Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 4, 39, 2001 ANALYSIS OF DYNAMIC BEHAVIOUR OF WAVY-PLATES WITH A MEZO-PERIODIC STRUCTURE Bohdan Michalak Department of Structural Mechanics, Technical University of Łódź e-mail: bmichala@ck-sg.p.lodz.pl The aimof the contribution is determination of such a formof themezo- shape function for amezostructuralmodel, which is suitable to quantita- tive analysis of dynamic behaviour of awavy-plate.Governing equations of the averaged theory of wavy-plates, as opposed to those obtained by Michalak et al. (1996), are obtained for different forms of the mezo- shape functions for in-plane and out-of-plane displacements of the plate. The mezo-shape functions of wavy-plates are determined from the so- lution to the eigenvalue problem of a periodic cell ∆ by making use of the finite element method. The comparison of free vibration frequen- cies obtained from the mezostructural model, the homogenized theory, orthotropic plate model and the finite element method is presented. Key words: shell, periodic structure, dynamic 1. Introduction The subject of this paper is determination of mezo-shape functions for a mezostructuralmodel of aperiodic shell-like structure,which is referred to as a wavy-plate and dynamic analysis of this structure (Fig.1). The exact analysis of periodic wavy-plates within the theory of thin elastic shells is too com- plicated to constitute a basis for investigations of most engineering problems related to those structures. The simplest model of a wavy-plate periodically waved in one direction is the orthotropic plate model e.g. by Troitsky (1976). Investigation of that problem incorporates known asymptotic homogenisation method by Lewiński (1992). However, within the orthotropic platemodel and homogenisation theory restricted to the first length scale effect, the effect of 948 B.Michalak periodicity of the cell size on the global responseof thewavy-plate becomesne- gligible. Such approach leads to substructuralmacrodynamics of wavy-plates, which is based on the modelling approach presented byWoźniak (1997). Fig. 1. Scheme of analysedmezo-periodic structure Themodel of a wavy-plate is a mezostructural model in which the gover- ning equations of the averaged theory of a wavy-plate depend on the mezo- structure length parameter l (l ≡ √ (l1)2+(l2)2, where the wave lengths l1, l2 are small enough compared to the minimum characteristic length di- mension L of the projection of the wavy-plate on the plane Ox1x2. The averaged models of this kind were applied to selected dynamic problems of periodic structures, see e.g. Baron and Woźniak (1995), Cielecka (1995), Ję- drysiak (1998), Wierzbicki (1995). The direct description of the wavy-plate is given within the approximated well-known linear theory of thin elastic shells (Green and Zerna, 1954). Theaimof this paper is, first, to obtain the averagedmodel of awavy-plate for different forms of local oscillations of in-plane and out-of-plane displace- ments. Second, to determine such a form of the mezo-shape function which is suitable to quantitative analysis of dynamic behaviour of the wavy-plate. The correctness of the assumed mezo-shape functions is to be verified by compa- ring free vibration frequencies obtained from the mezostructural model with those found from the homogenized theory, orthotropic plate model and finite element method. Throughout the paper, the indices i,j,k, ... run over 1,2,3, being related to the orthogonal Cartesian coordinates x1,x2,x3 with the base vectors ei. Analysis of dynamic behaviour of... 949 The indices α,β,γ, ... run over 1,2 and are related to the midsurface shell parameters θ1, θ2. 2. Modelling approach and choice of mezo-shape functions The modelling approach to the mezostructural theory of wavy-plates was presented byMichalak eta al. (1996). For the sake of self-consistency, we recall here its key concepts. Themodel is based on thewell-known, for thin elastic shells, linear appro- ximated theory including its strain-displacement equations, stress-strain rela- tions and equations of motion in the weak form. By u=ui(x, t)ei we denote the displacement vector field of the wavy-plate midsurface, by p= pi(x, t)ei the external forces, and by ρ themass density averaged over a shell thickness with respect to the midsurface. Let the midsurface of the undeformed plate be given by the parametric representation xi =Ri(θ1,θ2), where θ1, θ2 are the surface parameters. In the sequel, the above parameterisation is defined by x1 = θ1, x2 = θ2 and x3 = z(θ1,θ2) = z(θ1 + l1,θ 2) = z(θ1,θ2+ l2). By ∆≡ (o,l1)×(0, l2), we denote the basic cell of the periodic plate structure on the Ox1x2 plane. For an arbitrary integrable function f(z) defined on Π we denote its averaged value by 〈f〉(x)= 1 l1l2 ∫ ∆(x) f(z) dz1dz2 (2.1) For a ∆-periodic function f formula (2.1) yields a constant averaged value. Kinematics hypothesis Contrary toMichalak eta al. (1996), we assume in this approach, that the local in-plane and out-of-plane displacement oscillations have different forms. We restrict considerations to the motion, in which the macrodisplacements Ui(x, t) = 〈ui〉(x, t) describing the averaged motion of the wavy-plate and their derivatives are slow-varying functionsof x, i.e. canbe treatedas constant in calculations of the averages 〈·〉(x). The displacement field ui(x, t) of the wavy-plate we approximate by uα(x, t)=Uα(x, t)+h(x)Vα(x, t) u3(x, t)=U3(x, t)+g(x)V3(x, t) x=(x1,x2)∈Π t­ 0 (2.2) 950 B.Michalak where Ui(·, t), Vi(·, t) – slow-varying functions (basic unknowns). The func- tions h(·)Vα(·, t) and g(·)V3(·, t) describe local displacement oscillations, cau- sed by the mezostructure of the periodic plate. The functions h(·) and g(·) are referred to as the mezo-shape functions and the choice of these functions is obtained as an approximate solution to the eigenvalue problem of a perio- dic cell ∆ together with periodic boundary conditions. These functions are continuous functions defined on R2, having continuous derivatives of the first and second order. Moreover, values h(x) and g(x) satisfy the conditions h(x) ∈ O(l2), h,α(x) ∈ O(l), h|αβ(x) ∈ O(l), 〈ρh〉 = 0, g(x) ∈ O(l2), g,α(x)∈O(l), g|αβ(x)∈O(l), 〈ρg〉=0. The choice of these functions will be determined by analysis of free vibrations of the periodic cell ∆ with the use of the finite element method. The form of the mezo-shape functions is obtained as eigenvibration forms of the periodic cell ∆. From numerical calculations of the free vibrations with the use of the finite element method for the periodic cell ∆ descri- bed by the function z = f sin(2πx/l) (where we have assumed f/l = 0.1 and δ/l =0.1) together with the periodic boundary conditions, the forms of the eigenvibrations are determined. These forms are determined on the nodal displacements of the finite elements and can be approximated by analytical functions: for in-planevibrations h= l2 sin(2πx/l), for out-of-plane vibrations g= l2 sin(4πx/l). 3. Averaged description: mezo-structural theory (MST) Themacromodelling procedure proposed by Woźniak (1997) and the afo- rementioned kinematics hypotheses lead from the direct description of the wavy-plate to a system of equations within respect to themacrodisplacements Ui and correctors Vi, constituting the governing equations of the averaged theory of wavy-plates. The equations of motion presented below in the coordinate form are Miαβ,αβ −Miα,α−Niα,α+Ni+ 〈ρ̃〉Üi = p̃i Kγ +Lγ + 〈ρ̃hh〉V̈ γ = 〈p̃γh〉 (3.1) K3+L3+ 〈ρ̃gg〉V̈ 3 = 〈p̃3g〉 The constitutive equations have the form Analysis of dynamic behaviour of... 951 Niα =Diα|jβUj,β+H iα|µVµ+H iα|3V3 Ni =Di|jβUj,β +C i|µVµ+C i|3V3 Kα =Hα|jβUj,β+H α|µVµ+H α|3V3 K3 =H3|jβUj,β+H 3|µVµ+H 3|3V3 (3.2) Miαβ =Biαβ|jγδUj,γδ−Biαβ|jγUj,γ +Biαβ|µVµ+Biαβ|3V3 Mi|α =−Biα|jγδUj,γδ+Biα|jγUj,γ−Biα|µVµ−Biα|3V3 Lα =Bα|jγδUj,γδ −Bα|jτUj,τ +Bα|µVµ+Bα|3V3 L3 =B3|jγδUj,γδ −B3|jτUj,τ +B3|µVµ+B3|3V3 where we have denoted Diα|jβ ≡D 〈 HδαγβGiδG j γ √ a 〉 Hiα|µ =Hµ|iα ≡D 〈 HδαγβGiδG µ γh,β √ a 〉 Hiα|3 =H3|iα ≡D 〈 HδαγβGiδG 3 γg,β √ a 〉 Di|jβ ≡D 〈 Hαδγβ { λ αδ } GiλG j γ √ a 〉 Ci|µ ≡D 〈 Hαβγδ { λ αβ } GiλG µ γh,δ √ a 〉 Ci|3 ≡D 〈 Hαβγδ { λ αβ } GiλG 3 γg,δ √ a 〉 Hα|µ ≡D 〈 HτβγδGατG µ γh,δh,β √ a 〉 Hα|3 =H3|α ≡D 〈 HτβγδGατG 3 γg,δh,β √ a 〉 Biαβ|jγδ ≡B 〈 Hαβγδninj √ a 〉 Biαβ|µ =Bµ|iαβ ≡B 〈 Hαβγδh|γδn inµ √ a 〉 952 B.Michalak Biαβ|3 =B3|iαβ ≡B 〈 Hαβγδg|γδn in3 √ a 〉 Biαβ|jγ =Bjγ|iαβ ≡B 〈 Hαβµδ { γ µδ } ninj √ a 〉 Biα|jγ ≡B 〈 Hµδτν { α µδ }{ γ τν } ninj √ a 〉 Biα|µ =Bµ|iα ≡B 〈 Hβτγδ { α βτ } ninµh|γδ √ a 〉 Biα|3 = b3|iα ≡B 〈 Hβτγδ { α βτ } nin3g|γδ √ a 〉 Bα|µ ≡B 〈 Hγβτδh|γβh|τδn αnµ √ a 〉 Bα|3 =B3|α ≡B 〈 Hγβτδh|γβg|τδn αn3 √ a 〉 B3|3 ≡B 〈 Hαβγδg|αβg|γδn 3n3 √ a 〉 The above equations (3.1) and (3.2) represent a system of 9 differential equ- ations for the 3macro- displacements Ui and 3 internal variables Vi. 4. Applications To compare themezo-structural theory (MST), homogenized theory (HT), orthotropicplatemodel andfinite elementmethodweshall investigate a simple problemof cylindrical bendingof a rectangularwavy-plate (Fig.2). In this case the basic unknowns Ui and Vi depend only on the arguments x2 and t. Fig. 2. Simply supported wavy-plate Analysis of dynamic behaviour of... 953 4.1. Mezo-structural theory In this case, neglecting an external loading, the system of equations of motion will take the form M122,22−M12,2−N12,2+N1+ 〈ρ̃〉Ü1 =0 M222,22−M22,2−N22,2+N2+ 〈ρ̃〉Ü2 =0 M322,22−M32,2−N32,2+N3+ 〈ρ̃〉Ü3 =0 K1+L1+ 〈ρ̃hh〉V̈ 1 =0 (4.1) K2+L2+ 〈ρ̃hh〉V̈ 2 =0 K3+L3+ 〈ρ̃gg〉V̈ 3 =0 After substituting the right-hand sides of Eqs (3.2) into Eqs (4.1), we obtain a system of equations for Ui =Ui(x, t) and Vi =Vi(x, t). Let the wavy-plate midsurface be defined by z = f sin(2πx2/l) and the mezo-shape functions (obtained in Section 2) by h = l2 sin(2πx2/l), g = l2 sin(2πx2/l). Let us restrict the considerations to analysis of free vibrations of an unbounded wavy-plate. In this case, we shall look for solutions to Eqs (4.1) in the form U1 =0 V1 =0 U2 =A2 sin(kx2)cos(ω2t) V2 =C2cos(kx2)cos(ωt) U3 =A3 sin(kx2)cos(ωt) V3 =C3cos(kx2)cos(ωt) (4.2) where k := π/L is the wavenumber, L – vibration wavelength (L ≪ l). Substituting the right-hand sides of Eqs (4.2) into Eqs (4.1), we obtain non- trivial solutions if only ∣∣∣∣∣∣∣∣ ω2〈ρ̃〉−C33 C35 C36 C53 ω 2〈ρ̃hh〉−C55 C56 C63 C65 ω 2〈ρ̃gg〉−C66 ∣∣∣∣∣∣∣∣ =0 (4.3) where we have denoted C33 ≡B 〈 H2222(n3)2 √ a 〉 k4+ + [ B 〈 H2222 ({ 2 22 } n3 )2√ a 〉 +D 〈 H2222(G32) 2 √ a 〉] k2 954 B.Michalak C35 =C53 ≡ [ B 〈 H2222 { 2 22 } n2n3h,22 √ a 〉 −D 〈 H2222G22G 3 2h,2 √ a 〉] k C36 =C63 ≡ [ B 〈 H2222 { 2 22 } (n3)2g,22 √ a 〉 −D 〈 H2222(G32) 2g,2 √ a 〉] k C55 ≡D 〈 H2222(G22h,2) 2 √ a 〉 +B 〈 H2222(n2h,22) 2 √ a 〉 k (4.4) C56 =C65 ≡−B 〈 H2222n2n3h,22g,22 √ a 〉 −D 〈 H2222G22G 3 2h,2g,2 √ a 〉 C66 ≡D 〈 H2222(G32g,2) 2 √ a 〉 +B 〈 H2222(n3g,22) 2 √ a 〉 Let the amplitude of the shell midsurface is equal f = l/10. In this case, formulae (4.4), for the constant thickness δ and with the notation λ := δ/l, γ :=αl, yield C33 = E δ(1−ν2) (0.05641174λ4γ4+0.031026495λ4γ2+0.13539982λ2γ2) C35 = E δ(1−ν2) (0.36019325λ2γ+1.35399818γ) C36 = E δ(1−ν2) (4.58612251λ2γ−0.74263257γ) C55 = Eδ3 1−ν2 ( 4.21014436+13.53998280 1 λ2 ) (4.5) C56 = Eδ3 1−ν2 ( 53.60521952−7.42632580 1 λ2 ) C66 = Eδ3 1−ν2 ( 682.522787+10.08234978 1 λ2 ) FromEqs (4.3)we conclude that for the above formof vibrationswehave three free vibration frequencies: the lower vibration frequency ω1 and two higher one ω2, ω3 (which can be called the mezo-resonance frequencies) caused by themezo-periodic structure of the wavy-plate. 4.2. Homogenized theory (HT) Thehomogenizedmodel of dynamics of thewavy-plate canbederived from Eqs (4.1)-(4.5) by the asymptotic approximation in which the mezostructure Analysis of dynamic behaviour of... 955 of the wavy-plate is scaled down l → 0. Keeping in mind that δ/l = const, we shall neglect the mezoinertial terms 〈ρ̃hh〉 → 0, 〈ρ̃gg〉 → 0, and we can eliminate the correctors Vi in Eqs (4.1). Now, formula (4.3) leads to 〈ρ̃〉ω2 =C33− C66(C35) 2+C55(C36) 2+2C35C36C56 C55C66− (C56)2 (4.6) where ω is the lower free vibration frequency. 4.3. Orthotropic plate model Let us restrict the considerations to analysis of transverse vibrations of orthotropic plates. In this case, the equation ofmotion has the form (Troitsky, 1976) B22U3,2222+ ∂2 ∂t2 (ρ̃U3−J1U3,22)=0 (4.7) where B22 =B 1 1+ ( π f l )2 J1 = 1 l ∫ s (z2+x2)ρ ds (4.8) We shall look for a solution to Eqs (4.7) in the form U3 =A3 sin(kx2)cosωt). For the free vibration frequency, we obtain the following expression ω2 = B22k 4 ρ̃+J1k2 (4.9) 4.4. Finite element method Now we shall look for a solution to free vibrations of a simply suppor- ted wavy-plate with the use of the finite element method. The span of the wavy-plate is equal L = 10l = 10.0m, where l is the mezostructure length parameter (length of the periodic cell). From the solution obtained by ma- king use of the finite element method we have many free vibration frequen- cies for the corresponding form of the eigenvibrations. In Table 1 free vibra- tion frequencies correspnding to the eigenforms approximated by the function U3 =Asin(kx2), where k=π/10l, see Eqs (4.2), are presented. We have analysed free vibration frequencies for the above-mentioned mo- dels of the wavy-plates. In Table 1, the free vibration frequencies versus δ/l ratio found from the mezostructural theory, homogenized theory, orthotropic plate model and finite element method, are shown, where the amplitude of the wave is assumed as f = l/10. Table 2 presents the free vibration fre- quencies versus f/l ratio, where the thickness of wavy-plate is assumed as 956 B.Michalak delta= l/10. The free vibration frequencies in Table 1 and Table 2 are deter- mined for l=1.0m, E=210GPa and ρ=785kg/m2. Table 1.Free vibration frequencies versus ratio δ/l, f/l=1/10= const ω [1/s] δ/l 1/10 1/25 1/50 1/100 MST ω1 12.800 5.263 2.650 1.349 ω2 21.221 ·103 16.420 ·103 15.467 ·103 15.214 ·103 ω3 34.954 ·103 33.049 ·103 32.837 ·103 32.787 ·103 HT ω 12.803 5.265 2.654 1.339 Orth. ω 14.050 5.620 2.810 1.405 FEM ω 13.925 5.570 2.785 1.392 MST/FEM ω1/ω 91.9% 94.4% 95.0% 96.9% Table 2.Free vibration frequencies versus ratio f/l, δ/l=1/100= const ω [1/s] f/l 1/20 1/10 1/7 MST ω1 1.444 1.349 1.806 ω2 9.977 ·103 15.214 ·103 16.283 ·103 ω3 33.685 ·103 32.787 ·103 31.814 ·103 HT ω 1.441 1.339 1.813 Orth. ω 1.502 1.405 1.294 FEM ω 1.486 1.392 1.302 MST/FEM ω1/ω 97.2% 96.9% 138.7% 5. Conclusions In this paper we have applied the modelling approach, which leads to the length-scale model, and is different from that known from the homogenized theory, orthotropic plate theory and finite element method, because it takes into account the effect of the mezo-structure size on dynamic behaviour of the plate. The mezo-structural model is based on the assumption that the displacements of a periodic wavy-plate can be described by slowly varying macro-displacements, on which the oscillations are superimposed as a sum of productsofmezo-shape functions and internal variables.The internal variables are assumed to be slowly varying functions, and they play the role of unknown amplitudes for these oscillations. Themezo-shape functions have been derived Analysis of dynamic behaviour of... 957 as eigenvibration forms of a periodic cell ∆. In this paper the choice of these functions has beendeterminedbyanalysis of free vibrations of theperiodic cell ∆ with the use of the finite element method. In Section 4, different models of the wavy-plate have been discussed. On the basis of the results, we can formulate the following conclusions: • Free vibration frequencies can be successfully applied for determination of the form of the mezo-shape functions. • Analysing the results presented in Table 1, we can observe that the as- sumed approximate formof themezo-shape functions h= l2 sin(2πx/l), g = l2 sin(4πx/l) well describes dynamic behaviour of the wavy-plates for different ratios δ/l and wavy amplitude f ¬ l/10. • Analysing the results in Table 2, for the plates with wave amplitudes f > l/10, we conclude differences between the values of free vibration frequencies for the mezo-structural theory and finite element method, because the assumed form of the mezo-shape functions describes the free vibration frequencies for the periodic cell ∆ only in an approximate way. For the wavy-plate with amplitudes f > l/10 the form of the eigenvibrations of the periodic cell ∆ should be described by more accurate functions. • Only themezo-structural model gives us lower and higher free vibration frequencies for the assumed vibration form of the wavy-plate. References 1. BaronE.,WoźniakC., 1995,OnMicrodynamics ofComposite Plates,Arch. Appl. Mech., 65, 126-133 2. Cielecka I., 1995, On the Continuum Modelling the Dynamic Behaviour of CertainCompositeLattice-TypeStructures,J.Theor. Appl.Mech.,33, 351-360 3. GreenA.E.,ZernaW., 1954,Theoretical Elasticity,Oxford,ClarendonPress 4. Jędrysiak J., 1998, On Dynamics of Thin Plates with a Periodic Structure, Engng. Trans., 46, 73-87 5. LewińskiT., 1992,HomogenizingStiffnesses ofPlateswithPeriodicStructure, Int. J. Solids Structures, 21, 309-326 6. Michalak B.,WoźniakC., WoźniakM., 1996, TheDynamicModelling of ElasticWavy Plates,Arch. Appl. Mech., 66, 177-186 958 B.Michalak 7. Troitsky M.S., 1976, Stiffened Plates: Bending, Stability and Vibrations, Amsterdam-Oxford-NewYork, Elsevier Company 8. Wierzbicki E., 1995, Nonlinear Macro-Micro Dynamics of Laminated Struc- tures, J. Theor. Appl. Mech., 33, 1-17 9. Woźniak C., 1997, Internal Variables in Dynamics of Composite Solids with PeriodicMicrostructure,Arch. Mech., 49, 421-441 Analiza dynamicznych zachowań płyty pofałdowanej Streszczenie Celempracybyło znalezienie takichpostaci funkcjimezo-kształtu,któredawałyby poprawne wyniki w analizie ilościowej zachowań dynamicznych płyty pofałdowanej. Uśrednione równania opisujące płytę pofałdowaną w przeciwieństwie do przedsta- wionych w Michalak i inni (1996) są otrzymane dla różnych postaci oscylacji prze- mieszczeńwpłaszczyźnie płyty iwkierunkuprostopadłymdopłaszczyzny środkowej. Funkcje mezo-kształtu zostały określone w wyniku rozwiązania, przy pomocy meto- dy elementów skończonych, problemu drgań własnych komórki periodyczności ∆. Porównano częstości drgańwłasnych otrzymane zmodelumezostrukturalnego, teorii homogenizacji, modelu płyty ortotropowej i z metody elementów skończonych. Manuscript received November 23, 2000; accepted for print March 27, 2001