Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency–energy domain by computing forced frequency–energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.

[1]  O. Gendelman.  Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators , 2001 .

[2]  Gaëtan Kerschen,et al.  Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques , 2009 .

[3]  Leonid I. Manevitch,et al.  The Mechanics Of Nonlinear Systems With Internal Resonances , 2005 .

[4]  Claude-Henri Lamarque,et al.  Nonlinear energy pumping under transient forcing with strongly nonlinear coupling: Theoretical and experimental results , 2007 .

[5]  R. M. Rosenberg,et al.  On Nonlinear Vibrations of Systems with Many Degrees of Freedom , 1966 .

[6]  Oleg Gendelman,et al.  Isolated Resonance Captures and Resonance Capture Cascades Leading to Single- or Multi-Mode Passive Energy Pumping in Damped Coupled Oscillators , 2004 .

[7]  A. Vakakis,et al.  Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification , 2014 .

[8]  Leonid I. Manevitch,et al.  Complex Representation of Dynamics of Coupled Nonlinear Oscillators , 1999 .

[9]  Oleg Gendelman,et al.  Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture , 2001 .

[10]  Claude-Henri Lamarque,et al.  Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment , 2007 .

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

[12]  Mehmet Kurt,et al.  Frequency-energy plots of steady-state solutions for forced and damped systems, and vibration isolation by nonlinear mode localization , 2014, Commun. Nonlinear Sci. Numer. Simul..

[13]  Oleg Gendelman,et al.  Energy pumping in nonlinear mechanical oscillators : Part I : Dynamics of the underlying Hamiltonian systems , 2001 .

[14]  Alexander F. Vakakis,et al.  Periodic orbits, damped transitions and targeted energy transfers in oscillators with vibro-impact attachments , 2009 .