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Experimental study on slow flexural waves around the defect modes in a phononic crystal beam using fiber Bragg gratings
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Experimental study on slow flexural waves around the defect modes in a phononic crystal beam using fiber Bragg gratings
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Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, School of Aeronautics and Astronautics, Institute of Applied Mechanics, Zhejiang University, Hangzhou, 310027, China
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Kuo-Chih Chuang ∗ , Zhi-Qiang Zhang, Hua-Xin Wang
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Article history: Received 6 August 2016 Received in revised form 29 September 2016 Accepted 29 September 2016 Available online xxxx Communicated by R. Wu Keywords: Phononic crystals Band gaps Defect modes Fiber Bragg gratings (FBGs) Flexural waves
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This work experimentally studies influences of the point defect modes on the group velocity of flexural waves in a phononic crystal Timoshenko beam. Using the transfer matrix method with a supercell technique, the band structures and the group velocities around the defect modes are theoretically obtained. Particularly, to demonstrate the existence of the localized defect modes inside the band gaps, a high-sensitivity fiber Bragg grating sensing system is set up and the displacement transmittance is measured. Slow propagation of flexural waves via defect coupling in the phononic crystal beam is then experimentally demonstrated with Hanning windowed tone burst excitations. © 2016 Published by Elsevier B.V.
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1. Introduction
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Phononic crystals (PCs) have received considerable attention for their potential applications in various wave-manipulation devices. PCs are artificial structures consisting of alternating segments with large contrast in material or geometric properties [1]. Due to Bragg scattering or local resonance, PCs have frequency band gaps in which acoustic/elastic waves are strongly attenuated in a certain direction [2]. The existence of the band gaps in PCs is of interest to many researchers and has led to various practical applications such as vibration control, sound insulation, frequency filters, and so on [3–5]. In addition to the band gap phenomena, confinement of acoustic/elastic waves in localized modes is possible through the introduction of crystal defects. A point defect acts as a microcavity which locally disturbs the crystal periodicity and generates the defect modes to confine the acoustic/elastic waves inside the band gaps [6,7]. When line defects are created in a PC, waves are guided along the defects and thus they can be used as an efficient waveguide. Despite the fact that various properties or devices have been proposed relating to the defect modes [6–10], less attention has been paid to PC beam structures in consideration of the defect modes and their associated flexural wave propagations.
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Corresponding author. E-mail address: [email protected] (K.-C. Chuang).
http://dx.doi.org/10.1016/j.physleta.2016.09.055 0375-9601/© 2016 Published by Elsevier B.V.
Band structures of flexural waves in a PC with a point defect can be theoretically studied using the plane wave expansion (PWE) method with a supercell technique [11,12]. PC plates with line defects have also been studied using the finite difference time domain (FDTD) method [13,14]. Recently, Hou et al. investigated the band structures of surface acoustic waves on nanostructured PCs with defects based on Brillouin light scattering and finite element simulations [15]. For sound waves in phononic crystals, Robertson et al. studied slow group velocity propagation of sound due to defect coupling in an acoustic diameter-modulated waveguide [16]. For light in photonic crystals, the so-called slow light caused by the defect modes provides a way to achieve time delay for optical systems [17]. However, the group velocity of flexural waves around the defect modes in PC beams is seldom theoretically or experimentally studied. In this letter, propagations and transmission properties of the defect modes of flexural waves in a PC Timoshenko beam (hereafter simply referred as a PC beam) with a point defect are investigated. We specifically focus on experimental demonstration of the existence of the defect modes inside the band gaps and their influences on the group velocity of flexural waves. Band structures are obtained using the transfer matrix method (TMM) with a supercell technique. In addition, the group velocity of the defect modes is calculated from the dispersion curves. Particularly, two point-wise fiber Bragg grating (FBG) sensors are set up to measure the displacement transmittance at the two ends of the PC beam and to verify the existence of the defect modes. The FBG sensing system
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Fig. 1. Schematic diagram of the supercell of the PC beam.
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A transfer matrix method (TMM) is an efficient analytical method very suitable for calculation of dispersive curves or transmission properties of one-dimensional periodic structures such as rods or beams. In this section, the TMM with a supercell technique is adopted to analyze the defect modes in an infinite PC beam. A supercell is a large cell containing alternating unit cells with a defect in the center. Fig. 1 illustrates the nth supercell of the infinite PC beam. The supercell consists of 8 unit cells in which each unit cell is built of two segments with different elastic constants and the crystal defect is introduced by varying the height of the middle segment (i.e., labeled as (3) in Fig. 1). Before applying the TMM to the PC beam, the governing equation of the free bending vibration of a homogeneous Timoshenko beam segment with a constant cross-section is considered [18,19]:
∂ 2 w (x, t ) ∂ 4 w (x, t ) E ∂ 4 w (x, t ) ρS + EI − ρ I 1 + κ G ∂ x2 ∂ t 2 ∂t2 ∂ x4
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+
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ρ 2 I ∂ 4 w (x, t ) = 0, κG ∂t4
φ (4) (x) − α φ (2) (x) − βφ(x) = 0,
Next, we consider the supercell of the infinite PC Timoshenko beam with a point crystal defect illustrated in Fig. 1. The amplitude of the flexural deflection of the jth segment in the nth supercell can be described as: j
j
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(a).
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Then, the relationship between the adjacent cells can be established by a transfer matrix as:
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ρ 2 ω4 − . Eκ G
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where x j = x − nb − ( j − 1)a and nb + ( j − 1)a ≤ x ≤ nb + ja, j = 1, 2, 3, . . . , r. The four interfacial continuity conditions of the transverse displacement, the angle of rotation, the bending moment, and the shear force between jth and ( j − 1)th segment can be written as:
Ψ n1+1 = TΨ n1 ,
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φn (x j ) = An cos λ1 x j + B n sin λ1 x j + C n cosh λ2 x j j j + D n sinh λ2 x j ,
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where ρ , S, E, G, I , and κ respectively represents the density, the cross-sectional area, the Young’s modulus, the shear modulus, the cross-sectional area moment of inertia, and the Timoshenko shear coefficient. For a steady-state vibration response, the amplitude of the transverse deflection w (x, t ) = φ(x)e i ωt satisfies:
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is also employed to measure transient flexural wave propagations excited by N-cycle tone bursts for calculations of the group velocities. To the best of the authors’ knowledge, this is the first research in phononic crystals using a high-sensitivity point-wise FBG displacement sensing system. Due to its capability of performing simultaneous in-plane or out-of-plane point-wise displacement measurements, this research opens the possibility of employing the FBGs for experimental demonstrations of various phenomena (e.g., negative refraction, focusing, and so on) in periodic structures such as elastic phononic crystals or metamaterials.
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By solving the corresponding characteristic equations, the general solution for Eq. (1) can be expressed as a linear combination of cosine, sine, hyperbolic cosine, and hyperbolic sine functions as follows:
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(10)
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where T = K1−1 Hr Kr−1 Hr −1 · · · K2−1 H1 . Since now the flexural wave is propagated in an infinite periodic beam, the well-known Bloch theorem is satisfied and can be applied between the state vectors of the adjacent supercells as:
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(11)
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where q is the wavenumber. By solving the corresponding eigenvalue problem obtained by subtracting Eq. (11) from Eq. (10):
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Fig. 2. Dispersion curves of the PC beam: (a) without the point defect analyzed by the unit cell; (b) with (right plot) and without (left plot) the point defect analyzed by the supercell.
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the band structures of the PC Timoshenko beam can be obtained. Band gaps are the bands in which strongly attenuated evanescent waves are propagated when all the four roots of the wavenumber in Eq. (12) have a non-vanishing imaginary part. A larger imaginary part of the wavenumber results in stronger spatial decay of the flexural waves along the PC beam. In this letter, a PC beam consisting of two materials, 6061 aluminum (i.e., material (1) shown in Fig. 1) and acrylic (PMMA, material (2) shown in Fig. 1), is investigated. Each of the two materials has the size of 0.08 m (length) × 0.015 m (height) × 0.015 m (width). The middle acrylic (i.e., material (3) shown in Fig. 1) is introduced as a defect whose height is triple (i.e., 0.045 m) that of the rest of the segments. The elastic constants used in the TMM are ρ1 = 2735 kg/m3 , E 1 = 7.47 × 1010 Pa, ρ2 = 1142 kg/m3 , E 2 = 4.5 × 109 Pa, and κ = 1.2. The Poisson’s ratio of the two materials is 0.33. Note that the Young’s modulus of the acrylic (i.e., E 2 = 4.5 × 109 Pa) is experimentally obtained beforehand by matching the frequency responses of an acrylic plate subjected to steel ball-impacts to those predicted by the finite element simulations. Fig. 2 shows the results of the band structure of the PC beam with or without crystal defects calculated by the TMM. The wavenumbers in the band structures are normalized (i.e., q∗ = q(2a)/π for the unit cell and q∗ = qL /π for the supercell). Two band gaps can be seen in Fig. 2(a) within 0–4500 Hz (i.e., 561–935 Hz; 2523–3412 Hz) for the PC beam without the point defect. The right and left plots in Fig. 2(b) respectively shows the band structures of the PC beam using the supercell technique with and without the point defect. By comparing Fig. 2(a) with the left plot of Fig. 2(b), we can see that the supercell with 8 unit cells folds the first original dispersion curves 8 times below the first band gap. Clearly, we can observe that when a point crystal defect is introduced in a supercell, a flat defect band is generated within the band gap. The narrow pass band of the defect mode inside the band gap indicates slow group velocities of flexural wave propagations around the defect mode. Further analysis by the TMM and supercell method shows that, compared to the perfect PC beam, the 8th folding dispersion curve shifts toward the middle frequency of the band gap as the Young’s modulus or the height of the middle acrylic is increased. Interestingly, on the other hand, when reducing the Young’s modulus or the height of the middle acrylic, the 9th folding flexural dispersion curve shifts toward the lower edge of the band gap to form the defect mode. To save space in this letter, however, only the defect mode that will later be employed to verify the group velocity is illustrated in Fig. 2(b). It should be noted that the point crystal defect introduced in the supercell also results in the existences of additional narrower band gaps in the band structure. Before performing experiments on transmission properties and demonstrating the slow group velocity of flexural waves via de-
fect coupling, we briefly introduce the sensing principle of the FBG displacement sensing system in the next section. The band structure obtained from the TMM will finally be employed to verify the group velocity of flexural waves propagated in the PC beam in Section 4.
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3. FBG sensing system
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Most research for periodic structures such as phononic crystals or elastic metamaterials use non-contact laser Doppler vibrometers (LDVs) or accelerometers for experimental demonstrations. Whether or not the function of full-field scanning is built into the LDV system, the LDV system is costly and occupies a large physical space. As for the accelerometers, the measured physical quantities are accelerations instead of displacements which are of direct interest to the analytical analysis. In this letter, a fiber Bragg grating (FBG) displacement sensing system is set up (as shown in Fig. 3) to directly obtain displacement transmittance frequency response function (FRF) of a finite PC beam. Multiplexing and multidimensional measurements can easily be achieved by the proposed FBG sensors with a low cost and small physical space compared to the LDVs. The FBG sensing system consists of two point-wise FBG displacement sensors that are capable of simultaneously measuring the input and output displacements at the two ends of the PC beam. In this section, the sensing principle of the FBG displacement sensor is briefly introduced. An FBG is a segment in a singlemode fiber in which permanent periodic perturbation of the refractive index is formed in the fiber core. When irradiated with a broadband light source, a narrow Gaussian shape spectrum will be reflected by the FBG. The center wavelength in the reflected spectrum satisfies the Bragg condition:
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λ B = 2neff Λ,
(13)
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where λ B is the Bragg wavelength, Λ is the Bragg grating period, and neff is the effective refractive index of the fiber core. Any external applied strain or temperature variation would shift the Bragg wavelength by modifying the grating period or the effective refractive index. In this letter, the commonly used FBG strain sensor is set up as a point-wise displacement sensor in which one end of the FBG is glued to the sensing point and the other end is glued to a fixed translation stage [20]. Thus, shifts of the Bragg wavelength convert the strain of the FBG to the displacements of the sensing point. Neglecting the environmental temperature variations, the shift in Bragg wavelength can be expressed by:
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n2eff
λ B = 1+ ν ( p 11 + p 12 ) − p 12 ε, λB 2
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(14)
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Fig. 3. Experimental setup of the PC beam and the FBG displacement sensing system.
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where ε is the applied axial strain of the fiber, ν is the Poisson’s ratio, and p i j is the photo-elastic coefficient. A demodulation system with photodiodes is required to detect the Bragg wavelength shifts and convert them to electrical signals, especially for high-frequency transient measurements. In this letter, an FBG filter is employed to demodulate the Bragg wavelength shifts of the FBG sensor due to transient elastic wave propagations. Both the reflectance of the FBG sensor and the FBG filter can be approximated by a Gaussian function [21]:
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λ − λB
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(15)
∞
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σ
2
where A is the maximum reflectivity of the FBG and σ is the grating full-width at half maximum (FWHM). When the intensity of the broadband source transmitted into the FBG sensor is I i (λ), the total light power P d detected on the photodiode (e.g. PD1 in Fig. 3) can be expressed by:
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Pd = k
I i (λ) S (λ) F (λ)dλ,
(16)
−∞
where S (λ) is the spectrum of the reflected light by the FBG sensor, F (λ) is the reflected spectrum of the FBG filter, and k indicates the optical intensity losses in the circulator, the coupler, and the light path in the fiber. When S (λ) is shifted dynamically due to transient waves, or vibrations in the PC beam, it can be demodulated according to Eq. (16) provided that the two spectra of the FBG sensor and the filter are well matched.
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4. Experimental results and discussions
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beam and the all-fiber FBG displacement sensing system is illustrated in Fig. 3. The PC beam consists of a finite repetition of 16 alternating aluminum/acrylic segments in which the height of the middle acrylic is triple that of the rest of the segments. Each of the aluminum and acrylic segments are bonded together by epoxy glues. The material properties and geometric size of the PC beam have been described in Section 2. The PC beam is excited by a piezoelectric multilayered actuator (AE0505D16, Thorlabs, Newton, NJ, USA) on the right-hand end of the PC beam, whose elongation is approximately proportional to the driving voltage signal [23]. Two point-wise FBG displacement sensors are glued to the two ends of the beam by a mix of epoxy resin and hardener (i.e., FBG1 and FBG2 as shown in Fig. 3) as input and output displacement sensors, respectively. Other than the excitation point and the two sensing points on the beam, only two points on the PC beam are rested on soft rubbers (as shown in the inset photo in Fig. 3). In the FBG sensing system, a C-band amplified spontaneous emission (ASE) light source (China-Fiber Optics, Shanghai, China) is transmitted to a 1 × 2 coupler and then split into two paths for the two FBG sensors and corresponding demodulation systems. Each demodulation system consists of two 3-port directional optical circulators (OC’s) to guide the light beam. Photodiodes (PDA10CS, InGaAs amplified detector, Thorlabs, Newton, NJ, USA) are employed to convert the light to electrical signals and are connected to a dSPACE system (dSPACE GmbH, Paderborn, Germany) for the transmittance FRF measurement or an oscilloscope (LeCroy, HD04054, USA) for calculation of the group velocity. The Bragg wavelengths of the FBG sensors and FBG filters shown in Fig. 3 are FBG1 = 1550.356 nm, FBG filter1 = 1550.516 nm, FBG2 = 1550.492 nm, and FBG filter2 = 1550.464 nm, respectively. The grating lengths of the FBG sensors and FBG filters are both 10 mm. Details of setting up the point-wise FBG displacement sensor, for dynamic measurement, can be found in [20] proposed by the first author.
4.1. Experimental setup
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4.2. Experimental results First, the vibration transmittance FRF of the finite PC beam is measured to investigate the band gap phenomena predicted by the infinite PC beam [22]. In this letter, the FRF is defined as T d = 20 log10 (dout /din ) in which dout and din are respectively the displacements of the transmitted and incident waves, at the two ends of the PC beam. The experimental setup and photos of the PC
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The FRF in this study is obtained by a stochastic spectral estimation commonly used in the control system identification. White noise random signals, generated by the Simulink (The MathWorks, Natick, MA) and the dSPACE DS1104 system, with a sampling frequency 50 kHz, are sent to the piezoelectric multilayered actuator
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Fig. 4(c), indicating the existence of the defect modes in the PC beam. Compared to Fig. 2(b), in which band gaps are calculated by the TMM as 561–935 Hz and 2523–3412 Hz with a defect band from 665–702 Hz for the infinite PC beam, the displacement transmittance FRF of the finite PC beam is related to the band structures predicted in the infinite beam. However, there are some discrepancies mainly caused by the finite or infinite geometries [26]. By plotting the steady-state displacement mode shapes at the point crystal defect, we can see that the defect modes are localized vibration modes with the first and the third being bending modes and the second, rotational vibration mode. After verifying the transmission properties and the defect states of the PC beam, a tone burst signal is generated by the dSPACE system to excite transient wave propagations for determinations of the group velocity of flexural waves. A N-count Hanning windowed tone burst excitation can be expressed by:
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V in (t ) = H (t ) − H (t − N p / f c )
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Fig. 4. The FRF between signals from (a) FBG1 and the random excitations, (b) FBG2 and FBG1 at the two ends of the perfect PC beam, and (c) that with the defect introduced in the middle of the PC beam.
through a power amplifier (XE501-C, XMT, China) to excite flexural wave propagations in the PC beam. The FRF is obtained from the relationship T d (ω) = 20 log10 | S yu ( j ω)/ S uu ( j ω)| [25], where S uu ( j ω) is the auto-spectral density function of the input random displacement (detected by the FBG1) and S yu ( j ω) is the crossspectral density function between the input and transmitted displacements, respectively (i.e., responses of the FBG1 and FBG2). Dynamic sensing performance of the proposed point-wise FBG displacement sensor has been confirmed with a LDV system in a study regarding precision tracking control performed by the first author [24]. However, since the dynamics of the FBG displacement sensing system might be slightly influenced by various designs of the fixed-stage supporting the FBG displacement sensors, it is necessary to validate the FRF between the FBG1 input displacement sensor and the random excitations before band gap measurements. The transmittance FRF between the random excitations and the FBG1 input sensor, between FBG1 and FBG2 for a perfect PC beam, and that for the imperfect PC beam are respectively shown in Figs. 4(a), 4(b), and 4(c). In Fig. 4(a), we can see a rather flat spectrum in frequency ranges corresponding to three band gaps of the PC beam, indicating good dynamic measurement capability of the proposed FBG displacement sensing system. It is noted that the band gaps are defined as frequency ranges exhibiting low transmission below 0 dB and are marked in Fig. 4 in grey shaded regions. Despite the high frequency mismatch, above the upper edge of the third band gap, an obvious resonant frequency at 235 Hz and an anti-resonant frequency at 218 Hz can be observed. The local resonance and anti-resonance might be explained by the loading effects from the contact-type FBG sensor with the dynamics of its supporting fixed-stage. Since the focus of this study is about the defect modes inside the band gaps whose frequencies of interest are not overlapped with the local resonance and antiresonance of the sensing system, it is demonstrated in Fig. 4 that the proposed FBG displacement sensing system is suitable to investigate flexural wave propagations in the PC beam around the band gaps. In Figs. 4(b) and 4(c), experimental results coincide well with the finite element method (FEM) simulations performed by the COMSOL Multiphysics® software. In FEM simulations, the damping, mainly due to viscous epoxy between each segment, is modeled as a loss factor with a value of 0.01. As can be seen in Figs. 4(b) and 4(c), there are three deep band gaps within 0 to 8000 Hz (i.e., 458–750 Hz, 2062–3563 Hz, and 5212–6980 Hz). The three transmission peaks inside the band gaps are marked in
1 − cos
2π f c t Np
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sin 2π f c t , (17)
where N p is the peak number, f c is the central frequency of interest, and H (t ) is the Heaviside step function. In most experimental studies regarding calculations of group velocity in dispersive media, a 5-count Hanning windowed tone burst excitation signal (i.e., N p = 5) is usually chosen (e.g., such as studies of lamb waves in composites [27–29]) due to its excellent time and moderate frequency resolution. Fig. 5(a) shows a 5-count Hanning windowed tone burst excitation signal at 568 Hz (i.e., the first defect mode in this study) and the transmitted wave packets predicted by FEM at the tip of the PC beam. It should be noted that, other than the damping effect, the transmitted wave is attenuated due to the fact that the excitations realized by the piezoelectric multilayer actuator try to stop the vibration of the PC beam after a complete generation of a tone burst. We can see that the group velocity is difficult to be extracted simply by reading the time of flight of the 5-count-tone-burst excited wave packets. To have a better calculation of the time of flight for the wave packets, without resorting to complicated techniques such as Gabor wavelet transform, in this letter we choose a 21-count Hanning windowed tone burst to excite the beam. The frequency spectra of the 5-count and 21-count smoothed tone burst signals are shown in Fig. 5(b). It can be seen that the 21-count tone burst has a higher frequency resolution and stronger intensity. The increasing of the frequency resolution, however, sacrifices the time resolution due to the wellknown Heisenberg’s Uncertainty Principle [27]. Fig. 5(c) shows the FBG measurement results and the FEM simulations of the transmitted waves from excitations at the frequency of the defect mode (i.e., 568 Hz). Since the reflected wave packets, from the left-hand free end of the beam, are unavoidably superimposed coherently to the incoming wave packets, the number of the peaks in the transmitted wave packet is more than that of the excitation tone burst. The time of flight of the maximum peak, in the wave packet in the case of the 21-count smoothed tone burst, can now be more easily determined by using the cursor functions of the oscilloscope. The group velocities can then be obtained by dividing the length of the PC beam (i.e., approximately the distance between the two FBG displacement sensors) by the time of flight. From Fig. 5(c) we can also see that the proposed FBG displacement sensing system is capable of detecting transient wave propagations with the discrepancy due to difficulties of perfect estimation of the damping. Fig. 6(a) shows the group velocities predicted by the band structures in Fig. 2(b) obtained by the TMM by calculating C g = dω/dq ∼ = |(ω2 − ω1 )/(q2 − q1 )| between the adjacent frequencies around the first band gap. Group velocities evaluated from the TMM indicate that the defect mode possesses the slowest group velocity compared to other dispersion curves. Fig. 6(b) displays the transmittance FRF in the first band gap obtained numerically and
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Fig. 5. Excitations of the PC beam by the Hanning windowed tone bursts: (a) 5-count tone burst and simulation response, (b) frequency spectra of the 5-count and 21-count tone burst, and (c) the experimental and simulation results for the excitations at the frequency of the defect mode (i.e., 568 Hz).
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Fig. 6. (a) Group velocity calculated by the band structure obtained by the TMM; (b) comparison of the experimental result and the numerical prediction for the FRF; (c) group velocities in the band gap containing the defect mode.
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Fig. 7. Transient wave propagation at three time points corresponding to adjacent peak–valley–peak in the tone burst-induced responses: (a) inside the band gap at 480 Hz; (b) the defect mode at 568 Hz. (Note: In the FEM simulations, the left-hand end is the excitation point.)
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experimentally. The calculated group velocities from the FEM and the FBG measurements are shown in Fig. 6(c) where the group velocities are collected from excitations at 35 different frequencies in the frequency range of interest. The slow group velocity around the defect mode is clearly validated. Other flexural waves excited at frequencies corresponding to resonant peaks also have slow group velocity due to the effect of standing waves. However, it should be noted that the final zero excitation exerted by the 21-count Hanning windowed tone burst prevents the occurrence of “perfect” standing waves. It is also interesting to note that the group velocities inside the band gap away from the defect mode reach to higher values than those in other transmission regions. This result is similar to the conclusion for sound wave propagations via defect coupling in an acoustic array performed by Robertson et al. in [16]. In order to have a clearer physical understanding of the slow wave propagations in the PC beam with the point crystal defect, two transient displacement fields of the PC beam, excited at 480 Hz (i.e., an arbitrary chosen frequency inside the band gap)
and 568 Hz (i.e., the defect mode), are simulated by the FEM and the results are shown in Fig. 7. In FEM simulation shown in Fig. 7, the PC beam model is excited from the left-hand side and the beam is traction-free other than the excitation point. The transient wave propagations excited at the frequencies of interest are plotted at three different time points corresponding to those where the transmitted wave packet reaches adjacent peak–valley–peak. We can see that, unlike the evanescent wave propagations in the band gap, the localized defect mode causes a almost standing wave propagation in the PC beam, resulting in corresponding slow group velocity at the defect mode, a phenomenon that can also be seen in photonic crystals [30,31].
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5. Conclusions
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In summary, the transmission properties of a PC Timoshenko beam, with or without a point defect, are investigated using an FBG displacement sensing system. The slow group velocity of flex-
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ural waves via defect coupling is experimentally demonstrated. Based on the TMM with the supercell technique, the band structures of the PC beam with and without the point defect are analyzed. Good agreements can be obtained between the experimental measurements, the theoretical analysis, and the numerical simulations for the defect modes and the associated slow group velocity inside the band gaps. In addition, to the best of the authors’ knowledge, this is the first time the point-wise FBG displacement sensors are employed to obtain the displacement transmittance FRF and to measure the wave packet propagations in the PC beam. Compared to the commonly employed LDVs or accelerometers, the proposed FBG displacement sensing system opens a new experimental tool for research of phononic crystals or elastic metamaterials to achieve high-sensitivity and low-cost direct dynamic displacement sensing.
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Acknowledgements
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The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 11272282). K.C. Chuang appreciates the encouragement from Professor C.T. Sun during his stay as a visiting scholar at the Purdue University.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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References [1] M.F. Su, R.H. Olsson III, Z.C. Leseman, I. El-Kady, Appl. Phys. Lett. 96 (2010) 53111. [2] M.M. Sigalas, E.N. Economou, J. Sound Vib. 158 (1992) 377. [3] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289 (2000) 1734.
[25] [26] [27] [28] [29] [30] [31]
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M. Oudich, M.B. Assouar, Z. Hou, Appl. Phys. Lett. 97 (2010) 193503. D. Yu, Y. Liu, G. Wang, H. Zhao, J. Qiu, J. Appl. Phys. 100 (2006) 124901. M.M. Sigalas, J. Acoust. Soc. Am. 101 (1997) 1256. F. Wu, Z. Liu, Y. Liu, Phys. Rev. E 69 (2004) 66609. S. Benchabane, A. Khelif, A. Choujaa, B. Djafari-Rouhani, V. Laude, Europhys. Lett. 71 (2005) 570. J. Sun, T. Wu, Phys. Rev. B 71 (2005) 174303. S. Yang, J.H. Page, Z. Liu, M.L. Cowan, C.T. Chan, P. Sheng, Phys. Rev. Lett. 93 (2004) 24301. Z. Yao, G. Yu, Y. Wang, Z. Shi, Int. J. Solids Struct. 46 (2009) 2571–2576. G.J. Pearce, T.D. Hedley, D.M. Bird, Phys. Rev. B 71 (2005) 195108. J. Sun, T. Wu, Phys. Rev. B 76 (2007) 104304. Y. Pennec, B. Djafari-Rouhani, H. Larabi, J.O. Vasseur, A.C. Hladky-Hennion, Phys. Rev. B 78 (2008) 104105. C.G. Hou, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, J. Deng, S.J. Wang, Appl. Phys. Lett. 105 (2014) 243104. W.M. Robertson, C. Baker, C.B. Bennett, Am. J. Phys. 72 (2004) 255. H. Tian, Y. Ji, Mod. Phys. Lett. B 23 (2009) 1053. R. Zhu, X.N. Liu, G.K. Hu, C.T. Sun, G.L. Huang, J. Sound Vib. 333 (2014) 2759. Y. Liu, D. Yu, L. Li, H. Zhao, J. Wen, X. Wen, Phys. Lett. A 362 (2007) 344. K.C. Chuang, C.C. Ma, Appl. Opt. 47 (2008) 3561. A.S. Paterno, V. de Oliveira, T.S. Figueredo, H.J. Kalinowski, IEEE Sens. J. 6 (2006) 1662. Y. Xiao, J. Wen, G. Wang, X. Wen, J. Vib. Acoust. 135 (2013) 41006. J. Li, X. Zhou, H. Zhao, M. Shao, P. Hou, X. Xu, Smart Mater. Struct. 24 (2015) 65007. K.C. Chuang, C.C. Ma, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56 (2009) 2036. B.J. Huang, Y.C. Chen, J. Dyn. Syst. Meas. Control 124 (2002) 176. E. Baravelli, M. Ruzzene, J. Sound Vib. 332 (2013) 6562. L. Wang, F.G. Yuan, Compos. Sci. Technol. 67 (2007) 1370. M. Yang, P. Qiao, Smart Mater. Struct. 14 (2005) 1083. V. Giurgiutiu, J. Bao, W. Zhao, Exp. Mech. 43 (2003) 428. R. Ozaki, Y. Matsuhisa, M. Ozaki, K. Yoshino, Appl. Phys. Lett. 84 (2004) 1844. S. Furumi, H. Fudouzi, H.T. Miyazaki, Y. Sakka, Adv. Mater. 19 (2007) 2067.
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Experimental study on slow flexural waves around the defect modes in a phononic crystal beam using fiber Bragg gratings
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Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, School of Aeronautics and Astronautics, Institute of Applied Mechanics, Zhejiang University, Hangzhou, 310027, China
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Kuo-Chih Chuang ∗ , Zhi-Qiang Zhang, Hua-Xin Wang
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Article history: Received 6 August 2016 Received in revised form 29 September 2016 Accepted 29 September 2016 Available online xxxx Communicated by R. Wu Keywords: Phononic crystals Band gaps Defect modes Fiber Bragg gratings (FBGs) Flexural waves
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This work experimentally studies influences of the point defect modes on the group velocity of flexural waves in a phononic crystal Timoshenko beam. Using the transfer matrix method with a supercell technique, the band structures and the group velocities around the defect modes are theoretically obtained. Particularly, to demonstrate the existence of the localized defect modes inside the band gaps, a high-sensitivity fiber Bragg grating sensing system is set up and the displacement transmittance is measured. Slow propagation of flexural waves via defect coupling in the phononic crystal beam is then experimentally demonstrated with Hanning windowed tone burst excitations. © 2016 Published by Elsevier B.V.
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1. Introduction
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Phononic crystals (PCs) have received considerable attention for their potential applications in various wave-manipulation devices. PCs are artificial structures consisting of alternating segments with large contrast in material or geometric properties [1]. Due to Bragg scattering or local resonance, PCs have frequency band gaps in which acoustic/elastic waves are strongly attenuated in a certain direction [2]. The existence of the band gaps in PCs is of interest to many researchers and has led to various practical applications such as vibration control, sound insulation, frequency filters, and so on [3–5]. In addition to the band gap phenomena, confinement of acoustic/elastic waves in localized modes is possible through the introduction of crystal defects. A point defect acts as a microcavity which locally disturbs the crystal periodicity and generates the defect modes to confine the acoustic/elastic waves inside the band gaps [6,7]. When line defects are created in a PC, waves are guided along the defects and thus they can be used as an efficient waveguide. Despite the fact that various properties or devices have been proposed relating to the defect modes [6–10], less attention has been paid to PC beam structures in consideration of the defect modes and their associated flexural wave propagations.
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*
Corresponding author. E-mail address: [email protected] (K.-C. Chuang).
http://dx.doi.org/10.1016/j.physleta.2016.09.055 0375-9601/© 2016 Published by Elsevier B.V.
Band structures of flexural waves in a PC with a point defect can be theoretically studied using the plane wave expansion (PWE) method with a supercell technique [11,12]. PC plates with line defects have also been studied using the finite difference time domain (FDTD) method [13,14]. Recently, Hou et al. investigated the band structures of surface acoustic waves on nanostructured PCs with defects based on Brillouin light scattering and finite element simulations [15]. For sound waves in phononic crystals, Robertson et al. studied slow group velocity propagation of sound due to defect coupling in an acoustic diameter-modulated waveguide [16]. For light in photonic crystals, the so-called slow light caused by the defect modes provides a way to achieve time delay for optical systems [17]. However, the group velocity of flexural waves around the defect modes in PC beams is seldom theoretically or experimentally studied. In this letter, propagations and transmission properties of the defect modes of flexural waves in a PC Timoshenko beam (hereafter simply referred as a PC beam) with a point defect are investigated. We specifically focus on experimental demonstration of the existence of the defect modes inside the band gaps and their influences on the group velocity of flexural waves. Band structures are obtained using the transfer matrix method (TMM) with a supercell technique. In addition, the group velocity of the defect modes is calculated from the dispersion curves. Particularly, two point-wise fiber Bragg grating (FBG) sensors are set up to measure the displacement transmittance at the two ends of the PC beam and to verify the existence of the defect modes. The FBG sensing system
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λ1 =
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λ2 =
Fig. 1. Schematic diagram of the supercell of the PC beam.
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A transfer matrix method (TMM) is an efficient analytical method very suitable for calculation of dispersive curves or transmission properties of one-dimensional periodic structures such as rods or beams. In this section, the TMM with a supercell technique is adopted to analyze the defect modes in an infinite PC beam. A supercell is a large cell containing alternating unit cells with a defect in the center. Fig. 1 illustrates the nth supercell of the infinite PC beam. The supercell consists of 8 unit cells in which each unit cell is built of two segments with different elastic constants and the crystal defect is introduced by varying the height of the middle segment (i.e., labeled as (3) in Fig. 1). Before applying the TMM to the PC beam, the governing equation of the free bending vibration of a homogeneous Timoshenko beam segment with a constant cross-section is considered [18,19]:
∂ 2 w (x, t ) ∂ 4 w (x, t ) E ∂ 4 w (x, t ) ρS + EI − ρ I 1 + κ G ∂ x2 ∂ t 2 ∂t2 ∂ x4
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ρ 2 I ∂ 4 w (x, t ) = 0, κG ∂t4
φ (4) (x) − α φ (2) (x) − βφ(x) = 0,
Next, we consider the supercell of the infinite PC Timoshenko beam with a point crystal defect illustrated in Fig. 1. The amplitude of the flexural deflection of the jth segment in the nth supercell can be described as: j
j
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(a).
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(8)
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j −1 j −1 j −1 j −1 sin(λ cosh(λ sinh(λ a) a) a) a) 1 1 2 2 j −1 j −1 j −1 j −1 j −1 j −1 j −1 j −1 λ2 cosh(λ2 a) −λ1 sin(λ1 a) λ1 cos(λ1 a) λ2 sinh(λ2 a) j −1 j −1 j −1 j −1 j −1 j −1 −(λ1 )2 cos(λ1 a) −(λ1 )2 sin(λ1 a) (λ2 )2 cosh(λ2 a) (λ2 )2 sinh(λ2 a) j −1 3 j −1 3 j −1 3 j −1 j −1 3 j −1 (λ1 ) sin(λ1 a) −(λ1 ) cos(λ1 a) (λ2 ) sinh(λ2 a) (λ2 ) cosh(λ2 a) cos(λ
⎤ ⎥ ⎥. ⎦
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(9c)
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Then, the relationship between the adjacent cells can be established by a transfer matrix as:
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ρ 2 ω4 − . Eκ G
j
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where x j = x − nb − ( j − 1)a and nb + ( j − 1)a ≤ x ≤ nb + ja, j = 1, 2, 3, . . . , r. The four interfacial continuity conditions of the transverse displacement, the angle of rotation, the bending moment, and the shear force between jth and ( j − 1)th segment can be written as:
Ψ n1+1 = TΨ n1 ,
ρ S ω2
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⎢ 0 ⎢ Kj = ⎢ j ⎣ −(λ1 )2
where ρ , S, E, G, I , and κ respectively represents the density, the cross-sectional area, the Young’s modulus, the shear modulus, the cross-sectional area moment of inertia, and the Timoshenko shear coefficient. For a steady-state vibration response, the amplitude of the transverse deflection w (x, t ) = φ(x)e i ωt satisfies:
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is also employed to measure transient flexural wave propagations excited by N-cycle tone bursts for calculations of the group velocities. To the best of the authors’ knowledge, this is the first research in phononic crystals using a high-sensitivity point-wise FBG displacement sensing system. Due to its capability of performing simultaneous in-plane or out-of-plane point-wise displacement measurements, this research opens the possibility of employing the FBGs for experimental demonstrations of various phenomena (e.g., negative refraction, focusing, and so on) in periodic structures such as elastic phononic crystals or metamaterials.
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(10)
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where T = K1−1 Hr Kr−1 Hr −1 · · · K2−1 H1 . Since now the flexural wave is propagated in an infinite periodic beam, the well-known Bloch theorem is satisfied and can be applied between the state vectors of the adjacent supercells as:
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(11)
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Fig. 2. Dispersion curves of the PC beam: (a) without the point defect analyzed by the unit cell; (b) with (right plot) and without (left plot) the point defect analyzed by the supercell.
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the band structures of the PC Timoshenko beam can be obtained. Band gaps are the bands in which strongly attenuated evanescent waves are propagated when all the four roots of the wavenumber in Eq. (12) have a non-vanishing imaginary part. A larger imaginary part of the wavenumber results in stronger spatial decay of the flexural waves along the PC beam. In this letter, a PC beam consisting of two materials, 6061 aluminum (i.e., material (1) shown in Fig. 1) and acrylic (PMMA, material (2) shown in Fig. 1), is investigated. Each of the two materials has the size of 0.08 m (length) × 0.015 m (height) × 0.015 m (width). The middle acrylic (i.e., material (3) shown in Fig. 1) is introduced as a defect whose height is triple (i.e., 0.045 m) that of the rest of the segments. The elastic constants used in the TMM are ρ1 = 2735 kg/m3 , E 1 = 7.47 × 1010 Pa, ρ2 = 1142 kg/m3 , E 2 = 4.5 × 109 Pa, and κ = 1.2. The Poisson’s ratio of the two materials is 0.33. Note that the Young’s modulus of the acrylic (i.e., E 2 = 4.5 × 109 Pa) is experimentally obtained beforehand by matching the frequency responses of an acrylic plate subjected to steel ball-impacts to those predicted by the finite element simulations. Fig. 2 shows the results of the band structure of the PC beam with or without crystal defects calculated by the TMM. The wavenumbers in the band structures are normalized (i.e., q∗ = q(2a)/π for the unit cell and q∗ = qL /π for the supercell). Two band gaps can be seen in Fig. 2(a) within 0–4500 Hz (i.e., 561–935 Hz; 2523–3412 Hz) for the PC beam without the point defect. The right and left plots in Fig. 2(b) respectively shows the band structures of the PC beam using the supercell technique with and without the point defect. By comparing Fig. 2(a) with the left plot of Fig. 2(b), we can see that the supercell with 8 unit cells folds the first original dispersion curves 8 times below the first band gap. Clearly, we can observe that when a point crystal defect is introduced in a supercell, a flat defect band is generated within the band gap. The narrow pass band of the defect mode inside the band gap indicates slow group velocities of flexural wave propagations around the defect mode. Further analysis by the TMM and supercell method shows that, compared to the perfect PC beam, the 8th folding dispersion curve shifts toward the middle frequency of the band gap as the Young’s modulus or the height of the middle acrylic is increased. Interestingly, on the other hand, when reducing the Young’s modulus or the height of the middle acrylic, the 9th folding flexural dispersion curve shifts toward the lower edge of the band gap to form the defect mode. To save space in this letter, however, only the defect mode that will later be employed to verify the group velocity is illustrated in Fig. 2(b). It should be noted that the point crystal defect introduced in the supercell also results in the existences of additional narrower band gaps in the band structure. Before performing experiments on transmission properties and demonstrating the slow group velocity of flexural waves via de-
fect coupling, we briefly introduce the sensing principle of the FBG displacement sensing system in the next section. The band structure obtained from the TMM will finally be employed to verify the group velocity of flexural waves propagated in the PC beam in Section 4.
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Most research for periodic structures such as phononic crystals or elastic metamaterials use non-contact laser Doppler vibrometers (LDVs) or accelerometers for experimental demonstrations. Whether or not the function of full-field scanning is built into the LDV system, the LDV system is costly and occupies a large physical space. As for the accelerometers, the measured physical quantities are accelerations instead of displacements which are of direct interest to the analytical analysis. In this letter, a fiber Bragg grating (FBG) displacement sensing system is set up (as shown in Fig. 3) to directly obtain displacement transmittance frequency response function (FRF) of a finite PC beam. Multiplexing and multidimensional measurements can easily be achieved by the proposed FBG sensors with a low cost and small physical space compared to the LDVs. The FBG sensing system consists of two point-wise FBG displacement sensors that are capable of simultaneously measuring the input and output displacements at the two ends of the PC beam. In this section, the sensing principle of the FBG displacement sensor is briefly introduced. An FBG is a segment in a singlemode fiber in which permanent periodic perturbation of the refractive index is formed in the fiber core. When irradiated with a broadband light source, a narrow Gaussian shape spectrum will be reflected by the FBG. The center wavelength in the reflected spectrum satisfies the Bragg condition:
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where λ B is the Bragg wavelength, Λ is the Bragg grating period, and neff is the effective refractive index of the fiber core. Any external applied strain or temperature variation would shift the Bragg wavelength by modifying the grating period or the effective refractive index. In this letter, the commonly used FBG strain sensor is set up as a point-wise displacement sensor in which one end of the FBG is glued to the sensing point and the other end is glued to a fixed translation stage [20]. Thus, shifts of the Bragg wavelength convert the strain of the FBG to the displacements of the sensing point. Neglecting the environmental temperature variations, the shift in Bragg wavelength can be expressed by:
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Fig. 3. Experimental setup of the PC beam and the FBG displacement sensing system.
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where ε is the applied axial strain of the fiber, ν is the Poisson’s ratio, and p i j is the photo-elastic coefficient. A demodulation system with photodiodes is required to detect the Bragg wavelength shifts and convert them to electrical signals, especially for high-frequency transient measurements. In this letter, an FBG filter is employed to demodulate the Bragg wavelength shifts of the FBG sensor due to transient elastic wave propagations. Both the reflectance of the FBG sensor and the FBG filter can be approximated by a Gaussian function [21]:
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,
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where A is the maximum reflectivity of the FBG and σ is the grating full-width at half maximum (FWHM). When the intensity of the broadband source transmitted into the FBG sensor is I i (λ), the total light power P d detected on the photodiode (e.g. PD1 in Fig. 3) can be expressed by:
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4. Experimental results and discussions
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beam and the all-fiber FBG displacement sensing system is illustrated in Fig. 3. The PC beam consists of a finite repetition of 16 alternating aluminum/acrylic segments in which the height of the middle acrylic is triple that of the rest of the segments. Each of the aluminum and acrylic segments are bonded together by epoxy glues. The material properties and geometric size of the PC beam have been described in Section 2. The PC beam is excited by a piezoelectric multilayered actuator (AE0505D16, Thorlabs, Newton, NJ, USA) on the right-hand end of the PC beam, whose elongation is approximately proportional to the driving voltage signal [23]. Two point-wise FBG displacement sensors are glued to the two ends of the beam by a mix of epoxy resin and hardener (i.e., FBG1 and FBG2 as shown in Fig. 3) as input and output displacement sensors, respectively. Other than the excitation point and the two sensing points on the beam, only two points on the PC beam are rested on soft rubbers (as shown in the inset photo in Fig. 3). In the FBG sensing system, a C-band amplified spontaneous emission (ASE) light source (China-Fiber Optics, Shanghai, China) is transmitted to a 1 × 2 coupler and then split into two paths for the two FBG sensors and corresponding demodulation systems. Each demodulation system consists of two 3-port directional optical circulators (OC’s) to guide the light beam. Photodiodes (PDA10CS, InGaAs amplified detector, Thorlabs, Newton, NJ, USA) are employed to convert the light to electrical signals and are connected to a dSPACE system (dSPACE GmbH, Paderborn, Germany) for the transmittance FRF measurement or an oscilloscope (LeCroy, HD04054, USA) for calculation of the group velocity. The Bragg wavelengths of the FBG sensors and FBG filters shown in Fig. 3 are FBG1 = 1550.356 nm, FBG filter1 = 1550.516 nm, FBG2 = 1550.492 nm, and FBG filter2 = 1550.464 nm, respectively. The grating lengths of the FBG sensors and FBG filters are both 10 mm. Details of setting up the point-wise FBG displacement sensor, for dynamic measurement, can be found in [20] proposed by the first author.
4.1. Experimental setup
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4.2. Experimental results First, the vibration transmittance FRF of the finite PC beam is measured to investigate the band gap phenomena predicted by the infinite PC beam [22]. In this letter, the FRF is defined as T d = 20 log10 (dout /din ) in which dout and din are respectively the displacements of the transmitted and incident waves, at the two ends of the PC beam. The experimental setup and photos of the PC
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The FRF in this study is obtained by a stochastic spectral estimation commonly used in the control system identification. White noise random signals, generated by the Simulink (The MathWorks, Natick, MA) and the dSPACE DS1104 system, with a sampling frequency 50 kHz, are sent to the piezoelectric multilayered actuator
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Fig. 4(c), indicating the existence of the defect modes in the PC beam. Compared to Fig. 2(b), in which band gaps are calculated by the TMM as 561–935 Hz and 2523–3412 Hz with a defect band from 665–702 Hz for the infinite PC beam, the displacement transmittance FRF of the finite PC beam is related to the band structures predicted in the infinite beam. However, there are some discrepancies mainly caused by the finite or infinite geometries [26]. By plotting the steady-state displacement mode shapes at the point crystal defect, we can see that the defect modes are localized vibration modes with the first and the third being bending modes and the second, rotational vibration mode. After verifying the transmission properties and the defect states of the PC beam, a tone burst signal is generated by the dSPACE system to excite transient wave propagations for determinations of the group velocity of flexural waves. A N-count Hanning windowed tone burst excitation can be expressed by:
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V in (t ) = H (t ) − H (t − N p / f c )
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Fig. 4. The FRF between signals from (a) FBG1 and the random excitations, (b) FBG2 and FBG1 at the two ends of the perfect PC beam, and (c) that with the defect introduced in the middle of the PC beam.
through a power amplifier (XE501-C, XMT, China) to excite flexural wave propagations in the PC beam. The FRF is obtained from the relationship T d (ω) = 20 log10 | S yu ( j ω)/ S uu ( j ω)| [25], where S uu ( j ω) is the auto-spectral density function of the input random displacement (detected by the FBG1) and S yu ( j ω) is the crossspectral density function between the input and transmitted displacements, respectively (i.e., responses of the FBG1 and FBG2). Dynamic sensing performance of the proposed point-wise FBG displacement sensor has been confirmed with a LDV system in a study regarding precision tracking control performed by the first author [24]. However, since the dynamics of the FBG displacement sensing system might be slightly influenced by various designs of the fixed-stage supporting the FBG displacement sensors, it is necessary to validate the FRF between the FBG1 input displacement sensor and the random excitations before band gap measurements. The transmittance FRF between the random excitations and the FBG1 input sensor, between FBG1 and FBG2 for a perfect PC beam, and that for the imperfect PC beam are respectively shown in Figs. 4(a), 4(b), and 4(c). In Fig. 4(a), we can see a rather flat spectrum in frequency ranges corresponding to three band gaps of the PC beam, indicating good dynamic measurement capability of the proposed FBG displacement sensing system. It is noted that the band gaps are defined as frequency ranges exhibiting low transmission below 0 dB and are marked in Fig. 4 in grey shaded regions. Despite the high frequency mismatch, above the upper edge of the third band gap, an obvious resonant frequency at 235 Hz and an anti-resonant frequency at 218 Hz can be observed. The local resonance and anti-resonance might be explained by the loading effects from the contact-type FBG sensor with the dynamics of its supporting fixed-stage. Since the focus of this study is about the defect modes inside the band gaps whose frequencies of interest are not overlapped with the local resonance and antiresonance of the sensing system, it is demonstrated in Fig. 4 that the proposed FBG displacement sensing system is suitable to investigate flexural wave propagations in the PC beam around the band gaps. In Figs. 4(b) and 4(c), experimental results coincide well with the finite element method (FEM) simulations performed by the COMSOL Multiphysics® software. In FEM simulations, the damping, mainly due to viscous epoxy between each segment, is modeled as a loss factor with a value of 0.01. As can be seen in Figs. 4(b) and 4(c), there are three deep band gaps within 0 to 8000 Hz (i.e., 458–750 Hz, 2062–3563 Hz, and 5212–6980 Hz). The three transmission peaks inside the band gaps are marked in
1 − cos
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sin 2π f c t , (17)
where N p is the peak number, f c is the central frequency of interest, and H (t ) is the Heaviside step function. In most experimental studies regarding calculations of group velocity in dispersive media, a 5-count Hanning windowed tone burst excitation signal (i.e., N p = 5) is usually chosen (e.g., such as studies of lamb waves in composites [27–29]) due to its excellent time and moderate frequency resolution. Fig. 5(a) shows a 5-count Hanning windowed tone burst excitation signal at 568 Hz (i.e., the first defect mode in this study) and the transmitted wave packets predicted by FEM at the tip of the PC beam. It should be noted that, other than the damping effect, the transmitted wave is attenuated due to the fact that the excitations realized by the piezoelectric multilayer actuator try to stop the vibration of the PC beam after a complete generation of a tone burst. We can see that the group velocity is difficult to be extracted simply by reading the time of flight of the 5-count-tone-burst excited wave packets. To have a better calculation of the time of flight for the wave packets, without resorting to complicated techniques such as Gabor wavelet transform, in this letter we choose a 21-count Hanning windowed tone burst to excite the beam. The frequency spectra of the 5-count and 21-count smoothed tone burst signals are shown in Fig. 5(b). It can be seen that the 21-count tone burst has a higher frequency resolution and stronger intensity. The increasing of the frequency resolution, however, sacrifices the time resolution due to the wellknown Heisenberg’s Uncertainty Principle [27]. Fig. 5(c) shows the FBG measurement results and the FEM simulations of the transmitted waves from excitations at the frequency of the defect mode (i.e., 568 Hz). Since the reflected wave packets, from the left-hand free end of the beam, are unavoidably superimposed coherently to the incoming wave packets, the number of the peaks in the transmitted wave packet is more than that of the excitation tone burst. The time of flight of the maximum peak, in the wave packet in the case of the 21-count smoothed tone burst, can now be more easily determined by using the cursor functions of the oscilloscope. The group velocities can then be obtained by dividing the length of the PC beam (i.e., approximately the distance between the two FBG displacement sensors) by the time of flight. From Fig. 5(c) we can also see that the proposed FBG displacement sensing system is capable of detecting transient wave propagations with the discrepancy due to difficulties of perfect estimation of the damping. Fig. 6(a) shows the group velocities predicted by the band structures in Fig. 2(b) obtained by the TMM by calculating C g = dω/dq ∼ = |(ω2 − ω1 )/(q2 − q1 )| between the adjacent frequencies around the first band gap. Group velocities evaluated from the TMM indicate that the defect mode possesses the slowest group velocity compared to other dispersion curves. Fig. 6(b) displays the transmittance FRF in the first band gap obtained numerically and
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Fig. 5. Excitations of the PC beam by the Hanning windowed tone bursts: (a) 5-count tone burst and simulation response, (b) frequency spectra of the 5-count and 21-count tone burst, and (c) the experimental and simulation results for the excitations at the frequency of the defect mode (i.e., 568 Hz).
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Fig. 6. (a) Group velocity calculated by the band structure obtained by the TMM; (b) comparison of the experimental result and the numerical prediction for the FRF; (c) group velocities in the band gap containing the defect mode.
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Fig. 7. Transient wave propagation at three time points corresponding to adjacent peak–valley–peak in the tone burst-induced responses: (a) inside the band gap at 480 Hz; (b) the defect mode at 568 Hz. (Note: In the FEM simulations, the left-hand end is the excitation point.)
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experimentally. The calculated group velocities from the FEM and the FBG measurements are shown in Fig. 6(c) where the group velocities are collected from excitations at 35 different frequencies in the frequency range of interest. The slow group velocity around the defect mode is clearly validated. Other flexural waves excited at frequencies corresponding to resonant peaks also have slow group velocity due to the effect of standing waves. However, it should be noted that the final zero excitation exerted by the 21-count Hanning windowed tone burst prevents the occurrence of “perfect” standing waves. It is also interesting to note that the group velocities inside the band gap away from the defect mode reach to higher values than those in other transmission regions. This result is similar to the conclusion for sound wave propagations via defect coupling in an acoustic array performed by Robertson et al. in [16]. In order to have a clearer physical understanding of the slow wave propagations in the PC beam with the point crystal defect, two transient displacement fields of the PC beam, excited at 480 Hz (i.e., an arbitrary chosen frequency inside the band gap)
and 568 Hz (i.e., the defect mode), are simulated by the FEM and the results are shown in Fig. 7. In FEM simulation shown in Fig. 7, the PC beam model is excited from the left-hand side and the beam is traction-free other than the excitation point. The transient wave propagations excited at the frequencies of interest are plotted at three different time points corresponding to those where the transmitted wave packet reaches adjacent peak–valley–peak. We can see that, unlike the evanescent wave propagations in the band gap, the localized defect mode causes a almost standing wave propagation in the PC beam, resulting in corresponding slow group velocity at the defect mode, a phenomenon that can also be seen in photonic crystals [30,31].
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5. Conclusions
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In summary, the transmission properties of a PC Timoshenko beam, with or without a point defect, are investigated using an FBG displacement sensing system. The slow group velocity of flex-
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ural waves via defect coupling is experimentally demonstrated. Based on the TMM with the supercell technique, the band structures of the PC beam with and without the point defect are analyzed. Good agreements can be obtained between the experimental measurements, the theoretical analysis, and the numerical simulations for the defect modes and the associated slow group velocity inside the band gaps. In addition, to the best of the authors’ knowledge, this is the first time the point-wise FBG displacement sensors are employed to obtain the displacement transmittance FRF and to measure the wave packet propagations in the PC beam. Compared to the commonly employed LDVs or accelerometers, the proposed FBG displacement sensing system opens a new experimental tool for research of phononic crystals or elastic metamaterials to achieve high-sensitivity and low-cost direct dynamic displacement sensing.
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Acknowledgements
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The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 11272282). K.C. Chuang appreciates the encouragement from Professor C.T. Sun during his stay as a visiting scholar at the Purdue University.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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References [1] M.F. Su, R.H. Olsson III, Z.C. Leseman, I. El-Kady, Appl. Phys. Lett. 96 (2010) 53111. [2] M.M. Sigalas, E.N. Economou, J. Sound Vib. 158 (1992) 377. [3] Z. Liu, X. Zhang, Y. Mao, Y.Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Science 289 (2000) 1734.
[25] [26] [27] [28] [29] [30] [31]
7
M. Oudich, M.B. Assouar, Z. Hou, Appl. Phys. Lett. 97 (2010) 193503. D. Yu, Y. Liu, G. Wang, H. Zhao, J. Qiu, J. Appl. Phys. 100 (2006) 124901. M.M. Sigalas, J. Acoust. Soc. Am. 101 (1997) 1256. F. Wu, Z. Liu, Y. Liu, Phys. Rev. E 69 (2004) 66609. S. Benchabane, A. Khelif, A. Choujaa, B. Djafari-Rouhani, V. Laude, Europhys. Lett. 71 (2005) 570. J. Sun, T. Wu, Phys. Rev. B 71 (2005) 174303. S. Yang, J.H. Page, Z. Liu, M.L. Cowan, C.T. Chan, P. Sheng, Phys. Rev. Lett. 93 (2004) 24301. Z. Yao, G. Yu, Y. Wang, Z. Shi, Int. J. Solids Struct. 46 (2009) 2571–2576. G.J. Pearce, T.D. Hedley, D.M. Bird, Phys. Rev. B 71 (2005) 195108. J. Sun, T. Wu, Phys. Rev. B 76 (2007) 104304. Y. Pennec, B. Djafari-Rouhani, H. Larabi, J.O. Vasseur, A.C. Hladky-Hennion, Phys. Rev. B 78 (2008) 104105. C.G. Hou, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, J. Deng, S.J. Wang, Appl. Phys. Lett. 105 (2014) 243104. W.M. Robertson, C. Baker, C.B. Bennett, Am. J. Phys. 72 (2004) 255. H. Tian, Y. Ji, Mod. Phys. Lett. B 23 (2009) 1053. R. Zhu, X.N. Liu, G.K. Hu, C.T. Sun, G.L. Huang, J. Sound Vib. 333 (2014) 2759. Y. Liu, D. Yu, L. Li, H. Zhao, J. Wen, X. Wen, Phys. Lett. A 362 (2007) 344. K.C. Chuang, C.C. Ma, Appl. Opt. 47 (2008) 3561. A.S. Paterno, V. de Oliveira, T.S. Figueredo, H.J. Kalinowski, IEEE Sens. J. 6 (2006) 1662. Y. Xiao, J. Wen, G. Wang, X. Wen, J. Vib. Acoust. 135 (2013) 41006. J. Li, X. Zhou, H. Zhao, M. Shao, P. Hou, X. Xu, Smart Mater. Struct. 24 (2015) 65007. K.C. Chuang, C.C. Ma, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 56 (2009) 2036. B.J. Huang, Y.C. Chen, J. Dyn. Syst. Meas. Control 124 (2002) 176. E. Baravelli, M. Ruzzene, J. Sound Vib. 332 (2013) 6562. L. Wang, F.G. Yuan, Compos. Sci. Technol. 67 (2007) 1370. M. Yang, P. Qiao, Smart Mater. Struct. 14 (2005) 1083. V. Giurgiutiu, J. Bao, W. Zhao, Exp. Mech. 43 (2003) 428. R. Ozaki, Y. Matsuhisa, M. Ozaki, K. Yoshino, Appl. Phys. Lett. 84 (2004) 1844. S. Furumi, H. Fudouzi, H.T. Miyazaki, Y. Sakka, Adv. Mater. 19 (2007) 2067.
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