Nonlinear Acoustic Metamaterials:Theory of Elastic Wave Propagation and Applications on Vibration Reduction

Author:Fang Zuo

Supervisor:wen ji hong


Degree Year:2018





Modern equipment aims to be large-scale,super-speed,light-weight and integrated gradually.Those designing objectives raise imperious demands for the low-frequency,broad-band and highly efficient vibration and noise reduction techniques.Beams,plates and shells are fundamental structures of the equipment.Suppressing the vibration of these structures is an effective approach to reduce the vibration and noise of the whole equipment.Low-frequency and broad-band vibrations are main troubles.However,it is difficult to realize the low-frequency and broad-band vibration reductions with traditional absorbing and damping techniques.In recent years,massive theoretical and applied studies on the unusual features of the acoustic metamaterials provide new methods to go beyond the traditional techniques.Acoustic metamaterials represent the subwavelength structures/materials having unusual characteristics.Locally resonant structures are the most representative acoustic metamaterials,whose elastic waves’ frequency bandgaps can suppress the low-frequency vibrations effectively.At present,relevant studies are mainly focusing on the linear acoustic metamaterials(LAMs).Unfortunately,with the limited mass in LAMs,it is difficult to synchronously realize the ―low-frequency‖ and ―broad-band‖ vibration suppressions.The generalized width γ for the vibration suppression with bandgaps in LAM is less than 1,generally(the mass ratio is less than 50%).Moreover,the attached local resonators increase the number of resonances in the passbands of finite structures.Those disadvantages of LAMs limit their applications in equipment.Nonlinear effects in materials can open vast prospects and space for wave manipulations.The nonlinear electromagnetic metamaterials have obtained tremendous progress in the past two decades,and now their promising applications are expectable.In contrast,the wave dynamics and vibration of nonlinear acoustic metamaterials(NAMs)are not uncovered.Nonlinear acoustic metamaterials represent acoustic metamaterials having nonlinear dynamic effects of elastic waves.To meet the urgent requirements for the ―ultra-low frequency,ultra-broad band,and efficient vibration attenuation‖ techniques,this dissertation systematically investigates the theories of elastic wave propagation and nonlinear dynamics in nonlinear acoustic metamaterials(NAMs)and their applications to the vibration reduction of typical structures—beams,plates and shells.The main innovations are as follows:1.A number of theoretical methods are improved to analyze the dispersion properties and elastic wave propagation principles in typical NAMs.For infinite structures,the homotopy approach is adopted to solve the nonlinear dispersion relations,and analytical method is proposed to describe the couplings between the fundamental wave and the third harmonic in the NAM beam.For finite structures,an approach for analyzing the bifurcations of periodic solutions in frequency domain is put forward based on the harmonic average method and a perturbation continuation algorithm;moreover,dimension reduction algorithm is studied.2.This thesis initiatively reports and demonstrates a new regime that can broaden the band for low-frequency elastic wave attenuation/suppression— chaotic band.The transmissions,state transitions,bifurcations and chaotic attractors of elastic waves propagating in the passbands and the nonlinear locally resonant(NLR)bandgaps are investigated,which lead to the discovery of the chaotic band that can suppress the low-frequency and broadband waves with a light weight.New band structures containing bandgaps and chaotic bands are proposed and their manipulating rules are unveiled.Furthermore,we design strongly nonlinear sub-wavelength meta-cells to construct the NAM beam and NAM plate based on the chaotic band mechanisms.The experiments firstly verify that those NAMs suppress the elastic waves in an ultra-low frequency and ultra-broad band significantly,which is termed as ―double-ultra‖ effect.The double-ultra effect greatly breakthrough the bandwidth limit for wave suppression in LAMs: the NAM beam realizes γ=21 and meanwhile the NAM plate achieves the surprising goal γ=45.6;in such broad bands,the wave transmission decreases by 20-40 d B.3.The manuscript discovers and verifies an effective mechanism to manipulate the chaotic band—bridging coupling of NLR bandgaps.It is found that,the strongly nonlinear coupling between resonators makes the passbands between and near the NLR bandgaps become chaotic bands;by increasing the frequency distance between two NLR bandgaps,the total width of the chaotic band is broadened and the efficiency for elastic wave suppression is enhanced;the energy transferring between resonators in a nonlinear way shares the negative effective mass in the broad passband,that is the remote bridging coupling of NLR bandgaps.This mechanism is verified by designing a new NAM beam with nonlinearly coupled resonances.Moreover,the regimes of the double-ultra effect originated from the bridging coupling of multiple NLR bandgaps are elaborated,and the multi-state behavior of the elastic wave propagation in the NAM’s bandgap is demonstrated.4.Properties of high-order harmonics in NAMs are explored.Based on the half-finite NAM beam model,this manuscript studies the propagations,couplings and bifurcations of the fundamental wave and its third harmonic.In addition,we report the interesting properties of nonlinear resonances of the finite NAM plate including the generation of high-order harmonics,amplitude-dependent modes and damping mechanism;and interesting phenomena,such as the quiet mode,are found.5.A number of designing schemes of NAMs’ cells are put forward.This thesis applied the principle of the bridging coupling bandgaps to the NAM cylinder shells and achieves the low-frequency and broadband vibration reduction successfully.The influences of the attached numbers and positions of nonlinear resonators are studied.In summary,this thesis systematically investigates theories of the elastic wave propagation in the nonlinear acoustic metamaterials.A number of design techniques and analyzing approaches are proposed.This manuscript firstly discovers,unveils and demonstrates a series of new phenomena,new mechanisms,and new features of elastic wave propagations,which provide powerful principles to manipulate waves.The work preliminary realizes the ultra-low frequency,ultra-broad band and highly efficient vibration reduction/suppression of the beam,plate and cylinder shell.The results provide important theoretical and technical foundations for the front field ―nonlinear acoustic metamaterial‖,and will support the vibration reduction in equipment.