Numerical optimization of piezolaminated beams under static and dynamic excitations

Journal of Science: Advanced Materials and Devices (2017) 255e262 Contents lists available at ScienceDirect Journal of Science: Advanced Materials and Devices journal homepage: www.elsevier.com/locate/jsamd Original Article Numerical optimization of piezolaminated beams under static and dynamic excitations Rajan L Wankhade a, *, Kamal M Bajoria b a b Applied Mechanics Department, Govt College of Engineering Nagpur, Maharashtra, 441108, India Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India a r t i c l e i n f o a b s t r a c t Article history: Received 16 June 2016 Received in revised form 23 February 2017 Accepted 20 March 2017 Available online 30 March 2017 Shape and vibration controls of smart structures in structural applications have gained much attraction due to their ability of actuation and sensing The response of structure to bending, vibration, and buckling can be controlled by the use of this ability of a piezoelectric material In the present work, the static and dynamic control of smart piezolaminated beams is presented The optimal locations of piezoelectric patches are found out and then a detailed analysis is performed using finite element modeling considering the higher order shear deformation theory In the first part, for an extension mode, the piezolaminated beam with stacking sequence PZT5/Al/PZT5 is considered The length of the beam is 100 mm, whereas the thickness of an aluminum core is 16 mm and that of the piezo layer is of mm The PZT actuators are positioned with an identical poling direction along the thickness and are excited by a direct current voltage of 10 V For the shear mode, the stacking sequence Al/PZT5/Al is adopted The length of the beam is kept the same as the extension mechanism i.e 100 mm, whereas the thickness of the aluminum core is mm and that of the piezo layer is of mm The actuator is excited by a direct current voltage of 20 V In the second part, the control of the piezolaminated beam with an optimal location of the actuator is investigated under a dynamic excitation Electromechanical loading is considered in the finite element formulation for the analysis purpose Results are provided for beams with different boundary conditions and loading for future references Both the extension and shear actuation mechanisms are employed for the piezolaminated beam These results may be used to identify the response of a beam under static and dynamic excitations From the present work, the optimal location of a piezoelectric patch can be easily identified for the corresponding boundary condition of the beam © 2017 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Keywords: Piezoelectric Finite element method Higher order shear deformation theory Actuator and sensor Introduction Piezoelectric materials are often used in various structural applications, including structural health monitoring, precision positioning, aeronautical and mechanical structures Piezolaminated composite structures play an important role in controlling the response of structure to shape and vibration as these are relatively lightweight, strong, more stiff, and capable of sensing and actuating than that of regular composites Piezoelectric materials are able to produce an electrical response when mechanically stressed which is a direct effect, and inversely high precision stresses can be * Corresponding author E-mail address: rajanw04@gmail.com (R.L Wankhade) Peer review under responsibility of Vietnam National University, Hanoi obtained with the application of an electric field Hence this property of sensing and actuation is used for the effective active shape, vibration, and buckling control To achieve the significant response of the structure in shape and vibration control, a piezoelectric material with different modes can be considered Intelligent (smart) structures and systems have become an emerging research area that is multi-disciplinary in nature, requiring technical expertise from mechanical engineering, structural engineering, electrical engineering, applied mechanics, engineering mathematics, material science, computer science, biological science, etc The technology of smart structures is quite likely to contribute significant advancements in the design of highperformance structures, adaptive structures, high-precision systems and micro/nano-mechanical systems This emerging area has been rapidly gaining momentum in the last few decades Also it can be accepted that to some extent only initial, but highly feasible http://dx.doi.org/10.1016/j.jsamd.2017.03.002 2468-2179/© 2017 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) 256 R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 studies of the concepts of piezolaminated structures have been conducted [1] A mathematical model based on a layerwise theory is also developed for laminated composite beams with embedded piezoelectric actuators The one-dimensional beam formulation also accounts for lateral strains for which the finite element model has been developed [2] Application of the piezoelectric material to beams with considering Timoshenko vs EulereBernoulli theory was performed to study the distributed control [3] The interaction between active and passive vibration control characteristics was investigated numerically using the finite element approach These numerically obtained characteristics were further verified experimentally for carbon/epoxy laminated composite beams with a collocated piezoceramic sensor and an actuator It is observed that when the gain in velocity feedback control is small, the active control follows the trend of the passive control, but provides additional effects due to the active control Also for a large feedback gain, the active control is dominant over the passive control [5] A model simulating the effects of the control potential on the static configuration of a piezo-elastic structure has also been developed It is centered on the divergence free electric displacement field [6] The active vibration control under damping was investigated for the EulereBernoulli model with transverse vibrations of a cantilever beam It was extended to include viscous and KelvineVoigt (strain rate) damping Displacement and velocity feedback controls provided by full length or patch piezoelectric actuators and sensors bonded to the top and the bottom of the beam can be used to control the response of the beam [7] A method for finding distributed actuators or dense actuator networks such that a desired displacement field is tracked was investigated Dynamic deflection tracking of beams by a distributed actuation and the use of dense networks of actuator patches with a section-wise constant intensity to approximate the effectiveness of the distributed actuators were also studied [8] The under critical frequency behaviors related to deformed beams, showing the transition from softening to hardening effects, were studied for various levels of active voltage, static response and imperfection amplitudes of simply-supported sandwich piezoelectric beams [9] Free vibration and stability analysis of piezolaminated plates using the finite element method employing the higher order shear deformation theory were also performed Control of structure with its stability was examined for simply supported smart piezolaminated composite plates [10] Piezolaminated plates with cross ply and angle ply stacking sequences with both symmetric and anti-symmetric lay ups were studied [11,12] Theories for the accurate simulation of the shear-mode behavior of thin or thick piezoelectric sandwich composite beams have been developed considering both the electrically induced strain component and the transverse flexibility of structure [13] Vibrations of a cantilever piezolaminated beam greatly influenced the extension and shear actuation mechanisms of piezo actuators Also the proper placement of a piezo actuator changes the response of beams in vibration in extension mode as compared to that of the shear mode actuation [14] Vibration of FGM Piezoelectric Plate also can be controlled using the LQR Genetic Search technique [15] Shape control and vibration analysis of the piezolaminated plates subjected to electro-mechanical loading have been performed to study the effect of actuator voltages [16] Hence fabrication and characterization of PZT string based MEMS devices are mostly useful for this purpose The micro-fabrication and characterization of free-standing doubly clamped piezoelectric beams based on Pd/FeNi/Pd/PZT/LSMO/STO/Si heterostructures were conducted, in which the displacements in static and dynamic modes were investigated for string based MEMS devices [17] In this work, the finite element modeling for the extension and shear modes of actuation for piezolaminated beams are considered for shape control of the piezolaminated beams Further optimal location of the piezoelectric actuator is considered for shape control Different boundary conditions are considered in the piezolaminated beam analysis The optimal location of the piezoelectric actuator is obtained when stuying the response Equilibrium equations Equilibrium equations are obtained using the virtual displacement principle Equilibrium between the internal and external forces has to be satisfied If J represents the vector of the sum of the internal and external forces, then fJg ¼ fRg À fPg (1) where {R} represents the external forces due to imposed load and {P} is a vector of the internal resisting forces The equilibrium state is achieved when {J} ¼ Further, {J} can be written as fJg ẳ fRg ỵ Z V Z À fεgT fsgdV À  Z  p ÃT Ea fDa gdV Va ÃT Esp fDs gdV Z n o T N fs0 gdV ỵ Vs (2) V where V, Vs and Va are the area of the entire structure, the sensor layer, and the actuator layer, respectively Considering the work done by external forces due to the applied surface traction and applied electric charge on an actuator, the equation for the external work done can be written as Z fugT fsx; yịgdA ỵ fRg ¼ A Z f0a qea ðx; yÞdA (3) A Thus the internal potential energy can be written as P¼ Z È e ÉT d V À Z  à ½BŠT ½CŠ½BŠ de dV À È e ÉT d T T ½BŠ ½eŠ ½Bs Š È fes Z È e ÉT d va É dV Vs À Z È eÉ È e ÉT fa ½Ba ŠT ½eŠ½BŠ d dV À Va À È É ½BŠT ½eŠT ½Ba Š fea dV Z È fes ÉT È eÉ ½eŠ½bŠ d dV Vs Z È É È eÉ fa ½Ba ŠT ½gŠ½Ba Š fea dV Va À Z È É È e ÉT fs ½Bs ŠT fgg½Bs Š fes dV Vs (4) and Z Rẳ ẩ e ẫT d ẵNT fsgdA ỵ A Z ẳ fEa gT qea x; yịdA (5) Aa È e ÉT d A Z ½NŠT fsgdA À Z È e ÉT  ÃT fa Npa qea ðx; yÞdA: Aa Element stiffness matrix can be written as (6) R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 È e É Â ÃÈ e É Â Ã Â e à ½K e d ỵ Kse d ẳ F1e ỵ Fac (7)  à  e ÃÀ1  e àe à  e àe ÃÀ1  e à Kaa Kad ỵ Kds Kss Ksd in which ẵK e ẳ Kde ỵ Kda (8) ẩ ẫ e e ÃÀ1 È s É a ¼ Kda Kaa Qa Fac (9) where  à Kde ¼ Z  e à e T ẵBT ẵC ẵBdV; Kda ẳ Kad ẳ V Z ẳ e ẵBT ẵeẵBa dV Kaa Va Va  eà Kss ¼ Z where, u and w are the displacement of any point in the plate domain in lateral and transverse directions respectively u0 and w0 are the displacement of midpoint of normal qx, qy are the rotations of normal at the middle plane in the x and y directions about the y and x axes, respectively u*0 ; v*0 w*0 ; q*x and q*y are higher order terms 3.2 Strain within the element εp ẳ Lp ỵ N p |{z} ẵBT ẵeẵBs dV Vs Z (11) Strains associated with the displacement field can be written as follows: ½Ba ŠT ½ g Š½Ba ŠdV; à  e ÃT e ¼ Ksd ¼ Kds and Z * u ẳ u0 ỵ zqx ỵ z2 u*0 ỵ z3 qx w ẳ w0 257 (12) 31 T ẵBs Š ½ g Š½Bs ŠdV: Vs Finite elemnt modeling (10) Strains are related to displacements using a strain displacement matrix as ε ¼ B de |{z} |{z} 3.1 Displacement field Fig shows the models under investigation for a piezolaminated beam with embedded and/or surface mounted piezoelectric patches Displacement field for the beam is considered as ‘u’ and ‘w’ along the x and z directions, respectively The higher order shear deformation theory considering the effect of shear deformation is adopted in the displacement field Hence, the displacement field can be written as (13) 3.3 Electro-mechanical coupling For piezolaminated plates, two constitutive relationships exist which include the effect of mechanical and electrical loadings as given by eq (5) Temperature variation effect is neglected in the formulation Fig Models under investigation for the piezolaminated beam with extension and shear actuator mechanisms 258 R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 fDg ẳ ẵefg ỵ ẵgfEp g fsg ẳ ẵCfg ẵet fEp g sx0 > > > > sy0 > > < > > > > > > = Q10 10 Q20 10 6 Q30 10 sz ¼6 Q10 40 sx0 y0 > > > > > > > > 0 s > > > ; : xz > sy0 z0 6 À6 6 ex6 Q10 20 Q20 20 Q30 20 Q20 40 0 0 0 ey5 (14) Q10 30 Q10 40 Q20 30 Q20 40 Q30 30 Q30 40 Q30 40 Q40 40 0 Q50 50 0 Q60 50 ez1 p9 ez2 7> < Ex0 > = ez3 7 Ep0 y 7> : p> ; Ez0 38 ε x0 > > > > ε > > > 7> > y > > 7> 7< εz0 = 7> εx0 y0 > 7> > 0> > ε > Q50 60 5> > > > : xz > ; εy0 z0 Q60 60 0 0 Micro- and nano-mechanical systems have dimensions which range in length from mm to 1000 mm Such M/NEMS have thicknesses typically in the range of few micrometers down to 25 nm or even sub nm regime which are bonded with the host materials As a result, the physical properties of the piezolaminated beams such as mechanical, electrical, thermal and magnetic properties can be different from the bulk values This of course depends on limitation and an opportunity as we can use these micro/nanostructures, such as nanowires and nanobeams as the excellent systems for studying thickness effects in material properties and behavior at the small scale Among the different properties that arise when varying the size of any device we can find changes in mechanical properties of the structure The latter case includes an anomalous behavior of their static and dynamic responses, the appearance of nonlinear damping, and the variation of the Young's modulus Result and discussion Hence extension actuation mechanism 8 < D x0 = Dy0 ¼ : 0; Dz e0z1 0 e0z2 gx0 x0 ỵ4 0 0 e0z1 gy0 y0 εx0 > > > > > > > εy0 > > > 0 e0x6 > < > = ε z e0y5 > εx0 y0 > > > 0 > > εx0 z0 > > > > > > : ; εy0 z0 38 p > < Ex0 > = p Ey0 > > gz0 z0 : Ep0 ; z 4.1 Smart beam with extension and shear piezo actuators (15) For the piezoelectric layer polarized in the horizontal direction i e parallel to the x axis, the dielectric displacement vector using a direct piezoelectric equation is given as follows: Hence for the shear actuation mechanism ex1 < D x0 = Dy0 ¼ : 0; Dz e0x2 0 gx0 x0 ỵ4 0 e0x3 0 gy0 y0 ε x0 > > > > > > 3> εy0 > > > 0 > < > = εz 0 ey6 > ε x0 y > > > > e0z5 0 > > ε0 0> > > > : xz > ; εy0 z0 p > < Ex0 > = p Ey0 > > gz0 z0 : Ep0 ; z (16) where {D} is the electric displacement vector, [e] is the dielectric permittivity matrix, ε is the strain vector, {g} is the dielectric matrix {E} is the electric field vector, [s] is the stress vector and [C] is the elastic matrix for a constant electric field 3.4 Electrical potential function One electrical degree of freedom is used to consider piezoelectric response Both actuator and sensor layers are separately 0 considered in the formulation Hence fa and fs are the electric displacement at any point in the actuator and sensor layers, respectively, the electrical potential functions in terms of the nodal potential vector are written as  ÃÈ É f0a ¼  NpaÃÈ feaÉ fs ¼ Nps fes Ä (17) Ä Å where Npa and Nps are the shape function matrices for the actuator and sensor layers, respectively ffea g and ffes g are the nodal electric potential vectors for the actuator and sensor layers, respectively The analysis of a piezolaminated beam with thickness-shear and extension piezoelectric actuators is performed The models constructed are based on the higher-order shear deformation theory using eight node isoparametric elements Numerical examples of beams having piezoelectric actuators with different boundary conditions are presented The validity of the proposed models using extension as well as shear-mode actuators in smart beams is done by comparing the results available in literature [4] The significance of using the finite element model adopted in the present work is illustrated to show its efficacy and efficiency in terms of the analysis Further this adopted model can be effectively used to analyze beams with different boundary conditions and piezopatches with different shapes/sizes which are not possible by using the direct approach as reported in literature The models considered under investigation for the piezolaminated beam with extension and shear actuator mechanisms are shown in Fig In case of the extension mode, a three-layer piezolaminated beam with stacking sequence PZT5/Al/PZT5 is analyzed The beam is 100 mm in length, whereas the thickness of an aluminum core is 16 mm and that of the piezo layer is mm The PZT layers attached to the top and bottom of the plate acts as actuators in the extension mode The PZT actuators are positioned with an identical poling direction along the thickness and are excited by a direct current voltage of 10 V The deflection induced by the actuators is calculated for beams with different boundary conditions including cantilever, simply supported and clamped-hinged Dimensions of the beam are shown in Table Furthermore, in case of the shear mode a three-layer laminated beam with stacking sequence Al/PZT5/Al is analyzed The length of the beam is taken as 100 mm whereas the thickness of the aluminum core is mm and that of the piezo layer is mm These dimensions are taken to make a comparison with the extension mode The PZT layer acts as an actuator and operates in the shear mode Here the actuator is excited by a direct current voltage of 20 V The response of beams with varying boundary conditions is presented in Figs 2e5 Material properties of the beams are obtained: E ¼ 70:3 GPa; m ¼ 0:35 For PZT 5: Å C11 ¼ 126 GPa; C12 ¼ 79:5 GPa; G12 ¼ G13 ¼ G23 ¼ 24:8 GPa; m ¼ 0:29; r ¼ 7600 kg m3 ; d31 ¼ d32 ¼ À166 pm=V; R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 259 Table Dimensions of the smart beam with extension and shear piezoelectric actuators Stacking sequence Length Thikness of Aluminum core (tAl) Thickness of Piezoelectric layer (tp) Voltage Smart beam with extension piezo actuator Smart beam with extension piezo actuator PZT5/Al/PZT5 100 mm 16 mm mm 10 V Al/PZT5/Al 100 mm mm mm 20 V Fig Transverse deflection of the piezolaminated cantilever beam within the extension and shear modes Fig Transverse deflection of the simply supported piezolaminated beam within the extension mode The shear deformation theory employed to model the piezolaminated smart beams with extension and shear mode piezoelectric actuators Exact solutions for these beams with various boundary conditions are developed [4] Results obtained in the present work are in good tune with the exact solution Comparison of the transverse deflection of the cantilever beam with the extension and shear modes is done showing that the tip displacement of the beam within the extension mode is 74% higher than that of the shear mode Thus the shear mode gives less deflection than that of the extension mode Further, the piezolaminated beams with simply supported, clamped, clamped-hinged features are analyzed subjected to the piezoelectric effect 4.2 Control of the piezolaminated beam with an optimal location of actuator The piezolaminated beam made up of the aluminum core having a pair of linear actuators placed on either side is considered in the analysis to illustrate the optimization procedures with different boundary conditions The vibration equation of the beam is solved using the proposed finite element method Thickness of the beam is taken as mm, whereas the thickness of the piezoelectric actuator is taken as 0.4 mm The piezoelectric patch can be applied to each element The thickness of the bonding layer is not considered for simplicity The optimal location obtained from the procedure is used 260 R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 Fig Transverse deflection of the clamped piezolaminated beam within the shear mode Fig Transverse deflections of the clamped-hinged piezolaminated beam within the extension mode and shear modes to find the optimal shape using the finite element procedure Cantilever and fixedefixed boundary conditions are used in the analysis The properties of the aluminum beam are summarized in Table The best location of one piezoelectric actuator is found using the LQR based algorithm to be location The dynamic simulation of the beam with a uniformly distributed load of sint N/m applied through out the beam with a voltage of 100 sint V applied to the actuators placed at the optimal location is done for 10 s using the MATLAB function An optimal feedback controller is Table Material properties of the aluminum beam Property Aluminum PZT Young's modulus GPA Length m Density kg/m3 Piezoelectric constant m/V 79 1m 2500 e 63 0.1 m 7600 À254*10À12 designed using the gain obtained from MATLAB's LQR function Fig shows the first four modes for the controlled response of the cantilever piezolaminated beam when the actuator is placed at an optimal location PZT is a chemically inert material which exhibits a high sensitivity of about mV/Pa and a large range of linearity up to an electric field of kV/cm Hence it is possible to account for the fast response and long term stability high energy conversion efficiency, as observed in Fig 6, of the dynamic response to control vibrations of the beams using PZT The maximum deflection is observed at the free end of the beam It is observed that in the second mode, the deflection at the free end increased by 4.61% as compared to the first mode While in the third and fourth modes the deflection at the free end decreased by 3.22% as compared to the first mode Further, the piezolaminated beam having a piezoelectric patch at the same optimal location is also examined in vibration In this case, the boundary condition of the beam is fixed at both the ends Fig shows the first four modes for the controlled response of the R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 Fig Vibration modes of the cantilever piezolaminated beam Fig Vibration modes of the clamped piezolaminated beam 261 262 R.L Wankhade, K.M Bajoria / Journal of Science: Advanced Materials and Devices (2017) 255e262 clamped piezolaminated beam when the actuator is placed at the optimal location About 72% reduction in displacement is observed for the clamped beam as that of the cantilever beam for the same location of the piezoelectric patch In case of the clamped piezolaminated beam, the deflection in the second and fourth modes is increased by 5.5% compared to the first mode While in the third mode, there is no considerable change in the maximum deflection of the beam as compared to the first mode Considerable deflection is controlled by using the piezoelectric patch on the clamped beam as that of the cantilever beam The smart beam with different boundary conditions is studied to control response in vibration Cantilever and clamped boundary conditions are considered for the analysis Response in vibration for the cantilever beam is much faster than for the clamped beam The best location of one piezoelectric actuator is found using the LQR based algorithm to be location for which the response is studied in the analysis Conclusion Static and dynamic excitations are studied for the piezolaminated beams considering the numerical optimization technique The piezolaminated beams are effectively used for shape and vibration control in micro- and nano-mechanical systems due to their sensing and actuation properties In the present work, the shape control and vibration characteristics of the piezolaminated beams are studied considering the extension and shear actuation mechanisms The shear deformation theory is employed to model the piezolaminated smart beams with extension and shear mode piezoelectric actuators The obtained results are in good agreement with the exact solutions available in the literature A comparative analysis of the transverse deflection of the cantilever beam in the extension and shear modes shows that the maximum deflection for the cantilever beam in the extension mode is 74% higher than that in the shear mode Thus the shear mode gives less deflection than that of the extension mode The piezolaminated beams with simply supported, clamped, clamped-hinged features are analyzed subjected to the piezoelectric effect Analysis of the vibration characteristics of the piezolaminated beams is useful in identifying the response of beams under static and dynamic excitations The optimal location of the piezoelectric patch can be easily identified for the corresponding boundary condition of the beam This work can be further extended to verify the stability problem of beams with respect to piezoelectric actuation and sensing mechanism References [1] H.S Tzou, G.L Anderson, Intelligent Structural Systems, Kluwer, Norwell, MA, 1992 [2] P Dhonthireddy, K Chandrashekhara, Modeling and shape control of composite beams with embedded piezoelectric actuators, Compos Struct 35 (1996) 237e244 [3] O.J Aldraihem, R.C Wetherhold, T Singh, Distributed control of laminated beams: Timoshenko vs EulereBernoulli theory, J Intell Mater Syst Struct (1997) 149e157 [4] O.J Aldraihem, A.A Khdeir, Smart beams with extension and thickness-shear piezoelectric actuators, Smart Mater Struct (2000) 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Conclusion Static and dynamic excitations are studied for the piezolaminated beams considering the numerical optimization technique The piezolaminated beams are effectively used for shape and vibration... the vibration characteristics of the piezolaminated beams is useful in identifying the response of beams under static and dynamic excitations The optimal location of the piezoelectric patch can... properties of the structure The latter case includes an anomalous behavior of their static and dynamic responses, the appearance of nonlinear damping, and the variation of the Young's modulus Result and