Embed Size (px)
Citation preview
Ultrasonics 85 (2018) 39–48
Contents lists available at ScienceDirect
Ultrasonics
journal homepage: www.elsevier .com/locate /ul t ras
Estimation of dispersion curves of water-loaded structures by usingapproximated acoustic mass
https://doi.org/10.1016/j.ultras.2017.12.0090041-624X/� 2017 Elsevier B.V. All rights reserved.
⇑ Corresponding author at: School of Naval Architecture and Ocean Engineering,University of Ulsan, 93 Daehak-ro, Nam-gu, Ulsan 680-749, South Korea.
E-mail address: [email protected] (J. Ryue).
B.K. Jung a, J. Ryue b,⇑, C. Hong c, W.B. Jeong d, K.K. Shin a
aAgency for Defense Development, 645-016, South Koreab School of Naval Architecture and Ocean Engineering, University of Ulsan, 680-749, South Koreac School of Mechanical Engineering, Ulsan College, 680-749, South Koread School of Mechanical Engineering, Pusan National University, 609-735, South Korea
a r t i c l e i n f o a b s t r a c t
Article history:Received 29 January 2017Received in revised form 15 October 2017Accepted 29 December 2017Available online 29 December 2017
Keywords:Waveguide finite elementWaveguide boundary elementAcoustic massWave acoustic massRayleigh integral
For coupled structures surrounded by heavy fluids, it is difficult to obtain dispersion curves from aneigenvalue analysis, because the external fluid term in the coupled equation includes transcendentalfunctions for frequency and wavenumber. Thus, in this study, the acoustic mass of the external fluidwas approximated as a function of frequency or wavenumber only. The coupled equation can then beused to calculate eigenvalues, and can estimate dispersion curves from an eigenvalue analysis. Becauseof this assumption, those dispersion curves will contain errors. Accordingly, those errors were evaluatedin this study through a comparison with a dispersion curve derived from forced responses. The acousticmass was also evaluated for a water-loaded plate; this can be formulated theoretically. As a result, theacoustic mass is less sensitive to frequency changes than wavenumber changes, and using the fluid termdefined at a low frequency has advantages when estimating the dispersion curve. Finally, the generalityof the proposed method was identified through the application for a submerged cylindrical shell.
� 2017 Elsevier B.V. All rights reserved.
1. Introduction
To analyze structural vibration and noise by using a wavemethod, the types and propagation characteristics of waves inthe structure must be identified; this can be done by analyzingthe dispersion curves of the structure. The dispersion curves canbe obtained by solving an eigenvalue problem formulated forwaves in structures, in general. When the structures are sub-merged in external water, the eigenvalue problem becomes moredifficult to solve. Thus, several approaches have been studied toobtain the dispersion curves of water-loaded structures, theoreti-cally or numerically.
Theoretical studies on waves of a structure coupled with exter-nal fluid have been performed [1,2], but this theoretical approachis limited in applying for practical structures shaped arbitrarily.For these complex shaped structures, numerical methods arerequired in general. A typical numerical approach is to couple finiteelements with boundary elements for structures and fluids, respec-tively [3–5]. When using this approach, however, it is difficult toanalyze waves sustained in the structure, because the approach
calculates vibration and noise based on the superposition of eigen-modes. In other words, it requires a huge number of meshes anddegrees of freedoms to deal with waveguide structures having con-stant cross-sections and long lengths. Alternatively, waveguidefinite element (WFE) and waveguide boundary element (WBE)have been researched for their potential to effectively estimatewaves in waveguide structures.
The WFE method can effectively analyze elastic wave propaga-tion and waveguide structure vibration, with the assumption thatwaves of cross-sectional modes propagate harmonically along thelength direction of the waveguide. TheWBE method is used to ana-lyze unbounded problems such as external fluid fields. Nilsson andFinnveden [6] used the WFE approach to calculate the vibration ofa cylindrical shell filled with an FE-modeled fluid, and Nilsson et al.[7] used the WFE and the WBE method independently to predictthe radiated noise of a tram rail. Ryue [8] coupled the WFE andWBE to investigate the vibration of a submerged pipe. In addition,wave analysis has been performed for a fluid coupled waveguidestructure by coupling the WFE and the WBE method [9–11].
In this wave approach, it is essential to identify the types ofwaves sustained in the structures. The wave characteristics arerepresented by dispersion curves which can be found from aneigenvalue analysis for the governing equation of waveguide struc-tures. The eigenvalue analysis, however, cannot be performed at
40 B.K. Jung et al. / Ultrasonics 85 (2018) 39–48
this point, because the governing equation for structures coupledwith external heavy fluids becomes implicit. Therefore, dispersioncurves cannot be obtained directly from the WFE/BE equations ofmotion without repeatedly forming new WBE matrices. The inte-gral method studied by Beyn [12] could be the best solution forsolving this nonlinear eigenvalue problem. However, to obtaineigenvalues in this way, lengthy calculation time is required. Alter-natively, the dispersion characteristics of the coupled structure andfluid can be indirectly estimated using the resonance peaks offorced responses of the structure. It should be noted that theappearance of resonance peaks depends on the locations of excita-tion and response points and directions. This means that disper-sion curves of some waves cannot be found from their forcedresponses.
Accordingly, a new method is proposed in this paper that caneffectively estimate dispersion relationships for water-loadedwaveguide structures. In chapter 2, the WFE/BE method is intro-duced. We identify the characteristics of an acoustic mass of theexternal fluid term in the coupled equation of a water-loaded plateby fixing the frequency or wavenumber. Additionally, we establishthe method for readily estimating the dispersion curve by usingeigenvalue analysis. In chapter 3, Rayleigh integral [13,14] andwave acoustic mass are used to evaluate errors caused by theassumption that acoustic mass is a function of the frequency. Inchapter 4, the errors are estimated using the wave acoustic massmentioned above. Finally, in chapter 5, an application to a sub-merged cylindrical shell is introduced to verify the proposedmethod for calculating the dispersion curve of submergedstructures.
2. Waveguide finite element/boundary element modeling
In this chapter, the waveguide finite element and boundary ele-ment (WFE/BE) method is described briefly. Details of the deriva-tion of the governing equations are well presented in Ref. [7,15]so only a brief introduction of the method is given here. In thispaper, a software program, called WANDS [16], is used for theWFE/BE analysis.
2.1. Waveguide finite element (WFE)
The waveguide finite element method (WFEM) models only the2D cross section of a waveguide structure, with the assumptionthat the vibration mode of the cross section propagates harmoni-cally along the longitudinal direction. This method is also referredto as a semi-analytical FE (SAFE), because it uses a theoretical solu-tion along the longitudinal direction. This harmonic wave propaga-tion in the longitudinal direction is expressed as e�jjx, where x isthe axis of the longitudinal direction and j is the wavenumber ofthe waves propagating along the x direction.
The WFE equation of a waveguide structures described withplates strip element which has 4 degree (3 translations, 1 rotation)of freedom per node is given by
ðK4ð�jjÞ4 þ K2ð�jjÞ2 þ K1ð�jjÞ þ K0 �x2MÞU ¼ 0 ð1Þwhere K4, K2, K1, and K0 are matrices related to the stiffness of thestructure, M is a mass matrix, x is the frequency, j is thewavenumber, and U is the displacement vector of the cross-section. When the number of nodes of the system is n, a size ofthese matrices is 4n by 4n and a size of the vector is 4n by 1. Eq.(1) has two variables for x and j. If one of these two variables isknown, an eigenvalue analysis can be performed. If j is given, Eq.(1) becomes a general eigenvalue problem for finding x. Con-versely, if x is given, it becomes a polynomial eigenvalue problemfor finding j. Dispersion curves can be obtained via this eigenvalue
problem by finding x using real wavenumbers, which representpropagating waves.
2.2. Waveguide boundary element (WBE)
When a waveguide structure comes into contact with externalinfinite fluid, the external fluid can be modeled with boundary ele-ments. WBEs are also used for the cross-sectional modeling of fluidcoupled waveguide structures. If there is no external acoustic exci-tation, the WBE equation of motion for the fluid is expressed as
HW� Gð@W=@nÞ ¼ 0 ð2Þwhere W is the velocity potential vector, n is a normal vector ofWBE, and H and G are Green’s function matrices, which are depen-dent on the wavenumber j and the frequencyx. Since the WBE has1 degree of freedom per node, a size of the Green’s function matri-ces is n by n and a size of the vectors is n by 1 when the number ofnodes of the system is n. The acoustic pressure p and the normaldirection particle velocity vn are calculated from the velocity poten-tial and its derivatives as
vn ¼ �@W=@n; p ¼ jxq0W ð3Þwhere q0 is the density of the fluid. Substituting Eq. (3) into Eq. (2),the governing equation for WBEs is given by
Hðp=jxq0Þ þ Gvn ¼ 0 ð4Þ
2.3. Coupling of the WFE and WBE
With respect to the external forces applied to the structure andthe loads from the coupled fluid, the governing equation of thewaveguide structure is written as:
ðK�x2MÞU� C1p� Fe ¼ 0 ð5Þ
where K ¼ K4ð�jjÞ4 þ K2ð�jjÞ2 þ K1ð�jjÞ þ K0, C1 is a couplingmatrix between the structure and fluid. C1p is the fluid load andFe is the external force. In the coupled interface, continuity condi-tions should be satisfied. Because the normal velocity at the bound-ary must remain the same, the relationship between the coupledWFE and WBE is written as:
vn ¼ jxC2U ð6Þwhere C2 is a matrix used to designate coupled dofs and transformthe Cartesian coordinate system into the normal direction.
From Eqs (4)–(6), a fully coupled equation is expressed inmatrix form as:
K�x2M �C1
jxGC2 H=jxq0
" #Up
( )¼ Fe
0
( )ð7Þ
Eq. (7) can be rewritten with respect to variable U as
ðK�x2ðMþ q0C1H�1GC2ÞÞU ¼ Fe ð8Þ
In the left-hand side of Eq. (8), the second term within the innerbracket is obtained from the fluid coupling. This additional termactually behaves like mass and damping, because of the inertia ofthe fluid and the energy loss from the structure to the externalfluid. We are interested in identifying dispersion curves, whichare the frequency-wavenumber relationships of the propagatingwaves rather than the forced responses of a structure. Becausethe added mass effect is more concerning in terms of the disper-sion relationship in this study, the additional term is hereafterreferred to in this study ‘‘added mass” or ‘‘acoustic mass”, to makethe purpose clear. Note that the additional term has a damping
Fig. 1. Waveguide model of plate.
B.K. Jung et al. / Ultrasonics 85 (2018) 39–48 41
effect, but it does not create considerable variation, because thedamping influences the forced response spectrum. Thus, it is disre-garded in this study.
The acoustic mass matrix for the external fluid is defined as
MA ¼ q0C1H�1GC2 ð9Þ
Therefore, the coupled equation, which includes the massmatrix of the external fluid, becomes
ðK�x2ðMþMAÞÞU ¼ Fe ð10ÞTo obtain a dispersion diagram from Eq. (10), free vibration
must be solved for by setting Fe ¼ 0. Because MA is a function ofwavenumber and frequency, the eigenvalue analysis cannot beimplemented. An alternative way of solving Eq. (10) is to find jand x by gradually varying their values until the determinant ofthe matrix in Eq. (10) becomes zero. However, this method canrequire very long computation times to form dispersion curves.Another method for obtaining the dispersion curves of a coupledstructure using Eq. (10) is to use forced responses in the realwavenumber domain, which means propagating wave, at each fre-quency by applying an external force Fe. The response of the struc-ture excited by a unit impulse in the time domain, which equals to1 in the wavenumber domain, represents the eigenvalues as reso-nance points on the spectrum. The dispersion curves of the coupledstructure can be constructed by collecting the frequencies andwavenumbers of resonance peaks from the forced response spec-trum. However, this method is also time-consuming, becauseresponses must be found for all of the frequencies and wavenum-bers in the ranges of interest. Furthermore, some waves may notappear, depending on the locations of the excitation and responsepoints, which are close to the nodal points of the structure’s cross-sectional modes. Additionally, some waves with relatively smallamplitudes may not be detected when seeking resonance peaks.Even when multi-point excitation is applied and the excitationpoints are changed, a considerable effort is required to find wholewaves sustained in the structure, because of the reasons men-tioned above. Therefore, to obtain dispersion relationships moreeasily and effectively for structures coupled with heavy fluids(such as water) another solution method is attempted here.
In this study, by using the acoustic mass matrix MA at a specificfrequency x0, Eq. (10) can be rewritten as
ðK�x2ðMþMAjx¼x0ÞÞU ¼ 0 ð11Þ
where MAjx¼x0now becomes a function of wavenumber only. Thus,
if the wavenumber is given, the approximate natural frequency xcan be found with eigenvalue analysis for Eq. (11). Then, the disper-sion curves of the coupled structure can be predicted more easily.Similarly, by using added mass matrix MA at a specific wavenumberj0, Eq. (10) can be rewritten as
ðK�x2ðMþMAjj¼j0ÞÞU ¼ 0 ð12Þ
where MAjj¼j0now becomes a function of frequency only. In Eq.
(12), if a frequency is given, the approximate natural wavenumberj can be found through a polynomial eigenvalue analysis, and thedispersion curves for the coupled structure can also be predicted.
However, because the acoustic mass matrix is specified at a cer-tain frequency x0 or wavenumber j0, there may be a considerablediscrepancy in the predicted dispersion curves. Thus, we estimatedispersion curves using the eigenvalue analysis for Eq. (11) andpolynomial eigenvalue analysis for Eq. (12), and select a moreappropriate method for estimating the dispersion curve of the cou-pled structure by comparison those dispersion curves. This isdescribed in the next section.
2.4. Example of water-contacting plate
In Eqs. (11) and (12), a coupled equation with acoustic massmatrix MAjx¼x0
and MAjj¼j0were proposed, allowing eigenvalue
analysis or polynomial eigenvalue analysis to become available.In this section, dispersion curves for a plate contacting water onone side are predicted using the proposed equations. Then, the fea-tures of the obtained dispersion curves are presented and com-pared with the features of the curves obtained from the forcedvibration responses.
The specifications of the plate models used are presented inFig. 1 and Table 1. Fig. 1 illustrates the excitation and responsepoints, which are used for forced vibration analysis.
Prior to the estimation of dispersion curves of the coupledstructure, dispersion curves for a non-fluid-loaded structure arecalculated with the eigenvalue analysis for the function of fre-quency and the polynomial eigenvalue analysis for the functionof wavenumber. Fig. 2 shows the comparison of the dispersioncurves between the two methods. Despite the different methods,the dispersion curves of the non-fluid-loaded structure are identi-cal. Then, we will discuss the dispersion curves of the coupledstructure.
First, for verification, the dispersion curves of the coupled struc-ture obtained using Eqs. (11) and (12) are compared with the dis-persion curve obtained from the force vibration responses. Fig. 3compares the dispersion curves obtained by collecting the peaksfrom the forced vibration and eigenvalue analysis using the cou-pled equation in Eq. (11) with three different x0 frequencies: 1Hz, 500 Hz, and 1500 Hz. Fig. 4 compares the dispersion curvesobtained from the forced vibration and polynomial eigenvalueanalysis using the coupled equation in Eq. (12) with three differentj0 wavenumber: 1 rad/m, 10 rad/m, and 20 rad/m.
Through the comparison with the dispersion curves derivedfrom the forced vibration analysis, we definitely determine thatthe dispersion curves obtained from the eigenvalue analysis usingthe acoustic mass at specific frequencies are more appropriate thanthose obtained from the polynomial eigenvalue analysis using theacoustic mass at specific wavenumbers. In other words, withrespect to estimating appropriate dispersion curves for a coupledstructure, eigenvalue analysis of the coupled equation using acous-tic masses at specific frequencies is more appropriate than polyno-mial eigenvalue analysis of the coupled equation using acousticmasses at specific wavenumbers. Hereafter, we will use the disper-sion curves derived from the eigenvalue analysis using acousticmass at specific frequencies.
In the dispersion curves in Fig. 3, the bending waves in the plateare primarily plotted. By comparing the dispersion curves obtained
Table 1Specification of the water-contacting plate.
Specification of the plate Specification of the external fluid
Width 1 [m] Sound speed 1500 [m/s]Thickness 0.003 [m] Density 1000 [kg/m3]Elastic modulus 210 [GPa]Poisson ratio 0.3Density 7850 [kg/m3]
0 50 100 150 200 250 300
Frequency (Hz)
0
5
10
15
20
Wav
enum
ber (
rad/
m)
Eigenvalue analysisPolynomial eigenvalue analysis
Fig. 2. Comparison of dispersion curves of the non-fluid-loaded plate betweeneigenvalue analysis and polynomial eigenvalue analysis.
42 B.K. Jung et al. / Ultrasonics 85 (2018) 39–48
from the forced responses and the acoustic masses at specific fre-quencies, we identify that several curves from the former approachare not shown, because the selected excitation and response pointscoincide with the nodal points of these waves, as previouslymentioned.
The figure also shows that the dispersion diagrams have morediscrepancies as x0 increases. This error is most evident near thewavenumber corresponding to the acoustic wavenumberj ¼ x=c for the input frequency. Here, c is the acoustic wave speed.When the acoustic mass matrix of 1 Hz was used, the resulting dis-persion curves look more like those obtained via forced vibrationanalysis (i.e., the comparison target).
In the next section, the proposed method using the acousticmass matrixMAjx¼x0
is investigated for a plate coupled with water,which means that MA can be evaluated theoretically. From thisinvestigation, we will attempt to quantify the errors for the pro-posed method.
3. Characteristics of acoustic mass matrix for a plate coupledwith water
3.1. Theoretical acoustic mass matrix
For an infinite plate, the Rayleigh integral expresses the theoret-ical relationship between the acoustic pressure at the receivingpoint and particle velocity normal to the plate on the wetted sur-face, as given by
pðYÞ ¼ZC2jxq0UvnðXÞdC ð13Þ
where q0 is the density, r is the relative distance (jX � Y j) between asource location (X) on the plate and a response location (Y), k is theacoustic wavenumber given by k ¼ x=c, as a function of frequencyx and propagation velocity c. vn is the normal directional velocity ofthe fluid particle on the plate surface and p is the surface pressure of
the plate coupled with water, U is the solution of the Green’sfunction.
Because we deal with the acoustic radiation in 2D space (yz-plane) by an analytical method and the acoustic radiation in thelongitudinal direction (x-direction) by using a theoretical solution,the Green’s function excluding the longitudinal acoustic radiationis defined as follows [15],
@
@y2þ @
@z2
� �Uþ k2nU ¼ dðr � roÞ ð14Þ
where dðr � roÞ is delta function, r0 is the location of source, kn is theplanar wavenumber which is defined as follows.
kn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qð15Þ
For k2 > j2, kn becomes a real wavenumber, and this wave radi-
ates well into the fluid. Conversely, for k2 < j2, kn becomes animaginary wavenumber corresponding to a near-field wave. Thenthis wave does not create sound in the external medium. This fea-ture is expressed by
U ¼� j
4 Hð2Þ0 ðr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞ; k2 > j2
1; k2 ¼ j2
12pK0ðr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞ; k2 < j2
8>><>>: ð16Þ
where Hð2Þ0 ðr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞ is the 0-th order of a Hankel function of the
second kind and K0ðrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞ is the 0-th order of a modified Bes-
sel function. Now, the solution of the Green’s function is a function
of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
p, then U ¼ Uðr
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞ. The solution of the Green’s
function in Eq. (16) for the 2D plate has imaginary solutions where
k2 > j2, and the acoustic pressure in Eq. (13) becomes real, imply-ing that the plate adequately radiates sound to the outside. In con-trast, the solution of the Green’s function becomes real where
k2 < j2 and the sound pressure in Eq. (13) is purely imaginary. Thismeans that the sound generated by the plate decays quite rapidlyalong r and that sound is not radiated far away from the plate. If
k2 ¼ j2 where the acoustic wavelength becomes identical to thestructural wavenumber, the acoustic pressure becomes infinte.
Applying a discretization to Eq. (13),
pi ¼XNl¼1
2jxq0Uðri;lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞðvnÞlDCl
¼ 2jxq0½Uðri;1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞDC1Uðri;2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞ
� DC2 � � �Uðri;Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞDCN�vn
ð17Þ
where i and l are node indicators for the discretized plate, N is thetotal number of nodes, and DCl is the effective length of the elementat l-node. Eq. (17) is expressed in matrix form as
p ¼ ½2jxq0Uðrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞDC�vn ¼ �2jxq0Gvn ð18Þ
where G ¼ ½�Uðrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
pÞDC�. Using Eq. (18), the acoustic mass
matrix of the external fluid acting on the plate is given as
MA ¼ C1½�2q0Uðrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � j2
qÞDC�C2 ð19Þ
This acoustic mass matrix is complex where k > j2; the real andimaginary portions indicate the coupled physical acoustic massand damping from the fluid, respectively.
Fig. 3. Comparison of dispersion curves of the plate between forced vibration analysis and eigenvalue analysis using the acoustic mass of fixed frequencies; (a) x ¼ 2p 1, (b)x ¼ 2p 500, (c) x ¼ 2p 1500.
B.K. Jung et al. / Ultrasonics 85 (2018) 39–48 43
3.2. Wave acoustic mass
In Section 3.1, the theoretical acoustic matrix for a plate inwater was derived in Eq. (19) by using the Rayleigh integral. Theacoustic mass matrix which, depends on the frequency andwavenumber, reflects the effect of added mass on the waveguidestructure. Because the coupled equation in Eq. (10) includes thismatrix, a general eigenvalue analysis is not available. Therefore,the dispersion diagram cannot be obtained from Eq. (11).
By assuming that the acoustic mass matrix is not significantlysensitive to frequency changes, Eq. (10) is modified to Eq. (11),which makes the eigenvalue analysis possible. To validate the pro-posed method, it is necessary to evaluate the accuracy of the cou-pled motion equation, which includes a fluid matrix of a specificfrequency. To access the acoustic mass matrix, a wave mass is cho-sen (as an analogy to modal mass in modal analysis) for each wavein the structure. The wave mass is defined by
mr ¼ V�rMAVr ð20Þ
where Vr is a cross-sectional mode shape vector of the r-th wave;sin rp
L y, and the superscript ⁄ denotes the complex conjugate trans-pose. L is the length of the cross-section of the plate and y is the
direction of that length. The wave acoustic mass mr is the acousticmass that contributes to the r-th wave.
Using the wave mass, the difference betweenMA in Eq. (18) andMAjx¼x0
in Eq. (11) can be evaluated to check for the errors thattake place in the proposed approach. The wave mass of the r-thwave is calculated by
mr;f0 ¼ V�r ðMAjx¼x0
ÞVr ð21Þ
4. Error analysis for a fluid coupled plate
In this section, the wave mass of mr;f0 is calculated using threedifferent frequencies of x0 for several bending waves of the plateshown in Fig. 3 as curved dotted lines. The three selected frequen-cies are 1 Hz, 500 Hz and 1500 Hz.
A graph of wave acoustic masses against frequency is illustratedin Fig. 4 for the 1st, 5th 10th and 15th bending waves of the wettedplate. The cut-on frequencies of those waves are 0 Hz, 91.74 Hz,486.7 Hz, and 1250 Hz, respectively. It can be seen from Fig. 5 (a)that the wave mass m1;1 is in very good agreement with the theo-retical wave mass of m1, whereas m1;500 and m1;1500 are fairly dif-ferent from m1, especially for low wavenumbers. As the order
Fig. 4. Comparison of dispersion curves of the plate between forced vibration analysis and polynomial eigenvalue analysis using the acoustic mass of fixed wavenumbers; (a)j ¼ 1, (b) j ¼ 10, (c) j ¼ 20.
44 B.K. Jung et al. / Ultrasonics 85 (2018) 39–48
number of the waves increases, however, m1;500 and m1;1500 consec-utively approach m1 as shown in Fig. 5(b)–(d).
Using the wave acoustic mass for each wave shown in Fig. 5, theerrors are calculated using Eq. (22):
E ¼ avgrðjmr �mr;f0 j=mrÞ � 100 ð22Þ
where avgr denotes the average with respect to the number of waveof r. In this calculation, waves of 1 6 r 6 15 are regarded.
The average errors for wave masses contained in all wavesusing MAjx¼x0
are shown in Fig. 6, which reveals that MAjx¼x0with
a lower frequency of x0 produces a relatively small error. If a dis-crepancy of approximately 1% is accepted, MAjx¼x0
at 1 Hz wouldbe applicable to frequencies up to approximately 1 kHz. The fewesterrors occur when the input frequency of the acoustic mass isequal to the frequency of the external force. In contrast, the errorsis much larger at the cut-on frequencies of the waves. In addition,it shows that the errors over the 1 kHz become almost equalregardless of the input frequency of the acoustic mass. Therefore,it is better to use the MAjx¼x0
with a lower frequency of x0 whenestimating the dispersion curve of the fluid-coupled plate by usingthe eigenvalue problem.
5. Application example
We assume that the characteristics of the acoustic mass used toestimate the dispersion curves of the coupled structures areapproximately similar, regardless of the geometry of the waveg-uide structure. Thus, we apply the suggested method from theabove section to estimate the dispersion curve for a submergedcylindrical shell, which makes it difficult to theoretically handlean acoustic mass. Fig. 7 represents the application model; waveg-uide cylindrical shell, and excitation, response points for the forcedvibration analysis. Table 2 indicates the material properties of thecylindrical shell.
For the waveguide model in Fig. 7, the dispersion curves areestimated by the eigenvalue analysis of the coupled cylindricalshell using the acoustic mass at specific frequencies; 1 Hz, 500Hz, and 1500 Hz, and represented in the Fig. 8. Fig. 9 shows the dis-persion curves obtained by the polynomial eigenvalue analysis ofthe coupled cylindrical shell using acoustic mass at specificwavenumbers; 1 rad/m, 10 rad/m, 20 rad/m. Both figures are com-pared with the dispersion curves by collecting the peaks from theforced responses. The dispersion curves by using the acoustic massat specific wavenumbers are much different to that of forcedresponses regardless of the wavenumber change. The dispersion
Fig. 5. Comparison of r-th wave acoustic mass with respect to changes in frequency; (a) 1st wave acoustic mass, (b) 5th wave acoustic mass, (c) 10th wave acoustic mass, (d)15th wave acoustic mass.
Fig. 6. Comparison of the error with respect to changes in frequency. Fig. 7. Waveguide model of cylindrical shell.
B.K. Jung et al. / Ultrasonics 85 (2018) 39–48 45
Table 2Specification of the submerged cylindrical shell.
Specification of the cylindrical shell Specification of the external fluid
Diameter 1.35 [m] Sound speed 1500 [m/s]Thickness 0.008 [m] Density 1000 [kg/m3]Elastic modulus 210 [GPa]Poisson ratio 0.3Density 7850 [kg/m3]
46 B.K. Jung et al. / Ultrasonics 85 (2018) 39–48
curves using the acoustic mass at 500 Hz, 1500 Hz exhibit distor-tion near the wavenumber corresponding to the fixed frequencyof the acoustic mass (i.e. j ¼ 2p=3 and j ¼ 2p). These errors aremore evident as the fixed frequency increase like Fig. 8(c). Onthe contrary, the dispersion curve using the acoustic mass at 1Hz is the most similar to that obtained from the forced vibration.The overall tendencies for the dispersion curves of the coupledcylindrical shell are similar to those of the coupled plate in Fig. 3but the errors are represented more clearly. Accordingly, the sug-gested method, which readily estimates the dispersion curvethrough the eigenvalue analysis of the coupled equation thatincludes the acoustic mass at a low frequency, can be also appliedto determine the wave propagating characteristics of the cylindri-cal shell.
Fig. 8. Comparison of dispersion curves of the cylindrical shell between forced vibrationx ¼ 2p 1, (b) x ¼ 2p 500, (c) x ¼ 2p 1500.
6. Conclusion
A method for effectively estimating the dispersion curves offluid-coupled structures using an eigenvalue analysis of thewaveguide FE/BE model is described in this paper. In reality, thedispersion curve of the coupled structure cannot be accurately cal-culated from the eigenvalue analysis, because the fluid term of thecoupled equation is a function of frequency and wavenumber. Asan alternative, a dispersion curve of the coupled structure can beestimated by collecting the peaks from a forced vibration: how-ever, this is costly, and depending on the excitation and responsepoints, some dispersion curve data may be lost. Thus, an approxi-mated coupled equation having a fluid term that is a function ofwavenumber is suggested, by fixing the frequency.
By using the proposed method, we readily estimated the disper-sion curves of the coupled plate with the acoustic mass calculatedat specific frequencies; 1 Hz, 500 Hz, and 1500 Hz. We also calcu-lated the dispersion curves of the coupled plate having the acousticmass at specific wavenumbers, which is a function of the frequencyonly. Those were compared with the dispersion curves derivedfrom the proposed method for verifying of the assumption thatthe fluid term is insensitive to frequency changes. Through a com-parison with the dispersion curve derived by forced vibration, itwas demonstrated that the dispersion curves obtained from the
analysis and eigenvalue analysis using the acoustic mass of fixed frequencies; (a)
Fig. 9. Comparison of dispersion curves of the cylindrical shell between forced vibration analysis and eigenvalue analysis using the acoustic mass of fixed wavenumbers; (a)j ¼ 1, (b) j ¼ 10, (c) j ¼ 20.
B.K. Jung et al. / Ultrasonics 85 (2018) 39–48 47
eigenvalue analysis using the acoustic mass at a specific frequencywere much more accurate than those obtained from the polyno-mial eigenvalue analysis using an acoustic mass at a specificwavenumber. Among the dispersion curves derived by the pro-posed method, the dispersion curve obtained from the approxi-mated coupled equation using the acoustic mass at a frequencyof 1 Hz was the most accurate. The others dispersion curves exhib-ited the distortion near the acoustic wavenumber corresponding tothe frequency of the acoustic mass calculation.
To evaluate the errors caused by the acoustic mass assumption,we used Rayleigh integral and then calculated the wave acousticmass by using a theoretical acoustic mass matrix of the fluid-coupled plate. By comparing the wave acoustic mass and erroranalysis, it was verified that a lower input frequency for the acous-tic mass leads to better estimation of the dispersion curve of thecoupled plate.
Finally, the proposed method was applied to estimate the dis-persion curve of a submerged cylindrical shell, which makes it dif-ficult to handle the acoustic mass theoretically for verification. Thetendencies represented in the dispersion curves of the coupledcylindrical shell were similar to those observed for the coupledplate, but distortions in the dispersion curves were more apparent.Nevertheless, the dispersion curve of the coupled cylindrical shellwas also estimated effectively by the proposed method using an
acoustic mass of 1 Hz, and it was sufficiently accurate when com-pared with the dispersion curve obtained from the forcedresponses.
Acknowledgment
This study was supported by Agency for Defense Developmentin Korea [UD120002DD]. Also the authors are grateful to ISVR atUniversity of Southampton for allowing using theWANDS programthat they made for WFE/BE analysis.
References
[1] D.G. Crighton, Point admittance of an infinite thin elastic plate under fluidloading, J. Sound Vib. 54 (3) (1977) 389–391.
[2] D.G. Crighton, The free and forced waves on a fluid-loading elastic plate, J.Sound Vib. 63 (2) (1979) 225–235.
[3] C. Li, J. Wang, Y. Qu, Numerical and experimental investigation on vibro-acoustic response of a shaft-hull system, Eng. Anal. Bound. Elem. 71 (2016)129–139.
[4] S. Finnveden, Evaluation of modal density and group velocity by a finiteelement method, J. Sound Vib. 273 (2004) 51–75.
[5] C. Droz, Z. Zergoune, R. Boukadia, Vibro-acoustic optimization of sandwichpanels using the wave/finite element method, Compos. Struct. 156 (2016)108–114.
[6] C.M. Nilsson, S. Finnveden, Waves in thin walled fluid-filled ducts witharbitrary cross-sections, J. Sound Vib. 310 (2008) 58–76.
48 B.K. Jung et al. / Ultrasonics 85 (2018) 39–48
[7] C.M. Nilsson, C. Jones, D. Thompson, J. Ryue, A waveguide finite element andboundary element approach to calculating the sound radiated by railway andtram rails, J. Sound Vib. 321 (2009) 813–836.
[8] J. Ryue, A numerical method for analysis of the sound and vibration ofwaveguides coupled with external fluid, Acoust. Soc. Korea 29 (7) (2010) 448–457.
[9] H. Kim, J. Ryue, Wave propagation in a strip plate with longitudinal stiffeners,Trans. Korean Soc. Noise Vib. Eng. 23 (6) (2013) 512–519.
[10] B.K. Jung, C. Hong, J. Ryue, W.B. Jeong, K.K. Shin, Dispersion characteristics ofcylindrical shells submerged in the fluid, Trans. Korean Soc. Noise Vib. Eng. 25(8) (2015) 575–582.
[11] B.K. Jung, J. Ryue, C. Hong, W.B. Jeong, K.K. Shin, Dispersion diagrams of awater-loaded cylindrical shell obtained from the structural and acousticresponses of the sensor array along the shell, in: INTER-NOISE and NOISE-CONCongress and Conference Proceedings, Institute of Noise Control Engineering,vol. 249, no. 4, 2014, pp. 3176–3183.
[12] W. Beyn, An integral method for solving nonlinear eigenvalue problems, LinearAlgebra Appl. 436 (2013) 3793–4088.
[13] S.M. Kirkup, Computational solution of the acoustic field surrounding a baffledpanel by the Rayleigh integral method, Science and Research Institute,University of Salford, UK, 1994.
[14] A. Putra, N. Shyafina, D. Thompson, N. Muhammad, M. Nor, M. Jailani, Z.Nuawi, Modeling sound radiation from a baffled vibrating plate for differentboundary conditions using an elementary source technique, in: INTER-NOISEand NOISE-CON Congress and Conference Proceedings, Institute of NoiseControl Engineering, vol. 249, no. 1, 2014, pp. 5812–5819.
[15] C.M. Nilsson, C. Jones, Manual for WANDS 2.1: Wavenumber domain softwarefor sloid and fluids, Institute of Sound and Vibration Research, University ofSouthampton, UK, 2007.
[16] C.M. Nilsson, J. Ryue, C.J.C. Jones, Theory manual for WANDS 2.1 Wave numberdomain FE-BE software for structures and fluids, ISVR Technical MemorandumNo. 975, University of Southampton, 2010.
