Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 6 стр.

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6
ooo
z
dt
zd
dt
d
dt
dr
dt
d
rr
dt
d
r
dt
rd
w
2
2
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2
2
2
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2
¸
¸
¹
·
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¸
¸
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M
MMM
.
zr
FFF ,,
M
– ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ
ooo
zr ,,
M
ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ.
F
– ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɨɠɟɧɢɹ, ɫɤɨɪɨɫɬɢ, ɜɪɟɦɟɧɢ.
2
2
2
2
2
2
2
,, ,, , , ,
2,,,,,,,
,, ,, , , .
r
z
dr d dr d dz
mr Ftrz
dt dt dt dt dt
ddrd drddz
mr F tr z
dt dt dt dt dt dt
dz dr d dz
mFtrz
dt dt dt dt
M
MM
M
MM M
M
M
M
§·
§· § ·
°
¨¸
¨¸ ¨ ¸
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(2)
ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
III. ȿɫɬɟɫɬɜɟɧɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ
bn,,W
.
)(tSS
– ɞɭɝɨɜɚɹ
ɤɨɨɪɞɢɧɚɬɚ.
W
dt
dS
v
,
bn
v
dt
Sd
w 0
2
2
2
U
W
.
bn
FFF ,,
W
– ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ ɟɫɬɟɫɬɜɟɧɧɨɝɨ
ɬɪɟɯɝɪɚɧɧɢɤɚ.
2
2
,, ,
1
,, ,
0,,.
n
b
dS dS
mFtS
dt dt
dS dS
mFtS
dt dt
dS
FtS
dt
W
U
§·
¨¸
°
©¹
°
°
§·
®
¨¸
©¹
°
°
§·
°
¨¸
©¹
¯
ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɨɫɹɯ
ɟɫɬɟɫɬɜɟɧɧɨɝɨ ɬɪɟɯɝɪɚɧɧɢɤɚ.
      § d 2 r § dM · 2 · o § d 2M      dr dM · o d 2 z o
    w ¨ 2  r¨     ¸   ¸r  ¨ r
                            ¨ dt 2  2        ¸M  2 z .
      ¨ dt
      ©       © dt ¹ ¸¹     ©          dt dt ¸¹   dt
                                                 o   o   o
      Fr , FM , Fz – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ r , M , z ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ.
      F – ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɨɠɟɧɢɹ, ɫɤɨɪɨɫɬɢ, ɜɪɟɦɟɧɢ.
­ § d 2r         § dM · ·
                         2
                                  §             dr dM dz ·
° m ¨¨ 2  r ¨         ¸ ¸¸ Fr ¨ t , r , M , z , ,     , ¸,
° ©    dt        ©  dt ¹   ¹      ©              dt dt dt ¹
°                                                                   (2)
° § d 2M            dr dM ·         §              dr dM dz ·
®m ¨ r 2  2                 ¸ FM ¨ t , r , M , z , ,   , ¸,
° © dt              dt dt ¹         ©              dt dt dt ¹
° § d 2z ·
° m ¨ 2 ¸ F z ¨§ t , r , M , z , dr , dM , dz ¸· .
° © dt ¹            ©            dt dt dt ¹
¯
      ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
      III. ȿɫɬɟɫɬɜɟɧɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ W, n, b .
      S S (t )             –         ɞɭɝɨɜɚɹ
ɤɨɨɪɞɢɧɚɬɚ.
            dS
      v         W,
            dt
            d 2S       v2
      w         2
                   W  n  0˜b.
             dt         U

     FW , Fn , Fb – ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɫɢɥɵ ɧɚ ɨɫɢ ɟɫɬɟɫɬɜɟɧɧɨɝɨ
ɬɪɟɯɝɪɚɧɧɢɤɚ.
     ­ d 2S           §        dS ·
     ° m dt 2     FW ¨ t , S ,    ¸,
                               dt ¹
     °                ©
     ° dS 1             §       dS ·
     ®m            Fn ¨ t , S ,     ¸,
     ° dt U             ©       dt ¹
     °          §     dS ·
     ° 0 Fb ¨ t , S ,      ¸.
     ¯          ©      dt ¹
    ɗɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɜ ɨɫɹɯ
ɟɫɬɟɫɬɜɟɧɧɨɝɨ ɬɪɟɯɝɪɚɧɧɢɤɚ.



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