Nonlinear power flow analysis of the Duffing oscillator

Power flow characteristics of different forms of the Duffing oscillator, subject to harmonic excitations, are studied in this paper to reveal the distinct power input and dissipation behaviour arising from its nonlinearity. Power flow variables, instead of the displacement and velocity responses, are used to examine the effects of nonlinear phenomena including sub-/super-harmonic resonances, non-uniqueness of solutions, bifurcations and chaos. Both analytical harmonic balance approximations and Runge–Kutta numerical integrations are adopted to effectively address instantaneous/time-averaged power flows of the system with periodic/chaotic motions without losing the essential nonlinear characteristics. It is demonstrated that only the in-phase velocity component with the same frequency as the excitation contributes to the time-averaged input power (TAIP). It is shown that super-/sub-harmonic resonances may result in substantial increases in TAIP and the nonlinearity leads to varying time-averaged power flow levels sensitive to the initial conditions. The study reveals that bifurcations may cause large jumps in time-averaged input power. However, for bifurcations of periodic to chaotic motions encountered in the low-frequency range, the corresponding variations in TAIP of the double-well potential systems are small. For a chaotic response, the associated TAIP is insensitive to the initial conditions but tends to an asymptotic value as the averaging time increases, and thus can be used as a measure to quantify chaotic responses. The paper concludes some inherently nonlinear power flow characteristics which differ greatly from those of the linear systems, and provides useful information for applications.

[1]  B L Clarkson.  Estimation of the Coupling Loss Factor of Structural Joints , 1991 .

[2]  B. Mace,et al.  Energy flow models from finite element analysis , 2000 .

[3]  Alexander F. Vakakis,et al.  NONLINEAR NORMAL MODES , 2001 .

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

[5]  I. Kovacic,et al.  The Duffing Equation: Nonlinear Oscillators and their Behaviour , 2011 .

[6]  Yeping Xiong,et al.  Dynamics and power flow behaviour of a nonlinear vibration isolation system with a negative stiffness mechanism , 2013 .

[7]  Yeping Xiong,et al.  Power flow behaviour of the Duffing oscillator , 2012 .

[8]  J. M. Cuschieri,et al.  Structural power‐flow analysis using a mobility approach of an L‐shaped plate , 1990 .

[9]  Thomas J. Royston,et al.  OPTIMIZATION OF PASSIVE AND ACTIVE NON-LINEAR VIBRATION MOUNTING SYSTEMS BASED ON VIBRATORY POWER TRANSMISSION , 1996 .

[10]  Jing Tang Xing,et al.  Interactive power flow characteristics of an integrated equipment—nonlinear isolator—travelling flexible ship excited by sea waves , 2005 .

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

[12]  Seshadri Sankar,et al.  Modelling and analysis of non-linear orifice type damping in vibration isolators , 1995 .

[13]  W. G. Price,et al.  A power flow mode theory based on a system's damping distribution and power flow design approaches , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  R. G. White,et al.  Vibrational power flow from machines into built-up structures, part III: Power flow through isolation systems , 1980 .

[15]  Robin S. Langley,et al.  A wave intensity technique for the analysis of high frequency vibrations , 1992 .

[16]  Thomas J. Royston,et al.  Vibratory power flow through a nonlinear path into a resonant receiver , 1996 .

[17]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[18]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[19]  Stephen J. Elliott,et al.  ACTIVE ISOLATION OF STRUCTURAL VIBRATION ON A MULTIPLE-DEGREE-OF-FREEDOM SYSTEM, PART II: EFFECTIVENESS OF ACTIVE CONTROL STRATEGIES , 1997 .

[20]  Alexander F. Vakakis,et al.  Nonlinear normal modes, Part I: A useful framework for the structural dynamicist , 2009 .

[21]  Jon R. Pratt,et al.  A Nonlinear Vibration Absorber for Flexible Structures , 1998 .

[22]  G. Kim,et al.  A study of passive and adaptive hydraulic engine mount systems with emphasis on non-linear characteristics , 1995 .

[23]  R. G. White,et al.  Vibrational power flow from machines into built-up structures, part I: Introduction and approximate analyses of beam and plate-like foundations , 1980 .

[24]  D. Ewins,et al.  The Harmonic Balance Method with arc-length continuation in rotor/stator contact problems , 2001 .

[25]  W. G. Price,et al.  A power–flow analysis based on continuum dynamics , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[26]  John E. Mottershead,et al.  Identification of nonlinear bolted lap-joint parameters by force-state mapping , 2007 .

[27]  W. Price,et al.  POWER FLOW ANALYSIS OF COMPLEX COUPLED SYSTEMS BY PROGRESSIVE APPROACHES , 2001 .

[28]  Robin S. Langley,et al.  Analysis of power flow in beams and frameworks using the direct-dynamic stiffness method , 1990 .