Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 22 стр.

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ɂɡɨɛɪɚɡɢɦ ɩɨɪɲɟɧɶ ɫɦɟɳɟɧɧɵɦ ɢɡ ɧɭɥɹ ɧɚɩɪɚɜɨ ɧɚ ɯ, ɩɪɢ ɷɬɨɦ
ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɜɨɡɧɢɤɲɟɣ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F ɪɚɜɧɚ:
)( xdccdF
xx
. (8)
Ʉɪɨɦɟ ɬɨɝɨ, ɤ ɩɨɪɲɧɸ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɜɟɫ, N – ɧɨɪɦɚɥɶɧɚɹ
ɪɟɚɤɰɢɹ ɤɨɪɩɭɫɚ, S – ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɫɠɚɬɨɝɨ ɜɨɡɞɭɯɚ.
ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɪɲɧɹ D:
xx
FSxm
.
ɍɱɢɬɵɜɚɹ (1) ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ S, ɧɚɯɨɞɢɦ
.)3cos()cos(
310
cxcdptHptHHx
g
P
ɍɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ
ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ ɜ ɜɢɞɟ
)3cos()cos(
31
2
pthpthxkx
, (9)
ɝɞɟ
.,,
3
3
1
1
P
gH
h
P
gH
h
P
cg
k
ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2) ɜ ɜɢɞɟ
)3cos()3sin()cos()sin(
22112
ptBptAptBptAx
. (10)
Ʉɨɷɮɮɢɰɢɟɧɬɵ Ⱥ
1
, ȼ
1
, Ⱥ
2
, ȼ
2
ɧɚɣɞɟɦ, ɜɵɱɢɫɥɢɜ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ
ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɯ
2
ɢ ɩɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ (2), ɚ ɡɚɬɟɦ
ɩɪɢɪɚɜɧɹɜ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɫɬɨɹɳɢɟ ɜ ɩɪɚɜɨɣ ɢ ɥɟɜɨɣ ɱɚɫɬɹɯ ɭɪɚɜɧɟɧɢɹ ɩɪɢ
ɫɢɧɭɫɟ ɢ ɤɨɫɢɧɭɫɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
.
9
,0,,0
22
3
33
22
1
11
pk
h
BA
pk
h
BA
ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ (3), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ:
).3cos()cos(
22
3
22
1
2
pt
pk
h
pt
pk
h
x
ȼ ɫɥɭɱɚɟ k = p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ.
ȼ ɫɥɭɱɚɟ k = 3p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ.
Ɍɚɤ ɤɚɤ
P
cg
k
, ɬɨ ɩɨɞɛɨɪ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ
ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɨɞɢɬɶ ɬɚɤ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɪɚɜɟɧɫɬɜ k p ɢ
k 3p. ɉɪɢ ɷɬɨɦ ɩɨɪɲɟɧɶ ɧɟ ɛɭɞɟɬ ɩɨɩɚɞɚɬɶ ɜ ɪɟɡɨɧɚɧɫ.
     ɂɡɨɛɪɚɡɢɦ ɩɨɪɲɟɧɶ ɫɦɟɳɟɧɧɵɦ ɢɡ ɧɭɥɹ ɧɚɩɪɚɜɨ ɧɚ ɯ, ɩɪɢ ɷɬɨɦ
ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɜɨɡɧɢɤɲɟɣ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F ɪɚɜɧɚ:
                            Fx cd x c(d  x) .               (8)
     Ʉɪɨɦɟ ɬɨɝɨ, ɤ ɩɨɪɲɧɸ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɜɟɫ, N – ɧɨɪɦɚɥɶɧɚɹ
ɪɟɚɤɰɢɹ ɤɨɪɩɭɫɚ, S – ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɫɠɚɬɨɝɨ ɜɨɡɞɭɯɚ.
     ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɪɲɧɹ D:
mx S x  Fx .
     ɍɱɢɬɵɜɚɹ (1) ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ S, ɧɚɯɨɞɢɦ
     P
       x H 0  H 1 cos( pt )  H 3 cos(3 pt )  cd  cx.
     g
     ɍɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ
ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ ɜ ɜɢɞɟ
                  x  k 2 x h1 cos( pt )  h3 cos(3 pt ) ,   (9)
               cg          H1 g         H3g
ɝɞɟ    k          , h1          , h3          .
                P            P            P
     ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2) ɜ ɜɢɞɟ
         x2 A1 sin( pt )  B1 cos( pt )  A2 sin(3 pt )  B2 cos(3 pt ) . (10)
     Ʉɨɷɮɮɢɰɢɟɧɬɵ Ⱥ1, ȼ1, Ⱥ2, ȼ2 ɧɚɣɞɟɦ, ɜɵɱɢɫɥɢɜ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ
ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɯ2 ɢ ɩɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ (2), ɚ ɡɚɬɟɦ
ɩɪɢɪɚɜɧɹɜ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɫɬɨɹɳɢɟ ɜ ɩɪɚɜɨɣ ɢ ɥɟɜɨɣ ɱɚɫɬɹɯ ɭɪɚɜɧɟɧɢɹ ɩɪɢ
ɫɢɧɭɫɟ ɢ ɤɨɫɢɧɭɫɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
                          h1                           h3
      A1 0, B1                  , A3 0, B3                    .
                      k 2  p2                    k 2  9 p2
     ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ (3), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ
ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ:
              h1                 h
     x2            cos( pt )  2 3 2 cos(3 pt ).
          k 2  p2             k p
     ȼ ɫɥɭɱɚɟ k = p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ.
     ȼ ɫɥɭɱɚɟ k = 3p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ.
                      cg
     Ɍɚɤ ɤɚɤ k             , ɬɨ ɩɨɞɛɨɪ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ
                       P
ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɨɞɢɬɶ ɬɚɤ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɪɚɜɟɧɫɬɜ k  p ɢ
k  3p. ɉɪɢ ɷɬɨɦ ɩɨɪɲɟɧɶ ɧɟ ɛɭɞɟɬ ɩɨɩɚɞɚɬɶ ɜ ɪɟɡɨɧɚɧɫ.




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