The structure of an active acoustic metamaterial with tunable effective density

A new class of one-dimensional active acoustic metamaterials (AAMMs) with programmable effective densities is presented. The proposed AAMM is capable of producing densities that are orders of magnitudes lower or higher than the ambient fluid. Such characteristics are achieved by using an array of fluid cavities separated by piezoelectric diaphragms that are controlled to generate constant densities over wide frequency bands. The piezodiaphragms are augmented with passive electrical components to broaden the operating frequency bandwidth and enable densities higher than the fluid medium to be generated. The use of these components is shown to be essential to maintain the closed-loop compliance of the piezodiaphragm away from the zone of elastic instabilities. The values of the passive components are selected on a rational basis in order to ensure a balance between the frequency bandwidth and control voltage. With this unique structure of the AAMM, physically realizable acoustic cloaks can be implemented and objects treated with these active metamaterials can become acoustically invisible.

Recently, interest has been focused on the development of various metamaterials with acoustical properties that are unachievable with natural materials. For example, composites of periodically distributed sonic scatterers are considered to produce acoustical properties, such as the sound speed and dynamical mass densities, that can be controlled by proper selection of the volume fraction, the elastic properties, positions in the unit cell, scattering interactions and refractive properties of the scatterers [1]- [6]. The resulting effective properties are highly dependent on the acoustic frequency range and can only be tuned for a narrow frequency band based on the arrangement of the sonic crystals. Other attempts include efforts to synthesize negative dynamic acoustical densities in fluid domains through the use of spring-mass lattice systems [7]- [10]. In these attempts, it was found that near a local resonant frequency, spatial decay of wave amplitude occurs as the effective mass becomes negative. Effective acoustic properties for poroelastic composites consisting of fluid matrix-solid constituents have also been investigated by Berryman [11] in order to predict the wave properties in the long wavelength limit. Also, acoustic metafluids consisting of layered composite media have been considered. These metafluids have either anisotropic density and scalar bulk modulus [12] or anisotropic density and bulk modulus [13].
Unlike the extensive theoretical studies of acoustic metamaterials, experimental investigations are so far lacking. However, an important experimental study that is relevant to this paper is the work of Lee et al [14,15] which demonstrated the negative effective density characteristics of an acoustic metamaterial consisting of an array of cavities separated by thin In all the above studies, the focus has been placed on passive metamaterials with fixed material properties. This limits considerably the potential of these materials. In this paper, the emphasis is placed on investigating the details of the structure of a new class of active acoustic metamaterial (AAMM) introduced by Baz [16]. In the AAMM, the effective densities can be programmed to have increasing or decreasing variation patterns along and across the material volume. With these unique capabilities, physically realizable acoustic cloaks can be achieved and objects treated with these active metamaterials can become acoustically invisible.
This paper is organized in five sections. In section 1, a brief introduction is presented. In section 2, the configuration of the AAMM is introduced. In section 3, a lumped-parameter model of cavities with piezoelectric diaphragms is developed to motivate the need for the particular structure of the AAMM in order to achieve 'programmable' and 'tunable' characteristics. In section 4, numerical examples are presented to illustrate the different capabilities of the AAMM and the effect of its different components on the frequency bandwidth and control voltage. A brief summary of the conclusions and outline of future work are outlined in section 5. Figure 1 displays a configuration of a multi-cell AAMM which consists of an array of fluid cavities separated by piezoelectric boundaries. These smart self-sensing and actuating boundaries are used to control the stiffness of the individual cavity and in turn its dynamical density through direct acoustic pressure feedback. Mechanically, each unit cell of this array is identical to other cells which make the physical realization of this concept rather feasible. However, electrically, the piezo boundaries are controlled separately in order to achieve increasing or decreasing dynamical density distributions that can also vary by many orders of magnitudes along the array.

Basic structure
Consider the basic structure of an acoustic cavity with piezoelectric diaphragm which is shown schematically in figure 2(a). The corresponding electrical analogue is displayed in figure 2(b). The basic equations that govern the modeling of the acoustic cavity and piezoelectric diaphragm by their equivalent electrical analogues are given in appendices A and B, respectively. In figure 2(b), three components are added to the system to provide it with its desirable and tunable characteristics. These components are encircled with the dashed-line ellipses. The first of these components is active in nature which is the capacitance (−d 2 A G) that is generated by feedback control. The other two components are passive and include the capacitance C s and the inductance L P . The role of each of these components and their effect on the metamaterial performance will be discussed in great detail in what follows.

Basic equations
The basic constitutive equation for a piezoelectric material is given by where S = strain, D = electrical displacement, T = stress, E = electrical field, s E = compliance, d = piezoelectric strain coefficient and ε = permittivity. Equation (1) can be rewritten as where Vol = change in diaphragm volume, q =electrical charge, p P = pressure across piezodiaphragm and V P = voltage. Also, C D = diaphragm compliance and Z P = impedance of piezodiaphragm and attached passive elements, given by where C P = capacitance of piezodiaphragm = Aε/t with A = diaphragm area and t = diaphragm thickness. Also, L P denotes a shunted inductance in-parallel with the piezodiaphragm, and C s denotes a capacitance in-series with the piezodiaphragm. Using the piezodiaphragm as a self-sensing actuator, then the second row of equation (2) gives, for a short-circuit piezo-sensor, the following expression: Then, the voltage V P applied to the piezodiaphragm can be generated by a direct feedback of the charge q such that where G = feedback gain. Then, the first row of equation (2) yields where C DC = closed-loop compliance of piezodiaphragm. Equation (6) indicates that the closed-loop compliance of piezodiaphragm (C DC ) consists of two capacitances which are connected in parallel. The first capacitance can be identified as the passive (open-loop) compliance of the diaphragm whereas the second capacitance is generated by virtue of the feedback of the pressure difference across the diaphragm. Figure 3 displays a simplified electrical analogue of the acoustic cavity with closedloop piezodiaphragm with the electrical components converted to their equivalent acoustical analogues.
The transfer function of the controlled cavity system, relating the flow velocity u to the pressure drop P is given by where with where φ = −d A /C D = electro-acoustic transduction coefficient. Equation (7) yields the following expression for the effective density ρ eff : If ρ eff /ρ 0 is to assume a desired value = ρ d , then equation (10) yields the following control gain G:

Special cases
Several special cases of equation (10) are considered in order to demonstrate the importance of the selected structure of the AAMM shown in figure 2. Furthermore, the considered cases will emphasize the role that the added active and passive components (−d 2 A G, C S , L P ) have in shaping the spectral performance characteristics of the metamaterial. where For sinusoidal pressures, with a frequency ω, equation (12) becomes Equation (13) indicates that the relative effective density ρ eff /ρ 0 is influenced by both of the diaphragm stiffness k D and the frequency ω. Increasing the frequency ω from 0 to ∞ 8 makes ρ eff /ρ 0 vary from −∞ to a maximum value of 1. Similarly, as k D is varied from 0 to ∞, ρ eff /ρ 0 changes from a maximum value of 1 to −∞. Note that k D cannot assume negative values in order to make ρ eff /ρ 0 > 1 because this means that the diaphragm becomes elastically unstable. Hence, passive acoustic cavities with flexible diaphragms can only generate values of ρ eff /ρ 0 1 and are incapable of maintaining ρ eff /ρ 0 constant except at a particular frequency. These characteristics render this class of cavities unsuitable for the practical realization of acoustic cloaks where ρ eff /ρ 0 must assume and maintain any specific value between ∞ and 0 over broad frequency ranges. Such serious limitations of the passive acoustic cavities are avoided completely by adopting the concept of active acoustic cavities.
where C t = C PB C s /(C PB + C s ).
For sinusoidal pressures, with a frequency ω, equation (14) becomes If ρ eff /ρ 0 is to assume a desired value = ρ d , then two sub-cases are considered (a) ρ d < 1: Equation (15) can be rewritten as If the closed-loop compliance C DC is maintained >0, in order to avoid elastic instability of the diaphragm, then both the numerator and denominator of equation (16) are negative. This indicates that ρ d can be maintained constant for any positive and real frequency ω. Note that this conclusion is also valid even when C S = ∞ and L P = 0. Accordingly, only the active control component capacitance (−d 2 A G) is needed to maintain the value of ρ eff /ρ 0 = ρ d . (b) ρ d > 1: Equation (15) can be rewritten as Hence, (ρ d − 1) > 0 and the closed-loop compliance C DC is maintained > 0, requires that Equation (18) sets the bounds on the frequency range for which ρ d − 1 > 0. Note that the upper bound of this frequency ranges can be increased by reducing both L P and C t . The frequency band between the upper and lower bounds is defined as the 'bandwidth' of the AAMM.
Note that the presence of L P is essential for (ρ d − 1) > 0 as it makes the numerator of equation (18) positive. However, if L P = 0, the numerator will always be negative and ρ d > 1 cannot be achieved. Therefore, it is necessary that the structure of the AAMM must include the shunting inductance L P to ensure that ρ d > 1. Furthermore, the effective bandwidth of operation of the AAMM can be enhanced by the addition of the passive capacitance C S as implied by equation (18). Table 1 summarizes the role of the three components which are added to the structure of the AAMM shown in figure 2. Numerical examples are presented in section 4 to illustrate the influence of the different components of the AAMM on its overall performance.

Effect of structure of the AAMM on its performance
Consider an acoustic cavity with a piezodiaphragm that has the characteristics listed in table 2 (Parasad et al [17]). Figure 4 shows typical performance characteristics of the AAMM when ρ d = 20, L P = 50 H, and C S = 0.2 pf. Under these conditions, the AAMM has an 'operating frequency bandwidth' of 48 950 Hz as shown in figure 4(c). Beyond that bandwidth, the closed-loop compliance C DC , in m 4 s 2 kg −1 , becomes negative indicating that the AAMM becomes elastically unstable. Figure 4(a) indicates that the AAMM can maintain the relative density ρ d at 20 while requiring the input control voltage displayed in figure 4(b).
The effect of the inductance L P and the capacitance C S on the effective bandwidth of the AAMM is shown in figure 5. Figure 5(a) displays the contours of the iso-frequency bandwidth of the AAMM in the L P -C S plane, for ρ d = 20.
For example, an iso-frequency bandwidth line that has a value of 0.5 × 10 4 Hz defines all the designs of the AAMM that have frequency bandwidth of 0.5 × 10 4 Hz. Such designs can be obtained by providing the AAMM with L P = 1000 H and C S = 1 pf, or with L P = 200 H and C S = 5.2 pf.  The displayed results suggest that decreasing both L P and C S is essential to enhancing the frequency bandwidth of the material. However, decreasing either L P or C S can also improve the frequency bandwidth. Figures 5(b) and (e) show the frequency response profiles of the closed-loop compliance for values of L P (in H) and C S (in pf) of (1000,10), (50,1), (1000,0.2) and (50, 0.2), respectively. The effect of decreasing L P and C S on extending the bandwidth of operation of the metamaterial is evident. Table 3 summarizes a comparison between the upper and lower bounds of the effective frequency band of the metamaterial as predicted by equation (18) and by the zones for which the closed-loop compliance is positive in order to avoid elastic instability of the piezodiaphragm. Close agreement between these two methods is evident from table 3.  that decreasing L P increases the control voltage whereas C S has virtually no effect on the control voltage. Figures 6(b) and (c) show the frequency response profiles of the control voltage for values of L P (in H) and C S (in pf) of (1000,0.2) and (50,0.2), respectively. It is evident that decreasing L P reduces the control voltage of the metamaterial which is necessary to maintaining ρ d = 20.  Figure 5 indicates that increasing the frequency bandwidth of the metamaterial requires that L P to be small but at the expense of increasing the control voltage as shown in figure 6.
Considering a performance index (PI) which is defined as PI = Frequency bandwidth/(control voltage Pa −1 ).
Maximizing this PI ensures that the frequency bandwidth is maximized while the control voltage is minimized. Figure 7 displays the effect of the inductance L P and the capacitance C S on the iso-performance index contours. For the considered ranges of L P and C S , PI attains a maximum at L P = 1000 H and C S = 0.2 pf resulting in a frequency bandwidth of 11 254 Hz and control voltage of 64.3 V Pa −1 . Note that maximizing the frequency bandwidth alone results in a bandwidth of 50 000 Hz but a control voltage of 2345 V Pa −1 . This bandwidth is 4.47 times that resulting from maximizing the PI but the control voltage is 36.47 times that obtained by maximizing PI.

Conclusions
This paper has presented a detailed analysis of the structure of a new class of one-dimensional acoustic metamaterials with programmable densities. The AAMM is shown to be theoretically capable of generating densities that are orders of magnitudes lower and higher than the ambient fluid. Such capabilities are achieved by using an array of fluid cavities separated by piezoelectric boundaries which are controlled actively to generate constant densities over broad frequency bands. Furthermore, the piezo boundaries are augmented with passive electrical components to enhance the frequency bandwidth of operation and enable the generation of densities which are orders of magnitudes higher than the ambient medium. The addition of these components is found to be essential to avoid the elastic instability of the piezo boundaries. Selection of the values of the passive components, on a rational basis, is presented in order to ensure a balance between the frequency bandwidth and control voltage.
Comparisons are also presented between the characteristics of the active and passive metamaterials to emphasize the potential of the active metamaterials for physically generating a wide range of effective densities in a simple and uniform manner.
A natural extension of this work is to extend the presented 1D lumped-parameter model of the AAMM to a 2D model. Also, work is now in progress to include active control capabilities to tailor the bulk modulus distribution of the metamaterial. Combining the tunable density and bulk modulus capabilities will enable the physical realization of practical acoustic cloaks and objects treated with these active metamaterials can become acoustically invisible. Experimental prototypes of the AAMM are now being manufactured to demonstrate the feasibility of the concept and to evaluate its merits and limitations. The acoustic cavity, shown in figure (A.1), is modeled using its electrical analogue. This analogue can be easily coupled with the electrical analogue of the piezoelectric diaphragm. In this regard, the acoustic cavity model is developed using a lumped-parameter approach which is valid when the wavelength of the acoustic waves is much greater than the cavity dimensions [18]. The lumped-parameter approach suggests that the cavity is equivalent to a spring-mass system which electrically is equivalent to a capacitance C C and inductance L C . It can be easily shown that the capacitance C C and inductance L C are given by [18] C C = V /ρ 0 c 2 0 and L C = ρ 0 l/A (A.1) where V = volume of cavity, A = cross-sectional area of cavity, l = cavity length, ρ 0 = density of fluid inside cavity, and c 0 = speed of sound.

Appendix B. Electrical analogue of piezoelectric diaphragm
The constitutive equation of the piezodiaphragm alone without any added electrical components can be obtained from equation (2) as follows: Equation (B.1) can be differentiated with respect to time to yield where C PB = (1 − k 2 )C P = blocked capacitance of the piezodiaphragm with k 2 = d 2 A /C D C P . Note that k 2 is defined as the electro-acoustic coupling factor. Equation (B.3) represents the flow of the current (i + φ Q) through the capacitance C PB due to an applied voltage V P . This can be schematically represented by the electrical circuit shown in left half of figure (B.1) to describe the physics of the electrical domain of the piezodiaphragm.
Accordingly, the electrical analogue of the piezoelectric diaphragm can be simplified as shown in figure (B.2).
In this figure, it can be seen that the pressure difference P P across the piezoelectric diaphragm counterbalances the inertia of the diaphragm M D , its mechanical compliance C D , and the mechanical compliance (C PB /φ 2 ) which is equivalent to the piezoelectric capacitance C PB .
Note that the piezoelectric capacitance C PB is related to the voltage V P and current i P as follows: To determine the equivalent mechanical impedance, the basic equations of the electroacoustic transduction coefficient φ are used. These basic equations are given by In effect, these equations make the piezodiaphragm act as a transformer of the electric energy into acoustic energy with a transformer turning ratio = φ. During this energy transformation process, the voltage is amplified by φ and the current is reduced by φ while the total power V P i P = P P Q. Hence, equations (B.4) through (B.6) yield Equation (B.7) indicates that the mechanical compliance (C PB /φ 2 ) is equivalent to the piezoelectric capacitance C PB .