Механика.Методика решения задач
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ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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Ƚɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ – ɩɪɨɰɟɫɫ, ɩɪɢ ɤɨɬɨɪɨɦ ɮɢɡɢɱɟ- |
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ɫɤɚɹ ɜɟɥɢɱɢɧɚ [(t) ɦɟɧɹɟɬɫɹ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ |
(ɫɦ. |
ɪɢɫ. 8.3).
[(t) |
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Ɋɢɫ. 8.3. Ɂɚɜɢɫɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ [(t) ɨɬ ɜɪɟɦɟɧɢ ɜ ɫɥɭɱɚɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ
ɋɜɨɛɨɞɧɵɟ (ɫɨɛɫɬɜɟɧɧɵɟ) ɤɨɥɟɛɚɧɢɹ – ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ,
ɩɪɟɞɨɫɬɚɜɥɟɧɧɨɣ ɫɚɦɨɣ ɫɟɛɟ (ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɜɧɟɲɧɢɯ ɭɫɥɨɜɢɹɯ).
8.1.1. ɋɨɛɫɬɜɟɧɧɵɟ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɤɨɥɟɛɚɧɢɹ
ɍɪɚɜɧɟɧɢɟ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨ-
ɪɨɟ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ, ɢɦɟɟɬ ɜɢɞ:
[ Z2[ 0 |
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ɝɞɟ [ – ɨɞɧɚ ɢɡ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ – ɧɟɡɚɜɢɫɢɦɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɨɥɨɠɟɧɢɟ ɬɟɥ ɫɢɫɬɟɦɵ; Z0 – ɭɝɥɨ-
ɜɚɹ ɱɚɫɬɨɬɚ ɢ T |
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– ɩɟɪɢɨɞ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨ- |
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ɥɟɛɚɧɢɣ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɫɢɫɬɟɦɵ.
Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɩɪɢ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɹɯ (ɡɚɜɢɫɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɨɬ ɜɪɟɦɟɧɢ) – ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ: |
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[(t) Acos Z0t M0 . |
(8.2) |
Ɂɞɟɫɶ Z0t M0 – ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ; A – ɚɦɩɥɢɬɭɞɚ ɢ M0 |
– ɧɚ- |
ɱɚɥɶɧɚɹ ɮɚɡɚ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ – ɡɧɚɱɟɧɢɹɦɢ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ
[0 { [ t t0 ɢ ɫɤɨɪɨɫɬɶɸ ɟɟ ɢɡɦɟɧɟɧɢɹ [0 { [ t t0 ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0:
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɨɬɫɱɟɬɚ ɫɨɜɦɟɫɬɢɦ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɬɟɥɚ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɧɟɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɟ (ɪɢɫ. 8.4).
ɇɚ ɬɟɥɨ ɜ ɩɪɨɰɟɫɫɟ ɤɨɥɟɛɚɧɢɣ ɞɟɣɫɬɜɭɟɬ ɭɩɪɭɝɚɹ ɫɢɥɚ Fɭɩɪ ɫɨ
ɫɬɨɪɨɧɵ |
ɩɪɭɠɢɧɵ, |
ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ |
ɡɚɤɨɧɭ Ƚɭɤɚ (ɫɦ. |
ɩ. 2.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ |
ɦɚɬɟɪɢɚɥ ɜ Ƚɥɚɜɟ 2). |
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ |
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ɬɟɥɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢɦɟɟɬ ɜɢɞ: |
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kx , |
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ɝɞɟ m ɦɚɫɫɚ ɬɟɥɚ, k ɤɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ. ɉɪɟɨɛɪɚɡɭɟɦ (8.6) ɤ ɜɢɞɭ ɭɪɚɜɧɟɧɢɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚ-
ɧɢɣ:
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x m x 0 . |
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ɋɪɚɜɧɢɜɚɹ (8.7) ɫ (8.1), ɞɥɹ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɩɪɭ- |
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ɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ ɩɨɥɭɱɢɦ: |
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(8.8) |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɜɟɪɬɢɤɚɥɶɧɨɦ ɪɚɫɩɨɥɨɠɟɧɢɢ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ ɟɝɨ ɱɚɫɬɨɬɚ ɧɟ ɢɡɦɟɧɢɬɫɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ (8.7) ɩɪɢ ɜɵɛɨɪɟ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɜɟɪɬɢɤɚɥɶɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥɚ ɜ ɩɨɥɨɠɟɧɢɢ ɟɝɨ ɪɚɜɧɨɜɟɫɢɹ.
Ɂɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɬɟɥɚ, ɩɪɢɤɪɟɩɥɟɧɧɨɝɨ ɤ ɩɪɭɠɢɧɟ, ɢ ɢɡɦɟ-
ɧɟɧɢɹ ɟɝɨ ɫɤɨɪɨɫɬɢ ɚɧɚɥɨɝɢɱɧɨ (8.2) ɢ (8.5) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ: |
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Acos Z0t M0 , |
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AZ0 sin Z0t M0 . |
(8.10) |
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Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ ɤɢɧɟɬɢɱɟ- |
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ɫɤɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ, ɩɪɢɤɪɟɩɥɟɧɧɨɝɨ ɤ ɩɪɭɠɢɧɟ: |
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E k (t) |
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mA |
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sin2 Z0 t M0 |
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kA2 |
sin2 |
Z0t M0 . |
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ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ, ɪɚɫɩɨɥɨɠɟɧɧɨɝɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨ, ɪɚɜɧɚ ɷɧɟɪɝɢɢ ɭɩɪɭɝɨɣ ɞɟɮɨɪɦɚɰɢɢ ɩɪɭɠɢɧɵ:
E p (t) |
kx2 (t) |
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cos2 Z0t M0 . |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɢ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ ɢɡɦɟɧɹɸɬɫɹ ɜ ɩɪɨɬɢɜɨɮɚɡɟ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɫ ɱɚɫɬɨɬɨɣ 2Z0 ɢ ɨɞɢɧɚɤɨɜɵɦɢ ɚɦɩɥɢɬɭɞɚɦɢ (ɫɦ. ɪɢɫ. 8.5). Ɇɟɯɚɧɢɱɟɫɤɚɹ
ɷɧɟɪɝɢɹ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜ ɩɪɨɰɟɫɫɟ ɤɨɥɟɛɚɧɢɣ:
E E |
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Ɋɢɫ. 8.5. Ɂɚɜɢɫɢɦɨɫɬɢ ɤɢɧɟɬɢɱɟɫɤɨɣ Ek ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ Ep ɷɧɟɪɝɢɣ ɦɚɹɬɧɢɤɚ ɨɬ ɜɪɟɦɟɧɢ ɜ ɫɥɭɱɚɟ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ
Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ, ɩɨɞɜɟɲɟɧɧɚɹ ɧɚ ɧɟɜɟɫɨɦɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɢ ɜ ɩɨɥɟ ɫɢɥ ɬɹɠɟɫɬɢ (ɫɦ.
ɪɢɫ. 8.6). |
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Ɋɚɫɫɦɨɬɪɢɦ ɤɨɥɟɛɚɧɢɹ |
ɦɚɬɟɦɚɬɢ- |
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ɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧ- |
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ɬɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɜ ɨɞɧɨɣ |
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ɩɥɨɫɤɨɫɬɢ. |
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ɥɚɛɨɪɚɬɨɪɧɭɸ |
ɢɧɟɪɰɢ- |
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ɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɬɟ- |
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ɥɨɦ, ɤ ɤɨɬɨɪɨɦɭ ɩɨɞɜɟɲɟɧ ɦɚɬɟɦɚɬɢɱɟ- |
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ɫɤɢɣ ɦɚɹɬɧɢɤ. |
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɦɨ- |
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ɦɟɧɬɨɜ (6.39) ɞɥɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ |
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ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ |
Ɋɢɫ. 8.6. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ |
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ɬɨɱɤɭ ɩɨɞɜɟɫɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨ- |
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ɫɬɢ ɤɨɥɟɛɚɧɢɣ ɦɚɹɬɧɢɤɚ (ɫɦ. ɪɢɫ. 8.6):
dL |
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ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɝɞɟ L l(mlD) |
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D ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɨɬ- |
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ɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ, D ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, m ɢ l ɦɚɫɫɚ ɢ ɞɥɢɧɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ, M mg mgl sin D ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ
ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɣ ɠɟ ɨɫɢ.
ɉɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɭɪɚɜɧɟɧɢɟ (8.14) ɫɜɨɞɢɬɫɹ ɤ ɜɢɞɭ ɭɪɚɜɧɟɧɢɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ (8.1):
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mgl sinD , |
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ɋɪɚɜɧɢɜɚɹ (8.16) ɫ (8.1), ɞɥɹ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɦɚ- |
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ɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɩɨɥɭɱɢɦ: |
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Z0 |
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(8.17) |
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Ɂɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɢ ɢɡɦɟɧɟɧɢɹ ɟɝɨ |
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ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɚɧɚɥɨɝɢɱɧɨ (8.2) ɢ (8.5) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ: |
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D(t) |
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Acos Z0t M0 , |
(8.18) |
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AZ0 sin Z0t M0 . |
(8.19) |
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Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ ɤɢ- |
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ɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɩɨɞɜɟɲɟɧɧɨɣ ɧɚ ɧɢɬɢ: |
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mglA |
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sin2 Z0t M0 . |
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ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ ɷɧɟɪɝɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɂɟɦɥɢ. ȿɫɥɢ ɡɚ ɧɨɥɶ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɪɢɧɹɬɶ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɦɚɹɬɧɢɤɚ, ɬɨ ɟɝɨ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɪɢ ɨɬɤɥɨɧɟɧɢɢ ɧɚ ɭɝɨɥ D ɪɚɜɧɚ:
E p mgl(1 cosD) # |
mglA2 |
cos2 Z0t M0 . |
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Ʉɢɧɟɬɢɱɟɫɤɚɹ ɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ, ɬɚɤ ɠɟ ɤɚɤ ɢ ɜ ɫɥɭɱɚɟ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ, ɢɡɦɟɧɹɸɬɫɹ ɜ ɩɪɨɬɢɜɨɮɚɡɟ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɫ ɱɚɫɬɨɬɨɣ 2Z0 ɢ ɨɞɢɧɚɤɨ-
ɜɵɦɢ ɚɦɩɥɢɬɭɞɚɦɢ (ɫɦ. ɪɢɫ. 8.5). Ɇɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɤɨɥɟɛɚɧɢɣ ɢ ɪɚɜɧɚ:
276 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɏɢɡɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɟ ɬɟɥɨ, ɩɨɞɜɟɲɟɧ- |
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ɧɨɟ ɜ ɩɨɥɟ ɫɢɥ ɬɹɠɟɫɬɢ (ɫɦ. ɪɢɫ. 8.7). |
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Ɋɚɫɫɦɨɬɪɢɦ |
ɤɨɥɟɛɚɧɢɹ |
ɮɢɡɢɱɟɫɤɨɝɨ |
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ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ, |
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ɜ ɩɪɨɰɟɫɫɟ ɤɨɬɨɪɵɯ ɜɫɟ ɦɚɬɟɪɢɚɥɶɧɵɟ ɬɨɱɤɢ |
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ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɞɜɢɠɭɬɫɹ ɜ ɩɚɪɚɥ- |
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J0 |
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ɥɟɥɶɧɵɯ ɩɥɨɫɤɨɫɬɹɯ. |
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ȼɵɛɟɪɟɦ ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ |
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mg |
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ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɬɟɥɨɦ, ɤ ɤɨɬɨ- |
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ɪɨɦɭ ɩɨɞɜɟɲɟɧ ɮɢɡɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ. |
Ɂɚɩɢ- |
Ɋɢɫ. 8.7. Ɏɢɡɢɱɟɫɤɢɣ |
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ɲɟɦ ɭɪɚɜɧɟɧɢɟ |
ɦɨɦɟɧɬɨɜ (6.48) ɞɥɹ |
ɚɛɫɨ- |
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ɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨ- |
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ɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɤɨɥɟɛɚ- |
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ɧɢɣ ɦɚɹɬɧɢɤɚ (ɫɦ. ɪɢɫ. 8.7): |
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M mg . |
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(8.23) |
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JD |
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Ɂɞɟɫɶ D ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, J |
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ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ |
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ɨɫɢ, M mg |
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mgl sin D ɦɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ, |
ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ |
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ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɣ ɠɟ ɨɫɢ, m ɦɚɫɫɚ ɮɢɡɢɱɟ- |
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ɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɢ l ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɰɟɧɬɪɚ ɦɚɫɫ ɦɚɹɬɧɢɤɚ ɞɨ ɬɨɱɤɢ |
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ɟɝɨ ɩɨɞɜɟɫɚ. |
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ɉɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɭɪɚɜɧɟɧɢɟ (8.23) ɫɜɨ- |
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ɞɢɬɫɹ ɤ ɜɢɞɭ ɭɪɚɜɧɟɧɢɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ (8.1): |
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mgl sinD . |
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(8.24) |
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mgl |
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ɋɪɚɜɧɢɜɚɹ (8.25) ɫ (8.1), ɞɥɹ ɭɝɥɨɜɨɣ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɮɢ- |
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ɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɩɨɥɭɱɢɦ: |
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Z0 |
mgl |
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(8.26) |
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ɂɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ Ƚɸɣɝɟɧɫɚ ɒɬɟɣɧɟɪɚ (6.42), ɜɵɪɚɡɢɦ ɭɝɥɨɜɭɸ ɱɚɫɬɨɬɭ ɤɨɥɟɛɚɧɢɣ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɱɟɪɟɡ ɟɝɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ J 0 ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ ɩɚ-
ɪɚɥɥɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ:
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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(8.27) |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɥɭɱɚɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɢ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɨɜ ɜ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɜɵɫɬɭɩɚɟɬ ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ.
Ɂɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɢ ɢɡɦɟɧɟɧɢɹ ɟɝɨ ɭɝ-
ɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɢɞɟɧɬɢɱɧɵ ɫɥɭɱɚɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ: |
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D(t) |
Acos Z0t M0 , |
(8.28) |
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AZ0 sin Z0t M0 . |
(8.29) |
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D(t) |
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Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ (ɫɦ. (7.7) |
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ɜ ɩ. 7.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ Ƚɥɚɜɵ 7): |
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E k |
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sin2 Z0t M0 . |
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ȿɫɥɢ ɡɚ ɧɨɥɶ ɨɬɫɱɟɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɩɪɢɧɹɬɶ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ ɦɚɹɬɧɢɤɚ, ɬɨ ɟɝɨ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɪɢ ɨɬɤɥɨɧɟɧɢɢ ɧɚ ɭɝɨɥ D ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ:
E p |
mgl(1 cosD) # |
mglA2 |
cos2 Z0t M0 . |
(8.31) |
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Ɇɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɮɢɡɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ ɪɚɜɧɚ: |
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p |
mglA2 |
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(8.32) |
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8.1.2. ɋɨɛɫɬɜɟɧɧɵɟ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜ ɫɥɭɱɚɟ ɫɨɛɫɬɜɟɧɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨ-
ɥɟɛɚɧɢɣ ɢɦɟɟɬ ɜɢɞ: |
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[ (t) 2G[ Z02[ 0 , |
(8.33) |
ɝɞɟ G – ɤɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ (ɨɩɪɟɞɟɥɹɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɫɢɫɬɟɦɵ).
Ɋɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8.33) ɪɚɡɥɢɱɧɵ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ ɢ ɱɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ.
ɋɥɭɱɚɣ ɫɨɛɫɬɜɟɧɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɫ ɡɚɬɭɯɚɧɢɟɦ ɦɟɧɶɲɟ ɤɪɢɬɢɱɟɫɤɨɝɨ (G < Z0).
Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ (ɫɦ. ɪɢɫ. 8.8):
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
279 |
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Ⱦɨɛɪɨɬɧɨɫɬɶ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ Q ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɬ-
ɧɨɲɟɧɢɟɦ ɫɪɟɞɧɟɣ ɡɚ ɩɟɪɢɨɞ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɤ ɫɪɟɞɧɟɣ ɦɨɳɧɨɫɬɢ ɩɨɬɟɪɶ:
Q { 2S |
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ɋɥɭɱɚɣ ɚɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɡɚɬɭɯɚɧɢɟɦ ɛɨɥɶɲɟ
ɤɪɢɬɢɱɟɫɤɨɝɨ (G > Z0).
Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ:
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[ (t) A e © |
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ɝɞɟ A1 ɢ A2 – ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ A1 ɢ A2 ɦɨɝɭɬ ɛɵɬɶ ɤɚɤ ɨɞɧɨɝɨ, ɬɚɤ ɢ ɪɚɡɧɵɯ ɡɧɚɤɨɜ.
ɉɪɢ A1 ! 0 ɨɛɨɛɳɟɧɧɚɹ ɤɨɨɪɞɢɧɚɬɚ [(t) ɦɨɧɨɬɨɧɧɨ ɫɬɪɟ-
A2
ɦɢɬɫɹ ɤ ɧɭɥɸ ɩɪɢ t o f (ɫɦ. ɪɢɫ. 8.9).
[(t)
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Ɋɢɫ. 8.9. Ɂɚɜɢɫɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ [(t) ɨɬ ɜɪɟɦɟɧɢ |
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ɜ ɫɥɭɱɚɟ ɚɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɩɪɢ |
A1 |
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ɉɪɢ |
A1 |
0 ɨɛɨɛɳɟɧɧɚɹ ɤɨɨɪɞɢɧɚɬɚ [ (t) |
ɜ ɧɟɤɨɬɨɪɵɣ ɦɨ- |
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ɦɟɧɬ ɜɪɟɦɟɧɢ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ, ɡɚɬɟɦ ɞɨɫɬɢɝɚɟɬ ɥɨɤɚɥɶɧɨɝɨ ɷɤɫ-
280 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɬɪɟɦɭɦɚ ɢ ɞɚɥɟɟ ɦɨɧɨɬɨɧɧɨ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ ɩɪɢ t o f (ɫɦ.
ɪɢɫ. 8.10).
[(t)
t
Ɋɢɫ. 8.10. Ɂɚɜɢɫɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ [(t) ɨɬ ɜɪɟɦɟ-
ɧɢ ɜ ɫɥɭɱɚɟ ɚɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɩɪɢ A1 0
A2
ɋɥɭɱɚɣ ɤɪɢɬɢɱɟɫɤɨɝɨ ɡɚɬɭɯɚɧɢɹ (G = Z0). Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ:
[ (t) ( A A t)e G t , |
(8.42) |
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ɝɞɟ A1 ɢ A2 – ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ȼɨɡɦɨɠɧɵɟ ɜɢɞɵ ɡɚɜɢɫɢɦɨɫɬɢ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɨɬ ɜɪɟɦɟɧɢ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɢɡɨɛɪɚɠɟɧɵ ɧɚ ɪɢɫ. 8.11.
[(t)
t
Ɋɢɫ. 8.11. Ɂɚɜɢɫɢɦɨɫɬɶ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ [(t) ɨɬ ɜɪɟɦɟɧɢ ɜ ɫɥɭɱɚɟ ɤɪɢɬɢɱɟɫɤɨɝɨ ɡɚɬɭɯɚɧɢɹ
ɇɟɡɚɜɢɫɢɦɨ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ G ɢ ɱɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ Z0 ɨɛɨɛɳɟɧɧɚɹ ɤɨɨɪɞɢɧɚɬɚ [(t) ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ ɩɪɢ t o f .