On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

Applied Mathematical Modelling 37 (2013) 2503–2515

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On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid T.-P. Chang ⇑ Department of Construction Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan

a r t i c l e

i n f o

Article history: Received 20 May 2011 Received in revised form 24 May 2012 Accepted 4 June 2012 Available online 13 June 2012 Keywords: Vibration Magneto-electro-elastic (MEE) plates Added virtual mass Fluid-structure interaction Added virtual mass incremental (AVMI) factor Natural frequencies

a b s t r a c t In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction For a number of decades, there has been addressed general interest in fluid-structure interaction problem. It is quite essential to determine the natural frequencies of the structure in contact with fluid or immersed in fluid since that the natural frequencies of structures in the liquid are definitely different from those in the air. It is well known that the fluid motion will be induced by the vibrating movement of the water-contacting structure and thus a remarkable increase in the kinetic energy to the whole system will be provoked. The added virtual mass incremental factor (AVMI factor) is often used to denote the ratio between the kinetic energy of the fluid and that of the structure, furthermore, the natural frequencies of the structure coupled with fluid can be determined by using the added virtual mass, which can be calculated following the determination of AVMI factor. The analytical approach for the addressed problem was initiated by Rayleigh [1]. He calculated the increase of inertia of a rigid disk vibrating in a circular aperture. Lamb [2] calculated the change in natural frequency of a thin circular plate fixed along its boundary and placed in the aperture of an infinitely rigid wall in contact with water, and then Powell and Roberts [3] verified the theoretical results of Lamb’s work by conducting experiments. McLachlam [4] extended Lamb’s work and Peake and Thurston [5] generalized the work done by Lamb and McLachlam. Kwak and Kim [6,7] investigated the above problem and solve the mixed boundary problem by using Hankel Transform. Amabili and Kwak [8] performed the experiment verification about the validity of making the assumption that wet mode shapes are equivalent to dry ones. Amabili [9] researched into the effect of finite fluid depth on the hydroelastic vibrations of circular and annular plates. Amabili and Kwak [10] considered the effect of surface wave on the vibration of circular plates placing on a free fluid surface. ⇑ Tel.: +886 7 6011000x2111; fax: +886 7 6011017. E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.06.016

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515

Chang and Liu [11] studied the free vibration behavior of a rectangular isotropic plate in contact with liquid by using the double Fourier Transform. Amabili [12] discussed the solution to the fully coupled problem of the vibrations of circular plates resting on sloshing liquid. Later on, Chang and Liu [13] performed the forced vibration analysis of a rectangular composite plate in contact with fluid as well as discussed the variation of AVMI factor for different plates with various widths and length. Besides, Liu and Chang [14] studied the vibration behavior of a varying-thickness circular plate in contact with fluid by adopting the Galerkin’s method in conjunction with Hankel transform. Recently, many researchers have paid their attentions to the mechanics problems associated with the so-called magnetoelectro-elastic (MEE) materials. It is found that a wide class of crystals and emerging composite materials will possess simultaneously piezoelectric, piezomagnetic and magneto-electric effects, which are thereby classified as magneto-electro-electric solids. These kinds of composites permit the new properties of magneto-electricity with secondary pyroelectric effects, which cannot be found in the single-phase piezoelectric or piezomagnetic materials. In some cases, the magneto-electric effect of piezoelectric/piezomagnetic composites can be obtained a hundred times larger than that of a single-phase magneto-electric material. This property of composites offers great opportunities to engineer new materials to make efficient sensors and actuators for smart and intelligent structures, which may not be achieved otherwise. As far as the structure analysis is regarded, Pan [15] derived an exact three-dimensional solution of a simply supported multilayered orthotropic magneto-electro-elastic plate using a propagator matrix method. Wang et al. [16] extended the previous works on elastic and piezoelectric plates to study the bending of multi-layered orthotropic magneto-electro-elastic rectangular plates by adopting the state space formulations. The free vibration investigation was recently performed by Pan and Heyliger [17], who found that some natural frequencies of a multi-field plate were identical to the ones of the corresponding elastic plate. They argued that certain vibration modes of the plate were insensitive to the coupling effects among elastic, electric, and magnetic fields. Chen et al. [18] studied the free vibration problem of simply supported MEE rectangular plates with general inhomogeneous material property along the thickness direction. Annigeri et al. [19] investigated the free vibrations of simply supported layered and multiphase magneto-electro-elastic cylindrical shells. In addition, Milazzo et al. [20] obtained an analytical solution for the magneto-electro-elastic bimorph beam forced vibrations problem. Wang et al. [21] derived an analytical solution for a multilayered magneto-electro-elastic circular plate under simply supported lateral boundary. Moreover, three-dimensional exact solutions for free vibrations of simply supported magneto-electro-elastic cylindrical panels were introduced by Wang et al. [22]. Recently, a regular variational boundary element formulation for free vibrations of magneto-electro-elastic structures was developed by Davì et al. [23]. Furthermore, Liu and Chang [24] have offered a compact form expression for the transverse vibration of a magneto-electro-elastic (MEE) thin plate, in which a new and simplified differential equation is presented. The objective of this investigation is to calculate the natural frequencies of a transversely isotropic magneto-electro-elastic (MEE) rectangular thin plate with general boundary conditions, which is in contact with fluid. 2. Formulations Consider the physical model of a transversely isotropic magneto-electro-elastic (MEE) rectangular plate in contact with fluid as illustrated in Fig. 1, where 2a and 2b represent the width and length of the MEE rectangular plate, and h is the thickness respectively. F denotes the fluid domain, SF denotes the surface between the fluid and an infinite rigid wall and SB denotes the surface between the fluid and the plate, also SR denotes the surface at infinity. By neglecting the effects of rotatory inertia and transverse shear deformation, and utilizing the differential equation for MEE rectangular plates stated in Ref. [24], the governing equation of the undamped free vibration of a transversely isotropic magneto-electro-elastic (MEE) rectangular plate in contact with fluid can be written as follows:

ðD þ E þ MÞr4 w þ ðqP h þ Mf Þ

o2 w ¼ 0; ot 2

ð1Þ

where w is the transverse deflection of the plate, qp is the mass density of the plate, h is the thickness of the plate, Mf denotes 3

3

3

the fluid-added mass and D  c1112h , E  e3112h  DD1 , M  q3112h  DD2 represent the plate rigidity, effective rigidities due to the pres2

ences of electricity and magnetism, respectively. Here, D ¼ e33 l33  d33 , D1 = (e31l33  d33q31), D2 = (e33q31  d33e31), cij, eij, eij, qij, dij and lij are the elastic, dielectric, piezoelectric, piezomagnetic, magnetoelectric, and magnetic constants, respectively. For free vibration analysis in the air, Eq. (1) yields

ðD þ E þ MÞr4 w þ qP h

o2 w ¼ 0: ot 2

ð2Þ

By adopting the separation of variables, the solution of Eq. (2) can be expressed as follows:

wðx; y; tÞ ¼

1 X 1 1 X 1 X X W mn W mn ðx; yÞTðtÞ ¼ W mn X m ðxÞY n ðyÞTðtÞ; m¼1 n¼1

ð3Þ

m¼1 n¼1

where

TðtÞ ¼ sin xt

ð4Þ

T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515

2505

Fig. 1. The MEE rectangular plate in contact with fluid.

and Xm(x), Yn(y) are the orthogonal mode shape functions which satisfy the boundary conditions of magneto-electro-elastic (MEE) rectangular plate in the x and y directions individually. In addition, the natural frequency x can be computed from the boundary conditions of the MEE rectangular plate. In the present study, the MEE rectangular plate is considered to be in contact with fluid on one side only, and the fluid is assumed to be incompressible and inviscid. Furthermore, the fluid flow is considered as irrotational under plate vibration only so that its velocity potential can be represented as 

Uðx; y; z; tÞ ¼ /ðx; y; zÞ T ðtÞ;

ð5Þ

where / is the spatial distribution of the velocity potential and ‘‘’’ denotes the derivative with respect to time. Based on the assumption of the fluid, / has to satisfy the Laplace equation as follows:

r2 / ¼

o2 / o2 / o2 / þ þ ¼ 0 in F; ox2 oy2 oz2

ð6Þ

where F denotes the fluid domain. The condition of the rigid wall on SF, can be stated in the following:

 o/ðx; y; zÞ  ¼ 0 on SF : oz z¼0

ð7Þ

In addition, the interaction between the fluid and the plate can be expressed as follows:

 o/ðx; y; zÞ ^  ¼ Wðx; yÞ on SB ; oz z¼0

ð8Þ

^ denotes the ‘‘wet’’ mode shape of the MEE plate vibrating in contact with the fluid. Moreover, we must impose the where W conditions that the velocity potential / and the velocities o//ox, o//oy and o//oz approach zero on SR, that is,

/;

o/ o/ o/ ; ; ! 0asx; y; z ! 1 on SR : ox oy oz

ð9Þ

Based on the results verified by several researchers [25], it is assumed that the ‘‘wet’’ mode shape of the MEE plate in contact with the fluid is the same as the ‘‘dry’’ mode shape of the MEE plate when vibrating in the air. Hence, the approx^ imation Wðx; yÞ ¼ Wðx; yÞ will be used in the following derivations. First of all, denote the double Fourier transform in the following:

/ðu; v ; zÞ ¼

Z

1 1

Z

1

/ðx; y; zÞeþiðuxþv yÞ dxdy:

ð10Þ

1

Utilizing double Fourier transform on Eq. (6) and applying the boundary conditions stated in Eq. (9), then the velocity potential /(x, y, z) can be represented as

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515



2 Z 1 Z 1  2 Z 1 Z 1 1 1 2 2  v ; zÞeiðuxþv yÞ dudv ¼ 1 /ðu; Bðu; v Þeðu þv Þ2 z eiðuxþv yÞ dudv : 2p 2p 1 1 1 1

/ðx; y; zÞ ¼

ð11Þ

Applying the boundary conditions specified in Eqs. (7) and (8), B(u, v) in Eq. (11) can be determined in the following:

Bðu; v Þ ¼ ðu2 þ v 2 Þ1=2

Z

b

b

Z

a

X m ðxÞY n ðyÞeþiðuxþv yÞ dxdy:

ð12Þ

a

Generally B(u, v) is a complex function of both u and v. It is noted that the derivations after Eq. (11) are valid for a fixed mode Wmn(x, y). Based on the previous assumption that the wet mode shapes are almost the same as the dry mode shapes, the natural frequency of the MEE plate in contact with fluid xf can be evaluated as follows [6]:

xamn ; xfmn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ cmn

ð13Þ

where xamn is the natural frequency of the MEE plate in the air. Utilizing Eq. (2) in conjunction with the general boundary conditions for the MEE plate, it is quite feasible to determine xamn in the following form:

xamn ¼

ðD þ E þ MÞ 4 ðam þ 2a2m b2n þ b4n Þ qp h

ð14Þ

in which am and bn are the corresponding eigenvalues according to the MEE plate’s boundary conditions along the x and y directions, respectively. Also cmn in Eq. (13) is the AVMI factor that denote the ratio between the reference kinetic energy of fluid induced by the plate vibration and that of the plate which can be expressed as

cmn ¼

TF : TP

ð15Þ

The reference kinetic energy of the MEE plate can be calculated as

Tp ¼

1 qh 2 p

Z

Z

b

b

a

a

X 2m ðxÞY 2n ðyÞdxdy:

ð16Þ

Employing the assumption on the irrotational movement of the fluid flow, the reference kinetic energy of the fluid can be determined as follows from its boundary condition:

1 T F ¼  qF 2

Z

1

1

Z

1

1

o/ðx; y; 0Þ /ðx; y; 0Þdxdy: oz

ð17Þ

Substituting Eq. (7), (8) and (11) into Eq. (17) yields

TF ¼

 2 Z b Z a Z 1 Z 1 1 1 qF X m ðxÞY n ðyÞBðu; v Þeiðuxþv yÞ dudv :dxdy 2 2p b a 1 1

ð18Þ

To calculate the above multiple integral, reverse the order of integration, thus Eq. (18) can be simplified as follows:

#  2 Z 1 Z 1 "Z b Z a 1 1 iðuxþv yÞ TF ¼ qF X m ðxÞY n ðyÞe dxdy Bðu; v Þdudv 2 2p 1 1 b a  2 Z 1 Z 1 h  2 Z 1 Z 1 i 1 1 1 1 1 1 ¼ qF B ðu; v Þðu2 þ v 2 Þ2 ½Bðu; v Þdudv ¼ qF jBðu; v Þj2 ðu2 þ v 2 Þ2 dudv ; 2 2p 2 2 p 1 1 1 1

ð19Þ

where ⁄ is the complex conjugate and it is noted that B(u, v) can be evaluated from Eq. (12) as long as the dry mode shapes of the MEE plate are available. Hence, once TF and TP are calculated, cmn (AVMI factor) can be determined from Eq. (15) and finally the natural frequency xfmnof the MEE plate in contact with fluid are readily obtained from Eq. (13). 3. Numerical examples and discussions In this section, one performs the free vibration analysis of the MEE plate in contact with fluid by considering a bi-layered BaTiO3–CoFe2O4 composite with variable volume fraction (v.f.) of BaTiO3. The dimensions of the plate are assumed as follows: length a = 2 m, width b = 2 m and height h = 0.05 m. The density of the bi-layer plate is assumed to be proportional to the volume-faction of these two materials, i.e. qP ¼ qBaTiO3  v:f: þ qCoFe2 O4  ð1  v:f:Þ. Besides, the density of the fluid is assumed as qf = 1000 kg/m3. The numerical computations are carried out by selecting six different plates with volume fractions in steps of 20%, i.e. 0%, 20%, 40%, 60%, 80%, 100%. The magneto-electro-elastic material properties are listed in Table 1 with different volume fraction as given by Annigeri et al. [26]. The boundary conditions for the MEE plate are selected to be simplysupported on four sides (SSSS) or clamped on four sides (CCCC), nevertheless, any other kinds of boundary conditions can be considered without any difficulties.

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515 Table 1 Material constants for MEE BaTiO3  CoFe2O4 composite (cited from Ref. [20]). v.f.

0%

20%

40%

60%

80%

100%

C11 C12 C13 C33 C44 e31 e33 e15 e11 e33

286 173 170 269.5 45.3 0 0 0 0.08 0.093 5.9 1.57 580 700 560 0 0 5300

250 146 145 240 45 2 4 0 0.33 2.5 3.9 1.33 410 550 340 2.8 2000 5400

225 125 125 220 45 3 7 0 0.8 5.0 2.5 1.0 300 280 220 4.8 2750 5500

220 110 110 190 45 3.5 11 0 0.9 7.5 15 0.75 200 260 180 6.0 2500 5600

175 100 100 170 50 4 14 0 1.0 10 0.8 0.5 100 120 80 6.8 1500 5700

166 77 78 162 43 4.4 18.6 11.6 11.2 12.6 0.05 0.1 0 0 0 0 0 5800

l11 l33 q31 q33 q15 d11 d33

qP

Unit: elastic constants, Cij, in 109 N/m2, piezoelectric constants, eij, in C/m2, piezomagnetic constants, qij, in N/Am2, dielectric constants, eij, in 109 C2/Nm2, magnetic constants, lij, in 106 Ns2/C2 and magnetoelectric coefficients, dij, in 1012 Ns/VC.

To the best knowledge of the author, so far no one has ever studied the natural frequency of transversely isotropic magneto-electro-elastic (MEE) plates in contact with the fluid since it involves the complex properties of MEE plate and the determination of virtual added mass due to the fluid effect. Therefore, in order to show the validity of the present approach, first of all, one simplifies the material property of the MEE plate to a purely elastic thin plate without considering the effective rigidities due to the presences of electricity and magnetism. Therefore, the added virtual mass incremental (AVMI) factor for an isotropic square plate is calculated and listed in Table 2. The comparisons of the AVMI between the present study and the results shown in Table 7 of Ref. [11] are presented, and a good agreement is found in Table 2. Secondly, a purely piezoelectric plate made of the commonly used PZT-4 material without considering the fluid effect is selected as the comparison between the complete 3D analysis and the present 2D thin plate analysis in order to understand the applicable range for the proposed approach. The material constants for a PZT-4 ceramic plate are cited from Table 1 in Ref. [27], and the dimensions for the PZT-4 plate are selected as h = 0.05 m for the thickness, 2a = 0.50 m for the plate span in the x-axis, and 2b = 1.00 m for the one in the y-axis, respectively. It is noticed that the dimensions are selected to fit the parameter requirements in Ref. [28], which was investigated by Chen et al., in which the length-to-width ratio s1 = 2a/2b is fixed at 0.5. Meanwhile, for the thin plate theory to be applicable, the thickness-to-span ratio s2 = h/2a is chosen to be 0.1. In their investigation, non-dimensional natural frequencies for a piezoelectric rectangular plate are presented by adopting both the complete 3D state space method and the 2D plate theory, as shown in Figs. 2 and 3 of Ref. [28]. As it can be found from these figures, there are large differences between the 3D and 2D analysis, especially when the thickness-to-span ratio is getting larger. Nevertheless, some dimensionless frequencies are approximately extracted from these figures for both the 3D and 2D analysis in the case of s2 = 0.1, and the comparisons between these data and the results obtained by the present study are listed in Table 3. The qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi non-dimensional natural frequencies for the present study are formulated as Xmn ¼ xmn ð2aÞ qp =C max , where Cmax is the maximum value of the elastic constants, and are presented from first mode through third mode in order to describe the variations between 2D and 3D simulations. As it can be detected from Table 3, the non-dimensional frequencies calculated by the present study match with those by the 2D theory in Ref. [28], and the frequencies are always larger than the corresponding ones obtained by the 3D theory; these phenomena have also been stated in Ref. [28]. Last of all, as for the comparison

Table 2 The comparison of cmm (AVMI) between present study and Ref. [11] for an isotropic square plate. SSSS = simply supported along four edges, CCCC = clamped along four edges. a=b=1 m

SSSS (present study)

SSSS (Ref. [11])

CCCC (present study)

CCCC (Ref. [11])

c11 c22 c33 c44 c55 c66 c77 c88

6.9526 2.1696 1.3938 0.9939 0.7878 0.6449 0.5508 0.4775

6.9526 2.1696 1.3938 0.9939 0.7878 0.6449 0.5508 0.4775

5.9219 1.9393 1.3528 0.9510 0.7690 0.6265 0.5401 0.4671

5.9219 1.9393 1.3528 0.9510 0.7690 0.6265 0.5401 0.4671

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515 4

2

x 10

m=1 m=2 m=3 m=4 m=5 m=6

1.8

Natural frequency (rad/s)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

60

70

80

90

100

Volume fraction (%) Fig. 2. Natural frequencies xmm versus volume fractions of a MEE plate in contact with fluid for simply supported boundary condition.

4

2.5

x 10

m=1 m=2 m=3 m=4 m=5 m=6

Natural frequency (rad/s)

2

1.5

1

0.5

0

0

10

20

30

40

50

60

70

80

90

100

Volume fraction (%) Fig. 3. Natural frequencies xmm versus volume fractions of a MEE plate in contact with fluid for clamped boundary condition.

with the existing MEE literatures without considering the fluid effect, since most of the MEE plate analyses are performed in a complete 3D point of view, the results obtained by imposing the present 2D thin plate theory are quite different from those offered by the other research approaches such as in Refs. [18,29]. Although the dimensions for the MEE plate adopted in Refs. [18,29] do not actually ensure the thin plate theory requirement (mh/2a = nh/2b = 1.0 and h = 0.3 m in both papers), two special cases on the non-dimensional fundamental natural frequency for the MEE plates, the pure BaTiO3 laminate, and the pure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CoFe2O4 one are provided below with the dimensionless formula being X11 ¼ x11 ð2aÞð2bÞ qp =C max =h. The fundamental natural frequency by adopting the above dimensions for pure BaTiO3 plate is 5.724, while the natural frequency for pure CoFe2O4 plate is 5.719. Even though the two data are much higher than the values presented in Ref. [29], which are X1 = 2.30for pure BaTiO3 case and X1 = 1.97for pure CoFe2O4 case, the boundary conditions for the MEE plate on the top and bottom surfaces in Ref. [29] are assumed as an open circuit, i.e., Dz = 0 andBz = 0, which imposed two additional boundary conditions on the structure; this might be one of the reasons why lower frequency in Ref. [29] was obtained. On the other hand, as it is stated in the last paragraph, the 2D analysis will always get a larger frequency than the 3D theory that has been concluded in Ref. [28]. Despite the overestimate on the natural frequencies caused by imposing the thin plate theory to the MEE laminate, the proposed model provides a fast, concise, efficient and conservative prediction for the free vibration characteristic of a MEE rectangular bilayered thin plate in contact with fluid shown in Eq. (1).

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515

Based on Eq. (15), one first determines cmn (AVMI factor) of a MEE rectangular plate in contact with fluid. In particular, a bi-layered BaTiO3  CoFe2O4 MEE plate with 40% BaTiO3 is considered. Tables 4a and 4b show the values of added virtual mass incremental factor (AVMI factor) for such a MEE plate with the simply supported boundary conditions. In addition, Table 3 Non-dimensional natural frequencies for a PZT-4 rectangular plate. Non-dimensional natural frequencies qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xmn ¼ xmn ð2aÞ qp =C max

PZT-4: Data extracted from Figs. 2 and 3 in Ref. [28]

PZT-4: Data extracted from Figs. 2 and 3 in Ref. [28]

2a = 0.5 m, 2b = 1.0 m, h = 0.05 m

Mode m = 1, m = 1, m = 1, m = 2, m = 2, m = 2,

3D 0.2 0.4 0.6 0.8 1.0 1.2

2D 0.4 0.6 1.0 1.2 1.4 1.6

Present study 0.386 0.617 1.004 1.313 1.544 1.930

n=1 n=2 n=3 n=1 n=2 n=3

Table 4a The values of cmn for odd mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with simply supported boundary conditions.

cmn

m=1

m=3

m=5

n=1 n=3 n=5

3.084426 0.999956 0.578724

0.999956 0.618340 0.436316

0.578724 0.436316 0.349497

Table 4b The values of cmn for even mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with simply supported boundary conditions.

cmn

m=2

m=4

m=6

n=2 n=4 n=6

0.962513 0.581607 0.403931

0.581607 0.440930 0.341378

0.403931 0.341378 0.286101

Table 5a The values of cmn for odd mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with clamped boundary conditions.

cmn

m=1

m=3

m=5

n=1 n=3 n=5

2.627170 0.995919 0.593630

0.995919 0.600151 0.428420

0.593630 0.428420 0.341156

Table 5b The values of cmn for even mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with clamped boundary conditions.

cmn

m=2

m=4

m=6

n=2 n=4 n=6

0.860344 0.553081 0.395591

0.553081 0.421898 0.330997

0.395591 0.330997 0.276164

Table 6a Natural frequencies for odd mode of a pure CoFe2O4 square plate (0% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

256.25 1839.61 5396.66

1839.61 3689.09 7407.41

5396.66 7407.41 11247.57

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515 Table 6b Natural frequencies for odd mode of a bilayered square plate (20% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

293.27 2100.50 6154.17

2100.50 4207.55 8442.12

6154.17 8442.12 12813.44

Table 6c Natural frequencies for odd mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

267.96 1914.69 5602.99

1914.69 3831.34 7681.71

5602.99 7681.71 11654.26

Table 6d Natural frequencies for odd mode of a bilayered square plate (60% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

248.95 1774.77 5187.43

1774.77 3547.50 7107.96

5187.43 7107.96 10779.44

Table 6e Natural frequencies for odd mode of a bilayered square plate (80% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

209.91 1493.07 4358.90

1493.07 2981.40 5969.29

4358.90 5969.29 9049.30

Table 6f Natural frequencies for odd mode of a pure BaTiO3 square plate (100% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

193.22 1371.32 3998.78

1371.32 2735.40 5473.24

3998.78 5473.24 8294.03

Table 7a Natural frequencies for even mode of a pure CoFe2O4 square plate (0% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

928.68 3525.19 8817.33

3525.19 6960.17 11734.02

8817.33 11734.02 16602.54

Table 7b Natural frequencies for even mode of a bilayered square plate (20% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

1060.28 4020.15 10047.32

4020.15 7932.69 13366.69

10047.32 13366.69 18906.85

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T.-P. Chang / Applied Mathematical Modelling 37 (2013) 2503–2515

Table 5a and Table 5b present the values of added virtual mass incremental factor (AVMI factor) for the same MEE plate with the clamped boundary conditions. It can be detected from these tables that the values of c11 is much larger than those of the other modes for both boundary conditions. This implies that c11 plays a dominant role as far as AVMI factor is concerned; besides it can be found that cmn decreases with the mode order, meaning the effect of fluid also decreases with the mode order. Generally speaking, AVMI factor cmn of the lower mode number is larger than that of the higher mode number since the fluid movement stroke of the lower mode number is larger than that of the higher one. Hence, the fluid movement stroke

Table 7c Natural frequencies for even mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

966.42 3660.29 9141.04

3660.29 7218.29 12157.28

9141.04 12157.28 17191.09

Table 7d Natural frequencies for even mode of a bilayered square plate (60% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

895.73 3388.76 8456.81

3388.76 6679.25 11243.92

8456.81 11243.92 15894.99

Table 7e Natural frequencies for even mode of a bilayered square plate (80% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

753.50 2847.58 7101.23

2847.58 5609.44 9438.84

7101.23 9438.84 13339.51

Table 7f Natural frequencies for even mode of a pureBaTiO3square plate (100% of BaTiO3) in contact with fluid with simply supported boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

691.98 2612.19 6510.31

2612.19 5143.39 8650.96

6510.31 8650.96 12222.69

Table 8a Natural frequencies for odd mode of a pure CoFe2O4 square plate (0% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

616.75 2673.20 6716.93

2673.20 5050.27 9285.48

6716.93 9285.48 13652.99

Table 8b Natural frequencies for odd mode of a bilayered square plate (20% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

705.57 3052.25 7660.20

3052.25 5759.61 10582.22

7660.20 10582.22 15552.95

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will be reduced as the mode number increases, which will end up with the reduction of the added mass and AVMI factor. The similar phenomenon can be detected in the free vibration of a liquid-filled circular cylindrical shell [30]. One now calculates the natural frequencies of a MEE plate in contact with fluid by using Eqs. (13) and (14). Tables 6a-f and Tables 7a-f present the natural frequencies of the bi-layered MEE thin plate in contact with fluid subjected to different volume fraction of the PZ material BaTiO3, which are ranging from 0% to 100% with 20% offset. The boundary conditions of the MEE plate are considered as simply supported. The natural frequencies presented in Tables 6a–f are for odd modes; while

Table 8c Natural frequencies for odd mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

644.52 2782.30 6974.60

2782.30 5244.19 9628.66

6974.60 9628.66 14145.54

Table 8d Natural frequencies for odd mode of a bilayered square plate (60% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

598.66 2578.99 6457.57

2578.99 4855.53 8909.07

6457.57 8909.07 13083.05

Table 8e Natural frequencies for odd mode of a bilayered square plate (80% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

504.61 2169.64 5426.52

2169.64 4080.39 7481.83

5426.52 7481.83 10982.76

Table 8f Natural frequencies for odd mode of a pure BaTiO3square plate (100% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=1

m=3

m=5

n=1 n=3 n=5

464.41 1992.59 4978.51

1992.59 3743.69 6859.88

4978.51 6859.88 10065.87

Table 9a Natural frequencies for even mode of a pure CoFe2O4 square plate (0% of BaTiO3) in contact with fluid with clamped boundary conditions Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

2385.35 5546.47 10723.29

5546.47 8869.30 14159.56

10723.29 14159.56 19562.49

Table 9b Natural frequencies for even mode of a bilayered square plate (20% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

2722.53 6324.39 12218.91

6324.39 10107.61 16129.15

12218.91 16129.15 22276.80

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those showed in Tables 7a–f are for even modes. In Tables 8a–f and Tables 9a–f, the natural frequencies of the bi-layered MEE thin plate in contact with fluid, are presented for the clamped boundary conditions. It is noted that the natural frequencies

Table 9c Natural frequencies for even mode of a bilayered square plate (40% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

2480.86 5757.48 11116.18

5757.48 9196.54 14668.82

11116.18 14668.82 20253.84

Table 9d Natural frequencies for even mode of a bilayered square plate (60% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

2298.83 5329.89 10283.91

5329.89 8509.01 13566.34

10283.91 13566.34 18726.17

Table 9e Natural frequencies for even mode of a bilayered square plate (80% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

1933.35 4478.24 8635.17

4478.24 7145.66 11387.89

8635.17 11387.89 15714.75

Table 9f Natural frequencies for even mode of a pureBaTiO3 square plate (100% of BaTiO3) in contact with fluid with clamped boundary conditions. Omega (rad/s)

m=2

m=4

m=6

n=2 n=4 n=6

1775.10 4107.95 7916.23

4107.95 6551.47 10436.68

7916.23 10436.68 14398.17

800 SSSS CCCC

Natural frequency (rad/s)

700

600

500

400

300

200

100

0

10

20

30

40

50

60

70

80

90

100

Volume fraction (%) Fig. 4. Natural frequencies x11 versus volume fractions of a MEE plate in contact with fluid for SSSS and CCCC boundary conditions.

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listed in Tables 6a-9f are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid. In order to observe the trend of variation for the natural frequencies of a bi-layered MEE thin plate in contact with fluid, some particular data are chosen from Tables 6a-9f and plotted in Figs. 2 and 3. As it can be detected, the natural frequencies xmm of MEE plate with 20% BaTiO3 constituent always have the largest value in all the modes. These phenomena are very interesting and remarkable; those people who are working on the frequency problem of the MEE plate in contact with fluid, ought to pay attention to them and take the advantages of the findings. Based on the results shown in Figs. 2 and 3, the natural frequencies xmm of MEE plate with the simply supported boundary conditions are smaller than those with the clamped boundary conditions, which is quite reasonable and predictable. To actually see the discrepancies due to different boundary conditions, the natural frequencies x11 versus volume fractions of a MEE plate in contact with fluid are plotted in Fig. 4. for simply supported boundary condition (SSSS) and clamped boundary condition (CCCC) individually. As it can be found from Fig. 4, the natural frequencies x11 with CCCC boundary condition are much larger than those with SSSS boundary condition. 4. Conclusions In this paper, the fluid-structure interaction problem is studied. In particular, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. It is well known that the natural frequencies of the uniform plate in contact with fluid can be calculated by using the added virtual mass incremental (AVMI) factor, which represents the kinetic energy due to the fluid. In the present study, the mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. A recently proposed differential equation governing the dynamical responses of the MEE rectangular plates is introduced, and the attempt to extend the system into a fluid-interaction model in order to account for the water influence is achieved. On the fluid-structure interface, some techniques are adopted to deal with the relationships between the velocity potential and the mode shapes, and then by imposing the Fourier transform, one can derive the formulations of the reference kinetic energy for both the fluid and the plate itself. After these two energies have been computed, the added virtual mass incremental (AVMI) factor can be obtained rapidly and the added virtual mass can thus be acquired. Furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. From the engineering point of view, the proposed approach and some important findings can be adopted as a guide for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid. Acknowledgements This research was partially supported by the National Science Council in Taiwan through Grant NSC-95-2211-E-327-046. The author is grateful for the financial supports. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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