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  1. 1. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161155 | P a g eShape Control And Optimization Using Cantilever BeamHitesh R. Patel*, J. R. Mevada***Student, M.Tech, (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva-384012,** Associate Prof., (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva-384012,AbstractIn this study, Analytical work is carried out forstatic shape control of cantilever beam structurewith a use of laminated piezoelectric actuator(LPA). The mathematical modeling of beamelement covered with LPA based on Timoshenkobeam element theory and linear theory ofpiezoelectricity has been used. This work showshow number of actuators, actuator size, actuatorlocation on beam and control voltage aredepended on the desired shape of beam. Initialcondition of beam is taken as horizontal positionand three higher order polynomial curves aretaken as desired shapes for beam to achieve.Here error between desired shape and achievedshape is taken as an objective to minimize, size,location and control voltage of actuators taken asvariables. Genetic Algorithm for calculatingoptimum values of all variables is carried outusing Matlab toolKeywords – Shape control, Genetic Optimization,Cantilever beam,1. IntroductionNow a day due to requirement of highlyprecise structure works considerable attention hasbeen carried out behind shape control of structure.In some application areas like reflecting surfaces ofspace antennas, aerodynamic surfaces of aircraftwings, hydrodynamic surfaces of submarines andships where highly precise surfaces required.Recently, researcher giving considerable attentionon developing advanced structures having integratedcontrol and self monitoring capability. Suchstructures are called smart structure. Using directand converse effect of piezoelectric materials forsensing and control of structures, a considerableanalytical and computational modeling works forsmart structures have been reported in the reviewpaper of Sunar and Rao [10], most of the past workcarried out for the control of vibrationcharacteristics of structures, but on shape controlside fewer work is carried out. A review paper by H.Irschik [4] on static and dynamic shape control ofstructure by piezoelectric actuation show workcarried out for shape control and relevantapplication of shape control. Agrawal and Trensor[1] show in his work analytical and experimentalresults for optimal location for Piezoceramicactuators on beam here size of Piezoceramicactuators are not varied only optimization carriedout for optimal location of actuators and the desiredshape of the curve is taken as parabola which is asecond order polynomial curve. One moreconference report is made available by Rui Ribeiroand Suzana Da Mota Silva [8] for the optimal designand control of adaptive structure using geneticoptimization algorithm. They have been carriedshape control for beam in two different boundaryconditions like clamped - free and clamped –clamped on both the end of beam and carriedoptimization for location of piezo electric actuatorsand sensors. Also in paper on the application ofgenetic algorithm for shape control with piezoelectric patches and also show comparison withexperimental data by S daMota Silva and R Ribeiro[9]. Paper presented by author Osama J. Aldraihem[7] shows analytical results for optimal size andlocation of piezoelectric actuator on beam forvarious boundary conditions of beam at here desiredshape of beam is considered as horizontal and singlepair of actuator is used for his work. In the paper ofE.P. Hadjigeorgiou, G.E. Stavroulakis, C.V.Massalas [3] work carried out for shape control ofbeam using piezo electric actuator. Here all themathematical modeling is based on the Timoshenkobeam theory. They have been investigated asplacement of actuator near the fixed end of beamAnd optimization carried out for the optimal voltageof actuators. If control voltage is goes higher thenactuators get damaged so for minimization ofcontrol voltage and increase actuation force ofactuator, layers of laminated piezoelectric actuators(LPA) are mounted one over another so actuationenergy of patches are added and with lower voltagewe can get higher actuation force. This concept isused by author Y Yu, X N Zhang and S L Xie [11]in this work carried out for shape control ofcantilever beam and optimization done for optimumcontrol voltage.From all above work carried out earlier bydifferent authors are limited to vary only position ofactuators on the beam. Here in this work number ofactuators, size and location of actuators are going tobe varied, and also three types of desired shape ofbeam having higher order rather than parabolicshape is to be considered to control.
  2. 2. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161156 | P a g e2. Mathematical Modeling of Beam2.1 governing equation of beamMathematical modeling carried out basedon Timoshenko beam element theory and lineartheory of piezoelectricity.Figure 1. Beam with surface bonded LPA [11]Considering Cartesian coordinate system asshown in Fig. 1, here analysis is restricted in x-zplane only so displacements in all three directions(using Timoshenko beam theory) are as below [3].Where 𝜔 is transverse displacement of point oncentroidal axis and 𝜓 is the rotation of beam crosssection about positive y axis.𝑢1 𝑥, 𝑦, 𝑧, 𝑡 ≈ 𝑧𝜓 𝑥, 𝑡 , (1)𝑢2 𝑥, 𝑦, 𝑧, 𝑡 ≈ 0 , (2)𝑢3 𝑥, 𝑦, 𝑧, 𝑡 ≈ 𝜔 𝑥, 𝑡 , (3)Nonzero strain component of beam using aboveequations are as below:[3]𝜀 𝑥 = 𝑧𝜕𝜓𝜕𝑥, 𝛾𝑥𝑧 = 𝜓 +𝜕𝜔𝜕𝑥, (4)The linear piezoelectric coupling between elasticfield and electric field – no thermal effect isconsidered are as following [5].𝜎 = 𝑄 𝜀 − 𝑒 𝑇𝐸 , (5)𝐷 = 𝑒 𝜀 + 𝜉 {𝐸}, (6)The equation of laminated beam is derived with theuse of Hamilton principle as below.𝛿 𝐻 − 𝑊𝑒 𝑑𝑡 = 0𝑡2𝑡1(7)Where (.) denote first variation operator, T kineticenergy of beam, H enthalpy of beam with laminatedpiezoelectric actuator and We external work done.Now electric enthalpy of beam [5,6] by using aboveequations form (4) to (6) as𝐻 =12𝜀 𝑇𝜎 𝑑𝑉𝑉 𝑏+12𝜀 𝑇𝜎 − 𝐸 𝑇𝐷 𝑑𝑉𝑉𝑝=12𝜀 𝑇[𝑄] 𝜀 𝑑𝑉𝑉 𝑏+12𝜀 𝑇𝑄 𝜀 − 𝜀 𝑇𝑒 𝑇𝐸𝑉𝑝− 𝐸 𝑇𝑒 𝜀 − 𝐸 𝑇𝜉 𝐸 𝑑𝑉==12𝐸𝐼𝜕𝜓𝜕𝑥2+12𝐺𝐴 𝜓 +𝜕𝜔𝜕𝑥2𝐿 𝑒0− 𝑀 𝑒𝑙𝜕𝜓𝜕𝑥− 𝑄 𝑒𝑙𝜓 +𝜕𝜔𝜕𝑥𝑑𝑥−12𝐸𝑥2𝜉11𝑠 𝑝𝐿 𝑒0+ 𝐸𝑧2𝜉33 𝑑𝑠𝑑𝑥(8)Where,(𝐸𝐼) = 𝑧2𝑄11 𝑑𝑠 + 𝑧2𝑄11𝑝 𝑑𝑠𝑠 𝑝,𝑠 𝑏𝐺𝐴 = 𝑘 𝑄55 𝑑𝑠 + 𝑄55𝑝 𝑑𝑠𝑠 𝑝𝑠 𝑏𝑀 𝑒𝑙= 𝑧𝑒31 𝐸𝑧 𝑑𝑠𝑠 𝑝,𝑄 𝑒𝑙= 𝑒15 𝐸𝑥 𝑑𝑠 = 0𝑠 𝑝,Here Qel=0 because electric field intensity in xdirection is neglected and also value of k is taken as5/6 [3].Finally the work of external force is given by𝑊𝑒 = 𝑞𝜔 + 𝑚𝜓 𝑑𝑥,𝐿 𝑒0(9)2.2 Finite element formulation of beamConsidering a beam element of length Le having twodegree of freedom per node one in transversedirection W1 (or W2) and other is rotation degree offreedom 𝜓1 (or 𝜓2). As shown in Fig 2.Figure 2. Beam elementThe array of nodal displacement is defined as𝑿 𝑒= 𝜔1 𝜓1 𝜔2 𝜓2𝑇 (10)On the basis of Timoshenko beam elementtheory the cubic and quadratic Lagrangianpolynomial are used for transverse and rotationdisplacement where the polynomials are madeinterdependent by requiring them to satisfy the twohomogeneous differential equations associated withTimoshenko’s beam theory. The displacement androtation of beam element can be expressed as𝜔𝜓 =𝑁 𝜔𝑁 𝜓𝑿 𝑒 (11)𝜀 𝑥𝛾𝑥𝑧=0 𝑧𝜕𝜕𝑥𝜕𝜕𝑥1𝜔𝜓 =𝐵𝑢𝐵 𝜓𝑿 𝑒 (12)
  3. 3. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161157 | P a g eWhere, [Nω] and [Nψ] are cubic and quadratic shapefunctions of tiemoshenko beam element [11].Electric potential inside element at any arbitraryposition as𝑉 = 1 −𝑥𝐿 𝑒𝑥𝐿 𝑒𝑉1 𝑉2𝑇= 𝑁 𝑉 𝑉𝑒 (13)And also electric field intensity Ez can be expressedas𝐸𝑧 = −1𝑛𝑡 𝑝1 −𝑥𝐿 𝑒𝑥𝐿 𝑒𝑉1 𝑉2𝑇= 𝐵𝑉 𝑉𝑒(14)Substituting equations (10) – (14) in equations (8)and (9) and then substituting them in to equation (7)the equation for shape control is obtained as bellow[11].𝐾 𝑒𝑿 𝑒= 𝐹 𝑒+ 𝐹𝑒𝑙𝑒 (15)𝐾 𝑒= 𝐾𝑢𝑢𝑒+ 𝐾𝑢𝑉𝑒𝐾𝑉𝑉𝑒 −1𝐾𝑢𝑉𝑒 𝑇 (16)Where all matrices can be express as bellow𝐾𝑢𝑢𝑒=𝜕 𝑁 𝜓𝜕𝑥𝑁 𝜓 +𝜕 𝑁 𝜔𝜕𝑥𝐿 𝑒0×𝐸𝐼 00 𝐺𝐴×𝜕 𝑁 𝜓𝜕𝑥𝑁 𝜓 +𝜕 𝑁 𝜔𝜕𝑥𝑑𝑥,(17)𝐾𝑢𝑉𝑒= 𝐵𝑢𝑇𝑒31 𝐵𝑉 𝑑𝑠𝑑𝑥𝑠 𝑝𝐿 𝑒0, (18)𝐾𝑉𝑉𝑒= 𝐵𝑉𝑇𝜉33 𝐵𝑉 𝑑𝑠𝑑𝑥,𝑠 𝑝𝐿 𝑒0(19)To obtain stiffness matrix of beam withoutcoverage of laminated piezoelectric actuator puttp=0 in equation of 𝐾𝑢𝑢𝑒.And the external force matrix and electric forcematrix are as follow,𝐹 𝑒= 𝑁 𝜔 𝑁 𝜓𝑞𝑚𝑑𝑥,𝐿 𝑒0(20)𝐹𝑒𝑙𝑒=𝜕 𝑁 𝜓𝜕𝑥𝑁 𝜓 +𝜕 𝑁 𝜔𝜕𝑥𝑀 𝑒𝑙𝑄 𝑒𝑙 𝑑𝑥𝐿 𝑒0,(21)3. Shape control and optimization usinggenetic algorithmHere, study carried out for static shapecontrol, so transverse displacement vector X is timeindependent, in above equation (15) there is not anytime dependent vector so it is directly used in staticshape control problem. With fixed time independentvalues of electric voltage at various location ofactuator one can control shape of beam.3.1 Material properties of composite cantileverbeam and Piezoceramic actuatorMaterial selected for cantilever beam isgraphite epoxy composite because having lowerdensity and lower thermal coefficient of expansion,which will minimize deflection due to temperaturerise and also due to self weight, and materialselected for actuator is PZT G1195N, properties ofboth the material are specified in Table 1.[3]Here size of beam is considered as 300 mm long,40 mm width and 9.6 mm thick, also width andthickness of LPA layer is considered as 40 mm and0.2 mm respectively. Here length of LPA is going tobe varied.3.2 Shape controlFigure 3. Cantilever beam with surface bonded LPALayouts of beam with laminated piezoelectricactuators (LPA) are as shown in below Fig 3. Asshown in below figure beam divided in to 30 finiteelements having 10 mm elemental length. And oneLPA covers several elements of beam, equalamplitude voltage with an opposite sign areprovided to upper and lower LPA, in generally saysfor upward displacement of beam upper LPA neednegative voltage and lower LPA need positivevoltage.Table 1. Material properties of base beam and LPAProperties SymbolLPA materialPZT G1195NGraphite epoxy compositematerial T300/976Young modulus (GPa) E11 63 150Poisson Ratio 𝜈12 0.3 0.3Shear modulus (GPa) G12 24.2 7.1Density (Kg/m3) 𝜌 7600 1600Piezoelectric constant (C/m2) e13 17.584Electric permittivity (F/m)𝜉13 15.3 x 10-9𝜉15 15.0 x 10-9
  4. 4. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161158 | P a g eFor shape control work initial condition of beamis taken as horizontal beam. Here no other effect isconsidered like gravity or thermal expansion ofbeam. And for desired shape of beam is take as threehigher order polynomial curves having differentcurvatures as shown in figure 5.Figure 4. Desired shape of beamHere error between desired shape and achievedshape is to be considered as an objective function tominimize and in this study number, size, locationand Voltage provided to the LPA are considered asdesign variables. The error function used in thisstudy as under[8].𝐸𝑟𝑟𝑜𝑟 = 𝛾𝑖 − 𝑞𝑖2𝑛𝑖=1(22)Where 𝛾i is the pre-defined displacement at the ithnode and qi the achieved displacement.3.3.1 Optimization using Genetic algorithmFor varying no of actuators on the beam hereoptimization is carried out in four cases as bellowtable.Table 2. Design variable for different cases.Case Description1 Using single LPA2 Using two LPA’s3 Using three LPA’s4 Using four LPA’sFor Genetic algorithm population is set as 100for all three cases and max. Number of generation isallowed as 1000.Also algorithm having stochastic innature. Here crossover function is selected asheuristic in nature. Upper Control limit for voltageis specified as 400 V and lower limit is specified as400 V in reverse polarity (for reverse polarityvoltage shown as minus (-) sign), minimum lengthof LPA is considered as 30 mm because as lengthdecreased control force (Actuation force) of LPA onbeam is also decreased. And maximum lengthprovided as full length of beam, for empty lengthupper limit is considered as full length of beam andlower limit is considered as zero. Constraintequation is become as the summation of allActuators length and Empty length of beam is lessthan or equal to 300 mm.3.3.2 Optimization for first desired shape ofbeam.Results obtained after optimization for first desiredshape are shown in bar chart as bellow. Here voltageshown in negative direction means it is in reversepolarity.Figure 5. Optimization result for first shape
  5. 5. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161159 | P a g eFrom above results we can say that To achievingsuch desired shape one LPA is enough for optimalcontrol of shape, further more if number of actuatorsincreased we can obtain better fitness means bettercontrol of shape, also it is required to place actuatorsat higher strain area on beam, in this case higherstrain area is near the fixed end of beam.3.3.3 Optimization for second desired shape ofbeamResults obtained after optimization for second typeof desired shape are as under.For optimal control of such kind of desired shapewe need minimum 2 numbers of actuators, fromabove results to increase number of actuators forcontrol of such a kind of shape is meaningless, alsothere is two strain concentrated area in desired shapeof beam one at fixed end and other at the middleposition were beam is getting to deflect form lowerto upper position. Here form above results we canalso say that it is necessary to cover strainconcentrated area on beam by actuator for optimallycontrol the shape.3.3.4 Optimization for third desired shape ofbeam.Results obtained after optimization for third type ofdesired shape as following.From above result we can say that it is notpossible to achieve third kind of shape using twoLPA, also further using three and four LPA andobtained results as show in below figure 10.Figure 6. Optimization result for second shapeFigure 7. Optimization result using one and two patches for third shape
  6. 6. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161160 | P a g eFigure 8. Optimization result using three and four patches for third shapeFigure 9. Optimization result using three LPA (Varying number of layers) for third curveIn this third type of shape for cantilever beam thereare three strain concentrated area on the beam atvarious location. From the above results bettercontrol of shape obtained as compared to use of oneand two LPA, but voltage limit reached for LPA sofor better control of shape here multiple layers ofLPA on one over another is used, furtheroptimization being carried out for three LPA havingmultiple layers. And results obtained from that are asbelow shown in figure 11.For Optimal shape control of third kind threeLPA are enough, there is no need to increase LPAfurthermore, there are three strain concentrated areaon the beam, also from above charts there are twoareas on beam where strain concentration is zero andso there is no need to place LPA over there.
  7. 7. Hitesh R. Patel, J. R. Mevada / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.155-161161 | P a g e4 ConclusionHere mathematical modeling of cantileverbeam based on Timoshenko beam element theory.To minimize control cost of any structure it isrequired to optimize voltage, location and size ofpiezoelectric patch on the structure. Optimization iscarried out for the error minimization betweendesired and actual shape with the use of geneticoptimization algorithm for different three types ofhigher order polynomial curves. Here number ofactuators, voltage applied to the actuator, size ofactuator and location of actuator on beam carried asa variable. Number of actuators, size of actuators, locationof actuators and control voltage provided to theactuators are depending on the curvature ofdesired shape of beam. For optimum shape control, Minimum numberof actuator required to control the shape ofbeam is greater than or equal to number of strainconcentrated area formed during achievingdesired shape. As number of actuators increasedwe get better shape control. But if number ofactuators less than required then shape controlnot possible. For better shape control it is desired to coverhigher strain concentrated area by actuators andthere is no need to cover that portion of thebeam where strain concentration is remain zerowhile achieving desired shape. In case of cantilever beam strain concentratedarea is near the fixed end of beam so it isessential to cover fixed end of beam withactuator for better shape control. There is noneed to place actuators at the free end side ofbeam.References[1] Brij N Agrawal and Kirk E Treanor, Shapecontrol of a beam using piezoelectricactuators, Smart Mater. Struct. 8 (1999)729–740. Printed in the UK,[2] Crawley E F and Anderson E H 1985Detailed models of piezoelectric actuationof beams J. Intell. Mater. Syst. 1 4–25[3] E.P. Hadjigeorgioua*, G.E. Stavroulakisb,C.V. Massalasa, Shape control and damageidentification of beams using piezoelectricactuation and genetic optimization,International Journal of EngineeringScience 44 (2006) 409–421,[4] H. Irschik, A review on static and dynamicshape control of structures usingpiezoelectric actuation, Comput. Mech. 26(2002) 115–128.[5] H.F. Tiersten, Linear Piezoelectric PlateVibration, Plenum Press, New York, 1969.of adaptive structures, CMS ConferenceReport, 11th May 2000[6] H.S. Tzou, Piezoelectric Shells, KluwerAcademic Publishers, The Netherlands,1993.[7] Osama J. Aldraihem, Optimal Size andLocation of Piezoelectric Actuator/Sensors:Practical Considerations, journal ofguidance, control, and dynamics Vol. 23,No. 3, May–June 2000,[8] Rui Ribeiro and Suzana Da Mota Silva,Genetic algorithms for optimal design andcontrol of adaptive structures, CMSConference Report, 11th May 2000[9] S daMota Silva1, RRibeiro2, andJMMonteiro2, The application of geneticalgorithms for shape control withpiezoelectric patches — an experimentalcomparison, Smart Mater. Struct. 13 (2004)220–226,[10] Sunar M and Rao S 1999 Recent advancesin sensing and control of flexible structuresvia piezoelectric materials technology Appl.Mech. Rev. 52 1–16[11] Y Yu, X N Zhang1 and S L Xie, Optimalshape control of a beam using piezoelectricactuators with low control voltage, SmartMater. Struct. 18 (2009) 095006 (15pp),