Механика.Методика решения задач
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Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɧɢɣ ɞɥɢɧ ɢ ɦɚɫɫ, ɪɚɫɩɨɥɨɠɟɧɧɚɹ ɜɵɲɟ ɢɡɨɛɪɚɠɟɧɧɨɣ ɤɪɢɜɨɣ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɥɭɱɚɸ ɫɨɭɞɚɪɟɧɢɹ ɫɬɟɪɠɧɹ ɫ ɩɨɬɨɥɤɨɦ.
l/L |
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0.8 |
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Dmax |
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0.4 |
Dmax |
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0.2 |
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m/M
Ɋɢɫ. 7.3
ɇɚ ɪɢɫ. 7.4 ɢɡɨɛɪɚɠɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɭɝɥɚ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɨɬ ɨɬɧɨɲɟɧɢɹ ɞɥɢɧ ɦɚɹɬɧɢɤɚ ɢ ɫɬɟɪɠɧɹ Dmax l / L ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɨɬɧɨɲɟɧɢɹ ɢɯ ɦɚɫɫ
(m / M ) .
Dmax ,q |
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l/L
Ɋɢɫ. 7.4
Ʉɚɤ ɜɢɞɢɦ, ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɨɬɧɨɲɟɧɢɹ ɞɥɢɧ ɦɚɹɬɧɢɤɚ ɢ ɫɬɟɪɠɧɹ ɦɚɤɫɢɦɚɥɶɧɵɣ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɫɬɟɪɠɧɹ ɜɨɡɪɚɫɬɚɟɬ, ɩɪɢ-
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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1) ɫɪɚɡɭ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɲɚɣɛɚ |
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ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ, |
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2) ɲɚɣɛɚ ɩɟɪɟɞɚɟɬ ɫɬɟɪɠɧɸ ɦɚɤ- |
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ɫɢɦɚɥɶɧɵɣ ɢɦɩɭɥɶɫ, |
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3) ɫɤɨɪɨɫɬɶ ɤɨɧɰɚ ɫɬɟɪɠɧɹ (ɬɨɱɤɚ |
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A ɧɚ ɪɢɫ. 7.6) ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɪɚɜɧɚ |
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ɧɭɥɸ. |
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Ɋɟɲɟɧɢɟ |
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I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚ- |
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ɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬ- |
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ɫɱɟɬɚ. ɉɨɫɤɨɥɶɤɭ ɫɨɭɞɚɪɟɧɢɟ ɲɚɣɛɵ ɫɨ |
Ɋɢɫ. 7.6 |
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ɫɬɟɪɠɧɟɦ ɹɜɥɹɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɢɦ, ɚ |
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ɧɚ ɫɢɫɬɟɦɭ ɬɟɥ «ɫɬɟɪɠɟɧɶ + ɲɚɣɛɚ» ɧɟ ɞɟɣɫɬɜɭɸɬ ɜɧɟɲɧɢɟ ɫɢɥɵ ɜɞɨɥɶ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɨ ɜɵɩɨɥɧɹɸɬɫɹ ɜɫɟ ɬɪɢ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ: ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ, ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 7.6. Ɉɫɶ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜɪɚɳɟɧɢɟ, ɭɞɨɛɧɨ ɜɡɹɬɶ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɫɬɟɪɠɧɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɧɚɩɪɚɜɥɟɧɧɨɣ ɢɡ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ.
II. Ɂɚɩɢɲɟɦ ɬɪɢ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɞɥɹ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɞɥɹ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ ɞɨ ɫɨɭɞɚɪɟɧɢɹ – ɫɪɚɡɭ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ.
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɧɚ ɨɫɶ X ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
mX |
mXc MXcc. |
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Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧ- |
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ɧɨɣ ɨɫɢ: |
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mXl mX l J0Z . |
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Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ: |
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mXc2 |
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MXcc2 |
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J0Z2 |
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Ɂɞɟɫɶ Xc , Xcc – ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɟɣ ɲɚɣɛɵ ɢ ɰɟɧɬɪɚ ɫɬɟɪɠɧɹ ɧɚ ɨɫɶ X ɫɪɚɡɭ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ (ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɟɣ ɧɚ ɨɫɶ Y ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɵ ɧɭɥɸ), Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɫɬɟɪɠɧɹ ɜ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɬɟɪɠɧɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟ- |
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ɪɟɡ ɟɝɨ ɰɟɧɬɪ ɦɚɫɫ, ɪɚɜɟɧ (6.43): |
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ML2 |
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ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ (ɫɦ. |
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(1.26) ɜ Ƚɥɚɜɟ 1) |
ɫɤɨɪɨɫɬɶ XA |
ɬɨɱɤɢ Ⱥ ɫɬɟɪɠɧɹ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ |
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ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɢ ɫɤɨɪɨɫɬɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɷɬɨɣ ɬɨɱɤɢ ɜɨɤɪɭɝ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ:
XA |
Xcc Z |
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(7.38) |
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III. Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (7.34) – (7.38) ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ ɢɦɟɟɬ ɜɢɞ: |
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(m M )J |
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(m M )J0 Mml2 |
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(m M )J0 Mml |
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© M |
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Ɋɚɫɫɬɨɹɧɢɟ l ɨɬ ɬɨɱɤɢ ɫɨɭɞɚɪɟɧɢɹ ɞɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ, ɩɪɢ |
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ɤɨɬɨɪɨɦ ɲɚɣɛɚ ɨɫɬɚɧɨɜɢɬɫɹ ɩɨɫɥɟ ɭɞɚɪɚ, |
ɧɚɣɞɟɦ ɢɡ (7.39) ɩɪɢ |
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Xc 0 ɫ ɭɱɟɬɨɦ (7.37): |
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(7.43) |
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Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ (7.40), ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɫɬɟɪɠɧɹ ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ l = 0. ɉɪɢ ɷɬɨɦ ɭɫɥɨɜɢɢ ɲɚɣɛɚ ɩɟɪɟɞɚɫɬ ɫɬɟɪɠɧɸ ɦɚɤɫɢɦɚɥɶɧɵɣ ɢɦɩɭɥɶɫ.
Ɂɧɚɱɟɧɢɟ l, ɩɪɢ ɤɨɬɨɪɨɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ A ɫɪɚɡɭ ɩɨɫɥɟ ɭɞɚɪɚ ɛɭɞɟɬ ɪɚɜɧɚ ɧɭɥɸ, ɧɚɯɨɞɢɦ ɢɡ (7.42) ɫ ɭɱɟɬɨɦ (7.37):
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ɉɨɫɥɟ ɩɨɩɚɞɚɧɢɹ ɲɚɣɛɵ ɜ ɬɨɱɤɭ |
ɫ ɬɚɤɢɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ |
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ɫɬɟɪɠɟɧɶ ɫɪɚɡɭ ɩɨɫɥɟ ɭɞɚɪɚ ɛɭɞɟɬ ɫɨɜɟɪɲɚɬɶ ɬɨɥɶɤɨ ɜɪɚɳɚɬɟɥɶɧɨɟ
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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ɞɜɢɠɟɧɢɟ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ A.
Ɉɬɜɟɬ:
1) ɲɚɣɛɚ ɨɫɬɚɧɨɜɢɬɫɹ ɫɪɚɡɭ ɩɨɫɥɟ ɭɞɚɪɚ, ɟɫɥɢ l L |
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2)ɲɚɣɛɚ ɩɟɪɟɞɚɫɬ ɫɬɟɪɠɧɸ ɦɚɤɫɢɦɚɥɶɧɵɣ ɢɦɩɭɥɶɫ, ɟɫɥɢ ɨɧɚ ɩɨɩɚɞɟɬ ɜ ɰɟɧɬɪ ɦɚɫɫ ɫɬɟɪɠɧɹ (l = 0);
3)ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ A ɫɪɚɡɭ ɩɨɫɥɟ ɭɞɚɪɚ ɛɭɞɟɬ ɪɚɜɧɚ ɧɭɥɸ ɩɪɢ
ɭɫɥɨɜɢɢ l |
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Ɂɚɞɚɱɚ 7.3
Ⱦɜɚ ɨɞɢɧɚɤɨɜɵɯ ɨɞɧɨɪɨɞɧɵɯ ɜɪɚɳɚɸɳɢɯɫɹ ɬɟɥɚ ɫɮɟɪɢɱɟɫɤɨɣ ɮɨɪɦɵ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ r ɞɜɢɠɭɬɫɹ ɧɚɜɫɬɪɟɱɭ ɞɪɭɝ ɞɪɭɝɭ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɩɨ ɦɨɞɭɥɸ ɫɤɨɪɨɫɬɹɦɢ X0 . ɍɝɥɨɜɵɟ ɫɤɨɪɨɫɬɢ ɜɪɚ-
ɳɟɧɢɹ ɬɟɥ, Ȧ1 ɢ Ȧ2 , ɫɨɫɬɚɜɥɹɸɬ ɭɝɨɥ D ɢ ɪɚɜɧɵ ɩɨ ɦɨɞɭɥɸ Ȧ1 
Ȧ2 Z0 . ȼ ɪɟɡɭɥɶɬɚɬɟ ɥɨɛɨɜɨɝɨ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɨɝɨ ɫɨɭɞɚ-
ɪɟɧɢɹ ɨɛɪɚɡɭɟɬɫɹ ɨɞɧɨ ɬɟɥɨ ɬɨɣ ɠɟ ɩɥɨɬɧɨɫɬɢ, ɮɨɪɦɭ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɬɚɤɠɟ ɫɱɢɬɚɬɶ ɫɮɟɪɢɱɟɫɤɨɣ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ Ȧ ɜɪɚɳɟɧɢɹ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ ɢ ɢɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ
ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ǻE k .
Ɋɟɲɟɧɢɟ
I. ɋɢɫɬɟɦɚ ɞɜɭɯ ɬɟɥ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɢɡɨɥɢɪɨɜɚɧɧɨɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɭɦɦɚɪɧɵɣ ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɢ ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɫɨɯɪɚɧɹɸɬɫɹ. ɇɚɩɪɚɜɢɦ ɨɫɶ X ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜɞɨɥɶ ɥɢɧɢɢ, ɫɨɟɞɢɧɹɸɳɟɣ ɰɟɧɬɪɵ ɦɚɫɫ ɞɜɭɯ ɬɟɥ ɞɨ ɫɨɭɞɚɪɟɧɢɹ.
II. Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɧɚ ɨɫɶ X ɞɥɹ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ, ɜɤɥɸɱɚɸɳɟɝɨ
ɦɨɦɟɧɬ ɢɯ ɫɨɭɞɚɪɟɧɢɹ: |
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mX0 mX0 2mX , |
(7.45) |
ɝɞɟ X – ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ X ɫɤɨɪɨɫɬɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɬɟɥɚ ɦɚɫɫɨɣ 2m. Ʉɚɤ ɜɢɞɢɦ, X 0 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɜɢɠɟɧɢɟ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ ɹɜɥɹɟɬɫɹ ɱɢɫɬɨ ɜɪɚɳɚɬɟɥɶɧɵɦ.
246 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɯ ɨɛɳɟɝɨ ɰɟɧɬɪɚ ɦɚɫɫ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ, ɜɤɥɸɱɚɸɳɟɦ ɦɨɦɟɧɬ ɢɯ ɫɨɭɞɚɪɟɧɢɹ:
L1 L2 L , |
(7.46) |
ɝɞɟ L1 ɢ L2 – ɦɨɦɟɧɬɵ ɢɦɩɭɥɶɫɚ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɬɟɥ ɞɨ ɫɨɭɞɚ-
ɪɟɧɢɹ, L – ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ. ɉɨɫɤɨɥɶɤɭ ɫɤɨɪɨɫɬɢ ɬɟɥ ɞɨ ɫɨɭɞɚɪɟɧɢɹ ɧɚɩɪɚɜɥɟɧɵ ɜɞɨɥɶ ɥɢɧɢɢ, ɧɚ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ, ɬɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (6.27) Ƚɥɚɜɵ 6 ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɤɚɠɞɨɝɨ ɢɡ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɟɥ ɪɚɜɟɧ ɦɨɦɟɧɬɭ ɢɦɩɭɥɶɫɚ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɰɟɧɬɪɚ ɦɚɫɫ.
Ɇɨɦɟɧɬɵ ɢɦɩɭɥɶɫɚ ɤɚɠɞɨɝɨ ɢɡ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɵɯ ɬɟɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɯ ɫɨɛɫɬɜɟɧɧɵɯ ɰɟɧɬɪɨɜ ɦɚɫɫ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (6.32) Ƚɥɚɜɵ 6 ɪɚɜɧɵ:
L1 |
J0Ȧ1 , |
(7.47) |
L2 |
J0Ȧ2 , |
(7.48) |
L |
JȦ , |
(7.49) |
ɝɞɟ J 0 ɢ J – ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɤɚɠɞɨɝɨ ɢɡ ɫɨɭɞɚɪɹɸɳɢɯɫɹ ɬɟɥ ɢ
ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɯ ɫɨɛɫɬɜɟɧɧɵɯ ɨɫɟɣ ɜɪɚɳɟɧɢɹ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.45):
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(7.50) |
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Ɋɚɞɢɭɫ R ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ ɧɚɯɨɞɢɦ ɢɡ ɭɫɥɨɜɢɹ ɫɨɯɪɚɧɟ- |
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ɧɢɹ ɩɥɨɬɧɨɫɬɢ (ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɨɛɴɟɦɚ): |
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SR3 . |
(7.52) |
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ɋɨɝɥɚɫɧɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɦɨɞɭɥɢ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɜɪɚɳɟ- |
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ɧɢɹ ɬɟɥ ɞɨ ɢɯ ɫɨɭɞɚɪɟɧɢɹ ɪɚɜɧɵ: |
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(7.53) |
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ɂɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ
ɬɟɥ ǻE k ɜ ɪɟɡɭɥɶɬɚɬɟ ɢɯ ɚɛɫɨɥɸɬɧɨ ɧɟɭɩɪɭɝɨɝɨ ɫɨɭɞɚɪɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (7.6) ɪɚɜɧɨ:
Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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III. Ɋɟɲɚɹ |
ɫɢɫɬɟɦɭ |
ɭɪɚɜɧɟɧɢɣ (7.46) – (7.53), |
ɩɨɥɭɱɚɟɦ ɦɨ- |
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ɞɭɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɨɭɞɚɪɟɧɢɹ ɬɟɥɚ:
Z |
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cos |
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cos |
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(7.55) |
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ɉɨɫɤɨɥɶɤɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (7.46) – (7.49) |
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ɚ ɦɨɞɭɥɢ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ ɜɪɚɳɟɧɢɹ ɬɟɥ ɞɨ ɢɯ ɫɨɭɞɚɪɟɧɢɹ ɪɚɜɧɵ (7.53), ɬɨ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɨɛɪɚɡɨɜɚɜɲɟɝɨɫɹ ɬɟɥɚ Ȧ ɧɚɩɪɚɜɥɟɧɚ ɩɨ ɛɢɫɫɟɤɬɪɢɫɟ ɭɝɥɚ D, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɟɤɬɨɪɚɦɢ ɭɝɥɨɜɵɯ ɫɤɨɪɨɫɬɟɣ Ȧ1 ɢ Ȧ2 .
ɂɫɤɨɦɨɟ ɢɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ ɬɟɥ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɨɭɞɚɪɟɧɢɹ ɩɨɥɭɱɢɦ, ɩɨɞɫɬɚɜɥɹɹ (7.55) ɜ
(7.54) ɫ ɭɱɟɬɨɦ (7.50) – (7.52): |
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Ɂɚɞɚɱɚ 7.4
Ⱦɜɟ ɨɞɢɧɚɤɨɜɵɟ ɝɚɧɬɟɥɢ ɦɚɫɫɨɣ m ɜ ɜɢɞɟ ɲɚɪɢɤɨɜ, ɫɨɟɞɢ-
ɧɟɧɧɵɯ ɫɬɟɪɠɧɟɦ, ɫɤɨɥɶɡɹɬ ɩɨ ɝɥɚɞɤɨɣ |
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ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɧɚɜɫɬɪɟɱɭ |
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ɞɪɭɝ ɞɪɭɝɭ ɫɨ ɫɤɨɪɨɫɬɹɦɢ ȣ1 ɢ ȣ2 ɬɚɤ, ɤɚɤ |
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ɢɡɨɛɪɚɠɟɧɨ ɧɚ ɪɢɫ. 7.7. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ |
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ɤɚɠɞɨɣ ɝɚɧɬɟɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨ- |
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ɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ ɦɚɫɫ ɩɟɪɩɟɧɞɢɤɭ- |
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ɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ, ɪɚɜɟɧ J, ɚ ɪɚɫ- |
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ɫɬɨɹɧɢɟ ɦɟɠɞɭ ɰɟɧɬɪɚɦɢ ɲɚɪɢɤɨɜ ɝɚɧɬɟ- |
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ɥɢ – l. Ʉɚɤ ɛɭɞɭɬ ɞɜɢɝɚɬɶɫɹ ɝɚɧɬɟɥɢ ɩɨɫɥɟ |
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Ɋɢɫ. 7.7 |
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ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɨɝɨ ɫɨɭɞɚɪɟɧɢɹ? |
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248 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɞɜɭɯ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ: ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ, ɨɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɢɦ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 7.8, ɢ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɟɥ, ɫ ɨɫɶɸ X', ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫ. 7.9.
ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɝɚɧɬɟɥɢ ɞɜɢɠɭɬɫɹ ɩɨ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɟɥ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɞɜɭɯ ɝɚɧɬɟɥɟɣ, ɞɜɢɠɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɢ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɚɹ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ, ɹɜɥɹɟɬɫɹ ɢɧɟɪɰɢɚɥɶɧɨɣ.
D |
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Ɋɢɫ. 7.8 |
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Ɋɢɫ. 7.9 |
ɉɨɫɤɨɥɶɤɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɫɢɫɬɟɦɚ ɬɟɥ ɡɚɦɤɧɭɬɚ, ɚ ɫɨɭɞɚɪɟɧɢɟ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɨɟ, ɬɨ ɜɵɩɨɥɧɹɸɬɫɹ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ, ɢɦɩɭɥɶɫɚ ɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɞɥɹ ɷɬɨɣ ɫɢɫɬɟɦɵ ɜ ɥɸɛɨɣ ɢɡ ɜɵɛɪɚɧɧɵɯ ɫɢɫɬɟɦ ɨɬɫɱɟɬɚ.
II. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɝɚɧɬɟɥɢ ɞɜɢɠɭɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɹɦɢ X1 ɢ X2, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɧɚ ɨɫɶ X ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ
ɨɬɫɱɟɬɚ (ɫɦ. Ƚɥɚɜɭ 3) ɪɚɜɧɚ |
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X1 X2 , |
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ɚ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɟɣ ɰɟɧɬɪɨɜ ɦɚɫɫ ɝɚɧɬɟɥɟɣ u1 ɢ u2 |
ɧɚ ɨɫɶ X' ɨɬ- |
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ɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɰɟɧɬɪɚ ɦɚɫɫ ɨɛɟɢɯ ɝɚɧɬɟɥɟɣ ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɵɪɚɠɟɧɢɹɦɢ:
u1 |
X1 Xɰɦ |
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Ƚɥɚɜɚ 7. Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ |
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Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɢɦɩɭɥɶɫɚ ɧɚ ɨɫɶ X' ɞɥɹ ɫɢɫɬɟɦɵ ɞɜɭɯ ɝɚɧɬɟɥɟɣ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ, ɜɤɥɸɱɚɸɳɟɦ ɦɨɦɟɧɬ
ɢɯ ɫɨɭɞɚɪɟɧɢɹ, ɜ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɦɚɫɫ: |
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mu1 mu2 |
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ɝɞɟ u1c ɢ u2c – ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɨɜ ɦɚɫɫ ɝɚɧ- |
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ɬɟɥɟɣ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ. |
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ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ |
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ɞɜɢɠɟɧɢɟ ɤɚɠɞɨɣ ɝɚɧɬɟɥɢ ɛɭɞɟɬ ɫɭɩɟɪɩɨ- |
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ɡɢɰɢɟɣ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɟɟ ɰɟɧ- |
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ɬɪɚ ɦɚɫɫ ɢ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜɨ- |
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ɤɪɭɝ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ ɦɚɫɫ |
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ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɨ ɤɨɬɨ- |
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Ɋɢɫ. 7.10 |
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ɪɨɣ ɩɪɨɢɫɯɨɞɢɬ |
ɫɤɨɥɶɠɟɧɢɟ ɝɚɧɬɟɥɟɣ |
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(ɫɦ. ɪɢɫ. 7.10).
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɞɜɭɯ ɝɚɧɬɟɥɟɣ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ, ɜɤɥɸɱɚɸɳɟɦ ɦɨɦɟɧɬ ɫɨɭɞɚɪɟɧɢɹ, ɜ ɫɢɫɬɟɦɟ ɢɯ ɰɟɧɬɪɚ ɦɚɫɫ, ɩɪɢ ɷɬɨɦ ɭɱɬɟɦ, ɱɬɨ ɩɪɢ ɩɥɨɫɤɨɦ ɞɜɢɠɟɧɢɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɟɝɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪ-
ɦɭɥɨɣ (7.10):
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Ɂɞɟɫɶ Z1 ɢ Z2 |
– ɭɝɥɨɜɵɟ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɝɚɧɬɟɥɟɣ ɩɨɫɥɟ ɫɨɭɞɚ- |
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ɪɟɧɢɹ.
Ɂɚɩɢɲɟɦ ɬɚɤɠɟ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɞɜɭɯ ɝɚɧɬɟɥɟɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪ ɦɚɫɫ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɨ ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɫɤɨɥɶɠɟɧɢɟ ɝɚɧɬɟɥɟɣ, ɧɚ ɬɨɦ ɠɟ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ:
2l mu1 2l mu2 2l mu1c JZ1 2l muc2 JZ2 . (7.63)
ȼ (7.63) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɝɚɧɬɟɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (6.23) ɢɡ Ƚɥɚɜɵ 6 ɪɚɜɟɧ ɫɭɦɦɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɰɟɧɬɪɚ ɦɚɫɫ ɝɚɧɬɟɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɦɚɫɫ ɞɜɭɯ ɝɚɧɬɟɥɟɣ ɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɝɚɧɬɟɥɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ ɦɚɫɫ, ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɷɬɢɦ ɰɟɧɬɪɨɦ ɦɚɫɫ.
ɍɱɢɬɵɜɚɹ ɫɢɦɦɟɬɪɢɸ ɡɚɞɚɱɢ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɦɚɫɫ ɞɜɭɯ ɝɚɧɬɟɥɟɣ, ɡɚɩɢɲɟɦ ɨɱɟɜɢɞɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɭɝɥɨɜɵɦɢ ɫɤɨɪɨ-
250 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɫɬɹɦɢ ɜɪɚɳɟɧɢɹ ɝɚɧɬɟɥɟɣ ɢ ɫɤɨɪɨɫɬɹɦɢ ɢɯ ɰɟɧɬɪɨɜ ɦɚɫɫ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ (ɫɦ. ɪɢɫ. 7.11):
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(7.65) |
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Ɋɢɫ. 7.11 |
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III. Ɋɟɲɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (7.59) – (7.65) |
ɨɬɧɨɫɢɬɟɥɶɧɨ |
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u1c , u2c , Z1 ɢ Z2 : |
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4lm |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɝɚɧɬɟɥɢ ɩɪɢ ɡɚɞɚɧɧɨɣ ɦɚɫɫɟ m ɦɚɤɫɢɦɚɥɟɧ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɟɟ ɦɚɫɫɚ ɫɨɫɪɟɞɨɬɨɱɟɧɚ ɧɚ ɤɨɧɰɚɯ ɝɚɧɬɟɥɢ:
§ l |
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J d 2m¨ |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ |
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (7.66) |
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ɪɢɫ. 7.11). |
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ɋɤɨɪɨɫɬɢ ɰɟɧɬɪɨɜ ɦɚɫɫ ɝɚɧɬɟɥɟɣ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɜ ɥɚɛɨɪɚ- |
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ɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɪɚɜɧɵ: |
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X1c |
uc Xɰɦ |
X ml |
2 4X |
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(7.69) |
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X2c |
uc Xɰɦ |
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ml2 4X J |
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(7.70) |
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ɂɬɚɤ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɚɛɫɨɥɸɬɧɨ ɭɩɪɭɝɨɝɨ ɫɨɭɞɚɪɟɧɢɹ, ɞɜɢɠɟɧɢɟ ɝɚɧɬɟɥɟɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɩɟɪɩɨɡɢɰɢɸ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ