Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 17 стр.

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ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɸɬ ɜɢɞ
0
0
4
ɩɪɢ 0.
240 /
xx ɫɦ
t
x
x ɫɦ ɫ
®
¯

ɋɥɟɞɭɹ ɪɟɲɟɧɢɸ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ
02 kxxnx
, ɝɞɟ
.
2
,,
P
g
n
P
cg
kx
x
E
X
ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɚɟɦ k=10 c
-1
, n=26 ɫ
-1
, ɬɚɤɢɦ
ɨɛɪɚɡɨɦ, n>k (ɫɥɭɱɚɣ ɛɨɥɶɲɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ).
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
,02
22
kn
OO
ɟɝɨ ɤɨɪɧɢ
ɪɚɜɧɵ
.,
22
2
22
1
nknknn
OO
Ɍɚɤ ɤɚɤ n > k, ɬɨ ɤɨɪɧɢ Ȝ
1
ɢȜ
2
ɹɜɥɹɸɬɫɹ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɢ
ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ
tt
eceɫx
21
21
OO
. (9)
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɧɚɣɞɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:
21
001
2
21
002
1
,
OO
O
OO
O
xx
c
xx
c
.
ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (1) ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ:
>@
tt
exxexxx
12
)()(
1
002001
21
OO
OO
OO
. (10)
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɧɚɱɟɧɢɹɦɢ Ȝ
1
ɢȜ
2
ɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ
ɮɭɧɤɰɢɹɦɢ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜɜɢɞɟ
>@
tknchknxknshnxx
kn
e
x
nt
2222
0
22
00
22
)(
. (11)
Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɚɩɟɪɢɨɞɢɱɟɫɤɢɦ ɢ ɩɪɢɬɨɦ ɡɚɬɭɯɚɸɳɢɦ, ɬ. ɤ.
ɩɪɢ tĺ xĺ0.
ɉɨɞɫɬɚɜɢɦ ɜ (3) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ,
ɩɨɥɭɱɢɦ
)529(
6
1
242426 ttt
eeex
ɢɥɢ
)24172412(
3
1
26
tshtchex
t
ȼɵɹɫɧɢɦ, ɩɟɪɟɯɨɞɢɬ ɥɢ ɝɪɭɡ
ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ:
0)529(
6
1
242426
ttt
eee
. ȼɵɱɢɫɥɹɹ, ɩɨɥɭɱɚɟɦ t
1
=0,037 ɫ, t
2
=.
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
                                         cg       Eg
    x  2nx  kx 0 , ɝɞɟ X x x, k       , n      .
                                         P        2P
    ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɚɟɦ k=10 c-1, n=26 ɫ-1, ɬɚɤɢɦ
ɨɛɪɚɡɨɦ, n>k (ɫɥɭɱɚɣ ɛɨɥɶɲɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ).
                                              2        2
Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ O  2nO  k         0,  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  O 2
                    >                                        @
                  (O1 x 0  x 0 )e O2t  (O 2 x 0  x 0 )e O1t .                        (10)
    ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɧɚɱɟɧɢɹɦɢ Ȝ1 ɢ Ȝ2                                ɢ ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ
ɮɭɧɤɰɢɹɦɢ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ

     x
           e  nt
           2
          n k    2
                        >
                    ( x 0  nx 0 ) sh n 2  k 2  x 0 n 2  k 2 ch n 2  k 2 t .    @    (11)

    Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɚɩɟɪɢɨɞɢɱɟɫɤɢɦ ɢ ɩɪɢɬɨɦ ɡɚɬɭɯɚɸɳɢɦ, ɬ. ɤ.
ɩɪɢ tĺ’ xĺ0.
                                                 ɉɨɞɫɬɚɜɢɦ ɜ (3) ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ,
                                                                     1  26 t
                                                 ɩɨɥɭɱɢɦ x             e (29e  24t  5e 24 t )
                                                                     6
                                                               1  26t
                                                 ɢɥɢ x           e (12ch 24t  17 sh 24t )
                                                               3
                                                 ȼɵɹɫɧɢɦ, ɩɟɪɟɯɨɞɢɬ ɥɢ ɝɪɭɡ
                                                 ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ:


       1  26t
         e (29e  24t  5e 24t )      0 . ȼɵɱɢɫɥɹɹ, ɩɨɥɭɱɚɟɦ t1=0,037 ɫ, t2=’.
       6

                                               17

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