Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 4, pp. 919-930, Warsaw 2007 ADAPTIVE VIBRATION CONTROL THROUGH A SMA EMBEDDED PANEL Gianluca Diodati Salvatore Ameduri Antonio Concilio The Italian Aerospace Research Centre, CIRA ScpA, Italy e-mail: g.diodati@cira.it; s.ameduri@cira.it; a.concilio@cira.it Dynamic behavior of structural elements and their performance in terms of noise and vibration controlmay be remarkably affected by several pa- rameters, like geometry, material properties, stress field, etc. Ability of adaptively controllingoneormoreof theseparameters leads to a structu- re fitting different requirements in several working conditions. Research activities presented in this work are focused on the design of a fiber- glass laminate structural element with SMA wires embedded along the widest dimension. SMA contraction by the Joule effect heat adduction leads, if suitably constrained on the edges, to an internal stress field with a consequent over-all stiffness increase. The finite element code MSC.NASTRAN is used to simulate the whole system made of a plate and SMAwires controlling elements. The behavior of the SMA ismode- led through theNASTRAN cardCELAS that allows one to consider the additional SMA activation stiffness with a suitable ”spring” constant, depending on the wire temperature and a suitable discretization para- meter. The finite differencemodel of the SMA is achievedand integrated in the FEM solver. Key words: SMA, embedding, vibration 1. Introduction The ability of affecting the dynamic response of a structure, so that its per- formance in terms of noise and vibration control could fit several working con- ditions, represents a challenging objective. Due to the limitations of practical conventional solutions and due to the impossibility of producing significant 920 G. Diodati et al. changes of themain structural features, innovative materials and design stra- tegies have well done for themselves. According to this trend, the search for non-conventional materials oriented to the actuation to satisfy control perfor- mance requirements has been the main task during the past years. Among all, NiTi Shape Memory Alloys (SMAs) with their abilities to change their material properties such as Young’s modulus (Ford and White, 1996; Otsuka and Wayman, 1998), damping capacity (Gandhi and Wolons, 1999; Piedbo- euuf et al., 1998), and the generation of large internal forces (Rogers et al. 1989), have foundmany applications. The idea of embedding SMA actuators in a composite laminate for structural control was first introduced by Rogers and Robertshaw [Rogers et al. 1988]. Such a structure was termed a Shape Memory Alloy Hybrid Composite (SMAHC). The ”Smart Structure&Material” concept and related propertiesmay be used to make up for unsatisfactory dynamic characteristics, usually referred to natural frequencies and mode shapes; moreover, the design of a structure which is known to experience a dynamicworking environment needs to satisfy some defined criteria such as averting vibration resonances in a variable load environment. Shift of natural frequencies away from resonance conditions, shift of an antiresonance to a selected frequency for an assigned FRF, pole-zero cancel- lations, no resonance frequency range creation, may be achieved by precise tuning of SMA components in SMAHCs (Fuller et al., 1996; Rogers et al., 1989). Rogers et al. (1989) presented concepts of using SMA wires for control of natural frequencies andmodes of vibration of the simply supported plate but only examined SMA/Epoxy composite plates, for which the relative volume fraction of SMAwires was very high. Changes in natural frequencies of clamped-clamped composite beamswith SMA wires were investigated analytically and experimentally by Baz et al. (1995). Natural frequencies of composite beamsmodified in this manner were also significantly affected, butavery lowthickness-to-length ratiowasadopted. It was shown that for a beam with SMA fibres with a nitinol volume of 15% the first natural frequency of the beam increased from 21 to 62Hz, by heating the SMA wires from the room temperature up to 149◦C (Rogers et al., 1991). Ostachowicz et al. 1998 investigated more general SMAcomposites paying attention to factorswhich influence thecomposite performance: the ratio of the SMA/reinforcing fibresYoung’s modulus ratio, the relative volume fraction of theSMAcomponents, the relative volume fraction of the reinforcingfibres, the Adaptive vibration control through... 921 structure thickness-to-length ratio, the location and orientation of the SMA components within the structure, and so on. Two methods have been proposed for integrating SMA actuators into a composite: bonding the actuators within the composite matrix as a consti- tuent and embedding the actuators within sleeves through the laminate and attaching it at some convenient chosen point in order to eliminate high she- aring stresses arising from their activation process.Thefirstmethod gives rise, from a control point-of-view, to the Active Property Tuning method, which exploits only changes in the stiffness of SMA components during their acti- vation, while the second one to the so called Active Strain Energy Tuning method, which exploits the high recovery stress generated during activation of the SMA elements. Variousmethodology, in the framework of Finite ElementAnalysis (FEA), have been proposed to simulate the behaviour of the SMAHC, differing for the underlyingassumptions.Onemethod is to develop special composite elements. This has been done for multi-layered composite plates (Lagoudas et al., 1997; Ostachowicz et al., 1998; Zak et al., 2003) and for layered beams (Marfia et al., 2003). A more general way is to model the matrix and reinforcing members separately (Gao et al., 2004; Ghomshei et al., 2001; Sun et al., 2002). The present paper is organized as in the following. Section 2 describes a numerical specimen that has been simulated and analyzed through the MSC.NASTRANcode. Section 3 presents numerical models used in the simu- lation. Section 4 illustrates results concerning a preliminary estimate focused on the stiffness increase of a single wire due to activation. Lastly, Section 5 collects related results in terms of the dynamic response of the system with and without wires activation. Furthermore, the dependence of the dynamic behaviourwith respect to the number of SMAwires and different initial stress conditions are presented. 2. Main features of the numerical specimen Numerical investigations illustrated in the following paragraphs have been performed on a ply-angle symmetric hybrid composite panel with planar di- mension of 330× 210mm. Constituted by twelve 0.3mm thick glass fiber reinforced epoxy plies disposed according to a stacking sequence of [±45G3 ]S and characterized by a fiber volume ratio of 0.2, the specimen has been inte- grated by SMA wires inserted into sleeves embedded within the mid-surface and running along the plate widest dimension. 922 G. Diodati et al. The wires are free to move within the sleeves but fixed on both the ends. Due to this design choice, no shear stresses are transmitted by the structure to thewires,witha consequent easier activation, andno local instability problems (buckling, wrinkling,...)may occur, being laminate in-plane displacements not enforced bywires contraction. Thematerial and geometrical properties of the panel are summarized in Table 1. The properties of the SMA adopted are listed in Table 2. Figure 1 shows the physical arrangement considered in the numerical simulation. Table 1.Geometry and physical constants for the panel Parameter Value Parameter Value Dimensions Lx×Ly =330mm×210mm Poisson ratio resin 0.35 Thickness 12 ·0.3mm=3.6mm Mass density fibres 2250Kg/m3 Stacking sequence [±45G 3 ]S E fibres 65.5GPa Mass density resin 1250.0Kg/m3 Poisson ratio fibres 0.23 E resin 3.43GPa Vol. ratio of fibres 0.20 Table 2.Material properties of the nitinol alloy Modul, Transformation Transformation Maximum density temperatures constants residual strain EA =67.0GPa MF =9.0 ◦C CM =8.0MPa/ ◦C εL =0.067 EM =26.3GPa MS =18.4 ◦C CA =13.8MPa/ ◦C θ=0.55MPa/◦C AS =34.5 ◦C σS =100.0MPa ρ=6448.1Kg/m3 AF =49.0 ◦C σF =170.0MPa Fig. 1. The physical arrangement considered in simulations Adaptive vibration control through... 923 3. Modeling strategy and numerical tools Numerical investigations on theaforementioned specimenhavebeenperformed by adopting a FE approach. The modeling of the panel has been carried out through the FEMAP 9.0 preprocessor, while numerical predictions in terms of mode shapes and modal frequencies up to a value of 1000Hz have been achieved by the MSC/NASTRAN solver. A right-handedCartesian coordinate systemhas been chosen for the global FE space. The origin of the global coordinate system has been located on a panel corner. The X-direction has been chosen along the widest dimension and the Z-direction orthogonal to the specimenmid-surface. 6400 CQUAD elements (100 along X and 64 along Y directions) have been used to represent the plate. The 2D-orthotropicmaterial nature has been defined through the PCOMP element card. To take into account the additional mass and stiffness due to the SMA wires, 200 elements for each wire (in the simulation several numbers of wires havebeenconsidered)havebeenaddedto themodel, to subtract thecomposite properties and to add the SMA properties. Tomodel theSMAsactuators, aFiniteDifference (FD) scheme that canbe included in the Finite Element code through the CELAS card has been used. Themechanism responsible for the shape recovery is a phase transformation of the SMAmaterial frommartensite to austenite. In themartensitic phase, the SMA is relatively soft and can be plastically deformed with low stress levels. As the alloy is heated, austenitic transformation occurs between a certain temperature range. The phase transformation frommartensite to austenite is associated with a strain recovery process. If suitable constraint conditions are imposed, a stress field occurs inside the SMAelements during the recovery. To be able to predict the deformation behavior of a structure subjected to SMA- induced forces, the interaction of the SMA wire with the structure needs to be clearly understood as well as the thermo-mechanical constitutive modeling involved in the strain recovery of the SMA. The Brinson model (Brinson, 1993; Rogers et al., 1991) has been adop- ted as the thermo-mechanical model of the SMA. The constitutive law is the generalized Hooke law σ=σ0+D(ξ)ε−D(ξ0)ε0−εLD(ξ)ξs+εLD(ξ0)ξs0+θ(T −T0) (3.1) where σ, ε, T represent the actual stress, strain and temperature, the sym- bol (·)0 denotes related initial values, θ – the thermoelastic coefficient. It is 924 G. Diodati et al. assumed thatYoung’smodulus D is a function of themartensite volume frac- tion ξ. The function to describe the martensite volume fraction ξ is defined as the sum of two fractions ξ= ξS +ξT (3.2) where ξS and ξT describe stress and temperature-inducedmartensite volume fractions, respectively. Further details on themodel can be found in Zak et al. (2003). The dynamic properties of the plate can be predicted as a function of temperature of the SMA wire. Assuming the initial and total strain in the wire (ε0 and ε values), temperature of the wire T and the initial strain σ0, residual equation (3.1)maybe solved in termsof σ; related axial force F inside the wire may be derived (no interaction between the wire and the panel are allowed in the activation phase). From the tension F, the effect of actuators on the structure may be calculated. The wires within the sleeves behave like a string governed by the equation m d2w(x,t) dt2 −F d2w(x,t) dx2 = f(x,t)= fm(x,t)+fk(x,t) (3.3) where m is the wire mass per unit length, w – the vertical displacement of the wire and F – the tension. The force acting on the structure due to the SMAwire is represented by the term −f(x,t). Themass effects of the wire is taken into account by adding additional beam elements to the plate FEmodel (the term −fm(x,t) = −md 2w(x,t)/dt2), but the most important change in the structure is caused by themechanical coupling between the string and the plate subsystems (−fk(x,t)=Fd 2w(x,t)/dx2). Basing on the finite difference scheme, this force can be expressed by the following equation f plate i =−fk(xi, t)∆l= F ∆l (wi+1−wi)− F ∆l (wi−wi−1) (3.4) theMSC/NASTRANCELAS cardmay be used to add the two contributions of equation (3.4) if a suitable spring constant F/∆l is adopted. The FE model has been tested through a theoretical model developed for a beam-like structure clamped at both ends (see Diodati and Ameduri [4]). Neglecting the bending stiffness of the string, the natural frequencies of the system are the zeros of the following equation cos(k2L)− 1 cosh(k1L) −β2 sin(k2L)tanh(k1L)= 0 (3.5) InFig.2a, thefirst fourbeamnatural frequencies of the systemare plotted, while in Fig.2b the shift of these frequencies for SMA wire tension between Adaptive vibration control through... 925 0N and 110N is illustrated. The maximum relative error of 2.5%, when the internal force is 110N, shows a good agreement between the numerical and theoretical models. Fig. 2. Natural frequencies of the beam – comparison of numerical and theoretical results 4. Estimate of stiffness increase of a single wire To have an idea of the SMAwires ability to affect the dynamic response and increase stiffness, the abovementioned SMA model has been adopted to esti- mate the singlewiremodes shift, before evaluating its effect on the surrounding structure. More in detail, by hindering each deformation of thewire, the axial load P has been estimated vs. temperature. The frequency shift has been evaluated by Blevins et al. (1995) fi = ( i2π2 2πL2 ) √ 1+ PL2 i2π2EI √ EI m (4.1) where i, L, m, I, E represent the mode id., the wire length, the mass per unit length, the cross section inertia moment and the Young modulus. No initial stress has been imposed. The first 10modes shifts vs. temperature and axial stress have been plotted in Fig.3. A temperature increase of 40◦C, has determined an arise of the first mode from 7.48 us to 229.2Hz. 926 G. Diodati et al. Fig. 3. Frequency shift in a single wire 5. Numerical results Numerical calculations have been carried out for examination of SMA wires actuation effect on dynamic characteristics of the plate. As the first example, some results for the influence of SMA wires number are presented in Fig.4. Increasing the number of SMA wires results in increasing the weight of the plate that hinders the natural frequencies from shifting towards higher values. Fig. 4. Effect of increasing the number of SMAwires The study shows thatwith a very limited increase inweight, one can reach a sensible shift of all natural frequencies of the plate (up to 160Hz). All the modes are affected by the actuation, but the best performance is observed for modes for which the nodal lines are perpendicular to the orientation angle Adaptive vibration control through... 927 of SMA wires (Fig.4b). The sensitivity study has proved that the frequency shift is a quasi-linear function of the wires number: the SMA control system [df/d(weight)] does not undergo saturation (Fig.4a). The initial stress and strain conditions (Fig.5) have proved to strongly affect the SMAtemperature influence on dynamic characteristics: the larger is the initial stress, the lower is the frequency shift. The initial strain has to be kept as small as possible to have the final activation temperature lower than the resin cure one. Fig. 5. Effect of increasing the temperature of SMA wires for different initial conditions FRFcurves plotted inFig.6 show that large peaks andanti-resonance shift may be obtained only in the transformation temperature range (i.e. 36-74◦C and 45-75◦C for the initial stress of 0MPa (Fig,6a) and 128MPa (Fig.6b), respectively). Fig. 6. Effect of increasing the temperature of SMAwires: FRF data 928 G. Diodati et al. 6. Conclusions and further steps In the present paper, design and implementation of a numerical model aimed at describing the dynamic behavior of an anisotropic plate controlled by em- bedded SMA wires has been described. The stress field due to contraction of SMAwires constrained at the edges has been used for increasing the over-all stiffness. The numericalmodel has been obtained by adopting a finite differen- ce approach aimed at estimating the stiffness increase due to activation of the axial load, predicted by the Brinson model. The achieved results have been presented in terms of frequency shift of both the single wire and the entire structure. Further efforts will be spent on themodeling of structure controlled by SMAwires. The aforementionedmodelwill be comparedwith another one, describing the stiffness increase through an additional stiffness contribution (axial load geometric stiffness). Experimental campaigns are being carried out to validate the numerical predictions. References 1. BazA.,PohS.,RoJ.,GilheanyJ., 1995,Control of the natural frequencies of nitinol-reinforced composite beams, Journal of Sound andVibration, 185, 1, 171-185 2. BlevinsR.D., 2001,Formulas for Natural Frequency andMode Shape, Krieger Publishing Company,Malabar Florida, p.144 3. Brinson L.C., 1993, One-dimensional constitutive behavior of shapememory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable, Journal of Intelligent Material Systems and Structures, 4, 2, 229-242 4. Diodati G., Ameduri S., Natural Frequencies of a Coupled Beam-String System, Technical Report CIRA-TR-06-0187 5. Ford D.S., White S.R., 1996, Thermomechanical behaviour of 55Ni45Ti nitinol,Acta Materialia, 44, 6, 2295-2307 6. FullerC.R.,ElliottS.J.,NelsonP.A., 1996,ActiveControl ofVibration, Academic Press Ltd., London 7. Gandhi F.,WolonsD., 1999,Characterizationof the pseudoelastic damping behaviour of shape memory alloy wires using complex modulus, Journal of Smart Material Structures, 8, 49-56 Adaptive vibration control through... 929 8. Gao X., Burton D., Brinson L.C., 2004, Finite element simulation of a self-healing shapememory alloy composite,Mechanics of Materials, in press 9. Ghomshei M.M., Khajepour A., Tabandeh N., Behdinan K., 2001, Fi- nite elementmodeling of shapememory alloy composite actuators: theory and experiment, Journal of Intelligent Material Systems and Structures, 12, 11, 761-773 10. Lagoudas D.C., Moorthy D., Qidway M.A., Reddy J.N., 1997,Model- ling of the thermomechanical response of active laminates with SMA strips using the layerwise finite elementmethod, Journal of Intelligent Material Sys- tems and Structures, 8, 6, 476-488 11. Marfia S., Sacco E., Reddy J.N., 2003, Superelastic and shape memory effects in laminated shapememory alloy beams,AIAA Journal, 41, 1, 100-109 12. Ostachowicz W., Krawczuk M., Zak A., 1998, Natural frequencies of multi-layer composite plate with embedded shapememory alloywires, Journal of Intelligent Material Systems and Structures, 9, 3, 232-237 13. OtsukaK.,WaymanC.M., 1998,ShapeMemoryMaterials, CambridgeUni- versity Press, Cambridge 14. Piedboeuuf M.C., Gauvin R., Thomas M., 1998, Damping behaviour of shapememory alloys: strain amplitude, frequency and temperature effects, Jo- urnal of Sound and Vibration, 214, 5, 885-901 15. RogersC.A.,BakerD.K., JaegerC.A., 1989a, Introduction to SmartMa- terials and Structures, Smart Materials, Structures, and Mathematical Issues, Technomic Publishing Company, Inc. 16. Rogers C.A., Liang C., Baker D.K., 1989b, Dynamic Control Concepts Using ShapeMemory alloy Reinforced Plates, SmartMaterials, Structures, and Mathematical Issues, Technomic Publishing Company, Inc. 17. Rogers C.A., Liang C., Fuller C.R., 1991, Modelling of shape memory alloy hybrid composites for structural acoustic control, Journal of Acoustical Society of the America, 89, 210-220 18. Rogers C.A., Robertshaw H.H., 1988, Shape Memory Alloy Reinforced Composite, Engineering Science Preprints 25, Society of Engineering Science, Inc., ESP25.8027 19. Sun S.S., Sun G., Wu J.S., 2002, Thermo-viscoelastic bending analysis of a shape memory alloy hybrid epoxy beam, Smart Materials and Structures, 11, 6, 970-975 20. Zak A., Cartmell M., Ostachowicz W., 2003a, Static and dynamic beha- viour of composite structures with shapememory alloy components,Materials Science Forum, 440/441, 345-352 930 G. Diodati et al. 21. Zak A.J., Cartmell M.P., Ostachowicz W., 2003b, Dynamics of multi- layered composite plates with shape memory alloy wires, Journal of Applied Mechanics-Transactions of the ASME, 70, 3, 313-327 Adaptacyjne sterowanie drganiami w panelu zawierającym elementy ze stopu z pamięcią kształtu Streszczenie Właściwości dynamiczne elementów konstrukcyjnych oraz ich cechy rozważane pod kątem sterowalności drganiami i hałasem mogą być znacząco zmieniane takim parametrami jak geometria,właściwościmateriałowe, pole naprężeń, itp. Zdolność do modyfikacji jednego lubwięcej zwyżejwymienionychparametrów tworzykonstrukcję adaptowalnądo różnychwarunkówpracy.Badania opisanew artykule dotyczą panelu wykonanego z laminatuwzmacnianegowłóknemszklanym i dodatkowozawierającego druty ze stopu z pamięcią kształtu (SMA)umieszczonewzdłuż największegowymiaru panelu. Kurczenie się stopu wywołane efektem Joule’a przy obecności źródła ciepła wytwarzawewnętrzne pole naprężeń, które prowadziwprost do zwiększenia sztywno- ści panelu. Do analizy całego układu laminowanej płyty i zatopionych drutów SMA użyto pakietuMSC.NASTRAN. Zachowanie SMAsymulowanowpakiecie za pomocą karty CELAS, która pozwoliła na uwzględnienie dodatkowej sztywności od aktywacji SMA poprzez wprowadzenie nowej „sprężyny” o sztywności zależnej od temperatury oraz zadeklarowanego parametru dyskretyzacji. Model SMA wygenerowany metodą elementów skończonych poddano całkowaniu wewnątrz zastosowanego pakietuMES. Manuscript received February 21, 2007; accepted for print April 4, 2007