[american institute of aeronautics and astronautics 38th structures, structural dynamics, and...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1100-

VIBRATION ANALYSIS OF MULTIPLE PLATESINTERACTING WITH AIR

by

Naoyuki WATANABE* , Yasuyo TANZAWA^ and Hidehiko NARAHASHltDepartment of Aerospace Engineering, Tokyo Metropolitan Institute of Technology

6-6, Asahigaoka, Hino-shi, Tokyo 191, JAPANPHONE : Int. 81-425-83-5111 ext. 3509, FAX : Int. 81-425-83-5119

E-mail: [email protected]

AbstractThe layered structures vibrate differently

from a single structure because of the air containedbetween the structures. If the structures are light andexist adjacently to each other, it is possible that thenatural frequency becomes high. In this paper anumerical method for the vibration problems formultiple plates interacting with the air is formulatedin order to estimate both the added mass effect and thestiffness effect of the air. The behavior of the air isexpressed only in terms of pressure, and the motionof the structure is described by its displacements.Both areas of the coupled systems, structure and airregion, are discretized and analyzed by the finiteelement method. By using two dimensinal modelthe present FEM is applied to the vibration problemsthat a flat single plate or multiple plates aresurrounded with the air and rigid walls. Numericalresults demonstrate that the present FEM can estimatesufficiently both the added mass effect and stiffnesseffect of the air.

1 IntroductionRecently, solar arrays or deployable antennas

of artificial satellites become much larger and muchlighter than before. Therefore, they must be foldedup and put on satellites. The layered structuresvibrate differently from a single structure because ofthe air contained between the structure. Such aphenomenon, what is called as a fluid-structureinteraction problem, exists wherever relative motionof the two domains occurs. The effect of interactionis very important in the fields of aerospace, nuclear,mechanical, naval, and civil engineering.

*. Associate Professor, Member of AIAAPHONE: 0425-83-5111 ext.3509FAX: 0425-83-5119f. Graduate Studentcopyright © 1997 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

There are many studies about fluid-structureinteraction problems, but most of them discuss aboutthe interaction between a heavy fluid and structuresuch as water-structure interaction or oil-structureinteraction 1.There are various researches to analyze the coupledproblems. The Lagrangian approach-^ is that thefluid is treated as an elastic solid with a negligibleshear modulus, and its behavior is expressed in termsof the displacements. The another is the Eulerianapproach^ that the fluid is characterized by thepressure or velocity potential, and it is currently usedin many fluid-structure interaction problems. As forthe numerical analysis, there are some methods; thepurely analytical one, the method that the structure isanalyzed by the finite element method and theboundary element method is used in the fluid domainor the method that the finite element method is usedin the both5'6.

When the structure is partially or completelyfilled with the fluid, or surrounded with the fluid, thenatural frequencies become low^A However, if thestructures are light and placed adjacently to each other,it is possible that the natural frequencies becomehigh. According to the results of the vibration testfor SAR antenna that was mounted in JERS-1artificial satellite, the natural frequency for the lowestout-of-plane vibration was numerically predicted asabout 52.1 [Hz] while the value obtainedexperimentally is 61 [Hz]9. There are few studiesthat the structure contains a light fluid such as the air.Moreover, the analysis for multiple structuresinteracting with air has never been proposed yetwithin authors' knowledge.

The influence of the air to structure isclassified roughly into three effects : the added masseffect, stiffness effect and damping effect. Accordingto the relative location between the structure and air,the influence of the air becomes much different. Theclassical categories of such problems are shown inFig. 1. When the structure is surrounded with the airas shown in Fig. l(a), the natural frequency becomes

395American Institute of Aeronautics and Astronautics


low due to the added mass effect of the air. In another situation, which the structure contains theisolated air as shown in Fig. l(b), the naturalfrequency becomes high due to the stiffness effect ofthe air. In an another situation shown in Fig. l(c),both the added mass effects and stiffness effect areconsidered to appear and it differs in each situationwhich one is dominant.

1

1Air 1

Type(a) Air

Type(b)

1Air

rType(c)

Fig. 1 Category of air-structure interaction problem.

In this paper, a numerical method for thevibration problems of multiple plates interacting withthe air is formulated in order to estimate both theadded mass effect and stiffness effect of the air. Thebehavior of the air is expressed only in terms ofpressure, and the motion of the structure is describedby its displacements. Both areas of the coupledsystem, structures and air region , are discretized andanalyzed by the finite element method. By usingtwo-dimensional model the present FEM is applied tothe vibration problem that a single plate or multipleplates are surrounded with the air and rigid walls.Numerical results demonstrate the efficiency andaccuracy of the present FEM.

2 Fluid-Structure Interaction Problem2.1 Basic Equations for the Air———- Eulerian Pressure Formulation

To simplify the coupled problem, the dampingof the structure, the viscosity and variation oftemperature of the air, and the acoustic radiation ofthe systems are assumed to be neglected. In addition,the air is assumed to be irrotational. Figure 2illustrates the general geometry of the systemconsidered here. The structures are surrounded withthe air or walls, and the infinite air region is alsoincluded in this problem. The rectangular Cartesiancoordinate system, x, y and z are employed here.

Structure

Structure (V/all)

Fig. 2 Coupled system of structures and air

The continuity equation of the air is written asfollows:

If the convective acceleration and viscous effects areneglected, the Navier-Stokes equation is written asfollows:

pfv = - Vp (2)where pF,\, and p are the density, velocity andpressure, respectively. The relation among pF,\ andp is given as follows:

dtwhere c is the sound velocity

c2 = ̂ - (4)

(5)

and V is the Hamiltonian

V = 'A '— —dx dy dz

where i, j and k are unit vectors in direction x, yand z, respectively.

Assume that the perturbation of density andsound velocity with respect to time are negligible, weobtain from the above equations••» i

(6)--p = Q in DF

where

On the air-structure interface, the normalcomponent of the air velocity is equal to the timederivative of the normal displacement of thestructures. Thus

Vp n = -pii n onS° (8)

where n and u is the outer normal unit vector andnormal acceleration vector of the structure.



2.2 Variational Principlefor Air-Structure Interaction Problems

From the previous equations and boundaryconditions, the weak form of governing equation forthe air region becomes as follows:

P2 dt rf [_!_•• 2 \dDj'i ' H Sp[c*p~v p\

JUp \ i

+ Sp (vp + p'u\ • ndS 1 = 0 (ONJS° \ F I J

where the variation of /? satisfies the followingcondition:

Sp=0 at t = ti,t2 (10)Using the Gauss 's divergent theorem equation

(9) leads to

(U)

Above equation includes the effect of compressibilityof the air.

From Hamilton's principle '0\ the appropriatefunctional for the structure leads to

J>[f (J'i JDS V. .

--p uu--£E dD

where ps and E are a density of structure, andelasticity matrix. U, £, fs,and t are a displacement,strain, body force and external force vector,respectively. The first term represents the kineticenergy of the structure and the second term representsthe strain energy of the structure. The last threeterms express the external work done on the systemsby surface forces, body forces and dynamic pressure,respectively. On the geometrical boundary 5M; thedisplacement must satisfy the following equations:

u = u onS, (13)where u is the prescribed displacement.

Finally, adding together the air and structurefunctionals Eq.(l l) and (12), directly, we canestablish a variational principles for air-structureinteraction problems:

f* rf f ^ i 'sk *[ \--^^p-vp1 I i~\ iJLJp

2cdD

uds -

\dD

(14)

3 Finite Element Formulationfor the Air-Structure Interaction Problem

Both the structure and air region arediscretized by the finite element method'1. Thedisplacements and strain of the structure are expressedin terms of the nodal displacement:

u = Nuq (15)

e = Bq (16)where q, Nu and B are nodal displacement vector,displacement shape function for structure and strainmatrix, respectively. Similarly, the pressureperturbation in the air element is given as follows:

P = NpP (17)where Np and P are pressure shape function for fluidand nodal pressure perturbation vector, respectively.Here the pressure perturbation is measured from theatmospheric pressure.

For the structure region, let the mass matrixM, the stiffness matrix K and external force vector gbe defined as follows:

M = , ,dD

DBT.E.B do

idS

(18)

(19)

(20)

For the air region, let the fluid inertial mass matrixG, the volumetric fluid matrix H, and the couplingmatrix between air and structure L be defined asfollows:

Tp' P (21)

H=f 'JDF

,,-n dS

(22)

(23)

Using above notations, the variational equation (14)becomes



>r^{- ipJf, I 2

T • •

- (24)

and this leads to ordinary FEM equations which canbe written as

M 0

G

K Ll

0 H(25)

where the matrices are not symmetric and thus theproblem must be solved with full matrices. In thisstudy, natural frequencies and modes of this systemare investigated, so the term g related to the externalforce is deleted.

Calculations were carried out by usingFujitu VPP-500 vectorized super computer in TokyoMetropolitan Institute of Technology. Theseeigenvalue problems were solved by the QR methodwith the subroutine in SSLII Library of VPP-500.

4 Analysis of a Single Plate Interactingwith Air

4.1 Analyzed ProblemIn this study, the flexible plate is assumed to

be placed vertically to the plane of the sheet andcontinues infinitely along the out-of plane directionin order to simplify the analysis, so the plate can bemodeled as a beam in 2-dimensional planecorresponding to x-z plane as shown in Fig, 4.Therefore, 2-dimensional finite element model isavailable.

Rigid wall

1 nfiniteboundary

Hr-S.S.

* /Plate

s.s. lv

—— A I) *1 nfiniteboundary

Rigid wall

Fig. 3 A single plate surrounded withthe air and two walls.

The coupled problem that is discussed in thisstudy is illustrated in Fig. 3, where the beam isplaced at the center of the air region and simplysupported at the both ends. The wall is a very rigidplate, so this problem is considered to be the firstapproximation of the multiple plates interacting withthe air. The beam length and the air width are / and

da. A finite element model is illustrated in Fig. 4.Although a full model is used in the analysis, only anhalf region is shown here. The beam and air regionare discretized by 2-noded bending element and 8-noded rectangular quadratic element, respectively.The beam is usually discretized into twelve elements.As shown in Fig. 4, long and slender elements areplaced on the outskirts of the mesh. The side lengthof the elements is at least four times as long as thatof the beam elements, and the side length of innerelements is twice of beam elements.

d,Airwidth

Seam

Fixed boudaryLong and slender element

^

Fixed boudary

Virtualboundary

2 d=djl

Fig.4 FEM model for the problemwith a single plate.

With this model, both sides of the model are"infinite boundary ", but they are here approximatedas virtual boundary which is sufficiently far from thebeam. On the virtual boundary, it is assumed thatthe pressure perturbation can be neglected; namely, p=0, where the pressure perturbation is measured fromthe atmospheric pressure.

In order to evaluate the difference of pressureperturbation between upper and lower surfaces of thebeam, the nodes of air region and nodes of the beamare separated at the air-structure interface, as shown inFig. 5.

Nodes are separated betweenbeam element and air element

2 node beam element

No'de 8-node rectangular elementfor the air

Fig.5 FEM model around the beam



4.2 Two First Beam ModeThe characteristics of the analyzed system are

as follows:flexural rigidity of the beam, D - E Ixx :

D =2.2X101 0 [Kg-mm3/s2]density of the beam:

ps = 4.7X ID'5 [Kg/mm]length of the beam:

/= 0.5 [m]density of the air:

pF= 1.22X10-9 [Kg/mm3]and da ,/0 and/are the air width between fixed walls,the natural frequency of the beam without the airinfluence and that with the air influence, respectively.At first, the natural frequency of the first mode isinvestigated as the air width da is changed.

Figure 6 shows frequency changes of two themodes that includes the first bending beam modes.The ordinate is the normalized frequency / and theabscissa is the normalized air width da,where f-f/foand d~a=da/l. As for this beam the natural frequency/0 of the first bending mode is 135.94 Hz without theair effect. There is only one mode when da is largerthan 2.2, and the frequency decreases as da becomeslarge. Let call it as Mode-1C. On the other handthere are two kinds of modes that the beam vibrates asthe first bending mode when d~a is smaller than 2.2.In both the modes the deformation mode of the beamis almost the same. Let call them as Mode-1A andMode-IB. The frequency of the Mode-1A is nearlyconstant and almost 1.0, that means the frequencywithout the air effect, while that of the Mode-IBfairly increases as d~a becomes small. The differencebetween two modes is related to the airflow during thevibration.

2.5

1.5 -

1.0 -

0.5 -

0.0

Mode-1AMode-18Mode-IC

0.0 1.0 2.0 3.0 4.0 5.0 6.0da= djl

Fig.6 Normalized frequency of modes related to thefirst bending beam modes : / = 0.5[m].

0.09 -0.080.07 -

0.09

Beam

Fig.7 Pressure distribution of Mode-lA:/=0.5[m], Ja=l.O, f =1.011

Beam

Fig.8 Pressure distribution of Mode-IB: /=0.5[m], ^=1.0,7=1.572

Beam x

Fig. 9 Pressure distribution of Mode-1C:/=0.5[m], <fa= 1.0, 7=0.808



Wall

iam

[_^ (a) Mode-1 Ax

(b) Mode-IB (c)Mode-1C

Fig.10 Airflow in three modes, Mode-lA, -IB, and -1C.

The distribution of the pressure fluctuationis shown in Fig. 7 for Mode-1A and in Fig. 8 forMode-IB, where da=l.Q. In calculations thevibration mode is normalized so that the norm ofsigen mode is unit. The vertical axis is the pressurefluctuation, and x-z coordinates system is defined asshown in Fig. 4. For its anti-symmetric distributiononly a half region of z>0 is shown here. The beamis in the enter of the front edge and the wall iscorrespond to the line of p=0 on the rear face asshown in Fig. 7. The normalized frequency / is1.011 for Mode-1A and 1.572 for Mode-IB. Fromtaking inclination of the pressure, the velocity vectorof the airflow is obtained. The result isschematically illustrated in Fig. 10 (a) and (b) for twomodes. In Mode-1A two-dimensional airflow isseen, while the airflow is fairly similar to one-dimensional flow in Mode-IB. In Mode-1A the airmoves from one side to the other side of the beamnear the end of the beam, but there isn't suchphenomenon in Mode-IB. Although the pressureworking directly on the beam doesn't much differs inthese two modes, there is a large difference near therigid wall. In that region the maximum value is0.065 in Mode-1 A, and it is 0.95 in Mode-IB.Therefore, the frequency of Mode-IB prominentlyincreases as da becomes small.

In Fig. 9, the pressure distribution of Mode-1C is shown, when da=3.0. It is fairly differentfrom Mode-1A or Mode-1B. The pressure becomeshigh as the point goes from the beam to the wall.The maximum value is 0.04 at the enter of the beamand 0.075 on the wall. The airflow in this mode isschematically illustrated in Fig. 10(c). It is seen thatthe air moves from the upside to the bottom in the allregion. Since the added mass effect appearsprominently and the pressure change near the beam issmall, the frequency decreases largely in this case.

4.3 Influence of Beam LengthIn Fig. 11 the relation between the

normalized air width da and the normalizedfundamental frequency / is plotted for three beamlengths, 0.2[m], 0.5[m] and 0.8[m). Where / foreach length is normalized by each/o corresponding toits length. The result for /=0.5[m] is omitted heresince it is mentioned in the previous section.

At first the result for /=0.8[m] is discusses.The tendency is almost the same as that for /=0.5[m].There are two modes, Mode-1A and IB, which havethe first beam mode, in the region where the value ofda is smaller than 4.0, while there is only one mode,Mode-1C, in the left region from <Ja=4.0. Thenormalized frequency of Mode-1A is a very littlehigher than 1.0 and nearly constant in except for verysmall da region, while those of Mode-IB and 1Cbecome small as d~a becomes large.

^ 3.0

2.5

2.0

1.5

1 .0

0.5

0.00. 1.0 2.0 3.0 4.0 5.0 6.0

Fig. 1 1 Frequency change for different beam length.



On the other hand, the tendency for /=0.2[m]is fairly difficult from those for /=0.5[m] and/=0.8[m]. There is only one mode corresponding toMode-1C and its normalized frequency nerve exceed1.0. The pressure distribution is like that shown inFig. 9, and the added mass effect is prominent in thissituation

From this figure it is well seen that twoeffects of the air, the stiffness effect and the addedmass effect, prominently appear, and that theinfluence of the air strongly depends on the beamlength. Let us consider the reason why the air effectsappear in such way. To simplify the analysis andreduce the computational efforts, the analysis of thecoupled system is approximated as 2-dimensionalmodel. Therefore, the airflow along y-direction, out-of-plane direction, is restricted and the air cannotmove smoothly in comparison with the actual 3-dimensional situation. Consequently, the pressure inthe air is likely to change more than the actualsituation and the inertia forces of the air isn't easilygenerated. Then the added mass effect estimated lessthan the actual phenomenon, and the stiffness effect isestimated large. The region that the air can movefrom one surface of the beam to the other is restrictedto the region near the ends of beam, thus when thebeam is long the region relative to the beam lengthbecomes smaller in comparison with the case wherethe beam is short. Consequently, the behavior of thelong beam is likely to be governed by the stiffnesseffect of the air, while the behavior of the short beamis likely to be governed by the added mass effect.

4.4 Difference Between ModeFigure 12 shows the difference of the air

influence between on different bending beam modes.The beam with the same parameters as used inprevious sections is analyzed for the length / =0.8[m],and the beam is simply supported at the both ends.There are also three modes, Mode-A, -B and -C, thathave the second bending beam mode with the differentpressuredistributions. The value / of the secondmode is calculated by using/0 of the second bendingmode. It is observed that / for the first bendingmode is higher than that for the second bending modein all the region. When the vibration is the secondbending mode, the beam deforms as shown in Fig.13. In the region (A) and (D) the pressure becomeshigh, while it becomes low in the region (B) and (C).The change of the pressure is averaged in both theupper and lower air regions of the beam, then theaveraged change of the pressures is small when thevibration mode is higher order. Therefore, thestiffness effect is the largest in the first bendingmode.

,o 3.5"-V.

V 3.0^

2 C

2.0

1.5

1.0

0.5

0.00

-*

•

.— ---jie._. ^»

0 1

»%

Second

XI•-.I

0 2

^r****"•

ii

• •~ - •

0 3

1-

First

-• •.

0 4

- Mode-B 1- Mode-c|

\• »•;•» —— ••-

••-.

0 5 0 6.

<,= •«/

Fig. 12 Difference of air influence betweenon different bending beam mode.

Wall

B

WallFig. 13 Second bending mode of the beam.

5 Analysis of Multiple Plates Interactingwith Air

In this section the coupled problem that havetwo flexible plates between two rigid walls isinvestigated. This is the same as the problem shownin Fig. 3 except for the number of plates. The finiteelement mesh of this problem is shown in Fig. 14.The total air width is da, and the distance betweenplates and distance between plate and rigid wall are allassumed to be da /3.

Long and slender element

Air Widthd.

s*s/ss'Bearri

f

-

:

Fixed boundary /x/yyxyx/xx/>>vx/x

^™ Virtual

boundary

/p=o

/VT/XX/X/XX/XXXX/X/XX/X' x "

_ dad, =

Fig. 14 FEM model for the problem with two plates



There are two vibration modes in thissystem as shown in Fig. 15 when each plate vibratesin the first bending mode. One is the anti-symmetricmode that two plates vibrate in the same phase asshown in Fig. 15(a), and this is the lowest mode ofthe coupled system. Another is the symmetric modethat two plates vibrate in the reverse phase as shownin Fig. 15(b). This order is determined by thepressure change of the air region between two plates.

Wall Wal l

Beam Beam

Wal l Wal l(a) Anti-symmetric mode (b) Symmetric mode

Fig. 15 Anti-symmetric and symmetric modes for theproblem with two plates.

The changes of the Anti-symmetric andsymmetric natural frequencies are shown in Fig. 16,where the beam length is 0.8[m]. The abscissa isthe normalized air width and the ordinate is thenormalized frequency that is normalized by the firstnatural frequency of a single beam without the aireffect. It is because the Anti-symmetric andsymmetric natural frequency/0 of the coupled systemare the same if the air doesn't exist. There are alsothree modes, Mode-A, -B and -C, with the differentpressure distribusions in the both modes. As foreach plate and each mode the frequency decreases dueto the added mass effect of the air as Sa becomeslarge, and it increases due to the stiffness effect as da

becomes small. The frequency of the first modebecomes high due to the stiffness effect of the air asda becomes small. On the other hand that of thesecond mode doesn't increase more than a certainvalue even if da becomes smaller than approximately2.0. Generally speaking, which effect the air gives tothe system is considered to be almost the same as thesystem with only one plate.

6 ConclusionsA numerical method for the vibration

problems of multiple plates interacting with the air isformulated in order to estimate both the added masseffect and stiffness effect of the air. The behavior ofthe air is expressed only in terms of pressure, and themotion of the structures is described by itsdisplacements. Both areas of the coupled system,structure and air region , are discretized and analyzedby the finite element method. By using two-dimensional model the present FEM is applied to the

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

040.0 1.0 2.0 3.0 4.0 5.0 6.0

d =a a

Fig. 16 Normalized frequency for the problemwith two plates

vibration problems that one or two flat plates issurrounded with the air and rigid walls because theyall considered to be an approximation of multipleplate interacting with the air. Numerical resultsdemonstrate that the present FEM can estimatesufficiently both the added mass effect and stiffnesseffect of the air.

The following conclusions are obtained:1) For a short beam the added mass effect is

prominent, and the frequency becomes fairly lowwith increasing the air width. Contrarily, for along beam the stiffness effect is prominent, andthe frequency becomes fairly high with reducingthe air width.

2) The influence of the air becomes relatively smallas the order of the mode becomes high.

3) As for the system with a single plate, there are twodifferent modes that have the same first bendingdeformation of the beam and the completelydifferent airflow when the air width is narrow. Butthere is only one mode when the air width isbroad.

4) As for the system with two plates, it is confirmedthat there are two different modes which each platevibrates in the first bending mode when the airwidth is considerably broad. On the other hand,there are four modes due to the difference ofairflow when the air width is narrow. And theirfrequencies become high or low by the air effect.

At the last we think that the present FEMcan be developed into 3-dimensional model by somerefinements and that the vibration characteristics ofmultiple plates interacting with the air will besufficiently evaluated soon.



AcknowledgmentsThe authors wish to express their thanks to

Mr. Toshio Inoue, Kamakura Works of MitsubishiElectric Corporation, and Mrs. Keiko Yoshida,Central Research Laboratory of Mitsubishi ElectricCorporation, for their kind advices at the beginning ofthis study and for offering the valuable data of thevibration test for SAR antenna of JERS-1 artificialsatellite. The authors also wish to express theirthanks to Mr. Yasumoto Togashi, Toshiba TESCOCo., Ltd. for his efforts in calculating some examplesby the computer.

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9. Inoue, T. et al "The Results of the Vibration Testfor the exploitative model of SAR antenna that isloaded into ERS-1 (in Japanese)," Proc. ofJSASS32nd Space Sciences and TechnologyConference, "Technology Meeting, pp.616-617Oct. 1988.

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