Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 15 стр.

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Ɂɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɜ ɜɢɞɟ
)sin(
D
ktax
, (11)
ɝɞɟ
2
2
0
2
0
k
x
xa
– ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ,
0
0
x
kx
arctg
D
– ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ.
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (2), ɧɚɣɞɟɦ ɚɦɩɥɢɬɭɞɭ ɢ ɧɚɱɚɥɶɧɭɸ ɮɚɡɭ:
J
gS
P
v
k
v
a
0
0
,
0
D
. (12)
ɂɫɩɨɥɶɡɭɹ (6),(7) ɢ (5), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ
¸
¸
¹
·
¨
¨
©
§
t
P
gS
gS
P
vx
J
J
sin
0
. (13)
ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ
J
S
S
gs
P
k
T
2
2
.
§4. Ɂɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ
Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ ɜɟɫɨɦ Ɋ = 98 H, ɩɨɞɜɟɲɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ,
ɞɜɢɠɟɬɫɹ ɜ ɠɢɞɤɨɫɬɢ. Ʉɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ ɫ=10 ɇ/ɫɦ. ɋɢɥɚ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɝɪɭɡɚ:
R=ȕȣ, ɝɞɟ ȕ=1,6 ɇɫ/ɫɦ. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɝɪɭɡ ɛɵɥ ɫɦɟɳɟɧ ɢɡ ɩɨɥɨɠɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɜɧɢɡ
ɧɚ 4 ɫɦ
ɢ ɟɦɭ ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɧɢɡ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ȣ
0
=4 ɫɦ/ɫ.
Ɋɟɲɟɧɢɟ.
ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ
ɩɪɭɠɢɧɟ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜɨɡɶɦɟɦ ɜ
ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
0
0
0, 4 ,
4/.
txxɫɦ
x
x ɫɦ c

ɂɡɨɛɪɚɡɢɦ ɝɪɭɡ ɜ ɩɨɥɨɠɟɧɢɢ, ɤɨɝɞɚ
ɩɪɭɠɢɧɚ ɩɨɥɭɱɢɬ ɭɞɥɢɧɟɧɢɟ
D = d + x.
ɋɢɥɚ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ, ɧɚɩɪɚɜɥɟɧɧɚɹ
ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɪɚɜɧɚ
)( xdcF
x
. (1)
ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
xx
RFPxm
.
ɉɨɞɫɬɚɜɢɦ ɜ ɭɪɚɜɧɟɧɢɟ ɡɧɚɱɟɧɢɹ F
x
ɢ R
x
:
    Ɂɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɜ ɜɢɞɟ
                      x a sin(kt  D ) ,                                 (11)
                  2
             x                                     kx0
ɝɞɟ a    x 02    0
                 – ɚɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ, D arctg
                  2                                       – ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ.
             k                                       x 0
    ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ (2), ɧɚɣɞɟɦ ɚɦɩɥɢɬɭɞɭ ɢ ɧɚɱɚɥɶɧɭɸ ɮɚɡɭ:
                           v0        P
                      a         v0        , D 0.                          (12)
                            k       gSJ
    ɂɫɩɨɥɶɡɭɹ (6),(7) ɢ (5), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ
                                P       § gSJ ·
                      x v0         sin ¨¨    t ¸¸ .                       (13)
                               gSJ      ©  P    ¹
                                    2S       P
    ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ      T      2S       .
                                     k      gsJ
                          §4. Ɂɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ

    Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ ɜɟɫɨɦ Ɋ = 98 H, ɩɨɞɜɟɲɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ,
ɞɜɢɠɟɬɫɹ ɜ ɠɢɞɤɨɫɬɢ. Ʉɨɷɮɮɢɰɢɟɧɬ ɠɟɫɬɤɨɫɬɢ ɩɪɭɠɢɧɵ ɫ=10 ɇ/ɫɦ. ɋɢɥɚ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɜɢɠɟɧɢɸ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɫɤɨɪɨɫɬɢ ɝɪɭɡɚ:
R=ȕȣ, ɝɞɟ ȕ=1,6 ɇɫ/ɫɦ. ɇɚɣɬɢ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɝɪɭɡ ɛɵɥ ɫɦɟɳɟɧ ɢɡ ɩɨɥɨɠɟɧɢɹ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɜɧɢɡ ɧɚ 4 ɫɦ
ɢ ɟɦɭ ɛɵɥɚ ɫɨɨɛɳɟɧɚ ɜɧɢɡ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ȣ0=4 ɫɦ/ɫ.
    Ɋɟɲɟɧɢɟ.
                                ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ ɩɨ
                                ɩɪɭɠɢɧɟ, ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜɨɡɶɦɟɦ ɜ
                                ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
                                ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
                                 t 0, x x0 4 ɫɦ,
                                           x x0 4 ɫɦ / c.
                                    ɂɡɨɛɪɚɡɢɦ ɝɪɭɡ ɜ ɩɨɥɨɠɟɧɢɢ, ɤɨɝɞɚ
                                    ɩɪɭɠɢɧɚ ɩɨɥɭɱɢɬ ɭɞɥɢɧɟɧɢɟ
                                    D = d + x.
                                    ɋɢɥɚ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ, ɧɚɩɪɚɜɥɟɧɧɚɹ
                                    ɜɟɪɬɢɤɚɥɶɧɨ ɜɜɟɪɯ, ɪɚɜɧɚ
                                    Fx c(d  x) .                  (1)


    ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ:
                             mx P  Fx  Rx .
    ɉɨɞɫɬɚɜɢɦ ɜ ɭɪɚɜɧɟɧɢɟ ɡɧɚɱɟɧɢɹ Fx ɢ Rx:

                                       15

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