ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 28 Number 9 September 2011 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY Institute of Physics PUBLISHING CHIN. PHYS. LETT. Vol. 28, No. 9 (2011) 094301 Locally Resonant Gaps of Phononic Beams Induced by Periodic Arrays of Resonant Shunts * CHEN Sheng-Bing(陈圣W)**, WEN Ji-Hong(§-õ), WANG Gang(王f), HAN Xiao-Yun(¸小云), WEN Xi-Sen(温熙森) Laboratory of Science and Technology on Integrated Logistics Support, National University of Defense Technology, Changsha 410073 (Received 18 February 2011) Periodic arrays of shunted piezoelectric patches are employed to control the propagation of elastic waves in phononic beams. Each piezo-patch is connected to a single resistance-inductance-capacitance shunting circuit. Therefore, the resonances of the shunting circuits will produce locally resonant gaps in the phononic beam. However, the existence of locally resonant gaps induced by resonant shunts has not been clearly proved by experiment so far. In this work, the locally resonant gap in a piezo-shunted phononic beam is investigated theoretically and verified by experiment. The results prove that resonances of shunting circuits can produce locally resonant gaps in phononic beams. PACS: 43.20.+g, 43.40.+s, 02.60.−x, 77.65.−g DOI:10.1088/0256-307X/28/9/094301 Over the last decades, extensive efforts have been of resonant shunts with Antoniou’s circuit have been made to analyze the propagation of elastic or acoustic investigated theoretically and experimentally,[14] but waves in periodic composite materials called phononic the experimental results have not clearly proved the crystals (PCs).[1−6] Many works are particularly fo- existence of LR gaps. cused on the characteristics of so-called phononic band In this Letter, both theoretical analysis and exper- gaps (PBG), in which elastic wave propagation is com- imentation are based on a one-dimensional phononic pletely blocked. These are referred to as stop bands or beam with RLC shunts. The results of theoretical cal- band gaps. The specific frequency bands where waves culation and experimental test both prove that using can propagate without attenuation are referred to as resonant shunts to produce LR gaps is feasible and pass bands. valid. By the ratio of the wavelength to the lattice con- Piezoelectric patches with RLC shunting circuits stant, the band gaps are classified into Bragg gaps and are periodically bonded to the surfaces of a host beam, locally resonant (LR) gaps. Bragg gaps result from the forming one-dimensional phononic crystals as shown interaction between incident, reflected and transmit- in Fig.1. The parts with and without piezoelectric ted waves in scatter units and matrix. However, the patches are respectively denoted as A and B. resonances of scatter units are the dominant factor in The polarising direction of the piezoelectric the formation of LR gaps. LR gaps can exist in a fre- patches is along the z-axis and all surfaces of the patch quency range of two orders of magnitude lower than are free of constraint except along the x-axis. In the the one resulting from Bragg scattering. piezoelectric equation, the directions x, y and z in the With the development of smart materials, the de- coordinate system are denoted as 1, 2 and 3, respec- sign of intelligent PCs whose spectral width and band tively. Therefore, the piezoelectric equations can be gap location can be actively tuned has received con- reduced to[9] siderable attention.[7−13] The dual mode of piezoelec- [︂ S ]︂ [︂ sE d ]︂ [︂ T ]︂ tric materials enables them to be used both as sensors 1 = 11 31 1 ; (1) D d "T E and actuators. So, by using the resonant property of 3 31 33 3 resistance-inductance-capacitance (RLC) circuits and where S1 and T1 are the mechanical strain and stress E the electromechanical coupling of piezoelectric mate- in the x-axis direction, respectively, s11 is the piezo- rials, a new kind of PCs with tunable LR gaps can be electric material’s compliance coefficient at constant designed. By varying the values of inductance L and electric field intensity, d31 is the piezoelectric constant capacitance C of shunting circuits, the LR gap can be that couples the mechanical and electrical properties actively tuned. LR gaps induced by periodic arrays of the piezoelectric material, D3 is the electric dis- *Supported by the National Natural Science Foundation of China under Grant Nos 50905182 and 50875255. **Email: [email protected] ○c 2011 Chinese Physical Society and IOP Publishing Ltd 094301-1 CHIN. PHYS. LETT. Vol. 28, No. 9 (2011) 094301 T placement on the electrodes, "33 is the dielectric con- At the interfaces of A and B in the phononic beam, stant of the material and E3 is the electric field in the using the continuities of displacement, slope, bending patch. moment and shear force, the transfer matrix T can be The complex impedance of shunting circuits can extracted, which gives the relations be written as Ls + R T Z = ; (2) n = [ An Bn Cn Dn ] 1 + RCs + LCs2 T n+1 = [ An+1 Bn+1 Cn+1 Dn+1 ] : where s is the Laplace operator, L is the inductance of the shunting inductor and C is the capacitance of Therefore, using periodic boundary conditions and the shunting capacitor. the Bloch theorem, the band gaps of the phononic Applying the definitions of electric displacement beam shown in Fig. 1 can be calculated using a trans- and electric field, Eq. (1) can be solved as[13] fer matrix (TM) method.[15;16] The band gaps can be 2 −1 h E d31Ashp s(R − Ls) i described by the propagation constant. The real part S1 = s11 + 2 T1; 1 + R(C + Cp)s + L(C + Cp)s of the propagation constant, namely the attenuation (3) constant, indicates that amplitude attenuation occurs where As is the area of electrodes, hp is the thickness of as the elastic wave propagates from one cell to the piezoelectric patch and Cp is the inherent capacitance next. of the piezoelectric patch at constant stress, which can To validate the existence of LR gaps induced by the be expressed as oscillations of shunting circuits and evaluate the capa- T "33As Cp = : (4) bility of the periodic shunts to attenuate the propaga- hp tion of elastic waves in the LR gaps, a phononic beam If harmonic vibration is assumed, s = i! can be sub- consisting of epoxy host and PZT-5H patches was cho- stituted into Eq. (3). The equivalent elastic modulus sen to perform a numerical simulation. The geometric of the piezoelectric patch can be written as properties and material parameters employed in the 2 calculations are listed in Tables 1 and 2. Ep =[1 + iR(C + Cp)! − L(C + Cp)! ] n E E 2 −1 Table 1. Geometric and material properties of the host beam. · s11 + iR[s11(C + Cp) − d31Ashp ]! Density Elastic modules Width Thickness o−1 Material E 2 −1 2 (kg/m3) (GPa) (m) (m) − L[s11(C + Cp) − d31Ashp ]! ; (5) epoxy 1180 4.35 0.02 0.005 where ! is the angular frequency. Assuming an Euler-Bernoulli beam, the governing Table 2. Geometric and material properties of the PZT patch. differential equations of transverse vibration can be Here width, thickness, length and the lattice constant are in units of m. expressed as Material Density sE d "T @4휑(x; t) @2휑(x; t) 11 31 33 EI + 휌퐴 = 0; (6) (kg/m3) (m3/N) (C/m2) (F/m) @x4 @t2 PZT-5H 7500 1.65×10−11 −2:74×10−10 3.01×10−8 where EI, 휌 and A are the bending rigidity, density Width Thickness Length Lattice constant and cross section, respectively, at the coordinate x; 0.02 0.0002 0.04 0.08 휑(x; t) is the displacement of the beam at coordinate x in the z-direction. Table 3. RLC parameters of shunting circuits. The harmonic solution to Eq. (6) can be expressed Group R (Ω) L (mH) C (nF) feig(Hz) as 1 5 2.2 310 5189 휑(x; t) = Φ(x)ei!t; (7) 2 24 5 110 5413 where Φ(x) is the vibration amplitude that can be To obtain a direct comparison with the experi- written as mental results, the shunting circuit parameters cho- sen coincide with the parameters in the experiment. Φ(x) =A cos(휆푥) + B sin(휆푥) Two groups of RLC parameters, as shown in Table 3, + C cosh(휆푥) + D sinh(휆푥); (8) are calculated by the transfer matrix method and the where A, B, C and D are undetermined coefficients, eigenfrequency of shunting circuits can be expressed and 휆 is the wavenumber, which is given by as r 휌퐴휔2 1 휆 = 4 : (9) feig = p : (10) EI 2휋 L(C + Cp) 094301-2 CHIN. PHYS. LETT. Vol. 28, No. 9 (2011) 094301 Figure2 illustrates the band gaps calculated using The experimental results of the two groups of RLC the two groups of RLC parameters in Table 3. Com- parameters are shown in Fig. 4. From the compar- pared with shorted ones, the RLC shunting circuits ison of the two curves shown in Fig. 4(a), the local can produce a new gap. When the shunting circuits resonance of RLC shunting circuits obviously pro- are shorted, there are no electromagnetic oscillations duces an LR gap, which forms considerable attenu- produced in the circuits and no LR gap is formed. As ation between the frequencies of 5000 Hz and 5600 Hz.