Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 31 стр.

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31
ɝɞɟ
R
ɢ
M
0
– ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ ɫɢɥ;
Ɉ – ɩɪɨɢɡɜɨɥɶɧɚɹ ɬɨɱɤɚ ɬɟɥɚ.
Ɋɚɛɨɬɚ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ ɨɫɢ,
;
M
G
dMA
Z
³
2
1
12
M
M
M
dMA
Z
,
ɝɞɟ M
Z
– ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ Oz.
ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ.
Ɇɨɳɧɨɫɬɶ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɬɨɱɤɟ,
N
A
dt
Fv Fv Fx Fy Fz
xyz
G
W

.
Ɇɨɳɧɨɫɬɶ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ,
Z
OO
MvRN
.
Ⱦɥɹ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ
ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ,
Z
Z
MN
.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɛɨɬɚ ɫɢɥɵ ɧɚ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ ɡɚɜɢɫɢɬ ɨɬ
ɮɨɪɦɵ ɤɪɢɜɨɣ L, ɩɨ ɤɨɬɨɪɨɣ ɩɟɪɟɦɟɳɚɟɬɫɹ ɬɨɱɤɚ. ȿɫɥɢ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ
ɧɚ ɬɨɱɤɭ, ɬɚɤɨɜɵ, ɱɬɨ
x
F
y
F
y
x
w
w
w
w
,
x
F
z
F
z
x
w
w
w
w
,
y
F
z
F
z
y
w
w
w
w
,
ɬɨ ɪɚɛɨɬɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ ɢ ɩɨɥɟ ɫɢɥ ɧɚɡɵɜɚɸɬ
ɩɨɬɟɧɰɢɚɥɶɧɵɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
dɉA
G
; A
12
=ɉ
1
– ɉ
2
,
ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ;
ɉ
1
ɢɉ
2
– ɡɧɚɱɟɧɢɹ ɉ(x, y, z) ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɩɨɥɨɠɟɧɢɹɯ
ɬɨɱɤɢ.
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ
constPɉ
Zc
.
ȿɫɥɢ ɜɵɛɪɚɧɚ ɧɭɥɟɜɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɭɪɨɜɧɹ, ɬɨ ɩɨɥɭɱɢɦ
Phɉ r
, ɝɞɟ h –
ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɪɢɱɟɦ ɡɧɚɤ
«ɩɥɸɫ» ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɪɚɫɩɨɥɨɠɟɧ ɜɵɲɟ ɷɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ.
ɉɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɩɪɭɠɢɧɵ (ɥɢɧɟɣɧɨɣ ɢ ɫɩɢɪɚɥɶɧɨɣ) ɜɵɪɚɠɚɸɬ
ɮɨɪɦɭɥɨɣ
2
2
'
c
ɉ
,
ɝɞɟ ɞɥɹ ɥɢɧɟɣɧɨɣ ɩɪɭɠɢɧɵ:
c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɣ ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ
ɩɪɭɠɢɧɵ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ;
ɝɞɟ R ɢ M 0 – ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɢ ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɟɥɭ ɫɢɥ;
         Ɉ – ɩɪɨɢɡɜɨɥɶɧɚɹ ɬɨɱɤɚ ɬɟɥɚ.
    Ɋɚɛɨɬɚ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ ɨɫɢ,
                               M2

    GA   M Z dM ;        A12   ³M   Z   dM ,
                               M1
ɝɞɟ MZ – ɝɥɚɜɧɵɣ ɦɨɦɟɧɬ ɜɫɟɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ Oz.
     ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ.
     Ɇɨɳɧɨɫɬɶ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɬɨɱɤɟ,
          GA
      N           F ˜ v FW v Fx x  Fy y  Fz z .
           dt
     Ɇɨɳɧɨɫɬɶ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ,
     N R ˜ vO  M O ˜ Z .
     Ⱦɥɹ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɜɪɚɳɚɸɳɟɦɭɫɹ ɜɨɤɪɭɝ
ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ,
     N M ZZ .
     ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɚɛɨɬɚ ɫɢɥɵ ɧɚ ɤɪɢɜɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ ɡɚɜɢɫɢɬ ɨɬ
ɮɨɪɦɵ ɤɪɢɜɨɣ L, ɩɨ ɤɨɬɨɪɨɣ ɩɟɪɟɦɟɳɚɟɬɫɹ ɬɨɱɤɚ. ȿɫɥɢ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ
ɧɚ ɬɨɱɤɭ, ɬɚɤɨɜɵ, ɱɬɨ
      wFx wFy wFx wFz wFy wFz
                   ,          ,              ,
       wy     wx      wz   wx    wz       wy
ɬɨ ɪɚɛɨɬɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɬɪɚɟɤɬɨɪɢɢ ɬɨɱɤɢ ɢ ɩɨɥɟ ɫɢɥ ɧɚɡɵɜɚɸɬ
ɩɨɬɟɧɰɢɚɥɶɧɵɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
     GA dɉ ; A12=ɉ1 – ɉ2,
ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ;
     ɉ1 ɢ ɉ2 – ɡɧɚɱɟɧɢɹ ɉ(x, y, z) ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɩɨɥɨɠɟɧɢɹɯ
ɬɨɱɤɢ.
     ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɩɨɥɹ ɫɢɥɵ ɬɹɠɟɫɬɢ ɉ PZc  const .
     ȿɫɥɢ ɜɵɛɪɚɧɚ ɧɭɥɟɜɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɭɪɨɜɧɹ, ɬɨ ɩɨɥɭɱɢɦ ɉ r Ph , ɝɞɟ h –
ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɪɢɱɟɦ ɡɧɚɤ
«ɩɥɸɫ» ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɰɟɧɬɪ ɬɹɠɟɫɬɢ ɪɚɫɩɨɥɨɠɟɧ ɜɵɲɟ ɷɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ.
     ɉɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɩɪɭɠɢɧɵ (ɥɢɧɟɣɧɨɣ ɢ ɫɩɢɪɚɥɶɧɨɣ) ɜɵɪɚɠɚɸɬ
ɮɨɪɦɭɥɨɣ
          c'2
     ɉ          ,
            2
ɝɞɟ ɞɥɹ ɥɢɧɟɣɧɨɣ ɩɪɭɠɢɧɵ:
     c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɣ ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ
ɩɪɭɠɢɧɵ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ;


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