Dinamika i stabilnost hibridnih dinamičkih sistema
Simonović, Julijana D.
Faculty:University of Niš, Faculty of Mechanical Engineering
Stevanović Hedrih, Katica
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- Dynamics of hybrid systems with complex structures. Mechanics of materials. (MPNTR-ON 174001)
- Problemi teorijske i tehničke mehanike krutih i čvrstih tela.Mehanika materijala.
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This work dedicates to research results of dynamics and stability of dynamics hybrid systems. It contains the presentation and systematization of main results of the other authors which were the base for obtaining own new and original results. The following classes of hybrid systems are presented: coupled linear and nonlinear oscillators, coupled continual and discrete subsystems and coupled continual subsystems with different properties and constitutive relations of coupling elements. The analytical methods and results of mathematical models of presented hybrid systems dynamics are systematized and used for numerical quantitative analysis of dynamics and stability of presented hybrid systems and their subsystems in mutual inter-reaction. The phenomena of non-linearity in classes of hybrid systems with nonlinearities of third order are presented. Such a phenomenon are passage through the resonant regions in multi frequencies forced oscillation in stationary and no stationary regimes, the
appearance of amplitude and phase resonant jumps, mutual interaction of component harmonics in that regimes, intersection of stable and unstable manifolds of saddle nodes in phase plains of dynamics in such a models, all of them are the source of negative physical representation of nonlinearity which brings the no stability and no regular dynamics of models. We proposed approach of optimal control in hybrid systems, by optimizing the values of Melnikov’s functions that guarantee avoiding of such negative manifestation of nonlinearity in systems. On the base of numerical calculation and quantitative analyses of analytical forms of transcendent frequency equations for obtaining characteristic numbers and frequencies in hybrid systems of coupled continual and discrete subsystems on present characteristic phenomenon of discretization in frequencies spectra of frequency transcendent equations corresponding to continual subsystems because of coupling with discrete subsystems and vice versa continualization of frequencies spectra of discrete subsystems. General feature of dynamics systems coupling is multiplicity of oscillatory processes modes and their transformation under mutual interactions. The number of modes multiplicity depends of nature and number of subsystems while presence of non linearity generates the correspondence and mutual interaction of component subsystem dynamics creating the general hybrid systems dynamics. For example, the solutions of proper coupled differential equations on time functions of transversal oscillations of two plates coupled with viscous elastic nonlinear elements layer confide that presence of coupling layer causes doubling of circular frequencies in every of modes of own transversal oscillation of plate. Applying the classical theory of oscillation of deformable bodies the systems of partial differential equations of transverse oscillations of circular plates systems coupled with distributed layer of viscous elastic nonlinear elements were obtained. The method of Bernoulli’s particular integral is used for solving the systems of coupled non homogeneous partial differential equations (PDE), so that we practically integrate solutions in forms of infinite series of own amplitude functions satisfying proper boundary conditions and corresponding orthogonality conditions. Then we introduce that series into equations of motion and conditions of displacement compatibility and equating the coefficients beside equal own amplitude functions and obtained the system of ordinary differential equations (ODE) by unknown time functions. Thereat we use the orthogonally conditions of own amplitude functions like as the initial condition. The Lagrange’s method of variation constants or asymptotic method of nonlinear mechanics - Krilov-Bogoliubov-Mitroplski’s method is used to solve the systems of ODE depending of forms and features of obtained ordinary linear or nonlinear differential equations. The proper boundary and initial conditions and proper assumptions and limits are used for obtaining particular solutions of differential equations systems. The structural stability was investigated applying the Lyapunov’s method and for stationary regimes stability we used the theorems of stability by linearization the obtained systems of solutions for amplitudes and phases of component harmonics in the first asymptotic approximation in the vicinity of stationary solutions. The energy transfer between coupling subsystems of hybrid systems was identified. Using the obtained forms of reduced values of kinetic and potential energy of proper mode and proper subsystems, and potential like as kinetic energy of interaction between subsystems, and Rayleigh’s function of dissipation at the interconnected elements the energy exchange of hybrid systems was analyzed. While analyzing the energies transfer the expressions of Lyapunov’s exponents were obtained with negative values, so we conclude that process of oscillations like as sub-processes of interaction between subsystems are structuraly stabile, so like that we assign the Lyapunov’s exponents like measure of of integrity of dynamics-motion of hybrid systems and their subsystems. The synchronization in the hybrid systems is presented like form of time accordance in subsystems global dynamics. The results of own investigations of identical synchronization attractor shapes are presented in the classes of hybrid discrete systems with static and dynamics coupling elements. The values of static and dynamic coupling coefficients which guarantee identical synchronization of subsystems dynamics are determined for studied classes of hybrid systems. Those results are obtained from numerical calculating of proper differential equations systems for special choice examples of hybrid systems dynamics by using the software tools for continual multi parametric transformations of solutions and their visualizations; they are also original contributions of this dissertatView More
Keywords:hibridni dinamički sistem, podsistemi, komponentne dinamike, interakcija, nelinearnost, standardni visko-elastični nelinearni sprežući element, udvajanje kružnih frekvencija, Bernoulli-eva metoda partikularnih integrala, asimptotska metoda usrednjenja Krilov-Bogoliyubov-Mitropolskiy-kog, frekventne transcedentne jednačine, prenos energije, Lyapunov-ljevi eksponenti, asimptotska aproksimacija rešenja, strukturna i lokalna stabilnost, singularitet, fazni portret, prolazak kroz rezonantno stanje, rezonantni skok, numerički eksperiment, sinhronizacija, Melnikov-ljeva funkcija, optimalno upravljanje, naučno računanje, softverski alati, vizuelizacija, analogija.; hybrid dynamics system, subsystem, component dynamics, interaction, non linearity, standard viscous elastic nonlinear coupling element, doubling of circular frequencies, method of Bernoulli’s particular integrals, Krilov-Bogoliubov-Mitroplski’s asymptotic method, characteristic transcendent equation, energy exchange, Lyapunov’s exponents, asymptotic approximation of solutions, global and local stability, singularities, phase portrait, passing through resonant regimes, resonant jump, numerical experiment, Melnikov’s function, optimal control, synchronization, scientific calculation, software tools, visualization, analogy.