Механика.Методика решения задач
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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[(t, r) [0 (r) cos Z(t r / c) M0 |
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cos Zt kr M0 , |
(9.13) |
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ɝɞɟ A – ɜɟɥɢɱɢɧɚ, ɱɢɫɥɟɧɧɨ ɪɚɜɧɚɹ ɚɦɩɥɢɬɭɞɟ ɜɨɥɧɨɜɨɝɨ ɜɨɡɦɭɳɟɧɢɹ ɧɚ ɟɞɢɧɢɱɧɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɬɨɱɤɢ S.
ȼ ɫɥɭɱɚɟ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɝɨ ɡɚɬɭɯɚɧɢɹ (ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɡɚɬɭɯɚɧɢɹ G ) ɫɮɟɪɢɱɟɫɤɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ ɡɚɤɨɧ ɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ:
[ t, r [0 (r) cos Zt kr M0 |
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e Gr cos Zt kr M0 . (9.14) |
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9.1.4.ɋɤɨɪɨɫɬɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɢɯ ɜɨɥɧ
ɜɪɚɡɥɢɱɧɵɯ ɫɪɟɞɚɯ
Ⱥ. ɉɪɨɞɨɥɶɧɚɹ ɭɩɪɭɝɚɹ ɜɨɥɧɚ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ
ɋɤɨɪɨɫɬɶ ɜɨɥɧɵ:
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(9.15) |
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ɝɞɟ U – ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɬɟɥɚ, |
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– ɦɨɞɭɥɶ ɘɧɝɚ ɢɥɢ ɦɨ- |
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ɞɭɥɶ ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɪɚɫɬɹɠɟɧɢɹ (ɫɠɚɬɢɹ), V – ɩɪɨɞɨɥɶɧɨɟ ɧɚ-
ɩɪɹɠɟɧɢɟ, H – ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ. |
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Ɂɚɤɨɧ Ƚɭɤɚ ɞɥɹ ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɪɚɫɬɹɠɟɧɢɹ (ɫɠɚɬɢɹ): |
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Ɋɚɫɫɦɨɬɪɢɦ ɮɢɡɢɱɟɫɤɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ ɫɥɨɣ dx ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ
(ɫɦ. ɪɢɫ. 9.3).
[(x) [(x+dx)
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Ɋɢɫ. 9.3. ɋɦɟɳɟɧɢɟ ɝɪɚɧɢɰ [(x) ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɨɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɩɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɩɪɨɞɨɥɶɧɨɣ ɭɩɪɭɝɨɣ ɜɨɥɧɵ
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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Ɍɨɝɞɚ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ H ɪɚɜɧɚ |
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[(x dx) [(x) |
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ɢ ɡɚɤɨɧ Ƚɭɤɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ |
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V (x) E[x' . |
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ȿɫɥɢ S – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ |
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ɮɪɚɝɦɟɧɬɚ ɬɟɥɚ, ɚ U – ɟɝɨ ɩɥɨɬɧɨɫɬɶ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ, ɬɨ ɭɪɚɜɧɟ- |
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ɧɢɟ ɞɜɢɠɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɨɹ ɬɟɥɚ ɦɚɫɫɨɣ dm |
USdx ɢɦɟɟɬ |
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ɜɢɞ: |
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S V ( x dx) V ( x) . |
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USdx[ |
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ɉɪɟɨɛɪɚɡɭɟɦ (9.19) ɫ ɭɱɟɬɨɦ ɡɚɤɨɧɚ Ƚɭɤɚ (9.18) ɢ ɦɚɥɨɫɬɢ |
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ɬɨɥɳɢɧɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɨɹ: |
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U[ |
wV |
V x' , |
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(9.20) |
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ɋɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɜɨɥɧɨɜɵɦ ɭɪɚɜɧɟɧɢɟɦ (9.4), ɩɨɥɭɱɢɦ ɩɪɢɜɟɞɟɧɧɨɟ ɜɵɲɟ ɜɵɪɚɠɟɧɢɟ (9.15) ɞɥɹ ɫɤɨɪɨɫɬɢ ɩɪɨɞɨɥɶɧɨɣ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ.
Ȼ. ɉɨɩɟɪɟɱɧɚɹ ɭɩɪɭɝɚɹ ɜɨɥɧɚ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ
ɋɤɨɪɨɫɬɶ ɜɨɥɧɵ:
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(9.21) |
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ɝɞɟ |
G { W |
– ɦɨɞɭɥɶ ɫɞɜɢɝɚ, |
W – ɩɨɩɟɪɟɱɧɨɟ (ɤɚɫɚɬɟɥɶɧɨɟ) ɧɚ- |
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ɩɪɹɠɟɧɢɟ, J = tgD # D – ɬɚɧɝɟɧɫ ɭɝɥɚ ɫɞɜɢɝɚ D. Ɉɬɦɟɬɢɦ, ɱɬɨ ɜ ɨɞɧɨɪɨɞɧɨɦ ɢɡɨɬɪɨɩɧɨɦ ɬɜɟɪɞɨɦ ɬɟɥɟ E > G ɢ ɫɤɨɪɨɫɬɶ ɩɪɨɞɨɥɶɧɨɣ ɡɜɭɤɨɜɨɣ ɜɨɥɧɵ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɩɨɩɟɪɟɱɧɨɣ ɜɨɥɧɵ c|| ! cA .
Ɂɚɤɨɧ Ƚɭɤɚ ɞɥɹ ɫɞɜɢɝɚ: |
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W GJ . |
(9.22) |
Ɋɚɫɫɦɨɬɪɢɦ ɤɨɥɟɛɥɸɳɢɣɫɹ ɩɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɜɨɥɧɨɜɨɝɨ ɜɨɡɦɭɳɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɣ ɮɪɚɝɦɟɧɬ ɬɟɥɚ, ɡɚɤɥɸɱɟɧɧɵɣ ɦɟɠɞɭ ɤɨɨɪɞɢɧɚɬɚɦɢ x ɢ x + dx ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ
(ɫɦ. ɪɢɫ. 9.4).
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ɋɤɨɪɨɫɬɶ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɢɞɟɚɥɶɧɨɦ ɝɚɡɟ ɜ ɫɥɭɱɚɟ ɚɞɢɚɛɚɬɢ-
ɱɟɫɤɨɝɨ (ɛɟɡ ɬɟɩɥɨɩɟɪɟɞɚɱɢ) ɩɪɨɰɟɫɫɚ ɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ: |
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c |
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J |
P0 |
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(9.31) |
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ɝɞɟ J { cP / cV , ɫP ɢ cV – ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɞɚɜɥɟɧɢɢ ɢ ɨɛɴɟɦɟ ɝɚɡɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, P0 – ɞɚɜɥɟɧɢɟ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɨɥɧɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɨɣ dx ɢɞɟɚɥɶɧɨɣ (ɛɟɡ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ) ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɩɥɨɫɤɨɣ ɜɨɥɧɵ (ɫɦ. ɪɢɫ. 9.6).
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[(x) |
[(x+dx) |
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P(x) |
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P(x+dx) |
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x+dx |
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Ɋɢɫ. 9.6. ɋɦɟɳɟɧɢɟ ɝɪɚɧɢɰ [(x) |
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɨɹ ɢɞɟɚɥɶɧɨɣ |
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ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ ɩɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɭɩɪɭɝɨɣ ɜɨɥɧɵ |
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ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜɵɛɪɚɧɧɨɝɨ ɫɥɨɹ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ |
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ɢɦɟɟɬ ɜɢɞ: |
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USdx[ S P( x) P( x dx) S |
dx , |
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(9.32) |
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ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɦɚɬɟɪɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫɪɟɞɵ P |
P(U) . |
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ɉɪɢ ɦɚɥɵɯ ɜɨɡɦɭɳɟɧɢɹɯ ɩɥɨɬɧɨɫɬɢ ǻU |
ɢ ɞɚɜɥɟɧɢɹ ǻP , |
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ɩɪɨɢɫɯɨɞɹɬ ɜɫɥɟɞɫɬɜɢɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜ ɧɟɣ ɭɩɪɭɝɨɣ ɜɨɥɧɵ, ɡɚɩɢɲɟɦ:
ǻP |
wP |
ǻU . |
(9.33) |
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U |
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ɉɪɢ ɷɬɨɦ ɞɥɹ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɩɥɨɬɧɨɫɬɢ
ǻU
U
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ (ɫɦ. ɪɢɫ. 9.6):
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ǻU |
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[(x dx) [(x) |
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[x' . |
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(9.34) |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɜɵɛɪɚɧɧɨɝɨ ɫɥɨɹ ɠɢɞɤɨɫɬɢ |
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ɢɥɢ ɝɚɡɚ ɩɪɢɦɟɬ ɜɢɞ: |
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wP |
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w ǻP |
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w ¨ wP |
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w[ ¸ |
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w [ |
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¨ wU |
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U wx ¸ |
U wU |
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wx2 |
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(9.35) |
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ɋɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɫ ɜɨɥɧɨɜɵɦ ɭɪɚɜɧɟɧɢɟɦ (9.4), ɩɨɥɭɱɢɦ ɩɪɢɜɟɞɟɧɧɨɟ ɜɵɲɟ ɜɵɪɚɠɟɧɢɟ (9.30) ɞɥɹ ɫɤɨɪɨɫɬɢ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɢɞɟɚɥɶɧɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɟ.
Ⱦɥɹ ɫɥɭɱɚɹ ɚɞɢɚɛɚɬɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɢɞɟɚɥɶɧɨɦ ɝɚɡɟ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ ɫɨɫɬɨɹɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɥɨɹ ɝɚɡɚ ɨɛɴɟɦɨɦ dV = Sdx ɢ ɦɚɫɫɨɣ dm = UdV:
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dm |
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P dV J |
const { C , P |
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C dm J UJ . |
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¨ |
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C dm J JUJ 1 |
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(9.36) |
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ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɩɪɢɜɟɞɟɧɧɨɟ ɜɵɲɟ ɜɵɪɚɠɟɧɢɟ (9.31) ɞɥɹ ɫɤɨɪɨɫɬɢ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɢɞɟɚɥɶɧɨɦ ɝɚɡɟ ɜ ɫɥɭɱɚɟ ɚɞɢɚɛɚɬɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ.
9.1.5. ɗɧɟɪɝɟɬɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ
ɉɪɢ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɢ ɩɥɨɫɤɨɣ ɭɩɪɭɝɨɣ ɩɪɨɞɨɥɶɧɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ ɜɞɨɥɶ ɨɫɢ X ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɫɦɟɳɟɧɢɟ ɱɚɫɬɢɰ ɫɪɟɞɵ
ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ [(t, x) , ɢɯ ɫɤɨɪɨɫɬɶ [(t, x) ɢ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɞɟɮɨɪɦɚɰɢɹ ɬɟɥɚ H(t, x) ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ (ɫɦ. ɩɩ. 9.1.2 ɢ
9.1.4.Ⱥ): |
[0 cos Zt kx M0 , |
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[0Z sin Zt kx M0 , |
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[0k sin Zt kx M0 . |
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Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ɉɪɢ ɷɬɨɦ ɨɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫ-
ɬɢɰ ɬɟɥɚ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɜɨɥɧɨɜɨɦ ɞɜɢɠɟɧɢɢ, ɪɚɜɧɚ: |
(9.40) |
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wk (t, x) |
U[ |
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[0 UZ |
sin2 |
Zt kx M0 . |
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Ɉɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰ ɬɟɥɚ,
ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɜɨɥɧɨɜɨɦ ɞɜɢɠɟɧɢɢ:
wp (t, x) |
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ɉɨɫɤɨɥɶɤɭ ɫɤɨɪɨɫɬɶ ɭɩɪɭɝɨɣ ɩɪɨɞɨɥɶɧɨɣ ɜɨɥɧɵ ɜ ɬɜɟɪɞɨɦ |
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ɬɟɥɟ c |
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wp (t, x) |
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[02 UZ2 sin2 |
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Ʉɚɤ ɜɢɞɢɦ, ɞɥɹ ɛɟɝɭɳɟɣ ɜɨɥɧɵ ɦɚɤɫɢɦɭɦɵ ɢ ɦɢɧɢɦɭɦɵ ɤɢ- |
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ɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ ɫɨɜɩɚɞɚɸɬ. |
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Ɉɛɴɟɦɧɚɹ ɩɥɨɬɧɨɫɬɶ ɷɧɟɪɝɢɢ ɜɨɥɧɵ: |
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w(t, x) |
wk (t, x) wp (t, x) w sin2 |
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ɝɞɟ ɚɦɩɥɢɬɭɞɚ ɢɡɦɟɧɟɧɢɹ ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜɨɥɧɵ |
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ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɛɴɟɦɧɨɣ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɝɚɪɦɨɧɢ-
ɱɟɫɤɨɣ ɜɨɥɧɵ ɡɚ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ:
w(t, x) |
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ɉɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɜɨɥɧɵ – ɜɟɥɢɱɢɧɚ, ɱɢɫɥɟɧɧɨ ɪɚɜɧɚɹ ɷɧɟɪɝɢɢ, ɩɟɪɟɧɨɫɢɦɨɣ ɜɨɥɧɨɣ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɟɞɢɧɢɱɧɨɣ ɩɥɨɳɚɞɢ, ɨɪɢɟɧɬɢɪɨɜɚɧɧɨɣ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɜɨɥɧɵ. ȼ ɫɥɭɱɚɟ ɢɡɨɬɪɨɩɧɵɯ ɫɪɟɞ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɮɪɨɧɬɚ ɜɨɥɧɵ:
S (t, x) { |
w(t, x)cs d t |
w(t, x)c , |
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ɝɞɟ s ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɜɨɥɧɵ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȼɟɤɬɨɪ ɍɦɨɜɚ – ɜɟɤɬɨɪ, ɧɚɩɪɚɜɥɟɧɢɟ ɤɨɬɨɪɨɝɨ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɜɨɥɧɵ, ɚ ɦɨɞɭɥɶ ɪɚɜɟɧ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ. ȼ ɫɥɭɱɚɟ ɢɡɨɬɪɨɩɧɵɯ ɫɪɟɞ:
S (t, x) { w(t, x)c . |
(9.47) |
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɜɨɥɧɵ – ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɜɨɥɧɵ ɡɚ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ:
I { S (x,t) |
T |
w(x,t) |
T |
c |
w0 |
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɚɦɩɥɢɬɭɞɚ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɭɩɪɭɝɨɣ ɜɨɥɧɵ w0 [02 UZ2 (9.44), ɱɟɪɟɡ ɤɨɬɨɪɭɸ ɜɵɪɚɠɚɸɬɫɹ ɟɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ (9.45) – (9.48), ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɪɟɞɵ – ɟɟ ɩɥɨɬɧɨɫɬɢ U. ȼ ɫɥɭɱɚɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɭɩɪɭɝɨɣ ɜɨɥɧɵ ɜ ɫɬɪɭɧɟ ɩɨɞ ɩɥɨɬɧɨɫɬɶɸ ɷɧɟɪɝɢɢ ɩɨɞɪɚɡɭɦɟɜɚɟɬɫɹ ɟɟ ɥɢɧɟɣɧɚɹ ɩɥɨɬɧɨɫɬɶ, ɚ ɩɥɨɬɧɨɫɬɶ ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɨɣ ɩɥɨɬɧɨɫɬɶɸ Uɥ ɫɬɪɭɧɵ.
9.1.6. ɉɪɨɞɨɥɶɧɵɣ ɷɮɮɟɤɬ Ⱦɨɩɥɟɪɚ (ɤɥɚɫɫɢɱɟɫɤɢɣ)
ɉɭɫɬɶ Xs ɢ Xd – ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɡɜɭɤɨɜɵɯ ɝɚɪ-
ɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ ɢ ɞɟɬɟɤɬɨɪɚ, ɪɟɝɢɫɬɪɢɪɭɸɳɟɝɨ ɷɬɢ ɜɨɥɧɵ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɫɪɟɞɵ, c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɵ ɜ ɫɪɟɞɟ, ɤɨɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɜɨɣɫɬɜɚɦɢ ɫɪɟɞɵ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɟɣ ɢɫɬɨɱɧɢɤɚ ɢ ɞɟɬɟɤɬɨɪɚ (ɫɦ. ɪɢɫ. 9.7).
Xs |
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Ɋɢɫ. 9.7. ȼɡɚɢɦɧɨɟ ɪɚɫɩɨɥɨɠɟɧɢɟ ɜɨɥɧɨɜɵɯ ɮɪɨɧɬɨɜ ɩɪɢ ɢɫɩɭɫɤɚɧɢɢ ɢ ɪɟɝɢɫɬɪɚɰɢɢ ɡɜɭɤɨɜɨɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɜɨɥɧɵ
ɇɚ ɪɢɫ. 9.7 ɥɟɜɚɹ ɱɟɪɬɚ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚɹ ɨɫɢ ɏ, ɩɨɤɚɡɵɜɚɟɬ ɩɨɥɨɠɟɧɢɟ ɮɪɨɧɬɚ ɜɨɥɧɵ, ɢɫɩɭɳɟɧɧɨɣ ɞɜɢɠɭɳɢɦɫɹ ɢɫɬɨɱɧɢɤɨɦ, ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t. Ʉ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t + Ts (Ts – ɩɟɪɢ-
Ƚɥɚɜɚ 9. Ȼɟɝɭɳɢɟ ɢ ɫɬɨɹɱɢɟ ɜɨɥɧɵ. Ɇɨɞɵ ɢ ɧɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ |
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ɨɞ ɤɨɥɟɛɚɧɢɣ ɢɫɬɨɱɧɢɤɚ) ɮɪɨɧɬ ɫɦɟɫɬɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɟ cTs, ɚ ɢɫɬɨɱɧɢɤ – ɧɚ ɪɚɫɫɬɨɹɧɢɟ XsTs . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɢɧɚ ɜɨɥɧɵ ɜ ɫɪɟɞɟ
ɪɚɜɧɚ (ɪɢɫ. 9.7): |
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O cTs XsTs . |
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ɉɭɫɬɶ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t' ɩɪɢɟɦɧɢɤ ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɥ ɮɪɨɧɬ ɜɨɥɧɵ. ɋɥɟɞɭɸɳɢɣ ɮɪɨɧɬ ɜɨɥɧɵ, ɧɚɯɨɞɹɳɢɣɫɹ ɨɬ ɩɪɢɟɦɧɢɤɚ ɧɚ ɪɚɫɫɬɨɹɧɢɢ Ȝ, ɛɭɞɟɬ ɡɚɪɟɝɢɫɬɪɢɪɨɜɚɧ ɜ ɦɨɦɟɧɬ t' + Td (Td – ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɩɪɢɟɦɧɢɤɚ). ɉɨɫɤɨɥɶɤɭ ɡɚ ɜɪɟɦɹ Td ɩɪɢɟɦɧɢɤ ɫɦɟɫɬɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɟ XdTd , ɚ ɜɨɥɧɨɜɨɣ ɮɪɨɧɬ – ɧɚ ɪɚɫɫɬɨɹ-
ɧɢɟ cTd, ɬɨ |
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O XdTd cTd . |
(9.50) |
ɍɱɢɬɵɜɚɹ, ɱɬɨ Ts = 1/Ȟs ɢ Td = 1/Ȟd, ɩɨɥɭɱɚɟɦ ɢɡ (9.49) ɢ (9.50) ɫɜɹɡɶ ɱɚɫɬɨɬ ɤɨɥɟɛɚɧɢɣ ɞɥɹ ɢɫɬɨɱɧɢɤɚ Ȟs ɢ ɩɪɢɟɦɧɢɤɚ Ȟd:
v |
c Xd |
v . |
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9.1.7. ɋɨɛɫɬɜɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɫɢɫɬɟɦ
ɋɨɛɫɬɜɟɧɧɵɟ (ɫɜɨɛɨɞɧɵɟ) ɤɨɥɟɛɚɧɢɹ – ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ,
ɩɪɟɞɨɫɬɚɜɥɟɧɧɨɣ ɫɚɦɨɣ ɫɟɛɟ (ɩɪɢ ɩɨɫɬɨɹɧɧɵɯ ɜɧɟɲɧɢɯ ɭɫɥɨɜɢɹɯ). Ɋɚɫɩɪɟɞɟɥɟɧɧɚɹ ɫɢɫɬɟɦɚ – ɤɨɥɟɛɚɬɟɥɶɧɚɹ ɫɢɫɬɟɦɚ ɫ ɛɨɥɶ-
ɲɢɦ ɱɢɫɥɨɦ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ, ɯɚɪɚɤɬɟɪɧɵɟ ɪɚɡɦɟɪɵ ɤɨɬɨɪɨɣ
L ! cW , (9.52)
ɝɞɟ c – ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɜɨɥɧɨɜɨɝɨ ɜɨɡɦɭɳɟɧɢɹ, W – ɯɚɪɚɤɬɟɪɧɨɟ ɜɪɟɦɹ ɟɝɨ ɡɚɦɟɬɧɨɝɨ ɢɡɦɟɧɟɧɢɹ.
ɇɨɪɦɚɥɶɧɵɟ ɤɨɥɟɛɚɧɢɹ (ɦɨɞɵ) – ɫɨɛɫɬɜɟɧɧɵɟ ɝɚɪɦɨɧɢɱɟ-
ɫɤɢɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ. ɋɩɟɰɢɚɥɶɧɵɦ ɜɵɛɨɪɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɦɨɠɧɨ ɜɨɡɛɭɞɢɬɶ ɜ ɫɢɫɬɟɦɟ ɬɨɥɶɤɨ ɨɞɧɨ (ɥɸɛɨɟ) ɢɡ ɜɫɟɯ, ɫɜɨɣɫɬɜɟɧɧɵɯ ɫɢɫɬɟɦɟ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ. ɉɪɢ ɧɨɪɦɚɥɶɧɨɦ ɤɨɥɟɛɚɧɢɢ ɫɢɫɬɟɦɵ ɜɫɟ ɟɟ ɷɥɟɦɟɧɬɵ ɤɨɥɟɛɥɸɬɫɹ ɫ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɱɚɫɬɨɬɨɣ – ɧɨɪɦɚɥɶɧɨɣ ɱɚɫɬɨɬɨɣ.
ɇɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ – ɱɚɫɬɨɬɵ ɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ. ɇɨɪɦɚɥɶɧɵɟ ɱɚɫɬɨɬɵ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɟɟ ɩɚɪɚɦɟɬɪɚɦɢ (ɞɥɹ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɫɜɨɣɫɬɜɚɦɢ ɫɪɟɞɵ ɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ).
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȼɨɛɳɟɦ ɫɥɭɱɚɟ ɤɨɥɟɛɚɧɢɹ ɫɢɫɬɟɦɵ ɹɜɥɹɸɬɫɹ ɫɭɩɟɪɩɨɡɢɰɢɟɣ
ɟɟɧɨɪɦɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ɋɬɨɹɱɚɹ ɜɨɥɧɚ – ɩɟɪɢɨɞɢɱɟɫɤɨɟ ɜɨ ɜɪɟɦɟɧɢ ɫɢɧɮɚɡɧɨɟ ɤɨɥɟɛɚɧɢɟ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɢɫɬɟɦɵ ɫ ɯɚɪɚɤɬɟɪɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɚɦɩɥɢɬɭɞɵ ɷɬɢɯ ɤɨɥɟɛɚɧɢɣ – ɱɟɪɟɞɨɜɚɧɢɟɦ ɭɡɥɨɜ ɢ ɩɭɱɧɨɫɬɟɣ.
ɉɭɱɧɨɫɬɢ ɫɬɨɹɱɟɣ ɜɨɥɧɵ – ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɨɛɥɚɫɬɢ, ɜ ɤɨɬɨɪɵɯ ɱɚɫɬɢɰɵ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɥɟɛɥɸɬɫɹ ɫ ɦɚɤɫɢɦɚɥɶɧɨɣ ɚɦɩɥɢɬɭɞɨɣ.
ɍɡɥɵ ɫɬɨɹɱɟɣ ɜɨɥɧɵ – ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɨɛɥɚɫɬɢ, ɜ ɤɨɬɨɪɵɯ ɱɚɫɬɢɰɵ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɫɢɫɬɟɦɵ ɨɫɬɚɸɬɫɹ ɧɟɩɨɞɜɢɠɧɵ.
ȼɫɥɭɱɚɟ ɫɬɨɹɱɢɯ ɜɨɥɧ ɨɫɧɨɜɧɨɣ ɦɨɞɨɣ (ɬɨɧɨɦ) ɧɚɡɵɜɚɟɬɫɹ ɦɨɞɚ ɫ ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɥɢɧɨɣ ɜɨɥɧɵ ɢ ɦɢɧɢɦɚɥɶɧɨɣ ɱɚɫɬɨɬɨɣ. Ɉɫɬɚɥɶɧɵɟ ɦɨɞɵ ɧɚɡɵɜɚɸɬɫɹ ɨɛɟɪɬɨɧɚɦɢ.
ɋɬɨɹɱɚɹ ɜɨɥɧɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ ɫɭɩɟɪɩɨɡɢɰɢɢ ɜɨɥɧɨɜɵɯ ɩɨɥɟɣ (9.3) ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɫɭɩɟɪɩɨɡɢɰɢɢ ɞɜɭɯ ɛɟɝɭɳɢɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɜɨɥɧ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɱɚɫ-
ɬɨɬɨɣ Z , ɫɤɨɪɨɫɬɶɸ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ c ɢ ɚɦɩɥɢɬɭɞɨɣ [0 , ɪɚɫɩɪɨ-
ɫɬɪɚɧɹɸɳɢɯɫɹ ɧɚɜɫɬɪɟɱɭ ɞɪɭɝ ɞɪɭɝɭ (ɧɚɩɪɢɦɟɪ, ɩɚɞɚɸɳɚɹ ɢ ɨɬɪɚɠɟɧɧɚɹ ɜɨɥɧɵ):
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ɚɦɩɥɢɬɭɞɚ ɫɬɨɹɱɟɣ ɜɨɥɧɵ.
ɉɭɱɧɨɫɬɢ ɜ ɜɨɥɧɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (9.53) ɛɭɞɭɬ ɧɚɛɥɸɞɚɬɶɫɹ, ɜ ɬɨɱɤɚɯ (ɫɦ. ɪɢɫ. 9.8), ɤɨɨɪɞɢɧɚɬɵ ɤɨɬɨɪɵɯ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɸ:
kx \ 0 nS , n 1, 2, 3, ... . |
(9.54) |