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?????? 2008
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1.??????????? ??????? 1 1.???????? 3
2.????????? ????? 4
2.1 ????? ??????????? ?????? 4
2.1.1 ??????????? ????? 4
2.1.2 ????? ? ?????????????? ???????????? 5
2.2 ??????????? ?????????? ????????????? 7
3.2 ??????????? ??????11
3.?????15
4.?????????? 16 ????????
? ????????? ????? ???????????????? ?????????? ????? ?????????? ??????????????? ? ???????? ?????????? ? ????? ?????????? ?? ????????? ???????????? ?????????. ? ????? ???????? ?? ?????????? ????????? ? ?????????? ????????????? ?????????????, ??????????, ? ???????, ? ????????? ?????????? ??? ?????????? ?????? ???????? (?????????, ????????? ??????????? ? ?.?.), ? ????? ??????? ?????????. ????? ???????????, ?????? ??????? ????????????? ????????, ???????? ??????????. ??? ??????????? ??? ???????????? ???????? ????? ????? ???????? ????? ? ?????? ???????????? ??? ??? ?????? ?? ?????. ??????? ?????? ??????? ???????? ?????????? ??????, ????????? ??????? ??????????? ?? ?????????? ?? ?????????? ??? ?????????? ?????? ????????. ??? ???? ???????????? ??????????? ??????????????? ???????????. ?????????? ? ????? ?????? ??????? ?? ?????? ????????? ?????????? ????????. ???????????? ??????????? ???????? ????????? ????????????? ?????????, ???????????? ??? ????????, ? ????????? ?????????? ???????: n=a/g
??????????? - ??????? ??????? ????, ?? ??????? ??????????????? ?????? ??? ?????? ?? ??????? ???????????? ???????????. ??????????? - ???????? ???????????????? ????? ?????? ??? ?????????????? ???????????. ??????????? ???????????? - ????????, ?????? ????????? ????????? ???????? ? ????????? ?????. K_(? )=a_??/?_???? =n_??/n_???? ?????????? ??????????? ???????????? ???????????? ??? [?_? ]=([n])/n_? . ??? ?????????? ??????????? ??????????, ????? K? <[ K? ].
???????????? ????? ??????? - ????? ????????? ?? ??????????? ??????. ?? ??????????? ? ?????? ??????????? ?????? ????? ???????, ????? n ???? ??????? ?? ?????????? ????????. ????? ??? ??????? ???????? ??????? ??? ??????? ??????? ????????? ?????????????. ????????? ?????
????? ??????????? ??????
??????????? ?????
?????????? ??????????? ????? - ??????? ????? ???????.
???????? ????????? ???????? ???? ???????
(????????? ???????? 2-?? ????).
???????? ??????-????????????.
??? ??????????? ????????? ????????,??????
???????????? ???????
T=(m_(2 ) ?v_2?^2)/2+(I_(2 ) ??_2?^2)/2
?_2= f ?_2 ; v_2=1/2 af ?_2 ; I_2=(m?l_2?^2)/12=(m_2 ?(3a)?^2)/12=3/4 m_2 a^2
T=1/2(m_2 1/4 a^2 ?f ?_2?^2+3/4 m_2 a^2 ?f ?_2?^2=1/2 m_2 a^2 ?f ?_2?^2 ?T/f ?_2 =m_2 a^2 f ?_2 ; d/dt (?T/f ?_2 )=m_2 a^2 f �_2
?????? ????????????? ???????
?=1/2 ??_2?^2 ?_2+1/2 ??_3?^2 ?_3
?_2=?f_2 - ??????? ?????????????
?_3=?f_2 - ??????? ?????????????
??????
?=1/2 a^2 ?f^2 ??_2+1/2 a^2 f^2 ?_3
??/(?f_2 )=C_2 a^2 f_2+C_3 a^2 f_2=a^2 f_2 (C_2+C_3)
???????? ????????? ????????. ? ????? ????
d/dt (?T/f ?_2 )-?T/f_2 +??/(?f_2 )=0
m_2 a^2 f �_2+a^2 f_2 (C_2+C_3 )=0 |:a^2
m_2 f �_2+f_2 (C_2+C_3 )=0
??????? ???? ? ???? f_2=?_2 sinkt ; (f_2 ) �=?-??_2 k^2 sinkt ???????:
?-??_2 k^2 sinkt m_2+?_2 sinkt(C_2+C_3 )=0 |:?_2 sinkt
(C_2+C_3 )=k^2 m_2
????? ? ?????????????? ????????????
?????????? ??????? ? ??????? ???????????? ???????????? h=H*sin?t
?????? ???????????? ???????:
T=(m_(2 ) ?v �_2?^2)/2+(I_(2 ) ??_2?^2)/2
?_2= f ?_2; v �=1/2 af ?_2+(h ;) ? I_2=(m?l_2?^2)/12=(m_2 ?(3a)?^2)/12=3/4 m_2 a^2
T=1/2(m_2 (1/2 af ?_2+(h ) ? )^2+3/4 m_2 a^2 ?f ?_2?^2)
?T/f ?_2 =1/2 am_2 (1/2 af ?_2+(h ) ? )+3/4 m_2 a^2 f ?_2=m_2 a^2 f ?_2+1/2 am_2 (h ) ?;
d/dt (?T/f ?_2 )=m_2 a^2 f �_2+1/2 am_2 h � ????????????? ??????? ????? ??, ??? ? ? ?????????? ??????:
?=1/2 a^2 ??f_2?^2 ??_2+1/2 a^2 ?f_2?^2 ?_3
??/(?f_2 )=C_2 a^2 f_2+C_3 a^2 f_2=a^2 f_2 (C_2+C_3)
???????? ????????? ????????.
m_2 a^2 f �_2+1/2 am_2 h �+a^2 f_2 (C_2+C_3 )=0
? ?????? ????, ??? h �= -H?^2 sin?t; (C_2+C_3 )=?m_2 k?^2, ???????
m_2 a^2 f �_2+a^2 f_2 ?m_2 k?^2=1/2 am_2 H?^2 sin?t
????? ??????? ??????????? ?????????: Asinkt
??????? ??????? ????????????? ????????? ???? ? ????
f_2=B_2 sin?t ; (f_2 ) �=?-B?_2 ?^2 sin?t
?-m?_2 a^2 B_2 ?^2 sin?t+a^2 B_2 sin?t?m_2 k?^2=1/2 am_2 H?^2 sin?t | :am_2 sin?t
aB_2 k^2-aB_2 ?^2=1/2 H?^2 B_2 a?(k?^2-?^2)=1/2 H?^2
B_2/H=?^2/(2a?(k?^2-?^2))
??? ??? ??????? ?????????? ?? ?????????? r ?? ?????,?? a_??=r*f �+h �.
?? ???????????, K_(? )=a_??/?_???? , ? ?_????=h �. ???????
K_(? )=a_??/?_???? =(r*f �+h �)/h � =(r*f �)/h � +1
K_(? )=B_2/H*r+1
??? ??????????? ?????????? ?????????? K_(? )<[K_? ]=5/10=0.5
????,
|(1?^2 r)/(2a?(k?^2-?^2))+1|<[K_? ]
-[K_? ]<1-(r/2a ?^2)/(?^2-k^2 )<[K_? ]
1-(r/2a)/(1-[K_? ] )<k^2/?^2 <1-(r/2a)/(1+[K_? ] )
????? ????? ??????????? ??????????? ??????, ??? ??? ??? ?????? 0.
?k_2?^2<??_1?^2*(1-(r/2a)/(1+[K_? ] ))
k_2<?_1*v(+ 1-(r/2a)/(1+[K_? ] )� )
???????:
k_2=290 ???/?
??????????? k_1=k_2/2. ????? k_1=145 ???/?
? ?????, ??????????? ??????? :
k_1=145 ???/?
k_2=290 ???/?
II ??????????? ?????????? ?????????????
????????? ??????????? ???????? ??? ??????????? ????????? ????????.
?????? ???????????? ???????:
T=(m_(1 ) ?v_1?^2)/2+(I_(1 ) ??_1?^2)/2+(m_(2 ) ?v_2?^2)/2+(I_(2 ) ??_2?^2)/2
?_1= f ?_1; ?_2= f ?_2;; v_1=2af ?_1 ; v_2=1/2 af ?_2
I_1=(m?l_2?^2)/12=(m_1 ?(4a)?^2)/12=4/3 m_1 a^2
I_2=(m?l_2?^2)/12=(m_2 ?(3a)?^2)/12=3/4 m_2 a^2
T=1/2 (m_1 4a^2 ?f ?_1?^2+4/3 m_1 a^2 ?f ?_1?^2+m_2 1/4 a^2 ?f ?_2?^2+3/4 m_2 a^2 ?f ?_2?^2 )==1/2 a^2 (16/3 m_1 ?f ?_1?^2+m_2 ?f ?_2?^2)
?T/f ?_1 =16/3 m_1 a^2 f ?_(1 ); ?T/f ?_2 =m_2 a^2 f ?_2;
d/dt (?T/f ?_1 )=16/3 m_1 a^2 (f_1 ) �; d/dt (?T/f ?_2 )=m_2 a^2 (f_2 ) �; ?T/f_1 =?T/f_2 =0
?????? ????????????? ???????:
?=1/2 ??_1?^2 ?_1+1/2 ??_2?^2 ?_2+1/2 ??_3?^2 ?_3
????????? ??????:
?_1=af_1,
?_2=af_2-3af_1,
?_3=af_2,
???????
?=1/2(a^2 ??f_1?^2 ??_1+?_2 (af_2-3af_1 )^2+ a^2 ??f_2?^2 ??_3)
??/(?f_1 )=f_1 a^2 (?_1+9?_2 )-3a^2 f_2 ?_2
??/(?f_1 )=f_2 a^2 (?_2+?_3 )-3a^2 f_1 ?_2
???????? ????????? ????????:
{�(16/3 m_1 a^2 (f_1 ) �+f_1 a^2 (?_1+9?_2 )-3a^2 f_2 ?_2=0 |:a^2@m_2 a^2 (f_2 ) �+f_2 a^2 (?_2+?_3 )-3a^2 f_1 ?_2=0 |:a^2 )�
{�(16/3 m_1 (f_1 ) �+f_1 (?_1+9?_2 )-3f_2 ?_2=0 @m_2 (f_2 ) �+f_2 (?_2+?_3 )-3f_1 ?_2=0 )�
???? ??????? ? ????
f_1=A_1 sinkt ; f �_1=?-A?_1 k^2 sinkt
f_2=A_2 sinkt ; f �_2=?-A?_2 k^2 sinkt
??????????? ? ??????? ? ????? ??? ????????? ?? sinkt.
{�(A_1 (?_1+9?_2-16/3 m_1 k^2)-3A_2 ?_2=0 @-3A_1 ?_2+?A_2 (?_2+?_3-m?_2 k^2)=0 )�
??? ????????????? ?????????? ??????? ??????????, ????? ???????????? ??? ????? 0.
(?_1+9?_2-16/3 m_1 k^2) ?(?_2+?_3-m?_2 k^2)-9??_2?^2=0
k^4 (16/3 m_1 m_2 )-k^2 (16/3 m_1 (?_2+?_3 )+m_2 (?_1+9?_2 ))+?(??_2+?_3)( ?_1+9?_2)- 9??_2?^2=0
????????, ??? ?(??_2+?_3)( ?_1+9?_2)- 9??_2?^2=?_2 ?_1+?_3 ( ?_1+9?_2 ),??????? ?????????? ??????? ??????? ????? ??? ??????????? ?????????:
{�(16/3 m_1 m_2 (?k_1?^2+?k_2?^2 )=16/3 m_1 (?_2+?_3 )+m_2 (?_1+9?_2 )@16/3 m_1 m_2 4?k_1?^4=?_2 ?_1+?_3 ( ?_1+9?_2 ) )�
???????? ??????? ? ????? ???????????? (C_1,C_2,C_3) ? ????? ???????????.??????C_1=aC_2.
? ?????? ????, ??? k_2=2k_1 , ?????
{�(16/3 m_1 m_2 5?k_1?^2=16/3 m_1 (?_2+?_3 )+m_2 (?_1+9?_2 )@16/3 m_1 m_2 4?k_1?^4=?_2 ?_1+?_3 ( ?_1+9?_2 ) )�
?? ??????? ???????? ?_3:
?_3=m_2 5?k_1?^2-?_2-(3m_2)/(16m_1 ) ( ?_1+9?_2 )
????????? ?????????? ????????? ??? ?_3 ?? ??????.
64/3 m_1 m_2 ?k_1?^4=a?C_2?^2+?_2 (a+9) (m_2 5?k_1?^2-?_2-(3m_2)/(16m_1 ) ?_2 (a+9))
?????? z= a+9, ????? a=z-9.
64/3 m_1 m_2 ?k_1?^4=a?C_2?^2+?_2 z (m_2 5?k_1?^2-?_2-(3m_2)/(16m_1 ) ?_2 z)
?C_2?^2 (-z-(3m_2)/(16m_1 ) z^2+z-9)+?_2 zm_2 5?k_1?^2-64/3 m_1 m_2 ?k_1?^4=0
?C_2?^2 ((3m_2)/(16m_1 ) z^2+9)+64/3 m_1 m_2 ?k_1?^4-?_2 zm_2 5?k_1?^2=0
??????????, ????? ???????????? ??? ?????? 0:
D=?(zm_2 5?k_1?^2)?^2-4*64/3 m_1 m_2 ?k_1?^4 ((3m_2)/(16m_1 ) z^2+9)
D=25 z^2 ?m_2?^2 ?k_1?^2-768m_1 m_2 ?k_1?^4- 16?m_2?^2 z^2 ?k_1?^4=0
9?m_2?^2 z^2 ?k_1?^4=768m_1 m_2 ?k_1?^4
z^2=(768m_1)/(9m_2 )
Z=v((768m_1)/(9m_2 ))�20.655
a=v((768m_1)/(9m_2 ))-9�20.655 - 9�11.655
??????? ????????? ??? ?????? ?????????.
??_2?_1,2=(zm_2 5?k_1?^2�?k_1?^2 v(9?m_2?^2 z^2-768m_1 m_2 ))/(2((3m_2)/(16m_1 ) z^2+9))
???????? ????????? ??? C_3:
?_3=m_2 5?k_1?^2-?_2-(3m_2)/(16m_1 ) ?_2 (a+9) ?????????? ????????? ??? a=50
C_2=6969.5 H/M C_1=348450 H/M
C_3=-1362.7 H/M
C_2=1954.1 H/M
C_1=97700 H/M
C_3=14747.58 H/M
III ??????????? ??????
?????????? ???????, ??????? ? ??? ??????? ?????????????? ??????????? h=H*sin?t
T=(m_(1 ) (v_1 ) `^2)/2+(I_(1 ) ??_1?^2)/2+(m_(2 ) (v_2 ) `^2)/2+(I_(2 ) ??_2?^2)/2
?_1= f ?_1; ?_2= f ?_2;; (v_1 ) `=2af ?_1+h ? ; (v_2 ) `=1/2 af ?_2+h ?
I_1=(m?l_2?^2)/12=(m_1 ?(4a)?^2)/12=4/3 m_1 a^2
I_2=(m?l_2?^2)/12=(m_2 ?(3a)?^2)/12=3/4 m_2 a^2
T=1/2 (m_1 ?(2af ?_1+(h)) ??^2+4/3 m_1 a^2 ?f ?_1?^2+m_2 ?(1/2 a?f ?_2?^2+(h)) ??^2+3/4 m_2 a^2 ?f ?_2?^2 )==1/2 a^2 (16/3 m_1 ?f ?_1?^2+m_2 ?f ?_2?^2)
?T/f ?_1 =2m_1 a(2af ?_1+(h)) ?+4/3 m_1 a^2 f ?_1=16/3 m_1 a^2 f ?_1+2m_1 ah ?; ?T/f ?_2 =?1/2 am?_2 (1/2 a?f ?_2?^2+(h)) ?+3/4 m_2 a^2 f ?_2=m_2 a^2 f ?_2+?1/2 am?_2 h ?;
d/dt (?T/f ?_1 )=16/3 m_1 a^2 (f_1 ) �+2m_1 ah �; d/dt (?T/f ?_2 )=m_2 a^2 (f_2 ) �+?1/2 am?_2 h �; ?T/f_1 =?T/f_2 =0
????????????? ???????, ????? ?? ??? ? ? ?????????? ??????
?=1/2(a^2 ??f_1?^2 ??_1+?_2 (af_2-3af_1 )^2+ a^2 ??f_2?^2 ??_3)
??/(?f_1 )=f_1 a^2 (?_1+9?_2 )-3a^2 f_2 ?_2
??/(?f_1 )=f_2 a^2 (?_2+?_3 )-3a^2 f_1 ?_2
???????? ????????? ????????:
{�(16/3 m_1 a^2 (f_1 ) �(+f_1 a^2 (?_1+9?_2 )-3a^2 f_2 ?_2=-2m_1 ah � ) �@m_2 a^2 (f_2 ) �+f_2 a^2 (?_2+?_3 )-3a^2 f_1 ?_2=-1/2 am_2 h � )�
???? ??????? ? ????
f_1=B_1 sin?t ; f �_1=?-B?_1 ?^2 sin?t
f_2=B_2 sin?t ; f �_2=?-B?_2 ?^2 sin?t
??????????? ? ???????, ???????? h �= -H?^2 sin?t, ? ????? ??? ????????? ?? ?? Ha^2 sin?t.
{�(B_1/H (-16/3 m_1++ ?_1+9?_2 �)+B_2/H [-3?_2 ]=(2m_1 ?^2)/a@B_1/H [-3?_2 ]+B_2/H (-m_2 ?^2++ ?_2+?_3 �)=1/2a m_2 ?^2 )�
??????, ?? ??????? ???????:
B_2/H=(1/2a m_2 ?^2 (-16/3 m_1++ ?_1+9?_2 �)-3?_2 (2m_1 ?^2)/a)/((-16/3 m_1++ ?_1+9?_2 �)*(-m_2 ?^2++ ?_2+?_3 �)-9??_2?^2 )
K_(? )=B_2/H*r+1
?????
? ???? ?????? ???? ?????????? ????????? ?????????????, ? ????? ????????? ??????????? ??????? ???????. ???????????? ????????? ????????? ??????? ???????? ?? ???????? ????????? ?????? ?? ?????????? ????????, ?????? ????????? ???????? ?????????? ??? ????????? ????????? C_1=97370.1 H/M, C_2=1947.1,C_3=14768.58 H/M
? ???? ?????? ?? ???????? ????????? ?????? ?????????? ??????????? ???????????? ?? ?????? ?? ????????? 0.1 (1<K_?<0.99998), ? ?? ????? ??? ???????? ?????????? ???????? ?????????? 0.5 ???????????? ??????? ?????????? ?????? ??????? ???????? ?? ??? ?? ????????? ????????? ??????. ??????????
1 .?.?. ?????????, ?.?. ?????????, ?.?. ?????. ?????????????? ?????? ?????? ??? ?? ???????????? ??????????? - ?.:?????, 1992.
2 ???? ?.?. ??????? ???? ????????????? ????????. -?.:????. ??., 1986
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