Energy transmission in finite dissipative nonlinear periodic structures from excitation within a stop band

Abstract We study nonlinear transmission of wave energy in a finite dissipative periodic structure, which is harmonically driven at one end at a forcing frequency lying within its stop band. We show that there is a threshold for the driving amplitude above which there is a sudden increase in the energy transmitted across the finite structure. This generic phenomenon for discrete nonlinear periodic systems is due to a loss of stability of the periodic solutions that are initially localized to the driven end of the structure. The transmission threshold is therefore predicted analytically based on the corresponding saddle-node bifurcation. The influence of damping, strength of coupling and the type of nonlinearity (hardening or softening) are assessed. In particular, we show that damping may eliminate the transmission phenomenon within a frequency range in the stop band. Increasing the strength of coupling between the units is found to increase the minimum force required for the onset of transmission, and the type of nonlinearity determines on which side of the pass band the enhanced transmission may occur.

[1]  J. Leon,et al.  Cutoff solitons and bistability of the discrete inductance-capacitance electrical line: theory and experiments. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  G. Sen Gupta,et al.  Natural flexural waves and the normal modes of periodically-supported beams and plates , 1970 .

[3]  L. Brillouin,et al.  Wave Propagation in Periodic Structures , 1946 .

[4]  A. Vakakis,et al.  Nonlinear wave transmission in a monocoupled elastic periodic system , 1995 .

[5]  A. K. Mallik,et al.  Dynamics of a weakly non-linear periodic chain , 2001 .

[6]  S. Gonella,et al.  Spectro-spatial wave features as detectors and classifiers of nonlinearity in periodic chains , 2013 .

[7]  M. Ruzzene,et al.  Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook , 2014 .

[8]  J Leon,et al.  Energy transmission in the forbidden band gap of a nonlinear chain. , 2002, Physical review letters.

[9]  D. J. Mead.  Wave propagation and natural modes in periodic systems: II. Multi-coupled systems, with and without damping , 1975 .

[10]  Philip Holmes,et al.  A magnetoelastic strange attractor , 1979 .

[11]  D. J. Mead.  Wave propagation and natural modes in periodic systems: I. Mono-coupled systems , 1975 .

[12]  Jakob Søndergaard Jensen,et al.  Low-frequency band gaps in chains with attached non-linear oscillators , 2007 .

[13]  Alexander F. Vakakis,et al.  Forced localization in a periodic chain of non-linear oscillators , 1994 .

[14]  C. R. Willis,et al.  Discrete Breathers , 1997 .

[15]  Stefano Lepri,et al.  Nonlinear supratransmission and bistability in the Fermi-Pasta-Ulam model. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Takashi Hikihara,et al.  Coupled cantilever array with tunable on-site nonlinearity and observation of localized oscillations , 2009 .

[17]  A. Vakakis.  Relaxation oscillations, subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment , 2010 .

[18]  Massimo Ruzzene,et al.  A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures , 2010 .

[19]  J. E. Mac'ias-D'iaz,et al.  On the propagation of binary signals in damped mechanical systems of oscillators , 2007 .

[20]  G. Kopidakis,et al.  Energy dissipation threshold and self-induced transparency in systems with discrete breathers , 2006 .

[21]  Ramaz Khomeriki,et al.  Theory of a Josephson junction parallel array detector sensitive to very weak signals , 2006 .

[22]  J. Woodhouse,et al.  Vibration isolation from irregularity in a nearly periodic structure: Theory and measurements , 1983 .

[23]  S. Ruffo,et al.  Pattern formation and localization in the forced-damped Fermi-Pasta-Ulam lattice. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  S. Aubry.  Discrete Breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems , 2006 .

[25]  Renfrey B. Potts,et al.  Effect of Defects on Lattice Vibrations , 1955 .

[26]  Hadi Susanto.  Boundary Driven Waveguide Arrays: Supratransmission and Saddle-Node Bifurcation , 2008, SIAM J. Appl. Math..