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Albert C J Luo - One of the best experts on this subject based on the ideXlab platform.

  • a periodically forced Piecewise Linear System part ii the fragmentation mechanism of strange attractors and grazing
    Communications in Nonlinear Science and Numerical Simulation, 2007
    Co-Authors: Albert C J Luo
    Abstract:

    Abstract In the first part of this work, the local singularity of non-smooth dynamical Systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical Systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical Systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical System is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical Systems, which will help us better understand chaotic motions in non-smooth dynamical Systems.

  • a periodically forced Piecewise Linear System part i local singularity and grazing bifurcation
    Communications in Nonlinear Science and Numerical Simulation, 2007
    Co-Authors: Albert C J Luo
    Abstract:

    Abstract A methodology for the local singularity of non-smooth dynamical Systems is Systematically presented in this paper, and a periodically forced, Piecewise Linear System is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical Systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.

  • grazing and chaos in a periodically forced Piecewise Linear System
    Journal of Vibration and Acoustics, 2006
    Co-Authors: Albert C J Luo
    Abstract:

    The criteria for the grazing bifurcation of a periodically forced, Piecewise Linear System are developed and the initial grazing manifolds are obtained. The initial grazing manifold is invariant. The grazing flows are illustrated to verify the analytic prediction of grazing. The mechanism of the strange attractors fragmentation caused by the grazing is discussed, and an illustration of the fragmentized strange attractor is given through the Poincare mapping. This fragmentation phenomenon exists extensively in nonsmooth dynamical Systems. The mathematical structure of the fragmentized strange attractors should be further developed.

  • a global period 1 motion of a periodically excited Piecewise Linear System
    International Journal of Bifurcation and Chaos, 2005
    Co-Authors: Santhosh Menon, Albert C J Luo
    Abstract:

    The period-1 motion of a Piecewise-Linear System under a periodic excitation is predicted analytically through the Poincare mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonLinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the Piecewise-Linear System and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical Systems.

  • the mapping dynamics of periodic motions for a three Piecewise Linear System under a periodic excitation
    Journal of Sound and Vibration, 2005
    Co-Authors: Albert C J Luo
    Abstract:

    In this paper, the mapping dynamics of periodic motions for a three-Piecewise Linear System under a periodic excitation is developed, and the mapping structures for specified periodic motions are constructed. Based on the mapping structures, an analytical prediction of all possible, stable and unstable periodic motions is given. The symmetry for the stable, asymmetrical, periodic motions of such a System is observed. However, the unstable periodic motions do not have such symmetry. The methodology presented in this paper is applicable to other non-smooth Systems such as friction-induced vibration, impact oscillator and power control Systems. In addition, the mapping dynamics provides a useful and efficient tool for the co-existence of periodic motions and/or chaotic motions in nonLinear dynamical Systems.

J. White - One of the best experts on this subject based on the ideXlab platform.

  • A trajectory Piecewise-Linear approach to model order reduction and fast simulation of nonLinear circuits and micromachined devices
    IEEE ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE ACM Digest of Technical Papers (Cat. No.01CH37281), 2001
    Co-Authors: Michał Rewieński, J. White
    Abstract:

    In this paper we present an approach to the nonLinear model reduction based on representing the nonLinear System with a Piecewise-Linear System and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as Piecewise-Linear and then composing hundreds of components to make a System with exponentially many different Linear regions, we instead generate a small set of Linearizations about the state trajectory which is the response to a 'training input'. Computational results and performance data are presented for a nonLinear circuit and a micromachined fixed-fixed beam example. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with Linear or the recently developed quadratic reduction techniques. Finally, it is shown that the proposed technique is computationally inexpensive, and that the models can be constructed 'on-the-fly', to accelerate simulation of the System response.

K. V. Avramov - One of the best experts on this subject based on the ideXlab platform.

  • Numerical analysis of nonLinear modes of Piecewise Linear Systems torsional vibrations
    Meccanica, 2017
    Co-Authors: B. V. Uspensky, K. V. Avramov
    Abstract:

    The nonLinear modes of essentially nonLinear Piecewise-Linear finite degrees of freedom Systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the Systems motions in configuration space. The numerical method for the nonLinear modes of essentially nonLinear Piecewise-Linear Systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous System nonLinear modes. The nonLinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonLinear Piecewise-Linear System with fifteen degrees of freedom. The NNMs of this System forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the Systems with dissipation are close to NNMs of the System without dissipation.

  • NonLinear Normal Modes of Strongly NonLinear Periodically Excited Piecewise Linear Systems
    Journal of Mathematical Sciences, 2017
    Co-Authors: B. V. Uspensky, K. V. Avramov
    Abstract:

    A method for the numerical analysis of nonLinear normal modes of forced vibrations in strongly nonLinear Systems with Piecewise Linear elastic characteristics is proposed. The approach is based on the combination of the Shaw–Pierre method of nonLinear normal modes with the Rauscher technique. As a result of application of this approach, the nonautonomous Piecewise Linear System is transformed into an autonomous System. For this System, we determine the Shaw–Pierre nonLinear normal modes. We also study the nonLinear torsional vibrations of the power transmission in a three-cylinder transport engine.

Jiafu Wang - One of the best experts on this subject based on the ideXlab platform.

  • discontinuity induced limit cycles in a general planar Piecewise Linear System of saddle focus type
    Nonlinear Analysis: Hybrid Systems, 2019
    Co-Authors: Jiafu Wang, Chuangxia Huang, Lihong Huang
    Abstract:

    Abstract The aim of this paper is to deal with the problem of limit cycles for a general planar Piecewise Linear differential System of saddle–focus type. By using the Lienard-like canonical form with five parameters and dividing the total parameter space into several regions, the number of limit cycles is discussed in detail. In particular, we give parameter regions where there are at least two limit cycles. Moreover, we investigate the existence and stability of exactly two nested limit cycles in some parameter regions.

Angelo Luongo - One of the best experts on this subject based on the ideXlab platform.

  • A perturbation method for evaluating nonLinear normal modes of a Piecewise Linear two degree of freedom System
    Nonlinear Dynamics, 2008
    Co-Authors: Fabrizio Vestroni, Achille Paolone, Angelo Luongo
    Abstract:

    The classical Lindstedt-Poincaré method is adapted to analyze the nonLinear normal modes of a Piecewise Linear System. A simple two degrees-offreedom, representing a beam with a breathing crack is considered. The fundamental branches of the two modes and their stability are drawn by varying the severity of the crack, i.e., the level of nonLinearity. Results furnished by the asymptotic method give insight into the mechanical behavior of the System and agree well with numerical results; the existence of superabundant modes is proven. The unstable regions and the bifurcated branches are followed by a numerical procedure based on the Poincarè map.

  • a perturbation method for evaluating nonLinear normal modes of a Piecewise Linear two degrees of freedom System
    Nonlinear Dynamics, 2008
    Co-Authors: Fabrizio Vestroni, Angelo Luongo, Achille Paolone
    Abstract:

    The classical Lindstedt-Poincare method is adapted to analyze the nonLinear normal modes of a Piecewise Linear System. A simple two degrees-of- freedom, representing a beam with a breathing crack is considered. The fundamental branches of the two modes and their stability are drawn by varying the severity of the crack, i.e., the level of nonLinearity. Re- sults furnished by the asymptotic method give insight into the mechanical behavior of the System and agree well with numerical results; the existence of super- abundant modes is proven. The unstable regions and the bifurcated branches are followed by a numerical procedure based on the Poincare map.