Accepted Manuscript Experimental and Mathematical Analysis of a Piezoelectrically Actuated Multilayered Imperfect Microbeam Subjected to Applied Electric Potential Meisam Moory-Shirbani, Hamid M. Sedighi, Hassen M. Ouakad, Fehmi Najar PII: DOI: Reference: S0263-8223(17)32296-1 https://doi.org/10.1016/j.compstruct.2017.10.062 COST 9037 To appear in: Composite Structures Received Date: Revised Date: Accepted Date: 24 July 2017 13 October 2017 21 October 2017 Please cite this article as: Moory-Shirbani, M., Sedighi, H.M., Ouakad, H.M., Najar, F., Experimental and Mathematical Analysis of a Piezoelectrically Actuated Multilayered Imperfect Microbeam Subjected to Applied Electric Potential, Composite Structures (2017), doi: https://doi.org/10.1016/j.compstruct.2017.10.062 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Experimental and Mathematical Analysis of a Piezoelectrically Actuated Multilayered Imperfect Microbeam Subjected to Applied Electric Potential Meisam Moory-Shirbani1, Hamid M. Sedighi1*, Hassen M. Ouakad2*, Fehmi Najar3 1 Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz 61357-43337, Iran. 2 Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Kingdom of Saudi Arabia. 3 Applied Mechanics and Systems Research Laboratory Tunisia Polytechnic School University of Carthage B.P. 743 - 2078, La Marsa, TUNISIA, Email: fehmi.najar@ept.rnu.tn. * Corresponding author email: h.msedighi@scu.ac.ir. * Corresponding author email: houakad@kfupm.edu.sa. Abstract. In this paper we propose to fabricate a microcantilever beam actuator using thin layer PZT material. The multilayered microbeam is initially curved due to fabrication process. The device is also modeled using a the Euler-Bernoulli beam theory and takes into account the multilayer structure of the device. The inclusion of these imperfection into the theoretical model generates geometric nonlinearities. Static and dynamic experimental analysis is conducted to examine the physiomechanical behavior of the considered microstructure. To this end, The model is first compared and validated with experimental findings for an applied constant voltage. Very good agreement depicts the accuracy of the proposed mathematical model. When time varying voltage is applied, the nonlinear differential equation is solved using a perturbation technique. Again good agreement with experiments are found. Finally, a parametric study is carried out to explore the influences of different parameters on the frequency response of the piezoelectrically actuated imperfect microbeams. 1. Introduction Piezoelectric actuators are widely used as sensors, actuators and recently implemented in vibration energy harvesting devices. Thanks to their transduction mechanism very simple designs are needed to actuate these devices. Their implementation at the microlevel is easily obtained, such as for MEMS technology. Since 1996, Ishihara et al. reviewed several piezoelectric microdevices used essentially as actuators [1]. For example Higuchi et al. [2] introduced a driving micromechanism using PZT layers for microrobot arms. They reached 2mm/s as maximum velocity. Also, Fukuda et al. [3] utilized the same design to actuate micromobile robot in water. A more recent review by Tadigadapa and Mateti [4] on piezoelectric MEMS sensors reported several successful commercialized microdevices such as acoustic resonators, in inertial sensors, high-frequency oscillators and filters, microactuators for RF applications, chip-scale chemical analysis systems, etc. Several researchers developed analytical models to analyze the influence of geometrical and physical parameters on the general performance of mechanical systems [5-19]. DeVoe and Pisano [20] developed a model describing the static response of ZnO piezoelectric cantilever beam. The experimental validations of the actuator demonstrated the utility to optimize the layers position and thicknesses. A more detailed 1 model of multilayered beams is presented by Wolf et al. [21], in this model geometric nonlinearities due to relatively large displacements are also taken into account. Mahmoodi et al. [22] also take into account geometric nonlinearities for the equation of motion of a cantilever beam with a piezoelectric PZT patch. A clear softening behavior was observed for large displacements. The results were confirmed by experiments. Several devices were proposed for microscale applications where layered structures are preferred because of microfabrication restrictions. Piazza et al. [23] designed a microresonator by using a micromachined clamped-clamped beam which includes a ZnO layer used for actuations. The design, tested experimentally, showed very high mechanical quality factors with relatively low actuation voltages. Gerfers et al. [24] modeled and fabricated a AlN piezoelectric microaccelerometers and reported high signal-to-noise ratio and low dielectric loss, in addition to a good CMOS process compatibility. Thomas et al. [25] reported a high level of parametric amplification for microresonator using PZT patches deposited on a silicon clamped-clamped microbeam. The parametric amplification was observed up to a factor of 14 in terms of the quality factor. Hofmann and Twiefel [26] designed a self-sensing bending microactuator using a segmented electrodes. The proposed design was tested experimentally. Wang et al. [27] presented a AlN-based inplane resonant microaccelerometer. Experiments showed a sensitivity as high as 28.4 Hz/g which is 57% higher than previously reported data. Sarafraz and Seyed Roknizadeh [28] studied the shape influence of a bimorph piezoelectric beam using COMSOL Multiphysics software. Their results showed that due to the mechanical properties of the beams, the natural frequency of the triangular beam is higher than the others for all considered parameters. Cheraghbak and Loghman [29] investigated the magnetic field effects on the elastic response of polymeric piezoelectric cylinder reinforced with carbon nanotubes. Theconsidered cylinder was subjected to internal pressure, a constant electric potential difference at the inner and outer surfaces, thermal and magnetic fields. They indicated that considering magnetic field can reduce the stresses of nano-composite cylinders. The above mentioned devices are generally based on a multilayered structure having one or more active layers. Their geometric configurations and electrical connections are essential to ensure optimal working conditions. Zhu et al. [30] studied parallel and series connections for bimorph sensor with two piezoelectric layers. They found that adding a third layer reduces the performance of the device. Also within this context Jemai et al. [31] analyzed a multilayered cantilever beam with different number of PZT piezoelectric layers and connection configurations for energy harvesting applications, they deduced that an optimal number of layer should be used to increase the harvested energy. Using microfabrication techniques, interdigitated electrodes can be deposited on a piezoelectric layers, activating both the d31 and d33 modes. Using these electrodes, Michael and Kwok [32] designed a micro- lens actuator. For energy harvesting applications, Muralt et al. [33] demonstrated that the use of such electrodes can achieve high voltage generation. Using also similar electrode design Jemai et al. demonstrated that very good electromechanical coupling coefficient can be obtained if the geometrical parameter of the interdigitated electrode design are correctly tuned [34, 35]. Modeling and design of these devices needs accurate material properties to be correctly identified especially for thin film configuration. Since 1969, Holland and EerNisse [36] used a resonanceantiresonance method to determine piezoelectric coefficients. They reported several measurement problems such as nonlinear behavior, material inhomogeneity or frequency dependence coefficient. They proposed the measurement of the complex admittance at resonance and antiresonance to preserve phase information and eliminate most errors. Lately, Liu et al. reviewed several measurement techniques for 2 piezoelectric thin films [37]. Direct and indirect methods were reported for 1.0 to 100.0 µ m film thickness. They concluded that high resolutions in terms of displacement and charge are required to achieve acceptable measurements. Lefki and Dormans [38] measured the piezoelectric coefficients of PZT thin film using the normal load method with various compositions prepared by the sol-gel technique or by organometallic chemical vapor deposition. Ren et al. [39] used also the normal load method to measure piezoelectric coefficients of PZT thin films, but they proposed a composite tip to reduce the damage caused by the metallic tip. Using a different measurement technique, Kholkin et al. [40] deduced that a minimum of two microbeams should be used to obtain high-resolution interferometric measurements of displacements in piezoelectric thin films. Using quasi-static measurement, Dubois and Muralt [41] measured the effective piezoelectric transverse coefficient e31 for PZT and AlN using piezoelectric actuator to bend a cantilever microbeam with the piezoelectric layer to be characterized. Wooldridge et al. [42] used a vertical comb drive MEMS actuator to measure direct piezoelectric effect in PZT thin/thick films. [43] measured the PZT effective transverse piezoelectric coefficient on anisotropic substrate using a cantilever microbeam. Guo et al. used a mini impact hammer to measure d33 coefficient of a PZT thin film [44]. A noncontact technique using interferometry is used by Malakooti and Sodano [45] to determine the the d33 and d31 piezoelectric strain coefficients. Fialka and Beneš [46] compared several techniques for measurement of piezoeletric coefficients. They used the frequency method, the laser interferometer technique, and the quasi-static method. They concluded that the use of the frequency method is advantageous when we need to establish individual coefficients or the complete matrix of the piezoelectric material coefficients. Tsujiura et al. [47] compared direct and converse piezoelectric effects for the measurement of e31,f coefficient for PZT thin film using a unimorph cantilever microbeam. Polycrystalline and epitaxial PZT thin films were used. They deduced that the polycrystalline PZT thin films show a clear nonlinear piezoelectric contribution, however, the epitaxial PZT thin films depicted linear response. Multilayered microbeams are generally subject to several additional difficulties in terms of modeling because of their composite composition. In fact, multilayered microbeams are not symmetric in the transverse direction, which deviate its neutral axis to non-symmetric positions [18]. In addition, microfabrication techniques induce different residual stresses in the layers of the microbeam. This can lead to an inevitable initial curling or deflection [48]: these cases are generally referred to as imperfect microbeam microstructures. Initial deflection has been modeled by an initial axial force for piezoelectric microbeam by Weinberg [49]. He deduced that using a linear model one can deduce the stiffness variation due to this axial force. More recently, imperfect microbeams have been modeled using a bending deflection defined at the initial unforced state of the structure [50]. This approach is simple to implement and particularly valid for small curling of the beam. It has been also implemented for more complex composite materials such as functionally graded microbeams [51]. Barati and Zenkour [52] proposed a size-dependent nonlinear higher order refined beam model for post-buckling analysis of imperfect multiphase nanocrystalline nanobeams based on the modified couple stress theory. They employed a micromechanical model based on Mori-Tanaka scheme to include the size of nanograins/nanopores. In another research [53], they analyzed the post-buckling behavior of porous metal-foam nanobeams based on a refined nonlocal shear deformation theory by considering an initial imperfection. They found that the 3 porosities have a significant effect on the post-buckling configuration of imperfect nanobeams. For large initial bending deflections, curved beam and plate theories should be used [54]. In this paper we propose to fabricate an imperfect MEMS cantilever microbeam actuator using a PZT layer. The device is modeled using a the Euler-Bernoulli beam theory and takes into account the multilayer structure of the device. Due to fabrication residual stresses the microbeam is initially curved. The inclusion of these imperfection into the theoretical model generates geometric nonlinearities. Static and dynamic analyses from emprical and mathematical simulations well demonstrate the integrity of proposed mathematical modeling to capture the mechanical behavior of multileyered microbeams with a piezoelectric layer and initial imperfection. 2. Fabrication The cantilever beams shown in Fig. 1 were all fabricated using a multi-mask fabrication process. The bottom electrode has been spotted before growing the PZT. The beams have been fabricated in a 3 µm silicon-on-insulator (SOI) wafer with 0.5 µm buried oxide (BOX) layer. During the first step, a 670 nm silicon-oxide layer was grown. Then, a 10/100 nm Ti/Pt has been sputtered and patterned using the first mask. A (100)-dominant thin film (250 nm) PZT has been grown using PLD on LaNiO3 as a seed layer. All the films were cured at 600 C with an oxygen pressure of 0.1 mbar. On top of PZT, a 100 nm thick Pt has been sputtered. Using the second mask, the top Pt layer has been patterned, followed by sputtering a PZT using a wet etchant (the third mask). Around each fabricated beams, an area has been opened by reactive ion etching (RIE) of the SiO2/Si/SiO2 layer stack (fourth mask). Finally, using the fifth mask, the devices were released by isotropic etching of silicon, while the silicon device layer was protected by photoresist. As seen in Fig. 1, PZT step-coverage has isolated the top and bottom Pt layers to prevent any shortcut. The crystalline structure of the PZT thin-films was measured using a Philips XPert X-ray diffractometer (XRD). A typical XRD pattern of the optimized PZT thin-films grown on 4-inch Pt/Ti/SiO2/Si wafers, using large-scale PLD, is given in Fig. 2. The films were prepared at 600 ◦C with an oxygen pressure of 0.1mbar. The θ-2θ scan clearly indicates the growth of PZT thin-films with (100)preferred orientation and no pyrochlore phase is observed. On top of PZT, 100nm thick Pt has been sputtered. Using the second mask, the top Pt layer has been patterned, followed by patterning PZT using a wet etchant (the third mask). Around the devices, an area has been opened by reactive ion etching (RIE) of the SiO2/Si/SiO2 layer stack (fourth mask). Finally, using the fifth mask, the devices were released by isotropic etching of silicon, while the silicon device layer was protected by photoresist. In this fabrication process, the bottom Pt layer was etched under the top Pt contact-pads to minimize the parasitic capacitances. As seen in Fig. 3a, PZT step-coverage has isolated the top and bottom Pt layers to prevent the shortcut. 3. Mathematical Modeling and formulation 3.1 Geometry of the Multilayered Cantilever Microbeam A schematic representation of a cantilevered multilayer microbeam with piezoelectric layer is shown in Figure 3. The microbeam has length L and width w and is composed of a Silicon layer covered with two SiO2 supporting layers and a PZT piezoelectric layer covered with two platinum (Pt) layers on its top 4 and bottom surfaces. The piezoelectric layer is bonded throughout the microbeam length. The piezoelectric layer can be simultaneously actuated by a combination of DC and AC voltages. Stress gradient across the thickness in thin microstructures such as, such as cantilever beams, can cause an initial deformation in their geometry and hence may alter their functionality and may also lead to their failure through deforming them either upward or even downward and consequently sticking them to the substrate. In fact, such stress gradient across the thickness of a thin cantilever composite microbeam, initially designed to be straight (Fig. 4a), can be induced due to a thermal mismatch between the beam material and the substrate or among the various layers of the beam itself, in the case of a composite beam, during material deposition. This causes an initial curvature in cantilever microbeams also called curling [55] in cantilever beams, Fig. 4b. Therefore, a curled microbeam is a stress free structure that is brought to its deformed shape by a stress gradient across its thickness. 3.2 Basic concepts and governing equations According to the assumption of Euler-Bernoulli beam theory, the displacement field of a deflected beam can be expressed as: ∂w( x, t ) −z u x ∂x u = u y = 0 (1) u w( x, t ) + w ( x) 0 z in which, w (x ,t ) and w 0 ( x ) denote the displacement and initial imperfection in transversal direction associated with zero initial stress, respectively. It is assumed that x is the coordinate along the microbeam length and y , z are the distance of each point on the cross-section from the neutral axis. Referring to equation (1), the strain field of microbeam are written as [56]: ∂ 2 (w ) ε x = −z = −zw ,xx (2) ∂x 2 ε y = ε z = ε yz = ε xz = ε xy = 0 The stress-strain relation for the elastic layers are given by: σ x ,i = C xx ,i ε x ,i i = 1, 2,3, 4, 6 (3) where s x , e x and C xx represent the stress, strain and Young’s modulus in the axial direction (i.e., xdirection), respectively and i denotes the number of layer. The constitutive equations for the piezoelectric layer (fifth layer) incorporating the couple effects of electro-mechanical behaviors is represent as [57]: s x ,5 = C xx ,5e x ,5 - e zx E z (4) (5) D z ,5 = e zx ,5e x ,5 + h zz E z where Dz , eij and hij are the electric displacement in the transverse direction, piezoelectric and dielectric coefficient of the PZT piezoelectric layer. The material properties and geometrical dimensions of different layers is presented in table 1. Table1. Material properties and geometric dimensions of the multilayered piezoelectrically actuated cantilever microbeam 5 Assuming the uniform electric field E z throughout the constant PZT piezoelectric layer thickness h5 , then the electric field parallel to the poling direction E z , in terms of the applied DC voltage V DC across the thickness of the PZT piezoelectric layer, can be expressed as [57]: Ez = - VDC h5 (6) The location of neutral axis from the lower edge of the multilayered beam bender is denoted by z 0 which can be determined using the static equilibrium conditions for the resultant axial forces (in the presence of an acting external bending moment M) as [58]: 6 åF x ,i =0 (7) i =1 By considering equations (2) to (4), the following relation for the axial force in the ith layer can be obtained: zi Fx ,i = −b ∫C xx ,i wxx zdz (8) zi −1 and thereby, using equations (7) and (8) yields: 6 zi ∑∫C xx ,i zdz = 0 (9) i =1 z i −1 in which the lower and upper integration limits of integral for ith layer, i.e. z i −1 , z i , can be determined as follows: i −1 zi = z 0 − ∑h (10) j j =1 i z i −1 = z 0 − ∑h (11) j j =1 where h j is the height of ith layer. After some mathematical computations, equation (9) is re-written as follows: 6 6 ∑ i −1 6 C xx ,i [ zi 2 − zi −12 ] = 0 ⇔ ∑ i =1 C xx , i [( z 0 − i =1 ∑ i h j ) 2 − ( z0 − ∑h ) ]= 0 j =1 2 j (12) j =1 and thereby, the location of neutral axis from the bottom surface of multilayered microbeam can be obtained as: 6 2 ∑ i =1 z0 = 6 i C xx ,i hi ∑ hj − ∑C j =1 6 2 h2 xx ,i i i =1 ∑C (13) h xx , i i i =1 In order to derive the governing equation of motion, Hamilton’s principle is utilized. To this end, it is necessary to compute the strain and kinetic energies of each layer as well as the works done by external forces. The elastic strain energy of the multilayered microbeam Ps are given by [59]: zi i =6 ∑∫ ∫ 1 Πs = 2 i =1 Ai zi −1 1 σ x ,i ε x , i dzdA − 2A z5 ∫∫D 5 z ,5 E z ,5 dzdA5 (14) z4 Substituting equations (2) to (14) into (14) yields: zi i =6 ∑ ∫ ∫ zσ 1 Πs = 2 x ,i wxx dzdAi − Π E = − i =1 Ai zi −1 1 M x wxx dAi − Π E 2A ∫ (15) i in which the electrostatic energy of the PZT layer P E and bending moment M x are expressed by the following relations: z5 1 ΠE = − 2A ∫∫D e z ,5 zx ,5 5 z4 VDC dzdA5 h5 i = 6 zi Mx = ∑∫σ (16) i =6 zi x ,i zdz = − i =1 zi −1 ∑∫C z5 ∫ 2 xx ,i i =1 zi −1 wxx z dz + ezx ,5 z4 VDC zdz ⇔ M x = − Dwxx + M piezo h5 (17) in the above equations, the total flexural rigidity D and the piezoelectric bending moment of the multilayered system M piezo due to the applied voltage on the piezoelectric layer are calculated by: D= b 3 6 ∑C xx ,i ( zi 3 − zi −13 ) (18) i =1 e zxV DC b 2 ( z5 − z 4 2 ) 2 h5 M piezo = (19) The kinetic energy of the system Pk is also determined as: i =6 Πk = zi ∑∫ ∫ i =1 Ai zi −1 L ρi ( ∂w 2 ∂w ) dzdAi = m( ) 2 dx ∂t ∂t 0 ∫ (20) where m denotes the mass per unit length of microbeam which is given by: i =6 m= ∑ bρ h (21) i i i =1 The work done by the axial force N x can be obtained using the following relation [59]: L 1 ΠF = (N x )wx2 dx 20 ∫ (22) 7 in which N x is computed by: z5 z5 z4 z4 Nx = ∫ σ x,5bdz = ∫ (−Cxx,5w, xx z − ezx Ez )bdz = Nelectrical + Nunsymmetrical (23) where, N electrical and N unsymmetrical represent the axial forces due to the applied electrostatic voltage and asymmetric configuration of multilayered microbeam. By submitting equation (4) into equation (23), the exact expressions of the N electrical and N unsymmetrical are obtained as follows [59]: Nelectrical = be zx ,5 (V DC +V A C ) N unsymmetrical = − (24-a) 1 C xx ,5 b ( z5 2 − z 4 2 ) w, xx = K C w, xx 2 (24-b) where K C in equation (24-b) indicates the coupling stiffness caused by the asymmetric arrangement of piezoelectric layer. According to the Hamilton’s principle we have: t ∫ (δΠ k + δΠ F − δΠ s )dt = 0 (25) 0 Referring to equations (15), (20) and (22) and substituting into equation (25), one can obtain the governing equation and boundary conditions for transversal vibration of cantilevered microbeam as follows: ∂ 2M x ∂ 2w ∂2w + ( N electrical + Nunsymmetrical ) 2 + m 2 = 0 2 ∂x ∂x ∂t ∂M x ∂w + ( N electrical + N unsymmetrical ) x ) = 0 w = 0 or ( ∂x ∂x (26) − wx = 0 or (27) Mx = 0 (28) Using the relation described in equation (17), the governing differential equation in terms of w ( x ,t ) can be extracted: Dw xxxx + (N electrical + N unsymmetrical )w xx + mw = 0 (29) 3.3 Static analysis due to DC electric potential In order to examine the static behavior of the multilayered microbeam subject to DC electric potential load, the temporal terms in the governing equation (29) are omitted and then the linear static differential equation can be expressed as: Dws, xxxx + Nx ws, xx = 0 (30) It is assumed that the analytical solution of equation (30) can be described as: (31) ws ( x) = AeSx By substituting the static response (31) into (30), one can obtain the following auxiliary equation: N (32) S 4 + Ex S 2 = 0 D with roots: S1,2 = ± - N Ex , S 3,4 = 0 D (33) and finally, the general static solution can be expressed as: ws ( x) = A1eS1x + A2e- S1x + A3 x + A4 8 (34) where the different constants A1 to A 4 are determined using the boundary conditions which for the considered cantilevered microbeam are given by: w s (0) = 0 (35) (36) w s ,x (0) = 0 (37) M x (L) = 0 [ (38) ∂M x ∂w x + N Ex ] =0 ∂x ∂x x = L By considering the above boundary conditions in the solution of equation (34), one can obtained the coefficients as: M piezo (e- S1L - 1) (39) A1 = DS12 (2 - e- S1L - eS1L ) (40) M piezo (e S1L - 1) A2 = DS12 (2 - e - S1 L - e S1L ) (41) M piezo (eS1L - e- S1L ) A3 = DS1 (2 - e- S1L - eS1L ) (42) M piezo M piezo A4 = DS12 = N Ex 3.3 Dynamic analysis due to AC electric potential Owing to the initial geometric imperfection of microbeam in the z direction, the dynamic governing equation accounting for the initial rise w 0 ( x ) and subjected to the harmonic electric potential V (t ) =V AC sin(wt ) is re-written as follows: d + Dwd , xxxx + ( Nelectrical + KC wd , xx )[wd , xx + w0, xx ] = 0 Þ mw d + Dwd , xxxx + KC [wd , xx2 + wd , xx w0, xx ] + bezx,5VAC cos(w Et )[wd , xx + w0, xx ] = 0 mw with the corresponding dynamic boundary conditions at x=0, L as: w d (0,t ) = 0 w d , x (0,t ) = 0 (43) (44) (45) - Dwd , xx ( L, t) + M Piezo = 0 (46) Dw d ,xx x (L,t ) + N electricalw d ,xx (L,t ) = 0 (47) while M piezo = ezxVAC b 2 ( z5 - z4 2 ) sin(w t ) 2 h5 (48) Nelectrical (t) = bezx,5VAC cos(wEt) (49) In order to facilitate the analysis and obtain a generalized parametric solution, the following nondimensional parameters are defined as: t* = w w D x t , x* = , w* = d , w0* = 0 4 L L L mL (50) Substituting the dimensionless parameters into equation (43), the nondimensional nonlinear equation of motion becomes: 9 d* + wd*. xxxx + N * cos(w E*t * )[ w*d , xx + w0,* xx ] + l 2 [(w*d , xx )2 + w*d , xx .w0,* xx ] = 0 w (51) where new terms are defined as: bezx ,5VAC L2 KC L2 mL4 * (52) ,l = , w E* = w E , N = D D D In order to analytically study the dynamic behavior of multilayer microbeam, the Bubnov-Galerkin decomposition procedure is employed to reduce the order of partial differential equation of motion and discretize the governing equation. Therefore, the transverse displacement field can be assumed by the following approximate solution [60]: (53) w( x* , t* ) = f ( x* )h (t* ) where h (t * ) represents the generalized coordinate and f ( x * ) denotes the first eigenfunction for the transverse motion of cantilevered beam which can be expressed as [58]: cos a + cosh a (sinh a x * - sin a x * )], a = 1.8751 f (x ) = [(cosh a x * - cos a x * ) (54) sin a + sinh a By applying the weighted residual Bubnov-Galerkin method [61], the nonlinear governing equation of motion is reduced to: (55) h + wo2h = [ea2h 2 + e F1 cos(wE* t* )h + e 2 F0 cos(wE* t* )] in which different parameters appeared in the equation (55) are defined as follows: 1 ∫ ωo 2 = [φ (x ∗ ) 0 2 ∗ 2 ∗ d 4φ (x ∗ ) ∗ d φ (x ) ∗ ∗ d φ (x ) 2 x w x + λ φ ( ) ( ) ]dx ∗ 0 dx ∗4 dx ∗2 dx ∗2 1 ∫[(φ(x ∗ )φ (x ∗ ))dx ∗ 0 1 ∫ −λ 2 [φ (x ∗ ) 0 εα 2 = ∗ d φ (x ) d 2φ (x ∗ ) ∗ ]dx dx ∗2 dx ∗2 2 1 ∫[(φ(x ∗ )φ (x ∗ ))dx ∗ 0 (56) 1 d 2φ ( x∗ ) ∗ − [(φ ( x∗ )( N ∗ w0∗ ( x∗ )) ]dx dx∗2 0 ∫ ε 2 F0 = 1 ∫[(φ (x )φ(x ))dx ∗ ∗ ∗ 0 1 ∫ − [(φ ( x∗ )( N ∗ ) ε F1 = 0 d 2φ ( x∗ ) ∗ ]dx dx∗2 1 ∫[(φ (x )φ (x ))dx ∗ ∗ ∗ 0 3.4 The method of multiple scales The idea of the multiple scales method is to express the solution of ordinary differential equations as a function of multiple independent time-scales. In this procedure, the independent variables T n known as time-scales are defined by [61]: (57) Tn = e nt * , n = 1,2,3,.... 10 where ε << 1 is a small perturbation parameter and the derivatives with respect to nth time-scale is expressed by the symbolic operator D n as follows: ∂ = Dn ∂Tn (58) By considering the above assumptions, the derivatives with respect to the dimensionless time t * can be defined using the following expansions in terms of the partial derivatives with respect to time-scales T n : d = D0 + e D1 + e 2 D2 (59) * dt (60) d2 = D02 + 2e D0 D1 + e 2 ( D12 + 2 D0 D2 ) *2 dt and then the approximate solution of equation (55) can be assumed by an expansion series as a function of both t * and ε according to: (61) h (t* , e ) = h0 (T0 ,T1,T2 ) + eh1 (T0 ,T1 ,T2 ) + e 2h2 (T0 ,T1 ,T2 ) + O(e 2 ) Substituting (59) to (61) into (55) and equating the coefficients of ε n equal to zero, one gives the following linear ordinary equations: ε0 : ∂ 2η 0 + ω02η0 = 0 ∂T0 2 (62) ε1 : ∂ 2η1 ∂ 2η 0 + ω 02η1 = F1 cos(ω0 T0 + σ T2 )η 0 + α 2η 02 − 2 2 ∂T0 ∂T1 ∂T0 (63) ε2 : ∂ 2η 2 ∂ 2η1 ∂ 2η 0 2 + ω η = F cos( ω T + σ T ) + F cos( ω T + σ T ) η + 2 α η η − 2 − 0 2 0 0 0 2 1 0 0 2 1 2 0 1 ∂T0 ∂T1 ∂T12 ∂T0 2 (64) in which the dimensionless frequency of the AC voltage w E* is perturbed around the fundamental frequency w0 in the following from: (65) wE* = w0 +e2s where s denotes the detuning parameter. It is convenient to express the solution of the first equation (62) in terms of complex function A (T1 ,T2 ) and its complex conjugate A (T1,T2 ) by the following form: h0 = A(T1,T2 )eiw0 T0 + A(T1,T2 )e-iw0 T0 (66) Substituting equation (66) into (63) gives: ∂ 2η1 F ∂A + ω02η1 = 1 Aei (2ω0 T0 +σ T2 ) + α 2 A2 e 2iω0 T0 − 2i ω 0 eiω0 T0 + α 2 AA 2 ∂T1 ∂T0 2 (67) By vanishing the secular terms of the right hand side of equation (66), leads the function A to become independent of the time scale T1 . Consequently, one can obtain the particular solution for h 1 as: h1 = F1 Aei ( 2w 0 T0 +s T2 ) 2 F1 Ae - is T2 + 4a 2 AA a 2 A2 e 2iw 0 T0 + 6w 02 6w 02 3w 02 (68) Substituting for h 0 and h 1 from equations (66) and (68) into equation (64) and eliminating the secular terms in the right hand side of consequence equation leads to: −2iω0 5α F AA F F2A 5α F A2 10α 2 2 A2 A F12 A ∂A + [ 2 12 + 0 ]eiσ T2 + [ 1 2 ]e 2iσ T2 + [ 2 12 ]e − iσ T2 + + =0 ∂T2 2 3ω 0 4ω 0 6ω 0 3ω02 6ω 02 (69) To solve the above equation with respect to A (T1 ,T2 ) , it is suggested to assume A in the polar form as: 1 A = aeib (T2 ) 2 (70) 11 in which a and β are real functions. By introducing new parameter γ = σT 2 − β , after some complicated mathematical computations and separating the result into real and imaginary parts, for the steady state solution we have: a′ = aγ ′ = 1 1 [12 F0ω02 + 5α 2 F1a 2 ]sin(γ ) + [3F12 ]a sin(2γ ) = 0 24ω03 24ω03 (71) F1 5α 2 2 3 1 1 2 2 2 [12 + 15 ]cos( ) + [3 ] cos(2 ) + [ + ] + [ ]a = 0 F ω α F a γ F a γ σ a 0 0 2 1 1 24ω03 24ω03 12ω 03 12ω 03 (72) where the prime stands for the derivative with respect to the second order time-scale T2 . By solving the couple equations (71) and (72) for a and γ , and referring to equation (70), the approximate solution of the nonlinear governing equation (55) using the method of multiple scale can be concluded as follows: wd∗ ( x ∗ , t ∗ ) = [ a cos(ω E∗ t ∗ − γ ) − ε F1 a εα 2 a 2 εα 2 a 2 + ε F1 a cos(γ ) ∗ ∗ ∗ ∗ cos(2 t ) cos(2 t 2 ) ]φ ( x∗ ) ω − γ − ω − γ + E E 6ω 02 6ω02 2ω 02 (73) 4. Results and Discussion In this section, at first, the static and dynamic experimental analysis is conducted to verify the soundness of mathematical modelling and empirically extract the fundamental frequency of multilayered microbeam with initial geometric imperfection. Then, a parametric study is carried out to investigate the dynamic behavior of piezoelectrically actuated microbeam. 4.1 Experimental analysis Due to improper fabrication of microstructures in the manufacturing treatments, it is expected to create microbeams with initial geometric imperfection which affects the physiomechanical behavior of such systems. After some calibration methods, it was found that the considered multilayered microbeam contains an initial curvature proportional to its static deformation, i.e. w0 ( x) = W0 ws ( x) . The static results are reported in figure 5 for different lengths of microbeam. As evident from the presented findings, one can found that the theoretical results are in good agreement with those acquired by experimental analysis. Thus, it is concluded that the linear governing equation described in equation (30) as well as the analytical solution provided by equation (34) could predict the static behavior of the system with acceptable accuracy. It should be pointed out that the initial imperfection of the microbeam is shown with blue curves. To verify the accuracy of nonlinear modeling of vibrational equation of motion (55) and also the analytical approximation provided by (73) to capture the dynamic behavior and fundamental frequency of cantilevered multilayer microbeam, the theoretical results together with the experimental measurements are plotted in figure 6 for some specific microbeam's lengths. One can infer that the simulated dynamic results are in consistent with those of experimental findings as shown in figure 3. In addition, in Table 2, we especially compared the fundamental frequency obtained from theoretical and experimental analyses for different lengths. As can be seen, the maximum deviation is less than 1% which is worthy of acceptance. It is demonstrated that our proposed theoretical models can be genuinely employed in the design of imperfect multilayered microbeams in order to estimate their fundamental natural frequencies. Table 2. A comparison between fundamental frequencies of the multilayered microbeam obtained by theory and experimental methods Beam Length L=300µm L=210µm Measurement (kHz) 39.49 72.19 12 Theory (kHz) 39.20 72.15 Relative error % 0.74 0.06 L=170µm L=140µm L=120µm L=100µm 102.31 140.86 178.63 232.14 101.98 139.77 178.41 230.70 0.33 0.78 0.12 0.63 4.2 Parametric analysis To examine the performance of cantilevered multilayer microbeams as an energy harvester device, it is necessary to capture the vibrational characteristics and frequency response of the structure. To this end, the natural frequencies of the considered multilayer microbeam as function of its length for different values of the thickness ratio h5 / ht are presented in figure 7. It is exhibited that the fundamental frequency have a descending trend with respect to the beam length. On the other hand, it is shown that any increase in the thickness ratio leads to gradual increase in the fundamental frequency. As the beam length increases, the sensitivity of frequency to this dimensionless parameter is decreased. Moreover, figure 8 depicts the variations of the fundamental natural frequency versus the thickness ration for some assigned values of Young’s modulus of piezoelectric layer C x x ,5 . One can observe that any increase in the Young’s modulus results in a significant increase in the natural frequency of the multilayered microbeam. In addition, it is evident that there is a linear relationship between the natural frequency of the system and the thickness ratio h5 / ht . Figures 9 and 10 indicate the influence of excitation amplitude V A C and thickness ratio h5 / ht on the frequency response of the considered system. One concludes that as the amplitude of excitation increases the microbeam shows the higher amplitude of vibration. Moreover, according to the simulated results in figure 10, it is evident that decreasing the thickness ratio leads to the higher amplitude of vibration and therefore increase the efficiency of the system as an energy harvester. 5. Conclusion We designed and fabricated a microcantilever beam actuator using thin layer PZT material. The device is composed of 6 layers formed by the substrate materials, active layer and the electrodes. Due to fabrication residual stresses and dissymmetry of the microbeam, it was found initially curved. The device was modeled using the Euler-Bernoulli beam theory. The model takes into account the multilayer structure of the device. 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Comparison of analytical and experimental results of the dynamic transverse deflection with applied AC frequency for a) L=100µm b) L=120µm c) L=140µm d) L=170µm e) L=210µm f) L=300µm 31 Fig.7. Variation of natural frequency calculated using the theory model with beam length L for different values of h5 / ht 32 Fig. 8. Variation of natural frequency calculated using the theory model with thickness ratio h5 / ht for different values of C xx ,5 33 Fig. 9. Variation of maximum dynamic transverse deflection calculated using the theory model with applied AC frequency for different values of VAC and for when L=300µm 34 Fig. 10. Variation of maximum dynamic transverse deflection calculated using the theory model with applied AC frequency for different values of h5 / ht 35 36

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