Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 20 стр.

UptoLike

ВУЗ: 

ВГУ | Воронеж

Составители: 

Рубрика: 

20
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
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ
ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ Ɇ ɜɟɫɨɦ
Ɋ=196 Ƚ, ɞɜɢɠɭɳɟɣɫɹ ɜɞɨɥɶ ɨɫɢ ɯ
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F
ɢ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ S. ɉɪɨɟɤɰɢɢ
ɷɬɢɯ ɫɢɥ ɧɚ ɨɫɶ ɯ ɪɚɜɧɵ: F
x
= –cx,
S
x
= H·sin(pt), ɝɞɟ ɫ = 2 ɤȽ/ɫɦ,
ɇ = 1,6 ɤȽ, ɪ = 101 ɫ
-1
.
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɬɨɱɤɚ
ɧɚɯɨɞɢɥɚɫɶ ɜ ɩɨɤɨɟ ɜ ɧɚɱɚɥɟ
ɨɬɫɱɟɬɚ ɨɫɢ ɯ. ɋɢɥɨɣ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ
ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ. Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ Ɇ ɜ
ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ
xx
SFxm
ɢɥɢ
ptHcxxm sin
pthxkx sin
2
, (1)
ɝɞɟ
211 2
10000 , 101 , 8000 / .
cH
k ɫ p ɫ h ɫɦ ɫ
mm

.
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
pt
pk
h
kt
pk
h
k
p
x sinsin
2222
. (2)
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
.22,1 pkkp
k
p
|||
(3)
ɂɫɩɨɥɶɡɭɹ ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2),
)sin(sin
))((
ktpt
pkpk
h
x
|
.
ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ:
)sin(sin
)(2
ktpt
pkk
h
x
. (4)
ɉɪɟɨɛɪɚɡɨɜɚɜ ɜɵɪɚɠɟɧɢɟ, ɩɨɥɭɱɢɦ
pttax cos)(
, (5) 
    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
ɝɞɟ k
       2
                 10000 ɫ 1 , p 101 ɫ 1 , h                 8000 ɫɦ / ɫ 2 . .
           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         2      2
                                             sin kt  2        sin pt .        (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

Страницы