Free transverse vibration analysis of orthotropic multi-viscoelastic microplate system embedded in visco-Pasternak medium via modified strain gradient theory

Abstract

In this paper, an analytical process is proposed to investigate the size-dependent free vibration of orthotropic multi-viscoelastic microplate systems (OMVMPS) embedded in Kelvin–Voigt visco-Pasternak medium according to the modified strain gradient theory. Governing equations of motion in the partial form and the related boundary conditions are derived by utilizing the Kirchhoff plate theory and Hamilton’s variational principle. The two different sorts of “chain” boundary conditions like “clamped Chain” and “free chain” systems are considered for the ends of microplate system. Navier’s method, which convinces that the simply supported boundary conditions and trigonometric methods are applied to analytically investigate the size effect of the natural frequencies of OMVMPS. The numerical outcomes are offered to report the variation of OMVMPS natural frequencies with the numerous amounts of the microplate numbers, the length scale parameter, aspect ratio, visco-Pasternak foundation parameters, the thickness of microplate, and higher modes number. Several numerical outcomes of this research depict that when the number of microplates is low, there is a significant distinction between natural frequencies achieved for “clamped chain” and “free chain” systems. Also, it is demonstrated that by increasing the number of microplates, the effect of the visco-Pasternak substrate on the natural frequency of system vibration decreases.

Keywords

Orthotropic multi-viscoelastic microplate system modified strain gradient theory visco-Pasternak substrate free vibration exact solutions

Introduction

Micro- and nanostructures have numerous engineering applications due to their mechanical and physical properties. Experiments have proved that the effect of small size becomes important at micro/nanoscale. So, the classic continuum theory will not be able to predict the mechanical behavior of micro/nanostructures properly [1 –4]. Several atomistic methods and nonclassical continuation models have been proposed to capture the size effect at micro/nanoscale. Couple stress theory [5,6], strain gradient theory [7 –9], Eringen nonlocal elasticity theory [10,11] are all the samples of nonclassical continuum models. These higher order theories contain internal length scale parameters that are inevitable for micro/nanostructures. During the past years, many studies have been conducted on the micro/nanostructures via the aforementioned theories.

Fleck and Hutchinson [8] improved one of the higher order continuum theories called the strain gradient theory (SGT) and proposed new form of SGT. Later, by eliminating dilatation gradient tensor and deviatoric stretch gradient tensor in SGT presented by Fleck and Hutchinson [8], a modified couple stress theory (MCST) in which, according to it, the influence of small size is a function of one independent internal length scale parameter is suggested by Yang et al. [12]. Modified strain gradient theory (MSGT) developed by Lam et al. [13] is one of the most popular continuum theories. This higher order continuum approach has three material length scale parameters relevant to deviatoric gradient, dilatation gradient and symmetric rotation gradient tensors. In the framework of this theory, various studies have been reported where size-dependent continuum models are developed. For example, Li and Hu [14] investigated the influence of small-scale on the wave propagation in fluid-conveying viscoelastic single walled carbon nanotubes. Wang et al. [15] developed a nonclassical Kirchhoff plate model via MSGT. Sahmani and Ansari [16] considered the free vibration behavior of microplates made of functionally graded materials using parabolic shear deformation plate theory in conjunction with MSGT. This model considers both the microstructural and shear deformation effects without the need to any shear correction factor. Also, Akgöz and Civalek [17] introduced a new size-dependent thick plate model based on MSGT and sinusoidal shear deformation plate theory. They investigated the static bending, buckling, and free vibration analysis of simply supported square microplates. Ghorbanpour Arani and Shokravi [18] discussed the magneto-vibration of coupled double-layered viscoelastic graphene sheet systems embedded on elastic foundation based on MSGT. Also, Jamalpoor and Hosseini [19] have demonstrated closed-form solutions for biaxial buckling analysis of double-orthotropic microplate-systems subjected to in-plane magnetic field via MSGT. Moreover, according to the third-order shear deformable plate assumption and MSGT, Zhang et al. [20] proposed a novel size-dependent model to investigate free vibration, bending and buckling behavior of circular/annular microplates. Recently, Hosseini et al. [21] analyzed the size effect on the biaxial buckling behavior of an orthotropic multi-microplate system via MSGT. Quan and Duc [22] applied Reddy’s third-order shear deformation shell theory to investigate the effect of thermal loads on the nonlinear vibration and dynamic response of imperfect FGM thick double-curved shallow shells resting on elastic foundations. Furthermore, there are a number of literatures regarding complex shapes of plates and shells [23 –26].

Park and Gao [27] proposed a size-dependent Euler–Bernoulli beam model. They considered static response of beam under transverse loading. Lou et al. [28] analyzed the effect of the material length scale parameter to thickness ratio on the bending, buckling and free vibration behavior of functionally graded microplate using a unified higher order plate theory. According to MCST, Wang et al. [29] analyzed the nonlinear bending behavior of circular microbeam subjected to the uniformly distributed transverse loads. MCST is incorporated into the first-order shear deformation plate theory by Ansari et al. [30] to study the frequency response changes of FG microplate for different types of boundary conditions. Jung et al. [31] proposed a novel size-dependent model to capture size effect on sigmoid functionally graded material microplates using MCST. On the basis of MCST, MA et al. [32] developed a Timoshenko beam model to predict the size effect on the static bending and free vibration behavior of beam. By this model, Yin et al. [33] and Jomehzadeh et al. [34] investigated the free vibration of circular and rectangular microplates. Gupta et al. [35] applied MCST with the goal of numerically predict size effects on natural frequencies of cracked plate made of functionally graded materials. A size-dependent microplate model for the free vibration analysis of microplates is developed based on Mindlin plate theory and MCST by Ke et al. [36] to take into account the influences of transverse shear deformation, rotary inertia, and size effect. Li and Pan [37] studied static bending and vibration behavior of functionally graded piezoelectric microplate based on MCST and sinusoidal plate approach.

Viscoelastic substances present both viscous and elastic conducts when undergoing deformation. This sort of materials has many benefits such as isolating vibration, dampening noise, and absorbing shock. The elastic parts can be modeled as linear and shear springs and the viscous components can be modeled as dashpots. The Maxwell model, the Kelvin–Voigt model and the standard linear solid model are applied to forecast material’s response subjected to different loading conditions. According to the MSGT, Mohammadimehr et al. [38] carried out the free vibration of tapered viscoelastic microrod embedded on the Kelvin–Voigt visco-Pasternak foundation. Zenkour [39] accomplished the transient thermal analysis of single layer graphene sheet on viscoelastic substrate. Also, free transverse vibration analysis of coupled double viscoelastic graphene sheets by the visco-Pasternak medium is presented by Hashemi et al. [40]. On the basis of Reddy’s third-order shear deformation shell theory in conjunction with Sigmoid power law distribution for functionally graded materials, Duc [41] used the Galerkin method and Runge–Kutta approach for studying the nonlinear behavior of imperfect functionally graded circular cylindrical shells under thermal loading. Moreover, the influence of viscoelastic and elastic foundations on the plates and shells are reported by some researchers [42 –46].

The surveying of the literatures revealed the free vibration analysis of viscoelastic microplates based on the MSGT, and it is concluded that no investigation has been conducted on the free vibration analysis of orthotropic multi-viscoelastic microplates embedded in visco-Pasternak medium so far. In this study, we obtain the exact solution for the free transverse vibration of an OMVMPS embedded within a visco-Pasternak medium in the framework of MSGT. The two different types of “chain” boundary conditions such as “clamped chain” and “free chain” are supposed for ends of microplates system. Analytical solutions of homogenous set of equations are justified with the numerical results extracted from the same set of equations. The influence of the higher modes, structural damping of the microplates, aspect ratio, visco-Pasternak foundation parameters, and length scale parameter on the natural frequencies of the system is examined.

Modeling of the problem and formulation

Geometrical configuration

In the current study, a schematic diagram of a set of orthotropic multi-viscoelastic microplates in the rectangular shape embedded within a visco-Pasternak foundation is shown in Figure 1(a). Visco-Pasternak medium is presented by transverse, shear stiffness, and dashpot coefficients, which are denoted by k, k₁, and c, respectively. The assumed boundary conditions are simply supported in all four edges of each microplate. It is also assumed that geometrical and physical specifications comprising uniform thickness h, length a, width b, Poisson’s ratios $ϑ_{12}$ and $ϑ_{21}$ , mass density ρ, and Young’s modulus in the $x - y$ plane (E₁ and E₂) are same for every microplate. Furthermore, the transverse deflections of Λ-coupled orthotropic microplates are expressed by $w_{s} (x, y, t), s = 1, 2, \dots, Λ$ . As mentioned in the previous section, we investigate the size effect on the vibration response of multi-viscoelastic microplates for two different sorts of “chain” boundary conditions like “clamped chain” (could be observed from Figure 1(b), in this case the first and last viscoelastic microplates in the system are jointed with a fixed surface by visco-Pasternak medium) and “free chain” (as can be seen in Figure 1(c), first and last microplates are not connected with the fixed surface).

Figure 1.

Orthotropic multi-viscoelastic microplate system (OMVMPS): (a) the geometry of OMVMPS embedded in visco-Pasternak medium; (b) “clamped chain”; (c) “free chain”.

Constitutive relations for Kirchhoff orthotropic multi-microplate system

On the basis of the classical (Kirchhoff) plate theory, the displacement fields ( $u_{sx}$ , $u_{sy}$ , $u_{sz}$ ) of the plate in Cartesian coordinate for an arbitrary speck in the sth microplate can be expressed as follows

u_{sx} (x, y, z, t) = - {zw}_{s, x} u_{sy} (x, y, z, t) = - {zw}_{s, y} u_{sz} (x, y, z, t) = w_{s} (x, y, t)

(1)

where w_s is the mid-plane lateral displacement of sth microplate in z direction at time t, and

(, i)

expresses

\frac{\partial}{\partial i}

, (i = x, y). In addition, the linear strain–displacement relationships in terms of the displacement are obtained as

ɛ_{xx} = - {zw}_{s, xx} ɛ_{yy} = - {zw}_{s, yy} ɛ_{xy} = - 2 {zw}_{s, xy}

(2)

where

ɛ_{xx}

and

ɛ_{yy}

, respectively, display the normal strain members in x and y directions and

ɛ_{xy}

is the shear strain part in the x–y plane. With respect to plain stress state, the classical stress–strain relation for orthotropic viscoelastic microplate can be obtained considering Kelvin’s pattern of viscoelastic materials [47 –49]

[σ_{xx} σ_{yy} σ_{xy}] = (1 + χ \frac{\partial}{\partial t}) [\frac{E_{1}}{1 - ϑ_{12} ϑ_{21}} \frac{ϑ_{12} E_{2}}{1 - ϑ_{12} ϑ_{21}} 0 \frac{ϑ_{12} E_{2}}{1 - ϑ_{12} ϑ_{21}} \frac{E_{2}}{1 - ϑ_{12} ϑ_{21}} 000 G_{12}] [ɛ_{xx} ɛ_{yy} γ_{xy}]

(3)

where

σ_{xx}, σ_{yy}

, and

σ_{xy}

are the normal and shear stresses in the x–y plane. Also,

χ

refers to the viscoelastic structural feature. According to the concept of orthotropic materials, the mechanical elastic properties depend on orientation at a point in the body. Hence, the mechanical elastic characteristics including E₁, E₂,

ϑ_{12}

, and

G_{12}

have two perpendicular planes of symmetry, and it should also be pointed out that there exists the relation

E_{1} ϑ_{21} = E_{2} ϑ_{12}

for the orthotropic materials. Moreover,

G_{12}

is the shear modulus. In order to find the equilibrium equations of the motion and related boundary conditions along each edge of a microplate, Hamilton’s variational principle is applied in the analytical form as [50]

\int_{t_{1}}^{t_{2}} (δ T - δ U - {δ U}_{m}) d t = 0

(4)

where

δ T, δ U

, and

δ U_{m}

represent the variation of kinetic energy, variation of strain energy density of a microplate, and variation of strain energy induced by the visco-Pasternak medium, respectively.

Based on the facts mentioned about MSGT, it must be noticed that to take into account the influence of the small scale on the mechanical behavior of plate, the three independent higher order inner length scale parameters $(l_{0}, l_{1}, l_{2})$ should be regarded. MSGT expresses that whole strain energy of microplate in addition to the reliance on the classical strain tensor $ɛ_{ij}$ is related to the dilatation gradient vector $κ_{i}$ , the deviatoric stretch gradient tensor $η_{ijk}$ and the symmetric rotation gradient tensor $ϕ_{ij},$ and it can be presented as below

U = \frac{1}{2} \int (σ_{ij} ɛ_{ij} + d_{i} κ_{i} + τ_{ijk} η_{ijk} + m_{ij} ϕ_{ij}) d v, (i, j, k = x, y)

(5)

where

d_{i}, τ_{ijk}

, and

m_{ij}

express the higher order stress tensors, which are demonstrated for orthotropic material as [51]

d_{i} = 2 G_{12} l_{0}^{2} κ_{i} τ_{ijk} = 2 G_{12} l_{1}^{2} η_{ijk} g_{ij} = 2 G_{12} l_{2}^{2} ϕ_{ij}

(6)

where

κ_{i}, η_{ijk}

, and

ϕ_{ij}

are defined as

κ_{i} = ɛ_{mm, i} η_{ijk} = \frac{1}{3} (ɛ_{jk, i} + ɛ_{ki, j} + ɛ_{ij, k}) - \frac{1}{15} [δ_{ij} (ɛ_{mm, k} + 2 ɛ_{mk, m}) + δ_{jk} (ɛ_{mm, i} + 2 ɛ_{mi, m}) + δ_{ki} (ɛ_{mm, j} + 2 ɛ_{mj, m})] ϕ_{ij} = \frac{1}{4} (e_{ipq} ɛ_{qj, p} + e_{jpq} ɛ_{qi, p})

(7)

Here, $δ_{ij}$ and $e_{ijk}$ refer to the Kronecker delta and permutation badge, respectively. By using equations (2) and (7), the dilatation gradient vector $γ_{i}$ , the deviatoric stretch gradient tensor $η_{ijk}$ and the symmetric rotation gradient tensor $ϕ_{ij}$ , are resulted as

κ_{x} = - z (w_{s, xxx} + w_{s, xyy}) κ_{y} = - z (w_{s, yyy} + w_{s, yxx}) κ_{z} = - (w_{s, xx} + w_{s, yy}) η_{xxx} = \frac{1}{5} z (- 2 w_{s, xxx} + 3 w_{s, xyy}) η_{yyy} = \frac{1}{5} z (- 2 w_{s, yyy} + 3 w_{s, xxy}) η_{zzz} = \frac{1}{5} (w_{s, xx} + w_{s, yy}) η_{xxz} = η_{xzx} = η_{zxx} = \frac{1}{15} (- 4 w_{s, xx} + w_{s, yy}) η_{yyz} = η_{yzy} = η_{zyy} = \frac{1}{15} (- 4 w_{s, yy} + w_{s, xx}) η_{xyy} = η_{yxy} = η_{yyx} = \frac{1}{5} z (- 4 w_{s, xyy} + w_{s, xxx}) η_{xxy} = η_{xyx} = η_{yxx} = \frac{1}{5} z (- 4 w_{s, xxy} + w_{s, yyy}) η_{zxz} = η_{zzx} = η_{xzz} = \frac{1}{5} z (w_{s, xyy} + w_{s, xxx}) η_{zzy} = η_{zyz} = η_{yzz} = \frac{1}{5} z (w_{s, xxy} + w_{s, yyy}) η_{xyz} = η_{yxz} = η_{yzx} = η_{zxy} = η_{zyx} = η_{xzy} = - \frac{1}{3} w_{s, xy} ϕ_{xx} = w_{s, xy} ϕ_{yy} = - w_{s, xy} ϕ_{xy} = \frac{1}{2} (w_{s, yy} - w_{s, xx})

(8)

Based on the above-mentioned relations, the variation form of the total strain energy can be defined as

δ U = \int \int_{- \frac{h}{2}}^{\frac{h}{2}} (σ_{xx} δ ɛ_{xx} + σ_{yy} δ ɛ_{yy} + 2 σ_{xy} δ ɛ_{xy} + d_{x} δ κ_{x} + d_{y} δ κ_{y} + d_{z} δ κ_{z} + τ_{xxx} δ η_{xxx} + 3 τ_{xxy} δ η_{xxy} + 3 τ_{xxz} δ η_{xxz} + 3 τ_{xyy} δ η_{xyy} + τ_{yyy} δ η_{yyy} + 3 τ_{yyz} δ η_{yyz} + 3 τ_{xzz} δ η_{xzz} + 3 τ_{yzz} δ η_{yzz} + τ_{zzz} δ η_{zzz} + 6 τ_{xyz} δ η_{xyz} + m_{xx} δ χ_{xx} + m_{yy} δ χ_{yy} + 2 m_{xy} δ χ_{xy}) d z d A

(9)

Furthermore, the other terms of Hamilton’s variational method can be expressed as

δ T = - \int \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (u_{{sx}_{, tt}} δ u_{sx} + u_{{sy}_{, tt}} δ u_{sy} + u_{{sz}_{, tt}} δ u_{sz}) d z d A δ U_{m} = \int {k [(w_{s} - w_{s - 1}) - (w_{s + 1} - w_{s})] δ w_{s} + k_{1} [(w_{s, x} - w_{s - 1, x}) δ w_{s, x} + (w_{s, y} - w_{s - 1, y}) δ w_{s, y} - (w_{s + 1, x} - w_{s, x}) δ w_{s, x} - (w_{s + 1, y} - w_{s, y}) δ w_{s, y}] + c [(w_{s, t} - w_{s - 1, t}) - (w_{s + 1, t} - w_{s, t})] δ w_{s}} d A

(10)

Finally, by inserting relations $δ T, δ U$ , and $δ V$ into Hamilton’s variational method, the following governing equation for sth microplate is obtained

M_{{xx}_{, xx}} + 2 M_{{xy}_{, xy}} + M_{{yy}_{, yy}} - Ξ_{{xxx}_{, xxx}} - Ξ_{{xxy}_{, xxy}} - Ξ_{{xyy}_{, xyy}} - Ξ_{{yyy}_{, yyy}} - k (w_{s} - w_{s - 1}) + k (w_{s + 1} - w_{s}) + k_{1} (w_{s, xx} - w_{s - 1, xx}) + k_{1} (w_{s, yy} - w_{s - 1, yy}) - k_{1} (w_{s + 1, xx} - w_{s, xx}) - k_{1} (w_{s + 1, yy} - w_{s, yy}) - c (w_{s, t} - w_{s - 1, t}) + c (w_{s + 1, t} - w_{s, t}) = ρ {hw}_{s, tt} - \frac{ρ h 3}{12} (w_{s, xxtt} + w_{s, yytt})

(11)

where parameters

M_{xx}

M_{yy}

M_{xy}, Ξ_{xxx}, Ξ_{xxy}, Ξ_{xyy}

, and

Ξ_{yyy}

can be illustrated by the following integral phrases

M_{xx} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (σ_{xx} z + d_{z} + \frac{4}{5} τ_{xxz} - \frac{1}{5} τ_{yyz} - \frac{1}{5} τ_{zzz} + m_{xy}) d z M_{xy} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (σ_{xy} z + τ_{xyz} - \frac{1}{2} m_{xx} + \frac{1}{2} m_{yy}) d z M_{yy} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (σ_{yy} z + d_{z} - \frac{1}{5} τ_{xxz} + \frac{4}{5} τ_{yyz} - \frac{1}{5} τ_{zzz} - m_{xy}) d z Ξ_{xxx} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (d_{x} + \frac{2}{5} τ_{xxx} - \frac{3}{5} τ_{xyy} - \frac{3}{5} τ_{xzz}) z d z Ξ_{xxy} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (d_{y} + \frac{12}{5} τ_{xxy} - \frac{3}{5} τ_{yyy} - \frac{3}{5} τ_{yzz}) z d z Ξ_{xyy} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (d_{x} + \frac{12}{5} τ_{xyy} - \frac{3}{5} τ_{xxx} - \frac{3}{5} τ_{xzz}) z d z Ξ_{yyy} = \int_{- \frac{h}{2}}^{\frac{h}{2}} (d_{y} + \frac{2}{5} τ_{yyy} - \frac{3}{5} τ_{xxy} - \frac{3}{5} τ_{yzz}) z d z

(12)

The boundary condition obtained from Hamilton’s method along x and y directions with normal vectors r_x and r_y for sth microplate are also resulted as

w_{s} = 0 or r_{x} [M_{{xx}_{, x}} + M_{{xy}_{, y}} - Ξ_{{xxx}_{, xx}} - \frac{1}{2} Ξ_{{xxy}_{, xy}} - \frac{1}{2} Ξ_{{xyy}_{, yy}} + k_{1} (w_{s, x} - w_{s - 1, x}) - k_{1} (w_{s + 1, x} - w_{s, x}) + \frac{ρ h 3}{12} w_{s, xtt}] + r_{y} [M_{{yy}_{, y}} + M_{{xy}_{, x}} - \frac{1}{2} Ξ_{{xxy}_{, xx}} - \frac{1}{2} Ξ_{{xyy}_{, xy}} - Ξ_{{yyy}_{, yy}} + k_{1} (w_{s, y} - w_{s - 1, y}) - k_{1} (w_{s + 1, y} - w_{s, y}) + \frac{ρ h 3}{12} w_{s, ytt}] = 0 \frac{\partial w_{s}}{\partial x} = 0 or r_{x} (- M_{xx} + Ξ_{{xxx}_{, x}} + \frac{1}{2} Ξ_{{xxy}_{, y}}) + r_{y} (- M_{xy} + \frac{1}{2} Ξ_{{xxy}_{, x}} + \frac{1}{2} Ξ_{{xyy}_{, y}}) = 0 \frac{\partial w_{s}}{\partial y} = 0 or r_{x} (- M_{xy} + \frac{1}{2} Ξ_{{xxy}_{, x}} + \frac{1}{2} Ξ_{{xyy}_{, y}}) + r_{y} (- M_{yy} + \frac{1}{2} Ξ_{{xyy}_{, x}} + \frac{1}{2} Ξ_{{yyy}_{, y}}) = 0 \frac{\partial 2 w_{s}}{\partial x 2} = 0 or r_{x} Ξ_{xxx} + \frac{1}{2} r_{y} Ξ_{xxy} = 0 \frac{\partial 2 w_{s}}{\partial y 2} = 0 or \frac{1}{2} (r_{x} Ξ_{xyy} + r_{y} Ξ_{yyy}) = 0

(13)

By substituting equation (12) into equation (11) and with respect to equations (2), (3), and (6) to (8), we can achieve a clear expression of the equation of motion for sth viscoelastic microplate in terms of lateral displacements as

(1 + χ \frac{\partial}{\partial t}) {[D_{11} + G_{12} h (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] \frac{\partial 4 w_{s}}{\partial x 4} + [2 (D_{12} + 2 D_{66}) + G_{12} h (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] \frac{\partial 4 w_{s}}{\partial x 2 \partial y 2} + [D_{22} + G_{12} h (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] \frac{\partial 4 w_{s}}{\partial y 4} - G_{12} h 3 (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) (\frac{\partial 2 w_{s}}{\partial x 2} + \frac{\partial 2 w_{s}}{\partial y 2}) 3} - {kw}_{s - 1} + 2 {kw}_{s} - {kw}_{s + 1} + k_{1} (\frac{\partial 2 w_{s - 1}}{\partial x 2} + \frac{\partial 2 w_{s - 1}}{\partial y 2}) - 2 k_{1} (\frac{\partial 2 w_{s}}{\partial x 2} + \frac{\partial 2 w_{s}}{\partial y 2}) + k_{1} (\frac{\partial 2 w_{s + 1}}{\partial x 2} + \frac{\partial 2 w_{s + 1}}{\partial y 2}) - c \frac{\partial w_{s - 1}}{\partial t} + 2 c \frac{\partial w_{s}}{\partial t} - c \frac{\partial w_{s + 1}}{\partial t} + ρ h \frac{\partial 2 w_{s}}{\partial t 2} + \frac{ρ h 3}{12} (\frac{\partial 4 w_{s}}{\partial x 2 \partial t 2} + \frac{\partial 4 w_{s}}{\partial y 2 \partial t 2}) = 0

(14)

where

D_{11}, D_{12}, D_{66}

, and

D_{22}

are the in-plane bending stiffness of the orthotropic microplate, which can be written as

(D_{11}, D_{12}, D_{66}, D_{22}) = \int_{- \frac{h}{2}}^{\frac{h}{2}} (\frac{E_{1}}{(1 - ϑ_{12} ϑ_{21})}, \frac{ϑ_{12} E_{2}}{(1 - ϑ_{12} ϑ_{21})}, G_{12}, \frac{E_{2}}{(1 - ϑ_{12} ϑ_{21})}) z 2 d z

Exact solutions of the free vibration

As mentioned in the previous sections, this study focuses on the simply supported boundary conditions for all edges of every plate, thus the deflection and moment conditions can be written as

w_{s} (0, y, t) = w_{s} (a, y, t) = w_{s} (x, 0, t) = w_{s} (x, b, t) = 0, s = 1, 2, \dots, Λ M_{xxs} (0, y, t) = M_{xxs} (a, y, t) = M_{yys} (x, 0, t) = M_{yys} (x, b, t) = 0

(15)

It must be noticed that in addition to the classical boundary condition, nonclassical boundary condition on the basis of MSGT can be demonstrated as [21]

w_{s} = 0, w_{s, xx} = w_{s, yy} = 0, w_{s, xxxx} = w_{s, yyyy} = w_{s, xxyy} = 0 at = 0, a and y = 0, b

(16)

In order to analytically solve the partial differential equations of OMVMPS, Navier approach, which satisfies the simply supported boundary conditions, is utilized in the form of

w_{s} (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{smn} \sin (\frac{m π}{a} x) sin (\frac{n π}{b} y) e λ t, s = 1, 2, \dots, Λ

(17)

where m and n are half wave numbers, and complex eigenfrequency of the system is shown by λ. Also,

U_{smn}

displays an arbitrary parameter for sth orthotropic viscoelastic microplate. By applying the considered solution (17) into equation (14) and then implementing dimensionless phrases, a new dimensionless uniform algebraic equation for sth microplate can be obtained as

where

A = - K - K_{1} [(m π) 2 + R 2 (n π) 2] - C_{d} Γ B = {[1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (m π) 2 R 2 [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 4 R 4}{b 2}} \times (1 + η_{d} Γ) 2 K + 2 K_{1} [(m π) 2 + R 2 (n π) 2] + 2 C_{d} Γ + {1 + \frac{h 2}{12 a 2} [(m π) 2 + R 2 (n π) 2]} Γ 2

(19)

Here, dimensionless phrases employed in equation (19) are defined as follows

Z_{12} = \frac{D_{12}}{D_{11}}, Z_{66} = \frac{D_{66}}{D_{11}}, Z_{22} = \frac{D_{22}}{D_{11}}, R = \frac{a}{b}, K = \frac{Ka 4}{D_{11}}, K_{1} = \frac{k_{1} a 2}{D_{11}}, η_{d} = \frac{χ}{a 2} \sqrt{\frac{D_{11}}{ρ h}}, C_{d} = \frac{ca 2}{\sqrt{D_{11} ρ h}}, Γ = λ a 2 \sqrt{\frac{ρ h}{D_{11}}}

(20)

Clamped chain case

In this state, as shown in Figure 1(b), it is supposed that the first plate and the last plate in the system are linked on a fixed surface by visco-Pasternak foundation. Therefore, the following matrix shape of algebraic equation (18) is presented

[B A 0 \dots 000 \dots 000 A B A \dots 000 \dots 000 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 000 \dots B A 0 \dots 000000 \dots A B A \dots 000000 \dots 0 A B \dots 000 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 000 \dots 000 \dots A B A 000 \dots 000 \dots 0 A B] {U_{1 mn} U_{2 mn} U_{3 mn} \dots U_{s - 1 mn} U_{smn} U_{s + 1 mn} \dots U_{Λ - 2 mn} U_{Λ - 1 mn} U_{Λ mn}} = {000 \dots 0 00 \dots 0 00}

(21)

In addition, in order to acquire a clear closed-form statement to forecast the influence of small scale on the natural frequency of our system, we apply trigonometric method introduced and developed by Raskovic [52,53], Stojanović et al. [54], and Karličić et al. [55]. With respect to this method, we consider following solution of amplitude of sth algebraic equation for OMVMPS as

U_{smn} = Q cos (s θ_{CC}) + G sin (s θ_{CC})

(22)

where Q and G are nonzero constants, which should satisfy the boundary conditions and

θ_{CC}

shows an unknown parameter. By replacing the considered solution (22) into the sth microplate equation (18), we have

Q {A cos [(s - 1) θ_{CC}] + B cos (s θ_{CC}) + A cos [(s + 1) θ_{CC}]} = 0, s = 2, \dots, Λ - 1 G {A sin [(s - 1) θ_{CC}] + B sin (s θ_{CC}) + A sin [(s + 1) θ_{CC}]} = 0, s = 2, \dots, Λ - 1

(23)

Implementing a few trigonometric operations to equation (23) leads to following new forms

Q (B + 2 A cos θ_{CC}) cos (s θ_{CC}) = 0, s = 2, \dots, Λ - 1 G (B + 2 A cos θ_{CC}) sin (s θ_{CC}) = 0, s = 2, \dots, Λ - 1

(24)

Due to the fact that in equation (24), $Q, G, cos (s θ_{CC})$ and $sin (s θ_{CC})$ cannot be identical, we attain the following frequency equation

B + 2 A cos θ_{CC} = 0

(25)

However, to determinate unknown parameter $θ_{CC}$ , we use the first and the last equations of OMVMPS of equation (21) satisfied with assumed solutions (22). Thus, we have the following relations

Q (B cos θ_{CC} + A cos 2 θ_{CC}) + G (B sin θ_{CC} + A sin 2 θ_{CC}) = 0 Q {B cos (Λ θ_{CC}) + A cos [(Λ - 1) θ_{CC}]} + G {B sin (Λ θ_{CC}) + A sin [(Λ - 1) θ_{CC}]} = 0

(26)

On the basis of the nontrivial solutions obtained from parameters Q and G of equation (26) and after some trigonometric activities, the trigonometric equation is presented in the form

| 10 cos ((Λ + 1) θ_{CC}) sin ((Λ + 1) θ_{CC}) | = 0

(27)

Determinant of equation (27) gives us the unknown parameter

θ_{CC}

\sin [(Λ + 1) θ_{CC}] = 0 θ_{CC} = \frac{r π}{Λ + 1}, r = 1, 2, \dots, Λ

(28)

Inserting $θ_{CC}$ from relation (28) into relation (25), a dimensionless closed-form statement for complex frequency of multi-viscoelastic microplate system is driven as

Γ = - ω (ζ \pm i \sqrt{1 - ζ 2})

(29)

where the real component illustrates the damping effect and the imaginary component is the natural frequency of the system. In addition,

ω

and

ζ

demonstrate the dimensionless undamped frequency and damping ratio, respectively and can be written as

ω_{CC} = \sqrt{\frac{H 1_{CC}}{H_{2}}} H_{1 CC} = [1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (n π) 2 R 2 + [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 2 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(n π) 6 R 4}{b 2} + 2 K (1 - cos θ_{CC}) + 2 K_{1} [(m π) 2 + R 2 (n π) 2] (1 - cos θ_{CC}) H_{2} = 1 + \frac{h 2}{12 a 2} [(m π) 2 + R 2 (n π) 2], ζ_{CC} = \frac{H_{3 CC}}{H_{4 CC}} H_{3 CC} = η_{d} {[1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (n π) 2 R 2 + [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 2 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(n π) 6 R 4}{b 2}} + 2 C_{d} (1 - cos θ_{CC}) H_{4 CC} = 2 ω_{CC} H_{2}

(30)

Free chain case

In the present case, it is assumed that the first microplate and the last microplate are not coupled with the fixed surface. Matrix form of algebraic equation (18) that is adaptable with the “free chain” system can be expressed in the following form

[B + A A 0 \dots 000 \dots 000 A B A \dots 000 \dots 000 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 000 \dots B A 0 \dots 000000 \dots A B A \dots 000000 \dots 0 A B \dots 000 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots 000 \dots 000 \dots A B A 000 \dots 000 \dots 0 A B + A] \times {U_{1 mn} U_{2 mn} U_{3 mn} \dots U_{s - 1 mn} U_{smn} U_{s + 1 mn} \dots U_{Λ - 2 mn} U_{Λ - 1 mn} U_{Λ mn}} = {000 \dots 0 00 \dots 0 00}

(31)

Following the steps performed in the previous sections to obtain the closed-form expression for natural frequencies of the system, inserting the assumed solution (22) into sth equation of (18), we obtain

B + 2 A cos θ_{FC} = 0

(32)

where

θ_{FC}

represents the unknown parameter related to “free chain”, denoted from the first and the ultimate equation of system (31) as follows

\sin (Λ θ_{FC}) = 0 \Rightarrow θ_{Fc} = \frac{r π}{Λ}, r = 0, 1, 2, \dots, Λ - 1

(33)

Substituting equation (33) into equation (32), we achieve the closed-form statement for the natural frequencies of the system for “free chain” case as follows

ω_{FC} = \sqrt{\frac{H 1_{FC}}{H_{2}}} H_{1 FC} = [1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (n π) 2 R 2 + [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 2 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(n π) 6 R 4}{b 2} + 2 K (1 - cos θ_{FC}) + 2 K_{1} [(m π) 2 + R 2 (n π) 2] (1 - cos θ_{FC}) ζ_{FC} = \frac{H_{3 FC}}{H_{4 FC}} H_{3 FC} = η_{d} {[1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (n π) 2 R 2 + [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 2 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(n π) 6 R 4}{b 2}} + 2 C_{d} (1 - cos θ_{FC}) H_{4 FC} = 2 ω_{FC} H_{2}

(34)

Results and discussion

Comparison studies

To confirm the results obtained with the presented formulation, some comparative studies are done. Firstly, it is considered that in equations (30) and (34) that the number of orthotropic microplate $Λ$ tends to infinity and it is also supposed that the internal and external damping properties to be zero. Accordingly, we introduced the critical natural frequency of the system as follows

ω_{Λ \to \infty} = \sqrt{\frac{H}{L}} H = [1 + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (m π) 4 + [2 (Z_{12} + 2 Z_{66}) + \frac{G_{12} h}{D_{11}} (4 l_{0}^{2} + \frac{16}{15} l_{1}^{2} + 2 l_{2}^{2})] (m π) 2 (n π) 2 R 2 + [Z_{22} + \frac{G_{12} h}{D_{11}} (2 l_{0}^{2} + \frac{8}{15} l_{1}^{2} + l_{2}^{2})] (n π) 4 R 4 + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(m π) 6}{a 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 4 (n π) 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{2} + \frac{l_{1}^{2}}{5}) \frac{(m π) 2 (n π) 4 R 2}{b 2} + \frac{G_{12} h 3}{D_{11}} (\frac{l_{0}^{2}}{6} + \frac{l_{1}^{2}}{15}) \frac{(n π) 6 R 4}{b 2} L = 1 + \frac{h 2}{12 a 2} [(m π) 2 + R 2 (n π) 2]

(35)

It can be found from equation (35) that when the number of plate tends to infinity, the influence of the Pasternak-type foundation on the natural frequency eliminates. After introducing the critical natural frequency in the upper form, a comparison between the natural frequencies achieved for a single isotropic microplate based on MSGT against various nondimensional length scale parameter

(\frac{h}{l})

with those obtained by Akgoz and Civalek [17] is shown in Table 1. By briefly reviewing the data demonstrated in Table 1, the accuracy of our formulation will be proved.

Table 1.

Comparison of the critical natural frequencies of OMVMPS based on MSGT (m = 1, n = 1).

$\frac{h}{l}$	Present	Ref. [17]
1	6.3137	6.3137
3	0.8726	0.8726
5	0.4105	0.4105
7	0.2667	0.2667

Finally, the convergence of the numerical and analytical solutions based on Navier’s method is investigated in Table 2 by comparison of the eigenfrequencies obtained from two solutions. For this purpose, the first mode number

m = 1, n = 1

is considered and it is assumed that the multi-microplate system has the following material and geometrical properties [19,56,57]

E_{1} = 2.473 TPa, E_{2} = 2.433 TPa, ϑ_{12} = ϑ_{21} = 0.197, G_{12} = 1.033 TPa, ρ = 6316 \frac{kg}{m 3}, h = 17.6 μ m, a = b = 10 h, l_{0} = l_{1} = l_{2} = 0.1 h, K = 100, K_{1} = 20, R = 1, C_{d} = 15, η_{d} = 0.01, m = 1, n = 1

(36)

Table 2.

Comparison study of analytical and numerical solutions for eigenfrequencies of OMVMPS based on different chain systems.

No. of plates		“Clamped chain” system		“Free chain” system
No. of plates		N.S. of equation (21)	A.S. of equation (30)	N.S. of equation (31)	A.S. of equation (34)
$Λ$ = 2	1	−9.615284363 ± 29.01128925i	−9.615284362 ± 29.01128924i	−2.236657911 ± 21.03161771i	−2.236657910 ± 21.03161771i
$Λ$ = 2	2	−24.37253730 ± 36.24420026i	−24.37253728 ± 36.24420027i	−16.99391083 ± 33.64658388i	−16.99391082 ± 33.64658388i
$Λ$ = 3	1	−6.558957242 ± 26.25755198i	−6.558957221 ± 26.25755198i	−2.236657908 ± 21.03161772i	−2.236657910 ± 21.03161771i
	2	−16.99391071 ± 33.64658388i	−16.99391082 ± 33.64658388i	−9.615284368 ± 29.01128922i	−9.615284362 ± 29.01128924i
	3	−27.42886453 ± 36.83679769i	−27.42886444 ± 36.83679770i	−24.37253729 ± 36.24420027i	−24.37253728 ± 36.24420027i
$Λ$ = 5	1	−4.213754879 ± 23.66447464i	−4.213754913 ± 23.66447463i	−2.236657974 ± 21.03161771i	−2.236657910 ± 21.03161771i
	2	−9.615284352 ± 29.01128916i	−9.615284362 ± 29.01128924i	−5.055042177 ± 24.65178287i	−5.055042428 ± 24.65178296i
	3	−16.99391113 ± 33.64658432i	−16.99391082 ± 33.64658388i	−12.43366935 ± 31.07160719i	−12.43366889 ± 31.07160683i
	4	−24.37253690 ± 36.24419933i	−24.37253728 ± 36.24420027i	−21.55415240 ± 35.45629436i	−21.55415276 ± 35.45629494i
	5	−29.77406688 ± 37.11486398i	−29.77406674 ± 37.11486338i	−28.93277932 ± 37.03244084i	−28.93277922 ± 37.03244055i
$Λ$ = 10	1	−2.834429686 ± 21.87997731i	−2.834430344 ± 21.87997636i	−2.236640012 ± 21.03161697i	−2.236657910 ± 21.03161771i
	2	−4.579294006 ± 24.10204892i	−4.579319677 ± 24.10206788i	−2.958968603 ± 22.05053089i	−2.958929277 ± 22.05052030i
	3	−7.330217236 ± 27.01165962i	−7.329965350 ± 27.01138488i	−5.054988873 ± 24.65173586i	−5.055042428 ± 24.65178296i
	4	−10.86261323 ± 29.97198867i	−10.86352642 ± 29.97399618i	−8.319972527 ± 27.91837161i	−8.319815197 ± 27.91819453i
	5	−14.89479502 ± 32.57387942i	−14.89373477 ± 32.56548928i	−12.43297943 ± 31.07077361i	−12.43366889 ± 31.07160683i
	6	−19.09637330 ± 34.54413466i	−19.09408688 ± 34.56664060i	−16.99585548 ± 33.64992776i	−16.99391082 ± 33.64658388i
	7	−23.11275424 ± 35.96624031i	−23.12429524 ± 35.92466479i	−21.55123120 ± 35.44767339i	−21.55415276 ± 35.45629494i
	8	−26.65785631 ± 36.65833805i	−26.65785631 ± 36.71221346i	−25.66986553 ± 36.54129203i	−25.66800647 ± 36.52779432i
	9	−29.38894395 ± 37.12497738i	−29.40850199 ± 37.08141569i	−28.93290532 ± 37.02047259i	−28.93277922 ± 37.03244055i
	10	−31.16143531 ± 37.19307426i	−31.15339131 ± 37.20846777i	−31.02844840 ± 37.20656979i	−31.02889237 ± 37.20212836

N.S.: numerical solutions; A.S.: analytical solutions.

Also the number of plates is taken $Λ = 2, 3, 5, 10$ . From Table 2, it can be seen that in both chain boundary conditions, the results of numerical and exact solutions are in good agreement. These outstanding points can be concluded from this comparison: for different number of plates, as the number of plates increases, all eigenfrequencies decrease and the influence of visco-Pasternak medium on frequencies vanishes; the lowest frequency of multi-microplate system for “free chain” boundary condition is independent of the number of plates; the eigenfrequencies for “clamped chain” system are more than obtained for “free chain” system because stiffness causes higher frequency for “clamped chain” system.

Benchmark results

Afterward, to comprehend the behavior of multi-microplate system, some figurative results are given in the following. The values of parameters for all examples are taken as mentioned earlier. This procedure is carried out through different examples for two chain boundary conditions namely “clamped chain” and “free chain” conditions. Also, we consider $r = 1$ for all two chain systems.

Figure 2 shows the influence of the independent material length scale parameter on the variation of critical natural frequency (dimensional form of equation (35)) of OMVMPS for different values of microplate thickness. Here, we took $C_{d} = 0, η_{d} = 0$ . It can be seen from the figure that by increasing the microplate thickness, the critical natural frequency curves calculated based on modified strain gradient theory decrease towards the results of the classical theory (CT) $(l_{0} = l_{1} = l_{2} = 0)$ . Furthermore, by incrementing the independent material length scale parameter, the critical natural frequency increases.

Figure 2.

Variations of the critical natural frequencies of OMVMPS as a function of plate thickness for various values of the independent material length scale parameters ( $C_{d} = 0, η_{d} = 0$ ).

Figure 3(a) and (b) depicts the dimensionless natural frequencies of OMMPS forecasted by MSGT, MCST, and CT as a function of dimensionless length scale parameter. Here, we assumed $C_{d} = 0, η_{d} = 0$ for all chain systems. It is clear that an increase in the nondimensional length scale parameter $(\frac{h}{l})$ causes the natural frequency computed from MSGT and MCST to drop towards the results of the classical approach for the two chain systems. Furthermore, the natural frequency of OMVMPS reported by strain gradient theory is more than those predicted by the other two theories.

Figure 3.

Variations of the dimensionless frequencies of OMVMPS predicted by MSGT, MCST, and CT corresponding to various values of the nondimensional length scale parameter and different number of microplates ( $C_{d} = 0, η_{d} = 0$ ): (a) “clamped chain”; (b) “free chain”.

The variation of damping ratio versus damping coefficient of visco-Pasternak medium C_d for “clamped chain” and “free chain” conditions with different number of plates $Λ = 2, 5, 10, 50$ illustrated in Figure 4. This figure reveals that increase of damping coefficient of viscoelastic Pasternak medium results in increasing of damping ratio. Underdamped, critically damped, and overdamped vibration regions can be seen in the figure.

Figure 4.

Variation of damping ratio with damping coefficient of visco-Pasternak medium corresponding to different values of number of microplates: (a) “clamped chain”; (b) “free chain”.

Plotted in Figure 5 are the imaginary part of eigenfrequency against damping coefficient of visco-Pasternak medium C_d for the two different chain systems and different number of plates $Λ = 2, 5, 10, 50$ . It is apparent from this figure that the imaginary part of eigenfrequency reduces as damping coefficient increases. This reduction is considerable for the case of the two microplates, whereas for higher number of plates damping coefficient of visco-Pasternak medium causes a small effect on frequencies.

Figure 5.

Variation of imaginary part of eigenfrequency with damping coefficient of visco-Pasternak medium corresponding to different values of number of microplates: (a) “clamped chain”; (b) “free chain”.

Figure 6 presents the influence of damping coefficient of visco-Pasternak C_d on the real part of eigenfrequency for $Λ = 2, 5, 10, 50$ . The reduction of the real part of eigenfrequency by increasing damping coefficient is obviously shown for the two chain systems. The figure demonstrates that the damping coefficient has a major role on the real part of eigenfrequency for case of $Λ = 2$ . The intersection that is demonstrated in the figure shows the boundary between underdamped and overdamped regions.

Figure 6.

Variation of real part of eigenfrequency with damping coefficient of visco-Pasternak medium corresponding to different values of number of microplates: (a) “clamped chain”; (b) “free chain”.

In order to show how material length scale parameters influence on the behavior of multi-microplate system, the imaginary part of eigenfrequency is plotted versus $l_{0} = l_{1} = l_{2} = l$ in Figure 7. The results obtained for “clamped chain” and “free chain” conditions and for $Λ = 2, 5, 10, 50$ . It can be seen that an increase in the value of the independent material length scale parameters leads to an increase in frequency due to the increment of the stiffness of microplates. Furthermore, as the number of plates increases the value of frequency decreases.

Figure 7.

The effect of the independent internal length scale parameter on the damped frequencies vs different numbers of microplate: (a) “clamped chain”; (b) “free chain”.

To assess the role of isotropic and orthotropic properties on frequencies of OMVMS, the imaginary part of eigenfrequency versus the nondimensional length scale parameter $(\frac{h}{l})$ for both isotropic and orthotropic materials was plotted in Figure 8. As it is expected, the frequencies for isotropic material are more than the orthotropic one. This arises from this reality that isotropic characteristics gives more rigidity to microplate than orthotropic ones. Besides, the figure demonstrates that increasing the nondimensional length scale parameter makes the microplate less stiff, thus it causes high decrease in frequency.

Figure 8.

The separate influence of the orthotropic and isotropic materials on damped frequencies of OMVMPS vs nondimensional length scale parameter for different number of microplates: (a) “clamped chain”; (b) “free chain”.

Depicted in Figure 9 are the imaginary part of eigenfrequency as a function of dimensionless length scale parameter $\frac{h}{l}$ with different values of shear and transverse stiffness coefficients of the visco-Pasternak medium $K, K_{1}$ and with different values of plate dimensions $a, b$ for $Λ = 2$ . It can be found that by increasing $\frac{h}{l}$ the frequency significantly reduces. One can see from this plot that by increasing the length-to-thickness ratio, the frequency slightly increases also decreasing values of shear and transverse stiffness coefficients leads to a reduction in frequency.

Figure 9.

Variations of the damped frequencies of OMVMPS against the nondimensional length scale parameter for different values of shear and transverse stiffness coefficients K₁ and $K$ and various values of length-to-thickness ratio ( $Λ = 2$ ): (a) “clamped chain”; (b) “free chain”.

Figure 10 shows the imaginary part of eigenfrequency against aspect ratio with different values of nondimensional length scale parameter $(\frac{h}{l})$ for $Λ = 2, 5, 50$ . It is evident from the figure that increasing of aspect ratio leads to increase of frequency until it reaches at a maximum value and then frequency decreases. Also, it is found that this peak occurs in higher aspect ratio by increasing the ratio $(\frac{h}{l})$ .

Figure 10.

Variations of the damped frequency against the aspect ratio, nondimensional length scale parameter, and number of microplates: (a) “clamped chain”; (b) “free chain”.

Figure 11 displays how frequency varies by changing aspect ratio when damping coefficient of visco-Pasternak medium and viscoelastic structural property are zero. Influences of number of microplates and nondimensional length scale parameter $(\frac{h}{l})$ are also considered. From this figure, it can be found that when the influence of damping properties is supposed to be zero, the frequency monotonically increases as the aspect ratio increases.

Figure 11.

Variations of the undamped frequency against the aspect ratio, nondimensional length scale parameter, and number of microplates $(C_{d} = 0, η_{d} = 0$ ): (a) “clamped chain”; (b) “free chain”.

Figure 12 examines the vibration behavior of the system with respect to increase in the number of plates at higher modes as well as under various values of viscoelastic structural damping $η_{d}$ for two chain boundary conditions. The decreasing effect of viscoelastic structural damping on the frequency is observed from the figure for the two chain systems. It can be seen that at lower modes, the frequencies extracted for different values of $η_{d}$ are nearly coincident. Also for a definite number of plates, frequencies are independent of the number of plates. In the other words, by increasing the half wave numbers, the effect of $η_{d}$ on the variation of frequencies increases. Also, to better highlight the discrepancy between two cases of wave number for ( $m = 1, n = 2$ ) and ( $m = 2, n = 1$ ), Figure 13 is presented in the enlarged view of Figure 12. Due to the consideration of the orthotropic property, it can be concluded that the natural frequency for $m = 1, n = 2$ is not equivalent with the calculated frequency for $m = 2, n = 1$ .

Figure 12.

Coupled effect of the viscoelastic damping structural and higher modes on the damped frequency vs different number of microplates: (a) “clamped chain”; (b) “free chain”.

Figure 13.

An enlargement of Figure 12 for two cases of wave number ( $m = 1, n = 2; m = 2, n = 1)$ : (a) “clamped chain”; (b) “free chain”.

Conclusion

In this paper, vibration characteristics of an OMVMPS embedded in visco-Pasternak medium was studied based on the MSGT and Kirchhoff plate approach. Navier’s method and trigonometric method were used to determine an explicit closed-form expression for eigenfrequency of OMVMPS for the two chain conditions, namely “clamped chain” and “free chain” systems. Comparing the analytical and numerical solutions, the satisfactory agreement between them was observed. Some important conclusions can be drawn from the results: the number of microplates, the dimensionless length scale parameter $(\frac{h}{l})$ , aspect ratio, and higher modes significantly affect the vibration behavior of OMVMPS. Also, viscoelastic properties are important to analyze and control the vibrations. It was shown that the influence of visco-Pasternak medium on the natural frequencies vanishes as the number of microplates in OMVMPS increases. Moreover, it was seen that the frequencies for isotropic material is more than that of the orthotropic one because of more rigidity in isotropic case.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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