Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 16 стр.

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16
x
cxcdPx
g
P
EX
. (2)
ȼ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɤ ɝɪɭɡɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ –
ɟɝɨ ɜɟɫ, ɧɚɩɪɚɜɥɟɧɧɵɣ ɩɨ ɜɟɪɬɢɤɚɥɢ ɜɧɢɡ, ɫɬɚɬɢɱɟɫɤɚɹ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ
F
ɫɬ
=cd, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ. Ɍɚɤ ɤɚɤ ɝɪɭɡ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ, ɬɨ
0 cdP
. (3)
ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ
02
2
xkxnx
, (4)
ɝɞɟ
P
g
n
P
cg
kx
x
2
,,
E
X
, ɩɨɥɭɱɚɟɦ k = 10 c
-1
, n = 8 ɫ
-1
, ɬɚɤɢɦ
ɨɛɪɚɡɨɦ, n < k.
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4):
.,02
2222
2,1
22
nkinknnkn r r
OOO
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ
))sin()cos((
22
2
22
1
tnkctnkcex
nt
. (5)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɩɨɥɭɱɚɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
22
00
201
,
nk
nxx
cxc
.
ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɨɩɪɟɞɟɥɹɹ
DD
cos,sin
22
00
0
A
nk
nxx
Ax
. (6)
Ɍɟɩɟɪɶ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
)sin(
22
D
tnkAex
nt
. (7)
Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɡɚɬɭɯɚɸɳɢɦ (ɬ.ɤ. ɩɪɢ tĺ xĺ0) ɫ ɤɪɭɝɨɜɨɣ
ɱɚɫɬɨɬɨɣ
22
nkk
c
. ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɨɪɦɭɥɵ,
ɧɚɯɨɞɢɦ Ⱥ=7,2 ɫɦ, Į=0,59 ɪɚɞ, k
c
=6 ɫ
-1
.
ɂɬɚɤ, ɝɪɭɡ ɫɨɜɟɪɲɚɟɬ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ
ɫɦtex
t
)59,06sin(2,7
8
. (8)
ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ
ɫ
k
T
c
c
05,1
2
S
.
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 ȣ
0
=240 ɫɦ/ɫ.
Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ ɩɪɭɠɢɧɟ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ
ɜɨɡɶɦɟɦ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
                         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)
                 cg        Eg
ɝɞɟ X x x, k       , n            , ɩɨɥɭɱɚɟɦ k = 10 c-1, n = 8 ɫ-1, ɬɚɤɢɦ
                  P        2P
ɨɛɪɚɡɨɦ, n < k.
    Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4):
           O2  2nO  k 20, 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
                    x0 A sin D ,                        A cos 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
ɱɚɫɬɨɬɨɣ k c      k  n . ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɮɨɪɦɭɥɵ,
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
    ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ Tc            1,05 ɫ .
                                   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

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