Федоров М.П. и др. Информационно-компьютерные технологии в строительстве. Применение программы SCAD для расчета сооружений на динамические воздействия
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30
Ⱦɥɹ ɡɚɝɪɭɠɟɧɢɹ ɩɨɫɬɪɨɟɧɧɨɣ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ ɞɢɧɚɦɢɱɟɫɤɢɦ ɜɨɡɞɟɣɫɬɜɢɟɦ ɜ ɪɚɡɞɟɥɟ
«Ɂɚɝɪɭɠɟɧɢɹ» ɧɚ ɢɧɫɬɪɭɦɟɧɬɚɥɶɧɨɣ ɩɚɧɟɥɢ ɧɚɠɢɦɚɟɦ ɤɧɨɩɤɭ
. Ɉɬɤɪɨɟɬɫɹ ɞɢɚɥɨɝɨɜɨɟ
ɨɤɧɨ, ɜ ɤɨɬɨɪɨɦ ɜɵɛɢɪɚɟɦ «ɋɨɡɞɚɬɶ ɧɨɜɨɟ ɡɚɝɪɭɠɟɧɢɟ». Ⱦɚɥɟɟ ɜɵɛɢɪɚɟɦ «Ɇɨɞɚɥɶɧɵɣ
ɚɧɚɥɢɡ» ɢ ɡɚɞɚɟɦ ɬɪɟɛɭɟɦɨɟ ɱɢɫɥɨ ɋɎɄ: 1.
Ɂɚɬɟɦ ɨɬɤɪɵɜɚɟɦ ɨɤɧɨ
«ɂɧɟɪɰɢɨɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ» ɢ ɜɵɛɢɪɚɟɦ «Ɂɚɞɚɧɢɟ
ɦɚɫɫ». Ɂɚɩɨɥɧɹɟɦ ɞɢɚɥɨɝɨɜɨɟ ɨɤɧɨ (ɨɧɨ ɩɪɢɜɟɞɟɧɨ ɧɢɠɟ) ɩɨ ɡɚɞɚɧɢɸ ɜɟɫɚ
gmG
ɝ
=
= 6 · 9.81 = 58.86 ɤɇ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ ɜ ɭɡɥɟ 2. Ⱦɥɹ ɜɜɨɞɚ ɷɬɨɝɨ ɡɧɚɱɟɧɢɹ ɨɬɦɟɬɢɦ ɭɡɟɥ 2 ɢ
ɩɨɞɬɜɟɪɠɞɚɟɦ ɫɞɟɥɚɧɧɵɣ ɜɵɛɨɪ ɧɚɠɚɬɢɟɦ ɜ ɦɟɧɸ ɪɚɡɞɟɥɚ «Ɂɚɝɪɭɠɟɧɢɹ» ɡɟɥɟɧɨɝɨ ɰɜɟɬɚ
ɝɚɥɨɱɤɢ «ɈɄ».
Ɂɚɬɟɦ ɚɧɚɥɨɝɢɱɧɵɦɢ ɞɟɣɫɬɜɢɹɦɢ ɜɜɨɞɢɬɫɹ ɜɟɫ ɨɞɧɨɝɨ ɩɨɝɨɧɧɨɝɨ ɦɟɬɪɚ ɛɚɲɧɢ
1)(1
ɛɛ
P gq = 1)81.952.0( = 5.1 ɤɇ:
ɗɬɢ ɞɚɧɧɵɟ ɨɬɧɨɫɢɦ ɤ ɟɞɢɧɫɬɜɟɧɧɨɦɭ ɤɨɧɟɱɧɨɦɭ ɷɥɟɦɟɧɬɭ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɵ.
ȼ ɪɟɡɭɥɶɬɚɬɟ (ɩɪɢ ɧɚɠɚɬɵɯ ɤɧɨɩɤɚɯ
ɢ ) ɩɨɥɭɱɢɦ ɪɚɫɱɟɬɧɭɸ
ɫɯɟɦɭ ɜ ɜɢɞɟ ɢɡɨɛɪɚɠɟɧɧɨɦ ɧɚ ɪɢɫ. 1.14.
ɋɨɯɪɚɧɹɟɦ ɡɚɝɪɭɠɟɧɢɟ 1 ɫ ɧɚɡɜɚɧɢɟɦ «ɦɨɞ-ɜɷɭ1» ɢ ɜɵɩɨɥɧɹɟɦ
ɪɚɫɱɟɬ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɜ ɩɪɨɝɪɚɦɦɟ SCAD ɜ ɬɚɛɥɢɰɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɫɨɜ
ɦɚɫɫ ɩɨɥɭɱɢɦ ɜɟɫ
2.132
2
G ɤɇ ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɦɚɫɫɵ ɜ ɭɡɥɟ 2.
ȼɨ ɜɬɨɪɨɣ ɬɚɛɥɢɰɟ ɩɪɢ ɩɨɩɟɪɟɱɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɛɚɲɧɢ ɫ ɭɤɚɡɚɧɧɨɣ
ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɣ ɦɚɫɫɨɣ ɜ ɭɡɥɟ 2 ɩɨɥɭɱɢɦ ɤɪɭɝɨɜɭɸ ɫɨɛɫɬɜɟɧɧɭɸ ɱɚɫɬɨɬɭ
3.8 Z ɫ
-1
, ɱɚɫɬɨɬɭ 3.12/ SZ Q Ƚɰ ɢ ɩɟɪɢɨɞ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ
76.0/1/2 Q ZS T ɫ.
ɋɨɛɫɬɜɟɧɧɚɹ ɮɨɪɦɚ ɤɨɥɟɛɚɧɢɣ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɟɪɜɨɣ
ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɟ ȼɗɍ, ɩɨɥɭɱɟɧɧɨɣ ɩɨ ɜɵɛɪɚɧɧɨɣ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɟ,
ɩɨɥɭɱɚɟɬɫɹ ɩɪɢ ɪɚɫɱɟɬɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɪɨɝɪɚɦɦɵ SCAD ɜ ɜɢɞɟ,
ɢɡɨɛɪɚɠɟɧɧɨɦ ɧɚ ɪɢɫ. 1.13,
ɛ.
Ɂɧɚɱɟɧɢɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
T,, QZ , ɨɩɪɟɞɟɥɟɧɧɵɯ ɜɪɭɱɧɭɸ ɢ ɫ
ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɦɵ SCAD, ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ
ɬɨɱɧɨɫɬɶɸ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɨɜɩɚɞɚɸɬ.
D58.86
D5.1
1
2
Ɋɢɫ. 1.14
31
1.2. ɋɜɨɛɨɞɧɵɟ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ
ɍɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ. ȼ ɪɟɚɥɶɧɵɯ
ɫɨɨɪɭɠɟɧɢɹɯ ɷɧɟɪɝɢɹ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɧɟɨɛɪɚɬɢɦɨ ɬɟɪɹɟɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɩɨɬɟɪɶ ɧɚ ɩɪɟɨɞɨɥɟɧɢɟ ɜɧɭɬɪɟɧɧɟɝɨ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɚ,
ɩɨɬɟɪɶ ɜ ɦɟɫɬɚɯ ɫɨɟɞɢɧɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɢ ɨɩɨɪɚɦɢ, ɚ ɬɚɤɠɟ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɩɨɬɟɪɶ ɱɟɪɟɡ ɨɫɧɨɜɚɧɢɟ, ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ ɢ ɬ. ɞ. ɉɨɷɬɨɦɭ
ɫɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɨ ɜɪɟɦɟɧɟɦ ɡɚɬɭɯɚɸɬ.
ɉɪɨɛɥɟɦɟ ɭɱɟɬɚ ɪɚɡɥɢɱɧɵɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɩɪɢ ɤɨɥɟɛɚɧɢɢ ɫɨɨɪɭɠɟɧɢɣ
ɩɨɫɜɹɳɟɧɚ ɦɧɨɝɨɱɢɫɥɟɧɧɚɹ ɥɢɬɟɪɚɬɭɪɚ (ɫɦ. ɛɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɟ ɫɩɢɫɤɢ,
ɧɚɩɪɢɦɟɪ, ɜ ɪɚɛɨɬɚɯ [5, 11, 15, 23]). ȼ ɭɱɟɛɧɢɤɟ [11] ɪɚɫɫɦɨɬɪɟɧ ɫɩɨɫɨɛ ɭɱɟɬɚ
ɫɢɥ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜ ɡɚɞɚɱɚɯ ɞɢɧɚɦɢɤɢ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɝɢɩɨɬɟɡɟ «ɜɹɡɤɨɝɨ
ɬɪɟɧɢɹ», ɩɪɟɞɥɨɠɟɧɧɨɣ Ɏɨɣɝɬɨɦ, ɚ ɬɚɤɠɟ ɫɩɨɫɨɛ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɬɟɨɪɢɢ
ɧɟɭɩɪɭɝɨɝɨ ɩɨɝɥɨɳɟɧɢɹ ɷɧɟɪɝɢɢ ɤɨɥɟɛɚɧɢɣ, ɜ ɬɨɦ ɱɢɫɥɟ ɩɪɢ ɭɱɟɬɟ ɜɧɭɬɪɟɧɧɟɝɨ
ɬɪɟɧɢɹ ɜ ɤɨɦɩɥɟɤɫɧɨɣ ɮɨɪɦɟ ɩɨ ɬɟɨɪɢɢ ȿ.ɋ. ɋɨɪɨɤɢɧɚ.
ɂɫɫɥɟɞɨɜɚɧɢɹ, ɜɵɩɨɥɧɟɧɧɵɟ ɜ [11], ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ
ɨ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ
ɭɪɚɜɧɟɧɢɟ, ɨɩɢɫɵɜɚɸɳɟɟ ɷɬɢ ɤɨɥɟɛɚɧɢɹ, ɢ ɟɝɨ ɪɟɲɟɧɢɟ ɦɨɝɭɬ ɛɵɬɶ ɡɚɩɢɫɚɧɵ ɜ
ɜɢɞɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɢɫɩɨɥɶɡɨɜɚɧɢɸ ɝɢɩɨɬɟɡɵ «ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ» Ɏɨɣɝɬɚ, ɧɨ
ɡɚɬɟɦ ɫɤɨɪɪɟɤɬɢɪɨɜɚɧɵ ɫ ɭɱɟɬɨɦ ɢɦɟɸɳɢɯɫɹ
ɤ ɧɚɫɬɨɹɳɟɦɭ ɜɪɟɦɟɧɢ
ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɣ.
ɑɬɨɛɵ ɩɨɹɫɧɢɬɶ ɷɬɨ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɫɫɵ ɜ ɫɢɫɬɟɦɟ ɫ
ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɝɢɩɨɬɟɡɵ Ɏɨɣɝɬɚ ɜ ɬɟɯ ɠɟ ɞɜɭɯ
ɮɨɪɦɚɯ, ɱɬɨ ɢ ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ (ɫɦ. (1.7) ɢ (1.9)), ɧɨ ɫ
ɞɨɛɚɜɥɟɧɢɟɦ ɫɢɥɵ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
)(
ɧɧ
tRR
:
0
ɧ
KRJ
(1.57)
)(
ɧ
RJu G ;
(1.58)
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɢɩɨɬɟɡɨɣ «ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ» Ɏɨɣɝɬɚ ɫɢɥɚ ɧɟɭɩɪɭɝɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɤɨɥɟɛɚɧɢɹɦ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ:
ucR
ɧ
.
(1.59)
ɝɞɟ
c
– ɫɢɥɚ ɜɹɡɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɟɞɢɧɢɱɧɨɣ
ɫɤɨɪɨɫɬɢ (
1 u
) ɞɜɢɠɟɧɢɹ ɦɚɫɫɵ.
Ɂɧɚɤ «ɦɢɧɭɫ» ɜ ɜɵɪɚɠɟɧɢɢ (1.59) ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɫɢɥɚ ɜɹɡɤɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɬɨɱɟɱɧɭɸ ɦɚɫɫɭ ɩɪɢ ɟɟ ɞɜɢɠɟɧɢɢ, ɢɦɟɟɬ
ɧɚɩɪɚɜɥɟɧɢɟ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɧɚɩɪɚɜɥɟɧɢɸ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ.
ȼ ɪɟɡɭɥɶɬɚɬɟ, ɞɜɚ ɜɚɪɢɚɧɬɚ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ,
32
ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ ɧɟɜɟɫɨɦɨɣ ɥɢɧɟɣɧɨ ɞɟɮɨɪɦɢɪɭɟɦɨɣ ɛɚɥɤɟ (ɫɦ. ɪɢɫ. 1.1, ɚ),
ɩɪɢ ɩɨɩɟɪɟɱɧɵɯ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɫ ɭɱɟɬɨɦ ɫɢɥ ɜɹɡɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɜɦɟɫɬɨ (1.57) ɢ (1.58) ɫ ɭɱɟɬɨɦ (1.4) ɢ (1.5) ɛɭɞɭɬ ɢɦɟɬɶ ɜɢɞ
0
kuucum
;
(1.60)
)(
ucumu
G ɢɥɢ 0 G uucum
(1.61)
Ɋɚɡɞɟɥɢɜ (1.60) ɢ (1.61) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ
m ɢ mG ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ
ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɜɢɞɟ
02
2
ZH uuu
.
(1.62)
ɝɞɟ
2
Z – ɨɛɨɡɧɚɱɟɧɢɟ (1.13), ɩɪɟɞɫɬɚɜɥɹɸɳɟɟ ɫɨɛɨɣ, ɤɚɤ ɷɬɨ ɩɨɤɚɡɚɧɨ ɜ
ɩɨɞɪɚɡɞɟɥɟ (1.1), ɤɜɚɞɪɚɬ ɤɪɭɝɨɜɨɣ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɵ ɫɢɫɬɟɦɵ ɩɪɢ
ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ, ɨɛɨɡɧɚɱɟɧɢɟ
mc
/
2 H
(1.63
)
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɭɞɜɨɟɧɧɭɸ ɜɟɥɢɱɢɧɭ H , ɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ
ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ ɢɥɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɞɟɦɩɮɢɪɨɜɚɧɢɹ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (1.62) ɜɚɠɧɨ ɫɨɨɬɧɨɲɟɧɢɟ ɜɟɥɢɱɢɧ H ɢ Z . ȼ
ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɹɯ ɯɚɪɚɤɬɟɪɧɵɦ ɹɜɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟ
ZH
.
(1.64)
Ⱥɧɚɥɢɡ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (1.56) ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫ ɭɱɟɬɨɦ ɫɢɥ
«ɜɹɡɤɨɝɨ» ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɤɨɥɟɛɚɧɢɹɦ
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1.62) ɫ ɭɱɟɬɨɦ
(1.64) ɢɦɟɟɬ (ɫɦ. ɯɨɞ ɪɟɲɟɧɢɹ, ɧɚɩɪɢɦɟɪ, ɜ [4]) ɜɢɞ:
)sincos(
21
tAtAeu
DD
t
ZZ
H
;
(1.65)
)cos( DZ
H
teAu
D
t
;
(1.66)
)sin( MZ
H
teAu
D
t
.
(1.67)
ɝɞɟ ɩɨɫɬɨɹɧɧɵɟ
21
, AA ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɞɜɢɠɟɧɢɹ (1.16):
D
uu
AuA
Z
H
ooo
2
o
1
;,
(1.68)
ɚ ɩɨ ɧɢɦ ɢɡ ɮɨɪɦɭɥ (1.21), (1.22), (1.25) - ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ MD,,
A
.
Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɜɵɪɚɠɟɧɢɣ (1.65) - (1.67), ɫɜɨɛɨɞɧɵɟ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɫ
ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɩɪɢ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɞɜɢɠɟɧɢɹ ɜ ɜɢɞɟ (1.16)
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ (ɪɢɫ. 1.15), ɩɪɨɢɫɯɨɞɹɳɢɟ ɫ
ɤɪɭɝɨɜɨɣ ɱɚɫɬɨɬɨɣ
2
2
22
1
Z
H
Z HZ Z
D
, 1/ɫ,
(1.69)
ɢ ɚɦɩɥɢɬɭɞɨɣ )exp(
t
A
H , ɤɨɬɨɪɚɹ ɜ ɫɜɹɡɢ ɫ ɧɚɥɢɱɢɟɦ ɦɧɨɠɢɬɟɥɹ )exp(
t
H ɫɨ
33
ɜɪɟɦɟɧɟɦ ɭɦɟɧɶɲɚɟɬɫɹ, ɡɚɬɭɯɚɟɬ. ɑɚɫɬɨɬɚ ɩɪɨɰɟɫɫɚ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ
D
Q ,
Ƚɰ
,, ɢ ɩɟɪɢɨɞ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ
D
T
, ɫ, ɨɩɪɟɞɟɥɹɸɬɫɹ, ɤɚɤ ɢ ɩɪɢ
ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢɡ ɜɵɪɚɠɟɧɢɣ
DD
D
D
D
D
T
T
f
Q
Z
S
S
Z
12
;
1
2
.
(1.70)
Ɋɢɫ. 1.15
ɏɚɪɚɤɬɟɪɢɫɬɢɤɢ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ. ɗɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ
ɢɫɫɥɟɞɨɜɚɧɢɹɦɢ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ H (1.63), ɩɨɥɭɱɟɧɧɵɣ
ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɝɢɩɨɬɟɡɵ Ɏɨɣɝɬɚ, ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ. ȼɬɨɠɟ
ɜɪɟɦɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɩɨɥɭɱɟɧɵ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ,
ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɵɦɢ (ɱɚɫɬɨɬɧɨ-ɧɟɡɚɜɢɫɢɦɵɦɢ) ɞɥɹ
ɤɨɧɤɪɟɬɧɨɝɨ ɬɢɩɚ ɫɨɨɪɭɠɟɧɢɣ ɢɥɢ ɢɯ ɷɥɟɦɟɧɬɨɜ.
Ɉɞɧɚ ɢɡ ɬɚɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ –
ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɝɥɨɳɟɧɢɹ ɷɧɟɪɝɢɢ
ɤɨɥɟɛɚɧɢɣ
[4, 10, 14]
W
W
'
\ ,
(1.71)
ɝɞɟ
W' – ɩɨɬɟɪɹ ɷɧɟɪɝɢɢ ɤɨɥɟɛɚɧɢɣ ɡɚ ɨɞɢɧ ɰɢɤɥ (ɡɚ ɩɟɪɢɨɞ T ); W –
ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɭɩɪɭɝɨɣ ɞɟɮɨɪɦɚɰɢɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɚɦɩɥɢɬɭɞɟ
n
A ɜ
ɧɚɱɚɥɟ ɰɢɤɥɚ.
ȼɬɨɪɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ, ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ ɱɚɫɬɨɬɵ
ɤɨɥɟɛɚɧɢɣ, ɹɜɥɹɟɬɫɹ
ɥɨɝɚɪɢɮɦɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ,
ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
34
1
ln
G
n
n
ɥ
A
A
,
(1.72)
ɝɞɟ
1
ɢ
nn
AA ɹɜɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɞɪɭɝ ɡɚ ɞɪɭɝɨɦ ɫ ɩɟɪɢɨɞɨɦ
DD
T ZS /2
ɚɦɩɥɢɬɭɞɚɦɢ ɤɨɥɟɛɚɧɢɣ (ɫɦ. ɪɢɫ.1.15).
Ɇɟɠɞɭ ɩɨɫɬɨɹɧɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ
\G ɢ
ɥ
ɫɭɳɟɫɬɜɭɟɬ ɪɚɜɟɧɫɬɜɨ [5, 11]:
ɥ
G
\
2 .
(1.73)
ȼ ɞɢɧɚɦɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɢɫɩɨɥɶɡɭɸɬ ɬɚɤɠɟ
ɤɨɷɮɮɢɰɢɟɧɬ ɧɟɭɩɪɭɝɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɤɨɥɟɛɚɧɢɹɦ
S
G
S
\
J
ɥ
2
ɧ
.
(1.74)
Ɂɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɧ
,, J\G
ɥ
ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɪɚɡɥɢɱɧɵɯ ɪɚɛɨɬɚɯ,
ɧɚɩɪɢɦɟɪ [11, 15, 16]. ɂɡ ɩɪɢɜɟɞɟɧɧɵɯ ɬɚɦ ɞɚɧɧɵɯ ɫɥɟɞɭɟɬ, ɱɬɨ ɧɚɢɦɟɧɶɲɟɟ
ɡɧɚɱɟɧɢɟ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɦɟɸɬ ɫɬɚɥɶɧɵɟ ɤɨɧɫɬɪɭɤɰɢɢ (ɫɬɚɥɶɧɵɟ
ɞɵɦɨɜɵɟ ɬɪɭɛɵ –
11.0|\ , ɫɬɚɥɶɧɵɟ ɦɨɫɬɵ – 17.0|\ ). Ⱦɥɹ ɠɟɥɟɡɨɛɟɬɨɧɧɵɯ
ɩɟɪɟɤɪɵɬɢɣ
56.044.0 |\ , ɦɨɫɬɨɜ |0.63. ɇɚɢɛɨɥɶɲɟɟ ɡɧɚɱɟɧɢɟ \ ɢɦɟɟɬ ɞɥɹ
ɪɚɡɥɢɱɧɵɯ ɡɞɚɧɢɣ (
|0.66), ɠɟɥɟɡɨɛɟɬɨɧɧɵɯ ɮɭɧɞɚɦɟɧɬɨɜ ɩɨɞ ɝɟɧɟɪɚɬɨɪɵ
(
|0.8). ɉɪɢ ɷɬɨɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɧ
,
J
G
ɥ
ɦɨɠɧɨ
ɭɫɬɚɧɨɜɢɬɶ ɢɡ (1.73).
Ʉɨɪɪɟɤɬɢɪɨɜɤɚ ɝɢɩɨɬɟɡɵ Ɏɨɣɝɬɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ
ɡɚɬɭɯɚɧɢɹ
H (1.63) ɞɥɹ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɭɜɹɡɵɜɚɸɬ ɫ
ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
ɧ
,, J\G
ɥ
, ɧɟ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɢ ɫ ɱɚɫɬɨɬɨɣ
D
Z
. ɗɬɨ ɩɨɥɭɱɚɟɬɫɹ, ɧɚɩɪɢɦɟɪ, ɫ ɩɨɦɨɳɶɸ ɜɵɪɚɠɟɧɢɹ (1.72) ɢ ɫ ɭɱɟɬɨɦ
ɪɟɲɟɧɢɹ (1.66) ɢɥɢ (1.67). Ɍɨɝɞɚ ɩɪɢ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫ ɱɚɫɬɨɬɨɣ
D
Z :
D
D
Tt
t
T
eA
eA
n
n
Z
S
H H G
H
H
2
ln
)(
ɥ
.
(1.75)
Ɉɬɫɸɞɚ ɫ ɭɱɟɬɨɦ (1.74)
DDD
Z
J
Z
S
\
Z
S
G
H
242
ɧ
ɥ
.
(1.76)
ȼ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ [5] ɩɨɤɚɡɚɧɨ, ɱɬɨ ɜɦɟɫɬɨ ɜɵɪɚɠɟɧɢɹ (1.69) ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ
2
ɧ
2
2
ɥ
)
2
(1)
4
(1
)
2
(1
J
Z
S
\
Z
S
G
Z
Z
D
.
(1.77)
Ɍɚɤ ɤɚɤ ɞɥɹ ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ (ɫɨɨɪɭɠɟɧɢɣ) ɤɨɷɮɮɢɰɢɟɧɬɵ
35
ɧɥ
,, J\G ɦɚɥɵ (ɫɦ. ɩɪɢɜɟɞɟɧɧɵɟ ɜɵɲɟ ɡɧɚɱɟɧɢɹ \ ɞɥɹ ɪɚɡɥɢɱɧɵɯ
ɤɨɧɫɬɪɭɤɰɢɣ), ɬɨ
1)
2
()
4
()
2
(
2
ɧ
22
ɥ
J
S
\
S
G
(1.78)
ɢ ɜɬɨɪɵɦɢ ɫɥɚɝɚɟɦɵɦɢ ɩɨɞ ɡɧɚɤɨɦ ɤɨɪɧɹ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɜɵɪɚɠɟɧɢɣ (1.77)
ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɞɢɧɢɰɟɣ. Ɍɨɝɞɚ ɞɥɹ ɫɬɪɨɢɬɟɥɶɧɵɯ
ɤɨɧɫɬɪɭɤɰɢɣ (ɫɨɨɪɭɠɟɧɢɣ) ɩɨɥɭɱɚɟɦ:
TT
DDD
|Q|QZ|Z ;;.
(1.79)
ȼ ɞɚɥɶɧɟɣɲɟɦ ɬɚɤɚɹ ɩɪɢɛɥɢɠɟɧɧɚɹ ɡɚɦɟɧɚ ɱɚɫɬɨɬ ɢ ɩɟɪɢɨɞɨɜ
ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɨɬɚɦɢ ɢ ɩɟɪɢɨɞɚɦɢ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɞɥɹ
ɫɬɪɨɢɬɟɥɶɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ ɫɞɟɥɚɧɚ ɜɨ ɜɫɟɯ ɢɫɩɨɥɶɡɭɟɦɵɯ ɞɚɥɟɟ ɮɨɪɦɭɥɚɯ.
ɋɤɨɪɪɟɤɬɢɪɨɜɚɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ Ɏɨɣɝɬɚ H ɫ ɭɱɟɬɨɦ (1.76) ɢ
(1.79) ɛɭɞɟɦ ɩɪɟɞɫɬɚɜɥɹɬɶ ɜ ɜɢɞɟ [17]:
]Z H ,
(1.80)
ɝɞɟ
] – ɤɨɷɮɮɢɰɢɟɧɬ ɞɟɦɩɮɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɣ ɬɚɤɠɟ ɤɚɤ ɤɨɷɮɮɢɰɢɟɧɬɵ
ɧɥ
,, J\G ɹɜɥɹɟɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦ ɨɬ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɢ ɦɨɠɟɬ ɛɵɬɶ
ɩɪɟɞɫɬɚɜɥɟɧ ɥɸɛɵɦ ɢɡ ɪɚɜɟɧɫɬɜ:
242
ɧɥ
J
S
\
S
G
]
.
(1.81)
ɉɪɢɦɟɪ 1. Ɉɩɪɟɞɟɥɢɦ ɤɪɭɝɨɜɭɸ ɱɚɫɬɨɬɭ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɡɚɬɭɯɚɸɳɢɯ
ɤɨɥɟɛɚɧɢɣ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɜ ɜɢɞɟ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɦ ɧɚ ɪɢɫ. 1.1, ɚ,
ɩɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɭɩɪɭɝɚɹ ɥɢɧɟɣɧɨ-ɞɟɮɨɪɦɢɪɭɟɦɚɹ ɛɚɥɤɚ ɹɜɥɹɟɬɫɹ ɠɟɥɟɡɨɛɟɬɨɧɧɨɣ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɢɫɤɨɦɨɣ ɜɟɥɢɱɢɧɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɜɬɨɪɵɦ ɪɚɜɟɧɫɬɜɨɦ ɜ (1.77). Ɂɧɚɱɟɧɢɟ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɨɝɥɨɳɟɧɢɹ ɷɧɟɪɝɢɢ 56.0|\ ɜɨɡɶɦɟɦ ɢɡ ɫɩɪɚɜɨɱɧɢɤɚ [15]. Ɍɨɝɞɚ
Z|
Z
Z
S
\
Z
Z
002.01
)
14.34
56.0
(1)
4
(1
22
D
.
ɉɪɢɦɟɪ ɧɚɝɥɹɞɧɨ ɩɨɞɬɜɟɪɠɞɚɟɬ ɜɵɪɚɠɟɧɢɹ (1.78) ɢ (1.79). Ʉɪɭɝɨɜɚɹ ɫɨɛɫɬɜɟɧɧɚɹ ɱɚɫɬɨɬɚ
Z
ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɛɵɥɚ ɩɨɥɭɱɟɧɚ ɜ ɩɪɢɦɟɪɟ 1 ɩɨɞɪɚɡɞɟɥɚ 1.1.
ɉɪɢɦɟɪ 2. Ⱦɥɹ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ, ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜ ɩɪɟɞɵɞɭɳɟɦ
ɩɪɢɦɟɪɟ (ɫɦ. ɪɢɫ. 1.1,
ɚ) ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɟɪɟɦɟɳɟɧɢɟ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ ɜ ɧɟɤɨɬɨɪɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
k
tt W ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɫɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ ɧɚɱɚɥɢɫɶ ɨɬ
ɦɝɧɨɜɟɧɧɨɝɨ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɢɦɩɭɥɶɫɚ, ɩɪɢɥɨɠɟɧɧɨɝɨ ɤ ɬɨɱɟɱɧɨɣ ɦɚɫɫɟ: 1) ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0 t (ɪɢɫ. 1.16, ɚ); 2) ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ W t (ɪɢɫ. 1.16, ɛ).
ȼɚɪɢɚɧɬ ɫ ɢɦɩɭɥɶɫɨɦ ɩɪɢ t = 0 (ɫɦ. ɪɢɫ.1.16, ɚ). ɍɞɚɪɧɵɣ ɢɦɩɭɥɶɫ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
oo
muJ
p
, ɝɞɟ
oo
u
– ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɦɚɫɫɵ. Ɍɨɝɞɚ ɢɡ (1.16) ɩɨɥɭɱɢɦ:
mJuuuu
p
tt
/;0
oo
0
o
0
.
36
ɂɫɩɨɥɶɡɭɹ ɷɬɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɦɚɫɫɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɜɟɥɢɱɢɧ
1
A ɢ
2
A ɜ ɜɵɪɚɠɟɧɢɢ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɜ ɜɢɞɟ (1.65), ɢɡ (1.68) ɫ ɭɱɟɬɨɦ (1.79) ɧɚɣɞɟɦ:
Z Z Z mJmJuAuA
pDpD
///;0
oo
2
o
1
.
ɉɪɢ ɷɬɨɦ ɢɡ (1.65) ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɢɫɤɨɦɨɝɨ
ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ:
te
m
J
tu
t
p
Z
Z
H
sin)(
Ƚɪɚɮɢɱɟɫɤɢ ɢɡɦɟɧɟɧɢɟ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ
ɩɨɫɥɟ ɜɨɡɞɟɣɫɬɜɢɹ ɢɦɩɭɥɶɫɚ ɜ ɜɚɪɢɚɧɬɟ 1 ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 1.16, ɚ.
Ɋɢɫ. 1.16
ȼɚɪɢɚɧɬ ɫ ɢɦɩɭɥɶɫɨɦ ɩɪɢ t = IJ (ɫɦ. ɪɢɫ. 1.16, ɛ). Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɪɟɲɟɧɢɹ ɜ ɷɬɨɦ
ɜɚɪɢɚɧɬɟ ɜɜɟɞɟɦ ɧɨɜɭɸ ɩɟɪɟɦɟɧɧɭɸ
W
tt
1
. ɉɨɫɤɨɥɶɤɭ
W
ɧɟɤɨɬɨɪɵɣ ɮɢɤɫɢɪɨɜɚɧɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɚ
1
ɢ tt ɩɟɪɟɦɟɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɬɨ dtdudtdu //
1
ɢ
222
1
2
// dtuddtud .
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ (1.62) ɢ ɟɝɨ ɪɟɲɟɧɢɟ (1.65) ɩɨɫɥɟ ɜɜɨɞɚ
ɧɨɜɨɣ ɩɟɪɟɦɟɧɧɨɣ ɨɫɬɚɸɬɫɹ ɛɟɡ ɢɡɦɟɧɟɧɢɹ. Ɍɨɥɶɤɨ ɜ ɷɬɢɯ ɜɵɪɚɠɟɧɢɹɯ ɮɭɧɤɰɢɹ
)(tu ɢɟɟ
ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ t ɡɚɦɟɧɹɬɫɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ ɮɭɧɤɰɢɸ
)(
1
tu
ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ
1
t
. Ɍɨɝɞɚ
ɫ ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ
W
tt
1
ɩɨɥɭɱɢɦ
)(sin)(ɢɥɢsin)(
)(
11
1
WZ
Z
WZ
Z
WH
H
te
m
J
tute
m
J
tu
t
p
t
p
.
Ƚɪɚɮɢɤ ɷɬɨɣ ɮɭɧɤɰɢɢ ɢɡɨɛɪɚɠɟɧ ɧɚ ɪɢɫ. 1.16, ɛ. Ɉɧ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɝɪɚɮɢɤ ɨɬ
ɢɦɩɭɥɶɫɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
0 t (ɫɦ. ɪɢɫ. 1.16, ɚ), ɧɨ ɫɞɜɢɧɭɬɵɣ ɩɨ ɜɪɟɦɟɧɢ ɧɚ ɜɟɥɢɱɢɧɭ
W
.
37
ɉɪɢɦɟɪ 3. Ʉɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜɵɲɟ (ɫɦ. (1.79) ɢ ɩɪɢɦɟɪ 1), ɭɱɟɬ ɫɢɥɵ ɧɟɭɩɪɭɝɨɝɨ
(ɩɨ Ɏɨɣɝɬɭ – ɜɹɡɤɨɝɨ) ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɪɢ ɤɨɥɟɛɚɧɢɹɯ ɫɬɪɨɢɬɟɥɶɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɩɨ
ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɟ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɜɥɢɹɟɬ ɧɚ ɱɚɫɬɨɬɭ ɢɩɟɪɢɨɞ
ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɵɟ ɩɨɷɬɨɦɭ ɩɪɢɧɢɦɚɸɬɫɹ ɪɚɜɧɵɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ
ɜɟɥɢɱɢɧɚɦ ɩɪɢ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ.
Ɉɫɧɨɜɧɨɟ ɜɥɢɹɧɢɟ ɫɢɥɚ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɨɤɚɡɵɜɚɟɬ ɧɚ ɡɧɚɱɟɧɢɹ
ɩɟɪɟɦɟɳɟɧɢɣ ɦɚɫɫɵ, ɜ ɬɨɦ ɱɢɫɥɟ ɧɚ ɚɦɩɥɢɬɭɞɵ ɤɨɥɟɛɚɧɢɣ (ɫɦ. ɪɢɫ. 1.15 ɢ ɪɢɫ. 1.16).
Ɉɰɟɧɢɦ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɚɬɭɯɚɧɢɹ ɚɦɩɥɢɬɭɞ ɤɨɥɟɛɚɧɢɣ
ɦɚɫɫɵ
m ɜ ɪɚɫɱɟɬɧɨɣ ɫɯɟɦɟ ɠɟɥɟɡɨɛɟɬɨɧɧɨɣ ɛɚɥɤɢ, ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ
(56.0|\ ), ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɹɯ.
ɂɡ ɩɪɢɜɟɞɟɧɧɨɣ ɜ ɩɪɢɦɟɪɟ 2 ɮɨɪɦɭɥɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɹ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ
ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ, ɜɵɡɜɚɧɧɵɯ ɢɦɩɭɥɶɫɨɦ
p
J ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ
0 t (ɫɦ. ɩɪɢɦɟɪ 2 ɢ ɪɢɫ. 1.16, ɚ), ɜɢɞɧɨ, ɱɬɨ ɚɦɩɥɢɬɭɞɚ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ
ɢɦɟɟɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
Z
S
2/
1
tt , ɤɨɝɞɚ 1sin
1
Z t . ɉɪɢ ɷɬɨɦ ɜ
ɭɤɚɡɚɧɧɨɣ ɮɨɪɦɭɥɟ ɛɭɞɟɦ ɢɦɟɬɶ
07.0
8
56.0
24
1
Z
S
S
\Z
H t ɢ
Z
Z
Z
H
m
J
e
m
J
e
m
J
A
pp
t
p
93.0
07.0
max
1
.
Ɉɩɪɟɞɟɥɢɦ, ɜɨ ɫɤɨɥɶɤɨ ɪɚɡ ɭɦɟɧɶɲɢɬɫɹ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ
max
A ɱɟɪɟɡ 5 ɩɟɪɢɨɞɨɜ
ɩɨɫɥɟ ɦɚɤɫɢɦɚɥɶɧɨɣ ɚɦɩɥɢɬɭɞɵ. ɋ ɷɬɨɣ ɰɟɥɶɸ ɜ ɮɨɪɦɭɥɟ ɞɥɹ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɫɫɵ ɩɪɢɦɟɦ
Z
S
Z
S
Z
S
Z
S
5.10
2
5
2
5
2
1
Ttt .
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ
47.1
4
5.1056.05.10
4
1
Z
S
S
\Z
H t
ɢ
Z
Z
Z
H
m
J
e
m
J
e
m
J
A
pp
t
p
t
23.0
47.1
1
1
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɟɪɟɡ 5 ɩɟɪɢɨɞɨɜ ɩɨɫɥɟ ɩɨɹɜɥɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɨɣ ɚɦɩɥɢɬɭɞɵ ɡɚɬɭɯɚɸɳɢɯ
ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɛɚɥɤɢ ɩɨɥɭɱɢɦ ɚɦɩɥɢɬɭɞɭ ɦɟɧɶɲɭɸ, ɩɪɢɦɟɪɧɨ, ɜ
4 ɪɚɡɚ,
ɦɚɤɫɢɦɚɥɶɧɨɣ.
Ƚɪɚɮɢɱɟɫɤɢ
(ɪɢɫ. 1.17) ɷɬɨ ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɧɨ ɫ ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɦɵ Mathcad ɩɪɢ
1)/( ZmJ
p
,
08.72 Z
ɢ 56.0 \ .
Ɋɢɫ. 1.17
38
ɍɱɚɳɢɦɫɹ ɪɟɤɨɦɟɧɞɭɟɬɫɹ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɷɬɨɣ ɩɪɨɫɬɨɣ ɞɥɹ ɩɨɥɶɡɨɜɚɬɟɥɹ ɩɪɨɝɪɚɦɦɨɣ ɢ
ɢɫɫɥɟɞɨɜɚɬɶ ɜɥɢɹɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ \ ɧɚ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɚɬɭɯɚɧɢɹ ɚɦɩɥɢɬɭɞ ɤɨɥɟɛɚɧɢɣ.
1.3. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɫɢɥɨɜɨɦ ɜɨɡɞɟɣɫɬɜɢɢ
Ɋɚɫɫɦɨɬɪɢɦ ɞɟɣɫɬɜɢɟ ɧɚ ɬɨɱɟɱɧɭɸ ɦɚɫɫɭ ɫɢɫɬɟɦɵ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ
ɫɜɨɛɨɞɵ (ɪɢɫ. 1.18, ɚ) ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɥɵ
)(
t
P
, ɝɪɚɮɢɱɟɫɤɨɟ ɢɡɦɟɧɟɧɢɟ ɤɨɬɨɪɨɣ
ɜɨ ɜɪɟɦɟɧɢ ɢɡɨɛɪɚɠɟɧɨ ɧɚ ɪɢɫ. 1.18, ɛ [8].
Ɋɢɫ. 1.18
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɨ ɤɚɤ
ɭɪɚɜɧɟɧɢɟ ɟɟ ɪɚɜɧɨɜɟɫɢɹ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ )(
t
P
P
, ɭɩɪɭɝɨɣ ɫɢɥɵ )(
t
K
K
, ɫɢɥɵ ɧɟɭɩɪɭɝɨɝɨ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ
)(
ɧɧ
tRR ɢ ɫɢɥɵ ɢɧɟɪɰɢɢ )(
t
J
J
, ɤɨɬɨɪɚɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
ɩɪɢɧɰɢɩɨɦ Ⱦɚɥɚɦɛɟɪɚ ɬɚɤɠɟ ɫɱɢɬɚɟɬɫɹ ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɦɚɫɫɟ:
0
ɧ
PKRJ
.
(1.82)
ɋ ɭɱɟɬɨɦ (1.4), (1.6), (1.53) ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
P
kuucum
.
(1.83)
ɉɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɜɬɨɪɵɦ ɫɩɨɫɨɛɨɦ (ɤɚɤ ɜɵɪɚɠɟɧɢɹ
ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɟɪɟɦɟɳɟɧɢɹ ɭɩɪɭɝɨɣ ɫɢɫɬɟɦɵ ɜ ɦɟɫɬɟ ɪɚɫɩɨɥɨɠɟɧɢɹ
ɬɨɱɟɱɧɨɣ ɦɚɫɫɵ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɡɚɞɚɧɧɨɣ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ, ɫɢɥɵ ɢɧɟɪɰɢɢ ɢ
ɫɢɥɵ ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ), ɩɨɥɭɱɢɦ:
)(
ɧ
RIPu G
;
(1.84)
)( ucum
P
u
G ɢɥɢ
P
uucum G GG
.
(1.85)
Ɍɚɤɠɟ ɤɚɤ ɢ ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɹɯ, ɭɪɚɜɧɟɧɢɹ (1.83) ɢ (1.85)
ɩɪɢɜɨɞɹɬɫɹ ɤ ɨɞɧɨɦɭ ɬɢɩɭ:
mPuuu /2
2
ZH
,
(1.86)
ɝɞɟ ɫɨɛɫɬɜɟɧɧɚɹ ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ
Z
ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ
(1.14).
39
ɉɪɢɦɟɱɚɧɢɟ. Ɂɞɟɫɶ, ɜ ɨɛɟɢɯ ɮɨɪɦɚɯ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɚɤ
ɢ ɩɪɢ ɫɜɨɛɨɞɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ ɫ ɱɚɫɬɨɬɨɣ Z|Z
D
, ɫɢɥɚ
ɧɟɭɩɪɭɝɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ, ɤɨɬɨɪɵɣ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɝɢɩɨɬɟɡɟ «ɜɹɡɤɨɝɨ
ɬɪɟɧɢɹ» Ɏɨɣɝɬɚ. ȼɨɩɪɨɫ ɨ ɤɨɪɪɟɤɬɢɪɨɜɚɧɢɢ ɩɪɢ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɡɚɬɭɯɚɧɢɹ ɤɨɥɟɛɚɧɢɣ H ɫ ɭɱɟɬɨɦ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢɫɫɥɟɞɨɜɚɧɢɣ
ɪɚɫɫɦɨɬɪɢɦ ɩɨɡɠɟ.
Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ )(
t
uu ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ (1.86) ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ
)(
ɨɞɧɨɞɧ
tuu
(1.65)
ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (1.62) ɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ
)(
ɱɱ
tuu
.
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɚɣɞɟɦ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɜɵɪɚɠɟɧɢɟɦ, ɩɨɥɭɱɟɧɧɵɦ
ɜɵɲɟ (ɫɦ. ɩɪɢɦɟɪ 2 ɩɨɞɪɚɡɞɟɥɚ 1.2), ɩɪɢ ɞɟɣɫɬɜɢɢ ɧɚ ɦɚɫɫɭ ɦɝɧɨɜɟɧɧɨɝɨ
ɢɦɩɭɥɶɫɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ W (ɫɦ. ɪɢɫ. 1.16, ɛ). ȿɫɥɢ ɜɡɹɬɶ ɜɨɡɦɭɳɚɸɳɢɣ
ɫɢɥɨɜɨɣ ɢɦɩɭɥɶɫ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɜɟɥɢɱɢɧɵ WW dPdJ
p
)((ɫɦ. ɪɢɫ.1.18, ɛ), ɬɨ
ɨɬ ɧɟɝɨ ɩɨɥɭɱɢɦ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
t
ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɟ ɩɟɪɟɦɟɳɟɧɢɟ:
)(sin
)(
)(sin)(
)()(
WZ
Z
WW
WZ
Z
WHWH
te
m
dP
te
m
dJ
tdu
tt
p
.
(1.87)
ɉɪɨɫɭɦɦɢɪɭɟɦ ɞɟɣɫɬɜɢɟ ɜɫɟɯ ɢɦɩɭɥɶɫɨɜ ɨɬ 0
W
ɞɨ
t
W
. ȼ ɪɟɡɭɥɶɬɚɬɟ
ɩɨɥɭɱɢɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ
³
WWZ
WH
W
Z
t
dt
t
eP
m
tu
0
ɱ
)(sin
)(
)(
1
)( .
(1.88)
ɉɪɢ ɧɭɥɟɜɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɞɜɢɠɟɧɢɹ (1.16) ɩɨɫɬɨɹɧɧɵɟ
21
ɢ AA
ɜ
ɪɟɲɟɧɢɢ (1.65) ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (1.62) ɛɭɞɭɬ ɪɚɜɧɵ ɧɭɥɸ ɢ ɨɛɳɟɟ
ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1.86) ɛɭɞɟɬ ɪɚɜɧɨ ɱɚɫɬɧɨɦɭ ɪɟɲɟɧɢɸ ɜ ɜɢɞɟ (1.88):
)()(
ɱ
tutu
.
ȼɨɩɪɨɫ ɨ ɜɵɱɢɫɥɟɧɢɢ ɢɧɬɟɝɪɚɥɚ Ⱦɸɚɦɟɥɹ ɪɚɫɫɦɨɬɪɢɦ ɩɪɢ ɤɨɧɤɪɟɬɧɵɯ
ɞɢɧɚɦɢɱɟɫɤɢɯ ɜɨɡɞɟɣɫɬɜɢɹɯ, ɬ.ɟ. ɩɪɢ ɤɨɧɤɪɟɬɧɨɦ ɜɢɞɟ ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢ P(t).
1.4. Ⱦɟɣɫɬɜɢɟ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ
Ɋɚɫɫɦɨɬɪɢɦ ɜɚɪɢɚɧɬ ɜɨɡɦɭɳɚɸɳɟɣ ɧɚɝɪɭɡɤɢ ɜ ɜɢɞɟ tPtP T sin)(
o
, ɝɞɟ
Tɢ
o
P – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɚɦɩɥɢɬɭɞɚ ɢ ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ.
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɳɟɣɫɹ ɢɧɠɟɧɟɪɧɨɣ ɡɚɞɚɱɢ ɪɚɫɫɦɨɬɪɟɧɨ ɜ
ɪɚɡɥɢɱɧɵɯ ɭɱɟɛɧɵɯ ɩɨɫɨɛɢɹɯ, ɭɱɟɛɧɢɤɚɯ ɢ ɫɩɪɚɜɨɱɧɢɤɚɯ ɩɨ ɞɢɧɚɦɢɤɟ
ɫɨɨɪɭɠɟɧɢɣ, ɧɚɩɪɢɦɟɪ ɜ [5, 8, 10 - 15] ɢɬ. ɞ.
Ɂɞɟɫɶ ɪɚɫɫɦɨɬɪɢɦ ɬɨɥɶɤɨ ɪɟɡɭɥɶɬɚɬ ɪɟɲɟɧɢɹ ɩɪɢ ɬɚɤɢɯ ɡɧɚɱɟɧɢɹɯ t, ɩɪɢ
ɤɨɬɨɪɵɯ ɫɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɜɵɡɜɚɧɧɵɟ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɞɜɢɠɟɧɢɹ, ɚ
ɬɚɤɠɟ ɢ ɫɨɩɪɨɜɨɠɞɚɸɳɢɟ
ɫɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɫɜɹɡɚɧɧɵɟ ɫ ɢɯ ɜɨɡɦɭɳɟɧɢɟɦ