Механика.Методика решения задач
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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N |
m2g |
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F |
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Fɬɪ
Ɋɢɫ. 6.13
II. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɲɚɪɚ (ɫɨɜɩɚɞɚɸɳɟɝɨ ɫ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɰɟɧɬɪɨɦ) ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ X ɢ Y ɥɚɛɨɪɚɬɨɪ-
ɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɢɦɟɸɬ ɜɢɞ: |
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m2a2 Fɬɪ , |
(6.87) |
0 N m2 g , |
(6.88) |
ɝɞɟ Fɬɪ – ɦɨɞɭɥɶ ɫɢɥɵ ɬɪɟɧɢɹ ɩɨɤɨɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɲɚɪ ɫɨ ɫɬɨɪɨɧɵ ɞɨɫɤɢ, N – ɦɨɞɭɥɶ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɞɨɫɤɢ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɲɚɪ.
Ⱦɨɫɤɚ ɞɜɢɠɟɬɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ ɚ1 ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ X ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ F ɢ ɫɢɥɵ ɬɪɟɧɢɹ ɫɨ ɫɬɨɪɨɧɵ ɲɚɪɚ, ɦɨɞɭɥɶ ɤɨɬɨɪɨɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɪɟɬɶɢɦ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ ɪɚɜɟɧ ɦɨɞɭɥɸ Fɬɪ ɫɢɥɵ ɬɪɟɧɢɹ ɩɨɤɨɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɲɚɪ ɫɨ ɫɬɨɪɨɧɵ ɞɨɫɤɢ.
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɞɨɫɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X ɢɦɟɟɬ ɜɢɞ:
m1a1 F Fɬɪ . |
(6.89) |
ɍɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (6.47) ɞɥɹ ɲɚɪɚ ɡɚɩɢɲɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ ɦɚɫɫ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ
ɩɥɨɫɤɨɫɬɹɦ ɞɜɢɠɟɧɢɹ ɜɫɟɯ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɲɚɪɚ: |
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J0 E Fɬɪ R , |
(6.90) |
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ɝɞɟ J0 – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɲɚɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɧɧɨɣ ɨɫɢ, E |
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dZ |
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– ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɲɚɪɚ. ȼ (6.90) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬɵ ɫɢɥ ɬɹɠɟɫɬɢ ɢ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɞɨɫɤɢ ɪɚɜɧɵ ɧɭɥɸ, ɩɨɫɤɨɥɶɤɭ ɥɢɧɢɢ ɢɯ ɞɟɣɫɬɜɢɹ ɩɪɨɯɨɞɹɬ ɱɟɪɟɡ ɨɫɶ ɜɪɚɳɟɧɢɹ.
Ⱦɨɩɨɥɧɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɲɚɪɚ ɢ ɞɨɫɤɢ (6.87) – (6.90) ɭɪɚɜɧɟɧɢɟɦ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ, ɤɨɬɨɪɨɟ ɫɥɟɞɭɟɬ ɢɡ ɭɫɥɨɜɢɹ ɨɬɫɭɬɫɬɜɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɦɟɠɞɭ ɲɚɪɨɦ ɢ ɞɨɫɤɨɣ:
a2 a1 ER . |
(6.91) |
212 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɍɱɬɟɦ ɬɚɤɠɟ, ɱɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɞɧɨɪɨɞɧɨɝɨ ɲɚɪɚ ɨɬɧɨɫɢ-
ɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ ɦɚɫɫ (6.45) ɪɚɜɟɧ: |
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J0 |
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mR2 . |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.87), (6.89) – (6.92), ɩɨɥɭɱɚɟɦ |
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ɢɫɤɨɦɵɟ ɭɫɤɨɪɟɧɢɹ ɞɨɫɤɢ ɢ ɰɟɧɬɪɚ ɲɚɪɚ: |
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a1 |
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(6.93) |
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7m1 |
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a2 |
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(6.94) |
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Ɉɩɪɟɞɟɥɢɦ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɞɜɢɠɟɧɢɟ ɲɚɪɚ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ. Ⱦɥɹ ɞɜɢɠɟɧɢɹ ɲɚɪɚ ɩɨ ɞɨɫɤɟ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɫɢɥɚ ɬɪɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ⱥɦɨɧɬɨɧɚ–Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2 ɜ Ƚɥɚɜɟ 2) ɭɞɨɜɥɟɬɜɨɪɹɥɚ
ɧɟɪɚɜɟɧɫɬɜɭ: |
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Fɬɪ d PN . |
(6.95) |
Ɂɞɟɫɶ P – ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ.
Ɇɨɞɭɥɢ ɫɢɥ ɬɪɟɧɢɹ ɢ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɧɚɯɨɞɢɦ ɢɡ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɲɚɪɚ, ɡɚɩɢɫɚɧɧɨɝɨ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ X ɢ Y ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (6.87) ɢ (6.88) ɢ ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɧɟɪɚɜɟɧɫɬɜɨ (6.95). ȼ ɪɟɡɭɥɶɬɚɬɟ ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɨɝɨ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɲɚɪɚ (6.94) ɩɨɥɭɱɢɦ:
F d Pg |
7m1 2m2 |
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(6.96) |
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Ɉɬɜɟɬ: |
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7m1 2m2 |
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7F |
ɢ a |
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2F |
ɩɪɢ F d Pg |
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7m1 2m2 |
7m1 |
2m2 |
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Ɂɚɞɚɱɚ 6.6
ɋɢɫɬɟɦɚ ɬɟɥ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɞɜɭɯ ɝɪɭɡɨɜ, ɫɜɹɡɚɧɧɵɯ ɦɟɠɞɭ ɫɨɛɨɣ ɫ ɩɨɦɨɳɶɸ ɧɟɜɟɫɨɦɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɢ, ɢ ɞɜɭɯ ɨɞɢɧɚɤɨɜɵɯ ɛɥɨɤɨɜ, ɢɡɨɛɪɚɠɟɧɚ ɧɚ ɪɢɫ. 6.14. Ɉɫɶ ɥɟɜɨɝɨ ɛɥɨɤɚ ɡɚɤɪɟɩɥɟɧɚ, ɚ ɩɪɚɜɵɣ ɛɥɨɤ ɫɜɨɛɨɞɧɨ ɥɟɠɢɬ ɧɚ ɧɢɬɢ. ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɟɥ ɫɢɫɬɟɦɵ ɧɟ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɢɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɜɟɪɯɧɨɫɬɟɣ ɛɥɨɤɨɜ. ɋɱɢɬɚɹ ɡɚɞɚɧɧɵɦɢ ɦɚɫɫɵ ɝɪɭɡɨɜ m1 ɢ m2, ɦɚɫɫɵ ɛɥɨɤɨɜ M ɢ ɢɯ ɪɚɞɢɭɫɵ R, ɨɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɨɜ a1 ɢ a2, ɚ ɬɚɤɠɟ ɪɚɡɧɨ-
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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ɫɬɢ ɫɢɥ ɧɚɬɹɠɟɧɢɹ ɧɢɬɟɣ ɩɨ ɨɛɟ ɫɬɨ- |
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ɪɨɧɵ ɤɚɠɞɨɝɨ ɢɡ ɛɥɨɤɨɜ. Ɍɪɟɧɢɟɦ ɜ |
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ɨɫɢ ɛɥɨɤɚ ɩɪɟɧɟɛɪɟɱɶ. |
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Ɋɟɲɟɧɢɟ |
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T1 |
T3 |
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I. ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪ- |
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T4 |
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ɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ |
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T3 |
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ɫɜɹɡɚɧɧɭɸ ɫ ɬɨɱɤɨɣ ɩɨɞɜɟɫɚ ɨɫɢ ɥɟɜɨ- |
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ɝɨ ɛɥɨɤɚ, ɨɫɢ X ɢ Y ɞɟɤɚɪɬɨɜɨɣ ɫɢɫ- |
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T1 |
Mg |
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ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɢɡɨɛɪɚɠɟɧɵ |
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T2 |
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ɧɚ ɪɢɫ. 6.14. ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫ- |
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T2 |
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ɬɟɦɟ ɨɛɚ ɝɪɭɡɚ ɢ ɧɟɡɚɤɪɟɩɥɟɧɧɵɣ ɛɥɨɤ |
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ɞɜɢɠɭɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɜɞɨɥɶ ɨɫɢ X, |
X |
m1g |
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ɩɪɢ ɷɬɨɦ ɛɥɨɤɢ ɜɪɚɳɚɸɬɫɹ ɜɨɤɪɭɝ |
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m2g |
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ɫɨɛɫɬɜɟɧɧɵɯ ɨɫɟɣ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥ |
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ɬɹɠɟɫɬɢ (m1g, m2g, Mg) ɢ ɫɢɥ ɧɚɬɹɠɟ- |
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ɧɢɹ ɧɢɬɟɣ (T1, T2, T3 ɢ T4). ɉɨɫɤɨɥɶɤɭ |
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Ɋɢɫ. 6.14 |
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ɦɚɫɫɵ ɛɥɨɤɨɜ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɨɬ- |
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ɥɢɱɧɵ ɨɬ ɧɭɥɹ, ɬɨ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ ɨɬ ɛɥɨɤɨɜ |
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ɪɚɡɥɢɱɧɵ. |
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II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɨɜ ɢ ɰɟɧɬɪɚ ɦɚɫɫ ɧɟɡɚ- |
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ɤɪɟɩɥɟɧɧɨɝɨ ɛɥɨɤɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪ- |
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ɞɢɧɚɬ: |
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m1a1 |
m1g T1 , |
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(6.97) |
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m2a2 |
m2 g T2 , |
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(6.98) |
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Ma2 |
Mg T2 T3 T4 . |
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(6.99) |
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ɍɪɚɜɧɟɧɢɹ ɦɨɦɟɧɬɨɜ ɞɥɹ ɜɪɚɳɚɸɳɢɯɫɹ ɛɥɨɤɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɨɫɟɣ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɦɚɫɫ (ɫɦ. ɪɢɫ. 6.14), ɢɦɟɸɬ ɜɢɞ: |
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J0 E1 |
T1R T3R , |
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(6.100) |
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J0E2 |
T4 R T3R , |
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(6.101) |
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ɝɞɟ E1 ɢ E2 ɭɝɥɨɜɵɟ ɭɫɤɨɪɟɧɢɹ ɛɥɨɤɨɜ, ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ J0 ɤɨɬɨ- |
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ɪɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɵɯ ɨɫɟɣ ɪɚɜɧɵ (6.44): |
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J0 |
MR2 |
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(6.102) |
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Ⱦɨɩɨɥɧɢɦ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɭɪɚɜɧɟɧɢɹɦɢ ɤɢ- |
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ɧɟɦɚɬɢɱɟɫɤɢɯ ɫɜɹɡɟɣ, ɫɥɟɞɭɸɳɢɦɢ ɢɡ ɭɫɥɨɜɢɣ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ |
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ɧɢɬɟɣ ɢ ɨɬɫɭɬɫɬɜɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɧɢɬɟɣ ɩɨ ɛɥɨɤɚɦ: |
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214 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
a1 2a2 0 , |
(6.103) |
a1 E1R , |
(6.104) |
E1 E2 . |
(6.105) |
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɟɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɞɟɜɹɬɢ |
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ɧɟɢɡɜɟɫɬɧɵɯ ɜɟɥɢɱɢɧ: |
a1 , a2 , T1 , T2 , T3 , T4 , E1 , E2 ɢ J 0 . |
III. ȼɵɪɚɡɢɦ ɜɫɟ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɧɢɬɟɣ ɱɟɪɟɡ ɭɫɤɨɪɟɧɢɟ a1,
ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ (6.97), (6.98) ɢ (6.100) (6.105): |
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T1 m1g m1a1 , |
(6.106) |
T2
T3
T4
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m1g m1 M a1 . |
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(6.107)
(6.108)
(6.109)
ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɢɫɤɨɦɵɟ ɪɚɡɧɨɫɬɢ ɫɢɥ ɧɚɬɹɠɟɧɢɹ ɧɢɬɟɣ ɩɨ ɨɛɟ ɫɬɨɪɨɧɵ ɤɚɠɞɨɝɨ ɢɡ ɛɥɨɤɨɜ ɪɚɜɧɵ:
T T |
T T |
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a . |
(6.110) |
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ɉɨɞɫɬɚɜɥɹɹ (6.106) – (6.109) ɜ ɭɪɚɜɧɟɧɢɟ (6.99) ɧɚɯɨɞɢɦ ɢɫ- |
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ɤɨɦɨɟ ɭɫɤɨɪɟɧɢɟ ɩɟɪɜɨɝɨ ɝɪɭɡɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X: |
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a1 |
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4m1 m2 4M |
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ɂɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ (6.103), ɩɨɥɭɱɢɦ ɢɫɤɨɦɨɟ ɭɫɤɨɪɟɧɢɟ ɜɬɨɪɨɝɨ ɝɪɭɡɚ ɬɚɤɠɟ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X:
a2 |
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m2 M 2m1 |
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(6.112) |
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ɂɫɤɨɦɵɟ ɪɚɡɧɨɫɬɢ ɫɢɥ ɧɚɬɹɠɟɧɢɹ ɧɢɬɟɣ (6.110) ɫ ɭɱɟɬɨɦ |
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(6.111) ɪɚɜɧɵ: |
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2m1 m2 |
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T1 |
T3 |
T3 T4 |
Mg . |
(6.113) |
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4m1 m2 |
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Ɂɚɞɚɱɚ 6.7
ȼ ɫɢɫɬɟɦɟ ɬɟɥ, ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ. 6.15, ɢɡɜɟɫɬɧɵ ɦɚɫɫɚ ɝɪɭɡɚ m1, ɦɚɫɫɚ ɫɬɭɩɟɧɱɚɬɨɝɨ ɛɥɨɤɚ m2, ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɛɥɨɤɚ J0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ ɢ ɪɚɞɢɭɫɵ ɫɬɭɩɟɧɟɣ ɛɥɨɤɚ R1 ɢ R2 (R2 > R1). Ɇɚɫɫɚ
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
215 |
ɧɢɬɟɣ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɚ. ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɚ a1 ɢ ɰɟɧɬɪɚ ɦɚɫɫ ɛɥɨɤɚ a2 ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ.
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Ɋɟɲɟɧɢɟ |
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I. ȼɵɛɟɪɟɦ |
ɥɚɛɨɪɚɬɨɪɧɭɸ |
ɢɧɟɪɰɢɚɥɶ- |
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ɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɩɨ- |
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ɬɨɥɤɨɦ (ɫɦ. ɪɢɫ. 6.15), ɨɫɶ Y ɞɟɤɚɪɬɨɜɨɣ ɫɢɫ- |
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ɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɢɦ ɜɟɪɬɢɤɚɥɶ- |
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ɧɨ ɜɧɢɡ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ |
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ɦɚɫɫɚɦɢ ɬɟɥ ɫɢɫɬɟɦɵ ɛɥɨɤ ɦɨɠɟɬ ɤɚɤ ɜ ɩɨɥɨ- |
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T m2g |
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ɠɢɬɟɥɶɧɨɦ, ɬɚɤ ɢ ɜ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟ- |
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ɧɢɢ ɨɫɢ Y, ɫɨɜɟɪɲɚɹ ɩɪɢ ɷɬɨɦ ɱɢɫɬɨ ɜɪɚɳɚ- |
Y |
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ɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ |
ɨɬɧɨɫɢɬɟɥɶɧɨ |
ɦɝɧɨɜɟɧɧɨɣ |
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ɨɫɢ ɜɪɚɳɟɧɢɹ. ɉɨɫɤɨɥɶɤɭ ɧɢɬɶ, ɩɪɢɤɪɟɩɥɟɧɧɚɹ ɤ ɩɨɬɨɥɤɭ, ɧɟɪɚɫɬɹɠɢɦɚ, ɬɨ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ
ɜɪɚɳɟɧɢɹ ɛɥɨɤɚ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɭ A ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ ɛɥɨɤɚ ɢ ɷɬɨɣ ɧɢɬɢ. ɉɪɢ ɷɬɨɦ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ, ɚ ɟɟ ɜɵɛɪɚɧɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɭɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.15.
II. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ |
ɝɪɭɡɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ Y (ɫɦ. |
ɪɢɫ. 6.15) ɢɦɟɟɬ ɜɢɞ: |
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m1a1 m1g T , |
(6.114) |
ɝɞɟ T – ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ, ɧɚ ɤɨɬɨɪɨɣ ɩɨɞɜɟɲɟɧ ɝɪɭɡ. ɍɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ)
ɛɥɨɤɚ ɡɚɩɢɲɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ:
JE m2 gR T (R2 R1) . |
(6.115) |
Ɂɞɟɫɶ J – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɛɥɨɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ, E – ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɛɥɨɤɚ. ȼ (6.114) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ T0 ɜɟɪɯɧɟɣ ɧɢɬɢ, ɩɪɢɤɪɟɩɥɟɧɧɨɣ ɤ ɩɨɬɨɥɤɭ (ɪɢɫ. 6.15), ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɪɚɜɟɧ ɧɭɥɸ.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɛɥɨɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɵɪɚɡɢɦ ɱɟɪɟɡ ɡɚɞɚɧɧɵɣ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ J0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɦɨɣ Ƚɸɣɝɟɧɫɚ-ɒɬɟɣɧɟɪɚ
(6.42):
J J |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ⱦɨɩɨɥɧɢɦ ɭɪɚɜɧɟɧɢɹ (6.114) – (6.116) ɭɪɚɜɧɟɧɢɹɦɢ ɤɢɧɟɦɚ- |
ɬɢɱɟɫɤɨɣ ɫɜɹɡɢ, ɤɨɬɨɪɵɟ ɫɥɟɞɭɸɬ ɢɡ ɭɫɥɨɜɢɹ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ ɧɢɬɟɣ:
a1 E (R2 R1) , |
(6.117) |
a2 ER1 . |
(6.118) |
III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.114) – (6.118), ɩɨɥɭɱɚɟɦ ɜɵ- |
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ɪɚɠɟɧɢɹ ɞɥɹ ɢɫɤɨɦɵɯ ɭɫɤɨɪɟɧɢɣ ɝɪɭɡɚ a1 ɢ ɰɟɧɬɪɚ ɦɚɫɫ ɛɥɨɤɚ a2: |
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a1 |
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m1 |
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Ʉɚɤ ɜɢɞɢɦ, ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɚ ɢ ɰɟɧɬɪɚ ɛɥɨɤɚ ɧɚɩɪɚɜɥɟɧɵ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɩɪɢ ɥɸɛɨɦ ɫɨɨɬɧɨɲɟɧɢɢ ɦɚɫɫ ɝɪɭɡɚ ɢ ɛɥɨɤɚ (ɫɦ. (6.119) ɢ (6.120)), ɩɪɢ ɷɬɨɦ ɤɚɠɞɨɟ ɢɡ ɬɟɥ ɢɡɧɚɱɚɥɶɧɨ ɩɨɤɨɹɳɟɣɫɹ ɫɢɫɬɟɦɵ ɦɨɠɟɬ ɤɚɤ ɨɩɭɫɤɚɬɶɫɹ, ɬɚɤ ɢ ɩɨɞɧɢɦɚɬɶɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɦɚɫɫ ɬɟɥ ɫɢɫɬɟɦɵ ɢ ɪɚɞɢɭɫɨɜ ɫɬɭɩɟɧɟɣ ɛɥɨɤɚ.
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, ɬɨ ɝɪɭɡ ɛɭɞɟɬ ɨɩɭɫɤɚɬɶɫɹ ɫ ɭɫɤɨɪɟɧɢɟɦ a |
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(6.119), ɚ |
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, ɦɨɞɭɥɶ ɤɨɬɨɪɨɝɨ ɦɨɠɟɬ ɛɵɬɶ ɤɚɤ ɛɨɥɶɲɟ (ɩɪɢ |
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R2 2R1 ), ɬɚɤ ɢ ɦɟɧɶɲɟ ( R2 ! 2R1 ) ɦɨɞɭɥɹ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɚ a1 . |
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ɉɪɢ ɨɛɪɚɬɧɨɦ ɫɨɨɬɧɨɲɟɧɢɢ ɦɚɫɫ ɝɪɭɡ ɛɭɞɟɬ ɩɨɞɧɢɦɚɬɶɫɹ, ɚ |
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ɰɟɧɬɪ ɛɥɨɤɚ ɨɩɭɫɤɚɬɶɫɹ ɫ ɬɟɦ ɠɟ ɫɨɨɬɧɨɲɟɧɢɟɦ ɭɫɤɨɪɟɧɢɣ. |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɪɚɜɟɧɫɬɜɚ ɪɚɞɢɭɫɨɜ ɫɬɭɩɟɧɟɣ |
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ɛɥɨɤɚ R2 |
R1 ɜɧɟ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɦɚɫɫ ɝɪɭɡɚ ɢ ɛɥɨɤɚ |
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ɭɫɤɨɪɟɧɢɟ ɝɪɭɡɚ a1 |
ɪɚɜɧɨ ɧɭɥɸ, ɚ ɭɫɤɨɪɟɧɢɟ ɰɟɧɬɪɚ ɛɥɨɤɚ ɧɚɩɪɚɜ- |
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ɥɟɧɨ ɜɧɢɡ ɢ ɪɚɜɧɨ |
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ɢ ɭɫɤɨɪɟɧɢɟ ɟɝɨ ɰɟɧɬɪɚ ɛɭɞɟɬ ɪɚɜɧɨ |
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Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
217 |
Ɂɚɞɚɱɚ 6.8
ɇɚ ɥɟɠɚɳɭɸ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɚɬɭɲɤɭ ɦɚɫɫɨɣ m = 100 ɝ ɢ ɦɨɦɟɧɬɨɦ ɢɧɟɪɰɢɢ J0 = 400 ɝ ɫɦ2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɟ ɨɫɢ ɧɚɦɨɬɚɧɚ ɧɟɜɟɫɨɦɚɹ ɧɟɪɚɫɬɹɠɢɦɚɹ ɧɢɬɶ. ȼɧɟɲɧɢɣ ɪɚɞɢɭɫ ɤɚɬɭɲɤɢ ɪɚɜɟɧ R = 4 ɫɦ, ɚ ɜɧɭɬɪɟɧɧɢɣ – r = 1 ɫɦ. Ʉ ɤɨɧɰɭ ɧɢɬɢ ɩɨɞ ɭɝɥɨɦ D = 60q ɤ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɢɥɨɠɟɧɚ ɫɢɥɚ F = 0.2 ɇ (ɫɦ. ɪɢɫ. 6.16).
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Ɋɢɫ. 6.16
ɇɚɣɬɢ ɭɫɤɨɪɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɤɚɬɭɲɤɢ a ɞɥɹ ɫɥɭɱɚɹ, ɤɨɝɞɚ ɤɚɬɭɲɤɚ ɞɜɢɠɟɬɫɹ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɢ ɜɟɥɢɱɢɧɭ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɬɚɤɨɟ ɞɜɢɠɟɧɢɟ ɜɨɡɦɨɠɧɨ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɨɫɢ X, Y ɢ Z ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɧɚɩɪɚɜɥɟɧɵ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.16. ɉɨɫɤɨɥɶɤɭ ɞɜɢɠɟɧɢɟ ɤɚɬɭɲɤɢ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɚɪɚɥɥɟɥɶɧɵɦ ɩɥɨɫɤɨɫɬɹɦ, ɜ ɤɨɬɨɪɵɯ ɞɜɢɝɚɸɬɫɹ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ ɤɚɬɭɲɤɢ. ȼ ɨɬɫɭɬɫɬɜɢɟ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɦɝɧɨɜɟɧɧɚɹ ɨɫɶ ɜɪɚɳɟɧɢɹ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɬɨɱɤɢ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ ɤɚɬɭɲɤɢ ɫ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. Ɂɚɞɚɞɢɦ ɜ ɤɚɱɟɫɬɜɟ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ɨɫɢ ɜɪɚɳɟɧɢɹ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɨɫɢ Z ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɨɞɧɨɣ ɢɡ ɬɨɱɟɤ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɹ (ɪɢɫ. 6.16).
II. Ɂɚɩɢɲɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɤɚɬɭɲɤɢ ɜɦɟɫɬɟ ɫ ɧɚɦɨɬɚɧɧɨɣ ɧɚ ɧɟɟ ɧɟɜɟɫɨɦɨɣ ɧɢɬɶɸ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ
218 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɜ ɤɨɬɨɪɭɸ ɜɨɣɞɭɬ ɭɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜɨɤɪɭɝ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɢ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɤɚɬɭɲɤɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ X ɢ Y ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
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Ɂɞɟɫɶ J – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɚɬɭɲɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ, Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɤɚɬɭɲɤɢ, d – ɤɪɚɬɱɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ ɜɪɚɳɟɧɢɹ ɞɨ ɥɢɧɢɢ ɞɟɣɫɬɜɢɹ ɫɢɥɵ F (ɩɥɟɱɨ ɫɢɥɵ F), Fɬɪ – ɫɢɥɚ ɬɪɟɧɢɹ ɩɨɤɨɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɤɚɬɭɲɤɭ ɫɨ ɫɬɨɪɨɧɵ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, N – ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ.
ɍɪɚɜɧɟɧɢɹ (6.121) – (6.123) ɞɨɩɨɥɧɢɦ ɭɪɚɜɧɟɧɢɟɦ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ (ɜ ɫɢɥɭ ɨɬɫɭɬɫɬɜɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɪɢ ɞɜɢɠɟɧɢɢ ɤɚɬɭɲɤɢ), ɬɟɨɪɟɦɨɣ Ƚɸɣɝɟɧɫɚ-ɒɬɟɣɧɟɪɚ (6.42) ɞɥɹ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ J ɢ ɨɱɟɜɢɞɧɵɦ ɝɟɨɦɟɬɪɢɱɟɫɤɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ (ɫɦ. ɪɢɫ. 6.16):
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Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɨɦ ɜɨɡɦɨɠɧɨ ɞɜɢɠɟɧɢɟ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ
Ⱥɦɨɧɬɨɧɚ–Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2 ɜ Ƚɥɚɜɟ 2) ɡɚɩɢɲɟɦ: |
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Fɬɪ d PN . |
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III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (6.121) – (6.126) ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɫɤɨɦɨɝɨ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɤɚɬɭɲɤɢ a, ɩɨɥɭɱɚɟɦ:
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Ⱦɥɹ ɨɬɫɭɬɫɬɜɢɹ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɤɚɬɭɲɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɧɟɪɚɜɟɧɫɬɜɭ:
Ƚɥɚɜɚ 6. Ʉɢɧɟɦɚɬɢɤɚ ɢ ɞɢɧɚɦɢɤɚ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ |
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ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.128) ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ a ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɤɨɦ ɜɵɪɚɠɟɧɢɹ
§ |
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¸ . ȼ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɬɨɱɤɚ ɩɟɪɟɫɟɱɟɧɢɹ ɥɢɧɢɢ ɞɟɣɫɬ- |
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ɜɢɹ ɫɢɥɵ F ɢ ɨɫɢ Y ɥɟɠɢɬ ɥɟɜɟɟ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ,
ɤɚɬɭɲɤɚ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɫɥɟɜɚ ɧɚɩɪɚɜɨ, ɩɨɫɤɨɥɶɤɭ cosD Rr ! 0 . ȼ
ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɤɚɱɟɧɢɟ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫɩɪɚɜɚ ɧɚɥɟɜɨ. ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɡɚɞɚɧ-
ɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, ɩɨɥɭɱɢɦ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɤɚɬɭɲɤɢ ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ:
a = 0.4 ɦ/ɫ2, P t 0,075.
Ɂɚɞɚɱɚ 6.9
Ɉɫɢ ɫɩɥɨɲɧɨɝɨ ɢ ɬɨɧɤɨɫɬɟɧɧɨɝɨ ɰɢɥɢɧɞɪɨɜ ɫɨɟɞɢɧɟɧɵ ɧɟɜɟɫɨɦɨɣ ɲɬɚɧɝɨɣ. ɐɢɥɢɧɞɪɵ ɫɤɚɬɵɜɚɸɬɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D (ɫɦ. ɪɢɫ. 6.17). Ɋɚɞɢɭɫɵ ɰɢɥɢɧɞɪɨɜ ɨɞɢɧɚɤɨɜɵ ɢ ɪɚɜɧɵ R, ɩɪɢ ɷɬɨɦ ɦɚɫɫɚ ɫɩɥɨɲɧɨɝɨ ɰɢɥɢɧɞɪɚ ɪɚɜɧɚ m1, ɚ ɬɨɧɤɨɫɬɟɧɧɨɝɨ m2. ɇɚɣɬɢ ɭɝɨɥ D , ɩɪɢ ɤɨɬɨɪɨɦ ɰɢɥɢɧɞɪɵ ɛɭɞɭɬ ɫɤɚɬɵɜɚɬɶɫɹ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ.
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Ɋɢɫ. 6.17
220 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɤɥɢɧɨɦ, ɨɫɢ X ɢ Y ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɤɨɬɨɪɨɣ ɢɡɨɛɪɚɠɟɧɵ ɧɚ ɪɢɫ. 6.17. ɇɚ ɰɢɥɢɧɞɪɵ ɜ ɩɪɨɰɟɫɫɟ ɢɯ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɞɟɣɫɬɜɭɸɬ ɫɢɥɵ ɬɹɠɟɫɬɢ m1g ɢ m2 g , ɫɢɥɵ
ɬɪɟɧɢɹ Fɬɪ1 ɢ Fɬɪ2 , ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ N1 ɢ N2 , ɚ
ɬɚɤɠɟ ɫɢɥɵ ɪɟɚɤɰɢɢ ɲɬɚɧɝɢ T (ɫɦ. ɪɢɫ. 6.17).
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɨɜ ɦɚɫɫ ɰɢɥɢɧɞɪɨɜ ɜ
ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɢ X ɢ Y ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ: |
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m1g sin D T Fɬɪ1 , |
(6.130) |
m2a m2 g sin D T Fɬɪ2 , |
(6.131) |
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N1 m1g cosD , |
(6.132) |
0 |
N2 m2 g cosD , |
(6.133) |
ɉɪɢ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɣ (6.130) ɢ (6.131) ɭɱɬɟɧɨ, ɱɬɨ ɫɢɥɵ ɪɟɚɤɰɢɢ ɲɬɚɧɝɢ T, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɰɢɥɢɧɞɪɵ, ɪɚɜɧɵ ɩɨ ɦɨɞɭɥɸ. ɗɬɨ ɥɟɝɤɨ ɞɨɤɚɡɚɬɶ, ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɧɟɜɟɫɨɦɨɣ ɲɬɚɧɝɢ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X ɢ ɬɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ. ɍɫɤɨɪɟɧɢɹ ɰɟɧɬɪɨɜ ɦɚɫɫ a ɰɢɥɢɧɞɪɨɜ ɬɚɤɠɟ ɪɚɜɧɵ, ɩɨɫɤɨɥɶɤɭ ɲɬɚɧɝɭ ɫɱɢɬɚɟɦ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ.
ɍɪɚɜɧɟɧɢɹ ɦɨɦɟɧɬɨɜ ɞɥɹ ɰɢɥɢɧɞɪɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ ɜɪɚɳɟɧɢɹ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɦɚɫɫ, ɢɦɟɸɬ ɜɢɞ (ɫɦ. (6.47)):
J1E |
Fɬɪ1R , |
(6.134) |
J2 E |
Fɬɪ2 R . |
(6.135) |
Ɂɞɟɫɶ J1 ɢ J2 ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɫɩɥɨɲɧɨɝɨ ɢ ɬɨɧɤɨɫɬɟɧɧɨɝɨ ɰɢ-
ɥɢɧɞɪɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɦɚɫɫ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ; E – ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ, ɨɞɢɧɚɤɨɜɨɟ ɞɥɹ ɫɩɥɨɲɧɨɝɨ ɢ
ɬɨɧɤɨɫɬɟɧɧɨɝɨ ɰɢɥɢɧɞɪɨɜ ɜ ɫɢɥɭ ɭɪɚɜɧɟɧɢɹ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ, ɤɨɬɨɪɨɟ ɫɥɟɞɭɟɬ ɢɡ ɭɫɥɨɜɢɹ ɤɚɱɟɧɢɹ ɰɢɥɢɧɞɪɚ ɛɟɡ ɩɪɨɫɤɚɥɶɡɵɜɚɧɢɹ:
ER a . |
(6.136) |
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɢɡɜɟɫɬɧɵɦɢ ɜɵɪɚɠɟɧɢɹɦɢ ɞɥɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɨɞɧɨɪɨɞɧɵɯ ɫɩɥɨɲɧɨɝɨ (6.44) ɢ ɬɨɧɤɨɫɬɟɧɧɨɝɨ ɰɢɥɢɧɞɪɨɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɢɯ ɰɟɧɬɪɵ ɦɚɫɫ:
J1 |
m R2 |
, |
(6.137) |
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