stability theory methodology in problem of the motions separation for sgs
- Paper number
IAC-20,C2,2,14,x56551
- Author
Prof. Lyudmila Kuzmina, Russian Federation, Kazan National Research Technical University
- Year
2020
- Abstract
The subject of research is the developing of methods for modelling and analysis in problems of the complex mechanical systems dynamics. Here for aerospace systems the methods of classical stability theory are developed; the generalization of the reduction principle for the general qualitative analysis is obtained (A.M.Lyapunov, K.P.Persidskiy). On the base of the unified approach, connected with comparison method (V.M.Matrosov, R.Bellman ),with combining of traditional methods of stability theory and of perturbations theory it is introduced initial statement, that is allowing to reduce the solving of the problems of motions separation and modelling to regular scheme. The methodology and systematic procedures are formed for constructing of the comparison systems. Here as generating system and, accordingly, as generating solution are taken nonlinear (on the totality of all considered variables) approximate system and, accordingly, - approximate solution. Here it is developed non-traditional approach, when generating system is singularly perturbed one and generating solution is non-degenerate (in development of the ideology of A.M.Lyapunov stability theory it is s-approximation on variables part ). With reference to the structure of mechanical-mathematical models in dynamics of the systems of stabilizations, orientation and control, with taken in consideration of their typical peculiarities, with introducing of big and small parameters, it is given the development of the methods with building of the simplified models as working ones. The developed algorithm, carried to engineering level, allows on established scheme within the framework of considered dynamic problem to separate the parameters and variables in original system on essential and unessential, to reveal the unessential freedom degrees, with the following transition to correct shortened model (idealized one in corresponding sense), with determining of the influence of rejected terms (non-idealities) on dynamic characteristics. The problem about optimal mechanical-mathematical model, with obtaining of minimal model (on N.N.Moiseev sense) is considered. The obtained results are used in problem of dynamics of gyrostabilization-orientation systems. By developed methods the reduced models are constructed, with separation of the motions in original system on different-scale components in nonlinear statement, with possibility of separation of stabilization and control channels in dynamics of multi-axis systems, with separation of the cases of small and big stabilized objects (satellites, space stations, …). In regard to the stabilization and orientation systems with the gyroscopic controlling elements, it leads to the singularly perturbed problems with the different singularities types, with critical cases, with the nonlinear singular generating systems. From theoretical points the principal questions are discussed: -the methodology of the reduction-decomposition problems;-the development of the manners, of methods for both physical and mathematical decomposition;-the substantiation of legitimacy of decomposed models in dynamics and control problems; -the determination of the qualitative equivalence conditions and correctness… Here above formulated problems are solved by authors method, following to the ideology of stability theory. General approach, based by A.M.Lyapunov and N.G.Chetayev , is extended here. The understanding these problems via singularly perturbed systems approach gives the perspective results both for theory and for applications, with revealing a constructiveness of Lyapunov stability methods as effective unified mathematical tool. As illustration there is considered the family of the stabilization and orientation systems models with gyroscopic controlling elements (including the models for small satellites, for big stabilized objects, …). The cases of full mathematical decomposition for original model are examined.
- Abstract document
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