Механика.Методика решения задач
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Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
181 |
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Ɉɬɜɟɬ: tg- |
c |
3 |
tg- , L |
c |
L |
3 |
cos2 - |
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2 |
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2 |
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Ɂɚɞɚɱɚ 4
Ʉɨɫɦɨɧɚɜɬ ɫɩɭɫɬɹ ɜɪɟɦɹ W0 (ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɱɚɫɚɦ) ɩɨɫɥɟ
ɫɬɚɪɬɚ ɩɨɥɭɱɚɟɬ ɪɚɞɢɨɝɪɚɦɦɭ ɫ ɫɨɨɛɳɟɧɢɟɦ ɨ ɪɨɠɞɟɧɢɢ ɜɧɭɤɚ. Ɍɨɬɱɚɫ ɠɟ, ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɜɧɭɤ ɩɨɥɭɱɢɥ ɩɨɡɞɪɚɜɥɟɧɢɟ ɜɨɜɪɟɦɹ, ɨɧ ɩɨɫɵɥɚɟɬ ɨɬɜɟɬɧɭɸ ɪɚɞɢɨɝɪɚɦɦɭ, ɜ ɤɨɬɨɪɨɣ ɩɨɡɞɪɚɜɥɹɟɬ ɜɧɭɤɚ ɫ ɫɨɜɟɪɲɟɧɧɨɥɟɬɢɟɦ (ɜɨɡɪɚɫɬ ɪɚɜɟɧ Ɍ). Ʉɚɤɨɜɚ ɫɤɨɪɨɫɬɶ ɤɨɫɦɢɱɟɫɤɨɝɨ ɤɨɪɚɛɥɹ?
Ɉɬɜɟɬ: V Tc
4W02 T 2 .
Ɂɚɞɚɱɚ 5
ɋɢɫɬɟɦɚ ɨɬɫɱɟɬɚ S' ɞɜɢɠɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S ɫɨ ɫɤɨɪɨɫɬɶɸ V = c/2 ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɨɫɢ X. ȼ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ U = c/2 ɩɨɞ ɭɝɥɨɦ - ɤ ɨɫɢ X. ɇɚɣɬɢ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ U c ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S'. Ɉɩɪɟɞɟɥɢɬɶ, ɤɚɤɨɣ ɭɝɨɥ -c ɫɨɫɬɚɜɥɹɟɬ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɫ ɨɫɶɸ X' ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S'.
Ɉɬɜɟɬ: U c |
c |
8 8cos- sin |
2 |
- , tg-c |
3 sin- |
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4 cos- |
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2 1 cos- |
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Ɂɚɞɚɱɚ 6
Ⱦɜɚ ɤɨɫɦɢɱɟɫɤɢɯ ɤɨɪɚɛɥɹ ɥɟɬɹɬ ɜɞɨɥɶ ɨɞɧɨɣ ɩɪɹɦɨɣ ɜ ɨɞɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɫɨ ɫɤɨɪɨɫɬɹɦɢ V1 > V2 . ɋɨ ɜɬɨɪɨɝɨ ɤɨɪɚɛɥɹ ɜɞɨɝɨɧɤɭ ɩɟɪɜɨɦɭ ɩɨɫɵɥɚɟɬɫɹ ɞɜɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɢɦɩɭɥɶɫɚ ɫ ɢɧɬɟɪɜɚɥɨɦ ɜɪɟɦɟɧɢ W1 ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɋ ɤɚɤɢɦ ɢɧɬɟɪɜɚɥɨɦ ɜɪɟɦɟɧɢ W2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɣ ɠɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɨɧɢ ɜɟɪɧɭɬɫɹ ɧɚɡɚɞ ɩɨɫɥɟ ɨɬɪɚɠɟɧɢɹ ɨɬ ɩɟɪɜɨɝɨ ɤɨɪɚɛɥɹ?
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§ |
c V1 |
·§ |
c V2 |
· |
Ɉɬɜɟɬ: W2 |
¨ |
¸¨ |
¸ |
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W1¨ c V |
¸¨ c V |
¸ . |
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© |
1 |
¹© |
2 |
¹ |
Ɂɚɞɚɱɚ 7
Ⱦɜɚ ɫɨɛɵɬɢɹ ɫɨɜɟɪɲɚɸɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ l = 6 105 ɤɦ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɫ ɩɪɨɦɟɠɭɬɤɨɦ ɜɪɟɦɟɧɢ W = 1 ɫ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ
182 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɋ ɤɚɤɨɣ ɫɤɨɪɨɫɬɶɸ V ɞɨɥɠɟɧ ɥɟɬɟɬɶ ɤɨɫɦɢɱɟɫɤɢɣ ɤɨɪɚɛɥɶ, ɱɬɨɛɵ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɤɨɪɚɛɥɟɦ, ɷɬɢ ɫɨɛɵɬɢɹ ɫɬɚɥɢ ɨɞɧɨɜɪɟɦɟɧɧɵɦɢ?
Ɉɬɜɟɬ: V |
c2W |
1,5 108 ɦ/ɫ . |
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l |
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Ɂɚɞɚɱɚ 8
ȼ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ Sƍ, ɞɜɢɠɭɳɟɣɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ c / 2 ɜɞɨɥɶ ɨɫɢ X ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S, ɞɜɢɠɟɬɫɹ ɬɟɥɨ ɧɟɛɨɥɶɲɢɯ
ɪɚɡɦɟɪɨɜ ɫɨ ɫɤɨɪɨɫɬɶɸ c / 2 ɩɨɞ ɭɝɥɨɦ Dƍ ɤ ɨɫɢ X'. ɇɚɣɬɢ ɭɝɨɥ D, ɤɨɬɨɪɵɣ ɫɨɫɬɚɜɥɹɟɬ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɫ ɨɫɶɸ X ɜ ɫɢɫɬɟɦɟ S.
3 sin Dc
2 cosDc 2
Ɂɚɞɚɱɚ 9
Ⱦɜɟ ɱɚɫɬɢɰɵ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɫɤɨɪɨɫɬɹɦɢ V ɞɜɢɠɭɬɫɹ ɜɞɨɥɶ ɨɞɧɨɣ ɩɪɹɦɨɣ ɢ ɩɨɩɚɞɚɸɬ ɜ ɦɢɲɟɧɶ ɫ ɢɧɬɟɪɜɚɥɨɦ ɜɪɟɦɟɧɢ W ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɦɢɲɟɧɶɸ. ɇɚɣɬɢ ɪɚɫɫɬɨɹɧɢɟ l ɦɟɠɞɭ ɥɟɬɹɳɢɦɢ ɱɚɫɬɢɰɚɦɢ ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɱɚɫɬɢɰɚɦɢ.
Ɉɬɜɟɬ: l |
VW |
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1 V / c 2 |
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Ɂɚɞɚɱɚ 10
Ʉɨɫɦɢɱɟɫɤɢɣ ɤɨɪɚɛɥɶ ɭɞɚɥɹɟɬɫɹ ɨɬ Ɂɟɦɥɢ, ɞɜɢɝɚɹɫɶ ɫɧɚɱɚɥɚ ɫɨ ɫɤɨɪɨɫɬɶɸ X1 , ɩɨɬɨɦ ɫɨ ɫɤɨɪɨɫɬɶɸ X2 . ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɤɨɫɦɨɧɚɜɬɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɧɚ ɤɨɫɦɢɱɟɫɤɨɦ ɤɨɪɚɛɥɟ, ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɫɨ ɫɤɨɪɨɫɬɹɦɢ X1 ɢ X2 ɨɞɢɧɚɤɨɜɨ. Ʉɚɤɨɟ ɪɚɫɫɬɨɹɧɢɟ L ɩɪɨɥɟɬɢɬ ɤɨ-
ɪɚɛɥɶ, ɟɫɥɢ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɤɨɪɚɛɥɹ ɩɨ Ɂɟɦɧɵɦ ɱɚɫɚɦ ɪɚɜɧɨ Ɍ? (ɉɪɟɧɟɛɪɟɱɶ ɜɪɟɦɟɧɟɦ, ɡɚɬɪɚɱɟɧɧɵɦ ɧɚ ɢɡɦɟɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɤɨɪɚɛɥɹ.)
Ɉɬɜɟɬ: |
L |
X1J1 X2J 2 T , |
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J1 J 2 |
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ɝɞɟ J1 |
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1 |
ɢ J 2 |
1 |
. |
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X1 / c 2 |
1 X2 / c 2 |
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1 |
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Ƚɥɚɜɚ 5. Ʉɢɧɟɦɚɬɢɤɚ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ |
183 |
Ɂɚɞɚɱɚ 11
ɇɚ ɥɟɝɤɭɸ ɧɟɩɨɞɜɢɠɧɭɸ ɱɚɫɬɢɰɭ ɧɚɥɟɬɚɟɬ ɬɹɠɟɥɚɹ ɩɥɢɬɚ. Ɉɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ V, ɩɪɢɨɛɪɟɬɟɧɧɭɸ ɱɚɫɬɢɰɟɣ ɩɨɫɥɟ ɭɩɪɭɝɨɝɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɫ ɩɥɢɬɨɣ, ɞɜɢɠɭɳɟɣɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɦ ɫɜɨɟɣ ɩɥɨɫɤɨɫɬɢ ɫɨ ɫɤɨɪɨɫɬɶɸ X c
3 .
Ɉɬɜɟɬ: V |
2X |
0,6c . |
1 X / c 2 |
184 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȽɅȺȼȺ 6 ɄɂɇȿɆȺɌɂɄȺ ɂ ȾɂɇȺɆɂɄȺ ȺȻɋɈɅɘɌɇɈ
ɌȼȿɊȾɈȽɈ ɌȿɅȺ
6.1.Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ
6.1.1.Ʉɢɧɟɦɚɬɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
Ⱥɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ – ɬɟɥɨ (ɫɢɫɬɟɦɚ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ), ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɞɜɭɦɹ ɥɸɛɵɦɢ ɦɚɬɟɪɢɚɥɶɧɵɦɢ ɬɨɱɤɚɦɢ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɵɦɢ ɜ ɭɫɥɨɜɢɹɯ ɞɚɧɧɨɣ ɡɚɞɚɱɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S. Ⱦɥɹ ɷɬɨɝɨ ɠɟɫɬɤɨ ɫɜɹɠɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ S' ɫ ɷɬɢɦ ɬɟɥɨɦ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɧɬɟɪɟɫɭɸɳɟɟ ɧɚɫ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ ɜɵɫɬɭɩɚɟɬ ɜ ɤɚɱɟɫɬɜɟ ɬɟɥɚ ɨɬɫɱɟɬɚ ɫɢɫɬɟɦɵ
S'.
Ɂɚɩɢɲɟɦ ɮɨɪɦɭɥɵ, ɫɜɹɡɵɜɚɸɳɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɧɟɤɨɬɨɪɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɜɭɯ ɩɪɨɢɡɜɨɥɶɧɨ ɞɜɢɠɭɳɢɯɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ S ɢ S' (ɫɦ. Ƚɥɚɜɭ 4):
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r |
R r' , |
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(6.1) |
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ȣ |
V >Ȧr'@ ȣ' , |
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(6.2) |
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a |
A >Ȧr'@ >Ȧ>Ȧr'@@ 2>Ȧȣ'@ a' . |
(6.3) |
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Ɂɞɟɫɶ |
r(t) , r'(t) ɪɚɞɢɭɫ-ɜɟɤɬɨɪɵ, |
c |
ɫɤɨɪɨɫɬɢ ɢ a(t) , |
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ȣ(t) , ȣ (t) |
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c |
ɭɫɤɨɪɟɧɢɹ ɧɟɤɨɬɨɪɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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a (t) |
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ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ S ɢ S' ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ; R(t) , V (t) ɢ A(t) – ɪɚɞɢɭɫ-
ɜɟɤɬɨɪ, ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɧɚɱɚɥɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S', ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɢ ɧɟ ɫɨɜɩɚɞɚɬɶ ɫ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ; Ȧ(t) ɢ Ȧ(t) – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɢ ɭɝɥɨɜɨɟ
ɭɫɤɨɪɟɧɢɟ ɫɢɫɬɟɦɵ S' (ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ) ɜɨɤɪɭɝ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɚɱɚɥɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S' (ɪɢɫ. 6.1).
ȿɫɥɢ ɧɟɤɨɬɨɪɚɹ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ M (ɫɦ. ɪɢɫ. 6.1) ɩɪɢɧɚɞɥɟɠɢɬ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɦɭ ɬɟɥɭ (ɬɟɥɭ ɨɬɫɱɟɬɚ ɫɢɫɬɟɦɵ S'), ɬɨ, ɩɨɫɤɨɥɶɤɭ ȣc(t) ɢ ac(t) , ɞɥɹ ɷɬɨɣ ɬɨɱɤɢ:
r |
R r' , |
(6.4) |
ȣ |
V >Ȧr'@, |
(6.5) |
a A >Ȧr'@ >Ȧ>Ȧr'@@. |
(6.6) |
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
185 |
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Ȧ |
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S |
S' |
M |
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r |
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r' |
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R
O'
O
Ɋɢɫ. 6.1. Ɋɚɞɢɭɫ-ɜɟɤɬɨɪɵ ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ M ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɢ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S', ɫɜɹɡɚɧɧɨɣ ɫ ɬɟɥɨɦ
ɂɡ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɫɤɨɪɨɫɬɢ ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (6.5) ɫɥɟɞɭɟɬ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ (ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ) ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ.
ɉɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ – ɥɸɛɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɷɬɨɝɨ ɬɟɥɚ) ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɭɩɟɪɩɨɡɢɰɢɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɫɭɳɟɫɬɜɥɹɟɦɵɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɷɬɨɝɨ ɬɟɥɚ (ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɩɟɪɟɧɨɫɚ ɫɨ ɫɤɨɪɨɫɬɶɸ V ) ɢ ɩɨɜɨɪɨɬɚ ɜɨɤɪɭɝ ɨɫɢ ɜɪɚɳɟɧɢɹ (ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ ).
ɉɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ – ɞɜɢɠɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɹɦɚɹ, ɫɨɟɞɢɧɹɸɳɚɹ ɥɸɛɵɟ ɞɜɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɬɟɥɚ, ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɫɚɦɨɣ ɫɟɛɟ. Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɞɨɫɬɚɬɨɱɧɨ ɨɩɢɫɚɬɶ ɞɜɢɠɟɧɢɟ ɥɸɛɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɷɬɨɝɨ ɬɟɥɚ.
ɉɪɨɢɡɜɨɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ɢ ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɧɟɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S') ɨɞɧɨɡɧɚɱɧɨ ɡɚɞɚɟɬɫɹ ɡɚɤɨɧɨɦ ɞɜɢɠɟɧɢɹ ɥɸɛɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɬɟɥɚ (ɧɚɱɚɥɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S' ɫɨɜɩɚɞɚɟɬ ɫ ɷɬɨɣ ɬɨɱɤɨɣ ɬɟɥɚ) R(t) ɢ ɡɚɤɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɭɝɥɨɜɨɣ
ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɬɟɥɚ (ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ S') ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɬɨɱɤɢ Ȧ(t) .
ɑɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ – ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ, ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɨɥɨɠɟɧɢɟ ɬɟɥ ɫɢɫɬɟɦɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ.
186 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɍ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɲɟɫɬɶ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ. ɇɚɩɪɢɦɟɪ, ɬɪɢ ɤɨɨɪɞɢɧɚɬɵ ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɬɟɥɚ, ɞɜɚ ɭɝɥɚ, ɡɚɞɚɸɳɢɯ ɧɚɩɪɚɜɥɟɧɢɟ ɩɪɹɦɨɣ, ɫɨɟɞɢɧɹɸɳɟɣ ɞɜɟ ɬɨɱɤɢ ɢ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɬɟɥɚ ɜɨɤɪɭɝ ɷɬɨɣ ɩɪɹɦɨɣ.
ɉɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
ɉɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ – ɞɜɢɠɟɧɢɟ ɬɟɥɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɬɪɚɟɤɬɨɪɢɢ ɜɫɟɯ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɬɟɥɚ ɥɟɠɚɬ ɜ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɥɨɫɤɨɫɬɹɯ. ȼ ɫɥɭɱɚɟ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ ɢɦɟɟɬ ɬɪɢ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ.
ȼɪɚɳɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ – ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɬɟɥɚ ɞɜɢɝɚɸɬɫɹ ɩɨ ɨɤɪɭɠɧɨɫɬɹɦ ɫ ɰɟɧɬɪɚɦɢ, ɥɟɠɚɳɢɦɢ ɧɚ ɷɬɨɣ ɨɫɢ, ɧɚɡɵɜɚɟɦɨɣ ɨɫɶɸ ɜɪɚɳɟɧɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ ɨɛɥɚɞɚɟɬ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ.
ɉɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ ɫɤɨɪɨɫɬɶ ȣ(t) ɢ ɭɫɤɨɪɟɧɢɟ a(t) ɦɚɬɟ-
ɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɥɟɠɚɬ ɜɫɟ ɜɪɟɦɹ ɜ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ P ɷɬɨɣ ɬɨɱɤɢ (ɫɦ. ɪɢɫ. 6.2).
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P
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O'
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Ɋɢɫ. 6.2. Ʉɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ Ɇ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɪɢ ɟɝɨ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ n const |
– ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɧɨɪɦɚɥɢ |
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ɤ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ, ɬɨɝɞɚ |
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nȣ |
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dr · |
w nȣ |
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¸ 0 , |
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(6.7) |
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© |
dt ¹ |
wt |
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
187 |
ɍɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ Ȧ ɢ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ Ȧ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɫɥɭɱɚɟ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɜɫɟɝɞɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɭɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɜɡɚɢɦɨɫɜɹɡɢ ɫɤɨɪɨɫɬɟɣ (6.5) ɫɤɚɥɹɪɧɨ ɧɚ ɧɨɪɦɚɥɶ ɤ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ. Ɍɚɤ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜ ɩɪɢ ɥɸɛɵɯ V (t) ɢ r'(t) , ɬɨ:
nȣ nV n>Ȧr'@ |
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n AV , n || Ȧ ɢ Ȧ AV . |
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ɍɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɜɡɚɢɦɨɫɜɹɡɢ ɭɫɤɨɪɟɧɢɣ (6.6) ɫɤɚɥɹɪɧɨ ɧɚ ɧɨɪɦɚɥɶ ɤ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ. Ɍɚɤ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜ ɩɪɢ ɥɸɛɵɯ A(t) ɢ r'(t) , ɬɨ:
na nA n>Ȧr'@ n>Ȧ>Ȧr'@@ 0 , |
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(6.9) |
n A A ɢ n || Ȧ . |
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ɉɨɤɚɠɟɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɧɚɣɞɟɬɫɹ ɬɚɤɨɣ ɜɟɤɬɨɪ r0c , ɱɬɨ
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɟɨɛɪɚɡɭɹ (6.10), ɩɨɥɭɱɢɦ:
V >Ȧ, r' r0c@ 0 , d { r' r0c , >n,V >Ȧd @@ 0 .
ɉɪɢ ɷɬɨɦ ɛɭɞɟɦ ɢɫɤɚɬɶ ɬɚɤɨɣ ɜɟɤɬɨɪ d , ɱɬɨ d A n , ɬɨɝɞɚ:
>nV @ >n>Ȧd @@ >nV @ Ȧ nd d nȦ >nV @ d nȦ
(6.10)
0 ,
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>nV @ |
>nV @ |
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(6.11) |
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ɂ, ɧɚɤɨɧɟɰ, ɧɚɣɞɟɦ ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɧɨɜɨɝɨ ɧɚɱɚɥɚ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ R0 , ɞɥɹ ɤɨɬɨɪɨɣ ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɫɨɨɬɧɨɲɟɧɢɟ (6.10):
r R r' R0 r0c , |
>nV @. |
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R0 R r' r0c R d R |
(6.12) |
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ɂɬɚɤ, ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɬɟɱɟɧɢɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ, ɤɚɤ "ɱɢɫɬɵɣ" ɩɨɜɨɪɨɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɨɫɢ – ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ.
188 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɇɚ ɪɢɫ. 6.3 ɢɡɨɛɪɚɠɟɧɚ ɜɟɤɬɨɪɧɚɹ ɞɢɚɝɪɚɦɦɚ ɫɤɨɪɨɫɬɟɣ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɬɟɥɚ, ɥɟɠɚɳɢɯ ɜ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɦɝɧɨɜɟɧɧɭɸ ɨɫɶ ɜɪɚɳɟɧɢɹ.
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Ɋɢɫ. 6.3. ȼɟɤɬɨɪɧɚɹ ɞɢɚɝɪɚɦɦɚ ɫɤɨɪɨɫɬɟɣ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɪɢ ɟɝɨ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ
Ʉɚɤ ɜɢɞɧɨ ɧɚ ɪɢɫ. 6.4, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɜɵɩɨɥɧɹɟɬɫɹ ɫɨɨɬɧɨɲɟ-
ɧɢɟ:
Z |
X1 |
X2 |
Xi |
, |
(6.13) |
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R |
R |
R |
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ɝɞɟ Xi – ɫɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɚ
Ri – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɷɬɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɞɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ.
Ɇɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ – ɨɫɶ ɜɪɚɳɟɧɢɹ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɜ ɬɟɱɟɧɢɟ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɤɚɤ "ɱɢɫɬɵɣ" ɩɨɜɨɪɨɬ, ɬ.ɟ. ɩɪɟɞɫɬɚɜɢɬɶ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɬɟɥɚ ɜ ɜɢɞɟ (6.10).
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɨɥɨɠɟɧɢɟ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɢɡɦɟɧɹɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ – ɪɚɞɢɭɫ-ɜɟɤɬɨɪ R0 , ɡɚɞɚɸɳɢɣ ɩɨɥɨɠɟɧɢɟ ɦɝɧɨɜɟɧɧɨɣ
ɨɫɢ, ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ (ɫɦ. (6.12)).
Ɇɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɜɫɟɝɞɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɧɟɩɨɞɜɢɠɧɭɸ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɢɥɢ ɬɨɱɤɭ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɤɨɬɨɪɚɹ ɠɟɫɬɤɨ ɫɜɹɡɚɧɚ ɫ ɷɬɢɦ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ.
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
189 |
Ⱦɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɜ ɬɨɱɤɟ
ȿɫɥɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ (ɬɟɥɨ ɨɬɫɱɟɬɚ ɫɢɫɬɟɦɵ S') ɡɚɤɪɟɩɥɟɧɨ ɜ ɬɨɱɤɟ, ɩɨɤɨɹɳɟɣɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S, ɬɨ, ɫɨɜɦɟɫ-
ɬɢɜ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɨɛɟɢɯ ɫɢɫɬɟɦ ( R 0 , V |
0 , A 0 ), ɩɨɥɭɱɢɦ: |
|
r |
R r' , |
(6.14) |
ȣ |
>Ȧr'@, |
(6.15) |
a |
>Ȧr'@ >Ȧ>Ȧr'@@. |
(6.16) |
|
|
|
ɂɡ ɭɪɚɜɧɟɧɢɹ (6.15) ɞɥɹ ɫɤɨɪɨɫɬɢ ɩɪɨɢɡɜɨɥɶɧɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɜ ɬɨɱɤɟ, ɫɥɟɞɭɟɬ ɬɟɨɪɟɦɚ ɗɣɥɟɪɚ.
Ɍɟɨɪɟɦɚ ɗɣɥɟɪɚ – ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ, ɡɚɤɪɟɩɥɟɧɧɨɟ ɜ ɬɨɱɤɟ, ɦɨɠɟɬ ɛɵɬɶ ɩɟɪɟɜɟɞɟɧɨ ɢɡ ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɥɸɛɨɟ ɞɪɭɝɨɟ ɨɞɧɢɦ ɩɨɜɨɪɨɬɨɦ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɡɚɤɪɟɩɥɟɧɢɹ. ɉɪɢɱɟɦ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɤɚɤ ɞɥɹ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ, ɬɚɤ ɢ ɞɥɹ ɤɨɧɟɱɧɨɝɨ ɩɨɜɨɪɨɬɚ. Ɉɞɧɚɤɨ ɪɟɡɭɥɶɬɚɬ ɞɜɭɯ ɤɨɧɟɱɧɵɯ ɩɨɜɨɪɨɬɨɜ ɡɚɜɢɫɢɬ ɨɬ ɢɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɞɜɭɯ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɯ ɩɨɜɨɪɨɬɨɜ.
ȿɫɥɢ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ S1 ɫ ɨɛɳɢɦ ɧɚɱɚɥɨɦ ɫ ɫɢɫɬɟɦɨɣ S ɜɪɚɳɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɟ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ ȍ ɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ, ɡɚɤɪɟɩɥɟɧɧɨɟ ɜ ɧɚɱɚɥɟ ɨɬɫɱɟɬɚ ɷɬɢɯ ɫɢɫɬɟɦ, ɜɪɚɳɚɟɬɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ȧ1 ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S1, ɬɨ ɜ ɫɨɨɬɜɟɬ-
ɫɬɜɢɢ ɫ (6.5): |
>ȍr'@ >Ȧ r'@ |
>ȍ Ȧ , r'@ >Ȧr'@, |
|
|
ȣ |
V >ȍr'@ ȣ' |
(6.17) |
||
|
, |
1 |
1 |
|
|
0 |
|
|
|
Ȧ |
ȍ Ȧ1 . |
|
|
(6.18) |
Ɂɞɟɫɶ ȣ ɫɤɨɪɨɫɬɶ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ S, Ȧ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɣ ɠɟ ɫɢɫɬɟɦɵ.
Ʉɚɤ ɜɢɞɢɦ, ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ Ȧ ɜɪɚɳɟɧɢɹ ɞɚɧɧɨɝɨ ɬɟɥɚ, ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɜ ɬɨɱɤɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɜɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɛɭɞɟɬ ɪɚɜɧɚ ɫɭɦɦɟ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɜɬɨɪɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ Ȧ1 ɢ ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɫɢɫɬɟɦɵ Ȧ2 .
Ʉɚɤ ɢ ɜ ɫɥɭɱɚɟ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ, ɞɜɢɠɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɡɚɤɪɟɩɥɟɧɧɨɝɨ ɜ ɬɨɱɤɟ, ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ "ɱɢɫɬɵɣ" ɩɨɜɨɪɨɬ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ.
190 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
6.1.2. Ⱦɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ L ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ – ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɞɢɭɫ-
ɜɟɤɬɨɪɚ r |
ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɧɧɨɣ ɬɨɱɤɢ ɩɪɨ- |
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ɫɬɪɚɧɫɬɜɚ ɧɚ ɟɟ ɢɦɩɭɥɶɫ p ɜ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ: |
||
L >rp@. |
(6.19) |
|
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ L ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶ- |
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ɧɨ ɬɨɱɤɢ – ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɢɦɩɭɥɶɫɨɜ Li |
ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ, |
|
ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ: |
|
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L |
¦Li . |
(6.20) |
i
ɇɚɣɞɟɦ ɫɜɹɡɶ ɦɟɠɞɭ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ L ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɧɟɩɨɞɜɢɠɧɨɣ ɬɨɱɤɢ Ɉ ɢ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ ɷɬɨɣ ɫɢɫɬɟɦɵ L0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɟ ɰɟɧɬɪɚ ɦɚɫɫ O'
ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ (ɫɦ. ɪɢɫ. 6.4).
mi
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ri |
ri ' |
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rɰɦ |
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S' |
S |
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O' |
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||
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O |
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Ɋɢɫ. 6.4. ɋɜɹɡɶ ɦɟɠɞɭ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɚɦɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ S ɢ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɦɚɫɫ S'
Ɋɚɞɢɭɫ-ɜɟɤɬɨɪ i-ɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ri ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱ-
ɤɢ Ɉ ɪɚɜɟɧ: |
|
|
ri |
rɰɦ ric , |
(6.21) |
ɝɞɟ rɰɦ |
– ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɰɟɧɬɪɚ ɦɚɫɫ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ, ric – |
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ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɟ ɰɟɧɬɪɚ ɦɚɫɫ.
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ Ɉ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.19) ɢ (6.20) ɪɚɜɟɧ: