Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise

In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system

Periodically Forced Nonlinear Oscillators With Hysteretic Damping

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.Comment: Accepted for publication in the Journal of Computational and Nonlinear Dynamic

14th Conference on Dynamical Systems Theory and Applications DSTA 2017 ABSTRACTS

From Preface: This is the fourteen time when the conference “Dynamical Systems – Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and the Ministry of Science and Higher Education. It is a great pleasure that our invitation has been accepted by so many people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcome nearly 250 persons from 38 countries all over the world. They decided to share the results of their research and many years experiences in the discipline of dynamical systems by submitting many very interesting papers. This booklet contains a collection of 375 abstracts, which have gained the acceptance of referees and have been qualified for publication in the conference proceedings [...]

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Vibration analysis and intelligent control of flexible rotor systems using smart materials

Flexible rotor-bearing system stability is a very important subject impacting the design, control, maintenance and operating safety. As the rotor bearing-system dynamic nonlinearities are significantly more prominent at higher rotating speeds, the demand for better performance through higher speeds has rendered the use of linear approaches for analysis both inadequate and ineffective. To address this need, it becomes important that nonlinear rotor-dynamic responses indicative of the causes of nonlinearity, along with the bifurcated dynamic states of instabilities, be fully studied. The objectives of this research are to study rotor-dynamic instabilities induced by mass unbalance and to use smart materials to stabilise the performance of the flexible rotor-system. A comprehensive mathematical model incorporating translational and rotational inertia, bending stiffness and gyroscopic moment is developed. The dynamic end conditions of the rotor comprising of the active bearing-induced axial force is modelled, the equations of motion are derived using Lagrange equations and the Rayleigh-Ritz method is used to study the basic phenomena on simple systems. In this thesis the axial force terms included in the equations of motion provide a means for axially directed harmonic force to be introduced into the system. The Method of Multiple Scales is applied to study the nonlinear equations obtained and their stabilities. The Dynamics 2 software is used to numerically explore the inception and progression of bifurcations suggestive of the changing rotor-dynamic state and impending instability. In the context of active control of flexible rotors, smart materials particularly SMAs and piezoelectric stack actuators are introduced. The application of shape memory alloy (SMA) elements integrated within glass epoxy composite plates and shells has resulted in the design of a novel smart bearing based on the principle of antagonistic action in this thesis. Previous work has shown that a single SMA/composite active bearing can be very effective in both altering the natural frequency of the fundamental whirl mode as well as the modal amplitude. The drawback with that design has been the disparity in the time constant between the relatively fast heating phase and the much slower cooling phase which is reliant on forced air, or some other form of cooling. This thesis presents a modified design which removes the aforementioned existing shortcomings. This form of design means that the cooling phase of one half, still using forced air, is significantly assisted by switching the other half into its heating phase, and vice versa, thereby equalising the time constants, and giving a faster push-pull load on the centrally located bearing; a loading which is termed ‘antagonistic’ in this present dissertation. The piezoelectric stack actuator provides an account of an investigation into possible dynamic interactions between two nonlinear systems, each possessing nonlinear characteristics in the frequency domain. Parametric excitations are deliberately introduced into a second flexible rotor system by means of a piezoelectric exciter to moderate the response of the pre-existing mass-unbalance vibration inherent to the rotor. The intended application area for this SMA/composite and piezoelectric technologies are in industrial rotor systems, in particular very high-speed plant, such as small light pumps, motor generators, and engines for aerospace and automotive application

Vibration, Control and Stability of Dynamical Systems

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”