INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES C 0 U R S E S A N D L E C T U R E S - No. 57 PETER SAGIROW STUTTGART UNIVERSITY STOCHASTIC METHODS IN THE DYNAMICS OF SATELLITES COURSE HELD AT THE DEPARTMENT FOR GENERAL MECHANICS OCTOBER 1970 UDINE 1970 SPRINGER-VERLAG WIEN GMBH ISBN 978-3-211-81092-7 ISBN 978-3-7091-2870-1 (eBook) DOI 10.10071978-3-7091-2870-1 P R E F A C E This textbook is based partly on the author's leatures on Satellite Dynamias at the Uni versity of Stuttgart and partly on an advanaed seminar on Modern Stoahastia Methods held by Mr. L. Arnold and the author at Stuttgart during three semesters in 1969 and 19?0. The author is deeply grateful to the partiaipants of the seminar for disaussions, help ful remarks, and the reading of the manusaript. The author is very muah indebted to Professor Sobrero, CISM, for the invitation to deliver this leature at Udine and for understanding aoopera tion. Stuttgart, 14-9-19?0. 1. Introduction The librational motion of a satellite depends on the gravity gradient, aerodynamic, and mag netic torques, on the moments of inertia of the satel lite and on some less significant influences as solar radiation, the electrical field of the earth, the me teorite impacts. Only the gravity gradient torque can be considered with a great degree of accuracy as dete£ ministic. Aerodynamic and magnetic torques can be treated as deterministic quantities only in a very rough first approximation. In reality these torques always have stochastic components caused by the fluc tuations of the atmospheric density and the earth's magnetic field. Furthermore, the calculation of these torques for a given satellite is based on assumptions whose nature is indeed stochastic. The same is true for influences caused by solar radiation, meteorite impacts, and electrical field of the earth. The moments of inertia of a satellite are deterministic only for completely rigid bodies. The uncontrolled thermo-elas tic oscillations of the stabilizing rods, antennae, and sun cell panels, the motion of the crew, the wob- 6 Chap. 1. Introduction bling of liquid in the tanks etc. make the concept of an ideally rigid satellite questionable. Thus, the moments of inertia have to be regarded as stochastic quantities too. In consequence of the stochastical nature of the mentioned outer and inner influences the librational motion of a satellite is a stochastic process and has to be described by stochastic diffe rential equations. The first conception of these equa tions was given in the late forties by It& [1]. In the following twenty years this first conception was developed by the works of Skorokhod, Bucy, Kushner, Khasminski a.o. The first books on this subject ap peared in the last few years : Bucy (2], Kushner [3], Gikhmanand Skorokhod [4], Khasminski [5]. Today the Theory of Stochastic Differential Equations and the Theory of Stochastic Stability are powerful tools in the applications and especially in the dynamics of satellites. In the present contribution the stochas tic models of librating satellites are considered. As the stochastic methods are new, a brief survey on stochastic processes, stochastic differential equa tions and stochastic stability is given first. Then some less complicated problems of stochastic dynamics of satellites are discussed. The stochastic stability Introduction 7 of the considered systems is investigated as well by Liapunov techniques, as in first approximation, and in the mean square. With the exception of the last example in section 6.2 all problems are solved comple tely. Some known results on stochastic stability of satellites are improved and new results are obtained. In the last section 6.2 a satellite with variable mo ments of inertia is considered. This example shows the feasibilities of stochastic modeling and the difficul ties which can arise. This example is discussed only. 2. Stochastic Dynamical Systems The correct way from the intuitive idea of probability to the theory of stochastic stability is long and fatiguing : Measure Theory, Probability Theory, Theory of Stochastic Processes, Theory of Stochastic Differential Equations and finally Stocha stic Stability. This tour is seldom done completely by mathematicians and is unacceptable for a man in applications. On the other hand, just the results at the end of this cumbersome way - the criteria of stochastic stability - are of great interest in mecha~ ics and engineering. Therefore, here the attempt will be made to reach the correct results on stochastic stability by a non rigorous but brief treatment based more on faith and intuition than on an extensive mathe matical background and sophisticated proofs. Only some elementary knowledge on probability is supposed. Sec tion 2.1 contains notations and basic definitions on density functions, moments and stochastic processes. In 2.2 dynamic systems corrupted by noise are intro duced. Sections 2.3 to 2.6 give a hint of an outline of the theory of stochastic dif.ferential equations. Gaussian distribution 9 Finally in 2.7 the basic definitions and results on stochastic stability are collected. As the topics of 2.3 to 2.7 are relatively new and the few books treat ing them are written on a very high level, the formu lae and theorems most important for the applications in sections 3 to 6 are deduced by heuristic conside- rations. For strong proofs see [3], [4], [5], (6] and consult previously [8], [9], [10] - if necessary. 2. I Basic Definitions 2. 1. 1 Probability Density Functions With a scalar random quantity x is associated the probability density function f(x) defined by the property X2 j P(x1 ~ x < x2) = f(x)dx, Xt where P (:x1 ~ X < X2) denotes the probability of the event Xt Et X < X~ • For normally or Gaussian distributed random quantities the density is given by f(x) = (2.1) where x and a2 are the mean value and the variance of x, respectively (see 2.1.2). 10 Chap. 2. Stochastic Dynamical Systems 2. I. 2 Moments Random quantities can be described by means of their moments. The moments are defined by = k "" 1,2,3, ••• and by • j E<x - Ex)k = (x- Ex)kf(x)dx, k = 1,2,3, ... -CII The most important moments are the mean or average value or expected value (2.2) x = Ex and the variance - xf. _(2.3) E<x - For normally distributed random quantities all higher moments (K > 2) can be expressed by the first moment x and the second moment cr2• The quantity ~ is the standard deviation. 2. I. 3 Random Vectors The first and the second moments of a