Механика.Методика решения задач
.pdf
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
121 |
ɉɪɢɦɟɱɚɧɢɹ.
ȼ ɫɥɭɱɚɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɞɢɧɚɦɢɤɭ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɩɩ. I.3 – I.5 ɪɟɱɶ ɢɞɟɬ ɨ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɚ ɩ. II.2 ɧɚɞɨ ɨɩɭɫɬɢɬɶ.
ȼ ɫɥɭɱɚɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɧɚ ɞɢɧɚɦɢɤɭ ɩɪɨɫɬɟɣɲɢɯ ɦɟɯɚɧɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɜ ɩɩ. I.3 – II.2 ɪɟɱɶ ɢɞɟɬ ɨ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɢ ɭɪɚɜɧɟɧɢɹɯ ɞɜɢɠɟɧɢɹ ɬɟɥ ɢ ɫɢɥɚɯ (ɜ ɬɨɦ ɱɢɫɥɟ ɫɢɥɚɯ ɢɧɟɪɰɢɢ), ɞɟɣɫɬɜɭɸɳɢɯ ɦɟɠɞɭ ɬɟɥɚɦɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɵ.
ɉɭɧɤɬɵ II.1 – II.4 ɦɨɠɧɨ ɜɵɩɨɥɧɹɬɶ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɢ.
4.3. ɉɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
Ɂɚɞɚɱɚ 4.1
(ɉɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) Ⱦɜɚ ɧɟɛɨɥɶɲɢɯ ɲɚɪɢɤɚ ɫ ɨɞɢɧɚɤɨɜɨɣ ɦɚɫɫɨɣ m, ɫɨɟɞɢɧɟɧɧɵɟ
ɧɟɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɤɨɣ ɞɥɢɧɨɣ l0, ɥɟɠɚɬ ɧɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ɇɚ ɨɞɢɧ ɢɡ ɲɚɪɢɤɨɜ ɧɚɱɢɧɚɟɬ ɞɟɣɫɬɜɨɜɚɬɶ ɩɨɫɬɨɹɧɧɚɹ ɫɢɥɚ F, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɞɨɥɶ ɨɫɢ ɩɪɭɠɢɧɤɢ (ɫɦ. ɪɢɫ. 4.3).
Fɢɧ |
Fɢɧ |
F |
X, X'
Ɋɢɫ. 4.3
ɑɟɪɟɡ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ɞɥɢɧɚ ɩɪɭɠɢɧɤɢ ɫɬɚɧɨɜɢɬɫɹ ɦɚɤɫɢɦɚɥɶɧɨɣ ɢ ɪɚɜɧɨɣ lmax. Ɉɩɪɟɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɤɢ k.
Ɋɟɲɟɧɢɟ
I. ɉɪɢɥɨɠɢɦ ɫɢɥɭ F ɤ ɩɟɪɟɞɧɟɦɭ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɲɚɪɢɤɭ (ɫɦ. ɪɢɫ. 4.3), ɩɨɫɤɨɥɶɤɭ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɫɢɥɵ ɩɪɨɢɫɯɨɞɢɬ ɪɚɫɬɹɠɟɧɢɟ ɩɪɭɠɢɧɤɢ. ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɜɟ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ: ɥɚɛɨɪɚɬɨɪɧɭɸ ɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ, ɫɜɹɡɚɧɧɭɸ ɫ ɧɟɩɨɞɜɢɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɩɨ ɤɨɬɨɪɨɣ ɫɤɨɥɶɡɹɬ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɬɟɥɚ, ɢ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɭɸɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɭɸ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ ɫɢɫɬɟɦɵ «ɞɜɚ ɲɚɪɢɤɚ + ɩɪɭɠɢɧɤɚ».
ɇɚɩɪɚɜɢɦ ɨɫɶ X ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢ ɨɫɶ X' ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɞɟɣɫɬɜɢɹ ɫɢɥɵ
122 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
F (ɪɢɫ. 4.3). ȼ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɫɢɫɬɟɦɚ ɬɟɥ ɞɜɢɠɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɨɞɧɨɣ ɜɧɟɲɧɟɣ ɫɢɥɵ F . ȼ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤ ɭɤɚɡɚɧɧɨɣ ɫɢɥɟ ɞɨɛɚɜɥɹɸɬɫɹ ɞɜɟ ɩɟɪɟɧɨɫɧɵɟ ɫɢɥɵ ɢɧɟɪɰɢɢ Fɩɟɪ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
II. ɂɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ (ɫɦ. (3.6) ɜ Ƚɥɚɜɟ 3), ɧɚɣɞɟɦ ɭɫɤɨɪɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ «ɞɜɚ ɲɚɪɢɤɚ + ɩɪɭɠɢɧɤɚ» ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ:
aɰɦ |
F |
. |
(4.18) |
|
|||
|
2m |
|
|
ɉɟɪɟɧɨɫɧɵɟ ɫɢɥɵ ɢɧɟɪɰɢɢ (4.16), ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɤɚɠɞɵɣ ɢɡ |
|||
ɲɚɪɢɤɨɜ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɪɚɜɧɵ: |
|
||
Fɩɟɪ |
maɰɦ . |
(4.19) |
|
Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ «ɞɜɚ ɲɚɪɢɤɚ + ɩɪɭɠɢɧɤɚ» ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɞɨ ɦɨɦɟɧɬɚ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɫɬɹɠɟɧɢɹ ɩɪɭɠɢɧɵ (ɫɦ. (3.39) ɜ Ƚɥɚɜɟ 3):
|
k (l |
max |
l |
0 |
)2 |
c |
|
c |
|
|
|
|
|
|
|
Fɩɟɪǻx1 |
F Fɩɟɪ ǻx2 . |
(4.20) |
|
|
|
2 |
|
|
|||||
|
|
|
|
|
c |
c |
– ɢɡɦɟɧɟɧɢɹ ɤɨɨɪɞɢ- |
||
|
|
|
|
|
|
|
|||
Ɂɞɟɫɶ Fɩɟɪ – ɦɨɞɭɥɶ ɫɢɥɵ ɢɧɟɪɰɢɢ, ǻx1 |
ɢ ǻx2 |
||||||||
ɧɚɬ ɡɚɞɧɟɝɨ ɢ ɩɟɪɟɞɧɟɝɨ ɲɚɪɢɤɨɜ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɚɩɪɚɜɥɟɧɢɸ ɞɟɣɫɬɜɢɹ ɫɢɥɵ) ɡɚ ɭɤɚɡɚɧɧɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ. Ʌɟɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (4.20) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɢɡɦɟɧɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɭɩɪɭɝɨ ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɤɢ. ȼ ɦɨɦɟɧɬ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɪɚɫɬɹɠɟɧɢɹ ɩɪɭɠɢɧɤɢ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɨɜ ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɨɣ ɧɭɥɸ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɰɟɧɬɪɨɦ ɦɚɫɫ, ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɲɚɪɢɤɨɜ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ ɢ ɟɟ ɢɡɦɟɧɟɧɢɟ ɡɚ ɭɤɚɡɚɧɧɵɣ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ɬɚɤɠɟ ɪɚɜɧɨ ɧɭɥɸ. ɉɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (4.20) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɚɪɧɭɸ ɪɚɛɨɬɭ ɩɨɫɬɨɹɧɧɵɯ ɜɧɟɲɧɢɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɚ ɫɢɫɬɟɦɵ (ɜɤɥɸɱɚɹ ɫɢɥɵ ɢɧɟɪɰɢɢ).
III. Ɋɟɲɚɹ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ |
(4.18) – (4.20) |
ɫ ɭɱɟɬɨɦ |
||
ǻx2 ǻx1 |
lmax l0 , ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɵɣ |
ɤɨɷɮɮɢɰɢɟɧɬ |
ɭɩɪɭɝɨɫɬɢ |
|
ɩɪɭɠɢɧɤɢ: |
|
|
||
k |
F |
|
|
|
|
. |
|
(4.21) |
|
lmax l0 |
|
|||
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɠɟ ɡɚɞɚɱɢ ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɩɪɟɞɥɨɠɟɧɨ ɜ Ƚɥɚɜɟ 3 (ɡɚɞɚɱɚ 3.5).
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
123 |
ɗɧɟɪɝɟɬɢɱɟɫɤɢɣ ɩɨɞɯɨɞ, ɪɟɚɥɢɡɨɜɚɧɧɵɣ ɧɚɦɢ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ (3.5) ɢ (4.1), ɧɟ ɩɨɡɜɨɥɹɟɬ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɬɟɥ ɫɢɫɬɟɦɵ. ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɩɪɢ ɞɜɢɠɟɧɢɢ ɲɚɪɢɤɨɜ ɞɥɢɧɚ ɩɪɭɠɢɧɤɢ ɢɡɦɟɧɹɟɬɫɹ ɩɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ, ɩɟɪɢɨɞɢɱɟɫɤɢ ɞɨɫɬɢɝɚɹ ɫɜɨɟɝɨ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ. Ɂɚɤɨɧɵ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɨɜ ɢ ɢɡɦɟɧɟɧɢɹ ɞɥɢɧɵ ɫɜɹɡɵɜɚɸɳɟɣ ɢɯ ɩɪɭɠɢɧɤɢ ɛɭɞɭɬ ɩɨɥɭɱɟɧɵ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ (8.11) ɜ Ƚɥɚɜɟ 8.
Ɂɚɞɚɱɚ 4.2
(ɉɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) Ɇɚɬɟɦɚɬɢɱɟɫɤɢɣ ɦɚɹɬɧɢɤ ɞɥɢɧɨɣ l ɢ ɦɚɫɫɨɣ m ɩɨɞɜɟɲɟɧ ɤ
ɩɨɬɨɥɤɭ ɤɚɛɢɧɵ ɥɢɮɬɚ, ɨɩɭɫɤɚɸɳɟɝɨɫɹ ɜɧɢɡ ɫ ɭɫɤɨɪɟɧɢɟɦ a d g
(ɫɦ. ɪɢɫ. 4.4).
O |
|
X |
O' |
|
|
y0 |
|
X' |
|
|
|
D |
F |
|
|
ɩɟɪ |
|
T |
|
W |
Y'
mg
a
Y
Ɋɢɫ. 4.4
ɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɚɛɢɧɵ ɥɢɮɬɚ. Ɋɟɲɢɬɶ ɡɚɞɚɱɭ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɢ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɚɯ ɨɬɫɱɟɬɚ. ȼɥɢɹɧɢɟɦ ɜɪɚɳɟɧɢɹ Ɂɟɦɥɢ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ 1
I. ȼ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ X'O'Y', ɫɜɹɡɚɧɧɨɣ ɫ ɥɢɮɬɨɦ (ɪɢɫ. 4.4), ɧɚ ɦɚɹɬɧɢɤ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɫɢɥɚ ɧɚɬɹɠɟɧɢɹ ɧɢɬɢ T ɢ ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ
124 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Fɩɟɪ |
ma , ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ |
ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
II. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɦɚɫɫɨɣ m ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɤ ɬɪɚɟɤɬɨɪɢɢ ɨɫɶ IJ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɢɦɟɟɬ ɜɢɞ:
maW |
ma sin D mg sin D , |
(4.22) |
ɝɞɟ D – |
ɭɝɨɥ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬ |
ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ |
(ɪɢɫ. 4.4). |
|
|
Ɍɚɧɝɟɧɰɢɚɥɶɧɨɟ ɢ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɹ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ
(ɫɦ. (1.19) ɜ Ƚɥɚɜɟ 1):
|
|
(4.23) |
aW X |
Dl , |
ɝɞɟ X – ɦɨɞɭɥɶ ɥɢɧɟɣɧɨɣ ɫɤɨɪɨɫɬɢ ɦɚɹɬɧɢɤɚ.
III. ɂɡ ɭɪɚɜɧɟɧɢɣ (4.22) ɢ (4.23) ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟ-
ɧɢɹ ɦɚɹɬɧɢɤɚ: |
|
||
D |
g a |
sinD 0 . |
(4.24) |
|
|||
|
l |
|
|
ɉɨɥɭɱɟɧɧɨɟ ɧɟɥɢɧɟɣɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ D ɥɟɝɤɨ ɪɟɲɢɬɶ ɜ ɞɜɭɯ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ: ɩɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ ɢ ɩɪɢ ɞɜɢɠɟɧɢɢ ɥɢɮɬɚ ɫ ɭɫɤɨɪɟɧɢɟɦ, ɪɚɜɧɵɦ ɩɨ ɜɟɥɢɱɢɧɟ ɭɫɤɨɪɟɧɢɸ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ –
ag .
ɉɪɢ ɦɚɥɵɯ ɭɝɥɚɯ ɨɬɤɥɨɧɟɧɢɹ ɦɚɹɬɧɢɤɚ sinD | D ɭɪɚɜɧɟɧɢɟ (4.24) ɫɜɨɞɢɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ:
|
2 |
(4.25) |
D |
Z D 0 , |
|
ɡɞɟɫɶ Z |
(g a) / l |
– ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɚɹ ɨɩɪɟ- |
ɞɟɥɹɟɬɫɹ ɧɟ ɬɨɥɶɤɨ ɞɥɢɧɨɣ ɦɚɹɬɧɢɤɚ ɢ ɭɫɤɨɪɟɧɢɟɦ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ, ɧɨ ɢ ɭɫɤɨɪɟɧɢɟɦ ɥɢɮɬɚ.
ɇɟɬɪɭɞɧɨ ɭɛɟɞɢɬɶɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ
(4.25) ɹɜɥɹɟɬɫɹ ɝɚɪɦɨɧɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ |
|
D(t) D0 cos(Zt M0 ) , |
(4.26) |
ɝɞɟ ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ D0 ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ M0 |
ɨɩɪɟɞɟɥɹɸɬɫɹ |
ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ.
ȼ ɫɥɭɱɚɟ ɞɜɢɠɟɧɢɹ ɥɢɮɬɚ ɫ ɭɫɤɨɪɟɧɢɟɦ, ɪɚɜɧɵɦ ɩɨ ɦɨɞɭɥɸ ɭɫɤɨɪɟɧɢɸ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ, ɭɪɚɜɧɟɧɢɟ (4.24) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
|
(4.27) |
|
D |
0 . |
|
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
125 |
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɜɢɠɟɧɢɟ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɥɢɮɬɨɦ, ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɩɨɫɬɨ-
|
|
ɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ D , ɡɧɚɱɟɧɢɟ ɤɨɬɨɪɨɣ ɡɚɞɚɟɬɫɹ ɧɚɱɚɥɶɧɵ- |
|
ɦɢ ɭɫɥɨɜɢɹɦɢ. Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ: |
(4.28) |
|
|
D(t) D0 Dt , |
|
ɝɞɟ D0 – ɧɚɱɚɥɶɧɨɟ ɨɬɤɥɨɧɟɧɢɟ ɦɚɹɬɧɢɤɚ.
ɂɫɤɨɦɵɣ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɚɛɢɧɵ ɥɢɮɬɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ (4.24), ɤɨɬɨɪɨɟ ɞɨɩɭɫɤɚɟɬ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɜ ɞɜɭɯ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɧɚɦɢ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ (ɫɦ. (4.26) ɢ (4.28)).
Ɋɟɲɟɧɢɟ 2
ȼ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ XOY (ɫɦ. ɪɢɫ. 4.4) ɤɨɨɪɞɢɧɚɬɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɚɹɬɧɢɤɚ x, y ɫɜɹɡɚɧɵ ɫ ɟɝɨ ɭɝɥɨɦ ɨɬɤɥɨɧɟɧɢɹ D ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ X'O'Y' ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
x l sin D,
(4.29)
y y0 l cosD,
ɝɞɟ y0 – ɤɨɨɪɞɢɧɚɬɚ ɧɚɱɚɥɚ ɨɬɫɱɟɬɚ O' ɫɢɫɬɟɦɵ X'O'Y' ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ XOY .
ɉɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɧɚɯɨɞɢɦ, ɞɜɚɠɞɵ ɞɢɮɮɟɪɟɧɰɢɪɭɹ ɩɨ ɜɪɟɦɟɧɢ ɫɨɨɬɧɨɲɟɧɢɹ (4.29):
x lD 2 sin D lD cosD,
(4.30)
y y0 lD 2 cosD lD sin D.
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ X ɢ Y ɢɦɟɟɬ ɜɢɞ (ɫɦ. (2.2) ɜ Ƚɥɚɜɟ 2):
|
T sinD, |
|
mx |
(4.31) |
|
|
mg T cosD. |
|
my |
|
|
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɥɢɮɬ ɞɜɢ- |
||
|
|
a , ɩɨɥɭɱɚɟɦ ɢɡ ɭɪɚɜɧɟ- |
ɠɟɬɫɹ ɜɧɢɡ ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫɤɨɪɟɧɢɟɦ y0 |
||
ɧɢɣ (4.30) ɢ (4.31) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɚɹɬɧɢɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ:
|
g a |
sin D 0 . |
(4.32) |
|
l |
||||
D |
126 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
Ʉɚɤ ɜɢɞɢɦ, ɭɪɚɜɧɟɧɢɟ (4.32) ɫɨɜɩɚɞɚɟɬ ɫ ɭɪɚɜɧɟɧɢɟɦ (4.24), ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɜ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɫɨɜɩɚɞɟɬ ɫ ɪɟɲɟɧɢɟɦ (4.26) ɢ (4.28) ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɨɩɬɢɦɚɥɶɧɵɦ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɹɜɥɹɟɬɫɹ ɜɵɛɨɪ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ.
Ɂɚɞɚɱɚ 4.3
(ɉɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) ɇɟɛɨɥɶɲɨɟ ɬɟɥɨ ɩɨɦɟɫɬɢɥɢ ɧɚ ɜɟɪɲɢɧɭ ɝɥɚɞɤɨɝɨ ɩɨɥɭɰɢɥɢɧ-
ɞɪɚ ɪɚɞɢɭɫɨɦ R, ɧɚɯɨɞɹɳɟɝɨɫɹ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ (ɫɦ. ɪɢɫ. 4.5). ɉɨɥɭɰɢɥɢɧɞɪɭ ɫɨɨɛɳɚɸɬ ɩɨɫɬɨɹɧɧɨɟ ɝɨɪɢɡɨɧɬɚɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ à , ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɬɟɥɨ ɧɚɱɢɧɚɟɬ ɫɨɫɤɚɥɶɡɵɜɚɬɶ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ. Ɉɩɪɟɞɟɥɢɬɶ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ X0 ɬɟɥɚ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɭɰɢɥɢɧɞɪɚ ɜ ɦɨɦɟɧɬ ɨɬɪɵɜɚ ɢ ɜɵɫɨɬɭ H , ɧɚ ɤɨɬɨɪɨɣ ɩɪɨɢɡɨɣɞɟɬ ɨɬɪɵɜ.
Ɋɟɲɟɧɢɟ
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɟɣɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ, ɫɜɹɡɚɧɧɨɣ ɫ ɩɨɥɭɰɢɥɢɧɞɪɨɦ. Ɉɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɭɫɤɨɪɟɧɢɟ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɪɚɜɧɨ a . ɇɚ ɬɟɥɨ, ɧɚɯɨɞɹɳɟɟɫɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ ɞɟɣɫɬɜɭɸɬ ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ N ɢ
ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ Fɩɟɪ ma , ɢɡɨɛɪɚɠɟɧɧɵɟ ɧɚ ɪɢɫ. 4.5.
Y'
Fɩɟɪ
a
W |
X' |
|
n
Ɋɢɫ. 4.5
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
127 |
ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ. ɉɨɥɭɰɢɥɢɧɞɪ ɫɱɢɬɚɟɦ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ, ɚ ɫɨɫɤɚɥɶɡɵɜɚɸɳɟɟ
ɫɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɨ – ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɨɣ.
II. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɧɨɪɦɚɥɶɧɭɸ ɢ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɤ ɬɪɚɟɤɬɨɪɢɢ ɨɫɢ ɢɦɟɟɬ ɜɢɞ:
m |
X2 |
mg cos- N ma sin- , |
(4.33) |
|
R |
||||
|
mg sin- ma cos- , |
(4.34) |
||
|
|
|||
mX |
||||
ɝɞɟ X ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɭɰɢɥɢɧɞɪɚ, ɚ - ɭɝɨɥ, ɡɚɞɚɸɳɢɣ ɩɨɥɨɠɟɧɢɟ ɬɟɥɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɨ ɟɝɨ ɨɬɪɵɜɚ (ɫɦ. ɪɢɫ. 4.5).
ȼ ɦɨɦɟɧɬ ɨɬɪɵɜɚ ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ
ɬɟɥɨ ɫɨ ɫɬɨɪɨɧɵ ɩɨɥɭɰɢɥɢɧɞɪɚ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ: |
|
|
N |
0 . |
(4.35) |
Ⱦɨɩɨɥɧɢɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (4.33) – (4.35) ɧɚɱɚɥɶɧɵɦɢ ɭɫ- |
||
ɥɨɜɢɹɦɢ ɞɥɹ ɭɝɥɚ - ɢ ɫɤɨɪɨɫɬɢ ɬɟɥɚ X : |
|
|
-(t |
0) 0 , X(t 0) 0 . |
(4.36) |
ɉɨɥɭɱɟɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (4.33) – (4.36), ɩɨɡɜɨɥɹɸɳɚɹ ɨɩɪɟɞɟɥɢɬɶ ɧɟ ɬɨɥɶɤɨ ɫɤɨɪɨɫɬɶ ɬɟɥɚ, ɧɨ ɢ ɭɝɨɥ - .
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɨɩɵɬɤɚ ɩɪɹɦɨɝɨ ɪɟɲɟɧɢɹ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɤɨɪɨɫɬɢ X ɢ ɭɝɥɚ - ɩɪɢɜɨɞɢɬ ɤ ɝɪɨɦɨɡɞɤɢɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ, ɟɫɥɢ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɡɚɤɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ. ɂɡɦɟɧɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɬɟɥɚ ɧɚ ɢɧɬɟɪɜɚɥɟ ɜɪɟɦɟɧɢ ɨɬ ɧɚɱɚɥɚ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɞɨ ɦɨɦɟɧɬɚ ɟɝɨ ɨɬɪɵɜɚ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ ɪɚɜɧɨ ɪɚɛɨɬɟ ɫɢɥɵ ɢɧɟɪɰɢɢ (ɪɚɛɨɬɚ ɫɢɥɵ ɪɟɚɤɰɢɢ ɨɩɨɪɵ ɜ ɜɵɛɪɚɧɧɨɣ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɪɚɜɧɚ ɧɭɥɸ) ɧɚ ɷɬɨɦ ɢɧɬɟɪɜɚɥɟ:
|
mX02 |
|
- |
|
|
|
mgR(1 cos-0 ) |
³0 ma cos-R d- , |
(4.37) |
||
2 |
|||||
|
0 |
|
|||
|
|
|
|
||
ɝɞɟ X0 , -0 |
– ɫɤɨɪɨɫɬɶ ɢ ɭɝɨɥ - ɜ ɦɨɦɟɧɬ ɨɬɪɵɜɚ ɬɟɥɚ ɨɬ ɩɨɜɟɪɯɧɨ- |
||||
ɫɬɢ ɩɨɥɭɰɢɥɢɧɞɪɚ. |
|
|
|||
III. ɂɡ ɭɪɚɜɧɟɧɢɣ (4.33), (4.35) ɢ (4.37) ɩɨɥɭɱɚɟɦ ɞɜɚ ɫɨɨɬɧɨ- |
|||||
ɲɟɧɢɹ ɞɥɹ ɤɜɚɞɪɚɬɚ ɫɤɨɪɨɫɬɢ: |
|
||||
X2 |
R(g cos- a sin- ) , |
(4.38) |
|||
0 |
0 |
0 |
|
||
128 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
X2 |
2R(g(1 cos- ) a sin- ) . |
(4.39) |
|||||||||||
0 |
|
|
|
|
|
0 |
0 |
|
|
|
|
||
ɋɨɨɬɧɨɲɟɧɢɟ (4.39) ɩɟɪɟɩɢɲɟɦ ɜ ɜɢɞɟ |
|
|
|
||||||||||
X2 |
Rg Rg cos- Ra sin- . |
(4.40) |
|||||||||||
0 |
|||||||||||||
2 |
|
|
|
|
|
0 |
0 |
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
||
ɋɥɨɠɟɧɢɟ ɭɪɚɜɧɟɧɢɣ (4.38) ɢ (4.40) ɩɨɡɜɨɥɹɟɬ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ |
|||||||||||||
ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɭɰɢɥɢɧɞɪɚ ɜ ɦɨɦɟɧɬ ɨɬɪɵɜɚ: |
|
|
|
||||||||||
X0 |
|
2 |
|
gR . |
|
|
|
|
(4.41) |
||||
|
3 |
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
||
Ⱦɥɹ ɭɝɥɚ -0, ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɨɢɡɨɣɞɟɬ ɨɬɪɵɜ, ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ |
|||||||||||||
(4.38) ɢ (4.41) ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ: |
|
|
|
||||||||||
cos-0 |
|
|
2g 2 a 5g 2 9a2 |
|
|
|
|||||||
|
|
|
|
|
|
. |
|
(4.42) |
|||||
|
|
3(g 2 |
a2 ) |
|
|||||||||
|
|
|
|
|
|
|
|
|
|
||||
ɂɫɤɨɦɚɹ ɜɵɫɨɬɚ H, ɧɚ ɤɨɬɨɪɨɣ ɬɟɥɨ ɨɬɨɪɜɟɬɫɹ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ |
|||||||||||||
ɩɨɥɭɰɢɥɢɧɞɪɚ, ɪɚɜɧɚ: |
|
|
|
|
|
|
|
||||||
H |
R cos-0 |
2g 2 a |
5g 2 9a2 |
|
|
|
|||||||
|
|
|
|
R . |
(4.43) |
||||||||
|
3(g 2 |
|
|
||||||||||
|
|
|
|
|
|
a2 ) |
|
|
|
||||
ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɩɪɢ a 0 ɜɵɪɚɠɟɧɢɟ (4.43) ɞɚɟɬ ɡɧɚɱɟ- |
|||||||||||||
ɧɢɟ ɜɵɫɨɬɵ ɨɬɪɵɜɚ ɬɟɥɚ ɨɬ ɧɟɩɨɞɜɢɠɧɨɝɨ ɩɨɥɭɰɢɥɢɧɞɪɚ H |
|
2 |
R . |
||||||||||
3 |
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|||
Ɂɚɞɚɱɚ 4.4
(ȼɪɚɳɚɸɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) Ƚɨɪɢɡɨɧɬɚɥɶɧɵɣ ɞɢɫɤ ɜɪɚɳɚɟɬɫɹ ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨ-
ɪɨɫɬɶɸ Ȧ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ
(ɫɦ. ɪɢɫ. 4.6).
Z'
Z
Fɰɛ
FɄɨɪ |
V Y' |
mg
Ɋɢɫ. 4.6
Ƚɥɚɜɚ 4. Ⱦɜɢɠɟɧɢɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ |
129 |
ɉɨ ɨɞɧɨɦɭ ɢɡ ɞɢɚɦɟɬɪɨɜ ɞɢɫɤɚ ɜ ɫɬɨɪɨɧɭ ɨɬ ɰɟɧɬɪɚ ɞɜɢɠɟɬɫɹ ɧɟɛɨɥɶɲɨɟ ɬɟɥɨ ɦɚɫɫɨɣ ɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɢɫɤɚ ɫɤɨɪɨɫɬɶɸ V . ɇɚɣɬɢ ɫɢɥɭ F, ɫ ɤɨɬɨɪɨɣ ɞɢɫɤ ɞɟɣɫɬɜɭɟɬ ɧɚ ɬɟɥɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɨɧɨ ɧɚɯɨɞɢɬɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ r ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɧɟɢɧɟɪɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ X'Y'Z', ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɭɸ ɫ ɜɪɚɳɚɸɳɢɦɫɹ ɞɢɫɤɨɦ, ɩɪɢ ɷɬɨɦ ɧɚɩɪɚɜɢɦ ɨɞɧɭ ɢɡ ɨɫɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ Z' ɜɞɨɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ ɞɢɫɤɚ Z , ɚ ɞɪɭɝɭɸ Y' – ɜɞɨɥɶ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɢɫɤɚ V (ɪɢɫ. 4.6). ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ ɧɚ ɬɟɥɨ ɞɟɣɫɬɜɭɸɬ ɫɢɥɚ ɬɹɠɟɫɬɢ mg , ɫɢɥɚ ɪɟɚɤɰɢɢ ɞɢɫɤɚ F (ɧɟ ɢɡɨɛɪɚɠɟɧɧɚɹ ɧɚ ɪɢɫɭɧɤɟ),
ɩɟɪɟɧɨɫɧɚɹ ɫɢɥɚ ɢɧɟɪɰɢɢ, ɪɚɜɧɚɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɟ ɢɧɟɪɰɢɢ Fɰɛ (ɫɦ. (4.16)) –
Fɰɛ |
m>Ȧ>Ȧr@@, |
(4.44) |
ɢ ɫɢɥɚ ɢɧɟɪɰɢɢ Ʉɨɪɢɨɥɢɫɚ (ɫɦ. (4.17)) – |
|
|
FɄɨɪ |
2m>ȦV @. |
(4.45) |
ɋɢɥɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
II. ɉɨɞ ɞɟɣɫɬɜɢɟɦ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜɵɲɟ ɫɢɥ ɬɟɥɨ ɞɜɢɠɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ ɫ ɩɨɫɬɨɹɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɢɫɤɚ ɫɤɨɪɨɫɬɶɸ. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɜ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ ɜ
ɜɵɛɪɚɧɧɨɣ ɧɚɦɢ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ: |
|
0 mg FɄɨɪ Fɰɛ F . |
(4.46) |
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ (4.46) ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɫɢɫɬɟɦɵ X'Y'Z', ɭɤɚɡɚɧɧɵɟ ɧɚ ɪɢɫ. 4.6, ɢɦɟɟɬ ɜɢɞ:
0Fx' FɄɨɪ ,
0 Fy' Fɰɛ , |
(4.47) |
0Fz ' mg.
Ʉɚɤ ɜɢɞɢɦ, ɫɢɥɚ ɪɟɚɤɰɢɢ ɞɢɫɤɚ ɢɦɟɟɬ ɨɬɥɢɱɧɵɟ ɨɬ ɧɭɥɹ ɩɪɨɟɤɰɢɢ ɧɚ ɜɫɟ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ.
III. ɉɨɞɫɬɚɜɥɹɹ ɜ (4.47) ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɰɟɧɬɪɨɛɟɠɧɨɣ ɫɢɥɵ ɢɧɟɪɰɢɢ (4.44) ɢ ɫɢɥɵ Ʉɨɪɢɨɥɢɫɚ (4.45), ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɢɫɤɨɦɵɯ ɩɪɨɟɤɰɢɣ ɧɚ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɫɢɫɬɟɦɵ X'Y'Z' ɫɢɥɵ ɪɟɚɤɰɢɢ ɞɢɫɤɚ:
130 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
Fx' 2mZV , |
|
|
F |
mZ2r, |
(4.48) |
y' |
|
|
Fz ' |
mg. |
|
ɉɪɢ ɷɬɨɦ ɦɨɞɭɥɶ ɫɢɥɵ ɪɟɚɤɰɢɢ ɞɢɫɤɚ F ɪɚɜɟɧ |
|
|
F |
(mg)2 (mZ2r)2 4(mZV )2 , |
(4.49) |
ɚ ɧɚɩɪɚɜɥɹɸɳɢɟ ɤɨɫɢɧɭɫɵ ɫɢɥɵ F ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ X'Y'Z', ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɞɢɫɤɨɦ, ɪɚɜɧɵ:
cosD' |
|
|
|
|
2mZV |
, |
||||
|
(mg)2 (mZ2r)2 4(mZV )2 |
|||||||||
cos E ' |
|
|
|
|
mZ2r |
|
, |
|||
|
(mg)2 (mZ2r)2 4(mZV )2 |
|
||||||||
cosJ ' |
|
|
|
|
|
mg |
|
|
||
|
|
. |
|
|
||||||
|
(mg)2 (mZ2r)2 4(mZV )2 |
|
|
|||||||
|
|
|
|
|
|
|
Ɂɚɞɚɱɚ 4.5 |
|
|
|
|
(ȼɪɚɳɚɸɳɚɹɫɹ ɧɟɢɧɟɪɰɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɨɬɫɱɟɬɚ) |
|||||||||
ȼɢɧɬɨɜɤɭ |
ɫ |
ɨɩɬɢɱɟɫɤɢɦ |
|
|
||||||
ɩɪɢɰɟɥɨɦ ɧɚɜɟɥɢ ɧɚ ɜɟɪɬɢɤɚɥɶ- |
|
Z' |
||||||||
ɧɭɸ ɱɟɪɬɭ ɦɢɲɟɧɢ, ɧɚɯɨɞɹɳɭɸ- |
|
V |
||||||||
ɫɹ ɬɨɱɧɨ ɜ ɫɟɜɟɪɧɨɦ ɧɚɩɪɚɜɥɟ- |
|
Ȧ Fɝɪ |
||||||||
ɧɢɢ, ɢ ɜɵɫɬɪɟɥɢɥɢ. ɉɪɟɧɟɛɪɟɝɚɹ |
|
|||||||||
ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɜɨɡɞɭɯɚ, ɨɩɪɟ- |
|
|
||||||||
ɞɟɥɢɬɶ ɧɚ ɤɚɤɨɟ ɪɚɫɫɬɨɹɧɢɟ ɢ ɜ |
|
ij |
||||||||
ɤɚɤɭɸ ɫɬɨɪɨɧɭ ɩɭɥɹ, ɩɨɩɚɜ ɜ |
|
|
||||||||
ɦɢɲɟɧɶ, ɨɬɤɥɨɧɢɬɫɹ ɨɬ ɱɟɪɬɵ. |
|
|
||||||||
ȼɵɫɬɪɟɥ |
ɩɪɨɢɡɜɟɞɟɧ |
ɜɞɨɥɶ ɩɨ- X' |
|
|
||||||
ɜɟɪɯɧɨɫɬɢ |
Ɂɟɦɥɢ |
ɧɚ ɲɢɪɨɬɟ |
|
|
||||||
M 60q |
(ɫɦ. ɪɢɫ. 4.7), |
ɧɚɱɚɥɶɧɚɹ |
|
|
||||||
ɫɤɨɪɨɫɬɶ |
ɩɭɥɢ |
V |
900 ɦ/ɫ, ɪɚɫ- |
|
Ɋɢɫ. 4.7 |
|||||
ɫɬɨɹɧɢɟ ɞɨ ɦɢɲɟɧɢ s |
1ɤɦ. |
|
||||||||
Ɋɟɲɟɧɢɟ
(4.50)
FɄɨɪ
Fɰɛ
Y'
I. Ɂɚɞɚɱɭ ɪɟɲɚɟɦ ɜ ɧɟɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ X'Y'Z', ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ Ɂɟɦɥɟɣ, ɩɪɢ ɷɬɨɦ ɧɚɩɪɚɜɢɦ ɨɞɧɭ ɢɡ ɨɫɟɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ Z' ɜɞɨɥɶ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ ɜɪɚɳɟɧɢɹ Ɂɟɦɥɢ Ȧ , ɚ