Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker

In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties.

[1]  G. I. Schuëller,et al.  On the treatment of uncertainties in structural mechanics and analysis , 2007 .

[2]  Danuta Sado,et al.  Chaotic vibration of an autoparametrical system with a non-ideal source of power , 2007 .

[3]  José Manoel Balthazar,et al.  Nonlinear control in an electromechanical transducer with chaotic behaviour , 2014 .

[4]  José Manoel Balthazar,et al.  Nonlinear state estimation and control for chaos suppression in MEMS resonator , 2013 .

[5]  Kazuyuki Yagasaki,et al.  CONTROLLING CHAOS IN A PENDULUM SUBJECTED TO FEEDFORWARD AND FEEDBACK CONTROL , 1997 .

[6]  Thomas J. Bridges,et al.  A transverse spinning double pendulum , 2001 .

[7]  Bernard Friedland,et al.  Advanced Control System Design , 1996 .

[8]  Claude-Henri Lamarque,et al.  Numerical and Experimental Study of Regular and Chaotic Motion of Triple Physical Pendulum , 2008, Int. J. Bifurc. Chaos.

[9]  Khaled A. Alhazza,et al.  A hybrid command-shaper for double-pendulum overhead cranes , 2014 .

[10]  Rafael Castro-Triguero,et al.  Robustness of optimal sensor placement under parametric uncertainty , 2013 .

[11]  Claude-Henri Lamarque,et al.  Energy Pumping with Various Nonlinear Structures: Numerical Evidences , 2005 .

[12]  T. Nagamine,et al.  Stable Rotation of a Parametrically Excited Double Pendulum , 2007 .

[13]  Danuta Sado,et al.  ANALYSIS OF VIBRATION OF THREE-DEGREE-OF-FREEDOM DYNAMICAL SYSTEM WITH DOUBLE PENDULUM , 2008 .

[14]  Jerzy Warminski,et al.  Chaos in mechanical pendulum-like system near main parametric resonance , 2012 .

[15]  José Manoel Balthazar,et al.  Microcantilever chaotic motion suppression in tapping mode atomic force microscope , 2013 .

[16]  José Manoel Balthazar,et al.  On elimination of chaotic behavior in a non-ideal portal frame structural system, using both passive and active controls , 2013 .

[17]  José Manoel Balthazar,et al.  Nonlinear Control System Applied to Atomic Force Microscope Including Parametric Errors , 2013 .

[18]  B. R. Pontes,et al.  Statements on chaos control designs, including a fractional order dynamical system, applied to a “MEMS” comb-drive actuator , 2012 .

[19]  Ruiqi Wang,et al.  Chaos control of chaotic pendulum system , 2004 .

[20]  Steven R. Bishop,et al.  Rotating periodic orbits of the parametrically excited pendulum , 1995 .

[21]  Toshio Okada,et al.  A numerical analysis of chaos in the double pendulum , 2006 .

[22]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[23]  G. Kerschen,et al.  Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems , 2008 .

[24]  Stefano Lenci,et al.  Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective , 2011 .

[25]  Chunqing Lu.  Chaotic motions of a parametrically excited pendulum , 2006 .

[26]  Steven R. Bishop,et al.  Locating oscillatory orbits of the parametrically-excited pendulum , 1996, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[27]  J. D. Pearson.  Approximation Methods in Optimal Control I. Sub-optimal Control† , 1962 .

[28]  Takashi Hikihara,et al.  Tolerance of start-up control of rotation in parametric pendulum by delayed feedback , 2011 .

[29]  José Manoel Balthazar,et al.  TM-AFM nonlinear motion control with robustness analysis to parametric errors in the control signal determination , 2014 .

[30]  Zheng-Ming Ge,et al.  NON-LINEAR DYNAMICS AND CHAOS CONTROL OF A PHYSICAL PENDULUM WITH VIBRATING AND ROTATING SUPPORT , 2001 .

[31]  Ekaterina Pavlovskaia,et al.  Rotating solutions and stability of parametric pendulum by perturbation method , 2008 .

[32]  Matthew P. Cartmell,et al.  Rotating orbits of a parametrically-excited pendulum , 2005 .

[33]  José Manoel Balthazar,et al.  On energy transfer phenomena, in a nonlinear ideal and nonideal essential vibrating systems, coupled to a (MR) magneto-rheological damper , 2012 .

[34]  J. Cloutier,et al.  Control designs for the nonlinear benchmark problem via the state-dependent Riccati equation method , 1998 .

[35]  Mark J. Schulz,et al.  A refined nonlinear vibration absorber , 2000 .

[36]  Reyolando M. L. R. F. Brasil,et al.  On Saturation Control of a Non-Ideal Vibrating Portal Frame Foundation Type Shear-Building , 2005 .

[37]  Itamar Iliuk,et al.  Application of passive control to energy harvester efficiency using a nonideal portal frame structural support system , 2014 .

[38]  Oleg Gendelman,et al.  Targeted energy transfer in systems with non-polynomial nonlinearity , 2008 .

[39]  Ali H. Nayfeh,et al.  A Theoretical and Experimental Implementation of a Control Method Based on Saturation , 1997 .

[40]  Jakob Søndergaard Jensen,et al.  NON-LINEAR DYNAMICS OF THE FOLLOWER-LOADED DOUBLE PENDULUM WITH ADDED SUPPORT-EXCITATION , 1998 .

[41]  G. Cook,et al.  Suboptimal control for the nonlinear quadratic regulator problem , 1975, Autom..

[42]  M. Farid Golnaraghi,et al.  Vibration suppression of flexible structures using internal resonance , 1991 .

[43]  P. F. Pai,et al.  Structural Vibration Control Using PZT Patches and Non-Linear Phenomena , 1998 .

[44]  Chunqing Lu.  Chaotic Motions of a Parametrically Excited Pendulum With Non-Exponentially Small Parameters , 2005 .