Symbolic analysis of switching systems : application to bifurcation analysis of DC/DC switching converters

Author name used in this publication: Chi K. Tse2005-2006 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Some elements for a history of the dynamical systems theory

Leon Glass would like to thank the Natural Sciences and Engineering Research Council (Canada) for its continuous support of curiosity-driven research for over 40 years starting with the events recounted here. He also thanks his colleagues and collaborators including Stuart Kauffman, Rafael Perez, Ronald Shymko, Michael Mackey for their wonderful insights and collaborations during the times recounted here. R.G. is endebted to the following friends and colleagues, listed in the order encountered on the road described: F. T. Arecchi, L. M. Narducci, J. R. Tredicce, H. G. Solari, E. Eschenazi, G. B. Mindlin, J. L. Birman, J. S. Birman, P. Glorieux, M. Lefranc, C. Letellier, V. Messager, O. E. Rössler, R. Williams. U.P. would like to thank the following friends and colleagues who accompanied his first steps into the world of nonlinear phenomena: U. Dressler, I. Eick, V. Englisch, K. Geist, J. Holzfuss, T. Klinker, W. Knop, A. Kramer, T. Kurz, W. Lauterborn, W. Meyer-Ilse, C. Scheffczyk, E. Suchla and M. Wisenfeldt. The work by L. Pecora and T. Carroll was supported directly by the Office of Naval Research (ONR) and by ONR through the Naval Research Laboratory’s Basic Research Program. C.L. would like to thank Jürgen Kurths for his support to this project.Peer reviewedPostprintPublisher PD

On Hyperbolic Attractors in Complex Shimizu -- Morioka Model

We present a modified complex-valued Shimizu -- Morioka system with uniformly hyperbolic attractor. The numerically observed attractor in Poincar\'{e} cross-section is topologically close to Smale -- Williams solenoid. The arguments of the complex variables undergo Bernoulli-type map, essential for Smale -- Williams attractor, due to the geometrical arrangement of the phase space and an additional perturbation term. The transformation of the phase space near the saddle equilibrium "scatters" trajectories to new angles, then trajectories run from the saddle and return to it for the next "scatter". We provide the results of numerical simulations of the model and demonstrate typical features of the appearing hyperbolic attractor of Smale -- Williams type. Importantly, we show in numerical tests the transversality of tangent subspaces -- a pivotal property of uniformly hyperbolic attractor.Comment: 9 pages, 8 figure

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