Active feedback control of effective mass density and sound transmission on elastic wave metamaterials

Highlights

• The effective mass density is derived and the dynamic responses are presented by the dynamic effective medium method.

• The sound transmission loss (STL) is obtained based on the principle of virtual work.

• The negative mass density and STL in the lower frequencies can be adjusted by the active feedback control.

Abstract

With the active feedback control system on elastic wave metamaterials, this research is concentrated on the effective mass density and sound transmission by a harmonic incident sound pressure. The elastic wave metamaterials consist of double plates which are attached by the upper and lower four-link mechanisms being bonded with four lateral resonators. The vertical resonator in every unit cell is jointed by the active feedback control system which is connected by two four-link mechanisms. Using the dynamic effective medium method, the expressions of the effective mass density are obtained and the dynamic responses are presented. Based on the Bloch–Floquet theorem and Poisson summation formula, the sound transmission loss (STL) of this elastic wave metamaterial is shown by the principle of virtual work. This research also shows that the characteristics of the effective mass density and STL can be tuned by the acceleration and displacement feedback in the active control system. Furthermore, it is found that the dynamic response and STL are also changed obviously by different incident angles (including the elevation and azimuth angles), the periodicity spacing of local resonators and the structural damping.

Introduction

As a new type of periodic materials and structures, phononic crystals and elastic wave metamaterials play a significant role in the structural acoustics, and have been widely investigated for nearly twenty years [1], [2], [3], [4], [5], [6], [7], [8]. Based on their distinguishing features of elastic wave propagation and dynamic responses to the external medium and excitation, phononic crystals and elastic wave metamaterials can be extensively applied in many engineering fields, especially the automobile [9,10], aircraft manufacturing [11,12] and marine industries [13,14]. In addition, these superior periodic structures have shown tunable properties of elastic waves by shunted piezoelectric patches with the multiple fields coupling [15], [16], [17], [18], [19].

In recent years, elastic wave metamaterials have attracted a lot of attention because of their negative effective parameters and unusual wave properties [20], [21], [22], [23], [24], [25]. As a new type of artificial materials, the elastic wave metamaterials can be applied to the acoustic cloaking [26], [27], [28] and black holes [29], nonlinear metamaterials [30], electrochemically reconfigurable architected materials [31], piezoelectric phononic crystal nanobeams [32], multi-resonator metamaterials [33], low-frequency sound attenuation [34], [35], [36], [37], wave energy tunneling [38], [39], [40], [41] and sound radiation [11,42,43]. The effective mass density of metamaterials is widely used to present the underlying physical properties [20,44,45]. As the frequency of the excitation force approaches to the local resonance frequency, the effective mass density becomes a negative value [44,46,47] which cannot be found in the nature environment.

Huang and Sun [48] proposed an acoustic metamaterial model which exhibits simultaneously both negative mass density and negative Young's modulus. After that, some new discoveries initiated from this model have been reported [49]. A similar study shows the attenuation of transverse waves by a metamaterial beam with lateral local resonators [50]. This model can be further extended to the 3D metamaterial plates. These studies also show that elastic wave metamaterials have the ability to achieve a complete vibrational band gap in the low-frequency range which is due to the resonant properties of their constituent elements [51].

It is well known that the characteristic of sound transmission is a classical but still interesting and challenging problem in the structural and solid acoustics. Enhancing the capacity of the wave isolation and reducing the radiation of wave energy are important to improve the quality of life and work environment. Many researches have been reported to increase the sound transmission loss (STL) by different structures which are designed in the form of the rib-stiffened sandwiches [52], acoustic porous metasurfaces with composite structures [53], double-leaf aeroelastic plates [54] and attaching substructures [55,56], etc. It is denoted that higher sound transmission loss could be changed by arranging resonators vertically using the dynamic effective medium method. Xie et al. [57] presented the design and experimental demonstration of multiband asymmetric transmission of airborne sound and showed an alternative application on acoustic rectification and noise control. The influences of the external fluid on the STL have widely drawn close attention [12,58]. It was also demonstrated that the STL response is not only affected by the cell wall angle for sandwich panels with the honeycomb core, but depends on the number of unit cells in the horizontal and vertical directions [59].

In addition, we know that the active and passive control actions are two main approaches which are used to control the vibration and elastic wave propagation. Both methods can reduce the dynamic amplitude and absorb the corresponding energy [60,61]. Active control of the turbulent boundary layer induced sound transmission can achieve the large pressure reductions in a broad frequency band in the acoustic back cavity [62]. But for the passive control, its effectiveness is limited to the frequencies of short acoustic wavelength [62,63]. The passive noise treatment is mainly concentrated on the medium and high frequency region during the isolation performance, such as sound absorbing materials. Therefore, the active feedback control has been widely applied to the acoustic properties and vibration reduction. However, investigations about negative parameters and the acoustic-structure coupling with the active control system have received only a little attention.

In our previous work [64], a metamaterial with the active feedback control system is designed and its sound radiation property is discussed. The present work can be regarded as an extension of the studies on the active elastic wave metamaterials [5,61,[64], [65], [66], [67]]. The effective mass density and the sound transmission loss of the more complex metamaterial are derived and analyzed in this work. The structure consists of the double plates which are attached to the vertical and lateral resonators by the upper and lower four-link mechanisms. Based on the effective medium method, the expression of the dynamic effective mass density can be obtained. The physical and mechanics process of sound transmission through an infinite double plates model subjected to harmonic pressure is analytical formulated. Analysis is based on the space harmonic approach; and the principle of virtual work is used to calculate the STL.

Both the multi-directions resonators and double four-link mechanisms with the active feedback control system are simultaneously considered for this new metamaterial. We can find that the effective mass density for the in-phase mode and the STL can be tuned by the active control with the acceleration and displacement parameters. Furthermore, the upper surface of the metamaterial is immersed in the incident field with a fluid load. We can show the effects of the incident angle including the elevation and azimuth angles. And the influences of the periodicity spacing of local resonators and structural damping are also studied. In the field of acoustic-structure coupling, the thin plate problem has great significance in both the theory derivations and practical applications. Moreover, the metamaterials with the Mindlin–Reissner thick plate theory have also been discussed by different studies [68], [69], [70], [71], which can be applied to the future investigations about acoustic-structure coupling of elastic wave metamaterials with active feedback control.