɰɟɞɭɪɟ ɤɪɚɟɜɨɟ ɭɫɥɨɜɢɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ
73
ɋɬɟɩɚɯɧɨ Ɋ.Ƚ. ȿɳɟ ɪɚɡ ɨɛ ɭɪɚɜɧɟɧɢɢ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɇɨɦɨɬɨ
ɞɥɹ ɪɟɲɟɧɢɹ ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɧɨɜɨɣ ɡɚɞɚɱɢ. Ⱥ ɢɦɟɧɧɨ, ɢɞɟɧɬɢɮɢɰɢɪɨɜɚɬɶ ɩɚɪɚɦɟɬɪɵ ɜ ɦɨɞɟɥɢ
ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɤɨɬɨɪɭɸ ɨɛɵɱɧɨ ɡɚɩɢɫɵɜɚɸɬ ɜ ɜɢɞɟ:
T1 T2 d2Z / dt2 + (T1 + T2) dZ /dt + Z = K G.
(13)
ȼ ɷɬɨɣ ɦɨɞɟɥɢ ɭɠɟ ɬɪɢ ɩɚɪɚɦɟɬɪɚ Ɍ1, Ɍ2 ɢ Ʉ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɢɯ ɨɩɪɟɞɟɥɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɬɪɢ
ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɦɢ ɛɭɞɭɬ ɬɟɩɟɪɶ M(t3) = 0, Z(t4) = 0 ɢ M(Ɍ - t1) = G0. ɗɬɨ ɨɛɨɛɳɟɧɢɟ ɦɟɬɨɞɚ ɧɚ
ɦɨɞɟɥɶ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɪɟɚɥɢɡɨɜɚɬɶ ɜ ɞɚɥɶɧɟɣɲɢɯ ɪɚɛɨɬɚɯ ɩɨ ɩɪɨɛɥɟɦɟ
ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɨɛɨɛɳɟɧɧɨɣ ɦɨɞɟɥɢ ɇɨɦɨɬɨ.
6. Ɂɚɤɥɸɱɟɧɢɟ
ɂɡɥɨɠɟɧɧɵɣ ɜ ɪɚɛɨɬɟ ɧɨɜɵɣ ɩɪɢɟɦ ɨɛɪɚɛɨɬɤɢ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɬɭɪɧɨɝɨ ɦɚɧɟɜɪɢɪɨɜɚɧɢɹ ɬɢɩɚ
"ɡɢɝɡɚɝ" ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɚɡɥɨɠɟɧɢɹ ɡɚɤɨɧɚ ɩɟɪɟɤɥɚɞɤɢ ɪɭɥɹ ɜ ɪɹɞ Ɏɭɪɶɟ ɞɚɥ ɜɨɡɦɨɠɧɨɫɬɶ ɨɛɪɚɛɨɬɚɬɶ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɞɚɧɧɵɟ ɩɨ ɦɢɧɢɦɚɥɶɧɨɦɭ ɢ ɥɟɝɤɨ ɢɡɦɟɪɹɟɦɨɦɭ ɧɚɛɨɪɭ ɯɚɪɚɤɬɟɪɧɵɯ ɜɪɟɦɟɧ
ɩɪɨɰɟɫɫɚ: ɜɪɟɦɟɧɢ ɧɭɥɟɜɨɝɨ ɨɬɤɥɨɧɟɧɢɹ ɤɭɪɫɚ ɢɥɢ ɧɭɥɟɜɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɩɨɜɨɪɨɬɚ. ɗɬɨɝɨ
ɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɩɨɥɧɨɣ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ ɇɨɦɨɬɨ. Ȼɨɥɟɟ ɬɨɝɨ, ɩɨɥɭɱɟɧɨ
ɩɨɥɭɷɦɩɢɪɢɱɟɫɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɩɚɪɚɦɟɬɪɨɦ ɦɨɞɟɥɢ Ɍ1 ɢ ɦɨɦɟɧɬɨɦ ɜɪɟɦɟɧɢ t3 ɜ ɨɬɧɨɫɢɬɟɥɶɧɵɯ
ɜɟɥɢɱɢɧɚɯ. ɋɩɨɫɨɛ ɨɩɪɚɜɞɚɥ ɜɨɡɥɚɝɚɟɦɵɟ ɧɚ ɧɟɝɨ ɨɠɢɞɚɧɢɹ ɩɨ ɭɩɪɨɳɟɧɢɸ ɩɪɨɰɟɫɫɚ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɢ
ɩɨɤɚɡɚɥ ɩɪɢɧɰɢɩɢɚɥɶɧɭɸ ɜɨɡɦɨɠɧɨɫɬɶ ɩɪɢɦɟɧɢɬɶ ɬɨɬ ɠɟ ɩɪɢɟɦ ɞɥɹ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɨɛɨɛɳɟɧɧɨɣ ɦɨɞɟɥɢ
ɫ ɛɨɥɶɲɢɦ ɱɢɫɥɨɦ ɢɞɟɧɬɢɮɢɰɢɪɭɟɦɵɯ ɩɚɪɚɦɟɬɪɨɜ.
Ʌɢɬɟɪɚɬɭɪɚ
Nomoto K. and Norrbin N. A review of methods of defining and measuring the manoeuvrability of ships.
ITTC, Manoeuvrability Committee Report, 1969.
Nomoto K., Taguchi T. and Hirano S. On the steering qualities of ship. International Shipbuilding Progress,
v.4, N 35, p.56-64, 1957.
Ⱥɫɢɧɨɜɫɤɢɣ ȼ.Ⱥ. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɱɢɫɥɟɧɧɨɝɨ ɦɟɬɨɞɚ ɞɥɹ ɚɧɚɥɢɡɚ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɬɭɪɧɵɯ ɢɫɩɵɬɚɧɢɣ
ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɨɜ ɩɪɢ ɧɟɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɞɜɢɠɟɧɢɢ. ɋɛɨɪɧɢɤ ɫɬɚɬɟɣ ɦɨɥɨɞɵɯ ɧɚɭɱɧɵɯ
ɪɚɛɨɬɧɢɤɨɜ, ɱɚɫɬɶ VIII, 1969.
Ⱥɫɢɧɨɜɫɤɢɣ ȼ.Ⱥ., Ƚɨɮɦɚɧ Ⱥ.Ⱦ. Ɉɛ ɨɰɟɧɤɟ ɭɩɪɚɜɥɹɟɦɨɫɬɢ ɫɭɞɨɜ. ɇɚɭɱɧɨ-ɬɟɯɧɢɱɟɫɤɨɟ ɨɛɳɟɫɬɜɨ
ɢɦ. ɚɤɚɞ. Ⱥ.ɇ. Ʉɪɵɥɨɜɚ. Ʌ., ɋɭɞɨɫɬɪɨɟɧɢɟ, ɜɵɩ.90, 1967.
Ȼɚɫɢɧ Ⱥ.Ɇ. ɏɨɞɤɨɫɬɶ ɢ ɭɩɪɚɜɥɹɟɦɨɫɬɶ ɫɭɞɨɜ. Ɇ., Ɍɪɚɧɫɩɨɪɬ, ɫ.455, 1977.
Ʉɚɰɦɚɧ Ɏ.Ɇ., Ɇɭɡɵɤɚɧɬɨɜ Ƚ.Ɇ., ɒɦɟɥɟɜ Ⱥ.ȼ. ɗɤɫɩɥɭɚɬɚɰɢɨɧɧɵɟ ɢɫɩɵɬɚɧɢɹ ɦɨɪɫɤɢɯ ɫɭɞɨɜ. Ɇ.,
Ɍɪɚɧɫɩɨɪɬ, ɫ.272, 1970.
ɋɨɛɨɥɟɜ Ƚ.ȼ. ɍɩɪɚɜɥɹɟɦɨɫɬɶ ɤɨɪɚɛɥɹ ɢ ɚɜɬɨɦɚɬɢɡɚɰɢɹ ɫɭɞɨɜɨɠɞɟɧɢɹ. Ʌ., ɋɭɞɨɫɬɪɨɟɧɢɟ, ɫ.477, 1976.
74
Документ
Категория
Без категории
Просмотров
5
Размер файла
109 Кб
Теги
уравнения, номото, раз, управляемость
1/--страниц
Пожаловаться на содержимое документа