Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 13 стр.

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ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
ɉɪɢ
00
0: , 0
P
txx xx
c

,
cDF
, ɝɞɟ F – ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ,
(ɧɚɩɪɚɜɥɟɧɚ ɜɫɟɝɞɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ
ɫɦɟɳɟɧɢɸ); D – ɫɦɟɳɟɧɢɟ ɤɨɧɰɚ
ɩɪɭɠɢɧɵ ɢɡ ɧɟɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ,
ɬ. ɟ.
xdMMD
x
0
.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
)( xdcF
x
. (1)
ɋɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɜ
ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ:
x
FPxm
, (2)
ɢɫɩɨɥɶɡɭɹ (1), ɩɨɥɭɱɢɦ ɢɡ (2):
cxcdPx
g
P
. (3)
Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (3) ɜ ɜɢɞɟ
0
2
xkx
, (4)
ɝɞɟ
P
cg
k
– ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ (ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ).
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4)
0
22
k
O
.
Ʉɨɪɧɢ ɭɪɚɜɧɟɧɢɹ
ki
r
2,1
O
.
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
)sin()cos(
21
ktcktcx
. (5)
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɜɵɱɢɫɥɢɦ ɫɤɨɪɨɫɬɶ
)cos()sin(
21
ktkcktkcx
. (6)
ɉɨɞɫɬɚɜɢɦ (5) ɢ (6) ɜ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
00
0,0
P
txx xx
c

.
ɇɚɯɨɞɢɦ
0,
201
c
c
P
xɫ
.
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɩɪɢɦɟɬ ɜɢɞ
)cos()cos( t
P
cg
c
P
kt
c
P
x
.
ɉɨɞɫɬɚɜɥɹɟɦ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
ɫɦ
c
P
ɫ
P
cg
k 8,6,12
1
.
Ⱥɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ a = 6,8 cɦ;
ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ Į = –ʌ/2;
ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ k = 12 ɫ
-1
.
                             ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
                                                  P
                             ɉɪɢ t 0 :   x x0  , x x0 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 ,                                       (2)
ɢɫɩɨɥɶɡɭɹ (1), ɩɨɥɭɱɢɦ ɢɡ (2):
                       P
                           x P  cd  cx .                              (3)
                        g
Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (3) ɜ ɜɢɞɟ
                      x  k 2 x 0 ,                                     (4)
            cg
ɝɞɟ   k         – ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ (ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ).
            P
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4)             O2  k 2   0.
Ʉɨɪɧɢ ɭɪɚɜɧɟɧɢɹ           O1, 2 r ki .
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
                          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          P cg
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɩɪɢɦɟɬ ɜɢɞ x  cos(kt )  cos(           t) .
                                                  c          c P
                                      cg                  P
ɉɨɞɫɬɚɜɥɹɟɦ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ k             12 ɫ 1 ,         6,8 ɫɦ .
                                       P                  c
  Ⱥɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ       a = 6,8 cɦ;
  ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ Į = –ʌ/2;
  ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ         k = 12 ɫ-1.

                               13

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