Accurate damping analysis of viscoelastic composite beams and plates on suppressive foundation

Abstract

An accurate study for dynamic behavior of viscoelastic thin laminated composite beams and plates resting on viscoelastic foundations is presented. Boltzmann superposition integral based on the dynamic mechanical analysis results is adapted to accurately predict viscoelastic behavior of polymeric fiber-reinforced composite structures. Also, foundation viscoelasticity is described based on the Kelvin–Voigt model. Integro-differential governing equations of motion are derived based on classical lamination theory via Hamilton principle. Galerkin weighted residual method, iterative QZ algorithm, and Fourier transform are applied to obtain natural frequency, loss factor, and transient response of structures resting on suppressive foundations. Influence of foundation and geometrical parameters and also boundary conditions on the dynamic behavior is put into evidence via a parametric study, and pertinent conclusions are outlined. Due to the absence of similar results in the literature, this paper is likely to fill a gap in the state of the art of this problem.

Keywords

Transient vibration dynamic mechanical analysis results Boltzmann superposition integral polymeric matrix viscoelastic composite structures Kelvin–Voigt-type foundation

Introduction

Viscoelastic materials, such as polymers, exhibit both viscous and elastic behavior simultaneously; it means that energy dissipation occurs during deformation process. Vibrations are undesirable for structures, due to the need of structural stability, durability, reliability, performance in harsh environments, and noise reduction. The viscoelastic materials with damping property allow undesirable mechanical vibrations and wave propagation to be passively suppressed. These types of materials are extensively used in many modern engineering applications like aircrafts, ships, buildings, and space structures since they may eliminate the need for special energy absorbers and dampers to attenuate undesirable noise and mechanical vibration.

Viscoelasticity is an inherent property of the material, which normally appears as time dependency of material properties.^1–3 One of the most accurate models for linear viscoelasticity is the Boltzmann superposition theory in which time variation of all material properties is implemented in the constitutive relations.^4–6 However, many researchers assumed time-independent constants or simple complex viscoelastic properties.^7–10 The time dependency of material properties usually lead to complexity of the solution procedure so that some researchers challenged the problem by presenting various solution approaches.^11–17 Reddy¹¹ applied variational principles for integro-differential equations which were arisen from the time-dependent viscoelastic constitutive equations. Kaminskii and Selivanov¹² used the branching operator continued fraction approximation for the deformation analysis of cracked composite plates. Sladek et al.¹³ applied a meshless local Petrov–Galerkin method to viscoelastic Reissner–Mindlin composite laminated plates with constant bulk modulus under static and impact loads. The governing equations were transformed into local integral equation and a numeric example was presented by using moving least squares. Li et al.¹⁴ analyzed the free vibration behavior of isotropic viscoelastic Timoshenko beams using the Galerkin approach in conjunction with fractional derivative for three-dimensional constitutive relations. Finite element, finite difference, and differential quadrature methods were other prominent numerical techniques used by various researchers for viscoelastic problems.^15–18

Although viscoelasticity is internal material characteristic such as elasticity, plasticity, pseudo elasticity, etc. external damping can be also modeled by viscoelastic medium or foundation.^19–22 The viscoelastic foundation is composed of elastic and viscous parts. The elastic part can be modeled by linear spring or nonlinear spring known as hardening/softening foundations. Viscous part is also considered by dashpot as a dissipating factor in the foundation and may lead to suppress structural vibration. Babadzhanova et al.¹⁹ studied vibration of orthotropic viscoelastic laminated composite plates resting on simple complex viscoelastic foundation with simply supported boundary conditions (BCs). They used approximation technique of straight lines in conjunction with the averaging method to solve the problem. Some researchers investigated vibration suppression of isotropic homogenous elastic beams resting on viscoelastic foundations.^20–22

The literature survey reveals the fact that comprehensive studies were dedicated to investigate vibration damping of mechanical structures with inherent viscoelastic behavior, whereas few research works focused on the transient vibration of viscoelastic structures resting on viscoelastic foundations.¹⁹ Also, to prevent facing with some mathematical complexity, viscoelastic material behavior was usually simulated using simple constants or models.^{3,4,7–10,12,13} To the best of authors’ knowledge, there is no accurate vibration damping analysis of viscoelastic composite structures on viscoelastic foundations assuming time dependency for all material properties, in the open literature. In the present work, the main idea is arisen from a fact that the authors’ interest is to distinguish share of each viscosity on damping behavior of viscoelastic composite structures resting on viscoelastic foundation. In the other words, it is about comparison between internal damping effect due to accurate viscoelasticity and external damping effect due to foundation viscosity on the dynamic treatment of structures.

In the present work, an accurate dynamic analysis is developed to analyze free vibration and damping of fiber-reinforced composite beams and plates with polymer matrix resting on viscoelastic foundations as practical structural elements in dynamical environments. These simulations may be applied for railroad vibration with damping layer on the lateral surface of the rail for the beam and the polymeric composite flooring on the floor as a viscoelastic foundation for the plate.²³ The Boltzmann superposition integral theory based on the dynamic mechanical analysis (DMA) results and Kelvin–Voigt model are adapted to describe viscoelastic behavior of laminated composite structures and foundations, respectively. The displacement field of the viscoelastic laminated composite structures is assumed based on the classical lamination theory. Toward obtaining the governing equations of the vibrational response, the Hamilton principle is employed. Galerkin weighted residual method, Fourier transform, and iterative QZ algorithm are applied to extract natural frequency, modal loss factor, and transient response of freely vibrating viscoelastic laminated composite beams and plates resting on viscoelastic foundations. The solution procedure flowchart is depicted in Figure 1 for aim of illustration. Due to lack of similar results in the open literature, accuracy of the present solution procedure is verified by comparing the numerical results with existing results in the limited cases of viscoelastic plates and elastic plates resting on viscoelastic foundations. After validation study, a set of parametric study is performed to provide an insight into the influence of stiffness and viscosity of foundation, geometrical parameters, and BCs on the vibration characteristics of viscoelastic laminated composite structures resting on suppressive foundations.

Material and method

Constitutive equations

Consider a rectangular fiber-reinforced laminated composite plate with polymer matrix resting on a viscoelastic foundation as displayed in Figure 2(a). The length of plate is a and width and total thickness of the plate are denoted by b and h, respectively. The Cartesian coordinate system (x, y, z) is located on the middle surface of the composite plate. In addition, laminated composite beams with rectangular cross section on viscoelastic foundation can be considered with dimensions H and L in the x and z directions, respectively.

The infinitesimal deformation of thin monoclinic laminates is analyzed using the classical lamination theory. The linear viscoelastic constitutive equations for kth lamina in a plane stress state can be expressed based on the Boltzmann superposition theory as²

{σ x (t) σ y (t) σ xy (t)} (k) = \int_{- \infty}^{t} [\bar{Q} 11 (t - ξ) \bar{Q} 12 (t - ξ) \bar{Q} 16 (t - ξ) \bar{Q} 12 (t - ξ) \bar{Q} 22 (t - ξ) \bar{Q} 26 (t - ξ) \bar{Q} 16 (t - ξ) \bar{Q} 26 (t - ξ) \bar{Q} 66 (t - ξ)] (k) \times \frac{d}{d ξ} {ɛ x (ξ) ɛ y (ξ) ɛ xy (ξ)} (k) d ξ

(1)

where

σ i

and

ɛ i (i = x, y, xy)

are the components of stress and strain, and

\bar{Q} ij (i, j = 1, 2, 6)

denote time-dependent components of off-axis reduced relaxation matrix. Also, ξ represents the Boltzmann integral variable. The off-axis relaxation matrix can be expressed in terms of the orientation angle and on-axis relaxation matrix as

[\bar{Q}] (k) = [p 2 q 2 2 pq q 2 p 2 - 2 pq - pq pq p 2 - q 2] (k) - 1 [Q 11 Q 12 0 Q 12 Q 22 000 Q 66] (k) \times [p 2 q 2 2 pq q 2 p 2 - 2 pq - pq pq p 2 - q 2] (k)

(2)

in which $p = \cos θ$ , $q = \sin θ$ , and $+ θ$ is counterclockwise form the x axis. The plane stress-reduced matrix $[Q]$ for elastic material can be represented in terms of the engineering constants of the material as

[Q] (k) = [\frac{E L}{1 - υ LT υ TL} \frac{υ LT E T}{1 - υ LT υ TL} 0 \frac{υ LT E T}{1 - υ LT υ TL} \frac{E T}{1 - υ LT υ TL} 000 G LT] (k)

(3)

where E_L, E_T,

υ LT

, and

υ TL

are longitudinal Young modulus, transverse Young modulus, major and minor Poisson ratio, respectively, which are all defined as

E L = V m E m + V f E f

(4a)

E T = E m E f / (V m E f + V f E m)

(4b)

G LT = G m G f / (V m G f + V f G m)

(4c)

υ LT = V m υ m + V f υ f

(4d)

in which V_m, V_f, E_m, E_f, G_m, G_f, $υ m$ , and $υ f$ are matrix volume fraction, fiber volume fraction, matrix modulus, fiber modulus, matrix shear modulus, fiber shear modulus, matrix Poisson ratio, and fiber Poisson ratio, respectively. The minor Poisson ratio can be derived from reciprocation relation as

υ TL = \frac{E T}{E L} υ LT

(5)

Among the mentioned material properties, matrix and fiber volume fraction, Young and shear moduli also Poisson ratio of fiber are assumed to be constant or time independent, whereas shear and Young moduli, and Poisson ratio of matrix are considered to be time dependent.

By using Alfrey’s corresponding principle together with Laplace transform,² the time-dependent arrays of stiffness matrix can be expressed as

Q 11 (t) = L - 1 {\frac{\bar{E} L (s)}{1 - s 2 \bar{υ} LT (s) \bar{υ} TL (s)}}

(6a)

Q 12 (t) = L - 1 {\frac{s \bar{υ} LT (s) \bar{E} T (s)}{1 - s 2 \bar{υ} LT (s) \bar{υ} TL (s)}}

(6b)

Q 22 (t) = L - 1 {\frac{\bar{E} T (s)}{1 - s 2 \bar{υ} LT (s) \bar{υ} TL (s)}}

(6c)

Q 66 (t) = L - 1 {G LT (s)}

(6d)

where s is the Laplace variable and

L - 1

denotes the Laplace inverse operator. Superior bar presents average value of the quantity in the Laplace domain. It should be mentioned that Young modulus and Poisson ratio could not be derived directly from DMA results. Due to this fact, Young modulus and Poisson ratio of polymeric composite can be obtained from relations between elastic properties as

\bar{E} m (s) = \frac{9 \bar{K} m (s) \bar{G} m (s)}{3 \bar{K} m (s) + \bar{G} m (s)}

(7a)

\bar{υ} m (s) = \frac{3 \bar{K} m (s) - 2 \bar{G} m (s)}{6 s \bar{K} m (s) + 2 s \bar{G} m (s)}

(7b)

where

\bar{K} m

and

\bar{G} m

are the time-dependent relaxation bulk and shear moduli, respectively. These relaxation moduli can be expanded as the Prony–Dirichlet series.

Kinematics equations

Based on the classical laminated plate theory (CLPT), the displacement components of a material point within the laminate domain in Cartesian coordinates system can be written as

u (x, y, z, t) = u 0 (x, y, t) - z \frac{\partial w 0 (x, y, t)}{\partial x} v (x, y, z, t) = v 0 (x, y, t) - z \frac{\partial w 0 (x, y, t)}{\partial y} w (x, y, z, t) = w 0 (x, y, t)

(8)

where u₀, v₀, and w₀ denote the displacements of a point on the middle plane of the plate along x, y, and z coordinates, respectively.

The strain–displacement kinematic relations associated with the displacement field (8) can be expressed as

ɛ x = ɛ_{x}^{0} + z κ_{x}^{0} ɛ y = ɛ_{y}^{0} + z κ_{y}^{0} ɛ xy = ɛ_{xy}^{0} + z κ_{xy}^{0}

(9)

where

ɛ_{i}^{0} (i = x, y, xy)

denotes the membrane strains and

κ_{i}^{0}

is the flexural strains known as the curvatures

ɛ_{x}^{0} = \frac{\partial u 0}{\partial x}, ɛ_{y}^{0} = \frac{\partial v 0}{\partial y}, ɛ_{xy}^{0} = \frac{\partial u 0}{\partial y} + \frac{\partial v 0}{\partial x} κ_{x}^{0} = - \frac{\partial 2 w 0}{\partial x 2}, κ_{y}^{0} = - \frac{\partial 2 w 0}{\partial y 2}, κ_{xy}^{0} = - 2 \frac{\partial 2 w 0}{\partial x \partial y}

(10)

Governing equations of motion

According to the Hamilton principle, the motion equations of thin plate-like structures resting on viscoelastic foundations can be derived based on the displacement field (8) as²⁴

\frac{\partial N x}{\partial x} + \frac{\partial N xy}{\partial y} = I 0 \frac{\partial 2 u 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial x \partial t 2}

(11a)

\frac{\partial N xy}{\partial x} + \frac{\partial N y}{\partial y} = I 0 \frac{\partial 2 v 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial y \partial t 2}

(11b)

\frac{\partial 2 M x}{\partial x 2} + 2 \frac{\partial 2 M xy}{\partial x \partial y} + \frac{\partial 2 M y}{\partial y 2} = kw 0 + c 0 \frac{\partial w 0}{\partial t} + I 0 \frac{\partial 2 w 0}{\partial t 2} + I 1 (\frac{\partial 3 u 0}{\partial x \partial t 2} + \frac{\partial 3 v 0}{\partial y \partial t 2}) - I 2 (\frac{\partial 4 w 0}{\partial x 2 \partial t 2} + \frac{\partial 4 w 0}{\partial y 2 \partial t 2})

(11c)

where k and c₀ denote stiffness and damping coefficients of viscoelastic Kelvin-Voigt-type foundation. Also, the quantities N_i,

M i (i = x, y, xy)

, and

I i (i = 0, 1, 2)

are, respectively, in-plane force resultants and moment resultants, and inertia terms which are all defined as

(N i, M i) = \int_{- h / 2}^{+ h / 2} σ i (1, z) dz, (i = x, y, xy)

(12a)

(I 0, I 1, I 2) = \int_{- h / 2}^{+ h / 2} ρ (1, z, z 2) d z

(12b)

where ρ is mass density. Upon substitution of equation (9) into equation (1) and the resulted relation to equation (12a), and the subsequent results into equation (11), integro-differential equations of motion for viscoelastic laminated composite plates resting on suppressive foundations with similar formulation in elastic domain for linear case²⁴ are presented as

- \int_{- \infty}^{t} {A 11 (t - ξ) \frac{\partial 3 u 0}{\partial ξ \partial x 2} + A 12 (t - ξ) \frac{\partial 3 v 0}{\partial ξ \partial x \partial y} - B 11 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x 3} - B 12 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x \partial y 2} + A 26 (t - ξ) \frac{\partial 3 v 0}{\partial ξ \partial y 2} + A 66 (t - ξ) (\frac{\partial 3 u 0}{\partial ξ \partial y 2} + \frac{\partial 3 v 0}{\partial ξ \partial x \partial y}) + A 16 (t - ξ) (2 \frac{\partial 3 u 0}{\partial ξ \partial x \partial y} + \frac{\partial 3 v 0}{\partial ξ \partial x 2}) - 3 B 16 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x 2 \partial y} - B 26 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial y 3} - 2 B 66 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x \partial y 2}} d ξ = I 0 \frac{\partial 2 u 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial x \partial t 2}

(13a)

- \int_{- \infty}^{t} {A 12 (t - ξ) \frac{\partial 3 u 0}{\partial ξ \partial x \partial y} + A 16 (t - ξ) \frac{\partial 3 u 0}{\partial ξ \partial x 2} + A 22 (t - ξ) \frac{\partial 3 v 0}{\partial ξ \partial y 2} - B 16 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x 3} + A 26 (t - ξ) (\frac{\partial 3 u 0}{\partial ξ \partial y 2} + 2 \frac{\partial 3 v 0}{\partial ξ \partial x \partial y}) + A 66 (t - ξ) (\frac{\partial 3 u 0}{\partial ξ \partial x \partial y} + \frac{\partial 3 v 0}{\partial ξ \partial x 2}) - 3 B 26 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x \partial y 2} - (B 12 (t - ξ) + 2 B 66 (t - ξ)) \frac{\partial 4 w 0}{\partial ξ \partial x 2 \partial y} - B 22 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial y 3}} d ξ = I 0 \frac{\partial 2 v 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial y \partial t 2}

(13b)

- \int_{- \infty}^{t} {- B 11 (t - ξ) \frac{\partial 4 u 0}{\partial ξ \partial x 3} - B 12 (t - ξ) (\frac{\partial 4 u 0}{\partial ξ \partial x \partial y 2} + \frac{\partial 4 v 0}{\partial ξ \partial x 2 \partial y}) - B 22 (t - ξ) (\frac{\partial 4 v 0}{\partial ξ \partial y 3}) + D 11 (t - ξ) \frac{\partial 5 w 0}{\partial ξ \partial x 4} - B 16 (t - ξ) (3 \frac{\partial 4 u 0}{\partial ξ \partial x 2 \partial y} + \frac{\partial 4 v 0}{\partial ξ \partial x 3}) - B 26 (t - ξ) (\frac{\partial 4 u 0}{\partial ξ \partial y 3} + 3 \frac{\partial 4 v 0}{\partial ξ \partial x \partial y 2}) + 4 D 16 (t - ξ) \frac{\partial 5 w 0}{\partial ξ \partial x 3 \partial y} - 2 B 66 (t - ξ) (\frac{\partial 4 u 0}{\partial ξ \partial x \partial y 2} + \frac{\partial 4 v 0}{\partial ξ \partial x 2 \partial y}) + 2 (D 12 (t - ξ) + 2 D 66 (t - ξ)) \frac{\partial 5 w 0}{\partial ξ \partial x 2 \partial y 2} + 4 D 26 (t - ζ) \frac{\partial 5 w 0}{\partial ξ \partial x \partial y 3} + D 22 (t - ξ) \frac{\partial 5 w 0}{\partial ξ \partial y 4}} \times d ξ = kw 0 + c 0 \frac{\partial w 0}{\partial t} + I 0 \frac{\partial 2 w 0}{\partial t 2} + I 1 (\frac{\partial 3 u 0}{\partial x \partial t 2} + \frac{\partial 3 v 0}{\partial y \partial t 2}) - I 2 (\frac{\partial 4 w 0}{\partial x 2 \partial t 2} + \frac{\partial 4 w 0}{\partial y 2 \partial t 2})

(13c)

where the time-dependent material parameters

A ij, B ij

, and

D ij (i, j = 1, 2, 6)

are called extensional, bending-extensional coupling, and bending stiffnesses, respectively, which are defined in terms of the lamina stiffnesses

{\bar{Q}}_{ij}^{(k)}

(A ij, B ij, D ij) = \int_{- h / 2}^{h / 2} \bar{Q} ij (t - ξ) (1, z, z 2) d z = \sum_{k = 1}^{N} \int_{z k}^{z k + 1} {\bar{Q}}_{ij}^{(k)} (t - ξ) (1, z, z 2) d z, (i, j = 1, 2, 6)

(14)

Constitutive and kinematic relations of viscoelastic beams are similar to those for the plates but all terms involving displacement component in the y direction and differentiation with respect to the y variable are neglected. The equations of motion for thin beams resting on viscoelastic foundations can be derived as

\frac{\partial N x}{\partial x} = I 0 \frac{\partial 2 u 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial x \partial t 2}

(15a)

\frac{\partial 2 M x}{\partial x 2} - kw 0 - c 0 \frac{\partial w 0}{\partial t} = I 0 \frac{\partial 2 w 0}{\partial t 2} + I 1 \frac{\partial 3 u 0}{\partial x \partial t 2} - I 2 \frac{\partial 4 w 0}{\partial x 2 \partial t 2}

(15b)

By substituting the kinematic and constitutive equations into the motion equations (15), the integro-differential equations of motion for viscoelastic laminated composite beams resting on viscoelastic foundations are obtained as

- \int_{- \infty}^{t} {A 11 (t - ξ) \frac{\partial 3 u 0}{\partial ξ \partial x 2} - B 11 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x 3}} d ξ = I 0 \frac{\partial 2 u 0}{\partial t 2} - I 1 \frac{\partial 3 w 0}{\partial x \partial t 2}

(16a)

- \int_{- \infty}^{t} (D 11 (t - ξ) \frac{\partial 5 w 0}{\partial ξ \partial x 4} - B 11 (t - ξ) \frac{\partial 4 w 0}{\partial ξ \partial x 3}) d ξ = kw 0 + c 0 \frac{\partial w 0}{\partial t} + I 0 \frac{\partial 2 w 0}{\partial t 2} + I 1 \frac{\partial 3 u 0}{\partial x \partial t 2} - I 2 \frac{\partial 4 w 0}{\partial x 2 \partial t 2}

(16b)

where definition of the material parameters introduced in equation (16) for beams is similar to counterpart for plates defined in equation (14).

Equations (13) and (16) are, respectively, three and two highly coupled integro-partial differential equations of motion in terms of displacement components in the space and time domain. These equations will be solved to investigate transient response of vibrating viscoelastic laminated composite beams and plates resting on viscoelastic foundations in the next section.

Solution methodology

In this section, integro-differential equations (13) and (16) are solved based on the Galerkin weighted residual method using appropriate shape functions. The same procedure is considered for solving integro-differential equations in both cases of the viscoelastic beams and plates resting on viscoelastic foundations. In order to study free vibration behavior of the viscoelastic composite structures on viscoelastic foundations, a separable solution of the shape functions describing the modes of the vibration and a harmonic function of time can be adapted as follows

u 0 (x, y, t) = U (x, y) e i ω t

(17a)

v 0 (x, y, t) = V (x, y) e i ω t

(17b)

w 0 (x, y, t) = W (x, y) e i ω t

(17c)

where U, V, and W denote the spatial shape functions while ω is natural frequency of the viscoelastic structures on viscoelastic foundations. By substituting equation (17) in the governing equations of motion and applying some mathematic manipulations, the governing equations of motion of plates (13) and beams (16) can be obtained in the space and frequency domain.

For plates

i {\overset{\land}{A} 11 (ω) \frac{\partial 2 U}{\partial x 2} + \overset{\land}{A} 12 (ω) \frac{\partial 2 V}{\partial x \partial y} + \overset{\land}{A} 16 (ω) (2 \frac{\partial 2 U}{\partial x \partial y} + \frac{\partial 2 V}{\partial x 2}) + \overset{\land}{A} 26 (ω) \frac{\partial 2 V}{\partial y 2} + \overset{\land}{A} 66 (ω) (\frac{\partial 2 U}{\partial y 2} + \frac{\partial 2 V}{\partial x \partial y}) - \overset{\land}{B} 11 (ω) \frac{\partial 3 W}{\partial x 3} - (\overset{\land}{B} 12 (ω) + 2 \overset{\land}{B} 66 (ω)) \frac{\partial 3 W}{\partial x \partial y 2} - 3 \overset{\land}{B} 16 (ω) \frac{\partial 3 W}{\partial x 2 \partial y} - \overset{\land}{B} 26 (ω) \frac{\partial 3 W}{\partial y 3}} - I 0 ω 2 U + I 1 ω 2 \frac{\partial W}{\partial x} = 0

(18a)

i {\overset{\land}{A} 16 (ω) \frac{\partial 2 U}{\partial x 2} + \overset{\land}{A} 12 (ω) \frac{\partial 2 U}{\partial x \partial y} + \overset{\land}{A} 26 (ω) (\frac{\partial 2 U}{\partial y 2} + 2 \frac{\partial 2 V}{\partial x \partial y}) + \overset{\land}{A} 22 (ω) \frac{\partial 2 V}{\partial y 2} + \overset{\land}{A} 66 (ω) (\frac{\partial 2 U}{\partial x \partial y} + \frac{\partial 2 V}{\partial x 2}) - \overset{\land}{B} 16 (ω) \frac{\partial 3 W}{\partial x 3} - (\overset{\land}{B} 12 (ω) + 2 \overset{\land}{B} 66 (ω)) \frac{\partial 3 W}{\partial x 2 \partial y} - 3 \overset{\land}{B} 26 (ω) \frac{\partial 3 W}{\partial x \partial y 2} - \overset{\land}{B} 22 (ω) \frac{\partial 3 W}{\partial y 3}} - I 0 ω 2 V + I 1 ω 2 \frac{\partial W}{\partial y} = 0

(18b)

i {- \overset{\land}{B} 11 (ω) \frac{\partial 3 U}{\partial x 3} - \overset{\land}{B} 12 (ω) (\frac{\partial 3 U}{\partial x \partial y 2} + \frac{\partial 3 V}{\partial x 2 \partial y}) - \overset{\land}{B} 16 (ω) (3 \frac{\partial 3 U}{\partial x 2 \partial y} + \frac{\partial 3 V}{\partial x 3}) - \overset{\land}{B} 22 (ω) \frac{\partial 3 V}{\partial y 3} - \overset{\land}{B} 26 (ω) (\frac{\partial 3 U}{\partial y 3} + 3 \frac{\partial 3 V}{\partial x \partial y 2}) - 2 \overset{\land}{B} 66 (ω) (\frac{\partial 3 U}{\partial x \partial y 2} + \frac{\partial 3 V}{\partial x 2 \partial y}) + \overset{\land}{D} 11 (ω) \frac{\partial 4 W}{\partial x 4} + 4 \overset{\land}{D} 16 (ω) \frac{\partial 4 W}{\partial x 3 \partial y} + 4 \overset{\land}{D} 26 (ω) \frac{\partial 4 W}{\partial x \partial y 3} + \overset{\land}{D} 22 (ω) \frac{\partial 4 W}{\partial y 4} + 2 (\overset{\land}{D} 12 (ω) + 2 \overset{\land}{D} 66 (ω)) \frac{\partial 4 W}{\partial x 2 \partial y 2}} + (k + ic 0 ω - I 0 ω 2) W - I 1 ω 2 \frac{\partial U}{\partial x} - I 1 ω 2 \frac{\partial V}{\partial y} + I 2 ω 2 \frac{\partial 2 W}{\partial x 2} + I 2 ω 2 \frac{\partial 2 W}{\partial y 2} = 0

(18c)

For beams

i (\overset{\land}{A} 11 (ω) \frac{\partial 2 U}{\partial x 2} - \overset{\land}{B} 11 \frac{\partial 3 W}{\partial x 3}) - I 0 ω 2 U + I 1 ω 2 \frac{\partial W}{\partial x} = 0

(19a)

i (\overset{\land}{D} 11 (ω) \frac{\partial 4 W}{\partial x 4} - B 11 \frac{\partial 3 W}{\partial x 3}) + (k + ic 0 ω - I 0 ω 2) W - I 1 ω 2 \frac{\partial U}{\partial x} + I 2 ω 2 \frac{\partial 2 W}{\partial x 2} = 0

(19b)

where the frequency-dependent material parameters

\overset{\land}{A} ij (ω), \overset{\land}{B} ij (ω)

and

\overset{\land}{D} ij (ω) (i, j = 1, 2, 6)

are defined as

(\overset{\land}{A} ij (ω), \overset{\land}{B} ij (ω), \overset{\land}{D} ij (ω)) = ω \int_{0}^{\infty} (A ij (ϕ), B ij (ϕ), D ij (ϕ)) e - ω ϕ d ϕ, (i, j = 1, 2, 6)

(20)

The Galerkin weighted residual method together with appropriate shape functions is applied to treat the spatial dependency of the governing equations of motion of the viscoelastic composite structures resting on viscoelastic foundations. The resulted nonlinear algebraic equations of motion in the frequency domain is then solved using iterative QZ algorithm, which yields complex natural frequency in the form of $ω = ω Re + i ω Im$ . The natural frequency $ω n$ and the modal loss factors η are calculated as

ω n = ω Re η = \frac{2 ω Re ω Im}{(ω Re) 2 - (ω Im) 2}

(21)

The time histories of displacement components can be obtained by substitution of the complex natural frequencies into equation (17) for a multiterm Galerkin solution as follows

r 0 (x, y, t) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} R mn (x, y) \exp (- ω_{mn}^{Im} t) \times (A mn \sin ω_{mn}^{Re} t + B mn \cos ω_{mn}^{Re} t), r 0 = u 0, v 0, w 0, R = U, V, W

(22)

where

ω_{mn}^{Re}

and

ω_{mn}^{Im}

are real and imaginary parts of natural frequency for m and n longitudinal and transverse mode numbers, respectively. Unknown coefficients of sinus and cosine functions can be obtained by Fourier transform using initial conditions on displacement and velocity fields. Considering

r 00

and

\overset{\cdot}{r} 00

for

r 0 (x, y, t)

and

\overset{\cdot}{r} 0 (x, y, t)

at the initial moment (i.e.

t = 0

), the final form of dynamic response of viscoelastic composite structures on viscoelastic foundation can be obtained as

r 0 (x, y, t) = \sum_{m = 1, 2}^{\infty} \sum_{n = 1, 2}^{\infty} R mn (x, y) \exp (- ω_{mn}^{Im} t) \times (\frac{\overset{\cdot}{w} 00 + ω_{mn}^{Im} w 00}{ω_{mn}^{Re}} \sin ω_{mn}^{Re} t + w 00 \cos ω_{mn}^{Re} t)

(23)

Results and discussion

Due to lack of any results in the specialized literature, for the aim of comparison study, dynamic behavior of viscoelastic laminated composite plates without any foundation and isotropic elastic plates resting on viscoelastic foundations are analyzed. After validating the presented solution, the effects of material and geometrical parameters of structure and foundation on the dynamic behavior of symmetrically laminated viscoelastic beams and plates on viscoelastic foundations are investigated via a parametric study, and pertinent conclusions are outlined. In the present work, various combinations of simply supported and clamped BCs are considered (see Figure 2(b)). The edge conditions are tabulated in Table 1, for the beams and plates. The BCs of the plate are denoted by the capital letter so that the first, second, third, and the last letter rely on the condition at x = 0, y = 0, x = a, and y = b, respectively. For the beams, the first and second symbols rely on the condition at x = 0 and x = L, respectively. The associated shape functions are presented in Table 2 for plate structures. It is obvious that the appropriate shape functions for the beam can be obtained from those of plate neglecting the functions including y variable.

Table 1.

Boundary conditions.

Structure	Edges	Simply supported (S)	Clamped (C)
Plate	$x = 0 or a$	$M x = w = 0$	$w = \partial w / \partial x = 0$
Plate	$y = 0 or b$	$M y = w = 0$	$w = \partial w / \partial y = 0$
Beam	$x = 0 or L$	$M x = w 0 = 0$	$w 0 = \partial w 0 / \partial x = 0$

Table 2.

Plate shape functions for different boundary conditions.²⁵

BCs	W(x, y)
CCCC	$(1 - \cos \frac{2 m π}{a} x) (1 - \cos \frac{2 n π}{b} y)$
CCSS	$(\cos \frac{3 m π}{2 a} x - \cos \frac{m π}{2 a} x) (\cos \frac{3 n π}{2 b} y - \cos \frac{n π}{2 b} y)$
SSSS	$\sin \frac{m π}{a} x \sin \frac{n π}{b} y$

Comparative study

The first comparative study is dedicated to free vibration analysis of isotropic elastic plates resting on viscoelastic foundations. Natural frequencies and loss factor obtained from the present Galerkin solution are compared with counterparts reported by Leissa²⁵ based on the Ritz method. Material and geometry parameters are given in Table 3. The natural frequencies

(ω 11, ω 22, ω 33)

and relevant loss factors

(η 11, η 22, η 33)

for three different values of aspect ratios (a/b) of the elastic plate resting on viscoelastic foundation with 138 MN/m² stiffness and 100 kN s/m² viscosity are computed and compared with their counterparts in Table 4. As can be found from this table, the natural frequencies reported based on the Ritz method²⁵ are greater than those computed by the present Galerkin approach, as expected. Leissa mentioned that the Ritz method yields upper bound of natural frequency.²⁵ Due to this fact, the loss factors obtained by the Galerkin method are greater than other one, since the real part of natural frequencies is smaller. In spite of upper bound of the natural frequency computed by the Ritz method, the differences are not excessive than 1.91, 2.91, and 3.48% for CCCC, CCSS, and SSSS BCs, respectively. Moreover, the maximum differences for loss factor are 3.34, 3.94, and 5.06% for CCSS, CCCC, and SSSS BCs, respectively. Finally, Table 4 reveals that there is an acceptable correlation between the results computed by Galerkin and Ritz methods.

Table 3.

Material and foundation properties.

Item	Notation	Value
Mass density	ρ	7140 (Kg/m³)
Poisson ratio	υ	0.25
Young modulus	E	108 (GPa)
Shear modulus	G	43.2 (GPa)
Beam length	L	1.5 (m)
Thickness	h	20 (mm)
Plate length	a	1 m
Plate width	b	variable
Foundation stiffness¹⁹	k	138 (MN/m²)
Foundation viscosity¹⁹	c₀	1732.5 (kN s/m²)

Table 4.

The natural frequencies and loss factors of the elastic plate on viscoelastic foundation.

		ω			η
BCs	a/b	Present	Leissa²⁵	Disc. %	Present	Leissa²⁵	Disc. %
CCCC	0.5	1109.622^a	1128.213	1.67	0.0637^{d b}	0.0616	3.41
		3046.39^{b c}	3091.78	1.49	0.0084^{e d}	0.0081	2.97
		6645.32^{c e}	6746.99	1.53	0.0018^f	0.0017	3.69
	1	1363.388	1389.772	1.89	0.0422	0.0406	3.94
		4395.62	4470.78	1.71	0.0041	0.0039	3.36
		9746.44	10004.72	2.65	0.0008	0.0007	3.43
	1.5	1999.046	2037.918	1.91	0.0196	0.0189	3.70
		7315.69	7421.77	1.45	0.0015	0.0014	3.09
		16374.04	16601.64	1.39	0.0003	0.00029	3.21
CCSS	0.5	990.688	1016.803	2.57	0.0779	0.0759	2.63
		2298.63	2344.37	1.99	0.0148	0.0144	2.34
		4891.506	4994.72	2.11	0.0033	0.0032	3.43
	1	1170.453	1195.341	2.08	0.0573	0.0558	2.69
		3391.12	3461.65	2.08	0.0068	0.0064	2.26
		7442.793	7620.67	2.39	0.0014	0.0136	3.01
	1.5	1594.318	1642.137	2.91	0.0309	0.0299	3.34
		5499.99	5618.78	2.16	0.0019	0.0018	2.47
		12260.43	12538.74	2.27	0.0005	0.00048	3.48
SSSS	0.5	906.944	939.683	3.48	0.0944	0.0899	5.00
		1655.61	1707.09	3.11	0.0286	0.0274	4.06
		3324.84	3422.59	2.94	0.0071	0.0068	3.89
	1	1010.976	1035.816	2.39	0.0768	0.0731	5.06
		2435.87	2495.54	2.44	0.0132	0.0128	3.01
		5216.91	5373.94	3.01	0.0029	0.00278	4.15
	1.5	1248.701	1280.030	2.45	0.0503	0.0486	3.49
		3811.61	3910.33	2.59	0.0054	0.0052	3.24
		8409.98	8678.26	3.19	0.0011	0.00105	4.31

ω₁₁.

η₁₁.

ω₂₂.

η₂₂.

ω₃₃.

η₃₃.

In another comparison study, fundamental modal damping (

η %

) of simply supported viscoelastic plates computed by the present solution is compared with those reported by Hu and Dokainish.⁸ They investigated damped vibration of laminated composite plates with various layups and materials based on the first-order shear deformation theory (FSDT) using finite element method (FEM). They also employed specific damping capacity (SDC) model and viscoelastic damping model to simulate the damping behavior. Similar complex material features and geometrical properties together with staking sequences as reported in Hu and Dokainish⁸ for glass fiber reinforced plastic and carbon fiber reinforced plastic are used and listed in Table 5. Percentage of loss factor for the first natural frequency is listed in Table 6 and compared to those reported by Hu and Dokainish.⁸ Table 6 indicates that the present formulation based on the CLPT underestimates the modal damping in comparison with the FEM solution based on the FSDT. It is seen that there is a good agreement between the present results and those obtained by SDC. It should be mentioned that the most important reason for modal damping differences is arisen from transverse shear modulus

G TT

, which is neglected in CLPT relations.

Table 5.

The polymeric based plate data with $[θ / - θ / θ / - θ]$ layup.⁸

	E_L (GPa)	E_T (GPa)	$G LT$ (GPa)	$υ LT$	ρ(kg/m³)	$a \times b$ (mm)	h (mm)
GFRP	55	18.3(1 + 0.079i)	9.1(1 + 0.079i)	0.25	1993	1500 × 375	10
CFRP	211	5.3(1 + 0.079i)	2.6(1 + 0.079i)	0.25	1524	1500 × 375	10

Table 6.

Percentage of fundamental loss factor ( $η 11$ ).

θ	0	15	30	45	60	75	90
GFRP
VED⁸	50.68	47.46	37.89	25.09	13.87	6.479	3.663
SDC⁸	48.16	44.40	34.13	21.58	11.30	4.769	2.199
Present	47.61	43.03	33.09	21.02	10.73	4.354	2.141
CFRP
VED⁸	44.18	25.65	10.33	4.858	3.332	4.933	2.870
SDC⁸	42.92	24.61	9.665	4.489	3.174	4.807	2.861
Present	41.53	23.99	9.108	4.325	3.104	4.329	2.710

Parametric study

Viscoelastic plates

In this section, the effects of foundation parameters including stiffness and viscosity, BCs and geometrical parameters on the natural frequency, modal loss factor and dynamic response of viscoelastic plates resting on viscoelastic foundations with unit length and variable width are investigated. The material properties are given as

ρ f = 2500 kg / m 3, ρ m = 1200 kg / m 3, E f = 85 GPa, G f = 35.42 GPa, v f = 0.2, V f = 0.45, h = 20 mm

The prony series coefficients of bulk and shear moduli for polymeric material are given in Table 7. Based on the thin plate theory and symmetric lamination scheme of composite laminated plate, the coupling stiffness coefficients and in-plane motion of the plate become zero.

Table 7.

Prony–Dirichlet series coefficients.

i	τ_i (s)	G_i (MPa)	K_i (MPa)
1	0.017	85	197.5
2	0.142	85	197.5
3	1.202	83	197.5
4	10.206	83	175
5	86.67	82	46
6	735.99	82	30
7	6250	80	7.5
8	∞	180	1500

Influence of aspect ratio and fiber orientation on the natural frequency and modal loss factor of viscoelastic laminated composite plates are studied in Table 8 for fully simply supported edge conditions. As can be seen, increment of aspect ratio leads to increasing the natural frequency while a specific manner cannot be observed for loss factor variation. In all cases, the loss factors are much lower than unit values; it means the imaginary part of natural frequency is much lower than real part. The natural frequency and almost loss factor for

[0 / - 45] s

are less than counterparts for

[0 / 0] s

lamination scheme, as expected. Existence of layer with

- 45 \circ

orientation reduces the plate stiffness in comparison with plate embedded with

0 \circ

one, although this reduction is not significant. In both lamination schemes, the natural frequency is increased by aspect ratio increment whereas the regular variation of loss factor cannot be observed.

Table 8.

Natural frequency and loss factor of viscoelastic composite plates with various a/b.

Layup		$[0 / 0] s$		$[0 / - 45] s$
(m, n)	a/b	$ω n$ (rad/s)	$η (%)$	$ω n$ (rad/s)	$η (%)$
(1,1)	1	333.251	2.24	316.144	0.48
	2	560.616	2.50	545.564	0.87
	5	2415.627	0.24	2387.210	0.087
(1,2)	1	570.104	0.54	545.564	0.87
	2	1637.088	0.66	1588.48	0.19
	5	9727.050	0.02	9075.79	0.06
(2,2)	1	1313.508	0.04	1265.00	0.033
	2	2283.355	0.14	2182.82	0.055
	5	9961.512	0.01	9549.02	0.054
(2,3)	1	1660.547	0.12	1624.75	0.057
	2	3975.661	0.08	3890.95	0.027
	5	21566.46	0.13	20687.92	0.001
(3,3)	1	2930.457	0.0031	2846.31	0.006
	2	5237.658	0.003	4911.418	0.011
	5	23152.678	0.0032	21485.31	0.001

In order to grasp physical comprehension and damping conception of viscoelastic plate with represented vibrational characteristics in Table 8, the time history of central deflection of simply supported viscoelastic composite plates due to initial 1 mm/s velocity and 1 mm displacement with various aspect ratios is depicted in Figures 3 and 4 for $[0 / 0] s$ and $[0 / - 45] s$ layups, respectively. Importance of these figures and illustrated damping capability are discriminated obviously, when other source of energy dissipating such as foundation viscosity is added to the system for comparison of each damping factor effects. Figures 3 and 4 reveal that the aspect ratio has significant effects on the vibration amplitude and damping time for this set of parameters. It is observed that damping time of the viscoelastic plates attenuates considerably faster with the increase of aspect ratio. In addition, at some specific (not initial) times, vibration amplitude decreases intensely by aspect ratio increment. As can be found from Figure 3, the deflection amplitude almost attenuates to zero within 3.1, 1.6, and 3.9 s for $a / b = 1, 2$ , and 5, respectively. Comparing the presented results in Figures 3 and 4 reveals that the viscoelastic plate with staking sequence of $[0 / - 45] s$ has a less stiffness in comparison with $[0 / 0] s$ which affects vibration amplitude and damping time. Figure 4 shows that the oscillation amplitude almost attenuates to zero within 3.75, 2.1, and 5.5 s, respectively, for $a / b = 1, 2$ , and 5. In both stacking sequences, the damping time for $a / b = 2$ has lower value in comparison with other aspect ratio due to higher loss factor as presented in Table 8.

Foundation stiffness is prominent factor, which increases the structure stiffness; therefore, leads to the higher natural frequency of the structure. Some values of the stiffness increase the natural frequency to the extent that the loss factor approaches to the negligible value or in the other words viscoelasticity of the structures has no important role on the structure behavior.

Influences of elastic foundation stiffness on the natural frequency and modal loss factor of viscoelastic laminated composite plates with $[0 / - 45] s$ layup and 0.5 mm thickness for each layer are examined in Table 9 for various BCs and mode numbers. The preliminary conclusion drawn from Table 9 is the foundation stiffness increases the natural frequency, whereas it decreases the modal loss factor. Ascending the foundation stiffness up to 0.001 MN/m² leads to the increase of the natural frequency for various BCs. Increment of the stiffness more than the aforesaid value eliminates the BC effect as depicted in Figure 5. In the other words, the BCs have a tiny effect on the natural frequency and loss factor as foundation stiffness unfolds. It is seen that the effect of stiffness foundation is more pronounced on the increase of natural frequency and decrease of modal loss factor for the higher modes. Furthermore, the modal loss factor is increased by mode number increment, which is arisen from computational procedure of complex natural frequency for the system. From a physical point of view, it means that the vibration in the higher modes will be damped faster than vibration in the first complex natural frequency for system with this set of parameters.

Table 9.

The effect of foundation stiffness on vibration characteristics of viscoelastic composite plates on elastic foundations.

			k (MN/m²)
BCs	Mode		0.001	0.1	10	138
SSSS	(1,1)	$ω n$	24.151	236.688	2366.904	8792.66206
	(1,1)	η	0.046	$2.6 \times 10 - 5$	$9.77 \times 10 - 9$	$1.42 \times 10 - 12$
	(2,2)	$ω n$	47.651	236.798	2366.906	8792.66209
	(2,2)	η	0.797	0.0013	$1.3 \times 10 - 7$	$6.97 \times 10 - 10$
	(3,3)	$ω n$	102.732	240.319	2366.907	8792.66299
	(3,3)	η	0.968	0.03201	$3.4 \times 10 - 6$	$1.787 \times 10 - 8$
CCSS	(1,1)	$ω n$	25.966	236.688	2366.905	8792.66301
	(1,1)	η	0.181	$5 \times 10 - 5$	$2.7 \times 10 - 10$	$2.72 \times 10 - 12$
	(2,2)	$ω n$	69.889	237.409	2366.9052	8792.66303
	(2,2)	η	0.915	0.0068	$2.5 \times 10 - 6$	$3.63 \times 10 - 9$
	(3,3)	$ω n$	154.384	253.896	2366.921	8792.6631
	(3,3)	η	0.9869	0.1340	$1.77 \times 10 - 6$	$9.32 \times 10 - 8$
CCCC	(1,1)	$ω n$	29.735	236.693	2366.9051	8792.66302
	(1,1)	η	0.3827	$9.7 \times 10 - 5$	$5.1 \times 10 - 10$	$5.13 \times 10 - 11$
	(2,2)	$ω n$	95.378	239.417	2366.907	8792.66304
	(2,2)	η	0.955	0.0239	$6.92 \times 10 - 6$	$1.31 \times 10 - 8$
	(3,3)	$ω n$	212.382	284.138	2366.974	8792.6651
	(3,3)	η	0.993	0.3094	$6.4 \times 10 - 5$	$3.36 \times 10 - 7$

In addition, foundation stiffness increment up to the highest value results in neglecting plate treatment on system behavior. In the other words, the foundation stiffness and inertia of the plate are the main effective factors on the vibration behavior of the system with this set of parameters.

As can be found, SSSS BCs have less natural frequency and loss factor compared to the other BCs. In the other words, the stiffer BCs have higher natural frequency and loss factor. Moreover, for high values of foundation stiffness, natural frequency almost remains constant when the mode number increases. It means that the system with this set of parameters behaves like, not as an exact single degree of freedom (SDOF), so that the mode numbers variation of the structures has no obvious effect on the natural frequencies. Again, it is worth mentioning that this continuous system is not a SDOF, in spite of being similar (not the same) natural frequencies; as mentioned before, these values reveal remarks that the mode numbers and BCs have negligible effects on the system natural frequency for this set of parameters.

Influence of BCs on the central deflection time history of the viscoelastic plates resting on elastic foundation with 0.001 MN/m² stiffness is illustrated in Figure 6. This figure indicates that the vibration amplitude and damping time are significantly reduced when the edge conditions become more constraints. As can be seen in Figure 6, the SSSS, CCSS, and CCCC plates lose initial oscillation amplitude up to 38, 92, and 99% at $t = 1$ s. It is found that the deflection amplitude of CCCC and CCSS plates almost attenuates to zero within 1.2 and 2.5 s, respectively, whereas SSSS plate experiences zero amplitude at $t = 50$ s.

In Table 10, the effects of foundation viscosity on vibration damping behavior of the plate with 20 mm thickness on foundation with 138 MN/m² stiffness are studied. According to Table 10, the foundation viscosity affects strongly natural frequency and loss factor of viscoelastic plates with various BCs. Increment of the foundation viscosity up to 20 kN s/m² reduces the natural frequency, whereas it increases the loss factor in the same mode numbers for all the edge conditions. Moreover, mode numbers increasing with specific foundation viscosity lead to the same manner for natural frequency and loss factor while the loss factor values are negligible for foundation with 0.1 and 20 kN s/m² viscosities. Foundation viscosity variation from 0.1 up to 20 kN s/m² has no prominent influence on natural frequency decrement while it has obvious effect on trifling loss factor increment. Ascending the foundation viscosity more than the mentioned value up to 1732.5 kN s/m² results in the transition from harmonic motion to the over damp motion in which the real part of the natural frequency is negligible versus imaginary part. Due to this fact, the loss factor increases significantly. For this set of parameters, loss factor is increased surprisingly by foundation viscosity increment, so that the loss factor of the stiffer layup has demonstrated more values. Also, very high values of loss factor (more than unit) have represented the high sensitivity of this system to the foundation viscosity so that this value especially for the lower modes have depicted nonharmonic motion or highly over damp motion. This remark also can be discriminated from real and imaginary parts of complex natural frequency. In this realm of foundation viscosity variation, increment of mode numbers leads to the decrease of loss factor obviously and increase of the natural frequency especially in (3, 3) mode. From a mathematical point of view, it means that the real part of natural frequency has obvious increment in comparison with imaginary part for this mode especially in CCCC and CCSS edge conditions.

Table 10.

Natural frequency and loss factor of square plate with various foundation viscosities.

BCs	Mode		c₀ = 0.1 (kN s/m²)		c₀ = 20 (kN s/m²)		c₀ = 1732.5 (kN s/m²)
BCs	Mode		$[0 / 0] s$	$[90 / 0] s$	$[0 / 0] s$	$[90 / 0] s$	$[0 / 0] s$	$[90 / 0] s$
SSSS	(1,1)	$ω n$	1966.22	1966.20	1946.17	1946.15	0.11792	0.10639
	(1,1)	η	0.00016	0.00014	0.02436	0.02415	10007.12	9907.68
	(2,2)	$ω n$	1999.99	1999.43	1980.96	1977.29	0.6404	0.66868
	(2,2)	η	0.03415	0.03068	0.10829	0.10210	7797.422	8497.87
	(3,3)	$ω n$	2475.90	2438.55	2464.40	2426.72	0.9031	0.8890
	(3,3)	η	0.3707	0.35124	0.58819	0.52946	6330.68	5648.514
CCSS	(1,1)	$ω n$	1966.87	1966.78	1946.74	1946.04	0.2441	0.2392
	(1,1)	η	0.00086	0.00077	0.02941	0.02125	63025	62787
	(2,2)	$ω n$	2134.21	2118.54	2102.30	2086.74	0.9473	0.9114
	(2,2)	η	0.15225	0.13964	0.29969	0.26222	41025	40391
	(3,3)	$ω n$	3334.39	3259.38	3252.07	3179.11	371.298	350.362
	(3,3)	η	0.65338	0.63722	0.82691	0.78776	5328.86	5124.54
CCCC	(1,1)	$ω n$	1969.22	1968.87	1949.27	1948.91	0.32832	0.32808
	(1,1)	η	0.0033	0.0122	0.04068	0.03938	33223.05	33158.70
	(2,2)	$ω n$	2439.75	2403.68	2427.89	2391.46	0.6862	0.5989
	(2,2)	η	0.3516	0.3320	0.50922	0.48811	19203.9	16646.79
	(3,3)	$ω n$	4459.29	4344.66	4454.43	4339.65	737.678	695.750
	(3,3)	η	0.80639	0.79603	0.92645	0.91870	4851.24	4315.67

It is seen that, in all the BCs, $[90 / 0] s$ layup has less natural frequency than $[0 / 0] s$ due to the existence of the weaker layer in previous one. In addition, the natural frequencies and loss factors are almost the same values with foundation with 0.1 and 20 kN s/m² viscosities for both layups whereas these vibrational characteristics are different eminently for foundation with 1732.5 kN s/m² viscosity. Moreover, these differences become more obvious by mode number increment. As expected, the more rigid edge conditions have higher natural frequency and loss factors so that the CCCC, CCSS, and SSSS BCs have presented, respectively, more values for these vibrational characteristics. It is noteworthy that these characteristics with lower value of foundation viscosity are not affected eminently by variation of mode numbers and edge conditions.

Viscoelastic Beams

In this part, the vibration behavior of viscoelastic beams resting on viscoelastic foundations is investigated. The polymeric cured resin properties are considered as listed in Table 7. It is expected that the beam structures behave like to the plate one studied in proceeding section.

The first five natural frequency and loss factor of viscoelastic beams with 2 mm thickness, SS and CC BCs is presented in Table 11 for various beam lengths. As can be seen, in each mode, increasing the length leads to the increment of the loss factor and decrement of natural frequency while the variation of natural frequency is more prominent than loss factor. Fully clamped beams represent higher natural frequency and lower loss factor than simply supported one. The natural frequencies in the previous case are more than two times of later, although the loss factors have lower values at higher modes. Loss factor in the

[90 / 0] s

layup is more than

[0 / 0] s

layup since viscoelasticity behavior of the polymeric matrix is dominant in the first layup while in the second one, fibers behavior. Therefore, the lower real part and higher imaginary part of natural frequency in

[90 / 0] s

layup leads to the higher loss factor for this layup in comparison with

[0 / 0] s

. Moreover, the natural frequencies of fully clamped beams with the

[90 / 0] s

layup are similar to counterparts of fully simply supported beams with

[0 / 0] s

layup, whereas the loss factors of the earlier are higher than latter. In spite of the same natural frequency value, the dominant viscoelastic behavior of

[90 / 0] s

layup is emerged in the loss factor values. The interpretation of high values of loss factor can be referred to those have presented for Table 10.

Table 11.

The first five natural frequency (rad/s) and loss factor (%) of viscoelastic beams.

	CC				SS
	$[0 / 0] s$		$[90 / 0] s$		$[0 / 0] s$		$[90 / 0] s$
L	$ω n$	η	$ω n$	η	$ω n$	η	$ω n$	η
0.5	248.03	1.08	109.26	7.93	107.38	3.99	46.97	14.3
	992.18	0.74	438.71	0.73	429.62	0.40	189.72	3.4
	2232.3	0.15	987.30	0.16	966.63	0.78	427.41	0.77
	3968.3	0.047	1755.1	0.49	1718.4	0.25	759.97	0.23
	6199.9	0.019	2742.3	0.02	2684.8	0.1	1187.4	0.11
1	61.97	6.29	27.00	14.5	26.82	6.8	11.61	17.7
	248.03	1.08	109.26	7.9	107.34	3.9	46.97	14.2
	558.1	0.23	246.61	2.1	241.65	0.11	106.43	8.2
	992.17	0.74	438.71	0.73	429.62	0.38	189.73	3.4
	1550.2	0.031	685.59	0.31	671.28	1.61	296.71	1.5
1.5	27.53	6.8	11.93	17.5	11.92	8.04	5.12	19.8
	110.2	3.9	48.22	14.2	47.69	6.79	20.74	14.8
	248.03	1.08	109.26	7.9	107.36	4.01	46.97	14.2
	440.96	0.36	194.75	3.3	190.93	1.71	83.95	10.5
	689.01	0.15	304.56	1.4	298.34	0.77	131.57	6.2

The effects of foundation stiffness and viscosity on the natural frequency and loss factor of the viscoelastic composite beams are highlighted in Tables 12 and 13, respectively. The beam has 20 mm thickness, 1.5 m length, and

[0 / 0] s

layup. It is observed that enhancement of the stiffness leads to the increase and decrement of the natural frequency and loss factor, respectively. Increase of foundation stiffness has a more significant effect on the loss factor decrement especially in the lower mode than higher mode. In the other words, loss factors of the first mode are more sensitive to the foundation stiffness variation. Despite increasing of natural frequency with the increase of foundation stiffness up to 0.1 MN/m², the variation is not significant. Intensification of the foundation stiffness more than the mentioned value leads to the natural frequency increment prominently. In all cases, beam natural frequency almost has remained constant for foundation with 0.001 and 0.1 MN/m² stiffness while a development can be observed for it with 10 MN/m² stiffness. For foundation with 138 MN/m² stiffness, the natural frequency has no obvious difference in various mode numbers. In the other words, natural frequency is not affected by mode number variation for the highest values of foundation stiffness. It means that the beam structure on foundation with this high value foundation stiffness behaves like a SDOF with constant inertia and stiffness. Moreover, fully clamped beams have represented higher natural frequency and loss factor than fully simply supported beams. Similar interpretations can be expressed for high value (more than unit) of loss factor such as those are presented for Tables 10 to 12.

Table 12.

Effect of foundation stiffness k (MN/m²) on the natural frequency and loss factor of the viscoelastic composite beams.

BCs	Mode		0.001	0.1	10	138
SS	1	$ω n$	84.423	92.988	529.423	1966.102
	1	η	0.09996	0.06789	0.00006	3.4 × 10⁻⁷
	2	$ω n$	337.571	339.630	566.177	1966.947
	2	η	0.10003	0.09763	0.01264	0.00008
	3	$ω n$	759.513	760.426	856.519	1987.385
	3	η	0.10020	0.09954	0.06184	0.00213
CC	1	$ω n$	194.909	198.496	533.992	1966.192
	1	η	0.10004	0.09300	0.00177	0.00001
	2	$ω n$	779.564	780.454	873.875	1989.657
	2	η	0.10002	0.09956	0.06334	0.00235
	3	$ω n$	1754.009	1754.404	1794.369	2359.208
	3	η	0.10000	0.09992	0.09131	0.03055

Table 13.

Effects of foundation viscosity on the natural frequency and loss factor of the viscoelastic composite beams.

BCs	Mode		c₀ = 0.1 (kN s/m²)		c₀ = 20 (kN s/m²)		c₀ = 1732.5 (kN s/m²)
BCs	Mode		$[0 / 0] s$	$[90 / 0] s$	$[0 / 0] s$	$[90 / 0] s$	$[0 / 0] s$	$[90 / 0] s$
SS	1	$ω n$	1966.102	1966.098	1946.046	1946.042	0.0063	0.00225
	1	η	6.5 × 10⁻⁶	9.26 × 10⁻⁶	0.0213	0.0228	7.1 × 10⁸	8.97 × 10⁷
	2	$ω n$	1966.947	1966.100	1946.918	1946.045	0.1011	0.0340
	2	η	0.0009	0.00098	0.0302	0.0319	3.1 × 10⁶	3.52 × 10⁵
	3	$ω n$	1987.384	1966.43	1967.971	1946.387	0.4929	0.0741
	3	η	0.0215	0.0404	0.0548	0.0671	14767.81	6.5 × 10⁵
CC	1	$ω n$	1966.192	1966.096	1946.139	1946.040	0.0338	0.0119
	1	η	0.0001	0.0004	0.0237	0.0311	3.15 × 10⁶	2.5 × 10⁷
	2	$ω n$	1989.657	1966.471	1970.310	1946.429	0.5171	0.0733
	2	η	0.0238	0.0480	0.0592	0.0674	13413.6	6.67 × 10⁵
	3	$ω n$	2359.207	1977.051	2346.476	1957.357	127.207	1.4279
	3	η	0.3062	0.4116	0.4599	0.6361	1691.105	145000

Transient response of viscoelastic laminated composite beam with $[0 / 0] s$ layup on the elastic foundation is presented in Figure 7 for CC and SS BCs. Conclusions similar to those concluded from Figure 6 as stated earlier for plate structures can be achieved. As can be found from Figure 6, the oscillation amplitude of composite beams with CC and SS BCs approaches to zero in 51 and 59 s, respectively. It means the system is returned to the stable state sooner in fully clamped edge conditions.

Influence of foundation viscosity variation on the damping and vibration of viscoelastic composite beams resting on foundation with 138 MN/m ² stiffness is studied in Table 13. The results show that increment of foundation viscosity up to 20 kN s/m² has no significant effect on the natural frequency, although it leads to obvious increment of the loss factor. As can be found from this table, the system with 0.1 and 20 kN s/m² foundation viscosities almost has represented similar natural frequency so that the lamination scheme and edge conditions (SS and CC) have no obvious effect on the vibrational characteristics, although there are some tiny differences between natural frequencies of system with various layups in different edge conditions. In both SS and CC edge conditions, the first and second natural frequencies have values with tiny differences while the third natural frequency represents larger values especially in $[0 / 0] s$ layup with weaker viscosity. In addition, the third modal loss factors represent smoother manner in the both edge conditions for system with these foundation viscosity values. In the other words, these loss factors are less sensitive to foundation viscosity in this realm. Foundation with 1732.5 kN s/m² viscosity results in the transition of the harmonic motion to the inharmonic motion. This is due to the fact that the real part of natural frequency is negligible with respect to the imaginary part. Therefore, the loss factor increases significantly while natural frequency has an obvious decrement. As expected, the $[0 / 0] s$ layup has represented higher natural frequency for both SS and CC edge conditions in comparison with $[90 / 0] s$ layup while it has represented lower loss factor especially in CC edge conditions. In SS edge conditions the obvious overcoming cannot be seen for the first layup.

Concluding remarks

The present work is concerned with the dynamic response of freely vibrating viscoelastic laminated composite beams and plates resting on suppressive foundations. The constitutive relation of the fiber-reinforced composite laminates with polymer matrix is established by adopting the Boltzmann superposition principle. The polymeric matrix behavior is extracted from DMA results in terms of Prony–Dirichlet series. The Kelvin–Voigt model is considered to describe the foundation viscoelasticity. Employing the Hamilton principle, the governing equations of motion of viscoelastic laminated composite beams and plates resting on viscoelastic foundations are derived based on the displacement field of the classical lamination theory. Applying Galerkin weighted residual method, Fourier transform, and iterative QZ algorithm, the solution of unknown variables is obtained in the space and time domain. Due to lack of similar results in the specialized literature, the results of elastic isotropic plates on viscoelastic foundation and viscoelastic composite plates with constant complex stiffness and constant Poisson ratios are computed and compared with their counterparts in the literature. Then, a detailed analysis of the influence of stiffness and viscosity of foundation, geometrical parameters and BCs on the natural frequency, modal loss factor and transient response of freely vibrating viscoelastic laminated composite beams, and plates resting on viscoelastic foundations is carried out.

As revealed in the “Results and discussion” section, the viscoelastic beams and plates have internal damping capability due to existence of viscoelastic behavior of constitutive materials in the structures. Moreover, the effect of edge conditions, lamination schemes, stiffness, and viscosity of foundation can be applied simultaneously to study foundation–structure interaction and its vibrational and dynamic treatments. The following main conclusions can be drawn:

Foundation viscosity has significant influence on the dynamic behavior of the structure. It is observed that the high values of foundation viscosity may lead to the transition of harmonic motion into the inharmonic one.

Increment of the aspect ratio leads to ascending the natural frequency while a specific manner of loss factor cannot be observed.

Foundation stiffness increases the natural frequency of the plate to the extent that the loss factor approaches to the negligible value or in the other words viscoelasticity of the structures has no important role on the structure behavior.

The edge conditions have prominent effect on the natural frequency and loss factor of the viscoelastic plates resting on elastic foundation. However, it is found that increasing the foundation stiffness to the highest values debilitates BCs effects.

Stacking sequence $[90 / 0] s$ demonstrates more loss factor in comparison with $[0 / 0] s$ , since viscoelasticity behavior of polymeric matrix in the first layup is dominant and in the second one, fibers behavior.

Long beams represent low natural frequency and high loss factor while the short beams tend vice versa. Furthermore, natural frequency variation is more prominent than loss factor variation.

Significant descent of the loss factor is observed especially in the lower mode when the foundation stiffness is increased. Moreover, the loss factor of the first mode is more sensitive to the foundation stiffness variation.

The foundation with high values of viscoelasticity can be utilized to control the delirious vibration and fluctuation manner of beam and plate structures.

Finally, due to the absence of similar results in the specialized literature, the results of this research are expected to contribute to a better understanding of the vibrational characteristics of viscoelastic beams and plates resting on suppressive foundations. From a practical point of view, the results may be instrumental toward a reliable design of armored railroads with damping layer on the lateral surface and the polymeric composites flooring on the floor in dynamical environments.

Footnotes

Acknowledgement

The authors would like to express their gratitude to the anonymous reviewers who made valuable comments and suggestions to improve the paper.

Conflict of interest

None declared.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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