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EN
Dynamic properties of the three degrees of freedom autoparametric system with spherical pendulum including the magnetorheological (MR) damper are investigated. It was assumed that the spherical pendulum is suspended to the oscillator excited harmonically in the vertical direction. The influence of damping force described by Bingham’s model on the energy transfer can be modified by magnetic field. The equation of motion have been solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibrations there may also appear chaotic vibration. Results show that MR damper can be used to change the dynamic behavior of the autoparametric system with spherical pendulum giving semiactive control possibilities.
EN
In this work, a system with two degrees of freedom is studied, consisting of the two coupled pendulums, connected by the element characterized by linear elasticity of rigidity and linear damping. One of the pendulums is subjected to harmonic horizontal excitation. The equations of motion within the studied system are derived as Lagrange’s equations. For small vibrations it was assumed that displacement of the spring element is horizontal only. Exemplary results of the energy transfer, for different parameter values of the system are presented. Numerically, it was indicated that near the internal and external resonance, except different kinds of periodic vibrations, chaotic vibration may also occur. For characterizing an irregular chaotic response, the Poincare maps and maximal exponents of Lyapunov have been constructed.
EN
In the paper, the dynamics of a three degree of freedom vibratory system with a spherical pendulum in the neighbourhood of internal and external resonance is considered. It has been assumed that the spherical pendulum is suspended to the main body which is then suspended to the element characterized by some elasticity and damping. The system is excited harmonically in the vertical direction. The equation of motion has been solved numerically. The influence of initial conditions on the behaviour of the spherical pendulum is investigated. In this type of the system, one mode of vibration may excite or damp another one, and for different kinds of periodic vibrations there may also appear chaotic vibrations. For characterization of an irregular chaotic response, time histories, bifurcation diagrams, power spectral densities, Poincar´e maps and the maximum Lyapunov exponents have been calculated.
EN
The nonlinear response of a three degree of freedom vibratory system with spherical pendulum in the neighbourhood internal and external resonance is investigated. It was assumed that spherical pendulum is suspended to the main body which is suspended by the element characterized by elasticity and damping and is excited harmonically in the vertical direction. The equation of motion have bean solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibrations there may also appear chaotic vibration.
5
Content available remote Oscillations of an Autoparametrical Systems with the Spherical Pendulum
EN
Dynamic properties of the three degrees of freedom autoparametric system with spherical pendulum in the neighbourhood internal and external resonance are investigated. It was assumed that the spherical pendulum is suspended in the main body which is suspended by the element characterized by elasticity and damping and is excited harmonically in the vertical direction. The spherical pendulum is similar to the simple pendulum, but moves in 3-dimensional space, so the model with spherical pendulum is more similar to the real systems than the model with simply pendulum. In this paper the position of the main body is described by coordinate z and position of the pendulum is describe by the coordinate z and two angles: θ and φ in the vertical planes. This system has three degrees of freedom. Dynamic properties of the system described by three differential equations containing strongly nonlinear terms are investigated numerically. In autoparametric system one mode of vibration may excite or damp another one, and for except periodic or quasi-periodic vibrations there may also appear chaotic vibration. For characterizing an irregular chaotic response, time histories, bifurcation diagrams, power spectral densities, Poincaré maps and maximal exponents of Lyapunov have been developed.
6
Content available remote Vibration Control of Autoparametric System Using MR Dampers in the Pendula Joints
EN
In this paper a three-degree of freedom autoparametric system with a double pendulum including the magnetorheological (MR) dampers in the pendula joints is investigated numerically. The system consists of the two coupled pendula hangs down from the oscillator. Near the resonance regions except multiperiodic and quasiperiodic vibration, also chaotic motion may appear. For characterising a chaotic response the bifurcation diagrams, Poincaré maps and maximal exponent of Lyapunov for different magnetorheological damping parameters are constructed. The influence of damping moment in the pendula joins (described by Bingham’s model) on the phenomenon of energy transfer can be modified by magnetic field. Results show that MR dampers can be used to change the dynamic behavior of the autoparametric system giving semiactive control possibilities.
7
Content available remote Chaos in Autoparametric Three Degree of Freedom System with SMA Spring
EN
In this paper is studied a three degree of freedom autoparametric system with two pendulums connected by shape memory alloys (SMA) spring in the neighborhood internal and external resonance. The system consists of the body of mass mi which is hung on a spring and a damper, and two connected by SMA spring pendulums of the length l₁ and l₂ and masses m₂ and m₃ mounted to the body of mass m₁. It is assumed, that the motion of the pendulums are damped by resistive forces. Shape memory alloys have ability to change their material properties. A polynomial constitutive model is assumed to describe the behavior of the SMA spring (it was assumed that the uniaxial stress σ is a fifth-degree polynomial of the strain). The equations of motion have been solved numerically and there were studied pseudoelastic effects associated with martensitic phase transformations. It was assumed that SMA presents two stable phases: austenite and martensite. Solutions for the system response are presented for specific values of the parameters of system. It was shown that in this type system one mode of vibrations may excite or damp another mode, and that except different kinds of periodic vibrations there may also appear chaotic vibrations. For the identification of the responses of the system various techniques, including chaos techniques such as bifurcation diagrams and time histories, power spectral densities (FFT), Poincare maps and exponents of Lyapunov maybe use.
8
Content available remote Analysis of Vibrations of Autoparametric System with Springly Coupled Pendulums
EN
The nonlinear dynamics of a three degree of freedom autoparametrical vibration system with two coupled pendulums in the neighborhood internal and external resonances is presented in this work. It was assumed that the main body is suspended by an element characterized by linear elasticity and linear damping force and is excited harmonicaly in the vertical direction. The two pendulums connected by spring are mounted to the main body. It is assumed, that the motion of the pendulums are damped by viscotic resistive forces. Solutions for the system response are presented for specific values of the uncoupled normal frequency. Analytical solutions have been obtained by using multiple scales method. This method is used to construct first-order non-linear ordinary differential equations governing the modulation of the amplitudes and phases. Steady state solutions are computed for selected values of the systems parameters.
EN
The nonlinear response of a three-degree-of-freedom vibratory system with a double pendulum in the neighborhood of internal and external resonances has been examined. Numerical and analytical methods have been applied for these investigations. Analytical solutions have been obtained by using the multiple scales method. This method is used to construct first-order non- linear ordinary differential equations governing the modulation of amplitudes and phases. Steady state solutions and their stability are computed for selected values of the system parameters.
PL
W pracy przebadano drgania nieliniowego układu o trzech stopniach swobody z podwójnym wahadłem w otoczeniu rezonansów wewnętrznych i zewnętrznych. Badania przeprowadzono analitycznie i numerycznie. Rozwiązanie analityczne uzyskano przy użyciu metody wielu skali czasowych. Metoda posłużyła do zbudowania nieliniowych równań różniczkowych pierwszego rzędu opisujących modulację amplitud i faz. Rozwiązanie ustalone i jego stabilność zostały przedstawione dla wybranych wartości parametrów układu.
EN
This paper studies the dynamical coupling between energy sources and the response of a two degrees of freedom autoparametrical system, when the excitation comes from an electric motor (with unbalanced mass mo), which works with limited power supply. The investigated system consists of a pendulum of the length l and mass m, and a body of mass M suspended on a flexible element. In this case, the excitation has to be expressed by an equation describing how the energy source supplies the energy to the system. The non- ideal source of power adds one degree of freedom, which makes the system have three degrees of freedom. The system has been searched for known characteristics of the energy source (DC motor). The equations of motion have been solved numerically. The influence of motor speed on the phenomenon of energy transfer has been studied. Near the internal and external resonance region, except for different kinds of periodic vibration, chaotic vibration has been observed. For characterizing an irregular chaotic response, bifurcation diagrams and time histories, power spectral densities, Poincare maps and maximaI exponents of Lyapunov have been developed.
PL
W pracy uwzględniono wzajemne oddziaływania autoparametrycznego układu drgającego o dwóch stopniach swobody i układu wymuszającego, którym jest silnik elektryczny z niewyważoną masą o znanej charakterystyce. Układ podstawowy składa się z wahadła o długości l i masie m podwieszonego do ciała o masie M zawieszcnego na elemencie sprężystym. Uwzględniając nieidealne źródło energii dodaje się do badanego układu dodatkowy stopień swobody, bada się więc układ o trzech stopniach swobody, ale czas nie występuje w równaniach w postaci jawnej. Równania ruchu rozwiązywano numerycznie i badano drgania w pobliżu rezonansu wewnętrznego i rezonansu zewnętrznego. W tym zakresie parametrów oprócz różnego rodzaju drgań regularnych mogą wystąpić również drgania chaotyczne. Charakter drgań nieregularnych weryfikowano analizując diagramy bifurkacyjne, przebiegi czasowe, transformaty Fouriera, mapy Poincare oraz maksymalne wykładniki Lapunowa.
12
Content available remote The dynamic of a coupled three degree of freedom mechanical system
EN
In this paper, a nonlinear coupled three degree-of-freedom autoparametric vibration system with elastic pendulum attached to the main mass is investigated numerically. Solutions for the system response are presented for specific values of the uncoupled normal frequency ratios and the energy transfer between modes of vibrations is observed. Curves of internal resonances for free vibrations and external resonances for vertical exciting force are shown. In this type system one mode of vibration may excite or damp another one, and except different kinds of periodic vibration there may also appear chaotic vibration. Various techniques, including chaos techniques such as bifurcation diagrams and: time histories, phase plane portraits, power spectral densities, Poincare maps and exponents of Lyapunov, are used in the identification of the responses. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits. The results show that the system can exhibit various types of motion, from periodic to quasi-periodic and to chaotic, and is sensitive to small changes of the system parameters.
13
Content available remote Drgania autoparametrycznego układu z nieidealnym źródłem energii
EN
This work draws attention to the analysis of interactions between vibrations mode of autoparametric dynamical system with limited power supply. The system is called nonideal because the motion of the oscillatory system may affect the motor's speed. In this paper this phenomenon near the fundamental resonance region have been researched.
PL
Praca zawiera badanie wpływu nieliniowego tłumienia i nieliniowej sprężystości na drgania swobodne i wymuszone układu o trzech stopniach swobody. Wykazano, że nieliniowości te wpływają na cykle przenoszenia drgań między elementami układu oraz na przebiegi rezonansów wewnętrznych i zewnętrznych. Jest to istotne ponieważ w pobliżu rezonansów wewnętrznych i zewnętrznych oprócz różnego rodzaju drgań okresowych mogą pojawić się także drgania chaotyczne. Możliwość wystąpienia drgań o różnym charakterze przedstawiono na wykresach bifurkacyjnych, gdzie parametrem bifurkacyjnym jest amplituda wymuszenia.
EN
In this work the influence of nonlinear rigidity and nonlinear damping force for nonlinear dynamics three degree of freedom system with double pendulum is studied. Free vibrations and excitation vibrations are investigate. The equations of motion have been solved numerically. In this tape system nonlinearity has influence for cycles of vibration transfer between system's elements. Nonlinearity has influence for internal and external resonanses too. Near resonanses, besides periodic vibrations, can appear chaotic vibrations. To prove the character of this vibrations there have been constructed the bifurcation diagrams. This bifurcation diagrams show many sudden qualitative changes.
15
Content available remote The chaotic phenomenons of a system with inertial coupling
EN
The nonlinear response of two-degree-of-freedom vibratory beam-pendulum system in the neighbourhood internal and external resonances is investigated. The analysis was realised in the wide aspects of the influence of different kinds of nonlinearities, dampings and excitations. The equations of motion have bean solved numerically. The present paper is a continuation of the author's previous works, where it was shown that in this type system one mode of vibration may excite or damp another one, and that except different kinds of periodic vibration there may also appear chaotic vibration. To prove the character of this vibration and to realise the analysis of transitions from periodic regular motion to quasi-periodic and chaotic, there have been constructed the bifurcation diagrams and time histories, phase plane portraits, power spectral densities, Poincare maps and exponents of Lyapunov. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits.
PL
Rozpatrzono zjawisko występowania chaosu w nieliniowo sprzężonym autoparametrycznym układzie belka-wahadło obciążonym oprócz siły harmonicznej także stałą siłą. Stwierdzono, że stała siła przesuwa zakres występowania drgań chaotycznych. Chaotyczny charakter ruchu stwierdzono na podstawie przebiegów czasowych drgań, map Poincare, widm Fouriera oraz maksymalnych wykładników Lapunowa.
EN
This present work focuses on the chaos in non-linear coupled beam-pendulum system excited by both harmonic and constant forces. The constant force was shown to change the range of chaotic vibration. The chaotic character of the vibration was analysed with time histories, Poincare maps, the Fourier spectra as well as maximum exponents of Lyapunov.
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