JPRS ID: 8962 TRANSLATION ABSORPTION OF VIBRATION ON SHIPS BY A.S. NIKIFOROV

Document Type: 
Collection: 
Document Number (FOIA) /ESDN (CREST): 
CIA-RDP82-00850R000200030003-3
Release Decision: 
RIF
Original Classification: 
U
Document Page Count: 
246
Document Creation Date: 
November 1, 2016
Sequence Number: 
3
Case Number: 
Content Type: 
REPORTS
File: 
AttachmentSize
PDF icon CIA-RDP82-00850R000200030003-3.pdf10.15 MB
Body: 
APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 =1 ~w 0 4 DECEMBER 1979 BY S. V. CHEREMNYKH CFOUOy _ES i OF 3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02108: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USF. ONLY JPRS L/8794 4 December 1979 - Translation Stabilization afSpace Vehicles _ By S.V. Gheremnykh FBIS FOREIGN BROADCAST INFORMATION SERVICE FOR OFFICIAL USE ON LY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 NOTE JPRS publications contain information primarily from foreign " newspapers, periodicals and books, but also from news agency transmissions and broadcasts. Materials from foreign-language - sources are translated; those from English-language sources are transcribed or reprinted, with the original phrasing 3nd other characteristics retained. Headlines, editorial reports, and material enclosed in braGkets 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 information 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 contents of this publication in no way represent the poli- c ies, views or attitudes of the U.S. Government. For further information on report content call (703) 351-2938 (economic); 3468 (political, sociological, military); 2726 (life sciences); 2725 (physical sciences). COPYRIGHT LAWS AND REGULATIONS GOVERNING OWNERSHIP OF MATERIALS REPRODUCED HEREIN REQUIRE THAT DISSEMINATION OF THIS PUBLICATION BE RESTRICTED FOR OFFICIAL USE OiNI,Y. APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY JPRS L/8i94 4 December 1979 STABILIZATION OF SPACE VEHICLES - MosCOw STABILIZIRUYEMOST' KOSMICHESKIK.H LETATEL'NYKH APPARATOV in Russian 1978 signed to press 31 Oct 78 pp 1-208 Book by S. V. Cheremnykh, Mashinostroyeniye Publishing House, 1200 copies CONTENTS PAGE Foreword 2 Chapter l. Simplest IJonconservative Oscillatory Systems 6 1.1. Examples from Mechanics. Autonomous Systems with Ttao Degrees of Freedom 6 1.2. Oscillations of Unstable Systems lg 1.3. Effect of Dissipative Forces on Stability 32 1.4. Controllability of the Oscillatory System with Two Degrees of Freedom 40 Chapter 2. Stabilizability of Oscillatory Systems with One Input 52 - 2.1. Terminology: Stabilizability, Structural Properties, Stability. Formulation of the Basic Problems 52 2.2. Study of the Stabilizabality of Oscillatory Systems with One Input 63 2.3. Dynamic Instability as a Form of Nonstabilizability of the Object of Control 81 - 2.4. Investigation of the Dynamic Instability of an Object of ~ Control 83 Chapter 3. Stabilizability of Space Vehicles 102 3.1. Mathematical Models of Space Vehicles (Movement in the Active Segment 102 3.2. Simplest Cases of Investigatian of the Structural Stability of the Space Vehicle 113 ' a ' [2 - USSR - A FOUO] FOF: OFFICIAL U3E ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY 3.3. Stabilizability for the Classical System (Two 124 3.4. Fuel Compartments Space Vehicle with Engine on an Elastic Suspension 136 - 3 5 Other Examples of Investigation of the Stabilizability . . of a Space Vehicle 146 rhapter 4. Application o� the Theory of Stabilizability to the Problems of Space Vehicle Design 155 4 1 Investigation of the Structural Properties of the . . Designed Space Vehicles ' 155 4.2. Stabilizability Criterion as the Quality Criterion of 162 the Composxtional Layout of a Space Vehicle 4.3. Stabilization of Dynamically Unstable Space Vehicles (Calculation of the Damping Coefficients and the - Parameters of the Automatic Stabilization System) 170 4 4 Stabilization of Structurallq Unstable Space Vehicles . . Using a Discrete Statilization A].gorithm 194 Chapter 5. Stabilizability and Stability of Sg3ce Vehicles in 216 the !~ctive Segment of Flight Approximate Method of Investigating the 5tability of Space Vehicles. Amplitude and Phase Stabilization 216 Effect of the Fuel Mobility in the Compartments on the 21g Stability of a Space Vehicle ' Consideration of Elasticity of the Space Vehicle 223 Structure Stability of the Space Vehicle with 9ngular Position Control System (Nonlinear Case) 228 230 Bibliography - b - - FOR OFFICIAL USE ONLY  APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY PUBLICATION DATA English title : STABIZIZATION OF SPACE VEHICLES Russian title : STABII,IZIRUYEMOST' KOSMICHESKIKH LETATEL'NYKH APPARATOV AuLhor (s) . S. V. Cheremnykh - Ed:Ltor (s) , Publishing House : Mashinostroyeniye Place of Publication : Moscow ~ Date of Publication : 1978 Signed to press . 31 Oct 78 Copies : 1200 COPYRIGHT . Izdatel'stvo "Mashinostroyeniye", - 1978 - c - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY UDC 629.78.017.2.001 . STABILIZATION OF SPACE VEHICLES NEW PROBLEMS AND METHODS Moscow STABILIZIROYEMOST' KOSMICHESKIKH LETATEL'NYKH ApPARATOV in Russian 1978 signed to press 31 Oct 78 pp 1-208 [Book by S. V. Cheremnykh, Mashinostroyeniye, 208 pages, 1200 copies] [Text] A study is made of the problems of stabilizing spacecraft in the - active segments of the flight from the point of view of some new methods of motion control theory. A new approach to investigating the dynamic characteristics of a space vehicle as an object of control is discussed which is a development of the controllability and observability theory of Kalman as applied to the given class of obj ects. ihe study is made of various problems in tre analysis of the spacecraft dyanmics encountered in various planning and design stages. The book is intended for engiueerii.g and techr.ical workers involved in designing rockets and other flight vehicles. 1 p FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 ' FOR OFFICIAL USE ONLY - FOREWORD In this paper a discussion is presented of the methods of investigating the stabilizability of linear dynamic systems including an oscillating controlled target and controls : of given structure as applied to the problems of conCrolling a spacecraft with a liquid-propulsion rocket engine. - It is known that liquid-propellant rockets are very difficult subjects for stabilization because of the unfavorable dynamic characteristfcs arising from the mobility of the fuel in the tanks, the elasticity of structure and also the nonsteady-state nature of the characteristics of the vehicle and the environment. Therefore in spite of using the latest methads of _ synthesizing control systems for such objects, frequently the optimal quality indexes which could be achieved are not achieved. At the same time, in the design phase of the space vehicle as an object of control in practice there are always unused possibilities for selecting the structure and elements of the composite system basically determining its dynamic characteristics in the process of controlled movement. The problem of how these possibi].ities can be used is the starting point for the studies, the results of which are discussed in this paper. Of course, it would be desirable to solve the problem of optimizing the ~ dynamic characteristics of a;pace vehicle in the most general form, consider- - ing the cl-osed target-controller system as a whole, also taking into account ths ballistic, strength and other requireinents. For many reasons, primarily as a result of the "crrs,~ of size," as R. Bellman puts it, this is in - practice impossible an4 against oux wills it is necessary to limit ourselves to more modest goals. Let us note that in the liquid-propulsion rocket design developments the situation is typical where the structure of the controls is rigidly given for one reason or another. For example, a spacecraft for a different pur- pose designed on the basis of some basic version can have the same auto- matic stabilization system with respect to structure with, perhaps, only the values of the paraffieters altered. 2 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY In such situati.ons, when the structure of the controls is defined, it is natural to ask the following question: how are the structural parameter.s of the object of control selected so that it will have the best character- istics with respect to the controls of the given structure? To solve tliis problem i*_ is desirable to have an optimalneas criterion which does not _ depend on the specific parametere of the controls determined only by thc str,.:cture of the controls and the characteristic parameters of the ohject. Farmally, it is possible to construct such a criterion only after provirig the theoretical possibility of separating the regions of stability and space of the parameters of the object and the centrols (under assumptions that are reasonable for the investigated class of systems). So far as the author knows, this was done for the first time in the papers by B. I. Rabinovich [57], I. M. Sidorov, I. P. Korotayeva [67] for controlled oscillatory systems with a controller which slightly disturbs the natural frequencies of the auxiliary oscillators. There were other prerequisites for studying the problem of optimizing the dynamic properties of the object of control from general points of view. The fact is that in general control theory as a result of the work of R. Kalman and other researchers, there is a tendency at the present time to separate the investigation of the problems pertaining to the controlled system itself as the object of control into a separate region (the problems of controllability, observability [27], invariance [64], directivity [51], and so on). _ Although the structure of the control system itself is completely ignored - here, the corresponding criteria provide valuable information about the behavior of the object of control in the control process. - The methods discussed in this paper occupy an intermediate position between the corresponding methods of controllability (observability) theory and the classical theory of stability. The structure of the control system here is significant in contrast to the Kalman theory; at the same time the specific ' values of the parameters of the control system do not enter into various criterial relations, and the results of the investigations are formulated in terms of the regions in space of the parameters of the object of control itself. In order to emphasize this fact, and also considering that the concept of stability is too overworked, in this book, following the lead of reference [56], we shall call the criteria characterizing the object of control the - stabilizability criteria. The idea of the proposed approach consists in the following. Some formal analog a quadratic form with coefficients which depend on the parameters of the object of control is placed in correspondence to this object. The positive (or negative) definability of this form is identified with the concept of perfection of the object (for example, the space vehicle) in the dynamic sense. 3 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY Testing the Sylvester conditiona far the mentioned quadratic form and the construction of the corresponding regions in the space of the parameters of the object of control make up the content of the methods of investigating the stabilizability of controlled oscillatory systems. If the terminology adopted in space vehicle dynamics is used, the discussed = theory is the theory of "phase" stabilization of oscillatory systems. In pracrlice when "phase" stabilization of the space vehicle is impossible, usually the problems of "amplitude" stabilization are investigated which are essentially the classical problems of the analysis of the stability of moving objects. The methods of investigating stabiZizability are to some degree analogous to the above-mentioned methods of investigating controllability, obaerva- bility and so on in general control theory: both permit the general analysis of the properties of the object of control as the first step in solving the classical problems of stability of motion or various problems of optimal control. Chapter 2 of this book contains a discussion of stabilizabilit; theory. Chapter 1 is an auxiliary chapter. The simplest model including two connected oscillators is used to investigate some of the problems characteristic of the modern theory of linear controlled systems (the prob- - lems of dynamic instability, controllability, observability, modal control in various situations, and so on). Chapters 3 and 4 are of an applied nature. In these chapters a study is made of the problems of the stabilizability of various models of space vehicles and also adjacent problems pertaining to the design of optimal (in the dynamic sense) obj ects. The mathematical models of :space vehicles are used to the degree of complete- ness which corresponds to the level of the initial design phases of objects of this type: as a rule the equat'-ons are assumed to be linear, the coefficients are considered constant ("frozen" for some characteristic point in time T of the activ,. segment). The oscillatory natura c` the object of control in the given case comes from the presence of moving fuel components used to operate the sustainer - engines and also elasticity of the hull and other structural elements. Automatic stabilization systems (in the transverse oscillation mode) are used as the control systems here, and in the case of longitudinal oscilla- - tions, the engine is used directly. The perfoxmed studies of specific composite systems of space vehicles and the standard conditions of space ' vehicle movement provide a basis for considering that the discussed methods are a quite effective tool for investigating the dynamic properties of flight vehicles with liquid-propulsion rocket engines under the conditions 4 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE nNLY of incompleteness of a priori information about the stabilization system. It is appropriate to emphasize here that the effectiveness of the methods of inveatigating the stabilizability is especially noticeable when they are used in combination with the classical methods of analysis, preceding tttem in the general process of investigating the stability of the investigntc:d class of objects. Therefore, a discussion is presented below (see Chapter 5) of a number of the traditional methods of analyzing the stability of the _ closed system made up of the space vehicle and its control system, and t11e problems of amplitude stabilization are investigated. The author gives special attention here to the interpretation of the regions of stabilizabil- ity of space vehicles when investigating the stability of the control pro- cesses in the active segment. On the whole, in this book the author would like to attract the attention of the readers to the new possibilities which are offered by successive. (physical) analysis of such characteristics as controllability, observabil- ity, stabilizability, and so on as applied to dynam,ic systems of a large number of oscillatory degrees of freedom and with limited possiUilities of modal control. If we are talking about the general proUlem of stability, which in no way replaces the classical methods, this approach helps us to find the primary causes of instability and either to eliminate them or determine the direction of further research. - In conclusion, the author expresses his deep appreciation to doctor of _ technical sciences, Prof B. I. Rabinovich for valuable suggestions made when reviewing the manuscript of the book and also engineer Yu. V. Shchetinin for his assistance in preparing the manuscript for publication. It is requested that all critical comments and suggestions be sent to the - following address: Moscow, GSP-6, 1-y Basmannyy per., d.3, izd-vo "Mashinostroyeniye." 5 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY CHAPTER 1. SIMPLEST NONCONSERVATIVE OSCILLATORY SYSTEMS 1.1. Examples from Mechanics. Autonomous Systems with Two Degrees of - Freedom Aircraft Wing Flutter Flutter is the classical example of a phenomenon which cannot be explained by investigating the system with only one degree of freedom. This form of instability is characterized by intensive interaction of at Ieast two oscillatory elements under the effect of external nonpotential forces. The names given to the var.ious forms of flutter illustrate this fact: Bending-aileron wing flutter (bpnding vibrations of the wing combined with aileron vibrations); Bending-rudder flutter of the horcizontal empennage (bending vibrmtions of the fuselage in the vertical plane jointly with vibrations of the elevator around the axis of suspension); Torsional-rudder flutter of the horizontal empennage (torsional vibrations of the fuselage combined with vibrations of the elevator and tail aesembly), and 30 on. This is how this phenomEnon uppears to observers from the outside [30]: "While testing an expe-imental aircraft, a twin-engine monoplane, the wings _ began to vibrate unexpt,tedly. This occurred while the aircraft was flying a measured base line At -aaximum speed near the ground. The wing vibrations began abruptly and were of an antisymmetric nature, that is, if the right half of the wing went up, the left half went down at the same time. Power- - ful vibrations of the ailerons occurred at the same time, so that the controls were jerked out of the pilot's hands. In the given case the pilot made the right decision: he throttled down and in spite of the fact tbat the controls had been jerked out o� his hands, he suceeded in signif- - icantly taking up the elevator. As a result, the spee3 b.egan to drop sharply, and when it had decreased by about 20%, the vibration4 stopped almost as abruptly as they had begun. Five to eight seconds passed between the beginning and end of the uibrations. After the vibrations stopped, - the aircraft behaved normally, and af ter 5 minutes of flight the pilot - made a good landing at ':he airport. 6 FOR OFFICIAL USE OrfLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY c,(yt Zslt ZBl 6 ~ y al I---~ ( I I J cz . m~i B~2 B~Z - _ Figure 1.1. Force diagram on an inclined plate "Examination of the aircraft revealed that at soffie points in the structure of the center wing section rupture of the skin and cowlings had started - (cracks had formed), and some of the rivets holding the skin to the longerons, - stringers and ribs had sheared off. Residual 3eformations in the form of waves in the skin had appeared on the surface of the center wing section between the engines and the fuselage. Clay was detected in the suspension of the ailerons and the aileron servotabs. "However, flutter does not always end so favoYably. Cases are known where vibrations that began in flight have led to complete diaintegration of the _ aircraft in one or two seconds or less. It appears to observers of such an accident from the ground gs if the part of the aircraft where the _ vibrations started has exploded." Thus, observations indicate that under def ined flight conditions vibrations o� the fuselage and control surfaces which are extraordinarily intense can occur under def ined flight conditions. The complete theory of flutter is highly complex..[9, 25, 76]. Here, only = a suitable mechanical model [50] which can be used to explain the primary - aspects of the nature of this phenomenon which once was a threatening obstacle on the path of increasing the speed of aircraft, will be investigated. _ - Let the plate depicted in Fig 1.1 have two degrees of freedom. We shall characterize its position by two coordinates the angle of rotation ~ and vertical displacement y of the center of the plate. Horizontal displacements will be considered imposaible. Zet us write the equations def ining � and y as a�unction of time: _ �=0(t); Y=Y (t)� _ 7 - FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02148: CIA-RDP82-00850R040240030003-3 FOR OFFICIAL USE ONLY The tensions of the two springs cl and c2 will be assumed to be different, r and the mass of the plate, uniformly distributed over its entire surface. Let us denote by m the mass corresponding to a unit area of tlie central plane of the plate. I'uring its movement, the plate is acted on hy the list Y = da Q bltP, running at a distance a from the left edge of the plate and a7.so the force of reaction of the elastic supports proportional to the displacements of the long sides of the plate: 2 T )C11; Rs = - (y - 2- CP) c11. Reducing these reactions to the center of gravity of the platc:, we obtain the force R=Rl-f-R2=-(CI-4-C2)ly- 2 (c1-cs) t'f _ and the moment _ M = (y + 2 cpl c,/. l (y - i ~pl c2l ? , ~ J - tlow let us select the equations of motion of the plate. One of them describes the center of the gravity of the plate: . Y-{-R=mbl dd 2 where mbl is the mass of the erizlre plate and the other, the rotations of the plate around the horizontal axis z passing through the center of mass: b _al+. M=mb''1 d2~ , (1.2) 2 f 2 dt2 Substituting the expre6:.ions for Y, R and M in equations (1.1) and (1.2), we obtain the following system of differential equations: d~ ~aijy+aiz'?=0; (1.3) d2T . 72 ,ra2ly--a22'?=ol 8 ' FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY where 11 C2 3(cl-f- c2) dc4 v'' b- 2a ~ti = mb ~ a21- mb ^ da e 2 M2 ' ~ CC, _dcy va 1 6 (ct-c�t) t2 - 21n d~t ~ 2 m'~21= rnb2 ' (1.4~ When solving the problems of aeroelasticity, to which the discussed topic belongs, first the conditions are defined to which the static form of the stability loas, the so-called divergence, is poasibl.e. For this purpose it is propoaed that y and 0 are constants. Then the second derivatives vanish and the equations assume the form aiiy-i-a12?=0, a21Jd-a22?=0. (1.5) The condition of nonzero solutions of system (1.5) has the form ajjaza - ~j,_UZ1= 0. Substituting the expressions (1.4) for the coefficients, the following formula is obtained, which defines the critical divergence rate: ~ Ci v~=2 Q dcy 1- cl . da l cZ ) Key: 1. critical ~ y 9 ~ t -~`'t t (1.6) -y- ~ i-~C_======7C_~ _-s- -s ~ --Jiiii T -s-. i 1J771 d ''b .F Figure 1.2. Types of movements of the plate after in3.tia1 deflection: the axis of rigidity is denoted by the x, the center of mass of the plate is in the middle of the span. 9 FOR OFFICIAL liSE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY Divergence as a static form of loss of stability is possible obviously when - cl 0; . . . . (2.39) Substituting the values of u=0, u=-v12;...; u=-o 2 in the function gl(u), successively, we find that the inequalities [2A), and tagether with them, the conditions of stability of the given system (2.34) are satisf ied if: a) . Xo >O; xi >0 (2.40) 62 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY _ (the requirements on the parameters of the control system); b) bspl; >O; 1=1, 2,..., rn (2.41) ~ (the requirements on the parameters of the object of control). As is obvious, the situation is such that the stability of the given system (2.28)-(2.29) can be insured as a result of the successive performance of two operations: Ad3ustment of the cGntrol system parameters [in order that the conditions (2.40) be satisfied]; Selection of the parameters of the ob3ect of contirol in accordance with the inequalities (2.41) (having the senae of conditions of structural stability of the object). Let us introduce the space of the parametera of the object of control af dimensionality 2m( 2m (b,l, bem, ni', �..,'~m J+ , in which the conditions of uncontrollabi.lity or observability of the syatem bQJn;=O (j=1, 2,..., m) (2.42). are isolated by certain boundaries. The conditions of structural stability (2.41) of the.object of control provide the decoding of the regions on both sides of the boundary (2.42), which permits investigation of them 4s a generalization of the Kalman conditions as applied to the investigated apecial problem. The establishment of the conditiona of the type of (2.41) in the general case of oscillatory systema including n oscillatora and conatituting the object of comtrol with one input is the basic problem of further analysis. Here the central event is the theoretical possibility of the separation _ of investigation of the object of control from analysis of the closed system as a whule within the framework of reasonable assumptiona, which in _ the given case leads to the necessity for introducing the concept of controllability and observability of the system (problem 1), structural stability (problem 2), and then the concept of stabilizability of the object of control. _ Formalization of the Object of Control and the Control System. Statement of the Problem ` Let us consider the system of differential equations x =eBx-{-Ax+bu, (2.43) 63 FOR OFFICIAL USE aNLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFF'ICIAL USE ONLY where x is the vector of the generalized coordinates of the system: t is the control vector; u(t) is the control input; A is the matrix of dimensionality nXn, the elements of which depend, possibly, oii r parameters v1,v2,...9vr; e is a small parameter. - Let uH make the following assumptione: I. The system (2.43) is a set of connected oscillators characterized by - the frequency vi (i=1, 2,..., n), and it can be (for u(t)=0) both stable and unstable as a result of the effect of positional nonconservative forces. - H. The elements of the matrix B of generalized dissipative .4orces are - small, which is characterized by the introduction of the small parameter-E. III. The variability of the coefficients of the system (2.43) is small in the characteristic time interval T-2ff/a. - _ Then let us propose that the measuring device of the control system receives the signal , v(E)=(g, x), (2.44) ~ which is a physically observable value; the vector g=(gl,...,gn) is the observation vector for the investigated system. The equation of the control system will be assumed in the following form: L (P) u= Li (P) V, (L). - where L(p)=L1(p)/L0(p) is the transfer function of the control system given by its frequency characteristic _ L(tw)=A(w)(COSy(w)-}-isinp(u,)], With respect to the control ~ystam we shall assume the following: _ 1. The eigenvalues of the operator Lp(p) belongs to the region of stability QZ which dceG not intersect with the region of eigenvalues QA in the matrix A for ail the variations of the parameters vl, v2,...,vr in the given region. _ 2. The disturbances Apj (j=1,2,...,n) o� the eigenvalues (pi 2) of the - matrix A caused by the effect of the control system are small in the - sense that (Apjl�Q, where Q is the characteristic frequency. 64 . FOR OFFICIAL USE ONLY _ APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOA OFFICIAL USE ONLY Figure 2.1. Roots of the characteristic equation of an open system: � o-- object of control; x-- control system 3. The condition sign [Im L(iwj]=const is satisfied, where wj=lm pj; pj are the eigenvalues of the matrix A. The condition 1 means that the control system as an element of the cloaed automatic control circuit is asymptotically stable, and it retains this property under all conditiona of movement of the investigated system. _ Fig 2.1 shows the roots of the c.haracteristic equation of this system (2.43) including three connected oscillators characterized by the frequencies - al, 02, Qg (the symbols Q) (u(t)=0, e=0). On the same figure the multipli- cation symbol denotes the eigenvaluea of the operator Lp(p) satisfying the condition l. The condition "2" obviously imposes restrictions on the amplification - coefficient of the control systems and means tha.t the eigenvalues of the - closed system made up of the object of control and the control system are closed to their rated values (Fig 2.1) calculated in the open state of the system (u (t ) =0) . Condition "3" is the condition of "uniformity" of the phase shifts under the effect of the operator L for all of the eigenvalues pi of the object of control. � Fig 2.2 shows the amplitude an3 phase characteristics of the control - system satisfying condition 3. In the given case, as is obvious: sign (Im L (iwj)] - -~-1 ( j =1, 2, 3). ' Let us investigate the problem of the so-called phase [56] stabilization of ~ _ the object of control (D), and accordingly let us consider the following problems. 65 , , FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY Problem 2.1. Let the dynamic system x Ax+ bu, v=(g, x), (D) Lo(P)u = Li(P)v, (L) be given, where the object of control (L) is def ined by the properties I-III, the control system, by the properties 1-3. It is required that the regions in which the closed system (D)-(L) will be stable for any control system satisfying the conditions 1-3 be isolated in the space of the parameters of the object (D). Problem 2.2. Let the dynamic system be given: x-Ax+bu' ----v_(g'.x) (D) Lo(P)u = Li(P)v, (L) where the object of control is defined by the properties I-III, the control system, by the properties_1-2. ~ r-----~ - ~ 2 0 x' 1 Figure 2.2. Standard phase-amplitude characteristics of the control Gystem What should the reqtiicements be on the regulator as alternative (3) in order to insure stabil.Lty of the system (D)-(L) in the given region of variation of the par4^_!eters of the object (D)? Let iDo(P2)=aoPs"`+ al.p2(m-1)+...-}-u2(M-1)P# -}-u�tm (2.45) be the characteristic equation of the open system [for _ uy) = 0, V(t) = pJ; pj(I � 1, 2,..., n) , be its roots. 66 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY _ Then the case PJ= -wi (I =1, 2,. . n) (2,46) - corresponda to the dynamically stable object (D), the case pi,s=a � lcu; P3,4= (2.47) corresponda to its dynamic instability. Then let us denote l(P2E-A)Lo(P)-(bg)Li(P)I=0o(P2)Lo(A)+`Dk(P2)Li(P)�0 (2.48) as the characteristic equation of the closed system (D)-(L); wt ,p)=V/U the transmission function of the control object; ~ (D); v(p), u(p) the Laplacian transform of the general;ized coordinates - v(t), u(t), uk(k=1, 2,..., n-1) zeroes of the transmission function W(p); Xk(k=1. 2,..., n) the ; onest of the transmission function W(p) ; , K=($, Ab*-, An'1b) the controllability matrix of the system (D); - G=(J', A'l (A')1111) the obaervability matrix of the system (D). Simultaneously with the system (D)-(L) we shall also consider the system Y=AY-{-pu, v=h'7', u=L(P)v, . (2.49) Q3 (where A=diag {al, X2,..�, describing the movement of the dynamic system in the base made up opthe eigenvectors of the matrix A. Let us introduce the followfng definition. Definition 2.1. The object of control (D) will be called stabilized, if: a) The ones ak of the transmission function W(p2) of the ob3ect are _ prime,, real and negative; b) The zeroa Ak and the ones uk of.the txansmissian function W(p2) ~ are permutated in the fdllowing ordert ~l< N < Xz 1 or c>0, �0, i=1,2) or lag (sin �i I~t ttl Also assuming that the frequencies wi are numbered tn increasing order, - let us calculate the values of the pt5lynomial Ok(w2) 'at the points w32 _ (J=1,2,...,m): m (Dp(i"i) =aaja1j/(1,~ n (-u>i--o,2), 1tl For structuraZ stabiliry of the {-nvestigated system it is necessary and sufticient that the signs of k(w'Z) alternate on going from number j to number j+l: - (Dk (W1) > di 'Ok (0)2) < 0; ok (.)3) > 0; . pT ok (Wl) < Q+ e,~k (0)2~ > 0; ODk (W3) < 0. 4) (~,j)-aQ u~; (1=1, 2,..., m), Let us denote k jm 2 2 2 where 4I=W1 n (-�~I+01~~, /..i Since Di>0; A20; A4 0 (1=1~ 2,..., N). The characteristic equation of the system (3.44) will be represented in the form (3.32). Setting p=iw, we find: (Do (wZ)= -w2 -a9s,w2 -a9s2(j)2 -aasNCU2 -(L,,Aw2 --w2-{-wi 0 0 -QS,gtu2 n - - w2-~-wn 0. . . . . . -(L1Npro2 . . . . 0 . . . . 0 . . . . . ...-W2--roN -ae, -aesico2 -aes,(o2 , . . -llgsNtu2 ' 0 -w2-}-wi 0 0 Ok0) z)= 0 0 -w2 -}-w~... ~ . . . . . . . . . . . . . . . . 0 0 0 -w2-~'~"~r i On the basis of the definition, the system (3.44) is structurally stable if - the zeros and ones of the transmission.function W(pz) alternate. 123 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 FOR OFFICIAL USE ONLY Obviously, this condition will be satisfied if the signs of the function (po(w2) at the points w2=Wi2 (i=1,2,...,N) alternate. We have = - = - 2) (3.46) (D0(1�?~ =a6s,as,,iio~~u~i-u11~...(to)~._I-~'~i~~O1r,1.1-ui~~...~u~, - y-~~~i . On the basis of conservativeness of the system (3.44) aesiasi8>0' Assuming that w1262); 0=001=v2); 0=1(Q10 (a body ot relatively large elongation with equal partial oscillation frequenciea of *_he liquidint~oximated e b) the ellipseseg of _ the regions (3.68) are quit~ pre..ise1y aPP y the - (k-~- ;h j~ky)7i-}-(l �,k)ZiZ2-}-(1-C1' 0)Z2+ ~ /~.1.. -}-(21/k--CI, k),Z,-{-(--l l , C)Zs+(1+k~k- z+ (k-Gk6~kY)Zi-~r,Yyk(-l--h)Z,Z:~~ (1--CYT) Z0; (3.69) +(-21/-k-I-Ch)Zj-I'(2z2 -}'(1+ k)=0 (3.70) - - with centers at the points ~ r 1 ! k - -7 ; r.; y 2 � ' - -f- G; y- - The axes of the ellipses are rotared with respect to Z1 axis by the angle - 1 ;Y(t -k') , iil, 2. , 61= 2 arctg :V(~-t-k):F }"k(1-k) Let us investigate the problem of the mutual arrangement of the regions of ~ structural and natural dynamic instability of the space vehicle. Let us consider the simplest case of c-0; R=0 when the boundaries of the regions of structural and natural instability are defined by the equations ~ .(Zi kZsi IZ,-- kZ2-- C (1-- k) 11 - Y (1-' k)J j _O+ - [Zi-- kZz - f-( l +k)+C (Zl kZz))Z - 4k (ZZ - Zl?X x [ 1+ GY (Zi + kZz)] =0. i- d e Figure 3.16. Mutual arrangement of the regions of'dynamic double croashatching) and structural instability oi the space vehicle with two fuel tanks: a, b, c t r_ameter c>0; d, e, f-- parameter c=0 134 FOR OFFICIAL USE ONLY APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 APPROVED FOR RELEASE: 2007/02/08: CIA-RDP82-00850R000200030003-3 F'OR OFFICIAI. USE ONLY From the general theory discussed in Chapter 2 it follows that the regions of natural instability are located inside the regions of structural instability. ZZ ~ ' - ~ Z ~ ~ 17. i `r ~ _ . f /3-0 ~ o> ! ~ ~ I 75 j I I I ' -Z J �I ~ Q b Figure 3.17. Nature of the effect of various oscillation frequencies of the fuel in thz tank on the curvilinear boundaries of the regions of structural instability of a space vehicle (0=-1+Q22/a12); a parameter SfO; b-- parameter 0;0 Zz -3 _ 1 p 7~�lo 1 O, ,C J ~ 2 Z~ aQ~' ~ -2 Figure 3.18. MutLal arrangement of the regions of natural and structural stab:ility of a space vehicle for c