Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 18 стр.

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Ɂɧɚɱɟɧɢɟ t
1
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɟɪɟɯɨɞɭ ɝɪɭɡɚ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, t
2
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɚɬɭɯɚɧɢɸ ɞɜɢɠɟɧɢɹ. ɂɬɚɤ, ɜ
ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɝɪɭɡ ɩɪɨɯɨɞɢɬ ɨɞɢɧ ɪɚɡ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ
ɪɚɜɧɨɜɟɫɢɹ ɢ ɡɚɬɟɦ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɤ ɧɟɦɭ ɩɪɢɛɥɢɠɚɟɬɫɹ ɫ ɞɪɭɝɨɣ
ɫɬɨɪɨɧɵ.
§5. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ
Ɂɚɞɚɱɚ ʋ 1. ɇɚ ɪɢɫɭɧɤɟ
ɢɡɨɛɪɚɠɟɧɚ ɫɯɟɦɚ ɩɪɢɛɨɪɚ ɞɥɹ
ɢɡɦɟɪɟɧɢɹ ɞɚɜɥɟɧɢɹ. Ʉ ɩɨɥɡɭɧɭ Ⱥ
ɜɟɫɨɦ Ɋ=196 Ƚ ɩɪɢɤɪɟɩɥɟɧɚ
ɫɬɪɟɥɤɚ ȼ, ɨɬɦɟɱɚɸɳɚɹ
ɩɨɤɚɡɚɧɢɹ ɧɚ
ɧɟɩɨɞɜɢɠɧɨɣ
ɲɤɚɥɟ ɋ. ɉɨɥɡɭɧ Ⱥ,
ɩɪɢɤɪɟɩɥɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ
D, ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨ
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɢɞɟɚɥɶɧɨ
ɝɥɚɞɤɨɣ ɩɥɨɫɤɨɫɬɢ. Ʉ ɩɨɥɡɭɧɭ
ɩɪɢɥɨɠɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ
S = H·sin(pt), ɝɞɟ ɇ = 1,6 ɤȽ,
ɪ = 60 ɫ
-1
. Ʉɨɷɮɮɢɰɢɟɧɬ
ɭɩɪɭɝɨɫɬɢ ɪɚɜɟɧ ɫ = 2 ɤȽ/ɫɦ. ȼ
ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɩɨɥɡɭɧ
ɧɚɯɨɞɢɥɫɹ ɜ ɩɨɤɨɟ, ɜ ɩɨɥɨɠɟɧɢɢ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
Ɉɩɪɟɞɟɥɢɬɶ:
1) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɜ ɫɥɭɱɚɟ ɨɬɫɭɬɫɬɜɢɹ ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ;
2) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɩɪɢ ɧɚɥɢɱɢɢ ɫɢɥɵ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ,
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɩɟɪɜɨɣ
ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɩɨɥɡɭɧɚ
R=ȕȣ, ɝɞɟ ȕ=25,6 Ƚɫ/ɫɦ.
Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɜɩɪɚɜɨ, ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ
ɩɨɥɨɠɟɧɢɢ ɩɨɥɡɭɧɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɟ.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ
ɩɨɥɡɭɧɚ:
0,00ɩɪɢ
xxt
.
ɂɡɨɛɪɚɡɢɦ ɩɨɥɡɭɧ ɫɦɟɳɟɧɧɵɦ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɩɪɚɜɨ ɧɚ ɯ.
ɉɪɢ ɷɬɨɦ ɩɪɭɠɢɧɚ ɪɚɫɬɹɧɟɬɫɹ ɧɚ D = ɯ. Ʉ ɩɨɥɡɭɧɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ –
ɜɟɫ ɩɨɥɡɭɧɚ, N – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ, ɫɢɥɚ S, ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɪɚɫɬɹɧɭɬɨɣ
ɩɪɭɠɢɧɵ F.
ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ
ɯ:
xx
FSxm
ɢɥɢ
x
P
cg
pt
P
Hg
x sin
, ɨɬɤɭɞɚ
,sin
2
ptkxkx
(1)
ɝɞɟ
P
hg
h
P
cg
k ,
. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k=100 ɫ
-1
, h=8000 ɫɦ/ɫ
-2
.
Ɋɟɲɚɹ (1), ɧɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
    Ɂɧɚɱɟɧɢɟ t1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɟɪɟɯɨɞɭ ɝɪɭɡɚ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, t2 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɚɬɭɯɚɧɢɸ ɞɜɢɠɟɧɢɹ. ɂɬɚɤ, ɜ
ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɝɪɭɡ ɩɪɨɯɨɞɢɬ ɨɞɢɧ ɪɚɡ ɱɟɪɟɡ ɩɨɥɨɠɟɧɢɟ ɫɬɚɬɢɱɟɫɤɨɝɨ
ɪɚɜɧɨɜɟɫɢɹ ɢ ɡɚɬɟɦ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɤ ɧɟɦɭ ɩɪɢɛɥɢɠɚɟɬɫɹ ɫ ɞɪɭɝɨɣ
ɫɬɨɪɨɧɵ.

                        §5. ȼɵɧɭɠɞɟɧɧɵɟ ɤɨɥɟɛɚɧɢɹ

Ɂɚɞɚɱɚ ʋ 1. ɇɚ           ɪɢɫɭɧɤɟ
ɢɡɨɛɪɚɠɟɧɚ ɫɯɟɦɚ ɩɪɢɛɨɪɚ ɞɥɹ
ɢɡɦɟɪɟɧɢɹ ɞɚɜɥɟɧɢɹ. Ʉ ɩɨɥɡɭɧɭ Ⱥ
ɜɟɫɨɦ Ɋ=196 Ƚ ɩɪɢɤɪɟɩɥɟɧɚ
ɫɬɪɟɥɤɚ       ȼ,    ɨɬɦɟɱɚɸɳɚɹ
ɩɨɤɚɡɚɧɢɹ      ɧɚ   ɧɟɩɨɞɜɢɠɧɨɣ
ɲɤɚɥɟ       ɋ.     ɉɨɥɡɭɧ     Ⱥ,
ɩɪɢɤɪɟɩɥɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ
D,       ɩɟɪɟɦɟɳɚɟɬɫɹ         ɩɨ
                                         Ɉɩɪɟɞɟɥɢɬɶ:
ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ          ɢɞɟɚɥɶɧɨ
                                        1) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɝɥɚɞɤɨɣ ɩɥɨɫɤɨɫɬɢ. Ʉ ɩɨɥɡɭɧɭ
                                           ɜ ɫɥɭɱɚɟ ɨɬɫɭɬɫɬɜɢɹ ɫɢɥɵ
ɩɪɢɥɨɠɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ
                                           ɫɨɩɪɨɬɢɜɥɟɧɢɹ;
S = H·sin(pt), ɝɞɟ ɇ = 1,6 ɤȽ,
                                        2) ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ
ɪ = 60     ɫ-1.    Ʉɨɷɮɮɢɰɢɟɧɬ
                                           ɩɪɢ        ɧɚɥɢɱɢɢ         ɫɢɥɵ
ɭɩɪɭɝɨɫɬɢ ɪɚɜɟɧ ɫ = 2 ɤȽ/ɫɦ. ȼ
                                           ɫɨɩɪɨɬɢɜɥɟɧɢɹ,
ɧɚɱɚɥɶɧɵɣ       ɦɨɦɟɧɬ    ɩɨɥɡɭɧ
                                           ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ         ɩɟɪɜɨɣ
ɧɚɯɨɞɢɥɫɹ ɜ ɩɨɤɨɟ, ɜ ɩɨɥɨɠɟɧɢɢ
                                           ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɩɨɥɡɭɧɚ
ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
                                           R=ȕȣ, ɝɞɟ ȕ=25,6 Ƚ ɫ/ɫɦ.
Ɋɟɲɟɧɢɟ. ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɜɩɪɚɜɨ, ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ
ɩɨɥɨɠɟɧɢɢ ɩɨɥɡɭɧɚ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ ɩɪɭɠɢɧɟ.
    ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ: ɩɪɢ t 0 x 0, x 0 .
    ɂɡɨɛɪɚɡɢɦ ɩɨɥɡɭɧ ɫɦɟɳɟɧɧɵɦ ɢɡ ɩɨɥɨɠɟɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜɩɪɚɜɨ ɧɚ ɯ.
ɉɪɢ ɷɬɨɦ ɩɪɭɠɢɧɚ ɪɚɫɬɹɧɟɬɫɹ ɧɚ D = ɯ. Ʉ ɩɨɥɡɭɧɭ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ –
ɜɟɫ ɩɨɥɡɭɧɚ, N – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ, ɫɢɥɚ S, ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɪɚɫɬɹɧɭɬɨɣ
ɩɪɭɠɢɧɵ F.
  ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɥɡɭɧɚ ɜ ɩɪɨɟɤɰɢɢ ɧɚ
ɯ:
                                Hg            cg
     mx S x  Fx ɢɥɢ x           sin pt      x , ɨɬɤɭɞɚ
                                 P            P
                           x  k 2 x k sin pt ,                   (1)
              cg       hg
    ɝɞɟ k        , h      . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k=100 ɫ-1, h=8000 ɫɦ/ɫ-2.
               P        P
    Ɋɟɲɚɹ (1), ɧɚɣɞɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜ ɜɢɞɟ
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