Vibrational behavior of isotropic plate structures in contact with a bounded fluid via unif ied formulation

F.G. CANALES, J.L. MANTARI,b,*

a Faculty of Mechanical Engineering, Instituto de Investigacio′n en Ingenier?′a Naval (IDIIN), Universidad Nacional de Ingenieria (UNI), Lima 15333, Peru

b Department of Mechanical Engineering, University of New Mexico, NM 87131, USA

KEYWORDS Added mass;Fluid-structure interaction;Plate;Ritz method;Unif ied formulation;Vibration

Abstract This paper presents an analytical solution for free vibration analysis of thick rectangular isotropic plates coupled with a bounded fluid for various boundary conditions.In order to consider displacement theories of an arbitrary order, the Carrera Unif ied Formulation (CUF) is used. The eigenvalue problem is obtained by using the energy functional, considering plate and fluid kinetic energies as well as the potential energy of the plate. The Ritz method is used to evaluate the displacement variables, and the functions used in the Ritz series can be adjusted to consider any of the classical boundary conditions. The convergence of the solution is analyzed, and a validation of results considering open literature and 3D finite element software is performed.Parametric studies are carried out to obtain natural frequencies as a function of the side-to-thickness ratio, plate aspect ratio, fluid domain size, plate boundary conditions, and fluid-solid density ratio. Pressure and velocity in the fluid domain are evaluated in order to establish the consistency of the solution.Accurate results for thick plates are obtained with a much lower computational cost compared to that of 3D finite element solutions.

1. Introduction

Accurate modeling of the dynamical characteristics of a plate in contact with water is very important in ships, submarines,offshore structures, and fuel containers, among others. The natural frequencies of the plate decrease considerably compared to those in vacuum, due to an added mass effect of the water.The fluid-structure interaction signif icantly increases the complexity of the analysis, since a coupled hydroelastic solution is required. While finite element models can solve the interaction problem for any fluid-structure geometries, a high computational cost makes it inadequate for performing parametric studies. Analytical methods can obtain accurate results with a very low computational cost and are useful to obtain qualitative trends for cases with special geometries such as circular or rectangular plates coupled to cylindrical or parallelepipedic cavities.

Many references analyze circular plates due to the applicability in chemical industries. Compressibility and viscosity of fluid are considered using appropriate simplif ications in mathematical modeling.Jeong and Kim1developed a free vibration analysis of circular plates coupled with compressible fluid assuming that dry modal functions could approximate the wet dynamic displacements. Vibrational analysis of circular plates in asymmetric conditions has been presented by Tariverdilo et al.2. Finite element modeling of a plate coupled with fluid was developed by Kerboua et al.3. Chang4analyzed magneto-electro-elastic plates in contact with fluid.An analytical formulation for large-amplitude vibrations of plates coupled with fluid can be quite involved, and thus experimental studies are useful, as presented by Carra et al.5. An analysis of plate vibration in contact with viscous fluid has been developed by Kozlovsky6.Linearization of Navier-Stokes equations was required in order to obtain analytical results.

Analysis of rectangular plates is an important topic for structural design of fuel tanks and liquid containers. Cheung and Zhou7developed a vibration analysis of rectangular plates in contact with fluid using the Kirchhoff plate theory and Ritz method.Cho et al.8made further progress by using the Mindlin plate theory and considering stiffeners with various framing patterns.Liao and Ma9presented an analysis of plates in compressible inviscid fluid, showing that certain modes of vibration could not be predicted by an incompressible fluid theory. Hashemi et al.10developed an analysis of Mindlin plates on elastic foundations in contact with a fluid domain with a finite width and depth but an infinite length. Shahbaztabar and Ranji11presented an analytical formulation for free vibration of laminated plates on elastic foundations in contact with fluid considering in-plane loads. A forced vibration analysis of a plate coupled with fluid has been developed by Cho et al.12. Khorshidi and Farhadi13used a non-linear third-order shear deformation theory in order to analyze the free vibration of a composite plate in contact with fluid considering sloshing effects. Hydroelastic analysis of multiple plates in contact with fluid applied for fuel assemblies in a research reactor was presented by Jeong and Kang14. A comparison between experimental and analytical modal studies of plates in contact with a bounded fluid was given in Ref.15.The influence of the hydrostatic pressure on the vibrational response was presented in Refs.16,17. Plates with functionally graded materials have also been studied, as given in Ref.18.

Most of the aforementioned references used the Kirchhoff or Mindlin plate theory. More advanced theories known as Higher-order Shear Deformation Theories (HSDTs) can consider nonuniform shear distribution in the plate thickness and obtain more accurate results. However, analytical derivations can become quite lengthy as the complexity of an HSDT grows. In order to analyze HSDTs in a simple manner, the Carrera Unif ied Formulation (CUF) is very useful. This formulation is known to obtain results similar to those of 3D analysis with a much lower computational cost. The theoretical foundations were presented in Ref.19. This formulation has been applied for analysis of thermal stresses in plates20-22and general multifield problems23-26.Extension of the model to shells has been developed by Cinefra et al.27,28. The CUF has also been used for analysis of structures with functionally graded materials29.The core of the formulation was described in detail in Refs.30-32.

In order to approximate the displacement field of a plate,a Navier-type solution using trigonometric functions can be applied. However, this method is limited to simply supported plates. The Ritz method is a common choice in free vibration analysis due the capability to use displacement shape functions that can consider arbitrary boundary conditions. The method was described in Refs.33-35. The accuracy and convergence of results are highly dependent on the choice of shape functions used in the Ritz method.Free vibration analysis of plates using the CUF has been developed by Fazzolari and Carrera36-39.Trigonometric shape functions were used,with analysis limited to simply supported plates.Vibration analysis of shell elements using the Ritz method has been developed in Refs.40-42. In order to consider boundary conditions other than simply supported, many other shape functions can be used. Vibrational analysis of plates considering arbitrary boundary conditions has been developed by Dozio et al.43-48.

The CUF is a convenient formulation for defining displacement fields for plate structures beyond the classical plate theories, and does not def ine the method of solution, i.e., the method used to approximate displacement fields. If the CUF method is used with a 2D finite element discretization for a plate and is coupled with a 3D finite element discretization for a fluid domain, complicated geometries can be analyzed.However, the computational cost is high, albeit slightly lower than that of a full 3D fluid-structure solution, while maintaining a similar accuracy. In the present manuscript, the CUF method is used in conjunction with analytical methods for modeling a plate and a fluid. A closed-form solution for the velocity potential is obtained via solving the Laplace equation.This is used to def ine a fluid mass matrix F which contains all the influences of the fluid on the plate vibrational behavior.In this manner, no degree of freedom associated to the velocity potential or the pressure field is required in the eigenvalue problem, greatly reducing the computational cost. The main disadvantage of this analytical method is that it lacks generality in the types of geometries that can be analyzed,since a solution of the Laplace equation is required. This proves to be unfeasible for complicated geometries. However, even for the simple rectangular geometry considered in the manuscript,general characteristics of the fluid-structure coupling can be observed.

An analytical solution for free vibration analysis of thick rectangular isotropic plates in contact with a fluid domain is developed in the present manuscript. The main novelty of the present work relies on the analytical solution of the coupling between a hydrodynamic and a structural model that can consider displacement theories of arbitrary order by using the CUF. This allows a very accurate analysis of thick plates,and could be extended in further work to accurately analyze fluid-structure interaction using plates of advanced materials.The Ritz method is used to approximate the displacement field,with capability to account for any of the classical boundary conditions. A convergence analysis is carried out, and the validation of results with 3D finite element software is performed in order to demonstrate the capability of the present compact and generalized formulation.The influences of various parameters on the natural frequencies of the plate are studied. Consistency of the solution in the fluid domain is ascertained by analyzing the continuity of the pressure and velocity distributions.

2. Analytical modeling

A rectangular plate with length a, width b, and thickness h is considered, as shown in Fig. 1. The coordinate system x-y-z is used to describe the motion of the plate. It is assumed that the plate is in contact with an incompressible, inviscid, and irrotational fluid bounded by a rectangular domain with length c,width d,and depth e,as shown in Fig.2.The coordinate system x~-y~-z~is used to describe the fluid motion.Sloshing effects,i.e., free surface waves, are neglected, and only bulging modes are analyzed in the present study. Bulging modes, involving plate-dominated motion, are closely related to plate fatigue,and thus are deemed more important in the present work. In addition, the coupling between sloshing and bulging modes is very small49. Consequently, sloshing motion is neglected,and a simplif ied free surface condition is suff icient.The vertical walls and the bottom of the fluid domain are assumed to be rigid, except for the region in contact with the top surface of the plate at z~=0 or z=h/2. The plate is assumed to be linearly elastic, homogeneous, and isotropic, and the amplitude of motion is assumed to be small.The energy functional is used to obtain the eigenvalue problem. The structural model uses the plate potential energy U and the kinetic energy T with many similarities to the formulation given by Dozio and Carrera48.The hydrodynamic model uses the kinetic energy of the fluid TWin a similar way to that presented by Cheung and Zhou7. In this manner, the CUF structural model is coupled with a hydrodynamic model to develop an accurate analysis of the free vibration of the plate in contact with fluid.

2.1. Structural model

For convenience, the derivation of the structural model is given in a non-dimensional form. The x-y domain (0 ≤x ≤a,0 ≤y ≤b) of the rectangular plate is mapped into a computational ξ - η domain (-1 ≤ξ,η ≤1) by using a transformation of coordinates given by

The derivatives in the two coordinate systems are related by

Fig. 1 Coordinate frame of plate.

Fig. 2 Coordinate frames of fluid domain and plate.

A general displacement vector is introduced as

where u,v,and w are the displacement components in ξ,η,and z axes,respectively,and t is time.The stress and strain components are grouped as follows:

Considering small displacements, the strain-displacement relations are

where the linear differential operators Dp, Dnp, and Dnzare given by

and the derivatives must be evaluated using Eq.(2).The stress components are given by the following constitutive law:

Eq. (7) can be split by using Eq. (4) as follows:

where for an isotropic material,the matrices Cpp,Cpn,Cnp,and Cnnare given by

The stiffness coefficients Cijdepend on the elastic modulus E and Poisson’s ratio υ as follows:

The displacement field is expressed within the framework of the CUF as

where Fτare functions of the coordinate z on the thickness of the plate,N stands for the number of terms used in the expansion, uτis the vector of the generalized displacements, and the repeated subscript‘‘τ”indicates summation. A simple polynomial expansion is used to determine the functions Fτ. For example, the displacement field of the second-order (N=2)expansion model can be expressed as

Reduced-order plate theories can be obtained by using a suitable expansion order and eliminating certain displacement variables.For example,the Mindlin plate theory is obtained by imposing kinematic constraints in a first-order expansion(N=1), resulting in the following displacement field:

The potential energy U of the plate in the computational domain can be written as follows:

where J｜｜stands for the Jacobian of the coordinate transformation given by Eq. (1). Substituting Eqs. (8) and (5) into Eq. (14), the strain energy is given by

The kinetic energy T of the plate is given by the following expression:

where ρ is the mass density of the plate.

2.2. Hydrodynamic model

The fluid is considered to be incompressible,inviscid,and irrotational.A simplif ied surface condition is considered appropriate since sloshing effects are neglected. The velocity potential φ x,y,z（ ） is used to describe the motion in the fluid domain Ω. The Laplace equation associated to the velocity potential is given by

The boundary conditions in the fluid domain are given by a zero velocity normal to the rigid walls and the rigid bottom,except for the plate surface, and there are no surface waves on the top of the fluid domain.It is assumed that fluid particles are perfectly attached to the plate surface,with no diffusion or cavitation, i.e.,

where the plate displacement w couples the hydrodynamic model to the structural model. The amplitude W of the transverse plate displacement is defined as follows:

The solution of the Laplace equation Eq.(17)is assumed in the following form:

Substituting Eq.(23)into Eq.(17),the following three ordinary differential equations are obtained:

Solving Eqs.(24)-(26)and considering the boundary conditions given in Eqs.(18)-(20),the solution of the velocity potential of the fluid is given by

where

and non-dimensional coordinates have been introduced for the fluid domain as

Substituting Eq. (27) and Eq. (22) into Eq. (21), using the orthogonality of the trigonometric series, and since the fluid domain bottom surface is located at z=h/2 in the plate coordinates, the coefficient Apqis derived as follows:

where

The kinetic energy of the fluid is given by

where ρWis the fluid density, and ?is the gradient operator.By applying Green’s first identity to Eq. (34) in conjunction with Eq. (17), a surface integral involving the fluid domain boundary is obtained.By using the boundary conditions given by Eqs. (18)-(21), the integral is simplif ied to a single surface integral in the plate region ΓPas

Substituting Eqs. (21), (22), (27), (30), and (31) into Eq. (35), the fluid kinetic energy is given by

where

2.3. Ritz series

A harmonic motion is assumed for the generalized displacements given in Eq. (11) as

where M is the order of the Ritz expansion, cuτi, cvτi, and cwτiare unknown coefficients, and ψuτi, ψvτi, and ψwτiare assumed shape functions.These shape functions are assumed as follows:

where piis a generic element of a complete 2D polynomial space, andξ（ ） is a function used to satisfy the boundary conditions. A generic piterm is given by

where P is the degree of the polynomial space.The indices are related as follows:

The order M of the Ritz expansion is related to the polynomial degree P as follows:

The boundary-compliant functionis given by the following expression:

The expressions in parentheses are the equations of the plate boundaries. The exponents αiθτare chosen according to the boundary conditions on the edges. If the i-th edge has a clamped boundary condition, then the geometric boundary conditions are u=v=w=0, which implies that

αiθτ=1 （θ =u,v,w）. If the i-th edge is simply supported, then transverse and tangential displacements are imposed to be zero. For example, if it is a simply supported edge parallel to the x axis, then the geometric boundary conditions are u=w=0,which implies that αiuτ=αiwτ=1,and since the displacement in the y axis,i.e.,the displacement variable v,has no geometric restriction, then αivτ=0.

In matrix notation, Eq. (40) can be written as follows:

where

2.4. Stiffness and mass matrices

Substituting Eqs. (39) and (46) into Eq. (15) and noting that the linear operator Dnzonly affects the thickness function Fτz（ ）, the maximum plate potential energy is given by

where Kτsijis the stiffness nucleus, which is given by

A cross-sectional parameter has been used. A generic term is given by

Substituting Eqs.(39)and(46)into Eq.(16),the maximum plate kinetic energy is

where Mτsijis the solid mass nucleus, which is given by

The time-independent part of the transverse displacement w can be expressed in a convenient form using Eqs.(22)and(39)as follows:

Evaluating Eq. (46) for the w-τcomponent and substituting into Eq. (54) result in

Substituting Eq. (55) into Eq. (36), the maximum fluid kinetic energy is given by

2.5. Nuclei and matrix assembly

For simplicity of notation, the following generic term is introduced:

The stiffness and mass matrixes are constructed by expanding the nucleus over the indices τ, s, i, and j. The stiffness nucleus in Eq. (50) is given by

By varying the indices τ and s of the stiffness nucleus over the range τ,s=0,1,...,N, the following matrix is obtained:

The stiffness matrix is obtained by varying the indices i and j over the range i,j=1,2,...,M as follows:

The solid mass nucleus in Eq. (53) is given by

where

where

The solid mass matrix M and the fluid mass matrix F can be obtained by expanding the indices τ, s, i, and j of the mass nucleus, in a similar way to how the stiffness matrix is constructed.

2.6. Eigenvalue problem and solutions

The energy functional is given by

Substituting Eqs.(49),(52),and(56)into Eq.(69),and minimizing the functional with respect to the undetermined coefficients cτi, the following equation is obtained:

where c is a vector of unknown coefficients csjthat describe the plate displacement. By solving Eq. (70), the fundamental frequencies and mode shapes of vibration can be obtained. In order to obtain the j vibrational mode,it is required to perform a summation of the set of Ritz admissible functions evaluated at the point of interest times the term with indices τ and i of the eigenvector Xjfor each of the generalized displacement variables. Afterwards, the use of Eq. (11) is required in order to obtain the plate displacement amplitudes. For the displacement in the z axis, it is given as follows:

Values of the velocity potential in the fluid domain associated with the j mode shape can be obtained by substituting the coefficients of the Xjeigenvector in Eq. (55)and then combining with Eqs. (27) and (31). The coefficient Apqis derived as follows:

After obtaining each coefficient Apqand substituting into Eq. (27), the velocity potential can be evaluated. The amplitudes of pressure within the fluid are proportional to the velocity potential.In order to determine amplitudes of velocity, the velocity potential is differentiated with respect to the variable x, y, or z in order to obtain velocities in the X,Y, and Z axes respectively.

3. Numerical results and discussion

A computer program in MATLAB has been developed using the present formulation, and numerical results are given in the present section. In order to describe the boundary conditions of the plate, a four-letter symbolic notation is used. Letters ‘‘S” and ‘‘C” stand for simply supported and clamped edges. The boundaries are numbered starting from the edge at x=0 in a counterclockwise manner. Unless otherwise stated, the natural frequencies in the numerical results are given in the following non-dimensional form:

where ω is the natural frequency given in rad/s, and D is the flexural rigidity of the plate, i.e., D=Eh3/12 1-υ2（ ）. Unless otherwise stated, the non-dimensional parameters for the geometry of the plate and the fluid domain,the solid-fluid density ratio and Poisson’s ratio used are those given in Table 1.The plate geometric center is assumed to be coincident with that of the bottom of the fluid domain in all the numerical results.

3.1. Convergence study

The convergence of the results is analyzed in order to verify the stability of the results as the number of trigonometric terms p, q and the polynomial degree of the Ritz expansion P are increased. Table 2 presents the non-dimensional natural frequencies of a CCCC plate in contact with fluid as the polynomial degree P is increased,where the number of trigonometric terms used is p, q=10 and the parameters given in Table 1 have been used.A fast convergence is observed,with a slightly slower convergence for a higher expansion order N and a higher vibration mode.Results within 0.15%of the converged solution can be obtained by using a polynomial degree P=9.For the purposes of the present work, the polynomial degree P=13 is considered suff iciently accurate,and it is further used in the manuscript.

Table 3 presents the non-dimensional natural frequencies of a CCCC plate in contact with fluid as the number of trigonometric terms p, q is increased, assuming equal trigonometric terms in the x~and y~ directions and considering P=13. Convergence is fast and independent of the expansion order N.Results within 0.05% of the converged solution can be obtained by using as little as p, q=4 trigonometric terms.However,it has been considered that d/b=1,and the convergence of the trigonometric series is dependent on the fluiddomain size7. Table 4 presents the non-dimensional natural frequencies of a CCCC plate in contact with fluid as the number of trigonometric terms p, q is increased for various fluid domain width-plate width ratios. Larger fluid domains slow down the convergence considerably. The remainder of the paper considers numerical results with d/b ＜3, and thus the number of trigonometric terms used is p, q=16 in order to obtain accurate results.

Table 1 Non-dimensional parameters used in most of the numerical results.

Table 2 Convergence of the natural frequencies of a square CCCC plate with respect to the number of terms used in the Ritz series for various CUF expansion orders.

Table 4 Convergence of the natural frequencies of a square CCCC plate with respect to the number of terms used in trigonometric series for various fluid domain width-plate width ratios.

3.2. Validation of results

Validation of the results is performed by developing a numerical example and comparing the results from the present model with those from the Mindlin plate theory, as presented in Ref.8,and with those from finite element software.A 3D finite element solution for free vibration analysis of a plate coupled with a fluid domain has been developed using the ANSYS general purpose program. The three-dimensional model is composed of three-dimensional fluid elements (FLUID30) and solid elements (SOLID185) with an identical size, as shown in Fig. 3. The plate is segmented into 115200 (80×120×12)solid elements, and the fluid domain is divided into 384000(80×120×40) fluid elements. Viscosity and compressibility of the fluid are neglected,as in the present theoretical development. The boundary conditions for the fluid domain are the same as given in Eqs.(18)-(20).The fluid-structure interaction f lag is activated in fluid elements adjacent to solid elements,in order to satisfy Eq. (21).

Fig. 3 Finite element model used in ANSYS software.

Table 6 Parameters used in numerical example for validation of results, as used in Ref.8.

Table 5 presents the first three natural frequencies for free vibration of a plate in contact with fluid. The geometrical and material properties given in Table 6 have been used.Results obtained by the present theory for various expansion orders N, from Ref.8, and by a 3D finite element solution are reported.The maximum and average differences of results with respect to those from the 3D finite element solution are also given. Good agreement is observed between results from the present theory and those from Ref.8.It can be seen that results from Ref.8are accurate for an SSSS plate, but not for other boundary conditions. On the other hand, the results from the present theory with N=3 are fairly accurate for all the boundary conditions analyzed.

3.3. Parametric studies

In order to assess the effects of various parameters on the natural frequencies, a CCCC plate in contact with fluid using the parameters in Table 1 is considered as a baseline case. Results are obtained by varying the parameters one at a time. The solution from finite element software ANSYS is reported,obtained by using a three-dimensional model similar to the one used in the previous section. The total amount of used elements is between 300000 and 500000. The difference of the fundamental frequency with respect to that of the 3D FEM solution is indicated in Tables 7-12. The average difference for the first five natural frequencies is also given.

Table 7 Non-dimensional natural frequencies of a CCCC plate in contact with fluid for various plate thickness ratios.

Table 7 presents the first five non-dimensional natural frequencies of a CCCC plate in contact with fluid for various thickness ratios h/b,and corresponding 3D finite element solutions. Results from the present model with N=3 are in close agreement for thin plates, while slightly higher differences are obtained for thicker plates. However, arbitrarily lowererrors can be obtained by using a higher expansion order N.Even for very thick plates with h/b=0.50, the present model with N=4 has a maximum difference of 0.3% with respect to the 3D FEM results.Fig.4 shows the non-dimensional natural frequencies of plates in contact with fluid(wet frequencies)and in vacuum(dry frequencies)for a range of thickness ratios h/b using N=3. As the thickness ratio decreases, the added mass effect of the fluid becomes a dominant factor and considerably affects the natural frequencies.On the other hand,with higher thickness ratios, the frequencies are only slightly modified due to the presence of the fluid. It can be observed that the effect of the fluid on the natural frequencies becomes less important for higher vibrational modes.

Table 8 Non-dimensional natural frequencies of a CCCC plate in contact with fluid for various fluid domain depth ratios.

Table 9 Non-dimensional natural frequencies of a CCCC plate in contact with fluid for various plate aspect ratios.

Table 8 presents the first five non-dimensional natural frequencies of a CCCC plate in contact with fluid for various fluid domain depths. It can be observed that as the fluid depth increase, ω5and specially ω1monotonically decrease, while the other three frequencies approach constant values. This phenomenon has been reported in previous studies7. Fig. 5 shows the non-dimensional natural frequencies for a varietyof fluid depth ratios,obtained using N=3.It is observed that there is a small influence of the depth on ω2, ω3, and ω4for depth ratios up to e/b=1,while further increases of the depth are inconsequential on these three frequencies.In order to further study this point, the mode shapes for a depth ratio e/b=2 obtained by the present method and via 3D FEM software are shown in Fig. 6. The mode shape is evaluated in the mid-plane of the plate, i.e., at z=0. It can be seen that vibration modes with monotonically decreasing natural frequencies,i.e., ω1and ω5, correspond to doubly symmetrical modes. On the other hand, vibration modes with nearly constant natural frequencies with respect to the fluid depth, i.e., ω2, ω3, and ω4,correspond to antisymmetric modes.Arguably,in antisymmetric modes,the net vertical displacement of the fluid is minimal since upward and downward movements of the fluid are present simultaneously, and the influence of the fluid depth is expected to be low. On the other hand, doubly symmetrical modes have a net vertical fluid displacement, and the fluid depth is expected to have considerable repercussion on the natural frequencies.

Table 10 Non-dimensional natural frequencies of a plate in contact with fluid for various plate boundary conditions.

Table 11 Non-dimensional natural frequencies of a CCCC plate in contact with fluid for various solid-fluid density ratios.

Table 12 Non-dimensional natural frequencies of a CCCC plate in contact with fluid for various fluid domain width-plate width ratios.

Fig.4 Effect of plate thickness on non-dimensional dry and wet natural frequencies of a square CCCC plate using N=3.

Fig.5 Effect of fluid domain depth on non-dimensional natural frequencies of a square CCCC plate in contact with fluid using N=3.

Table 9 presents the first five non-dimensional natural frequencies of a CCCC plate in contact with fluid for various plate aspect ratios a/b. Results from the present theory with N=3 agree closely with those from the 3D FEM solution,with no signif icant influence of the aspect ratio on the accuracy. Fig. 7 shows the non-dimensional natural frequencies for a variety of plate aspect ratios. The frequencies monotonically decrease and approach a constant value,similar to what is observed for the non-dimensional frequencies in dry vibration. Table 10 presents the first five non-dimensional natural frequencies for plate boundary conditions different from CCCC. Certain vibration modes can be approximated with very high accuracy by using an expansion order of just N=1, see the 5th mode of an SSSS plate, the 4th mode of a CSCS plate, and the 4th mode of a CSSS plate. Table 11 presents the first five non-dimensional natural frequencies of a CCCC plate for various solid-fluid density ratios. With higher fluid densities, the ratio ρ/ρWdecreases, and the natural frequencies are lower, as expected.

Fig. 6 Mode shapes of vibration for a square CCCC plate with e/b=2.

Fig. 7 Effect of plate aspect ratio on non-dimensional natural frequencies of a CCCC plate in contact with fluid using N=3.

Fig. 8 Effect of plate thickness ratio and plate boundary conditions on the 1st natural frequency of a plate in contact with fluid using N=3.

Table 12 presents the first five non-dimensional natural frequencies of a CCCC plate in contact with fluid for various fluid domain width-plate width ratios. The natural frequencies increase as the fluid domain size grows. It can be argued that rigid side walls, by limiting fluid motion, contribute to the added mass effect of the fluid. As the fluid domain width increases, the fluid is free to flow sideways, resulting in a slightly lower influence on the natural frequencies. Fig. 8 shows the first non-dimensional natural frequency for a variety of thickness ratios and boundary conditions. The qualitative trend observed previously for a CCCC plate is seen to be unaffected by the boundary conditions. However, the difference in natural frequencies between thick (h/b=0.5) and moderately thick (h/b=0.1) plates is much higher for CCCC plates than for SSSS plates.

3.4. Pressure and velocity in fluid domain

Natural frequencies only give a global behavior of the system,but do not guarantee that the solution is correct in the entire fluid domain, i.e., errors in modeling could arise within the fluid and may not be detected by analyzing the accuracy of the natural frequencies. The consistency of the solution in the fluid domain is ascertained by analyzing the pressure and velocity in the fluid domain. A CCCC plate coupled to a fluid domain described by the parameters given in Table 1 and with b=1 is considered. Fig. 9 shows pressure amplitude plots on the planes=0,=0, and=1 of the fluid domain for the first five natural frequencies. The plot of pressure on the plane=0 has been rotated in order to give a developed view of the fluid domain boundaries.Colors are used to indicate the maximum and minimum values of the pressure amplitude within each plane individually. Continuity of the pressure on the plate and fluid domain boundaries is observed.In addition,the pressure distribution is in good agreement with the mode shapes of plate displacements shown in Fig. 6. It is important to note that the pressure distributions on theplane are evaluated on the top plate surface, i.e., at=0 or z=h/2.On the other hand, the mode shapes of the plate in Fig. 6 are evaluated in the mid-plane of the plate, i.e., z=0. Due to this, slight discrepancies between Figs. 6 and 9 are expected.

Fig. 10 shows velocity vector plots on the planes=0,=0, and=1 of the fluid domain for the first five natural frequencies.The continuity of the fluid velocity is evident from Fig. 10. In addition, since fluid movement is driven by differences in pressure, the coherence of the velocity plots can be conf irmed by comparing them with the pressure amplitude plots shown in Fig. 9. It can be concluded that, in addition to accurate natural frequencies of the fluid-plate coupled system,the present formulation can predict pressure and velocity amplitudes in the fluid too.

Fig. 9 Pressure amplitude plots in resonant modes of a CCCC plate.

Fig. 10 Velocity vector plots in resonant modes of a CCCC plate.

4. Conclusions

An analytical solution for free vibration analysis of thick rectangular isotropic plates coupled with a bounded fluid for various boundary conditions has been presented.The accuracy of the Carrera Unif ied Formulation in the structural model is evident from a comparison with 3D FEM solutions.Parametric studies have been carried out,and influences of various parameters on natural frequencies have been analyzed. Conclusions that emerge from this paper can be summarized as follows:

(1) Convergence of the present formulation is slightly dependent on the expansion order, and highly dependent on the fluid domain width-plate width ratio.

(2) Accurate results for natural frequencies are obtained by using an expansion order N =4,with a maximum difference of 0.3% with respect to 3D FEM results for very thick plates with h/b=0.5. However, for plates with h/b ＜0.25, an expansion order N =3 is suff iciently accurate.

(3) The depth of the fluid domain has a considerable influence only on certain modes of vibration. An analysis of the mode shapes indicates that vibrational modes with doubly symmetric mode shapes are the most affected, since the net vertical displacement of the fluid is higher than that for antisymmetric modes.

(4) The added mass effect of the fluid is of most importance for thin plates.As the thickness ratio increases,the influence of the fluid on the natural frequencies is reduced.

(5) A wider fluid domain reduces the added mass effect of the fluid, and the natural frequencies are seen to increase. Arguably, the presence of rigid walls near the plate, by limiting the fluid motion, increases the influence of the fluid on the natural frequencies.

Acknowledgments

This paper was written in the context of the project:‘‘Disen~o y optimizacio′n de dispositivos de drenaje para pacientes con glaucoma mediante el uso de modelos computacionales de ojos” founded by Cienciactiva, CON-CYTEC, under the contract number N° 008-2016-FONDECYT. The authors of this manuscript appreciate the fi-nancial support from the Peruvian Government.