Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 629-642, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.629 ON FREE VIBRATIONS OF THIN FUNCTIONALLY GRADED PLATE BANDS RESTING ON AN ELASTIC FOUNDATION Jarosław Jędrysiak, Magda Kaźmierczak-Sobińska Lodz University of Technology, Department of Structural Mechanics, Łódź, Poland e-mail: jarek@p.lodz.pl; magda.kazmierczak@p.lodz.pl In this note free vibrations of plate bandswith functionally graded properties, resting on an elastic foundation, are investigated. On the micro-level, these plate bands have a tolerance- -periodic structure. It can be shown that in dynamic problems of those objects, the effect of the microstructure size plays a role. This effect can be described in the framework of the tolerance model, which is presented here for these bands. Obtained results are evaluated introducing the asymptotic model. Both fundamental and higher free vibration frequencies of these plate bands are calculated using the Ritz method. The effects of differences of material plate properties in the cell on the microlevel and of the foundation are shown. Keywords: thin functionally gradedplate band,microstructure size, free vibrations,material properties, elasic foundation 1. Introduction In the civil engineering, plates interactingwith the subsoil are often used as elements of building foundations or reinforcements of roads foundations. In many cases, the first approximation of the subsoil can be a model ofWinkler’s foundation. In this paper, free vibrations of thin functionally graded plate bands with span L (along the x1-axis) interacting with elastic Winkler’s foundation are considered. It is assumed that these plate bands have the functionally graded structure along their span on the macrolevel, but on the microlevel their structure is, so called, tolerance-periodic in x1, i.e. nearly periodic, cf. Jędrysiak (2010), Jędrysiak and Michalak (2011), Kaźmierczak and Jędrysiak (2010, 2011, 2013). Hence, the plate bands are called thin functionally graded plate bands, cf. Jędrysiak (2010). The plate material properties are assumed to be independent of the x2-coordinate. The size of microstructure is determined by length l of “the cell”, being very small compared to the plate spanL. A fragment of the plate band is shown in Fig. 1. Fig. 1. A fragment of a thin functionally graded plate band interacting withWinkler’s foundation 630 J. Jędrysiak,M. Kaźmierczak-Sobińska Vibrations of such plates are described by a partial differential equation with highly oscilla- ting, tolerance-periodic, non-continuous coefficients. Because analysis of these plates is too com- plicated using the equation of the plate theory, different averaged models have been proposed. Thesemodels are usually described by partial differential equationswith smooth, slowly-varying coefficients. Functionally graded structures can be described by approaches applied to analyse macro- scopically homogeneous media, e.g. periodic, cf. Suresh and Mortensen (1998). Between these models it can bementioned those based on the asymptotic homogenization method for periodic solids, cf. Bensoussan et al. (1978). Models of such kind for periodic plates can be found in a series of papers, e.g. Kohn and Vogelius (1984). Other models are based on the microlocal parameters approach, cf.Matysiak andNagórko (1989), or the nonstandard homogenization, cf. Nagórko (1998). However, the effect of the microstructure size on the dynamic plate behaviour is neglected in the governing equations of those models. Composite plates can be also parts of more complicated structures such as thin-walled composite columns or beams, cf. Kołakowski (2009, 2012), Królak et al. (2009), Kubiak (2006). Moreover, nonhomogeneous Winkler’s type solids can be approximations of foam cores in three layered composite plates, cf. Magnucka- -Blandzi (2011). Theoretical and numerical results of different problems of functionally graded structures are presented in many papers. Jha et al. (2013) analysed free vibrations of functio- nally graded thick plates with shear and normal deformations effects. The static response of functionally graded plates and shells was investigated using higher order deformation theories byOktem et al. (2012). Vibrations of FG-type plates were analysed using a collocation method with higher-order plate theories by Roque et al. (2007). Free vibrations of shells were presented byTornabene et al. (2011). Problems of functionally graded plates resting on a foundationwere also considered, e.g. by Tahouneh and Naei (2014) with using the three-dimensional elasticity theory by Yajuvindra Kumar and Lal (2012), where vibrations of nonhomogeneous plates with varying thickness interacting with a foundation were analysed. A list of papers of various the- oretical and numerical results of thermomechanical problems of functionally graded structures can be found in Jędrysiak (2010), Woźniak et al. (2008, 2010). Unfotunately, the effect of the microstructure size is usually neglected in the governing equations of those models. This effect can be taken into account in the governing equations in the framework of the tolerance modelling, cf. Woźniak and Wierzbicki (2000), Woźniak et al. (2008, 2010). Various thermomechanical problems of periodic structures were investigated in a series of papers apply- ing this method, e.g. dynamics of periodic plane structures byWierzbicki andWoźniak (2000), vibrations of medium-thickness plates by Baron (2006), static problems of thin plates withmo- derately large deflections byDomagalski and Jędrysiak (2012), nonlinear vibrations of beams by Domagalski and Jędrysiak (2014), vibrations of thin plates resting on an elastic nonhomogene- ous foundation by Jędrysiak (1999, 2003), vibrations of medium-thickness plates resting on an elastic foundation by Jędrysiak and Paś (2005, 2014), vibrations with damping of plate strips with a periodic system of concentrated masses by Marczak and Jędrysiak (2014), vibrations of wavy-type plates by Michalak (2002), vibrations of thin plates with stiffeners by Nagórko and Woźniak (2002), vibrations of thin cylindrical shells by Tomczyk (2007, 2013). These papers show that the effect of the microstructure size plays a crucial role in nonstationary (and some stationary) problems of periodic structures. The tolerance modelling method is also applied to similar thermomechanical problems of functionally graded structures, e.g. Jędrysiak (2010),Woźniak et al. (2010). Someapplications to dynamic and stability problems for thin transversally graded plates with themicrostructure size bigger than the plate thickness were shownby Jędrysiak andMichalak (2011), Kaźmierczak and Jędrysiak (2010, 2011, 2013, 2014); for thin functionally graded plates with the microstructure size of an order of the plate thickness by Jędrysiak (2013, 2014), Jędrysiak and Pazera (2014); for functionally graded skeletonal shells byMichalak (2012); for heat conduction in functionally On free vibrations of thin functionally graded plate bands... 631 graded hollow cylinders byOstrowski andMichalak (2011); for thin longitudinally graded plates byMichalak andWirowski (2012),Wirowski (2012). An extended list of papers can be found in the books byWoźniak et al. (2008, 2010). The main aims of this paper are four. The first of them is to formulate the tolerance and the asymptotic models of vibrations for thin transversally graded plate bands. The second aim is to apply thesemodels to calculate free vibration frequencies of a simply supported plate band interacting withWinkler’s foundation using the Ritz method. The third is to analyse the effect of various distribution functions of material properties and the effect of the foundation on the frequencies. The fourth aim is to show the effect of differences in the cell between material properties (Young’s modulus andmass densities) on the frequencies. 2. Formulation of the problem Considerations are assumed to be independent of the x2-coordinate. Let us denote x = x1, z = x3, x ∈ [0,L], z ∈ [−d/2,d/2], with d as a constant plate thickness. The plate band is described in the intervalΛ=(0,L), with “the basic cell”∆≡ [−l/2, l/2] in the intervalΛ, where l is the length of the basic cell, satisfying conditions: d≪ l≪L. By ∆(x)≡ (x− l/2,x+ l/2) a cell with the centre at x ∈ Λ is denoted. It is assumed that the plate band is made of two elastic isotropic materials, perfectly bonded across interfaces. The materials are characterised byYoung’smoduliE′,E′′, Poisson’s ratios ν′, ν′′ andmass densities ρ′, ρ′′. Similarly, the elastic foundation is made of two various materials characterised by Winkler’s coefficients k′, k′′. Let E(x), ∆(x), k(x), x∈Λ, be tolerance-periodic, highly-oscillating functions in x, but Poisson’s ratio ν ≡ ν′ = ν′′ constant. AssumingE′ 6=E′′ and/or ρ′ 6= ρ′′, the plate material structure can be treated as functionally graded in the x-axis direction. Similarly, for k′ 6= k′′, the foundation structure is functionally graded.Let∂ denote thederivativewith respect tox, andw(x,t) (x∈Λ, t∈ (t0, t1)) be deflection of the plate band. Tolerance-periodic functions in x describe the plate band properties – the mass density per unit area of the midplane µ, the rotational inertia ϑ and the bending stiffnessB µ(x)≡ dρ(x) ϑ(x)≡ d3 12 ρ(x) B(x)≡ d3 12(1−ν2) E(x) (2.1) respectively. Free vibrations of thin functionally gradedplate bands, on thewell knownKirchhoff plate theory assumptions, canbedescribedby thepartial differential equation of the fourthorder with respect to the deflection w ∂∂[B∂∂w]+µẅ−∂(ϑ∂ẅ)+kw=0 (2.2) with highly-oscillating, non-continuous, tolerance-periodic coefficients. 3. Foundations of the modelling 3.1. Introductory concepts Following the book edited by Woźniak et al. (2010) and the book by Jędrysiak (2010), some introductory concepts of the tolerance modelling are used, i.e. the averaging operator, tolerance system, slowly-varying function SVαξ (Λ,∆), tolerance-periodic function TP α ξ (Λ,∆), highly oscillating function HOαξ (Λ,∆), fluctuation shape function FS 2 ξ(Λ,∆), where ξ is the tolerance parameter and α is a positive constant determining kind of the function. Some of these concepts are reminded below. 632 J. Jędrysiak,M. Kaźmierczak-Sobińska Acell at x∈Λ∆ is denoted by∆(x)=x+∆,Λ∆ = {x∈∆ : ∆(x)⊂Λ}. The basic concept of the modelling technique is the averaging operator, for an integrable function f defined by 〈f〉(x)= 1 l ∫ ∆(x) f(y) dy x∈Λ∆ y∈∆(x) (3.1) If f is a function tolerance-periodic in x, its averaged value by (3.1) is slowly-varying in x. Let h(·) be a highly oscillating function defined on Λ, h∈HO2ξ(Λ,∆), continuous together with the gradient ∂1h andwith a piecewise continuous and bounded gradient ∂2h. The function h(·) is the fluctuation shape function of the 2-nd kind, FS2ξ(Λ,∆), if it depends on l as a parameter and the condition 〈µh〉(x) ≈ 0 holds for every x ∈ Λ∆, where µ > 0 is a certain tolerance-periodic function, l is the microstructure parameter. 3.2. Fundamental assumptions Following thebooksbyWoźniak et al. (2010), Jędrysiak (2010) andapplying the introductory concepts, the following fundamental modelling assumptions can be formulated. The micro-macro decomposition is the first assumption, in which the deflection w appears in the form w(x,t) =W(x,t)+hA(x)VA(x,t) A=1, . . . ,N x∈Λ (3.2) with W(·, t), VA(·, t) ∈ SV 2ξ (Λ,∆) (for every t) as basic kinematic unknowns (W is called the macrodeflection;VA are called the fluctuation amplitudes) andhA(·)∈FS2ξ(Λ,∆) as the known fluctuation shape functions. In the tolerance averaging approximation, being the secondmodelling assumption, the terms of an order ofO(ξ) are treated as negligibly small in the course of modelling. 4. The tolerance modelling procedure The modelling procedure of the tolerance averaging technique was shown by Woźniak et al. (2010) and for thin functionally graded plates by Jędrysiak (2010). Below, it is outlined. The formulation of the action functional is the first step A(w(·)) = ∫ Λ t1∫ t0 L(y,∂∂w(y,t),∂ẇ(y,t), ẇ(y,t),w(y,t)) dt dy (4.1) where the lagrangean L is given by L= 1 2 (µẇẇ+ϑ∂ẇ∂ẇ−B∂∂w∂∂w−kww) (4.2) From the principle of stationary action to functional (4.1) combined with (4.2), after some manipulations, known equation (2.2) of free vibrations for thin functionally graded plate bands interacting withWinkler’s foundation is derived. Thenext stepof the tolerancemodelling is substitutingmicro-macrodecomposition (3.2) into lagrangean (4.2). Applying averaging operator (3.1) and the tolerance averaging approximation, the tolerance averaged form 〈Lh〉 of lagrangean (4.2) is obtained in the third step 〈Lh〉=− 1 2 {(〈B〉∂∂W +2〈B∂∂hB〉VB)∂∂W + 〈k〉WW +2W〈khB〉VB + 〈ϑ〉∂Ẇ∂Ẇ +(〈khAhB〉+ 〈B∂∂hA∂∂hB〉)VAVB −〈µ〉ẆẆ +(〈ϑ∂hA∂hB〉−〈µhAhB〉)V̇AV̇B} (4.3) On free vibrations of thin functionally graded plate bands... 633 where the macrodeflection W and the fluctuation amplitudes VA, A=1, . . . ,N, are new basic kinematic unknowns. The known fluctuation shape functions VA are introduced inmicro-macro decomposition (3.2). Using the principle of stationary action to the averaged functional Ah combined together with lagrangean (4.3), the system of governing equations is derived. 5. Model equations From the principle of stationary action applied to the averaged functional Ah with lagrangean (4.3), after somemanipulations, the following system of equations forW and VA is obtained ∂∂(〈B〉∂∂W + 〈B∂∂hB〉VB)+ 〈k〉W + 〈khA〉VA+ 〈µ〉Ẅ −〈ϑ〉∂∂Ẅ =0 〈B∂∂hA〉∂∂W + 〈khA〉W =−(〈B∂∂hA∂∂hB〉+ 〈khAhB〉)VB − (〈µhAhB〉+ 〈ϑ∂hA∂hB〉)V̈B (5.1) The above equations are a system of N+1 differential equations constituting the tolerance model of thin functionally graded plate bands. The underlined terms in these equations depend on themicrostructure parameter l. Hence, this model allows one to take into account the effect of the microstructure size on free vibrations of these plates. The coefficients of equations (5.1) are slowly-varying functions in x. It can be observed that boundary conditions for these plate bands (in Λ = (0,L)) are formulated only for the macrodeflection W (on edges x = 0,L) but not for the fluctuation amplitudes VA,A=1, . . . ,N. It can be observed that after neglecting terms with the parameter l in equations (5.1)2, the algebraic equations for fluctuation amplitudes VA are obtained VA =−(〈B∂∂hA∂∂hB〉)−1〈B∂∂hB〉∂∂W (5.2) Substituting formula (5.2) into (5.1)1, the following equation for W is derived ∂∂ ( (〈B〉−〈B∂∂hA〉(〈B∂∂hA∂∂hB〉)−1〈B∂∂hB〉)∂∂W ) + 〈k〉W + 〈µ〉Ẅ =0 (5.3) The above equation together with micro-macro decomposition (3.2) represents the asymp- totic model of thin functionally graded plate bands. Governing equation (15) with equations (5.2) of this model can be obtained using also the formal asymptotic modelling procedure, cf. Woźniak et al. (2010), Kaźmierczak and Jędrysiak (2011, 2013). It can be observed that this procedure leads to model equations without terms describing the effect of the microstructure size on free vibrations of these plates. Hence, in the framework of the asymptotic model, the macrobehaviour of these plate bands can be only investigated. 6. Example: free vibrations of plate bands 6.1. Introduction Free vibrationsof a simply supportedthinplatebandwith spanLalong thex-axis interacting withWinkler’s foundation are considered. The properties of the plate band are ρ(·,z),E(·,z) = { ρ′,E′ for z∈ ( (1−γ(x))l/2,(1+γ(x))l/2 ) ρ′′,E′′ for z∈ [0,(1−γ(x))l/2]∪ [(1+γ(x))l/2, l] (6.1) with a distribution function ofmaterial properties γ(x), see Fig. 2.Moreover, it is assumed that the foundation is homogeneous with theWinkler’s coefficient k= const. 634 J. Jędrysiak,M. Kaźmierczak-Sobińska Our considerations are restricted only to one fluctuation shape function, i.e. A = N = 1. Denote h≡ h1, V ≡ V 1. Hence, micro-macro decomposition (3.2) of the deflection w(x,t) has the form w(x,t) =W(x,t)+h(x)V (x,t) whereW(·, t),V (·, t)∈SV 2ξ (Λ,∆) for every t∈ (t0, t1), h(·)∈FS 2 ξ(Λ,∆). Fig. 2. “Basic cell” of the functionally graded plate band interacting withWinkler’s foundation The cell structure is shown in Fig. 2. Thus, the periodic approximation of the fluctuation shape function h(x) has the form h̃(x,z)=Λ2[cos(2πz/l)+ c(x)] z∈∆(x) x∈Λ where the parameter c(x) is a slowly-varying function in x and is determined by 〈µ̃h̃〉=0 c= c(x)= {sin[πγ̃(x)](ρ′−ρ′′)} { π{ρ′γ̃(x)+ρ′′[1− γ̃(x)]} }−1 where γ̃(x) is the periodic approximation of the distribution function of the material proper- ties γ(x). The parameter c(x) is treated as constant in the calculations of derivatives ∂h̃, ∂∂h̃. Under denotations: B̆= 〈B〉 B̂= 〈B∂∂h〉 B= 〈B∂∂h∂∂h〉 K̆ = 〈k〉 K̃ = l−2〈kh〉 K = l−4〈khh〉 µ̆= 〈µ〉 µ= l−4〈µhh〉 ϑ̆= 〈ϑ〉 ϑ= l−2〈ϑ∂h∂h〉 (6.2) tolerance model equations (5.1) can be written as ∂∂(B̆∂∂W + B̂V )+K̆W + l2K̃V + µ̆Ẅ − ϑ̆∂∂Ẅ =0 B̂∂∂W + l2K̃W +(B+ l4K)V + l2(l2µ+ϑ)V̈ =0 (6.3) however, plate band equation (5.3) has the form ∂∂[(B̆− B̂2/B)∂∂W ]+ K̆W + µ̆Ẅ − ϑ̆∂∂Ẅ =0 (6.4) Equation (6.4) describes free vibrations of this plate bandwithin the asymptotic model. All coefficients of model equations (6.3) and (6.4) are slowly-varying functions in x. 6.2. The Ritz method applied to the model equations Equations (6.3) or (6.4) have slowly-varying functional coefficients. Analytical solutions to them are too difficult to find. Hence, approximate formulas fo free vibrations frequencies can be derived using the known Ritz method, cf. Kaźmierczak and Jędrysiak (2010). In order to obtain these formulas, relations of the maximum strain energy Amax and the maximal kinetic energy Vmax have to be determined. On free vibrations of thin functionally graded plate bands... 635 Solutions to equation (6.4) and equations (6.3) are assumed in form satisfying the boundary conditions for the simply supported plate band W(x,t)=AW sin(αx)cos(ωt) V (x,t)=AV sin(αx)cos(ωt) (6.5) where α is the wave number, ω is the free vibrations frequency. Introducing denotations B̆= d3 12(1−ν2) L∫ 0 {E′′[1− γ̃(x)]+ γ̃(x)E′}[sin(αx)]2 dx B̂= πd3 3(1−ν2) (E′−E′′) L∫ 0 sin(πγ̃(x))[sin(αx)]2 dx B= (πd)3 3(1−ν2) L∫ 0 {(E′−E′′)[2πγ̃(x)+sin(2πγ̃(x))]+2πE′′}[sin(αx)]2 dx µ̆= d L∫ 0 {[1− γ̃(x)]ρ′′+ γ̃(x)ρ′}[sin(αx)]2 dx ϑ̆= d3 12 L∫ 0 {[1− γ̃(x)]ρ′′+ γ̃(x)ρ′}[cos(αx)]2 dx µ= d 4π L∫ 0 {(ρ′−ρ′′)[2πγ̃(x)+sin(2πγ̃(x))]+2πρ′′}[sin(αx)]2 dx + d π (ρ′−ρ′′) L∫ 0 c(x)[πc(x)γ̃(x)−2sin(πγ̃(x))][sin(αx)]2 dx +dρ′′ L∫ 0 [c(x)]2[sin(αx)]2 dx ϑ= πd3 12 L∫ 0 {(ρ′−ρ′′)[2πγ̃(x)− sin(2πγ̃(x))]+2πρ′′}[sin(αx)]2 dx K̆ = k L∫ 0 [sin(αx)]2 dx K̃ = k L∫ 0 c(x)[sin(αx)]2 dx= k(ρ′−ρ′′) π L∫ 0 sin(πγ̃(x)) ρ′γ̃(x)+ρ′′[1− γ̃(x)] [sin(αx)]2 dx K = 1 2 k L∫ 0 c(x)[sin(αx)]2 dx= k(ρ′−ρ′′) 2π L∫ 0 sin[πγ̃(x)] ρ′γ̃(x)+ρ′′[1− γ̃(x)] [sin(αx)]2 dx (6.6) and using (6.5), the formulas of the maximum energies – strain Emax and kinetic Vmax – in the framework of the tolerance model, take the form ETMmax = 1 2 [(B̆A2Wα 2−2B̂AWAV )α 2+ K̆A2W +2l 2K̃AWAV +(B+ l 4K)A2V ] VTMmax = 1 2 [A2W(µ̆+ ϑ̆α 2)+A2V l 2(l2µ+ϑ)]ω2 (6.7) 636 J. Jędrysiak,M. Kaźmierczak-Sobińska For the asymptotic model, they can be written as EAMmax = 1 2 [(B̆A2Wα 2−2B̂AWAV )α 2+K̆A2W +BA 2 V ] V AM max = 1 2 A2W(µ̆+ ϑ̆α 2)ω2 (6.8) The conditions of the Ritz method take the form ∂(Emax−Vmax) ∂AW =0 ∂(Emax−Vmax) ∂AV =0 (6.9) Using (6.9) to relations (6.7), after somemanipulations, the following formulas are obtained (ω−,+) 2 ≡ l2(l2µ+ϑ)(α4B̆+ K̆)+(µ̆+α2ϑ̆)(B+ l4K) 2(µ̆+α2ϑ̆)l2(l2µ+ϑ) (6.10) ∓ √ [l2(l2µ+ϑ)(α4B̆+ K̆)− (µ̆+α2ϑ̆)(B+ l4K)]2+4(l2K̃−α2B̂)2l2(µ̆+α2ϑ̆)(l2µ+ϑ) 2(µ̆+α2ϑ̆)l2(l2µ+ϑ) for the lower ω− and the higher ω+ free vibrations frequencies, respectively, in the framework of the tolerance model. For asymptotic model conditions (6.9) applied to equations (6.8) lead, after somemanipula- tions, to the following formula ω2 ≡ [(α4B̆+ K̆)B−α4B̂2][(µ̆+ ϑ̆α2)B]−1 (6.11) of the lower free vibration frequency ω. 6.3. Results Calculations aremade for the following distribution functions of thematerial properties γ(x) γ̃(x)= sin2 πx L γ̃(x)= cos2 πx L γ̃(x)= (x L )2 γ̃(x)= sin πx L γ̃(x)= 1 2 (6.12) where formula (6.12)5 determines an example of a periodic plate band. Let us also introduce dimensionless frequency parameters for the free vibration frequencies ω and ω−, ω+ determined by equations (6.11) and (6.10), respectively Ω2 ≡ 12(1−ν2)(E′) −1 l2ω2 (Ω−) 2 ≡ 12(1−ν2)(E′) −1 l2(ω−) 2 (Ω+) 2 ≡ 12(1−ν2)(E′) −1 l2(ω+) 2 (6.13) Moreover, a dimensionless parameter of the foundation is introduced κ≡ 12(1−ν2)(E′) −1 kd Results of calculations are shown in Figs. 3-6, where the results obtained by the tolerance or asymptotic models for plate bands with the simply supported edges are presented. Calculations are made for Poisson’s ratio ν = 0.3, wave number α = π/L, ratio l/L = 0.1, ratios of plate thickness d/l =0.1, 0.01 and ratios of the foundation κ= 5 ·10−5, 0.05. Figures 3 and 4 show plots of the lower frequency parameters versus both ratios E′′/E′ − ρ′′/ρ′, but Figs. 5 and 6 present diagrams of the higher frequency parameters versus these both ratios. Plots in Figs. 3a, 4a, 5a, 6a are made for κ = 5 · 10−5, but in Figs. 3b, 4b, 5b, 6b for κ = 0.05. Moreover, in Figs. 3 and 4, a comparison of the lower frequency parameters versus both ratiosE′′/E′−ρ′′/ρ′ calculated in the framework of the tolerance model (formulas (6.13)2 and (6.10)1) and of the asymptotic model (formulas (6.13)1 and (6.11)) is presented. Plots shown in Figs. 3 and 5 are made for d/l=0.1, but in Figs. 4 and 6 they are for d/l=0.01. From the results shown in Figs. 3-6 some remarks and comments are formulated. On free vibrations of thin functionally graded plate bands... 637 1◦ The lower frequency parameters calculated by the asymptotic model, (6.11), and the to- lerance model, (6.10)1, depend on the plate thickness ratio d/l and the parameter of foundation κ, see Figs. 3 and 4: • The lower frequency parameters calculated by the asymptotic model, (6.11), and the tolerance model, (6.10)1, are nearly identical for thicker plates, e.g. d/l = 0.1, and weaker foundations, e.g. κ=5 ·10−5, see Fig. 3a. • However, higher values of these parameters are found from the tolerance model for smaller thickness of plates, d/l < 0.1, and for stronger foundations, κ> 5 ·10−5, see Figs. 3b and 4. • Differences between these frequency parameters depend on the plate thickness ratio d/l and the parameter of foundation κ. They increase with a decrease in the plate thickness, d/l > 0, and an increase in the stiffness of foundation, κ > 5 · 10−5, e.g. d/l=0.01, κ=0.05, see Fig. 3b. 2◦ The effect of distribution functions of thematerial properties γ(x) on the lower frequency parameters for various ratios E′′/E′ ∈ [0,1], ρ′′/ρ′ ∈ [0,1] for the simply supported plate band can be observed in Figs. 3 and 4: • The highest values of these frequency parameters are obtained for all pairs of ratios (E′′/E′,ρ′′/ρ′) fromthe above intervals of the functionγ(x) by (6.12)2 and for smaller thickness of the plates, d/l < 0.1, or for stronger foundations, cf. κ > 5 · 10−5, see Figs. 3b and 4. • The highest values of these frequency parameters for thicker plates, e.g. d/l=0.1, or for weaker foundations, e.g. κ=5 ·10−5, see Fig. 3a, are obtained: – for γ(x) by (6.12)2 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ > (E′′/E′)0 > 0, ρ′′/ρ′ < (ρ′′/ρ′)0((E′′/E′)0) > 0, where (ρ′′/ρ′)0 de- pends on (E′′/E′)0, – for γ(x) by (6.12)4 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ < (E′′/E′)0 > 0, ρ′′/ρ′ > (ρ′′/ρ′)0((E′′/E′)0)> 0, where (ρ′′/ρ′)0 depends on (E′′/E′)0 (unfortunately, it is not visible in this form of these diagrams) Fig. 3. Plots of the dimensionless frequency parametersΩ andΩ− of lower free vibration frequencies versus ratiosE′′/E′−ρ′′/ρ′ by the asymptotic model (surfaces a), the tolerance model (surfaces b), made for: (a) d/l=0.1, κ=5 ·10−5; (b) d/l=0.1, κ=0.05 (1 – γ by (6.12)1; 2 – γ by (6.12)2; 3 – γ by (6.12)3; 4 – γ by (6.12)4; 0 – γ by (6.12)5; the grey plane is related to the frequency parameter for the homogeneous plate band, i.e.E′′/E′ = ρ′′/ρ′ =1) • The smallest values of these frequency parameters are obtained for all pairs of ratios (E′′/E′,ρ′′/ρ′) fromthe above intervals of the functionγ(x) by (6.12)4 and for smaller thickness of the plates, d/l < 0.1, or for stronger foundations, cf. κ > 5 · 10−5, see Figs. 3b and 4. 638 J. Jędrysiak,M. Kaźmierczak-Sobińska Fig. 4. Plots of the dimensionless frequency parametersΩ andΩ− of lower free vibration frequencies versus ratiosE′′/E′−ρ′′/ρ′ by the asymptotic model (surfaces a), the tolerance model (surfaces b), made for: (a) d/l=0.01, κ=5 ·10−5; (b) d/l=0.01, κ=0.05 (1 – γ by (6.12)1; 2 – γ by (6.12)2; 3 – γ by (6.12)3; 4 – γ by (6.12)4; 0 – γ by (6.12)5; the grey plane is related to the frequency parameter for the homogeneous plate band, i.e.E′′/E′ = ρ′′/ρ′ =1) • The smallest values of these frequency parameters for thicker plates, e.g. d/l = 0.1, or for weaker foundations, e.g. κ=5 ·10−5, see Fig. 3a, are obtained: – for γ(x) by (6.12)4 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ > (E′′/E′)1 > 0, ρ′′/ρ′ < (ρ′′/ρ′)1((E′′/E′)1) > 0, where (ρ′′/ρ′)1 de- pends on (E′′/E′)1, – for γ(x) by (6.12)2 and for pairs of ratios (E′′/E′,ρ′′/ρ′), such that E′′/E′ > (E′′/E′)2 > 0, ρ′′/ρ′ > (ρ′′/ρ′)2((E′′/E′)2) > 0, where (ρ′′/ρ′)2 de- pends on (E′′/E′)2 (unfortunately, it is not visible in this form of diagrams), – for γ(x) by (6.12)3 and for pairs of ratios (E′′/E′,ρ′′/ρ′), such that (E′′/E′)2 > E′′/E′ < (E′′/E′)3 > 0, (ρ′′/ρ′)2((E′′/E′)2) > ρ′′/ρ′ > (ρ′′/ρ′)3((E′′/E′)3) > 0, where (ρ′′/ρ′)2, (ρ′′/ρ′)3 depend on (E′′/E′)2, (E′′/E′)3, respectively (not visible in this form of diagrams), – for γ(x) by (6.12)5 (periodic plate band) and for pairs of ratios (E′′/E′,ρ′′/ρ′), such that (E′′/E′)1 > E′′/E′ > (E′′/E′)3 > 0, (ρ′′/ρ′)1((E′′/E′)1) < ρ′′/ρ′ < (ρ′′/ρ′)3((E′′/E′)3)> 0, where (ρ′′/ρ′)1, (ρ′′/ρ′)3 depend on (E′′/E′)1, (E′′/E′)3, respectively (not visible in this form of diagrams). 3◦ Figure 3 shows also an interesting feature that for the distribution functions of thematerial properties γ(x) used and for rather thicker plates, e.g. d/l=0.1, and weaker foundations, e.g. κ=5·10−5, the lower frequency parameters are higher or smaller than this parameter for the homogeneous plate bandmade of a stronger material, i.e. ρ′′/ρ′ =E′′/E′ =1 (the grey plane in Fig. 3a). 4◦ The effect of distribution functions of the material properties γ(x) on higher frequency parameters for various ratios E′′/E′ ∈ [0,1], ρ′′/ρ′ ∈ [0,1] for the simply supported plate band can be observed in Figs. 5 and 6: • The highest values of these frequency parameters for rather very thin plates, e.g. d/l=0.01, and for stronger foundations, e.g. κ=0.05, see Fig. 6a, are obtained: – for γ(x) by (6.12)3 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ > (E′′/E′)0 > 0, ρ′′/ρ′ < (ρ′′/ρ′)0((E′′/E′)0) > 0, where (ρ′′/ρ′)0 de- pends on (E′′/E′)0, – for γ(x) by (6.12)2 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ < (E′′/E′)0 > 0, ρ′′/ρ′ > (ρ′′/ρ′)0((E′′/E′)0) > 0, where (ρ′′/ρ′)0 de- pends on (E′′/E′)0. On free vibrations of thin functionally graded plate bands... 639 Fig. 5. Plots of the dimensionless frequency parametersΩ+ of higher free vibration frequencies versus ratiosE′′/E′−ρ′′/ρ′, made for: (a) d/l=0.1, κ=5 ·10−5; (b) d/l=0.1, κ=0.05 (1 – γ by (6.12)1; 2 – γ by (6.12)2; 3 – γ by (6.12)3; 4 – γ by (6.12)4; 0 – γ by (6.12)5) Fig. 6. Plots of the dimensionless frequency parametersΩ+ of higher free vibration frequencies versus ratiosE′′/E′−ρ′′/ρ′, made for: (a) d/l=0.01, κ=5 ·10−5; (b) d/l=0.01, κ=0.05 (1 – γ by (6.12)1; 2 – γ by (6.12)2; 3 – γ by (6.12)3; 4 – γ by (6.12)4; 0 – γ by (6.12)5) • The highest values of these frequency parameters for thicker plates, e.g. d/l > 0.01, and for weaker foundations, e.g. κ¬ 0.05, see Figs. 5 and 6a, are obtained: – for γ(x) by (6.12)4 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ < (E′′/E′)0 > 0, ρ′′/ρ′ > (ρ′′/ρ′)0((E′′/E′)0) > 0, where (ρ′′/ρ′)0 de- pends on (E′′/E′)0, – for γ(x) by (6.12)5 (periodic plate band) and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that (E′′/E′)1 > E′′/E′ > (E′′/E′)0 > 0, (ρ′′/ρ′)1((E′′/E′)1) < ρ′′/ρ′ < (ρ′′/ρ′)0((E′′/E′)0)> 0, where (ρ′′/ρ′)0, (ρ′′/ρ′)1 depend on (E′′/E′)0, (E′′/E′)1, respectively, – for γ(x) by (6.12)2 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ > (E′′/E′)1 > 0, (ρ′′/ρ′)2((E′′/E′)2) < ρ′′/ρ′ < (ρ′′/ρ′)1((E′′/E′)1) > 0, where (ρ′′/ρ′)1, (ρ′′/ρ′)2 depend on (E′′/E′)1, (E′′/E′)2, respectively, – for γ(x) by (6.12)3 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that (E′′/E′)2 > E′′/E′ > (E′′/E′)1 > 0, (ρ′′/ρ′)2((E′′/E′)2) > ρ′′/ρ′ < (ρ′′/ρ′)1((E′′/E′)1)> 0, where (ρ′′/ρ′)1, (ρ′′/ρ′)2 depend on (E′′/E′)1, (E′′/E′)2, respectively. • The smallest values of the higher frequency parameters for rather very thin plates, e.g. d/l=0.01, and for stronger foundations, e.g. κ=0.05, see Fig. 6b, are obtained for γ(x) by (6.12)4 and for all pairs of ratios (E′′/E′,ρ′′/ρ′). • The smallest values of these frequency parameters for thicker plates, e.g. d/l > 0.01, and for weaker foundations, e.g. κ¬ 0.05, see Figs. 5 and 6a, are obtained: 640 J. Jędrysiak,M. Kaźmierczak-Sobińska – for γ(x) by (6.12)3 and for all pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ > (E′′/E′)3 > 0, ρ′′/ρ′ < (ρ′′/ρ′)3((E′′/E′)3) > 0, where (ρ′′/ρ′)3 de- pends on (E′′/E′)3, – for γ(x) by (6.12)2 and for pairs of ratios (E′′/E′,ρ′′/ρ′) such that E′′/E′ < (E′′/E′)3 > 0, ρ′′/ρ′ > (ρ′′/ρ′)3((E′′/E′)3) > 0, where (ρ′′/ρ′)3 de- pends on (E′′/E′)3. 7. Remarks Using the tolerance modelling to the known differential equation of thin plates resting onWin- kler’s foundation, the averaged tolerancemodel equations of functionally graded plate bands are obtained. From the differential equation with non-continuous, tolerance-periodic coefficients, a system of differential equations with slowly-varying coefficients is derived. It should be noted that the tolerance model equations have terms dependent on the microstructure parameter l, and describe the effect of themicrostructure size on the behaviour of these plates. However, the asymptotic model equation neglects this effect. Free vibration frequencies of the simply supported plate band have been analysed in the example for various distribution functions of the material properties γ(x), different ratios of material propertiesE′′/E′, ρ′′/ρ′ and of the plate thickness d/l as well as various parameters of foundation κ. Analysing results of this example, it can be observed that: 1◦ Using both the presentedmodels – the tolerance and the asymptotic one, lower free vibra- tions frequencies can be analysed. 2◦ Lower and higher free vibrations frequencies decrease with an increase in the ratio ρ′′/ρ′, but they increase with the increasing ratio E′′/E′. 3◦ The asymptotic model cannot be applied to analyse lower free vibrations frequencies of rather very thin plates (with the ratio d/l=0.01) and rather strong foundations (with the foundation parameter κ=0.05), see Fig. 4b. 4◦ For thicker plates (e.g. d/l=0.1) and weaker foundations (κ=5 ·10−5), microstructured plates can bemade applyingdifferent distribution functions of thematerial propertiesγ(x) such that their lower fundamental free vibrations frequencies are smaller or higher than these frequencies for the homogeneous plate made of a stronger material (plates with the ratios E′′/E′ = ρ′′/ρ′ =1) for different pairs of the ratios (E′′/E′,ρ′′/ρ′). 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