Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 19 стр.

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19
)sin()cos(
211
ktcktcx
. (2)
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1) ɩɪɢɦɟɦ ɜ ɜɢɞɟ
)cos()sin(
2
ptBptAx
, ɬɨɝɞɚ
)sin(
22
2
pt
pk
h
x
. (3)
Ɂɚɩɢɲɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1), ɫɥɨɠɢɜ (2) ɢ (3):
)sin()sin()cos(
22
21
pt
pk
h
ktcktcx
. (4)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɵɟ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:
22
21
,0
pk
h
k
p
cc
.
ɂɬɚɤ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɟ ɫɬɪɟɥɤɢ
)sin()sin(
2222
pt
pk
h
kt
pk
h
k
p
x
. (5)
ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ
cɦttx ))60sin(25,1)100sin(75,0(
. (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.
xxx
RFSxm
ɢɥɢ
x
P
g
x
P
cg
pt
P
Hg
x
E
sin
, ɨɬɤɭɞɚ
,sin2
2
ptkxkxnx
(7)
ɝɞɟ
P
hg
h
P
g
n
P
cg
k ,
2
,
E
. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k = 100 ɫ
-1
, h = 80 ɫɦ ɫ
-2
,
n = 64 ɫ
-1
, ɪ = 60 ɫ
-1
. ɂɬɚɤ, n < k ɢ p < k.
8,0
4)(
8,76
22222
22
pnpk
h
ank
87,0
2
22
pk
np
arctg
H
.
Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɵɛɢɪɚɹ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ
)sin(
2
H
ptax
,
ɩɨɥɭɱɢɦ
).87,060sin(8,0)8,76sin8,76cos(
21
64
ttctcex
t
(8)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɨɩɪɟɞɟɥɢɦ ɩɨɫɬɨɹɧɧɵɟ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ: ɫ
1
= 0,62; ɫ
2
= 0,12. ɉɟɪɟɩɢɲɟɦ ɮɨɪɦɭɥɭ (9):
).87,060sin(8,0)8,76sin12,08,76cos62,0(
64
tttex
t
(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:
>
@
.)87,060sin(8,0)74,18,76sin(63,0
64
ɫɦttex
t
(10) 
                                x1 c1 cos(kt )  c2 sin(kt ) .                             (2)
    ɑɚɫɬɧɨɟ        ɪɟɲɟɧɢɟ          ɭɪɚɜɧɟɧɢɹ            (1)         ɩɪɢɦɟɦ         ɜ    ɜɢɞɟ
x2 A sin( pt )  B cos( pt ) , ɬɨɝɞɚ
                                          h
                                x2                sin( pt ) .                              (3)
                                      k  p2
                                        2

    Ɂɚɩɢɲɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1), ɫɥɨɠɢɜ (2) ɢ (3):
                                                                  h
                        x c1 cos(kt )  c2 sin( kt )  2               sin( pt ) .         (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
mx S x  Fx  R x ɢɥɢ x          sin pt         x        x , ɨɬɤɭɞɚ
                                  P              P         P
                       x  2nx  k 2 x k sin pt ,                                       (7)
            cg        Eg         hg
ɝɞɟ k          , 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
  k 2  n 2 76,8 a                            0,8 H arctg 2             0,87 .
                            2   2 2
                         ( k  p )  4n p2 2                k   p2
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)

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