79 ДИНАМИКА ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ВЕРТИКАЛЬНОЙ

15. Tewari U. S. Recent development in tribology of fibre reinforced composities with thermoplastic
and thermosetting matrices / U. S. Tewari, J. Bijwe // Advances in Composities Tribology. — 1993. —
ɋhap. 5.
16. Wang Y. AFM characterization of interfacial properties of carbon fibre reinforced composites
subjected to hygrothermal treatments / Y. Wang, T. H. Hahn // Composites science and Technology. —
2007. — ʋ 67.
17. Wilczewska I. Badanie mechanizmu tarcia i zuĪycia polimerów wzmacnianych wáóknami
wĊglowymi / I. Wilczewska // VII MiĊdzynarodowa Konferencja. — Tarcie, 2012.
ȁDzǸ 627.332, 519.63, 51.74
Ȁ. Ȍ. ǻȖȘȖȠȖțȎ,
ȎȟȝȖȞȎțȠ,
DZȁǺǾȂ ȖȚȓțȖ ȎȒȚȖȞȎșȎ ǿ. Ǽ. ǺȎȘȎȞȜȐȎ
ȾɂɇȺɆɂɄȺ ɉɈɉȿɊȿɑɇɕɏ ɄɈɅȿȻȺɇɂɃ
ȼȿɊɌɂɄȺɅɖɇɈɃ ɁȺɓȿɆɅȿɇɇɈɃ ȻȺɅɄɂ
DYNAMICS OF TRANSVERSE OSCILLATIONS OF A RIGID VERTICAL BEAM
ɋɬɚɬɶɹ ɩɨɫɜɹɳɟɧɚ ɜɨɩɪɨɫɭ ɭɫɬɨɣɱɢɜɨɫɬɢ ɤɨɧɫɬɪɭɤɰɢɣ ɢɡ ɬɨɧɤɢɯ ɜɟɪɬɢɤɚɥɶɧɵɯ ɫɬɟɧ. Ɋɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɭɩɪɭɝɨɣ ɛɚɥɤɢ ɫ ɡɚɳɟɦɥɟɧɧɵɦ ɧɢɠɧɢɦ ɤɨɧɰɨɦ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɣ ɫɢɥɵ. Ɉɩɢɫɚɧɵ ɚɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɨɣ ɛɚɥɤɢ ɫ ɩɪɢɦɟɧɟɧɢɟɦ «ɛɚɥɨɱɧɵɯ» ɮɭɧɤɰɢɣ Ⱥ. ɇ. Ʉɪɵɥɨɜɚ ɢ ɦɟɬɨɞɚ Ȼ. Ƚ. Ƚɚɥɟɪɤɢɧɚ ɜ ɜɚɪɢɚɧɬɟ ȼ. Ɂ. ȼɥɚɫɨɜɚ — Ʌ. ȼ. Ʉɚɧɬɨɪɨɜɢɱɚ ɉɪɢɜɟɞɟɧɵ ɪɟɡɭɥɶɬɚɬɵ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɜɵɩɨɥɧɟɧɧɨɝɨ ɜ ɩɪɨɝɪɚɦɦɟ Mathcad.
The article is devoted to the stability of rising walls structures. A mathematical model of the vertical transverse vibrations of an elastic beam clamped at the lower end by an external force is considered. The algorithms for
solving differential equations of transverse vibrations of an elastic beam with the use of “beam” functions of A. N.
Krylov, and B. G. Galerkin in the version of V. Z. Vlasov — L. V. Kantorovich are described. The results of numerical experiments performed in the program Mathcad are given.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɝɢɞɪɨɬɟɯɧɢɱɟɫɤɨɟ ɫɬɪɨɢɬɟɥɶɫɬɜɨ, ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɨɨɪɭɠɟɧɢɣ, ɩɨɩɟɪɟɱɧɵɟ ɤɨɥɟɛɚɧɢɹ ɭɩɪɭɝɨɣ ɛɚɥɤɢ.
Key words: hydraulic engineering, sustainability of structures, transversal vibrations of an elastic beam.
Выпуск 3
ȼɜɟɞɟɧɢɟ
ȼ ɩɨɪɬɨɜɨɦ ɫɬɪɨɢɬɟɥɶɫɬɜɟ ɞɥɹ ɫɨɡɞɚɧɢɹ ɩɪɢɱɚɥɶɧɵɯ ɢ ɜɨɥɧɨɡɚɳɢɬɧɵɯ ɫɨɨɪɭɠɟɧɢɣ ɩɪɢɦɟɧɹɸɬɫɹ ɤɨɧɫɬɪɭɤɰɢɢ ɫ ɷɥɟɦɟɧɬɚɦɢ ɬɨɧɤɢɯ ɜɟɪɬɢɤɚɥɶɧɵɯ ɫɬɟɧɨɤ. ɇɚ ɨɬɤɪɵɬɨɣ ɚɤɜɚɬɨɪɢɢ ɱɚɫɬɨ
ɩɪɢɦɟɧɹɸɬɫɹ ɛɨɥɶɜɟɪɤɢ (ɬɨɧɤɢɟ ɩɨɞɩɨɪɧɵɟ ɫɬɟɧɤɢ), ɜɵɩɨɥɧɟɧɧɵɟ ɢɡ ɲɩɭɧɬɨɜɵɯ ɫɜɚɣ ɢɥɢ ɫɜɚɣ
ɫɩɟɰɢɚɥɶɧɵɯ ɩɪɨɮɢɥɟɣ. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɬɚɤɢɯ ɫɨɨɪɭɠɟɧɢɣ ɫɜɹɡɚɧɨ ɫ ɩɪɨɛɥɟɦɨɣ ɜɨɡɧɢɤɧɨɜɟɧɢɹ
ɚɜɚɪɢɣɧɵɯ ɫɢɬɭɚɰɢɣ ɩɪɢ ɩɪɨɢɡɜɨɞɫɬɜɟ ɪɚɛɨɬ ɜ ɭɫɥɨɜɢɹɯ ɨɬɤɪɵɬɨɣ ɦɨɪɫɤɨɣ ɚɤɜɚɬɨɪɢɢ, ɚ ɢɦɟɧɧɨ:
— ɬɪɟɳɢɧɵ, ɡɚɥɨɦɵ ɷɥɟɦɟɧɬɨɜ ɫɬɟɧ, ɪɚɫɯɨɠɞɟɧɢɟ ɡɚɦɤɨɜ;
— ɨɬɤɥɨɧɟɧɢɟ ɫɬɟɧɤɢ ɨɬ ɩɪɨɟɤɬɧɨɝɨ ɩɨɥɨɠɟɧɢɹ;
— ɨɛɪɭɲɟɧɢɟ ɫɜɚɣɧɵɯ ɪɹɞɨɜ.
ɂɡ ɦɧɨɠɟɫɬɜɚ ɩɪɨɰɟɫɫɨɜ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɫ ɬɨɧɤɨɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɫɬɟɧɤɨɣ, ɡɚɝɥɭɛɥɟɧɧɨɣ ɜ
ɝɪɭɧɬ ɧɚ ɦɨɪɫɤɨɣ ɚɤɜɚɬɨɪɢɢ, ɩɪɟɞɫɬɚɜɥɹɸɬ ɢɧɬɟɪɟɫ [1]:
— ɤɨɥɟɛɚɧɢɹ ɫɬɟɪɠɧɹ ɩɨɞ ɜɨɡɞɟɣɫɬɜɢɟɦ ɜɨɥɧɨɜɨɣ ɧɚɝɪɭɡɤɢ;
79
— ɜɨɥɧɨɜɵɟ ɬɟɱɟɧɢɹ ɜɛɥɢɡɢ ɫɬɟɧɤɢ;
— ɪɚɡɠɢɠɟɧɢɟ ɢ ɜɵɦɵɜɚɧɢɟ ɱɚɫɬɢɰ ɝɪɭɧɬɚ.
Ʉɚɠɞɨɟ ɢɡ ɷɬɢɯ ɹɜɥɟɧɢɣ ɨɩɢɫɵɜɚɟɬɫɹ ɫɥɨɠɧɵɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɦɨɞɟɥɹɦɢ ɜ ɜɢɞɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. ȼɵɫɨɤɚɹ ɫɥɨɠɧɨɫɬɶ ɨɛɳɟɣ ɦɨɞɟɥɢ ɨɫɥɨɠɧɹɟɬ ɩɨɫɬɪɨɟɧɢɟ ɦɟɬɨɞɢɤ ɪɚɫɱɟɬɚ. ȼ ɩɪɨɟɤɬɧɨɣ ɩɪɚɤɬɢɤɟ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬ ɫɩɟɰɢɚɥɢɡɢɪɨɜɚɧɧɵɟ
ɜɵɱɢɫɥɢɬɟɥɶɧɵɟ ɫɪɟɞɫɬɜɚ ɞɥɹ ɩɪɨɱɧɨɫɬɧɵɯ ɪɚɫɱɟɬɨɜ ɧɚ ɨɫɧɨɜɟ ɫɟɬɨɱɧɵɯ ɢ ɤɨɧɟɱɧɨ-ɷɥɟɦɟɧɬɧɵɯ
ɦɟɬɨɞɨɜ. Ɉɞɧɚɤɨ ɷɬɢ ɫɪɟɞɫɬɜɚ ɨɬɥɢɱɚɸɬɫɹ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɨɢɦɨɫɬɶɸ ɢ ɹɜɥɹɸɬɫɹ ɢɡɛɵɬɨɱɧɵɦɢ
ɞɥɹ ɨɰɟɧɨɱɧɵɯ ɪɚɫɱɟɬɨɜ, ɨɬɥɢɱɚɸɳɢɯɫɹ ɭɦɟɪɟɧɧɵɦɢ ɬɪɟɛɨɜɚɧɢɹɦɢ ɤ ɬɨɱɧɨɫɬɢ ɪɚɫɱɟɬɚ. ɋ ɭɱɟɬɨɦ
ɩɨɝɪɟɲɧɨɫɬɟɣ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɩɨɝɪɟɲɧɨɫɬɶ ɨɰɟɧɨɱɧɨɝɨ ɪɚɫɱɟɬɚ ɜ 20 %, ɤɚɤ ɩɪɚɜɢɥɨ, ɜɩɨɥɧɟ
ɩɪɢɟɦɥɟɦɚ. ɉɨɷɬɨɦɭ ɚɤɬɭɚɥɶɧɵ ɪɚɡɥɢɱɧɵɟ ɭɩɪɨɳɟɧɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɩɪɨɰɟɞɭɪ, ɩɨɡɜɨɥɹɸɳɢɟ
ɩɪɢɦɟɧɹɬɶ ɩɪɢ ɫɨɡɞɚɧɢɢ ɬɪɚɧɫɩɨɪɬɧɨɣ ɢɧɮɪɚɫɬɪɭɤɬɭɪɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɟ ɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɞɨɪɨɝɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɩɚɤɟɬɵ, ɜ ɱɚɫɬɧɨɫɬɢ ɫɢɫɬɟɦɭ ɚɜɬɨɦɚɬɢɡɚɰɢɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ
Mathcad [2], ɤ ɩɪɟɢɦɭɳɟɫɬɜɚɦ ɤɨɬɨɪɨɣ ɨɬɧɨɫɹɬ ɟɫɬɟɫɬɜɟɧɧɵɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɹɡɵɤ ɢɧɮɨɪɦɚɰɢɨɧɧɨɝɨ ɨɛɦɟɧɚ ɢ ɞɪɭɠɟɫɬɜɟɧɧɵɣ ɢɧɬɟɪɮɟɣɫ. ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɨɬɦɟɬɢɬɶ ɧɚɥɢɱɢɟ ɫɜɨɛɨɞɧɨ ɪɚɫɩɪɨɫɬɪɚɧɹɟɦɨɝɨ ɚɧɚɥɨɝɚ ɷɬɨɣ ɩɪɨɝɪɚɦɦɵ.
Ɇɨɞɟɥɶ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜɟɪɬɢɤɚɥɶɧɨɣ ɭɩɪɭɝɨɣ ɛɚɥɤɢ ɫ ɡɚɳɟɦɥɟɧɧɵɦ ɧɢɠɧɢɦ ɤɨɧɰɨɦ (ɪɢɫ. 1) ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɣ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ — ɨɞɧɚ ɢɡ ɨɫɧɨɜɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɨɛɳɟɣ ɦɨɞɟɥɢ, ɨɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ [3–5] ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ (Ⱦɍɑɉ) ɞɥɹ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɨɣ ɛɚɥɤɢ (ɫɬɟɪɠɧɹ):
,
q — ɪɚɫɩɪɟɞɟɥɟɧɧɚɹ ɧɚɝɪɭɡɤɚ;
ȿ — ɦɨɞɭɥɶ ɘɧɝɚ;
J — ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɟɱɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɥɚɜɧɨɣ ɨɫɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɤɨɥɟɛɚɧɢɣ;
m — ɦɚɫɫɚ ɟɞɢɧɢɰɵ ɞɥɢɧɵ ɫɬɟɪɠɧɹ (ɩɨɝɨɧɧɚɹ ɩɥɨɬɧɨɫɬɶ);
c — ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɫɢɥɵ ɜɧɭɬɪɟɧɧɟɝɨ ɬɪɟɧɢɹ;
0 ” z ” l — ɤɨɨɪɞɢɧɚɬɚ ɜɞɨɥɶ ɞɥɢɧɵ ɫɬɟɪɠɧɹ.
Выпуск 3
ɝɞɟ
(1)
80
Ɋɢɫ. 1. ɋɯɟɦɚ ɞɟɮɨɪɦɚɰɢɣ ɲɩɭɧɬɨɜɨɣ ɫɬɟɧɤɢ ɩɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɜɨɥɧ
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɞɥɹ ɫɬɟɪɠɧɹ ɫ ɡɚɤɪɟɩɥɟɧɧɵɦ ɥɟɜɵɦ ɢ
ɫɜɨɛɨɞɧɵɦ ɩɪɚɜɵɦ ɤɨɧɰɚɦɢ ɜɵɪɚɠɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ.
ɍɫɥɨɜɢɹ ɡɚɤɪɟɩɥɟɧɢɹ ɫɬɟɪɠɧɹ ɜ ɬɨɱɤɟ z = 0:
.
ɍɫɥɨɜɢɹ ɧɚ ɫɜɨɛɨɞɧɨɦ ɤɨɧɰɟ (ɩɪɢ z = L):
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɩɪɢ t = 0:
ɝɞɟ ug(z), vg(z) — ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ.
ɋ ɰɟɥɶɸ ɜɡɚɢɦɧɨɣ ɩɪɨɜɟɪɤɢ ɩɪɚɤɬɢɱɟɫɤɢ ɡɧɚɱɢɦɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɪɟɲɟɧɢɹ Ⱦɍɑɉ (1) ɫ ɭɤɚɡɚɧɧɵɦɢ ɜɵɲɟ ɤɪɚɟɜɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɨɥɭɱɢɦ ɱɢɫɥɟɧɧɵɟ ɪɟɲɟɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɚɥɶɬɟɪɧɚɬɢɜɧɵɦɢ ɦɟɬɨɞɚɦɢ, ɚ ɢɦɟɧɧɨ ɦɟɬɨɞɚɦɢ Ɏɭɪɶɟ ɜ ɢɧɬɟɪɩɪɟɬɚɰɢɢ Ⱥ. ɇ. Ʉɪɵɥɨɜɚ ɢ Ȼ. Ƚ. Ƚɚɥɟɪɤɢɧɚ ɜ ɜɚɪɢɚɧɬɟ ȼ. Ɂ. ȼɥɚɫɨɜɚ — Ʌ. ȼ. Ʉɚɧɬɨɪɨɜɢɱɚ [3–7].
ɉɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
ɉɨɫɤɨɥɶɤɭ ɦɟɬɨɞ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ ɭɞɨɛɧɟɟ ɨɫɭɳɟɫɬɜɥɹɬɶ ɧɚ ɨɬɪɟɡɤɟ ɟɞɢɧɢɱɧɨɣ ɞɥɢɧɵ,
ɩɪɟɨɛɪɚɡɭɟɦ ɭɪɚɜɧɟɧɢɟ ɤ ɧɨɜɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ x = z / L, ɝɞɟ 0 ” x ”1. ȼ ɜɵɪɚɠɟɧɢɹɯ
ɞɥɹ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɧɚɝɪɭɡɤɢ ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɫɬɚɜɢɬɶ ɡɧɚɱɟɧɢɟ z = L ǜ x.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɡɚɦɟɧɵ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
(2)
,
ɝɞɟ Į = ɫ/ (2  m), ȕ2 = E  J  (m  L4) –1 ɢ F = q / m — ɩɚɪɚɦɟɬɪɵ ɢ ɜɨɡɦɭɳɟɧɢɟ.
Ⱦɚɥɟɟ ɛɭɞɟɦ ɪɟɲɚɬɶ ɭɪɚɜɧɟɧɢɟ (2).
Ɋɟɲɟɧɢɟ ɦɟɬɨɞɨɦ Ɏɭɪɶɟ
Ɋɟɲɟɧɢɟ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɨɫɭɳɟɫɬɜɢɦ ɜ ɞɜɚ ɷɬɚɩɚ. ɇɚ ɩɟɪɜɨɦ ɷɬɚɩɟ ɪɟɲɢɦ ɨɞɧɨɪɨɞɧɨɟ
ɭɪɚɜɧɟɧɢɟ
Выпуск 3
(3)
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (5) ɜ ɜɢɞɟ ɪɹɞɚ u (t, x) = ™Tk (t )  X k (x ), ɝɞɟ k = 0, 1, ... ’ — ɧɚk
ɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ. Ɍɨɝɞɚ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɞɥɹ ɤɚɠɞɨɝɨ ɧɨɦɟɪɚ k ɩɨɥɭɱɢɦ ɫɢɫɬɟɦɭ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ:
(4)
,
81
ɝɞɟ ı2 ɢ
— ɩɚɪɚɦɟɬɪɵ.
ɉɟɪɜɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ ɫɢɫɬɟɦɵ (4) ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɡɚɬɭɯɚɸɳɢɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ:
,
(5)
ɝɞɟ Ⱥ, ș — ɧɚɱɚɥɶɧɚɹ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ; Ȧ2 = ı2 – ɚ2 — ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ (4) ɜɵɪɚɡɢɦ «ɜɡɜɟɲɟɧɧɨɣ» ɫɭɦɦɨɣ «ɛɚɥɨɱɧɵɯ» ɮɭɧɤɰɢɣ
Ʉɪɵɥɨɜɚ:
,
ɝɞɟ
ɫ0, ɫ1, ɫ2, ɫ3 — ɩɨɫɬɨɹɧɧɵɟ «ɜɟɫɨɜɵɟ» ɤɨɷɮɮɢɰɢɟɧɬɵ;
S(x, K), T(x, K), U(x, K), V(x, K) — «ɛɚɥɨɱɧɵɟ» ɮɭɧɤɰɢɢ Ⱥ. ɇ. Ʉɪɵɥɨɜɚ [3].
ɋ ɭɱɟɬɨɦ ɫɜɨɣɫɬɜ «ɛɚɥɨɱɧɵɯ» ɮɭɧɤɰɢɣ ɨɩɪɟɞɟɥɢɦ ɡɧɚɱɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ, ɢɫɯɨɞɹ ɢɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɪɢ ɯ = 0 ɢ ɯ = 1. ɉɪɢ ɷɬɨɦ c0 = c1 = c2 = 0, ɬɨɝɞɚ
(6)
ɂɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ ɧɟɬɪɢɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɩɨɷɬɨɦɭ c3  0 ɢ ɭɪɚɜɧɟɧɢɟ (12) ɩɪɢ
L = 1 ɷɤɜɢɜɚɥɟɧɬɧɨ «ɱɚɫɬɨɬɧɨɦɭ» [4] ɭɪɚɜɧɟɧɢɸ: cos( K ) Â ch( K ) = –1. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ «ɫɨɛɫɬɜɟɧɧɵɯ ɱɚɫɬɨɬ» Ʉ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɚɩɩɪɨɤɫɢɦɚɰɢɟɣ [6, ɫ. 10–14]:
Ɋɟɲɟɧɢɹ ɱɚɫɬɨɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɨɧɭɦɟɪɨɜɚɧɵ ɧɨɦɟɪɚɦɢ n = 0, 1, 2 ..., ’.
ɉɪɢ ɢɡɜɟɫɬɧɵɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɚ K ɢ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ
ɦɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
ɪɟɲɟɧɢɟ ɜɪɟ-
(7)
ɝɞɟ Tk, n, An, șn — ɩɨɫɬɨɹɧɧɵɟ.
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɞɥɹ ɏn(x) ɩɪɢ K = Kn ɢ c3 = 1 ɟɫɬɶ ɫɨɛɫɬɜɟɧɧɚɹ ɮɨɪɦɚ:
.
(8)
Ɉɛɴɟɞɢɧɹɹ ɭɪɚɜɧɟɧɢɹ (7) ɢ (8) ɢ ɫɭɦɦɢɪɭɹ ɩɨ ɢɧɞɟɤɫɭ n, ɩɨɥɭɱɢɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ Ⱦɍɑɉ (3)
ɜ ɜɢɞɟ
(9)
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜɵɞɟɥɢɦ ɫ ɩɨɦɨɳɶɸ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ:
Выпуск 3
(10)
82
(11)
Ɋɚɡɥɚɝɚɹ ɜɵɪɚɠɟɧɢɹ (10), (11) ɜ ɪɹɞɵ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɨɪɦɚɦ ɩɨɫɥɟ ɜɵɱɢɫɥɟɧɢɣ ɨɩɪɟɞɟɥɢɦ
ɪɚɧɟɟ ɧɟɢɡɜɟɫɬɧɵɟ ɚɦɩɥɢɬɭɞɭ ɤɨɥɟɛɚɧɢɣ An, ɮɚɡɨɜɵɣ ɫɞɜɢɝ șn:
Ɋɚɡɥɨɠɢɦ ɩɪɚɜɭɸ ɱɚɫɬɶ ɧɟɨɞɧɨɪɨɞɧɨɝɨ Ⱦɍɑɉ (2) ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɨɪɦɚɦ:
,
ɝɞɟ qn ( t) — ɤɨɷɮɮɢɰɢɟɧɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɫ ɭɱɟɬɨɦ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɨɪɦ.
ɉɭɫɬɶ, ɤɪɨɦɟ ɬɨɝɨ, Un = Tn (t) Â ij(x, n) — ɪɟɲɟɧɢɟ Ⱦɍɑɉ (2), ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɧɨɦɟɪɭ n, ɬɨɝɞɚ,
ɩɟɪɟɯɨɞɹ ɤ ɩɨɱɥɟɧɧɵɦ ɪɚɜɟɧɫɬɜɚɦ, ɩɨɥɭɱɢɦ
Ɂɚɦɟɬɢɦ, ɱɬɨ
(ɷɬɨ ɫɥɟɞɭɟɬ ɢɡ ɜɵɪɚɠɟɧɢɹ (4)). ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɨɤɪɚɳɟ-
ɧɢɹ ɨɛɳɟɝɨ ɦɧɨɠɢɬɟɥɹ ij(x, n)  0 ɩɨɥɭɱɢɦ ɧɟɨɞɧɨɪɨɞɧɨɟ ɥɢɧɟɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ
(ɇɅȾɍ):
,
(12)
ɝɞɟ ȕ2 Â K4 = ı2 .
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɛɳɟɣ ɬɟɨɪɢɟɣ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
Tn = Tsn + Tvn ,
ɝɞɟ
Tsn — ɨɛɳɟɟ ɪɟɲɟɧɢɟ (7) ɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ;
Tvn — ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɇɅȾɍ (12).
ɉɪɢ ɝɚɪɦɨɧɢɱɟɫɤɨɦ ɜɨɡɛɭɠɞɟɧɢɢ
,
(13)
ɝɞɟ Umn, v, ș — ɧɟɢɡɜɟɫɬɧɚɹ ɚɦɩɥɢɬɭɞɚ, ɱɚɫɬɨɬɚ ɢ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ.
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (12) ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɢɞɚ [5]:
(14)
ɝɞɟ ȝ n, ș, ǻșn — ɚɦɩɥɢɬɭɞɚ, ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ ɢ ɫɞɜɢɝ ɮɚɡɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
ɉɨɞɫɬɚɜɢɦ (13) ɢ (14) ɜ (12), ɩɨɫɥɟ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɜɵɱɢɫɥɟɧɢɣ ɩɨɥɭɱɢɦ
Выпуск 3
(15)
83
ɂɬɚɤ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɇɅȾɍ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
(16)
Ɂɞɟɫɶ ɩɚɪɚɦɟɬɪɵ ȝ n ɢ ǻș ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ (15), ɚ ɩɨɫɬɨɹɧɧɵɟ An ɢ șn ɩɨɞɥɟɠɚɬ ɨɩɪɟɞɟɥɟɧɢɸ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨ ɮɨɪɦɭɥɚɦ:
(17)
Ⱥɥɝɨɪɢɬɦ ɪɚɫɱɟɬɚ ɦɟɬɨɞɨɦ Ɏɭɪɶɟ
ɉɨɥɭɱɟɧɧɵɟ ɜɵɲɟ ɜɵɪɚɠɟɧɢɹ ɩɨɥɨɠɟɧɵ ɜ ɨɫɧɨɜɭ ɚɥɝɨɪɢɬɦɚ, ɪɟɚɥɢɡɨɜɚɧɧɨɝɨ ɫɪɟɞɫɬɜɚɦɢ
ɫɢɫɬɟɦɵ ɚɜɬɨɦɚɬɢɡɚɰɢɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ Mathcad. Ⱥɥɝɨɪɢɬɦ ɜɤɥɸɱɚɟɬ ɫɥɟɞɭɸɳɢɟ ɨɩɟɪɚɰɢɢ:
1) ɜɜɨɞ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ, ɤ ɤɨɬɨɪɵɦ ɨɬɧɟɫɟɧɵ ɤɨɷɮɮɢɰɢɟɧɬɵ Ⱦɍɑɉ (2), (3), ɚ ɬɚɤɠɟ ɩɚɪɚɦɟɬɪɵ ɜɨɡɦɭɳɟɧɢɹ ɢ ɜɪɟɦɟɧɧɨ́ɣ ɢɧɬɟɪɜɚɥ, ɧɚ ɤɨɬɨɪɨɦ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɪɟɲɟɧɢɟ Ⱦɍɑɉ;
2) ɜɵɛɨɪ ɤɨɥɢɱɟɫɬɜɚ ɱɥɟɧɨɜ ɜ ɪɚɡɥɨɠɟɧɢɢ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɨɪɦɚɦ N;
3) ɪɚɫɱɟɬ ɩɨ ɮɨɪɦɭɥɚɦ (17) ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɹ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢ ɩɪɚɜɨɣ ɱɚɫɬɢ
Ⱦɍɑɉ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɨɪɦɚɦ ɫ ɧɨɦɟɪɚɦɢ ɜ ɞɢɚɩɚɡɨɧɟ 0 ” n ” N;
4) ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɢɫɤɨɦɨɝɨ ɪɟɲɟɧɢɹ ɩɭɬɟɦ ɩɨɞɫɬɚɧɨɜɤɢ ɜɵɱɢɫɥɟɧɧɵɯ ɜ ɨɩɟɪɚɰɢɢ (2) ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɜɵɪɚɠɟɧɢɟ (16).
Ɇɟɬɨɞ Ƚɚɥɟɪɤɢɧɚ–ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ
ɂɞɟɹ ɦɟɬɨɞɚ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ [7; 8] ɬɚɤɠɟ ɛɚɡɢɪɭɟɬɫɹ ɧɚ ɪɚɡɞɟɥɟɧɢɢ ɩɟɪɟɦɟɧɧɵɯ. Ɂɞɟɫɶ
ɪɨɥɶ ɮɭɧɤɰɢɣ ɮɨɪɦɵ ɢɝɪɚɸɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɩɪɨɛɧɵɟ ɮɭɧɤɰɢɢ, ɚ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɚɩɩɪɨɤɫɢɦɢɪɭɸɬ ɨɬɪɟɡɤɨɦ ɪɹɞɚ
,
(18)
ɝɞɟ
a(t) — ɤɨɷɮɮɢɰɢɟɧɬɵ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɪɟɦɟɧɢ;
ij(ɯ)n — ɫɢɫɬɟɦɚ ɢɡ (N + 1) ɩɪɨɛɧɨɣ ɮɭɧɤɰɢɢ (ɫɱɢɬɚɹ ɧɨɦɟɪ n = 0), ɩɪɢɱɟɦ ij(ɯ)0 ɭɞɨɜɥɟɬɜɨɪɹɟɬ
ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ;
ij(ɯ)m — ɝɞɟ m = 1, 2, …, N ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ij(0)m = ij(1)m = 0.
Выпуск 3
ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ ɜɵɪɚɠɟɧɢɹ (2), ɧɚɯɨɞɢɦ ɧɟɜɹɡɤɭ:
84
Ɇɢɧɢɦɚɥɶɧɨɟ ɚɛɫɨɥɸɬɧɨɟ ɡɧɚɱɟɧɢɟ ɧɟɜɹɡɤɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɧɚɢɥɭɱɲɟɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ
ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (2) ɮɭɧɤɰɢɟɣ (18), ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɧɟɜɹɡɤɢ R ɜɫɟɦ ɩɪɨɛɧɵɦ ɮɭɧɤɰɢɹɦ. ɍɫɥɨɜɢɟ ɬɚɤɨɣ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɜ ɜɢɞɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ:
ɩɪɢ ɥɸɛɨɦ 0 ” m ” n.
Ⱦɥɹ ɩɪɨɛɧɨɣ ɮɭɧɤɰɢɢ ijm(x), ɝɞɟ m ɩɪɨɢɡɜɨɥɶɧɵɣ ɧɨɦɟɪ, ɫɥɟɞɭɟɬ
(19)
ɝɞɟ Jm,n, JFm,n, JBm — ɤɨɷɮɮɢɰɢɟɧɬɵ.
Ⱦɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ m, ɩɨɥɭɱɚɟɦ ɦɚɬɪɢɱɧɨɟ ɭɪɚɜɧɟɧɢɟ:
(20)
ɋ ɭɱɟɬɨɦ ɨɫɨɛɟɧɧɨɫɬɟɣ ɫɢɫɬɟɦ ɤɨɦɩɶɸɬɟɪɧɨɣ ɦɚɬɟɦɚɬɢɤɢ ɩɪɢɜɟɞɟɦ ɭɪɚɜɧɟɧɢɟ (20) ɤ ɫɢɫɬɟɦɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ 1-ɝɨ ɩɨɪɹɞɤɚ.
(21)
ĺ
ĺ
ɇɚɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɤɬɨɪɨɜ a ɢ V ɨɩɪɟɞɟɥɢɦ ɪɚɡɥɨɠɟɧɢɟɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨ ɩɪɨɛɧɵɦ ɮɭɧɤɰɢɹɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɪɚɠɟɧɢɹɦɢ:
(22)
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɩɪɨɝɪɚɦɦɚ Mathcad ɩɨɡɜɨɥɹɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɫɬɨ ɜɵɱɢɫɥɹɬɶ ɡɧɚɱɟɧɢɹ ɢɧɬɟɝɪɚɥɨɜ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɞɢɧɚɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ (21), ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɫɨɞɟɪɠɢɬ ɪɚɡɜɢɬɵɟ ɫɪɟɞɫɬɜɚ
ɪɟɲɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɩɪɚɤɬɢɱɟɫɤɚɹ ɪɟɚɥɢɡɚɰɢɹ ɦɟɬɨɞɚ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ
ɨɤɚɡɵɜɚɟɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ ɩɪɨɳɟ. Ⱦɥɹ ɨɛɨɫɧɨɜɚɧɢɹ ɩɪɢɦɟɧɢɦɨɫɬɢ ɬɟɯ ɢɥɢ ɢɧɵɯ ɩɪɨɛɧɵɯ ɮɭɧɤɰɢɣ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɢɬɶ ɨɰɟɧɨɱɧɵɟ ɪɚɫɱɟɬɵ ɢ ɫɪɚɜɧɢɬɶ ɪɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɪɚɡɥɢɱɧɵɦɢ
ɦɟɬɨɞɚɦɢ.
(23)
Выпуск 3
Ⱥɥɝɨɪɢɬɦ ɪɚɫɱɟɬɚ ɦɟɬɨɞɨɦ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ
Ⱥɥɝɨɪɢɬɦ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ Ⱦɍɑɉ ɦɟɬɨɞɨɦ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ ɪɟɚɥɢɡɨɜɚɧ ɫɪɟɞɫɬɜɚɦɢ ɫɢɫɬɟɦɵ ɚɜɬɨɦɚɬɢɡɚɰɢɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ Mathcad. Ⱥɥɝɨɪɢɬɦ ɜɤɥɸɱɚɟɬ ɫɥɟɞɭɸɳɢɟ ɨɩɟɪɚɰɢɢ:
1) ɜɜɨɞ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ, ɤ ɤɨɬɨɪɵɦ ɨɬɧɟɫɟɧɵ ɤɨɷɮɮɢɰɢɟɧɬɵ Ⱦɍɑɉ (4), ɚ ɬɚɤɠɟ ɩɚɪɚɦɟɬɪɵ
ɜɨɡɦɭɳɟɧɢɹ ɢ ɜɪɟɦɟɧɧɨ́ɣ ɢɧɬɟɪɜɚɥ, ɧɚ ɤɨɬɨɪɨɦ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɪɟɲɟɧɢɟ Ⱦɍɑɉ;
2) ɪɚɫɱɟɬ ɦɚɬɪɢɰ J, JF ɢ ɜɟɤɬɨɪɚ JB ɩɨ ɮɨɪɦɭɥɚɦ (19) ɧɚ ɨɫɧɨɜɟ ɩɪɨɛɧɵɯ ɮɭɧɤɰɢɣ:
85
ɝɞɟ m = 1, 2, … N — ɧɨɦɟɪ ɩɪɨɛɧɨɣ ɮɭɧɤɰɢɢ;
3) ɪɚɫɱɟɬ ɜɟɤɬɨɪɨɜ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨ ɮɨɪɦɭɥɚɦ (22):
4) ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ Ʉɨɲɢ ɞɥɹ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (21) ɢ ɜɵɱɢɫɥɟɧɧɵɯ
ɜɵɲɟ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɫ ɩɨɦɨɳɶɸ ɜɫɬɪɨɟɧɧɵɯ ɮɭɧɤɰɢɣ ɩɪɨɝɪɚɦɦɵ Mathcad.
ȼɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ
ɋ ɰɟɥɶɸ ɩɪɨɜɟɪɤɢ ɪɟɚɥɢɡɚɰɢɢ ɨɩɢɫɚɧɧɵɯ ɜɵɲɟ ɚɥɝɨɪɢɬɦɨɜ ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢ
ɡɧɚɱɢɦɵɯ ɩɪɢɤɥɚɞɧɵɯ ɡɚɞɚɱ ɛɵɥ ɨɫɭɳɟɫɬɜɥɟɧ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ, ɡɚɤɥɸɱɚɸɳɢɣɫɹ ɜ
ɪɚɫɱɟɬɟ ɩɚɪɚɦɟɬɪɨɜ ɤɨɥɟɛɚɧɢɣ ɲɩɭɧɬɚ ɬɢɩɚ Ʌɚɪɫɟɧ ɩɨɞ ɜɨɡɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɨɜɨɣ
ɧɚɝɪɭɡɤɢ [1]:
ɝɞɟ
Ts(t), UM(x) — ɮɭɧɤɰɢɢ ɜɪɟɦɟɧɢ ɢ ɤɨɨɪɞɢɧɚɬɵ;
vz, șz — ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɜɨɡɦɭɳɟɧɢɣ;
ǻP — ɭɞɟɥɶɧɨɟ ɞɚɜɥɟɧɢɟ ɧɚ ɟɞɢɧɢɰɭ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ (ɜɟɥɢɱɢɧɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɡɧɨɫɬɶ ɞɚɜɥɟɧɢɣ ɧɚ ɜɟɪɬɢɤɚɥɶɧɭɸ ɫɬɟɧɭ, ɜ ɫɥɭɱɚɟ ɟɫɥɢ ɭɪɨɜɧɢ ɜɨɞɵ ɧɟɨɞɢɧɚɤɨɜɵ ɩɨ ɨɛɟ ɫɬɨɪɨɧɵ
ɨɬ ɧɟɟ);
xa, xb — ɤɨɧɫɬɚɧɬɵ, ɭɱɢɬɵɜɚɸɳɢɟ ɪɚɡɧɨɫɬɶ ɭɪɨɜɧɟɣ ɜɨɞɵ ɩɨ ɨɛɟ ɫɬɨɪɨɧɵ ɨɬ ɫɬɟɧɤɢ;
if () — ɭɫɥɨɜɧɚɹ ɮɭɧɤɰɢɹ [9].
Выпуск 3
ɇɚ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɦ ɷɬɚɩɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɛɵɥɨ ɜɵɩɨɥɧɟɧɨ ɫɪɚɜɧɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɫɱɟɬɨɜ
ɞɜɭɦɹ ɦɟɬɨɞɚɦɢ, ɩɨɤɚɡɚɜɲɟɟ ɯɨɪɨɲɟɟ ɫɨɜɩɚɞɟɧɢɟ ɪɟɡɭɥɶɬɚɬɨɜ ɩɪɢ ɧɟɫɤɨɥɶɤɢɯ ɬɟɫɬɨɜɵɯ ɧɚɛɨɪɚɯ
ɩɚɪɚɦɟɬɪɨɜ ɜɨɡɦɭɳɟɧɢɹ ɢ ɤɨɥɟɛɥɸɳɟɣɫɹ ɛɚɥɤɢ. ɉɪɢ ɷɬɨɦ ɜ ɪɚɡɥɨɠɟɧɢɹɯ ɭɞɟɪɠɢɜɚɥɨɫɶ ɩɨ ɩɹɬɶ
ɱɥɟɧɨɜ (N = 4). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɢɫɬɟɦɚ ɩɪɨɛɧɵɯ ɮɭɧɤɰɢɣ (23) ɩɨɞɯɨɞɢɬ ɞɥɹ ɪɟɲɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ.
Ⱦɚɥɟɟ, ɦɟɬɨɞɨɦ ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ ɛɵɥɢ ɩɨɫɬɪɨɟɧɵ ɬɚɛɥɢɱɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɨɝɢɛɚ ɨɬ
ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ ɛɚɥɤɢ (ɤɨɨɪɞɢɧɚɬɚ ɯ) ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ vz. ɇɚ ɡɚɜɟɪɲɚɸɳɟɦ ɷɬɚɩɟ ɡɚɜɢɫɢɦɨɫɬɢ ɩɪɨɝɢɛɚ ɛɚɥɤɢ ɨɬ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ ɛɵɥɢ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɧɵ ɩɨɥɢɧɨɦɚɦɢ ɬɪɟɬɶɟɣ ɫɬɟɩɟɧɢ (ɩɟɪɜɨɟ ɜɵɪɚɠɟɧɢɟ ɜ ɫɢɫɬɟɦɟ (24)). ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ
ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɫɤɨɦɨɝɨ ɩɨɥɢɧɨɦɚ ɬɚɤɠɟ ɛɵɥɢ ɩɨɫɬɪɨɟɧɵ ɩɨɥɢɧɨɦɵ (ɜɬɨɪɨɟ ɜɵɪɚɠɟɧɢɟ ɜ
ɫɢɫɬɟɦɟ (24)), ɨɩɢɫɵɜɚɸɳɢɟ ɢɡɦɟɧɟɧɢɹ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ
ɜɨɡɦɭɳɟɧɢɹ, ɜɵɪɚɠɟɧɧɨɣ ɜ ɞɨɥɹɯ ɱɢɫɥɚ Ⱥɪɯɢɦɟɞɚ ʌ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɡɚɜɢɫɢɦɨɫɬɶ ɚɦɩɥɢɬɭɞɵ ɩɪɨɝɢɛɚ ɛɚɥɤɢ ɨɬ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ ɢ ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ ɛɵɥɚ ɩɪɟɞɫɬɚɜɥɟɧɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɶɸ:
(24)
86
ɝɞɟ ȼ — ɦɚɬɪɢɰɚ ɤɨɧɫɬɚɧɬ.
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ Ⱦɍɑɉ ɫ ɩɚɪɚɦɟɬɪɚɦɢ, ɩɪɟɞɫɬɚɜɥɟɧɧɵɦɢ ɜ ɬɚɛɥ. 1, ɜ ɪɟɡɭɥɶɬɚɬɟ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɛɵɥɢ ɩɨɥɭɱɟɧɵ ɤɨɦɩɨɧɟɧɬɵ ɦɚɬɪɢɰɵ ȼ, ɩɪɢɜɟɞɟɧɧɵɟ ɜ ɬɚɛɥ. 2.
Ɍɚɛɥɢɰɚ 1
ɉɚɪɚɦɟɬɪɵ Ⱦɍɑɉ
ɉɚɪɚɦɟɬɪ
ɟɞ. ɢɡɦɟɪɟɧɢɹ
Ɂɧɚɱɟɧɢɟ
ǻP
ɤȽ
49,4
ǻz
ɪɚɞ
0
xa
—
0,8
xb
—
0,8
Į
—
0,0199
ȕ
—
7,112
Ɍɚɛɥɢɰɚ 2
Ɇɚɬɪɢɰɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ B
n
m
0
1
2
3
0
–5.306304E-4
0.0112193
0.1007814
–0.0426477
1
9.6927823E-6
–2.222159E-4
3.4844199E-3
–1.497188E-3
2
–1.9364113E-5
3.9119059E-4
6.364925E-3
–2.707281E-3
3
2.4506438E-6
–5.1477114E-5
–8.3594543E-5
3.4123641E-5
Ƚɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɚɦɩɥɢɬɭɞɵ ɩɪɨɝɢɛɚ ɛɚɥɤɢ ɨɬ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ ɩɪɢ ɪɚɡɥɢɱɧɵɯ
ɡɧɚɱɟɧɢɹɯ ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 2.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɚɧɧɵɣ ɚɥɝɨɪɢɬɦ ɩɨɡɜɨɥɢɥ ɨɰɟɧɢɬɶ ɩɪɨɝɢɛɵ ɫɜɚɣɧɨɝɨ ɪɹɞɚ ɞɥɹ ɪɚɡɥɢɱɧɵɯ
ɱɚɫɬɨɬ ɜɨɡɞɟɣɫɬɜɢɹ ɜɨɥɧɵ. Ƚɪɚɮɢɤɢ ɧɚ ɪɢɫ. 2 ɨɩɢɫɵɜɚɸɬ ɡɧɚɱɟɧɢɹ ɩɪɨɝɢɛɨɜ ɜ ɞɢɚɩɚɡɨɧɟ ɜɨɥɧ
ɫ ɩɟɪɢɨɞɨɦ ɨɬ 0,70 ɞɨ 20 ɫ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɩɪɚɤɬɢɱɟɫɤɢɦ ɭɫɥɨɜɢɹɦ ɷɤɫɩɥɭɚɬɚɰɢɢ ɩɨɪɬɨɜɵɯ ȽɌɋ.
ɚ
ɛ
Выпуск 3
Ɋɢɫ. 2. ɚ — ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ
ɚɦɩɥɢɬɭɞɵ ɩɪɨɝɢɛɚ max | U | ɛɚɥɤɢ
ɨɬ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɥɢɧɵ ɩɪɢ ɪɚɡɥɢɱɧɵɯ
ɡɧɚɱɟɧɢɹɯ ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ;
ɛ — ɝɪɚɮɢɤ ɡɚɜɢɫɢɦɨɫɬɢ ɚɦɩɥɢɬɭɞɵ
ɩɪɨɝɢɛɚ max | U | ɛɚɥɤɢ ɨɬ ɡɧɚɱɟɧɢɹ
ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ
ȼɵɜɨɞɵ
ɂɫɫɥɟɞɨɜɚɧɢɟ ɭɩɪɭɝɢɯ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɛɚɥɤɢ ɫ ɡɚɳɟɦɥɟɧɧɵɦ ɤɨɧɰɨɦ ɦɟɬɨɞɚɦɢ Ɏɭɪɶɟ ɜ ɢɧɬɟɪɩɪɟɬɚɰɢɢ Ⱥ. ɇ. Ʉɪɵɥɨɜɚ, ɢ Ȼ. Ƚ. Ƚɚɥɟɪɤɢɧɚ ɜ ɜɚɪɢɚɧɬɟ ȼ. Ɂ. ȼɥɚɫɨɜɚ — Ʌ. ȼ. Ʉɚɧɬɨɪɨɜɢɱɚ
ɩɨɤɚɡɚɥɨ, ɱɬɨ ɨɬɥɢɱɚɸɳɢɣɫɹ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɪɟɚɥɢɡɚɰɢɟɣ ɫɪɟɞɫɬɜɚɦɢ ɩɪɨɝɪɚɦɦɵ Mathcad ɦɟɬɨɞ
ȼɥɚɫɨɜɚ–Ʉɚɧɬɨɪɨɜɢɱɚ ɫ ɩɪɨɛɧɵɦɢ ɮɭɧɤɰɢɹɦɢ
(23) ɜɩɨɥɧɟ ɩɪɢɟɦɥɟɦ ɞɥɹ ɪɟɲɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɤɥɚɫɫɚ ɡɚɞɚɱ.
ɉɨɫɬɪɨɟɧɚ ɩɨɥɢɧɨɦɢɚɥɶɧɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ
ɦɨɞɟɥɶ, ɩɨɡɜɨɥɹɸɳɚɹ ɨɩɟɪɚɬɢɜɧɨ ɨɰɟɧɢɜɚɬɶ ɩɪɨɝɢɛ ɛɚɥɤɢ ɜ ɡɚɞɚɧɧɨɦ ɞɢɚɩɚɡɨɧɟ ɩɚɪɚɦɟɬɪɨɜ ɜɨɥɧɵ, ɧɟ ɪɟɲɚɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ.
Ⱥɧɚɥɨɝɢɱɧɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɦɨɠɟɬ
ɛɵɬɶ ɩɨɫɬɪɨɟɧɚ ɢ ɞɥɹ ɞɪɭɝɢɯ ɩɚɪɚɦɟɬɪɨɜ ɜɨɡɞɟɣɫɬɜɢɹ ɢ ɛɚɥɤɢ. ɉɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢ ɜɵɛɨɪɟ ɬɢɩɚ ɲɩɭɧɬɨɜɵɯ
ɫɜɚɣ ɫ ɭɱɟɬɨɦ ɜɨɥɧɨɜɵɯ ɩɚɪɚɦɟɬɪɨɜ.
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ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ
Выпуск 3
1. ɋɬɪɨɢɬɟɥɶɧɵɟ ɧɨɪɦɵ ɢ ɩɪɚɜɢɥɚ 2.06.04-82*. ɇɚɝɪɭɡɤɢ ɢ ɜɨɡɞɟɣɫɬɜɢɹ ɧɚ ɝɢɞɪɨɬɟɯɧɢɱɟɫɤɢɟ
ɫɨɨɪɭɠɟɧɢɹ (ɜɨɥɧɨɜɵɟ, ɥɟɞɨɜɵɟ ɢ ɨɬ ɫɭɞɨɜ). — Ɇ.: Ƚɨɫɫɬɪɨɣ ɋɋɋɊ, 1989. — 71 ɫ.
2. Ⱦɶɹɤɨɧɨɜ ȼ. ɉ. ɗɧɰɢɤɥɨɩɟɞɢɹ Mathcad 200i ɢ Mathcad 11 / ȼ. ɉ. Ⱦɶɹɤɨɧɨɜ. — Ɇ.: ɋɈɅɈɇɉɪɟɫɫ, 2004. — 384 ɫ.
3. Ʉɪɵɥɨɜ Ⱥ. ɇ. ɂɡɛɪɚɧɧɵɟ ɬɪɭɞɵ / Ⱥ. ɇ. Ʉɪɵɥɨɜ; ɫɬ. ɢ ɪɟɞ. ɘ. Ⱥ. ɒɢɦɚɧɫɤɨɝɨ, ɩɪɢɦɟɱ.
ɂ. Ƚ. ɏɚɧɨɜɢɱɚ. — Ɇ.: ɂɡɞ-ɜɨ Ⱥɇ ɋɋɋɊ, 1958. — (ɋɟɪ. «Ʉɥɚɫɫɢɤɢ ɧɚɭɤɢ»).
4. Ɋɚɛɨɬɧɨɜ ɘ. ɇ. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ / ɘ. ɇ. Ɋɚɛɨɬɧɨɜ. — Ɇ.: Ƚɨɫ. ɢɡɞ-ɜɨ ɮɢɡ.-ɦɚɬ.
ɥɢɬ., 1962. — 456 ɫ.
5. ɋɬɨɰɟɧɤɨ Ⱥ. Ⱥ. Ʉɭɪɫ ɬɟɨɪɢɢ ɫɨɨɪɭɠɟɧɢɣ. ɋɬɪɨɢɬɟɥɶɧɚɹ ɦɟɯɚɧɢɤɚ. / Ⱥ. Ⱥ. ɋɬɨɰɟɧɤɨ
[ɢ ɞɪ.]. — ȼɥɚɞɢɜɨɫɬɨɤ: ɂɡɞ-ɜɨ ȾȼȽɌɍ, 2007. — ɑ. 1: Ɍɟɨɪɢɹ ɫɨɨɪɭɠɟɧɢɣ ɜ ɢɧɠɟɧɟɪɧɨɦ ɞɟɥɟ. —
ɉɪɢɥ. 2: ɇɚɝɪɭɡɤɚ ɢ ɨɰɟɧɤɚ ɷɤɫɩɥɭɚɬɚɰɢɨɧɧɵɯ ɤɚɱɟɫɬɜ ɫɨɨɪɭɠɟɧɢɣ ɩɪɢ ɞɢɧɚɦɢɱɟɫɤɢɯ ɜɨɡɞɟɣɫɬɜɢɹɯ ɡɟɦɥɟɬɪɹɫɟɧɢɣ ɢ ɜɟɬɪɚ. — 80 ɫ.
6. ɉɚɯɨɦɟɧɤɨɜɚ Ɍ. ɘ. Ɉ ɩɪɢɦɟɧɟɧɢɢ ɱɢɫɥɟɧɧɨɝɨ ɦɟɬɨɞɚ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ⱥ. ȼ. Ⱦɪɚɝɢɥɟɜɚ
ɜ ɡɚɞɚɱɟ ɪɚɫɱɟɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɱɚɫɬɨɬ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɣ ɭɩɪɭɝɨɣ ɛɚɥɤɢ / Ɍ. ɘ. ɉɚɯɨɦɟɧɤɨɜɚ //
Ɏɗɇ-ɇɚɭɤɚ: ɩɟɪɢɨɞ. ɠɭɪɧ. ɧɚɭɱ. ɬɪ. — Ȼɭɝɭɥɶɦɚ, 2012. — 69 ɫ.
7. Ɏɥɟɬɱɟɪ Ʉ. ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɧɚ ɨɫɧɨɜɚɧɢɢ ɦɟɬɨɞɚ Ƚɚɥɟɪɤɢɧɚ: ɩɟɪ. ɫ ɚɧɝɥ. / Ʉ. Ɏɥɟɬɱɟɪ. — Ɇ.: Ɇɢɪ, 1988 — 352 ɫ.
8. ɂɥɶɢɧ ȼ. ɉ. ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɜ ɫɬɪɨɢɬɟɥɶɧɨɣ ɦɟɯɚɧɢɤɟ: ɫɩɪɚɜ. ɩɨɫɨɛɢɟ /
ȼ. ɉ. ɂɥɶɢɧ, ȼ. ȼ. Ʉɚɪɩɨɜ, Ⱥ. Ɇ. Ɇɚɫɥɟɧɧɢɤɨɜ; ɩɨɞ ɪɟɞ. ȼ. ɉ. ɂɥɶɢɧɚ. — Ɇɢɧɫɤ: ȼɵɫɲ. ɲɤ., 1990.
9. ɉɚɯɨɦɟɧɤɨɜɚ Ɍ. ɘ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɭɫɬɨɣɱɢɜɨɫɬɢ ɲɩɭɧɬɨɜɵɯ ɫɬɟɧ ɩɪɢ ɜɨɥɧɨɜɨɦ ɜɨɡɞɟɣɫɬɜɢɢ / Ɍ. ɘ. ɉɚɯɨɦɟɧɤɨɜɚ // Ƚɢɞɪɨɷɧɟɪɝɟɬɢɤɚ. ɇɨɜɵɟ ɪɚɡɪɚɛɨɬɤɢ ɢ ɬɟɯɧɨɥɨɝɢɢ: ɬɟɡ. ɧɚɭɱ.-ɬɟɯɧ.
ɤɨɧɮ. / ȼɇɂɂȽ ɢɦ. Ȼ. ȿ. ȼɟɞɟɧɟɟɜɚ, ɨɤɬɹɛɪɶ 2012 ɝ. — ɋɉɛ., 2012.
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