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Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Next 3 Page(s) In Document Denied Q Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 1d8THODS FO$ CONSTRUCTION OF PERIODIC LOTIONS IB PIECEWISE LINEAR SYSTEMS ( A suuvey of Soviet ~9orks ) The development of self oscillations and forced oeoillati- ons in non-linear systems comes to the statement of proper con- ditions of existence for stable periodic solutions of non-linear lifferential equatiens which govern the motion in question. Here we shall not discuss the problems connected with con- ?itions of periodic solutions stability. Our interests are to be restricted to the evaluation of periodic regimes or to the indi- .~ cation of conditions for their existence To construct the periodic regiotea a variety of approximate methods is widely used up to the date , these methods based an the assumption that the system under discussion slightly differs from some other systen which can be described by already known In this survey, only general problems of this kind are consi- dered, the order of equations not in view. Varions specific prob- lems for the second and the third order of equations were left ont of the scope. " generating " periodic solution. This assumption leads to the following mathematic,:. deduction ;the periodic regime to be found does Yorrcaliy slightly differ from generating, usually ga*.,~on~c solution. This statement though of use in different problems of mecha- nics, electrical oscillations and theory of autumati,: control turns out to appear not sufficiently accurate and ,:crrescoa- ding to the nature of phenomena in debate. Some attention was drawn in this sense to cart;:.:. methods of exact solutions evaluation ,for non-linear differential equations. Exact methods are in need for verification of a;.,~'i- cability and accuracy of the approximate ones as well as for the presentation oY periodic regimes when the assumption no- ted above is not the case and applicability of approximate methods cannot be proved. r'Yact methods were being developed only for special classes oY differential equations because of absence of re- gular ways to check up the periodic solutions for any genera- lization of the form of non-linear c~fferential equation. Methtlds of construction oY exact solution for piecewise linear differential equations have been worked nut in great detain. Three directions can be outlined here. These are I)1~~ethod of " sewing " together the solutions of separate linear systems which form the whole piece-wise problem. These solutions are presented eithor in the form of the sum of exponents or, if it is undesirable to seek the roo~s of some secular equations, in the form of power or trigonometric series. 2) Integral Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 equations method. 3) The periodic regimea'an presented in the form of trigonometric series and no garmonic _s negleted. The roe.'ficients of these series are found out from the considerations connected to the piecewise linea- rity of characteristics, t is uatural,iadeed, that all the methods described lead to the same resnlta. & T. Statement of the Problem, The Type of Periodic !'otion. 'Phis report concernes the dynamic systems whose potions are huided by the system of differential equati- ons ~~~6), 6=j`~ c 1.1 ) vector to be Found , 8 . 7L Xtl matriz of given constants, ~, and j _ ttt~ coloumos of constants know as yell, ~(6J denotes a piecewise linear function given as follows the equations ascertain r straight lines in the plane 6 , ~ ; on each of these lines thou are known the values GSK of these values being such that when during the motion of repn- :senting point along the 5-th line 6 becomes equal to 6~( just at this moment the representing point passes at once up to the k-th line and moves further along the ?.~ latter Fig,I shown the ezaIDples of piecewise charac_ teristics, these ezamplea met in non-linear oscillations problems, ~C6,~ I Fig. 2 presents an eza~ple of a characteristic conatnicted of three straight lines ;the inscriptions ahoy the mabers and corresponding values of g ,the arron denote the directions of transitions. ?~~ The possibility of this "j~p" ~ suimized,i,e, that the instantaneous transition of the representing point from the s-th to the k-th li.ne is not in contradiction with the equations considered, If,however, ~tK = MKS it may happen that just after the juap the velocity of the represe~ing point changes its siga,and the point does not actually move along the k-th line and glides dorm to the s-th, The "slippering" regimes like this are out of consideration hereafter; we put the motion of the npneemtiag point after the jump from the s-th line to the k-th to contimie along the latter in futon. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 When f(6), i.e.(I.2) and transition conditions, are know, various periodic solutions occur possible, these solutions differing by the order oY passing through separate branches of the characteristic is the course of one period of the periodic motion. The sequence oY straight lines which combine the characteristic for a period many be ascertained by set of numbers of the lines passed over in turn. For example, the sequence I232I denotes that in the course of a period the motion started on the line I pmloagea on 2 , pasaea up to 3, turves ever to 2 and then to I where it comes up to initial value of the coordinate and so on. But in the case of Fig. 2 except the noted regime the following regimes are possible ,for instance I2I, 2?2 etc. Let us call the inscription like this the formula of the periodic regime, this formula denoti7g the type of the regime. The regimes of the same formula are of equal tyre though they me~y differ ley the length of the period, the spectrum and so on. iPaen constructing the periodic solutions, their type have to be prescribed and car'^esponding regimes outlined,To examine all conceivable regimes, one moat separately deve- lop every regime of each type. Here we certainly come to the l,roblem of the limiting number of various types of periodic solutions Yor the s'stem ( I.I ) with given r . Some re- sults of this kind nay be found in `I2-I4~ . 69e shall further mark the straight lines according to their sequence in the periodic regime formula, so that the same line?may be meationed?aeveral times ;below from this point in the periodic regime formula the numbers follow the natural order I 2 3 ...Z ,the abacisaea of the transi- tion points having the indezation the numbers I ,'L to be related to the same straight line, T+or instance, the periodic regime of 21g. 3,Previously shown on Fig.2, will be presented now by the designation I2345 and the line I receives the nwabers I end 5 , the line II - numbers 2 and L? , III - 3 gquntions (I.I) - (I.2) do not contain t explicitly so that the origin for t mgybe chosen arbitrarily. Let the origin be placed in the moment when the representing mint starts its nation along the first (the r-th ) line cf Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 the periodic regime foimula, the value of ~ being initially 6s-+,t ,and the abaciases 6,s , ... , 6:-i,c-+ correspon- ding tTO the momenta ~, ... , tL_~ Up to thin notation, = I means the period of the regime in consideration. ~ha representing point moves along thle ~-St+( the ' t - th) line during the time interval ({~=0, l= ) ,the ~-~ -during the interval (t~~t3) ee+tc., the (Z-1 )th line is being passed for (ti-~ , tz= I .) tise. To summarize, the motion in the time interval (t~ , ~.~ ) is governed by the system aY the differential equa- bona The problem is to find the continuous periodic regime for (I.I) -~(I.2). Hence the solutions oY (I.3) are to be chose~n+ so t/hat the conditi+ons of con/jutnction t ~~ 1 ~=~I~~,~~~t1~'~~L1~..., ~v~lLt-I~a~~~L4-1~ (L4). are satisfied. The solutions of all the sys terra (I.3) pro+vide cer- tain relations among the vectors '~(11{)aad .L~ L~?~~ + s? that z4-z vector equations Yor just that mrmber oY unknowns Hill take place. Having deteained the noted unknowns sad expressed with their aid (L-1) quantities ]mown from the ,,;.li ~,.::,c ~.~r to ~,;o:r:e c~~,ua_i~-'~~. concai;.iu; L2 .. , `~ 'hest trrll ouL to ~~ file periu~i`~~. "he cunetr?;;r,tien of tilecc ~roblem to be %cl.vcre~5. 7a,~~=s the ?ain ~ , re exist the positive ;olutionu l 2~ tl~ .e '~(tl ~l~d ,;~~:~~~:~;~~ . ~ ~~~ t o.loald evaluate the quantity for every ~rouP of these :, utions. 'PP.:= :,, i:_ c'~ end of the story for one ~huuld verify chat ~~{)~'u does actually ?eet our su~,~estion ~:ot t~, "~-~ ~ ' a ,~G1Y, L.?,_;~g 6t,L.+ within the iutervai to , Q 4 so?e problems of suable regimas selectio.,, t':~e e~ac,.~s.~. of "sligpering " periodic solutiona a:.d ?~~ on? however, /caves oat of i^vs score tha ite;~:- j:;~-~ and we restrict ourselves only e:itih Cho :~~r.'?'+l '~" the question - Che Y:resartaton of r,or~e recE'! ? ~~- egUatlOnS CO1LStrUCtiCn. 4'ne process of evaluation fast describ:;d .'.~ ~ ., ~ :~r the ;;o,~ut~~,c.: conjunction condition:' (-~~ - ?art;5 of a i~ieca;ri:;e c'.1 r ctEr~. .. u:,,, ~ v it was au?eud;,~ :uentic;.~~ ~~ that Of ? SEWlIl~ ? afld ?ethod may be developed t'nroai'- ?~. .' .? ~.~ ~ ... tlus - according to certain 1'orrl cf the ;o~~,t!cn ~~. al '?lle ?'SeVyill~ ,~ Of dOllL t;.ese modifications are sed through tho roots oY t'le sec~~lar a:rlatlous ; 'serririL ,~ of Che CrlCcaoaletric ~.cr?i~''% o,` ' . . Dower cries. r::c:::;o,vu'. gludr, ti:e ~eddin" Ci. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ti'stam (l.I to some unegrai equation fors{fl, 1't;e ant o.' diiferentiai '~quatiens {I, 31 wit'~~iu sc~:arat~~ ;pis is r ,w xcluded, the conjunction coaditionr ~.,;) a'ato'~:at.cally :~e'~.ct, when the t,iecewise fw~ctior, f(o? ?,Li.ed cf faz~aliel line,, particularly as iC ie fcr sy;temc, c~hen it oerl:rs ~~~-_ no c~aouhie indeea to eva- ? - ~tCi out of the integral tcuation and to put doom the ~~__,.cd equations. In f~eneral however, it seems impcssibic :rcc;;e;i ;rith the "closed " :oiution of the resultant I're,~.c1m equation. i'ne Fourier series, ~,ehick: express the cclutions 1~(tl t!.ii: the luturvai l l~ , 1~.~ ). aro of the ~-erred and -_ . ~ ~.;rli ;;;~ shown beltev;, rresent the fu:.ctiocs of 6i;.;e, i ~shir~; .,,,en !] ~ ~ < LP and ~t a ~ ~ ~t ~ ~~ hence the sum ~~i1= ~2(t)+ , ~~l ~t) c- ?.es the general trigonor.,etr:c ~~~. _~;;entation of the ....~ .oiuticn for tine ~. ecenie~; L~:ecr ;rob~e~. 'P _e p~~rio;iic r:,;i.ce cor~ be i:: ~;;et ;ro~ected in iorx ..,_ ~,.~le Cri~.ar,,^,~etric ~erie:~~ Mitl.out 'semi; ~; ", u;ul . ._.utabie set ::f ~ curler ccefficier+te is e:,creaed ~~~,: tie 1...~te :~utcbrr ~f ,:.r:,:ocers tz .. ~ . _.a~ ,~ , - .~.,~~r it ~:a:,ed uPor: ~e~u~_cr .r'~: ^tic, c_ .~:?t::i~~ ~~e:i:;e ,..,.r:;c~___stic: ci i.o1 c,; ,,c cu,.~- oi' ':c... _, ~ ~. a~~.e cr tc;o ;-.ve:. s..,?rit;at lines 1 . ~u & 2. The `re~entaticn of Solutions for th~~ yn^c~,?.~^~~u~ Linear ;;stem of Differential Fgaatiear Witt Constant C1cef:'rcients, ~i'e shall ~~ow consider the homogeneous liu6ar syster~ of differential equations. Following the notation o.' "~ 1, we put x = Bx + ~Elj'x = (fix The determinants of the matrices P ~ - ~ end I F - Q will be assigned as gy(p)= ~pE-B~, o~P)=~pE-~1= IPE-B-~hj'~ ~z.z~ [uul the eigenvalues of the matrices a and ~ ,that is the roots of polynomials of the n-th r:ower correspondingly be ~a, and Il.~ . ~](~1, 3lD) ,will settle the formula connecting these polynomials let us introduce the coloumns H~O~, J(P1 oY the polynomials Hs~p! ,JSi~I ;the former being the determinants whic`a come out of API when the the numbers ~~ ,... , 4n ; with the s-th line instead Mlp)?'~InIP~~-Il Jtp)- SL,[~ RS~K~SKIP~ ~ 2.4 ~ s-th coloumn is substituted by the same way holds for the Latter of the coloumn and J+, . Iri instead oY ~ , ... , ~L2 . The result is n ""I /foe introducing the ;:alynomial a Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~ 11 i mi i~t J' msl~) ~2i1~~ m +5 rl~l (i ~4n1?-rnJS~n(Q-s) ~~ rriwii;u,t restltiu~ iro;r, `~(P) vr.:en c~,e rp 1T1 -th ?.i: ?:. -~,~ ~__.eu, S'tn ;.i.cU L -tl! , _.. ~1:J7I1 s ire thrown a n j ~~Siutl+d~ s ~m~Q~~~l~l~,~~(2.8 ) ~ m ~ ; I ~'-r~~br~lic complements e~ ~~~~~ (p?~?,flpl-~imflp~`~u(p!t ~lD'~ln~atigatien (we ~~ean tt~,at, it is the ;ualit;~tivc~ ir:ve:tiga- tion of tt:c ~iynan~icsl ::y:~tc~t, as much a. ~?ve . ~aLieti, .t i~ ae:t ~~~ct:::,ar? t;. a.. a ti 3t ~i~etie~n. ~> ur,d i, r~ ,_.. .t._e!, ix~. 1. 1tiwL .i ., .: 4' , '.?;f, Ir Yi.'. ,,,i' t}lc are interested only in those properties which are expressed by the qualitative prupert'? of the trajectories) very often allows to draw the non-trivial conelu~ions corcerniag the physical problem under consideration or to gi- ve original re?ommer,datior.s for the const*~c- tion of a technical device, The dynamical systems which are obtaired in considering physical and engineering prob- lems always contain a certain number of pa - rameters, Then there naturally arises the problem of investigating any possible than - ges of the qualitative structure of the phase- space that occurs with the changes o: the pa- rameters, This investigation represent the essence of the theory of bifurcations of the dynami- cal systems developed by A,A, Andronov ar.d his school, The role of the theory of bifur- cations, however, is not reduced to this only. The qualitative investigation of ttie sy- stem (A),~ is comparatively well developed for the cases I) when such system is clew to the linear conservative system (the method of small ,u.) or to the nonlinear conservative system, 2) when it is apiece-linear system, 3j when it has a particular form and graphi- cal methods (method by Lienard a~~' another) would be quite sufficient to invPsti;ate it. It is not quite u:>ual, however, that we can reduce the problem under consideration to one of the above cases, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ,.. Thus the probleu, that faces us is that of the development of regular methods or even sufficiently effective ways of the qualita- tive investigation of the general non-linear dynamical systems (A)~ These methods must give an opportunity of getting the partition of the parameter space into regions with. dif - ferent qualitative behavior of the trajecto- ries, corresponding to the different behavior of either physical or engineering problem un- der consideration, These methods may be developed by the theory of bifurcations. Before describing these methods we shall remind of the main facts of the theory of bifurcations (see [''? II~ ). In the system (A),t among all the traj- ectories - states of equilibrium, limit- cyc- les and separatrices (separatrices of the saddle points and the separatrices of the multiple states of equilibrium, position of the singular trajectories deter- mines the qualitative structure of the parti- tion of phase surface into trajectorics,7Phen considt:ring the changes (bifurcations) of the qualitative structure it is sufficient to follow the changes of the singular traj- cctorics,(See (,a-6] The main concept, on et;ich the theory of bifurcations or "the theory of the change of the dynamical systems with the change of the right-hand sides" is base/, is the concept of the structurally stable dynamical system in- troduced by A,A. Andronov and T? Pontrjagin [I_qJ and also the concept of the non-struc- turally at~ble systems with different degrees of non-structural stability[7J and, at last, the concept of bifurcations of the dynamical system [2,4), All these concepts are quite natural in their applications as Well as from purply mathematical point of viee.?) We do not give any exact definitions here; We only explain What these concepts mean in general, not aiming at exactness, First of all We shall remind that the dynamical system is called structurally stable if its qualita- tive structure does not change With all auf - ficiently small changes of its right-hand si- des With which there also occur rather small changes of the derivatives of these right - -hand aide o, If the system (A) is structu - rally stable then all the systems in Which the right-hand sides and their derivatives are sufficiently close to the right-hand si- des of the system and their derivatives are also structurally stable, As it is known, all states of equilibrium of structurally stable ? The exact definition of a structurally sta- ble system and systems With different degrees of non-structural stability is given by intro- ducing Banach space of the dynamical systems, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 systems are simple !i.e ?they are nodes,foci and saddle points, all limit cycles characte- ristic exponents do not vanish and the sepa- ratrices do not go from a saddle point, to the same saddle point or to another one. To- gether '.sith structurally stable systems the investigation of non-structurally stable sy- stems and their classification according to the degree of non-structural stability is of great importance for applied knowledge ~7,8J. Trc systems "of the first degree of non- structural stability", in particular, being relatively structurally stable among the set of non-structurally stable systems" are of great interest, It is quite natural to consi- der the conservative dynamical systems as sy- stems of an infinite degree of non-structural stability. p dynamical system of the first degree of non-structural stability must have the only singular trajectory of the first degree of non-structural stability, i.e. the traje- ctory of one of the following types; I) a double state of equilibrium 0(x.,y?) in which ~~'(x?,u?i.Qy~s=,y?) o. (the sad- dle-node ,,oint (see fig.I.); 2) a fine (mul- tiple) focus for which the first Liaponov's vogue is not vanishing (fig.2a); 3) a double limit-cycle (fig ? a),; 4) a seharatrix going rom one saddle point ~- u: ~~:.e: (: -~. ~ , a ~~L,~) t, this saddle point Pz ~-~.y?1 ~9y ~x?'y-~#0 (fig?a) Let us emphasize that wP.ile the con~?i- tions distinguishing among the dynamical sy- stems the structurally stable systems are conditions of the type of uuequalities ( a structurally stable systems may have onl; such of equilibrium for which tz ~x?~y?) ~~ (x>, y?~ there may be only such limit-cycles for which the characteristic exponent :%z*~ etc., the conditions distinguishing; the non-structural- ly stable systems are conditions of the type of equalities?).p non-structurally stable system must have either a state of equilibrium for which D = ~ or a fine (multiple) focus ..?or ?~xhich Qx(x?~y?)+Qy Cx?,y~~=o or limit- -cycles having the c,:ararteristic exponent tz =o or a separatr:x going from one saddle point to another orre or to the same saddle-point. Let us return to the system of the tyge (p)~ , i,a., the system containinG parame - ters and explain ~:shat is called "a bifurca- tion". ?) It should be noted that not for all these conditions there are analytical expressions; for instance, there is no such expression available for tha condition that the separat- rix goes from one waddle point to another, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 If the system (A),~ corresponding to a certain point ( ~ ?. ~2 , _ , .1 h ) of the parame- ter space is non-structurally stable and with certain change of the parameters there changes the qualitative structure, then it is said that a bifurcaticn takes place and the values ( ,{; , ,{~ , . , ,1n )are called "the bifurcatio- nal" values of the parameters, Thus, for example, there may be bifurca- tions at vhich the multiple states of equili- brium generate some states of equilibrium or disappear (see X2,4,8-II), bifurcations at which there appear or disappear limit-cycles etc. The bifurcations at which the system cor- responding to the values of ~t is a system of the first degree of nonctructural stability may De naturally called the main-:im~lest-bi- f_u_r_c_a_tion_s, The consideration of these bifurcations allows, in particular, to distinguish the sim- plest cases of the "generation" of limit - -cycles (B,IO,II~. As it is known, there are the a) c) The generation from a separatrix of the saddle point forming a loop(fig,5a, 5b) if in the saddle point the value The generation from a fins (multiple) Yocus (fig. 2a,2b) The generation from a double limit- cycle(fig,3a, 3b) In the last case the loop is either "stable" or "unstable" depending on the fact which of the uuequalities G0 holds, This when 6 < a from the loop there may generate the only stable limit-cycle and ~,vhen 6>o there may generate the only unstable limit cycle (see fig,5a, 5b } ?) d) The generation from the separatrix of the saddle-node point D(xo,ya) issuing from it and tending to it (fig,6a,6b). In this case there always appear the only li- mit-cycle the stable one if a ~ = px (xo, yo ~~,_ , ~~) f Qy (xo, yo, ~, . ,?)< o and the unstable one if 6-> o . Let us assume for the sake of simplicit3, ?) In case the dynamical system under consi- deration has the closed path consisting of separatrices of the saddle points and the sad- ly points and the path contains more than one saddle point,In thin case the corresponding dynamical system is evidently the' system of tY,e degree of non-structural stability higher than the first,Then the following statement holds: Let Dixt.Y,),~z(x=,yz),. 0" (x.Y") be the co- ordinates of the saddle points contained by the path d and s~-Px(z~,y~~t~~y~x,;y,~~, ~=,, r, ? In the case a'~ i=1. P,-.n "path is unstable",then in the first case from the path ti,ere may appear tic only stable li- mit-cycle and in the second case the only unstable limit cycle, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 that the dynamical system under consideration contains only two parameters ~c and ~P so that "the space" of the parameters is a plane and for this simple assumption let us illustrate some facts of the theory of bif urcations. If the problem under consideration is of such kind that in the corresponding dynamical system there are values of the parameters at which the system i3 structurally stable, then there is apparently a whole region of the valu- es of the parameters, the region of the plane ( ~,,~p )for which this system is also struc- turally stable (if follow from the very defi - nition of"the structural stability of a systems) The values of the parameters at which the system under consideration is non-structurally stable, i,e ? the bifurcational values of the parameters satisfy, as We have mentioned, the relations of the type of equalities, Each of these conditions determines a certain line in the parameter plane being, as a rule, a boun- dary of the regions t~ hic~~ correspond struc- turally stable systems with different quali- ?) There are problems which lead to the consi- deration of the dynae~ical systems which are nor-structurally stable at all values of the parameters of this system.?or example, this takes place xhen the problem is described by Aamiltoa'a systems, In this paper we consider only those dynamir,ai systems which ar certain values of the para~~eters and consequently for certain regions of tt;~ values of the parame- ters and conser;uentl;;~ Cor certain re~~ions o> the values cf tt^ ~?;ra,~atera are ;trur.turall ~~tablc, tative structures, :Vhcn u~e i_ztersec.t such a line in ~' ?~- rameter space, then. in cyna~:ical. syste^., the- re apt;arertl,y takes place a bi?urcation, To the points of such a line t}:ere ai;~~areutly correspond d3mamii~rii , ,temp of the fir;:, de- gree of non-structural stability or of a higher degree of non-structural ~tabilty (conservative system , in i;articuiar; coring to the depence of the right-har~ri sibs of the .dynamical system on the parameters ,?) But then to the points of intersections of two such line:: them certainly corm .pond the dynamical system of the degree of nou- -structural stability higher than the first, If t}!ere is kno~n ,the set of all the bifurcational-v31ue__of-the;-~arameters,or if ?) At the dependence of the right-hand sides of the dynamical syetem~ on the par:mnetera,it is quite natural to conai'er it most "gene - ral" or "normal" ~~~hen to the syste.ma of the first degr~c on non-structural stability,an~~ only to these systems, in the n- di~,.r~*::;ional parameter apace tt,ere correspond the points in the man:f~ld of n-I dimcr.sions, to the s - :,tems of i,he second degree of nnn-structural stability 'there correspond points in the ma- nifold of n-2 dimensions et s, Iii the concrete case:., however,whieh are considered i^ this paper the value of parame- ters fillia~; the manifold of the n-? dimansi- onal number often correspond to the .~ystam:; of the degree of non-structural ~tabilit~r higher than the first :nd to the conservative systems, in particular, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 it i:: proved that they are absent) and the character of all the bifurcations in passing through all the bifurcational values and be- sides if there is known the qualitative struc- ture of the dynamical system with same parti - culsr-values of the_-parameters, mYiFN,TAaI~?G INTO consideration the continuity, it is pos- sible to show that having these data, we can determine the qualitative structure for any point is the whole parameter space,) Thus the set of the bifurcational values evidently partitions the parameter space into regions of the values of the parameters corresponding to the same structurally stable qualitative structures, Thus if this situation. takes place in a concrete non-linear system then the qualitative investigation is conside- red to be complete, From what has been said it is clear that the development of the general methods of the establishment of the set of the bifurcational values of the parameters is a very important and actual problem in developing the methods of the qualitative investigation, As we have seen some bifurcational values can be defined from certain relations between the parameters ?) pne might Brae the analogies between the considerations given here and ~tors's ideas, The consideration of these analogies however would exceed the limit of this pa;,er, of the type of equalities. Analytical expressions, however, cannot be given for the .;hole set of the bifurcatio- nal values, These expressions exist only for the bifurcational values corresponding to the non-simple states of equilibrium, fievertheless it is possible sometimes to prove in the indi- rect way the existence or absence of the bifur- cations of one kind or another, As an example of one of such ways there may be regarded methods of establishing the existence of the bifurcations which take place with the change of the par~'~meter.s from those corresponding to one well-known qualitative structure to those corresponding to another well-known qualitative structure and the me- thods of reducing to contradiction the assum- ption of the existence of this or that bi - furcation as well. In the cases considered below such me- thods are used for the qualitative investi- gation of concrete systems in which the vec- tor field rotates to one side with the in- crease of one of tl~e parameters, for example .~~ For these systems the difference c+~ ~r~ ~= dy)~,? _~qyh,~~ ~a, ~~~ ~cjz.~ d.e' remains constant either on the whole phase surface, or on a certain part of it, !Yt this investigation of great impor- tance is the a'oility to e:~tablish uniquely the qualitative structure o1' the system Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 eVCn at Ccrtai[i ~~)6C'r'ctc: ValuBS Of t}le pflrame- ters, ?'or tt;is purt~o>e to dttermir.e the exis - tence and relativs position of the singular trajectories (and ir. a number of cases to pro- ve the absence of bifurcations) the well known classical methods of qualitative analysis, are of use. To these methods belongs the a;?plica ti on of d.~fferent criteria, on the e:~i~tence of limit-cycles (Bendixon :lulac's cr!terior, and their ~;eneralizaticns`; the con:>ideration of systems cl~~se to conservative, the investi- ~atico of t}:e uis.~esitio^ of the isoclines and the behavicr of the trajectoried ii: the regions betw~~~n t1!em and also dificrent versions of the investigation of tupogra;;hic systems and Jiff o- r~ntial equations of com;ariso~~ etc. Let us make some re?dr}:~ concerning the ca- ses wh~e~! the sets of the bifurcaticnal values of the parameters are given not :~nelytically , by the exlression o_' the tyre of the equality, but only by tt,e demonstration of the very fact of t6~ cxi~acnci: of tt~.ese set:. i^ so:re inter -, ~~; vals of the ct~angu of the parameters ~ ?~< I7) < 1, ~ -~. In t.i:ese cases mnrc exact quanta- tive evaluaticna for the ;.oirt:, o: the set of tt:e bif~urcational values of the parameters may be rec,eive~: ba the numerical u~ethu~?:,. Index d as it i., ..hover ir. ~I2J for teach str~icturally .:table :;y~tem ~;,",~ thErt i:~ not v:a,i~;;:i.r_g value ~o t'~~~ so-ca?E-? "geometrical ;:~~~,~:c:rc of tl~,c nor -::t rLrt.t.ral statil ;t~~" of structurally stable system and cor.sequer.tly , its qualitative structure may be determi~:ec by an approximatively cor_structiun of t}? gular trajectories and by the fire/ nur~b r o. operations. Sufficient exactness of the ap~~roxiira ti.on depends upon t};e value str?cturally stable system the value non .- r~,,= ? and it is impossible to determin the establiehment of the qualitative struct~.'rc of the non-structurally stable system b;; the ap- x:r~:>ximation of its singular trajectories. ?he qualitative structures of t}~~ "`'~'_ -.structurally st-,ble systems may be in a sen- se, "approximated" by the qualitative :;trur, - tures of the close structurally stable sys - tems with arbitrary :>mall geometrical pure of structural stability, The bifurcational value of the parame - tcr ~{,c of the system iA) a,K, the existence of ~vhich is established at some Interval of the < .~K _le) change of the parameter ~k, ~K d' when i#K are filed! , (then the values -~ ; may be approximated With the required exact - ness to the values of the parameters n,) 1 zi) 7 r~J 1 ~k~ I .~ ~`~ ~ ~ ~ ~K < llk, ~ flit J ~A~~o,l f the structurally stable systems ~y 0 ~A)~~'~i close to the non-structurally stable system;~,~~4(then the exactness of the "appro- ximation is characterized by the value of the ., i1o1 r' ) and ?measure of the struc- interval (!K ~ ~r tural stability" of structurally stable sy- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 be cycles, surrounding the cylinder (cycles of the 2nd lind), To the cycles of the second kind there correspond the solutions ylx) of the equation dx Pixy/ which are periodic in .x Some Nays facili- tatir;g the quantitative investigation of the non-linear system on the phase, are described ir, the work(27J. here we shall dwell on the systems (3) - - (6), tt~e system (3) being investigated in de- tail. Tn this investigation requiring the :study of all the main bifurcations there may be clearly shown the role of the theory of bi furcations in working out the effective me - thods of the qualitative investigation of con- crete nonlinear systems. ?or the system (4) and (5) we shall only enumerate any possible bifurcations, and for the system (6) we shall describe all possible qualitative structures and their change with the transition from one region of the parae~eter space into another, This system is equivalent to the equa nP 2pi-l-~P s~n`P) d~ P cosh u~re '~ i : the anp-le which the 'irection n;' v~_:rctt, of she centre o' ~~a~~it~ a,akes with the horizon, ~ i~ the value Proportional to the velocity square of the centre of gra- vity of the aeroplane, parameters ,~ and ~- are proportional to the values of the tracti- on of the propeller and the resistance of the medium respectively. According to the physical sense of the variables the rectilinear uniform motion of the aeroplane corresponds to the stable sta- tes of equilibrium of the system (3), the periodic mctions of the aeroplane on the wa- vy curves correspond to the limit-cycles of the first kir_d and the loops correspond to the cycles of the second kind. The equation (~) and the syste>r: (3) 'tle- re investigated in a number of works (T6,j,I17]~ ~28-i Q~. /;ere ve give a brief account of the full investigation 1171 of the system (3). Let us begin with the investigation of the main yualitative characteristics: states of equilit;ritr~,, :eparatrices of the saddles points and limp -^.ycles, having (prelimina- ry) singled out a conservative case of the system, ~~ there occurs a conserva - If ~=~' tive case of the ay stem, Zhukow~ky's case /22/. The general integral of ti;e :.yste'n has the form; f~'~~n~`?_ ~ ~~~.COnsf. ~, The state of eq~:i'_ibrium ( ;0 0 , P=~ i; the centre. The separat rix ~oe~ fro~~i t'^e '' o ) to t^e ,aJ~1e ~:nt sa~idla point ( Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 1+2, o) The other integral curves repre- sent closed curves surrounding either the sta- te of equilibrium, or the cylinder. This case is coastdered in details in [4], The non-conservative cases of the system (3) (~# a .~ # o ). TAE STATES OF E IIILIBRIUM. In the finite part of the phase cylinder P >o there are not more than four states of equilibrium of system -~tructurea of tt,t system (3) at any values of ~ and are establi- shed in [17~ by tho invcati~ation of the bi=ur- cational states of the system and coraideration of the continious dependence of the structure on the parameters, (gelow we take into ccnsida- ration the changes of the qualitative struct~.- re of the system on the phase aurface,a:; .;e did before,and ~inultaneously the notion of the point rep resenting t::^.is system in the parameter space), This ir.vesti~ation is possible due to the properties (I4) of the rotated vector fLeld of equation (7) :,pith the ir.creasa of the pare - ;ueter, The a;pli^ation of t}.c diffare:~tial eqn-,- tion comparisons and Ben^i::on-Durac' criterion on the cylinder a11o,~;s to distir~uish the re~~i- ons of the parameter plane (regions Gain fi~,IJ; with certain qualitative feature:~,For t}~- - nts of the region:: sand 6qualitari:~ :;tr~,.r. - tures of the system (3) are dat~rminidered separately, Let us fi% a certain ~~l and follo?rr the bifur- cutiona of the system ~nith the i..^.crease of the Far-n_terl~.from the points of the region G6 (`';..~;. T^ :,^? I2o.) tc the poi nta Of the region Gp rfig,T^ anal I2e). The system Passes the fist bi:'arcational state at ~~=~~ at the intersection i.~ the narsc,^ter plane of the boundary ~=~~rry~P of the region T (fig,tI) (see !IO) ),then from the ~m.~ltipl2 :Mate of equilibrium 03 t}=are ap - rcars the unstable limit-cycle (fig, I2 ). The >tce; has the second bifurcational state ~,ihen sy. f~=~uz at which the separatrices sr and ~Q merge rfig,I~~~, Then t';are are formed the closed un - ~tabl~~ paths~,aud 4Q from which according to the t}.cory (~e? (I) and (T3)'), only the unstable li- m:i.t-cycle can appear, Taking into consideration t}.is fact and the properties (Ta) of the ro- tated vector field we Nava the foil owing; Accarding to the statement al there is the rl,rve eQ (;u=we(~)~ of the parameter plane 'it b~~unds the region f on the right,-fig,iI), ~,r;; :J:~.~n r~a~,}~ing this curvy ^;ith the increase ~~~' u tt:e .rr,table limit-cycle of the first kind _ , ,, ^ith tt;o c1~~ red ^ath dr on the phase sur- in~-~ i:ao con~ide- -c~^1c; ~~nEra - t}~ trje^tories, ~' ~'~ ~ _tr~il r}It line e,, a ..otted curv.~ nn:i 1 i ^. ~ ~1= 3/p , ?) Then the qualitative atruot~.rres of the sy - stem (3) for certain region of t}~e parameter Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~~;ith the subsequent increase of~+c>u~ from the closed path d~(consisting of the separat - rices $ S2 and P=o (-x:~.-2 , Q ~~. ~r) there appears the only?nnstable cycle of the second kind (~fi~,I~2), The curvy ~p and also the bifurcat.ional cur- v::s,sho~,vn belovt ?), are drav;n through the points obtai~cd approximately from the qualitative in - t~.i.ration of the close structurally stable sy - ctr.e,s, Thus when +t>~tthe system has too cycles of the second kind (stable anr_' un;;tablel, :`'itt'i the i~~creasc of the parw;eter ~ the cycles a;~proach each other (according to the statement c) ) and oe~in~; to the existence of tha region Ge , them. is the only bifurcational value ujso that '.^hen ~~~j the t'.vo cycles merge into ahalf-:;table cyc- '-t' (fig,I2~), and when~ufu~ they disappear, The ~urv~-u?~,raJit exi;,t according to the statement b~ bounds the region 3 (fig, II) on the right , ?t is ira,~an by Weans of the major curve v;hich is ~,;resente9 in fig,Il by tha dotted line, In this: cased>1 tt.ere is u;;ed the same me - t,'~od of consi~erat.ion of the bi.?urcatfona, It i.~ stated that at the transition from region 'S :'~ the regions I~ and II of the value; of the ?~ Thew curves of the bifurcatior,al values ~~f the para~r,ctcrs arc given not analytically, oy the exr~r.~ ;pion of the type of the :quality, o~.t ~nl; by t'.~ ~:ier.on~tratiori of the eery fart .~i tir~ir xi.~t:~i~~~~s i-i ::ome intervals of the ,a~a:c~t~~r ,. , parameters (see fig,Il) tha qualitative str~~~- tures of the system change as it is aho'+~~? i'ig, 73a -13,x,, . gere besides the bifurcat;ioi~s e.~isting ?~~: the casc~~i there are four bif~zrcatior~.:>,as `ol- lov:s: I) ror the pow^ts of the e~zrve, 3E;~~-~ ~i - ting the regions 6 and 7 in the ;~arae;;r,ar ; la-~ RL', (flg II) the Se:p(lratrlCeS pa~it: S3 p:i~^?~, foroiing the closed path d~ surrou~~ci?i~ the ~.-~. of equilibrium (fi_g,13b); ~) Fer tiie ;Dints the curve separating the regions ? and ~ ~1' , ii) the separatrices S, and S~?cr~,~e !?:~,?~ - .~;; 3) For the points of the curve ~ei~aratin~ tr,o rerrions 8 and o the separatrices S~ and y~. ge forming the closed path ,~y surrou~r'in~,~ ti . cylinder (fig,13~c); 4) On the curve t+tiP_,Yr=~ separating the region II fig, II the states o`' equilibrium the saddle point and the node r[,:n- ge forming the multiple .Mate of equilibriums the saddle-node (fig,13~, ), ~'t:en this multi~:lr state of equilibrium disappears,for ~ > ti~y there appears Yrom the separatrix of the s~t~.i dle-node the limit-cycle of the second k~_~;? /fig, Ii,v), The change of tt~c qualitative atrurturc.~ of the System (3) with the increase of~~ r,ltf constant~l.1 is represented in fig,I~ with ~~'. - in fig, I;. To the boundary curves of thy, r~ gions I-II of the parameter ;lanes !`ig,T" there correspond the non-struct,uraiiy :~tAi,~: cases of the systen;, vrhich are al so r~~ rt.;~.r ted in fig, I2 and I3, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 S_Y S_T_E ~1__~'?Z The system (') is equivalent to the aqua- dv= -E (x+yYJ dx _~-A , y r+~Sx ~r~here o o a loop surrounding the cylinder , 'Kith the increase of the parameter T the on- ly stable limit-cycle of the second kind ap- pears from this loop. In this case the synchronous regime of work corresponds to the stable state of equi- librium, When the stable states of equilibri- uo disappear the synchronous motion of the machine is impossible, The asychronous mo - tion of the machine correspond to the stable cycle of the second kind. The investigation of the phase portait of the system (5) explains the presence and the condition of the existence of the asyn - chronous motion of the machine and also the possibility of the restoration of synchronism after the asynchronous motion; S Y S-T E M-~6~ This system is equivalent to the equa- tion c1~ jl-s~n`P-~(t-d cos~)~ d~- ~ where ~ is the difference of the phases of oscillations of the generators, ~' , ,~. , d are positive parameters proportional to initial difference of frequencies, damping and delay respectively, The system (6) in its form is similar to the system (5) With z =o The values of the parameter ~; ~ow- ever, which are physically significant allow the new qualitative structures X21]. In the given case to the stable state of equilibrium there corresponds t!:e r~g~i_ of synchronism, to the stable limit-cycJ.e of the first kind corresponds the stationary ~egime of pulsations in the system when the frequen- ce approach to the stable limit-cycle of the second kind corresponds the stationary regime of pulsations in the system, at which certain constant difference of frequencies is perio - dically repeated and the difference of phases is increased unrestrictedly, The system (5) with ~'onding to thew (fig,I51, Five bi,`.urcational surfaces , the plane ~."_? and the surfaces u, z!, u~,Dccr- resl,onding to the non-structurally :,table ca- ses of the system partition the par~~eter spa- r,; i~ito regions corres};ondiug to different structurally stable qualitativa structures ~~, ?` '^he qualitative structures oC the s?:t~n Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 (In fig, I4 the numbers of the correspon- ding regions of the parameter space are shown in small circles). For the points of the plane ~=1 (separa- ting the region I from the regions 2 and 4 (fiB,15) two states of equilibrium, the sad - dle point and the node merge into a multiple state of equilibrium the saddle-node (fig I4~and I4e;, With jf>t the state of equilibrium disap- pears. (Fig,I'+a). From the loop of the sops - ratrix of the saddle-node (fig.I43) at the in- tersection u~ith the increase of ~ of the boun- dary separating the regions I and 2 ~;fig,15) there appears the only stable limit-cycle of the second kind (fig,Ir+a), For the points of the surface W separating the regions 2 and 4 (see fig,15 and I4 q ), 5 and 3 (see fig.15 u and I4x) the separatrices of the saddle point serge forming in the half-plane zs o the clo- sed path surrounding the cylinder (the path of the second kind), At the intersection of the surface W with the increase of ~ (the transition into the region 4 or 5 corresponds to it) from the closed path there appears the only stable limit-cycle of the second kind (6) for the values of the parameters satis - fying the unequality a'>t are determined not taking into consideration the even number of cycles of the first kind vthich may to appear from the doubles limit-cycles generating with the thickening pf the trajectories. ig,j4~Kand ?~u.). The surfaces ~ ,u , V are situated in the part d >`1 of the parameter space, The surface ;~ is determined by the equality d = -- _-1-_-- For the points cf this 1 ; ~, 2 - surface of the state of equilibrium the fo- cus is fine (multiple). At tare intersection of the surface ~ with the increase of ~ and the con~~tants ~~ and ~ or ~~rith the r,ecrease of ~ and the constants ~. and cl (the tranai- tion into the red;ion 3 or 5 (fig.I;) corm - ;ronds to it) the focus from the stable beco- e.;;s unstable and the only at able limit-cycle of the first kind appears fro u: it (fig, I42 cr I4 u). Here the bcundary of the region of stability is "a safety boundary" [2uJ. r7ith this bifarcati.on which inevitably take:: pla- ce ~.+~~th tY;e increase of the parameterd!de - iay) in the ?ystem.gFCthc automatic osci].la- t:cns arice when the frequencies approach. For the points of the surface J ,:;c - _~aratir~~ th:: r~~~?ions 5 nn~i ~ of the ; arame- t._r ci nee t,t.c aci~,a: atrice, of tt.e saddles -~i:t: :.~r;~r foru;ir~~ tYc c1r=c! jatc ..r~rroun- ^111~ 1.lr :.t;t'_ Us F;(~'1.11J~1r'J!11:1 !~Jg,T.4.ti~, L1: LLS ~. ._~SC 0: wlt.`: ~-! C tl":if; ,C,lOn tii. tt ~ y,~, ~i.~tx^- .ra; n r:. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 o and ~ of the para>eter space the separa- trJccs of the saddle merge furntin~; the closed path of the second kind in ttie half-plane ~ ~ 0 (fig.~4H ), (pn the fig, 15 the sur- faces u and U are renre~erted approximate- ly). At the intersection of the surface u ?ith the decrease of ~' (i.e. at the transiti- on into the region ~) trop/ the loop of the second kind in the half-plane ~~~~1 same of whose coefficients may be zero except for the coef- ficient to the ~~" term, "Slow" motion is charac?erized ty the roots of equation /2,I/ whose approximate values are ~?termined ae the roots of the eouation ~v (p) . p, "1.c approximate values of the roots corresponding tc "?,as~" motion are found ea the roots of the equations derived from , ~ 2. I? when taking=p/m6 , removing negative powers of ltt and then setting m l=gyp Here 6 are rational numbers de- pendent on 1~ and /}r; they are determined according to the rules listed in Ref. II, Thua, the characteristic equation ie broken down into several independent small-degree equa- tions. This operation may be carried out without expanding the characteristic determinant along its raw. The characte- ristic determinant can also be broken down when /I.I/ conta- ine difference as xell as differential equations /for examp- le, when accounting for elastic hydraulic impact/. This method enabled ua to determined where the hypothe- sis of ideal regulators and the hypothesis of constant mecha- nical turbine torques are applicable in the study of small oscillations. The firat?hypotheais ie valid for sufficient- ly large time constants amounting to tens of seconds/ of derivational headwork structures in hydraulic system when the turbines have governors with follow-up /Ref, I2,Ij~'and rather fast so-called secondary regulation. With slow secon- dary regulation /Ref. Ij/ or transie tt feedback /Ref. I4,15/ tiQSu~fs the hypothesis of ideal regulatore~'in'stronger conditions for stability in the small sense for the steady-state opera- tion of hydraulic headwork syateia than they are in actuali- ty, The assumption of constant mechanical turbine torque is valid provided the power ~yatea is not operating near the so-called static limit of transfer capacity. Otherwise, tur- bine regulation hsa~abe accounted for when studying electro- mechanical processes, and if the ties oonatante of the head- work derivational structures are small /amounting to several seconds/, fructuatione of the liquid mass in these structures and in the equalizing reservoirs also have to be included. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ?'his may occur, for example, in channel installatiors v:hen s rather short penstock pipe in the body of the dam termi- nated in an equalizing reservoir before the power house affords headwater derivation. The equations for electrome- chanical,mechanical and hydraulic processes in this case cannot be separated into independent nets. We have not ae yet dealt with the possibility of brea- king down the equations for electromechanical processes them- selves into simple sets. This question was studied in Ref .16 and 17. In complicated cases this process of division is es- sentially simplified when the method applied is used in con- junction with matrix transformations of the initial lineari- zed equations fRef. 17~. Several studies were conducted Ref. I8, 19, 20, 2I, 22~ dealing with a particular aspect of the problem of simplify- ing linearized seta, namely, the reduction of similar type genera*.ors to a single "equivalent" machine. ".'heir quint es- sence in the final analysis also amounts to separating the motion of the system; more specifically, "fast" motion is ta- kea to be the power wings of the machines within one sta- tion or group of stations, while "slog" motion corresponds to power swings between the machines of different groups of stations. However, the various ingenuona attacks Wade in the- se studies are not generalized to a sufficient extent, which mould permit them to be applied in solving the problen as a whole. j. Dividing sets of non-linear equations into simple seta The lose of stability in the large sense, that is, the transient stability of a power system is associated with short circuits, which lead to instantaneous changes in the scheme and parameters of the system. After some time the r-t- ulted line is cleared; this results in a new cnan~~e in the scheme and parameters. The system must assume a ne?~ stead?- state under the latter conditions for normal operation tr, take place. Tn the above the perturbed values of the genera- lized coordinates are the values which they take cn at the moment when the short circuit is cleared. Therefore, the transient stability problem can be broken down into the f^?- iowing four stages: I, The set of equations for thr short cirruit transi:~~ ~~~ are simplified or separated, 2, T'he same ie done to the equations fir the ',rsnsirar~. after clearing the short circuit. 3, Regions of stability in the large sense are cc;a;- ructed for the steady-state operating conditions after ale~~- ring the fault, q. The perturbed values of the generalized coo rd`natr:: at the instant the fault is cleared are compared with t~u3 region of stability in the large sense for the steady-stnt~. operating conditions after clearing the fault, separation of the sets of equations into those for tror,~~chanical and merh,u~ir.a] ~-ocessee is dealt with in :~~ . ~'?, ,~~4 a~:i ?5. For the sake of illustration a si;,; '~. -a? wil':bec ..:~id~~red of two interconnected generator?. m:.e :~~~~?- U~,eeor Cra:~ ii~~alizsti~r. is not t~ a~'cm~nt. for ai tt,r .f) the results in Ref. 35 are applicable. If the function ~(d) can change in sign, self oscilla- tions of the first kind as well as other qualitative charac- teristics of the system appear. pith 1C=0 the conservative cane of equation /4.2/ holds true and /4.3/ has tiie following ger_eral solution: S'P=Pl7~+pco7s+ zco~ 2d'-c The equilibrium states of the system are either centers or saddle-points. With K * ~ the coordinates of the equilibrium states of equations /G,2j are independent of 'K. They are determined as in the case K = 0 by the equalities ~-sind-ztin2d--x(1-~,corP~)S=o, ,S~o !;,q/ "'he space of parameters k,2, ~ . B }e divided into regions ', ?I, lII ;fig. ?; at wi:oec points eq~~ations ~a.2J have G,2 and Q equilibrium states, respectively, with b=rjt and S=0 . ?./'~nese//rjegions are de?ined btu i.he ir~~~gUti;~.ties: 1 L'>~IIJ= -~?z?.YtP'vl +Ez~ '3+ F32=.~.' 3P z. _ __ Ii ~1>~~L2tc/` 1"iP2~ :~' aQ"> - r; `i~eCi) , The character of the en~~:i.~ltrium state; , . de,~>rr.i~;~ e~ the .~~:,ts of tht~ cnr~?esp:~n~i;a n.,Krscteris`ic e~~:A_';i; ~ -~f-hCa7~oi 1_~/K ;,-5cos2~ z ._zcosPa~ -~~n,h :.?ilf 9qui:;,.rii:IP ~,s`..,s 1G1,7) n.,;~ (~,~ 3 . ~ { P~.=-i, COS (St t~~CE~.f 0~'i ~~L a"F auicie pcir.:c; tk.a ?uil'~:r~:.r ..r:3c.ea c~ /when;u is the maximum rdinate of the isoclinic line /4.4~r. Tk:erefore, with x>D ar,d ,(~>~! the equations %4.2/ have a i~~order cycle. Thus, equations /4.2~~ nave only one stable ultimate .;role cf the second kind with>~(2) and with ~C~ko,O~c (~~ - :~o equilibrium states but a stable u.timate c,yele of the i~nd,..~.,. ?xis. , 2. X~k,,DsDd can occur in oft , dE reality. '"he conditions for small sense stability are the same as Tom's conditions for a simple cylindrical reservoir. Let us introduce a rase variable u = ~X!dE . '~~:'he prob- lem amounts to investigating the non-linear differential equation: N4_ l' :'/f~'^- c,~ -.,. ... ~t?'~~-~_-14u J_~4 (I SiJx T/j 4~~!`~Ti' ~X Z[.0, {. P, 3,.., n ~Dn~/7n,Yn~llpn,AJ-2Ct)Yn~1zn?n~~ , ?1 '! Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .^r a . er~ative q~.,;~:.tic form; 2n (/ //~~ ~:.,., 2 -t^=, :~r:en the 2n -F3rL imposed or. the ,r~itra- . ..- ii~~; ~.,, tae ~r~ parrr:etrrs ?_ ~ tisfied e ,total numher o` ~n e f ~ n' 2 ~, the T -~l.i ~~_ll./ 7~ ~ ,'~ j,i-~,,~1u: ui "t.is gives a total of 2n conditions. ? -i Pn-~ \ ~~ f:r:ctio;: Cr ',.ill have a positive sign, ani its derivative ~, ne,?~tive s~,ipn in the refiion ~ . In order to deter?ine ';e '._:rr._e sense stability region for steaiy-state opera'.ien, a fs~:ily of hypersurfaces ~ C and the hypersurface U'= ~ {(. are ;:unsucted with the latter tangent tc the hypersurface C~= Ca of the family. The region of the `fin- dimensio- nal phase spare inside the hypersurface t!"= Co will he ir:ciuied entirely in the large sense stability region for e'esdy-state operation. ~~odern high-speed electronic digi- t.;/ ca~n~;u'.ers may be used effectively for calcula!ing Co ~y uring i,i-~.punnv's toeorem on the unstability of ur;pertur- ' ei r:,tion, we can find in a siu.i tar ~~~ :y t?; ret^: ~n of a .cri ,..~'.,tility for steady-state cnndiliene. "'he dir~ict method cf Liapunoc can be sed t ~ :,`i-;vr tha~ Fin ta~'r~ is no additional captation throe:; zir. reservoir, back flo* in the nrc,rierc c. - ?~"> >? =~:... ~~en additional captatien at some initial sectione of the heH?- work sys?em stable back flow conditions can exist, It should be noted in conclusion that the stability problem fer steady-state operation of a hydroelectric sta- tion with a differential equalizing reservoir is a special case of the stability problem for steady-state operation with ?2 equalizing reservoirs whea rt=P,Bp~ooond~2--o. Isere, the first. reservoir must have a lumped resistance this explaine why the firnt 2-1 reservoirs were taken to have lumped resietances~. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 I. B. A. T a ~ r, 3aearpxcecxxe ueIIx c IIepxoAx- vecxH xaYexxngxWxcx napamerpaxtc a IIepexoAHae IIpouec- ca B cxBapoaBxx ramaxax, AH CCCP, 1958. 2. II. C. H(A a x o B, Ycro#RxBOCrb asexrpxuec- aax cacreti, I'ocaaeproaaAar, I948. 3. H. A. KaprBeaxmBxax, Biassxe saaYYOAe#crass rxApaBagvecxYx, nexaxxRecxxx g aaea- rpxvecxxx IIpoueccoa Ha ycro#~cBOCrb pa6ora aaesrpo- craa~#, flasecrxR AH CCCP, OTH, 2, I958. 4. t. B. A p o x o B x R, A. K. k n 6 x Yu a B, Baxssae xgepAxx BoAu B ryp6asxor rpy6oIIpoaoAe Ha yc- ro#gxBOCTb pa6oTH I'EC c ypasa$reabHbnia peaepByapaxa, 4H3Y3, PaAxocpxaxxa, 3, I960. 5. A. A. AxApoHOS, A. A. Barr, C. 3. % a # x x B, Teopxx xoae6axxli, ~xararrxa,I959. 6. H. A. ~e a e a ~o B, ]I. B. Po Ax rx x, K reopam cmrxerpxexoro xyabrxBa6paropa, AAH CCCP, BI, 3, I95I. 7. K. C. l p a A m r e# x, Heaxxe#ase xx~epea- uxaabeae ypaBxexHa c YaabDU~ msozxreaxYx IIpx xexoropb}x IIpomaao~Biu, AAH CCCP, 66, I949. 8. N. C. t p a A m T e# x, Aa~CpepeauxaabHxe ypaaaeama, B soropse rHOaxreas~ BxoAar paaaxu~e crenesx raaoro IIapareTpa, AAH CCCP, 82, I, I952. 9. JI. C. Il o a r p x r x H, AcaxIIrorgvecxoe IIo- BeAesme pemexx# cxcrem Amc~epeageaabHxx ypasxeHR# c Yg~b[Y IIapBYerp01[ IIPA BHCmxx IIpoH3BOABbIx, K3BeCTHH AH CCCP, cep. MareYar., 2I, 5, 2957. I0. A. M. T x x o x o s, Cxcreux Axc~epexux- aabHbnc ypaBxexx#, coAepxa~xe xaa~e IIapaxerpb~ npx IIpoxaBOAxbrx, MaremaT.c6., 3I /73/, I952. II. H. A. KaprH eaxmBxax, YcTO#ux- BocTb B Maaou Axxa1[HRecsxx excrex, coAepxa~xx re= axe napaNerpx, KaB.AH CCCP, OTA, 9, I957, I2. Y1. II. A H A, p e e B a, Hexoropb~e Bonpocx ycro#axHOCrx cragxoxapxxx pexxxos rxRpoaaexrpocrax- ux# x sxepreTxuecxxx cxcreY, TpyAbtid3Y1, I9, I956. I3. H. A. HapTBeaxmBxax, Baxxxxe BaaHNOAe#crBxx rx~pasaxaecxxx, ~exaxx~ecxxx x aaex- Tpxgecxxx IIpoueccoB Ha yCT01~4xBCCTb pa6oTE.~ aaexTno- craxux?, N3B. AH CCCP, OTH, 2, I953. I4. B. B x r e x, Baxsxxe IIaparerpoB peryax~,- pa cxopocrx rxAporyp6xHbt Ha xpxTxeecxym nao~a,gb ypasxxreabxoro peaepayapa, ?~aB. AH CCCF, 0TH, 3Heprerxxa x aBrorearxxa, 3, I960. I5. Y. Y i t e k, Yliv regulace turbin ne hv~1roulicK pieehodo~e bevy ve vyrovnavacich komorach vodnicn elek?.r - ren. Prace v~zkumnbho uetavu energrtick~ho, I, Prf:.:x. I6. H, B x '^ e x, YCT011KxHOCTb s ua::o~a rr.r;~ru- posaBxa Typ6xH B yczoaxRx zanaa,~erbxou {~aG~C?~ ~-ex- TpocTaxux#, ?iayvH,,goxa. Bxcm. mxoax, 3sc~, re~:~x ;,~ , I958. I7. H. A. a a p T B e a N m B x ~~, . ?~ rcZ~; HCCaeA0B8Hxa CTaTxueCxoN yCTO1~gxBOCT}} 3g;?. ^,. ;;r~,~.r.. xxx CHCTeLt IIpH CHabECL6 peryaxpOBdHx}i, T~.;;,~:~ ~tr;x;'; ~~, I959. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 I8. B. M. H a T n a H B, 0 craraaecao# ycro~- gNHOCTN 3deHTponepeABQI[ 8 CHSaY C HB~S~eY xeCEOHb- xax rexeparopoe xa nepe,~am,eit craa~ca, HaB. AH CCCP, 0TH, 7, I957. I9. B. Y. M a r n a a a, 06 ycro~~rsocrx cnos- xbI7C 3xeprocxcrer, H3B. AH CCCP, 0TH, ZI, I958. 20. T. B. Mx as e? a ~, T. ~. Ko a a o e- c x M #, YCTO~QHBOCTb x aaeecrBO aepeaoxebu npouec- coH cxcreYx perynapo?aaas Hoa6ysAeasx rxoroarperaT- HON 3dexTpOCTBHuMA, AH CCCP, I960. 2I. B. T. 6(o p o a o B c s a ~, 06 ycrox~- HocTx aapai=eabxo~ pa6oTS oAaoTanBb~c cxxapox~ re- xeparopoe, H38. AH CCCP, OTH, 3xepreTxsa x aBTOra- Txsa, 2, I959. 22. A. B. u y s e p x x $, YCTO~~IxBOCrb cea- 388EO~ CaCTetD;l sBTOraTHReC%oT0 pert'irposasYa npa BsyrprrpyIInoBO~ c$rretpas, HaH. AH CCCP, 0TH, 3xep- reraza a aeTOraraaa, 4, I959. 23. t.B. A p o x o B x a, $ onpeAeaeaxa~ Aocra- TOQH3DC yCAOBH~[ yCTO~REBOCTK a 6onbeor crauaoxapxbu peaxroe oAxoit npocrebseli axeprocacTer~r, YIBY3, Pa- Axo~Naxsa, I, I959. - 24.: T'. B. A p o x o B x a, H olipeAeaexxo Aoc- Ta-o~rx~~x ycnoHN~ ANxarxvecxois ycTnpuxBOCrx axepre- TN~reCKNI cc:cTe~a, Tpy~x BIiYr:`,5, 9, I959. :; ;;. :'. ... ;~ p o x o H rt ~;, .", oxpeAenexxn Aoc- TaTOuxx:.. ,rcTOHNH ANxa-aYUecxo~~ ycrnt'lgN3ocTZ-; cnozxoii axeprocs;cTe~ar1 xpa repeKexaa:~: a.,u.c., NBr'3,PaANO- j' 6 ~. - P>fe & ~B/f:-::Girl::^, ?p.::a7~ c_rr,as :oA rl~~~:;.~~. -,_ ?r~rraTenei: xpu peaxo- r~eper+errt~~r trar;: pax': - ~ _,,~_. nox3a?aT, I95~?. ;'3. ;. i;. _ o a r B K v, b: onpe~er,ex,rx~ yc- ToisuxBOCTrz n bo::Bmom; cTc.ur,orra~lrrrlx pe~aaE;oB t3C c ypasxxTenbxrar;.; nesepB;;apaMir, ?'BY3, FaA~to~';Nal;xa, 4, I96I. 29. 'i. C. m A a x o H, 0 craT:auecxoii ycTOito~rxxrx a~rexTp;rlrecxxx cxcTera, B c6. Pe6eAeH C. A., '',axoB Ti. C., I'opo~cxn>n p,. A., i{axTOp P.".., YcTOi~ur~BOCTb aarex.Tp~xecxrzx cl~creid /Tp.B3V1, 40/, I`J40. :30. II. `.,?. It p u ,i o B, H. H. E o r o x ro 6 0 ~, 0 xoneGaxxRr, cxx_r-poxx~x ua~l.x, C6 ;/crov~?u;HOCTU xa- pa~r.aeni:xox pa6oT~~: n crlxxpoxrrilx Mamxx, l~apbxoB-ItxeB, I932,. 3I. ii. r. B a a c o B, ABTOxone6axNq cxxxpox- xoro broTopa, Yx.aan.I'ophxoBCx. roc.yx-Ta, I3, I93~?. 32.J1. H. ;; e n ~ c T x x a, CG ycroiltl=zsocTN pe- xxraa pa6oTr>z axxo-non~crroro cuxxpcxrro; o AH!iraTena, YIaB. AH CCCP, 0TH, I0, I9~4. 33. JI. H. b e n ro c T N x a, Q6 o~xo;~l ypanxexsi.r N3 TeopNx anexTpNyecxvrx ntamux, C6. "!?arlxTa F. h. hxA- .oxona", !Hf CCCP, ii3~6. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 34? {{,Edgertoa,P.Fourmarier,The pulling into step of a salient-pole einchronous motor, AIEE, Trans., 54, 1931? 35. JI. H. B e n n c T a g a, 0ccneAosaxxe xe- dxxe~xon cxcre~ ~aaoso~+ aBTOnoAcTpo~ixx uacTOTx, MBY3, P8Ax0~lA3xxa, 2, I959. 36. II. H. B e n~ c T A x a, OnpeAenexxe xa- RecrBexxo~ cTpyxTypbc "rpy6o~" Axxaxxuecxox cxcxer~ sryzea npx6neaexxoro xocrpoexxs oco6bnc rpaexropxpi, flBY3, PaAao~xaxxa, 4, I959. 37. II. t. Mayxxoxsxu, ToxxxaoxeaTbt BpaAexxA, Boasxxan~A~ B cxxxpoxxo~l xamxxe npx Bxnn- ~exxx ee cnoco6or acaMOCxxxpoxxaauxx,Tp.A~3II,4,I956. 38. Ti. t. Mabxxoxxxu, ToxxaMOMexTbt acxxapoxxbtx x cxxxpoxxbu ~aamxx IIpg xas~exexxu cxo- pocxx xx Bpa~exxs, 3nexTpxuecTBO, 8, I958. 39. M. A. CapoxATxxxoB, Peax~pa- 60Tbi cxaapoxxba~ reaepaTOpoB, Tocaxeprox3Aar, I952. 40. B. A. A x A P e n g+ $btBOA AocTaTOexbu yc- AOBHYI yCTONgxHOCTH B 6oIIbIDOM cxc?eao3 cgxxpaxxbvc ecamxx, YIBB. HYM NocT. Toxa, 2, I95?. 4I. b. A. A x A p e A x, BbiBOA Aocrarouxoro yCJIdBxH yCTOV1uABOCTx B 60JIbmold Cx3ixpOHH0~1 HamxxbI, Tp.Jlexxxrp.i~onxTexx. xx-Ta, I9:;, I95B. 42. H. A. KapTH en xms xnx, iiepe:coA- xbTe npogeccx H axepreTx~ecxxx cxcTe-eax xax aaAaua o6ateH Teopxx xone6axxK, N3B? AH CCCP, 0TH, 3xepre- Txxa x aHTOxaxxxa, 2, I960. 4 3 t b. A ~~ o a o B x u, 3I . H. f e n n c T x- . . x a, 06 ycTOHaxBOCrr~ ~o:`:hs~txl~ ropxaoxra s ypaPxx- Teabxoli 6aIDxe, irxa. c6., I3, I95u. 44. H. A. K a p T H e n x m B x n x, YcTO1~9x- BOCTb B 6onbmoM cTagxoxapxbtx peax~+oH rxApocraxgxt~ c ypasxxTenbxbu~x peaepHyapaux, Yixac. c6?20, I954. 45. t. B. A p o x o B x a, YCTO~i~IxBOCTb xone- 6axxx ropxaoaTa H ypaHxxTeAbxoa peaepByape c con- porxBnexxeM, C6."IIaaxTx A.A.AxApoxoBa", Aft CCCP, I950. 46. H. H. fi a y T x x, IIoBeAexxe Axxar~xuecxxx cxcrea B6nxax rpaxxq o6nacTx ycroHUxBOCTx, I'ocrex- x3AaT, I949. 47, t. B. ApoxoBx x, OnpeAe.nexxe onac- xbnc x 6eaoxacxb~x rpaxxq o6nacrx ycro~igxaocTx Axxa- ~cxeecxoC~ cxcre~ B cnyeae ~oxyca, nexcagero xa nxxxx cxne~xx, YiBY3, PaAxoq~xaxxa, 2, I956. 48. q. K. n ro 6 x M g e B, YCTO~I~xBOCTb craux- oxapxux pe~sxMOB I'3C c ~x~;~,epexgxanbx~:a yparrr~rr.:;r;it,~.t peaepsyapoM, i~aH. AH CCCP, 0TH, I, I957. 49. fi. H. II m 6 x a g e B, K Hoapocy o6 ycTafi- ~xsocTx cragxoxapxux pexxMOH rxApoanexrpocraxgx~ c Axc~epexgxanbxxacx peaepsyapamx, i~I3BY3, PaAxoc~xaxxa, 2, I958. 50. N. t. M a a x x x, Teopxx ycrou~xBOCTx ABxaexxs, I'YITTA, I952. 5 I. t. B. ApoxoBx R, R. K. Il~ 6 x m g e B, OnpeAenexxe ycro~exsocTx cxcrentbt rxApaHnxgecxxx pe- aepHyapos aeTOAox rt -paa6xexxx, Nxx.c6.,2I.I955. 52. F. Vogt, Berechnung and Construction dea V~aeaer- achlossea, Stuttgart, I923. 53. H. A. ti a p T B e n x m B x ~~ x, I'xApaBnx- ~ecxax yCTOH4xBOCTb ypaBxxrenbxbrx 6amex, ;1aB. 51f~ri,`i, 26, I940, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ve .~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fts., Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~, ~ FM2 l7o,pocnpeo'eneNyou " meoPuu ~ , ycmouvu6o- cma /Io q~opMy~e ToMa r i X000 L, M Fig. 5 F~9. 6 Fig. 7 Ftg. B Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ,~~ ,~~ ~~ ~u N Ali ~I~ L'kr. SSR Acted. of Sci. Publ. Ilouac's Printcn Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 J. S. Arshanykh -Academe of Sciences of the l"`zbek SSR. Instiluie of Alathematics named alter t'. I. Romano~?ske, Tashkent 0 I~ECIHbIX CNCTEMAX TEOPHI~1 HEJINHE}~HbIX KOJIE6AHNI~ Non-linear oscilla'.i~ns are drfir.ed '~~~ the r,olutior.e o? the system of differential equ?t: one arising ;:om the correspor:ding physical laws, ifere wa cunsi~'er fcr tf,e sass of difinitneas the systems oonnected with Mechanics itne u::iversality of kinetic and canonical systems of diffsrer.- tial equations considered in the present paper will be seen from the note to the theorem 5 F; 2~, A'e call the chain system such system, the coordinates of which can be decomposed into groups YKI ~ ~KL ~ ~raK ~ k group so that its equations~lo motion ahonld have the form d ~JLrn _ CIL, ~rl - O~ 21 =1 2 , . ~Z 1 ~1 ~ ~~j1iy C~QSr;s of ~G cz) _ ~G ~~ d t c~' ~ ~~Z~Z . 0 ~ zL =1 z, ... hL l'l / !1' /'u) 7,1nt ~$! i ~22, 92nz d a~?~ _ ~~rKy -- =v I group ~2 group Kt where the functions K L r~l (/ = 12, ? ?. K> in neral case depend on tine, on all coordinates and on sli ve?~ locities: " ` ~'r~K ~~/ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 }~ , _i thr^e ~',,._ .Lon:; ...~.. ~. ._ ~__ , ,.entis:_ e..._ _. ~_~, si...,~t i:,.rodccti.n of ..~'~~ ~~-- ::ave a nuu,cer of c: ThE ~{pgpLBS OF THS C}IAIP SYS'P~IS ~,r 'tie rake of T,r,~t ciearress of the r:,,ce;:t o` the in uestecia ,r.d their tacts v.e sh::11 co::eider coae examn- ed ~-s s conditionally r,;,e-tacted ice. Yor exa:^cie: ., '.rirle a~.rtaeWatical :~endul~im rith a~asses of prime 1JZ K, ;~ ana ieng+.:.s of threads rcrmeeting them. it is !c.:o~s~n, nave the em. is one-tae?ed, bit in sore c,~es it r-.;~ be retire-- ~:eier::i~~ea in the piars motion be fi.ree angles +.' deviaticn cf the threads from the verticai. /,,rni~,n:.'_ tact cf tnis system is t'nree, .,r,cr the e:na- ,f c.^,ticn may be represented in the form ~q _ ~A - o 7 t C~c( rid G~ C~~ _ V ~ = C~ dt ~~ d~ d ~C _ ~~=0 dt ~~ ~~ ~n~:.ere ~= z (mfg+l~)pd~f (n~~)agcOS(d-~3~~~3t t~~accO,SC~c-~)oc~ +(m+n+~>o,~cosd, /~=Z (htp)~z~zt(ht/~)aBCOS(d~3)d,~i f ~= Z pc~` t~cco5(d-~"~a'vv~"f P~ccaS~-~"~~r~f l A l fP~~s~' 2~ the pendulum With length mass sliding along the inklined plane is determined by two coordinates: the length .S' of the path passed b;r the body acrd the angle real tact ie one, and of deviation from the vertical. The the conditional one is two, since be represented in t~~ c7s c~S _ ~~ dt ~~ ~~ ~' -z (17tm)s~tm~s~Dcos~~+o~f(111fm~~ss~~~~~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~~ A free rigid body may he considered as a conditio- .,~ll,y t+vo-tatted system with the following lick kinetic ao- tentials: z t ~~,,~,, ~~,~~, ~;, ~1 rwl_ 2 (A~~fB~~-~C2~1-1 (.~. after the work of generalized forcee along the virtual dis- Fiacementa has been reduced to the canonical fon: i Q~~~~ ~l=1 dt c~a~ - ~ =1, (/once, non-conservative, holonome system is a tro-tac- ted one. The tact two ie a real one here, if the field of pe- neralized forcee is a rotational one, i.e. under the condi- ~~ -~f ~ ~~~ ~~ Aa it was shown ~I/ a holonome conservative system with connections imposed has the same real tact as the considered one. A conditional. tact, the connections being, imposed, is equal to three. They connections ~R~A/~ fRf,. p, ~=1, ~~ ,m being imposed, the equations of motion will have the form d C7 ~L - f m ~ c%t a~ ~ ,c ~,+ ~r ~~'~~ If one reducos the elementary work of generalized for- ces to the canonical form and represents the eq~iationa of connections in the canonical form: +~Q ~=o, then, as it is easy to see, the equationa of Wotion will have the form d aL~9~ _ a~~~'_ o ~t ~P ~~ (fl c/ c~L~fJ _ ~L = o ~=1 dt ~~ ~f~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 d ~L`~' _ c~L r~~ _ ~ dt ~h~~ ~~?~,~ dt ~9~ r?.1 l ~ ~,u-1 /u nected fins}v with the interratir.v of ~iff~re;;:ial eouation, defininc ttese ,scillations. There ore are anal/ exarwi.^.e cer- tain. pro'~1er~s of the tneorv ,~~ ;ntegr2tina of the chain eye., terra 12~. Theorem I. ~l iet a kinetic ~otar,tial L be invariant with res- pect to Lic group with the operator k~ ~h~ The^ the chain system has an integral ~k. ~L ~L~bl _ coKSt. ih=f ~ h ~~ti~h Indeed, we have, according to the invariance of nh obi ~~ r~4j . ;~ ~aG uh~h + ~ . U~~h ) . o ~,~ ~~ `herefore the time derivative of the left-hand Bide of egas- ity ~5~, vhioh is equal to n~ ~l.`~~ ~ aL?~ ll uhzh ~9ihi } dt ~ hi uhihJ , iR=1 ~ h ~ h, ~G ~~l GonGequer,tiy we hays reduced non-holonome, non-conaerva- tive system to a chain system with four links. 2. Properties of the chair; system The investigation of the process of osciilntions is con- dt G~Qj/riy Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 'r'or more conveniencm we designate in the following the coor- dinates of eyaten b7 ~J ~ ~=1, 2, '~ N=ntfnZf'"nK, accepting; 8; ~d/ Theorem 2. the chain system may be reduced without introducing new coordinates to the form: ' ~- ~ ~ ~~~ ~ ~~ ~ ~F d~ ~ dt ~ ~1 j%. 1, 2, ... ~. -,;eed. a~e ecr,: but this~ezpreseion vanishes according to the invariancy N r ~ (~~a +u~ J=1 Theoren 4, Let the equations ~,.~ - ~~~ ~ P a~,rf~ :.ay ':e solved for ;,e re Y. aced by aii ~~imf~ ~ / ~nrt~ ~mfA eouivslvnt one _ ~R -~Fj~~7? ~m+,t ~Pinf.1 ~~ / (yf~m~~t ~ ~~fif.~ ~~= t, m _ ~' /~ / - _ ~ + ~ l Mfg ~iMt~l I ~p - `-'~' lndead we evaluate is two aesuWotions: 1; .~ '1 ~J 'r:^ ,, f ,~icrs :;f t ~~i ~~,_I J ~~y`~:. ~~ei:,. r"( 1 :, ar ? '_, :nay !~r,.~"1.Nrn1 ..tic:.- .. 4 4' ~~i~.M(r..; , V.P ?}~&~i Bet ~;rd the tr.,ti=u k'.r;er.. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 r~~ten. I:-~I`~/ i' evidently a senera:ization o.` the ~.outh sy:teme of 'he theory of eaus.tions of holonoc~e con- .ervetive sv~tems, _, _ r~s;r to Drove, t~:r.t if t'ne functions are invariant with respect to s groap. ..ith the o~-erator Mien tae enuations , 1 i have an ir,teF~ral - C'O/~S t . r. particular, if we :nave ~ - ~ ^cclic coordinates, ~~ . const sr~~i i t is necessary toaccept Q = L'~ in the em:atic:a A ,~ lti,e cyclic coordinates being disrekardedr, ~e solvable for velocities, the kinetic scstea is .-~t to the canonical systee of rank ~ /I?1 equira- ,nd tt,e co~,pleR,en- t=,ry ones /7~ are eznressed in te:?ms of verir.bles ne proof of this t:~eorem is the sayc as of the preceding one. 'ye'll r,.alce one reo-:.ark co~~nected with the universality of canonical system of rank ~~ore than zero. Let some s?: stem of even order ~x~ =X ~ t x~ ~~ ~ B~ ~Ce~eK acs) ~'x2N/J ~ 1'2Ni ~:ich d~scriles.`or instance, the nracesa of non-linear oscillations in the electrical circuit, Jerarate t~~e vyri;:'.- les ~~ ir.te two 3rnupa ~~ and ~~ and consider the sy~t.r: ~~=QJ~~i~i1, :wild the differential form ~~~;=~ ~Q~~P-p~~~J ~:rd re uce it tc the canenic:~l form ~~~~ _ ~'Nt~F ~H~ . P i:en tut Pi:~en r`"'-em t~ks-s t,hc canonical far r,e p:ass from ~ to v the es7i~nioal s?reten pill J' ' Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .-;;~,s`~? ~.~d to a kinetic one 'with the rarui a?re t:.an Caro. nir,il :-p: teT ,f ra^.k reare tnatl zero has a '~~.:~;,er c` . _.~~rt i^; !eee ~~~~, an;i ~,~a ~xill consii~ r two .f tr.e,^.~ being "hearem E. A canonical svst~~m with the rark more th,on zero i .r:ariar.t with reerect to the contact transformatiors. ;need let ue perform the tranc~for:aation L.. ~~~9J ~J~~~~ =CJl~~ j23/ holds, i.e. suppose", that the transformation j?2/ has the form ~Pne system /20/ may be written briefly in the form /24/ Q,~~ -p~J ~u~LF~1~ ~5 ~~/ / ~~ )- n ~~ i~ i it r ,~, _ ~~ ~~, v~l~ r.. ' ~ ,v ~ ~~- '_eplacin~ the left-nsnd side of the e;urlity right-nand aide of /26f, one obtains K=~I-~ ~t ' K~-k~ C~ tnat is eciciv=iler.t ~~ ~ svatF v ' ~ A _~+~ ~ ved theorem, on example of obvious interest for the problem of integratinr,. Theorem 7. Let th7e~ main canonical potential !7 and the complemen- tary ones rlp not depend on time and be in involution, i.e. the equalities (X,k~~=o, (K~ , X~z) / / 2= , To integrate such an involutioe system it is suffici= ent to obtain a total integral / ' of the compatible system of partial differential equations ~w aw ~ h t' ~ ~N~ r~~t' ~v~N Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ..en tiie in6e,{rsls of system Pill be where =t-t, ~=cl -coast .~=j,N-7-1 ,/ ~~ ~ ~C~ .t i ~ztt W=w~~lf...i~h~/1~~1~...~~jZC1,~,...~' nt~~' ,nd in order to obtair~all the rest it will be necessary to i inte~trate the system 0712 - ~~ t ~ ~ t4,: more than zero, ~~~ ir. order to prove this, perform atransformation /24/, having obtained b/ from the equations Hp=~,~ , ~= j-z ~J=~, ~~_C~=coast J=1 N /36/ T1th 'BPCBCt t0 ~.~? Cf.C`t1Ct ':"i^?9f02'?~fl'~-0n9 ~,_Ye~:i9 ~Re and the second ,group of the equations splits into three: ~' rr.is .. _~ .... .... ... ..r o,.e ~.. .. ~o rr.n,n ,., r,~c to^e. .vi.e^. I/ for ~=j, C1=-~ ~ ~t.n ~ J ~- I ~~~X, X~ A LL~~ e o/ ~ l e ~ ~ - _ - ~ ~_~> ~~/~a-2,~r, X, _ - ct~ _ ~ a tt ;~ , ;~ 1CC,ro- Xo~ ?AZ LC /.L o_Xnll, u L i ~ G -C_.D ,i- _ ~t 1 ~ ,,, ,3,,,~ ~ _xeap~ tf~,r- a, +.nare ,.t ard~u- are the roots of the characteristic equf:- z ticn ~~ ~ 2~~ +-a~= o ~e Po :real or r,omplez~, fl = 28 ' ~ ~ m ~ ?,^7 P--11o= ~. t trareformation ~C~ ;vn ~~ _,:ce+ upon rise ?r+~ verso ~' a;;l f rtea~sse rf th. ~r.e ui.ase space ,:,e s,vetFm .. ~ not cl;ar- :t~r:f at .er.c> r~' r,r.e _,~:t.a~:oke .~.~~': 5?retior~ of f::e :'pace r 0 _~ ~o t ~tr -~:r,not change the dynamic characteristics Nithin the :;?~ :' ;units as it t~~kes place in the chronometric escaper:c:~' model. For the symmetrical escapements we mall have -2 PQ f6f1~2-~Zf~~PQ-r~~-PQ~6~B~J - ~ r! 3_~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 where T ids the period, is the autaoecillation amplitude, Do and 8 are the coordinatee of the start and the end of the impulse for ~>O and ~ < 0 the start and the end ex- change their places~,PQ and /' are the constants, Let us denote ~ /~ f~~ ~ Sincep,-B = const, and ~odoes not depend upon Bo it is not difficult to estab- lish from the last expression that BT as a function of~ld,ie an odd function identically not equal to zero, For the fazed values of the parameters and /' with the symmetrical position of the impulse angle in relation to the balance equilibrium position ~for~u-=0~ the value of the period x111 be - as one can easily eee- minimal and hence by giving up the symmetry of the impulse angle one can attain only the increasing of the period, The peculiarities of the dynamics of the considered model are connected - ae it is not difficult to show - only xith those restrictions xhich the symmetry of the anchor fork imposes on the dynamics of the model, The close rela- tionship between the models cf the chronometric and free anchor escapements are revealed if we give up the symmetry in the fork-anchor design, If we neglect the mass of the anchor fork and the friction of e tooth with the pallets in the equations describing the motion of the anchor esca- pement model, we can treat the equations of the motion of the chronometric escapement model ~in some suitable idea- lization as?tge result of Limit process in the equa`ions of the anchor escapement motion, .. ~ i P a a. "t 1.4t ~G (~ ?'~ 1'r ass :,oc ~ u_: = anal ~ - ce egc:li_tr.um , ;sit_~~n. ';'he drpend~_noe ~` 'Y~e ~u' ..ii:ai~ upon t}:e haraaters~~Qand /" is st:^Kn o'. ',~. ~dodel of the_iectr;>a,ec~,sr''_..'.'_. elec.+r.~;r:echaricsl c~:ool i= ehs~~cr: :ur, t~erforn:., f__*~~e Htten ?a' ..~ a p~ ~:~n~ iT~~~ 1, e t~ th,o ne~l,~lizm '~:~ :a ~ n~~ : ~ ? s~: Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 2;~ The restoring force is linear. There_`ore tt.e ce,;='-- :ator moticn between. the impulses will be expressed ~;;`'~_ eYUetioi. ..~ .~capere:.t. '.t is .mown. ser _e'~,; _.. t+'o p;,~_- /YJ~SD-f,LfO t ~P,O= 0. _ _.e :rr,;:.,;; -~~ cl.~~.~.~it:~ tee vela?it,r ~: ~n ~;~iiF. T~ ;ui iastarte^.eous impulse. '~~itr~ t:1e fixed coor3ir~ate ,. c? .'~_e r,r;it , _ir:ked .?_ __. '.he r r'.~ana. ^, 'luring the taokr,~--.r,l ,.,,._~,er.t :,e ,~~r.',..~. t:.- ?c: ~a,~ _....., ,..~ .,.._ rtng ~ i and :`;orterr the alcctror~r.g.^.et cir- .._. the ..cond o~r'.ic:. 'a F:g,?i (D = Q' near the equ'_i'~riw;, posi`?.ion the cc;.e~'-art erer- Rl.ere V end '~' are cc^respondi^.pl~ the prei~pu:se ar~d postie,- nulse relocitiee and ~ is a constant characterizin< e._?.. `. e~ the power eupply feedi.nF the eiectroma~:.et. l,e? ~~>...> /-~~ be a seriee of '.he sncccsel~~;,, "left arc;,'_itn'.es" u~ith the Free at!enu~.tind; pF~:,ial~a mot`n^ tetween two successive i?p':'."yes. Considering the corctruc,tiop one can see *wc its Let us asr~.,.:,e t:u~t ir: ti_i= model: pecu'iarities: ~i':~e r~;ncuc?^r w?r+a ,;p e:+ery tics .,s t'_e 3Wx:?tide a~ The impulse place constancy. f1~ nets into t!.c L.ter~r^; The impulse is transmitted at ~ti certain angle of the perduln:~ deflect:en, t}.is angle iependine only npor. the constnc?ion of the triggering mechani.?n.. Tae :~ini:r~im er+clitude of an oscil'.ation durir.,; which 'ne elecUromagnet ^ircuit shor!en9, ^ar. re dif"crest. Let ~i_ __..ider *.he _ mpli:iel sode~, of ?}.e escape^U:.t i:a'ti:~g t !J' aS?7'tte ','.at ~.". ~".e mCiE.; '.ha err.rgy ._ _..,sipeted c. cn tse aeccuut o i ~ `!'. n it?lcn 'he energy ~osc __, ..., c,+.~ ar+~ ' e~:ec*ec,. wt;eie~''~~'.a "~e mrxia, acn:'ttij~? vs+.~:e .,...:itie` ~^,e tar.t~,r works +.p and /_- ~~4m,~e- W _ . 4''T. (5'.. ,. ~? .. _.e ilne l4-CY, ... C; ?L,...:~~ ~~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .r,.. the u:_tiai p~~iotWof the ioiow!~;q ,. ..~ tivo~ui. _.- ' Icfy rei;~tion~hic %~;, 6ccording ~ ~,.. ,-'~?fir,itic~: eef;- ~ent ., ie tr.e ;,i:a cst o.`z _., th? str~igFt 1ir,e fG=~'t,y the tipira: :+egc,en:. ~~ass~ng throagii its "ut~,~er" l;-^,ir,t; ~~s,r ~~,;:.- ', f'.'. t c 5 -eri. The cor-,?~.s;n~air,t, r Sri ~.--)L?~l.y,~ k ~ = /. ', N1 ~a ~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 no and , L - i9entica: trans.`crmation~ or of +.he trareforma - lion and wil: correscond to the ster~y periodical motion in which the a.terr,ation with the iepulees takes p: ace -these impulses ceing o` the series of ?~ ratiurs. For trans?oraatior, /22~, ..^or example, the series .~ ~ I cscil!at~nns.~;'~ 'he su^,ceecint impulse go one cv onpe ir. groups. at firctf?~-~, t~hen~Jz,...,~? am] at lastC Mmes. Each of these groups is separated frog tha :;they ore rY the series succession o? ~l esci:lations :a:tC: the folio - '"he stract~:ra o` the phase ourtrait o` the pura~eters ~a1~r~ ~ axis :r. the is!ervals, ;+~i',^h e,r^espc~rn; t, th~~ ca- rious types ,,. ~~';or., ._ raitk:er .^.o:c;>lioatc~. ?oR~ever, rho. most interesti::g rui:t for ractice ie. the ^e'.a~~tion of the simFlest metio?.a a~:~ tt:ue of ttcse intnrv,as ~!' thy: ci?.argee .._:~t. Va:ue~ ^. at. to . ?ivy' .r:~~.?,rr~at~.c:_ . i'a!. Va.'...^i ... ... ... t, r,r J:. .CCtrir ~, 'R'n; .. ?^. Ei~;.?' :..~,.._ ~.:r :r.ter~~ais ~ . , axi::h? C. 3. R a ~ $ x g, Teopxa aor.ebaaxla, 5,.-P..,OHTN, I937. 2. A. A. A x A p o a o B, ~. N. H e# x ap $, O AHx$eHxx xAeanba0~ xOAeax aaCOH, xlteAl$e# ,qBe CTe- nexe csoboy~. td~oAesb Ao-raBx~eeB~c aacoB, BAH CCCP, 5I, I~ I, I946, cTp. I7-20. 3. H. H. E a y T x a, Auaaaxaec$ax Teopxa aa- COBbI7C xoJ(OB be3 $oBCTpy$TxBaO~ OCT&EOB$x ROJ(OHOrO xo,neca, Naaeaepax~ ebopxus NxcTBTyTa uexaxxax AH CCCP, T. I6, I953. 4. H. H. B a y T u a, Teopxx cnyc$oaorc pery- IIHTOp1 C npysxaR$e~ IIIISCTHHE0~1, Nax. C6. ,~aCTKTyTa uezasasH AH CCCP, T.22, I955. 5. H. H. b a y T x a, AxxaMxaecxue aoAexH cao- boAa~x aacoaxx xoAos Cb, na~aaTU A. A.AHApoaoBa, AAH CCCP, I955. 6. H. H. b a y T H s, Ax~aeb$aa tioAeaa sxe$Tporezsaxaecaax aacoH c xoAOM I~IIna, 1lasecrxx AH CCCP 0TH, lF II, I957. 7. H. H. bay T x x, Axaa~rnaec$ue xoAe~a aec- BoboAH~ aacoaxx aoAoB, NasecTxg AH CCCP, 0TH, ~ I0, I955. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fig. i Eig.2 h ~0~ '-~' Ft g.3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ;'ly 5 Fi g. 4. Fig 8. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fr y. 9. i' tea. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 i 9. 15, Fr g t 5. Fi 9.17. T b=-OZS praQ Frq. 18.6. Frq. t9, xfr------ ~ -------------- ---- D ~~ ?r' 6ZB Fig? ftSa Frq.~G Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~'r9 ~'i. F~9.22 r"~ ~ ~~ 3_ ~,.,.,, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Next 1 Page(s) In Document Denied Q Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 :HTBUB-l C3iTe;&:-:;7s :i 5'i1~:i]L:i2Y PN.!~;CD:^1I ;uvT:tNS %r ;;UYs 2iG:i_,~:!;4AF SI6~BY5 lN~ :~:: Ai'P,.iCAT.UN NHTECPAJIhHblll KPNTGPNGI YCTONyNBOCTN CIEP~lOAY[~IECKNX ~B1~[iKGHNGi HEKOTOPb1X HEJIFIHF.FII{bIX C~ICTEAI N F.('0 tIPY(J10~1(EHYtA This paper shows that aka Savestigatlo~ o: stn:,. periodical soiutlons of a non-linanr diiferent~ai :qda- tions syatea r,ontair.ing .!tali paraaetic:r under de?a~- mined conditions co>Iea to ~he stationary points character The conditions oi' the aini nun of this function !u tae ,rt X stationary point (oti ~ , , ~ ar.~ )are indiepensabia (and in a nunber of casas aufficlent) conditions o1 stabi- lity a periodical solution correapondin3 to theel~'w~~F.~'~x'ry paraaeters vnluaa. Fora real plotting of function only a kaowledga of th^ ;venerating syaten periodical so'~a?? Lions set, depending rn tLe ;teutioued enova para~eetere~ is required. In tae simplest oasE; the function .c% !s aqua' :.~ to mean, far period value ei LaoranZ^. fuuctior: taican aiu`. an inverse si;n and calculated for a gsnarate solution. Tf ae consider tre system equilibrium pr its+'~ study of sons function +Z1, depending on ~1 ~ pareaeterr, of n "ganaratin~; solution". Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 then `un~t'_on ~ ;.s d3gandrata pariodicai ;-etion~~; into apste~ pct:~tia~ eaerg?1:.. Pius in this in conforaance rith (I.9} and (2.I) we obtnin ~{ M ~}_ ~ ~ ~ao~ ~oz ~.~~ x=1 ~,i ~ ~ ~`_ --~-s ------ 1 x) Iu I.G. ~6alicin's [4] monograph formulated theorf;m is accompanied 'oy raqu rements of unequal roots of eru&- tian (LIO), This requirement is yet not obligflto=',? and 1s only needed in the process of too thoorem %~-'con' stratlon by the aethod ohosen by L,p~alicintsae,for instance [6] and (~]~. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ,:;ar .can of ., ,cc~,'. order .a relatior. c~ x ~aot ion oG' in point. ~ ~i ~ , ~ . ~ ?~'s~ i it is enou5h the: the quad retie ;~nould bs a. positive xitn a deter- nine.. _ -? .~r", __. order 5c ac'Liaca tt_`:., it is z,! _~.our :, ~ 1 that a:~' t'~e root= of ~ order L IO) shoal.;,. a ~.a'-! oi. coinciding with equat.an :~o negetivo ;It is to ba added that all the rods cf t `uiS ~tiuation are real in so far as cn of arming to (2.I) d GSlt '-'v e~'.;hd ,. cV a:: :ai:J Gne ra at U~ aaa e`_1Cn (I. ii _? ~ ,;uadra~-ic forty is not wits a date~w'.~.~~? ;i,;~ a:~d the w~ a IIiguc~ o.-gar. ';has the ,.on~iitiona c. tea ~.t;s wrsxn ~- * k ~~77r~r,, ~;ra,:once in point ( ~basad ,;n :,.. .._.~lysis of saccIId order members in decomposition (2, 2;, caincid~; ;rich stability canditians cf a cor!yai~and- iaa grriodical solution cf the equations system (L I). Cnua in the prcblea of the existence an stabrlity pe- riodical solutions considered, the function /~ playa the of s stain 1 1 in the same rule as the potential energy 5 ,~roblam of seeking and iavastigation of stability equiiib- rlum positions o~ a conservative system with F.olonomic o,.yvizar ~.yu~ nadativa nG 38 nt. ~i1a Case :~~-~ arc roots equal .,c zero is dc:~~tfuli ;ncrge~,=~ ''~ edS ion Of d6o om". (;5 it~0 (L~"~';'~ stationary constraints. In otter words for the pariod!- cal solutions oY equations of the studied ty?e to?o"'=u? identical to the we 11 known theorems of Lagrange {L:jeune- Dirichlet) and A.t1. LjaPunov era true. The stability criterion formuQQl~~ated above is called integral in so far as function d/~ according to (2.I) and (I.9)~dependa on tho mean for theppp~~eriod characterie-? tics of the system motion. Function oaJ may ba called the potential of the moan ~ forces, The use of the integral criterion in the problem of mechanical vibrators syachro- nizatien. Insplte of the particular character of the ^on-/inset equations system considered abova,it met be pointed out to a series of mechanical systems the motion of which is described by means of exactly this typo of equations. The following problem C1~ is of interest for the dynamics of multi-vibrator vibrating machines drive as ~otth.er,, well as for`"ap~iications. II,, yet a certain number N of mechanical vibrators (unbalanced rotors brought to rotation by means of electric or other motors), be counted on one or several solid bodies connected between them and to a foundation by,menne of an arbitrary system of elastic element s, having V degrees of freedoID. The hard bodies deviations from a p'usi- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .b=d COOrn the _ ~ c .anctionu ~,~ and ~g the__ ~e'luas in 1u~ . _ .-w-, scluion (3.) and (i?iu) ha?~e :sea *.:a.'sen. ~. cs r,ri r ~._(ium ace" -.ons of tha mean _,., .n~~ .. - torques introcacod a'~,;~'a. for =u ea to v .t:o Jyr~,awa drw)e ~e;cr s a,? ess! n; the ~'U vbttll equal>t (a.ve'ra~;E~% uo~3Atio,,. :~ ?zhed?~rqu8 and ~L~e Torque rea4attu:ce fcroas, 2hnuld ba irl~ze. }ionce tho cihra- yy' .~r~~rated a~- additional torq- t10C1 tUr(f?e3 M9,~ - _:.. ;3.2i) i~.; 61 C fll l rFSents tG~ jean f~~ cee teriod ~~r..ve ~,` tha ue en r -, ~~a"cr ~. tha _..It.ns .., _ ~.,. , ~ u ,~, ~ , _, ;icn ~,=W (~~,...,~1~ ~_ tc file vir;rat-ons mauntad, c~,utor ~an~ - ~'s'. ca J a_ad ~~ ?s nntisa Gant __ ,~ ?.~ (3.251 y?,{,~ set be ,r seated ~a .: e; l!;1 n r 101? i C r;a'19. J C1.319t 1'.2~;> iii t0'C' ySrS a: i^T t;:~ -1 rat; par* t7'. ~11eaL' ~QP ~" valve ~P ,;ra:w, ,,,__ ~ p1 a; ~., rnie ;,e the period ~ .. . 'i)GL'~'. `1a1 e113 rnJ i1 nSt. %?',i ..?^,9n_i ,;:rC rE'Sl(!O1 .~ ti,'.b~ -~, _ ~': ;;ca:.~a of noacorse~vative ~;at;eralized i'oreas 'ti 'aeon Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 the vibration torque. The relationships (3.20), (3.24) and (3.25) allow to find vibration torques if the aeneratino solution (3.9) ~d (3.10} is known. Tor vibrators counted on a solid body Which can effect aflat-parallel motion end is bound by arbitrary eystea oY lineal erlastic elements With the ~a~obfol~ f~athenvibration ~~ the follovring ezQreeaion torque n co t oas o he work mentioned t R1 W = 0 dap ws=w sw,~atl,...,~x~= mtiEthcs~3w~r ~ tQ~ Ce~t~ d )~ zM PSZ ~" ~ ~c=~ t ' , D .p and Qst. Qv5 are values depend- ing on inerpia and emetic eystam parameters as Well as on the diaposltion and direction of vibrators rotation in the synchronous motion considered. The formulas for alg ~iven is pork ~ 1 psi and Qst Vibration torques reflect the presence in systems of eoaetiaea vary strong connections beteeen vibrators condi- tioned by oscillations vibrating bodies upon Which they are mounted. This connection explains many peculiar pheno- senra in those eyeteae,far instance the phenomen-a of vlbrat ors self~synchronisat *n. ~ A, ~/' / e~ ~ such Let exist function w W `~!)'?'1 x that ~s/~a ~s/~ P ~ o ar ~. ~a w zyl ~ ~ Q1 0 ~~g 2ft ~ o 0 d e -''~1 ~ JL~S - ~9 (3.28i -s _ ~5 > O be and let besides ail resis:~incc coca-`~='.~:~~~; ~r,~ntical and qus.~ ~ ~ ~Q ._;~ La ~;r toe is `ul- ~_~led fcr .d~... re 1; :~a ._ vie.awora. and tbere_`cra tna integral criterlor. ~'? stcbi',. ~ Pddu- ced iz 4 2,wi11 bs t^ae. Particularly siwplo is ins a,rmulation of roves*.iaa- tion results in tue case of a protler oe se lisynchroniza- tion of identical or nearly identica`,~~R~iors ahas it can ba cansidarad that the torques V.VS are equal to zero or are nagli~;ibly small while she forces resietan- DD (m) ce average torqueg 1]y snd the avorage rot ati*k; torq- ~ ?~ ues S for all vibrators are identical [ 9 ? Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 As far as, i'urther, due to the autoncwi~,~1 o: the initial equations system in sass of the se lfsynchroniza- function~ nd ti: !Y1 al, tion pro em a orques ~ aW ?-a~ mev :spend only on II s the difference of phases oLS- ola ~ but not cn the S itself (see, I'or instance the expression 3.27) then where oLO is an arbitrar;; aonatant. // Differentiating the last identity by oGO and ae- suming then that a{e ? ~ we obtain an w s~ a~ ""~W =D {.3.32) ? ss In reckoning {3.31) and (3.32) from equations (3.26) and relationship (3.28) it follows ?G! _~' 1 .Z.~/ J L ~~~1p OGL (3.34) 0 [~, In other worda,functioa ~J is the case considered is a mean for a period of oscillations x)~f Lagrange func- t:on value tade^ with ae opposite sign calculated for the x) Let us notice that the angle velocity of the synchro- nous rotation W in the particular caso of the self- synchronization considered, is determined from the first aqua-tion (3.33). Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 - 24 - q n it an oe Aritten 1 1 - -1 1 In these caees,obvlouely, Y o ? 4, Bzample of atabllity integral criterion use. ~e a concrete a:ample of the stability integral cri- terion use let us consider the problem of sel!'synchroni- zntion of two ide~ical vibrators mounted upon an elasti- cally supported solid body and having Parallel rues, sym- metrically situated is relation to the center of gravity sidered so soft, that the greatest of the natural frequencies of the body on shack absorbers is ueoli- gibly small i^ comparison with the f requency of forced. oacillat~ons. The motion equations of a vibrating body ac the ro- tation of vibrators, according to the laH, determined by equations i3.9) in the Qi?ren ,see bake := M u = -F' ls, sin, (~ t +~1) + ~ ~n(~~ I~=F~~, Harr x and ~ are the coordinate:: of tre ;ravi:y can- i tar of a body in relation to immebiie eras x ~~ system, is the rot et ion angle of tae vibrating mam'oar i. relation 1~1 a tt!o s.,~tem to t~nese axes counted ofi cio>~sq~sa, - 1). Nhile the elastic supports in this case are con- Gur'po T...~ ,. a. 2.nd ~,1 ie arelatice shift n 1l.ara ol."'~s - t~Z ~ C _ a value indepaudant ?ri:,rator ro~;etion phases aid 1 1.ron a':~.le ~~ ~ ~ to zerc wa coma to 3:lnatirg the der:ivetiv6lX+ /(,~~? g two substantislly differen' ~,rr.as'.:i~ar, sine=0 ,harin g) roc: s l~'~r ~ V ~'d ~~lg ' ~~ It is obvicus that in tea case ci vibraters' in identical directions (6i ~1 = 1~ ~~pti raspcncls to the first root provided that the }?.~;'~y a .CC ^: ..C E: ^~h .'i:e ,Main n tb - a,_~,sidrr~-'.. u a Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 -~ 7 - s;c;oc~ rcct. ni. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Next 1 Page(s) In Document Denied Q Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 I. I. I3LECHMAN, G. DZHAIELIDSE-Leningrad. Polvtechnical Institute. All-Union Scientific Research and Design Institute of :tilineral Dressing ('Mechanobr') ~, HEJIFIHENHb1E 3AAA4N TEOPNN BN6POTPAHCIIOPTA N BNfiPOCEIIAPAUYIH L Blechman, G. Dzhen!'.._ (Ieningrad USSR) Ths study of vi'oration transportation en3 vibretic.~ separation is a cardinal pro'o lam of CLa ut;e ~. ry o~ norr, 3,~ processes of s Ride Tango of vibration mschinary sic:: as vioration co.nveyars, acreans, feeders, unloadasy, dryers, concentrating tables, vibrasinkars etc. In turn, t:i,, central problem theory of vibrati r_ transportation and vibration separation is the probuss concerning the motion c~ a material particle or a .:~o"eis material. aedia layer upon r rough vibrating surface. So far, only the simplest poi,icular cases o~ ta:.s problem have bean investigated morn or less in detail. 'Phis is conditioned by an essential non-linoariaty of aquatiotts daecribin~ the vibrotranspertation processes. The most difficult questia^ concerning the motion of a loose materiel layer upon a vibrating surface which comes to the problem of dynamics of loose media complica- ted by the need to reckon upon the air resistance [9~, [2] However, the problem of a solid ai~;;le particle moti:^ too involves a loC o; difficdlties. The materiel na*.tic- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 2_ lee have in fact various and irregular shapes and that is the reason xby it is deeirsble to build np a tluary embracing a ride +nongh class of particles without a complete preciaioa concerning their shape. This can be achieved for "flat" particles which only slip dnrlag the movement upon the eurinee but do not roll from face [3J , [5~ .The movement of those particles is described noourately enough, as a moveimeat of mate- rial points differing only in friction coefficient value. In this case the shape is of no importance and the eapara tion at vibration Donors only according to friction ooef- ficienta. The movement of "romnded" particles (not "flat") is to be considered as the movement of solid bodies rolling upon a vibrating plane. Ia this case the shape of par- ticles is very important. Presently the that had been problem studied moat of e.11 is the problem concerning the movement of a material point upon a rough surface carrying out rectilinear iranalatad oscillations in conformance xith the simple harmonic lax. Tha xorra by G. Lindner [eJ , L. B.Lebenaon [3,4J , `.O.spivakovsky [10J, L. G. Lo~tzianaky [11J, d.:~.derg ~12~, H.P,Balxin [13J, u. D.Terekov [14,15 , +. k. Bauman ~16~, Y,l,Olevsky ~1T,18J, L I,Blechman [5-8~, 1. Xoung [20~ , B. Clockhouse ~21J , 3. BStcher [22~ , S.l. Os- ~adov [23J, ~.2aidei ~24~, D. D.dalkin [25J, Y./.Bruein ~26J rand others deal with this problem. However, dlld now this problem cannot ba considered ae exhaustively solved, particularly for motions with0u~ . '}}i! contact with a plane. The problem of a point motion up o., a vibrating surface inapite of its specific character has a great practical importance.. Tha results of its solution can be used in the problem of a ball and a cylinder notion upon a vibra- ting plane [5~ Teets have shown [2~ , ~8~ , [22~ that in many cases the mean velocity gf a materiel layer mo- tion is accurately enough determinated by formula for a particle. It last in the works [5-Z, 28, 29J it Las bean shown that many prroblema; and among them those not concerning the considered problem of uechanica, como to the same differential aqua tions system. The list oi' these problems will be expanded below. The problems concerning the motion of a material point upon a vibrating plane carrying out rectilinear non-harmonic oscillations, harmonic oacillationa in two mutually perpendicular directions (in particular-circular translated oscillations? as well as longitudinally. - transverse oscillations have been little iavestigsted. The particular cases of the first problem have been con- eidered in the works by G.7,i.ndner [9~ L,B.Levenson ~4J B. G,Yopilov [3G, 31~ and others. The case of circular oacillationa of a plane has bean analysed by L,B:Leven- aon [4J and ?Y. l.Olevsky[18J. Some proble'me concerning the Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 j Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ... o. points and _cu:.c. oc.iies luring, ~rans~~?ere and loa~:itudinelly transverse oscillations o: a ~,-1ana ware ?tu,.ed 'oy 'atz~a She-inua ~i3~. iris paper dea].r f'?rat whir L'ae ~_ robie~ ai a material zoint motion upc:. a rough vibratins ,Mono carryin~;~ cut ._arionic translated oscillatious in two nutuelly ger?,~an- di ;ular directleus with a diffarence in pirasas. hecti- ._.,car harmonic and circula. csci.llaticns arN speci.iic cases of ti.is law of motion. FurtLar tt,a paper deals ~ritc a detai_ad ^.:al;:sis of c'.e case o: the motion o_? a ~;~article in ats.a^v cf bounc- i:~ sad re.orts some new ra~aareh results ?~.~.-~"ins the macicr. with bourain,;. In conclusion it re,~orts a communica.io: ?onoer:rir~.. concrete applications o` theory to t..: nroble;rs o: calcu- lation of vibrating conveyers, vioroseparetors, vioro- sinkera etc. y 1. Equations o- a material point movewsr.c upon a rough plane carrpin,;~ out iraraonic translated oscillations in two mutually pr_;andizular directions let us consider tm~ material point (fla~ par~icle) relative motion upo^ a plane with a slope to the ncr!zont under some angle p(.carrying out translated oscil.la. ^ns upon elliptical tra~actoriea resulting iron th?3 s~.rmme- Lion of oscillation in two arbitrary mut:ally periendi- cuiar directions, lying In a vertical plane, parpaudi_ cular to the vibrating plane considered (Fig. ~.), At an arbitrary chcice of en immobile systom of ccor- dinates aces ~ O,~ displacement ~ and -' pcints of the plane along Lhasa axes in the problem axa?inad may ba presented in :arm: 1 (1. 1) here ~, and g .. era amplitudes of constituent oscil- lations, W - frequency, and E - difference in phase betsean constituents. The u~iuaticn of the elliptical trajectory of an ar- bitrary plane point A in the indicated mc,tlon in local immobile axes ~, OZ 1~ is obtained by means of exclusion of time ~ from equations (1.1): Q ~ .~. + ~ _ ~ ~ Cosh = Sin~~ a g ~a ~ The equation (1.2) does not change at substitution of E for - ~ . Practically speaking the eliiose shape at the substitution indicated as it was previous/? remains. However such substitution leads to a chance o. the motion direction of the plane points npan the e: :;r~.e, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ,N= ,II((f)= rn~coso(-m~co~sinwt The case of nncti;inear hara,nic ^.ncillations o.' a (1 5) plane uauar a^gla ~ ana ait'~~ an enplitu:a H is ob- and it ccatormanca u_'h Cculorbs lax the projection, of teiaed from (1.1) whamDD 11 tke friction force upon axis l~x ie (Z= ACOS~~ b= 11S(.2p and ~=O To the circular oscillations of a plane correspond a=~= Ao,~=~q (Qo-radius of circular trajectory oscillations). Let us introduce the Wobilu axes of coordinates x Oy invariably constrained with the oscillating plane and parallel to the immobile axes ~17t and ~ ~'h ~ ~0 Then the equations of the relative motio^ of narticlc of the mass 'llZ in erojactions oc axes xOy erili ha mx . maw~s~nc~t tE~- m~s~not + F my= m~WsS~Rwt -rn~coso~ +,,~ Hera I' -projection of tke force of Coulori's i'rlction on axes Ox , X -projection of the normal reaction upon axis Qy , - m.~= mQwz S~1t (Wt t j;~ and - 1R1~ c 'mGu)z StR CJt -pro jection of inertia ,~ on axis Ox and Oy . At the motion of the particle u, on the vibrating plane (y?4, x$~0) accardirg to equation (1.4. the projection of a normal reaction upon axis Q y is given by formula Suostituti;~ the axpreseic, (1.`;) i~ (;.E} and ;hen (1.6) ::n (", :.) ae ~,btair tr,: simple trana.,~sing the following equation of the notion sf a panicle upon e. plane ;:L12loC~P~k cis gg There ~=0.ZCt~~ COS ~ ahora 9esidea the followigg symools are brought in ~~. ~Q`'~:osz E~`I t ro5 di' t = _-_1 ti~Cus~ 511 ~~ ~ f 1~~2n~zs C~.C4S~'CaSr ~ ~;~~ 1 a r 4~,r4 ~? = u~ ~:.as ~ sirs Thereupon here anc further ~?,hiie dual sp~cbc~s a.c :resent in iormulao cya upper symbols sre ~;crres^_oadly; co the Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Blida of tide particle forward (JC ~0) and loner - to tiia slido bacYwards (x;, ee of the CoulolL's the presence of the fri-ctlc:, f~~;c and sucoad, by the ucilaterai constroiu,;, The basic difficulty of the preblew so'_cticr is or:--~ ditioned by the fact that the differential equat:oes rn: separate stegas (=or instance at slipping ~,r at f;.igh4 are written in a different alniytica f:om. At t:~e sa;.; time at a^ery equation stage it is easy i.ntagrate It is tho aaarch far-transition moments from our at age to the other that 1~ a very difficult pa:~ of ;._ problem. The basic investigation problem is to find _.. lotions for the mentioned shove squatlons system tc v._ correspond steady state regimes of the particle coot/:. upon a vibrating plane i.a. aotions established aft~~~: the passing of a great enough time interval. Treasitac;. r>~ ;. It the moment of attin~ jt_ , cc.s ,a~'.-^_le ~~~i11 immediately slip bacx ( x ~ ~ ). lnd the aubinterval of relative rest intervals ~~{ fetus/ II Inequalities at last et the moment is somewhere In I1p ~ the psrtlcle rill stay in a s~ ze till the beeinnin~ of ono of the sub- or ~t_ or til, the beginning o:' in- daiininr; intervals and subintervals, may be espreesed plainly prcvided the exspressionn far forces according to (1.5) and (1.131 ,JV ~t~ and F?~t~ would ba substituted. &eaulting relationships are liven in table 1. Table 1 ------------- ---- _ ------- i -- -?---------- i~-------------- Intervals ___----- n"~t e r w~a?1 s .C,p a"v~C (.~~ ~ Q Subintervals Subintervals Subintervals I.+ -----I'=------ ~ ,~-~Wt~o ~~+ b'LiL(W~f i ' In the table the following symbols besides (1.?) are presented: M,tw n -~ M,,. azcvsz~, fa6eos~~:n.Cp~ f~~~n=p, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 x'11. Cos ~'~+ _,~ cos P, mss ? ~~~~n i ~,~ fsvn.2~,, = acos ~, ~tn~ Hera the angia o: Lri~ticu ~~ _ ~.~ f; ;'~n~, wits for the intervais aa? suMniervals in~roduc%~.; ar- derived from equations ~n. Wto = .L`a (?. ,. }} , These equations have in:inita numbe+r of rods. We shall mean by ~o=W1` and ~13= Wt~i only roots defined by the expressions: ~o = f,UZC frL7l .~~ ~f ~""fir++ Ni.C fn.n ~ f, here NCZ~ltlti denotes the principal value o_' the 1 7 functions ranging in,the limits ~- ~ ~~~ ~~ Other roots of the equations (2.6), (2, 7) ara easily expreasad by means of theCC quantities Uo and ~~t Besides the phase angle Off- we shall introduce ~_ which is connected with ~~- by a relationshi; hU 'xe:_ as a_' other t?~,~l.atioas turn an~,o ~c_ra;~, values aa~ into thu formuia9 0: ~5~, /his is not dificnlc, to show, Tina: v ,..~ 3b 0':~i ~enLiOi.~fa partiri~iaT :8J8 a^CC, iL; t0 e,~._ ,1.9i an:1 ~2,5) ~iia foliotvinr; tcrmu:; M~=~ ~ ~c~st~+4)~~ M~~~=q~~os~l/fi~,) ~u 'llp u "`'` I. TT mpg ara true To divide ~ _ axis graphically into intervals any subintervals it is quite sufficient to plct the chart of the function sin W'C and sin (tat tc~'~t 1, The abscis_ sa of the intersection points of these curves xith the , corresponding, horisontal straight line y=~p and y=~~. will limit the intervals and subintervals. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 As is tha case of re~tiiinaar i~arwoaic oscille:ior.s ~S] some suSintarvals may not exist, This hnppans i" above ;aantione' subi:tervals are "ovarlappad" by tha intervals IJ +e ,gall not giva the detaiiel analysis here as this case is ~roerly discussed in article ~~~, Let's write ooth t!:e integrals of tha motion of the particle at diiiarent shafes and the equaticns de:ining :~a momants o'' transition from one stage oP motion to another. Ualocity projection and the dsplacemont of the par_ ticla a: the stage of splippia;; on the plane at initial couditious x(t"~, x~ x(t~) = Q ara given by t.._ :~llowin axprossioos ,ehicL are oatr..nad by r?_ ~ ~_ frt.ttlo(?~~ * xW rr x'.~t;- '~ C~S4 ~~-f ~' Co5~1C05!w4fdPf~- +~ bM7i CaCf Q/ ,s 9. ~f W Cos (Wt ~~~:~ xct)=-~ zC~sp (-t-t) t-- ~s~ -(t-~;- CAS (~t,?2(wt+dPf~-b-~.n(wtf~?t~,+x*(t-t"~ (2,12) _or:::ulas (~.';1) ani {2,1~) describe tae yotion of the ;'arti;~a until/ it slops or leaves file plane. in the last ~nsa .::y ~eTa..~ of Cra::,,_~io;.:ro,.. tae sta~?~ ,~ siippir~ U ~ .U ~araal Ii oapinnin~;, it ~,, ;t, ~., :,li, ~~in;; . ~,llo- in she+opposite dirac~icc, eaa Womei:C of :ansi~io:. ~= (,~* is :ieiined from aquation ,i?arch l; out~.ln;;l CAS( tdet)=COS(St+~?J_~~~~-(S~~J+Qt fit= M ~Q ~:n(~t ~ 4), Cif = x"coy y M ,. , (z. 15 ) Phrou~h there are sole roots in equation (2,13; only 1r' s the least root which is excaedin~ Uf has. In Casa whan them are no roots ;;mater than (~f the par- title would slip continuously in one direction. This ie obviously possible as in the easy c~ rectilinear csciila- is certain to stop after a certain lapse of time. As the result of the introduction oi' auxiliary variao- 1 f - 1 ~ ~ Kf ~ V i' - V ~~T K.i Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Way -' -:: :.acu: ~+ ic~''?] are omp;rtely applicable for U f - ~t +~'? she t~orui.on of tn~ "auutt,, ~ ~ ~~' ) n ,=..`3). Phe ahcve Waatiene.' rapes era ~ua~ plotted far 1 U OS Uau iv .,. .a 9-.,, - e rapes o: ti:is article e~~uatict ("19) o~ she article ~5~, ^1e`. is wily these cc sc _ "gaation fcr -.:a ~os~ _mportant particular case,. graphs are aYrli~ab.e to the sclu:irr, of eG~:ation (=.1~1). whe:. thv !;;itiai va_oeit~ equals taro and 43=0. For oo. ti:is it is sufficient to intreduco phase angles U t and ~r Let us considor the motion of the particle at the dafine3 by the relatianshi s ~ ~ P ~2,8,2.9~~it is assumed stage of flyi~k;'. Let us assume that 'the initial condi_ that ~ ~, ~ et us call the: steady stela regime in the large if the first condition and the relationships (4.3) are effected lndaPend oc the assum,:tion concernin,l the eampletion o? the condition (4.1). ':here may be cases wt~an the stability of the regime is interestird not for all typical transition roomer s ~o~ ~~ but only for some of team. Tien we shall soeaa snout a conditional stability on :orres~ondin~ transi- tion moments. 'iha requiremont for the stability of a steady state regime on the transition moments is less rigid than the requirement for stability in the sense of A. DE.Ljapunov. It is easy to refer to an example of emotion stably in the sense mentioned above, and unstable according to Ljapunov. The use in the problem considered of the conception concerning the stability, on transition moments allows in many cases to confine oneself to more simple calcula- tions and reasoning thnn those, that would be constrained with t.:e stability investigation in the sense of A.ld.Ljn- punov. Cm the other i:and for the ,:rcblam given where aseantielly only the Particle mean velocity and the-meet, acceleration represent interest, i~ is quite enough to prcv? the stsbility of the s:,aady state ra~ime o^ tran- eition moments. .., fact from the ccntinuc~s.characta: c~' the de~:en- dance of the Particle iisplacement fors steb;~ `rom the lniti;a and. the finite stage moments (see ~ ~i, it fol- lows :~at In case of a regular steady state regime stabi- lity an transition moments tie mean particle velocity for the s;li:.,icy period in a daturbed motion differs so sliall as desired (at a big enough t ) from the mean velocity in an undisturbed motion if only the disturbance is small enough. The same may be said in relation to the mean for a switching period acceleration of a particle in an accelerated steady state motion. It is to be emphasized, that the determination mentio- ned above may be useful at the investigation of proper- ties of solutions di2'ferential equations with right parts rhica in different regions of change of variables are set oy dii:erent annl}tical expressions. i,et us notice too, that when solving the problem con- sidered, and others close to it by means of the point transformation method ~26~ virtually the inverse method of looking for steady state motions is us3d and their sta- bility is investigated on transition moments, so that the __~ _._ t Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~~. ditfdrenca exists only in ~er~,,~l.uology. ~e snail mostly oe satisfied with consideration of stability "in .3w" thus the problem comes to the solution of a linear recurr~?nt equations system and to the study- ing of conditions for the realization of some relation- ships. S~aady states ~tion regimes in absence of bouncing. Case of rectilinear plane oscillations The motion of a particle without loss of contact with a vibratinb plane may occur , as it has been mentioned above only 'on condition ei execution o .. restitution coelficieht R at the in ~~he role -ainst tna ~~1ana is equal to ~._., siilne tlma aD is~l. t: .: LLSaU i. 'a ud ~Ur11C COAl1t10n .Or tna particle mu?~iun triecrr. ,..~..,. .. at )/ ~7 Q ti,:is cc,.ri :, is only >._ .yule at "IlonJ flClat'~~.. ~ ... 't-_ "., 3rd 2. 'ni++l state re;i~: , ~i ,.ur _,;ie motions ,v.iti: co~t.i::nous .~r*.,ainly , ~?~f ar_: .r~ctiral~f unstable as tae par roachin?; the p_as; s-.:~ a tr..nsverse component velocity ~.n:al ar near to zero et the execution of condition (1.9) rri,i nut bounce any ,,,ore. 2sin,; the iuvdrsu matu.,i .,.~ us oo, cider. the steady sea:e :notion rdoimes in abs~nar of bouncing. As it has been estaolisucd in '>_, t:;e ,,?" i:. which reached the vibra~in~; plane ',vita a zero t~?~ne:.r- se velocity component y and with ar. arbi~:ariiy lor~,i- tudinal velocity component x= xx Diener slips down- wards upon the plane without stop,,i:~ (this is possible only on condition IoCI ~ 4 ) or stops altar acerbain. iinita time interval. Thus, in absence of bouncin;;, twe ty'rds of ste&dy state ;article motions ago possible: a) uninterrupted slipping in one direction (down- wards). Such motion is acceldrated and has an iafinitd great switching; porici; b) regimes with stops which, naturally, are regular regimes since in accelerate) regimes a ?article cannot stop. Let us first study tiia ra?i,,,es with stops. 'Ne h~av^e to begin with a regime havin;; the switching period? ! , T a. I1 equal to the period of plane oscillations Io = w' 'or this purpose let us loox at 1'ig. 2 where at .~o>j the most general case of time axis division in subintar- vela is represented. At it has bean noticed this dTivi- aion is periodical: every other time interval t = to , time lengths corresponding to determined subintervals are recuring. i:ig. 2 shows moments ci' transition frog one stage of motion to another which in conformance vrith the princip- les of division may belong to ono or to another subinter- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 val. dt tai s, Ica moment of ,?~a:,~~-lion ~ro. any lasting step xl tc slipping forward is denoted by ~+ _ ~~ /wa ey ~-+ _ ~-t /W - from sipping Sac>wards to slip- ;ing :onward, '~y ~~_'=~o-/W../_fror -a{-lasting; stop to a slipping oacx. 1'h~ Foments t+o, Lt_ and t_o cor- res:or,d ~c ra:~erso t_ansitions respectively. if the trans iti~n moments het to the bounder; of two suoiut+}ervals as, for instance; it is lha .a:?e for ,moments (,p + and ~p_ it is canvenien'~ to relate them to a beginning; subinterval. The^ eaca transition momar.t wii~ balcu? only to one of tns subintervals. Farr insaa- ce mo::ant tt_ always belonbs to subinterval h_ ~:camication of .rig, 2 leac'.s to the conclusion that only the following steady state regimes ofTpartTicle mo?- tion with stops haviu,_; a sxitcnin~; period I = Io=~/W are possible. The arsons diractsd to the right on dig. 2 mars tiWa intervals c:rrac?cnd'..,g to '; be 'sta3os cf psr- ticle slipping forward; the arrows directed to the lest correspond to the stages of slipping oackwards; (the time intervals corresponding co relative rest are not marked). x) Under "lastin,; step" in contrast to an instantenuous stop is meant the stay of a particle in a state of relative rest during e:y. mgr, or less lasting, (hc'rever finila) time iatarval. Beu~ims 1. S1' ,-n of ..._ a..ic] r,: r backwards sith two :asti~~; sto,,s .n sve:? peric~t Ro~~imd __ Sli?Pick; i~,~rward a.: ~...?~,is ail,,. instanter,uous steps in every period; Regi:~. Slipping; forwards and ucek:var~~ ~ i t. lastic~ and ono irs::ar.:enuous sto. iu e-1_;' p ri rinds of such u reg~,c~ are ,~os~ib_ ;i~s,+ di:.: , the. ... ons case th, :astan~:nuous ~~ .~_._ .. pin,; forwards (regimo 36) ani in ':h' other - '~_~ ~~r_ tae slipping 'oar,~w.rds (regime 36 ) Reilma 4. Slip~~ing in one direotfen with onE, ~r,s~ step ie e?~erJ period. Tn~s reh;me nos also twc ca~:~~ corraspordi:~ to the slippin< foz^warns (raglm-~ ;., ,.. to tan slip pir~ backwards (re time 46 .. A further anal;rsis o? s~. stnto Ngim-;~, ~~~ licle motion. in absanca o~ .ru~. ,;zll .e Ld ~. taking place ur.dar agile ~3 As i; has b~~~n me?,: above even this relatively simplest probla~ iae-~.: the largo amount of investigations ?-ill rocartiy hr,; been fully enoui;h studied. I^ particalar the case ;1 unequal coefYicieata of cinet~c friction ~ a-~d ~. friolioa ~~ hsa not been ocnsidsred definiteh;. In the problem case considered in a1. relatien:,~:J of ~? 1 and 2 must be assumed Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 a=Qcosfi, ~. Q~n~,~=0 Than in conformance with (1.10), (2.5 ), (~. g) (?. 1C ), (2.1M1) and (2.17) wo shall have: ~ _ ~ cosd A wi &in~ ~-inld-~ P~) cos ~p f P,)~ ~ P bin (dt r ~a = ancsin ~o d, f = c~.csrn ,~' f, , r ~~ . - accsin ~,-, fable ti contairs summary of investigation re- sults of standy state regimes of particle motion without oi' particle motion without bouncing having a switching pariod equal to the period of plane oscillations or in- tiuitely large. If may be shown that other motions wit- hout bounciq; ara impossible. '1'he obtaining of necessary and sufficient conditions of existence and stability of regimes is carried out iu coniormanca witu the worn ~5~. In the present communica- tion thew conditiens ara given in a precised and simpli- fied form obtained by the author of the same worx. -, -s uuCe~.vcrthy, ihst in ti:-~ new i'orm a:- Bondi- , ~ion~ ace ea?ressed 'oy jeans of four ,arame';-: r; ~t+~ ~' r ~+ ~ i_ and ~- (or res;~ectively l,S~+ ~+~ ~_ r a,,d ~- )bound with the initial para~aeters by relation- si:ips (5.2). in cable 2 tiie following transcendental functions of ~::ese parawaters have been used ~(a, 6 ~~ . cos a? f cos c - (~rf c- B) S~:nQ , btin~` v't . ?I -1l_ = ?i bind ta~n~^ /~ 5in7~fi , (5.3) as well as functic.: ~~A,B) the ;ranks o!' wnich are Ic is to notice that the conditions of the existence c:' every raeime liven in the table, either desintegrate lntc two relationships groups which Curn into each other a~ the subltitution of pq} for ~_ , ~; for ~- , r ~- for (sz; and ~ for (r+ (or respectival.y ~9+ / / 1 for -,~~_ ~f for -~_ and inversely) or do not ciiaa3a r Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fora o1 stead, e~atence and stability of steacy state t t gim seers ?mo- tion regimes Pecessary and sufficient conditions of ~--- {- - S 2 ~_-, -- Gccela- forrarda ~ r ~tol~c~Pf I~~ ~ p P,egular regime 1 ~ z ~~b+,~,.,~~~a ~ - 4~LUs'L12 ~ )/ X11 ~ Regular regime z ~~ >-~ >: ucxoaae~,~u~x, Bflil. v. '~fIND~IbHHII CCP.apaTO". i:31;. 11H-ma .,~BXafiGO'.1. i`J.:~ . 4..,eaexcox ...L. ~,.a~zxx ;;;~~.t o6ora~exilr; aaresx:~~ ncxonaer~x. rocrta??,1emx3~aT. ;~.-?., I~~;iS. Vii. I.~rexaalt li.:', 4;ccaeRoHaxxe rmouecca an6pocenapalunr n 1=rrdpompaxcnopTatpo:,xtr. _'xxexepttuli cbor,xrr?:, 3A. s~I CCCP, mot.1 i~~;r. ~..~,ex;aax,i,h, iieJ4Dteiixxe 3a,uautl pyataentxt^ Btt6par~oxxxx :,ta~4m..yr.TOpe~:;~epam ~Ctccepmat~r xa cortex. ;,~tettoti cmenexn :r1,i(,. ~,i1;3.-Liar. }Ia,JIi, ..Qlll:kii'pa(~~H. ;;p~:};TeXiilal. 141~'I' :: , tu3.'B;liFixa, 1`.15;,. ~. I~~exn>ax ;..1'. 'feo~>rn xx6;oocena}~amopoP ;~ ee cxrta: c ~r~:- ;ixeii ttexomopxx upyt~-rx xoastx xl;bri;~loxxrl,~. nett. Coos;, " IF~~ maxa tt nacuem laa.nlx nn6paibloxxoro muna" uoT; peuarw. tuc:~li- l;,!!.ApmoSanei,cxoro, ca,;. All CCCP, ~ ~ , I954, ,s. 1.~lext,~1 4~.;~. O Bu6ope octtoHxxx na,,arnerpor Ht16n:x>}!oxi:~c ICOHBe1iepOB. LhUUfBT2x5 ?~)OOPalge'i3Ie IiYU'', x3u. 'vIHC1'limjrT ~ .-_`.?b- 'otip, .:exxxrp~, I'J;iJ. y. Lindner U. Fordarrinnen. Dis FordertACnai.:~, Heft 2, 1+abr. 1912. I0. C'nxaaxosct:xl'; A.O. lioxset~epxxe ycTaxoHxx, eacmb ii. i{auax~lecR xoxHeyiepns. I'oc.xay~txo~exxxe. x3uaTen>;cmBO Yrcpaxtflt; lapbxoH-;,rtetrponempoDCx, I9;~. ' II. ,io%4u4tcxxu ~,.P. x ~typtse A.H. TeopeTxxecxax r~exaltynsr, ~. Ill, OHT17, ,~.-I,1., I994. I2. bepr I;.A. ,,Bxxexxe taaTepxanbxo~ To~rxtl no xanecVtxWefdcrr xaxnoxxoH n~tocxocmx. c Tpexxe6l. L"6opx. "Teepxx,?xoxcmpy~.uls1 r; ]IpOH3B0J{CTBO CeJibCHO-X03A~ICTH. 61&1141}{", T. I, Cell]iX03PFi3: hi.-J[., I935. I;s. P~1aJxtxx H,P. 06 xxepi>Roxx~nc rpoxoTax. ;amlcxx ~Iexr~xrp. ropxoro xx~ra, T. X, Bxn. 3, I937. I4. TepcxoB P.J;,.:;Bmaexxe TeJ;a xa xaxnoxxolt xJtocxoeTx r, npouatbxar,41 xoJle6axxJ4~1. 1i3a. Taacxoro xF!nycTp. xx~'a xra. C.Iri,1(xpoBa, T. 50, Ban. IY, IJ;7. I5. TepcxoB I'.i,. ~xgexxe maTepxHna xa Tpaxcnopmepe c rap- 6lotxrtecxx6>x npoAOnax64Y4f xace6axxs4~u1. i3ecTxlut xxxexepoH rr me:;- x14(OH, !k I0, I940 x h ~, I94I. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 TNYJICD .. J1i:ii11'j~a;yi;1;~1`, ii0~!'h Rl ',ic~',2 . OTpo~auaT, 1J;i~. I7. ~:.e-Cffi:.`. !,..., r.Y`HeIJd:TS:FC~~' ,~.,. i94I, ~.~ i,ieTBrl'I;/DITI-;(aT, I'?5' Iy, I~41rOpbeB .,.. ":d~'.'!fciP.~' 91f6liat({i0Y.L1J::11'Pa'"`,.'I?]? ., xa40CTbe 3aTBUpO- u;j;~;Li9iili.X G;iliieruB ~~oE:CiLiii~ l:elf2 ;tE")2~6'! lt1lCi;epTill(4I;S ;,a CO}ICx. j"1BH. CTeA, IS8}j,q. '; (~;(fl, HLyd, J:!?:ii-,,^,1i. uCvIIlieYH. 41H-~P ti+1, '.. ~,:aST10-t', i 7.,'lOBCb, 1.5'L, :~C, Junb R, ;,'leitbewe~ung auf dar sc~wiejen,'an EUere. ~~orschun3 auf dem ~;ebiate das iti;aeiaurweseas, N. ~, 1952, ,.i. nlocahaus W. F6rdergeschwindl?ceit von Schuin;_ rinnen and schwindsiehen. nrdol and Kohla, N5 5, 1952. ~2,dbttcher S. ~leittrag zur lilarung der Gutbewegun~ auf Schwinfiexxx ~tacTx>$c xo ropxaoxTanbxoil Bx6pxpyr,~e~ xnoc- KOCTH. j`.3BeCTxH BHCI!1, yae6H. 3aBe~eHI4r1, I:TpOxTeJibCTBO x ap- XATeKTypa, ~6 ;i, Iy5E', 24. Seidel H. Die Wurfbewegung von Schiittgut auf der sah~ingenden 6bene. dergbautechnix, Nr 8, 9, 1958. 1C:i C ~( C1iiyVF:`C-, ,IN.hE,pc,iilhT, ,~..i:'S:, ':. .:i,., u ?~ieX;~lcd-i ii.:!, .i B~ ~~~~l;v v pdC~h:T~ - yexu.~ ;: r~a~ieverni>t ;La~;f :r ~'uiyx,~o-. CTBO, iN .,, I;15~, ,S~). 110II~iJiC7 .... .. 04~il1G TD.,}''i t,r:;; A;C'i~. ~; i i '.:OC. ,~ ]%2H?~dATj1a1~4?CiiYCiPO ~.TOJIa, li~~'[id~{1, ~k 4, 1:J'i, ,~I, iion~lo- }~ ; , tea, ter[ k-r~hEH ~r ,,er{t ,~x~r ~~ ? :.~ . 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'.~J. :'~.CBU.tiF1:10B il.... ,~,;~aP~J(OBLLu:a Cd;:GIIB:rC;iili!ra'HI1't RYiUDa- .1,4IOIiNOi L1il;6fFiH yIIaUTV;CIl11A I'!)yllT4. 11)y,yl 1Lt TOP~~CF01'0 POti- HO-h10TriUIyj11P11H. NH~I'~t 3?l, I....ii000H1, LNII, i,, 1JJu. ~, C2BIIBCH (i,u. 1. t~~, t..~YCFiNH. :S1l6JaI~riU}fllld~~i LIeTO,~~ [IOI'~1y1;+8- H11H CBd4: 1I C1'C IIP1i;d6HEII1I8 H CTPOHTE:;bCTHZ. I~OCCTP0~a13,~aT, ..BHItHP~11~ii, 1UvO. ,.`,'. i'CIIb]I11II ~:..,~. ~iCCIl2J(CB1H08 B C61L`ICTH TBOPHH ueHTpll~y- PC.:bFL'1X I;BO'.4e000B. ABTOpO(j!(:PdT J~31000PT21jYM H3 CORCft&}G~IB y9E- ![C.I CTQI;CICI ,(CHTO-A TeXH1P{. HBj~I(~ ,.eHHHPj)c'UICHNSS I;HCTRTyT XC- .'C;;Iu[bHO:: INOadLuI2iIHOCT;1, I~~~. -1' {0 w~ dt /z.I~ the following equations are obtained instead of system /z, I2/: dxs dys '~ ~z?fys ' ~z =FF,z,x~,y, , where t = ~ is a small parameter. /z.Ia/ To equations of type /2.II/ ie also reducible the more general case than that considered above, viz., when the ~onditiona of a dynamical ayetem are characterized by an angul~r.!sarieble d and n variables x,, x2 ,... , ,x? and described by the following system of equational Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 dxs-X at s (~,x,,-. dt=~,cJ(x,,.., ,?Cn~ lS= 1, P, .-,r xn)+;~ o~ x, xn) ~ ~ /2.I~/ xhere ~. is a large parameter,.~~ corresponds to the frequency of rotation oC; ,YS (a, :r,, , x2) , ~ (~, x,, . , x?) are periodic /or alnoat periodic/ functions of the angalsr variable oC of period P>2 , /Systems of type /2.15/ will be encountered when hygroecopical eyateme are studied, in the theory of accelerators, etc./. Introducing anew variable /2,15/ can be represented ae 1) Cs= 1, P, ... , n), = w (x~, , xn) f ~ A C~~ x~, , x~,) /2,16/ where ~ - ~ ie a small parameter, or eliiinating z , dx,~ X" ~ , Gt~~L' - ? S oC x! ..., xa E) (J'> 2 P, K~l w(x,, , x2)+d~{ (~r,xt, /2.17/ Rquatione of amore general type they /2,b/ can also be reduced to equations of type /2.II/, It is yell known ~3f that in studying non-etatibnary processes in nonlinear oscillatory eyateme we have to deal often xith systems of differential equations, au^h sa (1 ?1, 2, ,_~~ /219 f where ~t - generalized coordinates, E Qj - eztrrnai perturbing forces, c^^= Et -Blow time /slow ae compared to the natural unit of time - of the magnitude of the pe- riod of natural oaeillationa/, F. - a small positive pa- rameter, aij(z1= a~'x ~7 df _e.~o(x) ~fL~f~x~. /j.I4/ where ~ is a real vector function. Frem this system we cbtain the variables df~ ~, ~~-Ng=R R=R Introduce new variables n..eordint; to formulae /j.15/ where B and d are mutually conjugate, /j.I6/ Substituting /j,I6/ into /j,IQ/ and taking into con- sideration identities /j,I2/, /j,13/, we find: ~f~~)+Q (Ay~~)B ~A~~w)B)I ~da ~~*~~A(~r)(~~-H6~+ +A~~P)l~g-N~~~~ ~~X,~ft p (A(r1g+A(~)g~f -xo ~.~)- /j.17/ is satisfied, and, consequently, dg-HB=R 't schauld he noted that irsofar as the values of are rom;~lex, there is always a certain arbitrariness isl the choice of ;besides, choosing the additional co~di- tion in the form of /3.2^~ does rot appear to be neceasa- ril,y inns. Taking into considerations /3,20/, we obtain from system /j,IR/ a system of linear equations with r=al ~n>f- ficients in the variables of /j.19/: A~~)tA?J)(~f ~6)~ y. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Assume that the determinant of this system /3.22/ is different from zero for ~ = D *~ Then, by virtue of its continuity, it will be also continuous for some ~ -neighbourhood of the point B= o. Let Uj. denote the ~-neighbourhood of the point g=~ in which ~ (~ ~J# p and ~ Ua the region of variation of C~ B) for which ~E (~d. , Notice that it is always possible to find a small posi- tive S ,such that in the region ~ U~ the following inequality holds: Q~A~~)~fAl~)~la, ( -a ~. __ extends over the whole real axis b virt small number A _ /3.3a/ and inat expreasiana /j, j4/ and their partial derivatives with reaped to ~ and ~ up io 2 0 -order were bounded ir. the region ~_ ~ `r~t]~belonge tc some filed boundary dos+ain. Ia this perk the enfficient conditions of existence of D-property are gi- addition a behavior of solutions of ouch eyatet is studied. The previous achivements of the author ~2J, X31, and a theo?- re~ of S.A,Sam~edova ~4J are generalized in the work. ? 2. Basic lem0a. Let .x=(.xt,..., xn,~ E ~~' 1CCE"r ~Lx)? jJ,(x),..., f n(x)] E C!(3[). Let ua denote throhgh .~ (x) and ~ (x) respectively ~.h:. smallest and tLe largest characteristic numbers of a eym,~sr., led Jasohy's aatriz ,w ~ afi af~ 1 (f.f~ ~,(x~sP [.f (x)+f~"~~'Q ~a=,+ axe Then for a~ points aC and x + ~ such that ~=x+f~t. E,.d~ with ~ : t t ~ an inequalityl~ ie fulfilled, where ~~~ = fii1L ~~~) and ~~=nwx~~~~, proof . the hava fix+h) ~~)=!d f(x,~t~)dt= ~f'(f)hdt, /, where L : x + ~ ~1.' Aence ~f ~x+~)-~~x~, k)` I ~~(~~~,ol~ }t~= Frog the formula ~2.j/. evidently the in/equality ;2.2f follows. ^ Basic results of the work were reported at a Canferen~~ ce pf lioecow Mathematical Society on 2I/III I96T. ~ If JC= (,x~,?~?,1n) and y= (yt,???, yn) then we denote (x,tl)~E~i }yi the scalar product and ~~t= (~; xj l,p _ the Euclide norw. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~ j. The sufficient conditioae of a diaeipativabllity of the ayetes. Theorem I. Let where ~lz ~l E ~. ~Ee ~t -areal number for every t E 1+ d t ~J (x, ~J, or a symbol -~~, f (x,t,) a t. lG / ~ x t -the ]argent cha- and /~ 0 racterietic number of general~lized symmetrized Jacoby'a mat- riz ~(x'{~=P (tt f xlx.t~+LK7zl'x,~l!%} ~ where A = ~aij~ -ease positively defined constant n, x n, matriz. If : ,~ I~JI(x,{)~-dRa,tEI+ and 11(x,t)~ f3 }or I~c~ Q Ro /j.j/ P ~v`~A{r=.t)-A{(xy,{), x-xP~+(A{(xP,~J- Hence, using the basic le~oa and taking into account the conditions I~ and 2~ we get ?here ~ e ~/P a f,) p s IIA II ` ( ~,~ if only ~x~ ~ R t> Q R o /j.4/ . Frog inequality /i.4~ and on the bnaie of elementary reasoning, or by force of 2 Ioaidzava~e theory r6~, it follows that for each solution x = x (t) of the eyate^ ently large the inequality /3?I/ lxct~l 0 ) and "soft" ( ~"< 0 ) characteriatica of nonlinear restoring force in connection sith the specifics of the resonance pheno- menons in the nonlinear system separately. The system with "et if f" characteriatica of nonlinear restoring force (~`'>0) The results of modelation of the "stiff" nonlinear system are represented in Fig. i, 2. The resonance curve is constructed using dates of oscillograme of stationary stable regimes of motion. The points in Fig. 1 correspond to a calculation dates and to iuteraection characteristic Y (S2) with graph 5 (Q ) for a rather Bide region of values N (the shaded sector 1 in Fig. 2}. with the movement of the characteristic of energy source M (Q) from left to right the point F (point of intersection of characteriatica Y (S2) xith craph S~Q~ moves along the graph S (Q) from left to right in the direction, Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Point of the resonance curve moves is the same direction along branch BD. Near by point D the stationary regime braake and the system rather quickly passes to a new stationary regime of motion from the left to right branch of the resonance curve. This transition eeaentialy depends on incline N of characteristics M (Q) Stationary re- gimes has set at points I, II,III (Pig. I.) These points correspond to the points of intersection of characteris- tics Ml (S~) , ld2 ~Q) ,Ida (S~J with graph S ~,Sj~ (points I, II, III, Fig. 2). YVith decreasing of /N / the points of stationary oscil- lations will move-from left to right along the axis of S~ Oscillograms in Fig. 3a aad in Fig. 3b give us an obvious idea-about the transition process and stationary regimes at the points II, III, for Id2 (Q ) and Y3 (Q~ With the decreasing of S2, ,when characteristic g ( ~) moves from right to left along the axis ~ , the points of stationary regimes move in the same direction up till point E. ' At the point E the system passes to the points of the crossing characteristic M1 (Q) ,and the peak of the resonance curve is lost. The changes of amplitude %1 and velocity Z4 at the nonatationary pees of the system from the pout E to points IV, Y, VI are presented on the oscillograme in fig. 6, e,b, . I 1 I I i I~ ~ I u ~ oo ~D ~- CV ~ oo ~O ~- ~'V ni ~ ~.; ., .~ ~ p o 0 0 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .~ KPtt F~~. 3a F~. 38 m ~ Fig. 3 Oscillograms of nonstationary processes at frustra- M ~ tiona of oscillations in the system with ~`> ~ a) for the charac~ teristic ~ (S2 ) and b) for the charac- teristic M3 (52 )~ (SZ is increasing). Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 a? 2,0 /~ / ~, ~ Y ~ n F` B ..._ 0 40 1 0 J?0 !60 100 240 280 31 0 ~--~'to~i1.C Fig. 4 Resonance curve of the syste~ with y` > 0 ( Q ie decreasing) ~'~`(~) ~ ,~~ ~~ 0 Scs~} f'r ~ E -.-_ 20 40 60 80 100 f10 !40 !60 Id0 1001?0 240 260 ?BO 300 310 --- S2 tad~rac. Fig. 5 Graph S ( Q ) of system with ~`~ 0 ( Q ie decreasing) It was impossible to succeed in creating a stationary oscillation in the model for very alight elope and horizontal characteristic. Hatched sectors 2 in fig. 2 and fig. 5 represent the regions of these charactereitice. The oscillograms shaping the changes in magnitude of amplitude and frequency of oscillations for the characteristics ~ (Q ) within sector 2 Pig. 2 and fig. 5lare presented in fig. T. T h e s y s t e m wit h soft c h a r a c- teristic of nonlinear restoring f o r c e (~` 0, ~ a) for the characteriatlc M4 (g ), b) for charadteristic Y5 ( $~ ) c) for characteristic MG6 (~ ). (Q is Il~~~~~n~~u~i Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fig. 7 Oacillograma of nonatationary procesae for characteris- tic M (S2 } within sector 2, Fig. 2,5 , in the system with d'` >O >O 20 3v lip SD 60 TD 80 90 !00 1fo f~ p >30 X40 >So >6o I7o rBo f9o 200 Fig. 8 Graph S ( Q ) of the system with ~`ter~cTBne napa~ceTpxvecxo~l ~ HO7[BdaTedbH0~1 CNCTe-n~ C HCTOVHeHO~I BHepraa. H3B. AH CCCP, 0TH, ~ 5, 1960. &~-15881 Txp, 400 Ii poY3B06ICT88xxo-x3A8T@nbCY91~ YoM6xaaT BNHNTN Ab6epum, DYT66pbCYY~ ap? 403 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Next 1 Page(s) In Document Denied Q Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ,, ~~.. I. V. Ghteaok- MQscow Institute. of Energetics H. B. fJIATEHOK K B(?17PQCY OfiOGHOBAHHA METOAA 1'ApMOHH'~ECKOI'0 fiAdIAHCA To find periodic solutions in an analysis of automatic control systems one usually uses the epproaimate method of hanonic balance ~, 2]. About the permissibility of this method one judges comparing obtained result with results of enact analysis see, for example, j3J/. But such comparison one carries out only for the systems with piecewiae linear functions, because only for these systems one succeeds to find for example by the method of panting together ~nPunocoBaBas+u.2~ ~ the exact solution. In the author's thesis see ~4~ /there is a foundation of the method of harmonic balance in finding stable periodic solutions of the differential equation %I~ for some classes of functions ~(ii.~l Herewith onr- understards the foundation of this a:ethod not in the sense that one defines sequentially Niger approaimations accordixag to the method of harmonic balance and oae shows that they deecrihe the true solution more and more full, Here one sup pose that 1t ie found only the first approaimation y= cc.rtn cJt ! ~~ and shows that by some restrictions on ,~ ~ y. y~ not con- nected with its piecewise linear approaimation~ tha true pa- riodic solution of /I~ exists, it is stable and ie in soma neighbourhood of the approximate solution. When proving thnra are obtained estimates of size of this neighbourhood. Suppose that Pi n: f~~Ottnu,o~Jcosu~au=0. In this case the equations of harmonic balance have follaming P: 1.I~10Jtnt1,~~Jcoa'u~trnvclu, . - ~u~~ `J 1 J Pw I ~~~~Ufin4,l~CdCUJU)COSUc~Ll= 0. JI ~o Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 }et the system cif hat a solution cz,cJ , i.e, the method of harmonic balance yields the first approximation ~2/. For twice continuously differentiable functions f(y,y~ sufficient conditions of the existence of a stable periodic solution of ~I~ in some neighbourhood of the approximate so- lution ~2~ in essence mean that the Fourier coefficients of T (OSinu,au7casu), ~y (armu,arJcasu), ~y (prinu,o4JCOSV~ must be sufficientlq ~i.?? cl;Yr:f?ia* '::e variai;les :c reduoe '.he e?cuations ; I ?+unictr c: a.-~t~cteriue nor.;~tati~-nary _;'rati~;?,5 .L~cri; ..sae ~,. r:e :.ul'.i~iicc r...or, --- r;,p f:xr. 'y~;' P ~'' ale ~ l7ij2 a~ ,l"~'E Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~1'Clk._ oF~ll, _~ /~/~Gt~C~U~G~~, ~,-4~ ,~ Cl~ 9 _ /~ ~z. ~~ ~~ ~/ lK~~ ~ ~~~~7 /IP~ For the two-disc rotor the equations ~IB~ have been inte~*rated on the electronic analog [~'] H - 7 ""he curves in fig.6, 7 and 8 show the development of shaft-saggings ir. the resonance zone as a function of discs unbalance location. ;' Q. Let us consider applicability of the technique, described in chapter I, to the investigatior. of stabiiit~; cf ttre si:aft stationary movement. For comparison o= re~~a;.:: with those, published in literature ,~G1, we sna!1 sra~~~ stat~i- lity of elastic shat movr::ent, essu~r.ir= s if` sup?-art an. counting for w, roscopic soli;:;, ae a-~'_1 s~ `.: ert~=r:_zl sr.: resolved, ?'he use of complrx distil-.,cF?e:,.a ~'~ , t ?~ ' t~ _ ~ 2 will give us for the case of strsightfor- V-ard precession of the shaft following set statiorary oscilla- tion equations: ~e~,-a~~~~,-c~~ _ ~~~77, ~, t /Ih where aF = ~ 0 6 ~~~ 0 -~ ~ ~i~G ~~G// L ~~ ~ F- ~ Finding from /I9/ the value of CL and `~' far the shaft stationary oscillations and deriving the variation equations for amplitude and phase, we shall define, in accordance with jFi, the condition of stability for the shaft stationary mo- vement in the beyond - critical speed region: d`e ~ CJ _. S/ < CJK/~~f ~~ i `:'he condition ~2C~ is similar to one obtained by F.f.Diment- cerg ~E~, who investigated stability against perturbations of the generalized coordinates, but the technique proposed :.erE is much simpler, In conclusion it is necessary to note that the asympto- tic technique enal;les us to solve rroader range of nonline- ar problems of turbine rotor dynamics, than it was possible to show in this report. In ~Ql particularly the following ~:roblems sre discuseed: vibration of coaxial rotors, connep- ted by supporting elements and subjected tc double frrr,:er:cy ertar~oation; r,oupled vi~rratione of rotor and the c?.oe; viY,- -_,,~,^ ,~, +;ue -:.ass of n~hich is distribu?~~d r.ora +::e ri?cln, ~ r.d sn :. _. F :u ti~~or anal~~ .d _~:'~cillatior: o.~ ue Soaft i.n rearrn,,rs, tn.ins into account nonlinearity of the ~;/drodynai.,ic forces. It is shown that noniineari+.y of 'i:e hydrod,ynac:ic f~~rces, wrich arise in the oil '. lm of the tearing, leac:: to fraction resonance et an~~:it~r ._._-.- _ies euual to ior.Gle critical :'peed. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 I R E F r". R E N C R I. 6.I.Rrilov and N.N.Bogoljubov, Introduction to Non- linear kechanica, Kiev, 193T. 2, N.N.Aogoljubov and Ju.A.G:itropolslq , Asymptotic ~.:ethoda in the Theory of Nonlinear Oscillations, Phisical and watiumatical State Edition, IgSB. 3, Ju.3.lLitropolsky, Nonatationary phenomena in the Nonlinear Oscillating Systems, Edition of the Academy of Science of the UrSSR, I955. 4. V,A.Grobov, Aeyptotic Design Technique for Handing Gscillatione of the Turbine Shafts, Rdition of the Academy of Science of the USSR, 196I. 5, k.I.Kuahul, On the Nearly Periodical Solutions of Quasilinear Systems at kultiple Resonance. To the Theory of Robor Autoscillationa. News of the Academy ,f Science of the USSR ~mechanics~, issue I, 19F0. 6, F.k.Dimentberg, Bending Oscillations of the Rotating Shafts, Edition of the Academy of Science of the USSR, I958. 7. Y,A.Grobov, Oscillations of the Elastic Shaft Kith Unbalanced Discs at Pliability and Damping Qualities of tae Bearing Oil Film, Proceedings of the Nigh Aviation School, digs, issue II4, I960. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 =--~ 90 95 Fig . 2 !05 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 0 g r-.Tr ----~-- --_ ---+- $.O Fig` . 7 3>O Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 _~. .- i -~ Order w^1093. Ukr. 5SR Aced. of Sci Puhl. Huusc~, Prinlcn Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 OlY T$E dPPLICATIOIi OF TH8 SY6LL PdRdISBTER IIETROD FOR TRB DETSR1rIRATION OF DISCON!!1'il)- OOS PERIODIC SOLDTIOAS. M. Z. KOLOVS%Y -Leningrad, Polytechnical Institute 0 IIPI~IMEHEHI~H METOIIA MAJIOI'0 iIAPAMETPA AJIA OIIPEAEJIEHNSI PA3PbIBHbIX IIEPI~lOAH~IECKI~IX PEIDEHI~II~ Y.Z.Eolovelq (Leais~;rtd). Poinog1'e(C1]) has used the small parameter metho3 gar the systems of differential equations, the right-hand sidha of those being the anal~tioal iunotiona of the ualmowa quantities and the noel/ parameter. Further it was shown, that this method could De sso used is oases, when the right-hand sides oY the equations had the continuous derdvativea of the second arder(f2]), or trey rare the piece-continuous iaaationa([3J). For t;+e latter case rue integral form oY the necessary oo~ditioae of the eziatanoe of periodic solutions wan ?a- oeived in(C4~). The application of the small parameter method for the d?- termination of the periodic solutions, oloES to dlsoont;au- ous solutions of the generating system is discussed in this paper. The results received here can be used for the inve~ti- gation oY the syatebs, in which the impacts ooaur. 1.~.,x~rmulation of ~?oblem. The conditions of papa. Consider an equation (x and X are the ~ -diaenaional aalumn veators;,n ie the wall parameter). The right-hand aide of this equation satiafiea the fol- losinK conditions: 1) fnnotion X is simply defined with arDitraty real ~ , Yor aqY raluea of m from aegoent o~ t?' ~ ~~ and for all x item a certain n -dioeneional domain cam; 2) far all a from G and for o < 1'' ~ (~o is continuous and periodic with respect to with the period ~' X(x,trT ~.) ~ X(x,t,Ml Z) domain U can be divided Dy the smooth auriaoea i Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 referred furtl~ a a8 "dieaontiauity surfaoee" into domains vi,~~,,,.., deriratiree 2~ ,.a ~' and a~ .being oontlnu? ous is snob of them, the bcuadaries inc1J:::ig, It is aeoeanary to fled the periodic solution e1 the eq~'atioa {1.1) with the period T eatie~ing the following "ooaditiona of gape" on the surfaoee(1.2): where YK has the oontinuoae partial derirati!ea of the er oond order. It is supposed that the generating system c1z. -X ~.~ ~t,0~ dt has the lanily of periodio solutions with the perlod T P.elea~ ding on ~ independent parameters, all integral ourrea ct this family. rotor `.ing the same ensemble of discontinuity surface , ~Qrlx)= a irc:~,..,rt) In the points of intersections tL? following oonditions are fulfilled. l.> : mr fix; , o ) ( . 61 a~ dt (the line above means further a row reotor). Let vs be the pest o1 the domain G,contalaigg the ~~..?ts of the tra~eotories (1.5), comprised between the surface's `Ys(x) ~ 0 sad cQs.~(~-~? ~. ?s It has been shown is C4 1 ,the Keaeral solution cf the equation(1,1) xs= acs (t,Cs.~) (1.8) hating the oontiauoue partial derivatives with respect to r, ~s sad 1~' oan be found in ~5 with the accepted as~~;: n- tione. kith ;h~i! this solution is oontinuously turned ;n? , .one of the solutions of the equation (1.4). '"J~erNi~? SGWr Fa.~.+et .. , E'cr the dutern~ the periodic rolutl ~r. of t.,r e.;r.c,tion (1.1i .,, euCs..tt~:,: l~i,oj in (1,~~ ~~r, .._, 't'hen ne get the fallowing exoreeaiona: Ih'om these vn rector and Fn aoalar equations, the veo- tora ~, ,,, ~ C~, and the momenta `~, z aeotiona of the integral oru'~ea with the disoontinuityaeur_ faoea, oaa De defined. With ~ -o the gtatem of equations (1,10)-(1,12) hen the family of solutions depending on E par~etera, Therefore the Jaaoblan of this system Ss equal to zero if ~- 0 , and its matriz defeat is Q , It is easy to prove that with auffioiently small ;,, the system (1.10)-(1,12) has solution alone to the curves of the family (2.S), is whioh the values of parameters ~d satisfy some equations ld,, . P ~ ,?le)?0 If with these values of ~d a(P,,-.., Pal at =o the solution of the system (1,10)-(1.12) will be simple and the separate solution of the equation (1.1) will correspond tc it. If the generating equation has the only periodio solution, satisfying the oonditioae (1.4), the equation (1.1) has al- so the only solution satisfying the conditions of gape (1,3) sad being close to the generating one. The proof of these aesertiane ie based upon the ooneide- ration of the system (1.10){1,12) ea the transformation of points depending oa the parameter ,m , It follows lmmedl- ately from the results, obtained in (3], and therefore will not be discussed in this paper. If the generating solution satisfying the oonditione (1,13) is determined, the solution of the system (1.10) - (1.12) with ~ t 0 can be determined by means of the iteration procedure. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Now we ate to find such a torm of the conditions (1,13) that can be composed without determination of the general aoiutloaa (1.8), 2. The oonditiona of the exlatance of ceriodi~ solutions. Far the simplicity of writing we shall further write down the ezpreasion (1.11) in the form of (1.10) Mith k =1, xeep- ing is mind that ti, must be replaced by t,+ T in its left-hand side. . let the vnluea of ~, ,... , d~ be define' ?~tisfy- ing (t.13). Thus one of the solutions (1.5) of the egration (1.4) and all ~K ana 't K become known. For suffioiently small ~; there ie the solution of the system (1.10)-(1,12), is which the va:aes a' ~K and 'tK are close to those ai ~k and tK in generet+ng solution. Then the system (1.10)-(1.12) is egeisalent is the first approximation to the following; ax. ~~. a ~. JJ Jz aw. ~?`~ S ~K = d`nr ~~~ f~~K, ~c'k In these equations r= 0 ie substituted in the elements of matrixes, enoloeed ia~quare brackets, and in vectors, en- closed in round brackets. ~-. ~ b ~ ~ and ~ ~ ~ a ~. are the first approzlmatioas of the solution of tte equations (1.1c)-(1.12). The determinant of the left-hand side of the linear in- homogeaeoue equetlone (2.1)-(2.2) coinoidea with the Jacobiaa of the system (1.10)-(1.12) with r = 0 therefore ita mat- rix Aefeot is L , The system (2.1)-(2.2) has the solutions, if the veotor, formed by the right-hand aides of these equations, la orta- gonal to all E independent solutions of the system conJu- gated with the h~aogeneous one. Ito matrix :s farmed by the transposition of the matrix of the homogeneous system. It is evident that the conditions of the ortogonality are the same ae (1.13}. ~J- let ue transform the right-hand aides of the equatioro (2.1) sae (2.2). Funotioas ( ate) satisfy tae egnationa a Eaxk)_r~Xl~a_~+(ax) ou a~ lark at`^ lad that oan be received by differentiation of the equ3tioa (1,0). Ba appropriate homogeneous equation ~ (`x ]'dam (2.4) c1t Laxk which is a variational equation for the solution x,: , has the fuadameatel eyatem of aolutiona, forming the matrix y~(~,2)and these aolutiona can always be ohoaen eo that Ye ~T,T~= E E is the unitary matrix. In this case the function ( \ t X lad 1 t~ ~K~t:21~-~2)~"';~ aM satisfies the equation (2.3) what oan be easily proved by a substitution. It la evident that rK j (2.~1= 0. - ~~ r~ ~' 2K . refore T h - ~ ~}e~ l~T ry~ I > P(a,p,t,~)1 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 where q, p, ~ ~ aq ? ~~ P are n -dinenelonal Teotors; x ie the Hamiltonian of the ayateR, independent on tine. Strypoae also ghat the pe;ladio selntioa of the generating eyatem wlth the period T esists eatdafying the oonditiene of gape oa the enriaoes and with ~- 0 la (4.3). Ia other trorde, we have the oaaoaloal cystea, the motion of whioh is aooompanied b= the 'impaots' on the surfaoee (4.4) During these impaote the Hamiltonian, or the total energy in the oonservati~e system, my be changed. is the system (4.2) is the sutonomoas one in this case, it hoe the inGily of periodic solutions depending oa the ar- bitrary parameter, the latter being the phase. rhea, as it le known, the system of variational equations for the generating eyetena hoe also the family of periodic solutioaa? and the ooa~ngate ayatem hoe the iemily of eolutioae Tht conditions (2.27) are ful,iilled in this ease, as Consequently, generally speaking, the funotiona (4.7) satisfy the conditions (2.18) what becomes evident with the subatitutioa. Therefore the syetea (4.1) met hen the perlodia solution with the period T , oloae to one of the aolutione of the generating getem ii the equation far the parameter d. .JT1Q(yaP.,t~o19?-al9.,P,t,o1p.}dt ~~~k(po q.,o)ao - o (4.s) has the real roots. -l~ let the system (4.2) De the ooneervative one and ~(9~p.t~~)?0 Thee the integral in (4.8) determines in the first agp^o- rimatioa the work of the pertarbatians P in .the motion (4.3). The nun K k determines the variation of the kinetic energy during the impact. Therefore the condition (4.8) has in thin owes a simple physical meaning: the variation of the total energy of the system in the unknown?pe4lodic motion moat be equal to zero in the first approximation. 5. The eetimatio- oY the co~erRence of iterations. Having determined the parameters of the generating solu- tion from the equations (1.13) and having examined them by means o3 the condition (1.4), it is possible to look for the periodic solution of tl,e equation (1.1) natng the itera- tion procedure to the system (1.10)-(1.12). Let us estimate the influence of the number of the, discon- tinuities " on the meximuai value of the parameter ~~ , for which the convergence of the iterations can be guaranteed. Ifiark the unlmown scalar variables (tom and the components of the vectora~-. ) by I,,... ,f~? It ie evident that ~, ?l~n .,~ ,Let 2d- ~o for the generating solution. K'hen `?? o ,the eyetem (1.10)-(1.12) hoe the only solu- tion, as it is supposed. Consequently, it may be written down is the form: derhatives ~~ , ~3M being continuous in accordance with the auppositione formulated is (-1. Let us compose the lteratioas ~~,1. ?~'l,' i, ,{~) The fnnaiona 4d and 3~~ are limited, being continuous. Let $? ; r, az, I'c~~ - N Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 -(4-? ~~d't-~ ~d?'I'("~~j~*ly..,~ps~~)'~r(~x,,s.~~-,~vs.~.h]I~ . o, K, KZ-K,>o Where d 1 r~W2a~- ~ /~ ~ Kz=tim,'t (~e-IL)':o(~11)c~e+`~n4Z /,me+ Kj= C~~e-1Z)2~ a(~Je.Q)~Je J( 21LQL +N~,J + ~ ~ZS 2jo).Resona:,t2 curve a (n) of such system is pall known in literature, and the ezpression (3.6) represents the same resonanC2 curve. For further detailed erpoaition reaona-1Ce curves are not used, rather the graph S(n) p11~zQZ is used. The graphs S(1Z} - 21 - Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 end a(11) have similar force and oring to this fact, it is easy to use the notion of resonance curve. Graphic representation oY the possible scan- dard conditions of oscillations gives the const- ruction as it is shown in fig.8. Here the graph 8(R) is plotted for the aystea with ~'>O and the graph Y(~.), the reflecting property oY the actor, The intersection of these graphs eorree- ponde to the equation (3~5) and defines the points Cl, C2, C3, characterizing the possible standard conditions of oscillations. Therefore it ie possible to ezaaine the construction on fig.8.as the graphical solution of equation (3.5) and the abscissas Cl, C2, C3 , as its roots. For ezample, with the help of conditions of stability (3~7), it ie seen that the points C1 and C3 are the points of stability and CZ ie the point oY unstability. Frog the sane crite- rions, (7) it follows, es it is enown in fig.e, that the character of the mutual dispositions of stable and unstable states one preserved and for aqy other dedniahing cheracteristicsY(R ). Thus, Yor ezaaple, the point C1 will. be stable in all cases when the tangent 0~'tne characteris- tic Y(1Z ) at this point passes in the In~'cYtotof shaded quadrant. IY we take into account, that the characteristics Y(R) in this process of re- gulation of the notion regain parallel to itself, then on the branches BT and RD the points of sta- bility are placed and on the Drench TR the points unetability (dotted line) are placed. Disarrangements of oscillations under the changed frequency ~ are characteristic of non- linesr systea, and in our case, they will depend oa the characteristics of the actor. Having in vier, that the control of velocity of botatiaa corresponds to the shift characteristics Y (.~ ), it la not difficult to notice that when ~ in- ereesea, changes in amplitude take place along the continuous line, and rhea ~ decreases, then ahaagee of dmplitude take place along the dotted line, and shift takes place iraa the point B tc point P. (For the sake of siapli- city, it is again supposed here that the charac- teriatice Y( 2) shifts perallelly to itself). 2. The oeci Gating systea with "soft" characte- ristics of r3silient forces (~'< 0) , For this; system that very representation with the help oY graphs 8(.2.) and Y(11) is useful. The chat'acterietic peculiarity of the resonant curve s(11), when ~'< p and its in- clination to the aide of deminishing frequency is repeated again ae in graph B (1L) in fig.9, The conditions of stability (3,~) ellaw ~ to establish that with "soft" nonligear systea, the disposition is unstable with the change of hardness of the characteristic of the actor. It is interesting to note the following two posai- bilitieat If the characterietic~of tb: :~tion appear to be "steep", i.e. His gres? modulus, then points of stability are disposed on the branch TD and on the part BR. On the part TR (dotted - 23 - Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 line) points of unstable positions are placed. For erery point of stable part, it is possible to determine frog (3.~), the region of significance N, under Which the stability ie asintained. 'Phis region is represented in fig.9e by shaded sector. Por the point of stability C1 it is necessary that the tangent to the characteristics A(n )-r,assing through C, ahouldbc~ntain4ctirithin the iHferrOrof the shaded sector (fig.9a). If, on the other hand, the characteristics of the notion appear to be "gradual" i.e. N is of small modulus, then the points of stability are placed on the branch BT and in the part RD. Ia this case, poiata of unatability are placed on the part T8. The shaded sector represents the re- gion of aignificemce A, wader which, the point S eaa main as stable position. In Chia WB,y, due to the chengia9 of A, unstab- le states shift frog the growgr branch of resonance curve to its deminiehing branch. The disposition of the standard conditions of motion for the case, when the tangent to the cha- racteristic Y(.n ) passes through the inside of the shaded area of fig.9b, it is shown in fig.10. It is clear that S1 and E3 - ere the points'of sta- bility and E2 ' is unstable. Here it ie shown also the scheme of change of amplitude and the discuran- gement of oscillations under the growth of (continuous arrows) and under deminishing (dotted arrows). It is not difficult to see, that the schemes, described in literature, on the changes in aapli- tude under quasiatandard growth, and under demi- -2a- niahing ~ and the schemes of diacurangement of oscillatio~us for nonlinear systems with hard and soft characteristic of resilient forces 2,13], can be interpreted in exposition of higher repre- sentation as events oscillations, excited by sour- ce oY energy With infinitely big hardness of cha- racteristic. 'Phan these events Will correspond to those on the graph fig.8, 9, 10 to characteristics ~I(~ ), having vertical straight line position, i.e. the characteristic of ideal sources of ener- gY Angular velocity ?~~ ,except the basic aag- dt nitude ~ ,acquires a small harmonic compo- nents dt ~ 4J ~,7ncus(211t+,f)- 7n''casJlt. Coordinate of the oscillating motion, except- ing its own chief value x=occu a. K,=m- ~ , : ~ z P7m K3 - 27Q ~~e-J~ (~ -P s ~t / , The values ~, o, ~ suet ba substituted to these criterions for each of finding stationary solutions correaponding]y to points 1, 2, 3 is Fig.12. Thus the values l~,Q,y~ foand fraa equations (4.3) together dth the results of enalyais of their stability and dth the use criterions (4.6) give us the solution of the problem in first ap- prozimatioa. The stability of paremetrical oscillations depends one the properties of the source of ~ergy; on the value of paraaeter d' i on the amplitude of oscillationsi on the moment of iner- tia I sad on others parameters of system. These dependences era in the criterions (4.6) but their detailed analysis ie rather oompiica~ ted and it ie not considered here. We shall discuss only eaoh results Rhich determine the role of characteristics of the source of energy and the nonlinear parameter ~' . Characterietice of the source of ener? ie represented in the criterions of stability (4.6) as derivative 1;2 i.e. angle of bank of tangent Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 to the graph Y (-~) in the considering point is important. For real conditions H (~ ) more often is po- sitive decreasing function and its derivative N is negative value. The limit graphs for 6' (~) will be vertical straight line (/d2= -cY~) and no- ri2ontal straight line (l~Z = o ). ins tangents to all others characteristic and passing throw*h E will be arranged in the limits of forth and second quadrants (shading sectors in Fig.l;). Let us consider two typical cases for non- linear parametrical systems. 1. The case when oscillatory system has a "stiff" characteristics of nonlinear restoring force i. e. ~'~0 ~Phe graph 5 for this case shown in Fig.13 is similar tc graph of resonan- ce curve a2 ( -~ ). iAe shall use this similari- ty for the shortening of describing. Analysis of criterions (4.6) for this case gives the fol- lowing conclusion points at tae branch BT (so as correspondingly points at the resonance cur- ve correspond to stable regimes of motion when e tangent to characteristic g passes through point T, in the limit shading sectors (Fig.13) i.e. when characteristic ~ is decreasing. The point at the branch TR of graph S ( snd cor- respondingly points at resonance curve) will correspond to the unstable regimes of motion for the same characteristic K . As it is shown in Fig.12 there are three points of typical "equilibrium". The point 3 in Fig.12, so sa the point of croeaiug characteris- tics Y with aria (in Fig.13) correspond to stable non-oscillatory regisea of motion, i.e. rotation with the frequenby 12. The reciprocal disposition of stable end ua- atabla points of "equilibria!" determines the character of possible "frustration!" of oscilld- tions dth moving characteristics M at the pro- cesa of motor control, Let for the simplicity of aharacteriatica Y be rectilinear (Fig.13) sad at the motor control moves parallelly to itself. It ie evident that with increasing the point E dll move up along the Drench BT. !t the environment of point T the stability of stationary regime rill be lost end the system rill transfer to the new state, determined b7 point H. !t the point T parametrical oscillations will "frustrate" end disappear as the system will transfer from state T to state H. If than will decrease toe point, which represents the state of the system, it will move along the a=i~+ -~ from B to R. !t the point B parametrical oscillation will appear. They will be settled as the system pill transfer from point H to point P with subsequent decreasing of Che stable stations. ry oactllator= regime at the point P mp be chan- ged so 88 the point representing a process will move from P to B. 2. The case, when a system hen "soft" charac- teristic of restcring force, i.e. ~'< 0 .The graph B for this case is shown in Fig.14 and Fig.15, Graph B ie similar to resonance curve Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ( tl ), known from literature. For ~o , tl>/ac /. Under these conditions 1-~(Y x)}=K?+K, (v-z)+Kl (rl-z) +K~ (v z~, K',, xf, k:, ka are constants. The oscillations tend to be staple _3~_ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Harmonic, forces ~ x and T are small. It is clear that autooscillationa Would be possible only at decreasing part of the functions T((v x)f~~. In the first equation the right part expres- ses the algebraic sum of the moving moment L (~J end the moments of the resisting forces, 'Faking for granted that this sum is anal/, We are able to eey that the resulting acceleration ~ also Wiu be small. Considering the observations in the equation (5.1) We can introduce the small parameter ~ in the following manners here it 14 denoted ~`M, (qJ=CLIYI -KIyJI j ~ mt' m , ek' m ' ~Tf (ry-i)}'mTl ~ry-xJI~~ tl (''y-xJ}? j T~(r~"~1 8ubetitutione for the variables dt 9,x=Acas(Pt*?),dt?'A4uin(Pt*~~ (5.3) these equations aaa be ezpresaed is standard forms ae_E[M,CeJ-T fCrB+f},~s(n~Vl}J, `~'=pt+~, ~?-W~fiwAsin~+T((rO+Awsiny')f jriny~ dt-`~PWQ~hwAnn}vf'J'~(rO.A~r~n~')~cosy', (5.ui here P is the frequency of autooacillati- By means of tY~e theory of perturbation j9] it is possible to determine 6, f},~ in the first approximation consisting of two itemss 8=I1~~~,(t11,4 ~),A=a.E~Z(t,I1,4,fJ, ~=~~E~~rEI14~), Where 11, Q, f are constants at s1oWly changed values of the unknown variables Which represent the main pert of the solution. ~ ~, , E ZL , ~ ~ j -small periodical functions of t. dt-~ G^~(U)-rTh~J-2 (xz+3k,u)A2J dA, r (~+k,*2xrut3x,u1~~ xfAE)A,~ df Pin d~ _ dt3w ~ where. U=n r,,~l,_`')U,l`9~r~'~w R N Fry. 15 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Fiy. i~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~(~+r~:1 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Next 1 Page(s) In Document Denied Q Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 M. A. Krasnoselsky and A. I. Perov-Voronezh State University, Voronezh M. A. KPACHOCEJibCKHt~, A. H. IIEPOB 0 HEKOTOPbIX ~I]PN3HAKAX CYW,ECTBOBAHNR IIEPNOAN~IECKNX P~llIEHN)~ Y CNCTEM OfibIKHOBEHHbIX AN~~EPEHuNAJIbHbIX YPABHEHN~I :ti ~- , sent , v~ _ct iv cc..si^er sor.e Nererai ~cetnods ;.f ~tr,u1`.:~.in~ _; the Px.eter.ce - .i.enrems of per: ';~ ..~iutisns . ,. ,,, :ec~s ~f tue ordir~~sy .if`erer.ti~i. e;~-et: ~r,s c` ~nE- ,-:t cr.~~r It ie c:u:vrn;~~,.~ .o f'ozK~M,}o~~x,0~cc,`~? where ~~~~ is a positive function ado ~. ?l(u/ a `then the boundary problem /17~ i~as a solution. The boundary problem /I7/ may have several solutions. in some cases the number of those solutions can be estima- '.ed from bellow, Ccnsider one theorem of ~I~1. Assume the problem f17~ to have zero solution, i. e. ~~t `>> ~~- ~~ 'fr.eorem 5, Let the conditions of the theorem d be sa- tisfied and let the solution of the initial problem - - 4 + -- - -- U~ 0)= O~ - ~ ~ ~ pit ` ~ X c ~a c:/f. ~ ~ ~" have ~ zeroes in the in terval !~, ~ ~ and t~~ -2~~ #~~ Theo the number of nun-zero solutions of the problem /I'= nct lees than ~~E', The analysis of the two-point problem for systems wi'Y.: I2 degrees of freedom is more complicated, apparently, iLe simplest way o1' investigation of the problem /I7/ is a tras- eition to the eystea of nonlinear integral-differential equa- ~1 tions E c~X,/lJ d X~!!)) , x. ~~ ~~~~,5/~~tS,xrjj,.,Xrn~SJ - -, ~ --- /QJ~, r~ ds d5 where /~((fS)is Greens function of the operatorX with zero boundary conditions. Replace the system /I;% by the ryuiv,_.- lent system of the ietegrai equations zz~~t~_~p~ !> Ulf) rPt),~`~,l~~Lr(J~CZS, 0 R~~ are given in (' ~ . Some con.'.iiions of eol~bility of the tw^- point problem, based on the two-sided values, are given it ~ 3aJ. 7. One - aided values provided the existence of the periodic solutions for the system /2/. how take the ayata,~ 17 ,'neglecting /I6/. Then the following statement may 'oe obtained from the theorem . Theorem 6. If iueyuelities 2 r: Y Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 / where t~[- _ are positive, /n~~are satisfied, the system /2/ hoe a periodic solution. h. Features arising in the investigation of autonomous systems. llere we consider a system cl~X~ ~./,P _,Xn CYX, c~X.,~ /c=t,-,~~ /28/ c~tP c f' q't ' .~dt ; or in the vector fo n 2 X ~x It is difficult to obtain periodic solutions of such a ~~~atsn ae the period is a priori unknown. In other words, the problem of obtaining of periodic aolutiona of autonomous elstem is a problea with an arbitrary parameter, the value of which is to be chosen. Let t = [~z (O ~ 2's 1~, /291 Then the system /28/ is in the fora d{Z `~'`~~x'w dz)~ /30/ To obtain periodic aolutiona of the system /28/ is to obtain periodic aolutiona with some known period equal to I for the ayetem /j0/. The values of the parueter c~ ,when the ayetem /jo/ hoe one periodic solution, are the periods of eolutione for the ayetem /2B/. Greensfunctiona being used, the problem of I- periodic eolutione for the system /30/ may be rewritten ae a system of non-linear integral squationn in the vector for? Thus, the problem of periodic eolutione for autonomous system proves to be the problem of obtaining stationary pointaof a completely continuous operator A { x (~J, ~~ ,~ acting some space of the vector functions and deFending on the scalar parameter u~ To investigate the equation /31~ one should use all the methods of non-linear functional ana- ~33~e, lyais. Some results obtained in this way are given in There are some of the coneiderntiona associated with two problems only. Aasune the syete^ /28/ to have a zero-solution. Inves- tigate non-zero periodic eolutione with small amplitudes. For simplicity consider the case, when ~(,rt,.. , x,,, y? _, y?1--~(x,, , x,,,y,, ., y?~ ~l =1, -, ~J. 132/ Aa it was shown in q. the periodic solutions of the system /28/ nay problem, if conditions /j2/ are aatisfied.ittihis case ope- rator equation /31/ can be written in the form U(fl=~~~~AU:>ix aesTOpx~x xoaeV, T. I3I, N? 2, 1960 . 23. iG. t. fi o p x c o s x q, Bparexxe cna6o xexpepblexblx aexTOpxblx ronex, Tp. T6unxccxoro ~aare- n~aTUUecxoro xx-Ta uu. Faaraa~ae /HeuaTaeTCR/. 24. H'~. t. 5 o p u c o B x e, Cna6ax TouonorxA is nepxoAx~ecsxe pemexxx Aucpcyepeagxa:lbxxx ypaexexxx, ~,AH CCCP, I36, ~"' 6, I96I. 25. p~. C a x c c x e, 06xxxoBexxble Axc~epex- gxaabxtae ypasxexx$, 4iJ1., ~I., T.2, I954. 26. C. H. 5 e p x m T e ii x, 0~6 ypasxexxxx Bapxauxoxxoro xcuucnexxs, YMH, 8, 32, I942. 27.?;'.. A. Rpacriocenbcxuil, 06oAxo1~ spaeso~i aaAaee, 19as. AH CCCP, cep. raaTeMaT? 20, 24I, I956. 28. H a x ~{ a x X y x, CygecTSOSaxue u eAxxcT- B2HHOCTb pemexxx apae>3ux aaAa>; xeaxxelaxboc o6r3sxoeea- xxx Axc~repexr~xanblmnc ypasxexxu, i~AH CCCP, T.II3, h 6, I957. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 29, 3. D. C y p x a o B a, 0 xpaeBOi2 aa~a~te o6xxxoaexx~u ,gx~x~epexuxanbxxx ypaBxexxK, ~,xccepra- qxx, ~:., I954, 3C. A, li. P, e p o B, 0 AByxrageuxoff xpaeBOi+ aa~?ave, rAH CCCP, r, I?2, N~ 6, I958. 3i. iv. ~, H a o x o B, OAaa npeAeubxaa xpaesaa aaAa~ta Ana ypaBxexxa x~xf,c,~~+~YCz) ~, i".aH. BhSCIINX yue6x. aaBeAexxit, hiaTea~arxxa, '~? 6, I959. 32. i... C e~ e x o B, OAxocropoxxxe ouexxx B ycBOaxnx cy~ecraoBaxxa pemexxv xexoropxx xe.nxxeix- xux xpaeB~x aaRaq, iayQx. AOxnaAa BHCm. ms. ~~xa.-~aar. x,, 4~ 5, I958, crp. 53-56. 33.:'1. A, Iipacxoce:tbcxx%, uoxza? xa 3-x c~eaAe, Tp. TpeT~ero Bceconaaoro ecareuaTxaec- xoro c~eaAS, Rnp,, AH CCCP, M., I958, crp, 26I-2e8. 34, A. 4i. it e p o B, 0 apxxuxxe xenoABxacxox Toa- xx c AHyxcropoxxxxx ouexxaecx, p,AH CCCP, T.I24, I959. 35. 'r~. A e p e x 1;1. IQ a y A e p, Tononorxx x c~yxxuxoxanbxxe ypasxexxx, Ycn. mareu. x., I, '.P?+"'3-4, I946. 36. ~1. A. H p a c x o c e a b c x x K, 06 ype.B- xexxx A.11.HexpacoBa xa Teopx Borax xa aoBepxaocrx TH- aeAOr~ BxJjxOCTx, AAH CCCP, I09, R? 3, I956. 37. B.' S. ~1 e a a x e A, 0 axaxcnexxx xxAexca xenoABxaxox Toaxx Bnonxe xenpep~Bxoro Bexropxoro no- nA, AAH CCCP, h' 3, I26, I959. 36.61. A. I{pacxocenbcxx# x +. H. b a 6 p e H x o, O BH~iNCRexxA xxAexca oco6o~ Touxx Bexropxoro none, AAH CCCP /neuaraercR/. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 v' ~ i A1. Z 1.itri~iot - Mosaow State University named alter M' V. Lomonosov, blascorr M. 3.3IHTBI4H-CEAO>~1 :D C~~T`E3E KOFPEKTkIPYK?il.>(,NX uEIIEI~ ` A?-HE3~N'HEI~H~IX KOJIEfiATE1IbHbIX H PEFYJINPYEMbIX CNCTEMAX The oscillations in oscillating any re- gulated systems are often required to be li- mited in a definite gray around a stable equi- librium state r~hen the initial disturbances are bounded in a prescribed region. To realize the required lirJitation of the oscillations, active and yassive correcting devices are in - troduced in the system. Here ~e study some me- thocs of the construction of the correcting ~evicas eb.aracteri.stics in nog, li~~ear oscilla_ tin E; a;,d regulat~c .y_;~~;r,;, I. A Non_I,l,ncttr-S~_tem of-the-5econ_' order-tlct^' upon b~-an-.~,xternal Force, Let us consider an oscillating system , ::With i~ dc:>cribed by an ordinary ~iffera~~tial equation Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 M ~_ d which respect to a dimentionless value x , where 2 is a dimentionless time, ~ _ the characteristic of the damping force, Jz_ that of the restauring force and p _ that of the disturbing force, The functions ~x, x~) h ix) and P (2) are such that in a certain region ~ /x/ s X , /v/ . Y of the variables x and y , there exists a unique solution x (z) of the equation (I) , passing through the given point (in gion .D ) the re- and which is defined upon the whole positi- ve semiaxis of the independent variable z ; (3) In the mentioned class of function;, the problem is stated to synthesize the dam_ ping characteristic ~ , which would ensure the prescribed limitation of the oscillation of the system around the equilibrium point ~- = 0 0 y ( ~ ~ = o ) ; this point cor- responds to the case P = 0 and the oscilla- tions in question are induced by the action of the disturbing force and the initial con_ ditions (2), In this problem it is reasona _ ble to utilize a criterium of the ultir~ate 2 boundedness of the differential equation soj lutions, the importance of which was pointed out by V.V.i~iemytzky (Ij, ii.A,Antosie~zicz [~j d.:livered the follo- ;?ing criteriom of the ultimate boundedness of the solution, of the equation (I) upon the se- miaxis (3) , If 1. ~ (x, x')>p for all x , x~ ; s n. u(x~-fh(~~d~~o [~ 0 far all x#o; 0 then any solution of the equation (I) satis- fies ~x(rJ/< C, ~ coast. , ~x`(rJ/< C2=Cast as 2' -. oo . The direct application of this criterion to many physical systems is obstructed by the conditions I/ , p/ and IV .Really, even the characteristic sin x of the restau - ring force of the physical pendulum in a ho- mogenous gravity field does not satisfy the conditions 1/1 A lot of physical systems cease to be described by the equation (I) when /x/ ~ a ( q = cnnst, o ~ a , X ) due to the liaisons, imposed upon the coordinate x by limiting rods , boundary regulators etc. The condition 1 V excludes out of the consi- deration the vibration, which is on of im.. portant forcing actions, wherefore we shall Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 try to weaken the conditions jj , jl/ and Iv of the boudedness of the solution of the equation (I) in such a way as to render the given criterion useful for the investriga - tion of applied problems. Namely, let us substitute those conditions by the folio - wing ones s 11 ~ N ~x) > ~ as o ~ j~c/ < a 111( the function N(x) increases as o ~ f,~ ,~ i:> tier ccrr,;s ~ c- value of the coordinate x Ls fs,~r 11 /~~~ R ~ 0 , the oscillation:, o: r,r.~c .~~=. :; `:BP t~~e 11'~ILlul iLx1liE ~i (~~ ar'~ U,.t'?4R; ci, ~;ct ~o the limitation ,K' .~ : /ylr1;_ ~~C(2~~_ ~~~ m0 V3iG::~c tb~- aU;iLc~r Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 us make use of the inequality S ~ ~- s Thus /~~< ~? ,where .~,is the absolute va- lue of the greatest in absolute value root of the equation N o ,The ftinctio_n F ought to be negative, the function F - B, , B, / find that d > o when ~~2 2 ?C, Let us choose the value F as follow:, F =min ~~. ~f~~, J~?;~2. The function FrxJ=- 3P ~ x ~~-xJ satisfies to the condi~ion !11 a),b),c) aiu: to the condition 111 d) with respect to he function }~(x) (15). Thus the limitation /x' a, all the roots of equation //yn.t~ -NN=~ having negative real parts. Integration in (17) with respect to ~ is seen to be done between limits from -?? to hence the solutions fir, err being dependent on a , it is natural to extend the range of va- riation of fr, ~`~ over the whole complex plane in proving the lemma. That is the reason for re- placing system (10) oy system (13) which is being considered in the extended region. ,We now consider the functional equation F=SF, By applying the fixed point theorem, the exis- tence and uniqueness of the solution of this equation is established. Transforming equation (17) J~ (~i ~.) ~ ~~ ~ y(~~l! r l L ~r' . d 2 r (S . S ~ ~~, U~~t (~~ ~ * E~i M ~~t~~; ~ f (3`. ~ ~ ~~; fir, t ~~, ~'E);~(19 ) it is seen after a number of algebraic operati- ons that function (19) satisfies the equation Hence it follows that the manifold as determined by (19) is an integral one Ior system (13). - 10~ gystem (13) being equivalent to system (10) in the starting region, it follows that in this region represents the local integral manifold for system (10). rVe merely state the result which establist;es the property of the local integral manifold of system (10) to attract the trajectories of any so- lotions of system (10) issuing from some mensional region l,(~, Lemma II. Positive constants ~' E~ , ~ (E~`E~~ can be pointed out such that if all the characteri- stic numbers of matrix E/ corresponding to the equation lyR-~~-Nll=o have negative real parts, then for every E~f~ of any real t, and any ~, , ~` for which the dition is fulfilled, mere exists ana n-2 'dimensional region of initiall/ values h of uja such that if /lrEU~ for t = t, , then ~h N" f~t ~r .~` ~, cl~~ p(~~e~erlr- :l/he _ fit , ~ , ~", E~~, ) r /. (21 e ~ r for all t . where ~, , ~ + N h, represent ? N t = t, h N ~r , ~r + hr for ; ~ lution of system (10) not belonging to the integ- ral manifold II7r V (e, p~f is a positive constant depending on the perateters e and Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 w By extending the statement of properties of the solutions of system (10) to the solutions of the starting system, the result is obtained as stated in theorem I. 3. As this shown in the beginning of the pa' per, in investigating systems with many degrees of freedom by using the method of integral mani' folds, the stability property of an approximate two parametric family of solutions corresponding to the single frequency process can be establish- ed. Indeed, by taking a number of estimates it can be shown that the difference between the exact two'parametric family of solutions and its ~ -th approzi.mation /whose construction pre- sents no difficulty/ is a quantity of order ~ m . The integral surface ~S' /i.e. integral ma' nifold/ wnich is covered by the curves of the exact two parametric family of solutions attracts exponentially /as long as fit, ~~'E~j~ / the trajec' tories of any solutions issuing at the initial moment from points lying in the vicinity of this surface. Thus, in course of time these trajectories wfll also tend to the approximate two parametric family of solutions which is in the Em neigh- bourhood of tiie~exact two parametric family, or else tnis tendency will be effected according to the followinb law: ~x(l~-x,~(tl~`C~~~~~enrzl~s(f1=~Y(t,El __p (22) 4. As an example we consider the case ,~(x) = Px where P is a constrant nXn matrix. As a result, a system dt is obtained, which will be considered for any fi- nite values of x as t F R , ~ E ~c, In the case under consideration the locality condition falls away and theorem I holds for any finite x The property of attraction also exists for trajectories of az>,y solutions of system (23) who- se initial values belong now not to a small neigh- bourhood of the manifold but to any finite region. 5. By applying the method of integral mani- folds it seems possible to investigate the stabi- lity of the trivial solution x,=0 of system (3), acted upon by the functions F X (t.r,t~in critical cases. 'Ne shall consider the critical case only when there are purely imaginary roots. For the sake of simplicity we take up the case of two purely imaginary roots. It is evident that /x(t)-x./Jvenieat to take as given parameters ctie induction vat res cf ~o ar:d Bo . 'Phen the relations between o'~.er ai;rame- ters may be graphically represenaed ~y serves ~*_- a =-f ~6q k1, Using these curves we may choose ~~ne du- ty and plot changer characteristics, in parti- cular, the outer one. These relations for certain values of determinating parameters are represented gra- phically. For all relations cne value of re- lative induction of the ,fundamental Prequericy" Bp=6 is assumed. in determining the relations we vast coa- sider tihe constant induction compouer,ts of the core, which tau arise due to ~aa~rer,ic ?iux rec- tification, tr:is havi_a ?o~ noted py ~.~+.Rosen- olatt l3,4J. ~(? = 2~ 1r?x-,s(~Pllzx-, ~64~c~ ~2K- ~~ x For odd values 3 N Ao~ =10 ~BoJlo 16Q~+2~ 1x1(BPllx (eglc~>rz , ~nr - Q p~asiuili~,y of such rectification _U ;;:.e:. roa the recC:..ed ;;ii.e ei~.r,e ?i i:~!~ sit? ~~:~ zero ever: is t:.~ u~~::e:.ee ~: ,,:.e ou;,tr ~~ ce Ito ti;at iti, paysically is^rossi:.~1e. 1'r.e ~,c- oene,.nce of ~ni:. i;~te:aity frc,m pease a: ;le ? 1.5 3iS~i ~i.:j.=1:: F:ily is%pGSSi Dl. fi. -ice' a mal i;eT Oi :,~?_ .>:e i:.~.,._._t; v~.1ue i, must .;e equal to _ri, d.iLi~JUt _~.:eT ~e':_., _~._ ~ .. ](:re ~ ,. CC:.~ta]iL ..-? eid. It IIl~1St UB eOUal ~~"~~ ii a = .5;4 ~~ svittr Baer ma;~;r;itiaa~ion. ,..re ~n"la ttreCJre ii.4llC'.lOn CGi..~e;:,~ .^"t'vGneili ~e:;:it. on m. m.?'. o?' ~..e d7in~1;~, cr~?ulecc:a t Lhe co~~stant voltage supply. F~`am chess conditions the "imier" irduc- ~~ion component ~a is determined. iu the geaerai case Bo is detet~niaed from the expression 60 =60 - ~p th6a=-C,? C; -C2 Q2 , = -sh p / , "0 1/01102 /~4= "n /(? and ~~ ' 3essel functions series, For even values 5 q? =10 ~Bp~~a ~6Q~f7~ Izxs~6PlIPx (Bq;'~2Kx , Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 '.'ne sign in tae expression t~ Bo aepai;ds upon. ~ ~e direction of ttre induction Ba Nith tue positive sign 6, it is necessary to :ake tue plus sign and vice versa. Fig.4 represents the relatic~is net~xeen mul' tip'_ication frequency duty determining rarame- ters acid for tue even S = 2. rJithout outer mag' netization of the core by a constant field the mange is performed within the angle values 9o?~ne 5= e~i;ne ~ , 1 + ~'nnen here a is a small parameter: Its physical sense is a wave amplitude, The hn are unknown numbers. The functions ~l and C1 satisfy the linear problem ocgl = 0 in To ~Iz - fit g cp1A + vm t;l = 0 (2,5) It is required to find the periodic solutions in t withperiod 2n.lt is easily verified that these solutions are liven by the formulas ~1 = film ~(P) -fl NWm; ~~W m PETo; 1 m The function fl(e) satisfies the equation fl +t'1=0. (2,6) The functions W2 and ~y also satisfy the linear problem, but the conditions (2,5) will be not IromoReneous ~2z = t;2t + A2 (Wl, rl, ~, h ); (2,7) m here A2 and 82 are functions n, x and y which are determined by the first approximation. "hhe solution of the problem (2,7) may be riven in the following way Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 The functions f2k satisfy the following system of ordinary dif- ferential equations; ' # ?'; Q2 k + 2 fk = rl2k(B,ii), om m (2,9) The functions A2k have the form a2k = a2k cos !~ + b2k sin 2e, (2,10) where a k and t'2k are numbers, In or~er that the systems (2,9) and(2,10)hadhaveperiodicsolutions with the period 2~ it is necessary and sufficient, that a2m = Oo (~,11) The equation (2,11) determines the number is the first correction to the frequency. It appears that ht = 0. It is easy to show that the mentioned scheme allows to calculate the functions t;n and cpn of any index n. It should be noted that num- bers hl = 0 far all odd i. hence, the determination of the first cor- rection to the frequency requires calculation of the third approxima- tion. Thus, >"-th free oscillation has the following free surface cmt e,~m sin + e" (... ~+ ... (~,12) In this way the [.yapunov - l'oincare method may be formally exten- ?ha to the case of free oscillations of a fluid. li e m a r k s , 1. The theory developed is of a formal character, Ile brow nothing about the convergence of the series used, At the same time there is reason to believe w?e may be sure that the Por- n ula, ~lefinin~ the dependence of the amplitude on the frequency de- fines die dependence rshich exists in nature with mod accuracy, l~brrf~fore a rigorous analysis of the boundary layer problem (?, 1) rep- r~~.~~~nts ,1 u~r~~ important n~~athematical problem. 2, The problem (2,1) is a problem about ei~en-values, The theory described above allows one to define the structure of the spectrum and its dependence on the amplitude, The spectrum is an infinite union of finite intervals (see fib, 2). 11 11 \ 11 \ 111 1\ 111 i'1 I 11 1 1 1 1 \ I I ~ 1 ~ ~ ~, `- -' 02 03 bus, 2 In spite of the existence of the analogy between fluid oscilla- tions and free oscillations of conservative system which a finite num- ber of degrees of freedom there is an essential singularity in the fluid oscillation theory, Periodic motions of a fluid can exist only in the case of sufficiently low (small) amplitudes, 't'his fact is con- nectedwith the fact that the waves of a large slope destroy themselves. Let us consider the problem of the oscillations of a fluid mass under the action of an external force periodic in t and directly pro- portional to the mass and the tensity of the fluid, Let us designate by means of U(t, x, y) the potential of the perturbing force, Then it is necessary to replace the last to the conditions (2,1) by the fol- lowin~ one ~~t+~r+-(ow)e=i; (t, x,~~). Let us consider the most important case, when ? ~~, -7- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Dec~y... _____ . ..._.._a_ _ /ossified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 and rise the problem of tindin~ periodic solutions of the period T = ?w . hor ? = 0 the problen; wilt describe oscillations of some conserva- tivesystem and therefore a usual quasilinear treatment of the problem cannot be sufficient, [t is natural to consider our system as the ,ystem close to the l.yapunov system and to see the solutions which at ?+0 become the solutions of the problem (2,1). For ? = 0 in the system (2, 1) there exist periodic solutions, the period of which de- pends on the amplitude, In this case under consideration the period is given '2 ~ T=Tn=nw, where n is any integer..According to (2,]2) the period of oscillation is connected with the amplitude e by the following formula; i? _ 'hhus for the souKht for amplitude e we obtain the equation: 6 1+I:Ze~+i;4e~+???=nw e2 i: + e4t ~~ 4 (3,3) (3,4) 'this expression makes it clear that the problem posed Definitely does not have a unique solution. "1'he physical sense of this cir- cu,ustance is following. There can exist solutions, at ? -? 0 beco- n~in~, trivial, but there can exist solutions which at ? -? 0 become free oscillations, which were mentioneJ in the previous section. "fhe most interesting (aml the most difficult) case is the reso- nance case when the frequency of the disturbing force w is close to ~,ne of the eigen frequencies of the free oscillations. Lot us shoe, how the Lyapunov method can be used for finding solution; which hecnme trivial at ? -+0 under conditions that the nuniLers ~~.n are irrational.:lssuming the frequency w is close to >,,; let us put om=w2 -?a~ ~m ~m ~m [t is natural to seek solutions of the problem in the form of series `~= y.~n?na/b~ ~_ ~Sn?na~b (3,5) Itappearsthat the series of the form (3,5) define anon-trivial solu- tion only in the case when 3 = 3 . Thus 'the functions cpt and ~t will satisfy the linear problem w2 Ott + ~m Et = 0 ; t;tt = `~lz Supposing, as we did earlier, that ~t = ~ fnt~Wn , we obtain the following equation for the function f?+ An w2fnt = 0. nl ~m r-1,2,3... fnl (3,6) 'hhe only periodic solution of this equation having the period 2w is Cnt =- 0, if tt ~ m.; ~t =,tltsinwt + h'tcoswt. 'Ih~~ numbers ~1t and N't are unspecified constants. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 The functions cpt and ~1 will also satisfy a linear problem, but it will not be homogeneous one. [f we look solutions of this problem in the form of sums of the type (3,6), we shall obtain the fallowing equations for the functions f2 fn2+ ~ w2fn2 = rti ~sin2wt t ~t~ t cog2wt ~ m - (3,7) where ~fln) and ~(2 t are second degree polyr r~uals in Nl U~;d Ml , 13y hypothesis there do not exist eigen-values ~ and 1,m, that ~~" = 2, thus we can always construct periodic solutions with the required period fn= Ansin2wt + Bncos2wt; nom; fm= ~tmsin2wt + Bmcos2wt + M2sinwt + N2coswt, here An and Bn are constants, which depend upon Nl and Ml , The constants M2 and N2 are not specified, Equations of the third approximation are formed in exactly the same way, These equations will be analogous to the equations(3,7), but in their right members they will have terms of the kind ^sinlsin3wt + Tt"~cos3wt + R~n~coswt + Rlnlsin ~t, 11 12 1 2 where Mll]' ~l2' Rini' R2"1 are third degree in h'1 and btl and where C~ n) and R2nl do not depend on the numbers N2 and M2 , In order that this system have a solution it is necessary and suf- ficient that h'~m~(Nl, Ml) = 0; (3,8) 'f'he~ system (3,8) is a system of two cubic equations in ,1't and ,Ml. It is not difficult to see that we can continur this process indefinitely. 7~fie present method leas been used in several problems, In par- ticular, Cherkasov, a graduate student at the Steklov 1lathematical Institute of the AN USSR calculated the resonance oscillations of water induced by naves coming from the open sea into a part of rec- tangvlar shape. Besides that he made calculations of stationary waves which can be produced by a wave inducer in a canal of finite length if theoscillation frequencyof the wave inducer is close to one of the characteristic frequencies of the canal. 1. In previous sections of this paper the possibilities which are cfferedby the use of methods, generalizing, in the proper way, clas- sical methods of non-linear mechanics in investigations of ideal fluid oscillations. The possibilities of these methods are based on our ability to solve corresponding linear problems. It is extremely difficult to solve these auxiliary problems in the case of the viscous fluid oscillation theory. Therefore the investigation of the linear problem should procede the reduction of the non-linear problem to the linear one. Apparantly, from the applied point of view it is more interesting to study oscillations of a fluid of small viscousity, In this case essential simplifications may appear, In this paper the problem of oscillations of a gravitating fluid sphere rill be used as an example to show what simplification may arise from this supposition. '['he oscillation of a fluid sphere under the action of gravitational forces may be described by the following equations ~t=- p~F,+vUl+vG2+vau; (4,1) where Ul (x, u, z) is the potential of gravitational forces; and where lh = U`(x, t;, z)cos wt is the potential of di~lurbing forces. -ll- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 In the equilibrium position 0` = 0, u = 0, p = pp. The function v must satisfy the conditions of absence of strain on the surface of the sphere, To write these conditions we shall introduce a local system of Cartesian coordinates s , s , r. Then the conditions of absence of strain will be of the following form (hr =1): p31 - ?~H U3s1 t Ulr 1 p32- ?{H ~3s2+U2r 2 =0; (4,3) p~=-p+2?u3,.. v _I'2h2s2}=0~ 2 An unknown form of the free boundary r = f(t, sl, s2) is defined by the equation rt = U3 2. Let us assume t~ = py + u, where 4w=0; wt=vow; xc=vOx; u1=wZ-Xy; u2=y,~+yx; u3=-WX-Xy, (4,4) (4,5) (4,6) (4,7) (4,8) E3y such a choice of the function's cp, w and X we shall satisfy the equation of continuity (4,2). Remark . In the case of the arbitrary ortogonal system of the coordinates sl, s2, s3 the equation of continuity has the form (U1/(2N3)s] + (U2111y3)s2 + (U3htp2)s3 = 0, therefore the representation (4,8) should be written in the fore 1'2r~3u1 = ws - Y_s ; 3 2 N1/13 U2= Xs + Xs 1 3 li]l12u3= -cps -Xs . 1 2 - 12- (4,8') If we subject the function p to the condition pp =-wt+U]+U2, we shall also satisfy the equation of motion (4,1). (4,9) The problem of finding the three unknown functions cp, w and affords an opportunity to develop an asymptotical theory for small v. 3. We shall seek solutions in the following form: U = u'eot; p = pU] + p'e~t; f - ro = Feot, (4,10) where rp is the boundary of the sphere in the position of equilibrium. When the functions w and x will satisfy the equations 6w =vow; (4,11) aX = v~X. If v is small, then the functions w and X will noticeably differ from zero only in the neighbourhood of the boundary r = r. In this neigh- bourhood wecannot neglect the guaranties (values) w and X since the potential component of the velocity vector can satisfy only one of the conditions (4,3) and (4,5), for instance, the condition (4,5). Hence, on the boundary of the oscillating sphere the functions w and X will compensate the fact that there are too many boundary conditions. Since v is small, then it is reasonable to replace w and X by their asymptotic representations. The form of the equation (4,11) permits application of the standart methods for constructing asymptotics. In the neighbourhood of a point P lying on the boundary, we shall perform a stretching of the coordinate r For this we shall set r-ro= f Vv in equations (4,11). Then disregarding terms, containing v as a multiplier we shall bring the equations (4,11) to the form Qw - w~~, aX - XEE' -13- (4,12) ~~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 - Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 particular solutions of these equations are of the rollowing form: c ~,- ~ ~ ist, 52)exp ycrr - L 1 ,y' .ti ~ ~' ~ ~l V~ 3 C 1 where C and D are arbitrary functions ~t and ,~z. ICe must find solutions, disappearing at ~---~~. l,et us agree to take that value y'o for which R; I/6 > 0. 7~hen the sought for representations will be the Following: In order to define the functions C and D we use the condi- tions (4,;;)and (~,4), which will give us two equations in two unknown functions, It is easy to see, that C = 0(v); C = 0(v). = vC`; D = vD' and look for C`and D`. Usingthe expression(d,8'), w~e brim the conditions (d,;;) and (~},4) to the term c~~i]I au 1 ~,t + ~rS`~, Z 1)-~s (milt),-+0(~w)= I Iniiz)r ~ O( ~'v) - 01,15) Frrxr these Equations we immediately find the functions ~ `ond I "thus, the formulas (1,14) may be rewritten as - 14- rst+~.l~.,~~s ]r~T.~exp ~u(r-r'~)'Olv~~)i 3 y- v,a~t ~,.~ t ~1~~,y~ ;r~r f'xP ~~`-'(r-r )+O~v''), p _~ , ~ p ~ 0 tirhere ~1i7 is some known function of st and .52. 1lence, the functions ~7 and X are expressed by way of boundary values of derivatives of w.This makes it possible to exclude the functions '7 and y from the boundary condition (~1,5)~ Using (?1,9), where ~;~ (st, s2, f') is replaced by ~~p+ ~1 -~'o)~rlrp) we shall bring the condition (~,5) to the form: Irt;12 2 2 1 v L~~:i2 (~ll?0f5t t ~l~~V'S1)S1 here %~ is a constant. Intbis way the problem is reduced to satisfying Laplace equation together with a nonself-adjoint boundary condition ($,1~) and thus a principal simplification of the pmblem is made, In the case we have considered, oscillations of a sphere, thr~ problem may be completely solved since in this case it is possihle to separate variai~les and the solution !nay be explicitly obtained ~LV' alhl Ill I~~11 ~, ~~Il~d'I' riliti nut tU ('(11 LSIf~ef thr' .Sri~IlUull of .4~ir('Ifl(' rnncr~U~;~rnl.I~?n~~.l~nil~~,ai~t~~dhi indira(r~.,~,n~r~ E~n~~iLlr apG~nr~r~:r~s Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 _ - _. i Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 for investigations, It is clear that difficulties of ;he developed theory are such that many years will pass before the theory of oscil- lations of a continuous medium reaches the level of the oscillation theory of systems with a final number of degrees of freedom. Ready for t,rantan~ ~'3%G-196'1. Order :1Q Pranced on rotagrants an the Computanx Center of the USSR Academy o,' Scaence~ .Moscow B-333, 1-,Icademachesky ~roezd, 3~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 STAT Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 B. I Moseenkov- Kies State L'ni~ersil}' named after T. G. Shevchenk~ 'Cue t. ~e~ ~ ~ t: ., :'r. ,. _. ii.;aGec ,.~~u,+:~ucLi~ne _ ire; 6. H. MOCEEHKOB I~ICCJIEjIOBAHHE HECTAuKOHAPHbIX O~HO~IACTOTHbIX PE}KHMOB KOJIEbAHI~It~ B CHCTEMAX C PACiIPEAEJIEHHbIMI~I IIAPAMETPAMN al aaal~ is ;,s to t,.~ .., t, J1 .,~~r ~i~m;- .- o~' r,ne .,~:.st~~uct~ ~a -inn ~~~,az~'e n-,ltq, ~ ~!.g ~~~a i ntC1 account all the pu!?.II"ile~L.~ a;;pearir,~; ~ . process ~f vit~retion c~ t~..~._~ c~u;struc`o~..;; units and details. ~Nnen investi~atin~; non-stationary ~ib~a~, ons of such systems ?~itn a~stri.'outed :arame'~~~~..~ (if tie distributed masses are ::.'~:~^_ ir:`~ ~. Count) brt:at matiie.aatic~il uiif~z!~lties a_:::~ as corresponding partiaa uiff'eze~_;tial equati ~'~s /linear and non-linear' ~;ont~zi:~ variaole '~~e'`ii- cients depending upon t~eie. Ds Gr~:,~ obt~,i-i=,,~ o general solutions of taa ini_cat~d eauar_;,rr', i=~ impossiole in most cases, so ii; is of ~z .~at i..- terest to pay ettentior to the :;enst:uc~~o:~ `,v the asymptotical methods of ti,e tv~c-parama::_ic- al families of tae part!.al sol~~tions, wrsie~~.:;~r- respond to tue solo-i'repuent rE;~:i,,es of sc_ stationary vibrations in a defi:,ita form c ~:~e dynamical equilibrium. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 1. ,tie small con~iuer orief'ir ~,;ie r_?ocedure ~i ~nvesti;a~ion of ~~.. ~~.~lo-t~r~:ue:a viurations in the syscemc ;pith N degrees ~f !reedom. For the convenience of t?e ~innlicaCior. ~::',' the obtain- ed results to :ne sol~itian of practical problems, the yethod is ;corked out by Yu.A.~itropolsk~(41di- rectly for the systems of Lanorar.~;e equations of tue second order, wuich can be written as: ~~C7,~(IJQ,~+~B (r1~=EQ(I,BQ,,...,QH,Q?~ ~,~,v,f~. r=i t., r~ In eae equations (1.1) ~ is a small oositive parar.~eter, f -time z= Et -slow time, ~, ,..., ~N generalized coordinates, a~~C~=a~;(?J and ~,J~J=61;(7~infinicely derivable functions for all finite values of T Q~~T,B,4,,,..,QN Q,,...,QN.E~ are periodic in B (with the period 2~' ) functions which are infinitely derivable for all finite values of the variables and suffi- ciently small E ,and ~= Y(ZJ non negative infinitely derivable function for all finite values of For the perturbated system (1.1) asymptotic solution is being built, corresponding to solo- frequent vibrations,close for E sufficiently small to one of the normal vibrations of file unperturbated system /system (1.1) for E_ ~ /. Tf one assumes that the considered vibrations are close to the first normal unperturbated vib- ration and in urrpertui~oated system for all values of the parameter 2' ~0,,~ste;n (l.lj. the first approximation pas tue form: ne expressions for pproximatron equations: ~a=~':P,~~,a,~'~ ~j=~2,...,N~ (1.6J where ct and y are termined by the first a (1.'7) - ~rl ~,u,~r/and B~~~Q'~~being obtained as partial periodic solu- tions in ~ from Lhe following system: S~ a~, _ _ r a~ " ~, dr 2~??m ~ P X f~ Yq'N ' S-- oa ~ Irl rr f1~~o ~ F :sr J'i~r~.t~+y~JdBd~S!~ ' l Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 corresre:.ai:.~; to the v~,rl~:~~iui;s of arnnlitude and tu~ t.,o pr=ase ~f tce :first "nu~asol" vibration. .'fi Bfi, t~ Witi~i[1 vIle i1tLt Jr'u ci' terlll5, N _ (2.3) Let as uesio:;ate by ~}~ toe mean v;~lue of this avrk fill one whole cycle of vibration, teat is _ ~~? d'W -~7f d'Wd(~~?~~'J , ~rl ago courier douole series, (~.4) ~~=1, 2,..., NJ from (2.3) and r. zF,,, ~ ~ i6ryr m rU -16r r~W - Oil ~~ ~ ~~~ Qo ~ P ~Cd~1S?.y~lB(~~f?+~~'q_ 6=-~ r, ~~~, yr ~e fJ~Q~ ~ P ~ur~s~*~~aadif,~,~~~~ o e J` (2.5) Let urn use rag symoolic notations ~ , and~~~~`y' ~(~Q for the coefficients ~`{~ in the variations ~a and ~~ T~nen tiie system of equations (1.8) de- terraininn the functions II,~Z,p,~~ and B,~I,Q,~~ looks like: ~w,-rY~a~'-7ar~,8,=m, d`a , s a a d~m,w,l a(r~,_ryl~~,~7c~,~_-m~ d~ , Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~uow it is !,ot difficult to dive cue ~.ner,;etic- al interore?ation directly for t,.e eeu~tiiins of the first a>pruxim~aion. for this nur:~ose let us represe:,t toe mean vaiue of virtual. r:urk in t;.e i'orm of t:~e follo~uin~ suer.: 6-~ 6=-~ where ~Ws desi~;uates t~~e meau value of vir~aal work wnicu 6 -term of ttre Fourier cxoansion for excitative force in tree si:,usoidal regime would carry out at cycle of vior~~tions on the virtual deflections, corresponding to toe variation of the amplituae and posse ;;f t,ne vibration. Taking into consideration the notations (2.7) the functions ,~{~ and B, may ue easily obtain- ed from the system (2.6j as partial solution ne- riodic in ~ ds a result of this, tiie equations of the first aooroxin~ation are as followws: O lY~ 0 hr6 1 da_ EQ d(m,~l 2E ?? ~rw,-,rrJ6 ~a f2w, ~ a dt 2mw, a'T tm,~ 4~ Tiu,-sVJ~G2 ~t'GJ-S~+2E~ r m,a 4=-~ 6 ~rw,-svJ6 ~w a -2cv, ~ aW, -(rw,-s~)~s~ (2.8) Th?ss, when forming equations of the first approxi- mation it is necessary to find ~'{~/ and to ex- pand the obtained expression into Fourier series vnd~finally, to put the "partial derivatives" of 6 -term into the equations (2.8). For trig case when the oscillatory system is ~'Ws Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 only under tr,e action of potential excitotive forces N (2.9) where (/ is ,ne excitative potential energy. Hence it follows that it is necessary to find the mean value of the variation of tP~e excita- tive potential energy ~ V in order tc form the equations of the first approxi,aation and to s=pend the obtained expression into Fourier,se- ries, and after which those "partial derivati- ves" of the 6 -term taken with the opposite signs should be substituted into the equations (2.8) instead~of ~W6 and ~Ne ~' With the help of the euergetical interprita- tion the equations of the first (and second) ap- proximation are being formed when using direct- ly the form of work (or potential energy) and kinetic energy, This gives the possibility of obtaining the approximate solutions without pre- liminary forming the exact differential equati- ons of the problem, besides the possibility of the formal extension of tee worked out method for obtaining the approximate solutions of the partial differential equations. For linear oscillatory systems with, finite and infinite number of degrees of freedom the principle of superposition of the ~-;.br.a:.ions takes place, Tne principle of supe.,~osi~~_ion of the linear vibrations is clearl~~ elis~inated in the form of general solutions, As it is well 'drown tt>e latter have t1e "nrm ~~f fnnctiors m~'~.i e are finite and infinite sums (aa,or~:iz~ , t~.- t_e number oi' ti it ue~rte ~ :f ~, t do_~~ tf ~-arm. nir terms of tyre n~ ~Wal ._~~ ~ o~ ~;:~ co es~ r.:~+ ing confi;u:?etions ~1 Lne ?~ r i aF t~.~' ~'y.:a^ - equilibr.um. Besides tea ors:.~:~nla ;. .,ap~rpo:~~_:io:_ ration':; for the liner oa,,. 1.:'~.,~y ,y.,~-~ma ,,..~. , is a principle of so'_o-f.eq: ,;.cy of 6ua eir'_ :; regimes of vibrations. ke:zl_~ , in t~:~ ~reser.~.~:= ~f the external excite*ion of '~,,e ;;ei'iuite fregaa:.~- cy and the dissipative forces it is practica'_i,/ eery soon estaolisir;u fire forced vibrations this frequency in 'sue ri;so.~.a~~ce as well a:~ i~:r ~ _. non-resonance case. Consequently, all points :;i the system will pecfox~m solo-frequent vibrati~,< in definite confi~~u~~acion (for 11/ degrees o? freedom) or in tae ue.`~inite form ~f tae dyC:~:rai~al equilibrium (for .inf'initel,y m~~i~y deg^ees), fire principle, oY super.~osibion noes not aau~~ place for non-1ine~r oscil~_atary sy:ems, ~~,:at ~`__. principle of solo-f ~e:~uency o? one re~i;ne> of ~. .:~? nations ir. ma47 cas,s takes place ,and i.s k;2~in used in tee invc,sti..:ation, The validity of the principle of solo~-frec~.~-;~~ cy of the reei~~.es of vibrations for non-line~~ vibrational systems oesides of the linear anew testifies to ~.~s univ;;rsality, knalog~ica;l, ~~.;.th 'tee systems wJ_th.N de~_as of freedom, t_E ~-oill.ator.,v systems with the d;_st- ributed para~r ~ ~~escribe~~ ay the partial ..~' Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ferential equations _~rzit un1~r definite conditions t',e solo-frequent regime; of vibrations in corresporrding _~orms of the dynamical equilibrirm and in the case of their sta'oility the inve:;tigstion of ;uch regimes is of great physics/ ir:terest. That is why wnile investir;,:tir:~; solo-fre- quent regimes of the vibrations of Crie srstems with distributed parameters it is quite loei- Cally to use the method of forming the approxi- mate solutions paving oeen wor;ced out for tire oscillatory systems with N aegrees of free- dom ar.d to extend it to oscillatory systems with the infinite number of degrees of freedom. in the problems discussed below the procedu- re of tyre construction of the approximate solu- tions in its energetical interl,ritation is for- mally applied to the investigation of the solo- frequent vibrations of systems with distributed P~'ameters. 3? '1'he problem of the diametral vibrations of the pivot ~tinich is upper the action of the longitudinal sinusoidal force with the variable frequency is investigated by Yu. A.Niitropolsky(3~ who was tiie first to apply the above mentioned method while investigating non stationary solo- frequent vibrations of the systems urith distri- buted parameters. This problem deals with the diametrical vib- rations of the pivot of t`e lenot:' f tiie axis force ui' the following form acting on tue free end of the Divot: F(tJ=~1+g~iaB Y(7J ~ = and ~ ? (fig./) Let us designate the rigidity of the pivot by EJ .Let ~` be the weight of the unity of volume, ~ -the acceleration of gravity,n is the area of the diametrical section. Then for the potential energy of the curve disregarding the inertia of rotation and the cutting force we have the following expression: e V=2(E-n~~f~az~~dx-?Sf~ax)p~x (3.3) 0 0 and for kinetic energy f ~~or~ ay I pdx . T 2 ~ J at l (3.4) U=Vo+Eil,, (3.5) V = 2 (E-n~1 ~~~~x- Q sf ~axl dx (3.6) is the potential energy of the non~erturbated system e r E V, _- z F(t)fn f ~a y~dxf1~a y~2?'x~ (3.7) Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~ Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 is the "excitative" potential energy, In order to form the differential equations of the "non~erturbated" movement taking into consideration the expressions (3,4) and (3.6) use the princi~le of Ostrogradsky-Gamiltott As a result of tnis we have,nom-excited equa- l f n~~~+say+ ~ atf=o witY. the natural ~Ix,o -~ boundary conditions: ax x=o _~; axZ x_ e E 5; ~y -~ /'~ ax' Ix= e (3.8) _ ayl Having solved the equation (3,8) with bounda- conditions (3,9) we find the normal functions !tl !c/? ,c, rcl r,c ~x1=~.1, Ana, P+a~ ~Q'~r,1pF'lXco~a;`,'x-C/ra~~xJ_ r? where racteristic equation -- ~-~1nea rrom the cha- ~'ioing on to the investigation of the "excit- ed" movement, that is with the ace~untin~; of ti:e "excited" potential energy E ~ , it is necessa- ry to note tnat tue boundary conditions (3, y) don't give arly possibility by the correspondinU substitution of the variables to transform th2 equation of tae "excited" movement nary aifferential equation. un Cne to tue ordi- uase of the above stated method of fr~rmiug tiie eyuatior_s the first approximation for the arnplituie and of phase coming directly from tine expression of tha "excited" energy E ~ ,the solution of this problem causes no difficulties. Under the action of the raeriodic axis far- ce with the frequency which is approximately two times more than the fundamental main par- tial frequency cJ~ ,the intense vibrations appear (parametrical resonance), Starting to the investigation of these vib? rations one should seek the solution for the "excited" movement in the Yorm: y=`P!'~x~acoy~2 B+y~1 ~ (3, t2) where a and ~ must satisfy the first appro- ximation equations (2.8), gg (E-~JJ~rc/4+s'~r`/? w~_~ For their construction we find the veriatior of the "excited" potential e EV - o ner gy , and then ~ (3.11) ~ ' its mean value for the whole cycle cf vibr;_~tions ~ ~~ -~?,~..~are the fre quences of tare ,~_ mal vibration f th nor- 1N n ~ OVi=P I r~~~(78+~'l=- " ~ ~ ~fa_~'~ogp2r~ rom tine correspondi 7 a+ 8c :,; frequence equation. ~'w8 ~l~ ruaPB 2iy P ~a+~i e +~ ~' ~ ~~, r ~,~~ , 8 o Bt 8 ~t ~ ' p 8 ~~ ~T - V~a~t =o , Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 dx (3.14) ~ d' 6 l;~ Having substituted and with the (~Q opposite suns into the equations (2.8) we ob- tain the first approximation equations d_a_ _ E~ a8 a't 2m, Yl?J ~~~ cl~ _ W_ Y ~J_ ?~n~ dt - ' 2 2m,rv, ~,~ ~ jdP~'~x/)dxt f ~d ~"%xl P ~ dx ~ ,2 f ( dxr ~ + E~ I B 7m,V~J~`n2`~, (3,15) e m' - g f ~'P ?/~l JPdx . (3.16) From the equations (3.15) we can investigate the qualitative character of the solo-frequent regime of the vibrations. Particularly while in- vestigating stational regimes of the vibrations and their stability, it is easy to find a zone of the parametric resonance defined by the in- equality I W _ yr~~_ ~~";a I ~ E ~"'a ' 2 2m,W, 2n7,Vt7) (3.17) For the investigation of non"stationaryy~roy cess of the pivot oscillations with variable frequency of external force and passing the system through parametrical resonance a las of change V tiJ and numerical values of parame- ters of the oscillatory system are considered as data. Under some numerical values the first ap- proximation equations (3.15) are integrated by the numerical method and the resonance amplitu- de curve is built in the coordinates a ao,t where Q, is the initial value of the amplitu- de. This curve is characteristic for non-statio- nary process /~ig.2/. 4. The problem about the diametrical vibra' tions of the pivot which is under the action of the moving load and pulsative force, was inves- tigated by B.I.i~oseenkov (~ ~. This work deals with the investigations of non stationary vibrations of the pivot (beam) which are fastened swivelly on the ends, ~ be- ing the length of the beam and ~ the cons- tant diametrical section, with the assumption that some mass ?/y is moving along the beam (small in comparison with the mass of the beam). Besides, let us assume that the beam is under the action of the moving vertical quasi- periodical force EF(9)=EF,~+RB , the point of application of which coincides with the centre of gravity E/y /fig.3/. Let us introduce the following notations: p -the density of the material of the beam, E the Young modulus, ~ - the moment of iner- tia of the diametrical section. iVhile examining the diametrical vibrations which are being accomplished in vertical plane x~~ we ignore the inertia of rotation of the diametrical sections and cutting forces. Declassified in Part -Sanitized Co A roved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 ~ PY pp Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Kinetic and potential enemies of tiie exa- that are necessa mined system (the oeam and the load) are as f lows; - ol T=~f~ayJdxt~M aye o ~ ~at~x_~- T ~~T V 2Jf ~ ~ p J~/x_El~~~ylz_ = V + ~ U (4.1) The equation of p p diam (4.2) ro er etrical vibrati- ?~ of the beam, kinetic and potential energies of which are equal r and is the equation of " V respectively non excited" movement: dxy~`~~~=o E~ at and tue boundary conditions have the form l ylx_o=~ ~ d ~I 0=0 I fey' ax x. , yz-~=o =o ' a.2'PIa=P (4.3) (4.4) 1'he forms of the normal (principal) vibrations of "non excited" movement are determined from the equations (4 , 3) and the conditions ~ ~~~(x/= bra ~~x and their ~'~ - ; 2~y, ? . propez~frequencies (4.4), (4.5) ('~=12,3, ~ (4.6) ry for constructing asympto:'c solutions of "excited" movement. The excited force consists of the exter_sl pulsative force and others which are put on it with the account of the "excited" potential ener- gy ? (/~ and kinetic enemy E j, It has the form. ~'~(~,B,y~=~~F~aBfMr-^~afy~x=~, (4.7) where f=Ulis the current coordinate of its pout of application and U is the velocity of move- ment of load and pulsative force along the beam, If we consider the sag of the beam y as the generalized coordinate, the first approxima- tion of the solution corresponding to solo-fre- quent vibrations close to the first normal vibra? tion in the presence of the principal resonance in the "excited" movement will be as follows: y~,i bra ~-.za('o~(B+~) (4.a) The amplitude a and the phase ~i are termin- ed by the first approximation equations (2. S), In order to form them we find the expressi- on of the virtual work ~~~ ~,~'~d'ym=~f;~r2B tM~+MW~?~~ ~ ~aco~(g+~~~x Xlin ~ ~~acc~(9fv~~d'a-a~ra(9+w~fi~~. (4.9) This being done, we find the mean value of vir- tual work ~ and its 6 -th "partial ~ieriva- tives" with the help of the formula (2.4) and we also find the value Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 .._ .. Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Kinetic and potenti mined system al energies of the that are necessa lows; (the beam and the load) are T=~f~~~dx+~M~d x? - o ` ~ ~~=f-T +~T (4.1) ~-2~f ~~p~dx-EM~~ylxt = V GeV f + The equation of p (4.2) ?~ of the b roper diametrical vibrati- e~+ kinetic and potential energies of which are equal T and is the equation of '~ V respectively non excited" movement: dx~ }E~a =0 and the boundary conditions have the form ~ P (4.3) The fo (4.4) of " non eac tede normal (principal) vibrations movement .are determined from the equations (4.3) and the conditions (4.4), ~~~J(x/=~~t ~~x and their 1'C'12,~,.., J proper frequencies ~ = tir K (4.5) ry for constructing asympto~_c solutions of "excited" movement. The excited force consists of the exter:al pulsative force and others which are put on it with the account of the "excited" potential ener- gy ? V and kinetic energy ~ j, it has the form: Y ~~(z a, y~=~~F~ne+M~-Maty~~x-~, (4.7) where ~=this the current coordinate of its point of application and U is the velocity of move- ment of load and pulsative force along the beam, If we consider the sag of the beam y as the generalized coordinate, the first approxima- tion of the solution corresponding to sola-fre- quent vibrations, close to the first normal vibra- tion in the presence of the principal resonance in the "excited" movement will be as follows; yri ~~ ~.xaCc~(B+y~1 The amplitude a and the phase ~ are termin- ed by the first approximation equations (2, g), In order to form them we find the expressi- on of the virtual work X~~ ~ ~la~~~~+~1~a-a~~(9+w~~~~. (4.9) This being done, we find the mean value of vir- ` tual work ~ and its 6 -th "partial deriva- tives" with the help of the formula (2.4) and we also find the value (4. ~) Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 t ~~ ~~f rape xdx_/'~? (4.10) ~~~ are substituted into the equations (2,g), As a result of this We have the first approzima- tion equations for the solution (4,g); 'This system is integrated in quadratures ~=u+rU=aet~ X11 P(~, +~(~J) u" r ~,-v~J-fw,Mrrar~~rJ+ 2EF,~ra7~~J /?11 ~ ~~1P~w,-YIrJ~~ra ~~, (4.12) and by the substitution is reduced to the linear equation'~or a complex valued function ~ dt-I`wr-~(Tl- ~~,M-~~f(i/~__zfF,S~a~~/17 one 1- ,~~~~ ~~J . (4.13) Integrating (4.13) and then separating the real and imaginary parts of the solution We find the expression for c7 and ~ according to (4.'12). Having found the relation It=a(tJ we plotted the resonance amplitude curve of the principal tone of vibrations in non-stationary regime /fig.4/ for some numerical values. Non-stationarity of the oscillatory process chile crossing the principal resonance is cha- racterized by the finiteness of r~rplitudes of the transitional regime and by the displacement of maximum of the resonance curve to the right hand. 5? The problem about the bending vibrations of the pivot of double rigidity in transitional regime of rotation is ezaeined by B.I.Moseenkov ~ 8 ~, where the examined method is extended to more complicated case of spatial diametrical vib- rations of the pivot. Besides ordinary assump- tions about the rectilineasity of the pivot in an unstrained state and egnoring the influence of cutting forces and the inertia of rotation of the diametrical sections are assume the pivot does not twist in the process of vibrations, We think also that the pivot is unbalanced sta- tically, that is the line of centres of grairity of elements of the non-rotation pivot is dis- placed according to the rectilinear axis and is a plane curve, the form of the change of unba- lance along the pivot is considered ae datum. While examining the diametrical oscilla- tions are disregard the longitudinal vibrations. We suppose the conditions of fastening to be of isotropic snivel type. . Let us introduce taro coordinate systems O.xy,P~ and Oxy~ /fig.5/, the first being immovable, the second rotating arith the angu- lar velocity W Azes Ox, and Ox are di- rected along the axis of rotation, the direc- tion of which coincides With rectilinear axis of the non rotating pivot. The origin of the coordinates 0 is placed on the left support. da_ _ 2EF, Srn ~ f (T) Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified ion of the pivot the diametrical sec- , the axis of the movable s ny~ is directed ve rtically below. Ystem 'Then kinetic ever potential energy of d~oof bending vibrations and into acco rmation without taking unt their proper wei following form Kht will t accordin eke the Hate systems 6 to the movable coordi- in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 The otner axes of th ~y and ~I a movable coordinate system coincide by their directions the principal directions of xith t' of ~f~! ~y p r= o a~ - w(~~JI +~~+W(yt~~~? d~ ~~ e (5.1) (5.2) We h ave used the followin 6 not r p essions (.5,1) and ations in the ez- (5'2)r /I sity of the pivot 1 the linear den- of the const ant section, ~ - the length of the pivot ,~ ;;actions of vector of ec~ ~y~he pro- ~ ei ~ n i t ne of gravity, w - w ~~ y of the cent- th e augur, velocity of rotation, E -the Y - J -the princip~ m~~ a odulus, ,~ .J2 and area of the diametrical of ~ertia of the s the projections action, y and ~ ?f the of the vector of displacement Beometrical centre of the pivot's went. ele- Due to the fact that such asymmetry of the diem factors ae the etrical section or the "lack of coordination" of rigidity, the influen- ce of statical unbalancity az~e small choose in the , let us expressions T corresponding parts which t and V those se small factors eke into account the- and They will correspond to the "excited" kinetic and potential energies. Then the expressions (5, 1) and (5.2) will take the forms T= ~f~~ty ~'~`~I+ldt'w~~yJ~dX.21~,w~t~+p~z~t~ o z -~w~~;~~,(tyw~~~J~(df~tw~IyJl~dx=T *ET, , o ~l+(axJ~dx+E 4 1~~-~isyJ? 7 7 -(a J~dx =V +~~ r ?or the purpose of forming the di (5.4) equations of the ? iferential non excited" movement let us use again th~ principle of Ostrogradsl~y_O~ilton. f (T -~Jdt =o . t, a a result of this we have the system of two quatione concerning y and ~ . Having introduced complez-valued function ~(x,tJ=y(x,fJ+id(xf~ nd havin (5.5) g used the formula of transition from he coordinates of the movable system to the Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 coordinates of the immovable system ~(x t) =~(x t)e-rB ~lx,t)=J,(x,tl+iz,(x,t) rilzm which are the superpc~itions of the two cir- cular motions -the direct and the inverse with the angular velocity u7~ So the form of the norme' vibrations of "unper- turoed" movement are det ..wined by the real func- tions ~~'=1,2,3, ... (5.13) (5.7) xe obtain the equation of the "non-ezcited" move- ment in the immovable system of coordinate from the above mentioned system of the two equations pax+d =?~ and angular velocities are their proper frequen- ces (5.12). Going on to the investigation of the "excit- (5.8) ed" movement, we fora +he expression of exciting force, corresponding t~ the "excited" kinetic gyp E 2 ~1 ,the "excited" potential energy EV energy E T = m , (5.9) and small dissipating forces of external fric- The natural boundary conditions concerning the complex valued function ? have the form: (5.10) From the equations (5.8) and conditions (5.10) we deduces ?R (~' tJ=sra ,ri?~,/Q,~e'~~t~w,~~+ o e t ~w`t,wP~~ ' P ~`L r~ (5.11) (~'=1,1,3,...) (5.12) These solutions can be interpreted as the vibra- tions in the ~' -th forms of the dynamo equilib- tion. These exciting forces which act on the element of the pivot dx have the following complex form: EQ~B?,?`~~ 1~~~~?~~etef(I,-1,1~P2re~apldx ~~'/~ ? ax at/ (5.14 ) where , d~= ( ) ~ is the coef- ficient of the external friction and 7(x,t) is conjugated valued function to 7(x,ff. we consider that the angular velocity rotation m(Z-J on the interval of change (0n r,nr':* i. _ QtY;od? ?SG ",!4an.,:isi,ac~ said, _ tP~o_;e w~,o t:ad used od - _'.:o,Papal~:ri, A.I.rrti~^feld, and _?. 'a.~ i:rau ~_"cd ind_vidual casuisti- '.:. s.. c;n~~. .. t9er'~~ ., .. 'ir_ding ;,,, ,,c:~=t?or,, l;~avin, aside c,uesti~:ins i.it;; ar:d, in ~?,;neral, questiona ~ the ,., your c: ; ~luti.or ~ ~.rder other initial con- ir,; *.`_u,. ~..~s~ ~;rr:i~h cori?espou~ to the per.o- - mr;tio~". m,:a fit~:t concrf:te result, obtained in tY,is .. ' mGno~ra;,Y; by ,,~t.ai~;i~. "S'.:corY of oscillations" i~~n,ed in IQ3~. These results contain an exha,:..stive investigation of dyr_amics cs remain simplest ncn_lir:ear systems whi?~.h ~Nas per?'crmed by means of the method of point manp- ir~gs of a straight line into itself (t`~o prcblea. c?f~ t-~~ ?ri~}a~tir~r_ ~ ; a valve eneraiur a~~i Ch 7.- chc^arteri :tic a.r?ct t'~e simplest ?~ode1 ^f t?~^ c~~,eks :ins i:. ..'~e ~~,c_. ,~, ~,,re stat... i;~. ,",.:..r. .r.cn~?r ~. ~,."':e ~ . ec,.y c ?? ~ II ~, application of the mathcmati~al a~~naratus of the theory of point mappia~;3 ?rnd the theor;~ of the Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Le :rotle~, of stabilization of a plane by the autopilot i I2J,~13', the problem of yishnegradsky ~I4~, ~IjJ , and a number of others ~IS~-~20~ . The possibility of obtaining point mappings by direct integration and the consideration to- ~ether with the phase space of systems their spa- ce of parameters immensely contributed to an ef- fective application of the method of point mappings to piecewise linear systems, particularly in the questions of global behaviour, It would be wrong to consider the method of point mappings to be solely an effective method of the investigation of only piece-wise systems or a method based upon the possibility of step by step integration, It cannot also be considered to be a method avhich although is, ir. fact, the only r,et},od of the global investigation of dynamic sys- tuWs o_' more than second order, but is a special ot}:od ,and therefore it has a very limited region ,_~ application, The present paper aims at sho:^.ing first of all wide nor=sibilities and efficiency of the me- thod of poira mappings for t~~e ,solution of various problems of tte t}-eory of non-linear oscillation, side possitilities o`' t`.e met:~od o" point eappings not only for the investigation of concrete non- linear systems, but for the study of many general questions of the theory of non-linear dynamic sy- stems, The description of the method of point map- pings has to include the description of a collec- tion of mathematical means, the methods of con- crete investigation, and the types of non-linear systems which have been studied and can be studi- ed, the description of possible applications to the investigation of general questions of the theory of oscillations and a number of general questions which have Deen treated in one measure or another, This description represent consideral le difficulties not only because of vastness and variety of data, but also due to the fact that the method of point mappings keeps on developing, The method of point mappings is based upon the fact that it is possible to reduce the investiga~ tion of the motion of a dynamic system to the investigation of some point mappings, iri particu :, the theorem of the relationship of states o: - ~.: periodic motions with the fixed s of properly constructed point mappings, A general met had by which t}.e investigation of mo- Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3 tion of a dynamic system can be re- duced to the treatment of point tappings is described in ~ 2, Chap.II. p~hen the method of point mappings is employed in pra? ctice,it is very important to choose rro~arly tr:e -may of reduction to a fci_r.t :a_pin~ any: tr:e ~.aay of it> setting,The success and simplicity of solution of a problem depend largely on the- se ways,n general recommendation here is a. fnlr lows: on the basis of the treatment of tLe phone spade of a system one should try to em;;lob tha method of r ~ uction to point mappings of the le- ast dimension and to sat them in a parametric form.uonever,since these recommencations along explain too little,I should like to cite a few examples, ,rxample I. A relay control system is 4Fs^,ri- bed by a non-linear differential equation of the second order in the form: 2 cox ~l~ dr where (+t when Y'~d~and when -8