Pei‐Tai Chen

Complex power, reciprocity, and radiation modes for submerged bodies

This study of the surface interaction between a submerged body and the surrounding fluid begins by developing reciprocity relations between alternative pressure and normal velocity distributions on the wetted surface. A corollary of these principles is proof of the symmetry of the matrix representing the acoustic contribution to the structural impedance, even in situations where the acoustic relation between surface pressure and normal velocity is not symmetric. The reciprocity properties lead to two eigenvalue problems, whose solution yields velocity and pressure radiation modes, each of which decouples the complex surface acoustic power. The matrices required to obtain the eigensolutions are shown to arise in the ordinary course of modeling fluid–structure interaction. Further analysis reveals that the velocity and pressure modes occur in a one‐to‐one correspondence, with a relative phase angle that decreases monotonically as the radiated power increases relative to the reactive power. Using the radiati...

Analysis using variational principles of the surface pressure and displacement along an axisymmetrically excited disk in a baffle

A discussion of the singularities that arise in a variational principle for the surface pressure resulting from a specified harmonic motion of an arbitrary surface leads to the observation that essentially the same formulation may be employed to study a thin disk in an annular baffle and a disk in an infinite baffle. This variational principle is implemented jointly with Hamilton’s principle for structural displacement to study the response of an axisymmetrically excited flexible disk (membrane or elastic plate) in a baffle. Surface pressure, as well as displacement, are represented in a series of assumed modal functions. The system equations reduce to a set of simultaneous equations for the complex amplitudes of the assumed modes, in which the inhomogeneous terms are the modal generalized forces, whereas the coefficients of the homogeneous terms arise from the isolated responses of the fluid and structural media, and from the fluid–structure coupling. The coefficients associated with surface pressure are...

Variational formulation of acoustic radiation from submerged spheroidal shells

The surface variational principle (SVP) governing acoustic interaction between a vibrating surface and a surrounding fluid is combined with the dynamic equations governing response of a shell of revolution subjected to axisymmetric harmonic excitation. Ritz series representations are used to represent the spatial dependence of the surface pressure and shell displacement components. Two formulations are presented, with the difference being whether an intermediate computation of the in‐vacuo modes is performed. The direct approach, in which the Ritz coefficients are determined directly from the coupled formulation, is used to validate the SVP approach for the case of a spherical shell, whose response is known analytically. For the case of a slender spheroidal shell, the direct approach is compared to the results obtained from the modal approach, in which a truncated set of in‐vacuo modes forms the basis functions representing displacement. The aspect ratio and shell properties for this evaluation are select...

Modal properties and eigenvalue veering phenomena in the axisymmetric vibration of spheroidal shells

The axisymmetric free vibration properties of arbitrarily slender thin spheroidal shells are investigated by implementing the method of assumed modes in conjunction with energy functionals derived from classical linear bending theory. Loci depicting the dependence of the natural frequencies on aspect ratio are constructed for aspect ratios ranging from a sphere to a prolate spheroid whose length is seven times its diameter. At certain aspect ratios, loci associated with different eigensolutions come close, then veer away without intersecting. Modal properties, such as the number of nodes in the mode associated with increasing root number for the eigenvalue, are found to change irregularly when veering occurs. An analysis of the partitioning of strain energy between membrane and bending effects is performed in conjunction with an earlier general study of eigenvalue veering phenomena. The analysis demonstrates that veering results in mixing of membrane and bending effects, such that it no longer is meaningf...

VARIATIONAL FORMULATION OF INTERIOR CAVITY FREQUENCIES FOR SPHEROIDAL BODIES

This paper presents interior cavity frequencies for spheroidal bodies in cases of Neumann and Dirichlet homogenous boundary conditions. The Helmholtz wave equation is expressed in terms of prolate spheroidal coordinates from which a variational formulation is derived for the determination of the interior cavity frequencies. The search of stationarity is performed by means of a Rayleigh–Ritz type expansion of trial functions. The trial functions are expressed as a double summation of basis functions in radial and angular coordinates. A variable transformation is applied to the variational form in order to have homogenous boundary conditions, which are essential for establishing a complete function space of the associated boundary conditions. The present analysis is verified by comparisons of interior frequencies of a spherical body, of prolate spheroidal bodies, and of the interior characteristic functions with the radial and angular functions obtained from prolate spheroidal differential equations. Interi...

Responses of partially immersed elastic structures using a symmetric formulation for coupled boundary element and finite element methods

Using a coupled BEM/FEM, this work describes a numerical method to compute the response and acoustic radiation for structures partially immersed in fluid. The structures and their responses are assumed to be symmetric about a symmetric plane. A symmetric complex matrix derived from the BEM and a reciprocal principle for surface acoustics is also used to represent the acoustic loading against the structures. In addition, selecting a proper Greens function based on image source method satisfies the boundary conditions of pressure release on the fluid surface and null normal velocity on the symmetric plane. Moreover, a boundary integral equation emerges when the field point approaches the structural surface where the normal derivative of the Greens function over partial, infinitesimal spheres is evaluated. These limiting values depend on locations of the field point on the surface. Owing to the symmetry of the acoustic loading matrix, the matrix for the coupled BEM/FEM is a banded, symmetric one, thereby allowing us to employ a variable banded storage method and invert of the matrix. Doing so markedly increases computational efficiency. Furthermore, an analytical solution of a spherical thin shell with the lower semi-sphere immersed in water is carried out by characteristic function expansions for shell equation and acoustic loading. These analytical solutions compare with the results obtained from the proposed numerical method. A good correlation for low frequencies is obtained and minor discrepancies are observed with an increasing frequency.

Surface micromachined capacitive ultrasonic transducer for underwater imaging

Abstract This study presents the primary design, fabrication process and device measurement of a Capacitive Micromachined Ultrasonic Transducer (CMUT) for underwater acoustic imaging. Theoretical analysis and computer simulations of the CMUT are performed. The CMUT fabrication uses the full surface micromachining techniques of the Micro Electro Mechanical System (MEMS). These techniques are Low Pressure Chemical Vapor Deposition (LPCVD), photolithography, Reactive Ion Etching System (RIE) dry etching, sacrificial layer wet etching, metal thermal evaporation coating and Plasma‐Enhanced Chemical Vapor Deposition (PECVD). Several important issues regarding fabrication are discussed. The measured input impedance of the CMUT is in agreement with the theoretical prediction. The received signal has a 35 dB signal‐to‐noise ratio indicating that practical applications of the immersion CMUT are feasible and that the radiation pattern measurement of the CMUT array has good beamforming characteristics for underwater imaging.

Elucidation of the relationship between complex acoustic power and radiation efficiency for vibrating bodies

This investigation examines the physical meaning of surface complex acoustic power and its relationship to acoustic radiation efficiency. It is shown that the radiated power is the power radiating out of a far-field surface where the plane wave relationship between pressure and particle velocity holds. Meanwhile, the reactive power pertains to the difference between kinetic energy and potential energy. A stationary condition of the ratio between the radiated power to the reactive power yields an eigenvalue problem, subsequently decomposing the surface acoustics into a modal representation. Doing so further allows the examination of the relationship between acoustic radiation efficiency and power factor of the complex power. According to the results, the radiation efficiency of the first radiation mode is nearly equal to the square of the first modal power factor. The modes beyond the first of the modal radiation efficiencies are relatively larger than the corresponding squared modal power factors. Numeric...

A symmetrical formulation of coupled BEM/FEM for submerged structures based on acoustical reciprocity

The paper presents a structural acoustics formulation for elastic structures submerged in water. The structural equation is described by a finite element method where the linear displacement variables on the wetted surface are chosen locally as one normal displacement and the other two displacements tangent to the wetted surface, whereas the rest of rotational displacements or degrees of freedom not contacting with water are defined globally. A boundary element formulation describing the acoustic loading on the structure is expressed as a function of normal velocity or displacement on the wetted surface. The coupling of the FEM and BEM is through the normal velocity, or equivalently, the normal displacement on the wetted surface. An acoustical reciprocity is used to prove that the associated acoustic loading expressed as the normal displacement of the wetted structure is a complex symmetric matrix. This matrix can be viewed an acoustic element whose degrees of freedom are the normal displacements of the wetted surface. Thus, the coupled FEM and BEM becomes a symmetric banded matrix formulation, leading to an efficient numerical way to solve the equation. A capped cylindrical shell with periodical ring stiffeners and bulkheads submerged in water is used to demonstrate the present numerical method.The paper presents a structural acoustics formulation for elastic structures submerged in water. The structural equation is described by a finite element method where the linear displacement variables on the wetted surface are chosen locally as one normal displacement and the other two displacements tangent to the wetted surface, whereas the rest of rotational displacements or degrees of freedom not contacting with water are defined globally. A boundary element formulation describing the acoustic loading on the structure is expressed as a function of normal velocity or displacement on the wetted surface. The coupling of the FEM and BEM is through the normal velocity, or equivalently, the normal displacement on the wetted surface. An acoustical reciprocity is used to prove that the associated acoustic loading expressed as the normal displacement of the wetted structure is a complex symmetric matrix. This matrix can be viewed an acoustic element whose degrees of freedom are the normal displacements of the w...

On the study of vibrational interactions of an internal substructure with a main structure submerged in water and its acoustic radiations using admittance approach

It is important and practical to design an internal substructure for supporting machines which generates vibration sources for a main submerged stiffened shell structure. Vibration propagates from the internal supporting structure to the wetted shell structure, thus radiating acoustic energy into water. The present study can be divided into two categories: 1. the main wetted structure, including stiffeners and bulkheads, etc., radiates acoustic energy into water subject to forces which is the junction interacting forces arisen from the vibrating machine, 2. the interaction force pertains to the coupling dynamic characteristics between wetted structure and the internal structure. An admittance approach is adopted to characterize individual structures and the coupled equation is derived by continuity of displacement variables and equaling forces with opposite sign at the junction of connected structures. Admittance matrices are computed by using junction forces between the structures whereas the responded displacements at the junctions are the elements of the admittance matrices. The coupled admittance equation is complex symmetric matrix. An eigenvalue analysis is performed to investigate the interaction junction forces, accordingly, the radiation characteristics of the coupled main wetted structure under fluid loading in connection with supporting internal structures.It is important and practical to design an internal substructure for supporting machines which generates vibration sources for a main submerged stiffened shell structure. Vibration propagates from the internal supporting structure to the wetted shell structure, thus radiating acoustic energy into water. The present study can be divided into two categories: 1. the main wetted structure, including stiffeners and bulkheads, etc., radiates acoustic energy into water subject to forces which is the junction interacting forces arisen from the vibrating machine, 2. the interaction force pertains to the coupling dynamic characteristics between wetted structure and the internal structure. An admittance approach is adopted to characterize individual structures and the coupled equation is derived by continuity of displacement variables and equaling forces with opposite sign at the junction of connected structures. Admittance matrices are computed by using junction forces between the structures whereas the responded d...