Composite Structures 95 (2013) 254–263

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Composite Structures

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Free vibration of cantilevered composite plates in air and in water

Matthew R. Kramer a, Zhanke Liu b, Yin L. Young a,⇑a Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USAb P.O. Box 3263, Sugar Land, TX 77487

a r t i c l e i n f o a b s t r a c t

Article history:Available online 7 August 2012

Keywords:Added massCompositeHydroelasticityCantilevered plateFluid–structure interactionFree vibration

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.07.017

⇑ Corresponding author.E-mail addresses: mattkram@umich.edu (M.R. Kra

com (Z. Liu), ylyoung@umich.edu (Y.L. Young).

Composite materials are being used more frequently for marine applications due to the advantages of ahigher stiffness- and strength-to-weight ratio, and better corrosion resistance compared to metallicalloys. Many examples consist of cantilevered structures, such as hydrofoils, propeller and turbine blades,keels, and rudders. A wide range of analytical and numerical tools exist for the free vibration analysis ofcomposite structures in air due to their applicability to design problems in the aerospace industry, suchas airplane wings, turbofan and propeller blades, and flight control surfaces. For these aerospace struc-tures the inertial effects of the fluid are typically neglected due to the low relative density of air comparedto the structure. Contrarily, for marine structures, fluid inertial (added mass) effects cannot be neglected,especially for composites with much higher fluid-to-solid density ratios. The objective of this work is toinvestigate the effects of material anisotropy and added mass on the free vibration response of rectangu-lar, cantilevered composite plates/beams via combined analytical and numerical modeling. The resultsshow that the natural frequencies of the composite plate are 50–70% lower in water than in air due tolarge added mass effects. The added mass is found to vary considerably with material orientation dueto the bend-twist coupling of anisotropic composites, which affects the mode shapes and, consequently,the fluid inertial loads. The analytical method is found to yield accurate results for beam geometries andoffers significant savings in computational cost compared to the finite element method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The use of composite materials for marine applications isbecoming more prevalent due to the advantages of a generallyhigher strength- and stiffness-to-weight ratio, and better corrosionresistance compared to traditional metallic alloys such as steel,aluminum, or bronze. Additionally, composite materials offer theability to elastically tailor the deformation through the design ofthe material (e.g. via the laminate stacking sequence), which canyield better performance over a wider range of operating condi-tions [9,10,12,16–18,21,27].

Many marine applications for which composite materials havebeen used consist of cantilevered structures, including propellerand turbine blades, hydrofoils, keels, and rudders. One importantaspect of the design of these structures is free vibration analysis,where the natural frequencies and mode shapes are calculated inorder to predict the dynamic structural response, as well as toidentify potential instability limits such as resonance and flutter.Similar design problems may be found in the aerospace industry,including aircraft wings, turbofan and propeller blades, helicopter

ll rights reserved.

mer), zhankeliu2008@gmail.

rotors, and wind turbines, and many of these have benefittedgreatly from the use of composite materials. Consequently, manyanalytical and numerical tools have been developed and validatedfor composite structures in air.

Narita and Leissa [19] presented an analytical method for calcu-lating the dry natural frequencies and mode shapes of very thincomposite plates using the Ritz method. They analyzed a largenumber of configurations, varying both the aspect ratio and thelaminate layup sequence, and the results were shown to comparefavorably with both numerical (finite-element) and experimentalresults of Crawley [6]. Analytical methods have also been devel-oped for Timoshenko beams [4,22] for varying geometries andlaminate layup sequences. In each of these studies, the effects ofthe fluid inertia, i.e. added mass, of the surrounding fluid are ne-glected due to the relatively low density of air compared to thestructural density. This assumption, however, is not typically validfor marine structures, for which added mass effects are often sig-nificant. Consequently, methods for the prediction of the freevibration of composite structures must be developed with consid-eration for fluid inertial effects.

The effect of added mass on the natural frequencies increases asthe ratio of fluid-to-solid density increases, and consequently theadded mass of most structures in water may not be neglected,especially for composite structures, which generally have a lower

Fig. 1. Diagram of a typical rectangular composite cantilevered plate or beam.Aspect ratio is defined as = L/b, thickness-to-chord ratio is represented as t/b.

Table 1Dimensions of the plate and beam considered in this study.

Dimension Plate Beam

L 243.8 mm 243.8 mmb 92.50 mm 20.32 mmt 3.200 mm 4.063 mm

= L/b 2.6 12t/b 3.46% 20%

1 Additional geometries and material configurations are examined in Section 4.4 forvalidation studies with published results from the literature. However, all originaresults presented in this paper are for one or both of these two geometries.

M.R. Kramer et al. / Composite Structures 95 (2013) 254–263 255

effective density than metallic structures. The added mass effectsfor cantilevered isotropic plates have been investigated by severalauthors, including Marcus [14] and Yadykin et al. [26], to name afew. In [14], a finite element method is used, where the fluid effectsare modeled using inviscid, linearly compressible acoustic fluidelements. The results were found to compare well with the exper-imental results of Lindholm et al. [11]. In [26], an analytical formu-lation is presented for a submerged, cantilevered, isotropic plate inwater, where the fluid inertial effects are calculated based on po-tential flow theory. Although several methods for considering fluidinertial effects have been developed, they have previously been ap-plied to isotropic structures constructed of metallic alloys. Since, ingeneral, added mass effects are highly dependent on the nature ofthe vibrational modes (e.g. bending or twisting), the bend-twistcoupling that results from the material anisotropy present in com-posite materials introduces more complex added mass effects thatmust be quantified.

The objective of this work is to investigate the effects of mate-rial anisotropy and added mass on the free vibration response ofrectangular, cantilevered composite plates/beams via combinedanalytical and numerical studies. An analytical method is first pre-sented, followed by numerical modeling via finite element simula-tions. Convergence and validation studies are shown. The influenceof added mass and material anisotropy on the free-vibration re-sponse of composite plates and beams are examined, followed bya summary of the major findings.

2. Problem description

The current study is aimed towards cantilevered structuressuch as wings, hydrofoils, propeller and turbine blades, keels, andrudders. Due to the variability in the geometry of these structures,they are generalized in this study as rectangular, cantileveredplates and beams for simplicity. It is expected that the major phe-nomena exhibited by the rectangular plates/beams are representa-tive of those present for more complicated structures.

In this work, two fluid media are considered: air and water, de-noted as dry and wet configurations, respectively. For the drycases, the air is assumed to have a negligible effect and is ignored(i.e. fluid density, qf = 0). A schematic for a typical cantileveredplate or beam is shown in Fig. 1. The dimensions are defined byits length, width, and thickness, represented by L, b, and t, respec-tively. The aspect ratio is defined as = L/b, which is in line withits definition in aerodynamics, and the thickness-to-chord ratio isdefined as t/b. Results for two different geometries will be shownin this paper: one that is representative of a thin plate, with

= 2.6 and t/b = 3.46%, and one that satisfies beam assumptions,

with = 12 and t/b = 20%. The dimensions for each geometry areshown in Table 1. It should be noted that the length of both theplate and the beam, as well as the material properties, are identi-cal, and that only the aspect ratio and thickness ratio are varied.1

A right-handed coordinate system is placed with the x–y planeat the mid-plane of the plate, where the origin is located at the can-tilevered end and the z-direction is in the thickness direction (seeFig. 1). The principle axes of the composite material are denoted asthe 1–2 axes, where the 1-direction is along the effective fiber an-gle, which is oriented at an angle h with respect to the x–y axes.

In general, a composite plate/beam is composed of multiplelaminate plies, and each ply may be aligned at a different angle.It has been shown that, for symmetric layup sequences, an equiv-alent unidirectional fiber angle may be found to represent theoverall load-deformation characteristics, and hence simplify thecoupled fluid–structure interaction (FSI) analysis [17,28]. Althoughthe internal stress distributions of the structure with the equiva-lent unidirectional fiber angle will not match with the actual mul-ti-layered composite structure, the deformation behavior, andconsequently the natural modes and frequencies, will be equiva-lent. In this paper, the equivalent fiber angle is denoted as h andis used as a means for parametric studies to explore effect of mate-rial anisotropy more generally. It should be noted that both theanalytical and numerical methods presented in this paper arecapable of handling generalized composite layups.

The material is modeled as orthotropic, where the 1-direction isassociated with the fiber direction, and the stiffness in the 2- and3-directions is assumed to be equivalent. Under this assumption,the entire material behavior may be specified by five properties,E1, E2, G12, m12, and m23, where Ei is the Young’s modulus in the i-direction and Gij and mij are the shear modulus and Poisson’s ratioin the i–j plane, respectively. Symmetry considerations requirethe following: E2 = E3, m12 = m13, G12 = G13 and G23 = E2/2(1 + m23).The assumed material properties for the composite plate/beamare shown in Table 2.

3. Analytical formulation

The analytical solution for the free vibration of compositebeams with consideration for FSI is presented in this section. Themodel accounts for the coupling between bending and torsion in-duced by material anisotropy. The beam’s center of gravity is as-sumed to be collocated with the elastic axis to enableindependent study of material coupling effects. In line with Ber-noulli–Euler beam theory, the current model neglects shear defor-mation, rotary inertia, and warping effects. The added massformulas used in the analytical model are based on potential flowassumptions and strip theory.

Each longitudinal section is restricted to two degrees of free-dom: z-displacement and x-rotation about the reference axis, de-noted as h and /, respectively, as shown in Fig. 2. The governingpartial differential equations for such a structural system under

l

Table 2Assumed composite and fluid properties. It should be noted that, for dry cases, thefluid is neglected (i.e. qf = Kf = 0).

Material Item Symbol Value Unit

Composite Density qs 1500 kg/m3

Young’s modulus E1 171.42 GPaYoung’s modulus E2, E3 9.08 GPaShear modulus G12, G13 5.29 GPaPoisson’s ratio m12, m13 0.32 –Poisson’s ratio m23 0.29 –

Water Density qf 1000 kg/m3

Bulk modulus Kf 2.2 GPa

Fig. 2. Description of two analytical degrees of a freedom for a longitudinal strip ofthe beam. Note that the center of gravity (CG) and elastic axis (EA) are collocatedand assumed to translate in the z-direction only.

256 M.R. Kramer et al. / Composite Structures 95 (2013) 254–263

free vibration, ignoring both structural and fluid damping, havepreviously been given by others [2–4,7,8,13,23,24], and may be ex-pressed as:

EI@4h@x4 � K

@3/@x3 þm

@2h@t2 ¼ 0 ð1Þ

GJ@2/@x2 � K

@3h@x3 � Ix

@2/

@t2 ¼ 0 ð2Þ

where h = h(x, t) and / = /(x, t) are both a function of the longitudi-nal location x and time t. Here, EI is the bending stiffness parameter,GJ is the torsional stiffness parameter, K is the bending-torsion cou-pling parameter, m is the total mass per unit length, and Ix is the to-tal polar mass moment of inertia per unit length about the x-axis.The total mass and mass moment of inertia include both solid (ms

and Ix,s) and fluid (ma and Ix,a) components, and are defined as:

m ¼ ms þma ¼ qsbt þma ð3Þ

Ix ¼ Ix;s þ Ix;a ¼ qsbtðb2 þ t2Þ

12þ Ix;a ð4Þ

where qs is the solid density.The added mass terms are calculated based on potential flow

theory, as described in [5,20]:

ma ¼ qfp4ðb2 cos2 /þ t2 sin2 /Þ � qf

p4

b2 ð5Þ

Ix;a ¼ qfp

128b4 ð6Þ

where qf is the fluid density. It should be noted that, although theadded mass contains nonlinear terms in /, these terms may beapproximated based on small angle arguments for frequencyanalysis.

The three stiffness parameters (EI, GJ, and K) may be calculatedfollowing the work of Weisshaar and Foist [25] as:

EI ¼ b D11 �D2

12

D22

!ð7Þ

GJ ¼ 4b D66 �D2

26

D22

!ð8Þ

K ¼ 2b D16 �D12D26

D22

� �ð9Þ

where the expressions of bending stiffness terms D11, D22, D66, D12,D16, and D26 are dependent on the material properties and orienta-tion, as defined in Appendix A. Notice that the chordwise bendingmoment has been neglected because its contribution is assumedto be small in comparison to the spanwise moment. However, thereis no constraint to the chordwise curvature or camber bending, andhence it is different from the assumption of chordwise rigidity [25].

Using separation of variables, similar to Banerjee [3] and Wanget al. [23], it is assumed that h(x, t) = H(x)eixt and /(x, t) = U(x)eixt,where x is the eigenfrequency. Eqs. (1) and (2) may then be writ-ten in eigen format:

EIHð4Þ � KUð3Þ �mx2H ¼ 0 ð10ÞGJUð2Þ � KHð3Þ þ Ixx2U ¼ 0 ð11Þ

where f(n) = dnf/dxn represents the nth spatial derivative of a func-tion f. By eliminating either H or U from Eqs. (10) and (11), asixth-order differential equation is obtained:

W ð6Þ þ Ix � EI �x2

EI � GJ � K2 W ð4Þ � m � GJ �x2

EI � GJ � K2 W ð2Þ � mIxx4

EI � GJ � K2 W ¼ 0

ð12Þ

where W = H or U. Introducing the non-dimensional length n = x/Land letting D = d(�)/dn, Eq. (12) can be re-written as:

ðD6 þ �aD4 � �bD2 � �a�b�cÞW ¼ 0 ð13Þ

where

�a ¼ Ix � EI �x2L2

EI � GJ � K2 ð14Þ

�b ¼ m � GJ �x2L4

EI � GJ � K2 ð15Þ

�c ¼ 1� K2

EI � GJð16Þ

The general solutions of Eq. (13) takes the form [3,23]:

HðnÞ ¼ A1 cosh anþ A2 sinh anþ A3 cos bnþ A4 sin bnþ A5

� cos cnþ A6 sin cn ð17Þ

UðnÞ ¼ B1 cosh anþ B2 sinh anþ B3 cos bnþ B4 sin bnþ B5

� cos cnþ B6 sin cn ð18Þ

where

a ¼ f2ðq=3Þ1=2 cosðu=3Þ � �a=3g1=2 ð19Þb ¼ f2ðq=3Þ1=2 cos½ðp�uÞ=3� þ �a=3g1=2 ð20Þc ¼ f2ðq=3Þ1=2 cos½ðpþuÞ=3� þ �a=3g1=2 ð21Þq ¼ �bþ �a2=3 ð22Þu ¼ cos�1fð27�a�b�c � 9�a�b� 2�a3Þ=½2ð�a2 þ 3�bÞ3=2�g ð23Þ

The coefficients A1�6 and B1�6 are related and the relation-ship may be obtained by substituting Eqs. (17) and (18) intoEq. (10):

M.R. Kramer et al. / Composite Structures 95 (2013) 254–263 257

B1 ¼ kaA2=L; B2 ¼ kaA1=L ð24ÞB3 ¼ kbA4=L; B4 ¼ �kbA3=L ð25ÞB5 ¼ kcA6=L; B6 ¼ �kcA5=L ð26Þ

where

ka ¼EI � a4 �mx2L4

Ka3 ð27Þ

kb ¼EI � b4 �mx2L4

Kb3 ð28Þ

kc ¼EI � c4 �mx2L4

Kc3 ð29Þ

Following Banerjee [3] and Wang et al. [23], the expressions forthe bending rotation H(n), bending moment M(n), shear force S(n),and torsional moment T(n) may be obtained from Eqs. (17) and(18):

HðnÞ ¼1L

dHðnÞdn

¼ 1LðA1a sinh anþ A2a cosh an� A3b sin bn

þ A4b cos bn� A5c sin cnþ A6c cos cnÞ ð30Þ

MðnÞ ¼ � EI

L2

d2HðnÞdn2 ¼ � EI

L2 ðA1a2 cosh anþ A2a2 sinh an

� A3b2 cos bn� A4b

2 sin bn� A5c2 cos cn� A6c2 sin cnÞ ð31Þ

SðnÞ ¼ � 1L

dMðnÞdn

¼ EI

L3 ðA1a3 sinh anþ A2a3 cosh anþ A3b3 sin bn

� A4b3 cos bnþ A5c3 sin cn� A6c3 cos cnÞ ð32Þ

TðnÞ ¼GJL

dUðnÞdn

¼ GJ

L2 ðA1aka cosh anþ A2aka sinh an

� A3bkb cos bn� A4bkb sin bn� A5ckc cos cn� A6ckc sin cnÞð33Þ

To derive the frequency equation, the following boundaryconditions (at the clamped end n = 0 and at the free end n = 1,respectively), are applied:

Hð0Þ ¼ Hð0Þ ¼ Uð0Þ ¼ 0 ð34ÞMð1Þ ¼ Sð1Þ ¼ Tð1Þ ¼ 0 ð35Þ

By substituting Eqs. (17), (18), (30)–(32), and (33) into Eqs. (34)and (35), the following linear system of equations is obtained:

BA ¼ 0 ð36Þ

where A = [A1,A2,A3,A4,A5,A6]T and B takes the following form:

1 0 1 0 1 00 a 0 b 0 c0 ka 0 kb 0 kc

a2Cha a2Sha �b2Cb �b2Sb �c2Cc �c2Sc

a3Sha a3Cha b3Sb �b3Cb c3Sc �c3Cc

akaCha akaSha �bkbCb �bkbSb �ckcCc �ckcSc

26666666664

37777777775

where

Cha ¼ cosh a; Cb ¼ cos b; Cc ¼ cos cSha ¼ sinh a; Sb ¼ sin b; Sc ¼ sin c ð37Þ

The necessary and sufficient condition for a non-zero solutionto Eq. (36) is D = det[B] = 0, which yields the natural frequenciesof the composite beam.

The frequency, x, may be non-dimensionalized, as in [19]:

X ¼ xL2

ffiffiffiffiffiffiffiqstDo

rð38Þ

where Do = E1t3/12(1 � m12m21) corresponds to the reference bend-ing stiffness at h = 0�. All further results for frequency will be non-dimensionalized accordingly.

4. Numerical formulation

4.1. Governing equations

Numerical simulations have also been performed in order to (1)validate the results of the analytical model and (2) obtain resultsfor cases where the beam assumptions are not valid (e.g. cases withplate-like geometry with low aspect ratio or small thickness-to-chord ratio, where the chordwise bending moment is not smallcompared to the spanwise bending moment), or for cases wherehigher-order modes with combined bend, twist, and warpingbehavior develop, and the simplified expressions for the addedmass terms as shown in Eqs. (5) and (6) are no longer valid.

The commercial finite element code, Abaqus/Standard [1], hasbeen used to perform frequency analysis for both wet and dry con-figurations. For wet configurations, the fluid–structure interaction(FSI) is modeled by coupling an acoustic fluid domain to the frontand back faces of the plate via surface-based tie constraints on theshared boundaries, which enforce the displacement and pressurecompatibility conditions at the fluid–solid interface. The otherthree wetted faces were not coupled to avoid numerical instabilityissues. The effect of neglecting these faces should be negligible,however, since the front and back plates support the majority ofthe pressure forces.

Assuming the fluid to be inviscid, irrotational, linearly com-pressible, and neglecting gravitational effects, the equilibriumequation is:

@p@xþ qf

€uf ¼ 0 ð39Þ

where p is the fluid pressure, x is the positional vector, qf is the fluiddensity, and €uf is the fluid particle acceleration vector. The consti-tutive behavior of the fluid is assumed to obey the followingrelation:

p ¼ �Kf@

@x� uf ð40Þ

where Kf is the bulk modulus of the fluid and uf is the fluid particledisplacement vector. The fluid domain boundaries are assumed tobe infinitely far away, although the computational domain is finitein size. In order to represent an infinite domain on a finite numer-ical grid, a non-reflecting boundary condition is used on all exteriorboundaries, except the wall at the cantilevered end of the plate,where a fully reflective boundary condition is used.

In order to model the structural response, the nonlinear (i.e.large deformation) solid equation of motion is formulated as anequilibrium equation (i.e. no external forcing) and the eigenvaluesare extracted via the Lanczos method. The generalized finite ele-ment equilibrium equation for the structure, which assumes nostructural damping may be written as:

M€us þ Kus ¼ 0 ð41Þ

where M and K are the structural mass and stiffness matrices,respectively, and €us and us are the structural acceleration and dis-placement vectors, respectively. Further details of the formulationcan be found in [1].

4.2. Finite element mesh topology

The finite element mesh consists of two major element sets:one for the solid domain and one for the fluid domain. For dry sim-ulations, the fluid domain is not included in the model. Due to the

Fig. 3. Diagram of finite element mesh topology.

258 M.R. Kramer et al. / Composite Structures 95 (2013) 254–263

simplicity of the geometry, both domains consist solely of rectan-gular elements, and as such the domain may be completely speci-fied by a small number of parameters. The solid domain isdiscretized in each dimension using ni elements, where i corre-sponds to the x-, y-, or z-direction, a shown in Fig. 3a. Two differenttypes of solid elements were tested, including eight-noded solidelements with incompatible mode (C3D8I), and eight-noded con-tinuum shell elements with reduced integration (SC8R). Detailsof the formulation for each element may be found in [1]. The re-sults were found to agree favorably for both types of elements.Hence, for the results shown herein, the shell elements are chosenfor their computational efficiency.

The fluid domain is generated by extending the solid domain inall three spatial directions. This is done by specifying the numberof additional layers in each direction, nf,i, where i again correspondsto the x-, y-, and z-directions. For each of these parameters, the ex-tent of the fluid domain may be expressed as a length xB, yB, and zB.This nomenclature is defined in Fig. 3b, where roughly 1/4 of thetotal fluid domain is shown. It should be noted that Fig. 3b isshown for illustrative purposes and is not to scale. The completefluid domain may be constructed by reflecting the portion shownacross both the x–y and x–z planes. The objective is to place theboundaries of the fluid domain sufficiently far away from the platein order to eliminate boundary effects. The computational gridused in the current study is shown in Fig. 4.

The size of the fluid element faces with a surface normal pointingin the z-direction is made to be identical to the solid domain elementfaces and the elements were grown in the z-direction with a growth

rate c = 1.05 (i.e. Dzi+1 = cDzi). The thickness of the first layer of fluidelements is equal to the thickness of the plate. The fluid domain wasmodeled using eight-noded acoustic finite elements (AC3D8).

4.3. Convergence studies

As a measure of convergence, the normalized error of the non-dimensional frequency is chosen, which is defined as

e ¼ jXiðhÞ �Xi;finestðhÞjXi;finestðhÞ

ð42Þ

where Xi is the non-dimensional natural frequency correspondingto the ith mode. A sufficient level of convergence is accepted whenthe maximum error for the first two modes is less than 10�3. Theconvergence study was conducted for the plate with = 2.6 andt/b = 3.46%, and with a fiber orientation angle h = 30�. Additionalconvergence studies were conducted for other fiber orientation an-gles and aspect ratios with similar results, but the results are notpresented in this paper for the sake of brevity.

In order to determine the required level of solid domain meshrefinement, the three mesh parameters, nx, ny, and nz were system-atically varied for the dry case. The two parameters nx and ny werenot chosen independently and instead were chosen to yield ele-ment faces that were as close to a square as possible. The ratio ofcells is therefore dependent on the aspect ratio of the plate, i.e.nx/ny � . The absolute error of the dry natural frequency for aselection of grids is shown in Table 3. The final mesh was chosento be [nx,ny,nz] = [72,27,7].

Fig. 4. Converged finite element mesh for wet plate simulations showing thecomplete solid domain and roughly one half of the fluid domain (Fine mesh:[nx,ny,nz] = [72,27,7], and Large domain: [xB/L,yB/L,zB/L] = [0.50,0.51,1.24]).

Table 3Convergence of dry natural frequency for various structural mesh resolution for theplate with = 2.6, t/b = 3.46%, and at a fiber angle h = 30�. The second mode ischosen since it generally yielded higher levels of error than the first mode. The finemesh was chosen for the remainder of the analyses.

Mesh nx ny nz X2,dry e (�10�3)

Coarse 24 9 3 1363.4 8.964Medium 48 18 5 1364.9 1.273Fine 72 27 7 1364.6 0.412Finest 96 36 9 1364.3 0.000

Table 4Convergence of wet natural frequency for various fluid domain sizes for the plate with

= 2.6, t/b = 3.46%, and at a fiber angle h = 30�. The second mode is chosen since itgenerally yielded higher levels of error than the first mode. The Large domain waschosen for the remainder of the analyses.

Mesh xB/L yB/L zB/L X2,wet e (�10�3)

Small 0.33 0.34 0.17 437.81 41.08Medium 0.42 0.42 0.46 454.47 4.600Large 0.50 0.51 1.24 456.29 0.613Largest 0.58 0.59 2.89 456.57 0.000

M.R. Kramer et al. / Composite Structures 95 (2013) 254–263 259

Once convergence of the solid mesh for dry natural frequencywas demonstrated, the fluid mesh parameters were varied in asimilar manner and the absolute error of the wet natural frequencywas calculated for a selection of domain sizes. The results areshown in Table 4. The mesh was determined to have a sufficient le-vel of convergence with values of [nf,x,nf,y,nf,z] = [36,36,35], whichyields domain extents of [xB/L,yB/L,zB/L] = [0.50,0.51,1.24]. Theconverged mesh is shown in Fig. 4 for reference.

4.4. Validation studies

Once the converged mesh was identified, the geometry andmaterial properties were varied in order to compare the dry andwet frequencies with published results from open literature. Inorder to validate the numerical solution of submerged compositecantilevered plates, two separate validation studies wereperformed.

The first compares the results for a dry composite cantileveredplate to those of Narita and Leissa [19] in order to ensure that thesolid structural model is able to capture the effects of varying fiberangle and layup sequence. In Fig. 5a, the non-dimensional frequen-cies for a cantilevered, multidirectional composite plate (layupsequence [0/30/30/0]�) is compared to the analytic results of Naritaand Leissa [19], as well as with numerical and experimental resultsof Crawley [6]. The results are found to compare well, especially forthe first three modes. The current numerical study is thencompared to Narita and Leissa’s (N & L) analytical model [19] forunidirectional composite plates with a range of fiber angles.

Contours of the magnitude of the relative difference between theresults are shown in Fig. 5b, where it can be seen that the maxi-mum difference is 0.81%. Similar results were also found for twoother sets of material properties. Hence, the current finite elementmodel of the dry composite plates is assumed to be sufficientlyaccurate.

The second necessary validation study relates to the FSIcoupling between the fluid and solid domains. Comparisons ofthe predicted wet natural frequencies for a cantilevered, steel plate( = 2, t/b = 1.31%) with the numerical results of Marcus [14] andexperimental results of Lindholm et al. [11] are shown in Fig. 6.In general, good comparisons are observed. However, the frequen-cies tend to be slightly overpredicted for all modes compared tothe measured values, particularly for the higher modes. This maybe due to material or fluid damping effects, or measurement errors,particularly for the higher modes.

5. Results and discussion

5.1. Comparison of analytical and numerical methods

The in-air (dry) and in-water (wet) natural frequencies of thecomposite plate and beam with varying fiber angles, h, are shownin Fig. 7. The dimensions and material properties of the plate andbeam are shown in Tables 1 and 2, respectively. As shown inFig. 7b, both the analytical and numerical results agree well witheach other for the beam geometry. However, although both theanalytical and numerical results showed similar trends for theplate geometry (see Fig. 7a), the agreement between the two isnot as good because the beam assumptions are violated, andchordwise deformation and warping are no longer negligible.

The results suggest that the analytical solution is quite accuratefor beams with high and sufficiently high thickness-to-chord ra-tio t/b, and should be used for those cases because of the significantsavings in computational time. Each finite element computation ofthe wetted natural frequencies takes on the order of a few hours ofcomputing time on four processors (Dual socket six core Intel CoreI7 CPU nodes), but each analytical solution takes only a few sec-onds on a single processor. These computational savings offer hugebenefits, particular for early-stage design studies. Moreover, theanalytical solution allows easy derivation of the characteristic re-sponse for varying inputs, as well as critical scaling relations forexperimental studies.

5.2. Effects of added mass on natural modes and frequencies

The fluid inertial effects are highly dependent on the modeshapes, which are illustrated in Fig. 8 for the composite plate with

= 2.5 and t/b = 3.46% at select fiber angles h. The results showthat the dry and wet mode shapes are very similar, except for smalldifferences at the higher modes. For 0� < h < 90�, the mode shapesexhibit combined bend and twist deformation starting from thefirst mode because of the anisotropic material properties, andthe twisting deformation increases with h due to reduction of theeffective bending stiffness.

Fig. 5. Comparison of the predicted dry natural frequencies of composite platesobtained using the current finite element model with analytical predictions byNarita and Leissa [19], as well as with experimental measurements and numericalpredictions by Crawley [6].

Fig. 6. Comparison of current finite element model (FEM) results with numericalresults of Marcus [14] and experimental results of Lindholm et al. [11] for asubmerged, cantilevered, steel plate ( = 2, t/b = 1.31%).

Fig. 7. Comparison of analytical (curves) and numerical (symbols) results for thefirst two modes of (a) composite plates and (b) composite beams with varying fiberangle h when in air (dry) and when fully submerged in water (wet).

260 M.R. Kramer et al. / Composite Structures 95 (2013) 254–263

The non-dimensional wet and dry natural frequencies corre-sponding to the mode shapes seen in Fig. 8 are plotted in Fig. 9along with the wet-to-dry frequency ratios as a function of fiberorientation angle. It can be seen that in all cases, there is a 50–70% reduction in the natural frequencies due to added mass effects.The results also show that the wet-to-dry frequency ratios varywith fiber angle and mode number. At an angle of h � 20�, a cross-over behavior is observed for the second, third, and fourth modes,which is due to the change in added mass caused by the switchfrom bending dominated to combined bending and twisting modeshapes.

To better understand the dependence of the added mass (andhence the in-water frequencies) on the mode shapes, simplifiedformulas can be derived using two-dimensional (2-D) potentialflow theory for the added mass and isotropic beam theory for theresonance frequency. Assuming that the cantilevered beam onlyundergoes spanwise bending and twisting deformation with no

Fig. 8. Dry and wet mode shapes for the first five modes of the composite plate with= 2.6, t/b = 3.46%, and varying fiber angle h. Contours are colored by the

normalized z-deflection.

M.R. Kramer et al. / Composite Structures 95 (2013) 254–263 261

change in the cross sectional geometry, Eqs. (5) and (6) can be usedto estimate the sectional added mass and moment of inertia, whichyields the following simplified equations for the wet-to-dryfrequency ratios for pure bending and pure twisting modes:

Pure bending:

xwet

xdry¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffims

ms þma

r¼ 1þ p

4qf

qs

bt

� ��12

ð43Þ

Pure twisting:

xwet

xdry¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIx;s

Ix;s þ Ix;a

s¼ 1þ 3p

32qf

qs

bt

b2

b2 þ t2

" #�12

ð44Þ

The results of Eqs. (43) and (44) for the plate geometry are plot-ted as dashed lines in Fig. 9c. Note that the line with the largerXwet/Xdry value corresponds to the pure twisting mode while thelower value corresponds to the pure bending mode. As shown inthe figure, the wet-to-dry frequency ratios obtained using theseequations are slightly lower than the numerical values. This is be-cause Eqs. (5) and (6) assume simple harmonic motion and neglectshear deformation, rotary inertia, warping and damping effects,which leads to overestimates of the added mass/inertia. It shouldbe noted that Eqs. (43) and (44) are not valid for compositeplates/beams except for h = 0� and h = 90� because of coupledbend-twist deformations, and are not valid for higher modes wheresignificant warping occurs. Nevertheless, as shown in Fig. 9c, thesesimple equations do provide reasonable estimates of the lower andupper values, respectively, of the wet-to-dry frequency ratios forthe range of h and modes because the mode shapes are mostlydominated by combined bending and twisting patterns.

Eqs. (43) and (44) show that the wet-to-dry frequency ratio isinversely proportional to the fluid-to-solid density ratio (qf/qs).Hence, as the density of the fluid medium increases, the wetted

natural frequency will be reduced. The equations also explainwhy, for structures operating in air, inertial effects are often as-sumed to be negligible because qf/qs ? 0. For composite plates,added mass effects are much more significant than for steel plates,since the density of the composite is typically 4–6 times lower thansteel. Therefore, the wet-to-dry frequency ratios will also be muchlower for composite plates than for steel plates.

Further examination of both the mode shapes and frequency ra-tios helps to explain the cross-over behavior that is observed ath � 20�. In order to do so, it is useful to explore the limiting casesof h = 0� and h = 90�. At h = 0�, Modes 1 and 3 are seen to exhibitpure bending behavior while Modes 2 and 4 exhibit pure twistingbehavior. At h = 90�, Modes 1, 2 and 4 are pure bending modes be-cause of the much reduced effective bending stiffness, and Mode 3is a pure twisting mode. The values of the wet-to-dry frequency ra-tios for each of these modes for these two fiber angles are relativelyclose to the values given by Eqs. (43) and (44) for the associateddeformation pattern. As h increases, the effective EI, GJ, and Kchange, which leads to transition of the mode shapes, which inturn changes the added mass. For Modes 2, 4, and 5, the wet-to-dry frequency ratios decrease with increasing h because of the in-crease in added mass caused by greater bending deformationsresulting from the reduction in EI. For Mode 1, the frequency ratioincreases slightly with h because of small decreases in added masscause by the increasing twisting deformation near the tip of theplate. For Mode 3, which is a bending mode at h = 0�, the wet-to-dry frequency ratios increase with h because of the reduction inadded mass caused by the switch to a twisting dominated mode.Although the 3-D mode shapes are more complicated for compos-ite plates, the results show that Eqs. (43) and (44) can be use assimple effective bounds for the wet-to-dry frequency ratios forall modes and h.

6. Conclusions

This paper has presented results for the free vibration analysisof a composite cantilevered plate for both in-air and in-water casesvia combined analytical and numerical analysis. The effects ofmaterial anisotropy and fluid-to-solid density ratio were examinedin order to quantify the added mass effects for composite platesand beams vibrating in water.

The analytical model is based on composite beam theory for thestructural response, and strip theory with potential flow assump-tions for the added mass. Hence, the analytical model is able to ac-count for the effects of bend-twist coupling induced by materialanisotropy, as well as added inertial resistance provided by thesurrounding fluid. The numerical model is based on a finite elementformulation for both the solid and the fluid with tie-based displace-ment constraints at the fluid–solid interface. Convergence and vali-dation studies are shown for the numerical model. The finiteelement simulations compared well with published experimental,analytical, and numerical results for dry composite plates, and forwet steel plates. The finite element results also compared very wellwith analytical solutions of the in-water free vibration of compositebeams with varying fiber angles. The comparisons between the fi-nite element and analytical solutions are not as good for the compos-ite plate with low aspect ratio, and low thickness to chord ratio;nevertheless, the analytical method is able to capture the generaltrend of variation of the in-water resonance frequencies with fiberangle for the composite plate.

For all the cases investigated, the wetted frequencies wereshown to be significantly lower than the dry frequencies becauseof added mass effects, and this effect is more severe for lightweightcomposite plates than for heavier, metallic plates. Two simpleformulas, based on two-dimensional potential flow theory, were

0 15 30 45 60 75 900

5

10

15

20

25

30

35

40

45

0 15 30 45 60 75 900

5

10

15

20

25

30

35

40

45

0 15 30 45 60 75 900.000.050.100.150.200.250.300.350.400.450.50

Fig. 9. Comparison of numerical results for non-dimensional dry and wet frequencies for the first five modes of the composite plate with = 2.6, t/b = 3.46% for varying fiberangle h. Here, the symbols represent the discrete simulations, and the curves represent a fitted spline to better distinguish the curves and show the trends. In Fig. 9c, thehorizontal dashed lines represent the simplified estimations obtained using Eqs. (43) and (44).

262 M.R. Kramer et al. / Composite Structures 95 (2013) 254–263

derived for the wet-to-dry frequency ratios, and were found to beuseful as simple, effective bounds for the wet-to-dry frequency ra-tio for all modes and equivalent fiber angles for the compositeplate. The results show that the wet-to-dry frequency ratio is in-versely proportional to the fluid-to-solid density ratio, and henceis responsible for the 50–70% reduction in natural frequency whenthe composite plates are submerged in water as opposed to in air.The results also show that the added mass, and hence in-water nat-ural frequencies, are highly dependent on the mode shapes, whichin turn are highly affected by material anisotropy.

The objective of this paper has been to highlight the importanceof considering added mass effects for composite plates and beams,where the fluid-to-solid density ratio is large, and to quantify theseeffects for varying levels of material anisotropy by varying theequivalent fiber angle. While the current study has focused onthe simplified problem of a rectangular plate/beam, the overall ef-fects are applicable to other submerged cantilevered structures,such as hydrofoils, appendages, propeller blades, and fins.

The current study has focused on comparisons of the free vibra-tion response of cantilevered composite plates and beams in airand fully submerged in water. However, in order to better quantifyand characterize added mass effects for generalized composite mar-ine structures, varying levels of submergence must be considered.Motley et al. [15] presents numerical simulations for the free vibra-tion response of rectangular, cantilevered composite plates for var-ious plate aspect ratios and fiber angles with special focus on theeffects of (1) varying levels of submergence for surface-piercing(partially-submerged) plates, (2) varying submergence depth forfully-submerged plates near the free surface, and (3) tip gap effectsfor fully-submerged plates near a wall at the free end.

Acknowledgments

The authors are grateful for the financial support provided bythe Office of Naval Research (ONR) and Ms. Kelly Cooper (programmanager) through Grant Nos. N00014-10-1-0170 and N00014-11-1-0833. Matthew Kramer is supported by the Department of De-fense (DoD) through the National Defense Science & EngineeringGraduate Fellowship (NDSEG) Program.

Appendix A. Stiffness parameters as a function of fiber angle

For a laminate of N layers, the bending stiffness terms are de-fined as Dij � 1

3

PNk¼1ðQ ijÞkðz3

kþ1 � z3kÞ, where (zk,zk�1) defines the

thickness of the kth ply in the z-direction, and �Q ij are the trans-formed reduced in-plane stiffness coefficients for a single lamina:

Q 11 ¼ Q 11 cos4 hþ 2ðQ 12 þ 2Q66Þ sin2 h cos2 hþ Q 22 sin4 h

Q 22 ¼ Q 11 sin4 hþ 2ðQ 12 þ 2Q66Þ sin2 h cos2 hþ Q 22 cos4 h

Q 66 ¼ ðQ 11 þ Q22 � 2Q 12 � 2Q 66Þ sin2 h cos2 hþ Q66ðsin4 hþ cos4 hÞQ 12 ¼ ðQ 11 þ Q22 � 4Q 66Þ sin2 h cos2 hþ Q 12ðsin4 hþ cos4 hÞQ 16 ¼ ðQ 11 � Q12 � 2Q 66Þ sin h cos3 h

þ ðQ12 � Q 22 þ 2Q 66Þ sin3 h cos h

Q 26 ¼ ðQ 11 � Q12 � 2Q 66Þ sin3 h cos h

þ ðQ12 � Q 22 þ 2Q 66Þ sin h cos3 h

where h is the ply orientation angle measured positive counter-clockwise from the x coordinate to the principal material fiber coor-

M.R. Kramer et al. / Composite Structures 95 (2013) 254–263 263

dinate, and Qij are the reduced in-plane stiffness coefficients of indi-vidual laminae:

Q11 ¼E1

1� m12m21; Q 12 ¼

m12E2

1� m12m21

Q22 ¼E2

1� m12m21; Q 66 ¼ G12

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