Albert C. J. Luo

Fractional Dynamics and Control

Fractional Dynamics and Control provides a comprehensive overview of recent advances in the areas of nonlinear dynamics, vibration and control with analytical, numerical, and experimental results. This book provides an overview of recent discoveries in fractional control, delves into fractional variational principles and differential equations, and applies advanced techniques in fractional calculus to solving complicated mathematical and physical problems.Finally, this book also discusses the role that fractional order modeling can play in complex systems for engineering and science.

Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients

The stability problem for systems with distributed delay is considered using discretized Lyapunov functional. The coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform. The resulting stability criteria are written in the form of linear matrix inequality. Numerical examples are also provided to illustrate the effectiveness of the method. The basic idea can be extended to a more general setting with more involved formulation.

Discontinuous dynamical systems on time-varying domains

Flow Switchability.- Transversality and Sliding Phenomena.- A Frictional Oscillator on Time-varying Belt.- Two Oscillators with Impacts and Stick.- Dynamical Systems with Frictions.- Principles for System Interactions.

The Dynamics of a Bouncing Ball with a Sinusoidally Vibrating Table Revisited

The dynamical behavior of a bouncing ball with a sinusoidally vibrating table is revisited in this paper. Based on the equation of motion of the ball, the mapping for period-1 motion is constructured and thereby allowing the stability and bifurcation conditions to be determined. Comparison with Holmess solution [1] shows that our range of stable motion is wider, and through numerical simulations, our stability result is observed to be more accurate. The Poincaré mapping sections of the unstable period-1 motion indicate the existence of identical Smale horseshoe structures and fractals. For a better understanding of the stable and chaotic motions, plots of the physical motion of the bouncing ball superimposed on the vibration of the table are presented.

Chaotic motion in a micro-electro-mechanical system with non-linearity from capacitors

This investigation is to provide a possible prediction for design, manufacturing, testing and industrial applications of a simplified micro-electro-mechanical system (MEMS). The chaotic motion in a certain frequency band of such a MEMS device is investigated, and the corresponding equilibrium, natural frequency and responses are determined. Under alternating current (AC) voltage, the resonant condition for such a system is obtained. It is observed that the lower-order resonant motions can be easily converted to the mechanical force and sensed to the electrical signal. The chaotic motions in the vicinity of a specified resonant separatrix are investigated analytically and numerically. For given voltages, the AC frequency bands are obtained for chaotic motion in the specific resonant layers and resonant motions, and such chaotic motions can be very easily sensed by the output transducer in MEMS.

Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance

In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period-k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters.

An Unsymmetrical Motion in a Horizontal Impact Oscillator

Stability and bifurcation for the unsymmetrical, periodic motion of a horizontal impact oscillator under a periodic excitation are investigated through four mappings based on two switch-planes relative to discontinuities. Period-doubling bifurcation for unsymmetrical period-1 motions instead of symmetrical period-1 motion is observed. A numerical investigation for symmetrical, period-1 motion to chaos is completed. The numerical and analytical results of periodic motions are in very good agreement. The methodology presented in this paper is applicable to other discontinuous dynamic systems. This investigation also provides a better understanding of such an unsymmetrical motion in symmetrical discontinuous systems.

On the Mechanism of Stick and Nonstick, Periodic Motions in a Periodically Forced, Linear Oscillator With Dry Friction

In this paper the dynamics mechanism of stick and nonstick motion for a dry-friction oscillator is discussed. From the theory ofLuo in 2005 [Commun. Nonlinear Sci. Numer. Simul., 10, pp. 1-55], the conditions for stick and nonstick motions are achieved. The stick and nonstick periodic motions are predicted analytically through the appropriate mapping structures. The local stability and bifurcation conditions for such periodic motions are obtained. The stick motions are illustrated through the displacement, velocity, and force responses. This investigation provides a better understanding of stick and nonstick motions of the linear oscillator with dry friction. The methodology presented in this paper is applicable to oscillators with nonlinear friction forces.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

Global chaos in a periodically forced, linear system with a dead-zone restoring force

Abstract The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are introduced through switching planes pertaining to two constraints. The global periodic motions based on the Poincare mapping are determined, and the eigenvalue analysis for the stability and bifurcation of periodic motion is carried out. Global chaos in such a system is investigated numerically from the unstable global periodic motions analytically determined. The bifurcation scenario with varying parameters is presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed in this investigation.