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Fig. 3 Magnitude response for a water-loaded unsupportedbeam at k o hÄ0.05, MÄ0.01, 0.3 and 0.8Figure 3 is simply a slice through the wavenumber responsegiven in Fig. 2 within the wavenumber range from 100 to100 and at three different convected loading speeds, M0.01,0.3, and 0.8, respectively. For M0.01, the wavenumber responsepeaks are approximately symmetric at the wavenumber origin.Both of the peaks are identified as the propagating flexural waves.With the increased convected loading speed, the spectral responseof the unsupported beam changes in a predictable manner as describedpreviously. Clearly seen in this figure, there are four peaksin the wavenumber response for M0.3. Two of the peaks arereadily identified as the traveling flexural waves, one that is locatedapproximately at 30 is the positive-going flexural wave,the other is the negative-going flexural wave at 3. There arealso two flexural nearfields that emerge and become propagatingdue to the convected loading. One is located approximately at 20, the other almost merges with the negative-going flexuralwave at 4. For the case of M0.8, one of the propagatingflexural nearfield merges with one of the traveling flexural waves.The other flexural wave and flexural nearfield are out of the wavenumberrange, and so are not shown in this figure.Numerical solution of the infinite set of equations is required tobe truncated in Eq. 26. Unfortunately, no formula can give thenumber of terms required. The number of terms required dependson the support spacing, support stiffness, frequency of excitationand the speed of convected loading. In order to know how manyequations are of great importance in particular cases, the wavenumberdecomposition was performed. This was done by calculatingthe magnitude of each wavenumber component, ū n (),which was obtained by using Eq. 25. Shown in Fig. 4 are magnitudesof several components of the wavenumber response for awater-loaded beam with the spring support stiffness ratio S0.01, support spacing l/h100, Mach number M0.3 and excitedat the frequency k o h0.05. In this figure, it clearly demonstratesseveral facts. First, the largest peak was identified as thecomponent of wavenumber response corresponding to n0,which means no phase delay relative to the excitation. Second, asn increases or decreases, the magnitudes of ū n () get smaller.From the general trend indicated by these results, it is of considerableinterest to show that the magnitude of the largest peak ofū 50 () at 65 is much smaller than that of ū o () at 0.Hence, the wavenumber response series converges rapidly.Again, Fig. 5 illustrates magnitudes of several components ofthe wavenumber response for the same periodically supportedbeam described in Fig. 4 but subjected to a convected point loadingat speed M0.8. Comparing Fig. 5 with Fig. 4, contributionsFig. 5 Magnitude response for a water-loaded, periodicallyspring-supported beam, lÕhÄ100, SÄ0.01 at k o hÄ0.05, MÄ0.8Fig. 4 Magnitude response for a water-loaded, periodicallyspring-supported beam, lÕhÄ100, SÄ0.01 at k o hÄ0.05, MÄ0.3Fig. 6 Radiated sound power from a water-loaded, periodicallyspring-supported beam, lÕhÄ100, SÄ0.01 at k o hÄ0.05,MÄ0.01, 0.3 and 0.8Journal of Vibration and Acoustics JULY 2000, Vol. 122 Õ 277
from wavenumber components n2 to the total wavenumberresponse get progressively smaller with the increased convectedloading speed. Hence, only a few terms are required for Eq. 26to reach the convergent criteria. Note that no wavenumber components,except for n0, will exist for an unsupported beam. Thisimplies that the system behaves more like an unsupported infinitebeam while increasing the convected loading speed. Figure 5clearly demonstrates that the influences of spring supports decreasewhen the convected loading speed increases.Figure 6 is the sound power radiated from the specific modelwith spring support spacing l/h100, stiffness ratio S0.01 andat three different convected loading speeds, M0.01, 0.3, and0.8, respectively. Sound power radiated from an unsupportedbeam for M0.8 is also plotted together with other sound powercurves for the purpose of comparison. In the case of M0.01, it isinteresting to find that the radiated sound power shows pronouncedpeaks at a certain wavenumber ratios. Nonetheless, thesepeaks all occur at low wavenumber ratios. For the cases of M0.3 and 0.8, it is also clear that there are fluctuations in radiatedsound power at low wavenumber ratios. From the general trend ofthese curves, it shows that the influences of spring supports on theradiated sound power decrease while the wavenumber ratio andthe convected loading speed increase. Comparisons of the resultswith those obtained from an unsupported beam show that thepeaks in the radiated sound power greatly exceed those from theunsupported beam. From Eq. 11c, we observe that the reactionforces from elastic supports introduce numerous wavenumbercomponents in the wavenumber displacement Ū(,t). Althoughthe convected speed of loading is assumed to be subsonic, it ispossible that some of the wavelengths of the transformed displacementŪ(,t) are greater than the acoustic wavelength. Thisimplies that these wave components are supersonic, and are responsiblefor sound radiation. Of interest are the wavenumberratios at which the sound radiation curves have maxima. However,the locations of these peaks that are considered to be relatedto the nature of the wave propagation and attenuation in a periodicallysupported structure remain elusive.5 ConclusionAn alternative solution for the sound radiation from a fluidloading,periodic structure has been developed. The solution isobtained by Fourier wavenumber transforms of the whole structureinstead of reducing the analysis to a substructure. Hence, theperiodic boundary conditions and the phase relation between twosubstructures will not be used in this approach. It is worth notingthat the sound power radiated from a fluid-loaded periodic structureis calculated most conveniently if the surface motion of thisstructure can be expressed in a form in wavenumber domain. It isshown that the frequency-dependent propagating wavenumbercomponents of the responses are dependent on the velocity of theconvected loading. This phenomenon is attributed to the Dopplereffect. It is also clear that the influences of spring supports on theradiated sound power decrease while the wavenumber ratio andthe convected loading speed increase. The radiated sound power isseen to exhibit peaks at certain wavenumber ratios. Unfortunately,no formula so far could precisely help determine the locations ofthese peaks.AcknowledgmentsThis work was partially supported by the National ScienceCouncil of Taiwan, Republic of China under Contract No. NSC87-2212-E-194-019.Appendix AA Note on the Pressure Wavenumber Spectrum: Applyingspatial Fourier trans<strong>format</strong>ion to Eqs. 5 and 6, Eqs. 5 and 6becomen 2 2y 2 1 2 22 P˜ ,y,t0.C ot(A1) 2 ũ,t P˜ ,y,t ot 2 . (A2)yyoThe sound pressure equation isP˜ ,y,tn P n ,ye j2n/lVt . (A3)Substituting A3 into Eq. A1, the latter becomes 2 2y 2 1 2 22 pC ot n ,ye j2n/lVt 0.Equation A4 can be rewritten as 2nl 2 2y 2 2 V(A4)C o2p n ,y0. (A5)However, the radiation condition specifies that the sound pressureproduced by the beam motion must propagate away from thebeam and/or decay with the distance from the beam. It is easilyfound that the sound pressure of Eq. A5 isP n ,yS n e jK ny y ,(A6)wherej K ny ¦ 2 2nlfor 2 2nl 2nlfor 2 Mk o2 2nl Mk o2 Mk o2 2 Mk o2,k o and M are the acoustic wave number and the Mach number,respectively. Substituting Eq. A6 into Eq. A2, wefindS n ():j o 2nl2 VS n uK n . (A7)nyHence, the sound pressure P˜ (,y,t) is expressed asAppendix BP˜ ,y,tn S n e jK ny y e j2n/lVt .(A8)A Note on the Radiated Acoustic Power: The time averagedsound intensity is given by Morse and Ingard 19 asTĪ 1 PVdt,T0(B1)or Ī 1 Re Pu˙ *, (B2)2where Ī is the time-averaged sound intensity, P is the sound pressureon the beam surface and u˙ * is the beam surface velocity. Inorder to find the total acoustic power , the surface acoustic intensitydistribution may be integrated over the infinite length ofthe beam as278 Õ Vol. 122, JULY 2000 Transactions of the <strong>ASME</strong>
Page 2: P˜ 1,tk sn unl,te jnl . (9)Introdu
Page 5: tensity over the entire beam, the r

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