ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ ȽɈɋɍȾȺɊɋɌȼȿɇɇɈȿ ɈȻɊȺɁɈȼȺɌȿɅɖɇɈȿ ɍɑɊȿɀȾȿɇɂȿ ȼɕɋɒȿȽɈ ɉɊɈɎȿɋɋɂɈɇȺɅɖɇɈȽɈ ɈȻɊȺɁɈȼȺɇɂə «ȼɈɊɈɇȿɀɋɄɂɃ ȽɈɋɍȾȺɊɋɌȼȿɇɇɕɃ ɍɇɂȼȿɊɋɂɌȿɌ»
Ɋȿɒȿɇɂȿ ɁȺȾȺɑ ɉɈ ɌȿɈɊȿɌɂɑȿɋɄɈɃ ɆȿɏȺɇɂɄȿ. ɑɚɫɬɶ 3. Ⱦɢɧɚɦɢɤɚ
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ
ɋɨɫɬɚɜɢɬɟɥɢ: Ⱥ.ɋ. ɑɟɛɨɬɚɪɟɜ, ɘ.Ⱦ. ɓɟɝɥɨɜɚ
ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ 2007
ɍɬɜɟɪɠɞɟɧɨ ɧɚɭɱɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɦ ɫɨɜɟɬɨɦ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ 28 ɮɟɜɪɚɥɹ 2007 ɝ., ɩɪɨɬɨɤɨɥ ʋ 6
ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɨɞɝɨɬɨɜɥɟɧɨ ɧɚ ɤɚɮɟɞɪɟ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɢ ɩɪɢɤɥɚɞɧɨɣ ɦɟɯɚɧɢɤɢ ɮɚɤɭɥɶɬɟɬɚ ɉɆɆ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. Ɋɟɤɨɦɟɧɞɭɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 2-ɝɨ ɤɭɪɫɚ ɞɧɟɜɧɨɝɨ ɨɬɞɟɥɟɧɢɹ ɢ 3-ɝɨ ɤɭɪɫɚ ɜɟɱɟɪɧɟɝɨ ɨɬɞɟɥɟɧɢɹ.
Ⱦɥɹ ɫɩɟɰɢɚɥɶɧɨɫɬɢ: 010200 (010501) – ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚɬɢɤɚ
2
ɋɨɞɟɪɠɚɧɢɟ ȼɜɟɞɟɧɢɟ §1. Ⱦɢɧɚɦɢɤɚ ɬɨɱɤɢ. ɉɪɹɦɚɹ ɢ ɨɛɪɚɬɧɚɹ ɡɚɞɚɱɚ §2. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɵɯ ɬɟɥ §3. Ʉɨɥɟɛɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ɋɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ §4. Ɂɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ §5. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ §6. Ɋɟɡɨɧɚɧɫ §7. Ɍɟɨɪɟɦɚ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ §8. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ §9. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ §10. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ §11. ɍɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ §12. ɋɩɢɫɨɤ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ [2] §13. Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɧɚɦɢɤɢ ɋɩɢɫɨɤ ɪɟɤɨɦɟɧɞɭɟɦɨɣ ɥɢɬɟɪɚɬɭɪɵ
3
4 5 7 12 15 18 20 23 24 27 29 35 48 49 53
ȼɜɟɞɟɧɢɟ ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɫɩɟɰɢɚɥɶɧɨɫɬɢ 010501 (010200) «ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɢɧɮɨɪɦɚɬɢɤɚ», ɨɛɭɱɚɸɳɢɯɫɹ ɧɚ ɜɬɨɪɨɦ ɤɭɪɫɟ ɞɧɟɜɧɨɝɨ ɨɬɞɟɥɟɧɢɹ ɢ ɬɪɟɬɶɟɦ ɤɭɪɫɟ ɜɟɱɟɪɧɟɝɨ ɨɬɞɟɥɟɧɢɹ, ɩɨ ɞɢɫɰɢɩɥɢɧɟ ȿɇ.Ɏ.03.1 «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ». ɋɨɝɥɚɫɧɨ ɭɱɟɛɧɨɦɭ ɩɥɚɧɭ ɚɭɞɢɬɨɪɧɵɟ ɡɚɧɹɬɢɹ ɩɨ ɞɚɧɧɨɣ ɞɢɫɰɢɩɥɢɧɟ ɜɤɥɸɱɚɸɬ 2 ɱɚɫɚ ɥɟɤɰɢɣ ɢ 2 ɱɚɫɚ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɧɹɬɢɣ ɜ ɧɟɞɟɥɸ ɜ ɬɟɱɟɧɢɟ ɨɞɧɨɝɨ ɫɟɦɟɫɬɪɚ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɨɛɴɟɦ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ, ɨɬɜɨɞɢɦɨɣ ɧɚ ɨɫɜɨɟɧɢɟ ɩɪɟɞɦɟɬɚ, ɫɨɫɬɚɜɥɹɟɬ 68 ɱɚɫɨɜ (72 ɱɚɫɚ ɜ/ɨ). ɉɪɟɞɥɚɝɚɟɦɵɣ ɭɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɩɪɢɡɜɚɧ ɩɨɦɨɱɶ ɫɬɭɞɟɧɬɚɦ ɢɡɭɱɢɬɶ ɨɞɢɧ ɢɡ ɧɚɢɛɨɥɟɟ ɬɪɭɞɧɵɯ ɪɚɡɞɟɥɨɜ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ – ɞɢɧɚɦɢɤɭ. ɉɨɫɨɛɢɟ ɜɤɥɸɱɚɟɬ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɨɫɧɨɜɵ – ɨɩɪɟɞɟɥɟɧɢɹ ɢ ɩɪɚɤɬɢɱɟɫɤɢɟ ɩɪɢɦɟɪɵ ɜ ɜɢɞɟ ɪɟɲɟɧɢɹ ɧɚɢɛɨɥɟɟ ɬɢɩɢɱɧɵɯ ɡɚɞɚɱ ɞɢɧɚɦɢɤɢ. Ɍɚɤɠɟ ɜ ɩɨɫɨɛɢɢ ɫɨɞɟɪɠɢɬɫɹ ɫɩɢɫɨɤ ɜɨɩɪɨɫɨɜ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ɢ ɩɟɪɟɱɟɧɶ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ. ɋɩɢɫɨɤ ɨɫɧɨɜɧɵɯ ɮɨɪɦɭɥ ɞɢɧɚɦɢɤɢ ɢ ɥɢɬɟɪɚɬɭɪɧɵɟ ɢɫɬɨɱɧɢɤɢ ɩɨ ɞɚɧɧɨɣ ɞɢɫɰɢɩɥɢɧɟ ɞɨɥɠɧɵ ɧɚɰɟɥɢɬɶ ɱɢɬɚɬɟɥɟɣ ɧɚ ɩɪɨɞɭɤɬɢɜɧɭɸ ɫɚɦɨɫɬɨɹɬɟɥɶɧɭɸ ɪɚɛɨɬɭ.
4
§1. Ⱦɢɧɚɦɢɤɚ ɬɨɱɤɢ. ɉɪɹɦɚɹ ɢ ɨɛɪɚɬɧɚɹ ɡɚɞɚɱɚ ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ:
mw F . m – ɢɧɟɪɬɧɚɹ ɦɚɫɫɚ ɬɨɱɤɢ; w – ɭɫɤɨɪɟɧɢɟ; F – ɫɢɥɚ. ɉɪɹɦɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ ɬɨɱɤɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɩɨ ɢɡɜɟɫɬɧɨɦɭ ɡɚɤɨɧɭ ɞɜɢɠɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɫɢɥɭ, ɜɵɡɜɚɜɲɭɸ ɷɬɨ ɞɜɢɠɟɧɢɟ. Ɉɛɪɚɬɧɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ ɬɨɱɤɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɩɨ ɢɡɜɟɫɬɧɨɣ ɫɢɥɟ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ. ɑɬɨɛɵ ɪɟɲɢɬɶ ɩɪɹɦɭɸ ɢ/ɢɥɢ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ, ɧɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. I. Ⱦɟɤɚɪɬɨɜɚ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ xOyz . r xi yj zk , x x t ½ ° y j zk , y y t ¾ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ. v xi ɝɞe z z t °¿ w xi yj zk , F Fx i Fy j Fz k Fx , Fy , Fz – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ Ox, Oy , Oz . Ɍɚɤ ɤɚɤ ɫɢɥɚ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɨɠɟɧɢɹ ɬɨɱɤɢ, ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɢ ɜɪɟɦɟɧɢ, ɬɨ mx Fx (t , x , y , z , x , y , z ), ° (1) ® my Fy (t , x , y , z , x , y , z ), ° ¯ mz Fz (t , x , y , z , x , y , z ). ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ. II. ɐɢɥɢɧɞɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ r , M, z . r r (t ) ½ ° M M(t )¾ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ. z z (t ) °¿ v
dr o d M o dz o r r z M dt dt dt
r r
o
rMM
o
o z z ,
5
§ d 2 r § dM · 2 · o § d 2M dr dM · o d 2 z o ¸r ¨ r ¸M 2 z . w ¨ 2 r¨ 2 ¸ ¨ dt 2 ¨ dt dt dt ¸¹ dt © dt ¹ ¸¹ © © o
o
o
Fr , FM , Fz – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ r , M , z ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. F – ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɨɠɟɧɢɹ, ɫɤɨɪɨɫɬɢ, ɜɪɟɦɟɧɢ. 2 § d 2r dr dM dz · § dM · · § , ¸, ° m ¨¨ 2 r ¨ ¸ ¸¸ Fr ¨ t , r , M , z , , dt dt dt dt dt ¹ © ¹ © ° © ¹ ° (2) dr dM · dr dM dz · ° § d 2M § , ¸, ®m ¨ r 2 2 ¸ FM ¨ t , r , M , z , , dt dt ¹ dt dt dt ¹ © ° © dt ° § d 2z · ° m ¨ 2 ¸ F z ¨§ t , r , M , z , dr , dM , dz ¸· . dt dt dt ¹ ° © dt ¹ © ¯ ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ III. ȿɫɬɟɫɬɜɟɧɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ W, n, b . S S (t ) – ɞɭɝɨɜɚɹ ɤɨɨɪɞɢɧɚɬɚ. dS v W, dt d 2S v2 w W n 0b. 2 U dt
FW , Fn , Fb – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ ɟɫɬɟɫɬɜɟɧɧɨɝɨ ɬɪɟɯɝɪɚɧɧɢɤɚ. d 2S dS · § FW ¨ t , S , ¸, ° m dt 2 dt ¹ © ° ° dS 1 dS · § Fn ¨ t , S , ®m ¸, dt ¹ © ° dt U ° dS · § ° 0 Fb ¨ t , S , ¸. dt ¹ © ¯ ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɨɫɹɯ ɟɫɬɟɫɬɜɟɧɧɨɝɨ ɬɪɟɯɝɪɚɧɧɢɤɚ.
6
§2. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɵɯ ɬɟɥ Ɇɨɦɟɧɬɨɦ
ɢɧɟɪɰɢɢ
Jz
ɫɢɫɬɟɦɵ
N
ɦɚɬɟɪɢɚɥɶɧɵɯ
ɬɨɱɟɤ
N
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ J z
¦m r
2 k k
. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ
k 1
ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɦɚɫɫ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ ɨɫɢ. rk – ɪɚɫɫɬɨɹɧɢɟ ɨɬ Ʉ-ɣ ɬɨɱɤɢ ɞɨ ɨɫɢ z . Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ 2 ɜɟɥɢɱɢɧɚ J z t 0 . ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ >J z @ ɤɝ ɦ ɜ 2 ɋɂ, > J z @ ɤɝ ɦ ɫ ɜ ɬɟɯɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ.
Ⱦɥɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ȺɌɌ) § · ¨ M¸ 2 2 J z ³ r dm ³ U ( x, y, z )r dv ¨ U const ¸ V ¸ ¨
(M ) (V ) ȿɋɅɂ © ¹ ɝɞɟ M – ɦɚɫɫɚ, V – ɨɛɴɟɦ, dv – ɷɥɟɦɟɧɬ ɨɛɴɟɦɚ, M U – ɩɥɨɬɧɨɫɬɶ ɩɨɫɬɨɹɧɧɚɹ. V
M V
³r
(V )
2
dv ,
Ɍɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ (Ƚɸɣɝɟɧɫɚ): J z1 J zc Md 2 .
M – ɦɚɫɫɚ ɬɟɥɚ, z c – ɨɫɶ, ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ, z1 – ɨɫɶ ɩɚɪɚɥɥɟɥɶɧɚɹ z c , d – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ.
Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ R ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z – ɜɟɥɢɱɢɧɚ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ 2 ɪɚɜɟɧɫɬɜɭ: J z MR . ɐɟɧɬɪɨɦ ɦɚɫɫ ɫɢɫɬɟɦɵ N ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ (ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɩɨɥɹ ɬɹɠɟɫɬɢ) ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɚ ɋ, ɪɚɞɢɭɫ ɤɨɬɨɪɨɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ: rC
m1 r1 m2 r2 m N rN . m1 m2 m N 7
Ɂɚɞɚɱɚ 1.
Ɍɨɧɤɢɣ ɨɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɞɥɢɧɨɣ A ɢ ɦɚɫɫɨɣ M . ȼɵɱɢɫɥɢɦ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z , ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɫɬɟɪɠɧɸ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɤɨɧɟɰ Ⱥ. Jz
2 ³ U r dv (V )
A 2
2
U ³ r dv
U ³ x dx 0
(V )
U
x
3
A
U
A
3 0
3
3
M A
3
A 3
MA 3
2 ,
M – ɥɢɧɟɣɧɚɹ ɩɥɨɬɧɨɫɬɶ (ɦɚɫɫɚ ɟɞɢɧɢɰɵ A (V ) A . ɞɥɢɧɵ ɫɬɟɪɠɧɹ). r x ; dv dx
ɬ. ɤ. U
const
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɫɬɪɟɠɧɸ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ. A A 2 3 2 § · M A M § A3 A 3 · MA 2 2 ¨ ¸ ¸ ¨ J zC x dx U ³ A ¨© 8 3 8 3 ¸¹ 12 A ¨© 3 ¸¹ A 2 A 2 ɢ/ɢɥɢ ɩɨ ɬɟɨɪɟɦɟ ɒɬɟɣɧɟɪɚ:
J zC
MA 2 §A· M¨ ¸ 3 ©2¹
J z A Md 2
2 Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ R
2 Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ ɜɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ R
2
A2 R 3 A2 R 12
MA 2 MA 2 3 4
MA 2 . 12
A
. 3 A
2 3
.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɫɬɪɟɠɧɸ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ D.
Jz
MA 2 §A· M¨ ¸ 3 ©4¹
8
2
16 MA 2 3MA 2 48 48
13MA 2 . 48
Ɂɚɞɚɱɚ 2. Ɉɞɧɨɪɨɞɧɚɹ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɩɥɚɫɬɢɧɚ ɪɚɡɦɟɪɚɦɢ a u b ɢ ɦɚɫɫɨɣ M . ȼɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Ɉɯ. ɗɥɟɦɟɧɬɚɪɧɚɹ ɩɥɨɳɚɞɤɚ dS dxdy ɨɛɥɚɞɚɟɬ ɦɚɫɫɨɣ dm . ɉɥɨɬɧɨɫɬɶ ɜ ɩɥɨɫɤɨɦ ɫɥɭɱɚɟ M M U const – ɦɚɫɫɚ S ab ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ. r y – ɪɚɫɫɬɨɹɧɢɟ ɞɨ ɨɫɢ x . dv dS dxdy – ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ. V S – ɨɛɴɟɦ ~ ɩɥɨɳɚɞɶ. a b a b M M § 2 · ¨ y dy ¸dx J x ³ r 2 dm ³ Ur 2 dv ³ Uy 2 dS U ³ y 2 dS y 2 dydx ³ ³ ³ ¸ ab 0 0 ab 0 ¨© ³0 (M ) (V ) (S ) (S ) ¹
§ b· a a a a M ¨ y3 ¸ M b3 Mb 2 Mb 2 M b3 M b2 dx dx a dx x ¸ ¨ – ab ³0 ¨ 3 ¸ ab ³0 3 3a 3 ab 3 ³0 a 3 0 0 ¹ © ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Ox , ɫɨɜɩɚɞɚɸɳɟɣ ɫɨ ɫɬɨɪɨɧɨɣ ɩɥɚɫɬɢɧɤɢ. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Ɉɭ Ma 2 Jy . 3 ȼɵɱɢɫɥɢɬɶ ɩɨɥɹɪɧɵɣ ɦɨɦɟɧɬ J o – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɨɥɸɫɚ O (ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɩɨɥɸɫ Ɉ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ). ab
Jo
2 ³ U r dv
(V )
2
2
³³ U (x y )dydx 00
a b
º Mª 2 2 « ³ ³ ( x y )dydx » ab ¬ 0 0 ¼
M 2 (a b 2 ) , 3
Jx Jy
ɝɞɟ r – ɪɚɫɫɬɨɹɧɢɟ ɨɬ dv ɞɨ ɩɨɥɸɫɚ. ɋɚɦɨɫɬɨɹɬɟɥɶɧɨ ɜɵɱɢɫɥɢɬɟ J x1 , J y1 , J c ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ, ɢ ɩɨɥɹɪɧɵɣ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ C .
9
Ɂɚɞɚɱɚ 3. Ɍɨɧɤɨɟ ɤɪɭɝɥɨɟ ɨɞɧɨɪɨɞɧɨɟ ɤɨɥɶɰɨ ɪɚɞɢɭɫɨɦ R ɢ ɦɚɫɫɨɣ M . ȼɵɱɢɫɥɢɬɶ J z – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Cz , ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɤɨɥɶɰɚ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɝɨ ɰɟɧɬɪ Ɋɚɡɞɟɥɢɦ ɤɨɥɶɰɨ ɧɚ ɷɥɟɦɟɧɬɚɪɧɵɟ ɨɬɪɟɡɤɢ ɞɥɢɧɨɣ dl ɢ ɦɚɫɫɨɣ dmi. Ɋɚɫɫɬɨɹɧɢɟ ɨɬ ɤɚɠɞɨɝɨ ɬɚɤɨɝɨ ɨɬɪɟɡɤɚ ɞɨ ɨɫɢ ɪɚɜɧɨ ɪɚɞɢɭɫɭ ɤɨɥɶɰɚ Ri=R, ɬɨɝɞɚ ɩɨɥɭɱɢɦ n
Jz
Jc
lim ¦ Ri2 dmi nof
i 1
n
R 2 lim ¦ dmi nof
R2M .
i 1
Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ ɤɨɥɶɰɚ ɪɚɜɟɧ ɪɚɞɢɭɫɭ ɤɨɥɶɰɚ (ɞɥɹ ɨɫɢ z ). ɗɬɨ ɜɟɪɧɨ ɢ ɞɥɹ ɬɨɧɤɨɣ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɨɛɨɥɨɱɤɢ ɦɚɫɫɨɣ M ɢ ɪɚɞɢɭɫɨɦ R ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɟ ɨɫɢ. Ʉɪɭɝɥɚɹ ɨɞɧɨɪɨɞɧɚɹ ɩɥɚɫɬɢɧɚ (ɢ/ɢɥɢ ɰɢɥɢɧɞɪ) ɪɚɞɢɭɫɨɦ R ɢ ɦɚɫɫɨɣ M . Jz ȼɵɱɢɫɥɢɬɶ – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ Cz ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɥɚɫɬɢɧɵ ɢ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɟɟ ɰɟɧɬɪ. ɉɥɨɳɚɞɶ ɜɵɞɟɥɟɧɧɨɝɨ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɤɨɥɶɰɚ ɪɚɞɢɭɫɚ r ɢ ɲɢɪɢɧɚ dr ɪɚɜɧɚ dS 2Srdr , ɚ M ɦɚɫɫɚ dm U2Srdr , ɝɞɟ U – ɦɚɫɫɚ SR 2 ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ ɩɥɚɫɬɢɧɵ. 2 Ɍɨɝɞɚ, ɩɪɢɦɟɧɹɹ ɮɨɪɦɭɥɭ ɞɥɹ ɤɨɥɶɰɚ J z R M , ɩɨɥɭɱɢɦ ɞɥɹ ɜɵɞɟɥɟɧɧɨɝɨ 2 3 ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɤɨɥɶɰɚ dJ c r dm 2SUr dr , ɚ ɞɥɹ ɜɫɟɣ ɩɥɚɫɬɢɧɵ
R
Jc
Jz
2SU ³ r 3 dr 0
SU
R4 2
MR 2 ɢɥɢ ɢɫɩɨɥɶɡɭɹ ɞɜɨɣɧɨɣ ɢɧɬɟɝɪɚɥ. 2 Ɍɚɤ ɤɚɤ dS dxdy rdrdM , ɬɨ, ɩɟɪɟɯɨɞɹ ɤ ɰɢɥɢɧɞɪɢɱɟɫɤɢɦ ɤɨɨɪɞɢɧɚɬɚɦ, ɩɨɥɭɱɢɦ. 2
³ U r dv
Jz
(V )
M S R2
R 2S 2
³ ³ r rdI dr 0 0
R
M r 3 2S dr S R 2 ³0 10
M S R2
2
³ r ds
(S )
R 2S M § 3 · r dI ¸ dr 2 ³¨ ³ SR 0© 0 ¹
MR 2 . 2
Ɂɚɞɚɱɚ 4. ȼɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ x, y , z ɬɨɧɤɨɣ ɨɞɧɨɪɨɞɧɨɣ ɤɪɭɝɨɜɨɣ ɩɥɚɫɬɢɧɵ ɪɚɞɢɭɫɚ r , ɜɧɭɬɪɢ ɤɨɬɨɪɨɣ ɜɵɪɟɡɚɧ ɤɜɚɞɪɚɬ ɫɨ ɫɬɨɪɨɧɨɣ a , ɰɟɧɬɪɵ ɤɜɚɞɪɚɬɚ ɢ ɤɪɭɝɚ ɫɨɜɩɚɞɚɸɬ. M –ɦɚɫɫɚ ɩɥɚɫɬɢɧɵ ɫ ɜɵɪɟɡɨɦ. Ɉɩɪɟɞɟɥɢɬɟ J Y , J Z ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ. J x J x(1) J x( 2 ) ,
M ( 2) a 2 M (1) r 2 J x( 2) , . 6 2 ɇɚɣɞɟɦ ɦɚɫɫɵ ɤɪɭɝɚ ɢ ɤɜɚɞɪɚɬɚ M (1) , M ( 2) ɱɟɪɟɡ M :
(1)
ɝɞɟ J x – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɪɭɝɚ, J x( 2 ) – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɜɚɞɪɚɬɚ.
J x(1)
M
M (1) M (2)
M (1) (1 M (1) M ( 2)
Jx
M (1) M (1)
a2 S r2
a2 ), S r2
Sr 2 Sr 2 a 2 M (1)
a2 Sr 2
M, a2 M, Sr 2 a 2
3Sr 4 a 4 M. 6(Sr 2 a 2 )
ȿɳɺ ɧɟɫɤɨɥɶɤɨ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ: ɇɚɣɬɢ ɨɫɟɜɵɟ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ J x , J y ɞɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɬɨɧɤɨɝɨ ɤɪɭɝɥɨɝɨ ɞɢɫɤɚ ɪɚɞɢɭɫɚ R ɢ ɦɚɫɫɨɣ M . Ɉɫɢ ɋɯ ɢ ɋɭ ɩɪɨɯɨɞɹɬ ɱɟɪɟɡ ɰɟɧɬɪ ɞɢɫɤɚ ɢ ɥɟɠɚɬ ɜ ɟɝɨ ɩɥɨɫɤɨɫɬɢ.
ɇɚɣɬɢ J x , J y ɞɥɹ ɬɪɟɭɝɨɥɶɧɨɣ ɩɥɚɫɬɢɧɵ ɫ ɤɚɬɟɬɚɦɢ a ɢ b ɢ ɦɚɫɫɨɣ M , ɚ ɬɚɤɠɟ J x1 , J y1 .Ɍɨɱɤɚ ɋ – ɰɟɧɬɪ ɦɚɫɫ ɬɪɟɭɝɨɥɶɧɢɤɚ.
11
ɉɪɹɦɨɣ ɫɩɥɨɲɧɨɣ ɤɪɭɝɥɵɣ ɤɨɧɭɫ ɦɚɫɫɨɣ M ɢ ɪɚɞɢɭɫɨɦ ɨɫɧɨɜɚɧɢɹ R . Ɉɫɶ z ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɨɫɢ 2 ɫɢɦɦɟɬɪɢɢ. Ɉɬɜɟɬ J z 0,3MR .
ɋɩɥɨɲɧɨɣ ɲɚɪ ɦɚɫɫɨɣ M ɢ ɪɚɞɢɭɫɨɦ R ɨɫɶ z ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɞɢɚɦɟɬɪɚ. 2 Ɉɬɜɟɬ J z 0,4MR .
§3. Ʉɨɥɟɛɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ɋɜɨɛɨɞɧɵɟ ɤɨɥɟɛɚɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ ɜɟɫɨɦ Ɋ = 98 ɧ ɩɨɞɜɟɲɟɧ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ, ɧɚɯɨɞɢɜɲɟɣɫɹ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜ ɩɨɤɨɟ ɜ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ, ɢ ɨɬɩɭɳɟɧ ɛɟɡ ɬɨɥɱɤɚ. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɤɨɥɟɛɚɧɢɹ ɝɪɭɡɚ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɞɥɹ ɞɟɮɨɪɦɚɰɢɢ ɩɪɭɠɢɧɵ ɧɚ 1 ɫɦ ɧɚɞɨ ɩɪɢɥɨɠɢɬɶ ɤ ɧɟɣ ɫɢɥɭ, ɦɨɞɭɥɶ ɤɨɬɨɪɨɣ ɪɚɜɟɧ 14,4 ɧ. Ɋɟɲɟɧɢɟ: ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɧɢɡ. Ɍɨɱɤɚ Ɉ – ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɝɪɭɡɚ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɝɪɭɡ ɩɨɞɜɟɲɟɧ ɤ ɤɨɧɰɭ Ɇ0 ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɵ. ȼ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɤ ɝɪɭɡɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɟɝɨ ɜɟɫ, ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɧɢɡ, ɫɬɚɬɢɱɟɫɤɚɹ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ Fɫɬ=cd, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ. ɂɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɝɪɭɡɚ ɫɥɟɞɭɟɬ: P Fɫɬ 0 ɢɥɢ P cd 0 , ɨɬɤɭɞɚ ɧɚɣɞɟɦ d=Ɋ/ɫ – ɫɬɚɬɢɱɟɫɤɭɸ ɞɟɮɨɪɦɚɰɢɸ ɩɪɭɠɢɧɵ, ɫ – ɤɨɷɮɮɢɰɢɟɧɬ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ.
12
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ: P x x0 , x x0 0 , ɉɪɢ t 0 : c F cD , ɝɞɟ F – ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ, (ɧɚɩɪɚɜɥɟɧɚ ɜɫɟɝɞɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɫɦɟɳɟɧɢɸ); D – ɫɦɟɳɟɧɢɟ ɤɨɧɰɚ ɩɪɭɠɢɧɵ ɢɡ ɧɟɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɬ. ɟ. Dx MM 0 d x . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Fx c(d x) . (1) ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ:
mx P Fx , ɢɫɩɨɥɶɡɭɹ (1), ɩɨɥɭɱɢɦ ɢɡ (2): P x P cd cx . g Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (3) ɜ ɜɢɞɟ x k 2 x 0 , cg k – ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ (ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ). ɝɞɟ P Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4) O1, 2 r ki . Ʉɨɪɧɢ ɭɪɚɜɧɟɧɢɹ Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
O2 k 2
(2) (3) (4)
0.
x c1 cos(kt ) c2 sin( kt ) . (5) Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɜɵɱɢɫɥɢɦ ɫɤɨɪɨɫɬɶ x c1k sin(kt ) c2 k cos(kt ) . (6) P ɉɨɞɫɬɚɜɢɦ (5) ɢ (6) ɜ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ t 0 x x0 , x x0 0 . c P ɇɚɯɨɞɢɦ ɫ1 x0 , c 2 0 . c P cg P t) . ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɩɪɢɦɟɬ ɜɢɞ x cos(kt ) cos( c P c cg P Ⱥɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ a = 6,8 cɦ; ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ Į = –ʌ/2; ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ k = 12 ɫ-1.
ɉɨɞɫɬɚɜɥɹɟɦ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ k
13
12 ɫ 1 ,
P c
6,8 ɫɦ .
ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɝɪɭɡɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ T
2S k
0.52 ɫ .
Ɂɚɞɚɱɚ ʋ 2. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɜɟɪɬɢɤɚɥɶɧɵɯ ɤɨɥɟɛɚɧɢɣ ɫɭɞɧɚ ɜɟɫɨɦ Ɋ ɜ ɫɩɨɤɨɣɧɨɣ ɜɨɞɟ. ɉɥɨɳɚɞɶ ɟɝɨ ɫɟɱɟɧɢɹ ɧɚ ɭɪɨɜɧɟ ɫɜɨɛɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜɨɞɵ ɫɱɢɬɚɬɶ ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ ɤɨɥɟɛɚɧɢɣ ɢ ɪɚɜɧɨɣ S. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɰɟɧɬɪɭ ɬɹɠɟɫɬɢ ɋ, ɧɚɯɨɞɢɜɲɟɦɭɫɹ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɫɤɨɪɨɫɬɶ v0. ȼɹɡɤɨɫɬɶɸ ɜɨɞɵ ɩɪɟɧɟɛɪɟɱɶ. ɍɞɟɥɶɧɵɣ ɜɟɫ ɜɨɞɵ ɪɚɜɟɧ Ȗ= 1 T/ɦ3. Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ; ɬɨɱɤɚ Ɉ – ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ ɫɭɞɧɚ. ɉɪɢ ɷɬɨɦ ɜɵɫɨɬɚ ɩɨɞɜɨɞɧɨɣ ɱɚɫɬɢ ɫɭɞɧɚ ɪɚɜɧɚ d. Ʉ ɫɭɞɧɭ ɩɪɢɥɨɠɟɧɵ: Ɋ – ɜɟɫ ɜ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ ɋ ɫɭɞɧɚ, Rɫɬ – ɧɨɪɦɚɥɶɧɚɹ ɫɬɚɬɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɜɨɞɵ ɜ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ Ʉ ɨɛɴɟɦɚ ɜɨɞɵ, ɜɵɬɟɫɧɟɧɧɨɣ ɫɭɞɧɨɦ. Ɇɨɞɭɥɶ Rɫɬ ɪɚɜɟɧ ɜɟɫɭ ɨɛɴɟɦɚ V ɜɨɞɵ, ɜɵɬɟɫɧɟɧɧɨɣ ɫɭɞɧɨɦ, R J V J S d ɬ.ɟ. ɫɬ , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɟɬ ɜɢɞ P J S d 0. (7) ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ ɩɪɢ t 0 : x 0, x v0 . (8) ɂɡ-ɡɚ ɧɚɥɢɱɢɹ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ v0 ɫɭɞɧɨ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ. Ɉɛɴɟɦ ɜɨɞɵ, ɜɵɬɟɫɧɟɧɧɨɣ ɫɭɞɧɨɦ, ɪɚɜɟɧ S (d x) .Ɂɧɚɱɢɬ, ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ R ɪɚɜɧɚ Rx J S (d x) . (9) ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ ɜ mx Px Rx . ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ P x J Sx . Ɍɚɤ ɤɚɤ Ɋ=Ɋɯ, ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ (1) ɢ (3), ɩɨɥɭɱɢɦ g Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɤɚɧɨɧɢɱɟɫɤɨɦ ɜɢɞɟ: x k 2 x 0 , (10) gJ S k – ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ. ɝɞɟ P 14
Ɂɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɜ ɜɢɞɟ x a sin(kt D ) ,
(11)
2 0 2
kx0 x – ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ, D arctg – ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ. x 0 k ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (2), ɧɚɣɞɟɦ ɚɦɩɥɢɬɭɞɭ ɢ ɧɚɱɚɥɶɧɭɸ ɮɚɡɭ: v0 P a v0 , D 0. (12) k gSJ ɂɫɩɨɥɶɡɭɹ (6),(7) ɢ (5), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ § gSJ · P x v0 t ¸¸ . sin ¨¨ (13) gSJ P © ¹
ɝɞɟ a
x 02
P 2S 2S . k gsJ §4. Ɂɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ
ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ
T
Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ ɜɟɫɨɦ Ɋ = 98 H, ɩɨɞɜɟɲɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ, ɞɜɢɠɟɬɫɹ ɜ ɠɢɞɤɨɫɬɢ. Ʉɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ ɫ=10 ɇ/ɫɦ. ɋɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɝɪɭɡɚ: R=ȕȣ, ɝɞɟ ȕ=1,6 ɇɫ/ɫɦ. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɝɪɭɡ ɛɵɥ ɫɦɟɳɟɧ ɢɡ ɩɨɥɨɠɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɜɧɢɡ ɧɚ 4 ɫɦ ɢ ɟɦɭ ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɧɢɡ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ȣ0=4 ɫɦ/ɫ. Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ ɩɪɭɠɢɧɟ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜɨɡɶɦɟɦ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ. ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ: t 0, x x0 4 ɫɦ,
x x0 4 ɫɦ / c. ɂɡɨɛɪɚɡɢɦ ɝɪɭɡ ɜ ɩɨɥɨɠɟɧɢɢ, ɤɨɝɞɚ ɩɪɭɠɢɧɚ ɩɨɥɭɱɢɬ ɭɞɥɢɧɟɧɢɟ D = d + x. ɋɢɥɚ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɪɚɜɧɚ Fx c(d x) . (1) ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ: mx P Fx Rx . ɉɨɞɫɬɚɜɢɦ ɜ ɭɪɚɜɧɟɧɢɟ ɡɧɚɱɟɧɢɹ Fx ɢ Rx: 15
P x P cd cx EX x . (2) g ȼ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɤ ɝɪɭɡɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɟɝɨ ɜɟɫ, ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɧɢɡ, ɫɬɚɬɢɱɟɫɤɚɹ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ Fɫɬ=cd, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ. Ɍɚɤ ɤɚɤ ɝɪɭɡ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ, ɬɨ P cd 0 . (3) ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ x 2nx k 2 x 0 , (4) Eg cg , n ɝɞɟ X x x, k , ɩɨɥɭɱɚɟɦ k = 10 c-1, n = 8 ɫ-1, ɬɚɤɢɦ P 2P ɨɛɪɚɡɨɦ, n < k. Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4):
O2 2nO k 2
0, O1, 2 n r n 2 k 2 n r i k 2 n 2 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ x e nt (c1 cos( k 2 n 2 t ) c2 sin( k 2 n 2 t )) . (5) ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɩɨɥɭɱɚɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ x 0 nx 0 c1 x 0 , c 2 . k 2 n2 ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɨɩɪɟɞɟɥɹɹ x 0 nx0 A cos D . x0 A sin D , (6) k 2 n2 Ɍɟɩɟɪɶ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ x Ae nt sin( k 2 n 2 t D ) . (7) Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɡɚɬɭɯɚɸɳɢɦ (ɬ.ɤ. ɩɪɢ tĺ xĺ0) ɫ ɤɪɭɝɨɜɨɣ 2
2
k n . ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɨɪɦɭɥɵ, ɱɚɫɬɨɬɨɣ k c ɧɚɯɨɞɢɦ Ⱥ=7,2 ɫɦ, Į=0,59 ɪɚɞ, kc=6 ɫ-1. ɂɬɚɤ, ɝɪɭɡ ɫɨɜɟɪɲɚɟɬ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ x 7,2e 8t sin(6t 0,59) ɫɦ . (8) 2S 1,05 ɫ . ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ Tc kc Ɂɚɞɚɱɚ ʋ 2. Ɋɟɲɢɬɶ ɩɪɟɞɵɞɭɳɭɸ ɡɚɞɚɱɭ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɪɚɜɧɚ R=ȕȣ, ɝɞɟ ȕ=5,2 Hɫ/ɫɦ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɝɪɭɡ ɛɵɥ ɫɦɟɳɟɧ ɢɡ ɩɨɥɨɠɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɧɚ 4 ɫɦ, ɢ ɟɦɭ ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɜɟɪɯ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ȣ0=240 ɫɦ/ɫ. Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ ɩɪɭɠɢɧɟ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜɨɡɶɦɟɦ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ 16
ɧɚɱɚɥɶɧɵɟ
ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɸɬ ɜɢɞ x x0 4 ɫɦ ɩɪɢ t 0 ® . ¯ x x0 240 ɫɦ / ɫ ɋɥɟɞɭɹ ɪɟɲɟɧɢɸ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ Eg cg , n . x 2nx kx 0 , ɝɞɟ X x x, k P 2P ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɚɟɦ k=10 c-1, n=26 ɫ-1, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, n>k (ɫɥɭɱɚɣ ɛɨɥɶɲɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ). 2 2 0, Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ O 2nO k ɟɝɨ ɤɨɪɧɢ ɪɚɜɧɵ O1 n n 2 k 2 , O2 n k 2 n 2 . Ɍɚɤ ɤɚɤ n > k, ɬɨ ɤɨɪɧɢ Ȝ1 ɢ Ȝ2 ɹɜɥɹɸɬɫɹ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ x ɫ1e O1t c 2 e O2t . (9) ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɧɚɣɞɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ: O x x 0 O1 x0 x 0 c1 2 0 , c2 . O1 O2 O1 O2 ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (1) ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ: 1 x (O1 x 0 x 0 )e O2t (O 2 x 0 x 0 )e O1t . (10) O1 O 2 ɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɧɚɱɟɧɢɹɦɢ Ȝ1 ɢ Ȝ2 ɮɭɧɤɰɢɹɦɢ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ e nt x ( x 0 nx 0 ) sh n 2 k 2 x 0 n 2 k 2 ch n 2 k 2 t . (11) 2 2 n k Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɚɩɟɪɢɨɞɢɱɟɫɤɢɦ ɢ ɩɪɢɬɨɦ ɡɚɬɭɯɚɸɳɢɦ, ɬ. ɤ. ɩɪɢ tĺ xĺ0. ɉɨɞɫɬɚɜɢɦ ɜ (3) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, 1 26 t e (29e 24t 5e 24 t ) ɩɨɥɭɱɢɦ x 6 1 26t e (12ch 24t 17 sh 24t ) ɢɥɢ x 3 ȼɵɹɫɧɢɦ, ɩɟɪɟɯɨɞɢɬ ɥɢ ɝɪɭɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ:
>
@
>
1 26t e (29e 24t 5e 24t ) 6
@
0 . ȼɵɱɢɫɥɹɹ, ɩɨɥɭɱɚɟɦ t1=0,037 ɫ, t2=. 17
Ɂɧɚɱɟɧɢɟ t1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɟɪɟɯɨɞɭ ɝɪɭɡɚ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, t2 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɚɬɭɯɚɧɢɸ ɞɜɢɠɟɧɢɹ. ɂɬɚɤ, ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɝɪɭɡ ɩɪɨɯɨɞɢɬ ɨɞɢɧ ɪɚɡ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɢ ɡɚɬɟɦ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɤ ɧɟɦɭ ɩɪɢɛɥɢɠɚɟɬɫɹ ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ. §5. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ Ɂɚɞɚɱɚ ʋ 1. ɇɚ ɪɢɫɭɧɤɟ ɢɡɨɛɪɚɠɟɧɚ ɫɯɟɦɚ ɩɪɢɛɨɪɚ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɞɚɜɥɟɧɢɹ. Ʉ ɩɨɥɡɭɧɭ Ⱥ ɜɟɫɨɦ Ɋ=196 Ƚ ɩɪɢɤɪɟɩɥɟɧɚ ɫɬɪɟɥɤɚ ȼ, ɨɬɦɟɱɚɸɳɚɹ ɩɨɤɚɡɚɧɢɹ ɧɚ ɧɟɩɨɞɜɢɠɧɨɣ ɲɤɚɥɟ ɋ. ɉɨɥɡɭɧ Ⱥ, ɩɪɢɤɪɟɩɥɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ D, ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɢɞɟɚɥɶɧɨ ɝɥɚɞɤɨɣ ɩɥɨɫɤɨɫɬɢ. Ʉ ɩɨɥɡɭɧɭ ɩɪɢɥɨɠɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ S = H·sin(pt), ɝɞɟ ɇ = 1,6 ɤȽ, Ʉɨɷɮɮɢɰɢɟɧɬ ɪ = 60 ɫ-1. ɭɩɪɭɝɨɫɬɢ ɪɚɜɟɧ ɫ = 2 ɤȽ/ɫɦ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɨɥɡɭɧ ɧɚɯɨɞɢɥɫɹ ɜ ɩɨɤɨɟ, ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
Ɉɩɪɟɞɟɥɢɬɶ: 1) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ ɜ ɫɥɭɱɚɟ ɨɬɫɭɬɫɬɜɢɹ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ; 2) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ ɩɪɢ ɧɚɥɢɱɢɢ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɩɨɥɡɭɧɚ R=ȕȣ, ɝɞɟ ȕ=25,6 Ƚ ɫ/ɫɦ.
Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɜɩɪɚɜɨ, ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ ɩɨɥɡɭɧɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɟ. ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ: ɩɪɢ t 0 x 0, x 0 . ɂɡɨɛɪɚɡɢɦ ɩɨɥɡɭɧ ɫɦɟɳɟɧɧɵɦ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɩɪɚɜɨ ɧɚ ɯ. ɉɪɢ ɷɬɨɦ ɩɪɭɠɢɧɚ ɪɚɫɬɹɧɟɬɫɹ ɧɚ D = ɯ. Ʉ ɩɨɥɡɭɧɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɜɟɫ ɩɨɥɡɭɧɚ, N – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ, ɫɢɥɚ S, ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɪɚɫɬɹɧɭɬɨɣ ɩɪɭɠɢɧɵ F. ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɯ: Hg cg sin pt x , ɨɬɤɭɞɚ mx S x Fx ɢɥɢ x P P x k 2 x k sin pt , (1)
cg hg , h . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k=100 ɫ-1, h=8000 ɫɦ/ɫ-2. P P Ɋɟɲɚɹ (1), ɧɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
ɝɞɟ k
18
x1 c1 cos(kt ) c2 sin(kt ) . (2) ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɩɪɢɦɟɦ ɜ ɜɢɞɟ x2 A sin( pt ) B cos( pt ) , ɬɨɝɞɚ h sin( pt ) . x2 (3) 2 k p2 Ɂɚɩɢɲɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1), ɫɥɨɠɢɜ (2) ɢ (3): h sin( pt ) . x c1 cos(kt ) c2 sin( kt ) 2 (4) k p2 ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ: p h c1 0, c 2 . k k 2 p2 ɂɬɚɤ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɟ ɫɬɪɟɥɤɢ p h h x sin( kt ) 2 sin( pt ) . (5) k k 2 p2 k p2 ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ x (0,75 sin(100t ) 1,25 sin( 60t )) cɦ . (6) Ɋɟɲɢɦ ɷɬɭ ɡɚɞɚɱɭ ɫ ɭɱɟɬɨɦ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ. Ʉ ɫɢɥɚɦ, ɪɚɧɟɟ ɩɪɢɥɨɠɟɧɧɵɦ ɤ ɩɨɥɡɭɧɭ, ɞɨɛɚɜɥɹɟɬɫɹ ɫɢɥɚ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ R, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ. Hg cg Eg sin pt x x , ɨɬɤɭɞɚ mx S x Fx R x ɢɥɢ x P P P x 2nx k 2 x k sin pt , (7) cg hg Eg , n , h . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k = 100 ɫ-1, h = 80 ɫɦ ɫ-2, P 2P P n = 64 ɫ-1, ɪ = 60 ɫ-1. ɂɬɚɤ, n < k ɢ p < k. h 2np 0,8 H arctg 2 k 2 n 2 76,8 a 0,87 . 2 2 2 2 2 k p2 ( k p ) 4n p ɝɞɟ k
Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɵɛɢɪɚɹ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ x2 a sin( pt H ) , ɩɨɥɭɱɢɦ x e 64t (c1 cos 76,8t c 2 sin 76,8t ) 0,8 sin(60t 0,87). (8) ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ: ɫ1 = 0,62; ɫ2 = 0,12. ɉɟɪɟɩɢɲɟɦ ɮɨɪɦɭɥɭ (9): x e 64t (0,62 cos 76,8t 0,12 sin 76,8t ) 0,8 sin(60t 0,87). (9) ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ: 0,62=bsinĮ; 0,12=bcosĮ, ɩɨɥɭɱɢɦ b=0,63, Į=1,74. ɂɬɚɤ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ Ⱥ ɢ ɫɬɪɟɥɤɢ ȼ ɢɦɟɟɬ ɜɢɞ: x 0,63e 64t sin(76,8t 1,74) 0,8 sin(60t 0,87) ɫɦ. (10)
>
@
19
ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɹ ɨɩɪɟɞɟɥɹɬ ɤɨɥɟɛɚɧɢɟ ɫɬɪɟɥɤɢ ɫ ɱɚɫɬɨɬɨɣ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ, ɤɨɬɨɪɵɟ ɛɵɫɬɪɨ ɡɚɬɭɯɚɸɬ ɛɥɚɝɨɞɚɪɹ ɧɚɥɢɱɢɸ ɦɧɨɠɢɬɟɥɹ e-64t. ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɭɪɚɜɧɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬ ɜɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ ɫɬɪɟɥɤɢ ȼ. §6. Ɋɟɡɨɧɚɧɫ Ɂɚɞɚɱɚ ʋ 1. Ɉɩɪɟɞɟɥɢɬɶ ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɬɨɱɤɚ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɧɚɯɨɞɢɥɚɫɶ ɜ ɩɨɤɨɟ ɜ ɧɚɱɚɥɟ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ Ɇ ɜɟɫɨɦ ɨɬɫɱɟɬɚ ɨɫɢ ɯ. ɋɢɥɨɣ Ɋ=196 Ƚ, ɞɜɢɠɭɳɟɣɫɹ ɜɞɨɥɶ ɨɫɢ ɯ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F ɩɪɟɧɟɛɪɟɱɶ. ɢ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ S. ɉɪɨɟɤɰɢɢ ɷɬɢɯ ɫɢɥ ɧɚ ɨɫɶ ɯ ɪɚɜɧɵ: Fx = –cx, Sx = H·sin(pt), ɝɞɟ ɫ = 2 ɤȽ/ɫɦ, ɇ = 1,6 ɤȽ, ɪ = 101 ɫ-1. Ɋɟɲɟɧɢɟ. Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ Ɇ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ mx Fx S x ɢɥɢ mx cx H sin pt
x k 2 x
h sin pt , (1) c H 2 10000 ɫ 1 , p 101 ɫ 1 , h 8000 ɫɦ / ɫ 2 . . ɝɞɟ k m m ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɤɨɥɟɛɚɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜɛɥɢɡɢ ɪɟɡɨɧɚɧɫɚ (ɪɟɡɨɧɚɧɫ ɢɦɟɟɬ ɦɟɫɬɨ ɩɪɢ p=k) ɜ ɡɨɧɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɛɨɥɶɲɨɣ ɱɚɫɬɨɬɵ (p>k). Ɂɧɚɱɢɬ, ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ p h h x sin kt 2 sin pt . (2) 2 2 k k p k p2 Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ, ɢɫɤɨɦɨɟ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɧɚɥɨɠɟɧɢɹ ɞɜɭɯ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɫ ɩɨɱɬɢ ɪɚɜɧɵɦɢ ɤɪɭɝɨɜɵɦɢ ɱɚɫɬɨɬɚɦɢ ɫɜɨɛɨɞɧɵɯ k ɢ ɜɵɧɭɠɞɟɧɧɵɯ ɪ ɤɨɥɟɛɚɧɢɣ. Ɍ. ɤ. k§p, ɬɨ ɛɭɞɟɦ ɫɱɢɬɚɬɶ p | 1, p k | 2k | 2 p. (3) k ɂɫɩɨɥɶɡɭɹ ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2), h x| (sin pt sin kt ) . ( k p )( k p ) ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ: h x (sin pt sin kt ) . (4) 2k ( k p ) ɉɪɟɨɛɪɚɡɨɜɚɜ ɜɵɪɚɠɟɧɢɟ, ɩɨɥɭɱɢɦ x a (t ) cos pt , (5) 20
h pk sin t . ɉɪɢ k§p a(t) ɹɜɥɹɟɬɫɹ ɦɟɞɥɟɧɧɨ k (k p ) 2 ɢɡɦɟɧɹɸɳɟɣɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɫ ɩɟɪɢɨɞɨɦ 4S Ta . (6) pk Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ (5), ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɫ ɤɪɭɝɨɜɨɣ ɱɚɫɬɨɬɨɣ ɪ, ɩɨɷɬɨɦɭ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ 2S T . (7) p Ta 2p . Ɍɚɤ ɤɚɤ k § p, ɬɨ Ɍɚ >> T, ɩɪɢɱɟɦ T pk ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ t x a(t ) cos(101 t ), ɝɞɟ a(t ) 80sin Ɍ ɚ 12,56 ɫ, Ɍ 0, 063 ɫ. 2 ɝɞɟ
a (t )
Ɂɚɞɚɱɚ ʋ 2. ɉɧɟɜɦɚɬɢɱɟɫɤɢɣ ɜ ɤɨɪɩɭɫ ɦɨɥɨɬɤɚ. ɉɨɪɲɟɧɶ D ɨɬɛɨɣɧɵɣ ɦɨɥɨɬɨɤ ɫɨɟɞɢɧɟɧ ɲɬɨɤɨɦ ȿ ɫ ɛɨɣɤɨɦ Ɇ. ɩɪɢɜɨɞɢɬɫɹ ɜ ɞɜɢɠɟɧɢɟ ɫɠɚɬɵɦ ɜɨɡɞɭɯɨɦ, ɩɨɫɬɭɩɚɸɳɢɦ ɜ ɤɨɪɩɭɫ ɦɨɥɨɬɤɚ ɱɟɪɟɡ ɲɥɚɧɝ Ⱥ. Ⱦɚɜɥɟɧɢɟ ɜɨɡɞɭɯɚ, ɩɪɢɥɨɠɟɧɧɨɟ ɤ ɩɨɪɲɧɸ D ɦɨɥɨɬɤɚ, ɢɡɦɟɧɹɟɬɫɹ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɸ Ɉɩɪɟɞɟɥɢɬɶ ɭɪɚɜɧɟɧɢɟ S H 0 H1 cos( pt ) H 3 cos(3 pt ), ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ ɝɞɟ ɪ, ɇ0, ɇ1, ɇ3 – ɩɨɫɬɨɹɧɧɵɟ ɩɪɢ ɪɚɛɨɬɟ ɦɨɥɨɬɤɚ ɜɯɨɥɨɫɬɭɸ. ɜɟɥɢɱɢɧɵ. ȼ ɤɨɪɩɭɫ ɦɨɥɨɬɤɚ Ɇɚɫɫɨɣ ɲɬɨɤɚ ȿ, ɛɨɣɤɚ Ɇ ɢ ɜɦɨɧɬɢɪɨɜɚɧɚ ɩɪɭɠɢɧɚ ȼ ɫ ɩɪɭɠɢɧɵ ȼ, ɚ ɬɚɤɠɟ ɫɢɥɨɣ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɠɟɫɬɤɨɫɬɢ ɫ. ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɪɟɧɟɛɪɟɱɶ. ɉɪɭɠɢɧɚ ɭɩɢɪɚɟɬɫɹ ɥɟɜɵɦ ɤɨɧɰɨɦ ɜ ɩɨɪɲɟɧɶ, ɚ ɩɪɚɜɵɦ – Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɧɚɩɪɚɜɨ, ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɩɨɪɲɧɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɇ0 ɢ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ Fɫɬ . ȼ ɷɬɨɦ ɩɨɥɨɠɟɧɢɢ ɩɪɭɠɢɧɚ ɫɠɚɬɚ ɧɚ d ɫɢɥɨɣ ɇ0. ɉɪɢ ɷɬɨɦ ɜɨɡɧɢɤɚɟɬ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ Fɫɬ = ɫd. Ɂɚɩɢɲɟɦ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɨɪɲɧɹ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ: H 0 cd 0 . 21
ɂɡɨɛɪɚɡɢɦ ɩɨɪɲɟɧɶ ɫɦɟɳɟɧɧɵɦ ɢɡ ɧɭɥɹ ɧɚɩɪɚɜɨ ɧɚ ɯ, ɩɪɢ ɷɬɨɦ ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɜɨɡɧɢɤɲɟɣ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F ɪɚɜɧɚ: Fx cd x c(d x) . (8) Ʉɪɨɦɟ ɬɨɝɨ, ɤ ɩɨɪɲɧɸ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɜɟɫ, N – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɤɨɪɩɭɫɚ, S – ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɫɠɚɬɨɝɨ ɜɨɡɞɭɯɚ. ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɪɲɧɹ D: mx S x Fx . ɍɱɢɬɵɜɚɹ (1) ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ S, ɧɚɯɨɞɢɦ P x H 0 H 1 cos( pt ) H 3 cos(3 pt ) cd cx. g ɍɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ ɜ ɜɢɞɟ x k 2 x h1 cos( pt ) h3 cos(3 pt ) , (9)
H3g H1 g cg , h1 , h3 . P P P ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2) ɜ ɜɢɞɟ x2 A1 sin( pt ) B1 cos( pt ) A2 sin(3 pt ) B2 cos(3 pt ) . (10) Ʉɨɷɮɮɢɰɢɟɧɬɵ Ⱥ1, ȼ1, Ⱥ2, ȼ2 ɧɚɣɞɟɦ, ɜɵɱɢɫɥɢɜ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɯ2 ɢ ɩɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ (2), ɚ ɡɚɬɟɦ ɩɪɢɪɚɜɧɹɜ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɫɬɨɹɳɢɟ ɜ ɩɪɚɜɨɣ ɢ ɥɟɜɨɣ ɱɚɫɬɹɯ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɫɢɧɭɫɟ ɢ ɤɨɫɢɧɭɫɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ h3 h1 A1 0, B1 , A3 0, B3 . k 2 p2 k 2 9 p2 ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ (3), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ: h h1 cos( pt ) 2 3 2 cos(3 pt ). x2 k p k 2 p2 ȼ ɫɥɭɱɚɟ k = p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. ȼ ɫɥɭɱɚɟ k = 3p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ. cg , ɬɨ ɩɨɞɛɨɪ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ Ɍɚɤ ɤɚɤ k P ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɨɞɢɬɶ ɬɚɤ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɪɚɜɟɧɫɬɜ k p ɢ k 3p. ɉɪɢ ɷɬɨɦ ɩɨɪɲɟɧɶ ɧɟ ɛɭɞɟɬ ɩɨɩɚɞɚɬɶ ɜ ɪɟɡɨɧɚɧɫ. ɝɞɟ
k
22
§7. Ɍɟɨɪɟɦɚ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ Ɂɚɞɚɱɚ ʋ 1. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ, 1975. Ɍɨɧɤɢɣ ɨɞɧɨɪɨɞɧɵɣ ɫɬɟɪɠɟɧɶ ɈȺ ɞɥɢɧɨɣ l ɢ ɜɟɫɨɦ Ɋ ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ ɨɫɢ Ɉ1Ɉ2 ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z. Ɉɩɪɟɞɟɥɢɬɶ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ. Ɇɚɫɫɨɣ ɨɫɢ Ɉ1Ɉ2 ɩɪɟɧɟɛɪɟɱɶ. Ɋɟɲɟɧɢɟ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɦɨɣ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ n
e ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ Mwc = ¦ Fk
ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɝɥɚɜɧɨɝɨ ɜɟɤɬɨɪɚ
k 1
n
ɜɧɟɲɧɢɯ ɫɢɥ ɫɢɫɬɟɦɵ R =
¦F
e k
k 1
ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ MwC . Ɍɚɤ ɤɚɤ ɰɟɧɬɪ ɦɚɫɫ
ɫɬɟɪɠɧɹ ɧɚɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ ɋ ɧɚ ɪɚɫɫɬɨɹɧɢɢ l/2 ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɢ ɢɦɟɟɬ, ɜ ɫɢɥɭ Z, ɬɨɥɶɤɨ ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɨɟ ɭɫɤɨɪɟɧɢɟ ɩɨɫɬɨɹɧɫɬɜɚ ɜɟɤɬɨɪɚ 1 2 2 wn OCZ Z , ɤɨɬɨɪɨɟ ɧɚɩɪɚɜɥɟɧɨ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɨɬ ɋ ɤ Ɉ, ɬɨ ɝɥɚɜɧɵɣ 2 ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɫɢɫɬɟɦɵ R ɢɦɟɟɬ ɬɨ ɠɟ ɧɚɩɪɚɜɥɟɧɢɟ ɢ ɪɚɜɟɧ ɩɨ ɦɨɞɭɥɸ Pl 2 R MZ C Z . 2g ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɨɣ ɜɟɫɚ ɫɬɟɪɠɧɹ ɢ ɪɟɚɤɰɢɣ ɨɩɨɪ Ɉ1 ɢ Ɉ2. Ɂɚɞɚɱɚ ʋ 2. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975. Ʉɨɥɟɫɨ ɜɟɫɨɦ Ɋ ɤɚɬɢɬɫɹ ɫɨ ɫɤɨɥɶɠɟɧɢɟɦ ɩɨ ɩɪɹɦɨɥɢɧɟɣɧɨɦɭ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦɭ ɪɟɥɶɫɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ F, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɟɝɨ ɰɟɧɬɪɭ ɬɹɠɟɫɬɢ ɋ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɋ ɤɨɥɟɫɚ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɨɧɨ ɧɚɯɨɞɢɥɨɫɶ ɜ ɩɨɤɨɟ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɪɚɜɟɧ f. Ɉɫɢ x, y ɢɡɨɛɪɚɠɟɧɵ ɧɚ ɪɢɫɭɧɤɟ. 23
Ɋɟɲɟɧɢɟ. Ʉ ɤɨɥɟɫɭ ɩɪɢɥɨɠɟɧɵ ɜɧɟɲɧɢɟ ɫɢɥɵ: Ɋ – ɟɝɨ ɜɟɫ, F – ɞɜɢɠɭɳɚɹ ɫɢɥɚ, R – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɪɟɥɶɫɚ, Fɬɪ – ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɞɨɥɶ ɪɟɥɶɫɚ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɢɥɟ F. ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɯ ɢ ɭ: Mxc F Fɬɪ , Myc R P (1) ɉɪɢ ɞɜɢɠɟɧɢɢ ɤɨɥɟɫɚ y C r const . ɉɨɷɬɨɦɭ ɭC 0 , ɢ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭɟɬ R = P. Ɍɚɤ ɤɚɤ ɩɪɢ ɤɚɱɟɧɢɢ ɤɨɥɟɫɚ ɫɨ ɫɤɨɥɶɠɟɧɢɟɦ ɫɢɥɚ fR . ɂɫɩɨɥɶɡɨɜɚɜ ɬɪɟɧɢɹ ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɧɚɢɛɨɥɶɲɟɝɨ ɡɧɚɱɟɧɢɹ, ɬɨ Fɬɪ
F fP . P ɭɪɚɜɧɟɧɢɹ ɢɦɟɟɬ
ɷɬɨ ɡɧɚɱɟɧɢɟ Fɬɪ ɜ ɩɟɪɜɨɦ ɭɪɚɜɧɟɧɢɢ (1), ɢɦɟɟɦ xC
g
ɉɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɜɢɞ F fP xC g t C1 . P ɉɨɞɫɬɚɧɨɜɤɚ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ t = 0, ɯ 0 (ɤɨɥɟɫɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɧɚɯɨɞɢɥɨɫɶ ɜ ɩɨɤɨɟ) ɜ ɭɪɚɜɧɟɧɢɟ ɞɚɟɬ ɋ1=0. ȼɧɟɫɹ ɷɬɨ ɡɧɚɱɟɧɢɟ ɋ1 ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɣ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɋ F fP t . Ⱦɜɢɠɟɧɢɟ ɤɨɥɟɫɚ ɜɨɡɦɨɠɧɨ ɩɪɢ ɧɚɥɢɱɢɢ ɤɨɥɟɫɚ: x C g P ɧɟɪɚɜɟɧɫɬɜɚ F > fP. §8. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ Ɂɚɞɚɱɚ ʋ 1. ɉɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɚɬɮɨɪɦɟ Ⱥ ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɢɧɟɪɰɢɢ ɫɨ -0 , ɫɤɨɪɨɫɬɶɸ ɩɟɪɟɦɟɳɚɟɬɫɹ ɬɟɥɟɠɤɚ ȼ ɫ ɩɨɫɬɨɹɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ u 0 . ȼ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɬɟɥɟɠɤɚ ɛɵɥɚ ɡɚɬɨɪɦɨɠɟɧɚ. Ɉɩɪɟɞɟɥɢɬɶ: ɨɛɳɭɸ ɫɤɨɪɨɫɬɶ - ɩɥɚɬɮɨɪɦɵ ɫ ɬɟɥɟɠɤɨɣ ɩɨɫɥɟ ɟɟ ɨɫɬɚɧɨɜɤɢ. M – ɦɚɫɫɚ ɩɥɚɬɮɨɪɦɵ, m – ɦɚɫɫɚ ɬɟɥɟɠɤɢ. Ɋɟɲɟɧɢɟ. ɉɨ ɬɟɨɪɟɦɟ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ Ɍɚɤ ɤɚɤ ɬɨɪɦɨɠɟɧɢɟ ɜɧɭɬɪɟɧɧɟɣ ɫɢɥɵ, ɬɨ:
ɬɟɥɟɠɤɢ
24
ɩɪɨɢɫɯɨɞɢɬ
n (e) d Q ¦Fi . dt i 1 ɫ «ɩɨɦɨɳɶɸ»
"0" ɦɨɦɟɧɬ ɞɜɢɠɟɧɢɹ, d Q 0 Q const Q 0 Q1 ® dt ¯"1" ɦɨɦɟɧɬ ɨɫɬɚɧɨɜɤɢ. Ʉɨɥɢɱɟɫɬɜɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɫɨɯɪɚɧɹɟɬɫɹ. M - 0 m(- 0 u 0 ) M - m- , Q 0 M - 0 m(- 0 u 0 ), M - 0 m(- 0 u 0 ) m -0 u0. Q1 M - m- , mM mM Ɂɚɞɚɱɚ ʋ 2. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ,1999. Ɇɚɲɢɧɢɫɬ ɬɟɩɥɨɜɨɡɚ ɨɬɤɥɸɱɚɟɬ ɞɜɢɝɚɬɟɥɶ ɢ ɧɚɱɢɧɚɟɬ ɬɨɪɦɨɡɢɬɶ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɬɟɩɥɨɜɨɡ ɢɦɟɟɬ ɫɤɨɪɨɫɬɶ 90 ɤɦ/ɱ. ɑɟɪɟɡ ɫɤɨɥɶɤɨ ɜɪɟɦɟɧɢ ɬɟɩɥɨɜɨɡ ɨɫɬɚɧɨɜɢɬɫɹ, ɟɫɥɢ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ ɩɨɫɬɨɹɧɧɚ ɢ ɫɨɫɬɚɜɥɹɟɬ 0,12 ɟɝɨ ɜɟɫɚ, ɚ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦɭ ɢ ɪɨɜɧɨɦɭ ɭɱɚɫɬɤɭ ɞɨɪɨɝɢ? Ɋɟɲɟɧɢɟ. 1. Ɍɟɩɥɨɜɨɡ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɩɨɬɨɦɭ ɪɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɟɝɨ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ (ɰɟɧɬɪɚ ɦɚɫɫɵ), ɫɱɢɬɚɹ ɱɬɨ ɤ ɧɟɦɭ ɩɪɢɥɨɠɟɧɵ ɜɫɟ ɜɧɟɲɧɢɟ ɫɢɥɵ. 2. ɉɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɨɬɤɥɸɱɚɟɬɫɹ ɞɜɢɝɚɬɟɥɶ ɢ ɜɤɥɸɱɚɟɬɫɹ ɬɨɪɦɨɡɧɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɧɚ ɬɟɩɥɨɜɨɡ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ G, ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɪɟɥɶɫɨɜ R ɢ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ F. ȼ ɧɚɱɚɥɟ ɬɨɪɦɨɠɟɧɢɹ ɫɤɨɪɨɫɬɶ V0 = 90ɤɦ/ɱ = 25ɦ/ɫ, ɜ ɤɨɧɰɟ V = 0. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɭɬɶ S ɢ ɜɪɟɦɹ t, ɡɚ ɤɨɬɨɪɨɟ ɷɬɨɬ ɩɭɬɶ ɩɪɨɣɞɟɧ. 3. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ ɬɨɪɦɨɠɟɧɢɹ ɩɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ W
ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ: Q1 Q0
³ F (t )dt . 0
ɋɩɪɨɟɰɢɪɨɜɚɜ ɜɟɤɬɨɪɵ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɨɫɶ (ɨɫɶ ɯ), ɭɜɢɞɢɦ, ɱɬɨ ɩɪɨɟɤɰɢɢ ɫɢɥ G ɢ Rn ɪɚɜɧɵ ɧɭɥɸ, ɚ ɩɪɨɟɤɰɢɹ ɫɢɥɵ F ɩɨɥɭɱɚɟɬɫɹ ɪɚɜɧɨɣ ɟɟ ɦɨɞɭɥɸ, ɧɨ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ: ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ V0 ɬɚɤɠɟ ɪɚɜɧɚ ɟɟ ɦɨɞɭɥɸ, ɩɨɷɬɨɦɭ –Ft = –mV0.
25
4. Ɋɟɲɚɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ t: t = mV0/F. Ɍɚɤ ɤɚɤ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ F = 0, 12G = 0,12mg, ɬɨ ɨɤɨɧɱɚɬɟɥɶɧɨ V0 25 t 21, 2 ɫ . 0 ,12 g 0 ,12 9 ,81 Ɂɚɞɚɱɚ ʋ 3. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ,1999. Ʉɚɤɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɤɨɥɟɫ ɡɚɬɨɪɦɨɠɟɧɧɨɝɨ ɚɜɬɨɦɨɛɢɥɹ ɨ ɞɨɪɨɝɭ (ɫɱɢɬɚɬɶ, ɱɬɨ ɡɚɬɨɪɦɨɠɟɧɵ ɜɫɟ ɱɟɬɵɪɟ ɤɨɥɟɫɚ), ɟɫɥɢ ɜ ɦɨɦɟɧɬ ɜɵɤɥɸɱɟɧɢɹ ɞɜɢɝɚɬɟɥɹ ɢ ɧɚɠɚɬɢɹ ɬɨɪɦɨɡɚ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɚɜɬɨɦɨɛɢɥɹ V0 = 60 ɤɦ/ɱ ɢ ɚɜɬɨɦɨɛɢɥɶ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɱɟɪɟɡ 5 ɫ ɩɨɫɥɟ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ. Ɋɟɲɟɧɢɟ. 1. ȼ ɡɚɞɚɱɟ ɢɡɜɟɫɬɧɨ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɡɚɬɨɪɦɨɠɟɧɧɨɝɨ ɚɜɬɨɦɨɛɢɥɹ, ɬ. ɟ. ɢɦɟɟɬɫɹ ɜ ɜɢɞɭ ɢɦɩɭɥɶɫ ɫɢɥɵ, ɩɨɷɬɨɦɭ ɞɥɹ ɪɟɲɟɧɢɹ ɩɪɢɦɟɧɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ. 2. ɇɚ ɡɚɬɨɪɦɨɠɟɧɧɵɣ ɚɜɬɨɦɨɛɢɥɶ ɞɟɣɫɬɜɭɟɬ ɞɟɜɹɬɶ ɫɢɥ; G – ɜɟɫ ɚɜɬɨɦɨɛɢɥɹ, ɱɟɬɵɪɟ ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɞɨɪɨɝɢ, ɩɪɢɥɨɠɟɧɧɵɟ ɤ ɤɚɠɞɨɦɭ ɤɨɥɟɫɭ Ri , ɢ ɱɟɬɵɪɟ ɫɢɥɵ ɬɪɟɧɢɹ Ri f , ɬɚɤɠɟ ɩɪɢɥɨɠɟɧɧɵɟ ɤ ɤɨɥɟɫɚɦ. ɉɪɢɧɢɦɚɹ ɚɜɬɨɦɨɛɢɥɶ ɡɚ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ, ɫɱɢɬɚɟɦ, ɱɬɨ ɜɫɟ ɷɬɢ ɫɢɥɵ ɩɪɢɥɨɠɟɧɵ ɜ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ ɚɜɬɨɦɨɛɢɥɹ, ɢ ɬɨɝɞɚ, ɡɚɦɟɧɢɜ ɱɟɬɵɪɟ ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɢɯ ɫɭɦɦɨɣ Rn ɢ ɱɟɬɵɪɟ ɫɢɥɵ ɬɪɟɧɢɹ ɢɯ ɫɭɦɦɨɣ R f , ɩɨɥɭɱɢɦ ɬɨɥɶɤɨ ɬɪɢ ɫɢɥɵ G , Rn , R f . 3. ɋɢɥɵ G ɢ Rn ɱɢɫɥɟɧɧɨ ɪɚɜɧɵ ɞɪɭɝ ɞɪɭɝɭ ɢ ɜɡɚɢɦɧɨ ɭɪɚɜɧɨɜɟɲɢɜɚɸɬɫɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɩɭɥɶɫ ɫɨɡɞɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɨɣ ɬɪɟɧɢɹ R f Rn f Gf . 4. ɂɦɩɭɥɶɫ ɫɢɥɵ ɬɪɟɧɢɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɞɟɣɫɬɜɭɟɬ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɞɜɢɠɟɧɢɸ, ɩɨɷɬɨɦɭ ɬɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ t
ɞɜɢɠɟɧɢɹ ɞɥɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ mV1 mV0
³ G fdt . 0
26
ɇɨ ɚɜɬɨɦɨɛɢɥɶ ɱɟɪɟɡ t =5ɫ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ, mV0 Gft , ɩɨɷɬɨɦɭ G V0 fGt , Ɉɬɤɭɞɚ, ɢɦɟɹ ɜ ɜɢɞɭ, ɱɬɨ V0 = 60 ɤɦ/ɱ = 16,7 ɦ/ɫ, g V0 16,7 f 0,34 . gt 9,81 5 §9. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ Ɂɚɞɚɱɚ ʋ 1. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975. ɉɪɢ ɜɪɚɳɟɧɢɢ ɛɚɪɚɛɚɧɚ 1 ɜɟɫɨɦ Ɋ1 ɢ ɪɚɞɢɭɫɨɦ r ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ z ɧɚ ɟɝɨ ɛɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɦɚɬɵɜɚɟɬɫɹ ɧɢɬɶ, ɤɨɬɨɪɚɹ ɩɪɢɜɨɞɢɬ ɜ ɞɜɢɠɟɧɢɟ ɝɪɭɡ 2 ɜɟɫɨɦ Ɋ2, ɫɤɨɥɶɡɹɳɢɣ ɩɨ ɧɟɩɨɞɜɢɠɧɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɛɚɪɚɛɚɧɚ, ɟɫɥɢ ɤ ɧɟɦɭ ɩɪɢɥɨɠɟɧ ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ mɜɪ, ɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɝɪɭɡɚ ɨ ɩɥɨɫɤɨɫɬɶ ɪɚɜɟɧ f. ȼɵɫɨɬɨɣ ɝɪɭɡɚ ɩɪɟɧɟɛɪɟɱɶ. Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z, n dLZ momZ ( Fke ) . ɬ. ɟ. ¦ dt k 1 ɂɡɨɛɪɚɡɢɦ ɜɧɟɲɧɢɟ ɫɢɥɵ ɢ ɦɨɦɟɧɬɵ ɫɢɫɬɟɦɵ: mɜɪ – ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ; Ɋ1 – ɜɟɫ ɛɚɪɚɛɚɧɚ, Ɋ2 – ɜɟɫ ɝɪɭɡɚ, R1 ɢ R1' – ɫɨɫɬɚɜɥɹɸɳɢɟ ɪɟɚɤɰɢɢ ɨɫɢ ɛɚɪɚɛɚɧɚ, R2 – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɩɥɨɫɤɨɫɬɢ, Fɬɪ – ɫɢɥɚ ɬɪɟɧɢɹ ɩɪɢ ɫɤɨɥɶɠɟɧɢɢ ɝɪɭɡɚ ɨ ɩɥɨɫɤɨɫɬɶ. ɍɱɢɬɵɜɚɹ, ɱɬɨ R2 P2 , Fɬɪ = fP2, ɚ ɫɢɥɵ Ɋ1, R1 ɢ R1' ɩɪɢɥɨɠɟɧɵ ɜ ɬɨɱɤɟ, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ z, ɡɚɩɢɲɟɦ: (1) ¦ mom Z ( Fke ) mɜɪ fP2 r . Ƚɥɚɜɧɵɣ ɦɨɦɟɧɬ ɤɨɥɢɱɟɫɬɜ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ z
ɞɜɢɠɟɧɢɹ
LZ
ɞɚɧɧɨɣ
ɫɢɫɬɟɦɵ
P1 2 P2 2 P P1r 2 r 2 Zr r Z. 2g 2g g ȼɡɹɜ ɩɪɨɢɡɜɨɞɧɭɸ LZ ɩɨ ɜɪɟɦɟɧɢ ɫ ɭɱɺɬɨɦ ɬɨɝɨ, ɱɬɨ Z M , ɢɦɟɟɦ LZ
L(Z1) L(Z2)
J Z Z momZ (m2 v2 )
27
dL Z dt
P1 2 P2 2 r M . 2g
dLZ dt ɪɟɲɢɜ ɟɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ M , ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ: 2g M (mɜɪ fP2 r ) . ( P1 2 P2 )r 2 ɉɨɞɫɬɚɜɢɜ ɪɟɡɭɥɶɬɚɬɵ (1) ɢ (2) ɜ ɭɪɚɜɧɟɧɢɟ
(2) n
¦ mom
Z
( Fke ) ɢ
k 1
Ɂɚɞɚɱɚ ʋ 2. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975. ɇɚ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɪɭɝɥɨɝɨ ɤɨɧɭɫɚ ɫɢɦɦɟɬɪɢɱɧɨ ɪɚɫɩɨɥɨɠɟɧɵ ɞɜɚ ɝɪɭɡɚ, ɫɨɟɞɢɧɟɧɧɵɟ ɦɟɠɞɭ ɫɨɛɨɣ ɬɨɧɤɨɣ ɧɢɬɶɸ ɢ ɨɬɫɬɨɹɳɢɟ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ ɧɚ ɪɚɫɫɬɨɹɧɢɢ ɨɞɧɨɣ ɬɪɟɬɢ ɪɚɞɢɭɫɚ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ. Ʉɨɧɭɫ ɜɦɟɫɬɟ ɫ ɝɪɭɡɚɦɢ ɜɪɚɳɚɥɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z. ɉɨɫɥɟ ɜɧɟɡɚɩɧɨɝɨ ɪɚɡɪɵɜɚ ɧɢɬɢ ɝɪɭɡɵ ɧɚɱɚɥɢ ɨɩɭɫɤɚɬɶɫɹ ɩɨ ɧɚɩɪɚɜɥɹɸɳɢɦ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɧɭɫɚ. ȼɟɫ ɤɚɠɞɨɝɨ ɢɡ ɝɪɭɡɨɜ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɦɟɧɶɲɟ ɜɟɫɚ ɤɨɧɭɫɚ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɤɨɧɭɫɚ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɝɪɭɡɵ ɞɨɫɬɢɝɧɭɬ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ. ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɞɜɢɠɟɧɢɸ ɩɪɟɧɟɛɪɟɱɶ. Ƚɪɭɡɵ ɫɱɢɬɚɬɶ ɬɨɱɟɱɧɵɦɢ ɦɚɫɫɚɦɢ. Ɋɟɲɟɧɢɟ. ȼɡɹɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜ ɧɢɠɧɟɣ ɨɩɨɪɟ Ⱥ ɨɫɢ ɤɨɧɭɫɚ, ɧɚɩɪɚɜɢɦ ɨɫɶ z ɩɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ. Ɉɛɨɡɧɚɱɢɦ: Ɋ – ɜɟɫ ɤɨɧɭɫɚ, r –ɪɚɞɢɭɫ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ. ɂɡɨɛɪɚɡɢɦ ɜɧɟɲɧɢɟ ɫɢɥɵ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɤɨɧɭɫɚ ɢ ɞɜɭɯ ɝɪɭɡɨɜ: Ɋ – ɜɟɫ ɤɨɧɭɫɚ, Ɋ1 ɢ Ɋ2 – ɜɟɫɚ ɝɪɭɡɨɜ, RȺx, RȺy, RȺz, RBx, Rȼy – ɫɨɫɬɚɜɥɹɸɳɢɟ ɪɟɚɤɰɢɣ ɨɩɨɪ Ⱥ ɢ ȼ. ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z. n dLZ momZ ( Fke ) . ¦ dt k 1 28
Ɍɚɤ ɤɚɤ ɜɫɟ ɜɧɟɲɧɢɟ ɫɢɥɵ ɥɢɛɨ ɩɚɪɚɥɥɟɥɶɧɵ, ɥɢɛɨ ɩɟɪɟɫɟɤɚɸɬ ɨɫɶ ɜɪɚɳɟɧɢɹ z, ɬɨ ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɜɧɟɲɧɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɪɚɜɧɚ ɧɭɥɸ: n dLZ mom Z ( F ke ) 0 . Ɍɨ ɟɫɬɶ 0 ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, L1Z L2Z . ɂɬɚɤ, ¦ dt k 1 ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɭɱɚɣ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ. ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɨɣ LZ I Z Z , ɡɚɩɢɲɟɦ: L1Z I Z1 Z1 , L2Z I Z2Z2 , ɝɞɟ I Z1 , I Z2 – ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɜ 1 1 1 2 ɩɟɪɜɨɦ ɢ ɜɨ ɜɬɨɪɨɦ ɩɨɥɨɠɟɧɢɢ ɝɪɭɡɨɜ. Ɍɚɤ ɤɚɤ L Z L Z ɢ LZ I Z Z1 ,
I Z1 Z1 . I Z2 Ɉɫɬɚɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z. Ɍɚɤ ɤɚɤ ɦɚɬɟɪɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɫɨɫɬɨɢɬ ɢɡ ɤɨɧɭɫɚ ɢ ɞɜɭɯ ɝɪɭɡɨɜ, ɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɤɨɧɭɫɚ ɢ ɝɪɭɡɨɜ. 3 P 2 r . Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɨɧɭɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɪɚɜɟɧ 10 g ɋɱɢɬɚɹ ɝɪɭɡɵ ɬɨɱɟɱɧɵɦɢ ɦɚɫɫɚɦɢ, ɢɦɟɟɦ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɣ ɝɪɭɡɨɜ 2 3P 2 4P 2 P P §r· 3 P 2 16 P 2 r. r 2 r2 IZ2 I Z1 r 2 r , ¨ ¸ 10 g 4g 5g 10 g 4g © 3 ¹ 45 g 4 Z1 . ɂɫɩɨɥɶɡɭɹ ɧɚɣɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ Z 2 9 L2Z
1 I Z2Z2 , ɬɨ I Z Z1
I Z2Z 2 , ɨɬɤɭɞɚ Z 2
§10. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɸɬ ɮɨɪɦɭɥɨɣ n m v2 T ¦ k k . 2 k 1 Ⱦɥɹ ɫɢɫɬɟɦ, ɫɨɜɟɪɲɚɸɳɢɯ ɫɥɨɠɧɨɟ ɞɜɢɠɟɧɢɟ, ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ Ʉɺɧɢɝɚ Mv C2 T Tc, 2 ɝɞɟ T c – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɜ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɟɣɫɹ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɫ ɧɚɱɚɥɨɦ ɜ ɰɟɧɬɪɟ ɢɧɟɪɰɢɢ. (Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɩɨɞ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɩɨɧɢɦɚɟɦ ɰɟɧɬɪ ɦɚɫɫ). Ⱦɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, 29
Mv C2 , 2 ɝɞɟ M – ɦɚɫɫɚ ɬɟɥɚ; vC – ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ (ɢɥɢ ɥɸɛɨɣ ɞɪɭɝɨɣ ɬɨɱɤɢ). Ⱦɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɨɫɢ, J ZZ 2 T , 2 ɝɞɟ J Z – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ. Ⱦɥɹ ɨɛɳɟɝɨ ɫɥɭɱɚɹ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɞɥɹ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ) Mv C2 J C Z 2 T , 2 2 ɝɞɟ JC – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ (ɞɥɹ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɷɬɚ ɨɫɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ); Z – ɦɝɧɨɜɟɧɧɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ. ɗɥɟɦɟɧɬɚɪɧɭɸ ɪɚɛɨɬɭ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɟ, ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɨɩɪɟɞɟɥɹɸɬ ɮɨɪɦɭɥɨɣ T
GA F dr Fds cos( F , W ) Fx dx Fy dy Fz dz . Ɋɚɛɨɬɚ ɫɢɥɵ ɧɚ ɤɨɧɟɱɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
³ Fdr ³ FW ds ³ ( F dx F dy F dz) .
A
x
L
L
y
z
L
Ɋɚɛɨɬɚ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ ɧɚ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ s A Fs cos( F , s ) . Ɋɚɛɨɬɚ ɫɢɥ ɬɹɠɟɫɬɢ ɥɸɛɨɣ ɫɢɫɬɟɦɵ A12 ( P ) P( zC1 zC2 ) , ɝɞɟ P – ɜɟɫ ɜɫɟɣ ɫɢɫɬɟɦɵ; zC1 ɢ zC2 – ɚɩɩɥɢɤɚɬɵ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɩɨɥɨɠɟɧɢɹɯ ɫɢɫɬɟɦɵ. Ɋɚɛɨɬɚ ɭɩɪɭɝɨɣ ɫɢɥɵ Fx cx ɩɪɢ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɬɨɱɤɢ c 2 A12 x x 22 , 2 1 ɝɞɟ x1 x2 ɤɨɨɪɞɢɧɚɬɵ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɣ ɗɥɟɦɟɧɬɚɪɧɭɸ ɪɚɛɨɬɭ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɩɟɪɟɦɟɳɚɸɳɟɦɭɫɹ ɩɪɨɢɡɜɨɥɶɧɨ, ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ
GA
>R v
O
@
M OZ dt , 30
ɝɞɟ R ɢ M 0 – ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ ɫɢɥ; Ɉ – ɩɪɨɢɡɜɨɥɶɧɚɹ ɬɨɱɤɚ ɬɟɥɚ. Ɋɚɛɨɬɚ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ ɨɫɢ, M2
GA
M Z dM ;
A12
³M
Z
dM ,
M1
ɝɞɟ MZ – ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ Oz. ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ. Ɇɨɳɧɨɫɬɶ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɬɨɱɤɟ, GA N F v FW v Fx x Fy y Fz z . dt Ɇɨɳɧɨɫɬɶ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, N R vO M O Z . Ⱦɥɹ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ, N M ZZ . ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɛɨɬɚ ɫɢɥɵ ɧɚ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɤɪɢɜɨɣ L, ɩɨ ɤɨɬɨɪɨɣ ɩɟɪɟɦɟɳɚɟɬɫɹ ɬɨɱɤɚ. ȿɫɥɢ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɨɱɤɭ, ɬɚɤɨɜɵ, ɱɬɨ wFx wFy wFx wFz wFy wFz , , , wz wy wx wy wz wx ɬɨ ɪɚɛɨɬɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ ɢ ɩɨɥɟ ɫɢɥ ɧɚɡɵɜɚɸɬ ɩɨɬɟɧɰɢɚɥɶɧɵɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ GA dɉ ; A12=ɉ1 – ɉ2, ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ; ɉ1 ɢ ɉ2 – ɡɧɚɱɟɧɢɹ ɉ(x, y, z) ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɩɨɥɨɠɟɧɢɹɯ ɬɨɱɤɢ. ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ ɉ PZc const . ȿɫɥɢ ɜɵɛɪɚɧɚ ɧɭɥɟɜɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɭɪɨɜɧɹ, ɬɨ ɩɨɥɭɱɢɦ ɉ r Ph , ɝɞɟ h – ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɪɢɱɟɦ ɡɧɚɤ «ɩɥɸɫ» ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɪɚɫɩɨɥɨɠɟɧ ɜɵɲɟ ɷɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ɉɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɩɪɭɠɢɧɵ (ɥɢɧɟɣɧɨɣ ɢ ɫɩɢɪɚɥɶɧɨɣ) ɜɵɪɚɠɚɸɬ ɮɨɪɦɭɥɨɣ c'2 ɉ , 2 ɝɞɟ ɞɥɹ ɥɢɧɟɣɧɨɣ ɩɪɭɠɢɧɵ: c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɣ ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ ɩɪɭɠɢɧɵ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ; 31
' – ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ ɩɪɭɠɢɧɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɟ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɞɥɢɧɨɣ; ɞɥɹ ɫɩɢɪɚɥɶɧɨɣ ɩɪɭɠɢɧɵ: c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɦɨɦɟɧɬɚ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɝɨ ɡɚɤɪɭɱɢɜɚɧɢɟ ɩɪɭɠɢɧɵ ɧɚ 1 ɪɚɞɢɚɧ; ' – ɭɝɨɥ ɡɚɤɪɭɱɢɜɚɧɢɹ ɨɬ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ. ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ. T2 – T1 = A12. ȿɫɥɢ ɫɢɫɬɟɦɚ ɧɟɢɡɦɟɧɹɟɦɚɹ, ɬɨ T2 T1 A12( e ) , ɝɞɟ A12( e ) – ɫɭɦɦɚ ɪɚɛɨɬ ɜɧɟɲɧɢɯ ɫɢɥ. ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ dT N. ɦɨɳɧɨɫɬɟɣ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ. dt ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const. Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ Ɇ ɦɚɫɫɨɣ m ɩɨɦɟɳɟɧ ɧɚ ɧɟɝɥɚɞɤɭɸ ɧɚɤɥɨɧɧɭɸ ɩɥɨɫɤɨɫɬɶ, ɨɛɪɚɡɭɸɳɭɸ ɫ ɝɨɪɢɡɨɧɬɨɦ ɭɝɨɥ D , ɢ ɩɪɢɤɪɟɩɥɟɧ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ ɫ ɠɟɫɬɤɨɫɬɶɸ ɫ, ɞɪɭɝɨɣ ɤɨɧɟɰ ɤɨɬɨɪɨɣ ɡɚɤɪɟɩɥɟɧ ɧɟɩɨɞɜɢɠɧɨ. Ɉɩɪɟɞɟɥɢɬɶ ɦɚɤɫɢɦɚɥɶɧɨɟ ɪɚɫɬɹɠɟɧɢɟ S ɩɪɭɠɢɧɵ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɪɭɠɢɧɚ ɛɵɥɚ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɚ, ɚ ɝɪɭɡ ɨɬɩɭɳɟɧ ɛɟɡ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɬɟɥɚ ɨ ɩɥɨɫɤɨɫɬɶ ɪɚɜɟɧ f, ɩɪɢɱɟɦ f tgD . Ɋɟɲɟɧɢɟ. ɂɦɟɟɦ: ɧɚ ɝɪɭɡ Ɇ ɞɟɣɫɬɜɭɟɬ ɜɟɫ P , ɭɩɪɭɝɚɹ ɫɢɥɚ F , ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɩɥɨɫɤɨɫɬɢ N ɢ ɫɢɥɚ ɬɪɟɧɢɹ Fɬɪ , ɧɚɩɪɚɜɥɟɧɧɚɹ ɤɚɤ F . ɂɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɜɧɨ ɪɚɛɨɬɟ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ A12 ( F ) . Ɍɚɤ ɤɚɤ v1 v 2 0, ɬɨ T1 T2 0 , ɫɢɫɬɟɦɟ: T2 T1
¦
0, ɬ. ɟ. A12 A12 ( P ) A12 ( Fɬɪ ) A12 ( F ) A12 ( N ) fP cos D , F cx , P mg.
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, A12 Ɂɞɟɫɶ Fɬɪ
32
0.
Ⱦɚɥɟɟ ɧɚɣɞɟɦ A12 ( P ) PS sin D ; S
A12 ( Fɬɪ )
Fɬɪ S
fPS cos D ;
2
cS A12 ( N ) 0 , ɬɚɤ ɤɚɤ ɩɟɪɟɦɟɳɟɧɢɟ ɝɪɭɡɚ ; 2 0 ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɨɪɦɚɥɶɧɨɣ ɫɢɥɟ ɪɟɚɤɰɢɢ ɩɥɨɫɤɨɫɬɢ. cS 2 PS sin D PSf cosD 0, ɨɬɤɭɞɚ ɋɥɟɞɨɜɚɬɟɥɶɧɨ, 2 2mg (sin D f cos D ) . S c Ɂɚɞɚɱɚ ʋ 2. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ, 1999. Ɇɚɲɢɧɢɫɬ ɬɟɩɥɨɜɨɡɚ ɨɬɤɥɸɱɚɟɬ ɞɜɢɝɚɬɟɥɶ ɢ ɧɚɱɢɧɚɟɬ ɬɨɪɦɨɡɢɬɶ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɬɟɩɥɨɜɨɡ ɢɦɟɟɬ ɫɤɨɪɨɫɬɶ 90 ɤɦ/ɱ. ɉɪɨɣɞɹ ɤɚɤɨɣ ɩɭɬɶ, ɬɟɩɥɨɜɨɡ ɨɫɬɚɧɨɜɢɬɫɹ, ɟɫɥɢ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ ɩɨɫɬɨɹɧɧɚ ɢ ɫɨɫɬɚɜɥɹɟɬ 0,12 ɟɝɨ ɜɟɫɚ, ɚ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦɭ ɢ ɪɨɜɧɨɦɭ ɭɱɚɫɬɤɭ ɞɨɪɨɝɢ? Ɋɟɲɟɧɢɟ. 1. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɨɪɦɨɡɧɨɝɨ ɩɭɬɢ S ɩɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ T1 T0 ¦ A10 ( F ) . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ V1=0, A12 ( F ) ³ cxdx
A( Fɬɪ )
FS cos(D )
FS
0.12GS (ɭɝɨɥ D ɦɟɠɞɭ ɧɚɩɪɚɜɥɟɧɢɟɦ ɫɢɥɵ F ɢ
ɧɚɩɪɚɜɥɟɧɢɟɦ ɩɟɪɟɦɟɳɟɧɢɹ ɪɚɜɟɧ 180q), ɚ ɪɚɛɨɬɵ ɫɢɥ G ɢ Rn ɪɚɜɧɵ ɧɭɥɸ (ɷɬɢ ɫɢɥɵ ɞɟɣɫɬɜɭɸɬ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɤ ɧɚɩɪɚɜɥɟɧɢɸ ɩɟɪɟɦɟɳɟɧɢɹ), mV02 Fs . ɩɨɷɬɨɦɭ 2 mV02 V02 2. Ɋɟɲɚɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ S: S 2F 2 0,12 g (F =0,12G=0,12mg). ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜ ɷɬɭ ɮɨɪɦɭɥɭ ɱɢɫɥɨɜɵɯ ɡɧɚɱɟɧɢɣ 252 265 ɦ . S 2 0,12 9,81 Ɂɚɞɚɱɚ ʋ 3. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ, 1999. 33
Ɂɚ 500 ɦ ɞɨ ɫɬɚɧɰɢɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ ɩɪɢɝɨɪɤɟ ɜɵɫɨɬɨɣ 2 ɦ, ɦɚɲɢɧɢɫɬ ɩɨɟɡɞɚ, ɢɞɭɳɟɝɨ ɫɨ ɫɤɨɪɨɫɬɶɸ 12 ɦ/ɫ, ɧɚɱɢɧɚɟɬ ɬɨɪɦɨɡɢɬɶ. Ʉɚɤ ɜɟɥɢɤɨ ɞɨɥɠɧɨ ɛɵɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɨɬ ɬɨɪɦɨɠɟɧɢɹ, ɫɱɢɬɚɟɦɨɟ ɩɨɫɬɨɹɧɧɵɦ, ɱɬɨɛɵ ɩɨɟɡɞ ɨɫɬɚɧɨɜɢɥɫɹ ɭ ɫɬɚɧɰɢɢ, ɟɫɥɢ ɦɚɫɫɚ ɩɨɟɡɞɚ ɪɚɜɧɚ 106 ɤɝ, ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɪɟɧɢɹ 19600 ɇ. Ɋɟɲɟɧɢɟ. 1. Ɋɟɲɚɟɦ ɡɚɞɚɱɭ, ɢɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
T1 T0
¦A
10
( F ) , ɬɚɤ ɤɚɤ ɜ
ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɡɚɞɚɧɨ ɧɟ ɜɪɟɦɹ ɬɨɪɦɨɠɟɧɢɹ, ɚ ɬɨɪɦɨɡɧɨɣ ɩɭɬɶ S = 500 ɦ. 2. ɉɨɟɡɞ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɩɨɷɬɨɦɭ ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɞɜɢɠɟɧɢɟ ɟɝɨ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ.
ɉɪɢɥɨɠɢɦ ɤ ɬɨɱɤɟ ɋ ɜɫɟ ɞɟɣɫɬɜɭɸɳɢɟ ɫɢɥɵ. ȼɟɫ ɩɨɟɡɞɚ G ɪɚɫɤɥɚɞɵɜɚɟɦ ɧɚ ɞɜɟ ɫɨɫɬɚɜɥɹɸɳɢɟ G1 ɢ G2 . ɇɚ ɩɨɟɡɞ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɟɝɨ ɞɜɢɠɟɧɢɸ, ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɨɫɬɚɜɥɹɸɳɚɹ ɜɟɫɚ G2 , ɫɢɥɚ ɬɪɟɧɢɹ R ɢ ɢɫɤɨɦɚɹ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ F . 3. Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɷɬɢɯ ɫɢɥ ɪɚɜɧɚ ɢɯ ɫɭɦɦɟ (F+R+G2), ɧɚɩɪɚɜɥɟɧɢɟ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɫɤɨɪɨɫɬɢ, ɚ ɨɬɪɢɰɚɬɟɥɶɧɚɹ ɪɚɛɨɬɚ ɫɢɥ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɨɣ Ⱥ10 = –(F+R+G2)S. 4. Ɋɚɛɨɬɚ Ⱥ10 ɪɚɜɧɚ ɢɡɦɟɧɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɨɟɡɞɚ, ɧɨ ɬɚɤ ɤɚɤ mV02 F R G2 S ɤɨɧɟɱɧɚɹ ɫɤɨɪɨɫɬɶ ɩɨɟɡɞɚ V1=0, ɬɨ . 2 ɂɡ
F 5. G2
ɩɨɫɥɟɞɧɟɝɨ ɭɪɚɜɧɟɧɢɹ mV02 R G2 . 2S ɇɨ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ h G sin(D ) G . S
ɦɨɠɧɨ
ɧɭɠɧɨ
ɧɚɣɬɢ
ɨɩɪɟɞɟɥɢɬɶ
34
ɫɢɥɭ
ɬɨɪɦɨɠɟɧɢɹ
ɫɨɫɬɚɜɥɹɸɳɭɸ
ɜɟɫɚ
F:
G 2:
ɉɨɞɫɬɚɜɢɜ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ G2 ɜ ɮɨɪɦɭɥɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɫɢɥɵ F, mV02 h RG . ɩɨɥɭɱɢɦ F 2S S Ɂɚɬɟɦ ɜɵɱɢɫɥɹɟɦ ɜɟɥɢɱɢɧɭ ɫɢɥɵ F, ɭɱɢɬɵɜɚɹ, ɱɬɨ G=mg,
F
mV02 mgh R 2S S
10 6 12 2 10 6 9,81 2 19600 2 500 500
85100 ɇ .
§11. ɍɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ Ⱦɥɹ ɫɢɫɬɟɦɵ ɫ ɝɨɥɨɧɨɦɧɵɦɢ ɢɞɟɚɥɶɧɵɦɢ ɢ ɭɞɟɪɠɢɜɚɸɳɢɦɢ ɫɜɹɡɹɦɢ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɢɦɟɸɬ ɜɢɞ [1], [5] d §¨ wT ·¸ wT Q j (j = 1, 2, ... , s), dt ¨© wq j ¸¹ wq j ɝɞɟ q1, q2,..., qS – ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ; s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ; q1 , q 2 , …, q S – ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ; Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ; Q1, Q2, ..., QS – ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ. Ⱦɥɹ ɫɢɫɬɟɦ, ɞɜɢɠɭɳɢɯɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɛɭɞɭɬ d §¨ wL ·¸ wL 0, dt ¨© wq j ¸¹ wq j ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ. Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ: x ɭɫɬɚɧɨɜɢɬɶ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢ ɜɵɛɪɚɬɶ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ; x ɩɪɟɞɩɨɥɨɠɢɜ, ɱɬɨ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɭɜɟɥɢɱɢɜɚɸɬɫɹ, ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ, ɩɪɢ ɷɬɨɦ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɜɯɨɞɹɳɢɟ ɜ Ɍ, ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ, ɬ.ɟ. T T (q1 , q2 ,..., q S ; q1 , q 2 ,..., q S ; t ) (ɜ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɜɹɡɟɣ ɜɪɟɦɹ t ɧɟ ɜɯɨɞɢɬ ɜ ɜɵɪɚɠɟɧɢɟ T);
35
x ɨɩɪɟɞɟɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ
wT ; ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɢɡɜɨɞɧɵɟ wq j
d §¨ wT dt ¨© wq j
· ¸ , ɫɱɢɬɚɹ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ, ɜɯɨɞɹɳɢɟ ɜ wT , ɮɭɧɤɰɢɹɦɢ ¸ wq j ¹ ɜɪɟɦɟɧɢ t; x ɧɚɣɬɢ ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ; ɩɨɞɫɬɚɜɢɬɶ ɜɫɟ ɧɚɣɞɟɧɧɵɟ ɜɟɥɢɱɢɧɵ ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ. Ɂɚɞɚɱɚ ʋ 1. ɉɪɹɦɨɭɝɨɥɶɧɚɹ ɩɪɢɡɦɚ Ⱥ ɦɨɠɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. ɇɚ ɧɚɤɥɨɧɧɨɣ ɝɥɚɞɤɨɣ ɝɪɚɧɢ ɩɪɢɡɦɵ ɩɨɦɟɳɟɧ ɨɞɧɨɪɨɞɧɵɣ ɰɢɥɢɧɞɪ B, ɧɚ ɤɨɬɨɪɵɣ ɧɚɦɨɬɚɧɚ ɧɟɪɚɫɬɹɠɢɦɚɹ ɧɢɬɶ, ɩɟɪɟɤɢɧɭɬɚɹ ɱɟɪɟɡ ɢɞɟɚɥɶɧɵɣ ɛɥɨɤ ɋ. Ʉ ɤɨɧɰɭ ɧɢɬɢ ɩɪɢɤɪɟɩɥɟɧ ɝɪɭɡ D ɦɚɫɫɨɣ m. ɉɪɢɧɢɦɚɹ ɦɚɫɫɭ ɰɢɥɢɧɞɪɚ ɪɚɜɧɨɣ 2m, ɚ ɦɚɫɫɭ ɩɪɢɡɦɵ 3m, ɨɩɪɟɞɟɥɢɬɶ ɞɜɢɠɟɧɢɟ ɫɢɫɬɟɦɵ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɨɧɚ ɧɚɯɨɞɢɥɚɫɶ ɜ ɩɨɤɨɟ, ɚ ɭɝɨɥ D 30D . Ɋɚɡɦɟɪɚɦɢ ɢ ɦɚɫɫɨɣ ɛɥɨɤɚ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ. ɋɢɫɬɟɦɚ ɢɦɟɟɬ ɬɪɢ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ, ɬɚɤ ɤɚɤ ɞɥɹ ɨɞɧɨɡɧɚɱɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɨɠɟɧɢɹ ɜɫɟɯ ɟɺ ɷɥɟɦɟɧɬɨɜ ɧɟɨɛɯɨɞɢɦɨ ɬɪɢ ɧɟɡɚɜɢɫɢɦɵɯ ɩɚɪɚɦɟɬɪɚ. ȼɵɛɟɪɟɦ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ: q1 x – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɨɫɢ ɛɥɨɤɚ ɋ ɞɨ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɝɪɭɡɚ D; 36
q 2 y – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ ɛɥɨɤɚ ɋ ɢ ɰɢɥɢɧɞɪɚ ȼ; q3 z – ɩɟɪɟɦɟɳɟɧɢɟ ɩɪɢɡɦɵ. ɉɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢɡɦɟɧɹɸɬɫɹ ɜ ɫɬɨɪɨɧɭ ɢɯ ɭɜɟɥɢɱɟɧɢɹ. Ɉɩɪɟɞɟɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ. ɂɦɟɟɦ T T A TB TD , ɝɞɟ TA – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɢɡɦɵ; TB – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɰɢɥɢɧɞɪɚ; TD – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ. ɉɪɢɡɦɚ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ v3 z , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, m A 2 3m 2 TA z z . 2 2 Ƚɪɭɡ D ɭɱɚɫɬɜɭɟɬ ɜ ɞɜɭɯ ɩɨɫɬɭɩɚɬɟɥɶɧɵɯ ɞɜɢɠɟɧɢɹɯ: ɨɬɧɨɫɢɬɟɥɶɧɨɦ (ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ) ɫɨ ɫɤɨɪɨɫɬɶɸ v1 x ɢ ɩɟɪɟɧɨɫɧɨɦ (ɜɦɟɫɬɟ ɫ ɩɪɢɡɦɨɣ) ɫɨ ɫɤɨɪɨɫɬɶɸ v3 z . ɉɨɷɬɨɦɭ ɚɛɫɨɥɸɬɧɨɟ ɞɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɭɤɚɡɚɧɧɵɯ 2 2 2 2 2 ɫɤɨɪɨɫɬɟɣ, ɬ.ɟ. v D v1 v3 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, v D v1 v3 x z . m D v D2 m 2 ( x z 2 ) . 2 2 ɐɢɥɢɧɞɪ ɫɨɜɟɪɲɚɟɬ ɩɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ȿɝɨ ɤɢɧɟɬɢɱɟɫɤɭɸ m B v02 J 0Z B2 , ɷɧɟɪɝɢɸ ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ Ʉɺɧɢɝɚ: TB 2 2 ɝɞɟ v0 – ɚɛɫɨɥɸɬɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ; Z B – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɰɢɥɢɧɞɪɚ; J0 – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ Ɉ. ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ Ɉ ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɪɚɜɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɢ ɩɟɪɟɧɨɫɧɨɣ ɫɤɨɪɨɫɬɟɣ, ɬ. ɟ. v O v 2 v3 , ɝɞɟ v 2 y – ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, vO2 v 22 v32 2v 2 v3 cos D y 2 z 2 2 y z cos D . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, TD
Ɉɩɪɟɞɟɥɢɦ Z B . Ɍɚɤ ɤɚɤ ɞɜɢɠɟɧɢɟ ɩɪɢɡɦɵ Ⱥ ɹɜɥɹɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ, ɬɨ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ Z B ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ ɥɢɲɶ ɜ ɨɬɧɨɫɢɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɰɢɥɢɧɞɪɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ Ⱥ. ɂɡɜɟɫɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɞɜɭɯ r r ɬɨɱɟɤ ɰɢɥɢɧɞɪɚ Ɉ ɢ ȿ: v E x ; vO y.
37
Ɉɬɧɨɫɢɬɟɥɶɧɵɣ ɦɝɧɨɜɟɧɧɵɣ ɰɟɧɬɪ ɫɤɨɪɨɫɬɟɣ ɰɢɥɢɧɞɪɚ ȼ ɧɚɯɨɞɢɬɫɹ ɜ vOr v Er . ɋɨɫɬɚɜɥɹɹ ɩɪɨɢɡɜɨɞɧɭɸ ɩɪɨɩɨɪɰɢɸ, ɬɨɱɤɟ Ʉ, ɩɨɷɬɨɦɭ Z B OK EK vOr v Er y x , ɝɞɟ R – ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɚ. ɩɨɥɭɱɢɦ Z B OK EK R mB R 2 J mR 2 , Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ O 2 m 2 2 2 ɫɥɟɞɨɜɚɬɟɥɶɧɨ, TB m( y z 2 y z cos D ) ( x y ) . 2 Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɜɫɟɣ ɫɢɫɬɟɦɵ 3m 2 m 2 m T z ( x z 2 ) m( y 2 z 2 2 y z cos D ) ( x y ) 2 . 2 2 2 ɇɚɣɞɟɦ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ: wT mx m( x y ) ; w x
wT 2m( y z cosD ) m( x y ) ; w y wT 3mz mz 2m( z y cos D ) . w z Ɍɚɤ ɤɚɤ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ x, y, z ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɧɟ ɜɯɨɞɹɬ ɜ wT wT wT 0, ɞɚɥɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ T, ɬɨ wx w y wz
d §w T · ¸ ¨ dt ¨© w x ¸¹
mx m( x y)
d §w T · ¨ ¸ dt ¨© w y ¸¹
2m( y z cos D ) m( x y)
2mx my;
d §w T · ¸ 3mz mz 2m( z y cos D ) ¨ dt ¨© w z ¸¹ Ɉɩɪɟɞɟɥɢɦ ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ.
mx 3my 2mz cos D ;
2my cos D 6mz.
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ QX ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ Q X
GA1 , Gx
ɝɞɟ GA1 – ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ ɡɚɞɚɜɚɟɦɵɯ ɫɢɥ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ G x z 0; G y G z 0, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, GA1 PDG x mgG x, ɨɬɤɭɞɚ Q X mg.
38
GA2 , ɝɞɟ GA2 - ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ Gy ɡɚɞɚɜɚɟɦɵɯ ɫɢɥ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ G y z 0; G x G z 0. Ɉɱɟɜɢɞɧɨ, GA2 PBGy sin D 2mg sin DGy, ɬɨɝɞɚ QY 2mg sin D . G z z 0, ɬɨ GA3 0, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, QZ 0. ȿɫɥɢ ɠɟ G x G y 0; Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɣɞɟɦ
QY
Ɂɚɦɟɱɚɧɢɟ. Ɉɛɨɛɳɟɧɧɵɟ ɫɢɥɵ ɦɨɠɧɨ ɥɟɝɤɨ ɨɩɪɟɞɟɥɢɬɶ ɢ ɩɨ ɮɨɪɦɭɥɟ w3 Qj . ɉ PD x PB y sin D const , ȼ ɧɚɲɟɣ ɡɚɞɚɱɟ w qj ɫɥɟɞɨɜɚɬɟɥɶɧɨ, w3 QX PD mg ; wx
QY QZ
w3 wy w3 wz
PB sin D
2mg sin D ;
0.
ɉɨɞɫɬɚɜɥɹɹ ɜɫɟ ɧɚɣɞɟɧɧɵɟ ɜɟɥɢɱɢɧɵ ɜ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ, ɩɨɥɭɱɢɦ 2mx my mg;
mx 3my 2mzcos D 2mg sin D ; 2mycos D 6mz 0. D ɉɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ, ɱɬɨ D 30 , ɧɚɣɞɟɦ 2 x y g; x 3 y 3 z
g;
3 y 6 z 0. ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ. ɍɦɧɨɠɚɹ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚ 2 ɢ ɜɵɱɢɬɚɹ ɢɡ ɧɟɝɨ ɩɟɪɜɨɟ, ɩɨɥɭɱɢɦ 5 y 2 3 z g. Ɋɟɲɚɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɫɨɜɦɟɫɬɧɨ ɫ ɬɪɟɬɶɢɦ, ɧɚɣɞɟɦ 3 1 3 x g ; y g ; z g. 8 4 24 ɂɧɬɟɝɪɢɪɭɹ, ɩɨɥɭɱɢɦ 3 1 3 x gt C1 ; y gt C 2 ; z gt C3. 24 4 8 ɉɨɫɬɨɹɧɧɵɟ ɋ1, ɋ2, ɋ3 ɪɚɜɧɵ ɧɭɥɸ, ɬɚɤ ɤɚɤ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ x y z 0. 39
ɂɧɬɟɝɪɢɪɭɹ ɟɳɟ ɪɚɡ ɢ ɨɬɛɪɚɫɵɜɚɹ ɧɟɫɭɳɟɫɬɜɟɧɧɵɟ ɩɨɫɬɨɹɧɧɵɟ, ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ: 3 2 1 2 3 2 x gt ; y gt ; z gt . 16 8 48 Ɂɚɞɚɱɚ ʋ 2. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɈȺ ɩɥɚɧɟɬɚɪɧɨɣ ɩɟɪɟɞɚɱɢ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɜ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɂɭɛɱɚɬɨɟ ɤɨɥɟɫɨ 2 ɧɚɯɨɞɢɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɟɦ ɡɚɰɟɩɥɟɧɢɢ ɫ ɧɟɩɨɞɜɢɠɧɵɦ ɡɭɛɱɚɬɵɦ ɤɨɥɟɫɨɦ 1. Ʉɨɥɟɫɨ 2 ɩɪɢɜɨɞɢɬɫɹ ɜ ɞɜɢɠɟɧɢɟ ɩɨɫɪɟɞɫɬɜɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ, ɤ ɤɨɬɨɪɨɦɭ ɩɪɢɥɨɠɟɧ ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ m0. ȼɟɫ ɤɪɢɜɨɲɢɩɚ ɈȺ ɪɚɜɟɧ Ɋ, Ɋ2 – ɜɟɫ ɤɨɥɟɫɚ 2, r2 – ɪɚɞɢɭɫ ɤɨɥɟɫɚ 2, r1 – ɪɚɞɢɭɫ ɧɟɩɨɞɜɢɠɧɨɝɨ ɤɨɥɟɫɚ 1. Ʉɨɥɟɫɨ 2 ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ ɤɪɭɝɥɵɦ ɞɢɫɤɨɦ, ɚ ɤɪɢɜɨɲɢɩ ɈȺ – ɬɨɧɤɢɦ ɨɞɧɨɪɨɞɧɵɦ ɫɬɟɪɠɧɟɦ. ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɟɧɟɛɪɟɱɶ. Ɋɟɲɟɧɢɟ. ɍɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ M ɢɦɟɟɬ ɜɢɞ d wT wT QM . dt wM wM
ɉɥɚɧɟɬɚɪɧɚɹ ɩɟɪɟɞɚɱɚ ɢɦɟɟɬ ɨɞɧɭ ɫɬɟɩɟɧɶ ɫɜɨɛɨɞɵ, ɬɚɤ ɤɚɤ ɭɝɨɥ ɩɨɜɨɪɨɬɚ M ɤɪɢɜɨɲɢɩɚ ɈȺ ɨɩɪɟɞɟɥɹɟɬ ɩɨɥɨɠɟɧɢɟ ɜɫɟɯ ɬɨɱɟɤ ɦɟɯɚɧɢɡɦɚ. ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ ɜɵɛɢɪɚɟɦ ɭɝɨɥ M ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɨɬ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɨɫɢ ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ.
40
Ⱥɤɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɹɜɥɹɸɬɫɹ: Ɋ – ɜɟɫ ɤɪɢɜɨɲɢɩɚ, Ɋ2 – ɜɟɫ ɤɨɥɟɫɚ 2, m0 – ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ, ɩɪɢɥɨɠɟɧɧɵɣ ɤ ɤɪɢɜɨɲɢɩɭ ɈȺ. ȼɫɟ ɫɜɹɡɢ, ɧɚɥɨɠɟɧɧɵɟ ɧɚ ɫɢɫɬɟɦɭ, ɢɞɟɚɥɶɧɵ. Ⱦɚɞɢɦ ɤɪɢɜɨɲɢɩɭ ɈȺ ɜɨɡɦɨɠɧɨɟ ɭɝɥɨɜɨɟ ɩɟɪɟɦɟɳɟɧɢɟ GM ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɜɨɡɪɚɫɬɚɧɢɹ ɭɝɥɚ M , ɬ. ɟ. ɩɪɨɬɢɜ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɢ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɵ QM ɜɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɚɛɨɬ ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ GM : GA m0GM P OC cos M GM P2 OA cos M GM . Ɍɚɤ ɤɚɤ OA=OP-AP=r1-r2, ɚ OC
OA 2
r1 r2 , ɬɨ 2
1 2m0 P 2 P2 r1 r2 cosM GM . (1) 2 ɍɱɢɬɵɜɚɹ, ɱɬɨ GA QM GM , ɧɚɯɨɞɢɦ ɨɛɨɛɳɟɧɧɭɸ ɫɢɥɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ 1 2m0 P 2 P2 r1 r2 cosM . (2) ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ M : QM 2 ɉɟɪɟɯɨɞɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɦɟɯɚɧɢɡɦɚ, ɜ ɫɨɫɬɚɜ ɤɨɬɨɪɨɝɨ ɜɯɨɞɹɬ ɦɚɫɫɵ ɤɪɢɜɨɲɢɩɚ ɈȺ ɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 (ɡɭɛɱɚɬɨɟ (3) ɤɨɥɟɫɨ 1 ɧɟɩɨɞɜɢɠɧɨ), ɬ. ɟ. T T (1) T ( 2 ) . Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɤɪɢɜɨɲɢɩɚ ɈȺ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ Ɉ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɤ ɩɥɨɫɤɨɫɬɢ ɪɢɫɭɧɤɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ 1P 1P 1 IO OA 2 T (1) I M 2 , ɝɞɟ r r 2 – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ 3g 3g 1 2 2 O 1P (1) (r r ) 2 M 2 . ɤɪɢɜɨɲɢɩɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, T (4) 6g 1 2 Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ 1 P2 2 1 (2) v I Z2 . (5) ɞɜɢɠɟɧɢɟ, ɪɚɜɧɚ T 2 g A 2 A 2 ɇɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ Ⱥ, ɹɜɥɹɸɳɟɣɫɹ ɤɨɧɰɨɦ ɤɪɢɜɨɲɢɩɚ ɈȺ: v A OA M (r1 r2 )M . (6) Ɋɚɫɫɦɨɬɪɢɦ ɫɤɨɪɨɫɬɶ ɬɨɣ ɠɟ ɬɨɱɤɢ Ⱥ, ɩɪɢɧɚɞɥɟɠɚɳɟɣ ɡɭɛɱɚɬɨɦɭ ɤɨɥɟɫɭ 2, ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɝɧɨɜɟɧɧɨɦɭ ɰɟɧɬɪɭ ɫɤɨɪɨɫɬɟɣ P ɤɨɥɟɫɚ: v A r2Z2 . (7) r1 r2 M . ɋɨɩɨɫɬɚɜɥɹɹ ɮɨɪɦɭɥɵ (6) ɢ (7), ɧɚɯɨɞɢɦ: Z 2 (8) r2 Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɡɭɛɱɚɬɨɝɨ ɤɨɥɟɫɚ 2 ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ P2 r22 IA . (9) 2g
GA
>
@
>
41
@
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɡɧɚɱɟɧɢɣ v A , Z 2 ɢ I A ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɢɡ ɮɨɪɦɭɥ (6), 3 P2 (2) (r r ) 2 M 2 . (8) ɢ (9) ɜɵɪɚɠɟɧɢɟ (5) ɩɪɢɦɟɬ ɜɢɞ T (10) 4 g 1 2 ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (3), (4) ɢ (10), ɡɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɥɚɧɟɬɚɪɧɨɝɨ ɦɟɯɚɧɢɡɦɚ: 2 P 9 P2 T (r1 r2 ) 2 M 2 . (11) 12 g ȼɵɱɢɫɥɢɦ ɱɚɫɬɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɩɨ ɨɛɨɛɳɟɧɧɨɣ w T 2 P 9 P2 (r1 r2 ) 2 M ɢ ɜɨɡɶɦɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɫɤɨɪɨɫɬɢ M : 6g wM d w T 2 P 9 P2 (r1 r2 ) 2 M . (12) ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɡɭɥɶɬɚɬɚ ɩɨ ɜɪɟɦɟɧɢ: dt wM 12 g Ɂɚɦɟɬɢɜ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ Ɍ ɫɢɫɬɟɦɵ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɨɪɦɭɥɨɣ (11), ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ M , ɧɚɯɨɞɢɦ: wT 0. (13)
wM
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜɵɪɚɠɟɧɢɣ (2), (12), (13) ɜ ɭɪɚɜɧɟɧɢɟ Ʌɚɝɪɚɧɠɚ ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɦɟɯɚɧɢɡɦɚ ɞɥɹ ɨɛɨɛɳɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ M : 2 P 9 P2 1 2 r1 r2 M 2m0 P 2 P2 r1 r2 cosM , 6g 2 ɨɬɤɭɞɚ ɨɩɪɟɞɟɥɹɟɦ ɢɫɤɨɦɨɟ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ M ɤɪɢɜɨɲɢɩɚ ɈȺ:
M 3g
>
@
2m0 P 2 P2 r1 r2 cos M
. (14) 2 P 9 P2 Ɋɚɜɧɨɦɟɪɧɨɟ ɜɪɚɳɟɧɢɟ ɤɪɢɜɨɲɢɩɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ 1 m0 P 2 P2 r1 r2 cosM . ɭɫɥɨɜɢɹ: 2 Ɂɚɞɚɱɚ ʋ 3. Ʉ ɤɨɧɰɚɦ ɬɨɧɤɨɣ ɧɟɪɚɫɬɹɠɢɦɨɣ ɧɢɬɢ ɩɪɢɜɹɡɚɧɵ ɝɪɭɡ Ⱥ ɜɟɫɨɦ Ɋ1 ɢ ɝɪɭɡ ȼ ɜɟɫɨɦ Ɋ2. ɇɢɬɶ ɩɟɪɟɛɪɨɲɟɧɚ ɱɟɪɟɡ ɛɥɨɤɢ D ɢ ȿ ɢ ɨɯɜɚɬɵɜɚɟɬ ɫɧɢɡɭ ɩɨɞɜɢɠɧɨɣ ɛɥɨɤ Ʉ. Ʉ ɨɫɢ Ɉ5 ɩɨɞɜɢɠɧɨɝɨ ɛɥɨɤɚ Ʉ ɩɪɢɤɪɟɩɥɟɧ ɝɪɭɡ L ɜɟɫɨɦ Ɋ6; Ɋ3 – ɜɟɫ ɛɥɨɤɚ D, Ɋ4 – ɜɟɫ ɛɥɨɤɚ ȿ, Ɋ5 – ɜɟɫ ɛɥɨɤɚ Ʉ. Ƚɪɭɡɵ Ⱥ ɢ ȼ ɞɜɢɠɭɬɫɹ ɩɨ ɧɚɤɥɨɧɧɵɦ ɩɥɨɫɤɨɫɬɹɦ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɦ ɩɨɞ ɭɝɥɚɦɢ D ɢ E ɤ ɝɨɪɢɡɨɧɬɭ. Ɉɩɪɟɞɟɥɢɬɶ ɭɫɤɨɪɟɧɢɹ ɝɪɭɡɨɜ Ⱥ, ȼ ɢ L. Ȼɥɨɤɢ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦɢ ɤɪɭɝɥɵɦɢ ɞɢɫɤɚɦɢ. ɋɢɥɚɦɢ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɝɪɭɡɨɜ ɨ ɧɚɤɥɨɧɧɵɟ ɩɥɨɫɤɨɫɬɢ ɢ ɦɚɫɫɨɣ ɧɢɬɢ ɩɪɟɧɟɛɪɟɱɶ. 42
Ɋɟɲɟɧɢɟ. ɋɢɫɬɟɦɚ ɢɦɟɟɬ ɞɜɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ. ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɜɵɛɟɪɟɦ ɥɢɧɟɣɧɵɟ ɤɨɨɪɞɢɧɚɬɵ s1 ɢ s2, ɧɚɩɪɚɜɥɟɧɧɵɟ ɜɞɨɥɶ ɧɚɤɥɨɧɧɵɯ ɩɥɨɫɤɨɫɬɟɣ ɜɧɢɡ. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2: d wT wT QS 1 , dt w s1 w s1 d wT wT QS 2 . (1) dt w s2 w s2 Ⱥɤɬɢɜɧɵɦɢ ɫɢɥɚɦɢ ɹɜɥɹɸɬɫɹ: Ɋ1 – ɜɟɫ ɝɪɭɡɚ Ⱥ, Ɋ2 – ɜɟɫ ɝɪɭɡɚ ȼ, Ɋ3 – ɜɟɫ ɛɥɨɤɚ D, Ɋ4 – ɜɟɫ ɛɥɨɤɚ ȿ, Ɋ5 – ɜɟɫ ɛɥɨɤɚ Ʉ, Ɋ6 – ɜɟɫ ɝɪɭɡɚ L. Ɋɟɚɤɰɢɢ ɫɜɹɡɟɣ ɭɱɢɬɵɜɚɬɶ ɧɟ ɫɥɟɞɭɟɬ, ɬɚɤ ɤɚɤ ɜɫɟ ɫɜɹɡɢ, ɧɚɥɨɠɟɧɧɵɟ ɧɚ ɫɢɫɬɟɦɭ, ɢɞɟɚɥɶɧɵ (ɧɚɤɥɨɧɧɵɟ ɩɥɨɫɤɨɫɬɢ ɢɞɟɚɥɶɧɨ ɝɥɚɞɤɢɟ, ɬɪɟɧɢɟ ɜ ɨɫɹɯ ɛɥɨɤɨɜ ɨɬɫɭɬɫɬɜɭɟɬ, ɧɢɬɢ ɩɪɟɞɩɨɥɚɝɚɸɬɫɹ ɧɟɪɚɫɬɹɠɢɦɵɦɢ ɢ ɧɚɬɹɧɭɬɵɦɢ). Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɛɨɛɳɟɧɧɵɯ ɫɢɥ QS1 ɢ QS2 ɞɚɞɢɦ ɝɪɭɡɚɦ Ⱥ ɢ ȼ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɜɨɡɦɨɠɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ Gs1 ɢ Gs2 , ɧɚɩɪɚɜɥɟɧɧɵɟ ɩɚɪɚɥɥɟɥɶɧɨ ɥɢɧɢɹɦ ɧɚɢɛɨɥɶɲɟɝɨ ɫɤɚɬɚ ɧɚɤɥɨɧɧɵɯ ɩɥɨɫɤɨɫɬɟɣ ɜ ɫɬɨɪɨɧɭ ɜɨɡɪɚɫɬɚɧɢɹ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2. 43
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɵ QS1 ɞɚɞɢɦ ɫɢɫɬɟɦɟ ɨɛɨɛɳɟɧɧɨɟ ɜɨɡɦɨɠɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ Gs1 , ɫɱɢɬɚɹ ɩɪɢ ɷɬɨɦ Gs2 ɪɚɜɧɵɦ ɧɭɥɸ, ɬ. ɟ. Gs1 z 0; Gs2 0 . (ɗɬɨ ɨɫɭɳɟɫɬɜɢɦɨ, ɬɚɤ ɤɚɤ s1 ɢ s2 ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɨɛɨɛɳɟɧɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ.) Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɝɪɭɡ ȼ, ɛɥɨɤ ȿ ɢ ɩɪɚɜɚɹ ɜɟɬɜɶ ɧɢɬɢ ɨɬ ɝɪɭɡɚ ȼ ɞɨ ɬɨɱɤɢ N ɧɚɯɨɞɹɬɫɹ ɜ ɩɨɤɨɟ. ɉɪɢ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɝɪɭɡɚ Ⱥ ɜɧɢɡ ɧɚ Gs1 , ɜɜɢɞɭ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ ɧɢɬɢ, ɬɨɱɤɚ Ɇ ɧɢɬɢ ɩɨɥɭɱɢɬ ɜɨɡɦɨɠɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ GrM ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɜɟɪɯ, ɪɚɜɧɨɟ ɩɨ ɦɨɞɭɥɸ Gs1 . ɍɱɢɬɵɜɚɹ, ɱɬɨ ɬɨɱɤɚ N ɧɢɬɢ ɨɫɬɚɧɟɬɫɹ ɩɪɢ ɷɬɨɦ ɜ ɩɨɤɨɟ, ɨɩɪɟɞɟɥɢɦ ɜɨɡɦɨɠɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ ɨɫɢ ɛɥɨɤɚ GrO5 (ɪɢɫ. ɛ), ɪɚɜɧɨɟ ɩɨ ɦɨɞɭɥɸ ɩɨɥɨɜɢɧɟ ɦɨɞɭɥɹ ɜɨɡɦɨɠɧɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ GrM , ɬ. ɟ. G rM G s1 G rO 5 . (2) 2 2 ȼɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɚɛɨɬ ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɜɨɡɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɹɯ ɬɨɱɟɤ ɫɢɫɬɟɦɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɛɨɛɳɟɧɧɨɦɭ ɜɨɡɦɨɠɧɨɦɭ ɩɟɪɟɦɟɳɟɧɢɸ Gs1 ɝɪɭɡɚ Ⱥ:
GA P1 sin DG s1 ( P5 P6 )G rO 5 . ɉɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ ɮɨɪɦɭɥɭ (2), ɧɚɯɨɞɢɦ
GA
1 ª º «¬ P1 sin D 2 ( P5 P6 )»¼G s1 .
(3)
Ɋɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɋ2 ɪɚɜɧɚ ɧɭɥɸ, ɬɚɤ ɤɚɤ Gs2 0 , ɪɚɛɨɬɚ ɫɢɥ ɬɹɠɟɫɬɢ Ɋ3 ɢ Ɋ4 ɪɚɜɧɚ ɧɭɥɸ, ɬɚɤ ɤɚɤ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɷɬɢɯ ɫɢɥ ɧɟɩɨɞɜɢɠɧɵ. Ɉɛɨɛɳɟɧɧɨɣ ɫɢɥɨɣ QS1 ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ, ɫɬɨɹɳɢɣ ɩɪɢ ɨɛɨɛɳɟɧɧɨɦ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɜ ɭɪɚɜɧɟɧɢɢ (3), ɬ. ɟ.
QS1
1 P1 sin D ( P5 P6 ) . 2
(4)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɨɛɨɛɳɟɧɧɨɣ ɫɢɥɵ QS2 ɞɚɞɢɦ ɫɢɫɬɟɦɟ ɨɛɨɛɳɟɧɧɨɟ ɜɨɡɦɨɠɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ Gs2 , ɫɱɢɬɚɹ ɩɪɢ ɷɬɨɦ Gs1 ɪɚɜɧɵɦ ɧɭɥɸ: Gs2 z 0; Gs1 0 . ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɝɪɭɡ Ⱥ, ɛɥɨɤ D ɢ ɥɟɜɚɹ ɜɟɬɜɶ ɧɢɬɢ ɨɬ ɝɪɭɡɚ Ⱥ ɞɨ ɬɨɱɤɢ Ɇ ɧɢɬɢ ɧɚɯɨɞɹɬɫɹ ɜ ɩɨɤɨɟ. ɉɪɢ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɝɪɭɡɚ ȼ ɧɚ Gs2 ɜɧɢɡ, ɜɜɢɞɭ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ ɧɢɬɢ, ɬɨɱɤɚ N ɧɢɬɢ ɩɨɥɭɱɢɬ ɜɨɡɦɨɠɧɨɟ ɩɟɪɟɦɟɳɟɧɢɟ G rN ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɜɟɪɯ, ɪɚɜɧɨɟ ɜɟɥɢɱɢɧɟ ɦɨɞɭɥɸ ɜɨɡɦɨɠɧɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ G s 2 . ɍɱɢɬɵɜɚɹ, ɱɬɨ ɬɨɱɤɚ Ɇ ɧɢɬɢ ɨɫɬɚɟɬɫɹ ɩɪɢ ɷɬɨɦ ɜ ɩɨɤɨɟ, ɧɚɯɨɞɢɦ (ɪɢɫ. ɜ) 44
G rO 5
G rN
G s2
2
2
.
(5)
ȼɵɱɢɫɥɢɦ ɫɭɦɦɭ ɪɚɛɨɬ ɚɤɬɢɜɧɵɯ ɫɢɥ ɧɚ ɜɨɡɦɨɠɧɵɯ ɩɟɪɟɦɟɳɟɧɢɹɯ ɬɨɱɟɤ ɫɢɫɬɟɦɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɛɨɛɳɟɧɧɨɦɭ ɜɨɡɦɨɠɧɨɦɭ ɩɟɪɟɦɟɳɟɧɢɸ Gs2 :
GA P2 sin EG s2 ( P5 P6 )G rO 5 . ɍɱɢɬɵɜɚɹ ɮɨɪɦɭɥɭ (5), ɢɦɟɟɦ:
GA
1 ª º «¬ P2 sin E 2 ( P5 P6 )»¼G s2 .
(6)
Ɋɚɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɋ1 ɪɚɜɧɚ ɧɭɥɸ, ɬɚɤ ɤɚɤ Gs1 0 , ɪɚɛɨɬɚ ɫɢɥ ɬɹɠɟɫɬɢ Ɋ3 ɢ Ɋ4 ɪɚɜɧɚ ɧɭɥɸ, ɬɚɤ ɤɚɤ ɬɨɱɤɢ ɩɪɢɥɨɠɟɧɢɹ ɷɬɢɯ ɫɢɥ ɧɟɩɨɞɜɢɠɧɵ. Ɉɛɨɛɳɟɧɧɨɣ ɫɢɥɨɣ QS2 ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ, ɫɬɨɹɳɢɣ ɩɪɢ ɨɛɨɛɳɟɧɧɨɦ ɜɨɡɦɨɠɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɜ ɭɪɚɜɧɟɧɢɢ (6), ɬ. ɟ.
QS 2
1 P2 sin E ( P5 P6 ) . 2
(7)
ɉɟɪɟɯɨɞɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɲɟɫɬɢ ɦɚɫɫ: ɝɪɭɡɨɜ Ⱥ, ȼ ɢ L ɛɥɨɤɨɜ D, ȿ ɢ Ʉ:
T
T (1) T ( 2 ) T ( 3) T ( 4 ) T (5) T ( 6) .
(8)
Ƚɪɭɡɵ Ⱥ ɢ ȼ ɢɦɟɸɬ ɫɤɨɪɨɫɬɢ v A ɢ v B , ɧɚɩɪɚɜɥɟɧɧɵɟ ɩɚɪɚɥɥɟɥɶɧɨ ɥɢɧɢɹɦ ɧɚɢɛɨɥɶɲɟɝɨ ɫɤɚɬɚ ɧɚɤɥɨɧɧɵɯ ɩɥɨɫɤɨɫɬɟɣ. ɉɪɨɟɤɰɢɢ ɷɬɢɯ ɫɤɨɪɨɫɬɟɣ ɧɚ ɨɫɢ s1 ɢ s2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ s1 ɢ s2 . Ɉɛɨɡɧɚɱɢɦ ɪɚɞɢɭɫɵ ɛɥɨɤɨɜ D, ȿ ɢ Ʉ ɱɟɪɟɡ r3, r4 ɢ r5. ɉɪɢ ɷɬɨɦ ɭɝɥɨɜɵɟ ɫɤɨɪɨɫɬɢ ɛɥɨɤɨɜ D ɢ ȿ ɜɵɪɚɡɹɬɫɹ ɬɚɤ:
M3
s1 r3 ,
M4
s2 r4 .
(9)
ȼɜɢɞɭ ɧɟɪɚɫɬɹɠɢɦɨɫɬɢ ɧɢɬɢ ɫɤɨɪɨɫɬɶ v M ɬɨɱɤɢ Ɇ ɧɢɬɢ ɪɚɜɧɚ ɩɨ ɜɟɥɢɱɢɧɟ ɫɤɨɪɨɫɬɢ v A ɝɪɭɡɚ Ⱥ, ɬ.ɟ. v Mx s1 . Ⱥɧɚɥɨɝɢɱɧɨ v Nx s2 . ɇɟɬɪɭɞɧɨ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɪɢɫ. ɝ), ɧɚɣɬɢ ɫɤɨɪɨɫɬɶ ɨɫɢ Ɉ5 ɛɥɨɤɚ Ʉ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ:
vO 5 x
vMx v Nx 2
s1 s2 . 2
(10)
45
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ M5 ɛɥɨɤɚ Ʉ ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɬɨɱɤɢ v Mx v Nx (v MN ) x , ɬ. ɟ. Ɇ, ɩɪɢɧɹɜ ɡɚ ɩɨɥɸɫ ɬɨɱɤɭ N: (v MN ) x v Mx v Nx s1 s2 .
MN M5 2r5M5 , ɬɨ 2r5M5 s1 s2 2r5 .
Ɍɚɤ ɤɚɤ (v MN ) x ɫɤɨɪɨɫɬɶ ɛɥɨɤɚ Ʉ: M5
s1 s2 , ɨɬɤɭɞɚ ɭɝɥɨɜɚɹ (11)
Ʉɢɧɟɬɢɱɟɫɤɢɟ ɷɧɟɪɝɢɢ ɝɪɭɡɨɜ Ⱥ ɢ ȼ, ɫɨɜɟɪɲɚɸɳɢɯ ɩɨɫɬɭɩɚɬɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ, ɢɦɟɸɬ ɜɢɞ
T (1)
1 P1 2 ( 2 ) s , T 2 g 1
1 P2 2 s . 2 g 2
(12)
ȼɵɱɢɫɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɨɜ D ɢ ȿ, ɜɪɚɳɚɸɳɢɯɫɹ ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɵɯ ɨɫɟɣ:
T ( 3)
§ ¨¨ I O 5 ©
1 I O 5M 32 , T ( 3) 2
ɉɨɞɫɬɚɜɢɜ P3 r32 I O4 , 2g
T ( 3)
1 I O 4M 42 . 2
ɡɧɚɱɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɛɥɨɤɨɜ P4 r42 · ¸ ɢ ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (9), ɧɚɯɨɞɢɦ 2 g ¸¹
P3 s12 (4) , T 4g
P4 s22 . 4g
(13)
Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɛɥɨɤɚ Ʉ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɩɥɨɫɤɨɟ ɞɜɢɠɟɧɢɟ, ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
T (5)
1 P5 2 1 vO 5 I O 5M52 . 2 g 2
§ ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɧɟɪɰɢɢ ɛɥɨɤɚ Ʉ ¨¨ I O 5 ©
P5 r52 · ¸ 2 g ¸¹ ɢ
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɚɦɢ (10) ɢ (11), ɩɨɥɭɱɢɦ:
T ( 5)
3 P5 2 1 P5 s s . s1 s22 16 g 8 g 1 2
(14)
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ L, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɜɢɞ
T ( 6)
1 P6 2 v . 2 g O5
ɉɪɢɦɟɧɢɜ ɮɨɪɦɭɥɭ (10), ɨɩɪɟɞɟɥɹɟɦ 46
T ( 6)
1 P6 2 s1 s2 . 8 g
(15)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɩɨɞɫɬɚɜɥɹɟɦ ɜ ɮɨɪɦɭɥɭ (8) ɜɵɪɚɠɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɢɡ ɮɨɪɦɭɥ (12)–(15). ȼ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ:
1 P1 2 1 P2 2 1 P3 2 1 P4 2 s1 s2 s1 s2 2 g 2 g 4 g 4 g 3 P5 2 2 1 P5 1P ( s1 s2 ) s1s2 6 ( s1 s2 ) 2 , 16 g 8 g 8 g
T
ɬ. ɟ.
T
8 P1 4 P3 3 P5 2 P6 2 8 P2 4 P4 3 P5 2 P6 2 P5 2 P6 s1 s2 s1 s2 . (16) 16 g 16 g 8g
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫɥɟɞɭɟɬ ɜɵɱɢɫɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ Ɍ ɩɨ ɨɛɨɛɳɟɧɧɵɦ ɫɤɨɪɨɫɬɹɦ s1 ɢ s2
wT w s1 wT w s2
P 2 P6 8P1 4 P3 3P5 2 P6 s1 5 s2 , 8g 8g 8 P2 4 P4 3P5 2 P6 P 2 P6 s2 5 s1 8g 8g
ɢ ɜɡɹɬɶ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɩɨ ɜɪɟɦɟɧɢ
d wT dt w s1 d wT dt w s2
P 2 P6 ½ 8 P1 4 P3 3P5 2 P6 s1 5 s2 ° ° 8g 8g P5 2 P6 ¾ . 8P2 4 P4 3P5 2 P6 s2 s1 ° °¿ 8g 8g
(17)
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ Ɍ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɮɨɪɦɭɥɨɣ (16) ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2 , ɢɦɟɟɦ
wT w s1
0,
wT w s2
0.
(18)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɮɨɪɦɭɥ (4), (7), (17) ɢ (18) ɜ ɭɪɚɜɧɟɧɢɟ (1) ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ Ʌɚɝɪɚɧɠɚ ɜɬɨɪɨɝɨ ɪɨɞɚ ɞɥɹ ɨɛɨɛɳɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ s1 ɢ s2 :
47
8 P1 4 P3 3 P5 2 P6 P 2 P6 s1 5 s2 8g 8g 8 P2 4 P4 3 P5 2 P6 P 2 P6 s2 5 s1 8g 8g
1 P P6 , 2 5 1 P2 sin E P5 P6 . 2
P1 sin D
Ɋɟɲɚɹ ɷɬɭ ɫɢɫɬɟɦɭ, ɩɨɥɭɱɢɦ:
s1
8g
D(C B) C P2 sin E B P1 sin D , AB C 2
s2
8g
D(C A) A P2 sin E C P1 sin D , AB C 2
wO 5
4g
D(2C A B) ( A C ) P2 sin E ( B C ) P1 sin D , AB C 2
ɝɞɟ
8 P1 4 P3 3 P5 2 P6
A,
8 P2 4 P4 3 P5 2 P6
B,
P5 2 P6
C ɢ
P5 P6 2
D.
§12. ɋɩɢɫɨɤ ɡɚɞɚɱ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ [2]
26.1, 26.5, 26.6, 26.9. 27.1, 27.3, 27.12, 27.13, 27.34, 27.42, 27.53, 27.56. 28.1, 28.6, 28.7, 28.15, 28.17, 28.21. 29.7, 29.14. 30.4, 30.6, 30.11, 30.12, 30.15, 30.19, 30.22. 32.1, 32.2, 32.5, 32.16, 32.24, 32.26, 32.37, 32.53, 32.55, 32.57, 32.68, 32.70, 32.78, 32.79, 32.86, 32.93, 32.94, 32.96, 32.98, 32.99. 34.2, 34.3, 34.9, 34.11, 34.12, 34.15, 34.21. 35.3, 35.4, 35.10, 35.13, 35.14, 35.19. 48
36.3, 36.6, 36.9. 37.1, 37.3, 37.9, 37.14, 37.34, 37.39, 37.43, 37.46, 37.53, 37.56. 38.2, 38.4, 38.9, 38.20, 38.24, 38.27, 38.30, 38.40, 38.42, 38.44, 38.46, 38.47, 38.50. 48.5, 48.6, 48.12, 48.13, 48.28, 48.29, 48.30, 48.35, 48.44. §13. Ɉɫɧɨɜɧɵɟ ɮɨɪɦɭɥɵ ɞɢɧɚɦɢɤɢ
ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ:
mw F , ɝɞɟ m – ɢɧɟɪɬɧɚɹ ɦɚɫɫɚ ɬɨɱɤɢ, w – ɭɫɤɨɪɟɧɢɟ, F – ɫɢɥɚ. mx ° ® my ° ¯ mz
Fx (t , x , y , z , x , y , z ) Fy (t , x , y , z , x , y , z ) . Fz (t , x , y , z , x , y , z )
2 § d 2r dr dM dz · § dM · · § , ¸ °m¨ 2 r ¨ ¸ ¸ Fr ¨ t , r , M, z , , ¨ ¸ dt dt dt ¹ © dt ¹ ¹ © ° © dt ° dr dM · dr dM dz · ° § d 2M § ¸¸ F M¨ t , r , M, z , , , ¸ ®m¨¨ r 2 2 dt dt dt dt dt ¹ . dt © ¹ ° © ° § d 2z · dr dM dz · § ° m¨ ¸ F z ¨ t , r , M, z , , , ¸ dt dt dt ¹ ° ¨© dt 2 ¸¹ © ¯
d 2S dS · § FW ¨ t , S , ¸ °m 2 dt ¹ © ° dt ° dS 1 dS · § FW ¨ t , S , ¸ ®m dt dt ¹ . U © ° ° dS · § ° 0 Fn ¨ t , S , ¸ dt ¹ © ¯ 49
N
¦m r
2 k k
Jz
.
k 1
³r
Jz
2
³ U ( x, y , z ) r
dm
(M )
2
dv
(V )
J Z1
J Zc Md 2 .
Jz
MR 2 .
M V
³r
2
dv .
(V )
n
Mwc = ¦ Fke . k 1
n
W
d Q dt
¦F
dLZ dt
¦ mom
(e) i
;
Q1 Q0
i 1
³ F (t )dt . 0
n
Z
( Fke ) .
k 1
T1 T0
¦A
10
(F ) .
mk v k2 . 2 k 1 Mv C2 T , ɝɞɟ M – ɦɚɫɫɚ ɬɟɥɚ; vC – ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ. 2 J ZZ 2 T , ɝɞɟ J Z – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, 2 Z – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ. Mv C2 J C Z 2 T . 2 2 GA F dr Fds cos( F , W ) Fx dx Fy dy Fz dz . n
T
¦
A
³ Fdr ³ FW ds ³ ( F dx F dy F dz) . x
L
L
y
z
L
A Fs cos( F , s ) . A12 ( P ) P( zC1 zC2 ) . 50
Ɋɚɛɨɬɚ ɫɢɥɵ Fx
GA
>R v
GA
M Z dM ;
O
cx ; A12
@
c 2 x x22 . 2 1
M OZ dt ; M2
A12
³M
Z
dM .
M1
ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ. GA N F v FW v Fx x Fy y Fz z . dt N R vO M O Z . N M ZZ . wFx wFy wFx wFz wFy wFz , , – ɭɫɥɨɜɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɫɬɢ ɩɨɥɹ. wy wx wy wz wz wx GA dɉ ; A12=ɉ1–ɉ2 , ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ; ɉ r Ph , ɝɞɟ h – ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, c'2 ɉ . 2 ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ. T2 – T1 = A12. ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ ɦɨɳɧɨɫɬɟɣ dT N. ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ. dt ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const.
d §¨ wT dt ¨© wq j
· wT ¸ Q j (j = 1, 2, ... , s), ¸ wq j ¹ ɝɞɟ q1, q2,..., qS – ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ; s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ; q1 , q 2 , …, q S – ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ; Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ; Q1, Q2, ..., QS – ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ. d §¨ wL ·¸ wL 0 , ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ. dt ¨© wq j ¸¹ wq j Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ: 51
x ɭɫɬɚɧɨɜɢɬɶ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢ ɜɵɛɪɚɬɶ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ; x ɩɪɟɞɩɨɥɨɠɢɜ, ɱɬɨ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɬɚɤ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɭɜɟɥɢɱɢɜɚɸɬɫɹ, ɫɨɫɬɚɜɢɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ, ɩɪɢ ɷɬɨɦ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ ɜɟɥɢɱɢɧɵ, ɜɯɨɞɹɳɢɟ ɜ Ɍ, ɞɨɥɠɧɵ ɛɵɬɶ ɜɵɪɚɠɟɧɵ ɱɟɪɟɡ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢ ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ, ɬ. ɟ. T T ( q1 , q2 ,..., q S ; q1 , q 2 ,..., q S ; t ) (ɜ ɫɥɭɱɚɟ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɜɹɡɟɣ ɜɪɟɦɹ t ɧɟ ɜɯɨɞɢɬ ɜ ɜɵɪɚɠɟɧɢɟ T); wT x ɨɩɪɟɞɟɥɢɬɶ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ; ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɢɡɜɨɞɧɵɟ wq j
d §¨ wT dt ¨© wq j
· ¸ , ɫɱɢɬɚɹ ɜɫɟ ɩɟɪɟɦɟɧɧɵɟ, ɜɯɨɞɹɳɢɟ ɜ wT , ɮɭɧɤɰɢɹɦɢ ¸ wq j ¹ ɜɪɟɦɟɧɢ t; x ɧɚɣɬɢ ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ; ɩɨɞɫɬɚɜɢɬɶ ɜɫɟ ɧɚɣɞɟɧɧɵɟ ɜɟɥɢɱɢɧɵ ɜ ɢɫɯɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ.
52
ɋɩɢɫɨɤ ɪɟɤɨɦɟɧɞɭɟɦɨɣ ɥɢɬɟɪɚɬɭɪɵ Ɉɫɧɨɜɧɚɹ ɥɢɬɟɪɚɬɭɪɚ
1. Ɍɚɪɝ ɋ.Ɇ. Ʉɪɚɬɤɢɣ ɤɭɪɫ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ : ɭɱɟɛ. ɞɥɹ ɫɬɭɞ. ɜɬɭɡɨɜ / ɋ.Ɇ. Ɍɚɪɝ. – 12-ɟ ɢɡɞ., ɫɬɟɪ. – Ɇ. : ȼɵɫɲ. ɲɤ., 2002. – 416 ɫ. 2. Ɇɟɳɟɪɫɤɢɣ ɂ.ȼ. Ɂɚɞɚɱɢ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞ. ɜɭɡɨɜ, ɨɛɭɱ. ɩɨ ɬɟɯɧ. ɫɩɟɰɢɚɥɶɧɨɫɬɹɦ / ɂ.ȼ. Ɇɟɳɟɪɫɤɢɣ ; ɩɨɞ ɪɟɞ. ȼ.Ⱥ. ɉɚɥɶɦɨɜɚ, Ⱦ.Ɋ. Ɇɟɪɤɢɧɚ. – ɋɉɛ. : Ʌɚɧɶ, 2004. – 447 ɫ. 3. əɛɥɨɧɫɤɢɣ Ⱥ.Ⱥ. Ʉɭɪɫ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞ. ɜɭɡɨɜ, ɨɛɭɱ. ɩɨ ɬɟɯɧ. ɫɩɟɰɢɚɥɶɧɨɫɬɹɦ / Ⱥ.Ⱥ. əɛɥɨɧɫɤɢɣ, ȼ.Ɇ. ɇɢɤɢɮɨɪɨɜɚ. – 8-ɟ ɢɡɞ., ɫɬɟɪ. – ɋɉɛ. : Ʌɚɧɶ, 2001. – 763 ɫ. Ⱦɨɩɨɥɧɢɬɟɥɶɧɚɹ ɥɢɬɟɪɚɬɭɪɚ
4. ɋɛɨɪɧɢɤ ɡɚɞɚɧɢɣ ɞɥɹ ɤɭɪɫɨɜɵɯ ɪɚɛɨɬ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞ. ɜɬɭɡɨɜ / Ⱥ.Ⱥ. əɛɥɨɧɫɤɢɣ [ɢ ɞɪ.]. – Ɇ. : ɂɧɬɟɝɪɚɥ-ɉɪɟɫɫ, 2004. – 382 ɫ. 5. Ȼɚɬɶ Ɇ.ɂ. Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ : ɭɱɟɛ. ɩɨɫɨɛɢɟ ɞɥɹ ɫɬɭɞ. ɜɬɭɡɨɜ : ɜ 3 ɬ. / Ɇ.ɂ. Ȼɚɬɶ, Ƚ.ɘ. Ⱦɠɚɧɟɥɢɞɡɟ, Ⱥ.ɋ. Ʉɟɥɶɡɨɧ. – Ɇ. : ɇɚɭɤɚ, 1990. – Ɍ. 1 : ɋɬɚɬɢɤɚ ɢ ɤɢɧɟɦɚɬɢɤɚ. – 670 ɫ. 6. Ʉɪɚɬɤɢɣ ɫɩɪɚɜɨɱɧɢɤ ɞɥɹ ɢɧɠɟɧɟɪɨɜ ɢ ɫɬɭɞɟɧɬɨɜ. ȼɵɫɲɚɹ ɦɚɬɟɦɚɬɢɤɚ. Ɏɢɡɢɤɚ. Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ / Ⱥ.Ⱦ. ɉɨɥɹɧɢɧ [ɢ ɞɪ.]. – Ɇ. : Ɇɟɠɞɭɧɚɪ. ɩɪɨɝɪ. ɨɛɪɚɡɨɜɚɧɢɹ, 1996. – 431 ɫ. 7. Ȼɭɯɝɨɥɶɰ ɇ.ɇ. Ɉɫɧɨɜɧɨɣ ɤɭɪɫ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ : ɭɱɟɛ. ɞɥɹ ɝɨɫ. ɭɧ-ɬɨɜ / ɇ.ɇ. Ȼɭɯɝɨɥɶɰ; ɜ ɩɟɪɟɪɚɛɨɬɤɟ ɢ ɫ ɞɨɩ. ɋ.Ɇ. Ɍɚɪɝɚ. – Ɇ. : ɇɚɭɤɚ, 1972. – ɑ. 1 : Ʉɢɧɟɦɚɬɢɤɚ, ɫɬɚɬɢɤɚ, ɞɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ. – 467 ɫ.
53
ɍɱɟɛɧɨɟ ɢɡɞɚɧɢɟ
Ɋȿɒȿɇɂȿ ɁȺȾȺɑ ɉɈ ɌȿɈɊȿɌɂɑȿɋɄɈɃ ɆȿɏȺɇɂɄȿ. ɑɚɫɬɶ 3. Ⱦɢɧɚɦɢɤɚ ɍɱɟɛɧɨ-ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɭɡɨɜ
ɋɨɫɬɚɜɢɬɟɥɢ: ɑɟɛɨɬɚɪɟɜ Ⱥɧɞɪɟɣ ɋɟɪɝɟɟɜɢɱ, ɓɟɝɥɨɜɚ ɘɥɢɹ Ⱦɦɢɬɪɢɟɜɧɚ
ɉɨɞɩɢɫɚɧɨ ɜ ɩɟɱɚɬɶ 15.08.07. Ɏɨɪɦɚɬ 60×84/16. ɍɫɥ. ɩɟɱ. ɥ. 3,1. Ɍɢɪɚɠ 100 ɷɤɡ. Ɂɚɤɚɡ 1678. ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɢɣ ɰɟɧɬɪ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɩɥ. ɢɦ. Ʌɟɧɢɧɚ, 10. Ɍɟɥ. 208-298, 598-026 (ɮɚɤɫ) http://www.ppc.vsu.ru; e-mail:
[email protected] Ɉɬɩɟɱɚɬɚɧɨ ɜ ɬɢɩɨɝɪɚɮɢɢ ɂɡɞɚɬɟɥɶɫɤɨ-ɩɨɥɢɝɪɚɮɢɱɟɫɤɨɝɨ ɰɟɧɬɪɚ ȼɨɪɨɧɟɠɫɤɨɝɨ ɝɨɫɭɞɚɪɫɬɜɟɧɧɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ. 394000, ɝ. ȼɨɪɨɧɟɠ, ɭɥ. ɉɭɲɤɢɧɫɤɚɹ, 3. Ɍɟɥ. 204-133.
54
