Eigenvibrations of Kirchhoff Rectangular Random Plates on Time-Fractional Viscoelastic Supports via the Stochastic Finite Element Method

The present work’s main objective is to investigate the natural vibrations of the thin (Kirchhoff–Love) plate resting on time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM). The behavior of the supports is described by the fractional order derivatives of the Riemann–Liouville type. The subspace iteration method, in conjunction with the continuation method, is used as a tool to solve the non-linear eigenproblem. A deterministic core for solving structural eigenvibrations is the Finite Element Method. The probabilistic analysis includes the Monte-Carlo simulation and the semi-analytical approach, as well as the iterative generalized stochastic perturbation method. Probabilistic structural response in the form of up to the second-order characteristics is investigated numerically in addition to the input uncertainty level. Finally, the probabilistic relative entropy and the safety measure are estimated. The presented investigations can be applied to the dynamics of foundation plates resting on viscoelastic soil.


Introduction
Some stochastic numerical approaches, especially the Boundary and Finite Element Methods, have been intensively studied by many authors.Kami ński et al. applied the Finite Element Method (FEM) for the stochastic second-order perturbation technique [1] as well as the iterative scheme in the determination of the probabilistic moments of the structural response [2].Cheng et al. performed reliability analysis of plane elasticity problems via stochastic spline fictitious BEM [3] and a similar investigation for Reissner's plate bending problem [4] and fracture mechanics analysis of linear-elastic cracked structures [5].Christos et al. [6] used the Stochastic Boundary Element Method (SBEM) to analyze the dynamic response of tunnels.Karimi et al. [7] proposed coupled FEM-BEM with artificial neural networks (ANN) approach for the system identification of concrete gravity dams.A coupled FEM-BEM-least squares point interpolation for a structure-acoustic system with a stochastic perturbation technique was proposed by Zhang et al. [8].The stochastic Galerkin scaled boundary finite element approach was proposed by Duy Mihh et al. [9] for the randomly defined domain.The Monte Carlo simulation technique coupled with the scaled boundary FEM was applied by Chowdhury et al. [10] for probabilistic fracture mechanics.
Similarly, the same authors studied probabilistic fracture mechanics with uncertainty in the crack size and orientation in terms of the scaled boundary finite element approach [11].Luo et al. investigated the stochastic response determination of multidimensional nonlinear systems endowed with fractional derivatives [12].
Di Matteo et al. studied stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral [13].Łasecka-Plura and Lewandowski [14] investigated dynamic characteristics and frequency response function for frames with dampers using FEM with uncertain design parameters.Abdelrahman et al. studied the nonlinear dynamics of viscoelastic flexible structural systems using the finite element method [15].Ratas et al. investigated the solution of the nonlinear boundary value problems by applying the higher-order Haar wavelet functions [16], where the authors carried out a comprehensive analysis of the error and convergence of the obtained solution.Abdelfattah et al. investigated the generalized nonlinear quadrature for the fractionalorder chaotic systems using the SINC shape function [17] in which the authors applied the generalized Caputo definition of the fractional derivative to a study of fractional-order Lorenz oscillator describing a three-dimensional chaotic flow dynamical system.
The basis for considerations in the random approach is a finite set of solutions in the deterministic approach.The problem of deterministic damped vibrations employing viscoelastic dampers was the subject of Lewandowski's [18] and Pawlak with Lewandowski [19]'s research.Similarly, the dynamic characteristics of multilayered beams with viscoelastic layers described using the time-fractional Zener model have been investigated by Lewandowski and Baum [20].Łasecka-Plura and Lewandowski [21] applied the subspace iteration method to resolve nonlinear eigenvalue problems occurring in the dynamics of structures with viscoelastic elements.
Material parameters of the viscoelastic constraints may be easily described using fractional derivatives [22].Chang and Singh [23] investigated the seismic analysis of structures with a fractional derivative model of viscoelastic dampers.On the other hand, Kun et al. [24] applied fractional derivatives to analyze the stochastic seismic response of structures considering viscoelastic dampers.Next, Shubin [25] applied the generalized multiscale finite element method to an inverse random source problem.It is important to emphasize that all papers mentioned above studied the time-fractional viscoelastic dampers utilizing the "full memory" approach [22,25,26], i.e., the fractional operator governs the memory from initial time up to current time.Such an assumption may be generalized by using the "short memory" approach, in which memory is restricted to a particular time interval (time length scale) [22,27,28].
The present work's main objective is to investigate the natural vibrations of the thin (Kirchhoff-Love) plate resting on the time-fractional viscoelastic supports in terms of the Stochastic Finite Element Method (SFEM).The least squares method procedure enables the determination of random polynomials of eigenfrequencies, whose further integration with the Gaussian kernel returns probabilistic characteristics.Additionally, the stochastic perturbation technique (SPT) and the Monte Carlo simulation (MCS) approaches are all applied in the analysis to study the influence of various uncertainty sources on the basic eigenfrequencies of some more popular rectangular plates with viscoelastic supports.

Eigenvibrations Analysis Methodology of Deterministic Approach
The presented investigations aim to determine the first few natural frequencies of thin rectangular elastic and isotropic plates supported on the boundary and rested on the (classical) viscoelastic dampers.
A thin, elastic, and isotropic plate is considered, which can be supported classically in an ideal way, e.g., along the shore, and can also rest on a finite number of supports that have the nature of viscoelastic constraints.The plate may also rest solely on viscoelastic supports, which may be located on its edges or inside it.Due to the discrete and one-dimensional nature of additional supporting viscoelastic bonds, it is very easy to take into account their presence in the description of the deformation of the entire structural system, i.e., the plate-viscoelastic constraints.Thus, the presence of viscoelastic elements is taken into account in the boundary conditions while ensuring the kinematic invariance of the system.Applying the classical variational formulation to a plate single finite element, one obtains, e.g., [29].
udV (1) where δu and δε are column matrices of variations of small displacements and strains; F is the vector of volumetric forces (body forces); Φ is the column matrix of tractions acting on the surface; p i is the vector of concentrated loads acting at a selected point "i"; k d is a viscosity damping parameter; and finally, .u and .. u are the velocity and acceleration vectors, respectively [29].Equation ( 1) can be generalized and written for a set of finite elements, whereby the actual damping elements (viscoelastic elements) will be coupled with translational displacements perpendicular to the center surface of the plate.The above general formulation allows for a simple way of writing down the problem of system vibrations in a matrix manner by the FEM methodology.A description of the finite element analysis approach is provided below.
The Finite Element Method (FEM) with a regular discretization including 4-noded plate finite elements is applied to the numerical analysis (Figure 1).
A thin, elastic, and isotropic plate is considered, which can be supported classically in an ideal way, e.g., along the shore, and can also rest on a finite number of supports that have the nature of viscoelastic constraints.The plate may also rest solely on viscoelastic supports, which may be located on its edges or inside it.Due to the discrete and onedimensional nature of additional supporting viscoelastic bonds, it is very easy to take into account their presence in the description of the deformation of the entire structural system, i.e., the plate-viscoelastic constraints.Thus, the presence of viscoelastic elements is taken into account in the boundary conditions while ensuring the kinematic invariance of the system.Applying the classical variational formulation to a plate single finite element, one obtains, e.g., [29].
∫  T  + ∫  T  + ∑ ( T )     =1 = ∫  T  +  ∫  T ̈ + ∫  T   ̇     (1) where δu and δε are column matrices of variations of small displacements and strains; F is the vector of volumetric forces (body forces); Φ is the column matrix of tractions acting on the surface; pi is the vector of concentrated loads acting at a selected point "i"; kd is a viscosity damping parameter; and finally, ̇ and ̈ are the velocity and acceleration vectors, respectively [29].Equation ( 1) can be generalized and written for a set of finite elements, whereby the actual damping elements (viscoelastic elements) will be coupled with translational displacements perpendicular to the center surface of the plate.The above general formulation allows for a simple way of writing down the problem of system vibrations in a matrix manner by the FEM methodology.A description of the finite element analysis approach is provided below.
The Finite Element Method (FEM) with a regular discretization including 4-noded plate finite elements is applied to the numerical analysis (Figure 1).The displacement vector    of the -th corner node within the finite element  is defined as [30].
The field of displacements inside the element  is expressed as a linear combination of the shape functions    (, ). where The shape functions, the element stiffness matrix   and the element consistent mass matrix   are defined according to the standard FEM methodology [29,31].To conclude, the stiffness matrix is The displacement vector w e i of the i-th corner node within the finite element e is defined as [30].
The field of displacements inside the element e is expressed as a linear combination of the shape functions N e k (x, y).w e (x, where w e = w e 1 , w e 2 , w e 3 , w e

4
T and N e = N e 1 , N e 2 , N e 3 , . . .N e 12 .The shape functions, the element stiffness matrix K e and the element consistent mass matrix M e are defined according to the standard FEM methodology [29,31].To conclude, the stiffness matrix is derived analytically, whereas the mass matrix is calculated numerically using 16th point Gaussian quadrature.The discretization of a plate domain is shown in Figure 2, where the adopted method of numbering finite elements and their nodes is presented.The center plane of the plate with dimensions A × B is discretized with m × n finite elements with dimensions l x,i × l y,j (i = 1, 2, 3, . . ., m; j = 1, 2, 3, . . ., n).It was assumed that all finite elements have the same dimensions l x × l y .The discretized plate contains a total of (m + 1)(n + 1) nodes and 3(m + 1)(n + 1) degrees of freedom.
Gaussian quadrature.The discretization of a plate domain is shown in Figure 2, where the adopted method of numbering finite elements and their nodes is presented.The center plane of the plate with dimensions  ×  is discretized with  ×  finite elements with dimensions  , ×  , ( = 1,2,3, … , ;  = 1,2,3, … , ).It was assumed that all finite elements have the same dimensions   ×   .The discretized plate contains a total of ( + 1)( + 1) nodes and 3( + 1)( + 1) degrees of freedom.The matrix equation of the force balance for a structure with viscoelastic dampers can be written in the following form: where M and K denote mass and stiffness matrices of the plate, respectively; matrix  denotes the global plate damping matrix; () is the vector of additional forces resulting from the viscoelastic supports; and () is the excitation vector.An application of the Laplace transform with zero initial conditions leads to the following transform of Equation ( 4) where  ̅() is the ℒ-transform of (),  ̅() is the ℒ-transform of () and the vector  ̅ () is expressed as In Equation ( 6),   denotes the total number of dampers attached to the plate at selected nodes of a finite element mesh and   is a global matrix indicating the location of the -th damper within the plate domain.
A viscoelastic damper is described graphically in Figure 3, wherein k0, k1, c1, and αd describe material properties of viscoelastic constraints (damper); u is the force transferred by the damper; and  ̃ and  ̃ are displacements occurring at the end of viscoelastic constraints.The matrix equation of the force balance for a structure with viscoelastic dampers can be written in the following form: where M and K denote mass and stiffness matrices of the plate, respectively; matrix C denotes the global plate damping matrix; f(t) is the vector of additional forces resulting from the viscoelastic supports; and p(t) is the excitation vector.An application of the Laplace transform with zero initial conditions leads to the following transform of Equation ( 4) where q(s) is the L-transform of q(t), p(s) is the L-transform of p(t) and the vector f(s) is expressed as In Equation (6), n d denotes the total number of dampers attached to the plate at selected nodes of a finite element mesh and L r is a global matrix indicating the location of the r-th damper within the plate domain.
A viscoelastic damper is described graphically in Figure 3, wherein k 0 , k 1 , c 1 , and α d describe material properties of viscoelastic constraints (damper); u is the force transferred by the damper; and q j and q k are displacements occurring at the end of viscoelastic constraints.Consequently, quantities appearing in Equation ( 6) can be expressed as follows: where  1 =  1  1 ⁄ is the quotient of the stiffness and damping coefficients of the -th damper element.
Next, let the excitation vector () to be equal to zero.Hence, the relation described Consequently, quantities appearing in Equation ( 6) can be expressed as follows: where ν 1j = k 1j /c 1j is the quotient of the stiffness and damping coefficients of the j-th damper element.
Next, let the excitation vector p(t) to be equal to zero.Hence, the relation described by Equation (4) takes the following form: Now, the Laplace transform of Equation ( 8) leads to the following relation: where Equation ( 9) describes a nonlinear eigenproblem that is solved for the eigenvalues s and the eigenvectors q(s) using the subspace iteration method coupled with the continuation method.The continuation method was comprehensively described by Lewandowski in [20].
The obtained eigenvalues of Equation ( 9) are complex numbers of the form s j = µ j + iη j .This is the basis for the j-th natural frequency ω j of the structure and the non-dimensional damping ratio γ j of the j-th mode of vibration: The subspace iteration method makes it possible to calculate the first few natural frequencies and the corresponding non-dimensional damping ratios without solving the entire nonlinear eigenproblem.A general description of the subspace iteration method is given below-cf.[21].
A nonlinear eigenproblem (Equation ( 9)) can be rewritten as where X = [q 1 , q 2 , . . . , and n denotes the number of degrees of freedom of the considered structure.The first step is to find a solution of a linear eigenproblem in the following form: The solution Equation ( 14) consists of n real eigenvalues and corresponding eigenvectors.Then, the first m modes are taken as the initial approximation: Materials 2023, 16, 7527 6 of 32 where s (0) j = 0 + iω j and q (0) j = x j + i0 for j = 1, 2, . . ., m.In subsequent loops of the subspace iteration method, first, the following equation is solved with respect to X k : where 1) and k is the number of iterations.Next, the reduced nonlinear eigenproblem has to be solved: where  17)) is solved by the continuation method [21], which is successfully used in the analysis of systems with viscoelastic elements [19,20].
A new approximation of eigenvectors of the origin nonlinear eigenproblem (Equation ( 9)) is computed as follows: where Z k is formed as m .The iterative process is completed when the following conditions are fulfilled: where ε 1 and ε 2 are the assumed accuracies of calculations.In the current work, the required accuracy for eigenvalues and eigenvectors is 10 −4 .As mentioned above, the time-fractional viscoelastic support description is considered.The one-dimensional constitutive equation for viscoelastic constraints is introduced to the analysis [32] where σ and ε are the stress and the strain, E 0 and E ∞ are the relaxed and non-relaxed elastic moduli, and τ is the relaxation time.
Compared with the classical Zener model of damper, in the model shown in Figure 3, the element with only damping properties is replaced by a viscoelastic element, also the so-called Scott-Blair element (in Figure 3 shown as a rhombus).It is described by two parameters: c 1 -damping coefficient and α d -order of the fractional derivative (0 < α d ≤ 1).
The constitutive equation for the Scott-Blair element can be written as where D α d t (•) denotes the fractional derivate of the order α d with respect to the time and ∆q(t) = q k − q j .In this paper, the Riemann-Liouville definition [22,32] is applied, which is mainly used for the description of viscoelastic dampers [18,23,24]:

Structural Response Recovery and the Probabilistic Analysis
The eigenfrequencies ω i of the plate under consideration have been all found via the polynomial basis Applying the least squares method (LSM) based on several FEM deterministic solutions for varying values of the Gaussian input uncertainty source is denoted by ν.Then, approximations of various orders are subjected to some optimization procedure, where the deterministic search method enables a choice of the optimal polynomial, which minimizes the mean square error and maximizes the correlation coefficient for the structural output and input.
The basic probabilistic characteristics such as expectations, coefficients of variation, skewness, and kurtosis have been estimated via application of integral definitions: Moreover, let R denote the admissible limit of the given structure, and E denote its extreme effort.The previous engineering designing codes allow us to make the following interpretation in case of the eigenfrequency and extreme excitation: a distance in-between them cannot be smaller than a quarter of this eigenfrequency.This is to avoid a structural resonance.Therefore, the satisfactory safety of the given system may be measured with the use of the following FORM reliability index [33]: where ω i stands for each next eigenfrequency.Quite a similar calculation can be provided with the use of a relative entropy H quantifying a distance in-between two random distributions, which can be calculated due to the Bhattacharyya [34,35] theory as The above relation can be rewritten as Then, the final safety measure compensable to β FORM is obtained here as

Numerical Examples
Four square plates will be analyzed: the first one with asymmetric boundary conditions and asymmetrically placed viscoelastic supports; the second one, which is simply supported on three adjacent edges, with the fourth edge resting on viscoelastic constraints; the third one, which is simply supported on two opposite edges with the other two resting on viscoelastic constraints; and the fourth one, with all edges resting on viscoelastic constraints.Random moments and reliability measures for the appropriate random variables will be estimated for all plates.

Plate 1
The square isotropic plate fixed along the right vertical edge and simply supported along the horizontal lower edge has been discretized using the 15 × 15 plate rectangular finite elements, whose material properties are E = 205 GPa, ν p = 0.3, and ρ p = 7850 kg/m 3 .The plate dimensions are equal to l x × l y × H = (2.0 × 2.0 × 0.01) [m ].The set of viscoelastic supports (dampers) have been attached along one plate edge (Figure 4).Structural damping is neglected.The first four modes have been analyzed.It was assumed that the required accuracy for eigenvalues and eigenvectors is 10 viscoelastic constraints.Random moments and reliability measures for the appropriate random variables will be estimated for all plates.

Plate 1
The square isotropic plate fixed along the right vertical edge and simply supported along the horizontal lower edge has been discretized using the 15 × 15 plate rectangular finite elements, whose material properties are E = 205 GPa, νp = 0.3, and ρp = 7850 kg/m 3 .The plate dimensions are equal to   ×   ×  = (2.0 × 2.0 × 0.01) [m].The set of viscoelastic supports (dampers) have been attached along one plate edge (Figure 4).Structural damping is neglected.The first four modes have been analyzed.It was assumed that the required accuracy for eigenvalues and eigenvectors is 10 −4 .  1 and 2, respectively.1 and 2, respectively.The sample polynomial response functions for the first natural frequency and coefficient of damping have been obtained as the third-order polynomials listed below: The expected value E(ω i ) and coefficient of variation α(ω i ) of the first two natural frequencies (i = 1, 2) have been presented all in turn in Figures 5 and 6, both as the functions of the input random modulus coefficient of variation.How it could be expected from the data collected in Table 1, an impact of the input uncertainty on the expectations of the first eigenfrequency, has a rather limited character.The output uncertainty, although linearly dependent upon the input one, is definitely damped by this system (almost two times).
The expected value E(γ i ) and coefficient of variation α(γ i ) of the first two natural frequencies (I = 1, 2) are presented in turn in Figures 7 and 8. Contrary to the previous two figures, the expected values increase together with the input uncertainty level.All three numerical methods coincide perfectly for the entire range of fluctuations of the input coefficient of variation.The sample polynomial response functions for the first natural frequency and coefficient of damping have been obtained as the third-order polynomials listed below: The expected value (  ) and coefficient of variation (  ) of the first two natural frequencies (i = 1, 2) have been presented all in turn in Figures 5 and 6, both as the functions of the input random modulus coefficient of variation.How it could be expected from the data collected in Table 1, an impact of the input uncertainty on the expectations of the first eigenfrequency, has a rather limited character.The output uncertainty, although linearly dependent upon the input one, is definitely damped by this system (almost two times).The expected value (  ) and coefficient of variation (  ) of the first two natural frequencies (i = 1, 2) are presented in turn in Figures 7 and 8. Contrary to the previous two figures, the expected values increase together with the input uncertainty level.All three numerical methods coincide perfectly for the entire range of fluctuations of the input coefficient of variation.The expected value (  ) and coefficient of variation (  ) of the first two natural frequencies (i = 1, 2) are presented in turn in Figures 7 and 8. Contrary to the previous two figures, the expected values increase together with the input uncertainty level.All three numerical methods coincide perfectly for the entire range of fluctuations of the input coefficient of variation.The expected value (  ) and coefficient of variation (  ) of the first two natural frequencies (i = 1, 2) are presented in turn in Figures 7 and 8. Contrary to the previous two figures, the expected values increase together with the input uncertainty level.All three numerical methods coincide perfectly for the entire range of fluctuations of the input coefficient of variation.Similarly, as above, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 3 and 4, respectively.
There is no doubt that Poisson's ratio uncertainty induces almost no uncertainty in neither the first natural frequency nor in the coefficients of damping.This is confirmed in Figures 9 and 10, where expected values E(ω i ) and coefficients α(ω i ) of the circular frequencies (i = 3) and E(γ i ), α(γ i ) are presented.Even with marginal resulting uncer- tainty, a coincidence of all three probabilistic numerical strategies is perfect for the first two moments.Random distribution of the damper's parameter k0 is considered further, and the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given in Tables 5 and 6 below.Is it clearly seen in any column that the impact of this parameter statistical scattering is negligible; the same holds true for the damper's parameter k1; see Tables 7 and 8.  Random distribution of the damper's parameter k 0 is considered further, and the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given in Tables 5 and 6 below.Is it clearly seen in any column that the impact of this parameter statistical scattering is negligible; the same holds true for the damper's parameter k 1 ; see Tables 7 and 8.A negligible sensitivity of the solution value to any change in the design parameter for the first four natural frequencies can be observed here, and therefore, the first two probabilistic moments: E(γ i ) and α(γ i ) (i = 1) have been presented in turn in Figure 11.A negligible sensitivity of the solution value to any change in the design parameter for the first four natural frequencies can be observed here, and therefore, the first two probabilistic moments: (  ) and (  ) (i = 1) have been presented in turn in Figure 11.Similarly, as in previous computations, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.In this case, the sensitivity of the first two natural frequencies and all first four coefficients of damping to a change in the c 1 parameter can be observed; the results are summarized in Tables 9 and 10, respectively.It is possible to observe the practical insensitivity of the first, third, and fourth natural frequencies to changes in the design parameter.The probabilistic moments E(ω 2 ), α(ω 2 ), E(γ 2 ), and α(γ 2 ) are presented illustratively in turn in Figures 12 and 13   Finally, the analysis of the influence of the random parameter αd on dynamic characteristics will be considered.The results are summarized in Tables 11 and 12, respectively.Finally, the analysis of the influence of the random parameter αd on dynamic characteristics will be considered.The results are summarized in Tables 11 and 12, respectively.Finally, the analysis of the influence of the random parameter α d on dynamic characteristics will be considered.The results are summarized in Tables 11 and 12, respectively.To illustrate the random behavior of the solutions, the selected probabilistic moments E(ω 2 ), α(ω 2 ), E(γ 2 ), and α(γ 2 ) are presented in turn in Figures 14 and 15.To illustrate the random behavior of the solutions, the selected probabilistic moments ( 2 ), ( 2 ), ( 2 ), and ( 2 ) are presented in turn in Figures 14 and 15.Probabilistic relative entropy H and the safety measure βH are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 16 and 17, respectively.To illustrate the random behavior of the solutions, the selected probabilistic moments ( 2 ), ( 2 ), ( 2 ), and ( 2 ) are presented in turn in Figures 14 and 15.Probabilistic relative entropy H and the safety measure βH are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 16 and 17, respectively.Probabilistic relative entropy H and the safety measure β H are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 16 and 17  Quite expectedly, both relative entropies and the following reliability indices decrease in an exponential way while increasing the input uncertainty, which is typical for the FORM-based indices in engineering applications.The plate is quite rationally designed, even with the viscoelastic supports, because even with extreme uncertainty in Young's modulus reliability index is much larger than the widely used minimum values close to the interval [3.5, 5.2].A little bit larger safety is noticed for the first eigenfrequency than for the second one.

Plate 2
The plate simply supported on two opposite edges with the other two resting on viscoelastic constraints is considered (Figure 18).The dimensions of the plate are identical to those adopted in the previous section.The influence of random physical and geometric parameters will be presented in a similar way as above, but it has been contrasted against the well-known analytical solution available in the literature.The first four eigenfrequencies determined with the mean value of Young's modulus equal here in turn ω = {76.3133, 190.7834, 190.7834, 305.2534} [rad/s].It is documented that an introduction of the nonlinear supports decreases almost twice the basic eigenfrequencies.This means that modeling errors consisting of an assumption that simple supports are assumed instead of real viscoelastic may lead to the structural danger of a resonance.Quite expectedly, both relative entropies and the following reliability indices decrease in an exponential way while increasing the input uncertainty, which is typical for the FORM-based indices in engineering applications.The plate is quite rationally designed, even with the viscoelastic supports, because even with extreme uncertainty in Young's modulus reliability index is much larger than the widely used minimum values close to the interval [3.5, 5.2].A little bit larger safety is noticed for the first eigenfrequency than for the second one.

Plate 2
The plate simply supported on two opposite edges with the other two resting on viscoelastic constraints is considered (Figure 18).The dimensions of the plate are identical to those adopted in the previous section.The influence of random physical and geometric parameters will be presented in a similar way as above, but it has been contrasted against the well-known analytical solution available in the literature.The first four eigenfrequencies determined with the mean value of Young's modulus equal here in turn ω = {76.3133, 190.7834, 190.7834, 305.2534} [rad/s].It is documented that an introduction of the nonlinear supports decreases almost twice the basic eigenfrequencies.This means that modeling errors consisting of an assumption that simple supports are assumed instead of real viscoelastic may lead to the structural danger of a resonance.Quite expectedly, both relative entropies and the following reliability indices decrease in an exponential way while increasing the input uncertainty, which is typical for the FORM-based indices in engineering applications.The plate is quite rationally designed, even with the viscoelastic supports, because even with extreme uncertainty in Young's modulus reliability index is much larger than the widely used minimum values close to the interval [3.5, 5.2].A little bit larger safety is noticed for the first eigenfrequency than for the second one.

Plate 2
The plate simply supported on two opposite edges with the other two resting on viscoelastic constraints is considered (Figure 18).The dimensions of the plate are identical to those adopted in the previous section.The influence of random physical and geometric parameters will be presented in a similar way as above, but it has been contrasted against the well-known analytical solution available in the literature.The first four eigenfrequencies determined with the mean value of Young's modulus equal here in turn ω = {76.3133, 190.7834, 190.7834, 305.2534} [rad/s].It is documented that an introduction of the nonlinear supports decreases almost twice the basic eigenfrequencies.This means that modeling errors consisting of an assumption that simple supports are assumed instead of real viscoelastic may lead to the structural danger of a resonance.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically as for the plate presented in the preceding section.Identically, as in the previous examples, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 13 and 14, respectively.Probabilistic characteristics, namely expected values and coefficients of variation, ( 1 ), ( 1 ), ( 2 ), and ( 2 ) are presented illustratively in turn in Figures 19 and 20.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically as for the plate presented in the preceding section.Identically, as in the previous examples, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 13 and 14, respectively.Probabilistic characteristics, namely expected values and coefficients of variation, E(ω 1 ), α(ω 1 ), E(γ 2 ), and α(γ 2 ) are presented illustratively in turn in Figures 19 and 20.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically as for the plate presented in the preceding section.Identically, as in the previous examples, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 13 and 14, respectively.Probabilistic characteristics, namely expected values and coefficients of variation, ( 1 ), ( 1 ), ( 2 ), and ( 2 ) are presented illustratively in turn in Figures 19 and 20.Table 14.The results for the first four coefficients of damping and the random distribution of Young's modulus.A set of deterministic solutions for the first four natural circular frequencies and coefficients of damping was determined.The results are summarized in Tables 15 and 16, respectively.A set of deterministic solutions for the first four natural circular frequencies and coefficients of damping was determined.The results are summarized in Tables 15 and 16, respectively.The probabilistic moments E(ω 3 ), α(ω 3 ), E(γ 3 ), and α(γ 3 ) are presented illustratively in turn in Figures 21 and 22.The probabilistic moments ( 3 ), ( 3 ), ( 3 ), and ( 3 ) are presented illustratively in turn in Figures 21 and 22.The probabilistic moments ( 3 ), ( 3 ), ( 3 ), and ( 3 ) are presented illustratively in turn in Figures 21 and 22.   Identically, as in the previous examples, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 17 and 18, respectively.One can observe here the practical insensitivity of the solution to the design parameter k 0 .Furthermore, deterministic solutions for the first four natural circular frequencies and coefficients of damping are given.The results are summarized in Tables 19 and 20, respectively.One can observe the practical insensitivity of the solution for circular frequencies to the design parameter k 1 .For illustration, the probabilistic moments E(γ 1 ) and α(γ 1 ) have been presented in Figure 23.Deterministic solutions series for the first four natural circular frequencies and coefficients of damping are given.The results are summarized in Tables 21 and 22, respectively.18.The results for the first four coefficients of damping and the random distribution of the damper's parameter k 0 .20.The results for the first four coefficients of damping and the random distribution of the damper's parameter k 1 .One can observe here the total insensitivity of the solution for circular frequencies to the design parameter c1.For illustration, the probabilistic moments ( 1 ) and ( 1 ) are presented in Figure 24.Determination of the response polynomial bases for the first four natural circular frequencies and coefficients of damping is given using the data collected in Tables 23 and 24, respectively.One can observe here the total insensitivity of the solution for circular frequencies to the design parameter c 1 .For illustration, the probabilistic moments E(γ 1 ) and α(γ 1 ) are presented in Figure 24.Determination of the response polynomial bases for the first four natural circular frequencies and coefficients of damping is given using the data collected in Tables 23 and 24, respectively.
Probabilistic relative entropy H and the safety measure β H are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and (28), and presented in Figures 27 and 28, respectively.For an illustration of the structural random behavior, the probabilistic moments ( 2 ), ( 2 ), ( 2 ), and ( 2 ) are presented in Figures 25 and 26.Probabilistic relative entropy H and the safety measure βH are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and (28), and presented in Figures 27 and 28, respectively.Probabilistic relative entropy H and the safety measure βH are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) for selected random parameters according to the relations Equations ( 27) and (28), and presented in Figures 27 and 28, respectively.Probabilistic relative entropy H and the safety measure βH are estimated for the semianalytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and (28), and presented in Figures 27 and 28, respectively.Analogous observations hold true for Bhattacharyya entropy and the resulting reliability index as for the previous plate-both probabilistic methods return almost the same results-but now, both eigenfrequencies return almost the same target values and safety margin for the designed structure.Analogous observations hold true for Bhattacharyya entropy and the resulting reliability index as for the previous plate-both probabilistic methods return almost the same results-but now, both eigenfrequencies return almost the same target values and safety margin for the designed structure.

Plate 3
The square isotropic plate with all edges resting on viscoelastic constraints placed along all edges is considered (Figure 29).The influence of random physical and geometric parameters will be presented in similar way as above.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically for the plate presented in Section 4.1.For this plate, the influence of material parameters is expressed by Young's modulus E, Poisson's ratio v, and the parameter αd.Identically, as in the examples presented previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 25 and 26, respectively.
To illustrate the structural random behavior of a structure, the probabilistic moments ( 3 ), ( 3 ), ( 3 ), and ( 3 ) are presented in Figures 30 and 31.Quite typically, for marginal resulting uncertainty in this problem, the expected values slightly decrease together with an increase in the input coefficient of variation, whereas the coefficient of variation of the eigenfrequencies increases in a linear manner.

Plate 3
The square isotropic plate with all edges resting on viscoelastic constraints placed along all edges is considered (Figure 29).The influence of random physical and geometric parameters will be presented in a similar way as above.Analogous observations hold true for Bhattacharyya entropy and the resulting reliability index as for the previous plate-both probabilistic methods return almost the same results-but now, both eigenfrequencies return almost the same target values and safety margin for the designed structure.

Plate 3
The square isotropic plate with all edges resting on viscoelastic constraints placed along all edges is considered (Figure 29).The influence of random physical and geometric parameters will be presented in a similar way as above.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically as for the plate presented in Section 4.1.For this plate, the influence of material parameters is expressed by Young's modulus E, Poisson's ratio v, and the parameter αd.Identically, as in the examples presented previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 25 and 26, respectively.
To illustrate the structural random behavior of a structure, the probabilistic moments ( 3 ), ( 3 ), ( 3 ), and ( 3 ) are presented in Figures 30 and 31.Quite typically, for marginal resulting uncertainty in this problem, the expected values slightly decrease together with an increase in the input coefficient of variation, whereas the coefficient of variation of the eigenfrequencies increases in a linear manner.Probabilistic computations are carried out for the material properties of the plate and parameters describing viscoelastic supports whose values change randomly identically as for the plate presented in Section 4.1.For this plate, the influence of material parameters is expressed by Young's modulus E, Poisson's ratio v, and the parameter α d .Identically, as in the examples presented previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 25 and 26, respectively.To illustrate the structural random behavior of a structure, the probabilistic moments E(ω 3 ), α(ω 3 ), E(γ 3 ), and α(γ 3 ) are presented in Figures 30 and 31.Quite typically, for marginal resulting uncertainty in this problem, the expected values slightly decrease together with an increase in the input coefficient of variation, whereas the coefficient of variation of the eigenfrequencies increases in a linear manner.Probabilistic moments ( 3 ) , ( 3 ) , ( 3 ) , and ( 3 ) following polynomial response functions approximated with the least squares method from the data contained in Tables 27 and 28 are presented in Figures 32 and 33       Probabilistic moments ( 3 ) , ( 3 ) , ( 3 ) , and ( 3 ) following polynomial response functions approximated with the least squares method from the data contained in Tables 27 and 28 are presented in Figures 32 and 33   Probabilistic moments E(ω 3 ), α(ω 3 ), E(γ 3 ), and α(γ 3 ) following polynomial re- sponse functions approximated with the least squares method from the data contained in Tables 27 and 28 are presented in Figures 32 and 33       The same as previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 29 and 30, respectively.Now, the sensitivity of the eigenfrequencies is definitely higher than before for the Poisson ratio.To illustrate the structural random behavior of a structure, the preselected probabilistic moments ( 1 ) , ( 1 ) , ( 1 ) , and ( 1 ) are      The same as previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 29 and 30, respectively.Now, the sensitivity of the eigenfrequencies is definitely higher than before for the Poisson ratio.To illustrate the structural random behavior of a structure, the preselected probabilistic moments ( 1 ) , ( 1 ) , ( 1 ) , and ( 1 ) are The same as previously, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given.The results are summarized in Tables 29 and 30, respectively.Now, the sensitivity of the eigenfrequencies is definitely higher than before for the Poisson ratio.To illustrate the structural random behavior of a structure, the preselected probabilistic moments E(ω 1 ), α(ω 1 ), E(γ 1 ), and α(γ 1 ) are pre- sented in Figures 34 and 35.Contrary to Figures 32 and 33, fluctuations of the expected values with respect to the input uncertainty are remarkable, whereas the resulting uncertainty level is a little bit larger than the input one.Therefore, the damper material has remarkable importance in safe design and in reliability analysis or prediction for thin plates with viscoelastic supports.
Similarly, as in the previous examples, the probabilistic relative entropy H and the safety measure β H are estimated for the semi-analytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 36 and 37, respectively.Similarly, as in the previous examples, the probabilistic relative entropy H and the safety measure βH are estimated for the semi-analytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 36 and 37, respectively.Similarly, as in the previous examples, the probabilistic relative entropy H and the safety measure βH are estimated for the semi-analytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 36 and 37, respectively.Similarly, as in the previous examples, the probabilistic relative entropy H and the safety measure βH are estimated for the semi-analytical (SAM) and the stochastic perturbation technique (SPT) approaches for selected random parameters according to the relations Equations ( 27) and ( 28), and presented in Figures 36 and 37, respectively.Now, the probabilistic entropy and the reliability index are defined on their basis, although they have quite typical distributions widely seen in the reliability theory.Nevertheless, their values are so small that even for minimum input uncertainty in the damper's material parameter α d , these values are remarkably smaller than the values recommended in the engineering design as the admissible minimum.This once more confirms the paramount importance of this specific design parameter in the rational design of the plates under consideration.

1.
Fundamental eigenfrequencies of the Kirchhoff rectangular plates with various boundary conditions, including some viscoelastic supports, were studied in this paper.It was documented that viscoelastic supports dramatically decrease these eigenfrequencies with respect to plates having classical supports.The highest sensitivity of the solutions with respect to the viscoelastic support coefficients are documented; material parameters of the plate have remarkably smaller effect on the dynamic characteristics of the same plates.Therefore, uncertainty in the viscoelastic supports of the plates is decisive for their safety, reliability, and durability.The significance of viscoelasticity in supporting systems is so huge that all plates' eigenfrequencies are completely insensitive to their elastic parameters, unlike in the classical elasticity.It can also be noticed that the use of non-integer order calculus allows for a good approximation of the description of the behavior of the given constraints, but to solve the nonlinear eigenproblem, it is necessary to use the so-called continuation method.Hence, here, the approach with state variables by definition cannot be used.It gains remarkable importance in structural problems with a relatively large number of degrees of freedom.

2.
Theoretical and computational studies presented in this work clearly show that the common application of three probabilistic approaches, namely the semi-analytical (SAM), perturbation-based (SPT), and Monte-Carlo simulation technique (MCS) with continuation methods, allows for accurate determination of the probabilistic coefficients of free vibrations in the presence of input Gaussian In most cases, very good agreement in-between these three methods was noticed.Solving these problems consists primarily of determining the expected values and coefficients of variation for the circular natural frequencies along with the corresponding values of damping coefficients.Additionally, higher-order random moments are determined in selected problems, i.e., skewness and kurtosis, to verify if the desired output may have Gaussian PDF and ifverification is negative here.It must be mentioned that the third-degree response polynomials have been detected in all case studies as the most optimal approximation in-between structural output and input.This causes a maximum correlation coefficient and minimizes the mean square error of such a polynomial approximation.

3.
From the engineering point of view, an insertion of further flexible nodes significantly reduces the value of the basic dynamic characteristics, i.e., the natural frequencies of vibrations, compared to the system in which such constraints do not exist.The value of the first circular frequency of the third plate is more than twice as small as the corresponding value for the previous plate-Tables 13 and 25.As documented here, a change in the design parameter does not always result in a significant change in the solution, e.g., the results for the first four natural frequencies for the random distribution of the damper's parameter k 0 .The reliability measures expressed by the probabilistic relative entropy H and coefficient β H show that their values are decreasing deeply as the value of the uncertainty coefficient increases.The simplest random approach is the semi-analytical one, and it allows us to derive random moments in analytical terms.This is relatively easy for future implementations in any academic and commercial finite element or other kind of computational programs.

Figure 1 .
Figure 1.Finite element used for discretization of tested plate: node numbering and active degrees of freedom.

Figure 1 .
Figure 1.Finite element used for discretization of tested plate: node numbering and active degrees of freedom.

Figure 2 .
Figure 2. Plate discretization with rectangular plate finite elements.

Figure 2 .
Figure 2. Plate discretization with rectangular plate finite elements.

Figure 3 .
Figure 3.The model of viscoelastic damper.

Figure 4 .
Figure 4. Finite element mesh of the square isotropic plate simply supported on three adjacent edges, with the fourth edge resting on viscoelastic constraints.Probabilistic computations are carried out for material properties of the plate and parameters describing viscoelastic supports whose values change randomly.The plate Young's modulus ranges from 180 to 230 kN/m 2 with an increase of 5 kN/m 2 , and the plate Poisson's ratio varies from 0.25 to 0.35 with an increase of 0.01.The parameters describing the viscoelastic constraints change as follows: k0 ranges from 95 to 125 N/m with the increase of 3 N/m, k1 ranges from 17,500 to 22,500 N/m with the increase of 500 N/m, c1 ranges from 205 to 255 N•s/m with the increase of 5 N•s/m and finally αd ranges from 0.5 to 0.7 with the increase of 0.02.At the very beginning, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given; the results are summarized in Tables1 and 2, respectively.

Figure 4 .
Figure 4. Finite element mesh of the square isotropic plate simply supported on three adjacent edges, with the fourth edge resting on viscoelastic constraints.Probabilistic computations are carried out for material properties of the plate and parameters describing viscoelastic supports whose values change randomly.The plate Young's modulus ranges from 180 to 230 kN/m 2 with an increase of 5 kN/m 2 , and the plate Poisson's ratio varies from 0.25 to 0.35 with an increase of 0.01.The parameters describing the viscoelastic constraints change as follows: k 0 ranges from 95 to 125 N/m with the increase of 3 N/m, k 1 ranges from 17,500 to 22,500 N/m with the increase of 500 N/m, c 1 ranges from 205 to 255 N•s/m with the increase of 5 N•s/m and finally α d ranges from 0.5 to 0.7 with the increase of 0.02.At the very beginning, the set of deterministic solutions for the first four natural circular frequencies and coefficients of damping is given; the results are summarized in Tables1 and 2, respectively.

Figure 5 .
Figure 5. Probabilistic moments for the first natural frequency and the random distribution of Young's modulus.

Figure 5 .
Figure 5. Probabilistic moments for the first natural frequency and the random distribution of Young's modulus.

Figure 6 .
Figure 6.Probabilistic moments for the second natural frequency and the random distribution of Young's modulus.

Figure 7 .
Figure 7. Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 8 .
Figure 8. Probabilistic moments for the second coefficient of damping and the random distribution of Young's modulus.

Figure 6 .
Figure 6.Probabilistic moments for the second natural frequency and the random distribution of Young's modulus.

Figure 6 .
Figure 6.Probabilistic moments for the second natural frequency and the random distribution of Young's modulus.

Figure 7 .
Figure 7. Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 8 .
Figure 8. Probabilistic moments for the second coefficient of damping and the random distribution of Young's modulus.

Figure 7 .
Figure 7. Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 6 .
Figure 6.Probabilistic moments for the second natural frequency and the random distribution of Young's modulus.

Figure 7 .
Figure 7. Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 8 .
Figure 8. Probabilistic moments for the second coefficient of damping and the random distribution of Young's modulus.

Figure 8 .
Figure 8. Probabilistic moments for the second coefficient of damping and the random distribution of Young's modulus.

Figure 9 .
Figure 9. Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 9 .
Figure 9. Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 9 .
Figure 9. Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 10 .
Figure 10.Probabilistic moments for the first coefficient of damping and the random distribution of Poisson's ratio.

Figure 10 .
Figure 10.Probabilistic moments for the first coefficient of damping and the random distribution of Poisson's ratio.

Figure 11 .
Figure 11.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter k1.

Figure 11 .
Figure 11.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter k 1 . .

Materials 2023 , 32 Figure 12 .
Figure 12.Probabilistic moments for the second natural frequency and the random distribution of the damper's parameter c1.

Figure 13 .
Figure 13.Probabilistic moments for the coefficient of damping and the random distribution of the damper's parameter c1.

Figure 12 . 32 Figure 12 .
Figure 12.Probabilistic moments for the second natural frequency and the random distribution of the damper's parameter c 1 .

Figure 13 .
Figure 13.Probabilistic moments for the coefficient of damping and the random distribution of the damper's parameter c1.

Figure 13 .
Figure 13.Probabilistic moments for the coefficient of damping and the random distribution of the damper's parameter c 1 .

Figure 14 .
Figure 14.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter αd.

Figure 15 .
Figure 15.Probabilistic moments for the coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 14 .
Figure 14.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter α d .

Figure 14 .
Figure 14.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter αd.

Figure 15 .
Figure 15.Probabilistic moments for the coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 15 .
Figure 15.Probabilistic moments for the coefficient of damping and the random distribution of the damper's material parameter α d .

32 Figure 16 .
Figure 16.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 17 .
Figure 17.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 16 .
Figure 16.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 16 .
Figure 16.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 17 .
Figure 17.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 17 .
Figure 17.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 18 .
Figure 18.Finite element mesh of the square isotropic plate simply supported on two opposite edges, with the remaining edges resting on viscoelastic constraints.

Figure 19 .
Figure 19.Probabilistic moments for the first natural frequency and the random distribution of Young's modulus.

Figure 18 .
Figure 18.Finite element mesh of the square isotropic plate simply supported on two opposite edges, with the remaining edges resting on viscoelastic constraints.

Figure 18 .
Figure 18.Finite element mesh of the square isotropic plate simply supported on two opposite edges, with the remaining edges resting on viscoelastic constraints.

Figure 19 .
Figure 19.Probabilistic moments for the first natural frequency and the random distribution of Young's modulus.

Figure 19 .
Figure 19.Probabilistic moments for the first natural frequency and the random distribution of Young's modulus.

Figure 20 .
Figure 20.Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 20 .
Figure 20.Probabilistic moments for the first coefficient of damping and the random distribution of Young's modulus.

Figure 21 .
Figure 21.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 22 .
Figure 22.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 21 .
Figure 21.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 21 .
Figure 21.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 22 .
Figure 22.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 22 .
Figure 22.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 23 .
Figure 23.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter k1.

Figure 23 .
Figure 23.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter k 1 .

Figure 24 .
Figure 24.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter c1.

Figure 24 .
Figure 24.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's parameter c 1 .

Figure 25 .
Figure 25.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter αd.

Figure 26 .
Figure 26.Probabilistic moments for the second coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 27 .
Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 25 . 32 Figure 25 .
Figure 25.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter α d .

Figure 26 .
Figure 26.Probabilistic moments for the second coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 27 .
Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 26 .
Figure 26.Probabilistic moments for the second coefficient of damping and the random distribution of the damper's material parameter α d .

Figure 25 .
Figure 25.Probabilistic moments for the second natural frequency and the random distribution of the damper's material parameter αd.

Figure 26 .
Figure 26.Probabilistic moments for the second coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 27 .
Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 27 .
Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.Figure 27.The probabilistic relative entropy for the random distribution of Young's modulus.

Figure 28 .
Figure 28.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 29 .
Figure 29.Finite element mesh of the square isotropic plate resting on viscoelastic constraints placed along all edges.

Figure 28 .
Figure 28.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 28 .
Figure 28.The probabilistic safety measure for the random distribution of Young's modulus.

Figure 29 .
Figure 29.Finite element mesh of the square isotropic plate resting on viscoelastic constraints placed along all edges.

Figure 29 .
Figure 29.Finite element mesh of the square isotropic plate resting on viscoelastic constraints placed along all edges.

Figure 30 .
Figure 30.Probabilistic moments for the third natural frequency and the random distribution of Young's modulus.

Figure 31 .
Figure 31.Probabilistic moments for the third coefficient of damping and the random distribution of Young's modulus. below.

Figure 30 .
Figure 30.Probabilistic moments for the third natural frequency and the random distribution of Young's modulus.

Figure 30 .
Figure 30.Probabilistic moments for the third natural frequency and the random distribution of Young's modulus.

Figure 31 .
Figure 31.Probabilistic moments for the third coefficient of damping and the random distribution of Young's modulus. below.

Figure 31 .
Figure 31.Probabilistic moments for the third coefficient of damping and the random distribution of Young's modulus. below.

Figure 32 .
Figure 32.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 33 .
Figure 33.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 32 .
Figure 32.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 32 .
Figure 32.Probabilistic moments for the third natural frequency and the random distribution of Poisson's ratio.

Figure 33 .
Figure 33.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 33 .
Figure 33.Probabilistic moments for the third coefficient of damping and the random distribution of Poisson's ratio.

Figure 34 .
Figure 34.Probabilistic moments for the first natural frequency and the random distribution of the damper's material parameter αd.

Figure 34 .
Figure 34.Probabilistic moments for the first natural frequency and the random distribution of the damper's material parameter α d .

Figure 35 .
Figure 35.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 36 .
Figure 36.The probabilistic relative entropy for the random distribution of the damper's material parameter αd.

Figure 37 .
Figure 37.The probabilistic safety measure for the random distribution of the damper's material parameter αd.Now, the probabilistic entropy and the reliability index are defined on their basis, although they have quite typical distributions widely seen in the reliability theory.

Figure 35 .
Figure 35.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's material parameter α d .

Figure 35 .
Figure 35.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 36 .
Figure 36.The probabilistic relative entropy for the random distribution of the damper's material parameter αd.

Figure 37 .
Figure 37.The probabilistic safety measure for the random distribution of the damper's material parameter αd.Now, the probabilistic entropy and the reliability index are defined on their basis, although they have quite typical distributions widely seen in the reliability theory.

Figure 36 .
Figure 36.The probabilistic relative entropy for the random distribution of the damper's material parameter α d .

Figure 35 .
Figure 35.Probabilistic moments for the first coefficient of damping and the random distribution of the damper's material parameter αd.

Figure 36 .
Figure 36.The probabilistic relative entropy for the random distribution of the damper's material parameter αd.

Figure 37 .
Figure 37.The probabilistic safety measure for the random distribution of the damper's material parameter αd.Now, the probabilistic entropy and the reliability index are defined on their basis, although they have quite typical distributions widely seen in the reliability theory.

Figure 37 .
Figure 37.The probabilistic safety measure for the random distribution of the damper's material parameter α d .

Table 1 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 2 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 2 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 3 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 4 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 5 .
The results for the first four natural frequencies and the random distribution of damper's parameter k0.

Table 6 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter k0.

Table 5 .
The results for the first four natural frequencies and the random distribution of damper's parameter k 0 .

Table 6 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter k 0 .

Table 7 .
The results for the first four natural frequencies and the random distribution of the damper's parameter k 1 .

Table 8 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter k 1 .

Table 7 .
The results for the first four natural frequencies and the random distribution of the damper's parameter k1.

Table 8 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter k1.

Table 9 .
The results for the first four natural frequencies and random distribution of the damper's parameter c 1 .

Table 10 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter c 1 .

Table 11 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter αd.

Table 12 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter αd.

Table 11 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter αd.

Table 12 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter αd.

Table 11 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter α d .

Table 12 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter α d .

Table 13 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 14 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 13 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 13 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 14 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 15 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 15 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 16 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 16 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 16 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 17 .
The results for the first four natural frequencies and the random distribution of the damper's parameter k 0 .

Table 19 .
The results for the first four natural frequencies and the random distribution of the damper's parameter k 1 .

Table 21 .
The results for the first four natural frequencies and the random distribution of the damper's parameter c1.

Table 22 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter c1.

Table 21 .
The results for the first four natural frequencies and the random distribution of the damper's parameter c 1 .

Table 22 .
The results for the first four coefficients of damping and the random distribution of the damper's parameter c 1 .

Table 23 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter αd.

Table 24 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter αd.

Table 23 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter α d .

Table 24 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter α d .

Table 25 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 26 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 25 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 26 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 25 .
The results for the first four natural frequencies and the random distribution of Young's modulus.

Table 26 .
The results for the first four coefficients of damping and the random distribution of Young's modulus.

Table 27 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 28 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 27 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 28 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 27 .
The results for the first four natural frequencies and the random distribution of Poisson's ratio.

Table 28 .
The results for the first four coefficients of damping and the random distribution of Poisson's ratio.

Table 29 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter α d .

Table 30 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter α d .

γ 1 [-] γ 2 [-] γ 3 [-] γ 4 [-]
Figures 34 and 35.Contrary to Figures32 and 33, fluctuations of the expected values with respect to the input uncertainty are remarkable, whereas the resulting uncertainty level is a little bit larger than the input one.Therefore, the damper material has remarkable importance in safe design and in reliability analysis or prediction for thin plates with viscoelastic supports.

Table 29 .
The results for the first four natural frequencies and the random distribution of the damper's material parameter αd.

Table 30 .
The results for the first four coefficients of damping and the random distribution of the damper's material parameter αd.