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Natural frequencies of rectangular Mindlin plates coupled with stationary fluid
Applied Mathematical Modelling 36 (2012) 764–778
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Natural frequencies of rectangular Mindlin plates coupled with stationary fluid Shahrokh Hosseini-Hashemi a,⇑, Mahmoud Karimi a, Hossein Rokni a,b a b
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16848-13114, Iran School of Engineering, University of British Columbia (Okanagan), 3333 University Way, Kelowna, BC, Canada V1V 1V7
a r t i c l e
i n f o
Article history: Received 10 March 2009 Received in revised form 22 June 2011 Accepted 1 July 2011 Available online 13 July 2011 Keywords: Free vibration Rectangular plate Mindlin theory Fluid–structure interaction Added mass
a b s t r a c t The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Plate structures have been widely used in various engineering applications, from space vehicles to microelectromechanical system (MEMS) devices. Many researchers have worked on vibration of such plates. Excellent reference source may be found in the open literature (see, for example, [1,2]). According to the literature, it is found that many researchers have used the classical plate theory (CPT), overestimating the natural frequencies of the plate due to ignoring the effect of shear deformation through the plate thickness. In order to eliminate the deficiency of the CPT for moderately thick plates, the first-order shear deformation plate theory was proposed by Reissner [3], and developed further for the deformable plates in statics and dynamics by Mindlin et al. [4–6]. A very beneficial study for deriving the exact closed-form characteristic equations of vibrating moderately thick rectangular plates was carried out by Hosseini Hashemi and Arsanjani [7]. Plates coupled with fluid are of great practical significance in nuclear, ocean and naval engineering. Thus, a good understanding of the dynamic interaction between an elastic plate and fluid is necessary. It is well known that the vibrating plates in contact with fluid result in the fluid motion. In addition, the existence of the fluid around the plate causes the kinetic energy to considerably increase. Consequently, the natural frequencies of plate coupled with fluid significantly decrease in comparison with those of the plate in the air.
⇑ Corresponding author. Tel.: +98 21 7391 2912; fax: +98 21 7724 0488. E-mail address: [email protected] (Sh. Hosseini-Hashemi). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.07.007
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Many studies on the free and forced vibration analysis of plates, partially or totally submerged in the fluid, have been carried out. Lamb [8] calculated the natural frequencies of a thin clamped circular plate in an aperture of an infinite rigid plane wall coupled with water based on the assumption that the wet mode shapes are almost the same as those in vacuum. Powell and Roberts [9] established an experimental setup to verify the theoretical results reported by Lamb [8]. McLachlan [10] performed a similar analysis to Lamb’s work [8] except that free boundary conditions were applied at the circumferential edge of the circular plate. Peak and Thurston [11] generalized the work of Lamb and McLachlan. Added virtual mass (AVM) of rectangular plates was calculated by Kito [12] using Fourier series method. Lindholm et al. [13] reported the natural frequencies of cantilever plates in air and in water, whereas the fluid actions were evaluated by a strip-theory approach. Added mass of thin rectangular plates in infinite fluid was obtained by Meyerhoff [14] and dipole singularities were employed to model the potential flow around a flat rectangular plate. Muthuveerappan et al. [15] investigated the freely vibrating behavior of a cantilever square plate immersed in water. Kim et al. [16] presented both experimental and theoretical results to investigate the effect of supporting boundary conditions on the vibrating plates. Fundamental natural frequencies of vertical cantilever plates, obtained by the experiment and by the fluid finite element method, were presented by Volcy et al. [17] when the plates were partially or totally submerged in liquid. Espinosa and Gallego-Juarez [18] acquired the pressure distribution of water on the free-edge circular plate. Fu and Price [19] employed a finite element discretization to analyze the dry and wet dynamic characteristics of a vertical and horizontal cantilever plate. Robinson and Palmer [20] conducted a study on the modal analysis of a rectangular plate resting on an incompressible fluid. Kwak and Kim [21] studied on axisymmetric vibration of circular plates in the presence of fluid on the basis of the mixed boundary value problem. Free vibration of infinite elastic rectangular plate in contact with water was studied by Hagedorn [22]. Kwak [23] utilized a piecewise division to investigate the free vibrations of rectangular plates in contact with unbounded water on one side, while beam functions were used as admissible functions. Haddara and Cao [24] investigated dynamic responses of rectangular plates immersed in fluid. An approximate expression for the evaluation of the modal added mass was derived for cantilever and SFSF rectangular plates and the numerical results were verified by the experimental ones. The natural frequencies of annular plates in contact with a fluid on one side were theoretically obtained by Amabili et al. [25] using the added mass approach, whereas the coupled fluid–structure system was solved by adopting the Hankel transform. Meylan [26] employed an appropriate Green’s function to study the forced vibration of an arbitrary thin plate floating on the surface of an infinite liquid. Cheung and Zhou [27] also studied the case of a horizontal rectangular plate composing the base of a rigid rectangular container. The dynamic characteristics of a vertical cantilever plate partially in contact with fluid were investigated by Ergin and Ugurlu [28]. Liang et al. [29] adopted an empirical added-mass formulation to determine the frequencies and mode shapes of submerged cantilevered plates. Based on a finite Fourier series expansion, Jeong et al. [30] studied the wet resonance frequencies and associated mode shapes of two identical rectangular plates coupled with a bounded fluid. Tayler and Ohkusu [31] suggested expressions for the free–free rectangular plates in terms of the sinusoidal eigen-modes of a pinned–pinned beam and rigid body modes. Zhou and Cheung [32] employed an analytical-Ritz method to investigate a rectangular plate in contact with water on one side. Ugurlu et al. [33] investigated the effects of elastic foundation and fluid on the dynamic response characteristics of rectangular Kirchhoff plates using a boundary element method. Kerboua et al. [34] developed a combination of the finite element method and Sanders’shell theory to study the vibration analysis of rectangular plates in contact with fluid. Very recently, Hosseini Hashemi et al. [35,36] presented a comprehensive investigation on hydroelastic vibration analysis of horizontal and vertical rectangular plates resting on Pasternak foundation for different boundary conditions. To analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate were expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions. According to the above literature survey, no research work has been conducted on free vibration analysis of moderately thick rectangular plates coupled with quiescent fluid when the exact characteristic equations of the Mindlin rectangular plate are employed. Since an exact close-form solution exists only for the rectangular plates having at least two opposite simply-supported edges, six possible combinations of boundary conditions, namely SSSS, SCSS, SCSC, SSSF, SFSF and SCSF, are taken into account. First, results obtained by the present exact solution are compared with existing experimental and numerical data. Second, the effect of plate parameters such as aspect ratios, thickness to length ratios as well as boundary conditions, and fluid parameters such as fluid levels and fluid densities on natural frequencies of the plate is comprehensively investigated. Finally, some 3-D mode shapes of the rectangular Mindlin plates coupled with fluid are illustrated. 2. Plate governing equations A flat, isotropic and moderately thick rectangular plate of length a, width b, and uniform thickness h is depicted in Fig. 1. The Cartesian coordinate system (x1, x2, x3) is considered to extract mathematical formulations when x1 and x2 axes are located in the undeformed mid-plane of the plate. The displacements along the x1 and x2 axes are denoted by U1 and U2, respectively, while the displacement in the direction perpendicular to the undeformed middle surface is denoted by U3. In the Mindlin plate theory, the displacement components are assumed to be given by
U 1 ¼ x3 w1 ðx1 ; x2 ; tÞ; U 2 ¼ x3 w2 ðx1 ; x2 ; tÞ; U 3 ¼ w3 ðx1 ; x2 ; tÞ:
ð1Þ
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Fig. 1. Rectangular Mindlin plate, with coordinate convention.
where w1 and w2 are the rotational displacements about the x2 and x1 axes at the midsurface of the plate, respectively, w3 is the transverse displacement and t is the time variable. By neglecting the stress–strain relations containing e33, the general strain–displacement relations for small deformation are defined as
e11 ¼ x3 w1;1 ; e22 ¼ x3 w2;2 ; e12 ¼ x3
w1;2 þ w2;1 ; 2
e13 ¼
w1 w3;1 ; 2
e23 ¼
w2 w3;2 ; 2
ð2Þ
where eij i, j = (1, 2, 3) is the normal strain when i = j and the shear strain when i < j. Here, a comma-subscript convention represents the partial derivative. Based on the strain–displacement relations given in Eq. (2) and assuming a stress distribution in accordance with Hook’s law, the resultant bending moments, twisting moments, and the transverse shear forces, all per unit length in terms of w1, w2 and w3 are obtained by integrating the stresses and moment of the stresses through the thickness of the plate. These are given by
M 11 ¼ Dðw1;1 þ mw2;2 Þ; M 22 ¼ Dðw2;2 þ mw1;1 Þ; D M 12 ¼ ð1 mÞðw1;2 þ w2;1 Þ; 2 Q 1 ¼ j2 Ghðw1 w3;1 Þ; Q 2 ¼ j2 Ghðw2 w3;2 Þ
ð3Þ
in which j2 is the transverse shear correction coefficient, applied to the transverse shear forces due to the fact that the transverse shear strains (e13 and e23) have a nearly parabolic dependency to the thickness coordinate. D = Eh3/ [12(1 m2)] is the flexural rigidity, G = E/[2(1 + m)] is the shear modulus, m is the Poisson’s ratio and E is Young’s modulus of the plate. On the basis of the Mindlin plate theory, the governing differential equations of motion for the plate can be given in terms of the stress resultants by
1 qh3 w€ 1 ; 12 1 3€ M 12;1 þ M 22;2 Q 2 ¼ qh w 2; 12 €3 Q 1;1 þ Q 2;2 P ¼ qhw
M 11;1 þ M 12;2 Q 1 ¼
ð4Þ
where q is the mass density per unit volume and dot-overscript convention represents the differentiation with respect to the time variable t. Since the dynamic pressure of fluid is perpendicular to the plate surface, it appears only in x3 direction as an applied load P in Eq. (4) and can be expressed as
p ¼ m
@2 U3 € 3; ¼ m w @t2
ð5Þ
where m⁄, calculated in Section 3, is the added mass (AM) and dependent on fluid modeling and fluid–plate conditions. For a harmonic solution, the rotational and transverse displacements are assumed to be
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1 ðX 1 ; X 2 Þejxt ; w1 ðx1 ; x2 ; tÞ ¼ w 2 ðX 1 ; X 2 Þejxt ; w ðx1 ; x2 ; tÞ ¼ w
ð6Þ
2
1 jxt : w3 ðx1 ; x2 ; tÞ ¼ w 3 ðX 1 ; X 2 Þe a where x denotes the natural frequency of vibration in radians and j ¼ dimensionless. Introducing the non-dimensional parameters
X 1 ¼ x1 =a;
X 2 ¼ x2 =b;
d ¼ h=a;
pffiffiffiffiffiffiffi 1. Note that each parameter, having the over-bar, is
qffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ a=b; b ¼ xa2 qh=D; c ¼
m qh
ð7Þ
and using Eqs. (5)–(7), Eq. 4 can be rewritten in dimensionless form as 2 2 2 1;11 þ g2 w 1;22 þ t2 ðw 2;12 Þ 12j ðw ¼ 0; 1 w 3;1 Þ þ b d w w þ gw 2 t1 1;11 12 t1 1 d 2 2 2 2;11 þ g2 w 2;22 þ t2 gðw 2;22 Þ 12j ðw 3;2 Þ þ b d w ¼ 0; 1;12 þ gw 2 gw w 2 t1 12 t1 2 d 2 2 3;11 þ g2 w 3;22 ðw 2;2 Þ þ ð1 þ cÞ b d w 3 ¼ 0; 1;1 þ gw w 12K 2 t1
ð8Þ
where
t1 ¼
1v ; 2
t2 ¼
1þv : 2
ð9Þ
The boundary conditions along the edges X1 = 0 and X1 = 1, considered to remain simply supported, are as follows:
2 ¼ w 3 ¼ 0: M11 ¼ w
ð10Þ
The boundary conditions along the edges X2 = 0 and X2 = 1 are as follows: – for a free edge
M22 ¼ M12 ¼ Q 2 ¼ 0;
ð11Þ
– for a simply supported edge
1 ¼ w 3 ¼ 0; M22 ¼ w
ð12Þ
– for a clamped edge
1 ¼ w 2 ¼ w 3 ¼ 0: w
ð13Þ
The general solutions to Eq. (8) in terms of the three dimensionless potentials W1, W2 and W3 may be expressed as
1 ¼ C 1 W 1;1 þ C 2 W 2;1 gW 3;2 ; w 2 ¼ C 1 gW 1;2 þ C 2 gW 2;2 W 3;1 ; w w3 ¼ W 1 þ W 2 ;
ð14Þ
where
C1 ¼ 1
a22 a2 ; C2 ¼ 1 1 2 2 m1 a3 m1 a3
ð15Þ
in which a21 ; a22 and a23 are the coefficients that may be determined using equations of motion and can be given after mathematical manipulation by
2
3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u 2 u b d ð1 þ cÞ d ð1 þ cÞ 4ð1 þ cÞ7 a21;2 ¼ 6 þ1 t þ1 þ 4 5; 2 2 2 12 12 b2 k m1 k m1 ! 2 12k b2 d4 a23 ¼ 2 ð1 þ cÞ 1 : 2 d 144k m1 2
2
!
Eq. (8) can be restated in terms of the three dimensionless potentials as
ð16Þ
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W 1;11 þ g2 W 1;22 ¼ a21 W 1 ; W 2;11 þ g2 W 2;22 ¼ a22 W 2 ; 2
ð17Þ
2 3W 3:
W 3;11 þ g W 3;22 ¼ a
One set of solutions to Eq. (17) are taken as
W 1 ¼ ½A1 Sinðk1 X 2 Þ þ A2 Cosðk1 X 2 ÞSinðl1 X 1 Þ þ ½A3 Sinðk1 X 2 Þ þ A4 Cosðk1 X 2 ÞCosðl1 X 1 Þ; W 2 ¼ ½A5 Sinhðk2 X 2 Þ þ A6 Coshðk2 X 2 ÞSinðl2 X 1 Þ þ ½A7 Sinhðk2 X 2 Þ þ A8 Coshðk2 X 2 ÞCosðl2 X 1 Þ;
ð18Þ
W 3 ¼ ½A9 Sinhðk3 X 2 Þ þ A10 Coshðk3 X 2 ÞSinðl3 X 1 Þ þ ½A11 Sinhðk3 X 2 Þ þ A12 Coshðk3 X 2 ÞCosðl3 X 1 Þ in which l = mp (m = 1, 2, . . .), Ai are the arbitrary coefficients, kj and lj are related to the aj by
a21 ¼ l21 þ g2 k21 ; a22 ¼ l22 g2 k22 ; a23 ¼ l23 g2 k23 :
ð19Þ
On the assumption of simply-supported conditions at edges X1 = 0 and 1, Eq. (18) is given by
W 1 ¼ ½A1 Sinðk1 X 2 Þ þ A2 Cosðk1 X 2 ÞSinðlX 1 Þ; W 2 ¼ ½A5 Sinhðk2 X 2 Þ þ A6 Coshðk2 X 2 ÞSinðlX 1 Þ;
ð20Þ
W 3 ¼ ½A9 Sinhðk3 X 2 Þ þ A10 Coshðk3 X 2 ÞSinðlX 1 Þ; Introducing Eq. (20) in Eq. (14) and substituting the results into the appropriate boundary conditions along the edges X2 = 0 and 1, leads to six homogenous equations. To obtain non-trivial solution of these equations, the determinant of coefficients matrix must be zero, which yields characteristic equations for rectangular Mindlin plates with six combinations of boundary conditions, namely, SSSS, SSSC, SCSC, SSSF, SFSF and SCSF. 3. Fluid formulation The following assumptions are made to model dynamic behavior of the fluid as the mathematical formulations: (a) The amplitude of the vibrations is small (i.e., fluid motion is small). (b) The fluid is incompressible, inviscid and irrotational (i.e., the fluid flow is potential). The velocity potential function / must satisfy the Laplace equation throughout the fluid domain. This relation is expressed in the Cartesian coordinate system as:
r2 / ¼
@2/ @2/ @2/ þ þ ¼ 0: @x21 @x22 @x23
ð21Þ
Using the Bernoulli’s equation and ignoring the nonlinear terms, the fluid pressure at the plate–fluid interface (upper and lower surface of the plate) may be given by
@/ ; @t x3 ¼h 2 @/ ¼ qf ; @t h
pu ¼ pjx3 ¼h ¼ qf 2
pL ¼ pjx3 ¼h 2
ð22 - aÞ ð22-bÞ
x3 ¼ 2
where qf is the fluid density per unit volume. It is assumed that the out-of-plane velocity component of the fluid on the plate surface and the plate transversal velocity are identical due to fact that there exists permanent contact between the plate surface and the peripheral fluid layer. The impermeability condition of the plate’s surface can be expressed by
@/ @U 3 ¼ ; @x3 x3 ¼h @t 2 @/ @U 3 ¼ : @x3 x3 ¼h @t
ð23-aÞ ð23-bÞ
2
The method of separation of variables is adopted for the potential velocity function as follows:
/ðx1 ; x2 ; x3 ; tÞ ¼ Fðx3 ÞGðx1 ; x2 ; tÞ;
ð24Þ
where F(x3) and G(x1, x2, t) are two separate functions which must be determined. The velocity potential function on the upper and lower surfaces of the plate can be expressed by introducing Eq. (24) into Eq. (23) and then substituting G(x1, x2, t) into Eq. (24):
S. Hosseini-Hashemi et al. / Applied Mathematical Modelling 36 (2012) 764–778
/ðx1 ; x2 ; x3 ; tÞ ¼
/ðx1 ; x2 ; x3 ; tÞ ¼
Fðx3 Þ @U 3 ; @t
dF dx3
ð25-aÞ
x3 ¼2h
Fðx3 Þ @U 3 : @t
dF dx3
769
ð25-bÞ
x3 ¼2h
Substituting Eq. (25) into relation (21) leads to the following second-order differential equation: 2
d Fðx3 Þ 2
dx3
l2f Fðx3 Þ ¼ 0;
ð26Þ
where lf is a plane wave number and a real constant which must be specified. The general solution of Eq. (26) is written as:
Fðx3 Þ ¼ B1 elf x3 þ B2 elf x3 ;
ð27Þ
where B1 and B2 are constants which must be specified. By introducing Eq. (27) into Eq. (25), one gets the following expressions for the velocity potential function:
B1 elf x3 þ B2 elf x3 @U 3 ; dF @t dx3 h
ð28-aÞ
B1 elf x3 þ B2 elf x3 @U 3 /ðx1 ; x2 ; x3 ; tÞ ¼ : dF @t dx3 h
ð28-bÞ
/ðx1 ; x2 ; x3 ; tÞ ¼
x3 ¼2
x3 ¼2
3.1. Boundary conditions of a plate–fluid The boundary conditions at the fluid–structure interface along with the fluid extremity must be satisfied by adopting an appropriate velocity potential function. Fluid free surface, rigid wall and impermeability are the boundary conditions which are generally taken into account. In order to achieve a good understanding of the problem, a flexible rectangular plate submerged in fluid is studied, where the following conditions should be considered. 3.1.1. Plate–fluid model with free surface (BC1) At the fluid free surface, the following condition may be applied to the velocity potential (see Fig. 2), provided that free surface motion of the fluid creates insignificant perturbations,
@/ 1 @ 2 / ¼ @x3 x3 ¼h1 þh g @t 2 2 x
ð29Þ
; h 3 ¼h1 þ2
where g is acceleration due to gravity. Substituting the above boundary condition into Eq. (28-a), the following expression for the potential function is obtained:
Fig. 2. Plate–fluid model with free surface.
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elf x3 þ c1 elf ð2h1 þhx3 Þ @U 3 h ; h lf elf 2 c1 elf ð2h1 þ2Þ @t
/¼
where c1 ¼
lf g x2 : lf g þ x2
ð30Þ
The fluid pressure applying on the upper surface of the plate is obtained by introducing the above relationship into the Bernoulli’s equation:
pu ¼
qf 1 þ c1 e2lf h1 @ 2 U 3 @2 U3 ¼ m1 : lf 1 c1 e2lf h1 @t2 @t 2
ð31Þ
3.1.2. Plate–fluid model bounded by a rigid wall (BC2) The boundary condition at the wall, as shown in Fig. 3(a), was studied by Lamb [8] and referred to as the null-frequency condition. This rigid wall boundary condition is expressed as:
@/ ¼ 0: @x3 x3 ¼h2
ð32Þ
Similarly, by introducing Eq. (28-b) into relations (32), the following expression for the velocity potential is obtained:
elf x3 þ c2 elf x3 @U 3 ; h h lf elf 2 c2 elf 2 @t
/¼
where c2 ¼ e2lf h2 :
ð33Þ
The dynamic pressure for this case (lower surface of the plate) is determined as:
pL ¼
qf 1 þ c2 elf h @ 2 U 3 @2 U3 ¼ m2 : 2 l h lf 1 c2 e f @t @t2
ð34Þ
In case of totally submerged plate, as shown in Fig. 3(b), the total dynamic pressure will be a combination of the pressures corresponding to the fluid boundary conditions at both top and bottom surfaces of the plate.
@2 U3 @ 2 U3 p ¼ pu pL ¼ m1 m2 ¼ m : 2 @t @t 2
ð35Þ
3.1.3. Fluid bounded between two identical plates (BC3) Fig. 4 illustrates fluid surrounded between two parallel plates, which depict the side walls of reservoir. Impermeability conditions for both plates at x3 = h/2 and x3 = d + h/2 are considered. By introducing these boundary conditions into Eq. (28), the potential function is obtained as follows:
/¼
elf x3 þ c3 elf x3 @U 3 h ; h lf elf 2 c3 elf 2 @t
elf ðdþ2Þ elf 2 ; h h elf ðdþ2Þ elf 2 h
where c3 ¼
h
ð36Þ
where d is the distance between the two plates. The fluid pressure on each wall is obtained by introducing Eq. (36) into Bernoulli’s equation as:
p¼
qf 1 2elf d þ e2lf d @ 2 U 3 : lf @t 2 1 e2lf d
Fig. 3. Plate–fluid model bounded by a rigid wall. (a) A floating plate. (b) A submerged plate.
ð37Þ
S. Hosseini-Hashemi et al. / Applied Mathematical Modelling 36 (2012) 764–778
771
Fig. 4. Plate–fluid model representing two identical plates coupled with bounded fluid.
3.2. Determination of lf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The lf is net wave number which will be the magnitude of the wave motion in the x and x directions as l ¼ l21 þ l22 . 1 2 f pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kerboua et al. [33] used a simple form of lf ¼ ðp=aÞ 1 þ g2 for all boundary conditions. Though the value of lf varies for various boundary conditions, they ignored the effect of boundary conditions on the lf. In addition, the wave number presented in Ref. [33] is independent of air frequency and is constant when mode number changes. In this study, the wave number parameter proposed in Ref. [24] is modified by using Mindlin plate parameters. It is worthwhile to mention that as higher degrees of edge constraint (in the order from free to simply supported to clamped) are applied to the other two edges of the rectangular plate, the variation of the lf is more tangible (e.g., from SFSF to SCSC). The following wave number parameters are introduced when at least one of the rectangular plate edges can freely vibrate (i.e., SFSF, SCSF and SSSF):
l1 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np2 ; a2 a
l2 ¼
2p ; L
ð38Þ
where
2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi3 ! ! v !2 u 2 2 u d2 2 b d 1 1 4 6 7 t a2 ¼ a2 4 þ1 þ þ 1 þ 2 5; 2 2a 12 k2 m1 12 ba k m1 L is width of tank and ba is dimensionless frequency parameter in vacuum. For other three boundary conditions (i.e., SSSS, SCSC and SSSC), l2 in Eq. (38) is replaced with l2 = mp/a, while l1 will remain unchanged. Note that when l1 takes a comqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 must be used instead of that as l 1 ¼ a2 þ nap 2 . plex value, l 4. Comparison study In this paper, the results obtained by the present analytical solution for the rectangular plates coupled with fluid are compared with those acquired by the finite element method (FEM) [34] and experimental results [24] to show the applicability, reliability and effectiveness of the presented formulation. The natural frequencies of a steel rectangular SFSF thin plate completely surrounded by water are calculated while its two short edges are simply-supported. The mechanical and geometrical properties of plate are: Young’s modulus E = 207 GPa, material density q = 7850 kg m3, Poisson’s ratio t = 0.3, a = 0.655 m, b = 0.20165 m and thickness h = 9.36 mm. The fluid density is taken to be 1000 kg m3. The experimental apparatus was set up by placing the plate in a rectangular reservoir with dimensions 1.3 m 0.55 m 0.8 m. The total dynamic pressure is therefore the sum of lower and upper pressures and can be calculated using Eq. (35). In Table 1, natural frequencies of the plate in (Hz) unit obtained by the present theory are presented and compared with both experimental and numerical data described in Refs. [24, 34]], respectively. As can be seen, there is a good agreement between the present results and available data. However, it should be noted that the results of the present exact solution are much closer to the experimental results [24] than those of FEM [34]. A comparison study of the first six frequency parameters obtained by the CPT [27] and the present method is conducted in Table 2 for a square SSSS plate (d = 0.05 and k = 1) in contact with cubic volume of fluid, showing a good agreement between the results. The plate-water density ratio r is set to be 0.125. In view of Table 2, it is seen that all results obtained on the basis of the CPT are greater than those of the present method. It is worth noting from Table 2 that the CPT is unable to predict the wet frequency of the plate accurately for both d = 0.05 and the higher modes. This is attributed to the fact that the CPT overestimates the natural frequencies of the plate due to ignoring the effect of shear deformation and rotary inertia. These effects are more pronounced in the higher modes than in the lower modes.
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Table 1 Comparison of natural frequencies of a plate x(Hz), with S–F–S–F boundary condition, submerged in water. Mode no.
1 2 3 4 5
In vacuum
Completely submerged in water
Present theory
Present theory
Experiment [24]
FEM [34]
51.005 205.530 228.131 464.979 490.227
27.756 117.924 153.816 285.274 326.802
28.72 117.125 154.51 281.79 335.04
31.28 126.40 141.78 285.98 304.57
Table 2 Comparison of the first six frequency parameters for a square plate in contact with cubic volume of fluid.
a
B.C.
Method
b1
b2
b3
b4
b5
b6
SSSS
CPTa Present method
11.844 11.620
39.243 38.681
39.243 38.681
66.609 65.477
81.643 75.877
85.674 78.502
Results from Cheung and Zhou [27].
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffi First five frequency parameter b ¼ xa2 qh=D for a Mindlin plate submerged in water as a function of fluid level (d = 0.05, g = 2, h = 0.1 m and m = 0.3). Boundary conditions
Mode (m, n)
SFSF
(1, 1) (1, 2) (2, 1) (2, 2) (3, 1)
SSSF
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2)
SCSF
In vacuum 9.45825 26.4801 37.7831 61.6215 83.8573
h1 a
h1 a
¼0
¼ 0:1
h1 a
¼ 0:3
h1 a
¼ 0:5
h1 a
¼2
6.71860 23.7110 30.8581 54.9017 73.3207
6.33350 22.0905 28.7943 51.1114 68.1336
5.84300 21.6735 27.1502 50.0239 66.0611
5.62620 21.6657 26.8956 49.9987 65.9697
5.49410 21.6656 26.7929 49.9982 65.9660
15.8630 45.5242 72.3859 91.7889 104.764
12.1114 36.8051 67.5175 80.0587 96.2272
11.3383 34.2752 63.7460 74.3911 90.2427
10.5147 32.1789 63.5176 72.0377 89.4967
10.2633 31.7937 63.5174 71.9260 89.4934
10.1766 31.6093 63.5174 71.9103 89.4934
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2)
22.2469 49.0424 93.5256 93.9506 121.897
17.7375 39.4316 80.4486 80.7430 113.016
16.5479 36.7620 74.7767 75.0248 106.465
15.4747 34.4537 71.8909 72.1175 105.841
15.2413 33.9927 71.6879 71.9131 105.839
15.1076 33.9002 71.6748 71.8999 105.839
SSSS
(1, 1) (2, 1) (3, 1) (1, 2) (2, 2)
48.3006 76.3360 121.632 156.685 182.338
41.4293 64.5257 110.369 149.690 171.969
38.4638 59.9367 103.140 143.620 163.489
36.9582 57.1981 101.763 143.556 163.188
36.8523 56.9347 101.749 143.556 163.188
36.8455 56.9111 101.748 143.556 163.188
SCSS
(1, 1) (2, 1) (3, 1) (1, 2) (4, 1)
66.3511 89.7039 130.980 187.564 188.898
58.3467 77.0089 118.969 177.610 178.851
54.3177 71.6786 111.288 168.602 169.498
52.7481 68.8209 109.774 167.961 168.709
52.6826 68.6026 109.758 165.564 166.302
52.6801 68.5873 109.758 165. 543 166. 300
SCSC
(1, 1) (2, 1) (3, 1) (1, 2) (2, 2)
88.6635 106.9478 142.979 219.582 237.921
79.6484 93.4901 130.091 212.346 227.422
74.4777 87.1999 121.882 205.806 218.380
72.9584 84.2839 120.191 205.781 218.206
72.9222 84.1148 120.171 205.781 218.206
72.9214 84.1058 120.171 205.781 218.206
Table 4 Fundamental natural frequencies of a plate immersed in different fluid for six boundary conditions when h1/a = 0.1. Fluid
Density (kg m3)
Gasoline (16 °C) Alcohol ethyl (25 °C) Kerosene (16 °C) Water pure (4 °C) Sea water (25 °C) Milk (4% fat, 20 °C)
737.22 785.1 817.15 1000 1025 1033
Boundary conditions SFSF
SSSF
SCSF
SSSS
SSSC
SCSC
6.8491 6.7458 6.6792 6.3335 6.2902 6.2766
12.1447 11.9849 11.8814 11.3383 11.2697 11.2480
17.6128 17.4033 17.2673 16.5479 16.4564 16.4274
40.4576 40.0712 39.8187 38.4638 38.2891 38.2337
56.8364 56.3513 56.0335 54.3176 54.0951 54.0244
77.5371 76.9516 76.5669 74.4777 74.2051 74.1185
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Fig. 5. The fundamental frequency parameter b versus the thickness to length ratio d for the SCSS rectangular plate subjected to three boundary conditions of the fluid when g = 1.5, h1/a = 0.5 and d = 0.2.
Fig. 6. The fundamental frequency parameter b versus the aspect ratio g for the SCSS rectangular plate subjected to three boundary conditions of the fluid when d = 0.1, h1/a = 0.5, d = 0.2 and h = 0.1.
Fig. 7. The fundamental frequency parameter b versus the thickness to length ratio d for the SFSF, SSSS and SCSC rectangular plate when g = 2 and h1/a = 0.1.
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Fig. 8. The fundamental frequency parameter b versus the thickness ratio d for the SCSF rectangular plate (g = 2) with three different fluid levels.
Fig. 9. The fundamental frequency parameter b versus the aspect ratio g for the SFSF, SSSS and SCSC rectangular plate when d = 0.05, h1/a = 0.1 and h = 0.1.
Fig. 10. The fundamental frequency parameter b versus the aspect ratio g for the SCSF rectangular plate with three different fluid levels when d = 0.05 and h = 0.1.
5. Results and discussion In this section, natural frequencies of the rectangular Mindlin plates coupled with fluid are presented in tabular and graphical forms for different plate parameters. Note that except for Section 5.1, all results in this paper are presented
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for the plate submerged partially or totally in fluid (BC2) when the dimensions of the tank are assumed to be 5 m 5 m 5 m. The first five dimensionless natural frequencies of rectangular Mindlin plates (d = 0.05 and g = 2) with six combinations of boundary conditions are listed in Table 3 for different fluid levels when h = 0.1 m and m = 0.3. It is seen that regardless of boundary conditions, the frequency parameter b decreases with increasing fluid level (h1). It is interesting to note that the frequency parameter b is almost independent of h1 when the plate is immersed in a level higher than 50% of the plate length. As expected, the frequency parameter b enhances as the higher degrees of edge constraint (in the order from free to simply supported to clamped) are applied to the other two edges of the rectangular plate. Due to the practical applications, the fundamental natural frequency of rectangular Mindlin plate (d = 0.05 and g = 2) coupled with different fluids is listed in Table 4 for different boundary conditions when h1/a = 0.1 and h = 0.1. It is evident from Table 4 that when the plate is submerged in denser fluid, the fundamental natural frequency of the plate takes lower values.
5.1. Effect of different boundary conditions of the fluid on the frequency parameter In Fig. 5, the variation of the fundamental frequency parameter b against the thickness to length ratio d for the SCSS rectangular Mindlin plate is shown for g = 1.5, h1/a = 0.5 and d = 0.2. The plate under consideration is subjected to three
Fig. 11. Deformed mode shapes and frequency parameters of a SFSF submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 7.35783, (m, n) = (1, 1); (b) b = 9.97149, (m, n) = (1, 2) and (c) b = 14.8906, (m, n) = (1, 3).
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Fig. 12. Deformed mode shapes and frequency parameters of a SSSF submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 7.82004, (m, n) = (1, 1); (b) b = 12.4961, (m, n) = (1, 2) and (c) b = 18.4169, (m, n) = (1, 3).
boundary conditions of the fluid, including fluid with free surface (BC1), fluid bounded by a rigid wall (BC2) and fluid bounded between two identical plates (BC3). It is seen from Fig. 5 that with the increase of the thickness to length ratio d, the fundamental frequency parameter b generally decreases for any boundary conditions of the fluid. However, an contrary behavior is observed in BC1 and BC2 for d 6 0.075 and d 6 0.1, respectively. Fig. 6 shows the behavior of the fundamental frequency parameter b against the aspect ratio g for the SCSS rectangular Mindlin plate under three boundary conditions of the fluid when d = 0.1, h1/a = 0.5, d = 0.2 and h = 0.1. It is obvious that regardless of the fluid boundary conditions, the fundamental frequency parameter b increases as the aspect ratio g increases. It can also be inferred from Fig. 6 that for a given value of the aspect ratio g, the fundamental frequency parameter b of the plate coupled with the fluid, bounded between two identical plates (BC3), takes the largest value in comparison with two other fluid boundary conditions.
5.2. Effect of thickness to length ratio on the frequency parameter Fig. 7 displays the variation of the frequency parameter b versus the thickness to length ratio d for the SCSC, SSSS and SFSF rectangular Mindlin plate when g = 2 and h1/a = 0.1. It can apparently be observed that the frequency parameter b for the SSSS rectangular plate initially increases when the thickness to length ratio d varies between 0.05 and 0.1, and then decreases when d > 0.1. However, with the increase of the thickness to length ratio d, the frequency parameter b for
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Fig. 13. Deformed mode shapes and frequency parameters of a SCSC submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 11.1575, (m, n) = (1, 1); (b) b = 18.8624, (m, n) = (1, 2) and (c) b = 28.0825, (m, n) = (1, 3).
the SCSC rectangular plate continuously diminishes. Another interesting point about Fig. 7 is that the effect of the thickness to length ratio d on the frequency parameter b of the SFSF rectangular Mindlin plate is insignificant. The behavior of the frequency parameter b against thickness to length ratio d for the SCSF rectangular Mindlin plate (g = 2) is shown in Fig. 8 for three different fluid levels h1/a = 0, 0.1 and 0.5. It can be figured out from Fig. 8 that the maximum frequency parameter b lies between d = 0.1 and 0.125 for any values of fluid level within 0 and 0.5. It is worth noting that the maximum frequency parameter b shifts toward higher values of the thickness to length ratio d as the plate is submerged in deeper levels of fluid. 5.3. Effect of aspect ratio on the frequency parameter Fig. 9 displays the variation of the frequency parameter b versus the aspect ratio g for SFSF, SSSS and SCSC rectangular Mindlin plates (d = 0.05) when h1/a = 0.1 and h = 0.1. It can be seen that contrary to the SFSF rectangular plate, the frequency parameter b for the SSSS and SCSC rectangular plate increases as the aspect ratio g increases. Fig. 10 contains the plot of the frequency parameter b versus aspect ratio g for the SCSF rectangular Mindlin plate (d = 0.05 and h = 0.1) for three different fluid levels h1/a = 0, 0.1 and 0.5. It is obviously seen from Fig. 10 that with the enhancement of the aspect ratio g, the frequency parameter b increases for any fluid levels.
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5.4. Three-dimensional mode shapes To have a more appropriate sense of the transverse displacement w3, three first mode shapes of a submerged rectangular Mindlin plate for d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2 are illustrated in Figs. 11–13 for SFSF, SSSF and SCSC boundary conditions, respectively. 6. Concluding remarks In this paper, free vibration of totally or partially submerged rectangular Mindlin plates was investigated. The simplifying hypothesis that mode shapes are not to be modified by the fluid is not assumed in this study. In order to achieve an analytical solution, the plate must have two opposite simply-supported edges. Governing equations of plate coupled with stationary fluid were derived by combining Mindlin plate theory and potential flow theory. To verify the merit and accuracy of the present exact solution, a comparison was made with the experimental and numerical results reported in the literature. It was shown that the results obtained by the present method are an excellent agreement with the experimental results reported by Haddara and Cao [24]. Then, a detailed parametric study was conducted to show the influence of different fluid depths, fluid densities, aspect ratios and thickness to length ratios for six possible combinations of boundary conditions. Finally, some 3-D plots were shown for the mode shapes of the deflection in the submerged rectangular plates. Due to the inherent features of the present analytical solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
A.W. Leissa. Vibration of plates, NASA SP-160, Office of Technology Utilization, Washington, DC, 1969. J.N. Reddy, R.A. Arciniega, Shear deformation plate and shell theories: from Stavsky to present, Mech. Adv. Mater. Struct. 11 (2004) 535–582. E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech.-T ASME 12 (1945) 69–77. R.D. Mindlin, Influence of rotatory inertia and shear in flexural motion of isotropic elastic plates, J. Appl. Mech.-T ASME 18 (1951) 31–38. R.D. Mindlin, H. Deresiewicz, Thickness-shear and flexural vibrations of a circular disk, J. Appl. Phys. 25 (1954) 1329–1332. R.D. Mindlin, A. Schaknow, H. Deresiewicz, Flexural vibration of rectangular plates, J. Appl. Mech.-T ASME 23 (1956) 430–436. S.H. Hosseini Hashemi, M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, Int. J. Solids Struct. 42 (2005) 819–853. H. Lamb, On the vibrations of an elastic plate in contact with water, Proc. R. Soc. London, Ser. A 98 (1920) 205–206. J.H. Powell, J.H.T. Roberts, On the frequency of vibration of circular diaphragms, Proc. Phys. Soc. London (1923) 35170–35182. N.W. McLachlan, The accession to inertia of flexible discs vibrating in a fluid, Proc. Phys. Soc. London 44 (1932) 546–555. W.H. Peake, E.G. Thurston, The lowest resonant frequency of a water-loaded circular plate, J. Acoust. Soc. Am. 26 (7) (1954) 166–168. F. Kito, On the added mass of flat plates vibrating in water, Zoxen Kyokai Japan 266 (1944) 1–10 (in Japanese). U.S. Lindholm, D.D. Kana, W.H. Chu, H.N. Abramson, Elastic vibration characteristics of cantilever plates in water, J. Ship Res. 9 (1) (1965) 11–22. W.K. Meyerhoff, Added masses of thin rectangular plates calculated from potential theory, J. Ship Res. 14 (1970) 100–111. G. Muthuveerappan, N. Ganesan, M.A. Veluswami, A note on vibration of a cantilever plate immersed in water, J. Sound Vib. 63 (1979) 385–391. K.C. Kim, J.S. Kim, H.Y. Lee, An experimental study on the elastic vibration of plates in contact with water, J. Soc. Naval Archit. 16 (2) (1979) 1–7. G.C. Volcy, P. Morel, M. Bureau, K. Tanida. Some studies and researches related to the hydroelasticity of steel work, in: Proceedings of 122nd EUROMECH Colloquium on Numerical Analysis of the Dynamics of Ship Structures, Ecole Polytechnique, Paris, 1979, pp. 403–436. F.M. Espinosa, J.A. Gallego-Juarez, On the resonance frequencies of water-loaded circular plates, J. Sound Vib. 94 (2) (1984) 217–222. Y. Fu, W.G. Price, Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid, J. Sound Vib. 118 (1987) 495–513. N.J. Robinson, S.C. Palmer, A modal analysis of a rectangular plate floating on an incompressible liquid, J. Sound Vib. 142 (1990) 453–460. M.K. Kwak, K.C. Kim, Axisymmetric vibration of circular plates in contact with fluid, J. Sound Vib. 146 (3) (1991) 381–389. P. Hagedorn, A note on the vibration of infinite elastic plates in contact with water, J. Sound Vib. 175 (1994) 233–240. M.K. Kwak, Hydroelastic vibration of rectangular plates, J. Appl. Mech.-T ASME 63 (1996) 110–115. M.R. Haddara, S. Cao, A study of the dynamic response of submerged rectangular flat plates, Marine Struct. 9 (1996) 913–933. M. Amabili, G. Frosali, M.K. Kawk, Free vibrations of annular plates coupled with fluids, J. Sound Vib. 191 (5) (1996) 825–846. M.H. Meylan, The forced vibration of a thin plate floating on an infinite liquid, J. Sound Vib. 205 (5) (1997) 581–591. Y.K. Cheung, D. Zhou, Coupled vibratory characteristics of a rectangular container bottom plate, J. Fluids Struct. 14 (2000) 339–357. A. Ergin, B. Ugurlu, Linear vibration analysis of cantilever plates partially submerged in fluid, J. Fluids Struct. 17 (2003) 927–939. C.C. Liang, C.C. Liao, Y.S. Tai, W.H. Lai, The free vibration analysis of submerged cantilever plates, Ocean Eng. 28 (2001) 1225–1245. K.H. Jeong, G.H. Yoo, S.C. Lee, Hydroelastic vibration of two identical rectangular plates, J. Sound Vib. 272 (2004) 539–555. R.E. Tayler, M. Ohkusu, Green functions for hydroelastic analysis of vibrating free–free beams and plates, Appl. Ocean Res. 22 (2000) 295–314. D. Zhou, Y.K. Cheung, Vibration of vertical rectangular plate in contact with water on one side, Earthquake Eng. Struct. Dyn. 29 (2000) 693–710. B. Ugurlu, A. Kutlu, A. Ergin, M.H. Omurtag, Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid, J. Sound Vib. 317 (2008) 308–328. Y. Kerboua, A.A. Lakis, M. Thomas, L. Marcouiller, Vibration analysis of rectangular plates coupled with fluid, Appl. Math. Model. 32 (2008) 2570–2586. S.H. Hosseini Hashemi, M. Karimi, H. Rokni Damavandi Taher, Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method, Ocean Eng. 37 (2010) 174–185. S.H. Hosseini Hashemi, M. Karimi, H. Rokni Damavandi Taher, Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads, Soil Dyn. Earthquake Eng. 30 (2010) 1487–1499.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Natural frequencies of rectangular Mindlin plates coupled with stationary fluid Shahrokh Hosseini-Hashemi a,⇑, Mahmoud Karimi a, Hossein Rokni a,b a b
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16848-13114, Iran School of Engineering, University of British Columbia (Okanagan), 3333 University Way, Kelowna, BC, Canada V1V 1V7
a r t i c l e
i n f o
Article history: Received 10 March 2009 Received in revised form 22 June 2011 Accepted 1 July 2011 Available online 13 July 2011 Keywords: Free vibration Rectangular plate Mindlin theory Fluid–structure interaction Added mass
a b s t r a c t The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Plate structures have been widely used in various engineering applications, from space vehicles to microelectromechanical system (MEMS) devices. Many researchers have worked on vibration of such plates. Excellent reference source may be found in the open literature (see, for example, [1,2]). According to the literature, it is found that many researchers have used the classical plate theory (CPT), overestimating the natural frequencies of the plate due to ignoring the effect of shear deformation through the plate thickness. In order to eliminate the deficiency of the CPT for moderately thick plates, the first-order shear deformation plate theory was proposed by Reissner [3], and developed further for the deformable plates in statics and dynamics by Mindlin et al. [4–6]. A very beneficial study for deriving the exact closed-form characteristic equations of vibrating moderately thick rectangular plates was carried out by Hosseini Hashemi and Arsanjani [7]. Plates coupled with fluid are of great practical significance in nuclear, ocean and naval engineering. Thus, a good understanding of the dynamic interaction between an elastic plate and fluid is necessary. It is well known that the vibrating plates in contact with fluid result in the fluid motion. In addition, the existence of the fluid around the plate causes the kinetic energy to considerably increase. Consequently, the natural frequencies of plate coupled with fluid significantly decrease in comparison with those of the plate in the air.
⇑ Corresponding author. Tel.: +98 21 7391 2912; fax: +98 21 7724 0488. E-mail address: [email protected] (Sh. Hosseini-Hashemi). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.07.007
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Many studies on the free and forced vibration analysis of plates, partially or totally submerged in the fluid, have been carried out. Lamb [8] calculated the natural frequencies of a thin clamped circular plate in an aperture of an infinite rigid plane wall coupled with water based on the assumption that the wet mode shapes are almost the same as those in vacuum. Powell and Roberts [9] established an experimental setup to verify the theoretical results reported by Lamb [8]. McLachlan [10] performed a similar analysis to Lamb’s work [8] except that free boundary conditions were applied at the circumferential edge of the circular plate. Peak and Thurston [11] generalized the work of Lamb and McLachlan. Added virtual mass (AVM) of rectangular plates was calculated by Kito [12] using Fourier series method. Lindholm et al. [13] reported the natural frequencies of cantilever plates in air and in water, whereas the fluid actions were evaluated by a strip-theory approach. Added mass of thin rectangular plates in infinite fluid was obtained by Meyerhoff [14] and dipole singularities were employed to model the potential flow around a flat rectangular plate. Muthuveerappan et al. [15] investigated the freely vibrating behavior of a cantilever square plate immersed in water. Kim et al. [16] presented both experimental and theoretical results to investigate the effect of supporting boundary conditions on the vibrating plates. Fundamental natural frequencies of vertical cantilever plates, obtained by the experiment and by the fluid finite element method, were presented by Volcy et al. [17] when the plates were partially or totally submerged in liquid. Espinosa and Gallego-Juarez [18] acquired the pressure distribution of water on the free-edge circular plate. Fu and Price [19] employed a finite element discretization to analyze the dry and wet dynamic characteristics of a vertical and horizontal cantilever plate. Robinson and Palmer [20] conducted a study on the modal analysis of a rectangular plate resting on an incompressible fluid. Kwak and Kim [21] studied on axisymmetric vibration of circular plates in the presence of fluid on the basis of the mixed boundary value problem. Free vibration of infinite elastic rectangular plate in contact with water was studied by Hagedorn [22]. Kwak [23] utilized a piecewise division to investigate the free vibrations of rectangular plates in contact with unbounded water on one side, while beam functions were used as admissible functions. Haddara and Cao [24] investigated dynamic responses of rectangular plates immersed in fluid. An approximate expression for the evaluation of the modal added mass was derived for cantilever and SFSF rectangular plates and the numerical results were verified by the experimental ones. The natural frequencies of annular plates in contact with a fluid on one side were theoretically obtained by Amabili et al. [25] using the added mass approach, whereas the coupled fluid–structure system was solved by adopting the Hankel transform. Meylan [26] employed an appropriate Green’s function to study the forced vibration of an arbitrary thin plate floating on the surface of an infinite liquid. Cheung and Zhou [27] also studied the case of a horizontal rectangular plate composing the base of a rigid rectangular container. The dynamic characteristics of a vertical cantilever plate partially in contact with fluid were investigated by Ergin and Ugurlu [28]. Liang et al. [29] adopted an empirical added-mass formulation to determine the frequencies and mode shapes of submerged cantilevered plates. Based on a finite Fourier series expansion, Jeong et al. [30] studied the wet resonance frequencies and associated mode shapes of two identical rectangular plates coupled with a bounded fluid. Tayler and Ohkusu [31] suggested expressions for the free–free rectangular plates in terms of the sinusoidal eigen-modes of a pinned–pinned beam and rigid body modes. Zhou and Cheung [32] employed an analytical-Ritz method to investigate a rectangular plate in contact with water on one side. Ugurlu et al. [33] investigated the effects of elastic foundation and fluid on the dynamic response characteristics of rectangular Kirchhoff plates using a boundary element method. Kerboua et al. [34] developed a combination of the finite element method and Sanders’shell theory to study the vibration analysis of rectangular plates in contact with fluid. Very recently, Hosseini Hashemi et al. [35,36] presented a comprehensive investigation on hydroelastic vibration analysis of horizontal and vertical rectangular plates resting on Pasternak foundation for different boundary conditions. To analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate were expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions. According to the above literature survey, no research work has been conducted on free vibration analysis of moderately thick rectangular plates coupled with quiescent fluid when the exact characteristic equations of the Mindlin rectangular plate are employed. Since an exact close-form solution exists only for the rectangular plates having at least two opposite simply-supported edges, six possible combinations of boundary conditions, namely SSSS, SCSS, SCSC, SSSF, SFSF and SCSF, are taken into account. First, results obtained by the present exact solution are compared with existing experimental and numerical data. Second, the effect of plate parameters such as aspect ratios, thickness to length ratios as well as boundary conditions, and fluid parameters such as fluid levels and fluid densities on natural frequencies of the plate is comprehensively investigated. Finally, some 3-D mode shapes of the rectangular Mindlin plates coupled with fluid are illustrated. 2. Plate governing equations A flat, isotropic and moderately thick rectangular plate of length a, width b, and uniform thickness h is depicted in Fig. 1. The Cartesian coordinate system (x1, x2, x3) is considered to extract mathematical formulations when x1 and x2 axes are located in the undeformed mid-plane of the plate. The displacements along the x1 and x2 axes are denoted by U1 and U2, respectively, while the displacement in the direction perpendicular to the undeformed middle surface is denoted by U3. In the Mindlin plate theory, the displacement components are assumed to be given by
U 1 ¼ x3 w1 ðx1 ; x2 ; tÞ; U 2 ¼ x3 w2 ðx1 ; x2 ; tÞ; U 3 ¼ w3 ðx1 ; x2 ; tÞ:
ð1Þ
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Fig. 1. Rectangular Mindlin plate, with coordinate convention.
where w1 and w2 are the rotational displacements about the x2 and x1 axes at the midsurface of the plate, respectively, w3 is the transverse displacement and t is the time variable. By neglecting the stress–strain relations containing e33, the general strain–displacement relations for small deformation are defined as
e11 ¼ x3 w1;1 ; e22 ¼ x3 w2;2 ; e12 ¼ x3
w1;2 þ w2;1 ; 2
e13 ¼
w1 w3;1 ; 2
e23 ¼
w2 w3;2 ; 2
ð2Þ
where eij i, j = (1, 2, 3) is the normal strain when i = j and the shear strain when i < j. Here, a comma-subscript convention represents the partial derivative. Based on the strain–displacement relations given in Eq. (2) and assuming a stress distribution in accordance with Hook’s law, the resultant bending moments, twisting moments, and the transverse shear forces, all per unit length in terms of w1, w2 and w3 are obtained by integrating the stresses and moment of the stresses through the thickness of the plate. These are given by
M 11 ¼ Dðw1;1 þ mw2;2 Þ; M 22 ¼ Dðw2;2 þ mw1;1 Þ; D M 12 ¼ ð1 mÞðw1;2 þ w2;1 Þ; 2 Q 1 ¼ j2 Ghðw1 w3;1 Þ; Q 2 ¼ j2 Ghðw2 w3;2 Þ
ð3Þ
in which j2 is the transverse shear correction coefficient, applied to the transverse shear forces due to the fact that the transverse shear strains (e13 and e23) have a nearly parabolic dependency to the thickness coordinate. D = Eh3/ [12(1 m2)] is the flexural rigidity, G = E/[2(1 + m)] is the shear modulus, m is the Poisson’s ratio and E is Young’s modulus of the plate. On the basis of the Mindlin plate theory, the governing differential equations of motion for the plate can be given in terms of the stress resultants by
1 qh3 w€ 1 ; 12 1 3€ M 12;1 þ M 22;2 Q 2 ¼ qh w 2; 12 €3 Q 1;1 þ Q 2;2 P ¼ qhw
M 11;1 þ M 12;2 Q 1 ¼
ð4Þ
where q is the mass density per unit volume and dot-overscript convention represents the differentiation with respect to the time variable t. Since the dynamic pressure of fluid is perpendicular to the plate surface, it appears only in x3 direction as an applied load P in Eq. (4) and can be expressed as
p ¼ m
@2 U3 € 3; ¼ m w @t2
ð5Þ
where m⁄, calculated in Section 3, is the added mass (AM) and dependent on fluid modeling and fluid–plate conditions. For a harmonic solution, the rotational and transverse displacements are assumed to be
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1 ðX 1 ; X 2 Þejxt ; w1 ðx1 ; x2 ; tÞ ¼ w 2 ðX 1 ; X 2 Þejxt ; w ðx1 ; x2 ; tÞ ¼ w
ð6Þ
2
1 jxt : w3 ðx1 ; x2 ; tÞ ¼ w 3 ðX 1 ; X 2 Þe a where x denotes the natural frequency of vibration in radians and j ¼ dimensionless. Introducing the non-dimensional parameters
X 1 ¼ x1 =a;
X 2 ¼ x2 =b;
d ¼ h=a;
pffiffiffiffiffiffiffi 1. Note that each parameter, having the over-bar, is
qffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ a=b; b ¼ xa2 qh=D; c ¼
m qh
ð7Þ
and using Eqs. (5)–(7), Eq. 4 can be rewritten in dimensionless form as 2 2 2 1;11 þ g2 w 1;22 þ t2 ðw 2;12 Þ 12j ðw ¼ 0; 1 w 3;1 Þ þ b d w w þ gw 2 t1 1;11 12 t1 1 d 2 2 2 2;11 þ g2 w 2;22 þ t2 gðw 2;22 Þ 12j ðw 3;2 Þ þ b d w ¼ 0; 1;12 þ gw 2 gw w 2 t1 12 t1 2 d 2 2 3;11 þ g2 w 3;22 ðw 2;2 Þ þ ð1 þ cÞ b d w 3 ¼ 0; 1;1 þ gw w 12K 2 t1
ð8Þ
where
t1 ¼
1v ; 2
t2 ¼
1þv : 2
ð9Þ
The boundary conditions along the edges X1 = 0 and X1 = 1, considered to remain simply supported, are as follows:
2 ¼ w 3 ¼ 0: M11 ¼ w
ð10Þ
The boundary conditions along the edges X2 = 0 and X2 = 1 are as follows: – for a free edge
M22 ¼ M12 ¼ Q 2 ¼ 0;
ð11Þ
– for a simply supported edge
1 ¼ w 3 ¼ 0; M22 ¼ w
ð12Þ
– for a clamped edge
1 ¼ w 2 ¼ w 3 ¼ 0: w
ð13Þ
The general solutions to Eq. (8) in terms of the three dimensionless potentials W1, W2 and W3 may be expressed as
1 ¼ C 1 W 1;1 þ C 2 W 2;1 gW 3;2 ; w 2 ¼ C 1 gW 1;2 þ C 2 gW 2;2 W 3;1 ; w w3 ¼ W 1 þ W 2 ;
ð14Þ
where
C1 ¼ 1
a22 a2 ; C2 ¼ 1 1 2 2 m1 a3 m1 a3
ð15Þ
in which a21 ; a22 and a23 are the coefficients that may be determined using equations of motion and can be given after mathematical manipulation by
2
3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u 2 u b d ð1 þ cÞ d ð1 þ cÞ 4ð1 þ cÞ7 a21;2 ¼ 6 þ1 t þ1 þ 4 5; 2 2 2 12 12 b2 k m1 k m1 ! 2 12k b2 d4 a23 ¼ 2 ð1 þ cÞ 1 : 2 d 144k m1 2
2
!
Eq. (8) can be restated in terms of the three dimensionless potentials as
ð16Þ
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W 1;11 þ g2 W 1;22 ¼ a21 W 1 ; W 2;11 þ g2 W 2;22 ¼ a22 W 2 ; 2
ð17Þ
2 3W 3:
W 3;11 þ g W 3;22 ¼ a
One set of solutions to Eq. (17) are taken as
W 1 ¼ ½A1 Sinðk1 X 2 Þ þ A2 Cosðk1 X 2 ÞSinðl1 X 1 Þ þ ½A3 Sinðk1 X 2 Þ þ A4 Cosðk1 X 2 ÞCosðl1 X 1 Þ; W 2 ¼ ½A5 Sinhðk2 X 2 Þ þ A6 Coshðk2 X 2 ÞSinðl2 X 1 Þ þ ½A7 Sinhðk2 X 2 Þ þ A8 Coshðk2 X 2 ÞCosðl2 X 1 Þ;
ð18Þ
W 3 ¼ ½A9 Sinhðk3 X 2 Þ þ A10 Coshðk3 X 2 ÞSinðl3 X 1 Þ þ ½A11 Sinhðk3 X 2 Þ þ A12 Coshðk3 X 2 ÞCosðl3 X 1 Þ in which l = mp (m = 1, 2, . . .), Ai are the arbitrary coefficients, kj and lj are related to the aj by
a21 ¼ l21 þ g2 k21 ; a22 ¼ l22 g2 k22 ; a23 ¼ l23 g2 k23 :
ð19Þ
On the assumption of simply-supported conditions at edges X1 = 0 and 1, Eq. (18) is given by
W 1 ¼ ½A1 Sinðk1 X 2 Þ þ A2 Cosðk1 X 2 ÞSinðlX 1 Þ; W 2 ¼ ½A5 Sinhðk2 X 2 Þ þ A6 Coshðk2 X 2 ÞSinðlX 1 Þ;
ð20Þ
W 3 ¼ ½A9 Sinhðk3 X 2 Þ þ A10 Coshðk3 X 2 ÞSinðlX 1 Þ; Introducing Eq. (20) in Eq. (14) and substituting the results into the appropriate boundary conditions along the edges X2 = 0 and 1, leads to six homogenous equations. To obtain non-trivial solution of these equations, the determinant of coefficients matrix must be zero, which yields characteristic equations for rectangular Mindlin plates with six combinations of boundary conditions, namely, SSSS, SSSC, SCSC, SSSF, SFSF and SCSF. 3. Fluid formulation The following assumptions are made to model dynamic behavior of the fluid as the mathematical formulations: (a) The amplitude of the vibrations is small (i.e., fluid motion is small). (b) The fluid is incompressible, inviscid and irrotational (i.e., the fluid flow is potential). The velocity potential function / must satisfy the Laplace equation throughout the fluid domain. This relation is expressed in the Cartesian coordinate system as:
r2 / ¼
@2/ @2/ @2/ þ þ ¼ 0: @x21 @x22 @x23
ð21Þ
Using the Bernoulli’s equation and ignoring the nonlinear terms, the fluid pressure at the plate–fluid interface (upper and lower surface of the plate) may be given by
@/ ; @t x3 ¼h 2 @/ ¼ qf ; @t h
pu ¼ pjx3 ¼h ¼ qf 2
pL ¼ pjx3 ¼h 2
ð22 - aÞ ð22-bÞ
x3 ¼ 2
where qf is the fluid density per unit volume. It is assumed that the out-of-plane velocity component of the fluid on the plate surface and the plate transversal velocity are identical due to fact that there exists permanent contact between the plate surface and the peripheral fluid layer. The impermeability condition of the plate’s surface can be expressed by
@/ @U 3 ¼ ; @x3 x3 ¼h @t 2 @/ @U 3 ¼ : @x3 x3 ¼h @t
ð23-aÞ ð23-bÞ
2
The method of separation of variables is adopted for the potential velocity function as follows:
/ðx1 ; x2 ; x3 ; tÞ ¼ Fðx3 ÞGðx1 ; x2 ; tÞ;
ð24Þ
where F(x3) and G(x1, x2, t) are two separate functions which must be determined. The velocity potential function on the upper and lower surfaces of the plate can be expressed by introducing Eq. (24) into Eq. (23) and then substituting G(x1, x2, t) into Eq. (24):
S. Hosseini-Hashemi et al. / Applied Mathematical Modelling 36 (2012) 764–778
/ðx1 ; x2 ; x3 ; tÞ ¼
/ðx1 ; x2 ; x3 ; tÞ ¼
Fðx3 Þ @U 3 ; @t
dF dx3
ð25-aÞ
x3 ¼2h
Fðx3 Þ @U 3 : @t
dF dx3
769
ð25-bÞ
x3 ¼2h
Substituting Eq. (25) into relation (21) leads to the following second-order differential equation: 2
d Fðx3 Þ 2
dx3
l2f Fðx3 Þ ¼ 0;
ð26Þ
where lf is a plane wave number and a real constant which must be specified. The general solution of Eq. (26) is written as:
Fðx3 Þ ¼ B1 elf x3 þ B2 elf x3 ;
ð27Þ
where B1 and B2 are constants which must be specified. By introducing Eq. (27) into Eq. (25), one gets the following expressions for the velocity potential function:
B1 elf x3 þ B2 elf x3 @U 3 ; dF @t dx3 h
ð28-aÞ
B1 elf x3 þ B2 elf x3 @U 3 /ðx1 ; x2 ; x3 ; tÞ ¼ : dF @t dx3 h
ð28-bÞ
/ðx1 ; x2 ; x3 ; tÞ ¼
x3 ¼2
x3 ¼2
3.1. Boundary conditions of a plate–fluid The boundary conditions at the fluid–structure interface along with the fluid extremity must be satisfied by adopting an appropriate velocity potential function. Fluid free surface, rigid wall and impermeability are the boundary conditions which are generally taken into account. In order to achieve a good understanding of the problem, a flexible rectangular plate submerged in fluid is studied, where the following conditions should be considered. 3.1.1. Plate–fluid model with free surface (BC1) At the fluid free surface, the following condition may be applied to the velocity potential (see Fig. 2), provided that free surface motion of the fluid creates insignificant perturbations,
@/ 1 @ 2 / ¼ @x3 x3 ¼h1 þh g @t 2 2 x
ð29Þ
; h 3 ¼h1 þ2
where g is acceleration due to gravity. Substituting the above boundary condition into Eq. (28-a), the following expression for the potential function is obtained:
Fig. 2. Plate–fluid model with free surface.
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elf x3 þ c1 elf ð2h1 þhx3 Þ @U 3 h ; h lf elf 2 c1 elf ð2h1 þ2Þ @t
/¼
where c1 ¼
lf g x2 : lf g þ x2
ð30Þ
The fluid pressure applying on the upper surface of the plate is obtained by introducing the above relationship into the Bernoulli’s equation:
pu ¼
qf 1 þ c1 e2lf h1 @ 2 U 3 @2 U3 ¼ m1 : lf 1 c1 e2lf h1 @t2 @t 2
ð31Þ
3.1.2. Plate–fluid model bounded by a rigid wall (BC2) The boundary condition at the wall, as shown in Fig. 3(a), was studied by Lamb [8] and referred to as the null-frequency condition. This rigid wall boundary condition is expressed as:
@/ ¼ 0: @x3 x3 ¼h2
ð32Þ
Similarly, by introducing Eq. (28-b) into relations (32), the following expression for the velocity potential is obtained:
elf x3 þ c2 elf x3 @U 3 ; h h lf elf 2 c2 elf 2 @t
/¼
where c2 ¼ e2lf h2 :
ð33Þ
The dynamic pressure for this case (lower surface of the plate) is determined as:
pL ¼
qf 1 þ c2 elf h @ 2 U 3 @2 U3 ¼ m2 : 2 l h lf 1 c2 e f @t @t2
ð34Þ
In case of totally submerged plate, as shown in Fig. 3(b), the total dynamic pressure will be a combination of the pressures corresponding to the fluid boundary conditions at both top and bottom surfaces of the plate.
@2 U3 @ 2 U3 p ¼ pu pL ¼ m1 m2 ¼ m : 2 @t @t 2
ð35Þ
3.1.3. Fluid bounded between two identical plates (BC3) Fig. 4 illustrates fluid surrounded between two parallel plates, which depict the side walls of reservoir. Impermeability conditions for both plates at x3 = h/2 and x3 = d + h/2 are considered. By introducing these boundary conditions into Eq. (28), the potential function is obtained as follows:
/¼
elf x3 þ c3 elf x3 @U 3 h ; h lf elf 2 c3 elf 2 @t
elf ðdþ2Þ elf 2 ; h h elf ðdþ2Þ elf 2 h
where c3 ¼
h
ð36Þ
where d is the distance between the two plates. The fluid pressure on each wall is obtained by introducing Eq. (36) into Bernoulli’s equation as:
p¼
qf 1 2elf d þ e2lf d @ 2 U 3 : lf @t 2 1 e2lf d
Fig. 3. Plate–fluid model bounded by a rigid wall. (a) A floating plate. (b) A submerged plate.
ð37Þ
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771
Fig. 4. Plate–fluid model representing two identical plates coupled with bounded fluid.
3.2. Determination of lf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The lf is net wave number which will be the magnitude of the wave motion in the x and x directions as l ¼ l21 þ l22 . 1 2 f pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kerboua et al. [33] used a simple form of lf ¼ ðp=aÞ 1 þ g2 for all boundary conditions. Though the value of lf varies for various boundary conditions, they ignored the effect of boundary conditions on the lf. In addition, the wave number presented in Ref. [33] is independent of air frequency and is constant when mode number changes. In this study, the wave number parameter proposed in Ref. [24] is modified by using Mindlin plate parameters. It is worthwhile to mention that as higher degrees of edge constraint (in the order from free to simply supported to clamped) are applied to the other two edges of the rectangular plate, the variation of the lf is more tangible (e.g., from SFSF to SCSC). The following wave number parameters are introduced when at least one of the rectangular plate edges can freely vibrate (i.e., SFSF, SCSF and SSSF):
l1 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np2 ; a2 a
l2 ¼
2p ; L
ð38Þ
where
2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi3 ! ! v !2 u 2 2 u d2 2 b d 1 1 4 6 7 t a2 ¼ a2 4 þ1 þ þ 1 þ 2 5; 2 2a 12 k2 m1 12 ba k m1 L is width of tank and ba is dimensionless frequency parameter in vacuum. For other three boundary conditions (i.e., SSSS, SCSC and SSSC), l2 in Eq. (38) is replaced with l2 = mp/a, while l1 will remain unchanged. Note that when l1 takes a comqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 must be used instead of that as l 1 ¼ a2 þ nap 2 . plex value, l 4. Comparison study In this paper, the results obtained by the present analytical solution for the rectangular plates coupled with fluid are compared with those acquired by the finite element method (FEM) [34] and experimental results [24] to show the applicability, reliability and effectiveness of the presented formulation. The natural frequencies of a steel rectangular SFSF thin plate completely surrounded by water are calculated while its two short edges are simply-supported. The mechanical and geometrical properties of plate are: Young’s modulus E = 207 GPa, material density q = 7850 kg m3, Poisson’s ratio t = 0.3, a = 0.655 m, b = 0.20165 m and thickness h = 9.36 mm. The fluid density is taken to be 1000 kg m3. The experimental apparatus was set up by placing the plate in a rectangular reservoir with dimensions 1.3 m 0.55 m 0.8 m. The total dynamic pressure is therefore the sum of lower and upper pressures and can be calculated using Eq. (35). In Table 1, natural frequencies of the plate in (Hz) unit obtained by the present theory are presented and compared with both experimental and numerical data described in Refs. [24, 34]], respectively. As can be seen, there is a good agreement between the present results and available data. However, it should be noted that the results of the present exact solution are much closer to the experimental results [24] than those of FEM [34]. A comparison study of the first six frequency parameters obtained by the CPT [27] and the present method is conducted in Table 2 for a square SSSS plate (d = 0.05 and k = 1) in contact with cubic volume of fluid, showing a good agreement between the results. The plate-water density ratio r is set to be 0.125. In view of Table 2, it is seen that all results obtained on the basis of the CPT are greater than those of the present method. It is worth noting from Table 2 that the CPT is unable to predict the wet frequency of the plate accurately for both d = 0.05 and the higher modes. This is attributed to the fact that the CPT overestimates the natural frequencies of the plate due to ignoring the effect of shear deformation and rotary inertia. These effects are more pronounced in the higher modes than in the lower modes.
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Table 1 Comparison of natural frequencies of a plate x(Hz), with S–F–S–F boundary condition, submerged in water. Mode no.
1 2 3 4 5
In vacuum
Completely submerged in water
Present theory
Present theory
Experiment [24]
FEM [34]
51.005 205.530 228.131 464.979 490.227
27.756 117.924 153.816 285.274 326.802
28.72 117.125 154.51 281.79 335.04
31.28 126.40 141.78 285.98 304.57
Table 2 Comparison of the first six frequency parameters for a square plate in contact with cubic volume of fluid.
a
B.C.
Method
b1
b2
b3
b4
b5
b6
SSSS
CPTa Present method
11.844 11.620
39.243 38.681
39.243 38.681
66.609 65.477
81.643 75.877
85.674 78.502
Results from Cheung and Zhou [27].
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffi First five frequency parameter b ¼ xa2 qh=D for a Mindlin plate submerged in water as a function of fluid level (d = 0.05, g = 2, h = 0.1 m and m = 0.3). Boundary conditions
Mode (m, n)
SFSF
(1, 1) (1, 2) (2, 1) (2, 2) (3, 1)
SSSF
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2)
SCSF
In vacuum 9.45825 26.4801 37.7831 61.6215 83.8573
h1 a
h1 a
¼0
¼ 0:1
h1 a
¼ 0:3
h1 a
¼ 0:5
h1 a
¼2
6.71860 23.7110 30.8581 54.9017 73.3207
6.33350 22.0905 28.7943 51.1114 68.1336
5.84300 21.6735 27.1502 50.0239 66.0611
5.62620 21.6657 26.8956 49.9987 65.9697
5.49410 21.6656 26.7929 49.9982 65.9660
15.8630 45.5242 72.3859 91.7889 104.764
12.1114 36.8051 67.5175 80.0587 96.2272
11.3383 34.2752 63.7460 74.3911 90.2427
10.5147 32.1789 63.5176 72.0377 89.4967
10.2633 31.7937 63.5174 71.9260 89.4934
10.1766 31.6093 63.5174 71.9103 89.4934
(1, 1) (2, 1) (1, 2) (3, 1) (2, 2)
22.2469 49.0424 93.5256 93.9506 121.897
17.7375 39.4316 80.4486 80.7430 113.016
16.5479 36.7620 74.7767 75.0248 106.465
15.4747 34.4537 71.8909 72.1175 105.841
15.2413 33.9927 71.6879 71.9131 105.839
15.1076 33.9002 71.6748 71.8999 105.839
SSSS
(1, 1) (2, 1) (3, 1) (1, 2) (2, 2)
48.3006 76.3360 121.632 156.685 182.338
41.4293 64.5257 110.369 149.690 171.969
38.4638 59.9367 103.140 143.620 163.489
36.9582 57.1981 101.763 143.556 163.188
36.8523 56.9347 101.749 143.556 163.188
36.8455 56.9111 101.748 143.556 163.188
SCSS
(1, 1) (2, 1) (3, 1) (1, 2) (4, 1)
66.3511 89.7039 130.980 187.564 188.898
58.3467 77.0089 118.969 177.610 178.851
54.3177 71.6786 111.288 168.602 169.498
52.7481 68.8209 109.774 167.961 168.709
52.6826 68.6026 109.758 165.564 166.302
52.6801 68.5873 109.758 165. 543 166. 300
SCSC
(1, 1) (2, 1) (3, 1) (1, 2) (2, 2)
88.6635 106.9478 142.979 219.582 237.921
79.6484 93.4901 130.091 212.346 227.422
74.4777 87.1999 121.882 205.806 218.380
72.9584 84.2839 120.191 205.781 218.206
72.9222 84.1148 120.171 205.781 218.206
72.9214 84.1058 120.171 205.781 218.206
Table 4 Fundamental natural frequencies of a plate immersed in different fluid for six boundary conditions when h1/a = 0.1. Fluid
Density (kg m3)
Gasoline (16 °C) Alcohol ethyl (25 °C) Kerosene (16 °C) Water pure (4 °C) Sea water (25 °C) Milk (4% fat, 20 °C)
737.22 785.1 817.15 1000 1025 1033
Boundary conditions SFSF
SSSF
SCSF
SSSS
SSSC
SCSC
6.8491 6.7458 6.6792 6.3335 6.2902 6.2766
12.1447 11.9849 11.8814 11.3383 11.2697 11.2480
17.6128 17.4033 17.2673 16.5479 16.4564 16.4274
40.4576 40.0712 39.8187 38.4638 38.2891 38.2337
56.8364 56.3513 56.0335 54.3176 54.0951 54.0244
77.5371 76.9516 76.5669 74.4777 74.2051 74.1185
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Fig. 5. The fundamental frequency parameter b versus the thickness to length ratio d for the SCSS rectangular plate subjected to three boundary conditions of the fluid when g = 1.5, h1/a = 0.5 and d = 0.2.
Fig. 6. The fundamental frequency parameter b versus the aspect ratio g for the SCSS rectangular plate subjected to three boundary conditions of the fluid when d = 0.1, h1/a = 0.5, d = 0.2 and h = 0.1.
Fig. 7. The fundamental frequency parameter b versus the thickness to length ratio d for the SFSF, SSSS and SCSC rectangular plate when g = 2 and h1/a = 0.1.
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Fig. 8. The fundamental frequency parameter b versus the thickness ratio d for the SCSF rectangular plate (g = 2) with three different fluid levels.
Fig. 9. The fundamental frequency parameter b versus the aspect ratio g for the SFSF, SSSS and SCSC rectangular plate when d = 0.05, h1/a = 0.1 and h = 0.1.
Fig. 10. The fundamental frequency parameter b versus the aspect ratio g for the SCSF rectangular plate with three different fluid levels when d = 0.05 and h = 0.1.
5. Results and discussion In this section, natural frequencies of the rectangular Mindlin plates coupled with fluid are presented in tabular and graphical forms for different plate parameters. Note that except for Section 5.1, all results in this paper are presented
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775
for the plate submerged partially or totally in fluid (BC2) when the dimensions of the tank are assumed to be 5 m 5 m 5 m. The first five dimensionless natural frequencies of rectangular Mindlin plates (d = 0.05 and g = 2) with six combinations of boundary conditions are listed in Table 3 for different fluid levels when h = 0.1 m and m = 0.3. It is seen that regardless of boundary conditions, the frequency parameter b decreases with increasing fluid level (h1). It is interesting to note that the frequency parameter b is almost independent of h1 when the plate is immersed in a level higher than 50% of the plate length. As expected, the frequency parameter b enhances as the higher degrees of edge constraint (in the order from free to simply supported to clamped) are applied to the other two edges of the rectangular plate. Due to the practical applications, the fundamental natural frequency of rectangular Mindlin plate (d = 0.05 and g = 2) coupled with different fluids is listed in Table 4 for different boundary conditions when h1/a = 0.1 and h = 0.1. It is evident from Table 4 that when the plate is submerged in denser fluid, the fundamental natural frequency of the plate takes lower values.
5.1. Effect of different boundary conditions of the fluid on the frequency parameter In Fig. 5, the variation of the fundamental frequency parameter b against the thickness to length ratio d for the SCSS rectangular Mindlin plate is shown for g = 1.5, h1/a = 0.5 and d = 0.2. The plate under consideration is subjected to three
Fig. 11. Deformed mode shapes and frequency parameters of a SFSF submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 7.35783, (m, n) = (1, 1); (b) b = 9.97149, (m, n) = (1, 2) and (c) b = 14.8906, (m, n) = (1, 3).
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Fig. 12. Deformed mode shapes and frequency parameters of a SSSF submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 7.82004, (m, n) = (1, 1); (b) b = 12.4961, (m, n) = (1, 2) and (c) b = 18.4169, (m, n) = (1, 3).
boundary conditions of the fluid, including fluid with free surface (BC1), fluid bounded by a rigid wall (BC2) and fluid bounded between two identical plates (BC3). It is seen from Fig. 5 that with the increase of the thickness to length ratio d, the fundamental frequency parameter b generally decreases for any boundary conditions of the fluid. However, an contrary behavior is observed in BC1 and BC2 for d 6 0.075 and d 6 0.1, respectively. Fig. 6 shows the behavior of the fundamental frequency parameter b against the aspect ratio g for the SCSS rectangular Mindlin plate under three boundary conditions of the fluid when d = 0.1, h1/a = 0.5, d = 0.2 and h = 0.1. It is obvious that regardless of the fluid boundary conditions, the fundamental frequency parameter b increases as the aspect ratio g increases. It can also be inferred from Fig. 6 that for a given value of the aspect ratio g, the fundamental frequency parameter b of the plate coupled with the fluid, bounded between two identical plates (BC3), takes the largest value in comparison with two other fluid boundary conditions.
5.2. Effect of thickness to length ratio on the frequency parameter Fig. 7 displays the variation of the frequency parameter b versus the thickness to length ratio d for the SCSC, SSSS and SFSF rectangular Mindlin plate when g = 2 and h1/a = 0.1. It can apparently be observed that the frequency parameter b for the SSSS rectangular plate initially increases when the thickness to length ratio d varies between 0.05 and 0.1, and then decreases when d > 0.1. However, with the increase of the thickness to length ratio d, the frequency parameter b for
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777
Fig. 13. Deformed mode shapes and frequency parameters of a SCSC submerged rectangular Mindlin plate (d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2). (a) b = 11.1575, (m, n) = (1, 1); (b) b = 18.8624, (m, n) = (1, 2) and (c) b = 28.0825, (m, n) = (1, 3).
the SCSC rectangular plate continuously diminishes. Another interesting point about Fig. 7 is that the effect of the thickness to length ratio d on the frequency parameter b of the SFSF rectangular Mindlin plate is insignificant. The behavior of the frequency parameter b against thickness to length ratio d for the SCSF rectangular Mindlin plate (g = 2) is shown in Fig. 8 for three different fluid levels h1/a = 0, 0.1 and 0.5. It can be figured out from Fig. 8 that the maximum frequency parameter b lies between d = 0.1 and 0.125 for any values of fluid level within 0 and 0.5. It is worth noting that the maximum frequency parameter b shifts toward higher values of the thickness to length ratio d as the plate is submerged in deeper levels of fluid. 5.3. Effect of aspect ratio on the frequency parameter Fig. 9 displays the variation of the frequency parameter b versus the aspect ratio g for SFSF, SSSS and SCSC rectangular Mindlin plates (d = 0.05) when h1/a = 0.1 and h = 0.1. It can be seen that contrary to the SFSF rectangular plate, the frequency parameter b for the SSSS and SCSC rectangular plate increases as the aspect ratio g increases. Fig. 10 contains the plot of the frequency parameter b versus aspect ratio g for the SCSF rectangular Mindlin plate (d = 0.05 and h = 0.1) for three different fluid levels h1/a = 0, 0.1 and 0.5. It is obviously seen from Fig. 10 that with the enhancement of the aspect ratio g, the frequency parameter b increases for any fluid levels.
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5.4. Three-dimensional mode shapes To have a more appropriate sense of the transverse displacement w3, three first mode shapes of a submerged rectangular Mindlin plate for d = 0.2, g = 0.5, h1/a = 0.1 and h = 0.2 are illustrated in Figs. 11–13 for SFSF, SSSF and SCSC boundary conditions, respectively. 6. Concluding remarks In this paper, free vibration of totally or partially submerged rectangular Mindlin plates was investigated. The simplifying hypothesis that mode shapes are not to be modified by the fluid is not assumed in this study. In order to achieve an analytical solution, the plate must have two opposite simply-supported edges. Governing equations of plate coupled with stationary fluid were derived by combining Mindlin plate theory and potential flow theory. To verify the merit and accuracy of the present exact solution, a comparison was made with the experimental and numerical results reported in the literature. It was shown that the results obtained by the present method are an excellent agreement with the experimental results reported by Haddara and Cao [24]. Then, a detailed parametric study was conducted to show the influence of different fluid depths, fluid densities, aspect ratios and thickness to length ratios for six possible combinations of boundary conditions. Finally, some 3-D plots were shown for the mode shapes of the deflection in the submerged rectangular plates. Due to the inherent features of the present analytical solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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