Rib resonances present in the scattering response of a ribbedcylindrical shell

Martin H. Marcus and Angie SarkissianNaval Research Laboratory, Washington, DC 20375-5350

~Received 2 May 1997; revised 2 October 1997; accepted 19 December 1997!

The presence of ribs in an evacuated, finite cylindrical shell, placed in a fluid medium, produceshighlights in its scattering response at certain resonant frequencies. These resonances are identifiedfrom the end-incident bistatic response of a ribbed cylindrical shell and are shown to be producedby the flexural vibrations of individual ribs satisfying the boundary conditions of an annular plate.@S0001-4966~98!00804-2#

PACS numbers: 43.40.Dx@CBB#

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INTRODUCTION

The scattering response of an evacuated ribbed cylincal shell in an unbounded fluid medium contains, amomany features, highlights produced by the presence of ividual ribs. Rib resonances1,2 are different from Bloch waveresonances,1,3 which may also be produced by ribs if the ribhave regular spacing. Rib resonances do not require regspacing and may be produced by the vibrations of a sinrib present in the structure.

Using a finite elements/infinite elements computationthe scattering response of a ribbed cylindrical shell, rib renances are identified and shown to be produced by flexvibrations of individual ribs satisfying boundary conditionof an annular plate.

I. END-INCIDENT BISTATIC RESPONSE OF A RIBBEDCYLINDER

Figure 1 shows the end-incident bistatic response oribbed finite cylindrical shell with hemispherical end capThe cylinder, shown in Fig. 2, contains 85 ribs, all of whiare of equal size. The computations were performed usprogram Sara-2d4 which uses finite elements to model thcylindrical structure and finite and infinite elements to mothe unbounded fluid medium external to the structure.additional details on the finite elements and infinite elemeapproach, the reader is referred to Ref. 5 or Ref. 6.

Figure 3 shows enlarged version of the ribs. The cylder used for the computations has radiusa, a total lengthL513.7a and shell thicknesst50.0074a. The ribs havelength l 50.074a, thicknessh50.0065a, and rib spacing ofd50.14a. The elastic parameters used are that of nickel wmodulus of elasticityE52.131011 Pa, Poisson’s ration50.30, densityr58800 kg/m3, and a loss factorh50.005.The external fluid is water with sound speed ofc51500 m/s and density ofr151000 kg/m3.

The bistatic response is shown for a frequency range2.5<ka<10, wherek5v/c. Angles range from 0° representing end-incident forward scattering to 180° represenend-incident backscattering. The target strength plot is nmalized to reach a maximum value of 0 dB. The highligobserved nearka58 and higher is due to Bloch waves prduced by the periodicity of the rib spacing. A horizont

1864 J. Acoust. Soc. Am. 103 (4), April 1998

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highlight is observed nearka53. For this end-incident condition, the resonant behavior at this frequency seems inpendent of the scattered angle. This resonance may be itified most easily in the forward direction where it appearsa dip in the target scattering response.

Figure 4 shows plots of the end-incident forward sctered target strength as a function ofka for three different riblengths. Again, each curve is normalized to reach a mamum value of 0 dB. In each case a resonance may be idtified as a dip followed by a peak in the scattering respoas a function of frequency. The solid line in Fig. 4 represea line plot of Fig. 1 at an angle of 0° where the resonancondition nearka53 appears as a dip in the forward scatering response. It can be seen from the three curves in Fthat the forward target strength values drop by several dBthe rib resonances. We also observe from Fig. 1 thatresonances occur at all receiver angles.

II. RIB RESONANCES

To determine the frequencies where rib resonancescur we use a thin plate model which is applicable as longthe plate thickness does not exceedls/20, wherels is theshear wavelength.7 The results shown in this article are abelow ka510, whereh/ls50.0051, well within the limitwhere thin plate theory is applicable. The lateral displament field w on the surface of a rib, as shown in Fig.satisfies the plate equation8

D¹4w1rh]2w

]t2 50, ~1!

where

D5Eh3

12~12n2!.

Choosing the origin of the coordinate system to be atcenter of a rib and oriented such that the rib lies in thex-yplane, the solution to the above equation may be written

w~r !5 (n50

`

@anJn~kfr !1bnYn~kfr !1cnKn~kfr !

1dnI n~kfr !#einu, ~2!

1864

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-

ano

he

bth

n

ica

whereJn , Yn , Kn , and I n are the cylindrical Bessel func

tions, r 5Ax21y2 andkf is the flexural wave number,

kf5S rhv2

D D 1/4

.

The e2 ivt time dependence has been suppressed. Thesymmetric case is considered here where the field hasudependence in which case only then50 component of thesum in Eq.~2! contributes to the displacement,

w~r !5a0J0~kfr !1b0Y0~kfr !1c0K0~kfr !1d0I 0~kfr !.~3!

The boundary conditions require the vanishing of tbending momentM and the shearQ at the free end,

M ~b!52DS ]2w

]r 2 1n

r

]w

]r D Ur 5b

50, ~4!

whereb5a2 l ,

Q~b!52D]

]r¹2wU

r 5b

50, ~5!

while at the attached end, we approximate the plate toattached to fixed supports where the rotation is related tobending moment and shear by9

]w

]r Ur 5a

516.67M

Eh2 1~12n!Q

Eh Ur 5a

, ~6!

and the displacement vanishes,

w~a!50. ~7!

Satisfying the requirements of the boundary conditioresults in the matrix equations

Ai1a01Ai2b01Ai3c01Ai4d050, for i 51,2,3,4, ~8!

where

FIG. 1. End-incident bistatic scattering response of the ribbed cylindrshell.

FIG. 2. The ribbed cylinder used for the numerical simulations.

1865 J. Acoust. Soc. Am., Vol. 103, No. 4, April 1998

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A115J0~kfb!1n21

kfbJ1~kfb!,

A125Y0~kfb!1n21

kfbY1~kfb!,

A1352K0~kfb!1n21

kfbK1~kfb!,

A1452I 0~kfb!2n21

kfbI 1~kfb!,

A215J1~kfb!, A225Y1~kfb!,

A2352K1~kfb!, A245I 1~kfb!

A3152F12kf

2h2

12~11n!2

16.67h

12~11n!r GJ1~kfa!

216.67kfh

12~12n2!J0~kfa!,

A3252F12kf

2h2

12~11n!2

16.67h

12~11n!r GY1~kfa!

216.67kfh

12~12n2!Y0~kfa!,

FIG. 3. Enlarged version of the ribs shown attached to the cylinder.

FIG. 4. Forward normalized target strength as a function ofka for threedifferent rib lengths.

l

1865M. H. Marcus and A. Sarkissian: Scattering from ribbed shells

ject to ASA license or copyright; see http://asadl.org/terms

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A3352F11kf

2h2

12~11n!2

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12~11n!r GK1~kfa!

116.67kfh

12~12n2!K0~kfa!,

A345F11kf

2h2

12~11n!2

16.67h

12~11n!r G I 1~kfa!

116.67kfh

12~12n2!I 0~kfa!,

A415J0~kfa!, A425Y0~kfa!,

A435K0~kfa!, A445I 0~kfa!.

The resonant frequencies are obtained by requiringdeterminant of matrixA to vanish. The dashed line in Fig.shows the resulting lowest order resonant frequenciesfunction of rib length using dimensionless quantitieska andl /a. The solid line displays resonances numerically coputed using finite elements analysis. These resonancesobtained by examining a plot of the forward target strenas a function of frequency for each rib length, from whi

FIG. 5. Resonant frequencies computed numerically using finite elemanalysis, shown by the solid line, compared to values determined anacally, shown by the dashed line.

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the resonant frequency was determined by eye. Thecurves in Fig. 5 agree very closely.

Although rib resonances occur for all incident anglesthe acoustic field, the end-incident case is examined hwhere the incident field is axisymmetric. Since the rboundary conditions at the attached end are chosen toattached to fixed supports, no angular dependence appearib resonances. This is apparent in Fig. 1 where the rib renance occurs nearka53 for all receiver angles.

III. CONCLUSIONS

Using finite element computations of the scatteringsponse of a ribbed cylindrical shell, rib resonances are idtified. The resonances are produced by flexural vibrationsthe ribs which satisfy boundary conditions of an annuplate having one free end and one end welded to the stture. A specific algorithm is presented to compute the asymmetric resonances produced by the ribs. The resultthe algorithm are shown to agree very closely to rib renances determined by finite element computations.

ACKNOWLEDGMENT

This work was supported by the Office of Naval Rsearch Code 334.

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ntti-

1866M. H. Marcus and A. Sarkissian: Scattering from ribbed shells

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