Uncovering inner detached resonance curves in coupled oscillators with nonlinearity

Abstract Detached resonance curves have been predicted in multi-degree-of-freedom nonlinear oscillators, when subject to harmonic excitation. They appear as isolated loops of solutions in the main continuous frequency response curve and their detection may thus be hidden by numerical or experimental analysis. In this paper, an analytical approach is adopted to predict their appearance. Expressions for the amplitude-frequency equations and bifurcation curves are derived for a two degree-of-freedom system with cubic stiffness nonlinearity, and the effect of the system parameters is investigated. The interest is specifically towards the occurrence of closed detached curves appearing inside the main continuous frequency response curve, which may lead to a dramatic reduction of the amplitude of the system response. Both cases of hardening and softening stiffness characteristics are considered. The analytical findings are validated by numerical analysis.

[1]  Gianluca Gatti,et al.  On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system , 2010 .

[2]  Oleg Gendelman,et al.  Dynamics of a strongly nonlinear vibration absorber coupled to a harmonically excited two-degree-of-freedom system , 2008 .

[3]  Frank Schilder,et al.  Exploring the performance of a nonlinear tuned mass damper , 2009 .

[4]  Luc Masset,et al.  The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems , 2015, 1604.05621.

[5]  Ali H. Nayfeh,et al.  Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator , 2006 .

[6]  Gianluca Gatti,et al.  The characteristics of a nonlinear vibration neutralizer , 2012 .

[7]  Gianluca Gatti,et al.  On the effects of system parameters on the response of a harmonically excited system consisting of weakly coupled nonlinear and linear oscillators , 2011 .

[8]  Matthew S. Allen,et al.  Nonlinear normal modes modal interactions and isolated resonance curves , 2015, 1604.05567.

[9]  Granino A. Korn,et al.  Mathematical handbook for scientists and engineers , 1961 .

[10]  I. Kovacic,et al.  On the jump-up and jump-down frequencies of the duffing oscillator , 2008 .

[11]  Gianluca Gatti,et al.  On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator , 2010 .

[12]  B. Krauskopf,et al.  Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment , 2013, 1308.3647.

[13]  Chengwu Duan,et al.  Isolated sub-harmonic resonance branch in the frequency response of an oscillator with slight asymmetry in the clearance , 2008 .

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

[15]  Oleg Gendelman,et al.  Vibration absorption in systems with a nonlinear energy sink: Nonlinear damping , 2009 .

[16]  Keith Worden.  On jump frequencies in the response of the Duffing oscillator , 1996 .