Механика.Методика решения задач
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ɉɪɢ ɷɬɨɦ ɤɨɨɪɞɢɧɚɬɵ ɭɡɥɨɜ (ɪɢɫ. 9.8) ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɨɬɧɨ- |
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ɲɟɧɢɹɦɢ: |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɟɠɞɭ ɫɨɫɟɞɧɢɦɢ ɭɡɥɚɦɢ ɱɚɫɬɢɰɵ ɫɪɟɞɵ ɤɨɥɟɛɥɸɬɫɹ ɜ ɮɚɡɟ, ɩɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɭɡɟɥ ɮɚɡɚ ɤɨɥɟɛɚɧɢɹ ɫɤɚɱɤɨɨɛɪɚɡɧɨ ɢɡɦɟɧɹɟɬɫɹ ɧɚ S.
[(t, x) ɉɭɱɧɨɫɬɢ
t = 0
t = T/6
x
t = T/3
t = T/2
ɍɡɥɵ
Ɋɢɫ. 9.8. ɍɡɥɵ ɢ ɩɭɱɧɨɫɬɢ ɫɬɨɹɱɟɣ ɜɨɥɧɵ
ȿɫɥɢ [(t, x) – ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰ ɫɪɟɞɵ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɩɪɢ ɧɚɥɢɱɢɢ ɫɬɨɹɱɟɣ ɜɨɥɧɵ, ɬɨ ɞɥɹ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɪɟɞɵ X ɢ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɢ H ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ:
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[(t, x) CZ cos kx \ 0 sin Zt M0 |
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X0 cos kx \0 sin Zt M0 , |
(9.56) |
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[x' Ck sin kx \ 0 cos Zt M0 |
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H0 sin kx \ 0 cos Zt M0 . |
(9.57) |
Ʉɚɤ ɜɢɞɢɦ (ɫɪɚɜɧɢ (9.56) ɢ (9.57) ɫ (9.53)), ɭɡɥɵ ɢ ɩɭɱɧɨɫɬɢ ɞɥɹ ɫɤɨɪɨɫɬɢ ɢ ɫɦɟɳɟɧɢɹ ɫɨɜɩɚɞɚɸɬ, ɚ ɞɥɹ ɞɟɮɨɪɦɚɰɢɣ ɩɭɱɧɨɫɬɢ ɫɨɜɩɚɞɚɸɬ ɫ ɭɡɥɚɦɢ ɫɦɟɳɟɧɢɣ, ɚ ɭɡɥɵ ɫ ɩɭɱɧɨɫɬɹɦɢ ɫɦɟɳɟɧɢɣ. ɍɡɥɵ ɢ ɩɭɱɧɨɫɬɢ ɜ ɜɨɥɧɟ ɞɟɮɨɪɦɚɰɢɣ ɫɦɟɳɟɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɡɥɨɜ ɢ ɩɭɱɧɨɫɬɟɣ ɜ ɫɬɨɹɱɟɣ ɜɨɥɧɟ ɫɦɟɳɟɧɢɣ ɧɚ O/4.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɛɭɞɟɬ ɧɚɛɥɸɞɚɬɶɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɟ ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɪɟɞɧɟɣ ɷɧɟɪɝɢɢ ɜɨɥɧ – ɢɧɬɟɪɮɟɪɟɧ-
ɰɢɹ.
Ⱦɥɹ ɭɩɪɭɝɢɯ ɩɪɨɞɨɥɶɧɵɯ ɜɨɥɧ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɨɛɴɟɦɧɵɟ ɩɥɨɬɧɨɫɬɢ ɤɢɧɟɬɢɱɟɫɤɨɣ wk (t, x) ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ wp (t, x) ɷɧɟɪɝɢɣ
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ȿɫɥɢ ɝɪɚɧɢɰɚ ɫɜɨɛɨɞɧɚ (ɧɚ ɝɪɚɧɢɰɟ ɧɟɬ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɵ ɫɪɟɞɵ), ɬɨ ɧɚ ɧɟɣ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ H , ɧɚɩɪɹɠɟɧɢɟ V , ɢɡɦɟɧɟɧɢɟ ɞɚɜɥɟɧɢɹ ǻp ɢ ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ
ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ wp ɪɚɜɧɵ ɧɭɥɸ. ɉɪɢ ɷɬɨɦ ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰ ɫɪɟɞɵ [ , ɚ ɬɚɤɠɟ ɢɯ ɫɤɨɪɨɫɬɢ X ɢ ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɤɢɧɟɬɢ-
ɱɟɫɤɨɣ ɷɧɟɪɝɢɢ wk ɩɪɢɧɢɦɚɸɬ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ.
ɂɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɞɥɢɧɵ ɜɨɥɧ On ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ (ɦɨɞ), ɢ, ɡɧɚɹ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭ-
ɝɢɯ ɜɨɥɧ c, ɧɚɣɬɢ ɱɚɫɬɨɬɵ ɷɬɢɯ ɦɨɞ: Zn |
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ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɡɚɤɪɟɩɥɟɧɢɹ ɨɛɨɢɯ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ ɞɥɢɧɨɣ L ɩɪɢ ɜɨɡɛɭɠɞɟɧɢɢ ɜ ɧɟɦ ɩɪɨɞɨɥɶɧɵɯ ɢɥɢ ɩɨɩɟɪɟɱɧɵɯ ɭɩɪɭɝɢɯ ɜɨɥɧ.
ɋɦɟɳɟɧɢɟ ɱɚɫɬɢɰ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɪɟɞɵ ɩɪɢ ɧɚɥɢɱɢɢ ɫɬɨɹɱɟɣ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɡɚɤɨɧɭ (9.53). ɉɨɫɤɨɥɶɤɭ ɨɛɚ ɤɨɧɰɚ ɫɬɟɪɠɧɹ ɡɚɤɪɟɩɥɟɧɵ, ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰ ɧɚ ɝɪɚɧɢɰɚɯ ɫɪɟɞɵ
ɪɚɜɧɨ ɧɭɥɸ: |
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C cos \0 cos Zt M0 0 , |
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0) |
(9.64) |
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[(t, x |
L) |
C cos kL \0 cos Zt M0 0 , |
(9.65) |
ɍɫɥɨɜɢɹ (9.64) ɢ (9.65) ɞɨɥɠɧɵ ɜɵɩɨɥɧɹɬɶɫɹ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t, ɫɥɟɞɨɜɚɬɟɥɶɧɨ:
cos \0 0 ; |
(9.66) |
cos kL \ 0 0 . |
(9.67) |
Ɋɚɫɤɪɵɜɚɹ ɤɨɫɢɧɭɫ ɫɭɦɦɵ ɞɜɭɯ ɭɝɥɨɜ ɜ (9.67) ɫ ɭɱɟɬɨɦ (9.66), ɩɨɥɭɱɢɦ:
cos kL \ 0 |
cos kL cos \ 0 sin kL sin \0 rsin kL 0 . |
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ɂɡ ɩɨɥɭɱɟɧɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɟɬ ɜɡɚɢ- |
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ɦɨɫɜɹɡɶ ɱɚɫɬɨɬ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɟɪɠɧɹ ɫ ɟɝɨ ɞɥɢɧɨɣ: |
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kn L nS , |
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2SQn |
L nS , n = 1, 2, 3, …. |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɢɧɵ ɫɬɨɹɱɢɯ ɜɨɥɧ ɢ ɱɚɫɬɨɬɵ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ (ɦɨɞ) ɞɥɹ ɫɬɟɪɠɧɹ ɫ ɡɚɤɪɟɩɥɟɧɧɵɦɢ ɤɨɧɰɚɦɢ ɪɚɜɧɵ:
On |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɫɬɨɹɱɟɣ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɫɬɟɪɠɧɟ ɫ ɞɜɭɦɹ ɡɚɤɪɟɩɥɟɧɧɵɦɢ ɤɨɧɰɚɦɢ ɧɚ ɞɥɢɧɟ ɫɬɟɪɠɧɹ "ɭɤɥɚɞɵɜɚɟɬɫɹ" ɰɟɥɨɟ ɱɢɫɥɨ ɞɥɢɧ ɩɨɥɭɜɨɥɧ:
L n |
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(9.70) |
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Ⱥɧɚɥɨɝɢɱɧɨɟ ɪɚɫɫɦɨɬɪɟɧɢɟ ɞɪɭɝɢɯ ɫɥɭɱɚɟɜ ɡɚɤɪɟɩɥɟɧɢɹ ɤɨɧɰɨɜ ɫɬɟɪɠɧɹ ɩɪɢ ɜɨɡɛɭɠɞɟɧɢɢ ɜ ɧɟɦ ɩɪɨɞɨɥɶɧɵɯ ɢɥɢ ɩɨɩɟɪɟɱɧɵɯ ɫɬɨɹɱɢɯ ɭɩɪɭɝɢɯ ɜɨɥɧ ɩɪɢɜɨɞɢɬ ɤ ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɦɟɠɞɭ ɞɥɢɧɨɣ ɫɬɟɪɠɧɹ ɢ ɱɚɫɬɨɬɚɦɢ (ɞɥɢɧɚɦɢ ɜɨɥɧ) ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ.
Ɍɚɛɥ. 9.1. Ⱦɥɢɧɵ ɜɨɥɧ ɢ ɱɚɫɬɨɬɵ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɭɩɪɭɝɨɦ ɫɬɟɪɠɧɟ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
Ɉɛɚ ɤɨɧɰɚ |
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Ɉɞɢɧ ɤɨɧɟɰ |
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Ɂɚɤɪɟɩɥɟɧɢɟ |
ɉɪɨɞɨɥɶɧɚɹ |
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9.2. Ɉɫɧɨɜɧɵɟ ɬɢɩɵ ɡɚɞɚɱ ɢ ɦɟɬɨɞɵ ɢɯ ɪɟɲɟɧɢɹ 9.2.1. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɡɚɞɚɱ
Ȼɨɥɶɲɢɧɫɬɜɨ ɡɚɞɚɱ ɩɨ ɬɟɦɟ "Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ" ɦɨɠɧɨ ɭɫɥɨɜɧɨ ɨɬɧɟɫɬɢ ɤ ɫɥɟɞɭɸɳɢɦ ɬɢɩɚɦ ɢɥɢ ɢɯ ɤɨɦɛɢɧɚɰɢɹɦ. Ɂɚɞɚɱɢ ɧɚ:
1)ɛɟɝɭɳɢɟ ɜɨɥɧɵ ɢ ɷɥɟɦɟɧɬɵ ɚɤɭɫɬɢɤɢ;
2)ɷɮɮɟɤɬ Ⱦɨɩɥɟɪɚ;
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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3) ɫɬɨɹɱɢɟ ɜɨɥɧɵ, ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɦɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ.
Ʉɚɤ ɩɪɚɜɢɥɨ, ɨɞɢɧ ɢɡ ɬɢɩɨɜ ɡɚɞɚɱ ɢɦɟɟɬ ɨɫɧɨɜɧɨɟ, ɞɪɭɝɢɟ – ɩɨɞɱɢɧɟɧɧɨɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɡɧɚɱɟɧɢɟ.
9.2.2.Ɉɛɳɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
I.Ɉɩɪɟɞɟɥɢɬɶɫɹ ɫ ɦɨɞɟɥɹɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɹɜɥɟɧɢɣ.
1.ɇɚɪɢɫɨɜɚɬɶ ɱɟɪɬɟɠ, ɟɫɥɢ ɷɬɨ ɧɟɨɛɯɨɞɢɦɨ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ.
2.ȼɵɛɪɚɬɶ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ ɢ ɢɡɨɛɪɚɡɢɬɶ ɧɚ ɱɟɪɬɟɠɟ ɟɟ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ (ɢɡ ɫɨɨɛɪɚɠɟɧɢɣ ɭɞɨɛɫɬɜɚ).
3.ɂɡɨɛɪɚɡɢɬɶ ɢ ɨɛɨɡɧɚɱɢɬɶ ɤɢɧɟɦɚɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɥ ɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜɨɥɧ.
4.ȼɵɛɪɚɬɶ ɦɨɞɟɥɢ ɬɟɥ ɢ ɢɯ ɞɜɢɠɟɧɢɹ, ɦɨɞɟɥɢ ɜɨɥɧ ɢ ɯɚɪɚɤɬɟɪ ɢɯ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ (ɟɫɥɢ ɷɬɨ ɧɟ ɫɞɟɥɚɧɨ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ).
II. Ɂɚɩɢɫɚɬɶ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ.
1.Ɂɚɩɢɫɚɬɶ ɡɚɤɨɧɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɛɟɝɭɳɢɯ (ɫɬɨɹɱɢɯ): ɚ) ɜɨɥɧ ɫɦɟɳɟɧɢɹ, ɛ) ɜɨɥɧ ɫɤɨɪɨɫɬɢ, ɜ) ɜɨɥɧ ɭɫɤɨɪɟɧɢɹ,
ɝ) ɜɨɥɧ ɞɟɮɨɪɦɚɰɢɢ.
2.Ɂɚɩɢɫɚɬɶ ɧɚɱɚɥɶɧɵɟ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ.
3.Ɂɚɩɢɫɚɬɶ ɭɪɚɜɧɟɧɢɹ, ɫɜɹɡɵɜɚɸɳɢɟ ɪɚɡɥɢɱɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɜɨɥɧ.
4.ɂɫɩɨɥɶɡɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ ɪɚɧɟɟ ɪɟɲɟɧɧɵɯ ɡɚɞɚɱ ɢ ɨɫɨɛɵɟ ɭɫɥɨɜɢɹ ɡɚɞɚɱɢ (ɧɚɩɪɢɦɟɪ, ɡɚɞɚɧɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɫɢɫɬɟɦɵ).
III. ɉɨɥɭɱɢɬɶ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɢ ɱɢɫɥɟɧɧɨɦ ɜɢɞɚɯ.
1.Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɩɨɥɭɱɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ.
2.ɉɪɨɜɟɫɬɢ ɚɧɚɥɢɡ ɪɟɲɟɧɢɹ (ɩɪɨɜɟɪɢɬɶ ɪɚɡɦɟɪɧɨɫɬɶ ɢ ɥɢɲɧɢɟ ɤɨɪɧɢ, ɪɚɫɫɦɨɬɪɟɬɶ ɯɚɪɚɤɬɟɪɧɵɟ ɫɥɭɱɚɢ, ɭɫɬɚɧɨɜɢɬɶ ɨɛɥɚɫɬɶ ɩɪɢɦɟɧɢɦɨɫɬɢ).
3.ɉɨɥɭɱɢɬɶ ɱɢɫɥɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ.
ɉɪɢɦɟɱɚɧɢɹ.
346 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȼ ɫɥɭɱɚɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɷɮɮɟɤɬ Ⱦɨɩɥɟɪɚ ɩɩ. II.1, II.2 ɧɚɞɨ ɨɩɭɫɬɢɬɶ.
ɉɭɧɤɬɵ II.1 – II.3 ɦɨɠɧɨ ɜɵɩɨɥɧɹɬɶ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɡɚɞɚɱɢ.
9.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
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Ɂɚɞɚɱɚ 9.1 |
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ɉɥɨɫɤɚɹ |
ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ ɫ ɚɦɩɥɢɬɭɞɨɣ |
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[0 1 ɦɤɦ ɢ |
ɱɚɫɬɨɬɨɣ |
Q |
1 ɤȽɰ |
ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜ ɜɨɡɞɭɯɟ ɫ |
ɩɥɨɬɧɨɫɬɶɸ U |
1,3 ɤɝ/ɦ3 |
ɫɨ ɫɤɨɪɨɫɬɶɸ c 340 ɦ/ɫ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, |
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ɫɨɫɬɚɜɥɹɸɳɟɦ ɭɝɥɵ D |
60q |
ɢ E |
45q ɫ ɨɫɹɦɢ X ɢ Y ɞɟɤɚɪɬɨɜɨɣ |
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ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɇɚɣɬɢ ɪɚɡɧɨɫɬɶ ɮɚɡ ɤɨɥɟɛɚɧɢɣ ɱɚɫɬɢɰ ɜɨɡɞɭɯɚ
ɬɨɱɤɚɯ ɫ |
ɤɨɨɪɞɢɧɚɬɚɦɢ x1 1 ɦ , y1 1 ɦ , z1 1 ɦ ɢ x2 6 ɦ , |
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y2 6 ɦ , |
z2 |
6 ɦ , ɚ ɬɚɤɠɟ ɷɧɟɪɝɢɸ, ɩɟɪɟɧɨɫɢɦɭɸ ɡɜɭɤɨɜɨɣ ɜɨɥɧɨɣ |
ɡɚ ɜɪɟɦɹ W |
60 c ɱɟɪɟɡ ɷɥɟɦɟɧɬ ɩɥɨɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɨɳɚɞɶɸ |
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s 10 ɫɦ2 , ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ Z.
Ɋɟɲɟɧɢɟ
I. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɨɣ ɢ ɝɚɪɦɨɧɢɱɟɫɤɨɣ, ɥɚɛɨɪɚɬɨɪɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ ɢ ɫɜɹɡɚɧɧɚɹ ɫ ɧɟɣ ɞɟɤɚɪɬɨɜɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ X, Y, Z ɡɚɞɚɧɚ.
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɩɥɨɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ (ɫɦ. (9.9)):
[(t, r) [0 cos(Zt k r M0 ) , |
(9.71) |
ɝɞɟ r – ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ. ɉɨɫɤɨɥɶɤɭ ɧɚɩɪɚɜɥɟɧɢɟ ɜɨɥɧɨɜɨɝɨ ɜɟɤɬɨɪɚ k ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ c , ɬɨ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɷɬɢɯ ɜɟɤɬɨɪɨɜ ɪɚɜɧɨ
k r |
kx cosD ky cos E kz cosJ , |
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(9.72) |
ɝɞɟ x, y, z – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɧɚɛɥɸɞɟɧɢɹ, k |
Z |
ɢ |
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c |
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cosJ |
1 cos2 D cos2 E . |
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(9.73) |
Ɂɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɩɥɨɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɫɨɫɬɚɜɥɹɸɳɟɦ ɭɝɥɵ D E J ɫ ɨɫɹɦɢ X, Y, Z, ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
347 |
[ [0 cos(Zt kx cosD ky cos E kz cosJ M0 ) . |
(9.74) |
Ɏɚɡɵ ɤɨɥɟɛɚɧɢɣ ɜ ɬɨɱɤɚɯ ɫɪɟɞɵ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ x1, y1, z1 ɢ x2, y2, z2 ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ:
ĭ1 |
Zt k r1 |
Zt kx1 cosD ky1 cos E kz1 cosJ M0 (9.75) |
ɢ |
Zt k r2 |
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ĭ2 |
Zt kx2 cosD ky2 cos E kz2 cos J M0 . (9.76) |
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɜɟɤɬɨɪɚ ɍɦɨɜɚ (ɫɦ. ɩ. 9.1.5) ɢɫɤɨɦɚɹ ɷɧɟɪɝɢɹ, ɩɟɪɟɧɨɫɢɦɚɹ ɡɜɭɤɨɜɨɣ ɜɨɥɧɨɣ ɡɚ ɜɪɟɦɹ W ɱɟɪɟɡ ɷɥɟɦɟɧɬ ɩɥɨɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɨɳɚɞɶɸ s , ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɨɫɢ Z, ɪɚɜɧɚ:
E |
Sz T sW . |
(9.77) |
Ⱦɥɹ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɍɦɨɜɚ ɧɚ ɨɫɶ Z ɦɨɠɧɨ |
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ɡɚɩɢɫɚɬɶ: |
S(t, r) ez T S(t, r) T ez S(t, r) T cosJ . |
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Sz T |
(9.78) |
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ȼɵɪɚɡɢɦ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɦɨɞɭɥɹ ɜɟɤɬɨɪɚ ɍɦɨɜɚ ɱɟɪɟɡ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜɨɥɧɵ, ɢɫɩɨɥɶɡɭɹ
(9.47):
S (t, r) T |
{ w(t, r) T c . |
(9.79) |
ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪ- |
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ɝɢɢ ɜɨɥɧɵ (ɫɦ. (9.45)) ɪɚɜɧɨ: |
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w(t, r) T |
[ 2 UZ2 |
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0 |
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2 . |
(9.80) |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɟɧɵ ɞɜɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (9.73), (9.75), (9.76) ɢ (9.73), (9.77) (9.80) ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɨɣ ɪɚɡɧɨɫɬɢ ɮɚɡ ɤɨɥɟɛɚɧɢɣ ǻĭ ĭ2 ĭ1 ɢ ɩɟɪɟɧɨɫɢɦɨɣ ɷɧɟɪɝɢɢ E ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
III.Ɋɟɲɚɹ ɩɨɥɭɱɟɧɧɵɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ, ɧɚɯɨɞɢɦ ɢɫɤɨɦɵɟ
ɜɡɚɞɚɱɟ ɜɟɥɢɱɢɧɵ:
ǻĭ |
Z (x |
2 |
x ) cosD ( y |
2 |
y |
) cos E |
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c |
1 |
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1 |
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1 cos2 D cos2 E . |
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(z2 z1) |
(9.81) |
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E |
[ 2 UZ2csW |
1 cos2 D cos2 E . |
(9.82) |
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0 |
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2 |
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ɉɨɞɫɬɚɧɨɜɤɚ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɜɟɥɢɱɢɧ ɞɚɟɬ:
348 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ǻ) # S / 2 , |
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E |
2,6 10 4 Ⱦɠ . |
Ɂɚɞɚɱɚ 9.2
ȼ ɭɩɪɭɝɨɣ ɫɪɟɞɟ ɫ ɩɥɨɬɧɨɫɬɶɸ U 2 103 ɤɝ/ɦ3 ɜɞɨɥɶ ɨɫɢ X ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɩɥɨɫɤɚɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ ɫ ɡɚɤɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɫɪɟɞɵ X t, x X0 cos Zt kx . Ⱥɦɩɥɢɬɭɞɚ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ X0 1 ɫɦ/ɫ , ɚ ɫɤɨɪɨɫɬɶ ɜɨɥɧɵ c 2 ɤɦ/ɫ . ɇɚɣ-
ɬɢ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɥɧɵ, ɚ ɬɚɤɠɟ ɭɪɚɜɧɟɧɢɹ ɜɨɥɧ ɞɟɮɨɪɦɚɰɢɣ ɢ ɫɦɟɳɟɧɢɣ ɱɚɫɬɢɰ ɫɪɟɞɵ, ɫɱɢɬɚɹ, ɱɬɨ ɧɚɱɚɥɶɧɨɟ ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰ
[(t 0, x 0) 0 .
Ɋɟɲɟɧɢɟ
I. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɨɣ ɢ ɝɚɪɦɨɧɢɱɟɫɤɨɣ, ɤɨɬɨɪɚɹ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɜɞɨɥɶ ɨɫɢ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
II. Ɉɩɪɟɞɟɥɢɦ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɫɦɟɳɟɧɢɣ:
[(t, x) ³X(t, x)dt |
X0 |
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sin(Zt kx) C(x) , |
(9.83) |
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Z |
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Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɧɫɬɚɧɬɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ C( x) |
ɜ (9.83) ɡɚ- |
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ɩɢɲɟɦ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ ɫɪɟɞɵ ɜ ɬɨɱɤɟ
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ɢ (9.84) ɫɥɟɞɭɟɬ, ɱɬɨ ɤɨɧɫɬɚɧɬɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ |
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ɢ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɫɤɨɪɨɫɬɟɣ ɩɪɢɨɛɪɟɬɚɟɬ |
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ɜɢɞ: |
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ɚɦɩɥɢɬɭɞɚ ɫɦɟɳɟɧɢɣ ɱɚɫɬɢɰ ɫɪɟɞɵ. |
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Ⱦɥɹ ɜɨɥɧɵ ɞɟɮɨɪɦɚɰɢɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.17) ɢ (9.83) ɦɨɠ- |
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ɧɨ ɡɚɩɢɫɚɬɶ: |
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
349 |
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ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɢɫɤɨɦɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɧɟɩɨ- |
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ɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɟɬ ɢɡ (9.48) ɢ (9.44): |
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III. ɂɫɤɨɦɵɟ ɡɚɤɨɧɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧ ɫɦɟɳɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɸɬ ɢɡ (9.85) ɢ (9.86) ɫ ɭɱɟɬɨɦ ɜɵ-
ɪɚɠɟɧɢɹ ɞɥɹ ɚɦɩɥɢɬɭɞɵ ɫɦɟɳɟɧɢɣ ɱɚɫɬɢɰ ɫɪɟɞɵ [0 |
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ɂɫɤɨɦɚɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɥɧɵ, ɜɵɪɚɠɟɧɧɚɹ ɱɟɪɟɡ ɡɚɞɚɧɧɵɟ ɜ ɡɚɞɚɱɟ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ, ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɥɟɞɭɟɬ ɢɡ (9.87):
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ɉɨɞɫɬɚɜɢɜ ɜ (9.90) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɚɦɩɥɢɬɭɞɵ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ, ɩɥɨɬɧɨɫɬɢ ɫɪɟɞɵ ɢ ɫɤɨɪɨɫɬɢ ɜɨɥɧɵ, ɩɨɥɭɱɚɟɦ ɡɧɚɱɟɧɢɟ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɜɨɥɧɵ:
I200 ȼɬ/ɦ2 .
Ɂɚɞɚɱɚ 9.3
Ɍɨɱɟɱɧɵɣ ɢɡɨɬɪɨɩɧɨ ɢɡɥɭɱɚɸɳɢɣ ɢɫɬɨɱɧɢɤ ɢɫɩɭɫɤɚɟɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɡɚɬɭɯɚɸɳɭɸ ɝɚɪɦɨɧɢɱɟɫɤɭɸ ɡɜɭɤɨɜɭɸ ɜɨɥɧɭ ɫ ɱɚɫ-
ɬɨɬɨɣ Q = 1,45 ɤȽɰ. ɇɚ ɪɚɫɫɬɨɹɧɢɢ r0 = 5 ɦ ɨɬ ɢɫɬɨɱɧɢɤɚ ɚɦɩɥɢɬɭɞɚ ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰ ɫɪɟɞɵ [0 r0 50 ɦɤɦ , ɚ ɜ ɬɨɱɤɟ P, ɧɚɯɨɞɹɳɟɣɫɹ
ɧɚ ɪɚɫɫɬɨɹɧɢɢ r = 10 ɦ ɨɬ ɢɫɬɨɱɧɢɤɚ, ɚɦɩɥɢɬɭɞɚ ɫɦɟɳɟɧɢɹ [0 r ɜ K = 3 ɪɚɡɚ ɦɟɧɶɲɟ [0 r0 . ɇɚɣɬɢ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ ɜɨɥɧɵ G ɢ ɚɦɩɥɢɬɭɞɭ ɤɨɥɟɛɚɧɢɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɪɟɞɵ X0 r ɜ ɬɨɱɤɟ P.
Ɋɟɲɟɧɢɟ
I. Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɢɫɩɨɥɶɡɭɟɦ ɫɮɟɪɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɜ ɧɚɱɚɥɟ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɨɠɟɧ ɢɫɬɨɱɧɢɤ ɡɜɭɤɨɜɵɯ ɤɨɥɟɛɚɧɢɣ. ɉɨɫɤɨɥɶɤɭ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɧɟ ɨɝɨɜɚɪɢɜɚɟɬɫɹ ɢɧɨɟ, ɫɪɟɞɭ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɢɡɨɬɪɨɩɧɨɣ, ɬɨɝɞɚ ɡɜɭɤɨɜɚɹ ɜɨɥɧɚ, ɢɡɥɭɱɚɟɦɚɹ ɬɨɱɟɱɧɵɦ ɢɫɬɨɱɧɢɤɨɦ ɹɜɥɹɟɬɫɹ ɫɮɟɪɢɱɟɫɤɨɣ.
350 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɡɚɬɭɯɚɸɳɟɣ ɫɮɟɪɢɱɟɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ ɫɦɟɳɟɧɢɹ (ɫɦ. (9.14)):
[ t, r |
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e Gr cos 2SQt kr M0 . |
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ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɡɚɤɨɧ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɫɤɨɪɨɫɬɟɣ ɢɦɟɟɬ |
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ɫɥɟɞɭɸɳɢɣ ɜɢɞ: |
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X0 r sin 2SQt kr M0 , |
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ɝɞɟ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɪɟɞɵ X0 r ɪɚɜɧɚ |
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ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɚɦɩɥɢɬɭɞɚ ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰ ɧɚ ɪɚɫɫɬɨɹ- |
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ɧɢɢ r0 ɨɬ ɢɫɬɨɱɧɢɤɚ ɪɚɜɧɨ |
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ɚ ɜ ɬɨɱɤɟ P ɧɚ ɪɚɫɫɬɨɹɧɢɢ r |
ɜ K ɪɚɡ ɦɟɧɶɲɟ: |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (9.94) ɢ (9.95), ɩɨɥɭɱɢɦ ɢɫɤɨ- |
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ɦɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ ɜɨɥɧɵ G: |
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Ɉɩɪɟɞɟɥɢɦ ɚɦɩɥɢɬɭɞɭ ɤɨɥɟɛɚɧɢɣ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɪɟɞɵ ɜ ɬɨɱɤɟ P, ɪɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (9.93) ɢ (9.94) ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧ-
ɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ G (9.95): |
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X0 r 2SQ |
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ɉɨɞɫɬɚɜɥɹɹ ɜ (9.96) ɢ (9.97) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧɧɵɟ ɜ ɡɚɞɚɱɟ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ:
G # 0,08 ɦ-1 ,
X0 r #15 ɫɦ/ɫ .