О колебаниях в гироскопических системах при прохождении через резонанс
В настоящей работе рассматриваются колебания в гироскопических системах при прохождении через резонанс.
Збережено в:
| Дата: | 1953 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7739 |
| Теги: |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512731452407808 |
|---|---|
| author | Митропольский, Ю. А. Митропольский, Ю. А. |
| author_facet | Митропольский, Ю. А. Митропольский, Ю. А. |
| author_sort | Митропольский, Ю. А. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:12:02Z |
| description | В настоящей работе рассматриваются колебания в гироскопических системах при прохождении через резонанс. |
| first_indexed | 2026-03-24T03:33:27Z |
| format | Article |
| fulltext |
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336
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x 0 = -a1w1 sin (8+t,u1)-a2ro2 sin (r7+tf12),
yo=c 1a 1rv1 cos (O+t/J1)+c2a2 ro2 cos (O+tf·2).
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0 0 (J
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Bropol'I CJiyLiai1:
1{z:1) = 1"1 (7:1),
TpeTHH CJiyqafc
s
l' ( ri) = W2 ( r1 ), CL'1 ( r2) = r lfl2 ( 7:2), 1' ( 7:) =/= W 1 ( r).
CooTnercrBeHHO Ka)l<,li,OMY H3 3THX CJiylfaen onpe,ll,eJIHel\I BeJIHlfiiHbi
AF>(7:, a1, a2, lf'I> tp2) Af2>(7:, a1, a2, t/J1, 'I/J2),
B p> (7:, a 1, a2, 'I/J1, '1/'2) Bi2l (-r, at, a 2, '1/'1, '1/J,).
TioCJie pH,ll,a BbiKJia,ll,oK ,li,.!JH n epnoro cJiyLiaH no.!JylfHl\I CJie,ll,yioW.IIe Bbipa
)KeHHH :
Af!l('l:", al> a2 , t/J1 , 'h)= .2; A~~. Af2l(r, au a 2, 'f/J1 , 1/•2)= .2; A~.
!} ,,
(1 ,17)
B~~('r, a1, a2, V-'1> t/12)= 2: Bi~. Bj2l(-r, a 1, a2, '1/'t, 1/!2) = 2: BfJ,
!} <1
r,11,e A~~' A~~?, Bi~ 11 Bi~i HB.!J 5IIOTCH IICKOTOpbiWI n epiiO,li,HLiecKHl\!Il no
'1/h H 'f/J1 pe!l1eHH5IMH CIICTCJ\Ibl ypaBHeHHii:
2--r 2;-< 2:r
dA U> e-;o.p , f f f f . .
R aj d;" - L "piBiD = 4n 3 l h fi (-z:) f 1 (7:, A, x 0 , Xo, y0, Yo) COS (0 + tf')-
0 0 0
-gf/1:)/2 (7:, 0, x 0 , ;o, go, y0) sin(O+t/Ji)} eia'/J,dOd(A+tpJd(A+t/1~)+
2-"t l :t 2-"t (1,18)
+ e~;:,, J I J {h;i (r)/J(r , 0, x0 , ~o, Yo, Yo).cos (8+1f) -·
0 0 0
338
2:t 2ct 2:t
. . dB~~ Ul- e-N'P•j J Jf " . . .
R u;aJ dt- - L"iAlu-- 4:na lhlJ ('~)/1 (r, 0, x0, Xo, Ye• y 0) Sill (fi+tt'J)-
o 0 0
- g~j <~> f 2 ('c, 8, X o, ~o. Yo. Yo) cos ( (} + ?pj)} ei<r</•, d8 d (0+1t11) d (0 + 1/'2) -
2" 2:r 2:r ( 1, 18a)
- e~::,, J I I { h~J (~)! 1 ('r:, 0, Xo, ;o, y 0 , Yo) sin (0+ lf'J) -
0 0 0
(j= 1, 2).
3,LI,CCh BBC,ll;Cllhl 0603H3lfCHH51:
h~; ('l) = b22 ('l)-a22 ( 'l) [ u (•·- wi) + wJJ2,
h~/r) = bt2 (~)- a2~ ( c) [a(" - w2)+ W j]2,
g~j ('t-) = "/ (~) [ u (v-w 1 ) + wj],
gt ('l) =y(r) [a(,,-w2)+ roJ],
RaJ =a11 (~) {h~j ('l-) + h gi (~)}-a~2 ('l) C/'l) [gt{~) + g~i ('l)],
L a; = (2ai1 ( r) £tJ1 - }' (~) c/~)) [h~/~)+ h~i ('l)] -
- (2a22 (~) ci (") w 1 - y ('c)) (g~J ('c)+ g~J ( ~)) (j= 1, 2),
(1,1 9),
(1 ,20).
rrpHlJCM B C3,ll;C ,nucpcpe p e HI.I.HpOB3HHC H HI-ITCfpHpOB3!-IHC COBeprnaeM TOJihKO
no f, CO,ll;Cp)I\3UJ.CI\1YC51 HBHO, a 7: BC3,ll;C ClfHT3CM IlOCTOHIIHhiM rrap aMeTpOM.
)l.1H BTOj)OfO H TpCThCfO CJIYlJ3CB H 3 XO,ii,Hl\l:
1(il ( a a ··" ·•") = ~ '\:"'A(j ) BUl (~ a a ,,, ''' ) = ' "' ~ BUl (121)
.£. 1 ~, 1' ·2, 't' l' "t-'2 ..:::,; ~ l tJk ' 1 " ' 1 , 2' 't' l• 't'2 ...;., """"'-' } ak' ' '
<1 k (1 k
rJI.e AV~k H B\iJk (j= 1, 2) HB.:151IOTCH p e rne HHHMH CHCTe r-Ihi ypaBI·ICHHli:
dA1j l
p . -~- L .Bijl =
tuk7 df akj(t j l ok
2:-t: 2:-t 2:-t
- _!_ ei { u·(r t,'•,-sl(",)-k'/';} I I I h "k ( r)f (r 0 X ~ y y. ) X
- 4nS i j 1 • • o, •• o, o
0 0 0
2:-r: 2:t 2:T
- _!_ ei{O(r•;.·,-s•p,)-k'/';} f f f n"k (r.) / <~ 8 X X y y.) X
4 S u i j · 2 > > 0> 0 > 0 > 0 n .
0 0 0
(1 ,22)
339
()
R dBl~Tk L A(i) -
•1kjai dt - + '' k j 1r1k -
2:-t 2:'< 2:t
- 1 i{ u(r'/J,-•'/J,)-k'l';} fff/· "k( )f { 8 . . ) X -- 4ns e Iii r 1 r, ' Xo, Xo, y ' Yo
0 0 0
X e-i{a (ri/',-•>P,l-k'l';} sin (H+ 1/Ji) dli d (8+ t/Ji) d (fJ +t/J~ ) -
2:-t 2n: 2:<
1 i{ <T( r1p,-s.p, )-k'/•·} j' f f <Tk )/ ( 0 . . X - 4n 3 e ' gii i r 2 1:, , x0, Xo, 1{11 , .lio)
0 0 0
X e-i{u(r.p,-s.P.I-ki/Jt} cos (tJ+tJJ)d11d(f1+t/J1 )d(ll+ 11-·~ ·-
ld(IJ. de.) dw . de. - a J J "k . 1 , .1 a. 11 -d - y -d h .. (r)+a . a 22 c. - +2a22 w. -
; r r •J J '· J d r 1 a-c
<>k (t')
L i j , '
npHqellt Hl\Ieel\f o603Hal!eHHH:
h~k ('£') = b22 (-c)- a22 (-,;) [/l (,-wi) +a(rw1-sw2)+ . y,
07f(-c)= y (-,;-)[fl(Y-wi)+a(ruh-SW~)+wj] (j= 1, :2);
(1 ,23)
i = 1 COOTBCTCTByeT BTOpOMY CJiyqaiO, i = 2 - TpeTbeMy CJIYLI<J IO.
J<poMe TOrO, KaK B CHCTeMe ypaBHeHHH ( 1 ,18), T<JK H B ypamieHH51X
( 1 ,22) TIO)J, TIOJIHOH npOH3BO)J,HOH TIO BpeMeHH CJie.rr.yeT TIOH.Hl\WTb C~le:J.yiOlll,ee:
dAI1 oA~~~ oAi~i
- dl = - - (rol-l')+ -- (ro2- J•). (1,24)
iJt/J1 iN'2
ITocJie Toro KaK HaMH Hafr.rr.eHhi HBHhie Bhipa:t~<eHHH Jl.JIH ne.!JHljHJ-J
A~l (T, al, a" t/Jl, tf12), J3(,il(-,;-, al> a2, 1/Jl, '1/!2). a TaiOKe )J,JI51 <PYHI<UH}i llj c~. 0,
.al, as, B+tpl, 0+1p2) H V1 ('£', 8, a11 a~, O+tp1, O+ l/12), Mbi li!O}I\C\1 HarmcaTb
PHC. 1
peweHHe CHCTeMhi ypamrer-mii ( 1,1) B n epno 111 npH-
6JIH)KeHHH II B ,yJiyqmeHIIOM" nepBOM TipH6JIH:ti\C
HHH, T. e. C yqeTOM Bb!CUIHX r a pli!OHHK KO.TJe6aHH51.
2. ITepeH:.rr.eM K paccl\1oTpeHmo, c noMOlll.hiO
H3JIO)KCHHOrO MeTO)J,a, I<OJie6ai-IHH TipH npOXO:ti\)J,e!HIH
qepe3 p e30H aHC B I<OHKpeniOi·i rHpOCKOTIHLfCCKOii
CHCTel\IC.
PaccMoTpHM cxe~r y rHpocKon a, rnre1omero npH
!\ICHeHHe B HeKOTOphiX KOHCTPYKUHHX ue1rrpmpyr,
cenapaTOpOB H T. )1,. (pH C. 1) .
MaTepHa: IhJ-Ia5I och rHpocKona AD y,TJ:ep}IUI
DaeTCH B Bepn!K<JJ!hl-!0111 TIOJI()}({('HHH JlBY \15! OTIOp :HIH
C H D, npHl!eM nepxHHH onopa C ynpyraH c O)J,H
Hai<oBoii )KeCTKOCThiO I? no nce~I pa,!l.Hyca\r. M mKHO
6b! .T!O 6bl paCCl\lOTpeTh CJlyl!a ii, KOJ',Ua ;.KCCTKOCTb
BC p X!-ICH OnOpbl 3<JBHCHT OT nepeMClll,CHH5! OCH H npHtJC~ ! 3Ta 3<JBHCHMOCTb
IleJIHHef:'ma ; O,li.HaKO l\Ihl He 6y)J,e:\l YCJIO)KJ-!51Th 3<l ,'I,aqy.
YnpyrocThiO ocu mpoci<ona 6yneM npene6peraTh, cqHTaH ee a6co.TJIOTHO
)!{eCTI<OH.
340
PoTop mpocKona B npe,ll,nOJIO)KHM Hc ypaBI-IOBerneHHbiM KaK cTaTH4eCKH,
TaK H ,li,HH3MH4eCKH, KpOMe TOfO, M3CC3 pOTOpa 3 H34HTeJibHO npeBOCXO,li,HT
Macey ocH mpocKona, 4TO ,11,aCT Hai\1 npan o npe11e6peraTh nocJie,li,Heif.
06o3Ha4HM qepc3 a H fJ yrJI'bi, onpc ,ll,eJHUOill,He nOJIQ)Ke~me OCH rHpo
CKona OTHOCHTCJ!biiO HellO,li,BH)KIJOH CHCT CMbl KOOp,li,HHaT XljZ C H343JIOM
oTc4eTa B T04Ke D, oco Dz HanpanJieHa Be p THKaJibHO nBepx. ITy cTb e H :J -
JIHHeHHbiH H yrJIOBOH 3KCl1,eHTpHCHTe Tbl p oTopa; /1 - MOM eHT l iHCpl1,HH po
TOpa OTHOCHTeJ!bHO nonepe4HOH OCH, npo Be,ll,eHIIOH n e pneH,li,H KYJI51pHO OCH
CHMMe TpHH H npOXO,li,51Ill,e H 4epe3 HJUKHIO~ OllOpy D; f -r p a CCT051HHC OT
no,li,IlHTHHKa JlO u.c~ITpa TmKCCTH poTopa; L - paccTOHHHe or no,D.IlHTHHKa
)lO B e pxHe(I OllOpb! C; f - MOMeHT HIIC])U.HH pOTOpa OTHOCHTeJibHO OCH CHM
MeTpHii; J - MOMCHT H!IepU:HH poTOpa OT HOCHTCJibHO nonepe4HOH OCH, npo
Be,li,CHHOi1 n e pnCH,li,HKYJIHpHO OCH CHi\IMeTpHH H npOXO)lfllll,CH 4 epe3 l1,Cl-!Tp
noi<eCTH poropa; P - Be e poropa; m - M acca poropa.
Tor)la ,li,H¢¢epenu.uaJibHbie ypaBHeHHH )lBmKeHHH ocH mpoci<ona 6y ,11,yT
HMCTb CJie,ll,yiOilJ,HH BH,ll,:
d~!X dO df3 [ (d0)1 d 20 . ] d 20
I1 dtt +I -dt. dt + b!X=A dt cos 8+ dtt sm 0 -I dt2f3,
f,li,C BBe,ll,eHbl 0603H34CHH51:
(I - J).'J-+ mle = A,
KpoMe Toro, B 3THX ypaBHeHHHX HMee M:
kU-Pl = b.
dO
dt = v(~)=vo + rt,
(2,1)
(2,2)
y - riOCTOHHHOe y rJIOBOe ycrwpeHHe (HJIH 33Me,li,JieHHe) C06CTBCHH0f0 Bpa
Ill,CHHH pOTOpa, Bbl3 B3HHOe riOCTOHHHbiM MOMeHTOM, BHC33IlHO npHJIO){<eHHbiM
K OCH rHpOCKO!la. 3-aMCTHM, tiTO l ' (~) MO)KCT 6b!Tb npOH3BOJihi-!OH ¢YHKl1,HeJt
BpeMeHH, H 3TO He llbi30BeT IlpH perueHHif 33)l34Jf HHK3KHX ,li,OllOJIHHTCJihHhiX
33TPYJlHeHHH.
ITo,11,po6H'hiH BbiBO,ll, npHBe,ll,eHHhiX 3,li,CCb ypaBHeHHif (2, 1) ,li,JIH cxeMhi
rHpociwna, npHBe,ll,eHHOH Ha pHc. 1, HMeeTCH B pa6oTe A. A. CMeJIKOBa [3] .
ITocTpOHM npH6JIH)!<eHHoe pewe1-me CHCTeMhi ypaBHCHHH (2, 1) npH no
MOill,H npHBe)le HHbiX Bbirue ¢opMyJI. YpaBJJeHHH (2, 1) - JIHHeHHbie, npH4eM
nepeMCHHh!MH K03<lJ¢HU.HeHTaMH 51BJI51JOTCH TOJihKO K03<lJcpHl1,HCHThl npH rHpO
CKOilH4eCKHX 4JieHaX, a T3JOKC 3MllJIHTY,ll,b! H 43CTOThl BHCWHHX nepHO,li,H4C
CKHX CHJI. DJI31'0)l3p5! 3TOMY Mbl IlOJIY4HM 04eHb npOCTbJe ypaBHeHHH nepBO!'O
I1pH6JimKeHH51, 4HCJieHHOe IfHTerpHpOB3HHC KOTOph!X He npe,li,CT 3BHT HHK3KHX
,li,OllOJIHHTCJihHbiX 33Tpy ,li,He J-!HH.
flpe,li,IlOJIQ)KHM, 4TO npH paCCMOTpe HHif HCCT3l1,HOHapHOfO npou.ecca
13 rHpOCKOilH4CCKOH CHCTCMC, OllHCh!B3CMOH JlHcpcpe p e Hl1,H3JihHbiMH ypaBHC
HHHMH (2, 1). Mhl HMCCM nepBhiH CJJyqaif, npiiBe,ll,eHHbiH Ha CTp. 338, T. e. ,li,JIH
HCKOTOpb!X 3H34eHHH 'C H3 paCCM3TpHBaeMOM HHTCpBaJie 0-< 'C -< L B03·
MQ)KHhl CJIC,li,YIOill,He COOTHOWeHHH MC)!{,ll,y l' ('C), fo1 (r) H W~ (r):
s
v (T1) = W1 (T), l ' (c2)= W2 (r), w1 (r) =F- w2 (r).
r
7. YKpau:rrcKutt MaTer.ta.T. >KypHali, T. V. N11 3. 341
Tor.u.a .U.JISI rroCTpoemm p e rueHHSI HaM Heo6xo.U.HMO BOCITOJib30BaTbCSI cpopMy
Ji aMH (1,8), (1,4), (1,5), (1,17) H ypaBHeHHSIMH (1,9) H ( 1,18).
CorrocTaBJISISI ypaBHeHHSI (2, I) c ypaBHemmMH (I, I), HMeeM
a 11('£)= a22('£) =/1> y ('£)=h('£), b11 (r)=b~2 ('£ )=b .
.[l. aJiee, COrJI3CHO (I ,4), COCTaBJISieM CJi e.U.y iOW:ee xapaKTepHCTH'leCKOe
ypaBHeHHe )I.JISI orrpe)l.eJieHHSI ,co6CTBeHHb!X" qaCTOT:
(2,3)
OTKy.U.a H3XO)I.HM
( ) _ V ftv2 ('t-) + 4b/1 - f,, (r )
w1 r - 2/1 , (2,4)
1
/1 T (b + l v (r) cos)- b
1 = 1
/v (r)w2 ·
(2,5)
lhaK, p e rne HHe CHCTeMbl )I.HcpcpepeHU.H3JibHbiX ypaBHeHHH (2, I) Mbl MO
)I(eM 33ITHCaTb B BH)I.e:
a=a1 cos (A+V'1)+a2 cos (A+tp2),
{J =a1 ~in (6+ 1/11)+ a2 sin (A+V;2).
d8
(2,6)
r.u.e dt = l'o+ rt= v(-r) (Mbi rrHrueM v(r), TaK KaK npe.u.noJiaraeM, 'ITO
y - BeJIH'lHHa ITOpSI)I.Ka MaJIOCTH e), a BeJIH'lHHbl a 1, a 2, l/J1 H 1/J2 )I.OJI)!{Hbl
6b!Tb orrpe.u.eJieHbl H3 ypaBHeHHH n epBOfO rrpH6JIH)!{eHHSI, K COCT3BJi eHHIO
KOTOpb!X Mbl ceifqac H rrpHCTYITHM.
3'aMeTHM, 'ITO B CHCTeMe )I.HcpcpepeHU.H3JibHbi X ypaBHeHHH, OITHCbJBaJOru:e ii
II,BH)KeHHe OCH rupocKorra, He Y'!TeHo TpeHHe. n ocJie)I.Hee MO)!{HO yqeCTb,
paccMaTpHBaH, HarrpHMep, CJie.u.yJOW:YIO .u.uccunaTHBHyio cpyHKU.HIO PeifJieH :
(2,7)
r.u.e b1, bz, h - HeKOTOpbie KOHCTaHTbl.
EcJIH Mbi yqTeM TpeHHe, TO rroJiy'lHM CJie.U.yioru:He Bbi pa)!<eHHH .U.JISI B03-
Myw:mou.I.HX CHJI , )l.ei.fcTBY!OillHX Ha Hawy rHpOCKOITH'l eCKY!O CHCTeMy:
( drx d~) e/ 1 r , 0, rx, dt, (1, dt =
=A [t~2 ('£ ) cos A+v' (r) sin 8] -lv' (r){J -b1 ~;-h ~~,
( drx d~) E/ 2 '£, 0, a, dt ' {J, dt =
(2,8)
= A [v ~ (-r) sin 0-y' ( r) cos A]+ fy' (r)a-h :; -- b2 ~:.
342
drx dfl
no)l.CT3BJI5151 B rrpaBbie 43CTH 3TlfX Bbipa)KeHHH 3H34eHH51 a, {J, dt , df ,
H3H)l.eHHbie H3 (2,6) , IIOJIY4HM:
ef1 (~, 8, rxo, ~:), {10 , ~o)=A [1'2(<) cos 8+y' (~)sin 8]
- b/ (~) [a1 sin (8+tp1) +a2 sin (8+ tJ.'t] +
+b1 [a1 w1 sin (8+tp1)+a2w2 sin (8+tp2)]-
- h [a1w1 cos (8 +tJ11)+a2(V2 cos (8+tp2)];
E/ 2 (-r, 8, a0, ~o, flo , ~0 ) =A [yz (r) sin 8-y' (~)cos 8] +
+ lv' c~> [al cos (8+tpl)+a2 cos (8+11'2)]+
+h[a1w1 sin(8+11'1)+a2w2 sin (8+?J12)]
-b2 [atw1 cos(8+1/'1)+a2w~ cos (O+tp2)].
(2,9)
nepeH)l.eM Terrepb K COCT3BJieHHIO ypaBHeHHH ( 1, 18). )lJUI 3TOrO He06XO
)l.HMO IIO)l.C4HT3Tb HHTerpaJibl, CT0511ll.He B rrpaBbiX 43CT51X ypaBHeHJi51 (I , 18),
rrpe)l.BapHTeJibHO IIO)l.CT3BHB 3Ha4eHH51
. . . .
f 1 (~, 8, rxo, a0, flo, flo), f 2 (~, 8, ao, ao, {Jo, {Jo). (2,9')
3TH HHTerpa.Jibl 6y)l.yT OT.JIH4Hhl OT HYJUI )l..JIH C.Jie)l.yiOIIJ:HX 3Haqe~mH a= 0,
+ I, - I. nocJie p51)1.3 Bb!K.Jl3)1.0K IIOJiyLJaeM ypaBHeHH51, orrpe)l.eJI51IDill.He
a= 1:
(j=l, 2, a=O, +1, -1):
a=O:
dAVJ . .
3!1 (if- 3(2/1wi- !Y(~)) ai~~=ha/JJi,
dB(j)
311aj d;o + 3 (2f1uJj -lv (~)A\{!=
dw .
= -[(b1 +2b2)(oi-3fy'(~)]ai- 3f1ai d~J (j= 1, 2);
dA(i )
a=-1· I 1'-1 - (2/ro.-/v(r))a.BU) =
. 1 dt 1 J J 1, -1
1 ., ( -i•P · 1 , ( ) - IV• ·
= 2 AI'" ~) e ' + 2i A1' r e ',
dB(j)
fa . 1· - 1 +(2/w.- !Y(·r:))a .AU> =
I J df 1 J J 1, -1
1 2() -i11' ' 1 '() -i'/J· = 2i A1' ~ e ' - 2 Av ~ e ' (i= 1, 2).
(2,10)
(2,11)
(j=1, 2);
(2,12)
343
Pemas:r 3TH ypasHeHHSI, Haxo.n:HM :
I dwi
[ (bl + 2b2) (t) j - 3h' ('~) ] 1 ---;;:;
AU> - - --=-=-o-~--;---,--,-,--- a· - ---- ai·
10- 3(211£oj-h(r)) J 211wi-h '
1 .> hwj BJ =- .
10 3(2/li.Vj-/v(,;))'
~Av2 ( ,;) e;11'i + .!_ Av' (,;) ei1f';
A<i> __ 2t 2
1,1- 11 ((oj+v(,;))- /J.(,;)
BU) =
1, 1
1 A 2 ( ) -i•P· 1 A '( ) - i•P· - ,, ,;e , _ _ v re '
A<i> = 2i 2
1, -1 ldwi+v('c))-b('r)
A • ( ) i•P,· 1 A I ( ) - il/1.
2 , .• ,; e - 2i ,, "" e 7
BU> =
1, -1 ai [11 (roJ +,')- 11' (r)J
(j= 1, 2) .
•.
(2,13)
DocJie 3TOfO, BOCITOJ!b30BaBWHCb ¢ o pMyJioi1 (2, 17), lv!Q)KeM H3ITHC3Tb y p aB
HCHl15! rrepsoro rrp116JIH)KCHHH, o npe.neJrmow.He a 1, a2 , t/J 1, 1/'2:
1 dw1
da1 [(b1 + 2b2) w1 - 3lv' ('z;)] 1 d r
- =- a - a-
dt 3(2/1w1 -h(-r)) 211w1 - h(r) 1
A1'2 (,;) sin t/'1 - .AY'('r) cos 1/-'t.
l 1(ro1+ l' ('£))-/J.(,;) '
1 dw2
[(b1+2b2)(1>2- 3/JI (r)] 1 d·c
=--=-;.;-~;-=------=----,-~.,---.:---'-" ao - a.2 -
3(211w2 - I~'(T)) " 211ru2 -/J.(,;)
Av~ ( r) sin 1/'2 -A v' ( ,;) cos t/J2 •
11(w2+ Y(,;)) -I~~~'
(2,14)
3a~reTHM, lJTO .nmKe rrpH .nocnno:nro 6o.llblUHX crwpocTHX npoxon<..rr.eHHH
4 CpC3 {)C30H<l i!C a6COJ!IOTH35! BC.!!HlJH l l3 /I- ' (·c),l1r•1/(r) << b1Wj('l.), 11, KpO ll lC
Toro, ,,, (,;) < < ''2 (,;) . TaimM o6pa30M, 11e uapyma5I TO'lHOCTH Bbrcnrc.rr eHHH,
Mbl MQ)I\C~.I neJIHlJHIIbl /J/ (1:), 11 ro/ (r), ,,r (·c) OTHCCTH 1<0 B'rOpO\IY npH6JIH·
)KeHmo 11 B rrepsoM np116JIH)KeHm1 nx sosce He Y4HTbiBaTb . Kpo liie Toro,
34'4
,D,JHI ynpow.eHHH noJIOlliHM B ,D,HCCHnaTHBHOH cpyHKU.HH P eliJieH: h = 0,
br = b2 = <5, TOr,D,a ypaBHeHHH nepBOrO npH6JimKeHHH rrpHMYT CJie,D,yiO
W.HH BH,D,:
owla1
2/1 (1)1 - 1v ('r;)
da2 O'ro2a2 At'2 (r;) s in 1/!2
dt 2/1£v2-/J, (r) f 1(£o2+1'('r;))-/v(r)'
(2, 15)
di./Jr A1•2 ( r) cos 1/.!1
dt = ro1(r)-l' (',;-) - ad/1 (w1+v(r))-lv('r;))'
d'!fJ2 = w2(-r;)-Y (r) _ Av2(r) c os 1!J2
dt Ct2 [/1 (1v2 + l' (r)) - !J, (")) ·
l.J HC.TieHIWe HI-!Te rpHpOB3HHC JlOJIY4eHI-IOH CHCTeMbl ypamre r-!HH rr e pBOrO I1pH-
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,D,aeTCH Ha ,D,Be He33BHCHMb!e ,D,pyr OT Jlpyra CHCTeMbl ,D,BYX ypaBHeHHii, 'ITO
ew.e 6oJrhwe o6JJert.IaeT miTerpHpOBaJUie.
0CT3HOBHMC51 Tenepb Ha BbiBO,D,e COOTIIOUJCHHH , npH flOMOlUH KOTOpbl X
l\JQ)KHO f10CTp011Tb 33Bl1CI1MOCTb 3Mf1JII1T)',ll,bl KOJJE'6am151 OT 4 3CTOTbl BHeLUHeH
CHJibl (pe30H3HCH)' IO KpHB)'IO) npH CT3U.110H3pHOM pe)J(HMe. Ll,JJH 3TOfO HC06-
XO,ll,HMO rrpHpaBHHTb npaBbie t.IaCTH ypaBHeHI1~i (2,15) HyJiro H 113 noJryt.IeH
Hbi X I3blp3)I<C!l11H HCIUI!OtiHTb cpa3bl I<OJJ C631UIH I./J1 11 1/J2 . DocJJe 3TOrO flOJiy-
4HM CJJe,ll,)'IOLUHe COOTHOWeHHH, )1,3 10lli,I1C B03MO)IU!OCTb ITOCTp011Tb pe30H3HC
H)'IO KpHB)'IO ,ll,JIH 3Mf1JJHTY,D,bl K3)J<}:l0j:l OT,ll,CJ i bi!O~I rapi\10HHliCC!<OH COCT3 -
BJ!51IOW.eH:
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Hbl ~IH, 3 3313 HC51T OT V COrJ!3CHO Cj:>OpM yJiaM (2,4) .
3a~ICTI1\1, tiTO Bb!p3)KCHHH (2,6) J\I Q)I< l-10 rrpe,D,CT31311Tb 13 BH,D,e:
u = U cos (U+x),
(J=U sin (8+ x),
(2,16)
ITOCTOHH-
(2, 17)
f,ll,C U H X -·- 3Mf1JII1T)',ll,3 C)'J\ I M3pl-!Of0 KOJie6aHH51 H Cp330BblH C,ll,BHf - onpe
,D,eJI51IOTCH q:>Op M)'JiaMH:
U= V (a1 cos 1/.!1 +a2 cos 1p~)~+(a1 sin 1/J1 +a2 sin t/J2)~
a , sin 1/•1 + a 2 sin tfi2
t <r x= - -------- .
"' a1 cos 1/.!1 + a2 cos 1/'2
DpHMeHHM rroJIYtiCHHbiC cbopMyJibl K 411C.JICIIHO~ I Y rrpuM epy.
(2 ,18)
(2,19)
Ll,onycTHM, rrapaMeTpbi paccMaTpusa e~wli Hal\IH cxc\Ibl nrpocKorra HMeJoT
CJJC,ll,)'!OLUHC 3HatieHH51:
345
nee poTopa nrpoc 1mrra P =-= 5 Kc;
paCCT051HHe OT 1-IIUKHe H 0110pbl ,LI,O U e i-ITpa TmKeCTH pOTOpa rHpOCKO!la
l = 25 eM;
paccTomme ~ I e)K.UY HH)KHe ii H nepx1-1ei1 onopaMH ocu nrpocKorra
L = 15 eM;
)Ke CT KOCTb Be pXHCH 0110ph1 fl = 5 Kc/C~t ;
MOMeHT mre pUHH pOTOpa OTHOCHTeJlbl-10 OCH CH ~!MeTpHH J =0, 1 Kc•C,\t •CeK2;
MOM e i-IT HHe pUHH poTopa OTHOCHTeJlbl-10 rr onepe •-IHOH OCH, rrpoBe,LI,eiiHOH
rreprreH,LI,HKYJ151pHO OCH CHMMeTpHH H rrpOXO,ll,51lll,e H 4epe3 HH)KHIOIO Orropy,
/1 = 0,2 Kc • CM. ceK2 •
/
/
/
0 50 100 150 "
PHc. 2.
Dpn peweHHH 3a,LI,a'IH orpann<IHMCH TOJibKO noCTpoe~mellr pe3oHaHCHhiX
KpHBbiX K3K CT3UHOH 3 pHbiX, TaK H rrpH p33JlH'IHhiX pe)JU!M3X rrpOX Q)K,LI,eHH51
q epe3 pe3onaHc. B 3TOI\1 cJiyqae HeT H e o6xo,LI,H M OCTH Haxo,LI,HTb a6co.niOTHbie
3H3'leHH51 3M il.11HTY.U a1 H a2 , ,LI,OCTaTO'IHO llOCTpOHTb rpacjmKH ,li,J151 OTHOWe -
al a2
HHi-'1 A H A (3TO B03 MO)KIIO C,LI,eJ13Tb B UIJlY JlHHei-iHOCTH ypaBHe HHH
( 2, I)). Do3TOMY MmiOIO H e 3 a .uanaTh 'IHCJi e i-IIIble 3 H a 'leHHH napa M eTpon
rHpoc Korra, x apmne pH3 Y IOIUHX n eJm<Inny A, T . e . 'IHCJleHHbie 3Ha<Iemm J, e
H 8. Ko3cpcj:mune HT Tp e HHH npnMe M (5 = 0,02.
D o .ucTa B.rmH npHBe.ue HHbie 'IHCJieHHbie 3 HatJ e HHH B cjJOpMyJihi (2,4),
H3XO,li,H M
\'800 + 0 b-2 - 0 2J·
() - ' ' wl ,. - 0 4 '
'
~ ---------.,
w2 (r)= }800 + Q,l v"+ 0,2 ,,
0,4
(2,20)
rrpH J' = 0, DJI = ·W2 = 70,7 1/ceK. 3 a i3HCHMOCTh fo 1 II W 2 OT ~' I1pH13CJI:CHa Ha
pHC. 2, !1 3 KOTOp OrO 51 CHO, 'ITO npH H3 1\1 CHeHHH 1-' - qaCTOTbl B HCWHCJ'i
CHJlhl - OT 0 JI:O 100- 11 0 1 I ceK n c HcTe i\re .una)I<JLhi i\rmr<eT B03 1lHKHY Tb
p e30Hanc: n e pnbii-i p a3 npH ~, = 59 1/ce" 11 nTo p o i1 pa3 npH ,, = 101 1/ ceK.
BocrroJlb 30BaBmHCb cpopMyJraMn (2, 16), noJiy'IHi\1
(j = 1, 2), (2,21) 1/ ? 50cu'
(foj+ 0,5v) t (crlj - 1')" + (2wj-O'~S J•) 2
A 1 =10A.
346
npH DOMOlli.H KOTOpb!X CTpOHM pe30H3HCHhie KpHB'hie npH CT3U.HOHapHOM
p eMHMe, xapaKTepH3YIOW:He H3MeHeHHe 3MDJIHTYJI. K3)K}I.OH OT}I.eJibHOH rapMO
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1
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a2
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1
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8
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PHc. 3.
PHC. 4.
,llJIH 3H3JIH33 DOBe}l.eHHH HCCJJe.n.yeMOH rHpOCKODHtJ:eCJ<OH CHCTeMbl npH
pa3JIHtJ:HbiX pe)!<HM3X npoxo.n.a qepe3 p e30H3HC He06XOJI.HMO tJ:HCJieHHO npo
HHTerpHpOB3Tb ypaBHeHHH n epBOfO npH6JIHMeHHH (2,15) }I.JIH pa3JIHtJ:HbiX
CKOpOCTeH H3 MeHeHHH qacTOTbl B HeiiiHeH CHJ!bl v(-t) . Mbl Y)Ke OTMetJ:aJIH, 'ITO
'l ' (-t) MO)!<eT H3MeHHTbCH DO JII060MY 33KOHy, MOMHO }l.a)Ke 33KOH H3MeHe HHH
1• (-t) 3 3}1.3Tb C DOMOW:blO rpa¢H'IeCKOH xapaKTepHCTHKH. ,llJJH ynpomeHHH
npHMeM, 'ITO l' ('t) H3 MeHHeTC5I no JIHHeHHOMY 33KOHY 'I' ('t) = 1•0 +rt H HCCJJe
.n.yeM nose.n.etme cHcTeMhi JI.JJH 3HatJ: eHHH y = 8 1jceK2, 20 1/ceK2, 50 1/ceK2 .
.3TH 3 HatJ:eHH5I y COOTBeTCTBYtOT pa3rOHY rHpOCKODHtJ:eCKOH CHCTeMbl.
f10JI.CT3BJIH51 tJ:HCJi e HHbie 3HatJ:eHHH B ypaBHeHHH (2, 15), l!MeeM
d(a;)
A 14,14wi (aj) 10,2 (•) •
-cit= - 4£vj-J!('L) A - (2cu;+v('~')) 8m1pi'
d!./Ji l01•' ('r) - = Wj - v (r) - COS 1/J. , (2,22)
dt (:)(2cui+v('t)) J
347
r.ue j = 1, 2, v (r) = vo + rt,
HIITerpnpyH CHCTeMbl ypaBHeHI1i':i (2,22), ITO.!IycweM pe301IaHCHbie KpHBbie
rrpoxOJI<~TJ.eHim q epe3 pe3oHaHc .U.JIH y.cKopeHHH ?' = 8 1/ceK2, 20 1/ ceK2, 50 1 (ceK~.
Ha pnc. 3 3THM ycKopeHHHM cooTBeTCTBYIOT KpHBbie il'ftLtNt, MtFtDt
H il'ftGtTt, Ha pHC. 4- KpHBbie MzLzN2, M2F2D2 H MzG2T2.
u .
l I
A~~--~----~~~~~~----~1~;o~~~~~--~~J~0-----~~J~0--
PHc. 5.
u D,
PHC. 6.
Ha pnc. 5 rrpnse.ueiibi KpHBbie 3 aBHCHI\IOCTH U oT qacTOTbi ( corJiacHo
cpopMyne (2,18)) KaK .UMI cTaUHOHapiioro p O KHMa - KpHBaH ABCDE,
TaK 11 .U.TIH pa3.!IHIIHbix ycKopei-mif- KpHBaH AtBtCtDtflt- .U.TIH l' = 20 1/ceK1,
KpHBaH A 2B 2C2D 2E2 - .U.TIH )' = 50 1/ ceK2• ll:JJH IHirJIH.UHOCTH Ha p11c. 6 npH·
a
Be,UeHa 33BHCHMOCTb A
1
= U cos (O+x) OT spe M e HH, ,ll..TJH oTpe3Ka speMeHH,
B Teti CHHC KOTOpOrO HCCJJ e.UyeMa5I KOJJ e 6aTeJJbHa5I CHCTe Ma ITpOXO,UHT qepe3
o6a pe30HaHCHbiX 3 H3tiCI!HH.
KpnBbie pnc. 7 xapaKTepn3yioT 3aBHCHMOCTb x OT ~' rrpn CTaUHOHapHol\1
pe.iKH~Ie H pa3JIH4Hb!X CKOpOCTHX rrpOXO.iK,UeHH5I qepe3 pe3 0H8HC.
Jl:JJ5I npaKTI!liCCKHX U CJJei{ BITOJ!He ,UOCTiJTO'IHO orpaHHl!HTbCH ITOCTpOe
I·fHCM rpacpHKOB, rrpHB C~'l,CHHbiX Ha pHC. 5 H 7, KOTOpbie ITO.TII·!OCTbiO xapaKTC-
348
· pH3YJOT llOBe)l,eHHe CHCTeMbl npH npOXO)K)l,eHHH qepe3 pe30H3HC - CMelll.eHHe
H CHHMeHHe M3KCHMYMOB pe30H3HCHOH Kpr!BOH npH npOXO:tK)l,eHHH qepe3
pe30HaHCHbie 3Ha'IeHH5I 'I3CTOTbl, B03HHKHOBeHHe 6HeHHH, H3MeHeHHe C)l,BHrot
MeM.ll.Y 4Ja30H B03Mylll.a!Olll.eH CHJibl H 4Ja30H KOJI•e6aHH5I H T. )1,.
Jl!
200'1-------,------,--------,
0 100 150 v
PHc. 7.
Ha no)l,po6HoM aHaJIH3e noJiy'leHHbiX pe3oHaHCHbiX KpHBbiX Mbl He ocTa
HaBJIHBaeMCH, TaK K3K OH npOH3Be)l,eH B HaiiieH pa6oTe [4).
J111TEP ATYP A
I. H. M . K p bi JI o B H H. H. 5 o r o JI 10 6 o B, BBeAeHHe B HeJIHHei!HyiO Mtxa
'HHKy, KueB, 1937.
2. IO. A. M H T p o TI o .nbc K H il, M eAJieHHbie Tipoueccbi n HeJIHne i1HbiX KOJie6aTenb
·HbiX cHCTeMax co MHQrHMH CTeTieiHIMH cno60Abl, )I(ypHaJI flMM, T. XIV, B. 2, 1950.
3. A. A. C Me JI K 0 B, 0 KOJie6aHH51X B rHpOCKO!lH'lCCKHX CHCTeMaX llpH 60JibWHX
·cKopocTHx npaweHHH, 11mKeHepHbiH c6opHHK, T. VIII, 1950.
4. IO. A. M n T p o TI o JIb c K H i1, BbiHY)I{ACHHbie KOJie6aHHH B HeJinHei1HbiX cHcTe
.Max TIPH llpOXQ)KA CHHH tJepe3 pe3oHaHc, 11IDKeHepHbiH c6opHHK, T. XV, 1953.
TIOJI)''ICIIa 31 fiiiBapH 1953 r.
Kueu.
0334
0335
0336
0337
0338
0339
0340
0341
0342
0343
0344
0345
0346
0347
0348
0349
|
| id | umjimathkievua-article-7739 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:27Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1c/f934600960e89a8ea7e41873c1aa7f1c.pdf |
| spelling | umjimathkievua-article-77392023-08-15T10:12:02Z О колебаниях в гироскопических системах при прохождении через резонанс О колебаниях в гироскопических системах при прохождении через резонанс Митропольский, Ю. А. Митропольский, Ю. А. В настоящей работе рассматриваются колебания в гироскопических системах при прохождении через резонанс. В настоящей работе рассматриваются колебания в гироскопических системах при прохождении через резонанс. Institute of Mathematics, NAS of Ukraine 1953-08-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7739 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 3 (1953); 333-349 Український математичний журнал; Том 5 № 3 (1953); 333-349 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7739/9436 Copyright (c) 1953 Ю. А. Митропольский |
| spellingShingle | Митропольский, Ю. А. Митропольский, Ю. А. О колебаниях в гироскопических системах при прохождении через резонанс |
| title | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_alt | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_full | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_fullStr | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_full_unstemmed | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_short | О колебаниях в гироскопических системах при прохождении через резонанс |
| title_sort | о колебаниях в гироскопических системах при прохождении через резонанс |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7739 |
| work_keys_str_mv | AT mitropolʹskijûa okolebaniâhvgiroskopičeskihsistemahpriprohoždeniičerezrezonans AT mitropolʹskijûa okolebaniâhvgiroskopičeskihsistemahpriprohoždeniičerezrezonans |