Elastic wave propagation in periodic structures has been researched for several decades.^[1–3] The study of wave motion in periodic structures has been exerted primarily on pass band and stop band analyses. In recent years, the propagation of elastic waves in periodic composite materials, named phononic crystal (PCs), has received much attention.^[4–10] The pioneering work of Liu et al. opened new fields of PCs.^[6] They examined three-dimensional PCs consisting of cubic arrays of coated lead spheres immersed in an epoxy matrix, and proposed a new kind of gap formation mechanism, i.e., locally resonant (LR) band gap. More recently, the development of LR PCs has extended to newly emerging field: acoustic metamaterials.^[11–13] Acoustic metamaterials are generally regarded as materials with artificial microstructures that possess unusual physical properties such as band gaps, negative refraction, acoustic cloaking, etc.

Structures periodically shunted by electrical resonant circuits can also produce locally resonant band gaps. Moreover, the band gaps can be tuned over desired frequency ranges conveniently through adjustment of the circuit parameters. Thorp et al. utilized an array of resonantly shunted piezoelectric patches mounted on a rod to create band gaps centered at the tuning frequencies of the shunting circuits.^[14] Airold et al. proposed using multi-resonant shunts to form multiple band gaps in a beam.^[15] Subsequently, they proposed to design one-dimensional tunable acoustic metamaterials through periodic arrays of resonant shunts.^[16] The authors also did a lot of theoretical and experimental investigations on wave propagations in structures with shunting circuits,^[17–21] and tried to utilized shunting arrays to isolate vibrations propagating in flexible structures. In all these studies, symmetric scheme is generally adopted, i.e., the piezoelectric patches are oppositely mounted on the top and bottom surfaces of the substrate structure. The resulting structure is symmetric with respect to the neutral surface, in which elongation and bending deformations are decoupled, so flexural and longitudinal waves are uncoupled. Consequently, asymmetric schemes lack of elaborate exploration, and some valuable properties need to be dug out.

In this paper, we propose an anti-symmetric scheme of shunting arrays instead of the conventional symmetric layout, and obtain the dispersion relation of coupled waves in the composite beam. The propagation constants are evaluated by transfer matrix method, which is adapted for anti-symmetric scenario. Compared with the symmetric scheme, the anti-symmetric configuration demonstrates some new properties. Specifically, the attenuation of flexural dominated wave in anti-symmetric layout is much stronger than that in symmetric one. However, the attenuation of longitudinal dominated wave shows the converse feature. Finally, the theoretical predications are verified by finite element simulations.

The real structural configuration of the beam with antisymmetric piezoelectric shunting arrays is schematically shown in Fig. 1. Pairs of piezoelectric patches are periodically bonded to the surfaces of the substrate beam with infinite length, forming a superlattice structure. Each pair of piezo-patches is anti-symmetrically mounted on the top and bottom surfaces, respectively. Moreover, every piezo-patch is independently connected to a serial resistive-inductive (RL) shunting circuit with identical parameters.

Fig. 1.

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	Fig. 1. Layout of the beam assembly. (a) Front view and (b) top view.

On the analogy of lattice dynamics, the wave motions in the above beam assembly can be derived from one single periodic element as shown in Fig. 2 (shunting circuits are omitted), combined with application of the Floquet theorem, i.e.,^[22]

Fig. 2.

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	Fig. 2. Boundary forces acting on the beam element.

Both the beam segments A and B are asymmetrical with respect to the neutral surface of the substrate beam. As a result, the longitudinal and flexural motions in the segments are coupled. For simplicity, Euler–Bernoulli beam assumption is adopted. Consider an infinitesimally small element of segment A with length dx, as shown in Fig. 3. F(x), M(x), and Q(x) are axial force, bending moment, and shear force on the cross-section, respectively. The deflection on the cross-section (ε) can be considered as the superposition of a pure elongation (ε₁) and a bending (ε₂) with respect to neutral surface of the substrate beam.

Fig. 3.

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	Fig. 3. Deformation in the cross-section of segment A.

Hence, the strain on the cross-section can be written as

Piezoelectric patches on the beam surfaces are polarized through z-axis, and assuming that all the surfaces are free of constraint except those along the x-axis. For simplicity, the directions along x, y, and z axes are represented by 1, 2, and 3 in the piezoelectric equations. Hence, the piezoelectric equations can be reduced to^[14]

Solving equation set (5) yields

Each piezoelectric patch is attached to a serial RL shunting circuit as shown in Fig. 4. Hence, the electric current I_s can be given by

Substituting Eq. (9) into Eq. (8) yields

On the other hand, the amount of electrical charge on the electrode can also be given by

Substituting Eq. (6) into Eq. (11) yields

Utilizing Eq. (4), the strain in the thin piezoelectric patches can be expressed as

Substituting Eq. (13) into Eq. (12), the integrating result is

Combining Eqs. (10) and (14), the electric field intensity can be solved as

Substituting Eq. (15) into Eq. (5), the stress in the piezoelectric patch can be solved as

On the assumption of Euler–Bernoulli beam, the coupled longitudinal and flexural motions in the beam segment can be given by

If harmonic motion at frequency ω is assumed, equation (19) can be reduced to

Applying Laplace transformation into Eq. (21) and eliminating U(x) yield characteristic equation

The solution of Eq. (21) can be written as

Substitution of Eq. (24) into Eq. (21) yields

On the other hand, the axial force F(x), bending moment M(x), and shear force Q(x) can be given by

By analogy, the counterpart vector Z₂(x) in segment B can also be derived from the aforementioned procedures, and given by

The continuity between the segment A and B in the n-th periodic element can be expressed as

Therefore,

On the other hand, utilizing Eqs. (1) and (2), we have

Substituting Eq. (35) into Eq. (34) yields

Consequently, the propagation constant μ can be obtained by solving the eigenvalue of transfer matrix T.

In the numerical calculation, epoxy (Young’s modulus E = 4.35 × 10⁹ Pa, Poission’s ratio v = 0.37, and density υ = 1180 kg/m³) and PZT-5H (compliance coefficient piezoelectric constant d₃₁ = −2.74 × 10⁻¹⁰ C/N, dielectric constant F/m, and density ρ_p = 7500 kg/m³) are selected as the materials of substrate beam and piezoelectric patches, respectively. The geometrical parameters are as follows: l = 0.03 m, b = 0.02 m, h = 0.005 m, and h_p = 0.0002 m. The shunting circuit parameters are selected at R = 200 Ω and L = 0.6 H. Employing the foregoing transfer matrix method, the propagation constants of the beam with anti-symmetric piezoelectric shunting arrays are examined.

Figure 5 shows the propagation constant of waves propagating in the beam with the prescribed parameters. There are two types of coupled waves propagating in the beam. The first type is quasi-longitudinal wave mode (QLM), whose dispersion is weak. The other one is quasi-flexural wave mode (QFM), of which the dispersion with frequency is quite strong. Both the two wave modes show an attenuation region (called band gap), of which the locations are quite close. Notice, however, that the band-gap extent and attenuation magnitude of QFM are much larger than those of the QLM. As a result, the decay of QFM by shunting arrays are much more efficient.

Fig. 5.

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	Fig. 5. (a) Attenuation constant and (b) phase constant of the wave motions.

As the considered wave motions are coupled, it is of interest to draw a comparison with uncoupled ones. The conventional configuration, which is generally symmetric, also bears two types of wave motions, viz. flexural wave and longitudinal wave. However, these two waves propagate independently in the beam. The comparison of attenuation constant between coupled and uncoupled wave motions are illustrated in Fig. 6. In Fig. 6(a), one can find that the attenuation of uncoupled (symmetric) longitudinal wave in the band gaps is much larger than that of coupled (anti-symmetric) longitudinal wave. In contrast, the attenuation of uncoupled (symmetric) flexural wave in the band gap is much smaller than that of coupled (anti-symmetric) flexural wave, as shown in Fig. 6(b). As a result, the attenuation of flexural wave is enhanced by substituting anti-symmetric configuration for the convention symmetric one. This anti-symmetric configuration can be utilized to suppress flexural wave propagation in beams much more efficiently. As piezoelectric patches are mounted on the surfaces of the substrate beam, more mechanical energy will be concentrated in the patches when the beam subjected to bending deformation. As a result, more mechanical energy can be transformed and coupled in the shunting circuits, resulting in more efficient attenuation of the flexural wave.

Fig. 6.

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	Fig. 6. Comparison between symmetric and anti-symmetric configuration. Propagation constant of longitudinal wave (a) and flexural wave (b).

In order to verify the above theoretical analyses, two finite element models with anti-symmetric and symmetric shunting arrays, consisting of 12 periods, are built in COMSOL, respectively. When longitudinal excitation is applied on the left end of the beam, the transmission properties of longitudinal (symmetric) and quasi-longitudinal (anti-symmetric) waves propagating in the beam are demonstrated in Fig. 7(a). In order to characterize the band gap location, the transmission property of the longitudinal wave in the beam without internal resonance (L = 0 H) is also plotted in the diagram. Comparing the three curves, it is observed that a transmission dip is generated when the piezoelectric patches are shunted by resonant circuits. However, the shunting arrays with symmetric configuration show a stronger attenuation in the transmission dip. This result is in accordance with the theoretical analysis in Fig. 6(a). The transmission properties of flexural (symmetric) and flexural dominated (anti-symmetric) waves are presented in Fig. 7(b), when the left end of the beam is exerted on a transverse excitation. Comparing the different curves, one can find that resonant shunting circuits induce a transmission dip in the flexural wave as well. Contrary to the longitudinal wave, the shunting arrays with anti-symmetric configuration demonstrate much stronger attenuation ability in the band gap, which agree well with the theoretical predication in Fig. 6(b). In a word, the finite element simulations validate the theoretical predications, and prove correctness of the adapted transfer matrix derivation.

Fig. 7.

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	Fig. 7. Longitudinal (a) and flexural (b) wave transmission properties of the beam.

Wave propagation and attenuation in beams with anti-symmetric piezoelectric shunting arrays are investigated in this paper. Transfer matrix method is adapted for characterizing wave motions in the anti-symmetric model. In contrast to symmetric scenario, the longitudinal and flexural waves are coupled in anti-symmetric situation. As a result, some distinctive features are observed. First of all, the vibration energy is transformed between longitudinal and transverse degrees of freedom. Second, the attenuation of flexural wave is much stronger if the anti-symmetric configuration is adopted, which can be utilized to enhance vibration isolation capability of shunting arrays. Finally, the theoretical predications are verified by finite element simulations in COMSOL.