Механика.Методика решения задач
.pdfȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɝɞɟ xɩɪ – ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ ɤɪɟɩɥɟɧɢɹ ɥɟɜɨɣ ɧɢɬɢ ɤ ɩɪɭɠɢɧɟ, xɩɪ,0
– ɤɨɨɪɞɢɧɚɬɚ ɬɨɣ ɠɟ ɬɨɱɤɢ ɩɪɢ ɧɟɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɟ.
ɉɨɫɤɨɥɶɤɭ ɧɢɬɢ ɩɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɧɟɪɚɫɬɹɠɢɦɵ, ɢɡɦɟɧɟɧɢɟ ɭɝɥɚ ɩɨɜɨɪɨɬɚ ɛɥɨɤɚ ɢ ɢɡɦɟɧɟɧɢɟ ɤɨɨɪɞɢɧɚɬ ɬɟɥɚ ɢ ɬɨɱɤɢ ɤɪɟɩɥɟɧɢɹ ɧɢɬɢ ɤ ɩɪɭɠɢɧɟ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɹɦɢ:
ǻD |
ǻx |
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ǻD |
ǻxɩɪ |
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(8.117) |
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Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (8.116) ɩɨ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɭɝɥɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ɛɥɨɤɚ ɢ ɭɫɤɨɪɟɧɢɹ ɬɟɥɚ:
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(8.118) |
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ɂɫɤɥɸɱɚɹ ɢɡɦɟɧɟɧɢɟ ɭɝɥɚ ɩɨɜɨɪɨɬɚ ɛɥɨɤɚ ǻD ɢɡ (8.116) ɢ (8.117), ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ ɢɡɦɟɧɟɧɢɣ ɤɨɨɪɞɢɧɚɬ ɬɨɱɤɢ ɤɪɟɩɥɟɧɢɹ ɥɟɜɨɣ ɧɢɬɢ ɤ ɩɪɭɠɢɧɟ ɢ ɬɟɥɚ:
ǻx |
ɩɪ |
ǻx |
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(8.119) |
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ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ (8.119), ɩɪɟɨɛɪɚɡɭɟɦ (8.115) ɤ ɜɢɞɭ: |
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ɝɞɟ x0 – ɤɨɨɪɞɢɧɚɬɚ ɬɟɥɚ ɜ ɩɨɥɨɠɟɧɢɢ, ɤɨɝɞɚ ɩɪɭɠɢɧɚ ɧɟ ɪɚɫɬɹɧɭ-
ɬɚ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɡɚɩɢɫɚɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (8.113), (8.114), (8.118) ɢ (8.120), ɤɨɬɨɪɚɹ ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɟɥɚ.
III. ɂɫɤɥɸɱɚɹ D , T1 ɢ T2 ɢɡ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (8.113),
(8.114), (8.118) ɢ (8.120), ɩɨɥɭɱɚɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ ɤɨɨɪɞɢɧɚɬɵ ɬɟɥɚ x :
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(x x0 ) |
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(8.121) |
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¸x |
K x mg FȺɪɯ |
r 2 |
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ɇɚɣɞɟɦ ɤɨɨɪɞɢɧɚɬɭ ɬɟɥɚ |
xðàâí |
ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ, ɩɪɢ |
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ɤɨɬɨɪɨɦ ɨɬɫɭɬɫɬɜɭɸɬ ɤɨɥɟɛɚɧɢɹ ( x |
ɢ x |
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x (mg F ) |
r 2 |
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(8.122) |
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ɪɚɜɧ |
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Ⱥɪɯ |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɋɞɟɥɚɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ, ɨɡɧɚɱɚɸɳɭɸ ɜɜɟɞɟɧɢɟ ɤɨɨɪɞɢ- |
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ɧɚɬɵ ɬɟɥɚ [ , ɨɬɫɱɢɬɵɜɚɟɦɨɣ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ: |
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[ x xɪɚɜɧ . |
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ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɡ (8.121) ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɤɨɨɪɞɢɧɚɬɵ |
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ɬɟɥɚ [ : |
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Kr 2 |
kR 2 |
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(8.124) |
mr 2 J |
mr 2 J |
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ɤɨɬɨɪɨɟ ɢɦɟɟɬ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ (ɫɦ. (8.33) ɜ ɩ. 8.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ).
ɋɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ (8.33), ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɡɚɬɭɯɚɧɢɹ G |
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ɢ ɱɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ Z0 |
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ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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G |
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Z0 |
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(8.126) |
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mr 2 J |
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Ɋɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (8.124) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ |
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[ (t) |
Ⱥe G t cos(Zt M0 ) , |
(8.127) |
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ɝɞɟ Z |
Z02 G 2 ɱɚɫɬɨɬɚ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ, ɨɩɪɟɞɟɥɹɟɦɚɹ |
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ɩɚɪɚɦɟɬɪɚɦɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ, A ɚɦɩɥɢɬɭɞɚ ɢ M0 ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨ-
ɜɢɹɦɢ.
ɉɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɜɵɛɨɪɟ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
x(t) |
xɪɚɜɧ Ⱥe G t cos(Zt M0 ) , |
(8.128) |
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ɋ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɧɧɵɯ ɜ ɡɚɞɚɱɟ, |
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x(t |
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xɪɚɜɧ , |
(8.129) |
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(8.130) |
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ɧɚɯɨɞɢɦ ɚɦɩɥɢɬɭɞɭ ɤɨɥɟɛɚɧɢɣ ɬɟɥɚ A ɢ ɧɚɱɚɥɶɧɭɸ ɮɚɡɭ M0 : |
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A |
V0 |
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(8.131) |
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Z2 |
G 2 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
x(t)
xɪɚɜɧ |
t |
Ɋɢɫ. 8.27
Ɂɚɞɚɱɚ 8.7
(ɋɜɨɛɨɞɧɵɟ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ)
Ɍɨɧɤɢɣ ɨɞɧɨɪɨɞɧɵɣ ɞɢɫɤ ɦɚɫɫɨɣ m ɢ ɪɚɞɢɭɫɨɦ R, ɩɨɞɜɟɲɟɧɧɵɣ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦ ɩɨɥɨɠɟɧɢɢ ɤ ɭɩɪɭɝɨɣ ɧɢɬɢ, ɨɬɤɥɨɧɢɥɢ ɧɚ ɭɝɨɥ D0 ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɢ ɨɬɩɭɫɬɢɥɢ ɫ ɧɭɥɟɜɨɣ ɧɚɱɚɥɶ-
ɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ. Ⱦɢɫɤ ɫɨɜɟɪɲɚɟɬ ɤɪɭɬɢɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɜ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ (ɫɦ. ɪɢɫ. 8.28). ɋɢɥɚ ɜɹɡɤɨ-
ɝɨ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ |
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ɩɨɜɟɪɯɧɨɫɬɢ ɞɢɫɤɚ ɫɨ ɫɬɨɪɨɧɵ ɠɢɞɤɨɫɬɢ, |
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ɪɚɜɧɚ fɜ |
Kȣ , ɝɞɟ K |
const , X – ɫɤɨɪɨɫɬɶ |
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ɞɚɧɧɨɝɨ ɷɥɟɦɟɧɬɚ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɠɢɞ- |
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ɤɨɫɬɢ. Ɇɨɦɟɧɬ ɭɩɪɭɝɢɯ ɫɢɥ ɫɨ ɫɬɨɪɨɧɵ ɧɢɬɢ |
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Ɋɢɫ. 8.28 |
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ɪɚɜɟɧ M ɭɩɪ |
DD , ɝɞɟ |
D – ɩɨɫɬɨɹɧɧɵɣ ɤɨɷɮ- |
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ɮɢɰɢɟɧɬ, D – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ. ɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ.
Ɋɟɲɟɧɢɟ
I. ɂɫɩɨɥɶɡɭɟɦ ɞɢɧɚɦɢɱɟɫɤɢɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ. Ⱦɢɫɤ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ. ɇɚ ɧɟɝɨ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ, ɭɩɪɭɝɚɹ ɫɢɥɚ ɫɨ ɫɬɨɪɨɧɵ ɧɢɬɢ ɢ ɫɢɥɚ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɫɨ ɫɬɨɪɨɧɵ ɠɢɞɤɨɫɬɢ. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɭɤɚɡɚɧɧɵɯ ɫɢɥ ɞɢɫɤ ɫɨɜɟɪɲɚɟɬ ɡɚɬɭɯɚɸɳɢɟ ɤɪɭɬɢɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɞɢɫɤɚ.
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (ɫɦ. (6.48) ɜ ɩ. 6.1.2. Ƚɥɚɜɵ 6) ɞɥɹ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ:
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M ɜ , |
(8.137) |
JD M ɭɩɪ |
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɝɞɟ J – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, Mɜ – ɦɨɦɟɧɬ ɫɢɥ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ. Ɇɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ ɪɚɜɟɧ ɧɭɥɸ.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɞɢɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɨɫɶɸ ɜɪɚɳɟɧɢɹ, ɪɚɜɟɧ (ɫɦ. (6.44) ɜ Ƚɥɚɜɟ 6):
J |
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mR2 |
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(8.138) |
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Ɂɚɩɢɲɟɦ ɦɨɦɟɧɬ dMɜ ɫɢɥɵ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɤɨɥɶɰɟ- |
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ɨɛɪɚɡɧɵɣ |
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ɷɥɟɦɟɧɬ ɩɨɜɟɪɯɧɨɫɬɢ |
ɞɢɫɤɚ ɪɚɞɢɭɫɨɦ |
r ɢ ɩɥɨɳɚɞɶɸ |
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dS = 2Srdr: |
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dM |
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ɜ 2SrdrKXr 2Sr KdrD . |
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ɍɱɢɬɵɜɚɹ, ɱɬɨ ɫɢɥɚ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ ɞɟɣɫɬɜɭɟɬ ɧɚ ɨɛɟ ɩɨɜɟɪɯɧɨɫɬɢ ɞɢɫɤɚ, ɧɚɣɞɟɦ ɫɭɦɦɚɪɧɵɣ ɦɨɦɟɧɬ ɫɢɥ ɬɪɟɧɢɹ, ɢɧɬɟɝɪɢɪɭɹ ɩɨ ɨɛɟɢɦ ɩɨɜɟɪɯɧɨɫɬɹɦ ɞɢɫɤɚ:
R |
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M ɜ 2 2SKD ³r3dr SKR4D . |
(8.140) |
0 |
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III. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ |
ɩɨɥɭɱɚɟɦ ɩɨɞɫɬɚɧɨɜɤɨɣ |
(8.140) ɜ (8.137) ɫ ɭɱɟɬɨɦ (8.138) ɢ ɡɚɞɚɧɧɨɝɨ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɨɦɟɧɬɚ ɭɩɪɭɝɢɯ ɫɢɥ:
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ɋɪɚɜɧɢɜɚɹ (8.141) ɫ ɭɪɚɜɧɟɧɢɟɦ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ (8.33), ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ G ɢ ɱɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɞɢɫɤɚ Z0 :
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ɫɥɚɛɨɝɨ ɡɚɬɭɯɚɧɢɹ ( G Z0 ) |
ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ |
ȼ |
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(8.141) ɢɦɟɟɬ ɜɢɞ (ɫɦ. (8.34)): |
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Z2 G 2 |
– ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ |
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ɞɢɫɤɚ, A – ɚɦɩɥɢɬɭɞɚ, M0 – ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ.
ɋ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɡɚɞɚɧɧɵɯ ɜ ɡɚɞɚɱɟ,
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɧɚɯɨɞɢɦ ɚɦɩɥɢɬɭɞɭ A ɢ ɧɚɱɚɥɶɧɭɸ ɮɚɡɭ M0 ɤɨɥɟɛɚɧɢɣ ɞɢɫɤɚ: |
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A |
D0 , |
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0 . |
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ɂɫɤɨɦɵɣ ɜ ɡɚɞɚɱɟ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ ɨɩɢɫɵɜɚɟɬ ɡɚɬɭ- |
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ɯɚɸɳɢɟ |
ɤɨɥɟɛɚɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ |
ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ (ɫɦ. |
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ɪɢɫ. 8.29): |
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D(t) D0e G t cos Z02 G 2 t , |
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Ɋɢɫ. 8.29
ɉɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɩɪɢ ɦɚɥɨɦ ɡɚɬɭɯɚɧɢɢ ɤɨɥɟɛɚɧɢɣ, ɤɨɝɞɚ G Z0 (ɫɦ. ɩ. 8.1.2. ɋɨɛɫɬɜɟɧɧɵɟ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟ-
ɛɚɧɢɹ). ȿɫɥɢ ɧɟɪɚɜɟɧɫɬɜɨ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (8.141) ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ (8.41)
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ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ A1 ɢ A2 ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ
(8.145) ɢ (8.146):
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ɉɪɢ ɷɬɨɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɞɢɫɤɚ ɜ ɠɢɞɤɨɫɬɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ȼɵɪɚɠɟɧɢɟ (8.153) ɨɩɢɫɵɜɚɟɬ ɚɩɟɪɢɨɞɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ (ɫɦ. ɪɢɫ. 8.30), ɩɪɢ ɤɨɬɨɪɨɦ ɜ ɫɢɫɬɟɦɟ ɧɟ ɜɨɡɧɢɤɚɟɬ ɤɨɥɟɛɚɧɢɣ, ɨɧɚ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɩɨɥɨɠɟɧɢɸ ɪɚɜɧɨɜɟɫɢɹ.
D(t)
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Ɋɢɫ. 8.30 |
Ɂɚɞɚɱɚ 8.8
(ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɪɟɡɨɧɚɧɫ)
Ɍɟɥɨ ɦɚɫɫɨɣ m = 100 ɝ, ɩɨɞɜɟɲɟɧɧɨɟ ɧɚ ɥɟɝɤɨɣ ɩɪɭɠɢɧɟ ɠɟɫɬɤɨɫɬɶɸ k = 40 ɇ/ɦ, ɫɨɜɟɪɲɚɟɬ ɭɫɬɚɧɨɜɢɜɲɢɟɫɹ ɤɨɥɟɛɚɧɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɟɪɬɢɤɚɥɶɧɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ F F0 cos pt , ɱɚɫɬɨɬɚ
ɤɨɬɨɪɨɣ p = 25 ɪɚɞ/ɫ ɢ ɚɦɩɥɢɬɭɞɚ F0 1Í . ɋɦɟɳɟɧɢɟ ɬɟɥɚ ɢɡ ɩɨ-
ɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɨɬɫɬɚɟɬ ɩɨ ɮɚɡɟ ɨɬ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɧɚ M 3S / 4 . Ɉɩɪɟɞɟɥɢɬɶ ɞɨɛɪɨɬɧɨɫɬɶ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ Q , ɚ
ɬɚɤɠɟ ɪɟɡɨɧɚɧɫɧɭɸ ɱɚɫɬɨɬɭ pɪɟɡ , ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɪɟɡɨɧɚɧɫɭ ɫɦɟɳɟɧɢɹ, ɢ ɚɦɩɥɢɬɭɞɭ ɫɦɟɳɟɧɢɹ ɩɪɢ ɪɟɡɨɧɚɧɫɟ Aɪɟɡ .
Ɋɟɲɟɧɢɟ
I. ɇɚ ɬɟɥɨ, ɩɨɞɜɟɲɟɧɧɨɟ ɧɚ ɩɪɭɠɢɧɟ ɞɟɣɫɬɜɭɸɬ ɱɟɬɵɪɟ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ, ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɫɨ ɫɬɨɪɨɧɵ ɩɪɭɠɢɧɵ, ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɢ ɜɵɧɭɠɞɚɸɳɚɹ ɫɢɥɚ F F0 cos p t . Ʉɚɤ ɛɵɥɨ ɨɬɦɟ-
ɱɟɧɨ ɜ ɩ. 8.1.1, ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ ɬɹɠɟɫɬɢ ɧɟ ɜɥɢɹɟɬ ɧɚ ɱɚɫɬɨɬɭ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɨɧɚ ɥɢɲɶ ɫɦɟɳɚɟɬ ɩɨɥɨɠɟɧɢɟ ɪɚɜɧɨɜɟɫɢɹ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɉɨɷɬɨɦɭ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɨ ɤɚɤ ɩɪɢ ɜɟɪɬɢɤɚɥɶɧɵɯ, ɬɚɤ ɢ ɩɪɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɬɟɥɚ ɧɚ ɩɪɭɠɢɧɟ. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɩɪɭɠɢɧɚ ɥɟɝɤɚɹ, ɟɟ ɦɚɫɫɨɣ ɩɪɟɧɟɛɪɟɝɚɟɦ, ɫɱɢɬɚɹ ɟɟ ɪɚɜɧɨɣ ɧɭɥɸ.
II. ɂɫɤɨɦɚɹ ɞɨɛɪɨɬɧɨɫɬɶ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (8.40):
Q |
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Ɂɞɟɫɶ Ȧ – ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɬɟɥɚ, ɤɨɬɨɪɚɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩ. 8.1.2 ɪɚɜɧɚ:
Z |
Z02 G 2 . |
(8.155) |
ɑɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ Z0 |
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ɧɟɜɟɫɨɦɨɣ ɩɪɭɠɢɧɟ (ɫɦ. (8.8)) ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɚɫɫɨɣ ɬɟɥɚ m ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ k:
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Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ G, ɜɯɨɞɹɳɢɣ ɜ ɮɨɪɦɭɥɵ (8.154) ɢ (8.155), ɨɩɪɟɞɟɥɹɟɬ ɡɚɞɚɧɧɵɣ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɮɚɡɨɜɵɣ ɫɞɜɢɝ M ɦɟɠɞɭ ɫɦɟɳɟɧɢɟɦ ɢ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɨɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɪɚ-
ɠɟɧɢɟɦ (8.46):
tg M |
2Gp |
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p2 Z02 . |
(8.157) |
ɂɫɤɨɦɚɹ ɪɟɡɨɧɚɧɫɧɚɹ ɱɚɫɬɨɬɚ ɩɪɢ ɪɟɡɨɧɚɧɫɟ ɫɦɟɳɟɧɢɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (8.48) ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
pɪɟɡ |
Z02 2G 2 . |
(8.158) |
ɉɪɢ ɪɟɡɨɧɚɧɫɧɨɣ ɱɚɫɬɨɬɟ ɢɫɤɨɦɚɹ ɚɦɩɥɢɬɭɞɚ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ (ɫɦ. (8.49)) ɪɚɜɧɚ:
Aɪɟɡ |
A( pɪɟɡ ) |
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F0 |
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(8.159) |
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2Gm |
Z02 G 2 |
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ɉɨɥɭɱɟɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (8.154) – (8.159) ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɡɜɟɫɬɧɵɯ ɜ ɡɚɞɚɱɟ ɜɟɥɢɱɢɧ – ɞɨɛɪɨɬɧɨɫɬɢ Q , ɪɟɡɨɧɚɧɫ-
ɧɨɣ ɱɚɫɬɨɬɵ pɪɟɡ ɢ ɚɦɩɥɢɬɭɞɵ ɫɦɟɳɟɧɢɹ ɩɪɢ ɪɟɡɨɧɚɧɫɟ Aɪɟɡ .
III. ɋɨɜɦɟɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ (8.154) – (8.157) ɞɚɟɬ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɞɨɛɪɨɬɧɨɫɬɢ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ:
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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309 |
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Z02 p2 |
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kmp2 |
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(8.160) |
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k) |
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tg M |
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Z0 ) |
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tg |
M(mp |
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ɂɫɤɨɦɭɸ ɪɟɡɨɧɚɧɫɧɭɸ ɱɚɫɬɨɬɭ |
pɪɟɡ |
ɧɚɯɨɞɢɦ, ɪɟɲɚɹ ɫɢɫɬɟɦɭ |
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ɭɪɚɜɧɟɧɢɣ (8.156) – (8.158): |
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tg2M mp2 k 2 |
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pɪɟɡ |
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Z02 |
tg2M( p2 Z02 )2 |
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k |
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(8.161) |
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2 p2 |
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2m2p2 |
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Ⱥɦɩɥɢɬɭɞɭ ɫɦɟɳɟɧɢɹ ɩɪɢ ɪɟɡɨɧɚɧɫɟ |
Aɪɟɡ |
ɨɩɪɟɞɟɥɹɟɦ, ɪɟɲɚɹ |
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ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (8.156), (8.157) ɢ (8.159): |
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Aɪɟɡ |
A( pɪɟɡ ) |
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F0 p |
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m( p2 Z02 )tgM Z02 |
tg2M( p2 Z02 )2 |
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4 p2 |
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F0 p |
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(8.162) |
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(mp2 k )tgM |
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tg2M mp2 k 2 |
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ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɡɚɞɚɧɧɵɯ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ |
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ɜɟɥɢɱɢɧ ɜ ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ (8.160) (8.162), ɩɨɥɭɱɚɟɦ: |
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Q | 2,17 ; |
pɪɟɡ |19,0 ɪɚɞ/ɫ ; |
Aɪɟɡ |
| 5,7 ɫɦ . |
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Ɂɚɞɚɱɚ 8.9
(ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɪɟɡɨɧɚɧɫ) Ƚɨɪɢɡɨɧɬɚɥɶɧɵɣ ɩɪɭɠɢɧɧɵɣ ɦɚɹɬɧɢɤ ɫɨɜɟɪɲɚɟɬ ɜɵɧɭɠɞɟɧ-
ɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɫɢɥɵ F (t) F0 cos(pt) . Ʉɨɷɮɮɢɰɢɟɧɬ ɡɚɬɭɯɚɧɢɹ ɦɚɹɬɧɢɤɚ ɪɚɜɟɧ G , ɚ ɱɚɫ-
ɬɨɬɚ ɟɝɨ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ – Z0 . ɇɚɣɬɢ ɨɬɧɨ-
ɲɟɧɢɟ ɫɪɟɞɧɟɣ ɡɚ ɩɟɪɢɨɞ ɦɨɳɧɨɫɬɢ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ F(t) ɩɪɢ ɱɚɫɬɨɬɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɪɟɡɨɧɚɧɫɭ ɫɦɟɳɟɧɢɹ, ɤ ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɪɟɞɧɟɣ ɦɨɳɧɨɫɬɢ ɷɬɨɣ ɫɢɥɵ.
Ɋɟɲɟɧɢɟ
I. Ɋɚɫɫɦɨɬɪɢɦ ɤɨɥɟɛɚɧɢɹ ɦɚɹɬɧɢɤɚ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ F (t) F0 cos(pt) ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟ-
ɠɢɦɟ, ɤɨɝɞɚ ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɚɬɭɯɚɸɳɢɦɢ ɤɨɥɟɛɚɧɢɹɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ (ɫɦ. ɩ. 8.1.3).
310 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
II. ȼ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɤɨɨɪɞɢɧɚɬɚ ɢ ɫɤɨɪɨɫɬɶ ɦɚɹɬɧɢɤɚ ɦɟɧɹɸɬɫɹ ɩɨ ɡɚɤɨɧɚɦ (ɫɦ. (8.47) ɢ (8.53)):
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a cos( p t M) , |
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ap sin( p t M) . |
(8.164) |
Ɂɚɩɢɲɟɦ ɷɥɟɦɟɧɬɚɪɧɭɸ ɪɚɛɨɬɭ dA |
ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ F, |
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ɫɨɜɟɪɲɚɟɦɭɸ ɡɚ ɮɢɡɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ:
dA F (t)dx F (t)X(t)dt , |
(8.165) |
ɝɞɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɜɵɧɭɠɞɚɸɳɚɹ ɫɢɥɚ ɪɚɜɧɚ: |
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F (t) F0 cos(pt) . |
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ɋɭɦɦɚɪɧɭɸ ɪɚɛɨɬɭ ɷɬɨɣ ɫɢɥɵ ɡɚ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ T ɧɚɯɨ- |
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ɞɢɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɷɥɟɦɟɧɬɚɪɧɨɣ ɪɚɛɨɬɵ: |
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A ³F (t)X (t)dt . |
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Ɂɚɩɢɲɟɦ ɫɪɟɞɧɸɸ ɡɚ ɩɟɪɢɨɞ ɦɨɳɧɨɫɬɶ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ:
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F (t)X (t)dt . |
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ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (8.163) – (8.168) ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɫɪɟɞɧɟɣ ɦɨɳɧɨɫɬɢ Pɫɪ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɨɬ ɱɚɫɬɨɬɵ
p. Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɢɫɤɨɦɨɝɨ ɜ ɡɚɞɚɱɟ ɨɬɧɨɲɟɧɢɹ ɫɪɟɞɧɟɣ ɡɚ ɩɟɪɢɨɞ ɦɨɳɧɨɫɬɢ ɫɢɥɵ F ɩɪɢ ɱɚɫɬɨɬɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɪɟɡɨɧɚɧɫɭ ɫɦɟɳɟɧɢɹ, ɤ ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɪɟɞɧɟɣ ɦɨɳɧɨɫɬɢ ɷɬɨɣ ɫɢɥɵ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɦɚɤɫɢɦɭɦ ɫɪɟɞɧɟɣ ɦɨɳɧɨɫɬɢ, ɚ ɬɚɤɠɟ ɞɨɩɨɥɧɢɬɶ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɜɵɪɚɠɟɧɢɹɦɢ (8.45), (8.46) ɢ (8.48) ɞɥɹ ɚɦɩɥɢɬɭɞɵ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ a( p) , ɮɚɡɵ M( p) ɢ ɪɟɡɨɧɚɧɫɧɨɣ
ɱɚɫɬɨɬɵ pɪɟɡ |
ɩɪɢ ɪɟɡɨɧɚɧɫɟ ɫɦɟɳɟɧɢɹ: |
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a( p) |
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F0 |
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m Z02 p2 2 |
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4G 2 p2 |
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tgM( p) |
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2Gp |
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p2 Z02 |
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pɪɟɡ |
Z02 2G 2 . |
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III. ɂɧɬɟɝɪɢɪɭɹ (8.168) ɫ ɭɱɟɬɨɦ (8.167) ɢ ɡɚɞɚɧɧɨɝɨ ɜ ɡɚɞɚɱɟ ɡɚɤɨɧɚ ɢɡɦɟɧɟɧɢɹ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ F(t), ɩɨɥɭɱɚɟɦ: