Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 37 стр.

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37
yq
2
– ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ ɛɥɨɤɚ ɋ ɢ ɰɢɥɢɧɞɪɚ ȼ;
zq
3
– ɩɟɪɟɦɟɳɟɧɢɟ ɩɪɢɡɦɵ.
ɉɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢɡɦɟɧɹɸɬɫɹ ɜ ɫɬɨɪɨɧɭ
ɢɯ ɭɜɟɥɢɱɟɧɢɹ.
Ɉɩɪɟɞɟɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ. ɂɦɟɟɦ
DBA
TTTT
,
ɝɞɟ
T
A
– ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɢɡɦɵ;
T
B
– ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɰɢɥɢɧɞɪɚ;
T
D
– ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ.
ɉɪɢɡɦɚ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ
zv
3
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
.
2
3
2
22
z
m
z
m
T
A
A
Ƚɪɭɡ D ɭɱɚɫɬɜɭɟɬ ɜ ɞɜɭɯ ɩɨɫɬɭɩɚɬɟɥɶɧɵɯ ɞɜɢɠɟɧɢɹɯ: ɨɬɧɨɫɢɬɟɥɶɧɨɦ (ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ) ɫɨ ɫɤɨɪɨɫɬɶɸ
xv
1
ɢ ɩɟɪɟɧɨɫɧɨɦ (ɜɦɟɫɬɟ ɫ ɩɪɢɡɦɨɣ)
ɫɨ ɫɤɨɪɨɫɬɶɸ
zv
3
. ɉɨɷɬɨɦɭ ɚɛɫɨɥɸɬɧɨɟ ɞɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ
ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɭɤɚɡɚɧɧɵɯ
ɫɤɨɪɨɫɬɟɣ, ɬ.ɟ.
31
vvv
D
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
.
222
3
2
1
2
zxvvv
D
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
)(
22
22
2
zx
m
vm
T
DD
D
.
ɐɢɥɢɧɞɪ ɫɨɜɟɪɲɚɟɬ ɩɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ȿɝɨ ɤɢɧɟɬɢɱɟɫɤɭɸ
ɷɧɟɪɝɢɸ ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ Ʉɺɧɢɝɚ:
22
2
0
2
0 BB
B
Jvm
T
Z
,
ɝɞɟ
v
0
– ɚɛɫɨɥɸɬɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ;
B
Z
– ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɰɢɥɢɧɞɪɚ;
J
0
– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ Ɉ.
ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ
Ɉ ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɪɚɜɧɚ
ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɢ ɩɟɪɟɧɨɫɧɨɣ ɫɤɨɪɨɫɬɟɣ, ɬ. ɟ.
,
32
vvv
O
ɝɞɟ
yv
2
– ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
DD
cos2cos2
22
32
2
3
2
2
2
zyzyvvvvv
O
.
Ɉɩɪɟɞɟɥɢɦ
B
Z
. Ɍɚɤ ɤɚɤ ɞɜɢɠɟɧɢɟ ɩɪɢɡɦɵ Ⱥ ɹɜɥɹɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ,
ɬɨ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ
B
Z
ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ ɥɢɲɶ ɜ ɨɬɧɨɫɢɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ
ɰɢɥɢɧɞɪɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ
Ⱥ. ɂɡɜɟɫɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɞɜɭɯ
ɬɨɱɟɤ ɰɢɥɢɧɞɪɚ
Ɉ ɢ ȿ:
xv
r
E
;
.yv
r
O
     q 2 y – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ ɛɥɨɤɚ ɋ ɢ ɰɢɥɢɧɞɪɚ ȼ;
     q3 z – ɩɟɪɟɦɟɳɟɧɢɟ ɩɪɢɡɦɵ.
     ɉɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɜɫɟ ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɢɡɦɟɧɹɸɬɫɹ ɜ ɫɬɨɪɨɧɭ
ɢɯ ɭɜɟɥɢɱɟɧɢɹ.
     Ɉɩɪɟɞɟɥɢɦ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ. ɂɦɟɟɦ
     T T A  TB  TD ,
ɝɞɟ TA – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɩɪɢɡɦɵ;
     TB – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɰɢɥɢɧɞɪɚ;
     TD – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɝɪɭɡɚ.
     ɉɪɢɡɦɚ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɫɨ ɫɤɨɪɨɫɬɶɸ v3 z , ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
           m A 2 3m 2
     TA       z       z .
            2       2
     Ƚɪɭɡ D ɭɱɚɫɬɜɭɟɬ ɜ ɞɜɭɯ ɩɨɫɬɭɩɚɬɟɥɶɧɵɯ ɞɜɢɠɟɧɢɹɯ: ɨɬɧɨɫɢɬɟɥɶɧɨɦ (ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ) ɫɨ ɫɤɨɪɨɫɬɶɸ v1 x ɢ ɩɟɪɟɧɨɫɧɨɦ (ɜɦɟɫɬɟ ɫ ɩɪɢɡɦɨɣ)
ɫɨ ɫɤɨɪɨɫɬɶɸ v3 z . ɉɨɷɬɨɦɭ ɚɛɫɨɥɸɬɧɨɟ ɞɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ
ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ ɫɨ ɫɤɨɪɨɫɬɶɸ, ɪɚɜɧɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɭɤɚɡɚɧɧɵɯ
                                               2  2    2   2    2
ɫɤɨɪɨɫɬɟɣ, ɬ.ɟ. v D v1  v3 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, v D v1  v3 x  z .
                               m D v D2 m 2
    Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, TD                     ( x  z 2 ) .
                                  2     2
     ɐɢɥɢɧɞɪ ɫɨɜɟɪɲɚɟɬ ɩɥɨɫɤɨɩɚɪɚɥɥɟɥɶɧɨɟ ɞɜɢɠɟɧɢɟ. ȿɝɨ ɤɢɧɟɬɢɱɟɫɤɭɸ
                                                           m B v02 J 0Z B2
ɷɧɟɪɝɢɸ ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ Ʉɺɧɢɝɚ: TB                                   ,
                                                             2        2
ɝɞɟ v0 – ɚɛɫɨɥɸɬɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ;
     Z B – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ ɰɢɥɢɧɞɪɚ;
     J0 – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɰɢɥɢɧɞɪɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɟɝɨ ɨɫɢ Ɉ.
     ɋɤɨɪɨɫɬɶ ɬɨɱɤɢ Ɉ ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɪɚɜɧɚ
ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɫɭɦɦɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɢ ɩɟɪɟɧɨɫɧɨɣ ɫɤɨɪɨɫɬɟɣ, ɬ. ɟ.
     v O v 2  v3 ,
ɝɞɟ v 2 y – ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɨɫɢ ɰɢɥɢɧɞɪɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
     vO2 v 22  v32  2v 2 v3 cos D y 2  z 2  2 y z cos D .
     Ɉɩɪɟɞɟɥɢɦ Z B . Ɍɚɤ ɤɚɤ ɞɜɢɠɟɧɢɟ ɩɪɢɡɦɵ Ⱥ ɹɜɥɹɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɵɦ,
ɬɨ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ Z B ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ ɥɢɲɶ ɜ ɨɬɧɨɫɢɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ
ɰɢɥɢɧɞɪɚ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɢɡɦɟ Ⱥ. ɂɡɜɟɫɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɫɤɨɪɨɫɬɢ ɞɜɭɯ
                         r       r
ɬɨɱɟɤ ɰɢɥɢɧɞɪɚ Ɉ ɢ ȿ: v E x ; vO y.




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