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Simonović, Julijana D. 1975-

Dinamika i stabilnost hibridnih dinamičkih sistema

Autorstvo-Nekomercijalno-Bez prerade 3.0 Srbija (CC BY-NC-ND 3.0)

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Academic metadata

Doktorska disertacija

Tehničko-tehnološke nauke

Univerzitet u Nišu

Mašinski fakultet

Katedra za mašinske konstrukcije, razvoj i inženjering

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Dynamics and Stability of Dynamics Hybrid Systems

[J. D. Simonović]

VIII, 337 str.

Biobibliografija: str. 335-337

Datum odbrane: 07.12.2012.

Acknowledgment: Parts of this research were supported by the Ministry of Sciences and Environmental Protection of Republic of Serbia through Mathematical Institute SANU Belgrade Grant ON144002 “Theoretical and Applied Mechanics of Rigid and Solid Body. Mechanics of Materials” and Faculty of Mechanical Engineering University of Niš. Special thanks to my supervisor prof.dr Katica R.(Stevanović) Hedrih.

Stevanović Hedrih, Katica (mentor)

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 dissertation. The identical synchronization of coupled nonlinear oscillatory hybrid system is investigated too where the possibilities of subsystems synchronization are much bigger. For example it is need ten time smaller coefficient of static coupling of two nonlinear oscillators with chaotic attractors then in the case of coupling linear and nonlinear oscillators with static coupling element. When the linear and nonlinear oscillators are coupled with dynamic coupling element the synchronization is not possible instead of that there exists attractor of synchronization with initially rather big values of synchronization error but after time of synchronization that error is the periodic function with small amplitude. The influence of nonlinearity of coupling elements, of dissipations and of external forces amplitude values on synchronization features are investigated too in the hybrid systems of plates coupled with viscous elastic nonlinear layer. The certain visual similarities of Lissajous’s figures and harmonograph diagrams with diagrams of synchronization in hybrid systems are identified for proper values of system and coupling element parameters.
The model of two circular plates coupled with piezo-visco-elastic nonlinear layer is present like a new model of active structure. The system of coupled partial differential equations of such a model is derived and the methods of its solving and obtaining the results are proposed by analogy with ones used in the other models of hybrid systems. Here the exchange of polarization voltage on piezo element may present the control signal for real control required.
All presented models and proposed methodology of their describing by its abstracting from real systems and dynamics, like as their solving has the analogy mathematical approaches and phenomenological mappings. Since in the end of this dissertation on presents the systematization and overall resume of methods and methodologies of modeling, solving and studying the rare properties and phenomenon of dynamics in classes of hybrid systems. Introspection of all the results which are possible to get with proposed methodologies present one more original and meaningful contribution to studying not only the proposed classes of hybrid systems but also wider class of hybrid systems in which mathematical models possess properties of non linearity.

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.

531+534+517. 93(045)=111

531.3:531.3.011:532.511(045)=111

531.3+534+531.513+517.939(043.3)

Serbian

533542294

Elektronska teza

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 dissertation. The identical synchronization of coupled nonlinear oscillatory hybrid system is investigated too where the possibilities of subsystems synchronization are much bigger. For example it is need ten time smaller coefficient of static coupling of two nonlinear oscillators with chaotic attractors then in the case of coupling linear and nonlinear oscillators with static coupling element. When the linear and nonlinear oscillators are coupled with dynamic coupling element the synchronization is not possible instead of that there exists attractor of synchronization with initially rather big values of synchronization error but after time of synchronization that error is the periodic function with small amplitude. The influence of nonlinearity of coupling elements, of dissipations and of external forces amplitude values on synchronization features are investigated too in the hybrid systems of plates coupled with viscous elastic nonlinear layer. The certain visual similarities of Lissajous’s figures and harmonograph diagrams with diagrams of synchronization in hybrid systems are identified for proper values of system and coupling element parameters.
The model of two circular plates coupled with piezo-visco-elastic nonlinear layer is present like a new model of active structure. The system of coupled partial differential equations of such a model is derived and the methods of its solving and obtaining the results are proposed by analogy with ones used in the other models of hybrid systems. Here the exchange of polarization voltage on piezo element may present the control signal for real control required.
All presented models and proposed methodology of their describing by its abstracting from real systems and dynamics, like as their solving has the analogy mathematical approaches and phenomenological mappings. Since in the end of this dissertation on presents the systematization and overall resume of methods and methodologies of modeling, solving and studying the rare properties and phenomenon of dynamics in classes of hybrid systems. Introspection of all the results which are possible to get with proposed methodologies present one more original and meaningful contribution to studying not only the proposed classes of hybrid systems but also wider class of hybrid systems in which mathematical models possess properties of non linearity.