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Analysis of the measured response of a near-periodic framed shell to a point force reveals that the higher-order azimuthal modes (n>10) on the shell are significantly localized. A quantitative analysis of the data yielding the spatial attenuation rates, the measured group speeds, and the total measured damping factors for various azimuthal orders i...
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... of modes which are acoustically cut off in the frequency band of interest, the experimentally derived ‘‘spatial attenuation due to damping’’ does not account for the observed localization of the response. And further, the very strong fluctuations of the observed attenuation factor of these modes strongly suggest that a simple dissipation mechanism is not the source of the observed localization phenomena. Based on this analysis, and the comparison of the theoretical predictions of the localization lengths due to irregularity with experiment, we have concluded that Anderson localization is the dominant spatial attenuation mechanism in the framed shell system. The term Anderson localization is here meant to convey the observation that most of the modes of the system in this frequency range, almost all the large n modes, have become localized resonant states as a result of the irregularity. Recall from the above discussion that in the analogous exactly periodic system, the corresponding states, the Bloch or Floquet modes, are all extended. Thus as a result of the small amount of irregularity in our system, the qualitative character of the measured response of the system to a localized excitation has dramatically changed. The excitation levels near the drive point have become greatly enhanced and the resonant local modes dominate the drive point admittance and the nearby response. A spatial measurement of the ‘‘effective damping’’ of the system yields very large attenuation rates. But damping is in fact not the mechanism behind this spatial falloff in the response. Instead, one is simply observing a buildup of trapped reverberant vibrational energy. The observations described above are not in fact suffi- cient to indicate Anderson localization in a strict sense—the localization of all the resonant modes of the system by irregularity. The low-frequency modes whose wavelengths are of the order of the size of the system are of course never localized by weak irregularity. Nor in the case which we examine are the majority of the small n extended modes substantially localized; the influence of irregularity on these modes is typically too weak to produce localization lengths smaller than either the length of the structure or the radiation damping length. Nevertheless, the physical mechanism producing the localization of most of the resonant states of the system in the mid-frequency range is localization by irregularity. We loosely describe this as Anderson localization. The outline of the remainder of the paper is as follows. In Sec. I, we first present some theoretical background and previously obtained results. In particular, the general formulas for the localization lengths of oscillator chains obtained by Herbert and Jones 11 and Thouless 12 are given. We then employ an approximation technique which retains only those resonances of the framed shell system which lie within a particular frequency pass band of interest, the single band approximation. This approximation results in describing the system effectively as a linear chain of coupled oscillators, and the formulas due to Herbert and Jones 11 and Thouless 12 may be immediately applied. In Sec. II, the experimental results are presented. The measurements consist of NAH measurements of the vibratory response of a framed cylindrical shell to a local excitation. A substantial portion of the mid-frequency range was covered, roughly c Ͻ Ͻ 4 c , with c ϭ c p / a the ring frequency of the cylinder ͑ a is the radius of the shell and c p is the longitudinal wave speed ͒ . As mentioned above, an investigation of the response of this system in wave number space was recently published by the authors. 1 Here, we examine the spatial response of the system and focus on the localized modes, primarily those modes with relatively large azimuthal mode number, n у 11. In Sec. III, the results are discussed and analyzed, and the theoretical approach described in Sec. I is applied to the case at hand. The predictions of the theory are compared to the measurements along with predictions of the attenuation arising from the measured dissipation. We conclude with a discussion of the generality of these results and some applications to current research efforts in this area. The goal of this section is to provide a simple theoretical framework to explain our observations of the spatial localization of the large order azimuthal modes of the shell. The approximations we describe below rely on a number of known theoretical results concerning the behavior of simple oscillator chains. Here, we review some of the important facts we will need later. The one-dimensional coupled oscillator system is governed by Ϫ m 2 x i ϭϪ k ͑ 2 x i Ϫ x i ϩ 1 Ϫ x i Ϫ 1 ͒ Ϫ k i x i . ͑ 1 ͒ The individual oscillators have mass m and stiffness k i , while the springs coupling the oscillators to one another are taken to have the uniform stiffness value k . A schematic of the oscillator chain is shown in Fig. 1. One quickly ...
Citations
... These modes are physically quite different from the Bloch modes that span the entire domain in a periodic media [3][4][5][6]. Topological metastructures are novel materials that are capable of demonstrating energy localization [7][8][9][10][11][12][13][14][15] and wave transport mechanisms [16][17][18][19][20][21][22] that are immune to imperfections. Quasiperiodic structures have emerged as a type of topological metastructures that are characterized by deterministic disorder. ...
Quasiperiodic metastrucures are characterized by edge localized modes of topological nature, which can be of significant technological interest. We here investigate such topological modes for stiffened and sandwich beams, which can be employed as structural members with inherent vibration localization capabilities. Quasiperiodicity is achieved by altering the geometric properties and material properties of the beams. Specifically, in the stiffened beams, the geometric location of stiffeners is modulated to quasiperiodic patterns, while, in the sandwich beams, the core’s material properties are varied in a step-wise manner to generate such patterns. The families of periodic and quasiperiodic beams for both stiffened and sandwich-type are obtained by varying a projection parameter that governs the location of the center of the stiffener or the alternating core, respectively. The dynamics of stiffened quasiperiodic beams is investigated through 3-D finite element simulations, which leads to the observation of the fractal nature of the bulk spectrum and the illustration of topological edge modes that populate bulk spectral bandgaps. The frequency spectrum is further elucidated by employing polarization factors that distinguish multiple contributing modes. The frequency response of the finite stiffened cantilever beams confirms the presence of modes in the non-trivial bandgaps and further demonstrates that those modes are localized at the free edge. A similar analysis is conducted for the analysis of sandwich composite beams, for which computations rely on a dynamic stiffness matrix approach. This work motivates the use of quasiperiodic beams in the design of stiffened and sandwich structures as structural members in applications where vibration isolation is combined with load-carrying functions.
... Des analyses numériques et expérimentales sont ensuite menées pour des coques avec une répartition irrégulière des raidisseurs [KAPM05], [PhHo99]. En dépit de la bonne concordance des résultats numériques et expérimentaux, ces études restèrent limitées au domaine des BF, du fait que l'augmentation des fréquences nécessitait la prise en compte d'un nombre accru de degrés de liberté. ...
... La résolution est réalisée grâce à des distributions continues de séries harmoniques qui permettent de relier l'expression de la pression moyenne appliquée par la coque sur les raidisseurs au déplacement radial de cette dernière. Tran Van-Nhieu démontra ainsi, à l'appui d'expérimentations réalisées dans [PhHo99], que lorsque l'interaction entre les ondes de Bloch-Floquet et les raidisseurs est suffisamment faible, une approximation habituellement utilisée pour des modèles de coques infinies peut-être employée. La comparaison en basses et hautes fréquences a démontré une bonne concordance des résultats. ...
La sûreté des réacteurs nucléaires constitue une condition primordiale et nécessaire à leur industrialisation. Pour les réacteurs à neutrons rapides refroidis au sodium, cette sûreté passe par la possibilité de détection d’une fuite d’eau dans le sodium au niveau du générateur de vapeur à des stades très précoces de leur déclenchement. Le présent travail de thèse s’inscrit dans le cadre du développement d’une technique non intrusive pour la détection d’une réaction sodium-eau dans le générateur de vapeur d’un réacteur nucléaire refroidi au sodium. On désire identifier le bruit vibratoire de cette réaction à partir de mesures d’accélération sur la virole externe du générateur. Cependant, les vibrations dues à la fuite peuvent être noyées dans le bruit généré par l’écoulement du sodium ou par d’autres sources d’excitation. Afin d’augmenter, le rapport signal à bruit, on s’intéresse à l’apport d’un filtrage spatial par formation de voies pour ce type de système. Ce dernier se caractérise notamment par la présence d’un couplage structure-fluide lourd (c.-à-d. virole-sodium) et des modes vibroacoustiques qui apparaissent aux mêmes fréquences d’intérêt pour la détection de la fuite. Pour réaliser cette étude, nous considérons une maquette expérimentale composée d’une conduite cylindrique reliée à un circuit hydraulique par deux brides très rigides. La source à identifier est simulée par un hydrophone de taille réduite en mode émetteur positionné à l’intérieur de la conduite par un dispositif mécanique dédié à cet effet. Le bruit perturbateur est induit par l’écoulement de l’eau, supposé turbulent à partir d’un certain débit. L’antenne est composée de vingt-cinq accéléromètres placés sur la conduite d’essai. L’objectif consiste à traiter simultanément les signaux accéléromètriques pour faire ressortir le signal de la fuite tout en rejetant le bruit dû à l’écoulement. Deux types de traitement par formation de voies sont considérés : la formation de voies conventionnelle (dites de Bartlett) et une formation de voies optimisée qui vise à maximiser le rapport signal à bruit de la fuite.
... Des analyses numériques et expérimentales sont ensuite menées pour des coques avec une répartition irrégulière des raidisseurs [KAPM05], [PhHo99]. En dépit de la bonne concordance des résultats numériques et expérimentaux, ces études restèrent limitées au domaine des BF, du fait que l'augmentation des fréquences nécessitait la prise en compte d'un nombre accru de degrés de liberté. ...
... La résolution est réalisée grâce à des distributions continues de séries harmoniques qui permettent de relier l'expression de la pression moyenne appliquée par la coque sur les raidisseurs au déplacement radial de cette dernière. Tran Van-Nhieu démontra ainsi, à l'appui d'expérimentations réalisées dans [PhHo99], que lorsque l'interaction entre les ondes de Bloch-Floquet et les raidisseurs est suffisamment faible, une approximation habituellement utilisée pour des modèles de coques infinies peut-être employée. La comparaison en basses et hautes fréquences a démontré une bonne concordance des résultats. ...
This study takes part of the R&D framework on sodium-water steam generators, mainly used in the cooling of Fast Sodium Reactors. The aim is to develop a vibro-acoustic monitoring technique that can detect a leak of water into sodium from a defective tube inside the heat exchanger. In the presence of a significantly high background noise and a rather large bandwidth for the leak signature, threshold-detection methods might be insufficient. A beamforming technique is thus considered in order to enhance the leak’s signature against the nuclear facility’s background noise. The present study is carried out on a laboratory test case composed by a cylindrical shell filled with water which is coupled to a hydraulic circuit with two axisymmetric stiffeners. Two kind of excitations are considered: first is a monopole radiating inside the fluid-filled cylinder, the second is a well-established turbulent boundary layer at the neighborhood of the shell’s walls. Numerical simulations are performed to estimate the shell acceleration field used as input of the beamforming technique. Additionally, an experiment is carried out on a mock-up to study the correlations between the numerical and experimental results associated with each excitation. The theoretical and experimental performances of the beamforming are also compared.
... Experimental and numerical work are conducted to evaluate the effect of an increasing structural complexity of the submerged cylindrical shells: irregularly spaced ribs are considered [59,60], internal degrees of freedom are added through resonators [61]. ...
Many works can be found in the literature concerning the vibroacoustic modelling of submerged stiffened cylindrical shells, because of high interest in the industrial domain, in particular for aeronautical or naval applications. However, only a few of them take into account non-axisymmetric internal frames, as for instance engine foundations or floor partitions, that can play a role on the vibroacoustic behavior of the system. That is why a substructuring approach called the Condensed Transfer Function (CTF) approach is proposed in the first part of this thesis. The aim is to take advantage of both analytical models and element-based models, in order to be able to deal with the geometrical complexity, and to calculate at higher frequencies than with element-based methods only. The substructuring method is developed in the general case of thin mechanical structures coupled along curves. A set of orthonormal functions called condensation functions, which depend on the curvilinear abscissa along the coupling line, is considered. This set is then used as a basis for approximating and decomposing the displacements and the applied forces at the line junctions. Thanks to the definition and calculation of condensed transfer functions for each uncoupled subsystem and by using the superposition principle for passive linear systems, the behavior of the coupled subsystems can be obtained. The method is first developed and validated for plates and convergence criteria are defined in relation with the size of the basis of condensation functions. The CTF method is then applied to the case of a submerged stiffened cylindrical shell with non-axisymmetric internal frames. The system is partitioned in 3 types of subsystems: the submerged shell, the axisymmetric frames (stiffeners, bulkheads) and the non-axisymmetric frames. The submerged shell is described by a semi-analytical method based on the Flügge equations in the spectral domain. The axisymmetric frames are described by axisymmetric Finite Element models and the non-axisymmetric frames by Finite Element models. The CTF method is applied to different test cases in order to highlight the influence of non-axisymmetric internal frames on the vibroacoustic behavior of a submerged stiffened cylindrical shell, for different excitations particularly relevant in the naval domain: a point force, an acoustic plane wave, and a random pressure field (such as a diffuse sound field or a turbulent boundary layer for instance).
... They show that, in addition of the Bragg and Bloch-Floquet scattering due to the stiffeners, the effect of regularly spaced bulkheads can be seen. Experimental and numerical work are conducted to evaluate the effect of an increasing structural complexity of the submerged cylindrical shells: irregularly spaced 60 ribs are considered, 15,16 internal degrees of freedom are added through resonators, 17 and the axisymmetry of the system is broken by internal structures. 18 The theoretical formalism for these problems is given by Tran- Several conclusions can be drawn from these studies. ...
... The shell and the frames are supposed to be axisymmetric and the fluid domain is infinite, so that the circumferential orders are independent one from the others. The circumferential Ỹ shell +Ỹ f rame F f rame = −W shell (15) The free displacements are calculated from the Flügge equations using on the right hand term the blocked pressure due to the oblique plane wave: ...
The scattered pressure from a stiffened axisymmetric submerged shell impinged by acoustic plane waves has been investigated experimentally, analytically and through numerical models. In the case where the shell is periodically stiffened, it is shown that helical, Bragg, and Bloch-Floquet waves can propagate. The influence of non-axisymmetric internal frames on these scattering phenomena is nevertheless not well known, as it can considerably increase the computational cost. To overcome this issue, the condensed transfer function (CTF) method, which has been developed to couple subsystems along linear junctions in the case of a mechanical excitation, is extended to acoustical excitations. It consists in approximating transfer functions on the junctions and deducing the behavior of the coupled system using the superposition principle and the continuity equations at the junctions. In particular, the CTF method can be used to couple a dedicated model of an axisymmetric stiffened submerged shell with non-axisymmetric internal structures modeled by the finite element method. Incident plane waves are introduced in the formulation and far-field reradiated pressure is estimated. An application consisting of a stiffened shell with curved plates connecting the ribs is considered. Supplementary Bloch-Floquet trajectories are observed in the frequency-angle spectrum and are explained using a simplified interference model.
... Added mass and radiation effects related to the surrounding exterior fluid medium were taken into account, but the radiated pressure in the fluid was not estimated. Photiadis et al. have published numerical and experimental results concerning scattering from a ribbed cylindrical shell [16][17][18][19][20][21]. The model of an infinite ribbed cylindrical shell is based on the Bloch-Floquet theorem in [16]. ...
... The model of an infinite ribbed cylindrical shell is based on the Bloch-Floquet theorem in [16]. They also analysed the effect of an irregularly ribbed fluid loaded cylindrical shell using an axisymmetric finite element / infinite element model [20]. Calculations were achieved for a frequency range up to three times the ring frequency. ...
This paper describes the development of a numerical model to predict the vibro-acoustic behavior of an externally fluid loaded shell with non-uniformly space stiffeners and transversal bulkheads. This model constitutes an extension of the existing semi-analytic capability in predicting the acoustics of axisymmetric structures. It is based on the circumferential admittance approach (CAA) which consists in substructuring the problem so that the fluid loaded shell constitutes one subsystem and the frames constitute other independent subsystems. These subsystems are coupled together by assembling the circumferential admittances that characterize each uncoupled subsystem. Different numerical approaches can be used to estimate these admittances. The standard finite element code is well adapted for evaluating the admittances of the internal frames whatever their cross-section geometries and material properties. Classical discretization methods such as finite elements and boundary elements are too time-consuming for the fluid loaded shell. To avoid this obstacle, three different approaches with different degrees of approximation are proposed to estimate the shell admittances. Comparisons with a reference case are proposed to evaluate the accuracy and the efficiency of each of these three approaches. With the optimal approach, CAA gives very good results in satisfactory computing time. It is well-adapted for analyzing the behavior of a submarine pressure hull in a wide frequency range of interest.
... Added mass and radiation effects related to the surrounding exterior fluid medium were taken into account but the radiated pressure in the fluid was not estimated. Photiadis et al. [13][14][15][16] analysed the effect of irregularly ribbed fluid loaded cylindrical shell using an axisymmetric finite element/infinite element model. Calculations were achieved for a frequency range up to three times the ring frequency. ...
... Calculations related to the radiation of a flat plate are easier than those of a cylindrical shell. Moreover, idealisation of the submarine hull as flat plate, periodically stiffened by identical ribs, can provide a feeling for the physical involved, can give reasonable quantitative predictions and can help the understanding of more complicate models like those considering cylindrical shell [6][7][8][9][11][12][13][14][15][16]. ...
An efficient technique to estimate the vibration of and sound radiation from a stiffened fluid-loaded plate excited by a mechanical point force is presented. The process is based on the formulation in the wavenumber domain of the transversal displacement of the plate and of the acoustic pressure in the fluid domain. Analytical expressions can be easily obtained for these fields in the case of an infinite plate with stiffeners uniformly spaced. The approach consists then on using the discrete Fourier Transform to deduce the associated fields in the physical space. This transform necessitates truncation of the wavenumber domain and sampling of these wavenumber fields with adequate criterions to avoid loss of information and aliasing. These criterions are discussed in this paper concerning displacement point results on the plate and pressure point results in a part of the fluid domain “close to” the plate. This technique requires relatively small computing time and provides an efficient tool to analyse the vibro-acoustic behaviour of the stiffened plate, both, in the physical and the wavenumber spaces. A comparison between a case discussed in the literature and some examples are presented to illustrate the validity and interests of this approach.
... Added mass and radiation effects related to the surrounding exterior fluid medium were taken into account but the radiated pressure in the fluid was not estimated. Photiadis et al. [13][14][15][16] analysed the effect of irregularly ribbed fluid loaded cylindrical shell using an axisymmetric finite element/infinite element model. Calculations were achieved for a frequency range up to three times the ring frequency. ...
... Calculations related to the radiation of a flat plate are easier than those of a cylindrical shell. Moreover, idealisation of the submarine hull as flat plate, periodically stiffened by identical ribs, can provide a feeling for the physical involved, can give reasonable quantitative predictions and can help the understanding of more complicate models like those considering cylindrical shell [6][7][8][9][11][12][13][14][15][16]. ...
... Finally, we observe that we have a complete description of the response of a ribbed fluid-loaded elastic plate or membrane in the time-harmonic steady state. In passing, we have derived a means to produce similar results in any one-dimensional wavebearing system, and we see that this work could be trivially extended to consider, say, a ribbed elastic cylinder (Peake 1997;Photiadis & Houston 1999). ...
The problem of a ribbed membrane or plate submerged in a fluid with mean flow is studied. We first derive a method which can be used to reduce this, and similar problems to a band matrix inversion. We then find the pass and stop band structure found in the case of static fluid persists when a mean flow is introduced, and we give an explanation in terms of the eigenvalues of the transfer matrix of the system. We then study disordered structures and observe the phenomenon of Anderson localization. In some parameter régimes the addition of disorder causes significant delocalization.
... The confinement of vibration energy in vibrating structures has been studied deeply both for linear and nonlinear systems for slender periodic structures. In this field the phenomenon is called localization, see for example [34], [35] and [19]. One intentional form of building up one or various local modes, which significantly extend the usable frequency range in loudspeakers, is the use of a double cone or a whizzer. ...
The most important modes for a direct acoustic radiator are the axial modes, which are axisymmetric circular modes of a high temporal and spatial coherence [38]. Numeric modal analysis and measurement of the free and forced accelerations and displacement responses of the moving assemblies are performed to establish the main modes involved in the acoustic response. The axial modes had been identified by measurements (within the intrinsic degree of uncertainty). The experiences show evidence of clearly nonlinear normal modes (NNM) [18] and [19], justifying the high complexity of mode finding in loudspeaker cones. Based on the axial modes, a three degrees of freedom model is proposed, where only one of the masses is externally forced. The modal analysis of a double cone speaker has been treated in short form.
