SCIENTIFIC ABSTRACT BELETSKIY, V.V.  BELETSKIY, Z.M.
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AUTHOR
I
BXLXTSM VOITY 2 PA  3o12
THE
H
Into abi ity of Equations of motion of a Solid Around a Fixed
Point
wider the Action of a Central Newtonian field of Fore*.
(Ob
intogriruysmosti uravneniy dvizhoniya tvordigo tola okolo zakre
plennoy tochki. pod daystviyom tsontraltnogo n1yutonovskogo pola
ail.
Russian)
IMIODICAL
Doklady Akadomii Nauk S88R91957, Vol 113,
Ur 2, pp 28729o, (U.S.S.R.)
Received 6/1957 Reviewed 7/1957
ABSTRACT
The present paper investigates the problem mentioned in
the title un
der the assumption that the immovable point of the
solid is situated
in a distance R from the center of gravity
large enough with respect
to the dimensions of the solid. The
equations oftmotion of the solid
under the action of the above
mentioned forces are put down explicitly,
they are a
generalization 'of the equation of the classical problem of
motion of a heavy solid round a fixed point. The theory of the
last
multiplier of JAKOBI can be appleid to the equation hors
given and the
refore the integration of those equations is under
certain conditions
reduced to quadratures. 1) In the general case
this system of the equa
tions ofmotion has three primary
independent integrals, namely the en
ergy integral, the integral
of the moment of the momenta and a relation
betweenthe
direc;tion cosines. 2) If the solid has a total kinetic sym
metry, the system of equations has a fourth integral in
addition. There
is also a four~th integral, if the solid has a
kinetic symnetry around
Card 1/2
any principal axis of inertia or
if the solid is fixed in the center of
Integrability of Equations of Motion of a dolid IPA  3o12
Around a Fixed point under the Action of a Central Newtonian
Field
of Force.
gravity. The presence of certain terms in the equations of
motion
and in the primary integrals modfies the graph of motion in
some ca
seB in comparison with the conditions in a planeparaUel
field. A
solution for the twodimensional motion of the solid is put
down.
(Generalization of the problem of the phyaical pendulum). If
an ini
tial angular velocity is lacking, a solid fixed in the center
of mass
(as in the planeparallel homogeneous field) will not be in
equilibrium,
but will move periodically. Then the special case of LACRIANGE
is redu
ced to a quadrature. (Without illustrations).
ASSOCIATION Department for Appleid Mathematics of the
Mathematical Institute of
the AWemy of Science of the U.d.ti.R.
M123ZNTED BY ' *. ' ' L' K=18H, Mov*.*
dUBMITTIO 9.9;19~6
AVAILABIJ& Library of Congress
Card 2/2
AUTHORt Beletekiyj V.V, (140scow) 402163/18
TITLE: 8ome qu estions of the Motion of a So2td Body in a
Newton
Field of Forces (141ekoturz7.~v voprosy dvizheniya tverdogo tela
v n1yutonovBkom. pole sil)
PERTODICAL: Prikladnaya MatematAlka i Mekhanika, 1957* Vol 21,
Nr 6,
PP 749758 (USSR)
ABSTRAM The motion of a gyroscope in a radialsymmetric field
of gra
vity is investigated. Instead of the rigorous equations there
is carried out an approximation so that the measurement3 of
the body are assumed to be small compared with the distance
of the body from the center of attraction. For practical
applications of gyroscopes on the surface of the earth this
condition is always satisfied. As in the calculation of the
motion of a gyroscope in a parallel field of gravity, also
in a radialsymmetric field of gravity different integrals
of motion can be found. it is possible to reduce the problem
to quadratures similarly as it is done in the cases of the
gyroscopic theory calculated by Eitler and Lagrange. Diffe
rences against the wellknown gyroscopiO theory are obtained
in investigating the stability of a gyroscope which is sup
Card 1/2
Some Questions of the Motion of a S*:Ud Body in a 402163118
Newton Field of Forces
ported in the center of mass. For very small speeds of ro
tation of the gyroscope there result unstable motions even
when the gyroscope rotates around the axes of the greatest
or smallest principal moment of inertia. Of course, the gy
roscope becomes already stable for extraordinarily small
speeds of rotation, so that the wellknown stability behavior
of the unsymmetric gyroscope results. The revolutions around
the axis of the medium principal moment of inertia are always
unstable even in aradially symmetric field of gravity. The
author investigates the case of the spherical gyrbscope and
of the symmetric gyroscope. The results formerly obtained
can be simplified in both cases. There are 4 figures and 9
references, 5 of which are Soviet, 3 French, and I Swedish.
SUBMITTEDs April 29, 1957
AVAILABLE: Library of Congress
1. Bodies of revolutionMotion 2. GravityApplications
Card 2/2
A
Ski
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Libration. of a satellite. Isk. sput. zem. no3:1331
159.
(nU 12:12)
(Artificial satellites) (Mechanics, Celestial)
VAMIN, V.Ht; INIXTSKITe V.V.
..wow
'"'a
Using the anticipation method In observing an
artificial satellite.
loke spate soms uo*3t4753 159. (WRA M12)
(Artificial satellites)
to,
25984
8/560/61/000/006/oO2/010
E032/E114
AUTHOR: Beletakiy, V.V.
TITLEs Classification of the motions of an artificial earth
satellite about its centre of mass
PERIODICAL: Akademiya nauk SSSR. Iskunstvannyye sputniki
Zemli.
No. 6. moscow, 1961. pp. 1132
TEXT: In a previous paper (Ref.l: same journal, No.1, idzvd
AN SSSR, 1958, p.25) the present author considered the
motion of a
satellite about the centre of mass without taking into
account
various effects leading to energy dissipation. In the
present
paper the theory is generalised by the inclusion of
gravitational
perturbations, aerodynamic perturbations and also orbit
regression.
The notation employed is said to be the same as that used
in Ref.l.
The paper is divided into the following sections: 1)
equations for
the secular motion; 2) interaction between aerodynamic and
gravitational perturbational 3) effect of orbit regression.
The motion is classified in terms of the locus of the
andpoint of
the angular momentum vector L on a unit,sphere. Detailed
classification is given of the various satellite
trajectories in
Card 1/ 4
25984
Classification of the motions S/56o/61/000/006/002/010
E032/E114
terms of the above locus. This classification depends on the
given
initial conditions and also the conditions under which the
motion
takes place. Aerodynamic effects, which would slow down the
angular motion of the satellite, are not included. These will be
considered in a future publication. The present analysis is
based
on the following considerations. The motion of the axis of the
satellite in space is determined largely by the motion of the
angular momentum vector. It was shown in Ref.1 that the
equations
for the secular motion of the angular momentum vector L can be
written down in the form
d?,
sin a Wn 0
sin 0 d~
IcTn = 
where 2 X SiU2 e +
sin Y (e, sin OdO
Y
 sin 0 ~kncos X'sin w sin i + sin X(kw + kj)cos i)j (1.2)
Card 2/4  coo 0 ks, coo w sin i
25984
Classification of the motions 0060 s/56o/6i/ooo/oo6/002/010
E032/E114
Theme equations describe the motion of the vector L relative to
a set of coordinates which is attached to the perigee of the
orbiti
0 representn the angle between L and the velocity vector at the
perigee, and X tht rotation of L about the velocity vector at
the perigee. This angle in measured from the plane of the orbit.
In these equations the term containing represents the secular
effect of gravitational moment. The secular effect of aerodynamic
moments is represented by the function 9(e, %). The remaining
terms in (1.2) depend on orbit regression, and hence the rotation
of the above set of coordinates in absolute space. It is shown
that the equation of the trajectory of the and point of the
angular
momentum vector on a unit sphere, whose centre coincides with a
centre of mass of the satellite, in given byi
1 'V.cOs2 d"com 0 POC023 0  kn con Oo
F Y
This equation in derived subject to certain assumptions which,
however, hold in the came of the Soviet satellites. In Eq.(1.8),
e in the angle between the angular momentum vector and the
Card 3/4
1
Classification of the motions .....
25984
s/s6o/6l/ooo/oo6/oO2/0l0
E032/Ell4
direction towards the earth's north pole, 01 is the angle
between
the angular momentum vector and the normal to the plane of
the
orbit, a and 0 depend on the specific form assumed for the
function (p (Ref.1), k(I in the velocity of the node, and k.
is the velocity of the perigee. Eq. (1.8) includes effects
associated with the simultaneous action of aerodynamic
moments,
gravitational moments and regression. The entire
classification
scheme put forward in the present paper is based on the
detailed
analysis of Eq. (1.8).
There are 9 figures, I table and 3 Soviet references.
SUBMITTEDs August 1, 1959
Card 4/4
26659
S15 6o/61/000/007/003/010
0 a (11 E032/EI14
AUTHORS: Beletskiyo V.V., and Zonov, Yu~V.
,rITLE: kot'a,tion and orientation of the third Soviet satellite
34,RIODICAL: Akademiya nauk SSSR. Iskusetvennyye sputniki Zemli,
No.7, Moscow, 1961, pp. 3255
.%XT: The third Soviet artificial Earth satellite carried a
'Is,~tlforientaringll magnetometer whose function waq to
measure the
Earth's magnetic field (S..Sh. Dollinor, L.N. Zhuzzov.
iN.V. Pushkov, this journal, No.2', Izdvo AN SSSR, .1956, v.50).
,rhe magnetometer Incorporates a movable frame whose normal is
kept
parallel to he magneticfield vector by special probcs and
the
tracking system. The rotation of the frame relatlVe tO fh.~ body
of the satellite was measured by two probes and telemetereJ to
the
j,arth. The motion of the satellite about its cenlr~ of trwii
and
also ite orientation in space can bf ~.ktermined from the tmue
d,rpendence of these angles. The prosent paper descr:P.bes the
method
t1sed to solve this problem and also tbr reiuLts ebitained for
the
rotation and orientation of the satellifi up to the li,)4th
orbit.
rhe rotational parameters were determLriRd using these an.1 later
otbit data. rhe first part of the prP!,,cn1 paper gives an
account
(,It I d 1/ 11
Rotation and orientation of the .... i1560/61/000/007/003/010
E032/E114
of the theory of the method. The mntirin of a satellite about the
oentre of mass is affectod by Sravit.it.lonal and aerodynamic
moments (Ref.2i V.V. Bels~tskiy, thiv, Nc. 1, AN SSSR,
1958, p 215. Ref.3; V.V. Deletskiy, thts io,trnaI, Nr_ 3,
1rd.vo
AN SSSR, 1959, P13. Ref.11:~ V.V. this iournal, No. 6,
izdvo AN SSSR, 1961, p. 11), m,7jm^n14.T (Ref.5:
Yu.V. Zonov, this journal, No. 3, AN SS.SR, 1959, P. 118),
possible interactions between magnet.tc womenti associated witl
currents within the socellite itself and the
v,arth's magnetic field, e'tc. The mof irwn of the satellite is
therefore rather complicated, Othc_~uch ~. pr.:c~t.ice the
rotational
kinetic energy is very much greater than tlto~ work done by the
external forces so that in a finite incervai Gf time (for
example,
one complete orbit) the effect of the perturbing forces is small.
Hence,in the first approximation it may be assumed that within
such limited interval of time the motion of th satellite about
its centre of mass is identical with the motion of a free solid
body upon which no external forces are act)ng. In particular, in
the case of the third Soviet satellite which had two equal
principal central moments of inertia, the motion of the centre of
Card 2/11.
26659
Rotation and orientation of Che third..,5/56o/6,1/000/007/003/010
E032/Eii4
mass on this approximation wan found to take the form of a
regular
precession. The satellite's axis z'j which is assumed to coincide
with the dynamic symmetry axis, executed a uniform rotation with
a
tonstant precessional angular velocity tV about the angular
momentum vector L which remained fixed In absolutp space
%'Fig.1).
The nutation angle 0 between z and L wan r c, rt,~ 1 1 n t .
Furthermore, the sAtellite rotated aboot uritjl~ i tonstant
, , XYZ is thf. abcolti artesian frame
angular velocity In F ix. I
s,icb that the Z axi.s points in the direciion cf ibe earth's
pole,
X points towards the Sprinz P'~)int L) iA 11e eing.'~ bptween L
.).nd the Y axis, and jo Is the anje betureati ~hs LY and
XT planes. The problem can fl,.en be redl;ced f.( !1,i,
detc~rmination
of the parameters 9, f n r e a .b and also
the determination of the angles 10 Inij 0(1 c. f7 r . ii ci n
a PA
precession as functions of time. The Anfl.aiinrIr o:f 1)e
magneto
ineter probes can be used to provAde i1.1 these. porm('TPts,
FiS.2
shows the arrangement of the mairritttomotor frampr.. The
C,,:C,,, where A' B and C, ara..,principal central. moments :b fL
i~.L.4"
?
4 A r.,"
J
.2 3
4L
Il
f~Q
ACCESSION.NRe. AT3006839~
inertii:~ o f' th e body., The.', results
obtained are
formulated. as f o I..I
I equilibrium o f  a: bo dy
For the re ative
W
on a circular orbii
_be_s_ti6le,.'it_.is
:a Ne
~
that
. suf f icient
T~:

'
o
the.
durin :the,,, irection o
~nonpertur e ;nb t
r,
ax is
f
the maj
'
ellips6.1d of, 'i.nertta coi:ncidi ''with, the
direction of rtho
'
iad iuz'~ve c to r'. of ~the: orbit. that th'arminor axi
 coincide with th
s
to...
iract on' ot `:t he normal.,.,to the. 6.rb it plane, ',and that the intermAdia
'di
_q tang
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the. t s isplarte and: the Axi
er
11
the body coincides
.
normal to~ th
he e. '*rb it plane's is studied
t
Ori g', art.,~~h
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'ACCESSION IN41: A00 S 1%9/63 /000/016/0094/0323
AVMOR: 0khotsimskiyj D, Beletahly.p V, Vo
TITLE Use of an earth,~oriented satellite for'oolar investigation
SOURCE: AN SSSR. Iskunst, sputniki Zemli., no. 16,, 1963., 94123
tru
TOPIC TAGS; satellite attitudep 6rbital element,, solar
investigation., ins
ment 13lumir ion., satellite instrumentation
ABSTRACT: An analysis is made of solar Ilumination of instruments
mounted on
a seellite with triaxial stab to\'one axis oriented to the earth.,
the
iluat
second along the normal to the orbitalplane., and the third along
the trans
versal. A slight change occurs in the attitude of the orbit
relative to the
sun owing to the yearly motion of the earth around the sun and the
regression
of the orbital node of the satellite due to the oblateness of the
earth. The
problem of determining the total time of illiination is solved 1)
by deeterm1
ing the illumination time at, a constant angle v 1~eAw6en the
direction to the
sun and the normal to the orbital plane, and 2) by considering
changes in'
angle V vith time. Mumination time is the time the sun remains
within the
Card 1/3
L 1827563
ACCESSION DR: AT3006840 61
i angle of view of the instrument. To solve the first part.. the
maximal time of
instrument illumination during one orbit of the satellite at given
values of
angle V and instrument angle.of view p and with a varying angle
between the
optical axis of the instrument and the axis o:r the vate=te is
sought, It is
found that the maximal illumination time increases as angle V
decreases. The
determinati)n of angle V as a 'function of time makes it possible
to establIsh
its dependence on the angle of orbital inclination to the
equatorial plane and
on the initial conditions (hour and date) of satellite launching.
For's, typical
orbit (inclination to equator i = 650; period of rotation T  90
min) the ;otal~
0
time of Ulliiation during a satellite lifetime can reach about
60 hr under
optimal conditions and only about 15 hr for an instrument, vith an
angl~w of viev
of 5". Increasing the angle of view increases the iUumination ti=.
The tot~_I.
time of illumination depends on the hour and date of lemnehing.,
the positiop of
the.optuical axis of the instrument relative to the satellite.,
and the inclina
tion of the. orbit to the equatorial plane.  Optimization of the
orbital elements',
and progre d control of the position of the axis of the instrument
can increase
illumination time 210 times. The analysis rapports the
feasibility of using
earthoriented satellites ford solar investigations. Orig. art.
has: 25 figures'
and 23 formulas.
Card 213
ACCESSION NR: Ap4oo9621
S/0293/63/001/003/0339/0386
AUTHOR: Beletskly, V. V.
iTITLE., Evolution of spin In a.dynamIca Ily symmetrical
satellite
t SOURCE: Kosmichesklye Issledovanlya, v. 1, no. '3, 1963, 339386
TOPIC TAGS: satellite, dynamically symmetrical satellite,
satellite motion,
1i satellite spin, aerodynamic friction, aerodynamic pressure,
satelliteshell eddy
current, satellite magnetic field, light pressure, satellite
shell magnetization,
~i artificial earth satellite
ABSTRACT: A complete1system of equations in osculant elements it
presented, as a
continuation and expansion oY.prevlous reports (V. Ve Seletskly,
Sb. lilskus3t
!i venny*ye sputniki Zemli", lzdvo AN SSSR,.No. 6, 1961, 1332;
lbid, No. 1. 1958,
2643; ibld, No. 3. 1959, 1332). to describe the spin of a
dynamically symmetri
cal eatth satellite. The. osculAnt elements (see Fig. I In the
Enclosure) are used!
as system i n,
or
where moments of Interference forces are, respectively,
indVendentof cc dependent
C,
ACCESSION NR: AP40096il'
on the angular velocity of Inherent spin Y A method of averaging based
on one
or two fast variables Is employed for simplification. The concept of
"second'
approximation" Is introduced and defined as a solution of equations In
osculant
elements averaged only for the precession period. The evolved theory
Is employed I
to analyze the motion of a satellite, as affected by gravitational
forces, aero
dynamic friction and pressure forces, Its own magnetic field and
magnetization of
the satellite shell In the Earth's magnetic field, eddy currents in
the shell, and!
light pressure on a cosmic vehicle travelling In an orbit around the
Sun. "rho
author expresses gratitude to Do Ye. Okhotsimskly for his
evaluatlon*of this
study". OrIg. art. has: i4%fLSures,, I tableg and 236 formulas.
ASSOCIATION: none
SUBMITTED: OIjul63. ATD PRESSs 3047 t ENCL: 01
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BELETSKY. V.V. (Moscow)
"Motion of an artificial satellite about its center of mass"
report presented at the 2nd AllUnion Congress on Theoretical
and Applied
Mechanics, Moscow 29 Jan  5 Feb 64.
GOLUBKOV., V. Vq YEGOROV~ V. A.; YERSHOV., V. G. (Moscow)
"Investigation of flight trajectories with low thrust"
report presented at the 2nd AllUnion Congress on Theoretical
and Applied
Mechanics, Moscow., 29 Jan  5 Fab 19649
S/0293/64/002/003/0360/0391
NR: A?4041562
AUTHOR: Beletskiy. V& V.; Yegorov, V. A.
TITLE: Interplanetary flights wish constantpower engines
!SOURCE: Kosmicheskiye issledovaniya, v. 2, no. 3, 1964, 360391
'!TOPIC TAGS. interplanetary flight, space flight, space flight
trajectory, constant power flight, interstellar reaction vehicle
.!;'ABSTRACT: The problem of interplanetary flight,(between the
grivi
itational fields of planets) of a reaction vehicle withan ion or
!plasmaengine is studied, assuming that the'power input for
generating
ithe jet reaction is constant. A method for linearizing the
equationsi
~of motion relative to some suitable known trajectory is used in the
This method is called the "method of transporting
inVestigation4,
trajectories" and was initiated by'T. H. Eneyev. A"transporting
;Coordinate system" moving translationally along the transporting
1: trajectory is used$, thus taking hecount,Ln the first approxima'
Ition, of solar Sravitaition* Plights..with optimum control of the
Card
ACCESSION NR: AP4041562
reactive acceleration and flights with,a constant pcceleration
vector's
changing its direction in space by a sinRle jump are discussed. The
speed and high accuracy achieved in computing a large class of
interesting tra~cczories are the advantages of this
method. The proposed,.method can be used for any type of control of i
reactive acceleration. The authors express a sincere gratitude to
D.'Ye. Okhotaimskiy and T. It. Eneyev for their assistance and N. D.
Hyshetskiy, N. A. Malinina,*,hnd Ye. A. Sidorova for carrying
out*the,
computg.tiono, Orlg,, art. hos, '11 figures, 81 formulas, and I
table,,*
ASSOCIATION*. none
ENCL: 00
iJ5UbKLTTEDz 20Feb64 A TD PRESS 1 3052
NO REY SOVt 003 OTUERt 001 1
SUB CODE:. 5V, PR
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'!!ACCESSION NRI AP4041563..,. S/0293/64/0021003/03i2/0407,1
'AOTHORI Belsiskiy~ A
Ve Vo;.Xegor 'V.
'TITLE: Acceleration of a'space vehicle in.the gravitational field
of a planet
SOURCHt KosmicheskLye; 1saVodoviniya; V, .2, io. 3, 1964, 392407
TOPIC TAGSt space vehicle.'space vehicle.trajoeto yq reactive
P T
accelerailons acceior&tLon control, c.onstant, tangcnt'ial
acceleration.
ABSTRAM Trajectoftes of ii'spaci'.vehicle moving with a'small re
I active acceleration wLthLn.the gravitational. field of a planet are
IdLecuseedo An * approximate a o'Lu ti dn'is presented for the
variational
problem of optimum control' of'..,the reactive accoliratLon to
obtain
optimum traJectories after'liftoff'until the parabolic speqd is
reachad*
Some of these trajectories are L'I sttenfton',beLhg'~
on"alyzed, with.aptc a
aid to thos 19 CIO9G.T. to the optimal trajectories* Formula@ arb
given
p
_
foi calculating thw, paraU4tAirs,4of*',ths, trajectory at the and
of Its
acceleration sectLons .,RssuIts'of'thig c9uputation of sous
trajectorLos'
with constant taigentLa"Vrea'iti4ei acceleration up to~tha p6Lnt at
1/2'
i C~rd
N NRI AP404150.3. 
:i which the parabolic, sp4d .4s' ached 'argi
presented@., "
The aetbo
lexpress their gratituda, to 0 8, Ry.xhinq,
for'carrying
out the* pro
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grammLng and nuu*rLcal(1calcu0tLanm;.ou
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ACbESSION NR: AP4041564 S'/'O'"293/'64/002/003/0408/0413.
AUTHOR:
TITLEs Trajectories of cosmic flights with a constant vector of
reactive acceleration
SOURCE: Kosmicheskiya iss.ledovaniyai v. 2, %~o. 3, 1964, 408413
TOPIC TAGS: cosmic flightp cosmictrajectoryp cosmic fligh.t tra
jectory, Newtonian gravitational field, planstrajactory, three
dimeniional trajectory
ABSTRACT: The possible trajectories of flight.of a cosmic vehicle'
within a Newtonian gravitational field with a single'attracting
center are discussed. The vehicle.is said'to be propelled by 'an ion.,
plasma, or similar engine ensuring thrust for a long period: The' I
vector of reactive acceleration is assumed to be constant. Through
the integration of equations of motion the problem is completely
reduced to quadratures, and parametric equations of a trajectory are!
derived. The plane trajectories obtained are of four kindst l)Un,
bounded., selfcrosaing, notenclosing the center of gravitation;
Card' 1 1/2
ME M__
ACCH9SION Ni: AP4041'564*'
2) unbounded, selfcrossing, enclosing the center of gravitation;
3) unbounded, not selfcrossing; and 4) bounded trajectories* Three
dimensional trajectories can be classified in the same ways, Ori
art.. has3 2 figures And 14 formula&.
ASSOCIATION: none
SUBHITTEDt l4Fob64 ATD PRZSSS,~ 3049 ENCLt 00
SUB COM SV No' REP SOVI 000 OTHERI '001
BELETSKIY, Vladimir Vasillyevich; ABASHEVA, D.A.,,
red.
(Motion of an artificial satellite relative to the
cantor
of'M'Age] Dvizhenie iskusstvennogo sputnika,
otnositallno
I tsentra mass. Moskvap Nauka,_196~. 416 p. (MIRA
19il)
tcoe'd
ACC NRa AN60122.00 Hionograph
''Beletskiy,, Vladizrdr Vasillyevich
~bvement of an artivicial satellite relative to its center of mms
(Dvizheniye
iskusstvennogo sputnika. otnositell nogo tsentra mass Moscow,
Izdvo "Ka%W',
65. 0416 p. illus.., biblio.
1,000 copies printed.
Series note: Mkhanika kosadcheskogo poleta
...TOPIC TAGS: artificial Earth satellite, scientific satellitep
satellite motion,,
ortellite navigation.. Earth satellite orbit., elliptic
orbite4Vatorial orbit,,
orbit perturbation., orbital aircraft,, artificial satellite
PURPOSE AID) CCM=: After the launching of the first artificial
satellite
and consequent success in the conquest of space, the interest
rose sharply to
other prdblems connected with ftuther space evlorations, in
particular,, the
important problems are those dealingwith the motion theory of
artificial
satellites. This book deals vith one pert of the cosmic Mght
dynamics  the
motion of artificial satellite relative to its center Of mass.
The problems
discussed in this book axe limited to the dynamics of solids,
This book is
based on other vowlis published cc presented by the author at the
m*echanical and
mathematical faculty of Ibscow UhIversity, The results obtained.
by so= other
authors ere also used in this book,
Tipc,629.195.1
ACC NRt A~fO12200
TABIE CF CWBWTS (abridged):
roraword  8
Introduction 9
Cho 1, AraLlysis of force moments affecting the satellite 17
Ch. II. Stabilization arA libration motion of satel1ite in. Newton
field of
force  58
Ch. III, Effects of additional factors on the satellite stabi3ization
and
libration  122,
Ch. 1V. Interrelation of, forword and rotation motion of solids in a
Newton
field offorce  145
Ch. V. Satellite's ratatign motion and equations of targent elements 
175
Ch. VI, Effects of gravitation pertitbationB on rotation motion of a
satellite
191
Ch. VII. Effects of aerodynamic perturbations on rotation motion  229
Ch. VIII. Analvois of secular perturbations together vith the effeats of
'gravitational and aerodynamic moments and arbit, evolution  251
Q. IX. Effects of a magnetic field arA force moments of light pressure
on the
satell.ite's ratation and orientation  290
Cho X. The motim of some launched artificial oetellites aroxmd their
center
0
Maso  317
~L 53
Ch. X1. Use of an Earthcriented satelUte for expb=tion of the Sun
rd
ACC NRt AIAS01 22W
;,pp. I. Motion of a so2id. eround a fixed point in Newton field of
force 379
App. II. Or'bits of an equatorial artificial sateLlite  400
Bibliograj*W 411
SUB CODE SIMM DATE; t 090ct65/ CRIG MW% 073/
Card 3A
ULYUYEV, D.I., inzh.; BOLOTIN, V.I.,, insh., retaenaent;.JZj&jWWj,
a&&, inzh., retsenzent; SERGEYEVA, A.I., Inih., red.;
KHITROVA, N.A., tekhn. red.
I
(Handbook for the track maintenance worker] Posobis putevoma
rabochemu. Moskva,Transzheldorizdat, 1963. 322 p.
(MIRA 16:8)
(RailroadsTraft)
(RailroadaEquipment and aupplies)
DANILOV, Dmitriy Ivanovich, inzh.; PET.EMKTY
VsAyDjQdjUj%dimjrpjjO,
inzb.; GORYANSKIY., Yu.V., kand. tsk~n. nauk,,
retsenzent;
ORALOV, V.A.,_insh., retsenzent; YFMROV, SiLyInsh.,
nauchny7
red.; SOSIPATROV, O.A., red.; CHISTYAKOVA, R.K.,
tekhn. red.
[Trailer and container vessels) Treilernye i
konteine*ye suda.
Leningrad, Sudpromgiz, 1963. 235 P. (MIRIL 16:5)
(Ferries) (Unitized cargo systems)
'rIe4
USW/Gears~ Design. Jun 1947
Mathematics
"Certain Problems in the Manufacture of the
Crank
gear," V. Ya. Beletskly, 5 pp
"'Vestnik Itzhencrov i Tekhuikov" No6
Mathematical treatment, with diagrams of the
follov
Ing problems: 1) Designing a crankgear
according
to a given motion of the slide and. a
coefficient
of the change of speed in that motion. 2)
Designing
a crankgear in vhich the angle of
transmission of
preasure at the time of operation, vith a
given
motion of the slide, is not less in advance
of the
..61yea quantity...,.,3) Designing a
crankgear with given,:,,,
ior66
cmtd) jul 1941
Mathematics
=tion of the slide, coefficient of change of
sp6ed,
and a minimum angle of pressure transmission.
IOT66'
y
I V. a.
teletskiy) V. Y&.  "MovOment of grain in a horizontal sowing
bolter", Trudy Inta (Odes.
int inzhenerov mukomol. promsti i elevator. khozva im.
Stalim), Vol. 111 1948,
P. 5350,
SO: U3042, March 11, 1953, (letopis Inykh Statey, No. 10, 1949)
BMISKIYO V. Y&.
Beletskiyj V. Ya.  "Kinatostatics of selfbalancing sowing",
Trudy Inta (Odes. int
inzhencrov mukomol. promsti elevator. khozva, im. Stalin4p
Vol II, 19ap P. 88103
SO: U3042, 11 March 1953, (letopis Inykh Statey, No. 10,
1949):
BELETSKIY, V. Ya.
Beletskiy, V. Ya.  "The kinetostatics of selfbalancing
sifters," Trudy Vsesoyuz.
nauch.issled.inta zerna i produktov ego pererabotki,
Issue 16, 190, p. 607h.
SO: U4110, 17 July 53, (Letopis tZhurnal Inylch Statey,
No. 19, 1949).
BmsrsKIY, v. *.
K raschetu potrebnoi moshchosti kolebatelinykh
mekhanizmov. (Vestn
Mash. 1951, no. 3, p. 1317)
Includes bibliography.
Calculations of the required power of oscilatirg
mechanisms.
DLC: TN4V4
SO: Manufacturing and Mechanical Engineering in the
Soviet Union, Library
of Congress, 1953.
1.
B=MKH, V.Ya.
2.
USSR (600)
4.
Flywheels ,
7. Calculating flywheel masses of oscillating mechanism* with
s,,all amplitudes,
Sellkhozmashina no. 4, 1953.
9. Monthly List of Russian Accessions, Library of Congress, APEIL
1953, Uncl.
**u Ruzs~i;6~, Tmdl OdessIL
to, 1w, Rev.
phas a
formula was obtalued fat
codculAtiag the. osi*t1gi of th'. aeoced.
Cal I~cxfilq of the cam, A
number of *iiiubt. cises wtt

7 7
iyi z6r~l' us
Reffroov.
1245826344
Translation from: Refertivnyy zhurnal, Mekbanika, 1958, Nr
6, p 9 (USSR)
AUTHOR: B e I e t s k LY__V__Ya_
TITLE: Designing Slidingblock (Slidercrank) Linkages in
Accordance
With Given Laws of Motion for the Driving and Driven Links
(Proyektirovaniye krivoshipnoshatunnogo mekhanizma po
zadannym zakonam dvizheniya vedushchego i rabochego
zvenlyev)
PERIODICAL: Tr. Odessk. tekhnol. inta, 1957, pp 5767
ABSTRACT Bibliographic entry. See RzhMekh, 1958, Nr 6,
abstract 6343.
1. Mechanical drivesDesign
Card I j I
1245866343
Translation from: Referativnyy zhurnal, Mekhanika. 1958, Nr 6, p 9
(USSR)
AUTHOR: Beletskiy, V. Ya.
TITLE: Rendering More Precise the Dimensions of Plane Mechanisms
With Lower (Closed) Pairs Which Reproduce a Given Law of
Motion (Utochneniye razmerov ploskikh mekhanizmov s nizshi
mi parami, vosproizvodyashchikh zadannyy zakon dvizheniya)
PERIODICAL. Tr. Odessk. tekhnol. inta, 1957, Nr 8, pp 3747
ABSTRACT: A solution. is examined for the problem of designing a
hinged
four bar linkage mechanism intended to achieve a desired re
lationship between the movements of the driving link and the
driven (operating) link. In order to calculate three and four para
meters, multiple interpolation is performed with one coupling,
the interpolation factor of which equals three and four
respectively.
In the calculation of five parameters one parameter is determined
from the condition of zero deviation Afrorn the given function,
and the remaining four parameters are found from the minimum
meansquare value of the second derivative of a function which
expresses approximately the deviation A. This method for calcula
Card 1/2 ting five parameters is erroneous, because it does not
assure at
1245866343
Rendering More Precise the Dimensions of Plane Mechanisms (cont.
even a single point a zero value of the first derivative of the
deviation A; in
other words, when this method is used, the direction of the
tangents to the
given function does not coincide with that of the tangents to
the approximate
function.
N. 1. Levitskiy
1. MechanicsTheory
Card 2/2
1245866345
Translation from: Referativnyy zhurnal, Mekhanika, 1958, Nr
6, p 9 (USSR)
AUTHOR: Beletskiy, V. Ya.

TITLE: On the Designing of Plane Mechanisms With Lower
(Closed)
Pairs Which Reproduce Desired Trajectories (K
proyektirovani
yu ploskikh mekhanizmov s nizshimi parami, vosproizvodyash
chikh zadannyye trayektorii)
PERIODICAL: Tr. Odessk. tekhnol. inta, 1957, Nr 8, pp,
4956
ABSTRACT: A solution is given for the problem of
calculating three and
four parameters for a hinged fourbarlin~ap mechanism and
for a slidingblock (slidercrank) mechanism to satisfy the
con
dition of the minimum meansquare deviation oi the
connecting
rod curve from a desired trajectory. This method differs
from
the wellknown method with respect to the function which
charac
terizes the deviation from the desired trajectory.
1. Mechanical drivesDesign N. I. Levitskiy
Card 1/1
BL.LETSKIYS V. Ya.
RARER, G.O.AZAZJUOJATat;VORONKOV, P.I.;IZHIWV, P.G.;IIZTAI)ZIO, A.M.;
DOMMOVSKIT, G.D.*,ZOIDPARET, S.K.;MVCHENKO, I.K.;PIA.TONOV, P.N.;
PAY1,113KO6 A.V,;UGOLIK, N.F,
V. IA. Girahson. Muk.elev. prom, 23 no.4:23 Ap 157. (KLRL 10:5)
(Girsheon, Vasilii IAkovlevich, 18801957)
MEMSKIT &~;i doktor takhn. nauk, Prof.
designing crankgoare with given coefficient of
reverserunning
speed change and limit transmission angle. Izv. vys.
ucheb. sav.;
mashinostr. no.3/4:38 158. (MIRA 1225)
1.Odesskiy takhnologicheskiy institut iment I.V.
Stalina.
(Cranks and crankshafts)
B313TSKIY .0 doktor tekhn.nauk, prof.
Synthesis of crankshaft mechanisms at
approximately steady
speed of the driver. I%7.vys.uchob.zav.;
mashinostr. no.6:
1o14 '58. (MIRA 12:8)
1. Odeaskiy takhnoloelchaskly Institut ims
I.V.Stalinae
(Cranks and crankshafts)
25 (1) BOV/145587/82/24
AUTHOR: Beletgjay..,~Xa. Professor, Doctor of Technical
'M6r6ne es
TITLE: Estimation of Five Parameters for CrankConnecting
Rod Transfer Mechanisms
PERIODICAL: Izvestiya vyeshikh uchbenykh zavedeniy  Mashino
stroyeniye, 19589 Nr 78, pp 1116 (USSR)
ABSTRACT: The problem of estimation of transfer mechanism five
parameters has been analyzed by Professor, Doctor of
Technical Sciences, X.I. Levitekiyj in his work "Do
i ding Mechahismswith Lower Pairs", ASUSSR, 1950p
and by Docent, Candidate of Technical Scienceep
P. Novodvorskiy'*in his work "One Method of Syn
th;6is of Mechanisms". Proceedings of the Seminar'on
Theory of Machines and Mechanisms, Volume XI, Issue
42, AS USSR, 1951, fZ7. The present article deals
with the estimation of five parameters by applying
the same method as it was used by the author when
Card 1/3 estimating three and four parameters. Reference ZJ7
SOV/145587/82/24
Estimation of Five Parameters for CrankConnecting Rod
Transfer
Mechanisms
"Designing CrankConnecting Rod Mechanism on the Ba
sis of Given Law of Driving and Working Link Move
ment". Proceedings of the Odessa Technological Insti
tute imeni I.V. Stalin, Volume VIII, 1957. In Fig 1,
the author gives a diagram of transfer mechanism and
denotes the sought for parameters by r, 1, a, Xo and
C~k 00 The final values of parameters are determined
P1
by the following expressions: r =  i
sin M 0
1 = V(X 0+Xs)2 2r(X0+Xs)Cos(,D(0 +o~s)+2arsin(cKo+%)+r2+a2
a = P3 _p0 L; X0 = Po; tgoto = Pl, where X s is the
P2 P2 P2
relative value of the slide initial displacement;(Xs
angle of the crank initial turn; r  relative length
Card 2/3 of the crank, a  relative value of displac t. The
7
SOV/145587/82/24
Estimation of Five ParameterBfor CrankConnecting Rod
Transfer
Mechanisms
interrelations between the coefficients po9 P19 P2 and
P3 are expressed by the functions: p0 = Xo; p, = rein
0(0; P2 = rcoscxo; P3 = r(acosw_0+Xosino(o). There are 1
figure and 4 references, 3 of which are Soviet and 1
German.
ASSOCIATION: Odesskiy tekhnologicheskiy institut imeni I.V.
Stalina
(Odessa Technological Institute imeni I,V. Stalin)
SUBMITTED: November 19, 1958.
Card 3/3
SOV/145587/82/24
Estimation of Five Parameters~for CrankConnecting Rod Transfer
Mechanisms
interrelations between the coefficients po9 P19 P2 and
P3 are expressed by the functions: p0 = Xo; p, = rsin
0%; P2 = rco'so(0;P3 = r(acostebix0sinoto). There are 1
figure and 4 references, 3 of which are Soviet and 1
German.
ASSOCIATION: Odesskiy tekhnologicheskiy institut imeni I.V.
Stalina
(Odessa Technological Institute imeni I.V. Stalin)
SUBMITTED: November 19, 1958.
Card 3/3
BILETSKIT, T.Ta., prof., doktor tekhn.nank
Calculating five parameters of hinged fourbar
tranonission
linViages. Izv.vys.ucheb.zav.; mashinostr. no.6:39
159.
(MIRA 13:5)
1. Odesakly tekhnologichookly Institut.
(Links and link motion)
BEIMSKIY, V.Ya.I doktor tekhn.naukp prof. ,
04culating six parameters of threedimensional cra;kgearo
having
an appro3dmately uniform motion. Izv.vyv.ucheb,zav*
mashinootr.
no.1:2023 160o WRL 1415)
1. Odesskiy takhno.logicheakiy institut imeni I.V.Stalina.
(Cranks and arankshafte)
BILITSKIY. V.Ya.. doktor tekhn.nauk prof.
, ~ :.  ,
Determining the reduced dynamic coefficient of the
external
friction of bulk mixtures. Trukt.i s el'khos;maeb
no1:32
ja 6o. (KM 13.4)
(Friction)
30257
B/145/60/000/009/003/017
D221/D304
AUTHOR: Beletskiyq V.Ya.9 Doctor of Technical Sciencesq
lrafe_~~~
TITLE: The analytical method of designing fourbar mechan1sms
PERIODICAL: Izvestiya vyeshikh uchebnykh zavedeniy. Mashino
stroyeniyeg no* 99 1960, 29  33
TEXT: The author exposes the analytical solution for
determining
a family of fourbar mechanisms which ensures that the limit
angle
of pressure should be below a certain value. It is based on the
synthesis of fourbar linkages advanced by the author
previously.
The obtained equations permit computation of mechanism
parameters,
and thus relieve the designer of selecling linkages by trial
me
thods. The extreme two positions of the fourbar mechanism
(AB1 C 1D
and AB 2C2D) are shown in Fig. 1. The instantaneous ratio of
driven
and driving linkages is i = A'2 Applying the theorem of sines
to
Card 110 2 ~B2'
30252
S/145/60/'000/009/003/017
The analytical method of designing ... D221/D304
the triangle DC2B2t the author deduces equations for lengths of
cranks, CD and AB9 as well as conrod BC. Repeating the same for
the other position of the mechanism and designating by 1 1 the
in
stantaneous ratio of speeds of driven and driving linkages,
author
deduces further equations for the above arms.After mathematical
elaboration,,
12 = 0,5 (k + 1)
is obtainedp where
(14)
sin 70 11211 Yl
2 .  (15)
k g:,nW1 
sin T.
which is a known quantity. Substituting 'Y 29 y. and '2 in the
ex
pres'sion of linkageeg it is possible to compute the required
lengths
of linkages. These calculations are insufficient for the
four7bar
mechanisms. Thereforep in practice It is simpler to assume
some va.
lue of V1 in the limits of 0