Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 51 стр.

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51
Ɋɚɛɨɬɚ ɫɢɥɵ
Fcx
x
;

A
c
xx
12 1
2
2
2
2
.
>@
GZ
ARv M dt
OO
;
;
M
G
dMA
Z
³
2
1
12
M
M
M
dMA
Z
.
ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ.
N
A
dt
Fv Fv Fx Fy Fz
xyz
G
W

.
Z
OO
MvRN
.
Z
Z
MN
.
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
– ɭɫɥɨɜɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɫɬɢ ɩɨɥɹ.
dɉA
G
; A
12
=ɉ
1
–ɉ
2
, ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ;
Phɉ r
, ɝɞɟ h – ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ,
2
2
'
c
ɉ
.
ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ
ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ
ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ. T
2
– T
1
= A
12
.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ ɦɨɳɧɨɫɬɟɣ
ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ.
N
dt
dT
.
ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ
ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const.
j
jj
Q
q
T
q
T
dt
d
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
(j = 1, 2, ... , s),
ɝɞɟ
q
1
, q
2
,..., q
S
– ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ;
s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ;
,
1
q
,
2
q
…,
S
q
– ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ;
Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ;
Q
1
, Q
2
, ..., Q
S
– ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ.
0
w
w
¸
¸
¹
·
¨
¨
©
§
w
w
jj
q
L
q
L
dt
d
, ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ.
Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ:
                                          c 2
Ɋɚɛɨɬɚ ɫɢɥɵ Fx            cx ; A12         x  x 22 .
                                          2 1
GA     >R ˜ v   O          @
                     M OZ dt ;
                                     M2

GA     M Z dM ;                A12   ³M   Z   dM .
                                     M1

ɋɭɦɦɚ ɪɚɛɨɬ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɪɚɜɧɚ ɧɭɥɸ.
     GA
 N          F ˜ v FW v Fx x  Fy y  Fz z .
      dt
N R ˜ vO  M O ˜ Z .
N M ZZ .
 wFx wFy wFx wFz wFy wFz
             ,          ,               – ɭɫɥɨɜɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɫɬɢ ɩɨɥɹ.
  wy     wx     wz   wx    wz       wy
GA dɉ ; A12=ɉ1–ɉ2 , ɝɞɟ ɉ(x, y, z) – ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɬɨɱɤɢ;
ɉ r Ph , ɝɞɟ h – ɜɵɫɨɬɚ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɭɥɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ,
     c'2
ɉ         .
       2
ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ
ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ
ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ. T2 – T1 = A12.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ ɦɨɳɧɨɫɬɟɣ
                                                  dT
ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ.            N.
                                                   dt
ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ
ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const.

d §¨ wT      · wT
             ¸             Q j (j = 1, 2, ... , s),
dt ¨© wq j  ¸ wq
             ¹        j

ɝɞɟ q1, q2,..., qS – ɨɛɨɛɳɟɧɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɫɢɫɬɟɦɵ;
       s – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ;
       q1 , q 2 , …, q S – ɨɛɨɛɳɟɧɧɵɟ ɫɤɨɪɨɫɬɢ;
       Ɍ – ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ;
       Q1, Q2, ..., QS – ɨɛɨɛɳɟɧɧɵɟ ɫɢɥɵ.
 d §¨ wL ·¸ wL
                           0 , ɝɞɟ L = T – ɉ – ɮɭɧɤɰɢɹ Ʌɚɝɪɚɧɠɚ.
 dt ¨© wq j ¸¹ wq j
       Ⱦɥɹ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɣ Ʌɚɝɪɚɧɠɚ ɫɥɟɞɭɟɬ:

                                                     51

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