Механика.Методика решения задач
.pdfȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
281 |
8.1.3. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ. Ɋɟɡɨɧɚɧɫ
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜ ɫɥɭɱɚɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɞ
ɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɢɦɟɟɬ ɜɢɞ: |
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[ 2G[ Z02[ B cos( pt) , |
(8.43) |
ɝɞɟ B cos( pt) – ɨɛɨɛɳɟɧɧɚɹ ɜɵɧɭɠɞɚɸɳɚɹ ɫɢɥɚ, B ɢ p – ɟɟ ɚɦ-
ɩɥɢɬɭɞɚ ɢ ɱɚɫɬɨɬɚ.
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɩɪɭɠɢɧɧɨɝɨ ɦɚɹɬɧɢɤɚ ɜ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɜɵɫɬɭɩɚɟɬ ɨɬɧɨɲɟɧɢɟ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɬɟɥɨ, ɩɪɢɤɪɟɩɥɟɧɧɨɝɨ ɤ ɩɪɭɠɢɧɟ, ɤ ɦɚɫɫɟ ɷɬɨɝɨ ɬɟɥɚ.
Ʉɨɥɟɛɚɧɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɩɪɢ G < Z0 ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɫɨɛɫɬɜɟɧɧɵɯ ɢ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ. Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ
ɤɨɨɪɞɢɧɚɬɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ: |
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[(t) [ɫɨɛ (t) [ɜɵɧ (t) [ɫɨɛ (t) A( p) cos pt M( p) . |
(8.44) |
Ɂɞɟɫɶ [ɫɨɛ (t) – ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɩɪɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɹɯ ɜ ɨɬɫɭɬɫɬɜɢɢ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ, [ɜɵɧ (t) – ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɩɨɫɥɟ ɡɚɬɭ-
ɯɚɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ, A(p) – ɚɦɩɥɢɬɭɞɚ ɢ M(p) – ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ [ɜɵɧ (t) , ɤɨɬɨɪɵɟ
ɡɚɜɢɫɹɬ ɨɬ ɱɚɫɬɨɬɵ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ (ɫɦ. ɫɩɥɨɲɧɵɟ ɥɢɧɢɢ ɧɚ ɪɢɫ. 8.12 ɢ 8.13):
A( p) |
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B |
, |
(8.45) |
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Z02 p2 2 4G 2 p2 |
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tgM( p) |
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2Gp |
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. |
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(8.46) |
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p2 Z02 |
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ɇɚ ɪɢɫ. 8.12 ɢ ɪɢɫ. 8.13 ɲɬɪɢɯɨɜɵɦɢ ɥɢɧɢɹɦɢ ɢɡɨɛɪɚɠɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɚɦɩɥɢɬɭɞɵ ɢ ɮɚɡɵ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɞɥɹ ɭɞɜɨɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ 2G.
ɉɪɢ |
t >> 1/G, |
ɫɨɛɫɬɜɟɧɧɵɦɢ ɡɚɬɭɯɚɸɳɢɦɢ |
ɤɨɥɟɛɚɧɢɹɦɢ |
[ɫɨɛ (t) ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ: |
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||
[ (t) |
[ɜɵɧ (t) |
A( p) cos pt M( p) . |
(8.47) |
282 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
A(p)
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G |
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2G |
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Aɫɬ |
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0 |
pɪɟɡ Z02 2G 2 |
p |
Ɋɢɫ. 8.12. Ɂɚɜɢɫɢɦɨɫɬɶ ɚɦɩɥɢɬɭɞɵ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ A(p) ɨɬ
ɱɚɫɬɨɬɵ p ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ ɡɚɬɭɯɚɧɢɹ G |
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M(p) |
Z0 |
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0 |
p |
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–S/2 |
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G |
2G |
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–S |
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Ɋɢɫ. 8.13. Ɂɚɜɢɫɢɦɨɫɬɶ ɧɚɱɚɥɶɧɨɣ ɮɚɡɵ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ M(p) ɨɬ ɱɚɫɬɨɬɵ p ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɚɯ ɡɚɬɭɯɚɧɢɹ G
Ɋɟɡɨɧɚɧɫ ɫɦɟɳɟɧɢɹ (ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ) – ɹɜɥɟɧɢɟ ɪɟɡɤɨɝɨ ɜɨɡɪɚɫɬɚɧɢɹ ɚɦɩɥɢɬɭɞɵ A( p) ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ
ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɱɚɫɬɨɬɵ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ (ɪɢɫ. 8.12).
ȼ ɫɥɭɱɚɟ ɪɟɡɨɧɚɧɫɚ ɫɦɟɳɟɧɢɹ ɪɟɡɨɧɚɧɫɧɚɹ ɱɚɫɬɨɬɚ pɪɟɡ ɜɵ-
ɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɧɚɯɨɞɢɬɫɹ ɢɡ ɭɫɥɨɜɢɹ |
d A( p) |
0 : |
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d p |
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pɪɟɡ Z02 2G 2 . |
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(8.48) |
ɉɪɢ ɪɟɡɨɧɚɧɫɧɨɣ ɱɚɫɬɨɬɟ ɚɦɩɥɢɬɭɞɚ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɪɚɜɧɚ:
Aɪɟɡ |
A( pɪɟɡ ) |
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B |
. |
(8.49) |
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2G |
Z02 G 2 |
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ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
283 |
ɉɪɢ ɩɨɫɬɨɹɧɧɨɣ ( p 0 ) ɨɛɨɛɳɟɧɧɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɟ ȼ ɨɛɨɛɳɟɧɧɚɹ ɤɨɨɪɞɢɧɚɬɚ [ ɛɭɞɟɬ ɬɚɤɠɟ ɩɨɫɬɨɹɧɧɚ ɢ ɪɚɜɧɚ:
A A(0) |
B |
. |
(8.50) |
ɫɬ |
Z2 |
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0 |
ɉɪɢ ɫɬɪɟɦɥɟɧɢɢ ɱɚɫɬɨɬɵ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ (ɩɪɢ p !! Z0 ) ɚɦɩɥɢɬɭɞɚ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ (ɪɢɫ. 8.12):
A( p) ~ |
B |
p ofo0 . |
(8.51) |
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p2 |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɨɛɪɨɬɧɨɫɬɶ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɦɨɠɟɬ ɛɵɬɶ ɜɵɪɚɠɟɧɚ ɱɟɪɟɡ Aɪɟɡ ɢ Aɫɬ . ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (8.40), (8.49) ɢ
(8.50): |
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Aɪɟɡ |
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Q |
Z |
# |
(ɩɪɢ Z !! G ). |
(8.52) |
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2G |
0 |
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Aɫɬ |
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Ɂɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɨ ɜɪɟɦɟɧɟɦ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɜ ɫɥɭɱɚɟ ɜɵɧɭɠɞɟɧɧɵɯ ɭɫɬɚɧɨɜɢɜɲɢɯɫɹ ɤɨɥɟɛɚɧɢɣ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɚɪɦɨɧɢ-
ɱɟɫɤɨɣ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɢɦɟɟɬ ɜɢɞ: |
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[(t) [ɜɵɧ (t) A( p) p sin pt M( p) |
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A( p) p cos pt M( p) S / 2 . |
(8.53) |
Ɂɞɟɫɶ A( p) p – ɚɦɩɥɢɬɭɞɚ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ (ɫɦ.
ɫɩɥɨɲɧɭɸ ɥɢɧɢɸ ɧɚ ɪɢɫ. 8.14): |
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A( p) p |
Bp |
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Z02 p2 2 4G 2 p2 . |
(8.54) |
ɒɬɪɢɯɨɜɨɣ ɥɢɧɢɟɣ ɧɚ ɪɢɫ. 8.14 ɢɡɨɛɪɚɠɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɚɦɩɥɢɬɭɞɵ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɩɪɢ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɜ ɫɥɭɱɚɟ ɭɞɜɨɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɡɚɬɭɯɚɧɢɹ 2G.
Ɋɟɡɨɧɚɧɫ ɫɤɨɪɨɫɬɢ – ɹɜɥɟɧɢɟ ɪɟɡɤɨɝɨ ɜɨɡɪɚɫɬɚɧɢɹ ɚɦɩɥɢɬɭɞɵ A( p) p ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ [(t) ɩɪɢ ɢɡɦɟɧɟɧɢɢ
ɱɚɫɬɨɬɵ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ (ɪɢɫ. 8.14).
ȼ ɫɥɭɱɚɟ ɪɟɡɨɧɚɧɫɚ ɫɤɨɪɨɫɬɢ ɪɟɡɨɧɚɧɫɧɚɹ ɱɚɫɬɨɬɚ ɧɚɯɨɞɢɬɫɹ
ɢɡ ɭɫɥɨɜɢɹ |
d( A( p) p) |
0 ɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (8.54) ɪɚɜɧɚ: |
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dp |
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pɪɟɡ |
Z0 . |
(8.55) |
284 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
A(p)p
G
2G
0 |
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pɪɟɡ Z0 |
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p |
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Ɋɢɫ. 8.14. Ɂɚɜɢɫɢɦɨɫɬɶ ɚɦɩɥɢɬɭɞɵ ɢɡɦɟɧɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ A(p)p ɩɪɢ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɹɯ ɨɬ ɱɚɫɬɨɬɵ p ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɡɚɬɭɯɚɧɢɹ G
ɉɪɢ ɩɨɫɬɨɹɧɧɨɣ ( p 0 ) ɜɵɧɭɠɞɚɸɳɟɣ |
ɫɢɥɟ ɨɛɨɛɳɟɧɧɚɹ |
ɫɤɨɪɨɫɬɶ [(t) ɛɭɞɟɬ ɪɚɜɧɚ ɧɭɥɸ (ɪɢɫ. 8.14): |
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Ap ɫɬ 0 . |
(8.56) |
ɉɪɢ ɱɚɫɬɨɬɟ ɜɵɧɭɠɞɚɸɳɟɣ ɫɢɥɵ ɦɧɨɝɨ ɛɨɥɶɲɟ ɱɚɫɬɨɬɵ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ( p !! Z0 ) ɚɦɩɥɢɬɭɞɚ ɢɡɦɟɧɟ-
ɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɤɨɪɨɫɬɢ ɛɥɢɡɤɚ ɤ ɧɭɥɸ: |
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A( p) p ~ |
B |
p ofo0 . |
(8.57) |
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p |
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8.2. Ɉɫɧɨɜɧɵɟ ɬɢɩɵ ɡɚɞɚɱ ɢ ɦɟɬɨɞɵ ɢɯ ɪɟɲɟɧɢɹ
8.2.1. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɡɚɞɚɱ
Ȼɨɥɶɲɢɧɫɬɜɨ ɡɚɞɚɱ ɩɨ ɬɟɦɟ "ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦ ɫ ɨɞɧɨɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ. Ɋɟɡɨɧɚɧɫ" ɦɨɠɧɨ ɭɫɥɨɜɧɨ ɨɬɧɟɫɬɢ ɤ ɫɥɟɞɭɸɳɢɦ ɬɢɩɚɦ ɡɚɞɚɱ ɢɥɢ ɢɯ ɤɨɦɛɢɧɚɰɢɹɦ. Ɂɚɞɚɱɢ ɧɚ:
1)ɫɜɨɛɨɞɧɵɟ ɧɟɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ,
2)ɫɜɨɛɨɞɧɵɟ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ,
3)ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ, ɪɟɡɨɧɚɧɫ.
ȼɨɡɦɨɠɧɵ ɞɜɚ ɦɟɬɨɞɚ ɪɟɲɟɧɢɹ – ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɞɢɧɚɦɢɱɟɫɤɢɣ ɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɦɟɬɨɞɵ. Ⱦɢɧɚɦɢɱɟɫɤɢɣ ɦɟɬɨɞ ɩɪɟɞɩɨɥɚɝɚɟɬ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ, ɚ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ – ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɤɨɥɟɛɥɸɳɟɣɫɹ ɫɢɫɬɟɦɵ ɬɟɥ.
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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8.2.2. Ɉɛɳɚɹ ɫɯɟɦɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
ȿɫɥɢ ɡɚɞɚɱɚ ɫɜɨɞɢɬɫɹ ɤ ɤɨɥɟɛɚɧɢɹɦ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɬɨ ɨɫɧɨɜɧɵɟ ɷɬɚɩɵ ɪɟɲɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬɫɹ ɨɛɳɢɦɢ ɫɯɟɦɚɦɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱ, ɨɩɢɫɚɧɧɵɦɢ ɜ Ƚɥɚɜɟ 2 (ɞɢɧɚɦɢɱɟɫɤɢɣ ɦɟɬɨɞ) ɢ Ƚɥɚɜɟ 3 (ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɦɟɬɨɞ). ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨ ɤɨɥɟɛɚɧɢɹɯ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɯɟɦɵ, ɨɩɢɫɚɧɧɵɟ ɜ Ƚɥɚɜɟ 6 (ɞɢɧɚɦɢɱɟɫɤɢɣ ɦɟɬɨɞ) ɢ Ƚɥɚɜɟ 7 (ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɦɟɬɨɞ). Ʉɚɤ ɩɪɚɜɢɥɨ, ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɨɛɨɢɯ ɦɟɬɨɞɨɜ ɧɚ ɩɨɫɥɟɞɧɟɦ ɷɬɚɩɟ ɪɟɲɟɧɢɹ ɩɨɥɭɱɚɸɬɫɹ ɭɪɚɜɧɟɧɢɟ ɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ. ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɟɚɥɢɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɬɪɢ ɨɫɧɨɜɧɵɯ ɷɬɚɩɚ.
I. Ɉɩɪɟɞɟɥɢɬɶɫɹ ɫ ɦɨɞɟɥɹɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɨɛɴɟɤɬɨɜ ɢ ɹɜɥɟɧɢɣ.
II. Ɂɚɩɢɫɚɬɶ ɩɨɥɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɢɫɤɨɦɵɯ ɜɟɥɢɱɢɧ.
III. ɉɨɥɭɱɢɬɶ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ ɜ ɚɧɚɥɢɬɢɱɟɫɤɨɦ ɢ ɱɢɫɥɟɧɧɨɦ ɜɢɞɚɯ.
8.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ |
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Ɂɚɞɚɱɚ 8.1 |
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(ɋɜɨɛɨɞɧɵɟ ɧɟɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ) |
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ɋɩɥɨɲɧɨɣ ɨɞɧɨɪɨɞɧɵɣ ɰɢɥɢɧɞɪ ɦɚɫɫɨɣ |
m ɢ ɪɚɞɢɭɫɨɦ |
R , |
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ɲɚɪɧɢɪɧɨ ɡɚɤɪɟɩɥɟɧɧɵɣ |
ɜ ɧɢɠɧɟɣ |
k |
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ɬɨɱɤɟ, ɫɨɜɟɪɲɚɟɬ ɦɚɥɵɟ |
ɤɨɥɟɛɚɧɢɹ |
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ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɞɜɭɯ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ |
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ɨɞɢɧɚɤɨɜɵɯ ɥɟɝɤɢɯ ɩɪɭɠɢɧ, ɠɟɫɬ- |
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ɤɨɫɬɶ ɤɚɠɞɨɣ ɢɡ ɤɨɬɨɪɵɯ |
ɪɚɜɧɚ k |
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(ɪɢɫ. 8.15). ɉɪɭɠɢɧɵ ɩɪɢɤɪɟɩɥɟɧɵ ɤ |
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ɜɟɪɯɧɟɣ ɬɨɱɤɟ ɰɢɥɢɧɞɪɚ ɢ ɧɟɪɚɫɬɹ- |
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ɧɭɬɵ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɰɢ- |
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ɥɢɧɞɪɚ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɱɚɫɬɨɬɭ |
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Ɋɢɫ. 8.15 |
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ɦɚɥɵɯ ɤɨɥɟɛɚɧɢɣ ɰɢɥɢɧɞɪɚ. |
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Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɞɢɧɚɦɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɨɩɨɪɨɣ ɰɢɥɢɧɞɪɚ. Ɉɫɶ X ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɚɩɪɚɜɢɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨ. ɇɚɱɚɥɨ
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɨɬɫɱɟɬɚ ɨɫɢ X ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɥɨɠɟɧɢɸ ɬɨɱɤɢ ɲɚɪɧɢɪɧɨɝɨ ɡɚɤɪɟɩ- |
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ɥɟɧɢɹ ɰɢɥɢɧɞɪɚ. ɐɢɥɢɧɞɪ ɫɱɢɬɚɟɦ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ. ɇɚ |
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ɧɟɝɨ ɞɟɣɫɬɜɭɸɬ ɱɟɬɵɪɟ ɫɢɥɵ (ɫɦ. ɪɢɫ. 8.16): |
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ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɭɩɪɭɝɢɟ ɫɢɥɵ ɫɨ ɫɬɨɪɨɧɵ |
2Fɭɩɪ |
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ɞɜɭɯ ɩɪɭɠɢɧ 2Fɭɩɪ ɢ ɫɢɥɚ ɪɟɚɤɰɢɢ ɨɩɨɪɵ, ɧɟ |
D |
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ɢɡɨɛɪɚɠɟɧɧɨɣ ɧɚ ɪɢɫɭɧɤɟ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ |
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ɩɪɟɧɟɛɪɟɝɚɟɦ. ɉɪɭɠɢɧɵ ɫɱɢɬɚɟɦ ɧɟɜɟɫɨɦɵ- |
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ɦɢ, ɢɯ ɞɟɮɨɪɦɚɰɢɢ – ɦɚɥɵɦɢ. |
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II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (ɫɦ. |
mg |
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(6.48) ɜ Ƚɥɚɜɟ 6) ɞɥɹ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɨɫɢ (ɪɢɫ. 8.16), ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɟɝɨ |
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ɲɚɪɧɢɪɧɨɝɨ |
ɡɚɤɪɟɩɥɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ |
Ɋɢɫ. 8.16 |
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ɩɥɨɫɤɨɫɬɹɦ |
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ɤɨɥɟɛɚɧɢɣ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ |
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ɰɢɥɢɧɞɪɚ: |
mgR sinD 2kx(2R) | mgRD 4kxR . |
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(8.58) |
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Ɂɞɟɫɶ J – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ ɨɫɢ, D – ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɰɢɥɢɧɞɪɚ (ɪɢɫ. 8.16), ɯ – ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ ɤɪɟɩɥɟɧɢɹ ɩɪɭɠɢɧ ɤ ɰɢɥɢɧɞɪɭ. ɉɪɢ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ (8.58) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬ ɫɢɥɵ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɪɚɜɟɧ ɧɭɥɸ, ɢ ɩɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɩɨɜɨɪɨɬɚ ɰɢɥɢɧɞɪɚ ɩɥɟɱɨ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ ɪɚɜɧɨ 2R , ɚ sinD | D .
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ – ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɤɨɨɪɞɢɧɚɬɭ ɬɨɱɤɢ ɤɪɟɩɥɟɧɢɹ ɩɪɭɠɢɧ ɤ ɰɢɥɢɧɞɪɭ ɢ ɭɝɨɥ ɟɝɨ ɩɨɜɨɪɨɬɚ:
x 2RD . |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɟɝɨ ɲɚɪɧɢɪɧɨɝɨ ɤɪɟɩɥɟɧɢɹ, ɧɚɯɨɞɢɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
ɬɟɨɪɟɦɨɣ Ƚɸɣɝɟɧɫɚ – ɒɬɟɣɧɟɪɚ (ɫɦ. (6.42) ɜ Ƚɥɚɜɟ 6): |
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mR2 |
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III. ɉɨɞɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɹ (8.59) ɢ (8.60) ɜ (8.58), ɩɨɥɭɱɚɟɦ |
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ɭɪɚɜɧɟɧɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ: |
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɚɹ ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɧɟɡɚɬɭɯɚɸɳɢɯ ɤɨɥɟɛɚɧɢɣ ɪɚɜɧɚ
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɧɢɣ ɫɩɪɚɜɟɞɥɢɜɨ ɩɪɢ 8mk ! Rg . ȿɫɥɢ 8mk d Rg , ɬɨ ɜɟɪɬɢɤɚɥɶɧɨɟ ɪɚɜɧɨɜɟɫɧɨɟ ɫɨɫɬɨɹɧɢɟ
ɹɜɥɹɟɬɫɹ ɧɟɭɫɬɨɣɱɢɜɵɦ ɢ ɤɨɥɟɛɚɧɢɹ ɜ ɫɢɫɬɟɦɟ ɧɟ ɜɨɡɧɢɤɚɸɬ (ɫɦ. ɜ ɩ. 8.1.1 ɧɟɨɛɯɨɞɢɦɵɟ ɭɫɥɨɜɢɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ).
Ɂɚɞɚɱɚ 8.2
(ɋɜɨɛɨɞɧɵɟ ɧɟɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ)
Ɍɨɧɤɚɹ ɨɞɧɨɪɨɞɧɚɹ ɩɚɥɨɱɤɚ ɫɨɜɟɪɲɚɟɬ ɦɚɥɵɟ ɤɨɥɟɛɚɧɢɹ ɜɧɭɬɪɢ ɝɥɚɞɤɨɝɨ ɩɨɥɭɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɨɦ R , ɨɫɬɚɜɚɹɫɶ ɜ ɩɥɨɫɤɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ ɰɢɥɢɧɞɪɚ (ɪɢɫ. 8.17).
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Ɋɢɫ. 8.17
Ⱦɥɢɧɚ ɩɚɥɨɱɤɢ ɪɚɜɧɚ ɪɚɞɢɭɫɭ ɩɨɥɭɰɢɥɢɧɞɪɚ. ɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɩɚɥɨɱɤɢ, ɫɱɢɬɚɹ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɧɚ ɩɨɤɨɢɥɚɫɶ ɢ ɛɵɥɚ ɨɬɤɥɨɧɟɧɚ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɧɚ ɦɚɥɵɣ ɭɝɨɥ D0 .
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɞɢɧɚɦɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɩɨɥɭɰɢɥɢɧɞɪɨɦ. ɉɚɥɨɱɤɭ ɫɱɢɬɚɟɦ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ. ɇɚ ɧɟɟ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ – ɫɢɥɚ ɬɹɠɟɫɬɢ mg ɢ ɫɢɥɵ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ N1 ɢ N2 (ɪɢɫ. 8.17). ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɦɨɦɟɧɬɨɜ (ɫɦ. (6.48) ɜ Ƚɥɚɜɟ 6) ɞɥɹ ɩɚɥɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɨɫɶɸ ɩɨɥɭɰɢɥɢɧɞɪɚ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Ɉ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ
(ɪɢɫ. 8.17):
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mgl sinD , |
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JD |
ɝɞɟ J – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɩɚɥɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ, D – ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɩɚɥɨɱɤɢ ɨɬ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ, l – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɞɨ ɰɟɧɬɪɚ ɦɚɫɫ ɩɚɥɨɱɤɢ. ɉɪɢ ɡɚɩɢɫɢ (8.63) ɭɱɬɟɧɨ, ɱɬɨ ɦɨɦɟɧɬɵ ɫɢɥ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ N1 ɢ N2 ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɪɚɜɧɵ ɧɭɥɸ.
ɉɨɫɤɨɥɶɤɭ ɞɥɢɧɚ ɩɚɥɨɱɤɢ ɪɚɜɧɚ ɪɚɞɢɭɫɭ ɰɢɥɢɧɞɪɚ, ɬɨ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɞɨ ɰɟɧɬɪɚ ɦɚɫɫ ɩɚɥɨɱɤɢ ɪɚɜɧɨ:
l |
R sin(S / 3) |
R 3 |
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(8.64) |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɩɚɥɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɤɚɡɚɧɧɨɣ ɨɫɢ ɧɚɯɨ- |
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ɞɢɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɦɨɣ Ƚɸɣɝɟɧɫɚ – ɒɬɟɣɧɟɪɚ (6.42): |
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Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɨɧɤɨɣ ɩɚɥɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨ- |
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ɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ ɦɚɫɫ, ɪɚɜɟɧ: |
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J0 |
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mR2 |
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III. ɉɪɟɨɛɪɚɡɭɹ |
ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ |
(8.63) (8.66) ɫ ɭɱɟɬɨɦ |
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ɦɚɥɨɫɬɢ ɭɝɥɚ ɨɬɤɥɨɧɟɧɢɹ ɩɚɥɨɱɤɢ ɨɬ |
ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ |
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(sinD | D) , ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ: |
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3 |
3g |
D 0. |
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5R |
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Ʉɚɤ ɜɢɞɢɦ (ɫɪ. ɫ (8.1)), ɱɚɫɬɨɬɚ ɫɨɛɫɬɜɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɚɥɨɱɤɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ
Z0 |
3 3g |
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(8.68) |
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ɚ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ (ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ (8.67)) ɢɦɟɟɬ ɜɢɞ:
D(t) |
Acos(Z0 t M0 ) . |
(8.69) |
Ⱥɦɩɥɢɬɭɞɚ |
A ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ M0 |
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɹɦɢ |
ɡɚɞɚɱɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɭɝɥɚ ɨɬɤɥɨɧɟɧɢɹ ɢ ɫɤɨɪɨɫɬɢ ɟɝɨ ɢɡɦɟɧɟɧɢɹ:
ȽɅȺȼȺ 8. ɋɜɨɛɨɞɧɵɟ ɢ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ |
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D(0) |
D0 |
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AsinM0 . |
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(8.70) |
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ɋɨɜɦɟɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɣ (8.70) ɞɚɟɬ ɡɧɚɱɟɧɢɹ ɚɦɩɥɢ- |
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ɬɭɞɵ ɢ ɧɚɱɚɥɶɧɨɣ ɮɚɡɵ: |
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A D0 , M0 |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɫɤɨɦɵɣ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ ɩɚɥɨɱɤɢ |
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ɩɪɢɧɢɦɚɟɬ ɜɢɞ: |
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3g t |
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D(t) |
D0 cos ¨ |
¸ . |
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Ɂɚɞɚɱɚ 8.3 |
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(ɋɜɨɛɨɞɧɵɟ ɧɟɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ) |
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ɇɚ ɬɟɥɟɠɤɟ ɦɚɫɫɨɣ Ɇ, ɫɬɨɹɳɟɣ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɵɯ ɪɟɥɶɫɚɯ, |
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ɩɨɞɜɟɲɟɧ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɞɥɢɧɨɣ l ɢ ɦɚɫɫɨɣ m. Ɍɟɥɟɠɤɚ |
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ɦɨɠɟɬ ɤɚɬɢɬɶɫɹ ɩɨ ɪɟɥɶɫɚɦ ɛɟɡ ɬɪɟɧɢɹ. Ɍɟɥɟɠɤɟ ɫɨɨɛɳɢɥɢ ɧɚɱɚɥɶ- |
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ɧɭɸ ɫɤɨɪɨɫɬɶ V0 ɬɚɤ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɧɢɬɶ ɦɚɹɬɧɢɤɚ ɨɫɬɚɥɚɫɶ ɜɟɪɬɢ- |
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ɤɚɥɶɧɨɣ. ɇɚɣɬɢ ɡɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɢ ɬɟɥɟɠɤɢ ɨɬɧɨɫɢɬɟɥɶ- |
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ɧɨ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɩɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ ɧɢ- |
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ɬɢ ɦɚɹɬɧɢɤɚ ɨɬ ɜɟɪɬɢɤɚɥɢ. Ɉɩɪɟɞɟɥɢɬɶ, ɩɪɢ ɤɚɤɢɯ ɫɨɨɬɧɨɲɟɧɢɹɯ |
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ɦɚɫɫ ɦɚɹɬɧɢɤɚ ɢ ɬɟɥɟɠɤɢ ɚɦɩɥɢɬɭɞɵ ɢɯ ɤɨɥɟɛɚɧɢɣ Am ɢ AM ɛɭɞɭɬ |
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ɦɚɤɫɢɦɚɥɶɧɵɦɢ. |
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Ɋɟɲɟɧɢɟ |
ɡɚɞɚɱɢ |
ɢɫ- |
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I. ɉɪɢ ɪɟɲɟɧɢɢ |
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ɩɨɥɶɡɭɟɦ |
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ɞɜɟ |
ɫɢɫɬɟɦɵ |
ɨɬɫɱɟɬɚ: |
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D |
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ɢɧɟɪɰɢɚɥɶɧɭɸ ɥɚɛɨɪɚɬɨɪɧɭɸ ɫɢɫ- |
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ɬɟɦɭ, ɫɜɹɡɚɧɧɭɸ ɫ ɪɟɥɶɫɚɦɢ, ɢ ɧɟ- |
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Ɍ |
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ɢɧɟɪɰɢɚɥɶɧɭɸ, ɫɜɹɡɚɧɧɭɸ ɫ ɬɟ- |
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ɥɟɠɤɨɣ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɏ ɢɧɟɪɰɢ- |
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Fɩɟɪ |
mg |
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ɚɥɶɧɨɣ |
ɫɢɫɬɟɦɵ |
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ɨɬɫɱɟɬɚ |
ɜɞɨɥɶ |
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ɪɟɥɶɫɨɜ, ɩɨ ɤɨɬɨɪɵɦ ɤɚɬɢɬɫɹ ɬɟ- |
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ɥɟɠɤɚ (ɫɦ. ɪɢɫ. 8.18), ɧɚɱɚɥɨ ɨɬ- |
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ɏ |
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ɫɱɟɬɚ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɨɠɟ- |
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ɧɢɟɦ ɦɚɹɬɧɢɤɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨ- |
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Ɋɢɫ. 8.18 |
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ɦɟɧɬ ɜɪɟɦɟɧɢ. |
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ɉɨɫɥɟ ɫɨɨɛɳɟɧɢɹ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ V0 ɬɟɥɟɠɤɟ ɦɚɹɬɧɢɤ ɜ |
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ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɛɭɞɟɬ ɤɨɥɟɛɚɬɶɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ |
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ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ O, ɜ ɬɨ ɜɪɟɦɹ |
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ɤɚɤ ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɟɝɨ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɫɭɩɟɪ- |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɩɨɡɢɰɢɟɣ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜɦɟɫɬɟ ɫ ɬɟɥɟɠɤɨɣ ɢ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɥɟɠɤɢ.
Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɞɢɧɚɦɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ. ɇɚ ɦɚɹɬɧɢɤ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ (ɪɢɫ. 8.18) – ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ Ɍ ɢ ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ Fïåð . ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
II. ɉɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɦɚɹɬɧɢɤ, ɜ ɫɨ-
ɨɬɜɟɬɫɬɜɢɢ ɫ (4.16) |
ɜ ɩ. 4.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ Ƚɥɚɜɵ 4 ɪɚɜɧɚ: |
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(8.73) |
Fɩɟɪ mxM |
ɝɞɟ xM – ɭɫɤɨɪɟɧɢɟ ɬɟɥɟɠɤɢ (ɢ ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɧɟɣ ɧɟɢɧɟɪɰɢ-
ɚɥɶɧɨɣ ɫɢɫɬɟɦɵ) ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ (ɭɪɚɜɧɟɧɢɟ
ɦɨɦɟɧɬɨɜ; ɫɦ. (6.48) ɜ ɩ. 6.1.2 Ƚɥɚɜɵ 6) ɦɚɹɬɧɢɤɚ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ ɩɨɞɜɟɫɚ O ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɱɟɪɬɟɠɚ (ɪɢɫ. 8.18):
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ml |
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mgl sinD mxM l cosD , |
ɝɞɟ D – ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬ ɜɟɪɬɢɤɚɥɢ (ɫɦ. ɪɢɫ. 8.18). ɉɪɢ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ (8.74) ɭɱɬɟɧɨ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɵɛɪɚɧɧɨɣ
ɨɫɢ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɦɚɹɬɧɢɤɚ ɪɚɜɟɧ ml 2 , ɚ ɦɨɦɟɧɬ ɫɢɥɵ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ ɪɚɜɟɧ ɧɭɥɸ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɢɠɟɧɢɣ ɤɨɨɪɞɢɧɚɬɚ ɦɚɹɬɧɢɤɚ xm (t) ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟ-
ɬɚ ɪɚɜɧɚ: |
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xm (t) xM (t) l sinD(t) , |
(8.75) |
ɝɞɟ xM (t) – ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ ɩɨɞɜɟɫɚ ɦɚɹɬɧɢɤɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ
ɫ ɬɟɥɟɠɤɨɣ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ. ɍɪɚɜɧɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɫɜɹɡɢ (ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ
ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɦɚɹɬɧɢɤɚ D(t) ɢ ɭɫɤɨɪɟɧɢɟ ɬɟɥɟɠɤɢ xM (t) )
ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɪɚɫɫɦɚɬɪɢɜɚɹ ɞɜɢɠɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɟɥ «ɦɚɹɬɧɢɤ + ɬɟɥɟɠɤɚ» ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ʉɨɨɪɞɢɧɚɬɚ |
ɰɟɧɬɪɚ ɦɚɫɫ xɰɦ ɫɢɫɬɟɦɵ ɬɟɥ |
(ɫɦ. (3.1) ɜ |
|
ɩ. 3.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ Ƚɥɚɜɵ 3) ɪɚɜɧɚ: |
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x |
mxm MxM |
. |
(8.76) |
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ɰɦ |
m M |
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