Effect of damping on the nonlinear modal characteristics of a piecewice-smooth system through harmonic forced response

Abstract Several engineering systems present piecewise-linear characteristics, among them, the damaged beams with breathing cracks are of particular interest. The dynamics of such systems exhibits bifurcations at internal resonances characterized by the onset of superabundant nonlinear normal modes with their individual modal shapes. In this paper, a 2-DOF system with piecewise linear stiffness, representative of a damaged system with a breathing crack, is analyzed by means of a numerical and theoretical investigation. The oscillator is forced by a harmonic base excitation and the role of damping on the modification of the nonlinear modal characteristics is investigated. The outcomes are compared with the reference behavior of the undamped system which allows for semi-analytical solution. It is found that the damping, on one hand, softens the abrupt transitions from one behavior to another, typical of undamped systems, on the other, affects the bifurcations of the nonlinear modes causing some of them to completely disappear and leaving others largely unaffected.

[1]  Christophe Pierre,et al.  Normal modes of vibration for non-linear continuous systems , 1994 .

[2]  E. Butcher.  CLEARANCE EFFECTS ON BILINEAR NORMAL MODE FREQUENCIES , 1999 .

[3]  Alain Curnier,et al.  Non-Linear Real And Complex Modes Of Conewise Linear Systems , 1994 .

[4]  Fabrizio Vestroni,et al.  A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system , 2008 .

[5]  Steven W. Shaw,et al.  Normal modes for piecewise linear vibratory systems , 1996 .

[6]  C. H. Pak.  On the coupling of non-linear normal modes , 2006 .

[7]  A. H. Nayfeh,et al.  On Nonlinear Normal Modes of Systems With Internal Resonance , 1996 .

[8]  R. M. Rosenberg,et al.  The Normal Modes of Nonlinear n-Degree-of-Freedom Systems , 1962 .

[9]  Gerard Olivar,et al.  Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems , 2008 .

[10]  A. H. Nayfeh,et al.  Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems , 2003 .

[11]  Christophe Pierre,et al.  The construction of non-linear normal modes for systems with internal resonance , 2005 .

[12]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[13]  Fabrizio Vestroni,et al.  Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems , 2011 .

[14]  P. Casini,et al.  Non-linear dynamics of a cracked cantilever beam under harmonic excitation , 2007 .

[15]  Oleg Gendelman,et al.  A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities , 2003 .

[16]  S. Natsiavas,et al.  Dynamics of Multiple-Degree-of-Freedom Oscillators With Colliding Components , 1993 .

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

[18]  Claude-Henri Lamarque,et al.  Bifurcation and Chaos in Nonsmooth Mechanical Systems , 2003 .

[19]  R. M. Rosenberg.  On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems , 1961 .

[20]  G. V. Anand.  Natural modes of a coupled non-linear system , 1972 .

[21]  P. Casini,et al.  Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems , 2011 .

[22]  S. Mukherjee,et al.  MODAL ANALYSIS OF A CRACKED BEAM , 1997 .

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

[24]  Fabrizio Vestroni,et al.  Persistent and ghost nonlinear normal modes in the forced response of non-smooth systems , 2012 .

[25]  C. Pierre,et al.  Large-amplitude non-linear normal modes of piecewise linear systems , 2004 .

[26]  Alexander F. Vakakis,et al.  Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment , 2005 .

[27]  Alexander F. Vakakis,et al.  NON-LINEAR NORMAL MODES (NNMs) AND THEIR APPLICATIONS IN VIBRATION THEORY: AN OVERVIEW , 1997 .