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Computational FEA Model of a Coupled Piezoelectric Sensor and Plate Structure for Energy Harvesting

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Australian Journal of Mechanical Engineering, 2008
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  199 © Institution o Engineers Australia, 2008 * Reviewed paper srcinally presented at the FirstWorld Congress on Engineering Asset Management(1 st WCEAM), Gold Coast, Australia, 2006.† Corresponding author Dr Ian Howard can becontacted at I.Howard@curtin.edu.au. Computational FEA model of a coupled piezoelectricsensor and plate structure for energy harvesting * MF Lumentut, KK Teh and I Howard † Department o Mechanical Engineering, Curtin University o Technology SUMMARY: This paper presents a mathematical model o a piezo-plate energy-harvestingscheme. An analytical method is used to generate a fnite element model o the coupled piezoelectricsensor element using Love-Kirchho’s plate theory. Constitutive equations or a single layer plate element are ormulated. The polarisation o the piezoelectric sensor bounded on the upper plate structure is due to ambient vibration exerted on the structure. Forced vibration o the smartstructure will create strain energy within the crystalline structure o the piezoelectric material.The resulting electric feld generated by the sensor element was modelled using a linear thicknessinterpolation unction and the meshed plate elements were modelled using our-node rectangularelements with three degrees o reedom or each node. The structural eigenmodes and dynamicresponse o the coupled piezo-plate system were solved by using modal analysis and Newmark-integration methods respectively. The analysis is demonstrated with both dynamic displacementand electric voltage responses to an applied step orce. Further modelling o the smart structure isaimed at maximising the power generation capability. F orce, NKE Kinetic Energy, JPE Potential Energy, JPEE electrical energy, JK stiness matrix, N/mM consistent-mass matrix, kg Greek Letters strain vectors σ stress vectors, N/m 2   dielectric matrix at constant strain, F/m   poison ratio  shape unction    z ( ) electric potential, Volts   nodal vectors, m  dierential operator o shape unction   p density o plate, kg/m 3   s density o sensor, kg/m 3  total o energy, Nm NOMENCLATURE u displacement o plate element inx direction, mv displacement o plate elementin y direction, mw displacement o plate element inz direction, mC modulus o elastic, N/m 2 D electric displacement vectors, C/m 2 d piezoelectric constant, m/Ve piezoelectric stress coefcient, C/m 2 E electric feld, V/mD m stiness coefcient, N/m 2 t thickness, ml length, mw width, m  Australian Journal o Mechanical Engineering, Vol 5 No 2  200“Computational FEA model o a coupled piezoelectric sensor and plate …” – Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2 Ω work done o the system, J Subscripts x relative to x-axisy relative to x-axisz relative to z-axisn reer to joints o plate element θθ reer to plate Φθ reer to sensor–plate θΦ reer to plate-sensor Φ reer to sensor θ reer to plate Superscripts s reer to sensor elementp reer to plate elementps reer to plate-sensor elementT matrix transpose 1 INTRODUCTION The investigation o piezoelectric elements as smartstructures has many wide-ranging applications inthe engineering industry. The ability o piezoelectricmaterials to transorm mechanical strain, induced by ambient vibration, to electrical potential is wellknown, as evidenced in the direct mode by using thematerial as a sensor. On the other hand, piezoelectricmaterial which has the ability to transorm electricalfeld to mechanical strain is also well-known, as theconverse mode o using the material as an actuator.Numerical analysis o these smart structures usinga three dimensional fnite element model was frstpresented by Allike and Hughes in 1970 (Allik,1970). However, a three dimensional fnite elementanalysis o the integrated piezo and plate structurewas challenging and generally demanded largecomputational requirements. It was realised thatthe piezoelectric patches were too thin and twodimensional modeling o these patches was sufcient.Other numerical models using laminated rectangularplates with piezoelectric material bonded on thesurace have been previously developed (Moita et al,2004; Tzou & Tzeng, 1990; Fernandes & Pouget, 2003),with other researchers using multilayer compositeplates (Saravanos et al, 1996; Kogl & Bucalem, 2005).Several publications, (Sekouri et al, 2004; Liu et al,2002; Taciroglu et al, 2004) have reported the use o numerical models to simulate the dynamic responseo the coupled elastic and electric feld o the smartstructures.The previous scenarios, mostly, discussed static anddynamic systems using the piezoelectric sensor andactuator bounded on the structures. Moreover, recentdevelopments o piezoelectric technology involve theusage o sensors patched on the structure subject toambient mechanical vibration. The induced strainenergy can be converted to useul electrical energycapable o being stored on electrical devices such as batteries or capacitors. Such a technique is generallyreerred to as the energy harvesting technique. Thedevelopment o energy harvesting techniques orpowering smart structures and embedded sensorshas received increased attention over the past decade(Sodano et al, 2004). Analytical and experimentalanalyses have been used to investigate powerharvesting rom PZT elements to power electricdevices or or recharging o batteries. EmbeddedPZT materials in a vibrating machine environmentcan be used as the required power source providedthe vibration source does not stop. A more useuldevelopment would be or the PZT element tostore its energy into a rechargeable battery or lateruse. Further analytical and experimental studieswith optimising power ow or adaptive energyharvesting and sel-power harvesting have beendeveloped   (Ottman et al, 2002; James et al, 2004;Roundy et al, 2003; Glynne-Jones, 2004).The main objective o this paper is to presenta mathematical model o an energy harvestingtechnique with bonded piezo-sensors on a plateelement. Mechanical strain energy inducespolarisation in the piezo-sensor thus creating anelectric feld. The mathematical model introducesa our-node non-conormed quadrilateral elementwith a total o twelve nodal degrees o reedom,ormulated using Love-Kirchho’s plate theory.The paper presents results rom a two-sided fxed-fxed plate using numerical algorithms solved usingMATLAB, based on the suggested ormulation. 2 CONSTITUTIVE EQUATIONOF PLATE ELEMENT Classical plate theory can be used to derive theequations o motion or plates by assuming thatthe shear deormation eect is negligible. Thedisplacement vectors u , v , and w o an arbitrary pointin the deormed element can be written in terms o the mid-surace o the plate, as described in fgure1a and fgure 1b and equations 1, 2 and 3. u =  z  w   x  ,(1) v =  z  w   y ,(2) w = w (  x  ,  y )   (3)  201“Computational FEA model o a coupled piezoelectric sensor and plate …” – Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2  xw ∂∂  xw z u ∂∂−= Figure 1a: The deormations o plate elementwith respect.  yw ∂∂  yw z v ∂∂= - Figure 1b: The deormation o plate elementwith to x axis respect to y axis.The corresponding strain-displacement vectorrelationship in the x and y directions, can beormulated as,   { } =    x     y     xy  =  z  2 w   x  2  2 w   y 2 2  2 w   x    y  .(4)A state o plane stress is assumed and the correspondingstress-strain relationship is,    { } = C  1  v 2 1 v 0 v 1 00 01  v 2     x     y     xy  ,(5)or    { } = D m    { } .(6) 3 CONSTITUTIVE EQUATIONOF PIEZO-SENSOR The piezoelectric-sensor has a direct mode electricallyon its material due to the exerted mechanical strainon the structure. The matrix equation relating themechanical and electrical quantities can be writtenas (Naillon et al, 1983),  D { } = e    { } +    E  { } , (7)where D { } , e  = d   D m  ,   { } ,    , d   ,  D m  and  E  { }  are the electric displacement vector, piezoelectricstress coeicient, strain vector, dielectric matrixat constant strain, piezoelectric constant, stinesscoefcient and electric feld vector respectively. Inthis case, the converse mode gained by the actuatoris neglected.Discritised electric feld {E} induced by ambientvibration generates polarisation in the piezo-sensormaterial, in the z- direction along the sensor platethickness. The subsequent electrical potential isassumed linear and is ormulated as,    z ( ) { } =   z ( ) s     z ( ) s { } ,(8)where   z ( ) s =  z  t  p 2 t s is the shape unction over theinterval t p /2 < z < t p /2 + t s .The electrical feld is a unction o the electricalpotential with negative gradient operator,  E  { } =     z ( ) { } =    z ( ) s     z ( ) s { } ,(9)where  is a gradient operator, frst derivative o theshape unction with respect to thickness direction,giving,   z ( ) s  = 0 0 d    z ( ) s dz  T  = 0 01 t  s  T  .(10) 4 FINITE ELEMENT FORMULATION4.1 Discretised element plate matrix As mentioned previously, the classical plate theorycan be urther dealt with by establishing a 12-termpolynomial unction to model the non-conormingtransverse displacement, w (x,y) . The three degrees o reedom at each node w n(x,y) , θ xn  and   θ  yn can then beormulated as shown in fgure 2.The nodal degrees o reedom or the our-nodedrectangular element, can be expressed in the vectororm, {   n x  ,  y ( )  p }  {   i x  ,  y ( )  p ,    j x  ,  y ( )  p ,   k x  ,  y ( )  p ,   l x  ,  y ( )  p } and  { w n,   θ xn, θ  yn }. All variables o displacement obtainedwill be substituted into a polynomial unction wherethe results will give the consistent displacementunction as,  202“Computational FEA model o a coupled piezoelectric sensor and plate …” – Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2 w  x  ,  y ( ) =  n (  x  ,  y )  p    n x  ,  y ( )  p { } ,(11)where   n x  ,  y ( )  p   is the shape unction o thedisplacement feld or the non-conorming plateelement with reerence to transversal displacementunction,  n x  ,  y ( )  p  =      n  x  ,  y ( )  p   1 ,   n    { i,j,k,l } .(12) (xy) w Zx tsd PlatePiezo-sensor  dx,ady, b t  2  p  p t  y jik θ  x θ  y Figure 2: Geometry element o the coupledpiezo-sensor and plate element.The strain-displacement relationship rom equation(4) can be expanded by orming the second orderpartial dierential equation o shape unction,equation (12), with respect to x and y axes o theelement plate in each joint as equation (13).Equation (13) can be simply rewritten as ollows, ¿¾½¯®-»»»»»»»»»»»»¼º««««««««««««¬ª ∂∂ »¼º«¬ª Φ∂∂∂ »¼º«¬ª Φ∂∂∂ »¼º«¬ª Φ∂∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂∂ »¼º«¬ª Φ∂−= ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© § ¸ ¹ ·¨© §   p y xn y x p y xl  y x p y xk  y x p y x j y x p y xi y p y xl  y p y xk  y p y x j y p y xi x p y xl  x p y xk  x p y x j x p y xi z  , į ,22,22,22,222,22,22,22,22,22,22,22,2 ε  . (13) = – z  n x  ,  y ( )  p    n x  ,  y ( )  p { } , n    { i,j,k,l }.(14) 4.2 Governing differential dynamic equations By invoking Hamilton’s principle or a conservativedynamic system and applying Lagrange’ssimplifcation, the general dynamic equation can beexpressed as, d dt   u  .  u   F     = 1 s  = 0, (15)where,     = KE – PE + PEE,     . n x  ,  y ( )  p    n x  ,  y ( )  p { } ,    z ( ) s ( )  ; F η    {{Fs},{Fc},{ q }}The kinetic energy term is given by, KE  = 12    p w .  x  ,  y ( )  T  w .  x  ,  y ( )  d  ( vol )   + 12   s w .  T  w .  x  ,  y ( )  d  ( vol )    (16)and the potential energy can be written, PE  = 12   { } T    { } d(vol)   . (17)The electrical energy term can be ormed as, PEE  = 12  E  { } T   D { } d  ( vol )   . (18)Work done on the system is due to the surace orce,concentrated orce and electrical charge density andcan be written as,  = w  x  ,  y ( ) { }   T  Fs { } d area ( ) + w  x  ,  y ( ) { } T  Fc { }  w  x  ,  y ( ) { } T  q { } d area ( )   .(19)The solution procedure involves substitutingequations (6), (7), (9) and (14) into related expressionso equations (16) to (19) and dierentiating withrespect to all displacement vector parameters o the coupled sensor and plate structure. As given inequation (14), the resulting non-homogenous matrixdierential equation or the smart structure can beormed as, M  psn x y pzs ÈÎ »½ ÈÎÍÍÍ »½¼¼¼ ÏÌÔÓÔ¸ ( )( ) 00 0 d d  ..,.. ¿¿ Ô À Ô+ÈÎ »½ ÈÎ »½ ÈÎ »½ ÈÎ »½ ÈÎÍÍÍÍ »½ K K K K  T  ff fq qf qq  ¼¼¼¼¼ ÏÌÔÓÔ¸ ¿ Ô À Ô=ÏÌÔÓÔ¸ ¿ Ô À Ô ( )( ) d d  f q  n x y pzs F F  , (20)where:  M   ps  =   i  n p x  ,  y ( )    T   n p x  ,  y ( )  d  ( vol i )       i = 1  ,(21)  203“Computational FEA model o a coupled piezoelectric sensor and plate …” – Lumentut, Teh & Howard Australian Journal o Mechanical Engineering Vol 5 No 2 K    = z 2  n p x  ,  y ( )    T   D i  n p x  ,  y ( )  d  ( vol i )       i = 1  ,(22) K      = K      T  =  z 2  n p x  ,  y ( )    T  e  T    z ( ) s  d  ( vol s )       s = 1  , (23) K     =    z ( ) s    T    s   z ( ) s  d  ( vol s )       s = 1  ,(24) F   { } =  n p x  ,  y ( )  T    Fs i { } d  ( area i ) +  n p x  ,  y ( )  T  Fc i { }       i = 1  ,  (25) F    { } =    z ( ) s  T  q i { } d  ( area i )         i = 1   (26)To obtain the eigenmodes, dynamic displacementand electric voltage, the matrix dierential equation(20) can be solved using Guyan Reduction toseparate the degrees o reedom o the variabledisplacements where the equations yielded reectmore appropriately the independent equations orelectric voltage and linear dynamic displacement.At this point, the Newmark- β method was used tosolve or the dynamic response o the piezo-platevibration system. 4.3 Application of boundary condition Equations (21) to (26) can be solved by using integralalgebra and by incorporating appropriate geometric boundary conditions. In view o the dierentgeometrical boundary conditions or the sensorsand the base plate, special attention must be takenwhen modiying equations (21) to (24) to reect theseconditions. 5 RESULT AND DISCUSSION5.1 Numerical example A MATLAB program was written based on theproposed theory to predict eigenmodes, dynamicdisplacement, and electrical voltages under transientresponse conditions. The characteristics or the piezo-sensor using PZT PSI-5A4E (Piezo Systems, INC) andthe plate are listed in table 1 as ollows.The example used a two-sided fxed-fxed rectangularplate with piezo-sensors bonded on the upper suraceas shown in fgure 3. The plate was subjected to anexternally applied dynamic step orce o 1 Newtonat node 2, at time t=0 sec.Figure 3 shows the mesh pattern or the piezo-platestructure, consisting o 16 elements (4x4) and 16elements (4x4) o the piezo-sensors. The Piezo-sensor was bonded over the upper plate surace.The numerical methods outlined above were usedto solve the resulting fnite element model usingthe Newmark- β method with the application o theMATLAB program. Figure 3: Geometry o the coupled piezo-sensorand plate element. Table 1: Characteristic data.ItemPiezo-sensorPlateC (GPA)66 207   0.30.3   (kg/m 3 )78007870t (m)0.0002670.002d 31 (m/V)-190e-12–d 33 (m/V)390e-12–   33 (F/m)1.602e-8–l x w (m 2 )0.25 x 0.2 (element)1 x 0.8 5.2 Eigenmodes Figures 4a to 4d show the eigenvectors correspondingto the frst our natural requencies o the compositeplate-piezo structure. The eigenvector correspondingto the natural requency o 11.22 Hz shown in fgure4a indicates clearly the frst transverse vibrationo a ixed-ixed plate. Figure 4b shows the irsttwisting vibration mode at 14.31 Hz. The next twoeigenvectors shown in fgures 4c and 4d indicatethe second bending and twisting modes at 27.20 Hzand 31.23 Hz respectively. To optimise the locationo the piezo patch on the plate, modal analysis can be used to understand the prediction o deected
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