Danuta Sado

Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker

In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.

Note on Chaos in Three Degree of Freedom Dynamical System with Double Pendulum

The nonlinear response of a three degree of freedom vibratory system with double pendulum in the neighbourhood internal and external resonances is investigated. The equations 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 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, the following have been constructed: bifurcation diagrams and time histories, phase plane portraits, power spectral densities, Poincaré 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.

Nonlinear dynamics of a non-ideal autoparametric system with MR damper

The nonlinear response of a three degree of freedom autoparametric system with a double pendulum, including the magnetorheological (MR) damper when the excitation comes from a DC motor which works with limited power supply, has been examined. The non-ideal source of power adds one degree of freedom which makes the system have four degrees of freedom. The influence of damping force in MR damper on the phenomenon of energy transfer has been studied numerically. Near the internal and external resonance region, except periodic vibration also chaotic vibration has been observed. Results show that MR damper can be used to change the dynamic behavior of the autoparametric system.

Effect of Damping on the Periodic and Chaotic Vibration of System With Double Pendulum

The nonlinear response of three degree of freedom vibratory system with double pendulum in the neighborhood internal and external responses have been researched. Numerical and analytical methods have been applied for these investigations. Analytical solutions have been obtained by using multiple scale 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 and their stability are computed for selected values of the system parameters. Numerically it was shown, that near internal and external resonances except different kinds of periodic vibrations there may also appear chaotic vibration. For characterizing an irregular chaotic response bifurcation diagrams and time histories, Poincare maps and maximal exponents of Lyapunov have been constructed.Copyright

Pseudoelastic effect in autoparametric non-ideal vibrating system with SMA spring

Abstract In this paper a three degrees of freedom autoparametric system with limited power supply is investigated numerically. The system consists of the body, which is hung on a spring and a damper, and two pendulums connected by shape memory alloy (SMA) spring. Shape memory alloys have ability to change their material properties with temperature. A polynomial constitutive model is assumed to describe the behavior of the SMA spring. The non-ideal source of power adds one degree of freedom, so the system has four degrees of freedom. The equations of motion have been solved numerically and pseudoelastic effects associated with the martensitic phase transformation are studied. It is shown that in this type system one mode of vibrations might 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 systems various techniques, including chaos techniques such as bifurcation diagrams and time histories, power spectral densities, Poincare maps and exponents of Lyapunov may be used.

Nonlinear Oscillations of an Autoparametrical System With Two Coupled Pendulums

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 non-linear elasticity and non-linear damping force and is excited harmonicaly in the vertical direction. The two connected by spring pendulums characterized are mounted to the main body. It is assumed, that the motion of the pendulums are damped by nonlinear resistive forces. 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 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 to chaotic, and is sensitive to small changes of the system parameters.Copyright

The Periodic and Chaotic Vibration of Dynamical System With Elastic Pendulum

The nonlinear damping effect on response of coupled three degree-of-freedom autoparametric vibration system with elastic pendulum attached to the main mass is investigated numerically. It was assumed that the main body is suspended by an element characterized by non-linear elasticity and non-linear damping force and is excited harmonically in the vertical direction. The elastic pendulum characterized also by -linear elasticity and non-linear damping. 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 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 to chaotic, and is sensitive to small changes of the system parameters.Copyright

ON REGULAR AND IRREGULAR NONLINEAR VIBRATIONS IN TORSIONAL DISCRETE-CONTINUOUS SYSTEMS

The paper deals with regular and irregular nonlinear vibrations of discrete-continuous systems torsionally deformed. The systems consist of an arbitrary number of shafts connected by rigid bodies. In the systems, a local nonlinearity having a soft type characteristic is introduced. This nonlinearity is described by the polynomial of the third degree. General governing equations using the wave approach are derived for a multimass system. Detailed numerical considerations are presented for a two-mass system and a three-mass system. The possibility of occurrence of irregular vibrations is discussed on the basis of the Poincare maps and bifurcation diagrams.

Chaotic Vibrations in Multi-Mass Discrete-Continuous Systems Torsionally Deformed with Local Nonlinearities

The paper deals with nonlinear vibrations in discrete-continuous mechanical systems consisting of rigid bodies connected by shafts torsionally deformed with local nonlinearities having hard or soft characteristics. The systems are loaded by an external moment harmonically changing in time. In the study the wave approach is used. Numerical results are presented for three-mass systems. In the study of regular vibrations in the case of a hard characteristic amplitude jumps are observed while in the case of a soft characteristic an escape phenomenon is observed. Irregular vibrations, including chaotic motions, are found for selected parameters of the systems.

Vibration Control of Autoparametric System Using SMA Spring and MR Dampers in the Pendula Joins

This work is concerned with the problem of nonlinear dynamical motion of a non-ideal autoparametric system with two active elements — shape memory alloys (SMA) spring and the magnetorheological (MR) pin joint dampers in the neighbourhood internal and external resonance. A polynomial constitutive model was assumed to describe the behaviour of SMA’s spring and there was observed the pseudoelastic effects associated with martensitic phase transformations. The simplified resistance moment generated by the MR pin joint can be approximated using the hyperbolic tangent function. The influence of damping forces in MR dampers on the phenomenon of energy transfer and pseudoelastic effects associated with the martensitic phase transformation was studied. It was shown that in this type system one mode of vibrations might 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 may be used. The SMA spring and MR damper can be used to change the dynamic behaviour of the autoparametric system giving reliable semiactive control possibilities.Copyright