ON FRACTIONAL ORDER AND NONLINEAR PHENOMENA IN A MAGNETO-ELASTIC DYNAMIC SYSTEM X Conferência Brasileira de Dinâmica e Controle, Águas de Lindóia, Brasil
S. A. David - Universidade de São Paulo, Pirassununga, Brasil,
[email protected] J. M. Balthazar - Universidade Estadual Paulista, Rio Claro, Brasil,
[email protected] B.H.S.Julio - Universidade de São Paulo, Pirassununga, Brasil,
[email protected] Simulation results
Abstract
Numerous researchers have studied and taught about nonlinear phenomena that exhibit periodic, nonperiodic and chaotic solutions in several areas of science and engineering. However, in most cases, these concepts have been explored mainly from the standpoint of analytical and computational methods involving integer order calculus (IOC). In this paper we have examined the dynamic behavior of an elastic wide plate induced by two electromagnets and submitted to external excitation of a point of view of the fractional order calculus (FOC). The primary focus of this study is on to help gain a better understanding of nonlinear and possibly chaotic phenomena in fractional order dynamic systems.
We would like to illustrate some simulations results for non-integer order equation (2) in comparison to integer order for F = 0.18 , ω=1rad/s and = *0.8(red line); 1.0(blue line); 1.2(green line)].
Mechanical-system theorical model
Many technical devices such as motors, generators, transformers, and fusion reactors are known to employ wide elastic plates in magnetic fields. A system involving a flexible rod subjected to magnetic forces that can bend it while simultaneously subjected to external excitations produces complex and nonlinear dynamic behavior [2-4], which may present different types of solutions for its different movement-related responses [1]. This fact motivated us to analyze such a mechanical system based on modeling and numerical simulation involving both, (IOC) and (FOC), approaches. A continuum model based on linear elastic and nonlinear magnetic forces was developed, which can be reduced to an oscillator model with a single degree of freedom using the Lagrangian formalism [5] and the Galerkin's method [3-5]. The mechanical system whose theoretical model is developed is shown in Figure 1. A flexible rod is clamped in a rigid base. The electromagnets pull the beam in opposite directions and intensity of the field is strong enough to deflect the rod from one side to another. The electromagnets generate a magnetic field that induces a magnetization M per unit volume in the solid.
We have illustrated too some simulations results for non-integer order equation (2) in comparison to integer order for F = 4 , ω =1.12 rad/s and = [0.8(red line); 1.0(blue line); 1.2(green line)].
Conclusions
We can write the equation of motion this system with the dimensionless form [1], (1) where: δ >0 is the damping constant , F is the forcing strength and ω is the forcing frequency. In this article we investigate, by means of numerical simulations using Matlab/SimulinkTM (ninteger toolbox [6]), the modified magneto-elastic dynamic equations (2) , i.e., (2) We presented the time responses, pseudo phase portraits and Fourier spectra for some values of λ , ω, F and for δ = 0.2.
In this paper we have proposed some versions of the modified magneto-elastic dynamic equation like presented by equation (2). Such modifications consisted in the introduction of a non-integer order time derivative in the standard equation (1). The results reveal that the noninteger order systems can exhibit different and curious behavior from those obtained with the standard magnetoelastic dynamic system. The non-integer order can be useful for a better understanding and control of such nonlinear systems. References [1] David, S. A. ; Rosario, J. M. ; Machado, J.. Investigation about chaos in a magneto-elastic dynamical system. Computational Fluid and Solid Mechanics – Solids and Structures , Multi-Physics, vol.2, K.J.Bathe editor, pp. 11201123, Elsevier Science Ltd., England,ISBN 0-08043944-6. (2001). [2] Feynman, R. P., Leighton, R. B., Sands, M. "The Feynman Lectures on Physics", Addison-Wesley, Reading, M.A. (1965). [3] Goldstein, H. "Classical Mechanics", Reading Mass., Addison Wesley Publishing Company, Second Edition (1981). [4] Guckenheimer, J., Holmes, P. "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields ", Springer - New York (1983). [5] Hagedorn, P. Oscilacoes Nao Lineares. Sao Paulo: Editora Edgard Blucher LTDA, (1984). [6] Valerio, D.P.M.O.: Toolbox ninteger for Matlab, v. 2.3 (2005). http://mega.ist.utl.pt/~dmov/ninteger/ninteger.htm
Dincon 2011 – Águas de Lindóia, Brazil
