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Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www...
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  1. 1. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945940 | P a g eA Finite Element Model And Active Vibration Control OfComposite Beams With Distributed Piezoelectrics Using ThirdOrder TheoryPratik K. Gandhi*, J. R. Mevada***Student, M.Tech, (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva)** Associate Prof., (Department of Mechanical & Mechatronics Engineering, Ganpat University, Kherva)AbstractIn this study, A finite element modelbased on the third order theory is presented forthe active vibration control of composite beamswith distributed piezoelectric sensors andactuators. For calculating the total charge onthe sensor, the direct piezoelectric equation isused and the actuators provide a damping effecton the composite beam by coupling a negativevelocity feedback control algorithm in a closedcontrol loop. A modal superposition techniqueand a Newmark-β method are used in thenumerical analysis to compute the dynamicresponse of composite beams. Algorithm forcalculating the various values of variables iscarried out using Matlab toolKeywords- Active vibration control, cantilevercomposite beam, Newmark-β method1. IntroductionEngineering structures work frequentlyunder dynamic excitations. The type of excitationmay vary but the results of these excitations areshown generally in the form of the vibrations.Vibrations can be attributed as an unwanted formany engineering structures due to precisionlosses, noise, waste of energy, etc. and should bekept under control for lightweight structures.Attempts at solving these problems have recentlystimulated extensive research into smart structuresand systems. A smart structure can be defined as astructure with bonded or embedded sensors andactuators with an associated control system, whichenables the structure to respond simultaneously toexternal stimuli exerted on it and then suppressundesired effects. Smart structures have foundapplication in monitoring and controlling thedeformation of the structures in a variety ofengineering systems. Advances in smart materialstechnology have produced much smaller actuatorsand sensors high integrity in structures and anincrease in the application of smart materials forpassive and active structural damping. Severalinvestigators have developed analytical andnumerical, linear and non-linear models for theresponse of integrated piezoelectric structures.These models provide platform for exploring activevibration control in smart structures. Theexperimental work of Bailey and Hubbard, 1985[1] is usually cited as the first application ofpiezoelectric materials as actuators. Theysuccessfully used piezoelectric sensors andactuators in the vibration control of isotropiccantilever beams. P. R. Heyliger and J. N. Reddy,1988 [2] had presented higher order beam theoryfor static and dynamic behaviour of rectangularbeams using finite element equations. Ha, Keilersand Chang, 1992 [3] developed a modal based onthe classical laminated plate theory for the dynamicand static response of laminated composites withdistributed piezoelectric. H.S. Tzou and C. I.Tseng, 1990 [4] had presented finite elementformulation of distributed vibration control andidentification of coupled piezoelectric systems.Chandrashekhara and Varadarajan, 1997 [5] gave afinite element model based on higher order sheardeformation theory for laminated composite beamswith integrated piezoelectric actuators. Woo-seokHwang, Woonbong Hwang and Hyun Chul Park,1994 [6] had derived vibration control of laminatedcomposite plate with piezoelectric sensor/actuatorusing active and passive control methods. X. Q.Peng, K. Y. Lam and G. R. Liu,1998 [7] gave finiteelement model based on third order theory for theactive position and vibration control of compositebeams with distributed piezoelectric sensors andactuartors. Bohua Sun and Da Huang, 2001 [8] hadderived analytical solution for active vibrationcontrol and suppression of smart laminatedcomposite beam with piezoelectric damping layer.V. Balamuragan and S. Narayanan, 2002 [9]proposed finite element formulation and activevibration control on beams using smart constrainedlayer damping treatment (SCLD). S. Narayananand V. Balamuragan, 2003 [10] gave finite elementmodal of piezolaminated smart structures for activevibration control with distributed sensors andactuators.2. Mathematical modelling of beam2.1 Piezoelectric equationsAssuming that a beam consists of anumber of layers (including piezoelectric layers)and each layer possesses a plane of material
  2. 2. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945941 | P a g esymmetrically parallel to the x-y plane and a linearpiezoelectric coupling between the elastic field andthe electric field the constitutive equations for thelayer can be written as, [12]15 111 1 133313 5 30 000 kkk k keD EeD E                        (1)31111 1 155 155 5 3000 0k kk k keQ EQ e E                         (2)The thermal effects are not considered in theanalysis.The piezoelectric constant matrix  e can beexpressed in terms of piezoelectric strain constantmatrix  d as    e d Q (3)Where,  153100ddd    (4)2.2 Displacement field of third order theoryThe displacement field based on the third orderbeam theory of Reddy [2] given by      3 00, , , ,x xwu x z t u x t z x t zx        (5)   0, , ,w x z t w x t (6)Where, 243h  and h is the total thickness of thebeam.The displacement functions are approximated overeach finite element by [7]     201, i iiu x t u t x  (7)     21,x i iix t t x    (8)     401, i iiw x t t x  (9)Where, i are the linear Lagrangian interpolationpolynomials and the i are the cubic Hermitinterpolation polynomials. 1 and 3 representnodal values of 0w , whereas 2 and 4 representnodal values of 0wx.We define   Tu u w   1 1 1 2 2 2 3 4Tu u u     (10)Using finite element equations (5) and (6) can beexpressed as    u N u (11)The strain-displacement relations are given by    15B u    (12)Where,  N and  B are as follows   1313 113 222323 333 440000Tz zzxzxNz zzxzx                         (13)       13 21123 21 1 1223 22 2 2223 22223 23 3 3223 24 4 4201 3 03301 3 033xz z zxz zx x xz zx x xBxz z zxz zx x xz zx x x                                                             T(14)2.3 Dynamic EquationsThe dynamic equations of a piezoelectric structurecan be derived by Hamilton’s principle [7]. 210ttT U W dt    (15)
  3. 3. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945942 | P a g eWhere T is the kinetic energy, U is the strainenergy, and W is the work done by the appliedforces.In this, the kinetic energy at elemental level is   12 eTeVT u u dV    (16)Where, eV is the volume of the beam element.The strain energy at elemental level is   12 eTeVU dV   (17)And work done by the external forces is           1eT T Teb s cV SW u f dV u f dS u f   (18)Where,  bf is the body force, 1S is the surfacearea of the beam element,  sf is the surface forceand  cf is the concentrated load.2.4 Equation of motionTo develop the equation of motion of the system,consider the dynamic behavior of the system.These equations also provide coupling betweenelectrical and mechanical terms. The equation ofmotion in the matrix form can be written as, [7]     e e ee e e euvM u K u F K V            (19)Where,   eTeVM N N dV     (20)    eTeVK B Q B dV     (21)     eT Teuv vVK B e B dV     (22)             1eT T Teb s cV SF N f dV N f dS N f   (23)In order to include the damping effects, Rayleighdamping is assumed. So equation (19) modified as,       e e ee e e e e euvM u C u K u F K V                  (24)Where,eC   is the damping matrix, which ise e eC a M b K            (25)Where, a andb are the constants can bedetermined from experiments [11].Assembling all elemental equations gives globaldynamic equation is as follows:            VM u C u K u F F    (26)Where,  F is the external mechanical forcevector and  VF is the electrical force vector.    v uvF K V (27)2.5 Sensor equationsSince no external electrical field is applied to thesensor layer and a charge is collected only in thethickness direction, only the electric displacementis of interest and can be written as [7]3 31 1D e  (28)Assuming that the sensor patch covers severalelements, the total charge developed on the sensorsurface is          11 1 31112sk kjNjz z z zj Sq t B B e dS u       (29)Where,  1B is the first row of B .The distributed sensor generates a voltage when thestructure is oscillating; and this signal is fed backinto distributed actuator using a control algorithm,as shown in fig. 1. The actuating voltage under aconstant gain control algorithm can be expressedas,ei s i cdqV GV GGdt  (30)Fig.1 Block diagram of feedback control systemThe system actuating voltages can be written as
  4. 4. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945943 | P a g e     vV G K u  (31)Where,  G is the control gain matrix andi cG G G .In the feedback control, the electrical force vector vF can be regarded as a feedback force.Substituting equation (31) into equation (27) gives,      v uv vF K G K u  (32)We define,   uv vC K G K     (33)Thus, the system equation of motion equation (26)becomes           M u C C u K u F      (34)As shown in equation (34), the voltagecontrol algorithm equation (30) has a dampingeffect on the vibration suppression of a distributedsystem.For obtaining a dynamic response under a givenexternal loading condition, a modal analysis isused, and the nodal displacement  u isrepresented by [11]    u x  (35)Where  x are referred to as the generalizeddisplacements.  is the modal matrix and has theorthogonal properties as:     2TK       ,       TM I  (36)Where,2   is a diagonal matrix that stores thesquares of the natural frequencies i .Substituting equation (35) into (34) and thenmultiplying equation (34) gives                      T T T TM x C C x K x F             (37)Substituting equation (36) into (37) gives             22T Tx C x x F          (38)The initial condition on  x is      0 0Tx M u  (39)      0 0Tx M u   (40)3. Material properties of cantilevercomposite beamA cantilever composite beam with bothupper and lower surfaces symmetrically bonded bypiezoelectric ceramics, shown in fig. 2Fig. 2 A cantilever composite beamThe beam is made of T300/976graphite/epoxy composites and the piezoceramic isPZT G1195N. The adhesive layers are neglected.The material properties are given in table 1. Thestacking sequence of composite beam is0 045 / 45s   . The total thickness of thecomposite beam is 9.6 mm and each layer has thesame thickness (2.4 mm). and the thickness of eachpiezo-layer is 0.2 mm. The lower piezoceramicsserve as sensors and the upper ones as actuators.The relative sensors and actuators formsensor/actuator pairs through closed control loops.Table 1. Material properties of PZT andT300/976 graphite/epoxy compositeProperties Symbol PZT T300/976Young moduli(GPa)E11E22=E3363.063.0150.09.0Poisson’s ratio v12=v13v230.30.30.30.3Shear moduli(GPa)G12=G13G2324.224.27.102.50Density(Kg/mg3)Ρ 7600 1600Piezoelectricconstants (m/V)d31=d32 254×10-12Electricalpermittivity(F/m)ε11=ε22ε3315.3×10-915.0×10-94. Active vibration suppressionHere size of the beam is considered as 600mm long, 40 mm width and 9.6 mm thick, alsothickness of each piezo layer is 0.2 mm.In the analysis, the composite beam isdivided into 30 elements. The piezoelectriccomposite beam given in figure 2 is considered to
  5. 5. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945944 | P a g esimulate the active vibration suppression through asimple S/A active control algorithm (negativevelocity feedback). The piezoceramics on the lowersurface serve as sensors,and on the upper surfaceare actuators. In the analysis the beam is dividedinto 30 elements so each S/A pairs covers sixelements.First, modal superposition technique isused to decrease the size of the problem. thedifferent modes are used in the modal spaceanalysis and transient response of the cantileverbeam is computed by the Newmark directintegration method [11]. The parameters α and βare from Rayleigh method.Results obtain after these methods are as underFig. 3 The effect of negative velocity gain onthe beam subjected to first mode vibration. (Gain=-2 V/A)Fig. 4 The effect of negative velocity gain on thebeam subjected to first mode vibration. (Gain=-5V/A)Figure 3 and 4 shows the vibration controlperformance for PZT fully covered beam withcontrol gain -2 V/A and -5 V/A for first value ofalpha and beta of Rayleigh damping.Fig. 5 The effect of negative velocity gain on thebeam subjected to second mode vibration. (Gain=-1V/A)Fig. 6 The effect of negative velocity gain on thebeam subjected to second mode vibration. (Gain=-5V/A)Figure 5 and 6 shows the vibration controlperformance for PZT fully covered beam withcontrol gain -1 V/A and -5 V/A for the secondvalue of alpha and beta of Rayleigh damping.Fig. 7 The effect of negative velocity gain on thebeam subjected to third mode vibration. (Gain=-1V/A)Fig. 8 The effect of negative velocity gain on thebeam subjected to third mode vibration. (Gain=-5V/A)Figure 7 and 8 shows the vibration controlperformance for PZT fully covered beam withcontrol gain -1 V/A and -5 V/A for the secondvalue of alpha and beta of Rayleigh damping.As shown in figures, the vibrations decaymore quickly when higher control gains areapplied. However, it must be noted that the gainsshould be limited for the sake of the breakdownvoltage of the piezoelectrics.0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam0 20 40 60 80 100 120-0.1-0.0500.050.1Time (sec)TipDisplacement(m)Tip Displacement of cantilever beam
  6. 6. Pratik K. Gandhi, J. R. Mevada / International Journal of Engineering Research andApplications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.940-945945 | P a g e5.ConclusionA finite element model and computercodes in Matlab, based on third order laminatetheory, are developed for the cantilever compositebeam with distributed piezoelectric ceramics.Equation of motion of composite beam is derivedby using Hamilton’s principle. The S/A pairs must be placed in highstrain regions and away from the area of low strainregions for maximum effectiveness. More S/A pairs can generally induce moreefficiency on the active vibration suppression. Sothe number of S/A pairs has a great effect on theperformance of smart structures. As the feedback gain increases, thedamping ratios are large and as the feedback gaindecreases, the damping ratios are smaller. Changes in stiffness and the location andthe number of S/A pairs also affect the dynamicresponse of the composite structure.So, based on this work, active vibration controlwith finite element model is reliable and fail-safe.References[1] Bailey T. and Hubbard J.E., Distributedpiezoelectric-polymer active control of acantilever beam, Journal of Guidance,Control and Dynamics, vol. 8, pp. 605-611, (1985).[2] P. R. Heyliger and J. N. Reddy, A higherorder beam finite element for bending andvibration problem, Journal of Sound andVibration 126, 309-326 (1988).[3] S. K. Ha, C. Keilers and F. K. Chang,Finite element analysis of compositestructures containing distributedpiezoceramics sensors and actuators,American Institute of Aeronautics andAstronautics Journal, pp. 772-780 (1992).[4] H. S. Tzou and C. I. Tseng, Distributedpiezoelectric sensor/actuator design fordynamic measurement/control ofdistributed parameter system: apiezoelectric finite element approach,Journal of Sound and Vibration 138,17-34(1990).[5] K. Chandrashekhara and S. Varadarajan,Adaptive shape control of compositebeams with piezoelectric actuators,Journal of Intelligent Material Systemsand Structures, vol. 8, pp. 112-124,(1997).[6] Woo-Seok Hwang, Woonbong Hwangand Hyun Chul Park, Vibration control oflaminated composite plates withpiezoelectric sensor/actuator: Active andpassive control methods, MechanicalSystems and Signal Processing, vol. 8(5),571-583, (1994).[7] X. Q. Peng, K. Y. Lam and G. R. Liu,Active vibration control of compositebeams with piezoelectrics, Journal ofSound and Vibration, 209, 635-650,(1998).[8] Bohua Sun, Da Huang, Vibrationsuppression of laminated composite beamswith piezoelectric damping layer,Composite Structures vol. 53, 437-447,(2001).[9] V. Balamurugan and S. Narayanan, Finiteelement formulation and active vibrationcontrol study on beam using SCLDtreatment, Journal of Sound andVibration, vol. 249(2), 227-250, (2002).[10] S. Narayanan and V. Balamurugan, Finiteelement modeling of piezolaminated smartstructures for active vibration control withdistributed sensors and actuators, Journalof Sound and Vibration, vol. 262, 529-562,(2003).[11] K. J. Bathe, Finite element procedures inengineering analysis. Englewood Cliffs,New Jersey: Prentice- Hall.[12] IEEE Standard on Piezoelectricity ANSI-IEEE Std 176-187, (1987)

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