Acoustical And Vibratory Phenomena Resulting From Local Resonance

Local resonance can be modeled as a spring oscillator excited by an external driving force, and it is always spatially localized, but for some special cases, such as Helmholtz resonance, there is no spatial distribution of local resonance. The composite structures consisting of local resonant elements, such as the nonlinear chains, have been applied to describe soliton transmission and energy localization, etc. For the composite structures with non-local resonance, such as periodic structures, the propagation of linear waves is forbidden in the band gap induced by Bragg resonance. Several acoustical and vibra-tory phenomena resulting from local resonance have been studied in this dissertation, especially for acoustic resonant transmission and soliton generation.In chapter 1 of the dissertation, a brief review is given with regard to linear and non-linear phenomena in resonant structures. Since local resonance is closely bound up with non-local resonance, we mainly focus on phenomena resulting from the two resonances and give some emphasis on several acoustical and vibratory phenomena resulting from local resonance that we study in the dissertation. A brief summary of linear phenomena is given, such as band gap and resonant transmission, where acoustic resonant tunneling in chapter 3 is related to resonant transmission. For nonlinear phenomena, the role of resonance is illustrated by the nonlinear vibration response, the physical mechanism for soliton existence and soliton generation, where soliton generation by local resonance in chapter 4 is related to the mechanism of soliton generation and a acoustic soliton model in chapter 5 is related to the mechanism of soliton existence.In chapter 2 of the dissertation, several types of local resonance have been dis-cussed. Firstly, the response of a membrane under plane wave incidence is illustrated by the amplitude and phase of the transmission coefficient and group delay. Secondly, parallel resonance induced by the coupling between local resonant units has been ana-lyzed from the view point of acoustic impedance with two examples, one is the coupling between two different Helmholtz resonators connected to a tube at the same site, the other one is the coupling between the two same Helmholtz resonators connected to a tube in a small space interval. As a result of parallel resonance, acoustic tunneling can happen via an interaction between local resonance in time domain and length resonance in the spatial domain (see chapter 3). Finally, we discuss local resonance in the band gap. The vibration of the localized mode could be amplified very much due to the lack of radiation damping. Different from parallel resonance induced by coupling, this kind of resonance can be used to generate solitons, which is discussed in chapter 4 of the dissertationIn chapter 3 of the dissertation, we discuss the acoustic resonant tunneling phe-nomenon in local resonant structures, such as the array of Helmholtz resonators. This phenomenon results from parallel resonance. Since the frequencies of resonant trans-mission peaks lie in the band gap of a single Helmholtz resonator, this kind of resonant tunneling phenomenon means that acoustic waves tunnels the band gap of a single Helmholtz resonator. Different from the resonant tunneling phenomenon caused by the half wavelength resonance, this resonant tunneling phenomenon results from the interaction between local resonance in time domain and Bragg resonance in the spatial domain, and could even be induced by two different local resonances. Since local res-onance is the resonance in time domain, the group velocity of the resonant tunneling phenomenon has no dependence on barrier thickness as in quantum mechanics, and it is determined by the resonant frequency difference of Helmholtz resonance and Bragg resonance, particularly, it could even be determined by the two local resonant frequen-cies in the array of two different Helmholtz resonators. Further more, we investigate this phenomenon from the view point of acoustic impedance. When the frequency of an incident wave is above or below the resonant frequency of a Helmholtz resonator, the impedance of a Helmholtz resonator exhibits the property of reactance or capacitance, as a result, a phase difference has been formed, which is not caused by the distance of wave propagation, but by local resonance. This phase difference compensates with the phase diffference caused by the distance of wave propagation and then the resonant tunneling phenomenon happens. It should be noted the coupling between Helmholtz resonators not only induces the resonant tunneling phenomenon but also forms wide band gap between two different local resonant frequencies.In the chapter 4 of the dissertation, we propose a resonant mechanism to generate discrete solitons. This method is manageable and high efficient in generating solitons. By introducing a mass impurity in a semi-infinite nonlinear chains, a localized mode has been formed around the impurity. Due to the resonance with this localized mode, the localized mode absorbs energy from the driving end and stores in it. With the increasing the amplitude of the localized mode, the vibration of the localized mode enters into the nonlinear state. When the absorbed energy arrives a threshold, the stored energy is radiated in form of solitons. It is interesting to observed that the energy stored in the localized mode is almost transferred to the propagating solitons, so local resonance in generating solitons is more efficient. Due to the process of absorption, the averaged emission interval of solitons has strong dependence on the driving amplitude, so local resonance is more manageable in generating solitons. We also discuss local resonance in other discrete systems. Further more, by extending the concept of local resonance to continuous systems and making a simplification of the driving boundary condition, we obtain a model which is realizable in physics. At the end, we proposed a model to generated water solitons by this concept.In the chapter 5 of the dissertation, a acoustic soliton model has been proposed. With well understanding the mechanism of generating the non-propagating soliton in the water trough and the progress in acoustic composite structures, we build an acoustic composite structure, which has a cut-off frequency at low frequency and the nonlinear response at this cut-off frequency exhibits "soft spring" property. So this acoustic com-posite structure is similar to the water trough. It is interesting that there is no transverse mode at the cut-off frequency, the order of velocity disturbances is second order, and the order of density disturbances is first order. If the rigid boundary condition is im-posed, we could neglect the effect of velocity disturbances, so this acoustic composite structure only exhibits density disturbances and we call acoustic soliton as density type in our dissertation. In order to compensate the dissipation in the structure caused by the radiation loss and viscosity loss in holes, density flow is introduced in our model, and then we obtain the damped-driven nonlinear Schrodinger equation. By estimating the physical parameters, we thought it is possible to generate acoustic solitons in this acoustic composite structure. The principal contributions of the present study are summarized as below:1. The acoustic resonant tunneling phenomenon produced by local resonance in time domain and Bragg resonance in the spatial domain are given in chapter 3. We discuss the corresponding property of the resonant tunneling phenomenon and illustrate differences with the resonant tunneling phenomenon caused by Fabry-Perot resonance.2. We propose a resonant mechanism to generate discrete solitons. This method is manageable and high efficient in generating solitons. we also extend this mechanism to continuous systems and a physical model based on this mechanism has been given to generate water solitons.3. We propose a soliton model in an acoustic composite structure and give some evidences for the existence of this soliton.