Optimization of an acoustic rectifier for uni-directional wave propagation in periodic mass–spring lattices

Abstract We perform optimization studies on the construction of acoustic rectifiers, which allow uni-directional propagation of acoustic waves, from a periodic array of masses and springs arranged in one- and two- dimensions. An acoustic rectifier is achieved by pairing a nonlinear material, which can up-convert an input excitation frequency to a higher harmonic, with a bandgap material whose dispersion relation has a bandgap region for the input frequency range but a bandpass region at the higher harmonic. First, we analyze the mass and stiffness parameters that lead to acoustic rectification in infinite mass–spring arrays with the largest possible range of working frequencies. A combination of analytical techniques, numerical simulations, and particle swarm optimization is used to identify the optimal acoustic rectifier. Next, we study the practical working range of acoustic rectifiers of finite size and examine how the rectification properties change as a function of the lattice size and damping. Finally, we perform numerical simulations of an acoustic rectification device in which a Duffing oscillator is attached to the end of a tri-atomic mass–spring chain.

[1]  Xu Ni,et al.  Tunable unidirectional sound propagation through a sonic-crystal-based acoustic diode. , 2011, Physical review letters.

[2]  Mahmoud I. Hussein,et al.  Band structure of phononic crystals with general damping , 2010 .

[3]  S. Mukherjee,et al.  Dispersion relations and mode shapes for waves in laminated viscoelastic composites by variational methods , 1978 .

[4]  Leung,et al.  Photonic band structure: The face-centered-cubic case employing nonspherical atoms. , 1991, Physical review letters.

[5]  B. Alder,et al.  Radial Distribution Functions and the Equation of State of Fluids Composed of Molecules Interacting According to the Lennard‐Jones Potential , 1952 .

[6]  M. Leamy,et al.  Analysis of Bloch’s Method in Structures with Energy Dissipation , 2011 .

[7]  D. M. Mead,et al.  WAVE PROPAGATION IN CONTINUOUS PERIODIC STRUCTURES: RESEARCH CONTRIBUTIONS FROM SOUTHAMPTON, 1964–1995 , 1996 .

[8]  L. Rayleigh,et al.  XVII. On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure , 1887 .

[9]  A. Hladky-Hennion,et al.  Localized modes in a one-dimensional diatomic chain of coupled spheres , 2005 .

[10]  C. Elachi,et al.  Waves in active and passive periodic structures: A review , 1976, Proceedings of the IEEE.

[11]  G. Theocharis,et al.  Bifurcation-based acoustic switching and rectification. , 2011, Nature materials.

[12]  Eleftherios N. Economou,et al.  Elastic and acoustic wave band structure , 1992 .

[13]  Vincent Laude,et al.  Guiding and bending of acoustic waves in highly confined phononic crystal waveguides , 2004 .

[14]  Guoliang Huang,et al.  Band Gaps in a Multiresonator Acoustic Metamaterial , 2010 .

[15]  L. Brillouin.  Wave propagation in periodic structures : electric filters and crystal lattices , 1953 .

[16]  W. Cao,et al.  Acoustic bandgap formation in a periodic structure with multilayer unit cells , 2000 .

[17]  D. J. Mead.  A general theory of harmonic wave propagation in linear periodic systems with multiple coupling , 1973 .

[18]  F. Montero de Espinosa,et al.  Sonic Band Gaps in Finite Elastic Media: Surface States and Localization Phenomena in Linear and Point Defects , 1999 .

[19]  P. Yu,et al.  Complex elastic wave band structures in three-dimensional periodic elastic media , 1998 .

[20]  Sonic band structure and localized modes in a density-modulated system: Experiment and theory , 2002 .

[21]  Real-time imaging of acoustic rectification , 2011 .

[22]  P. Deymier,et al.  Elastic and viscoelastic effects in rubber/air acoustic band gap structures: A theoretical and experimental study , 2008 .

[23]  Davide Bigoni,et al.  Band-gap shift and defect-induced annihilation in prestressed elastic structures , 2009 .

[24]  Richard A. Scott,et al.  Dispersive elastodynamics of 1D banded materials and structures: Design , 2007 .

[25]  Eli Yablonovitch.  Photonic Band Structure , 1993 .

[26]  G. Wang,et al.  Flexural vibration band gaps in Timoshenko beams with locally resonant structures , 2006 .

[27]  Bin Liang,et al.  Acoustic diode: rectification of acoustic energy flux in one-dimensional systems. , 2009, Physical review letters.

[28]  A. Shanin,et al.  Resonances in a discrete system of variable length , 2009 .

[29]  John H. Page,et al.  Elastic Wave Scattering by Periodic Structures of Spherical Objects: Theory and Experiment , 2000 .

[30]  A. Hladky-Hennion,et al.  Quantitative analysis of the vibration modes in a finite set of coupled spheres , 2004 .

[31]  P. Sheng,et al.  Focusing of sound in a 3D phononic crystal. , 2004, Physical review letters.

[32]  R. A. Scott,et al.  Dispersive elastodynamics of 1D banded materials and structures: analysis , 2006 .

[33]  E. Yablonovitch,et al.  Inhibited spontaneous emission in solid-state physics and electronics. , 1987, Physical review letters.

[34]  Russell C. Eberhart,et al.  Parameter Selection in Particle Swarm Optimization , 1998, Evolutionary Programming.

[35]  B. Djafari-Rouhani,et al.  Sound isolation from cubic arrays of air bubbles in water , 1998 .

[36]  R. Martínez-Sala,et al.  Refractive acoustic devices for airborne sound. , 2001 .

[37]  J. Strutt.  Scientific Papers: On the Maintenance of Vibrations by Forces of Double Frequency, and on the Propagation of Waves through a Medium endowed with a Periodic Structure , 2009 .

[38]  Jan Awrejcewicz,et al.  Continuous models for chain of inertially linked masses , 2005 .

[39]  R. Eberhart,et al.  Comparing inertia weights and constriction factors in particle swarm optimization , 2000, Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512).

[40]  M. Makela,et al.  Phononic band structure in a mass chain , 1995 .

[41]  A. Hladky-Hennion,et al.  Experimental validation of band gaps and localization in a one-dimensional diatomic phononic crystal. , 2007, The Journal of the Acoustical Society of America.

[42]  Michael J. Leamy,et al.  Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices , 2012 .

[43]  Anne-Christine Hladky-Hennion,et al.  Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method , 1995 .

[44]  P. Kuchment,et al.  An Efficient Finite Element Method for Computing Spectra of Photonic and Acoustic Band-Gap Materials , 1999 .

[45]  B. Liang,et al.  An acoustic rectifier. , 2010, Nature materials.

[46]  R. Rechtman,et al.  Acoustic gaps in a chain of magnetic spheres. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Bruce L. Davis,et al.  Analysis of Periodicity Termination in Phononic Crystals , 2011 .

[48]  V. Romero-Garc'ia,et al.  Tunable wideband bandstop acoustic filter based on 2D multi-physical phenomena periodic systems , 2011, 1102.1593.

[49]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[50]  Riccardo Poli,et al.  Particle swarm optimization , 1995, Swarm Intelligence.

[51]  P. Sheng,et al.  Ultrasound tunneling through 3D phononic crystals. , 2002, Physical review letters.

[52]  P. Sheng,et al.  Locally resonant sonic materials , 2000, Science.

[53]  C. Daraio,et al.  Pulse propagation in a linear and nonlinear diatomic periodic chain: effects of acoustic frequency band-gap , 2009 .

[54]  Jakob Søndergaard Jensen,et al.  Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures , 2003 .

[55]  I.E.Psarobas,et al.  Phononic crystals with planar defects , 2000 .