JPRS ID: 9740 USSR REPORT ENGINEERING AND EQUIPMENT

• Document Type: CREST
• Collection: JPRS
• Document Number (FOIA) /ESDN (CREST): CIA-RDP82-00850R000400010036-7
• Release Decision: RIF
• Original Classification: U
• Document Page Count: 103
• Document Creation Date: November 1, 2016
• Sequence Number: 36
• Content Type: REPORTS

APPROVED FOR RELEASE: 2047102108: CIA-RDP82-00850R000400010036-7
FOR OI~FICIAI, USE UNLI'
JPRS L/9740 ~
- 19 May 1981
~ USSR Re ort
~
, .
ENGINEERING AND EQUIPMENT
_ (FOUO 3/81)
FB~$ FOREIGN BROADCAST INFORMATION SEI~VICE
- FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2047102108: CIA-RDP82-00850R000400010036-7
NOTE
JPRS publications contain information prima.ril'y from foreign
newspapers, periodicals and books, but also from news agency
transmissions and broadcasts. Materials froni foreign-language
sources are translated; those from Fnglish-language sources
are transcribed or reprinted, with the original phrasing and
other characteristics retained.
Headlines, editorial reports, and material enclosed in brackets
are supplied by JPRS. Processing indicators such as [Text)
or [Excerpt] in the first line of each item, or following the
last line of a brief, indicate how the original informa.tion was
processed. Where no processing indicator is given, the infor~
mation was summarized or extracted.
Unfamiliar names rendered phonetically or transliterated are
enclosed in parentheses. Words or names preceded by a ques-
tion mark and enclosed in parentheses were not clear in the
original but have been supplied as appropriate in context.
Other unattributed parenthetical notes within the body of an
item originate with the source. Times within items are as
given by source.
The conten*_~ of this publication in no way renresent the pol.i-
cies, views or at.titudes of the U.S. Government.
COPYRIGHT LAWS AND REGULATIONS GOVERNING OW~IERSHIP OF
MATERIALS REPRODUCED HEREIN REQUIRE THAT DISSEMINATION
- OF THIS PUBLICATION BE RESTRICTED FOR QFFICIAL 'JSE ONI,Y.
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL USE ONLY
- ~ ~TPRS L/9740
19 May 1981
USS R REPORT
ENGINEERING AND EQUIPMENT
(FOUO 3/81)
CONTENTS
~1EROP?PUTICl~L AND SPACE
rlight Tests of Helicopters 1
Aeroelastic Stability of Flight Vehlcles 7
~limination o~ Vibrations in Aviation Piping 11
SURF~'iCE iRAIvSPO~T~"�_TIOPi
Theory of ~lectrody-namic Levitation, i~lain Results and Further
Problems 1~~
Optimization of t�Iagnetic Suspension and Analysi~ of Vehicle
Dynar.?ics 41
Electrcmagnet Control in the Suspension Systems of High-Speed
Transport 51
i~Iathematical ~iodel of a?tail Car With Electromagnetic Suspension.. "58
~ynanics o~ Transport F.olling Stock on :lagnetic Suspension........ 66
I�tAR.Ii~T'~ AITD SHIPE;;ILDIi~G
Vihration :bsorption on Ships 77
Automated Ship Power Plants 79
PdON ~NCL :A? L~1EP.~Y
Iiodernization of Turbogenerators 84
- -a- [III-L~SSR- 21.FS&TFOUOJ
FOR OFFICIAL US~ ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
I
FOR OFFQCIAL USE ONLY 4
NAVIGATION AND GUIDANCE SYSTEMS ~
Gyro Stabilizers of Inertial Control Systems 88
Astroorientation Methods and Instrumentation Examined 96
- b -
FOR OFFICIAL USE ONLY -
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-00850R040440010036-7
FOR OF~ICIAL USE ONLY
AERONAUTICAL AND SPACE
- FLIGHT TESTS OF IiELICOPTERS
- P9oscow LETNYYE ISPYTANIYA VERTOLETOV in R~ssian 1980 (signed to press 28 Auq 80)
pp 2-3, 396-399
~ [~1nno~ation, foreti;ord and table of contents from book "Flight Tests of Helicoptersh,
~ by Aleksandr Ivano~~ich Akimov, Leonid Mikhaylovich Berestov and Rc~stislav
= Aleksandrovich D4ikr?eyev, Izdatel'stvo "Mashinostroyeniye", 2,700 copies, 400 pages]
[Text] The problems of flight tests and their postulation are considered in the
book and the me~thods of flight tests to determirie and reduce 'to controlled cor.di-
tions the flight data of helicopters with various types of engines are justi.fied.
I~lethods of determining characteristics of stability and controllability during
flight are outlined. 'I'he theoretical bases and methods of flight-strength tests
are considered. Attention is dev~ted to processing and analysis of the test
results.
The book is intended for engineering and technical personnel engaged in development,
tests, finishing and operation of helicopter technology.
Foreword
?lmong all t}~~es of experimental investigations of aviation equipment~ fliqht tests
~ reveal most fully the principles of complex processes of the interac',:ion of the
flight vehicle (LA), its units and systems, the medium in which t:ie LA functions
and of the person controlling this vehicle. Therefore, flight tests are not only
important in themselves as a means of ch~cking and developing aviation equipment,
without which introduction into operation is impossible, but they are a~so an ex-
ceptionally effective means which contributes to proper.formation of the engineer-
ing ideology of snecialists involved in both development of LA and their operation
~nd in scientific research.
The sections of the book devoted to flight tests and aerodynamic investigations
(Chantesr 1-7) were written by A. I. Akimov, those on flight dynamics (Chapters ~
8-14) were written by L. M. Berestov and those on strength (Chapters 13-18) were
. written by R, A. P-likheyev. -
The authors feel it their duty to note especially the important role of V. V. f
- Vinitskiy, Yu. A. Garnayev, A. A. Dokuchayev, Ye. F. Milyuticliev, A. I. Mukhi and
others in the inv~stigations. ~
1
FOR OFFICIAL USE ONLY _
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
� F(~dt OFFICIAL USE ONI.Y
_ Th~ authors express gratitude to S. B. Bren, who was responsible for general scien-
tific supervision of the investigations and who rendered important assistance in ~
the w~rk on the book, and to reviewer P_. V. Nekrasov for a ntunber ~f waluable
cofiments. -
'
Contents Paqe =
Fo reero r~3 3 -
Introduction 4
Section 1. Determination of the Flight Data of Helicopters ~
Chapter 1. ~lethod of Determining Fliqht Altitude and Speed 9
1.1. Physical .parameters of a?= g
1.2. Variation of the parameters of air in a real atmosphere. Condi- -
tional atmospheres 13
1.3. Principles of ineasuring flight altitude and speed 16
1.4. Relationship between measured and actual values of fli.ght speed
and altitude 20
1.5. Method of determining instrument corrsctions and corrections
for laq of altitude and speed indicaf:ors ~1
1.6, t~lethods of determining aerodynamic corrections 23
1.7. Effect of flight mode on aerodynamic corrections of speed
indicator 29
Chapter 2. Eff.ect of Flight Conditions on Power of Helicopter Turboprop
and Piston Engines 30
2.1. General propositions 30
_ 2.2, Similarity of operating modes of two-stage turboprop engine 30
2.3. Adjusting the power of a two-stage turboprop engine to the given -
flight c~~nditions by the differential corrections method 37 -
2.4. Adjustin~ the power of piston engines to given conditions 47 '
Chapter 3. Determinir~g the Thrust Characteristics and the Hover Ceiling
of Helicopters 52
3,1. Flight characteristics of helicopters in the hover mode 52
3.z. Similarity of hover modes of turboprop helicopters 54
3.3. biethod of determining thrust characteristics of helicopters in
the hover mode 56
3.4. Determining the thrust characteristics of helicop~ers using
ground ties 58
- 3.5. f~ethod of determining the hover ceiling 59 ~
3.6. Adjustirtg the thrust characteristics and hover ceiling of
turbopro1~ helicor~ters to given conditions using dimension.al analysis 62
- 3.7. Aajusting t[ie thrust characteristics and ho~~er ceiling or
; turboprop helicopters to given conditions usinq dimEnsional analysis _
' by the differential correccion method 64
3.8. Ilormalizing the thrust characteristics and the hover ceiling .
of helicopters with piston engines to given conditions by the ~
differential correction method 70
2
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
I
FOR OFFECIAL USE ONLY
3.9. Determi.niny the polar of the helicopter and main rotor in the
hover mode 74
3.10. The effect of some atmospheric factors and design measures on
the thrust and hover ceiling of the helicopter 77
Chapter 4. Determining the Characteristics of Rate of Climb and Practical -
Flight Ceiling of Helicopters 81
4.1. General pro~ositions gl -
~ 4.2. Similarit,y of steady-state straight flight modes of a turboprop
helicopter 83
4,3. Determining the most advantageous mode of gaining altitude 85
4,4. Determining the maximum rate of climb and the practical flight
ceiling of the helicopter 88
4.5. Determining the time required to gain altitude 90
4.6. Adjusting the maximum rate of climb of turboprop helicopters
to given flight conditions 90
4,7. Adjusti:ig tne naximum rate of climb of helicopters with piston
engines to given conditions~ g7
Chapter S. Determining the High-Speed Flight Characteristics of Helicopters 99
5.1. Determining the mini.mum horizontal flight speed 99
5.2. Determining the maximucs horizontal flight speed 101
5.3. Adjusting the rainimum and maximum horizontal flight speeds of
helicopters to given conditions 103
5.4. Determining and ad~usting the deccent characteristics in
windmilling mode of the ma,in rotor to given conditions 105
- 5.5. DeterMining the hicrh-speed flight characteristics of a multi-
engine helicopter upon engine failure 109
Chanter 6. Deterr~ining the Characteristics of Flight Range and Duration
of H?J.icopters 110
6.1. Characteristics of fliqht range and duration of the helicopter 110 -
6.2. Generalized characteristics of fuel consumption of turboprop
helicopters in horizontal flight 113
6,3. Method of determining the generalized characteristics of fuel
consu.^~;~tion of turboprop helicopters in horizontal flight 114
6,4. Deternlining hourly fuel constunption of turboprop helicopters
under qiven flight cenditions by functions Qpr = f(mprr Vpriv.
n~,r) or q= f(mpr, Vpriv and npr) 119
- 6.5. Determining the fuel constunption of turboprop helicopters
during flight for given conditions by the altitude selection
method 122
' 6.6. riethod of determining and reducing tn given conditions the
_ characteristics of horizontal flight range and duration of
piston-engine helicopters 124
6.7. Determining fuel consumpts.on under given conditions when
gaininc7 altitude and during descent and during enaine opera-
tion on the ground 12g
Chapter 7. Determining the Take~ff and Landing Characteristics of
_ Helicopters 130
3
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
I
- FOR OFFTCIAL USE ONLY
7.1. The takeoff and landing properties of helicopters during
normal operation of the power plant 130 ;
7.2. The takeoff and landi.ng characteristics of helicopters during
engine failure 133 ;
7.3. Problems solved in determination of takeoff and landing char- -
acte.ristics of helicopters during flight tests 143 !
_ 7.4. :~tethods of ineasuring the parameters of takeoff and landing ,
, trajectories of helicopters 144 ;
� 7.5. 5electing efficient takeoff characteristics of helicopters 151 ,
7.6. Developing the piloting technique and the method of perform-
ing takeoffs 155
_ 7.7. Developing piloting technique and methods of making landings 157
7.8. A4ethod of determining the landing characteristics of the
heliconter during engine failure and determining the zones
of dangerous altitude and flight speed combinations H-V 160
7.9. Method of determining the characteristics of interrupted and
pro~anged takeoffs 167
7.10. Method of reducing the flight characteristics of r.elicopters
to given flight conditions 175
7.11. ~;ormalizing of takeoff distance of the helicopter with respect
_ to wind speed 180
7.12. Comments on the method of ad~usting the takeoff distance of
the helicopter during a running takeoff 181
Section 2. Determination of the C`haracteristics of Helicopter Stability
and Controllability During Flight
Ctiapter 8. Matheinatical Model of a Helicopter 184
8.1. Equations of motion 184
8.2. Transfer functions ~ 188
8,3. Frequency characteristics 189
8�4. Pulse transfer function 190
_ 8.5. Characteristics of the linear mathematical model of the
helicopter 191
Chapter 9. Determination of the Balan~e Characteristics During Flight
and Inve~tigation of Free riotion 198 =
4.1. Balance characteristics 198
9.2. Balance characteristics ac:cording to flight speed 199
- 9.3. Balance characteristics according to slip 206 ,
~.4. The complex approach to determination of balance characteris-
tics by speed and slip 2~8
9.5. Balance characteristics by g-force 209
9.6. Investigation of the characteristics of �ree motion of a
helicopter during flight 210
Cha.pter 10. Identification o.f the Mathematical Model of the Helicopter 216
10.1. Determining the coefficients of linearized equations of motion 216
10.2. Determining the coefficients of equations of motion with large
variations of parameters ~ 242
Chapter 11. Evaluating the Controllability of a Helicopter Durinq Flight 244
- 4 ~
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
NOR OFFICIAL USE ONLY
9
11.1. The at-~oroach to evaluation of controllability 244 _
11.2. Flight simulation during investigation of controllability 245
11.3. Flight investigations of the controllability of an experi-
- mental heTicopter 252
Chapter 12. The ?lpproach to Investigati~n of Helicopter Maneuverability 255
12.1. The motion of a helicopter as a mass point 255
12.2. Restrictions placed on motion of a helicopter as a mass point 259 _
12.3. rharacteristics of standard maneuvers 262
Section 3. Flight and Strength Tests
- Chapter 13. Goals, Problems and TheoreticaZ Bases of Flight and Strength
Test~ 264
13.1. Prob~ems and goals of tests 264
13.2. Interaction of an elastic body and the medium 267
13.3. Characteristics of free and forced oscillations of a
helico~>ter 276
~.3.4. Similarity of modes and selecicion of them 281
13.5. Characteristics of loading helicopter units 286
Chapter 14. NIethod of Strain Measrrements During Flight and Strength Tests 291
14.1. Strair. measurements 293
14.2. Strain resistors 295
14.3. The measuring bridge 299
14.4. Oscillographs 304
14.5. Amplifier apparatus 306
- 14.6. Magnetographs 308 ~
14.7. Supplementary units 310
14.8. Special cases of strain measurements 312 1
14.9. Calibrations 316 '
_ 14.10. Preparation and conducting of ineasurements 318
14.11. MeasureMent errors 324
Chapter 15. Processinq. Statistical Methods 336
15.1. Means of processing oscillograms and maqnetic recordings 336
15.2. Software 340
15.3. Deter-~ination of damageability 341 -
15.4. The accuracy of evaluations 343
15.5. Other types of evaluations 348
15.6. Confidence ranges. Checking of hypotheses. Least squares
method 353
15.7. Determination of special characteristics 357
Chapter 16. Determining the Stresses, Forces and D4oments and Motions.
Other Measurements 362
16.1. i~eternination of stresses 362
1G.2. Determination of the elastic forces and moments in the cross
sectioiis and the responses of other units 366
1G.3. Measuring the total external loads and inertial forces 369 -
16.4. Determining the aerodynamic loads 372
5
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2047102108: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
16.5. Determining relative motions 375
16.6. Other ne3surements 3~~
Chapter 17. Investigating Helicopter Vibrations 379
17.1. Goals and methods of investigations 379
17.2. Measurement of vibrations 381
Chapter 18. Investigation of Autooscillations ~86
18.1. General data 386
18.2. t~tethod of testing 387
18.3. Approach of blad~s 390
18.4. Restrictions during Flight and strength tests 391
Blbliography 393
COPYRIGHT: Izciatel'stvo "Plashinostr~yeniye", 1980
[75-6521]
6521
CSO: 1861
6
FOR OFFICIAL iJSE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE aNLY
AERUELASTIC STABILITY C~F FLIGHT VEHICLES
Moscow AEROUPRUGAYA USTO~'CHIVOST' LETATEL'NYI~I APPARAT~V in Russian 19$0
(signed to press 30 Jul 80) pp 2-5, 230-231
[Annotation, foreword, introduction and table of contents from book "Aeroelastic
Stability of rlight Vehicles" by Aleksandr Ivanovich Smirnov, Izdate~'stvo
"riashinostroyeniye", 1220 copies, 232 pages]
[Text] The dynar,ic problems of aeroelasticity--classical f lutter of high- and
low-aspect wings (straight and swept), cf the tailplane, fuselage and of the en-
tire vehicle as a cahole and also flutter of thin panels and shells--are considered
in the book. Problems of stalling f lutter are analyzed.
_ The book is intended for engineering and technical ~ersonnel involved in problems
of aeroelasticity. _
Foreword ,
_ ~
This book is a logical continuation of one published previously by the author
[A. I. Smirnov, "Aerouprugost'. Chast` I. Staticheskiye zadachi aerouprugosti"
(Aeroelasticity. Part I. Static Problems of Aeroelasticity), Moscow Avidtion
Institute, 1971] and is devoted to the dynamic problems of aeroelasticity in ap-
plications to flight vehicles.
' A~reat deal of attention is devoted in the book to the physical pattern of the
_ phenomena under discussion and to postulation of the problem. However, unlike the
cited ref erence, main attention is turned to analys.'s and discussion of the results ,
of solving individual problems rather than to a description of different methods of
solving one or another problem. Each problem is usually formulated in the form and
is solved by the method which the author feels are most cunvenient for one reason
or another.
The limitations imposed by the framework of the book's volume and by the author's -
scienti=ic interests were naturally reflected both in the s~yle of the exposi-
_ tion and in selection of th e dynamic problems under discussion.
The autiior is grateful to the collective of the Department of the Moscow Aviation
Institute imeni S. Ordzhonikidze and its head Academician I. F. Obraztsov for -
advice and friendly as~istance in the book.
The author is also grateful to the collective of the Department of Design and
Strength of Flight Vehicles of the Moscow Institute of Civil Aviation Engineers,
who sent their comments on the nanuscript.
7
FOR OFFICIAL USE ONLY
~ ~
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
FOR UFFICIAL USE ONLY
Doctor of Tec~inical Sciences, Professor V. I. Protopopov performed important work
in review of the book, for which the author expresses his deep gratitude.
Inttoduction
Interest in dynamic problems has intensified appreciably during the past few years
- in all iields af technology and specifically in aviation. One should apparently ;
seek ~n er.planation to this on the one hand in the desire to develop optimum de-
signs and on the other hand to improve the mechanical properties of materials, to
increase the dimensions of flight vehicles and to intensify operating modes.
A modern flight vehicle is L~ually a combination of sufficiently flexible structur-
al components (wing, fuseluge and tailplane), the natural motions and deformations
of wl~ich determine to a significant degree their load during operatian. As a result
= it becomes necessary to take these motions into account, which determines the form-
ulation of the dynat~;ic problem.
Taking the dynamic effects int~ account appreciably complicates the equations of
elastic equilibrium ~f flight vehicle components due to th~ presence of yet another
independent variable--time t. Thus, even for the simplest one-dimer~sional struc-
tural component the differential equations of euilibrium will no longer be ordinary
equations, as in the case of the static problem, but equations in partial deriva- -
tives, the methods of solving which have still not been adequately worked out. For
problems of aeroelasticity, the latter circumstance is also comp:.icated by the fact
that the corresponding differential or integral operators will be non-self-adjoint
operators with complex eigenvalues, while the i.heory of these operators has been
rather poorly worked out.
~ystematic theoretical and experimental investigations of the dynamic problems and
s;~ecifically of the flutter probleM, were first begun in the Soviet Union in the _
early 1930s. Con ~ersion from the biplane to the monoplane aircraft had begun by
� this time ir~ the aviation of almost all countries, flight speeds increased appre- _
ciably and the num!~er of accidents for unexplained reasons increased significantly.
Investigations were carried out primarily by workers of TsAGI [Central Institute of
Aer.ohydrodynamics imer.i N. Ye. Zhukovskiy] and the main results w~re obtained by
this institute wh ich made it possible by the end of the 1930s to construct reliable
- flight vehicles safe from the viewpoint of loss of dynamic stability. _
T~i~ ma.in contribution to stud~~ of these problems was introduced at that time by
the papers of M. V. Keldysh, M. A. Lavrent'yev, A. I. Makarevskiy, A. I. Nekrasov,
L. t. Sedov, Ya. i~. Parkhomovskiy, L. S. Popov, Ye. P. Grossman and many others. _
A large part of the difficulties related to solution of dynamic problems G~as caused
k;y tre need to tal-e aerodynamic loads into account. These nonconservative condi-
tions are t~hat deteruiines the non-self-adjoint nat~ure of the boundary value
problem. Under these conditions the successful selection of the aerodynamic oper-
ator accuires e5per.ially important significance. The operator should meet two main -
requirements--the simplicity of the analytical structure and the adequacy of the
- real physical pattern of the effect of air flow on the supporting components of the
flight vehicle in the sense of objective reflection. Otherwise, the process of
8
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02/48: CIA-RDP82-00850R000400014436-7
FOR OFFICIAL USE ONLY
pro'~lem-solving may be rather complex and the results of calculating critical
parameters will be far from real values. ~
The aerodynamics of *ran~ient flo~~ are complex and are as yet a stiZl inadequately
developed section of fluid dynamics. The equations of motion of a viscous com-
- pressible fluid (the Navier-Stokes equation) are rather complex. Precise solu-
tions of. these eqi.iations were found only for special cases, of low interest to
aviation practice. In view of this, it becomes riecessary to si.mplify the initial
equations of motion. The simplifications reduce to adoption of a~ditional hy-
potheses w.ith resrect to the physical properties of floia, the range of velocities,
the nature of. perturbed motion, the configuration of the flight vehicle and so on.
As a result numerous and more limited ranges of transient flow theary arise which -
describe the behavior of an ideal compressible or noncompressible gas (devoid of
viscosity) at subsonic, near-sonic, supersonic or hypersonic speeds, the aerody--
- namics of narrow wings and low-aspect wings, the aerodynamics of quasi-steady flow, _
the aerodynamics of thin bodies and many others.
Additional t:ypotheses, significantly constricting the framework of the investiga-
tion, permit one in a numk~er of cases to find a comparatively simple expression for
the aerodynamic effect of transient flow on the components of the fligiZt vehicle
suitable for practical calculations. An example may be exter.sive use of the hy-
pothesis of quasi-steady flow or piston theory and its various modifications.
Probler?s of experimental invest:igation of the aeroelastic stability of flight ve-
hicles, which comprise an extensi.ve independent section, are not considered in the
book.
Contents Page
Foreword 3
Introduction 4
Chapter 1. Some Data on the F erodynamics of Transient Flow 6
1.1. General comments 6
1.2. Main ecuations. The Lagrange integral 6
1.3. The ve!ocity potential 8
1.~~. Tlie acceleration potential 10
1.5. Initial and l~oundary conditions 11
1.6. IncompressiblP flow 13
1.7. Compressible flow 36 .
l.b. Some other met;iods of calculating the aerodynamic character-
istics of airfoils 46
1.9. Complex representation of parameters in problems of aero-
dynamics and aeroelasticity S1
Chapter 2. Flutter 53
2.1. General comments 53
2.2. Main equations 56
2.3. Flexure-torsicn flutter of a wing section 62
2.4. Flexure-torsion flutter of a finite wing 78
9
FOR OFFtC[AL USE ONLY
i
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL USE ONLY ~
'
2.5. The criterion of stability of an elastic structure in gas flow 92 ;
2.6. Flutter of a swept wing 9a '
_ 2.7. Flutter of a low-aspect wing 102
2.8. Flutter of the tailplane 110 '
2.9. Flutter with one degree of freedom 120
2.10. Fluter oF a flightcraft without constraints 125 ;
2.11. Effect of air compressibility on the characteristics of flutter 134 ~
2.12. The role of structural damping in flutter problems 136 '
2.13. Effect of drag on wing flutter 142 '
2.14. Smooth solid of revolution with tail fins 146 '
2..15, flethods of improving the aeroelastic characteristics of flight
vehicles 149
Chapter 3. Panel Flutter 154 `
3.1. General comments 154
3.z. Flat plates 156 ~
3.3. Cylindrical panels 1~9 ~
3:4. Cylindrical shells 186
3.5. Effect of various parameters on the characteristics of flutter
of cylindrical panels and shells 19z
3.6. ~lonlinear problems 199 '
~ Chapt~r 4. Stall Flutter 211
4.1. General comments 211
4.2. The physical pattern of stall flutter 212
4.3. Criteria of analysis 214
Appendix. Adjoint and Non-Self-Adjoint Differential and Integral Operators
. and Boundary-Value Problems 219
Bibli.ography 225
COPYRIGHT: Izdatel'stvo "P~ashinostroyeniye", 1980
[76-6521)
6521
CSO: 1861
10
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
FLI~:INATIODI OF VIBRATIQNS IN AVIATION PIPING
Moscow liSTRANENIXA KOLEBAP7IY V P.VIATSIONNYKH TRUBOPROVODAKH in Russian 1980 (signed
to press 14 Dec 79) pp 2-4, 155-156
' [Annotation, forew-ord and table of contents from book "Elimination of Vibrati~ns in
, Aviation Piping" by Vladimir Pa~~lovich Shorin, Izdatel~stvo "i~iashinustroyeniye",
~ 1,070 copies, 16Q r~ages]
[Textj Probl~ns of designing acoustic dampers intended to eliminate vibrations
of the workir,g me~liuri in aviation piping systems are considered in the book. Me-
thods of calculating tl~e efficiency of dampers, of optimizing their characteristics -
and metr.ods of experimental investigations of dampers are outlined. The k~ook is
intended for engineers involved in design of hydraulic and fuel systems of flight
vehicles and engines, piping and automatic hydraulic systems of production units
and transport vehicles. It will also be useful to scientific workers, students and
graduate students of the corresponding specialties.
Foreword
Ar: increase of flight speed, load-carrying capacity and maneuverability of flight
vehicles is accompanied by a significant expansion of functional problems and by an
in~rease of the output of hydraulic and fuel systems. The specific parameters in-
crease and processes in automatic devices and hydraulic couplings are intensified
wlien the structur~s of the systems are complicated and when the number of units and ~
the len~th of pipi:~q are increased. At the same time, ever more rigid requirements
on reliability are placed on hydraulic and fuel systems throughout the service life
of the flight vehicle. In this regard the problem of prevention and elimination of
vibrations of the ~�~orking medium in piping systems becomes ever more timely.
Reducin~ the vibrational intensity of the working medium not only provides opera-
tional reliability of crucial assemblies of flight vehicles and engines, but is in
some cases a necessary condition of their functioning.
It has been established th~t the main type of failure of piping is vibrational
failure and one of the mair. sources of excitation of inechanical vibrations is pul-
_ sating flow of the working mediLUn. Reduction of the pressure fluctuation ampli-
tudes permits a reduution of strength reserves of piping and consequently a reduc-
tion of the mass of syste:r,s.
Flow fluctuation of. the working medium is one of the causes of seal failure of
' connections.
11
FOR OFFICIAL USE 6NLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2047102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
7'lie functional and parametric reliability of hydraulic systems is reduced to a sig-
nificant deqree duF~ to the effec~ of pulsation on the sensing elements and actuat-
ing member, of units. Variable pressure causes undamped vibrations of valves,
slide valves and servo pistons, which in turn leads to premature wear of them and
to the appearance of cold hardening and scoring on the working surfaces. The
interaction of periodic processes in piping with the working members of inechanisms
is one of the factors affecting their operating stability. Variable pressure in
piping is a source of servo cantrol unit errors and tracking system errors and is
the cause of disruption of their initial adj ustment.
Vibrational processes have a significant effect on the characteristics, efficiency _
and reliability o~ p~~s.
Ir~s tabili ty of the combustion process in liquid-fuel rocket engines at low and ~
intermzdiate frequencies is related to periodic processes in the piping of the
fue 1-feed system. Vibrational processes in fuel-feed systems of gas turbine en- ~
c~ines affect the ti~orking process in the combustion chamber, leading to deteriora-
ticn of engine economy.
The probl em of eliminating vibrations of the working medium is also timely for
~ other systEms, for example, for ship piping, the hydraulic systems of machine
tools, qround transport and power engineering units, the pipeline systems of the _
- oil and gas industry and heating and ventilatior. systems.
There are now sevaral directions in solving the problem of prevention and elimina-
tion of vibrations of tl:e working medium. One of. them--the us.e of special dampers
is reflected in the book. This method ~of elminating vibrations is more promising
and is gaining ever wider distribution in the systems of flight vehicles and
engines.
The author is grateful te Candidate of Technical Sciences A. G. Gimadiyev, Candi-
date of Technical Scierices L. I. Brudkov and Candidate of Technical Sciences V. I.
5anchugov, who participated in solution of some specific problems. The author is
al so grateful to P.octor of Technical Sciences B. F. Glikman for a number of useful
comments made in review of the manuscript.
Contents Page
Fo r eword 3
~iai r. Notations 5
Chapter l. Principles of Construction and Possible Structures of Vibra-
, tional Dampers of the Working rtedium ~
l.l. htethods of eliminating vibrations of the w~orking medium in
hydraulic circuits ~
1�'� Reactive dampers 10
1,3. Active dampers 16 '
Ct-eapter 2. Dynamic Characteristics of Hydraulic Circuits and Dampers 21
_ 2,1. Frequency characteris.*.ics of cylindrical piping 21
12
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
2.2. Frequency characteristics of components with concentrat~d
parameterr~ 26
2.3. Application of electric analogies to calculation of dampers 35
2.4. Calculating the characteristics of dampers 38
2.5. t~�~lathematical models of' boun~ary-value conditions 43
Chapter 3. Efficiency of Dampers
3.1. Criteria and methods of estimating the efficiency of dampers 47
3.2. Efficiency of the damper in the inlet section 49
3.3. Location of reactive dampers in piping circuit 54
3.4. Efficiency of damper in the output section 58
3.5. Comparing the methods of estimating the efficiency of dampers
in the output section 63
3.6. Efficiency of damper with effective wave in~pedance 66
3.7. Efficiency of resonance circuits 67
3.8. Efficiency of resonance circuit with semi-harmonic law of fluid
vibrati.ons 73
3.9. Efficiency of damper of arbitrary structure with single resonance
circuit 75
3.10. Using dampers to ensure stability of systems containing hydraulic
circuits in their structure 77
Chapter 4. Calculation and Design of Dampers 84
4.1. The simplest type of dampers 84
4.2. Low-frequency acoustic f ilters 88
4.3. Effective wave impedance dampers 97
4.4. Resonance circuit dampers 112
_ ~ 4.5. Low-frequency filters containing resonance circuits in their
structure 133
Appendix. Nomograms for Calculation of a Branched Cavity 146
Eibliograpry 152
COPYRIGHT: Izdatel'stvo "Mashinostroyeniye", 1980
[74-6521]
6521
CSO: 1861
13
~ FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
~
FOR OFFICIAL U~E ONLY
I
i-
SURFACE TRANSPORTATION
i
UDC 538.31.401.2 . ~
THEORY QF ELECT%JDYNAMIC LEVITATION, MAIN RESULTS AAID FURTHER PROBLEMS ~
Moscow IZVESTIYA AKADEMII NAUK SSSR: ENERGETIKA I~RANSPORT in Russian No 1, '
Jan-Feb 81 pp 72-91 , .
[Article by V. M. Kochetkov, K. I. Kim and I. I. Treshchev, Leningrad and Moscow]
[Text] The principles of electrodynamic suspension (EDP) are well known [1]. A
number of approximate and precise methods have now been worked out for calculating ~
the characteristics of suspension. The results of calculations presented in exist-
ing publications show in their entirety the achievable characteristics of IDP and ,
the advantages and disadvantages of this method compared to other noncontact
methods of suspension of transport vehicles. Specifically, the prospects for use
of IDP at high speeds (in the range ot 250-500 km/hr) has been determined.
The literature on EDP and related problems is very extensive and includes no less
_ than 250-300 titles. Attempts to systemize the results have been undertaken by
several authors; we note first the extensive technical survey [2] and a number of
paper.s of a survey nature devoted to individual problems of IDP [3, 4]. However,
the indicated papers mainly have a technical orientation and the ca?.culating-
methodical aspect is not the main one for them. At the same time, a need has
arisen for a survey where the presently known calculating methods would be sum-
marized and problems of the theory~ of electrodynamic levitation would be considered ~
from a unified viewpoint, in a sequential manner and in a unified system of nota-
tions with regard to the continuing investigations and optimization of suspension -
systems based on calculation of the forces of levitation, braking and stabiliza-
tion acting on suspension. The compilers of this survey had in mind satisfaction
of these requests.
Main attention was devoted to an outline of the calculating procedures, which in
the authors' opinion, are rather extensive and effective. The details of consider-
inq the calculating methods are different--from a brief inention with reference ta
- the publication to an outline expanded with respect to the original.
The given graphs are mainly illustrative in nature and one should turn to the pri-
mary source articles indicated in the bibliography for more complete calculating-
numerical information. Due to the limited volume of the survey and the number of
- literary references, it was not possi.ble to compile any kind of complete biblio-
graphy and far from all the authors wrorking actively in the field of EDP theory
and not all the articles and books which deserve mention are cited in the survey.
However, the bibliography may be supplemented with inclusion of the bibliographies
of those articles and collections which 3re indicated in this survey.
14
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPR~VED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USH: ONLY
l. Levitation abovF an infinitely wide bed. The starting point for the entire
subsequent outline is P4axwe1l equations without bias current:
roti E=-aB/ac, ci. i~
rot B=�oj, (1. 2)
where B and E are the magnetic and electric vectors, j is current density and u~
is the permeability of free space.
_ Further, nonmagnetic media are considered for simplicity (the generalization for
the case af inedia with constant permeability through bodies encounters no difricul-
ties [5j). Having selected a coordinate system with quiescent bed for certainty,
we have f~r current density
] =QE, (1. 3 )
where Q is the specific conductivity of the material.
~ The boundary conditions on the surface of a conductor for the magnetic vector have
the form [6]
B;=B,, ~ anB~,- \ ~3nB/; -I�,rotB;j, (1.4)
Here a~an is the derivative in the direction of the outer normal end (with respect
to the conducting zone) and the subscrir~ts correspond to the maximum values in the
domain of conductivity (i) and in the external space (e).
The following model corresponds to the suspension system: the conducting bed occu-
pies the domain -T < z< 0 in the Cartesian coordinate system (x, y, z) and the sys-
tem of conductors (electromagnets) which creates a magnetic field and which moves
parallel to the boundary of the bed in the direction of the x axis at velocity v is
located in domain z> H> 0.
In the domain that does not include the interfaces and sources, the following dif-
- fusion equation is valid for the magnetic field
dB=�oaBB/8t. (1. 5 )
At constant velocity v we have 4/at =-v~/~x; therefore, (1.5) can be rewritten in -
the form [7]
~B=-�oQVBB/8x. (1. 6)
Solution of equatiori(1.6) with boundary conditions (1.4) permits one to find the
- field in the entire space and also the density of the eddy currents in the conduct-
ing domain. The force acting on the conducting circuit of the electromagnet can be
determi.ned either by integration of the forces acting on the elementary magnetic
dipole
15
FOR OFF[CIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2047/02/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
r=r f f ~n,v~Bas, c~.~~
- $M
or by the reac.tive force with which the current circuit acts on the bed:
~
F=- ~ds f f [J~~,y,z)B�~x,U,Z)~dS. (1~8)
&
The following notations are used in formulas (1.7) and (1.8): I is the currer_~ of
the electromagnet circuit, Sk is the surface subtended by this circuit, S is the
cross-sectional area of the track bed, J is the density of the eddy currents in the
bed, B is the field of eddy currents of the bed actinq on the suspension electro-
magnet and B~ is the field of the electromagnet acting on the eddy currents in the
bed.
The value of B~ in (1.8) is assumed to be known and the values of B and J in (1.7)
and (1.8) can be determined from solution of (1.6). Besides B~ itself, its normal
derivative is also continuous for component B normal to the boundary of the conduc-
tor, accordinq to (1.4). For this reason solution of equation (1.6) can be found
for normal component BZ and the remaininq co~sponents of th~s sector can be deter-
mined outside the conductor from the equation rot B= 0.
Turning from the magnetic vector to the Fourier representationl
m
9(k,z)=(2~)-= f ~ exp(-ikr)B(r,z)dr, (1.9)
_m
we find from (1. 6 )
dZ~/dzz=aZB. (1.10 )
The following notations are used in (1.9) and (1.10): r= xeX + yey, k= kYeX +
+ kyep (here and further ex~ ey and ez are the unit veators in the direction of the
corresponding axes), a=(k2 - iupQk~)1~2 for the domain of cond~tctivity and q, = k
for a free space, k= Ik~.
Equation (1.10) has solutions of type exp az) and the coefficients in front of
these partial solutions are found frc.m the boundary conditions. The procedure of
. determining the coefficients which reduces to solution of a system of linear alge-
- braic equations (inhomogeneous--due to consideration of the boundary ~onditions
whic~z include the source field) is very simple [8]. Let us devote main attention
to the other problem, less illuminated in the literature--that of finding the
Fourier transform of the source field which determines the inhomogeneous part of
the mentioned system of equations.
With Coulomb calibration div A= 0, the vector potential of the currents ha~ the
known expression
16
FOR OFF[CIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02/48: CIA-RDP82-44850R000400014436-7
FUR UFF'ICIAL USE ONLY
~ ~r, z) _ (�o/4n) I~ ) ~r~, z~) ~ Ir~-rl= (z'-z) z]-~i~ dr' dz',
y
where j is the current density vector and integration is carried out by the volume
of the conductor with current.
Making use of the identity
.
f f eap(-ikr)[~r'-rl'+(z'-z)ZJ-'~~dr=~,: Exp(-ikr')X
` 2n
X f rl,(kr)[rt-~-(z'-z)=]-'~'dr= k exp(-ikr'-kla'-zl),
0
- for Fourier representation of the vector potential we have
4I(k,z)=(�o/$n=k) f j(r',z')exp(-ikr'-k~a'-z~)dr'dz'. (1.11)
v
Hence, the Fourier tr~nsform of the magnetic field of the source is found from the
- equality B= rot A and for Fourier representations in the rot operator, instead of
derivatives 8/ ax and 8/8y, one should substitute the multipliers ikX and iky.
Let the source be compiled from a set of conducting circuits Ln, each of which sub-
- tends some surface Sn and carries current In (n = 1, M). The integral throuqh
the volume in (1.11) now reduces to the sum of the integrals through the conducting
circuits. Using the identity valid for an arbitrary continuously differentiable
function
~ f dl = f f[ n', grad f~ dS,
L~ 8.
where n' is the unit vector of the normal to surface Sn at the point of integration
and introducing the vectors
1\'� = f j n' exp(-ikr'-kz')dS, (1.12)
s~ .
x=ik,~e=+ik�ey+kes, (1.13 )
we find for z< H, i.e., at a positi~~~ value of (z' - z) in (1.11),
17
FOR OFFICIAL USE OyLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPR~VED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
. !
r N i
41~k,Z)=(�oexp(kz)/8nZk) Lx,~l�N�~. ~
n~! ~
The Fourier representation of the magnetic field of the source can now be wri.ttert j
in the fot-m ;
Y
o _ �o exp (kz) �o exp (kz) \ ~ } (1.14 ~
8n~k ~ x' ~ g~2k � x~ InN� n. ,
n~!
Hence, it follows that For a source of arbitrary form
d8�/dz=k8�, (1.15 )
if the source is located above the conducting bed.
The values of Nn given by formula (1.12) are aomparatively easily calculated for_ a
wide range of circuit shapes. Thus, for a horizontally arranged rectangular cir-
cuit measuring 2b at z= h in the direction of motion and 2a in the transverse
direction, we find [7]
- sin b)~= sin aky
N= 4 exp (-kh) k k e=. (1.16 )
= y
The expressions for N at some other shapes of the circuit are indicated in [5, 9].
Finding the field of eddy currents is essentially simple with the known expressions
for the rourier transform of the source field. The component ~s of the Fourier
transform of field ~g is determined from boundary conditions at z= 0 and z=-T and
the inhomogeneous part of the derived system of equations contains the z-components
of vectors and ~o/dZ determined by (1.14) and (1.15). The remaining components
are found from the equation rot B= 0 for the field of eddy currents at known value
of ~ which yields ~9=~=x'/k , wherel~* is a vector complex conjugate to (1.13) .
The described procedure of determining the Fourier transform of the field of eddy
currents leads to the result
9=x'~ (k) exp (-kz) ~8,� ~ ,_o/k, (1.17 )
where
~ _ k-a ~ 1- exp (-2aT) ] / ~ q _ / k-a \ = exp ~-2aT ) ~ . (1.18 )
k-f-a ~ k+a ~
Now, turning from the Fourier transform of the field (1.7) to the original, we find
the force acting on the electrodynamic suspension from formula (1.7). The corre-
sponding transforms lead to the expression
18
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7
_ H'OR OFFI('IAI. USE ONLY
.
_ F Sn= J J k~~k~ (x' ~1"N" ) dk ~ d� J J exp (ikr-kz) ~n, -x') dS,=
-m n~! n..l _m
M t (1.19)
8nz JJ k�~~~~ I \~'~I"N"1I ~
-m n~l
Formula (1.19) is a generalization of the known formulas for force F given in
[5, 7-9].
Specific results were found in the greatest volu. > > >
y,=hZ+B, y.=h,-~-hZ-Fd, y,�h,-~d, y~=S,
(5)
42
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-00850R040440010036-7
FOR OFF[C[AL USE ONLY
a~-a, a~-Fa, a,-a, a,-F-a, (6)
z,~- 2-~- e, z, ~ 2--~e, z,= 2-I- e, z,=- Z -I-e,
where J is the magnetic moment of unit volume and b is the length along the direc-
tion of motion (the remaininq notations are ,qiven in Figure 1).
The results of calculating the components of the forces of interaction with differ-
ent arrangement of the magnets and also the results of experimentsl are shown in
Figure 1. The results of calculating the forces when the magnetic strips are re-
placed by dipole filaments arranged in the geometric centers of the strips are
shown in the same fiqure. It is obvious that the difference is small.
Relation (1) can also be integrated for magnets in the form of bars of finite
length [3]. However, in view of the cumbersomeness of the expressions for the
first estimates of the properties of systems, one can use the model of a local di-
pole for which
F=�oI,12S,S,b,f~2S',
(7)
~v=2y ~3z -yZ) l ~y2-t-i
(8)
1.=aZ=~~ -3y )~cy
. ~9~
y=y/S, i=z/S, (10)
where S1, 52, y and z are the cross-sectional areas of the bars and the vertical
and horizontal dimensions, bl is the lenqth of the bar along the direction of mo-
tion and S is the clearance.
~ 1 4'~~~
I, 0 � ~ _
~ hI
\ h,' } ~
- OS ~ /
, \
~ \ fy fZ z
\
\ ~
qS J,O !,5 E
r~~
I'igure l. Calculation of Components of Forces for the Case al = a2 =
= hl = h2 and S/a = 0.2: 1--calculation for magnetic dipoles;
2--experiment; 3--calculation for magnetic strips
The relations can be refined by modelling the fields [7] or by other methods if
there are ferromagnetic components near the permanent magnets [7). Some general
proper.ties of the force field can be found from consideration of the expression
� 43 -
F'OR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
~ FOR OFFICIAL USE ONLY
�or force. Consideration of these properties permits one to constrict the area
of searching for the optimum.
Thus, not only the impossibility of providing stability, which follows from Brown-
beck's theorem but also the nondivergence of the force field created by the perma-
nent magnets ensue from relation (1):
~F=~[ (p~)B]=(~p) (OB)~'p(VV)B~-(p~) (OB)~0, (11)
i.e., for elongated systems---the equality to zero of the sum of the diagonal c~m-
ponents of the magnetic stiffness tensor:
aFyiay=-aF./az. ci2 ~
Thus, an increase of magnetic stiffness in the vertical direction results in an
. increase of the gradient of the horizontal destabilizing force.
It is obvious from expressions (1)-(3) that the values of lift and lateral forces
change places when one of the magnetization vectors is rotated by 90�.
It is obvious from the dependence of .forces on lateral displacement (Figure 1)
that the lift decreases rapidly with an increase of the shift of the magnets with
respect to each other. The same is true of the lateral force after it passes
through a maximum. Each magnetic strip thus interacts with practically only two
adjacent strips and only this interaction can be taken into account for the first
estimates.
Finding the optimum configuration. There is a large number of possible configura-
tions of a magnetic system. To reduce the area of search, one may take into ac-
count the preferable execution of the block system mentioned above and use a
model of dipole filaments durina the initial calculations and one may also compare
some main configurations, for example, with vertical sign-constant orientation of
the magnetization vectors (canonical), vertical sign-variable (sign-variable orien-
tation), vertical sign-variable in the track and horizontal sign-variable in the
vehicle (checkerboar~?~ configuration). Checkerboard configuration of a magnetic
system is shown in Figure 2.
e
0 0 ~ ~ ~
~ 0~~ `
D
Figure 2
Comparison of the systems (Table 1) shows that th~ maximum possible values of lift
with sig;~-variable and chec}:erboard systems almost coincide upon variation of the
p~rameters named above, while those of a canonical s,ystem are less by a factor of
1.5. The value of the lateral force g~adient during maxir,ium lift is 33 percent
higher in a sian-variahle system than in a checkerboard system. If the value of -
the lateral force qradient must be limite~, the checkerboard system pe~cnits one to
- do this with far fewer losses of lift than the remaining systems. The checkerboard -
system also has shoYter distance between st~ips, correaponding to maximum lift, and '
thus lift is higher per unit width of the strip.
44
FOR OFFICIAL USE ~NLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
- FOR OFFICIAL IUSE ONLY _
Further dei:ailed optimization can also be carried out tor this system.
The variable parameters here may b e the characteristics of system configuration:
the width of one group of strips ql = D, the ratio of the width of a magnetic strip
to its height q2, the number of s trips of track q3 and of the vehicle q4 and so on.
Table 1. Comparison of Suspension Systems
BNACHCTEM6I ~2~
_ IIoi~aaaTenr, ,
~ . ' Kaxo~~rKBR I3H8KOiIC~`MEHHBft ma~5$TxaR
( i 1
MaxcaManhx~e axa4esax J�(F ) 1,04 1,64 l,52
PBCCTOAHH6 MB3Kjjy IIOJIOCBbiH IIyTH ~7 ~ � , �
H 8KHII8S)fd IIP8 MSKCHMBIIbHOH
noAveMaoH cane, oTaeceaaoe x hi 3,0 !,2 0,65 -
I'paQ$esz oTaomeunR 6oxoBOic can~ ~g �
K MBXCHMBJIbH01i [i0Jj7~QMHOS
a~~:~~v KaNC~~a~ IIpH IIOJ~'bQMHOH
cane, pasdoH Ma~:caManbao~ 2,2 4,0 3,0
a~HBYQHHe iy, IIPH KOTOpOM M03KeT
6~rao6ecIIegeaoe/,/Bz�i ~9~ 0,47 0,30 0,90
IIpo~eaTaoe cenx:eaHe IIop~es~aoH
cSa~ npa 6ohosoM cMe~esaII, ~ln )
pasaoM 20~ aaaopa 0,24 0.25 i 0,06
Key:
1. Indicator
2. Type of system
3. Canonical .
4. Siqn-variable
5. Checkerboard
6. rlaximum values of fy
7. Distance between strips of track and vehicle at maximum lift, related to hl
8. Gradient of ratio of lateral force to maximum lift a(fZ/fy m~s)/~z with -
lift equal to maximum
9. Value of fy at which ~fZ/ az < 1 can be provided
10. Percentage reduction of lift during lateral displacement equal to 20 percent
of clearance
A number of indicators may be limited, having selected for example, the maximum
variations of the gradients:
m< (-BFvlay) 0).
- In this algorithm, the frequency characteristic Wy2h(iw) in the high-frequency ~
range varies as 1/w2, which is quite adequate for satisfactory filtration of the
high-frequency noise on ttie track. However, the stability reserves according to
the parameters of the object ayy and ays are impermissibly low in this algorithm,
while coefficients ap, al and a2 are impermissib?.y high.
To explain the foregoing, let us write the expression for the free term bp of
- equation (2) with regard to (4), equal in turn to the product of the roots pi
(i = 1, 2, 3, 4) :
~ . a,a~,- (1+k,) ~a~y-~+:Z) '
- b0 ~ TT - ~P''
~ t_+
_ 53
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
~
FOR OFFICIAI, USE ONLY
Due to the fact that quite specific frequency properties m~sst: be assigned to the
system, the value 4 - ~
~pr=const.
I~et us write the equation for the coefficient of the algorithm ap: �
~
(i+kf~ IaYY-~12=)-~-TT ~ p~
Qo ~ .
aya
Let us introduce the relative deviation of ~a1,~, of coefficient ayl from the nominal -
ayy, i. e. , a~,~, = a~ (1 + ~a~,~,) , and let us express the required condition of sta- ,
bility of the system bp > 0 by ~a~,y:
~ a
TT~p, .
Da�u < ~i+k?) ~avv�-cilfz~) ~
we note that the value of kI ~ 50; therefore, the coefficients of the algorithm
increased 50-fold, which is impermissible from the viewpoint of the noise sta-
bility of the system, while the stability reserves decrease by a factor of 50 com-
pared to an algorithm ~aithout continuous current feedback.
r,o
Aytn y
~
1,5 N
5
J �
" 1,0
QS
~
0 ED 160 140 t"' 310
Figure 2. Amplitude-~'requency Characteristics -
An algorithm of the follo~ring type is free of the indicated deficiencies:
Le ~P) =an-Fa, p-I-aZ p2, (5 )
L, ~ P ) =-k,T.~rPI (T.K p-I-1) .
Introduction of feedback with respect to I*, i.e., with respect to the derivative
of the control force, is actually the basis of algorithm (5) at 'tM =
= TI (P1/ (TMp + 1) = I* (P~ -
54 _
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFIC'IAL USE ONLY
The stability zones and the amplitude-frequency characteristics were calculated for
system (1) H~ith algorithm (5).
The stability zone in the plane of deviation of the parameters of the object ayy
and ay~S from their nominal (calculated) values is presented in Figure 1. The set
of amplitude-frequency characteristics corresponding to Wy2h(iw) for five versions
of a1,~, and ayd : Aay~, = 0 (nominal ~ay~ = 0; ~ayy = 15 percent, ~apd = 15 per-
cent; L~a~,~, = 15 percent, ~ay~ _-15 percent; a~,~, _-15 per..~n�:, day~ _-15 per-
~ cent; and ~a~,~, _-15 percent, ~a~,~ = 15 percent, is presented in Figure 2.
� The notations in Figure 2 are explained below. Analysis of the results shows that
algorithm (5) satisfies all the requirements formulated above in the range of var-
iation oi the object parameters. 41e note that the given calculations were made by
, the compl.ete dynamic scheme with regard to the effect of the vehicle mass. As can
be seen, resonance phenomena at frequencies close to partial suspension frequencies
of the vehicle w~ wl are considered with the selected structure and at the param-
eters of the algorithm with respect to the electromagnet coordinate Y2.
The roots of the characteristic equation of a rail car with electromagnetic ski
stabilized by a proposed algorithri were calculated to justify ~he permissibi].itX of
synthesizing the algorithm on the basis of a simplified mechanical model. In view
of the large dimensionality of the system and the dense frequency spectrum, the
program de~cribed in [3), which is specially oriented toward calculation of sys-
tems of this class, was used in calculating the roots of the characteristic
equation.
/1 ~ 3f~aqeerfe xopxeN pa Ic-~), cooreercreyroulaz aoMepp xopess k t2~
Hon~ep noA-
~ cNCTeM~ S L
1: 2 I 3 ( i I 5 I' 6; 7
' ! -i,055ti7,782 -40 -9,834 -161,3 -54,S2tt77,03
? -l,139ti8,002 -40 -9,660 -164,6 -53,35~i77,25
3 - -40 -9,155 -l63,8 -53,92tt81,52
4 - -40 -8,370 -i67,! -52,64tt87,78
5 - -40 -7,445 -171,5 -50,87ti96,38
b - -40 -6,980 -174,0 -49,86tsl0i,40
Key:
~ 1. Number of subsystem S
2. Value c,f roots ps(s-1) corresponding to number of root k
Roots ps (s-1) are presented in the table. The subscript S= 1 correspon3s to a
subsystem with degrees of freedom corresponding to (~1. P0~ ~0~ ~2~ % S= 2-
- PO ~ ~p ) . S = 3 - (Pl ? d 1) ? S = 4 - (Pl . ~l ) . S = 5 - (P2 ~ S2 ) and S = 6 -
-(p2, 02). The notations correspond to those used in the indicated paper.
The roots of a two-mass system with algorithm (5) were calculated in the same man-
ner upon variation of parameters a~, ayd on the scattering plane around a circle .
with radius of 20 percent, inside which the system retains satisfactory frequency
properties (see Figure 2}, The rssults of calculation are represented in the form
of "root locus" curves in the upper hdlf-plane of tze complex variable in Figure 3,
where the "root locus"are ~lso shown according to the table upon transition from
~ 55
~ FOR OFFICIAL USE ONLY
i
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL USE ONLY
one clegree of freeciom to another (there is c~mplete agreement in number of the
versions in P'igures 2 and 3). The following notations were used here: I is the
_ var.iant of variation by parameters oc~,~, and Cc~,S and II is the variant o~ taking .into
- accouiit the multi~iimensionality of the system. The numbers of the roots are pre-
sented in parentheses. The root which remains constant from variant to variant,
as .roots p~1~2~, corresponding to forward and angular displacements of the rail
car with respect to the ski, is denoted by p~3~ _-1/'[M =-40 s'1. The roots p6~~
and ps correspond to the nominal values of t~ie parameter (point N in Fiqures 2
and 3). The height of the columns for real roots is arbitrary in nature.
.
o � ay~ I (ml, N~
_ ~ ~ \
~~6''~ rzo
3 2 ~ 8, p(6,
6 100
N ~ay� P6~
-40 8 40 % ~ ~
~ \ 80
5 \ -
6 5
\ 60
\ ~
-GO 2
4 40
I7 (5)
- ~ Il 4)
~ P3` I(S) j(y~ P (31
Z~
3
Re psk~ P~,Z~
-t00 � . -160 -120 - 80 -40 n
Figure 3. "Root Locus" Curves
Based on the fact that the roots of a multimass car-ski system are close to the
roots of a two-mass system, one can state that algorithm (5) should satisfactorily
solve the problem of stabilization for the entire system as a whole and selection
of the algorithm based on the simplified mechanical model is sufficiently justified. _
Conclusions. 1. The law of control which provides satisfactory solution of the -
- stabilization problem for the car-ski system can be synthesized as a whole on the
basis of a simplified two-mass model of the control object.
2. The stabilization algorithm is synthesized on the basis of signals prot~ortional
to variations of the gap and current and their derivatives up to second order.
BIBLIOGRAPHY _
1. Raybakov, S. N., B. I. Rabinovich and Yu. D. Sokolav, "The Dynamics of Trans-
port Rolling Stock on Magnetic Suspension," IZVESTIYA AN SSSR, ENERGETIKA I
TRANSPORT, No 1, 1981.
56
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
2. Bagryantsev, V. I., V. S. Nevarko and. B. I. Rabinovich, "The Mathematical
Model of a Ftail Car With Electromaqnetic Suspension," IZVESTIYA AN S~SR,
ENERGETIKA I TRANSPORT, No 1, 1981.
~ 3. Kalinina, V., V. G. Lebedev and B. I. Rabinovich, "Recursion Iteration
Algorithm for Solving the Characteristic Equation in Prcblems of the Dynamica
of Structures and its Realization on the BESM-6 and YeS-1040 Computers," in
"Issledovaniya po teorii sooruzheniy" [Investigations on the Theory of Struc-
tures] , Dioscow, Stroyizdat, 1979.
COPYRIGI~'i': Izdatel'stvo "Nauka", "Ixvestiya AN SSSR, energetika i transport", 1981
[62-6521)
6521
CSO: 1861
~
i
~ 57
, FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
UDC: [538. 31:621.333] 001.2
MATHEMATICAL MODEL OF A RAIL CAR [4ITH ELECTROMAGNETIC SUSPENSION
Moscow IZVESTIYA AKADEMII NAUK SSSR: ENERGETIKA I TRANSPORT in Russian No 1,
Jan-Feb 81 pp 101-107 -
_ [Article by V. I. Bagryantsev, V. S. Nevarko and B. I. Rabi.novich, Moscow)
[Text] Investigating the dynamics and stability of the rorling stock of high-speed
transport on magnetic suspension is a very comple~c problem [1-3]. The success of
solving it at the design stage is determined largely by the adequacy of mathematical
models of the "object-regulator" system used. A mathematical model of a rail car -
with suspension in the form of two multilink jointed kinematic chains with rigid
modules carrying two electromagnets each with independent control systems suspen3ed
to the body by viscoelastic couplings is synthesized below [1]. In this case the -
disturbed motion of the rail car in the vertical plane is considered at which one
equivalent "electromagnetic ski" with tw~o double electromagnets in each module can -
be considered. Horizontal motion with constant velocity v and constant gap s� is
taken as undisturbed motion. -
Let us use the linearized mathematical model of an electromagnet [1-3]. 'For an
, example let us consider a simplified stabilization a2c;orithm in which only the gap
is use~ as the observable coordinate: Let us introduce the following coordinate
systems.
1. The starting system Oxyz connected to an ideal track structure.
2. An absolute system G*x*y*z* connected to the "car-ski" system hardened in un-
disturbed motion (C* is the center of mass of the system). _
3. A correlated system Gxyz rigidly attached to the hardened "car-ski" system dur- ~
ing disturbed motion with respect to G*x*y*z*.
The position of coordinate system Gxyz with resp~ct to G*x*y*z* in the vertical
plane is determinec? by generalized coordinates r1 and ~(Figure 1), which are as-
sumed to be small in the sense that linearization by these coordinates is permis-
- sible. The position of the ski in coordinate system Gxyz is determined by the
deflection function f(x, t). Deflection of the ferromagnetic rail surface from
. the horizontal plane is characteriz~d by the function ~(x). The gap s(x, t)
duz~ing disturbed motinn is related to the nominal gap s� by the relations -
s(~, t)=s�+s(x, t), s(x, t)=}~(x)-[r~+$~+f(~, t)]. (1)
58
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFtCIAL USE ONLY
Let us introduce tl-,e following local coordinate systems to describs the motion of
an individual electromagnet (Figure 2): O*x*y*z* and Oxyz whose axes are directed
the same as the axes o� coordinate systems G*x*y*z* and Gxyz. The origins O* and
O lie in the plane o� the pole sl~oes of the electromagnets.
Let us consider an electiromagnet of unit length in the direction of the Ox axis.
The linearized system of equations of the dynamics of a controlled electromagnet
- with the simplest law of control (stabilization algorithm) with respect to the gap
and its =i.rst tw~ derivatives has the form [1-3]
F=c,1-c,s,
ct't+Rl-c,'s=U,
(2)
. U=aos~-a,s-f-a.~s, .
where U, I and F are the voltaqe, current and vertical force developed by the
electromagnet, cI, cI, cS and c~ are c~efficients dependent on the parameters of
the electromagnet and the undisturbed value of the gap s�, where clcs = cl~s and
cs = c2 and R is effective resistance. The dot indicates the time derivative.
~ After elimination of variable I and some transformations [1], one can impart the
' following form to system (2):
F=c,s-I-S,
, ~od+b-~-Ce~~~S+Ca.~S=O, ~3 )
i '
where
Qo~f ~I.
R ~6= R ,
(4)
� a:cl , Qs~r
_ c s-
e, , , , ~a~ = aCr- . .
Cr Cf Cf~Cn
The first term in the expression of variation of force (3) corresponds to quasi-
static control in which variation of current is proportional to variation of the
gap and consequently the total variation of the electromagnetic force is propor-
tional to variation of the gap s.
y y'~
~
y
~
A ~ ~ s
R
S
s
D 2~ _ 0 _ h
S*
~ Q
Figure l. Rail Car With Suspension in the Form of an Electric Ski with
Discrete Modules.
59
FOR OFFICIAL USE ONLY '
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/48: CIA-RDP82-00850R000400014436-7
FOR OIFFICIAL USE ONLY
The equivalent electromagnetic ski under consideration consists of identical hinge-
coupled modules which carry two double electromagnets each and two gap sensors
each.
Each n-th module considered separately from the remaining ones is a rigid body with _
two degrees of freedom ~excluding longitudinal displacement), to which the gener-
alized coordinaies ~n and ~n+l Presented in Figure 3 can be set into agreement (the
following notations are also presented in Fiqure 3: a/2 is the distance from the '
center to the end of the module, b/2 is the distance to the line of action of the
elastic components which couple the module to the body, d/2 is the distance to the _
line of action of the equivalent electromagnetic forces during quasi-static control
and e/2 is the distance to the axis of sensitivity of the gap sensor). Having com-
piled a Lagrangian equation of s~scond kind, we find the following equations of dis-
turbed motion of the ski ti~ith quasi-static control of each of the electromagnets
0)' 2(~'~~+a'S2Z~~)~"8`~:~'~~~Z~z=~, -
Ill~~~n"~'C~~~2~n~ i-a~~~n+i+bn-i~+k'~'~'Z\bntf+Sn-l~=~~ (5,
2 \ I�bN.r.f~a~~lGabHtl~ ~~~~N'F'~~JG2gN_O' -
b
where C Ro
SZ== ~=-1-I-8: +8,', 8; ~ ~ ,
- K,
a~'= ~ ,
a'= f -1-I-~o'(1+b2)+ ~ b!�(1+de) ] ~ , (6)
. [-1+ 2 80' (1-bZ) -F 2 8,' (1-de) J ~ ,
~'=1+E/~~; S'=1-e/2, e=12{s=-1,
b =b/a, d=d/a, e=e/a, ~5= p/a,
N is the number of modules, ri is the total mass of the ski, K~ is the total stiff-
ness of the springs which link the ski to the body, G is the modulus of the total
gradient of forces of uncontrolled electromagnets through the gap, K1 is the total
gra3ient of contrcl forces of electromagnets through the gap during quasi-static
control and p is the radius of inertia of the module with respect to its main
central axis.
Having set ~n =~neiwt, let us reduce system (5) to an ordinary boundary~~alue
probleir. for ei en-values whose solutions are essentially eigen-vectors with
components ~i~~, ~2~~, ~N+1 and eigenvalues (j = 1, 2, N + 1).
Since the gap sensor measures linear displacements, the amplitication factor of
the sensor with respect to angular displacement of the module varies upon varia-
' tion of its distance e/2 from the center of the module (I'igure 3). This leads
to the dependence of. the natural frequencies of angular oscillations of the mod-
ule from the position of the gap sensor. The amplification factor of the sensor
remains constant during forward motion of the module regardless of the coordinate
of e/2. The positi.on of the sensor does not affect the frequency of the
60 ~
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
corrpsponding oscil.lations of the module. Thus, the range of natural frequencies
- of the ski is totally dependent on parameter e. Specifically,at a"critical"
value of the coordinate of the sensor (e
, (7,
e 3d l1+ g E+ b1.~1-3b2) ~
some "maximum ski" is realized in which a single natural freguency w= SZ corre-
sponds to all the harmonics of natural oscillations. The ski makes forward and
angular oscillations like a rigid body at this same frequency. Let us assume for
further analysis that
e�1, So'/8,'>, ~N+1',We find the harmonics
of the natural oscillations of the ski in the form of piecewise linear functions
61
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2047102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
_ ~~(x) (j = 1, 2, N+ 1) corresponding to natural frequencies G.~j. The latter
form a dense spectrum which approaches point w~ = SZ at e-~ e� in view of the last
of conditions (8).
The mathematical model of an electromagnetic ski with complete law of control (3)~
i.e., at 0, is constructed by the Bubnov-Galerkin method. Following the ordin-
ary procedure of this method and utiliziny functions ~j(x), let us represent the
functions f(x, t) ,~S (x, t) , u(x, t) and i(x, t) , orthogonal on segment (-1/2, 1/2) ,
as coordinate functions, in the form of the following series:
x+i x+i
f(x~t)= ~fi(t)~,(x), s(x,t)= ~b~~t)~s(x),
(9)
N+l X}!
u~x,t)=~ U~~t)~r~x), i~x~t)� ~I~~t)~i~x)~
i-s
The mathematical model of the ski acquires the following form upon consideration of
(3), (5), (8) and (9):
�i ~~i+' ~~f
J+~i=li ) =bs-I-Ff ( t) ,
~ (10)
- ~aa,+a,- a ai(ca,"}~~+~e.f~)=~i(t)
(j=1, 2, . . . , N+1).
Here
�j=MNi=~21N, ~;=b;/�;,
~
~ N'2 = 3 ~~n+~3 +~�"'~)+S'~�+,~�cr~
n~1
L X (11' (1)' Z (n -g (1) tq 1 ~11. ~
u~-~ ~~n+t ) ~1+b )+2~n-Ff bn ~1-U ~ J .
~ N
cii' - v~' ci~ - ci~
~ 3~ (~n+~ +~n +~n.fi~n
n~!
i/i
F~~t)= ~ c~~~x).~~~x)~+
_,i:
c;= ~ (K,-C),
62
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFICIAL USE ONLY
' / 2 '~r3 p/
~!\~~-__'CO~U~J jr~/~S~b11z~~_
Q'
-1/t
2 ~i:
, __'ce~~v f ~~~x)~i~x)~,
a
. -?i_
- M is the mass of the ski and 6 is the coefficient of natural dampi.ng of visco-
elastic couplinqs of the ski to the body.
- The equations for ~eneralized coordinates U~(t) and Ij(t), which follow from (2),
are independerit and will not be given here.
Due to the symm~try of the ski and the boundary-value conditions with respect to
poiat x= 0, the set of functions ~j(x) reduces to two subsets: functions ~j(x)
symmetrical with respect to the point x= 0 and t~j(x) antisymmetrical with respect
to this point. Functions ~j(x) and nj(x) are mutually orthogonal or_ segment
(-1/2, 1/2). Let us ascribe the index j= 0 to functions ~p(x) = 1 and r~p(x) = x
and let us establish independent indexing and the following normal~zation for the
remaining functions:
~s~-1/2)=1; ~it~-l/2)=1.
Let N be an odd number, then j= 1, 2, (N - 1)/2. Let us denote the general-
ized coordinates corresponding to eigen functions (x) and r~~ (x) by p~ (t) and
_ q~ (t) and let us denote the natural frequencies and damping coefficients t~p~, c,,~~
and (3Pj, respectively. Let us retain the notations of ~5~ for the expansion
coefficients of function ~(x, t) (9), corresponding to the symmetrical harmonics
of oscillations and let us denote the same coefficients for antisymmetrical har-
monics by ~j. Let us ascribe subscripts s and a, respectively, to the disturbing
functions.
Returning to coordinate systems G*x*y*z* and Gxyz an.d using theorems of the momen-
- tum and kinetic moment, we find the following mathematical model of the rail car
as a whole.
1. Forward motion ir_ the direction of the G~y* axis
~ (M�+M) (r~-I-c~~z~)-F~Mpo-}-enPo=Bo-f-Py~t)~ �
A7 (po-f-~ppP~~ppy~IO~ -I-1VI t~-F-ea~l=ao-f-Pa ~t~ ,
~e8o-f-8o- Q eno~~e. ~~1~'Pu~~'~a,~~rl~"Po~~~~.o~t~, (13)
d~ (p~+~a~Pt+~r,ZPs) =b~+P, ~t~ ;
: ~e$~-~Sf - Z ee, ~~e."P~~"~e.~Pt) =~�i ~t)
a
~l=i, 2, . . . , (R'-1)/2).
� 63
. FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
NOR CFFICIAL USE ONLY
~ ~
J Here `
c~n M�+M ' e,,=K,-C; eno~l, ~
- u:
- pv ~t)�=c~ f ~ ~x) dx-I-P~ ~t),
-t/s -
,n . (14)
P; (t) =e~ ~ ~ lx) ~i ~x) d~; ~ ~i= ~�d~~
_u: _~i:
m.t a I ca."v= f (x) (x) -
~
_~yz
~s
-F-Ce.~U ~x) ~f ~ ,
- ~1=0, i, . . . , (N-i)/2),
M� is the mass of the body and M is tl:e mass of ski; thP appearance of multiplier
2/a in equations (13) and subsequently in equations (15) is related to the fact
that two electromagnets are located on a single module.
2. Rotary motion around the Gz axis
"~'~s~~ 1$'~We=$~'~'~sQo-~'eeQo=~o'~"'Ma:~t~,
~=~40+~~~o-I-u~ve'Qu) -Fls$-I-es$=0o-f-Oo (t), ~
2 � _
~~S~-I-~o--eoo~~e. ~$~'9o~+Ce�~~$~"9o~~-~ao~t~,
- . (15 )
~ b~~4i~'~ar4l+'~vi24i~ �~i~"Qi(t~, -
Z . -
~0~;-4-0~- - eet ~~e. Q~-~ei4~) =~ai (t) -
a
~7=1, 2, . . . , (N-1)~2).
Here
ee ed ~N-sii: ed l~ r
c,?ez= , . e,=4k~a= f 4+2 ~ Cn 4) J-G 12 '
J: ' ~ (16 )
n~t
l/2
apCl r ~
k. = R, eeo = 12 ' ep~ - J~liZ dx=d�
_u:
64
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7
- 1~'OR O~'Fl('IAL USE ONLY
~i: ~~Z
1V1~~=c~ ~ ~~x)adx+M~.�, Qs~t)=c{ f ~~x)~li~~)dx,
-,i:
- r~:
~a~~~)~- a [ca."U= f ~"~x)~l~~x)d~-I-cei'v f }~'~x)~li~x)dx
_riz ~
~1=0, i, (N-i)/2),
JZ is the moment of inertia of the body without a ski with respect to the Gz axis,
JZ is the moment of. i~~~rtia of the ski with respect to the Gz axis, JZ is the
moment or inertia of the ski with respect to the main central axis parallel to Gz
and MrZ is the projection of the main moment of ~he system of external forces ap-
plic~d to the car w.ith respect to point G onto the Gz axis.
Knowing the solvtions of equations (13) and (15) one can find the variation of the
gap at each point of the ski and the variation of current and voltage for each of
the elecLromagnets bp usinq ecruations (2) and expressions (1) and (9). In ~e
special case of .Fip - 0, - 0, ~p = 0 and 0~ - 0, equations (13) and (15) change
to a mathematical model of a rail car with permanent repulsion magnets [5] with
linear gradient ci (the absolute value) per unit length of the ski,
Conclusions. Plane disturbed motian of a rail car with an"electromagnetic ski"
can be described by two independent systems of ordinary differential equations of
order 2(N + 2) each, where N is the number of ski modLles, with the simplest law
of control with the two first derivatives of the gap. '
BIBLIOGRAPHY
1. Eaybakov, S. N., B. I. Rabinovich and Yu. D. Sokolov, "'I'hc- Dynamics of Trans-
port Rolling Stock on Mag~netic Suspension, " IZVESTIYA AN SSSR, ENERGETIIC~~ I ~
TRANSPORT , r1o 1, 1981.
2. Brock, K. H., L. Gottzein, E. Mannlein and J. Pfefferl, "Control Aspects of
Txacked Magnetic Levitati~n High-Speed 'Jehicles," Sixth IFAC Sympositun on
Autonatic Control in Space, Boston/Cambridge, Massachusetts, August 1g75,
3. Gottzein, E, and B. Lange, "Magnetic Suspension Control Systems for the Mgg
Hiqh-Speed Trai.n," ALTTOMATIKA, Vol 11, 1975.
4. Panovko, Ya. G. and I. I. Gubanova, "Ustoychivost' i kolebaniya uprugikh
= sistem" [ThE Stability and Oscillations of Elastic Systems], Moscow, Nauka,
- 1967.
5. Rabinovich, F.3. I., "Prikladnyye zadachi ustoychivosti stabilizirovannpkh `
ob"yektov" [Applied Problems of the Stability of
Stabilized Objects] , l~oscow,
Mashinostroyeniye, 1978.
COPYRIGf~T: Izdatel'stvo "Nauka", "Izvestiya AN SSSR, energ~tika i tr.ansport~~~ 1981
[62-6521)
6521 -
CSO: 1861 65
FOR OFFICIAL USE ONLY
-
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL USE ONLY
UDC [621.333+538.31):625.033.3.001
DYT1[a1~tICS OF TRAIdSPORT ROLLI?~IG STOCK OPI P7AGNETIC SUSPENSION
Moscow IZVESTIYA AKADEMII NAUK SSSR: ENERGETIKA I TRANSPORT in Russian No 1,
Jan-Feb 81 pp 92-100
[Art: nle by S. N. Baybakov, B. 2. Rabinovich and Yu. D. Sokol.ov, Moscow]
- ['rext] Introduction. INigh-speed passenger transport on magnetic suspension has
' won c~ver greater recognition durinq the past few years. There are a number of test
benches on which the speed of experimental vehicles to 400 km/hr has been achieved
[1, 2J. We shall dwell here on low-clearance suspension systems, on the order of
10-15 mm, realized on the basis of controlled attraction electromagnets. One of
the central problems related to development of this class of systems is that of
~ providing stability of motion with retention of t.ze clearance close to nominal, _
with simultaneous provis ion of a high level of filtr~tion of disturbances related
to unevennesses of the track to ensure the necessary conditions of comfort for
passengers and also to protect the system against interference. One must encounter
some of the problems aris ing here when considering suspension systems on permanent
magnets constructed on the repulsion principle and also electrodynamic suspension
systems. A number of as pects of these problems are discussed .in [1-8] .
Some shift of emphasis in determination of rational areas of the applicability of -
passenger transport syst ems on electromagnetic suspension has recently bEen ob-
_ served on the trend from mainline to "city-airport" and "city-satellite city" sys-
tems. Acccrdingly, the optinium range of speeds subject to consideration is reduced
to 100-250 km/hr, where the lower values correspond to intracity transport and the
larger values correspond to "city-airport" and"city-satellite city" transport,
compared to mainline systems. -
When developing a suspens ion with favorable dynamic characteristics, it is rational _
to strive for systc~ms having the hi.ghest functional reliability with simple coinpil-
ation of ineasurements and the absence of high-precision sensors having movable
components !gyroscopic sttitude and angular-rate sensors, integrating accelerome-
ters with movable mass and so on). In this case the on-board control system should
include a central processor for overall program control of the electromagnets,
processing information about the status of all systems and diagnosis of malfunctions _
and failures and micropro cessors or analog subsystems at a lower hierarchical level
_ --in the self-contained electromaqnet control circuits .
A suspension system should undoubtedly have adaptability to track unevennesses so
as not to place special requirements on the precision of formin_q the track
66
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000400010036-7
N'OR OF'FI('IAL USE ONI.Y
structure compared to existing transport systems. At the same time, it should
prov=de minimum energy expendit:ires to correct the corresponding disturbances and
should have a high level of comfort in the passenger section.
Solution of all these problems requires complex consideration of the mechanical
suspension system and the electromagnet control system.
Adaptation to track unevenness and filtration of disturbances. Let us consider a
system based on attraction electromagnets distributed along the length of the
body, naturally stable in the lateral direction due to the U-shaped cross-section
of the pole shoes and ferromagnetic rails with automatic control of each of the
electromagnets, which ensures its stability in the vertical direction, as a spe-
cif ic variar.t of the suspension system .
- : .f/, y fZ
~
~'~~~%~%'i ~
. ~ ; fz
- , s ~ ~ o ~i~.'' -
o ~ F z
~
. z
O f o ,~a
z
~o
~
- Figure 1. Diagram of Electromagnet With Two Degrees ~f Freedom
Let the suspension system be realized in the shape of two parallel elastic "magnetic
skis," along which are arranged attraction electromagnets (electromagnetic skis).
These skis have transverse elasticity in two mutualiy perpendicular planes and ~re 4
connected to the body by viscoelastic couplings oriented in the vertical and later-
al directions (Figures 1 and 2). The same design can also be used for a system
with permanent repulsion magnets interacting with the permanent magnets of the
track structure. T;ze spECific block diagram of the ski may be different; specif-
ically, this may be a multilink kinematic chain~aith rigid links--modules which
carry two electromagnets each, and elastic spherical joints which connect the mod-
ules to each other. The elastic couplings of the skis with the body, which create
viscous damping, play the role of a secondary suspension. This suspension scheme
has a number of advantages which make it possible to find principal solution of _
the problems formulated above:
67
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL U5E ONLY
1. The presence of. two levels of dynamic couplings (rail-skis and skis-body)
perznits firm tracking of the skis over track unevennesses with slight deviations of
the clearance from the nominal and soft suspension of the body to the skis when -
sel~cting rational values of stiffness and the degree of damping of the mechanical
part of the suspension system and the dynamic amplificatiora factor of the electro-
magnet control circui.t.
2. Elastic suspension of electromagnets to the body leads to the fact that the
controi system should stabilize the mass which comprises 10-20 percent of the total
mass of the car and essentially completely suppresses the excitation of the high- -
frequency harmonics of elastic oscillations of the body itself by the control sys-
- tem. The result is expansion of the zones of stability and a sharp reduction of
energy expenditures to correct track disturbances.
3. The elasticity of the skis in the lateral direction permits easy tracking of
unevennesses and deviations of the track structure and of each individual rail in
the lateral direction and also makes it possible to enter the curvilinear sections
of the route.
y y y~
~ ~ .
0 T
o ~
b
~1, s Q; S~
- Z'
Figure 2. Diagram of Suspension With Electromagnetic Skis
4. Continuous distribution of the electromagnets along the body permits a reduc- _
tion of their cross-section and accordingly of the cross-section of the ferromag-
_ netic rails, a reduction of linear loads on the supporting frame of the car and
the track structure, a reduction of losses to magnetic resistance during motion,
i.e., in the final analysis a reduction of the mass of the car and track structure.
S. The operating reliability of the suspension system structure is increased
sharply due to significant functional redundancy: the capability of operation upon
failure of any single electromagnet is retained and an emergency situation with.
simultaneous failure of several electromagnets is prevented since adjacent electro-
magnets take on their functions.
� 6. The system under consideration of course is not devoid of specific deficiencies,
namely: the lower absolute lift of the electromag:~et reduces its specific mass
characteristics, while the presence of an individual stabilization system fAr each
electromagnet complicates the control system as a whole. However, the advantages =
of the described configuration scheme undoubtedly exceed the disadvantages. ~
68
FOR OFTICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/48: CIA-RDP82-00850R000400014436-7
FOR OFFICIAL USE ONLY
- There are specific problems which require careful consideration in analyzing the
dis turbed motion of the skis and of the entire "car-ski-track structure" closed -
sys tem, which will be discussed below.
Mathematical mc3e1 of a sin�le electromagnet with clearance and current control.
Let us consider an electromagnet extended in the direction of the longitudinal
_ axis of the car having U-shaped cross-section in the plane perpendicular to this
- axis. Let us assume that it is attached to the car by viscoelastic couplings
_ which provide two degrees of freedom to it with respect to the car: motions in
the vertical and lateral directions (Figure 1). Let us assume that the car having
- significantly greater mass than the electromagnet is immobile or movable at con-
otant velocity v. Let us take the state in which there is a small permanent gap
s between the electromagnet poles and the ferromagnetic rail of the U-shaped cross-
section and in which there is no deflection of the electromagnet in the lateral
directi~~n (its plane of symmetry coincides with that of the rail), as steady un-
disturbed motion (or equilibrium) of the system. With regard to the plane of sym-
metry of the rail, let us also assume that in this state the rail has no devia-
tions from the nominal pasition either in the vertical or lateral directions. Let -
us introduce right-handed coordinate systems: Oxyz bound to an ideal system (hav-
ing no adjustment errors and undeformed by the track structure) and Oxyz, rigidly
- connected to the car. The Ox and Ox axes of these coordinate systems are directed
along the longitudinal axis of the electromagnet in the direction of motion. The
origin O is located at an arbitrary point, as which the end of acceleration to
velocity v may be taken, so that x= xp + x= vt + x. Point 0 lies on the plane
, of the electromagnet poles. Assuming that deviations of all the generalized co-
ordinates and of their first derivatives during disturbed motion are small compared
, to the corresponding unperturbed values, let us make use of the general equations
of disturbed motion of an electromechanical system with m independent electromag-
nets having n degrees of freedom [9]:
r m
~ ~Q,rq~+bn4;-f'(c;;-d~;)4r)-~ etkla=Q~,
h~!
m n ~1~ _
~ (L~,�~.+R~.S~rI.)+~ e~9r=U~
- s-~
~J=1, 2, . . n; k=1, 2,. . . , m),
~ where
1 m ~ V~Ur~ t o
d�-d" ( ~ Ir�I'�'
2 ~ aq;aq;
r,~~t
e'"-~ ~ aaq; ~ 01'~- a
qr, I'~~ (2 ~
.v,
~ at k~r
a~,=
1 at k=r;
69
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFI('IAL USE (~NLI~'
Q~ is the generalized force correspo~nding to the j-th coordinate, Uk is the voltage
applied to the k-th winding, Ir is the current in the r-th electromagnet, q~ is the
j-th generalized coordinate, Lkr (r ~ k) is the mutual inductance coefficient, Lkk
and Rkk are the self-induction and effective resistance coefficient of the winding
of the k-th el ectromagnet and aj i, c~ i and b~ i are the matrix components of the
generalized mass and stiffness and of the coefficients of the mechanical dissipativie
funr.tion. The superscript � c~rresponds to the parameters of unperturbed motion.
Let us apply equations (1) to the case of a single electromagnet with two degrees
of freedom and let us reduce the equation of current and clearance control to it:
- U=.sa~ (s~) +5~ (I), ~3 ~
where,sat and5~ are linear differential operators which we shall subsequently take as
secc~nd and first order, respectively. Using the more meaningful notations, we find
as a result the follow~.ng equations of perturbed motion:
m (~v+~v~~~"Qvzfv) =Fv-~-Fy (t~ ;
m ~=f:~-a.'f �F,+F,� ~t) ;
Cr I -F'RI-c,~Sy=U; ~
U=aos�+a,s�-I-a2s�-}-bol ~i-b,l ; ~ 4 ~
Fv=cll-c,,,sv; F==-c�szj
S~=-f~+'~v~x); S:=f=-}':~x),
where
(s~) , o ~~s (4)
Lu ~ o , C, =L� = o ,
s +s� s -
1 82Lu 1�Z
~~:=-d==--- p
2 BsZ~>=~'
s�
~,=c.'=e~~=L��(1'/s�), ~,,,=d>>=L��(Io/s�)2~0,
d~==d~~=U, e,Z=es~=0, (S)
:
a~ =c~�/m, Q=z-~~=/m~
m is the mass of the electramagnet, gy, 3y, cey and ceZ are the damping and stiff-
ness coefficients of the viscoelastic couplings, F and U are the ponderomotive
force. and voltage, sy and sZ is variation of the vertical and lateral clearance,
respectively, fy, fZ and fy(x), fZ(x) are the generalized coordinates which char-
acterize the positiori of the electromagne~ and the rail, Fy (t) and FZ (t) are the
external perturbinq f.orces reduced to the center of mass of the electromagnet Gp
(Figure 1); coefficients djj (j = 1, 2), LZ1 and ell depend on the ronfiguration of
the system and can be determined by means of known methods of simulation of electro-
magnetic: fields [10] .
Let us introduce a new variable related to Fy by the relations :
70
FOF OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
F'C1R OFFICIA~I. USE ONLY
F~ =C,~Sy'~' b,
(6)
c,ao
c;,,=k,-c.,+; k,= R >c.,,.
At ~ = 0 we find the quasi-static electromagnet control by the law
U=a,s,,, I =aos�/R. ~ .
Therefore, S has the meaning of "control" with respect to a variable control ob- -
ject in ~ahich an electromagnet with quasi-static control is transformed, making it
statically stable in the vertical direction (ciy > 0). Static stability is pro-
vided in the lateral direction by the edge effect intensified by the ii-shaped
cross-section of the poles and rails (cSZ > 0).
After expressions (6) are substituted into equations (4) and after 2 is eliminated
from the third equation, one can find
m( j'~+~~ f y+~~= j~) =d+F~' (t),
m (~'=+~=f.-~~:=f:) =F=' ~t) ,
~ea~'b-~-Qei~s~-~-�e,Sy=aauU, ~8 )
U=aos�-4-a,s�-~-a=s�-~bol-F b,l,
~ 1=aiad-~ar,sv,
Sy=-fr+l ~~x~� .
Fiere
_ / = 1
~Y - m \C~Y+~iY~ = m ~~�v-~.r+k+~ i (il: _ ~~rx+Ci:~ ~
m
~e=cr /II, aeu=Cr/R, �1e=1/cf, ~ ~9)
.
~ �~f ao
Qo. =c;y R, ae.=k� ai.= R,
F~'(t)=Fv�~t)+~rv~v~x), ~''z'~t)=F,�(t)-I-c.t~:(s).
The system of equations (7) is a variant of the mathematical model of an electro-
magnet convenient for investigation of the dynamics of an electromagnetic ski.
Mathematical model of an automatically stabilized electromagnetic ski, Let us con-
sider an elastic beam on a linear viscoelastic base (mechanical suspension) with
electromagnets continuously distributed along its length that do not interact with
eaah other as an idealized model of an electromagnetic ski. Let us describe the
latter by mathematical model (8), assuming that all its characteristics (4) and (9)
are related to the unit of length in the directian of the yongitudinal axis of the
electromagnet.
71
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-44850R000400014436-7
FOR OFFICIAL USiE ONLY
- I,et us now take int.o account the perturhed motion of the car. Let us use the co-
ordinate systems o;~yz an~3 ~xyz introduced above, to which we add the intermediate
coordinate system 0*x*y*z* connected to the car during undisturbed motion. Let
us place the origin O on tr.e line of intersection of the plane passing through the
center of mass of tne "car-ski" system and the end plane of the pole shoes (Fig-
ures 1 and 2). .
- The disturbed motion of the ski consists of its transient motion together with co-
ordinate system Oxyz (the disturbed motion of the car) and relative motion with
respect to coordinate system Oxyz (Figure 2).
Let us ascribe the same properties to the disturbed motion of the car as a rigid
solid as to the disturbed motion of each electromagnet and of the entire ski.
LinE~ar functions f* (x, t) and f* (x, t) of the displacements of the ski points
with respect to x correspond to Zthis transient motion.
1111 the values which characterize the mathematical model of the electromagnet in-
� troduced in the previous section now become functions of two independent variables
x and t and they may be dependent on time both explicitly and by means of x(t) as,
for example, ~`y (x) and ~Z (x) .
Let us assume that the track structure and the trestle at a given stage of investi-
gation are absolutely rigid. Taking kinematic relation~ into account
_ Sv~~~ t)=-fv t)-fv~~~ t)~'~v~x), (10)
s=(x, t)=f,'(x, t)+ff(x, t)-}'j(x)
and using equations (8) and also the boundary-value conditions for a beam with .free
ends, we find the following systems o~f equations in partial derivatives and boundary-
value conditions:
a~ ( aZfY~ aZfy a~v
a~ ~E~' 8x2 +m at2 +$y at +aYfv�S+F�' (x, t),
as as
~8 ac +8+a,,~~
a~ +a,,s~=a,uu,
- (11 ~
_ u(x, t)~�~sY-~-a!'
~3tY-I-Q= aZtz -f-boi+b, 8t'
i t~ =dlps-~-�f.Sri
SY ~x~ t~ fY+~Y ~x /-fY ~xt t~ f
a~ tE~i a2yZ ~=o; E~. a~ =o at x=t 2;
z Z z c~2 ~
a~2 ,~E~v ax3 at2 +~saat+�`�fj=E,�(x,t),
ax ~EJ" axZ ~=o, E~y a~s =o at x=f
~z
FOR OFFICIAL USE ONLY
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000400010036-7

APPROVED FOR RELEASE: 2007102/08: CIA-RDP82-00850R000400010036-7
FOR OFFIC'[AL USE ONLY
Here
Fy' t~ =fv ~x, t~ -m ~Zf
~t
t) +C'vsv~ ~x, t~ ~
Fi� (x' t) = f=� (x' c) -m a f atx t~ -~.:S=' (x, t),
- \ { c~3)
SY�~xi t/-~V~x~-lV ~~f t~~
S=' (x, t) =-I = (x) ~'f=' t) ,
IXv=Cty+Cnr, a:=C�:~C~:;
1 is the length of the ski, EJy an~3. EJZ are the beriding stiffness in two planes of
the ski as an elastic beani and u(x, t), i(x, t) , fy (x, t) and fZ (x, t) correspond
to U, I, Fy and FZ contained in (8) aad (9) .
i9e found ~~ao independ~nt boundary-value problPms (11) and E12) which toqether with
additional relations (13) form a closed mathematical model of an automatically
stabilized electromaqnetic ski. If one assumes that csZ =-csz? where csZ > 0 and '
0, then this nodel will describe a ski with permanent repulstion magnets (with-
out a control system). _
The equations in partial derivatives (10) and (11) can be reduced by the Bubnov-
Galerkin method to an infinite system of ordinary differential equations which is
reduced upon transition to one or another discrete model to a finite system of
linear equations. It is convenient to select the harmonics of the natural oscilla-
tions of the ski at 0, = 0 and 0 as the coordinate functions.
Some problems of the dynamics of the "car-ski" system. Let us consider the dis-
, turbed motion of a ski in the vertical plane, taking as undisturbed motion the mo-
t10I1 of the ski with constant velocity v in the direction of the nx axis and with
permanent gap s�. Let us disregard damping in the elastic couplings and let us
appxoximate by sine waves the harmonics of natural oscillations of the ski in the
first approximation. Let the track structure have sinusoidal disturbances ~y(x) -
with period ~ with respect to x(wavelength) . Introducing into consideration the
speed o.f propagation of the j-th travelling wave v~ = J~~Q~/2 , where ~j and Q~ are
the length of the standing wave which approximates the ~-th harmonic of natural
oscillations of the ski and the natural frequency corresponding to it, we find the
critical travellinc~ speed of the ski at which resonance corresponding to excita-
tion of the j-th harmonic of its natural oscillations begins at aj . The fol-
lowinq inequality must be fulfilled for nortna:. functioninq of the ski
v