(SANITIZED)UNCLASSIFIED SOVIET BLOC PAPERS ON METHODS FOR CONSTRUCTION OF PERIODIC MOTIONS IN PIECEWISE LINEAR SYSTEMS(SANITIZED)

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• Document Page Count: 747
• Document Creation Date: December 22, 2016
• Document Release Date: December 7, 2011
• Sequence Number: 1
• Publication Date: September 1, 1961
• Content Type: MISC

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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
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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.
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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
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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.
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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
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~ 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,
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,..
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,
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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,
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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,
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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,
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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
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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-
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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 (
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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
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~~;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,
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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
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(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:.
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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.
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?'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 '!
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 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.
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 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.
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 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,
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 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3
ve
.~
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 Declassified in Part -Sanitized Copy Approved for Release 2011/12/07 :CIA-RDP80T00246A018900360001-3
Fts.,
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 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
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,~~
,~~
~~
~u
N Ali
~I~
L'kr. SSR Acted. of Sci. Publ. Ilouac's Printcn
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STAT
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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 ~~/
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}~ , _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~~~~~
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~~ 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~
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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
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'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..
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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' '
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.-;;~,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
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..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_~
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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~:
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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,...:~~ ~~
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.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
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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.
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Fig. i
Eig.2
h ~0~ '-~'
Ft g.3
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;'ly 5
Fi g. 4.
Fig 8.
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Fr y. 9.
i'
tea.
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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
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~'r9 ~'i.
F~9.22
r"~ ~ ~~ 3_
~,.,.,,
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STAT
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: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".
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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 (~]~.
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,:;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-
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.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
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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 ?
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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).
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- 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
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-~ 7 -
s;c;oc~ rcct. ni.
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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-
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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
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... 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,
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,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
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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,
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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.
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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
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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
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~~.
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-
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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
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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
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 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.
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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 ~~ ? :.~ .
Ala, iia~rna.; lfl II, 'Iy5b,
~2, Pa~xxoBxq :',ii? lilene~x ?l_. r[sl~c ~ta~ 7, r ~o.~a _
pa6oTe B~;Gpa;~{ci~tx;, 6yxitiepl~: 3arpyso~xxx ycmpoCara. i~.::.~~i,
~aiu~ciu{ ~1-BObcxoro no.>IaTex~~x~. m{.tea, s~;n, ,,'~, I`~51.
3;i. il3H I.~-Fya~. HexomopHe aauaqu o ~xxmexvcl ~raep:~xx ~~:,~ ~~
wepoxoBaTa% n:.ocxocTx, coBepreaaue~ saua~x{oe ,uBxmeHZ{e. db^o?:.
e~epaT ,>~ccepTa~{ xa coxcxaxi{e yxeHOi3 cTexe{ix xat~. ixx~,?1:~?,:'^~
S{ayx ieHVlHPpaP,CxiS11 II0.16!TeYJ{4PieCxx~ xHCTftTyT SiM, I'~1,ii.ilaJlxill,~13,
'~59.
,---.
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~,ShICCI4IR ;~P;b:~Ciilt:~' liI;C9~0;j;,Ii111{~IX CIiCTC:.I. ? 3Bd...1__
"~~ .BR3IIHISa .. ~1326:i1IX;T'OCH'JC, .. .,.~..
'.~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.
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.'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
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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.
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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.
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~ 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,
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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
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.~
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).
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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
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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
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STAT
Next 1 Page(s) In Document Denied
Q
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,,
~~..
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
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}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
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~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.
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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.
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=--~
90 95
Fig . 2
!05
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0
g r-.Tr ----~-- --_
---+-
$.O
Fig` . 7
3>O
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_~. .- i
-~
Order w^1093.
Ukr. 5SR Aced. of Sci Puhl. Huusc~, Prinlcn
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STAT
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OlY T$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
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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.
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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
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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
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-(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}
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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
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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
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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
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( 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~_
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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
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Fiy. i~
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~(~+r~:1
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STAT
Next 1 Page(s) In Document Denied
Q
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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
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/
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.
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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/.
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STAT
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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
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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
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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
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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
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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 ,
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'.'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
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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-
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 Dec~y... _____ . ..._.._a_ _
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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.
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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-
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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)
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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
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 _ - _. i
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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~
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STAT
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 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.
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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
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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~ ,
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~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
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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 ..~'
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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)
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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 ,
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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.
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PY pp
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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
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 .._ ..
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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. ~)
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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)
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 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-
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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
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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