1 Introduction

Many efforts have been devoted in recent years to study propagation of elastic waves in phononic crystals (PCs)^[1-2]. The PC is an artificial crystal in which materials and structures are periodically arranged. The elastic waves are modulated by the periodic materials and structures of the PC, and thus elastic waves within some frequency ranges named band gaps are forbidden or attenuated to spread. The main interest for proposing new PCs is to explore their potential applications in engineering, such as resonators^[3], frequency filters^[4], vibration isolator^[5], transducers^[6]. Meanwhile, micro-vibration on the order of nanometers poses challenges for the improving precision of ultra-precision machine tools and metrology equipment^[7]. Owing to the band gap, PCs are expected to be used for the micro-vibration isolation.

However, to be a vibration isolator of the ultra-precision mechanical system, a PC still needs to be improved on the following properties: (i) miniaturization; (ii) low-frequency band gap; (iii) broad band gap; (iv) frequency range of band gap can be tuned; (v) strong vibration attenuation in band gap for a PC with a few unit cells^[8-9].

Based on the generation mechanisms, band gaps are divided into Bragg band gap and locally resonant band gap^[10]. Many studies have shown that the spatial modulation must be of the same order as the wavelength in the Bragg band gaps. This would require a big size PC to isolate low-frequency vibration. In other words, a normal size PC only can bring Bragg band gaps with high frequencies, which are far beyond the mechanical vibration frequency. However, for a PC with the same size, locally resonant band gaps can appear in a frequency range of two orders lower than the Bragg band gaps^[10-12]. Although with the merit of low-frequency, locally resonant band gaps are too narrow to make the PCs as low-frequency sound and vibration shelters in engineering. Recently, smart materials and structures have been applied in PCs to control the propagation of vibration^[13-14]. By replacing a commonly heavy mass-spring oscillator, a piezoelectric patch and a resonant shunting circuit act as an inductor-capacitor oscillator to compose a locally resonant unit. Hence, periodic piezoelectric structures shunted by electrical resonant circuits can also produce locally resonant band gaps.

Besides the light weight, the piezoelectric PCs have another significant advantage that the band gaps can be easily tuned by the circuits without necessity to reconfigure the structure. To obtain low-frequency band gaps, researchers have proposed different piezoelectric PCs. Wang et al.^[15] experimentally investigated low-frequency locally resonant band gaps induced by arrays of resonant shunts with Antoniou’s circuit. Thorp et al.^[16] applied a periodic array of resistor-inductor-shunted piezos-mounted on fluid-loaded shells to create band gaps. Chen et al.^[17] studied propagation and attenuation of elastic wave in plates with periodic arrays of shunted piezo-patches. Shu et al.^[18] presented a circular plate PC placing periodically with piezoelectric rings along the radial direction. Among these proposed PCs, piezoelectric bimorph beam and the bending mode are the most commonly used structure and operational mode respectively^[19-21].

Furthermore, compared with a long straight beam, a folding structure owns a low natural frequency but with a smaller size. Hence the folding structure is always employed by the low-frequency actuator or isolator^[22-23]. But to the best of our knowledge, little research has been done on the PCs with folding structures. We propose a new piezoelectric PC with a folding structure in which bimorphs are joined by masses periodically. The folding structure extends the propagation path of the elastic wave, which generates low frequency band gaps while the structure size remains quite small. Coupled extensional and flexural motion of the folding structure is studied. The band structure and vibration transmission are calculated by transfer matrix method. Vibration transmission is further examined by the finite element method (FEM) to confirm the existence of band gaps predicted by the transfer matrix method. We aim to reduce the center frequency of the band gap, broaden the band gap and enhance the vibration attenuation.

The paper is organized as follows: the mathematical model of the new PC is established in Section 2. A theoretical analysis on the structure is conducted, and analytic solution is obtained in Section 3. The numerical results are presented in Section 4. Finally, some conclusions are drawn in Section 5.

2 Mathematical model of folding beam-type piezoelectric PC 2.1 Governing equations

A folding beam-type piezoelectric PC model consists of bimorphs and masses arrayed periodically as illustrated in Fig. 1. One bimorph is composed of a host beam and a pair of piezoelectric patches attaching on its upper and lower surfaces. The pair of piezoelectric patches is polarized along the x₃ direction, and is connected by a resistive-inductive shunting circuit independently. Every periodic unit comprises of two masses, and two bimorphs which are denoted by Ⅰ and Ⅱ respectively. 2c and h represent thicknesses of host beam and a piezoelectric patch of the bimorph respectively. l and b represent the length and the width of the bimorph. The mass has the same width as the bimorph, and its length and thickness are denoted by a₁ and a₂.

Consider flexural motion of the bimorph in the x₃ direction and the extensional motion in the x₁ direction. From the Euler beam theory on laminated beam in coupled vibration between extension and flexure, the axial strain S₁ (x₁, t) is expressed as

(1)

where u₃ (x₁, t) and u₁ (x₁, t) represent flexural and extensional displacements of the bimorph respectively.

The constitutive relations of the bimorph are

(2)

where s₁₁ is axial elastic compliance constant in fixed electric field, d₃₁ piezoelectric strain constant, ε₃₃ dielectric constant, , E Young’s modulus of the host beam of the bimorph. D₃ is electric displacement, T₁ axial stress. The electric fields E3 in piezoelectric patches are^[24]

(3)

The bending moment, shear force, and axial force are given by

(4)

where , and with and D₂₁=. V is the voltage of top and bottom electrodes of the bimorph, and two electrodes in the middle are grounded.

From (2) and (3), the electrical displacement of two piezoelectric patches of the bimorph can be obtained

(5)

The charges on the top and bottom electrodes of the bimorph can be written as

(6)

The current can be calculated as

(7)

The complex impedance of each shunting circuit is defined as

(8)

where i, ω, L, and R represent the imaginary unit, angular frequency, inductance, and resistance respectively. Thus, the current and the voltage satisfy Ohm’s law

(9)

Equations of motion of a slim beam are expressed in terms of the bending moment and axial force as follows:

(10)

where mass per unit length m=ρ2cb + 2ρ′hb. ρ and ρ′ are densities of the host beam and piezoelectric patches. From (4), equations of motion become

(11)

2.2 Continuity and boundary conditions

As shown in Fig. 1, an arbitrary nth unit is circled by red dashed lines. All left and right masses are assumed to be rigid bodies by ignoring its deformation. The forces and displacements of left and right masses are illustrated in Figs. 2 and 3 respectively. Bimorphs Ⅰ in (n-1) th unit and Ⅱ in nth unit being joined by one left mass, extensional displacement, flexural displacement, rotation angle, extensional force, shear force, and bending moment satisfy continuity condition. Therefore, the rigid body motion of the left mass is governed by

(12)

where superscripts (n) represent nth periodic unit. The mass is made of the same material as the host beam, and hence m₀=ρa₁a₂b. J_c=m₀(a₁² + a₂²)/12 is centroidal moment of inertia of the mass. The relationship between displacements of the centroid of the left mass and the bimorph Ⅰ is

(13)

Similarly, the rigid body motion of the right mass is governed by

(14)

where

(15)

Additionally, for a PC in engineering with a finite periodicity number, the boundary conditions are considered to calculate vibration transmission. On one end of the proposed PC, a harmonic displacement excitation with amplitude A is considered at the x₃ direction. As a response end, the other end is free. Therefore, boundary conditions on two ends of the proposed PC with periodicity number J are

(16)

3 Band structure of folding beam-type piezoelectric PC

For harmonic motions, we use the complex notation

(17)

Then equations of motion (11) become

(18)

The general solutions of (18) can be written as

(19)

where , and are undetermined constants.

Substitution of (4) and (19) into (14) yields the matrix form of the continuity conditions between the undetermined constants of two bimorphs in an arbitrary nth unit

(20)

where K₁ and K₂ are coefficients matrices of φ₁⁽ⁿ⁾ and φ₂⁽ⁿ⁾ respectively.

(21)

where j=1, 2 correspond to bimorphs Ⅰ and Ⅱ as shown Fig. 1. Superscript T represents matrix transposition. Similarly, substitution of (4) and (19) into (12), the continuity conditions between the undetermined constants of two bimorphs in two neighboring units can be written as matrix form

(22)

where H₁ and H₂ are also coefficients matrices. From (20) and (22) transmission relationship between undetermined constants of bimorphs Ⅱ in two neighboring units becomes

(23)

where transfer matrix T=K₂^-1K₁H₁^-1H₂. From the Bloch theorem, we have

(24)

where k is wave vector, and a=2l is the lattice constant of the PC. From (23) and (24), nontrivial solutions may exist if the determinant satisfies

(25)

where I is an identity matrix. Thus, the band structure of the proposed PC is obtained. For a given ω, (25) determines 6 roots of k. These 6 roots correspond to 6 mode functions with respect to x₁: 2 extensional modes with trigonometric functions, 2 flexural modes with trigonometric functions, and 2 flexural modes with exponential functions. The real part of k is the wave number, which can be used to describe phenomena of wave propagation. When k is a pure real number but not 0 or π/(2a), a flexural wave can propagate without attenuation. On the other hand, a complex k means the frequency of the wave lies in a band gap. The attenuation constant µ is defined by the product of the imaginary part of k and lattice constant a, which indicates the attenuation of the amplitude when the elastic wave propagates from one period to the next.

4 Numerical results and discussion

In order to illustrate the vibration attenuation characteristic of the proposed PC, we choose epoxy as the material of the host beams and the masses, and PZT-5H as the material of piezoelectric patches of the bimorph. The material parameters of the epoxy are: density ρ=1 185 kg/m³, Young’s modulus E=4.35 GPa. PZT-5H has the following material parameters: ρ′=7 500 kg/m³, s₁₁=1.65×10^-11 m²/N, d₃₁=-2.74×10^-10 C/N, and ε₃₃=3.01×10^-8 F/m.

The geometrical parameters of the unit cell are: l=60 mm, b=10 mm, h=0.2 mm, c=1.5 mm, a₁=10 mm, a₂=10 mm. The inductance and resistance of shunting circuits are L=0.06 H and R=1 Ω. These parameters are fixed unless otherwise stated.

Firstly, the shunting circuits are shorted to consider the effect of the coupling vibration between extension and flexure on band structure. The band structure of the folding beam-type PC is illustrated in Fig. 4 as frequency versus wave numbers Re (k) and attenuation constants µ. As we know, a nontrivial attenuation constant µ means the corresponding frequency locates in a band gap. It is observed from Fig. 4(b) that frequency ranges below 4 000 Hz all are band gaps. In addition, these band gaps belong to evanescent wave since the wave number is also not equal to 0 or π/(2a). In this case, flexural wave and extensional wave spread along the folding beam-type structure as transverse wave and longitudinal wave respectively. However, we still cannot tell which kind of wave a certain dispersion curve belongs to.

The transmission factor is an important measure on performance of vibration attenuation of the PC with a finite periodicity number. We focus on discussing transmission factor of flexural wave which can demonstrate vibration attenuation in x₃ direction. Periodicity number is fixed at J=4. One attenuation constant curve of the infinite PC and transmission factor of flexural wave of the finite PC are demonstrated in Figs. 5(a) and 5(b), respectively. The transmission factor curve illustrates that vibration attenuation is obvious within the band gap. From Fig. 5(a), we know these two band gaps (abbreviated by BG in the figure) both belong to Bragg band gap since the attenuation constant curve has no sharp peak. As shown in Figs. 5(a) and 5(b), the first Bragg band gap exists from 388 Hz to 1 772 Hz for the infinite PC, while it appears from 369 Hz to 1 687 Hz for the finite PC. Furthermore, the starting frequencies of the second Bragg band gap are 2 261 Hz and 2 127 Hz, respectively. Therefore, vibration attenuation can be achieved within band gaps by the finite folding beam-type PC just with 4 unit cells. It is worth noting that the width reaches 1 318 Hz and the starting frequency lowers to 369 Hz for the first band gap even though as a Bragg band gap.

In order to validate the theoretical model, the transmission characteristic of the folding beam-type PC is also calculated by the FEM. The FEM is carried using the ANSYS software. SOLID45 is chosen as three-dimensional element for the host beams and masses. SOLID5 is chosen for piezoelectric patches with keyopt (1)=3 so that SOLID5 has a voltage degree of freedom. The resistors and inductors in the shunting circuits are defined by element CIRCU94 with keyopt (1)=0 and 1, respectively. Harmonic response analysis is conducted. Convergent solutions are obtained by refining the mesh. The comparison on the transmission factor curves between theoretical result and FEM result is presented in Fig. 6. We can find that the theoretical results are in agreement with those of FEM.

As the next step, we investigate the influence of parameters of resonant shunting circuits on the locally resonant band gaps (abbreviated by LR BGs in Fig. 7). First, the effects of shunting inductances on transmission factor are shown in Fig. 7. The frequency of the locally resonant band gap decreases with increasing of the inductance. It indicates that a required locally resonant band gap can be achieved conveniently by just tuning the inductances without necessity to reconfigure the structure. Second, we study the effect of the resistance on band gaps by fixing the inductance L=0.06 H. Figure 8 illustrates the transmission factor versus frequency for different resistances. The transmission factor amplitude of the frequency within the locally resonant band gap is reduced dramatically by decreasing the resistance. But for both locally resonant and Bragg band gaps, the frequencies almost have no change with the resistance. It is also worth mentioning that the transmission factors are all less than zero for the frequency from 1384Hz to 2204Hz when the resistance increases to 100Ω. Thus, the frequencies above 369Hz all become band gap since the first two Bragg band gaps are joined by the locally resonant band gap.

Finally, we investigate the effect of geometrical parameters on vibration transmission. Transmission factors versus frequency with different bimorph lengths l are illustrated in Fig. 9. It can be seen that frequencies of locally resonant band gap and the Bragg band gap both decrease with increase of bimorph length l. Moreover, when the length l increases, the Bragg band gap becomes narrower, and the attenuation is slightly enhanced. The minimum value of transmission factor reaches -134 dB. But it has little effect on the width of locally resonant band gap.

5 Conclusions

A folding beam-type piezoelectric phononic crystal is proposed to control the propagation of vibration. A theoretical solution for coupled motions of extension and flexure of piezoelectric bimorph beams is obtained. Theoretical results are verified by the FEM operated on ANSYS software. The numerical results have demonstrated that Bragg and locally resonant band gaps are both achieved. Moreover, low-frequency and broad band gaps with strong attenuation are obtained. The effects of circuit and geometrical parameters on band gaps and vibration attenuation are also investigated.