Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 3, pp. 641-664, Warsaw 2011 VIBRATION ANALYSIS OF A TRIMORPH PLATE FOR OPTIMISED DAMAGE MITIGATION Akuro Big-Alabo University of Port Harcourt, Department of Mechanical Engineering, Rivers State, Nigeria e-mail: bigalabo@yahoo.com Matthew P. Cartmell University of Glasgow, School of Engineering, Systems, Power and Energy Research Division, Glasgow, Scotland, UK; e-mail: matthew.cartmell@glasgow.ac.uk The dynamic response of a viscously damped rectangular trimorph plate subjected to a sinusoidally distributed load was investigated for simply- supported boundary conditions. The governing equation for the nonlinear deflection of the plate, which is first introduced in this paper, was deri- ved based on the classical plate theory (CPT) and the classical laminate theory (CLT). The governing equation was solved using the Navier me- thod and direct numerical integration. Optimised time-domain response plots for a trimorph plate made up of aluminium (Al), polyvinylidene fluoride (PVDF) and lead zirconate titanate (PZT) layers revealed that only three out of the six possible layer configurations are necessary for de- termining the best layer-stacking. In determining the best layer-stacking for the optimised dynamic response, three factors were considered name- ly: the stiffness, natural frequency and damping constant. Both of the Al/PVDF/PZT or Al/PZT/PVDF configurations were found to produce the best response qualities i.e. high elastic stiffness, high natural frequen- cy and low viscous damping. Frequency-domain plots were generated to compare the nonlinear and linear responses and it was discovered that the effect of the nonlinearity predictably reduces the natural frequency of the trimorph plate. This study can be applied to the analysis of optimised damage mitigation of intelligent car bodies and safety critical structures which are subject to potentially destructive loading conditions. Key words: trimorph, layer-stacking, reference layer, classical plate theory 1. Introduction In recent times, research on the vibration of plates has inclined towards lami- nated plates. This is because desirable operational qualities of materials such 642 A. Big-Alabo, M.P. Cartmell as strength, stiffness, low-weight, wear resistance andacoustical insulation etc. can be achieved by using laminates (Jones, 1999), especially composite lami- nates. Lee (1990) developed the theory of laminated piezoelectric plates (TLPP) for the design of distributed sensor/actuators. He developed the governing equations and the reciprocal relationships of distributed piezoelectric sensors and actuators for the laminated piezoelectric plate. Liew et al. (2004) stu- died the vibration control of a laminated composite plate with piezoelectric sensor/actuator patches. They developed an algorithm with which they stu- died the dynamic response of the laminated plate and illustrated the effect of stacking and positioning of sensor/actuator patches on the response of the laminated plate. Zenkour (2004) examined the response of cross-ply laminated plates subject to thermo-mechanical loading. He demonstrated the influence of material anisotropy and stacking sequence, among others, on the thermally induced response of the plate. Ganilova and Cartmell (2010) explored the vi- bration control of a shapememory alloy (SMA) integrated laminated sandwich platebymeansof a controllable activation strategy.Theirvibrationmodel con- tains time-dependent coefficients and hence requires careful selection of input conditions to obtain practically realistic results. In the literature, theword trimorphhasbeenused todescribe thekindof la- minate studied.For instance,Craciunescu et al. (2005) used theword trimorph to denote a substrate sandwiched between two film layers. The substrate used was silicon while the films were Nickel-Titanium (NiTi) shape memory alloy (SMA).ChangandLin (2003) in the vibration analysis of a ring used theword trimorph to represent a three-layer ring with an elastic material (substrate) laminated in between two piezoelectric layers. Papila et al. (2008) making re- ference to the configuration described by Chang and Lin considered it as a bimorph. Roytburd et al. (1998) and Erturk and Inman (2008) used the same architecture tomean bimorph just asPapila et al.. Hence, there is no standard definition as to the use of theword trimorph. In this study, theword trimorph is used in a general sense to mean a three-layer laminate. Also, layer-stacking is used to mean the different possible permutations of the layers, which are obviously six for a three-layer laminate. This paper investigates thedynamic responseof a trimorphplate subjected to forced vibration. Vibration analysis for nonlinear and linear responses for the different layer-stackings has been undertaken in order to determine the layer-stacking that produces the best desirable material characteristics for an optimised response. In developing the governing equations, classical laminate theory (CLT) has been applied and the governing equations were solved to Vibration analysis of a trimorph plate... 643 simulate the trimorph plate response using a combined solution of the Navier method and direct integration in the time domain usingMathematicaTM. 2. Governing equations of the trimorph plate The system (trimorph plate) whose dynamic behaviour is modelled is a rec- tangular plate having three layers bonded adhesively together so that there is negligible relative motion between the layers. The layers are of different materials and all the material properties are isotropic. The system is simply supported on all four edges, under-damped and subjected to a sinusoidally distributed load. Figure 1 provides a sketch of the system. The layers in Fig.1 have different shading patterns suggesting that they aremade of differentma- terials. Fig. 1. Diagrammatic description of the system The trimorph plate is modelled as a thin plate and therefore all the as- sumptions for modelling thin plates apply. The equations ofmotion for a thin rectangular plate subject tomembrane force basedon the classical plate theory (CPT) are derived inWhitney (1987) and are shown below ∂2Mx ∂x2 +2 ∂2Mxy ∂x∂y + ∂2My ∂y2 (2.1) = ρh ∂2w ∂t2 + c ∂w ∂t −q(x,y,t)+Nx ∂2w ∂x2 +2Nxy ∂2w ∂x∂y +Ny ∂2w ∂y2 and ∂Ny ∂y + ∂Nxy ∂x = ρh ∂2v ∂t2 ∂Nx ∂x + ∂Nxy ∂y = ρh ∂2u ∂t2 (2.2) 644 A. Big-Alabo, M.P. Cartmell In equation (2.1), the terms on the left hand side represent the effect of bending moments on the lateral displacement, while the last three terms on the right hand side represent the effect of membrane forces on the lateral displacement. The first three terms on the right hand side are the lateral acceleration, the classical linear viscousdamping force, and theexcitation force acting on the plate respectively. The excitation function is a time-dependent sinusoidally distributed load and can be expressed as q(x,y,t)= q(t)sin πx a sin πy b (2.3) where a and b are the in-plane dimensions of the plate and q(t) is a time- variant forcing function. For the purpose of this study, q(t) is assumed to be a sinusoidal function with amplitude q i.e. q(t) = q sin(ωt). In equations (2.2), the left hand side represents the effect ofmembrane forceswhile the right hand side represents the accelerations in the in-plane directions. The formulation of the classical laminate theory (CLT) is extensively di- scussed in some text on composites (Vinson and Sierakowski, 2004; Reddy, 2004; Voyiadjis andKattan, 2005 etc). Lee (1990) developed the TLPP using the CLT. Lu and Li (2009) compared results obtained by various plate the- ories, including CLT, and the CLT results were in good agreement with other theories. Ganilova and Cartmell (2010) applied a dynamic model based on CLT in themodelling and vibration control of an active sandwich plate. For thegeneral case of anunsymmmetric laminatewith isotropic layers, the constitutive laminate equations based on the CLT are given by Jones (1999) as      Nx Ny Nxy      =    A11 A12 0 A12 A11 0 0 0 A66         ε0x ε0y ε0xy      +    B11 B12 0 B12 B11 0 0 0 B66         k0x k0y k0xy      (2.4)      Mx My Mxy      =    B11 B12 0 B12 B11 0 0 0 B66         ε0x ε0y ε0xy      +    D11 D12 0 D12 D11 0 0 0 D66         k0x k0y k0xy      where {N}, {M}, {ε}, and {k} are the membrane force, bending moment, elastic strain, and surface curvature vectors, respectively. [Aij], [Bij] and [Dij] are the extension, bending-extension coupling, and bending stiffness matrices respectively, and the elements of the matrices are given by Vibration analysis of a trimorph plate... 645 [Aij] = N ∑ k=1 Qijk(zk−zk−1) [Bij] = 1 2 N ∑ k=1 Qijk(z 2 k −z 2 k−1) (2.5) [Dij] = 1 3 N ∑ k=1 Qijk(z 3 k −z 3 k−1) where Qijk is the transformed reduced stiffness of the kth layer of the lami- nate, k is the layer index, and N is the total number of layers. The kth layer is bounded by surfaces at zk and zk−1 at the top and bottom respectively. For the case of an unsymmetric laminate considered here, Qijk is related to the material properties of the kth layer as shown in equations (2.6) below (Q11)k =(Q22)k = Ek 1−ν2 k (Q16)k =(Q26)k =0 (Q12)k = νkEk 1−ν2 k (Q66)k = Ek 2(1+νk) (2.6) By substituting equations (2.4) in equations (2.1) and (2.2) and applying Kirchhoff’s strain-displacement relationships (Voyiadjis and Kattan, 2005), the following equations are obtained B11 ∂3u ∂x3 +(B12+2B66) ∂3u ∂x∂y2 +(B12+2B66) ∂3v ∂x2∂y +B11 ∂3v ∂y3 − [ D11 ∂4w ∂x4 +2(D12+2D66) ∂4w ∂x2∂y2 +D11 ∂4w ∂y4 ] = ρh ∂2w ∂t2 + c ∂w ∂t −q(x,y,t)+ ( A11 ∂2w ∂x2 +A12 ∂2w ∂y2 )∂u ∂x + ( A12 ∂2w ∂x2 +A11 ∂2w ∂y2 )∂v ∂y +B11 [(∂2w ∂y2 )2 + (∂2w ∂x2 )2] +2A66 (∂u ∂y + ∂v ∂x ) ∂2w ∂x∂y −4B66 ( ∂2w ∂x∂y )2 (2.7) A66 ∂2v ∂x2 +A11 ∂2v ∂y2 +(A12+A66) ∂2u ∂x∂y −B11 ∂3w ∂y3 −(B12+2B66) ∂3w ∂x2∂y = ρh ∂2v ∂t2 A11 ∂2u ∂x2 +A66 ∂2u ∂y2 +(A12+A66) ∂2v ∂x∂y −B11 ∂3w ∂x3 +(B12−2B66) ∂3w ∂x∂y2 = ρh ∂2u ∂t2 646 A. Big-Alabo, M.P. Cartmell Equations (2.7) are PDEs inmore than one independent variables, which can be reduced to ODEs using the Navier Method (Ganilova and Cartmell, 2010; Zak et al., 2003). For a time-variant sinusoidally distributed load as expressed in equation (2.3), the appropriate solution for the displacements based on the Navier method can be deduced fromGanilova and Cartmell (2010) as follows u(x,y,t) =u(t)sin πx a sin πy b v(x,y,t) = v(t)sin πx a sin πy b (2.8) w(x,y,t) =w(t)sin πx a sin πy b Substituting equations (2.8) and their derivatives into equations (2.7) and simplifying the resulting expressions, we obtain −π3 (B11 a3 + B12+2B66 ab2 ) u(t)cot πx a −π3 (B11 b3 + B12+2B66 a2b ) v(t)cot πy b −π4 (D11 a4 + 2(D12+2D66) a2b2 + D11 b4 ) w(t)−2π3A66 ( 1 ab2 u(t)cot πy b + 1 a2b v(t)cot πx a ) w(t)cos πx a cos πy b = ρhẅ+ cẇ−q(t)−π3 (A11 a3 + A66 a2b ) u(t)v(t)cos πx a sin πy b −π3 (A12 a2b + A11 b3 ) v(t)w(t)sin πx a cos πy b +π4 [ B11 ( 1 b4 + 1 a4 ) sin πx a sin πy b −4B66 1 a2b2 cot πx a cot πy b cos πx a cos πy b ] [w(t)]2 (2.9) π ab (A12+A66)u(t)cot πx a cot πy b −π2 (A66 a2 + A11 b2 ) v(t) −π3 (B11 b3 − B12+2B66 a2b ) w(t)cot πy b = ρhv̈ −π2 (A11 a2 + A66 b2 ) u(t)+ π ab (A12+A66)v(t)cot πx a cot πy b +π3 (B11 b3 − B12−2B66 ab2 ) w(t)cot πx a = ρhü Neglecting the in-planedisplacements i.e. u(t)= v(t) = 0, then fromequations (2.9)2,3, it is clear that x a = y b = 1 2 (2.10) Vibration analysis of a trimorph plate... 647 Usingequation (2.10) in equation (2.9)1, neglecting the in-planedisplacements, and simplifying the resulting expression, the equation for lateral displacement of the trimorph plate is derived as shown in equation (2.11) ẅ+Cẇ+Dw(t)+B[w(t)]2 =Qsin(ωt) (2.11) where w(t) is the time-dependent lateral displacement of the trimorph plate, C = c ρh is the damping constant per unit mass, D= π4 ρh [D11 a4 + 2(D12+2D66) a2b2 + D11 b4 ] is the bending elastic stiffness per unit mass, B= π4 ρh B11 ( 1 b4 + 1 a4 ) is the bending-extension elastic stiffness per unit mass, Q= q ρh is the magnitude of the excitation force per unit mass. Equation (2.11) is a nonlinear ODE which models the lateral response of a rectangular laminated plate acted upon by a sinusoidally varying and sinu- soidally distributed excitation force and subjected to linear viscous damping. The nonlinearity in the equation of lateral motion of the trimorph plate is in- troduced by themembrane forces. This equation is different from the common Duffing’s equation for nonlinear vibration because the nonlinear term has a power of two, and the equation is introduced for the first time. If the nonlinear term is neglected, equation (2.11) reduces to ẅ+Cẇ+Dw(t) =Qsin(ωt) (2.12) Equation (2.12) models the linear lateral response of the same system. Equ- ations (2.11) and (2.12) can be solved by direct integration using software pac- kages. BespokeMathematicaTM codes have been used to solve these equations and simulate graphically the time-domain and frequency-domain responses of the trimorph plate for the different layer-stacking. 3. Determination of the elastic coefficients of the trimorph plate The response of the plate is largely dependent on the elastic stiffness of the plate.The trimorphplate stiffness is determinedby D and B for thenonlinear response and just D for the linear response. From equation (2.11), D is de- pendent on the stiffness coefficients D11,D12, and D66 while B is dependent 648 A. Big-Alabo, M.P. Cartmell on the stiffness coefficient B11. A more generalised approach for the deter- mination of the stiffness coefficients of laminates was provided by Noor and Tenek (1992). Expressions for the determination of thermoelastic coefficients and the derivatives of coefficients with respect to material properties of the laminae were developed. They also illustrated the effect of stacking sequence and fibre orientation on the coefficients and for the first time provided sensiti- vity derivatives. Here an attempt has beenmade to simplify algebraically and to presentmore concise equations for determining the stiffness coefficients for the trimorph plate (see appendix). 4. Numerical integration and discussion of the results Numerical integration routines, as are found in many standard software pac- kages, can be used to solve PDEs and ODEs directly, and Israr (2008) used the NDSolve solver in MathematicaTM for the solution of the governing equ- ation for vibration of a cracked aluminiumplate.A similar approachwas taken by Atepor (2008) and Ganilova and Cartmell (2010). Optimised time-domain responses have been simulated here, using the NDSolve integrator. Optimisa- tion of the trimorph plate response involves solving the objective functions, which are equations (2.11) and (2.12), within the optimisation constraints. Two outputs have been constrained in this study to obtain an optimised re- sponse. The first constrained output is the time take for the plate response to go into steady-state as the initial response is transient. This transient behavio- ur, though it cannot be eliminated in reality, can only be allowed for a short period since it is not desirable and hence is the reason for constraining it. A steady-state response time of not more than thirty seconds is used here. The second constrained output is the amplitude of the steady-state response. It is desirable in many applications of vibration to keep the maximum amplitude of vibration within certain acceptable limits. This means that the response has to be constrained and for the purpose of this study, a response range of ±10 percent of one-tenth of the plate thickness is deemed to be accepta- ble. The objective function and optimisation constraints are summarised as follows: — objective functions (1) ẅ+Cẇ+Dw(t)+B[w(t)]2 =Qsin(ωt) nonlinear (2) ẅ+Cẇ+Dw(t)=Qsin(ωt) linear Vibration analysis of a trimorph plate... 649 —constraints Ts ¬ 30 0.09h¬ [w(t)]opt ¬ 0.11h where Ts is the time in seconds taken for the plate response to go into steady- state and [w(t)]opt is the optimum response inmetres. The limit of the steady- state response time is chosen arbitrarily. As for the response limit, the normal case for small deflections of thin plates (Timoshenko andWoinowsky-Krieger, 1959) is applied, i.e. the lateral displacement is in the order of one-tenth of the plate thickness. For the purpose of simulating the temporal response of the trimorph plate, three different materials were used namely: Aluminium (Al), Polyvinylidene Fluoride (PVDF) and Lead Zirconate Titanate (PZT). Properties of these materials and other input parameters are given in Tables 1 and 2 respectively. Table 1.Material properties of the plate layers Material Poisson’s ratio Modulus Density ν [–] E [GPa] ρ [kg/m3] Aluminium, Al 0.33 70 2700 PVDF 0.44 1.1 1770 PZT 0.3 64 7600 The properties of PVDF were obtained from Zhang et al. (1993) and the website, http://www.texloc.com/closet/cl pvdf properties.htm while the pro- perties of PZT were obtained from Malic et al. (1992) and Yimnirun et al. (2004). Table 2.Other inputs used for simulating trimorph plate response Input Value Unit Length, a 2 m Width, b 0.5 m Thickness of Al, δAl 1 mm Thickness of PVDF, δPVDF 0.7 mm Thickness of PZT, δPZT 0.3 mm Exciting force, q 10 N/m2 From Table 2, it can be seen that a rectangular plate of unit area and total thickness of 2mm has been chosen. Hence the thickness to length 650 A. Big-Alabo, M.P. Cartmell ratio (h/L) is 1/1000, which means the plate is defined as a thin pla- te. Since the trimorph plate comprises of three different layers, then the stacking of the layers can be carried out in six different permutations namely: Al/PVDF/PZT, Al/PZT/PVDF, PVDF/Al/PZT, PVDF/PZT/Al, PZT/PVDF/Al, and PZT/Al/PVDF. Response plots for the different stac- king arrangement are shown below. 4.1. Nonlinear time-domain response Fig. 2. Optimised time-domain response of Al/PVDF/PZT (a) and Al/PZT/PVDF (b) stacking with ξ=0.00018 Fig. 3. Optimised time-domain response of PVDF/Al/PZT (a) and PVDF/PZT/Al (b) stacking with ξ=0.01 Fig. 4. Optimised time-domain response of PZT/PVDF/Al (a) and PZT/Al/PVDF (b) stacking with ξ=0.0022 Vibration analysis of a trimorph plate... 651 4.2. Linear time-domain response Fig. 5. Optimised time-domain response of Al/PVDF/PZT (a) and Al/PZT/PVDF (b) stacking with ξ=0.00018 Fig. 6. Optimised time-domain response of PVDF/Al/PZT (a) and PVDF/PZT/Al (b) stacking with ξ=0.018 Fig. 7. Optimised time-domain response of PZT/PVDF/Al (a) and PZT/Al/PVDF (b) stacking with ξ=0.0022 In simulating the optimised response to satisfy the optimisation constra- ints, the damping of the systemwas varied by varying the damping ratio. This was so because for a given set of layer thicknesses and in-plane dimensions, the plate stiffnesses (D and B) cannot be altered to satisfy the optimisation con- straints, therefore leaving only the option of varying the damping constant C, 652 A. Big-Alabo, M.P. Cartmell which can easily be done through the damping ratio ξ. The results obtained from simulating the optimised response of the trimorph plate for the different layer-stacking are summarised in Table 3 and 4 for the nonlinear and linear analysis respectively. Table 3. Nonlinear analysis of the trimorph plate response for the different layer arrangement S / N Layer Plate Plate Plate Opera- Dam- Steady- Steady- arrangem. den- stiffness stiffness ting ping -state -state 1-2-3 sity (B) (BE) freq. const. response set-in respectively [kg/m3] [N/mkg] [N/m2kg] [rad/s] [Ns/mkg] [m] time[s] 1 Al/PVDF/PZT 3109.50 2.229·107 2.006·107 4721.08 1.700 2.004·10−4 6 2 Al/PZT/PVDF 3109.50 2.228·107 1.641·107 4719.77 1.699 2.005·10−4 6 3 PVDF/Al/PZT 3109.50 2.439·105 3.367·107 493.83 9.877 2.040·10−4 3 4 PVDF/PZT/Al 3109.50 2.458·105 3.430·107 495.74 9.915 2.024·10−4 3 5 PZT/PVDF/Al 3109.50 1.846·106 3.067·107 1358.64 5.435 2.175·10−4 3 6 PZT/Al/PVDF 3109.50 1.811·106 1.703·107 1345.61 5.921 2.017·10−4 3 B denotes bending stiffness while BE denotes bending-extension stiffness Table 4.Linear analysis of the trimorph plate response for the different layer arrangement S/N Layer Plate Plate Natu- Dam- Steady- Steady- arrangem. den- stiffness ral ping -state -state 1-2-3 sity (B) freq. const. response set-in respectively [kg/m3] [N/mkg] [rad/s] [Ns/mkg] [m] time [s] 1 Al/PVDF/PZT 3109.50 2.229·107 4721.08 1.700 2.003·10−4 6 2 Al/PZT/PVDF 3109.50 2.228·107 4719.77 1.699 2.005·10−4 6 3 PVDF/Al/PZT 3109.50 2.439·105 493.83 17.778 1.831·10−4 3 4 PVDF/PZT/Al 3109.50 2.458·105 495.74 17.847 1.817·10−4 3 5 PZT/PVDF/Al 3109.50 1.846·106 1358.64 5.435 2.178·10−4 3 6 PZT/Al/PVDF 3109.50 1.811·106 1345.61 5.921 2.012·10−4 3 Tables 3 and 4 provide a comprehensive summary of the results obtained from simulating the time-domain responses of the different layer-stacking for nonlinear and linear analysis respectively. It can be observed from column three of both tables that the density of the trimorph plate is not affected by different layer-stacking since the material properties and layer thicknesses are unchanged. The last column of the tables shows when the steady-state Vibration analysis of a trimorph plate... 653 response develops noticeably and definitively. It can be seen from the tables that the steady-state responsewas achieved in less than 10 seconds in all cases. The results also reveal that the responses for rows 1 and 2 are similar. So also are rows 3 and 4 and rows 5 and 6. The implication is that the response is dependent on the first layer (layer 1), which is called the reference layer in this study. The reference layer is the layer directly experiencing the lateral load and could be either of the top-most or bottom-most layer. This redu- ces the problem of optimising the trimorph plate response to simulating only three different configurations instead of the six possible configurations. The optimised responses of the three necessary configurations are given in rows 1, 3 and 5. In determining the best possible layer-stacking from the three configura- tions necessary for testing/simulation, the trimorph plate stiffness, natural frequency and damping coefficient (or ratio) are worth considering. It is requ- ired formost applications, for example in damagemitigation, to use amaterial with high resilience to load. Also, a better material would have a high natural frequency to avoid the occurrence of resonance in operation andwould require lower damping to reduce the cost of vibration andnoise control. From thenon- linear analysis (Table 3) it can be seen that row 1 has the best combination of these considerations compared to rows 3 and 5. It is important that the nonli- nearity in the response is minimised asmuch as possible. Although, row 2 has the least value for the coefficient of the nonlinear term (bending-extension stif- fness coefficient), row 1 still produces the best response characteristics after considering the other factors discussed above. Row 1 has the highest ben- ding stiffness (22.289MN/mkg), the highest natural frequency (4721.00rad/s = 751.29Hz –This value is obtained from the nonlinear frequency-domain re- sponse plot of the configuration Al/PVDF/PZT) and the second lowest dam- ping (1.700Ns/mkg). Similarly, from the linear analysis (Table 4) row 1 has the best combination of response characteristics as follows: bending stiffness= 22.289MN/mkg, natural frequency=4721.08rad/s=751.29Hz, anddamping constant = 1.700Ns/mkg. Hence, it can be concluded that the configuration with the best dynamic response is either ofAl/PVDF/PZTorAl/PZT/PVDF since both give very similar responses. 4.3. Frequency-domain response The frequency-domain (FD) plots for nonlinear and linear responses have beensimulatedunder identical condition i.e. the samemagnitudeof force, dam- ping factor, in-plane dimensions and materials. The FD plots (Figs.8 to 13) reveal approximately equal responses forAl/PVDF/PZT andAl/PZT/PVDF 654 A. Big-Alabo, M.P. Cartmell Fig. 8. Frequency-domain response of Al/PVDF/PZT trimorph plate configuration Fig. 9. Frequency-domain response of Al/PZT/PVDF trimorph plate configuration Fig. 10. Frequency-domain response of PVDF/Al/PZT trimorph plate configuration Vibration analysis of a trimorph plate... 655 Fig. 11. Frequency-domain response of PVDF/PZT/Al trimorph plate configuration Fig. 12. Frequency-domain response of PZT/PVDF/Al trimorph plate configuration Fig. 13. Frequency-domain response of PZT/Al/PVDF trimorph plate configuration 656 A. Big-Alabo, M.P. Cartmell trimorph configurations, slight difference in the responses for PVDF/Al/PZT andPVDF/PZT/Al trimorphconfigurations, andapproximately equal respon- ses forPZT/PVDF/AlandPZT/Al/PVDFtrimorph.Theplots also showthat the peak response for the linear analysis is higher than the peak response for the corresponding nonlinear analysis in the cases where there are differences. Also, in the cases were the linear and nonlinear responses differ, the natural frequency of the linear response is higher than the corresponding natural fre- quency of the nonlinear response (Table 5). Hence, it is concluded that the effect of the nonlinear term in the equation of motion is such as to reduce the natural frequency of the trimorph plate and this is expected in principle. Although, the reduction in natural frequency as shown inTable 5 is not large, it is believed that this reduction can be quite significant depending on the dimensions of and thematerials for the trimorph plate. Frequency-domain plots have been produced using the same system inputs for both analyses in order to compare the nonlinear and linear responses. The plots are shown in Figs. 8 to 13 below. Table 5.Comparison of natural frequencies for linear and nonlinear response obtained from frequency-domain plots S/N Layer arrangement Linear response Nonlinear response 1-2-3 respectively [rad/s] [rad/s] 1 Al/PVDF/PZT 4721.0 4721.0 2 Al/PZT/PVDF 4719.8 4719.8 3 PVDF/Al/PZT 493.2 493.7 4 PVDF/PZT/Al 495.2 495.6 5 PZT/PVDF/Al 1358.6 1358.6 6 PZT/Al/PVDF 1345.6 1345.6 Table 5 summarises the natural frequencies of the nonlinear and linear responses obtained from the frequency-domain plots for the different layer- stacking of the trimorph plate. 5. Conclusions The dynamics of a trimorph plate have been investigated in this paper. The trimorph plate is a three-layer laminated plate with each layer made of a different material. The plate is subjected to a sinusoidally distributed and varying load and a linear viscous damping force. Vibration analysis of a trimorph plate... 657 In studying the trimorphplate, the governing equations for deformation of the platewere developed using the classical plate theory and classical laminate theory as in equations (2.7). Thegoverning equationswere reduced to ordinary differential equations (ODEs) as in equations (2.11) and (2.12) for modelling the lateral deformation of the trimorph plate using the Navier method. The ODEs were used to simulate the time-domain and frequency-domain respon- ses of the plate by direct integration using bespokeMathematicaTM code. The time-domain plots revealed that the responseswere initially transient and set- tled into steady-state after some time. To optimise the responses, the steady- state commencement time was constrained to 30 seconds and the acceptable maximum deflection at steady-state was constrained to ±10 percent of one- tenth the thickness of the plate. Thematerials used to simulate the trimorph plate responses were aluminium (Al), polyvinylidene fluoride (PVDF) and le- ad zirconate titanate (PZT). From the response simulations, it was discovered that only three out of the six possible different layer-stacking is necessary to determine the best stacking for the trimorph plate. Considering the trimorph plate stiffness, fundamental natural frequency and damping coefficient, either one of the Al/PVDF/PZT or AL/PZT/PVDF configurations was found to produce the best response given that both stackings produced very similar responses. Frequency-domain responseplotswere also generated for the nonlinear and linear responses for the purpose of comparison. Frequency-domain responses were generated over a chosen frequency interval, and the data were plotted. Theplots reveal that for configurations inwhich the nonlinear effectwas signi- ficant, the peak response and the natural frequency of the nonlinear response are lower than that of the corresponding linear response for the same system inputs. Therefore, the plots show that nonlinearity in the responses acts to reduce the natural frequency of the trimorph plate. This studyhasbeen limited to a rectangular trimorphplate thathas isotro- pic layers subjected to a sinusoidally distributed load and is simply-supported on all four edges. This leaves room for further investigation of other loading and boundary conditions. It is intended that the results obtained here will be validated experimentally and numerically using software packages such as ABAQUS. This study can be applied to the study of optimised damage mi- tigation in structures subjected to potentially dangerous loads by considering one of the layers of the trimorph configuration as the hostmaterial, one as the sensor and the other as the actuator, and using the vibration analysis above to determine what arrangement of the layers will produce the best structural integrity under the operating constraints of the material. 658 A. Big-Alabo, M.P. Cartmell A. Appendix The stiffness coefficients for the trimorph plate have been presented in a sim- plified form based on the geometry of the layers. In order to accomplish this, the reference layer labelled layer 1 is an outer layer as shown in Fig.14. The reference layer is the layer directly experiencing the lateral load and could be either of the top-most or bottom-most layer, but for the purpose of illustration the bottom-most layer is used. Other layers are labelled accordingly as shown in Fig.14. Fig. 14. Sketch illustrating vertical deometry of laminae From Fig.14, the following can be deduced geometrically zk−zk−1 = δk zk = k ∑ i=1 δi k> 0 or (A.1) zk−1 = k−1 ∑ i=1 δi k> 1 else zk−1 =0 where δk is the thickness of the kth layer and k=1,2,3. Aij coefficients For the kth layer, (Aij)k =(Qij)k(zk−zk−1)= (Qij)kδk. So that (A11)k =(Q11)kδk = Ek 1−ν2 k δk (A12)k =(Q12)kδk = νkEk 1−ν2 k δk (A.2) (A66)k =(Q66)kδk = Ek 2(1+νk) δk where k=1,2,3 for layer 1, layer 2 and layer 3, respectively. Vibration analysis of a trimorph plate... 659 Hence, the extensional stiffness coefficients are given by A11 = k ∑ i=1 (A11)k = E1 1−ν21 δ1+ E2 1−ν22 δ2+ E3 1−ν23 δ3 A12 = k ∑ i=1 (A12)k = ν1E1 1−ν21 δ1+ ν2E2 1−ν22 δ2+ ν3E3 1−ν23 δ3 (A.3) A66 = k ∑ i=1 (A66)k = 1 2 ( E1 1+ν1 δ1+ E2 1+ν2 δ2+ E3 1+ν3 δ3 ) Bij coefficients For the kth layer, (Bij)k =(Qij)k(z 2 k −z2 k−1), z2 k −z2 k−1 =(zk +zk−1)(zk−zk−1). Substituting equations (A.1), z2k −z 2 k−1 =(δk +2zk−1)δk = δk(δk+2 ∑k−1 i=1 δi. Therefore, (Bij)k =(Qij)kδk(δk +2 ∑k−1 i=1 δi). So that, — for layer 1 (k=1) (B11)1 =(Q11)kδ 2 1 = E1 1−ν21 δ21 (B12)1 =(Q12)1δ 2 1 = ν1E1 1−ν21 δ21 (A.4) (B66)1 =(Q66)1δ 2 1 = E1 2(1+ν1) δ21 — for layer 2 (k=2), (Bij)2 =(Qij)2δ2(δ2+2δ1) (B11)2 =(Q11)2δ2(δ2+2δ1)= E2 1−ν22 δ2(δ2+2δ1) (B12)2 =(Q12)2δ2(δ2+2δ1)= ν2E2 1−ν22 δ2(δ2+2δ1) (A.5) (B66)2 =(Q66)2δ2(δ2+2δ1)= E2 2(1+ν2) δ2(δ2+2δ1) 660 A. Big-Alabo, M.P. Cartmell — for layer 3 (k=3), (Bij)3 =(Qij)3δ3[δ3+2(δ2+ δ1)] (B11)3 =(Q11)3δ3[δ3+2(δ2+ δ1)] = E3 1−ν23 δ3[δ3+2(δ2+δ1)] (B12)3 =(Q12)3δ3[δ3+2(δ2+ δ1)] = ν3E3 1−ν23 δ3[δ3+2(δ2+δ1)] (A.6) (B66)3 =(Q66)3δ3[δ3+2(δ2+ δ1)] = E3 2(1+ν3) δ3[δ3+2(δ2+ δ1)] Hence, the bending-extension stiffness coefficients are given by B11 = 1 2 k ∑ i=1 (B11)k = 1 2 ( E1 1−ν21 δ21 + E2 1−ν22 δ2(δ2+2δ1)+ E3 1−ν23 δ3[δ3+2(δ2+ δ1)] ) B12 = 1 2 k ∑ i=1 (B12)k (A.7) = 1 2 ( ν1E1 1−ν21 δ21 + ν2E2 1−ν22 δ2(δ2+2δ1)+ ν3E3 1−ν23 δ3[δ3+2(δ2+ δ1)] ) B66 = 1 2 k ∑ i=1 (B66)k = 1 4 ( E1 1+ν1 δ21 + E2 1+ν2 δ2(δ2+2δ1)+ E3 1+ν3 δ3[δ3+2(δ2+ δ1)] ) Dij coefficients For each layer, (Dij)k =(Qij)k(z 3 k −z3 k−1), z3k −z 3 k−1 =(zk −zk−1) 3+3(zk −zk−1)zkzk−1 = δ 3 k +3δk ∑ k i=1δi ∑k−1 i=1 δi. Therefore, (Dij)k =(Qij)k(δ 3 k +3δk) ∑k i=1δi ∑k−1 i=1 δi — for layer 1 (k=1), (Dij)1 =(Qij)1(δ 3 1 +3δ 2 1) (D11)1 =(Q11)1(δ 3 1 +3δ 2 1)= E1 1−ν21 (δ31 +3δ 2 1) (D12)1 =(Q12)1(δ 3 1 +3δ 2 1)= ν1E1 1−ν21 (δ31 +3δ 2 1) (A.8) (D66)1 =(Q66)1(δ 3 1 +3δ 2 1)= E1 2(1+ν1) (δ31 +3δ 2 1) Vibration analysis of a trimorph plate... 661 — for layer 2 (k=2), (Dij)2 =(Qij)2[δ 3 2 +3δ2δ1(δ1+ δ2)] (D11)2 =(Q11)2[δ 3 2 +3δ2δ1(δ1+ δ2)] = E2 1−ν22 [δ32 +3δ2δ1(δ1+ δ2)] (D12)2 =(Q12)2[δ 3 2 +3δ2δ1(δ1+ δ2)] = ν2E2 1−ν22 [δ32 +3δ2δ1(δ1+ δ2)] (A.9) (D66)2 =(Q66)2[δ 3 2 +3δ2δ1(δ1+ δ2)] = E2 2(1+ν2) [δ32 +3δ2δ1(δ1+ δ2)] — for layer 3 (k=3), (Dij)3 =(Qij)3[δ 3 3 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] (D11)3 =(Q11)3[δ 3 3 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] = E3 1−ν23 [δ33 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] (D12)3 =(Q12)3[δ 3 3 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] (A.10) = ν3E3 1−ν23 [δ33 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] (D66)3 =(Q66)3[δ 3 3 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3) = E3 2(1+ν3) [δ33 +3δ3(δ1+ δ2)(δ1+ δ2+ δ3)] Hence, the bending stiffness coefficients are given by D11 = 1 3 k ∑ i=1 (D11)k = 1 3 ( E1 1−ν21 (δ31 +3δ 2 1) + E2 1−ν22 [δ32 +3δ2δ1(δ1+ δ2)]+ E3 1−ν23 [δ33 +3δ3(δ1+ δ2)(δ1+ δ2+δ3)] ) D12 = 1 3 k ∑ i=1 (D12 = 1 3 ( ν1E1 1−ν21 (δ31 +3δ 2 1) (A.11) + ν2E2 1−ν22 [δ32 +3δ2δ1(δ1+ δ2)]+ ν3E3 1−ν23 [δ33 +3δ3(δ1+ δ2)(δ1+ δ2+δ3)] ) D66 = 1 3 k ∑ i=1 (D66)k = 1 6 ( E1 1+ν1 (δ31 +3δ 2 1) + E2 1+ν2 [δ32 +3δ2δ1(δ1+ δ2)]+ E3 1+ν3 [δ33 +3δ3(δ1+δ2)(δ1+ δ2+ δ3)] ) 662 A. 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Zhang H., Galea S.C., Chiu W.K., Lam Y.C., 1993, An investigation of thin PVDF films as fluctuating-strain-measuring and damage-monitoring devices, Smart Material Structures, 2, 206-216 Analiza drgań płyty trimorficznej pod kątem zoptymalizowanej ochrony przed uszkodzeniem Streszczenie Przedstawionobadaniadotyczącewiskotycznie tłumionej płyty trimorficznej pod- danej sinusoidalnie rozłożonemu obciążeniu przy warunkach brzegowych typu swo- bodne podparcie. Wyprowadzono równanie ruchu płyty, pierwszy raz w tym arty- kule, opisujące nieliniowy efekt ugięcia na podstawie klasycznej teorii płyt (CPT) oraz klasycznej teorii laminatów (CLT). Równanie to rozwiązano metodą Naviera oraz bezpośrednim całkowaniem numerycznym. Zoptymalizowane wykresy odpowie- dzi czasowych płyty wykonanej z aluminium (Al), polifluorku winylidenu (PVDF) oraz spieków cyrkonu i tytanu (PZT) wykazały, że tylko trzy z możliwych sześciu 664 A. Big-Alabo, M.P. Cartmell konfiguracji warstw trimorfu wystarczają do określenia najlepszego ułożenia warstw. Przy znajdowaniu najlepszej aranżacji warstwpod kątem zoptymalizowanej odpowie- dzi dynamicznej płyty wzięto pod uwagę trzy czynniki: sztywność, częstość własną i stałą tłumienia.Wtymkontekścienajlepszeokazały się konfiguracjeAl/PVDF/PZT orazAl/PZT/PVDF–uzyskałynajwyższą sztywność, największą częstośćdrgańwła- snych i najmniejszy współczynnik tłumienia. Wykresy otrzymane w dziedzinie czę- stości pozwoliły na porównanie odpowiedzi układu nieliniowego i zlinearyzowanego, ujawniając, zgodnie z przewidywaniami, że efekt nieliniowy zmniejsza częstośćwłasną trimorfu. Wykazano także, że zaprezentowane badania mogą zostać zastosowane do analizy „inteligentnych” nadwozi samochodowych celującej w zoptymalizowane wła- ściwości ze względu na ochronę przed uszkodzeniami oraz do krytycznych elementów bezpieczeństwa narażonych na potencjalne ryzyko zniszczenia. Manuscript received February 4, 2011; accepted for print April 27, 2011