Sensitivity Analysis of Vibrating Laminated Composite Rectangular Plates in Interaction with Inviscid Fluid Using EFAST Method

K. Khorshidi^*, M. Taheri, M. Ghasemi

 Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 38156-88349, Iran.

1.        Introduction

The extent to which a fluid can play a role in vibrational behavior of a structure continues to be an issue of interest to researchers. The existence of fluid around a structure changes the kinetic energy of the system and influences the natural frequencies and mode shapes[1, 2]. Hence, knowing about the dynamic response of the structure in contact with the fluid is necessary for various areas of engineering such as submarines, shipbuilding, nuclear, hydrodynamics and ocean engineering. Commonly, to analyze the frequency response of a structure in contact with a fluid is generally known as fluid-structure interaction (FSI). Widely analyzing FSI problems can be categorized into three methods: firstly, numerical methods such as the boundary element method and secondly, the fluid finite element method which needs a large number of computations and can be applied to numerous FSI problems [3-19]. In addition, sensitivity analysis (SA) investigates how the variation in the output of a numerical model can be attributed to variations of its input factors [20]. Sensitivity analysis is increasingly being used in environmental modeling for a variety of purposes, including uncertainty assessment, model calibration and diagnostic evaluation, dominant control analysis, and robust decision-making. Furthermore, sensitivity analysis is increase also being used in statics and dynamics modeling for a variety of scientific and engineering purposes, including vibrational behavior of structures, control analysis of systems, evaluation of material strengths in different conditions, etc. Numerous studies have been performed to investigate the sensitivity analysis of vibrating structures. Khedmati et al. [21] studied the sensitivity analysis of elastic buckling of a cracked plate with simply supported plates, subjected to an axial compressive edge load using the finite element method. Afonso and Hinton [22] used an automated approach to carry out sensitivity analysis and to obtain optimum shapes for plates and shells in which the natural frequencies were maximized. Akoussan et al. [23] proposed a high order continuous sensitivity analysis of the damping properties of viscoelastic composite plates according to their layers thicknesses. Chen and Tan[24], Fung and Chen [25], and Chen and Hsu [26], employing Galerkin and the Runge–Kutta method, presented the imperfection sensitivity of nonlinear vibration of a simply supported ceramic/metal functionally graded plate in a general state of arbitrary initial stresses. Łasecka-Plura and Lewandowski [27], using Euler-Bernoulli theory considered the sensitivity analysis of dynamic characteristics of composite beams with viscoelastic layers. In this study, the fractional Zener model was used to express the viscoelastic material properties.in another study, Kotełko et al. [28], presented a sensitivity analysis of thin-walled box-section girders subjected to pure bending based on the methodology of the Monte-Carlo method. Lima et al. [29], also developed the formulation of first-order sensitivity analysis of complex frequency response functions (FRFs) for composite sandwich plates composed by a combination of fiber-reinforced and elastomeric viscoelastic layers, in arrangements that were frequently used for noise and vibration attenuation. Research by Takezawa and Kitamura [30] studied the sensitivity analysis of objective functions including the eigenmodes of continuum systems using scalar Helmholtz equations. Additionally, Li et al. [31] proposed the sensitivity-analysis of vibro-acoustic systems by using the interval perturbation method compared with the Monte Carlo method. Li and Liu [32], developed a three-dimensional semi-analytical model for the static response and sensitivity analysis of the composite stiffened laminated plate with interfacial imperfections. Li et al. [33], investigated the free vibration analysis and eigenvalues sensitivity analysis of composite laminates with interfacial imperfection based on the radial point interpolation method (RPIM) in the Hamilton system. Liu [34], described an analytical method to calculate the sensitivity of the frequencies and modes with respect to fiber Vol. fractions and orientations, for the large-scale composite laminated structures with complex boundaries. Furthermore, Hu et al. [35], studied an explicit time-domain method for sensitivity analysis of structural responses under non-stationary random excitations. Yan and Cheng [36], analyzed the sensitivity of residual vibrations of structures subject to impacts using an adjoint method. In other publications, Choi and Byun [37], presented an effective sensitivity analysis algorithm for free vibration of a rectangular plate structure by using the finite element-transfer stiffness coefficient method. Liu and Paavola [38] also described a general analytical sensitivity analysis method for the composite laminated plates and shells, which was applied to both classical and first-order shear deformation theories and based on the finite element methods. Li et al. [39], studied the linear statics and free vibration sensitivity analysis of the composite sandwich plates based on a layerwise/solid-element method (LW/SE).

The main purpose of this paper is to present an analytical model for sensitivity analysis of vibrating laminated composite rectangular plates in interaction with inviscid fluid using modified higher-order shear deformation plate theory. Various plate theories are used to model the structure. Governing equations are derived using Hamilton’s principle and solved with the Galerkin method. Using the EFAST method sensitivity of fundamental natural frequency is obtained. After validation, influences of the aspect ratio, thickness ratio and material orthotropy orientation of the plate, depth ratio and width of the fluid on the wet fundamental natural frequencies of the plate are illustrated.

2.        Constitutive equation and modified shear deformation plate theory

Consider a laminated composite rectangular plate with length , width , total thickness , and  elastic orthotropic layers, which is a part of the vertical side of a bounded rigid tank filled with a fluid, as shown in Fig. 1. The tank contains fluid that has width , depth  and mass density of fluid . The fluid is considered incompressible, inviscid and irrotational. A Cartesian coordinate system is used to describe governing equations. The coordinate system is placed so that the origin is located in the corner of the studied plate on its middle surface, while axes  and lie on the plate’s edges and axis z is perpendicular to the middle plane.

Fig. 1. Plate-fluid interaction, Coordinates and dimensions

where superscript (k) refers to the k-th layer within a laminate,  and  are the normal stresses and strains, respectively,  and  are the shear stresses and strains, respectively, and s are transformed material constants and defined by

in which:

where  is the lamina material orthotropy orientation, G₁₂, G₁₃ and G₂₃ are the shear moduli in 1-2, 1-3 and 2-3 directions, respectively, and the coefficients c_ij are given by

Equation (1) is obtained (i) under the transverse isotropy assumption with respect to planes parallel to the 2-3 plane, i.e., assuming fibers in the direction parallel to axis 1, so that E₂ = E₃, G₁₂ = G₁₃ and n₁₂ = n₁₃, and (ii) solving the constitutive equations for e_zz as function of e_xxand e_yy and then eliminating it. Three independent displacements variables u₀, v₀ and w₀ in x, y and z directions, respectively, are used to describe middle surface deformations of the plate. The displacements , ,  of a generic point of the plate at distance  from the  plane (see Fig.1) are related to the middle surface displacements u₀, v₀ and w₀ by:

where  and  are the rotations of the transverse normals about the y and x axes, respectively, and  for exponential, trigonometric and hyperbolic shear deformation plate theories are given by

The linear strain-displacement equations for the modify higher order shear deformation plate theory are given by

The elastic strain  and kinetic  energies of the plate, including rotary inertia, are given by

where  is the mass density of the k-th layer of the plate and the overdot denotes time derivative. The total kinetic energy of the fluid with respect to the bulging modes of the plate and the fluid sloshing can be written[1, 2]

where  and  are fluid potential velocity relating to bulging modes and sloshing modes, respectively.

where  is the unknown coefficients related to fluid sloshing;  and  express the number of necessary terms to get desirable accuracy. Furthermore,  are Fourier coefficients associated with the bulging modes

, ,  and  are nonnegative integers. Moreover, Using the assumption of linearized sloshing [1, 2] at the free surface of fluid leads to

In which g is the acceleration of gravity and  is the wet natural frequency of plate-fluid interaction.

3.     Governing equations of fluid-structure

Hamilton’s principle is employed to obtain the governing equations and associated boundary conditions of the system as follows

In which  is variation operator. Now by inserting Eqs. (17), (18), (19) and (20) into Eq. (25) and then integrating by parts and setting the coefficients of , , ,  and  to zero, boundary conditions and governing equations in terms of stress resultants can be derived as follows

where

4.     Solution procedure using Galerkin approach

Governing equation systems (26)-(30) cannot be solved unless unknown coefficients  are determined. Hence, linear sloshing equation (31) must be added to Eqs. (26)-(30) as the seventh equation. For the sake of simplicity, the left-hand side of Eqs. (26)-(31) are denoted as  (i=1,2,..6) respectively. So applying the Galerkin method to Eqs. (26)-(31) leads to

According to the Galerkin method, unknown functions of , , ,  and  propose in the following form

where  and  indicate truncated orders of series. , , ,  and  are undetermined coefficients which are calculated after minimizing residuals. , , ,  and  are independent functions which satisfy at least essential boundary conditions. For simply supported boundary conditions , , ,  and  can be written as

m and n are numbers of half-waves along x- and y- axis, respectively. After solving the above equations, one can find natural frequencies and mode shapes which are related to the eigenvalues and eigenfunctions of the system.

5.     Sensitivity analysis

One of the parts of general sensitivity analysis techniques that has attracted more attention is the variance-based techniques. In these techniques, the sensitivity index is computed as the share of each parameter in the overall output variance of the model. The general sensitivity analysis techniques are implemented in four steps: (1) defining the inputs and the type of distribution of each input, (2) generating the samples for the input values, (3) computing the model’s output for each set of input samples and (4) determining the effect of each input factor on the output [40]. In this section, the variance-based sensitivity analysis techniques have been reviewed. The variance-based general sensitivity analysis approaches can be used to obtain the first-order effect and the second-order effect (which include the interaction between other parameters) [41].

Moreover, the Sobol method [42] is a model-independent general sensitivity analysis technique which is based on variance analysis. This method can be used for nonlinear and non-uniform functions and models. For the model defined by function , where  is the model output and  is the vector of input parameters, Sobol’ suggested to decompose the function f into summands of increasing dimensionality, where the integral of each term over its own input variables is zero. Sobol showed that, when all the inputs were perpendicular to one another, this resolution was unique and the output variance of the model (V) was the set of variances of each resolved term [42]:

In Eq. (77),  denotes the first-order effect for each input factor  and  to indicate the interactions between n factors. Therefore, the shares allocated to parameters, and the interactions of parameters can be determined from the total output variance. The sensitivity index is obtained as the ratio of each order’s variance to the total variance variance ( denotes the first-order sensitivity index,  represents the second-order sensitivity index, and so on). The total sensitivity index (i.e., the overall effect of each parameter) is obtained as the summand of all the orders of sensitivity index for that parameter [42]:

The general equation has been presented by Sobol in 1990, which will be given as follows [43]:

f₀ can be defined as:

Sobol showed that all the terms of f(X) could be calculated through multiple integrals:

Sensitivity index based on variance can be obtained as follows:

In which D is the variance of .

Sensitivity value can be obtained by dividing of every group of variables variance to total variance.

It should be noted that any professional and official titles or academic degrees (Dr, chief, sir) must not show in Author details. In addition, first name should be written in abbreviation and family name in complete form.

 is the first order sensitivity index for factor which presents the sensitivity effect of  parameter on output and is the second order sensitivity index which shows the interaction effect on total variance.

Total sensitivity index is a summation of all the sensitivity indices.

The EFAST method was presented by Cukier et al.[44] and was later improved by Saltelli et al. [45] Like the Sobol method, this approach is also based on variance and it is independent of any assumption of linearity and uniformity between inputs and output(s). Contrary to the Sobol method, which uses multidimensional integrals to obtain the total variance and the partial variances, this method converts the multidimensional integrals to one-dimensional ones by defining a transfer function and simplifies the procedure for the calculation of sensitivity indexes.

The EFAST method searches the n-dimensional space of the input factors (Unit Hypercube ) by using a Search Curve defined by a set of parametric equations [45]:

Where  is the frequency related to factor , s is a variable that changes from –π to +π, and  specifies the starting point of the curve. The output variance of the model is approximated by means of Fourier analysis:

In the above equation , G(s) are the transfer functions, and  and  are the Fourier coefficients . By calculating the Fourier coefficients for the basic frequency  and its higher harmonics , the partial first-order input variance  can be obtained.

Also, like the Sobol method, the ratio of the first-order partial variance to total variance is used to compute the main sensitivity index. The total sensitivity index is obtained from Eq. (82)[46]:

variance  is obtained by changing all the parameters except parameter .

The Sobol method employs the Monte Carlo integral to obtain each partial variance; and in comparison, with the EFAST method, it does not use a transfer function; that is why, it has a low computational efficiency.

6.     Numerical result

In this section, the wet natural frequency parameters are obtained from the Galerkin method for the free vibrations of a laminated composite rectangular plate interacting with the bounded fluid. Numerical results have been performed for simply supported boundary conditions. Materials chosen for the plate is graphite/epoxy with the material properties; Poisson’s ratio , , ,    and . Calculations have been performed for 4 layers symmetric laminated plates ( / / / ). Presented results are obtained for the thickness ratios , aspect ratios , material orthotropy orientation , depth of the fluid ratios  and width of the tank . Also, fluid density is considered as .

The comparison of the fundamental wet natural frequencies of the simply supported laminated composite plates obtained by Galerkin method are compared with those of Khorshidi and Farhadi [2] which are based on third order shear deformation plate theory (TSDT) and finite element analysis are presented in Table 1 for different thickness ratios including , 0.1 and 0.2. In Table 1, the numerical results are obtained for a four layer (0º/90º)s squared plate with different depth of the fluid  which is varies from 0 to 1. This comparison confirmed reliability of the proposed method and those that were presented by Khorshidi and Farhadi [2].

In this section, the effects of the various parameters such as aspect ratio (a/b), thickness ratio (h/a), material orthotropy orientation ( ), depth of the fluid ratio ( ) and width of the tank  are illustrated in Figs. 2-6. according to the EFAST model that is used for sensitivity analysis, for each parameter in x-axis there exists several outputs in the y-axis because when a parameter has a particular value other parameter can be changed. For example, in Fig. 2 when the aspect ratio is equal to 1 other parameter such as thickness ratio (h/a) and... can be changed.

Figure. 2 illustrates the fundamental natural frequency versus the aspect ratios parameter. It is realized that aspect ratios parameter is the most sensitive parameter, since its changes have the steepest slope relative to other parameters.

Table 1. Comparison study of fundamental natural frequencies of a simply supported square laminated composite plate between analytical and FEM results.

Fig. 2. Fundamental natural frequency versus the aspect ratios

Fig. 3. Fundamental natural frequency versus the material orthotropy orientation

At roughness values close to zero, the slope of this diagram is too sharp; but as the roughness values increase, the critical force of movement approaches a constant number; thus, this parameter is very sensitive at near-zero values, and from values in the vicinity of 2, its sensitivity diminishes considerably.

In view of Fig. 3, the material orthotropy orientation can be introduced as the second sensitive parameter after aspect ratios. With the increase of this parameter, first the fundamental natural frequency increases and then decreases, with a sharp, albeit milder, slope relative to aspect ratios.

The diagram of the fundamental natural frequency versus depth of the fluid ratios has been shown in Fig. 4 with a positive, and near zero, slope. With the change of depth of the fluid ratios in its range of variations, a minor change is observed in the fundamental natural frequency, and so this parameter is not considered as a sensitive parameter for the fundamental natural frequency. Figure 5 shows the fundamental natural frequency versus the thickness ratios parameter. The parameter of thickness ratios is an input parameter with negligible effects on the fundamental natural frequency; and with the changes of this parameter in its relevant intervals, no tangible change is observed in the fundamental natural frequency.

The diagram of the fundamental natural frequency versus width of the tank has been shown in Fig. 6 with a near-zero slope. With the change of width of the tank in its range of variations, a minor change is observed in the fundamental natural frequency, and so this parameter is not considered as a sensitive parameter for the fundamental natural frequency.

Figure. 7 shows the percent sensitivity of fundamental natural frequency to parameters of aspect ratios, material orthotropy orientation, depth of the fluid, thickness ratios and width of the tank, which have been obtained by means of the EFAST method. According to this figure, the aspect ratios parameter (with 70% sensitivity) is the most important parameter, and the material orthotropy orientation (with 25% sensitivity) is the other effective parameter.

Fig. 4. Fundamental natural frequency versus depth of the fluid ratios

Fig.5. Fundamental natural frequency versus the thickness ratios parameter

Fig.6. Fundamental natural frequency versus width of the tank

7.        Conclusions

In the present paper, sensitivity analysis of vibrating laminated composite rectangular plates in interaction with inviscid fluid has been presented. For the formulation, the modified higher order shear deformation plate theory is employed. Differential equations of motion were derived using Hamilton’s variational principle. Natural frequencies of the plate were calculated using the Galerkin’s method and the boundary conditions of the plate was simply supported. For sensitivity analysis, the EFAST model was utilized. This method was based on variance and it was independent of any assumption of linearity and uniformity between inputs and output(s).

Fig.7. Percent sensitivity of fundamental natural frequency to parameters

The EFAST model was used to qualitative and quantitative analysis of the effect of 5 parameters includes aspect ratio (a/b), thickness ratio (h/a), material orthotropy orientation (θ), depth of the fluid ratio  and width of the tank  on the natural frequency of plate. It was observed that the aspect ratios parameter (with 70% sensitivity) was the most important parameter and after that material orthotropy orientation (with 25% sensitivity) was the second effective parameter on the natural frequency of plate. Also, results showed that other parameters such as thickness ratio and depth of the fluid ratio did not have a significant effect with 4% and 1% sensitivity respectively. Finally, the width of the tank with 0% did not have any effect on the natural frequency of the plate.

Acknowledgements

The authors gratefully acknowledge the funding by Arak University, under grant No 97/5115.

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