Vol.3, No 14, 2003 pp. 881-890
TO MODELLING PROBLEM IN MECHANICS
(THEORETICAL AND APPLIED ASPECTS)
Lyudmila K. Kuzmina
Kazan Aviation Institute, Adamuck, 4-6, Kazan-15, 420015, RUSSIA
Abstract. The research is devoted to the development of approximate
methods in the complex systems dynamics on base of Lyapunov's stability
theory methods for singular perturbations problems and their applications.
It allows to obtain a solving urgent mathematical modelling problems in
Mechanics; to work out the reduction principle in general qualitative analysis
of complex systems.
The research subject is complex large-scale systems, for which the
original mathematical model, adequate real object, is extremely complex.
The principal tasks are the elaborating universal methods of modelling;
the constructing correct simplified models; the rigorous substantiating
these reduced models in dynamics; the estimating errors and admissible
parameters domains under using these reduced models (with keeping of qualitative
Suggested method is combining the stability theory and perturbations
theory methods based on two postulates (stability and singularity ones).
It allows to work out the effective manners of rigorous analysis with general
methodology of constructing simplified models and their analysing; with
dividing of original problem on separate particular ones; with decomposing
of original model and its dynamic characteristics; with building shortened
models hierarchy; with revealing essential variables and freedom degrees.
Besides the elaborated approach is extending the traditional statements
of stability problems and perturbations theory, with the treating modelling
problem as stability problem (in specific sense) under singular perturbations,
with the extension of N.G.Chetayev's, K. P. Persidskiy ideas. Moreover
the research results will allow to come nearer to fundamental modelling
problem in complex systems dynamics. This problem is concerned with reduction
principle in general qualitative analysis and with singularity problem
in Mechanics. The results allow to get the developing theory of approximate
methods for rigorous analysis; to give substantiation of approximate theories
and models in Mechanics (both for traditional ones, well-known in engineering
practice, and for new models, that are constructed by this developed method;
including non-Newton models of Mechanics).
MODELIRANJE PROBLEMA U MEHANICI
(TEORIJSKI I PRIMENJENI ASPEKTI)
Istraživanje je posvećeno razvoju aproksimativnih metoda u kompleksnim
sistemima dinamike na bazi metoda Ljapunovljeve teorije stabilnosti za
singularne perturbacione probleme i njihove primene. Taj razvoj uvek zahteva
neophodna rešenja modeliranja problema mehanike. Rad se bavi i redukcijom
principa u generalizaciji kvalitativne analize kompleksnih sistema. Predložena
metoda je kombinacija metoda teorije stabilnosti i teorije perturbacija
(poremećaja) bazirana na dva postulata (stabilnosti i singularnosti). Pristup
se sastoji u dekompoziciji originalnog modela i njegovih karakteristika,
kao i gradjenju skraćenih modela hijerarhije sa odgovarajućim promenljivim
i stepenima slobode.