ultrasound propagation in …...a temporal approach of sound transmission through layered rigid...
TRANSCRIPT
A Temporal Approach of Sound Transmission ThroughLayered Rigid Frame Porous Media
Z.E.A. Fellaha, S. Bergera, C.Depollier a, M. Fellah b and B. Castangèdea
aLaboratoire d’Acoustique, UMR-CNRS 6613, IAM,Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans cedex 09, France
bLaboratoire de Physique ThéoriqueInstitut de Physique, USTHB BP 32 El Alia Bab Ezzouar 16111, Algeria
A temporal approach of the propagation of pulses in rigid porous media is treated. A model of equivalent fluid is considered in whichthe interactions solid/fluid are modeled by the fractional calculus based method. The transmission and reflection scattering operatorsare derived given a good agreement with experimental results in the case of a slab and a two-layers porous medium.
MODEL
The determination of the properties of a medium fromwaves that have been reflected by or transmitted throughthe medium is a classical inverse scattering problem.Such problems are often approached by taking a physi-cal model of the scattering process generating a syntheticresponse for some assumed values of the parameters, ad-justing these parameters until reasonable agreement is ob-tained between the synthetic response and the observeddata. Most publications concerned with such acousticalinvestigations are devoted to frequency-domain methods.However, because of the transient nature of signals andto avoid the computation of a numerous of Fourier trans-forms, it is more appropriate to compare the synthetic sig-nal and the data in the time domain. There are severalother relevant reasons to deal with time domain technics:i) they allow the rapid acquisition of data over a largeband width; ii) they allow the separation of differentsevents by time gating in the time domain; iii) a time do-main model is often the must natural description of theway in which the actual experiment is performed.In the acoustics of porous materials, one distinguishestwo situations according to whether the frame is movingor not. In the first case, the dynamics of the waves dueto the coupling between the solid skeleton and the fluidis well described by the Biot theory [1]. In air-saturatedporous media the structure is generally motionless and thewaves propagate only in the fluid. This case is describedby the model of an equivalent fluid which is a particularcase of the Biot model, in which the interactions betweenthe fluid and the structure are taken into account in twofrequency dependent response factors: the dynamic tor-tuosity of the medium α
ω given by Johnson [2] and the
dynamic compressibility of the air included in the porousmaterial β
ω given by Allard [3]. In the frequency do-
main, these factors multiply the density of the fluid andits compressibility respectively and represent the devia-
tion from the behaviour of the fluid in free space as thefrequency increases. In the time domain, they act as op-erators and in the high frequency approximation their ex-pressions are given by Fellah [4]:
αt α∞
δt 2
Λ ηρ f π 1 2
t 1 2 (1)
βt
δt 2
γ 1 Λ η
Prρ f π 1 2t 1 2 (2)
In these equations, δt is the Dirac function, Pr is the
Prandtl number, η and ρ f are respectively the fluid vis-cosity and the fluid density and γ is the adiabatic con-stant. The relevant physical parameters of the model arethe tortuosity of the medium α∞ and the viscous and ther-mal characteristic lengths Λ and Λ introduced by John-son [2] and Allard [3]. In this model t 1 2 is interpretedas a semi derivative operator following the definition ofthe fractional derivative of order ν given in Samko andcoll [5],
Dν x t 1Γ ν t ∞
t u ν 1x
u du (3)
where Γx is the gamma function.
In this framework, the basic equations of our modelcan be written as
ρ f αt ∂vi
∂t ∇i p and
βt
Ka ∂p
∂t ∇ v (4)
where * denotes the time convolution operation, p is theacoustic pressure, v is the particle velocity and Ka is thebulk modulus of the air. The first equation is the Eulerequation, the second one is a constitutive equation ob-tained from the equation of mass conservation associatedwith the behaviour (or adiabatic) equation .
DIRECT PROBLEM
The direct scattering problem is that of determiningthe scattered field as well as the internal field, that ariseswhen a known incident field impinges on the porous ma-terial with known physical parameters. To compute thesolution of the direct problem one need to know theGreen’s function of the modified wave equation in theporous medium. In that case, the internal field is givenby the time convolution of the Green’s function with theincident wave and the reflected and transmitted field arededuced from the internal field and the boundary condi-tions. The generalized lossy wave equation in the timedomain is derived from the basic equations (4) by ele-mentary calculation in the following form
∂2 p∂x2 A
∂2 p∂t2 B t ∞
∂2 p ∂t 2t t dt C
∂p∂t
0 (5)
where the coefficients A, B and C are constants respec-tively given by ;
A ρ f α∞
Ka
B 2α∞
Ka
ρ f η
π 1Λ γ 1
PrΛ (6)
C 4α∞γ 1 η
KaΛΛ Pr
RESULTS
Let a homogeneous porous material occupies the re-gion 0 x L. This medium is assumed to be isotropicand having a rigid frame. A short time sound pulse islaunched in the region exterior to the material. Thisimpiges normally on the medium. The kernels of the re-flection and transmission operators are deduced from theGreen function of Equation (5) in the following form:
Rt φ α∞
φ α∞ ∑n 0 φ α∞
φ α∞ 2n G t 2n
Lc G t 2n 2 L
c (7)
Tt 4φ
α∞
α∞ φ 2∑n 0 φ α∞
φ α∞ 2n
G t Lc0
2n 1 Lc (8)
where G is the Green function [6] of the propagationequation (5). Theses expressions take into account the
0
0
3e-05
0.05
Time (s)
Am
plitu
de (
V)
Foam F1 F1-F2
Am
plitu
de (
V)
Time (s)
3e-05
0.0.2
FIGURE 1. simulated signal (dashed line) and experimentalsignal (solid line) for the foam F1 and the two-layers medium.
multiple reflections in the material.As an application of our model, some numerical simula-tions are compared to experimental results Fig.1 showsexperimental results for a plastic foam F1 and a layeredmedium compared with synthetic results. The syntheticresults are calculated by the time convolution of the inputsignal given out by a tranducer, with the transmission op-erator given in (8) ( T1 for F1 and T1 T2 for F1-F2), Froma such approach the reciprocity is easily verified The pa-rameters of foams F1 are: thikness 2cm, α∞ 1 055,Λ 234µm, Λ 702µm, σ 9000Nm 4s and φ 0 97,and those of F2 are thikness 2cm, α∞ 1 04, Λ 200µm,Λ 600µm, σ 9000Nm 4s and φ 0 97,
REFERENCES
1. M.A. Biot, J. Acoust. Soc. Am, 28, 168 (1956); 28,179 (1956).2. D.L. Johnson, J. Koplik and R. Dashen, “Theory ofdynamic permeability and tortuosity in fluid saturatedporous media”, J. Fluid Mech. 176, 379-402 (1987).3. J.F. Allard, Y.Champoux, J. Acoust. Soc. Am, 913346-3353.4. Z.E.A. Fellah and C. Depollier, J. Acoust. Soc. Am,107 (2), 2000.5. S. G. Samko, A.A. Kilbas and O.I. Marichev, Frac-tional Integrals and Derivatives: Theory and Applica-tions, Gordon and Breach Science, Amsterdam, 1993.6. Z.E.A. Fellah, C. Depollier and M. Fellah, “An Ap-proach to direct and inverse time-domain scattering ofacoustic waves from rigid porous materials by a fractionalcalculus based method “, Journal of Sound and Vibration,To appear (2001).
Parametric antenna in a granular mediumV. Tournata, V. E. Gusevb and B. Castagnèdea
aLaboratoire d’Acoustique de l’Université du Maine UMR-CNRS 6613,bLaboratoire de Physique de l’Etat Condensé UMR-CNRS 6087,
Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
A parametric antenna emits low-frequency acoustic signals due to the demodulation of amplitude-modulated high-frequency acousticwaves in a nonlinear medium. This method is applied to a granular medium (glass beads of average diameter 150 µm in air) whereextremely high acoustic non-linearity is associated with the existence of Hertz contacts among the beads. For frequency of thecarrier wave 40 kHz fh
300 kHz in our experiment, the attenuation highly depends on fh, both due to absorption and scatteringaugmentation versus fh. Strong increase in attenuation causes transformation (shortening when increasing fh) of emitting “body” ofthe parametric antenna. This, in turn, influences diffraction of the demodulated (low-frequency) waves even when the distance is fixedbetween an emitter of high-frequency waves and a receiver of the low-frequency waves. We observed experimentally remarkablechanges in the demodulated wave profile with increasing fh. Our observations indicate that it is possible to use nonlinear parametriceffects for the investigation of propagation of highly attenuated acoustic waves.
Parametric antennae have been widely studied in un-derwater acoustics [1]: a low-frequency acoustic wave isemitted due to the demodulation of high-frequency car-rier wave that is amplitude modulated. Important featuresof granular media compared to homogeneous media areusually dispersion, scattering and very high non-linearity.They ensure different behaviors compared to the classicalparametric antenna in water [2, 3, 4].
We present a theoretical way to describe influences ofthese specific features on the parametric processes. Ex-periments in agreement with the theory are performed inthe case of carrier wave ballistic propagation.
We show that the profile of the low-frequency demod-ulated wave is related to the carrier wave frequency. Thecarrier wave penetration distance is highly shortened be-cause of the attenuation increase versus fh (absorptionand scattering). This influences diffraction of the demod-ulated wave, and cumulative effect of velocity dispersionbetween carrier wave and demodulated wave. As a result,two temporal derivatives of the demodulated wave profileare observed.
THEORETICAL RESULTS
Assuming the medium to be macroscopically nonlin-ear due to the Hertz contacts among the beads, an ap-proached stress-strain relation can be derived from theHertz contact model:
σ C
ε0 ∂Ux
∂x 3 2(1)
where σ is the stress, C is a constant, ε0 is the static strainand Ux is the dynamic displacement along x.
For a pre-stressed sample, the static strain ε0 is muchgreater than the dynamic one ∂Ux
∂x . A Taylor expansion ofthis expression gives rise to quadratic non-linearity.
The obtained stress is substituted in one dimensionalequation of motion to obtain a propagation equation forUx. Writing Ux Uω
x UΩx with Uω
x the high-frequencydisplacement and UΩ
x the low-frequency component, con-sidering that in the region of interaction U Ω
x Uωx and
applying an averaging over one period of the high-frequency term, we derive:
∂2
∂t2 c20
∂2
∂x2 UΩx εc2
0∂∂x ∂Uω
x
∂x 2 (2)
With ε the nonlinear parameter of the granular medium,and c0 the phase velocity of the demodulated wave. Wecan rewrite this equation by introducing the energy den-sity profile of the high-frequency carrier wave W ω :
∂2
∂t2 c20
∂2
∂x2 UΩx ε
ρ0
∂∂xW ω (3)
With ρ0 the density of the granular medium. According tothis equation, the demodulated wave of low-frequency isexcited by the gradient of energy density profile of carrierwave. We can now describe the carrier wave transportwith both a propagation equation or a diffusion equation.Analytical solutions in the frequency domain exist.
When the carrier wave is attenuated mostly by absorp-tion than by scattering, the only contribution for the de-modulated signal comes from carrier wave ballistic prop-agation.
Case of Carrier Wave Ballistic Propagation
In the case of carrier wave ballistic propagation, wehave the following solution outside the excitation region:
UΩz ε
2ρ0c20Iω
fΩ
1 iΩ 1 cωc0 1
τ 1 iΩ 1 cω
c0 1τ e
i Ωc0
z(4)
with τ the attenuation time,fΩ the modulation func-
tion in the frequency domain, cω the group velocity ofthe carrier wave, c0 the velocity of the low-frequencydemodulated wave. The carrier wave has a sufficientlysmall spectral width to neglect influence of dispersion onit. However, the difference in velocities of carrier and de-modulated waves is not negligeable. Because of this ve-locity dispersion, the profile of the demodulated wave canbe integrated. In equation (4), integration of U Ω
z occurs ifboth factors in brackets are proportional to 1
iΩ . This is the
case if Ω 1 cωc0 1
τ .
EXPERIMENTS
A wide bandwidth ultrasonic transducer is at the bot-tom of a cylindrical tank. This recipient is filled with 100kilograms of 150 µm diameter glass beads. At the top ofthis granular medium, is placed an accelerometer for thedetection. The recepted signals are thus proportionnal tothe second derivative of the displacement U Ω
x . The carrierwave is modulated in amplitude with a gaussian functionof fixed duration.
Results and Discussions
For three different frequencies of the carrier wave (re-spectively 50 kHz, 100 kHz, and 150 kHz), in figure (1),the demodulated wave profile is close to the first, secondand third derivative of the gaussian function of modula-tion. Two derivations with increasing carrier wave fre-quency fh are thus observed.
At 50 kHz frequency, (figure 1:a, 1:b), the carrier waveis weakly attenuated: the cumulative effect of dispersionimplies an integration of the demodulated wave profile.This is why an acceleration that is proportional to the firstderivative of a gaussian has been detected at the top ofthe granular medium. With increasing frequency the at-tenuation becomes stronger. This decreases the penetra-tion distance of the carrier wave. The cumulative effectof dispersion on the demodulated wave disappears, im-plying derivation of the demodulated wave profile (figure1:c, 1:d).
0 0.2 0.4 0.6 0.8 1
x 10−3
−1
−0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
x 10−3
−1
−0.5
0
0.5
1Nor
mal
ized
am
plitu
de 0 0.2 0.4 0.6 0.8 1
x 10−3
−1
−0.5
0
0.5
1
2 4 6 8
x 10−4
−1
−0.5
0
0.5
1
2 4 6 8
x 10−4
−1
−0.5
0
0.5
1
2 4 6 8
x 10−4
−1
−0.5
0
0.5
1
a b
c d
e f
Time (s) Time (s)
Nor
mal
ized
am
plitu
de
FIGURE 1. a, emitted signal (centered at 50 kHz) and receiveddemodulated signal. b, theoretical results corresponding to a. cand d, same figures at 100 kHz. e and f, idem at 150 kHz.
The second derivation obtained with increasing fre-quency to 150 kHz, (figure 1:e, 1:f), is due to the diffrac-tion of the demodulated wave. The carrier wave, confinednear the ultrasonic emitter, plays the role of sources forthe demodulated wave. The effective diameter of thesesources is small (3 cm) compared to the wavelength ofthe demodulated wave in the granular medium ( 10 cm),which ensures a small diffraction length for the demodu-lated wave. If the diffraction length becomes smaller thanthe fixed distance between emitter and receiver, diffrac-tion occurs. Diffraction implies a derivative of the signalon the axis [1], which leads in our case to the third deriva-tive of a gaussian envelope.
ACKNOWLEDGMENTS
This work is supported by DGA (contract no
00.34.026).
REFERENCES
1. Novikov B.K., Rudenko O.V., Timoshenko V.I., NonlinearUnderwaterAcoustics, ASA, New-York (1987).
2. V.YU. Zaitsev, A.B. Kolpakov and V.E. Nazarov, Acousti-cal Physics 45, 3 (1999).
3. V.YU. Zaitsev, A.B. Kolpakov and V.E. Nazarov, Acousti-cal Physics 45, 2 (1999).
4. A. Moussatov, B. Castagnède and V. E. Gusev, Phys. Lett.A, 283, 216-223 (2001).
Nonlinear Acoustic Vibrations in Piezoelectrics withCantor-like Structure
C. Chiroiua
aInstitute of Solid Mechanics, Romanian Academy, P. O. Box 1-863, Bucharest 70701, Romania
The purpose of this paper is to explain the existence of multiple fracton and multiple phonon-mode regimes in thedisplacement field for a piezoelectric plate with Cantor-like structure (Cantor plate). An anharmonic coupling between theextended-vibration (phonon) and the localized-mode (fracton) regimes explained the subharmonic generation of ultrasonicwaves (SWG) in the Cantor plate.
METHOD AND RESULTS
We solve the nonlinear equations, which govern theSWG phenomenon by a generalization of Fourierseries, which uses the cnoidal wave as the fundamentalbasis function. The nonlinear wave motion is a linearsuperposition of cnoidal waves plus additional terms,which include nonlinear interactions among the waves.The nonlinear interactions among the cnoidal wavesare significant for the explanation the draining of theenergy away from the input wave towards lowfrequencies spectrum. We consider a composite plate(fig.1) formed by alternating elements of nonlinearisotropic piezoelectric ceramics (PZ) and epoxy resin(ER), following a triadic Cantor sequence (Craciun etal.[1] and Alippi et al.[2-3]).We consider the same sample using a triadic Cantorsequence up to the fourth generation (31 elements).The length of the plate is l , the width of the smallestlayer is 81/l and the thickness of the plate is h .Resonant vibration modes are excited by applying anexternal electric field )exp( 0
0 tiEE on both
sides of the plate with n .The undetermined system parameters are computed byusing a genetic algorithm. The agreement between theeigenfrequencies given by this curve and by theeigenvalue problem is noted to be excellent. If 0E is
increases above a threshold value 0thE = 5.27 V the
2/ subharmonic generation is observed [4].The amplitude of waves is calculated at the surface ofthe plate as a function of 0E .
In figure 2 the admittance curve ( /k vs. 2/ )
in the linear regime ( VE 1.00 ) marks by peaks the
frequencies n of the modes.
FIGURE 1 The plate with Cantor-like structure. Thedashed regions are occupied by piezoelectric ceramicand the white regions are occupied by epoxy-resin.
The "measured" eigenfrequencies are computablefrom the direct problem. A "noise" has been artificiallyintroduced by multiplication of the data values by
r1 , r being random numbers uniformly distributedin ],[ , with 321 10,10,10
.The agreement between the eigenfrequencies given bythis curve and by the eigenvalue inverse problemresults is noted to be excellent.On comparison of the results given by the admittancecurve with the results obtained from the similarexperimental curve derived by Craciun and al. [1], thedeviation between them is found to be 5-15% for lownatural frequencies, and less than 4% for high naturalfrequencies.Two kind of vibration regimes are found: a localised-mode (fracton) regime represented in figure 3 for
2/ =1169 kHz, 2672 kHz and 3340 kHz and anextended-vibration (phonon) regime represented infigure 4 for 2/ = 4175 kHz and 4250 kHz. Asketch of the plate geometry is given on the abscissa(dashed, piezoelectric ceramic and white, epoxy resin).The fracton vibrations are mostly localised on a fewelements, while the phonon vibrations essentially
extend to the whole plate. For a Cantor plate, we haveobtained the same result as Craciun and Alippi [1,2]:given a normal mode n , for excitation at n ,
the value of the expected threshold thE i. e. the ability
of generating the 2/ subharmonic, is determined bythe existence of a normal mode with: (i) smallfrequency mismatch 2/ n , and, (ii) largespatial overlap between the fundamental andsubharmonic displacement field.
FIGURE 2 The admittance curve ( /k vs. 2/ )
FIGURE 4 The normal amplitudes for two extendedvibration modes ( 2/ =4175 kHz and 2/ =4250 kHz).
ACKNOWLEDGEMENTS
Support for this work by The National Agency forScience, Technology and Innovation (ANSTI)Bucharest, Grant nr.6109/2000-B5, and Grant NATO2000-2001 PST.CLG. 976864/2000 is gratefullyacknowledged
REFERENCES
1. F. Craciun, A. Bettucci, E. Molinari, A. Petri, A. Alippi,Direct experimental observation of fracton modepatterns in one-dimensional Cantor composites, Phys.Rev. Lett., 68, 10 1555-8 (1992).
2. A. Alippi, G. Shkerdin, A. Berttucci, F. Craciun, E.Molinari, A. Petri, Threshold lowering for subharmonicgeneration in Cantor composite structures, Physica A,191, 540-8 (1992).
3. A. Alippi, F. Craciun, E. Molinari, Stopband edges inthe dispersion curves of Lamb waves propagating inpiezoelectric periodical structures, Appl. Phys. Lett. 53,19 1806-8 (1988).
4. C. Chiroiu, P. P. Delsanto, M. Scalerandi, V. Chiroiu,T. Sireteanu, Subharmonic generation in piezoelectricswith Cantor-like structure, Journal of Physics D:Applied Physics, 34, 3, 1579-86 (2001).
FIGURE 3 The normal amplitudes for three localisedvibration modes ( 2/ =1169 kHz, 2/ =2672kHz and 2/ =3340 kHz).
Acoustical propagation in glass beads under compression
J. Anfosso, L. Forest and V. Gibiat
Laboratoire Ondes et Acoustique, C.N.R.S. UMR 7587, E.S.P.C.I., 10 rue Vauquelin, 75352 Paris Cedex 05, France
We present preliminary results upon the acoustic propagation in a granular medium under uni-axial compression and for differentgazes filling the interstitial space. Our results should be interpreted in a different way than the classical one connected withmultiple scattering.
INTRODUCTION
Wave propagation in granular media reveals a trickyproblem. It is difficult to choose between a model wherethe wave propagation is concerned by an homogenousmedium filled with scatters [Jia et al., 1998] and anotherone, the so-called Biot theory, that takes into accountthe effective rules of both the rigid skeleton and theinterstitial fluid. First we give a brief review of thetheories used for the calculation of the velocity when astress is applied with a given gaseous atmosphere. Thenwe describe our experimental set-up. We briefly discussthe presented results and give some hints for aconclusion that may lead to new physical interpretationsfor wave propagation in granular media that are stronglyconnected with a recent work [De Billy 2000].
HOMOGENEISATION THEORY
In the case of a granular medium of glass beadsunder stress, Biot’s theory, as well as otherhomogeneisation theories, can be applied. Consideringthe beads packing as a solid frame and considering afluid phase much more compressible than the frame,Biot’s theory leads to the following formula concerningthe fast longitudinal wave [Forest et al., 1998]:If the skin depth is larger or lower than the pore size, wehave respectively :
4( )
3v
fl
fl
KK µ
φρ φρ
+ +=
+ (low frequency regime) (1).
4
3v1
(1 ) fl
K µ
ρ ρα ∞
+=
+ − (high frequency regime) (2).
and flK K are the bulk moduli of the aggregate and of
the fluid and µ is the shear modulus of the aggregate.
ρ and flρ correspond respectively to the bulk density of
the aggregate and the fluid density, since α ∞ and φ
correspond respectively to the tortuosity and the
porosity of the medium. The Effective Medium Theory(EMT) by focusing on the solid phase, suggests a
variation of K as 1
3P by introducing the Hertz-Mindlinforce law at each grain contact.
EXPERIMENTAL SET-UP
A plexiglas container is filled with mono-disperse glassbeads. Four different diameters (1, 3, 4 and 6 mm) havebeen used. The compactivity of the medium is around63 %. The inner dimensions of the container arerespectively 70 and 90 mm. The distance between thetwo transducers, Panametrics V101 0.5 MHz, is40 mm. One transducer excites the medium withcompressional wave of central frequency chosenbetween 10 kHz and 500 kHz. The opposite transduceris used to measure the transmitted signal. A static load,that does not exceed 1000 N is applied on the mobileupper piston.. A vacuum chamber is used to allowchange of interstitial gaze in it (air, carbon dioxide,helium, vacuum). Figure 1 shows a schematicrepresentation of the experimental vessel.
FIGURE 1. Scheme of the experimental vessel.
EXPERIMENTAL RESULTS
For a sample of beads under a given constraint, vacuumis done in the interstitial space up to approximately 200mbar. The acquired signal is our reference signal. Thenair from atmosphere is injected under ambient pressure.Finally vacuum is done again. This last point allows usto control that there is no rearrangement of the mediumwhen the latter operations have been realised. Moreoverwhen all the experimental process is shorter in time than
F0
the ageing of the medium only the change of gaze canhave an influence on the acoustic propagation,especially on the first part of the signal. The same workhas been executed with carbon dioxide and helium.If carbon dioxide is used to fill the interstitial space, asmall time delay between the two recorded signals isobserved. A frequency analysis shows a strongerdependence with the different signal’s components. Aninterpretation, which would deal only with stress path,cannot be justified. It is more evident if we observe thedelay between both signals for vacuum and carbondioxide and signals for vacuum and helium. Indeed, ifthis observation is done roughly at the same timeposition on the signals, there is a difference between thetwo delays (Figures 3). This difference can beinterpreted in term of Biot’s theory : for both gazes, theskin depth being smaller than the pore size, the highfrequency limit of the velocity for the fast longitudinalwave can be applied (equation 2). Then, for a givenframe, the acoustic velocity decreases as the fluiddensity increases due to the drag of the fluid in theporous network. This evolution is consistent with ourexperimental results.
FIGURE 3. Received signals for medium in vacuum orunder carbon dioxide and in vacuum or under helium with a100 kHz emitted signal. Granular medium with 4 mm glassbeads with an applied stress of 640 N. First vacuum (solid
line), carbon dioxide or helium (dashed line), secondvacuum (dotted line).
Several works, [Royer, 1988] and [De Billy, 2000], forone bead or a one dimensional chain of beads, deal withthe propagation of surface wave. the measured acousticwaves could be the effect of the coherent summation ofsurface waves as it has been shown in other works [DeBilly, 2000]. We have done preliminary experiments onsmall geometrical packing of steel beads of 12 mmdiameter made of two layers of beads. The wholepacking is under a constraint of about 300 N. Thesignal’s frequency in emission is 0.5 MHz. The two
transducers are in contact with the opposite part of thetwo layers. We notice a high powered low frequency(around 15 kHz) as it has been measured in 1D latterworks. All the results we have obtained for these kind ofpacking show the same phenomena that clearlycorrespond to the existence of surface waves. In theseexperiments, the extracted ToF leads to a velocitycorresponding to the theoretical velocity of Rayleighwaves (3800 m/s). As in [Royer, 1988] the satellitepeaks in the spectrum correspond also to those obtainedwhen Rayleigh waves appear. Our “extreme hypothesis”is so experimentally validated and seems to be a goodcandidate to explain contradictions in latter workswhere the ToF measured cannot correspond for thecoherent part to a compressionnal wave.
FIGURE 4. Received signals for two layers of steel beadsof diameter 12 mm for a emitted signal centred at .5 MHz.
Real signal (left). Low filter signal at 30 kHz (right).
CONCLUSION
The results we find here are independent with the wayto calculate the acoustic velocity, group or phasevelocity, in the medium. Our experiment allows us toemphasise the influence of the gaseous phase on theacoustic propagation. In this way, viscous phenomenaor surface propagation have probably to be taken intoaccount.
ACKNOWLEDGMENTS
We would like to thank Dominique Clorennec andMichel De Billy for useful discussion.
REFERENCES
M. De Billy, “Experimental study of sound propagation in a chain ofspherical beads”, J. Acoust. Soc. Am ., 108 , 1486-1495, (2000).
L. Forest, V. Gibiat and T. Woignier, “Biot's theory of acousticpropagation in porous media applied to aerogels and alcogels”,Journal of Non-Crystalline Solids 225 , pp. 287-292, (1998).
X. Jia, C. Caroli and B. Velicky, “Ultrasound propagation inexternally stressed granular media”, Phys. Rev. Lett. 82, 1863-1866,(1999).
D. Royer, Y. Shui, X. Jia, and E. Dieulesaint, “Surface waves onspheres”, the 3rd Western Pacific Regional Acoustic Conference, 2-4November 1988, Shangai, China.
2.7 µs2 µs
He
Co2
2.7 µsCo2
He
Free Wave Motion in Periodic Systems With One of the Constraints Displaced
A.S. Bansala , V. Aggarwalb
aGuru Nanak Institute of Management and Technology, Ludhiana-141002, India. bLG Electronics India Limited , Greater Noida-201306, India.
General expressions that govern free wave propagation in an infinite mono-coupled continuos periodic system with one of its periodic constraints displaced have been set up and applied to study free flexural waves in beam-type system resting on simple supports with one of the periodic supports displaced to different locations. Attenuation of free waves that occur even in the propagation zones varies with the frequency and the support displacement and it is very large near the bounding frequencies of the propagation zones, in general. Variation of resonance frequencies ( in various resonance zones) with support location has also been studied and the conditions have been identified under which the system with displaced support behaves like a periodic system and does not attenuate free waves. The study is quite useful in introducing disorders for the control of vibrations.
INTRODUCTION AND ANALYSIS
Free wave motion in periodic and periodically disordered systems [1,2] and their normal modes [3] have been studied earlier. Free wave motion in infinite periodic systems with single disorder has also been studied before [4] , where one of the elements was different from the rest. In this paper, the effect of displacement of one of the periodic constraints on the free wave propagation is presented. An infinite continuous uniform periodic system is shown in Figure 1(a) with the individual elements shown separated and the constraints indicated at the coupling coordinates. Periodic elements are denoted by E. When one of the constraints together with the corresponding coupling coordinates is displaced from its periodic location, two elements surrounding that displaced coordinate become unequal. Now the system may be assumed to consist of two semi-infinite left and right periodic systems connected through two different elements E1 and E2, as shown in Figure 1(b). Due to a disturbance in the LP system a free incident wave will travel towards and impinge upon the disorder. Here it will be partially reflected back towards the point of disturbance and partially transmitted across the disorder. The transmitted wave propagates as a free wave along the RP system. In the absence of the disorder, that is, when the constraint is not displaced from its periodic location , the incident wave would propagate through the periodic system as an FIGURE 1. Block diagram of mono-coupled infinite system with (a) periodic constraints, (b) one of the periodic constraints displaced.
unobstructed free wave. This also applies to the beam-type system which consists of infinite uniform beam resting on periodic simple supports, when the displacement of one of the supports renders the periodic beam disordered , such that two elements on either side of the displaced support become unequal and different from rest of the periodic beam elements.
The generalized displacements and forces at the coupling co-ordinates (or the slopes and moments at the simple supports in the case of the beam system) at A and B due to the forces (moments) to their left and right can be found by invoking the conditions of continuity of displacement (slope) and equilibrium of forces (moments). The expressions can be written in terms of the end receptances (α , β) of the elements E1
and E2 and the characteristic wave receptances (αw+ , αw- & αw+) of the semi-infinite periodic system due to the free waves traveling towards right & left in the L.P. System and towards the right in the R.P. System. The normalized moments at A and B associated with the reflected and transmitted waves in a beam-type system are finally expressed as:
Mr/Mi= - ( Ψφ- β12.β2
2 ) /(ξφ-β12. β2
2) (1)
Mt/Mi= - [(α1+α2 )2.αwβ1β2]/ (ξφ-β1
2 β2 2) (2)
Where, Ψ = (α1-αw)(α1+α2)- β2 2, (3)
φ = (α2+αw)( α1+α2)- β2 2 , (4)
ξ = (α1+αw)( α1+α2)- β1 2 . (5)
α1 , α2 and β1, β2 are the direct and cross end receptances of beam elements E1 and E2, respectively, and αw is the characteristics wave receptance.
(a)
(b)
E E E E E E
EE E E E1 E2
As expected from equations (1) & (2), the response will be large at frequencies which satisfy: ξ.φ -β1
2 . β2 2 = 0 (6)
In an undamped system α and β are always real. The equation (1) and (2) will be satisfied only when αw is purely real and this can be so only in the attenuation zone. The frequencies at which this occurs , the mass of the disordered elements resonates against the stiffness of the combined system. The combined system can thus behave like a spring-mass system in an attenuation zone.
COMPUTER INVESTIGATIONS AND DISCUSSION OF RESULTS
Computer investigations have been carried out over a wide range of parameters but only the most significant results are presented & discussed here. Moment transmitted across the disorder has been studied over a wide frequency range Ω= 0 to 60, where Ω is the non-dimensional frequency, defined as Ω=ω.l2(mb/EI)1/2 , where ω is the angular frequency (rad/sec) and mb is the mass per unit length l of the beam having flexural rigidity EI. It has been found that in the propagation zones, the displacement of the support always causes attenuation that depends upon the amount of displacement and the frequency. Close to the bounding frequencies of the propagation zones (PZs) the attenuation is very large, in general, except when the support is displaced to certain locations that coincide with the nodal points of the infinite periodic beam vibrating in flexure. Under these conditions the adjacent elements have some specific length ratios as compared to the periodic elements like (1/3,5/3), (2/3,4/3), (3/4,5/4) etc, depending upon the frequency PZ.
FIGURE 2. Variation of natural frequencies with support displacement. - - - - -, first mode and . . . . ., second mode of various groups of natural frequencies.
Variation of resonance frequencies of the beam system when one of the periodic supports is displaced to different locations (X) has also been studied and presented in Figure 2. In the first AZ, the system has only one natural frequency, whereas in the higher AZs it depends upon the displacement of the support. In the subsequent higher AZs there are two natural frequencies. The frequency curves in each group of natural modes attain some maximum and minimum values at certain specific support locations as it is displaced between its extreme left and right locations. Peaks of the second modes of all the groups of resonance frequencies coincide with the lower bounding frequencies of the PZs. For example, it occurs at (X,Ω) = (0.5, 39.47). The troughs of the first mode of all the groups of natural frequencies coincide with the higher bounding frequencies of the PZs. For example, it occurs at (X,Ω) = (0.66,61.67). Under these conditions, the disordered system behaves like a periodic one and the disorder caused by the displacement of the support does not interfere with the propagation of free waves. It may further be pointed out that the resonance frequencies corresponding to the peaks and troughs, that lie inside the attenuation zones, correspond to the support locations that always attenuate the waves propagating across the displaced support, just as at other frequencies which lie inside AZs. This study of wave motion in periodic systems with one of the periodic supports displaced should be quite useful in studying the effect of displacement of the support and the way such a disorder can be introduced to control transmission of vibration.
ACKNOWLEDGMENTS
The support from the Department of Mechanical Engineering, Punjab Agricultural University, Ludhiana -141004, where this work was conducted, is thankfully acknowledged
REFERENCES 1. D. J. Mead, J.Sound Vib. 11, 181 - 197, 1970.
2. A. S. Bansal, J.Sound Vib. 207, 365 - 382, 1997.
3. A. S. Bansal, J. Acoust. Soc. Am. 102, 3806 - 9, 1997.
4. D. J.Mead and A. S. Bansal, J. Sound Vib. 61, 481 - 496, 1978
Support Location, X
The Direct Superposition Method to Predict WoodAnomalies in Acoustic Diffraction Spectra
N. F. Declercq, R. Briers, O. LeroyInterdisciplinary Research Center, KULeuven Campus Kortrijk, E. Sabbelaan 53, 8500 Kortrijk, Belgium
Scholte – Stoneley waves (SSTW) exist at the interface between a solid and a liquid and are important for NDT of near surfacedefects over long ranges. The frequencies at which SSTW can be generated on a grating through diffraction using normallyincident plane waves are calculated using a new direct superposition method (DSM) that superposes evanescent plane wavesdirectly on the corrugations. The calculated frequencies agree very well with miniature scale experiments reported by Claeys etal1, measuring reflection spectra when a corrugated stainless steel/water interface is struck by longitudinal ultrasound in theliquid, as well as with large scale experiments on the sea bottom reported by Chamuel and Brooke2, measuring Braggfrequencies. The results are better than those obtained from the mode conversion theory1 (MCT).
THEORETICAL ACHIEVEMENTS
Let us consider an infinite surface that is periodicallycorrugated in one dimension, which is the crossingfrom an isotropic solid to an ideal fluid, described by
0),()(),( zxpzxfzxp , (1)
being the periodicity of the corrugation. The DSMassumes potentials, describing the acoustic field of asurface wave, of the form
tfzxki
ytfzxki
tfzxki
x
x
x
Ce
Be
Ae
2
1
1
eψ (2)
with lower index 1 in the solid, 2 in the liquid. Thecircular frequency is , kx is assumed to beindependent of x. respectively is the imaginary z-component of the wave vector, related with kx throughthe longitudinal phase velocity in the solid respectivelyin the liquid, while through the shear phase velocityin the solid. A, B and C are amplitudes.The novelty of the potentials (2) is that they containinformation about the corrugation f(x). If f(x)=0 then(2) is equal to the potentials at a plane interface. The MCT is only valid for
incidence
incidencexf
)(max (3)
and describes the diffraction of incident plane waves ata corrugated surface described by (1), supposingreflected and transmitted pure plane waves fordifferent orders m, i.e. plane waves of the form (2) butwith f(x)=0. It is known that the MCT is only valid in
the acoustic far field, while SSTW are typicallyacoustic near field phenomena; hence the DSMsuggests potentials as in (2). When continuity conditions are applied for normalstresses and normal displacements, we get the simpleScholte - Stoneley equation3, i.e. the determinant of acontinuity matrix, that must be zero, for planeinterfaces and f(x)=0 in (2), but we get a far morecomplicated expression5 when periodically corrugatedsurfaces are considered for potentials as in (2), i.e. thedeterminant of a complicated continuity matrix.Padilla et al3 define ‘the Scholte – Stoneley zero’ for aplane interface as the zero with lowest frequency foran arbitrary fixed kx. For corrugated surfaces, wedefine the Scholte – Stoneley frequency as the lowestfrequency for which the determinant of the continuitymatrix shows a distinct minimum for a certainarbitrary chosen kx and for a certain arbitrary chosen xcoordinate. It is also observed numerically that thisfound frequency is independent of the chosen x.
NUMERICAL VERSUSEXPERIMENTAL RESULTS FOR
DIFFRACTION
It is found5 from symmetry reasons that if a SSTW isgenerated by diffraction using normal incidence, that
SSTWx
mvfreqmk 2 (4)
for ,..2,1,1,2.., m ,which is the classical grating equation with m theorder of the generated surface wave. Equation (4) isalso used in MCT, which often leads to themisconception that SSTW at corrugated surfaces arebuilt up by simple plane wave potentials as for planeinterfaces. The DSM is applied for a stainless steel ( =
FIGURE 1. The zero order reflection spectrum for astainless steel water interface, showing Wood anomaliescorresponding to generated SSTW
3/7850 mkg , dv = sm /5790 , sv = sm /3100 ) –
water ( = 3/1000 mkg , dv = sm /1480 ) interfacefor a rounded sawtooth profile described by
xhxf 2sintan)1tan(2
)( (5)
with mh 500 and m1500 and for kx asdefined in (4) for order m=1,..,4. The results are foundin Table 1 and are compared with the experimentallyfound1 results, see Fig. 1, for Wood anomalies in thezero order reflection spectrum for the same interface. It is seen that the DSM is more accurate, even infrequency ranges where it is impossible to find areasonable solution with the MCT. Calculations werealso performed using a PVC ( sm /1360 ,
smvd /2268 , smvs /1100 ) – water interface andit is found that the SSTW velocity is higher forcorrugated surfaces than for plane interfaces (1479.6m/s for stainless steel- water and 909m/s for PVC -water interface)
NUMERICAL VERSUSEXPERIMENTAL RESULTS FOR A
LOAD IN THE SURFACE
Here we don’t need to use (4). Therefore weconsider one frequency and calculate the SSTWvelocity or vice versa. As an example, we havecalculated the velocity vSSTW of a SSTW traveling atthe Bragg frequency
2.. SSTWv
freqbragg (6)
Table 1. SSTW frequencies (freq) and velocities (v)
interface method m=1 m=2 m=3 m=4stainless exp* freq 1.17 2.22 3.31 -steel - MCT freq 1.22 2.50 - -water DSM freq 1.097 2.194 3.292 4.390
DSM v 1645.4 1645.9 1646.0 1646.1PVC - DSM freq 0.741 1.483 2.225 2.967water DSM v 1112.0 1112.2 1112.5 1112.6
* Ref. 1
Table 2: Calculated and experimental values for the Braggfrequency and the corresponding SSTW velocity for a sine, arounded sawtooth (rst) shaped surface and the mean values.
corrugation Bragg freq (kHz) SSTW velocity (m/s)sine 165.0 1320.0rst 184.7 1477.6
mean value 174.8 1398.8Experiment* 178.0 1423.0
*Ref. 2
for a corrugated dense limestone – water interface2
with mmh 5.0 and mm4 . The calculations wererealized using a sine and a rounded sawtooth profileand we have taken the mean velocities and meanfrequencies to compare with the experimental values2
for a surface shaped in between a sine and a roundedsawtooth. The results are shown in Table 2 and agreevery well with the experiments. Brooke et al4 have realized experiments in thecorrugated Barrow Strait of the Canadian Arctic where
mh 4 , m75 and the SSTW velocity is measuredand is 1460 m/s while 1444.6 m/s on a plane surface.The DSM predicts a SSTW velocity of 1477 m/s for arounded sawtooth profile. This deviation is due to boththe unknown exact bulk properties in the seawater andthe exact corrugation shape. Nevertheless, both thefield data and the DSM show an increased SSTWvelocity in the presence of roughness.
ACKNOWLEDGEMENTS
Sponsored by the Interdisciplinary Research Center,KULeuven Campus Kortrijk, E. Sabbelaan 53, 8500Kortrijk, Belgium
REFERENCES1. J.M. Claeys, O. Leroy, A. Jungman, L. Adler, J. Appl.
Phys. 54(10), 5657-5662 (1983)2. J. R. Chamuel, G. H. Brooke, J. Acoust. Soc. Am. 83(4),
1336-1344 (1988)3. F. Padilla, M. de Billy, G. Quentin, J. Acoust. Soc. Am.
106(2), 666-673 (1999)4. G.H. Brooke, D. J. Thomson, R.F. McKinnon, DREP
Tech. Memo. 85, 18 (1985)5. Article submitted to J. Acoust. Soc. Am.
Anomalous Acoustic Wave –A Precursor to GujaratEarthquake
B. S .Gera, V.Mohanan, G.Singh AND V.K.Ojha
National Physical LaboratoryDr.K.S.Krishnan Marg, New Delhi-110012, India
E-mail: bsgera @csnpl.ren.nic.in
A major devastating earthquake occurred in Kutch district, Gujarat on 26 January 2001. Sodar detected an anomalous lowfrequency wave in the atmospheric boundary layer at Vapi, a day in advance. Such a wave has not been seen during the past 20years of sodar observation in India. The wave is considered a precursor to the earthquake.
INTRODUCTION
The occurrence of earthquake is a globalphenomenon. In spite of advancements in science andtechnology the earthquakes can still be better explainedthan predicted. A reliable early prediction is crucial insounding alert to adopt safety measures. As suchstudies of earthquake precursors is important.Precursory phenomena may occur hours to days beforeearthquakes but only a few instruments have detectedunusual premonitory behavior. Analysis of thefrequency and amplitude spectra of several majorearthquakes [Cifuentes and Silver, 1989] revealprecursors as unusual signal in low frequency region ofspectra. Studies of precursors are mainly focussed onanalysis of seismograph, describing history of freeoscillations of the earth. However, possibilities ofacoustic wave emissions, at initial stages of faultdevelopment, being coupled to the lower atmosphericboundary layer (ABL), through air-earth interaction,can not be ruled out. We report sodar detection of ananomalous wave in the ABL, a day in advance to themain shock.
DESCRIPTION OF GUJARATEARTHQUAKE
The Kutch is a seismogenic domain and the mostseismically active part lies between Island belt fault andKutch main land fault, where the recent intraplateearthquake occurred on 26 January, 2001, at 08:46:42.9AM IST. The quake epicentre ( 23o40” N, 70.28 E) wasnear Bhuj (Fig.1). The intensity measured was 6.9 onRichter scale, it lasted for 30 seconds.nd the focal depthwas 15 Km. It had a seismic moment of 4 x 1020 andcorner frequency of 0.06 Hz. The main shock wasfollowed by another tremor measuring 5.3 after 20 min.and 140 after shocks. Bhavnagar had been feeling the
tremors for several months. However, they were nottaken seriously.
FIGURE 1. Geographical location of earthquakeepicentre (Bhuj) and sodar site, Vapi (Valsad)
EARTHQUAKE PRECURSORS
As per ancient Greece notion of 737 BC, unusualbehaviour of the animals and birds give an alarm of animpending major earthquake likely to occur. It is seenso prior to several earthquakes. Chinese used suchalarm successfully to evacuate the city of Haichengseveral hours before a 7.3 magnitude earthquakeoccurred on 4 Feb. 1975. Japanese also monitor animalbehaviour for earthquake warning. However, it is not ascheme for predicting earthquakes. A new study hasfound that rumblings can be forecasted from the bowelsof the earth a day in advance of earthquake. In aninnovative study, Russian physicists have found that
FIGURE 2. Sodar record of anomalous waves seen on (A) 28th December, 2000 and (b) 25th January, 2001
earth gives out warning signals in the form of flashes ofneutron radiation just before increase in the earthseismic activity. Thus a sudden burst of neutrons mayprovide key warning signals of a major earthquake.
SODAR OBSERVATIONS
Monostatic sodar, capable of probing the dynamicsof ABL thermal structures in real time upto a height of1Km, was in operation at Vapi ( Fig.1). The site iswithin a radial distance of about 400 Km from theepicentre (Bhuj) and closer to Bhavnagar wheretremors were felt several days in advance. The main objective of sodar operation was inversionstudies as part of EIA for Sabero Organics Ltd.Continuous observations for about 10 days ( except forholidays) were made every month. It was in operationupto 25 January, 2001- a day prior to the occurrence ofearthquake. Unfortunately, the system was not inoperation on the following day , for being a nationalholiday, when the earthquake occurred. Sodar studies, across the world, have revealed theoccurrence of several common features such as thermalplumes, ground based and elevated stable layers,oscillating layers (waves) etc. according to theprevailing meteorological conditions. However, studies(Aggarwal et al., 1980) have revealed that the wavesoccurring in the ABL have time period of 1-15 minutes. A anomalous wave period of more than 3 hours hasbeen seen at Vapi. Low frequency waves with highamplitude have been seen on certain days during theperiod 23 December, 2000 to 18 January, 2001. Atypical wave (Fig.2) seen a day prior to the main shockhad a frequency of 1.6 x 10- 4 Hz and a peak-to- peakamplitude of about 300m. Observation of anomalous wave prior to theoccurrence of a major earthquake and the observationsof similar waves on several days in advance inconjunction with the fact that tremors were also felt atBhavnagar ( Fig.1) suggest that observed anomalous
wave is a precursor to the earthquake Furthermore, thewave is seen to gain intensification (in terms of recordimpression ) with progressing days and the maximumintensity was on 25 January, 2001, just a day prior tothe occurrence of earthquake. Moreover, the occurrenceof anomalous waves has been seen to be mainlyconfined to the periods just prior and after theoccurrence of main shock. These considerations furthersupport that the observed anomalous wave is aprecursor. However, not being earthquake expert,detailed theoretical investigations have not been carriedout. The observation is reported for interest ofconcerned scientific community. It may be mentionedthat waves are not seen on all days as it will be seenonly if it happens to pass through the atmosphericvolume being probed by sodar. Therefore, having giventhe benefit of doubt, further studies using sodarnetwork in high risk zones are needed for ameaningful conclusion.
CONCLUSIONS
A very low frequency (1.6 X 10- 4 Hz) andhigh amplitude of about 300m wave, seen in the ABL,a day before the occurrence of earthquake in Gujarat isbelieved to be a precursor.
REFERENCES
Aggarwal,S.K., Singal. S.P., and Srivastava., S.K. (1980)“sodar studies of gravity waves in the planetary boundarylayer at Delhi” Mausam, (31), 373-384.
Cifuentes and Silver, (1989) “Low-frequency SourceCharacteristic of the Great1960 Chilean Earthquake”,volume94(B1), 643-663.
Sonar Ring Transducers with Periodic Structure.
L. B. Nikitin , V. B. Zhukov , R. P. Pavlov
Central Research Institute “Morphyspribor”, 46 Chkalovsky Prospect, St. Petersburg, 197376, Russia
In practical experience of using ring sonar transducers often arises the necessity in transmitting low frequency signals when the dimensions of the transducers are limited. Attempts to transmit acoustical signals in the frequency region well below the resonance lead to decrease of radiated power due to restrictions o f driving voltage. This drawback may be overcome by using the ring vibrating system with periodic structure in which part of active material (ceramic) is substituted by passive staves. Vibrating mechanical system of periodic structure with alternating ceramic and passive staves of various materials is analyzed. It is shown that passive staves of textolite lower the resonance frequency and mechanical stresses of vibrating system at great extent. Some practical results comparing ceramic and periodic structure rings are presented.
The aim of this paper is to analyze the possibilities that are given by usage of ring transducers with periodic structure [1,2]. Mechanical vibrating system in the form of a thin low height closed ring in which parts of piezoceramic agitated in phase and staves of passive material alternate with each other with constant angle step is considered (figure 1)
FIGURE 1. Vibrating system with periodic structure; 1-piezoceramic, 2-passive stave. Despite the fact that the properties of materials in the tangential direction are changing in a leap and assuming that the period of alternating passive staves is considerably less than the wavelength we shall perform the analysis using the mean parameters of mechanical system. These parameters are determined by following expressions: ñm=ñ[1-â(1-ñp/ñ)]; sm=s33
e[1-â(1-1/s33eYp)];
km2=k33
2(1-â)/[1-â(1-1/s33eYp)]; fm=1/ðD(ñmsm)1/2,
where signed: ñm–mean density; sm-mean elastic compliance modulus; km–mean coupling coefficient; fm–resonance frequency; D–mean diameter of the ring; ñp, Yp–density and Young’s modulus of passive materials respectively; ñ, s33
e–density and compliance modulus of piezoceramic respectively; â-relative volume of passive material in mechanical system (0≤â≤1). On the base of these formulas the parameters of mechanical system dependent on the relative volume of passive material (â) were calculated for some well known passive materials. The results are shown in
FIGURE 2. Relative to ceramic ring values of resonance frequency (a), coupling coefficient (c), dynamical mechanical stresses (c) of a ring with periodic structure. 1-textolite, 2-plumbum, 3–glass textolite, 4–steel, 5-aluminum. Figure 2(a,b,c). It can be observed that the most interesting passive materials are plumbum and
0,0 0,1 0,2 0,3 0,4 0,5 0,60,4
0,6
0,8
1,0
1,2
β
1
2
3
54
a
Rel
ativ
e re
sona
nce
frequ
ency
0,0 0,1 0,2 0,3 0,4 0,5 0,60,2
0,4
0,6
0,8
1,0
Rel
ativ
e m
echa
nica
l stre
sses
Rel
ativ
e co
uplin
g co
effic
ient
4
352
1
b
0,0 0,1 0,2 0,3 0,4 0,5 0,60,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
4
5
3
21
c
1 2
textolite. However the calculations for plumbum make only theoretical sense because it is cold flow and for this reason can’t be used in high power sonar transducers. Thus the material with good prospects for the usage in ring transducers is textolite. It enables to lower resonance frequency of mechanical system and provides the reduction of mechanical stresses to a great extent. This result is most valuable in the case of ring transducers working at acoustic impedance with small resistance and large reactance e.g. free flooding rings. It is worth mentioning also that for existing PZT ceramic voltage strength reserve due to high electromechanical efficiency is quite large. At the same time the piezoceramic being a fragile material greatly depends on dynamic mechanical stresses especially when the long term reliability is required. When the mean parameters are used it is assumed that there is no angular dependence of mechanical stresses and radial velocities. To check the correctness of this supposition the calculations concerning the influence of distribution of passive material on the mechanical stresses were performed by FE method. The mechanical system with textolite was chosen for analysis. Three cases with constant volume of passive material (â=0,13) were considered with recurrence period of staves: 7.20, 14.40 and 900. Figure 3 shows the distribution of tangential components of mechanical stresses and their relative to ceramic ring values in 450- part of the ring at resonant frequency. Considering the curves in Figure 3 it is seen that for recurrence period of staves up to 150 the distribution of mechanical stresses differs from the even distribution not more than by 20%. At the same time mean value of mechanical stresses amounts 0.4 of stresses in ceramic ring when the vibration velocities are equal.
FIGURE 3. Distribution of mechanical stresses on the part of the ring. 1 – ceramic ring, 2, 3, 4 – periodic structure with recurrence periods of passive staves 7.2o , 14.4 o and 90o respectively.
Quite different results are obtained when the recurrence period of passive staves is 900. In the passive stave itself the mechanical stresses are ten times less than in ceramic ring, but in the center of ceramic part they are close to the stresses in the ceramic ring. More over great unevenness of displacement distribution is observed. However the resonance frequency in all cases is lowered to the same extent against the resonance frequency of ceramic ring. In conclusion we will give an example of practical realization of a free flooding ring transducer with periodic structure. Figure 4 demonstrates the acoustical performance - the transmitting voltage response (TVR) of ceramic transducer (1) and transducer with periodic structure (2). The transducers posses the same dimensions (mean diameter – 385mm, height – 225mm). Active material is piezoceramic of PZT type, passive staves are made of textolite (â=0.13), recurrence period – 7.20.
FIGURE 4. TVR of ceramic ring (1) and the ring of periodic structure (2). It is seen that the periodic structure makes it possible to raise the level of radiated sound in the low frequency region. At the same time dynamical mechanical stresses in periodic structures are considerably less than the stresses in ceramic ring when vibration velocities are equal.
REFERENCES 1. Harris, W. T., Ring-shaped Transducer,
U.S. Patent No 3.142.035, 21 July 1964 2. Church, D. R., Mass-loaded Electromechanical
Transducer, U.S. Patent No 3.139.603, 30 June 1964. 0 10 20 30 400,0
0,2
0,4
0,6
0,8
1,0
2
34
1
Rel
ativ
e m
echa
nica
l stre
sses
angle (deg.)
1,0 1,5 2,0 2,5 3,0 3,5130
135
140
145
150
TV
R, d
B r
el. µ
Pa/
V a
t 1m
frequency, kHz