Механика.Методика решения задач
.pdfȽɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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Ɂɚɞɚɱɚ 9.4 |
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Ⱥɦɩɥɢɬɭɞɚ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɞɚɜɥɟɧɢɣ ǻP0 10 ɉɚ. |
ɇɚɣɬɢ |
ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ J, ɩɨɩɚɞɚɸɳɟɝɨ ɜ ɭɯɨ ɱɟɥɨɜɟɤɚ. ɋɱɢɬɚɬɶ ɩɥɨɳɚɞɶ ɭɯɚ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ, s = 4 ɫɦ2. ɉɥɨɬɧɨɫɬɶ ɜɨɡɞɭɯɚ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ U = 1,3 ɤɝ/ɦ3, ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɜ ɜɨɡɞɭɯɟ c = 334 ɦ/ɫ.
Ɋɟɲɟɧɢɟ
I. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ. ȼ ɤɚɱɟɫɬɜɟ ɦɨɞɟɥɢ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ ɜɵɛɟɪɟɦ ɩɥɨɫɤɭɸ ɝɚɪɦɨɧɢɱɟɫɤɭɸ ɜɨɥɧɭ, ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɭɸɫɹ ɜɞɨɥɶ ɨɫɢ X ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɉɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɫɱɢɬɚɟɦ ɚɞɢɚɛɚɬɢɱɟɫɤɢɦ.
II. ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ J, ɩɚɞɚɸɳɟɝɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɭɯɚ ɩɥɨɳɚɞɶɸ s, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ
(9.48) ɪɚɜɧɨ:
J s w(t, x) T c . |
(9.98) |
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Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ (9.45), ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɩɥɨɫɤɨɣ |
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ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ ɪɚɜɧɚ: |
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w(t, x) T |
[ 2 UZ2 |
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0 |
(9.99) |
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2 . |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ |
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ɧɟɨɛɯɨɞɢɦɨ ɜɵɪɚɡɢɬɶ ɚɦɩɥɢɬɭɞɭ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ [0 |
ɱɟɪɟɡ ɚɦɩɥɢ- |
ɬɭɞɭ ɜɨɥɧɵ ɞɚɜɥɟɧɢɣ ǻP0 , ɡɚɞɚɧɧɭɸ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ.
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ (9.8): |
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[(t, x) [0 cos Zt kx M0 . |
(9.100) |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.34) ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɢɡɦɟɧɟɧɢɟ ɩɥɨɬɧɨɫɬɢ |
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ɜɨɡɞɭɯɚ, ɜɵɡɜɚɧɧɨɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟɦ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ, ɪɚɜɧɨ |
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ǻU |
[xc , |
(9.101) |
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U |
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ɝɞɟ [xc ɧɚɯɨɞɢɦ, ɢɫɩɨɥɶɡɭɹ (9.100): |
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[xc |
[0k sin(Zt kx M0 ) . |
(9.102) |
ɉɪɢ ɚɞɢɚɛɚɬɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɜ ɝɚɡɟ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɢɡɦɟɧɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɫɪɟɞɵ ɢ ɞɚɜɥɟɧɢɹ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ (ɫɦ. (9.33) ɢ (9.36)):
352 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ǻP |
J |
ǻU |
, |
(9.103) |
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P |
U |
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ɝɞɟ P ɢ U – ɞɚɜɥɟɧɢɟ ɢ ɩɥɨɬɧɨɫɬɶ ɜɨɡɞɭɯɚ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɡɜɭɤɨɜɨɝɨ ɞɚɜɥɟɧɢɹ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɚɤɠɟ ɮɨɪɦɭɥɨɣ (9.31) ɞɥɹ ɫɤɨɪɨɫɬɢ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɜ ɜɨɡɞɭ-
ɯɟ: |
P |
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c2 J |
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(9.104) |
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U |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (9.101) – (9.104), |
ɧɚɯɨɞɢɦ ɢɡ- |
ɦɟɧɟɧɢɟ ɞɚɜɥɟɧɢɹ ɜɨɡɞɭɯɚ, ɜɵɡɜɚɧɧɨɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟɦ ɜ ɧɟɦ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ:
ǻP |
c2 U[0k sin(Zt kx M0 ) . |
(9.105) |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɚɦɩɥɢɬɭɞɚ ɡɜɭɤɨɜɨɝɨ ɞɚɜɥɟɧɢɹ ɪɚɜɧɚ: |
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ǻP |
c2 U[ |
k |
cU[ Z . |
(9.106) |
0 |
0 |
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0 |
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ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ J, ɩɚɞɚɸɳɟɝɨ ɩɟɪɩɟɧɞɢɤɭ- |
ɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ ɭɯɚ ɩɥɨɳɚɞɶɸ s, ɩɨɥɭɱɚɟɦ ɩɨɞɫɬɚɧɨɜɤɨɣ ɜ (9.98) ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜɨɥɧɵ (9.99) ɫ ɭɱɟɬɨɦ (9.106):
J s w(t, x) |
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c |
s |
[ 2 UZ2 |
c s |
ǻP2 UZ2 |
c s |
ǻP2 |
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0 |
0 |
0 |
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(9.107) |
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2 cUZ 2 |
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T |
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2 |
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2cU |
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ɉɨɞɫɬɚɜɢɜ ɜ (9.107) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:
J = 4.6 10–5 ȼɬ.
Ɂɚɞɚɱɚ 9.5
Ɍɨɱɟɱɧɵɣ ɢɡɨɬɪɨɩɧɨ ɢɡɥɭɱɚɸɳɢɣ ɢɫɬɨɱɧɢɤ ɡɜɭɤɚ S ɧɚɯɨɞɢɬɫɹ ɧɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟ ɤ ɩɥɨɫɤɨɫɬɢ ɤɨɥɶɰɚ, ɩɪɨɯɨɞɹɳɟɦ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ P (ɫɦ. ɪɢɫ. 9.9).
U+dU |
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D U |
S |
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R P |
L |
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Ɋɢɫ. 9.9 |
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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Ɋɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɨɣ P ɢ ɢɫɬɨɱɧɢɤɨɦ S ɪɚɜɧɨ L = 1 ɦ, ɪɚɞɢɭɫ ɤɨɥɶɰɚ – R = 0,5 ɦ. ɇɚɣɬɢ ɫɪɟɞɧɢɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ ɱɟɪɟɡ ɩɥɨɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɤɨɥɶɰɨɦ, ɟɫɥɢ ɜ ɬɨɱɤɟ P ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ I0 = 30 ɦɤȼɬ/ɦ2. Ɂɚɬɭɯɚɧɢɟɦ ɜɨɥɧ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ
I. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ P, ɹɜɥɹɸɳɟɣɫɹ ɰɟɧɬɪɨɦ ɤɨɥɶɰɚ.
ɉɨɫɤɨɥɶɤɭ ɢɫɬɨɱɧɢɤ ɡɜɭɤɚ S ɹɜɥɹɟɬɫɹ ɬɨɱɟɱɧɵɦ ɢ ɢɡɨɬɪɨɩɧɨ ɢɡɥɭɱɚɸɳɢɦ, ɬɨ ɨɧ ɜɨɡɛɭɠɞɚɟɬ ɫɮɟɪɢɱɟɫɤɭɸ ɜɨɥɧɭ, ɚɦɩɥɢɬɭɞɚ ɤɨɬɨɪɨɣ ɢɡɦɟɧɹɟɬɫɹ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɪɚɫɫɬɨɹɧɢɸ ɨɬ ɢɫɬɨɱɧɢɤɚ (ɫɦ. (9.13)), ɚ ɡɧɚɱɢɬ, ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɞɨ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ.
II. Ɋɚɡɨɛɶɟɦ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚ ɤɨɧɰɟɧɬɪɢɱɟɫɤɢɟ ɤɨɥɶɰɟɜɵɟ ɡɨɧɵ, ɡɚɤɥɸɱɟɧɧɵɟ ɦɟɠɞɭ ɨɤɪɭɠɧɨɫɬɹɦɢ ɫ ɪɚɞɢɭɫɚɦɢ U ɢ U+dU (0 d U d R) ɫ ɰɟɧɬɪɚɦɢ ɜ ɬɨɱɤɟ P.
ɉɨɫɤɨɥɶɤɭ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɞɨ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ, ɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɥɧɵ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɤɨɥɶɰɟɜɭɸ ɡɨɧɭ ɪɚɞɢɭɫɨɦ U (ɫɦ. ɪɢɫ. 9.9), ɪɚɜɧɚ:
I U |
L2 |
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I0 U2 L2 . |
(9.108) |
ɉɨɬɨɤ ɷɧɟɪɝɢɢ dJ ɱɟɪɟɡ ɜɵɞɟɥɟɧɧɭɸ ɮɢɡɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨ ɬɨɧɤɭɸ ɤɨɥɶɰɟɜɭɸ ɡɨɧɭ (ɫɦ. ɪɢɫ. 9.9) ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɜɨɥɧɵ (ɫɦ. ɩ. 9.1.5), ɪɚɜɟɧ:
dJ I (U) cosD(U)2SUdU I (U) |
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L |
2SUdU . |
(9.109) |
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U |
2 |
2 |
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L |
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III. ɂɫɤɨɦɵɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ J ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɤɨɥɶɰɨɦ ɪɚɞɢɭɫɨɦ R, ɨɩɪɟɞɟɥɢɦ, ɢɧɬɟɝɪɢɪɭɹ (9.109) ɫ ɭɱɟ-
ɬɨɦ (9.108):
R |
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L |
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R |
SL3 |
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d U |
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J ³I (U) |
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2SUdU |
³I0 |
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2 |
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U |
2 |
2 |
U2 L2 |
3 |
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0 |
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L |
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0 |
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2 |
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354 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
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· |
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¨ |
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1 |
¸ |
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2I SL2 |
¨1 |
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¸ . |
(9.110) |
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0 |
¨ |
R |
2 |
¸ |
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¨ |
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1 ¸ |
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© |
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L |
¹ |
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ɉɨɞɫɬɚɜɢɜ ɜ (9.110) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɪɚɫɫɬɨɹɧɢɹ L ɦɟɠɞɭ ɬɨɱɤɨɣ P ɢ ɢɫɬɨɱɧɢɤɨɦ S, ɪɚɞɢɭɫɚ ɤɨɥɶɰɚ R ɢ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ I0 ɜ ɬɨɱɤɟ P, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:
J = 20 ɦɤȼɬ.
Ɂɚɞɚɱɚ 9.6
ɇɚ ɨɫɢ X ɧɚɯɨɞɹɬɫɹ ɩɪɢɟɦɧɢɤ D ɢ ɢɫɬɨɱɧɢɤ S ɡɜɭɤɨɜɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ ɫ ɱɚɫɬɨɬɨɣ Qs = 2000 Ƚɰ. ɂɫɬɨɱɧɢɤ ɭɫɬɚɧɨɜɥɟɧ ɧɚ ɬɟɥɟɠɤɟ, ɫɨɜɟɪɲɚɸɳɟɣ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɜɞɨɥɶ ɷɬɨɣ ɨɫɢ ɫ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɨɣ Z ɢ ɚɦɩɥɢɬɭɞɨɣ Ⱥ = 50 ɫɦ. ɋɤɨɪɨɫɬɶ ɡɜɭɤɚ ɫ =340 ɦ/ɫ. ɉɪɢ ɤɚɤɨɦ ɡɧɚɱɟɧɢɢ Z ɲɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɡɜɭɤɚ, ɜɨɫɩɪɢɧɢɦɚɟɦɨɝɨ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ, ɛɭɞɟɬ ɫɨɫɬɚɜɥɹɬɶ 'Q = 20 Ƚɰ?
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɚ ɨɬ ɢɫɬɨɱɧɢɤɚ ɤ ɩɪɢɟɦɧɢɤɭ (ɫɦ. ɪɢɫ. 6.1). ɉɨɫɤɨɥɶɤɭ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɧɟ ɨɝɨɜɚɪɢ-
ɜɚɟɬɫɹ ɢɧɨɟ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɪɟɞɚ, ɜ |
Xs |
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ɤɨɬɨɪɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ |
ɡɜɭɤɨɜɚɹ |
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ɜɨɥɧɚ, ɧɟɩɨɞɜɢɠɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨ- |
S |
c |
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ɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. |
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ɂɡɦɟɧɟɧɢɹ ɱɚɫɬɨɬɵ ɡɜɭɤɨɜɨɣ ɜɨɥ- |
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ɧɵ, ɜɨɫɩɪɢɧɢɦɚɟɦɨɣ ɧɟɩɨɞɜɢɠɧɵɦ |
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ɩɪɢɟɦɧɢɤɨɦ, ɨɛɭɫɥɨɜɥɟɧɨ |
ɷɮɮɟɤɬɨɦ |
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Ɋɢɫ. 9.10 |
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Ⱦɨɩɥɟɪɚ (ɫɦ. ɩ. 9.1.6).
II. ɒɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ 'Q ɡɜɭɤɨɜɵɯ ɜɨɥɧ, ɜɨɫɩɪɢɧɢɦɚɟɦɵɯ ɩɪɢɟɦɧɢɤɨɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɡɧɨɫɬɶɸ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɦɢɧɢɦɚɥɶɧɨɣ ɱɚɫɬɨɬ ɷɬɢɯ ɜɨɥɧ.
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɢɫɬɨɱɧɢɤ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɩɪɢɟɦɧɢɤɭ, ɦɚɤɫɢ-
ɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ Qmax |
ɪɟɝɢɫɬɪɢɪɭɟɦɨɣ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ |
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ɜɨɥɧɵ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.51), ɪɚɜɧɚ |
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Qmax |
cQs |
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(9.111) |
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c Xs0 |
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
355 |
ɝɞɟ Xs0 – ɚɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ
ɫɪɟɞɵ, c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɜ ɫɪɟɞɟ.
ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥɟɠɤɢ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɚɦɩɥɢɬɭɞɚ ɟɟ ɫɤɨɪɨɫɬɢ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɚɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɢɫɬɨɱɧɢɤɚ,
ɪɚɜɧɚ: |
AZ . |
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(9.112) |
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Xs0 |
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ȼ ɫɥɭɱɚɟ ɭɞɚɥɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɨɬ ɩɪɢɟɦɧɢɤɚ ɱɚɫɬɨɬɚ Qmin , |
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ɜɨɫɩɪɢɧɢɦɚɟɦɚɹ ɩɪɢɟɦɧɢɤɨɦ, ɛɭɞɟɬ ɦɢɧɢɦɚɥɶɧɨɣ ɢ ɪɚɜɧɨɣ |
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Qmin |
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cQs |
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(9.113) |
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c Xs0 |
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III. ɂɫɤɨɦɚɹ ɲɢɪɢɧɚ ɱɚɫɬɨɬɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɡɜɭɤɚ 'Q, ɜɨɫɩɪɢɧɢɦɚɟɦɨɝɨ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ, ɫɨɝɥɚɫɧɨ (9.111) ɢ (9.113) ɪɚɜɧɚ:
cXs0 |
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ǻQ Qmax Qmin 2Qs c2 Xs02 . |
(9.114) |
ɂɫɩɨɥɶɡɭɹ ɜɡɚɢɦɨɫɜɹɡɶ ɚɦɩɥɢɬɭɞɵ ɫɤɨɪɨɫɬɢ ɢɫɬɨɱɧɢɤɚ ɫ ɚɦɩɥɢɬɭɞɨɣ ɟɝɨ ɤɨɥɟɛɚɧɢɣ ɜɦɟɫɬɟ ɫ ɬɟɥɟɠɤɨɣ (9.112), ɩɨɥɭɱɚɟɦ:
ǻQ |
2Qs |
cAZ |
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(9.115) |
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c2 A2Z2 |
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Ɋɟɲɚɹ ɭɪɚɜɧɟɧɢɟ (9.115) ɨɬɧɨɫɢɬɟɥɶɧɨ Z , ɧɚɯɨɞɢɦ ɟɟ ɜɟɥɢɱɢɧɭ: |
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Z |
Qsc r c Qs2 ǻQ 2 |
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(9.116) |
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AǻQ |
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ɉɨɫɤɨɥɶɤɭ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɹɜɥɹɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɬɨ ɞɥɹ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɬɟɥɟɠɤɢ ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ:
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Qsc |
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§ |
ǻQ |
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Z |
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¸ |
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1¸ . |
(9.117) |
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Qs |
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AǻQ ¨ |
© |
¹ |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ǻQ Qs , ɫɥɟɞɨɜɚɬɟɥɶɧɨ (9.117) ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ, ɨɝɪɚɧɢɱɢɜɚɹɫɶ ɜ ɪɚɡɥɨɠɟɧɢɢ ɤɜɚɞɪɚɬɧɨɝɨ
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ǻQ |
·2 |
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ɤɨɪɧɹ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ ɦɚɥɨɝɨ ɩɚɪɚɦɟɬɪɚ |
¨ |
¸ |
ɥɢɧɟɣɧɵɦ ɱɥɟ- |
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¨ |
Qs |
¸ |
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¹ |
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ɧɨɦ ɪɹɞɚ: |
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356 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Z # |
cǻQ |
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(9.118) |
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ɉɨɞɫɬɚɧɨɜɤɚ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɯ ɜ ɡɚɞɚɱɟ, ɞɚɟɬ:
Z # 3,4 ɪɚɞ/ɫ .
Ɂɚɞɚɱɚ 9.7
ȼ ɭɩɪɭɝɨɣ ɨɞɧɨɪɨɞɧɨɣ ɫɪɟɞɟ ɫ ɩɥɨɬɧɨɫɬɶɸ U ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɞɜɟ ɩɥɨɫɤɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɩɪɨɞɨɥɶɧɵɟ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ ɫɨ ɫɤɨɪɨɫɬɶɸ c, ɨɞɢɧɚɤɨɜɵɦɢ ɚɦɩɥɢɬɭɞɚɦɢ a ɢ ɱɚɫɬɨɬɚɦɢ Z , ɨɞɧɚ – ɜɞɨɥɶ ɨɫɢ X, ɞɪɭɝɚɹ – ɜɞɨɥɶ ɨɫɢ Y ɧɟɤɨɬɨɪɨɣ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɬɢ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY, ɫɱɢɬɚɹ ɨɞɢɧɚɤɨɜɵɦɢ ɧɚɱɚɥɶɧɵɟ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɢɰ ɫɪɟɞɵ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ, ɨɛɭɫɥɨɜɥɟɧɧɵɯ ɤɚɠɞɨɣ ɜɨɥɧɨɣ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
Ɋɟɲɟɧɢɟ
I. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɚɞɚɧɚ ɞɟɤɚɪɬɨɜɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ, ɜɞɨɥɶ ɨɫɟɣ X ɢ Y ɤɨɬɨɪɨɣ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɞɜɟ ɩɥɨɫɤɢɟ ɩɪɨɞɨɥɶɧɵɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɜɨɥɧɵ.
II. Ɂɚɩɢɲɟɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɡɚɤɨɧɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɛɟɝɭɳɢɯ ɩɥɨɫɤɢɯ ɩɪɨɞɨɥɶɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ
ɫɦɟɳɟɧɢɣ (ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.8)): |
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ȟ1 t, x |
aexcos Z t kx M0 , |
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(9.119) |
ȟ2 t, y |
aeycos Z t ky M0 , |
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(9.120) |
ɝɞɟ ex ɢ e y ɟɞɢɧɢɱɧɵɟ ɜɟɤɬɨɪɵ ɜɞɨɥɶ ɨɫɟɣ X ɢ Y, k |
Z |
ɜɨɥ- |
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ɧɨɜɨɟ ɱɢɫɥɨ ɞɥɹ ɨɛɟɢɯ ɜɨɥɧ, M0 ɧɚɱɚɥɶɧɵɟ ɮɚɡɵ ɤɨɥɟɛɚɧɢɣ ɱɚɫ-
ɬɢɰ ɫɪɟɞɵ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ, ɨɛɭɫɥɨɜɥɟɧɧɵɯ ɤɚɠɞɨɣ ɜɨɥɧɨɣ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
Ɉɩɪɟɞɟɥɢɦ ɚɦɩɥɢɬɭɞɭ A ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ
ɫɦɟɳɟɧɢɣ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY: |
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ȟ t, x, y |
ȟ1 t, x ȟ2 t, y |
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aex cos Zt kx M0 aey cos Zt ky M0 |
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aex cos Zt kx M0 aey cos Zt kx M0 |
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a ex ey cos Zt kx M0 . |
(9.121) |
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
357 |
ɇɚɩɪɚɜɢɦ ɜɫɩɨɦɨɝɚɬɟɥɶɧɭɸ ɨɫɶ ī ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY ɢ ɨɛɨɡɧɚɱɢɦ ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɜɞɨɥɶ ɷɬɨɝɨ ɧɚɩɪɚɜ-
ɥɟɧɢɹ ɤɚɤ eJ |
. ɉɨɫɤɨɥɶɤɭ ex ey |
2eJ , ɬɨ |
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ȟ |
a |
2eJ |
cos Zt kx M0 |
AeJ cos Zt kx M0 , |
(9.122) |
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ɝɞɟ ɚɦɩɥɢɬɭɞɚ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ A ɜɞɨɥɶ ɨɫɢ ī ɪɚɜɧɚ: |
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2a . |
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(9.123) |
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ɉɨɞɫɬɚɜɥɹɹ ɜ ɜɵɪɚɠɟɧɢɟ (9.122) ɞɥɹ ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰ ɫɪɟɞɵ |
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ɜɞɨɥɶ ɨɫɢ ī |
ɫɨɨɬɧɨɲɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɜɞɨɥɶ ɨɫɟɣ X ɢ ī |
x |
J |
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ɩɨɥɭɱɚɟɦ: |
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ȟ |
Ae cos¨Z¨t |
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¸ M |
¸ . |
(9.124) |
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¨ |
© |
2c ¹ |
0 ¸ |
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¹ |
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Ʉɚɤ ɜɢɞɢɦ, |
ɜɨɥɧɨɜɨɟ ɩɨɥɟ ɜɞɨɥɶ ɨɫɢ ī ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢ- |
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ɪɨɜɚɬɶ ɤɚɤ ɛɟɝɭɳɭɸ ɩɪɨɞɨɥɶɧɭɸ ɜɨɥɧɭ ɫɦɟɳɟɧɢɣ ɫ ɚɦɩɥɢɬɭɞɨɣ A |
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(9.123) ɢ ɫɤɨɪɨɫɬɶɸ |
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cJ |
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2c . |
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(9.125) |
ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɜɨɥɧɨɜɨɝɨ ɜɨɡɦɭɳɟɧɢɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.48) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
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A2 UZ2 |
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I S (x,t) |
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cJ . |
(9.126) |
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III. ɉɨɞɫɬɚɜɢɜ ɜ (9.126) ɚɦɩɥɢɬɭɞɭ ɤɨɥɟɛɚɧɢɣ A (9.123) ɢ ɫɤɨ- |
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ɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ cJ |
(9.125), ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɫɪɟɞɧɟɟ |
ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɜɨɥɧɨɜɨɝɨ ɩɨɥɹ ɜɞɨɥɶ ɩɪɹɦɨɣ y = x ɜ ɩɥɨɫɤɨɫɬɢ XY:
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2a2UZ2c . |
(9.127) |
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Ɂɚɞɚɱɚ 9.8 |
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ɂɫɬɨɱɧɢɤ ɡɜɭɤɨɜɵɯ ɤɨɥɟɛɚɧɢɣ S ɫ ɱɚɫɬɨɬɨɣ Q 0 |
1700 Ƚɰ ɧɚ- |
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ɯɨɞɢɬɫɹ |
ɦɟɠɞɭ ɩɥɨɫɤɢɦ ɨɬɪɚɠɚɬɟɥɟɦ ɢ ɩɪɢɟɦɧɢɤɨɦ D (ɫɦ. |
ɪɢɫ. 9.11). ɂɫɬɨɱɧɢɤ ɢ ɩɪɢɟɦɧɢɤ ɧɟɩɨɞɜɢɠɧɵ ɢ ɪɚɫɩɨɥɨɠɟɧɵ ɧɚ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɧɨɪɦɚɥɢ ɤ ɨɬɪɚɠɚɬɟɥɸ, ɤɨɬɨɪɵɣ ɭɞɚɥɹɟɬɫɹ ɨɬ ɢɫɬɨɱɧɢɤɚ ɫɨ ɫɤɨɪɨɫɬɶɸ u = 6 ɫɦ/ɫ. ɋɤɨɪɨɫɬɶ ɡɜɭɤɚ ɫ = 340 ɦ/ɫ. ɇɚɣɬɢ ɱɚɫɬɨɬɭ ɛɢɟɧɢɣ, ɪɟɝɢɫɬɪɢɪɭɟɦɵɯ ɩɪɢɟɦɧɢɤɨɦ.
358 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
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X
Ɋɢɫ. 9.11
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɫɨɜɩɚɞɚɸɳɢɦ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɞɜɢɠɟɧɢɹ ɨɬɪɚɠɚɬɟɥɹ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 9.11. ɉɪɢɟɦɧɢɤ ɪɟɝɢɫɬɪɢɪɭɟɬ ɫɭɩɟɪɩɨɡɢɰɢɸ ɞɜɭɯ ɡɜɭɤɨɜɵɯ ɜɨɥɧ: ɢɫɩɭɳɟɧɧɨɣ ɢɫɬɨɱɧɢɤɨɦ ɢ ɨɬɪɚɠɟɧɧɨɣ ɨɬ ɞɜɢɠɭɳɟɝɨɫɹ ɨɬɪɚɠɚɬɟɥɹ.
II. ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ (9.51), ɫɜɹɡɵɜɚɸɳɟɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɞɜɢɠɭɳɢɯɫɹ ɢɫɬɨɱɧɢɤɚ ɢ ɩɪɢɟɦɧɢɤɚ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ:
Q |
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c Xd |
Q |
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, |
(9.128) |
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d c Xs |
s |
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ɝɞɟ Xs |
ɢ Xd – ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɢ ɩɪɢɟɦɧɢɤɚ ɨɬɧɨɫɢ- |
ɬɟɥɶɧɨ ɫɪɟɞɵ, c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɜ ɫɪɟɞɟ, Qs – ɱɚɫɬɨɬɚ ɢɡɥɭɱɚɟɦɨɣ ɢɫɬɨɱɧɢɤɨɦ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ, Qd – ɱɚɫɬɨɬɚ ɜɨɥ-
ɧɵ, ɤɨɬɨɪɭɸ ɪɟɝɢɫɬɪɢɪɭɟɬ ɞɟɬɟɤɬɨɪ. ȼɫɟ ɫɤɨɪɨɫɬɢ ɧɚɩɪɚɜɥɟɧɵ ɜ ɨɞɧɭ ɫɬɨɪɨɧɭ.
ɑɚɫɬɨɬɚ ɜɨɥɧɵ, ɤɨɬɨɪɭɸ ɡɚɮɢɤɫɢɪɨɜɚɥ ɛɵ ɞɟɬɟɤɬɨɪ, ɧɚɯɨɞɹɳɢɣɫɹ ɧɚ ɞɜɢɠɭɳɟɦɫɹ ɨɬɪɚɠɚɬɟɥɟ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
(9.128) ɜɵɪɚɠɟɧɢɟɦ: |
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Q1 |
Q0 |
c Xd |
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Q0 |
c u |
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(9.129) |
c |
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c |
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ɋɱɢɬɚɹ, ɱɬɨ ɨɬɪɚɠɚɬɟɥɶ ɧɟ ɦɟɧɹɟɬ ɱɚɫɬɨɬɭ ɜɨɥɧɵ ɩɪɢ ɨɬɪɚɠɟɧɢɢ, ɡɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɱɚɫɬɨɬɵ ɜɨɥɧɵ, ɨɬɪɚɠɟɧɧɨɣ ɨɬ ɨɬɪɚɠɚɬɟɥɹ ɢ ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧɧɨɣ ɧɟɩɨɞɜɢɠɧɵɦ ɩɪɢɟɦɧɢɤɨɦ ɫɨɝɥɚɫ-
ɧɨ (9.128):
Q |
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c u |
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(9.130) |
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1 c Xs |
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0 c u |
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Ɂɧɚɤ ɩɥɸɫ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɮɨɪɦɭɥɵ (9.130) ɨɛɭɫɥɨɜɥɟɧ ɬɟɦ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɤɨɪɨɫɬɶ ɨɬɪɚɠɚɬɟɥɹ (ɢɫɬɨɱɧɢɤɚ ɨɬɪɚɠɟɧɧɨɣ
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
359 |
ɜɨɥɧɵ) ɢ ɫɤɨɪɨɫɬɶ ɨɬɪɚɠɟɧɧɨɣ ɜɨɥɧɵ ɧɚɩɪɚɜɥɟɧɵ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɬɨɪɨɧɵ.
Ⱦɥɹ ɱɚɫɬɨɬɵ ɛɢɟɧɢɣ (ɫɦ. ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ 8.10 ɜ Ƚɥɚɜɟ 8), ɜɨɡɧɢɤɚɸɳɢɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɜɨɥɧɵ ɫ ɱɚɫɬɨɬɨɣ Q0 , ɢɫɩɭ-
ɳɟɧɧɨɣ ɧɟɩɨɞɜɢɠɧɵɦ ɢɫɬɨɱɧɢɤɨɦ, ɢ ɜɨɥɧɵ ɫ ɱɚɫɬɨɬɨɣ Q 2 , ɨɬɪɚ-
ɠɟɧɧɨɣ ɨɬ ɞɜɢɠɭɳɟɝɨɫɹ ɨɬɪɚɠɚɬɟɥɹ, ɡɚɩɢɲɟɦ: |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (9.130) ɢ (9.131), ɧɚɯɨɞɢɦ ɢɫ- |
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ɤɨɦɭɸ ɜɟɥɢɱɢɧɭ ɱɚɫɬɨɬɵ ɛɢɟɧɢɣ: |
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Q |
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c u |
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2u |
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(9.132) |
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ɛɢɟɧ |
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ɉɨɞɫɬɚɜɥɹɹ ɜ (9.132) ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɩɨɥɭɱɢɦ ɡɧɚɱɟɧɢɟ ɱɚɫɬɨɬɵ ɛɢɟɧɢɣ, ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧɧɵɯ ɩɪɢɟɦɧɢɤɨɦ:
Qɛɢɟɧ 0,6 Ƚɰ .
Ɂɚɞɚɱɚ 9.9
ɋɬɚɥɶɧɚɹ ɫɬɪɭɧɚ ɞɥɢɧɨɣ L = 110 ɫɦ, ɩɥɨɬɧɨɫɬɶɸ U = 7,8 ɝ/ɫɦ3 ɢ ɞɢɚɦɟɬɪɨɦ d = 1 ɦɦ ɧɚɬɹɧɭɬɚ ɦɟɠɞɭ ɩɨɥɸɫɚɦɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɚ. ɉɪɢ ɩɪɨɩɭɫɤɚɧɢɢ ɩɨ ɫɬɪɭɧɟ ɩɟɪɟɦɟɧɧɨɝɨ ɬɨɤɚ ɱɚɫɬɨɬɨɣ Q = 256 Ƚɰ ɜ ɧɟɣ ɜɨɡɛɭɠɞɚɟɬɫɹ ɭɩɪɭɝɚɹ ɩɨɩɟɪɟɱɧɚɹ ɜɨɥɧɚ, ɩɪɢɱɟɦ ɧɚ ɞɥɢɧɟ ɫɬɪɭɧɵ "ɭɤɥɚɞɵɜɚɟɬɫɹ" n = 5 ɩɨɥɭɜɨɥɧ. ɇɚɣɬɢ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ.
Ɋɟɲɟɧɢɟ
I. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɢɡɦɟɧɟɧɢɟ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ, ɜɵɡɜɚɧɧɨɟ ɭɩɪɭɝɨɣ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɨɣ, ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨ (ɫɦ. ɩ. 9.1.4.Ȼ). Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɢɦ ɜɞɨɥɶ ɫɬɪɭɧɵ (ɫɦ. ɪɢɫ. 9.12).
0 |
L X |
Ɋɢɫ. 9.12
360 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
II. Ɂɚɩɢɲɟɦ ɜɡɚɢɦɨɫɜɹɡɶ ɫɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɩɨɩɟɪɟɱɧɵɯ ɜɨɥɧ ɜ ɫɬɪɭɧɟ ɢ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ (ɫɦ. (9.27) ɜ
ɩ. 9.1.4.ȼ)
c |
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ɝɞɟ ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɪɭɧɵ S ɪɚɜɧɚ: |
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Sd 2 |
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(9.134) |
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Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɨɣ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɩɨɩɟɪɟɱɧɵɯ ɜɨɥɧ ɜ ɫɬɪɭɧɟ (ɫɦ. (9.133)).
ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɧɚ ɞɥɢɧɟ ɫɬɪɭɧɵ "ɭɤɥɚɞɵɜɚɟɬɫɹ" n ɩɨɥɭ-
ɜɨɥɧ: |
n O . |
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(9.135) |
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ɑɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ ɢ ɞɥɢɧɚ ɜɨɥɧɵ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ: |
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O |
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c |
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(9.136) |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (9.133) (9.136), |
ɩɨɥɭɱɢɦ ɢɫ- |
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ɤɨɦɭɸ ɫɢɥɭ ɧɚɬɹɠɟɧɢɹ ɫɬɪɭɧɵ: |
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T |
Sd 2 L2Q 2 U . |
(9.137) |
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n2 |
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ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɜɯɨɞɹɳɢɯ ɜ (9.137), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:
T # 77,7 ɇ .
Ɂɚɞɚɱɚ 9.10
ɇɚɣɬɢ ɱɚɫɬɨɬɵ Qn , ɧɚ ɤɨɬɨɪɵɯ ɛɭɞɟɬ ɪɟɡɨɧɢɪɨɜɚɬɶ ɬɪɭɛɚ
ɞɥɢɧɨɣ L = 1,7 ɦ, ɡɚɤɪɵɬɚɹ ɫ ɨɞɧɨɝɨ ɤɨɧɰɚ, ɟɫɥɢ ɫɤɨɪɨɫɬɶ ɡɜɭɤɚ ɜ ɜɨɡɞɭɯɟ ɪɚɜɧɚ c = 340 ɦ/ɫ.
Ɋɟɲɟɧɢɟ
I.ɑɚɫɬɨɬɵ, ɧɚ ɤɨɬɨɪɵɯ ɛɭɞɟɬ ɪɟɡɨɧɢɪɨɜɚɬɶ ɬɪɭɛɚ, ɫɨɜɩɚɞɚɸɬ
ɫɱɚɫɬɨɬɚɦɢ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɢɰ ɜɨɡɞɭɯɚ ɜ ɬɪɭɛɟ, ɨɛɪɚɡɭɸɳɢɯ ɫɬɨɹɱɢɟ ɜɨɥɧɵ.