Phillip B. Abraham

A Poynting vector formulation for thin shells and plates, and its application to structural intensity analysis and source localization. Part I: Theory

This paper deals with a formulation of the Poynting vector (structural intensity) for thin shells and plates, and its application to structural intensity analysis and source localization. The procedure begins with the insertion of a Taylor series expansion of the displacement components (about the middle surface of the shell) directly into the three‐dimensional representation of the Poynting vector. From this representation, an average power flow per unit length, or equivalently an intensity resultant, is derived, whose form permits expressibility in terms of force and moment resultants. The corresponding equations of continuity for energy are derived for both body and surface forces, and the time integral of the net outflow is developed, yielding a technique for source localization. This technique offers a method for the determination of the structural intensity of thin, elastic shells and plates, and is successful for source localization.

Determination and analysis of guided wave propagation using magnetic resonance elastography

We present a novel extension of standard magnetic resonance elastography (MRE) measurement and analysis methods, which is applicable in cases where the medium is characterized by waveguides or fiber bundles (i.e., muscle) leading to constrained propagation of elastic waves. As a demonstration of this new method, MRI is utilized to identify the pathways of the individual fibers of a stalk of celery, and 3D MRE is then performed throughout the volume containing the celery fibers for a measurement of the displacements. A Helmholtz decomposition is performed permitting a separation of the displacements into longitudinal and transverse components, and an application of a hybrid Radon transform permits a spectral decomposition of wave propagation along the fibers. Dot product projections between these elastic displacements measured in the global coordinate system and three Frenet vectors representing the tangent and two corresponding orthogonal vectors along any particular fiber orientation yield the displacement contributions to wave propagation along the fiber as if it were a waveguide. A sliding window spatial Fourier transform is then performed along the length of each fiber to obtain dispersion images that portray space–wavenumber profiles. Therefore, this method can permit localized tracking and characterization of wave types, velocities, and coupling along arbitrarily oriented fibers. Magn Reson Med, 2005. Published 2005 Wiley‐Liss, Inc.

Estimation of source motion from time delay and time compression measurements

Expressions are derived for estimating the cross line‐of‐sight speed (CLSS) component of velocity from relative time compressions (RTC’s) and time delays. Theoretical expressions for the rms error in radial speed and CLSS are given and have been corroborated by simulation experiments. These expressions show the relative difficulty of estimating radial speed information as compared with CLSS.

Detection and localization of inclusions in plates using inversion of point actuated surface displacements

A numerical simulation is carried out demonstrating the use of plate surface vibration measurements for detecting and locating inclusions within the structure. A finite element code is used to calculate normal surface displacement for both steel and mortar plates subjected to a monochromatic point force. The data is generated for the homogeneous plate and the identical plate within which exists a small rectangular inclusion. It is observed that when the elastic modulus of the inclusion is orders of magnitude lower than the base material, resonances of the inclusion produce large local displacements that are readily observed in the raw displacement data. For more modest moduli differences, there are no such directly observable effects. In this case, three inverse algorithms are used to process the displacement data. The first two are local inversion techniques that each yield a spatial map of the elastic modulus normalized by density. These algorithms successfully detect and localize the inclusion based on its modulus difference from that of the base plate. The third technique uses a form of the inhomogeneous equation of motion to obtain the induced force distribution connected with the inclusion. The spatial mapping of this force also successfully detects and localizes the inclusion.

Detection and localization of rib detachment in thin metal and composite plates by inversion of laser Doppler vibrometry scans

The laboratory implementation of a fault detection and localization method based on inversion of dynamic surface displacements measured by a scanned laser Doppler vibrometer (SLDV) was investigated. The technique uses flexural wave and generalized force inversion algorithms which have previously been demonstrated using simulated noise-free vibration data generated for thick plates with a finite element model. Here these inversion algorithms to SLDV measurements made in the laboratory on a thin nickel plate and a thin carbon fiber composite plate, both having attached reinforcing ribs with intentional de-bonding of the rib/plate interface at a specific location on each structure are applied. The inverted displacement maps clearly detect and locate the detachment, whereas direct observation of the surface displacements does not. It is shown that the technique is relatively robust to the choice of frequency and to the presence of noise.

Inversion methods for the detection and localization of inclusions in structures utilizing dynamic surface displacements

In this paper, we investigate the feasibility of both detecting and localizing inclusions in structures given a knowledge of dynamic surface displacements. Provided with such displacement information, the equations of motion are utilized to estimate local material parameters through inversion, as well as to indicate the locations of inclusions using a novel generalized force mapping technique. Using a finite element code, numerical simulations were carried out for the determination of the normal surface displacements for both steel and mortar rectangular plates subject to monochromatic point actuation. The data is generated for both homogeneous plates and inhomogeneous plates within which a small rectangular inclusion of differing material parameters is present, and three algorithms are applied to the calculated displacement data. The first two are local inversion techniques which provide a spatial map of the elastic modulus normalized by density, while the third technique utilizes the inhomogeneous form of the equations of motion to obtain an induced force distribution caused by the inclusion. It will be demonstrated that the algorithms can both detect and locate inclusions in structures even when the materail parameter difference of the inclusions and the background medium is relatively low.

Fault detection and localization using measured surface vibration and local inversion

We present progress we have made in developing a structural acoustic-based methodology allowing interior fault detection and localization in plate-like structures using only processed vibration data readily available on the structures surface. Our methods use measurements of surface displacement associated with vibration of the structure caused by externally applied forces. These forces can be created simply by a local actuator in direct contact with the structure or in some cases by an incident airborne acoustic wave. The measured normal surface displacements, uz(x, y), are then inverted locally using various mathematically optimized algorithms in order to obtain a desired material parameter, for example, the elastic modulus, whose spatial variation then serves to detect and localize the fault. This structural acoustic approach is not limited to any particular length scale requiring only that the structure be mechanically excited at frequencies for which the structural wavelength is within an order of magnitude of the fault dimension and that the dynamic surface displacements be mapped with a spatial resolution smaller than the fault size. We present the results of deploying the structural acoustic technique in the US Capitol Building to locate faults within plaster walls and ceilings bearing large expanses of precious nineteenth century frescoes, in composite airframe skins in laboratory experiments to detect and locate de-bonding of thin (~1mm) stiffeners and frames, and in micro-structures to detect and locate faults in silicon micro-oscillators and their supporting structures with resolutions approaching 1μm.

Conformal Fourier wavenumber decompositions on continuous differentiable surfaces.

In this manuscript, a method is introduced for the evaluation of Fourier wavenumber decompositions on C(1) vibrating surfaces for spatial-spectral analysis. Whereas typical Fourier analysis is restricted to geometries that are separable for meaningful interpretations of the corresponding wave motion, this approach allows for conformal spectral analysis along curved surfaces. This is accomplished by restricting the wavevectors to lie within the local tangent to the surface and to be spatially dependent. The theoretical development is presented and it is demonstrated that commonly utilized kernels appropriate for some simple geometries can be recovered. Additionally, this approach is applied in the analysis of the vibration and radiation of a point driven, fluid loaded cone, where the displacements and pressures have been obtained using the finite element method.

The Poynting vector and source identification for thin plates and shells

The Poynting vector I, well known in the three‐dimensional theory of elasticity, represents the three‐dimensional power flow in a solid. It is shown how I can be used to derive the power flow in a thin flat plate, and a thin cylindrical shell. The divergence of I provides the rate of change of total energy density (the Hamiltonian density,∂H/∂t) in the shell. Its time average must vanish for steady‐state conditions. This leads to the conclusion that power flow in and out of the structure is given by Π = ∮〈I2D〉t n dl, a line integral (over the shell surface) of the in‐plane, average intensity in the shell. If the shell is point driven, then Π is large at that point, and can be used to locate the position and strength of the mechanical source. This provides an excellent tool for mechanical source identification. These concepts will be demonstrated and verified using synthetic and real data on point driven cylindrical shells in water. In the synthetic data, the full displacement vector (u,v,w) is specified ...

Reflection tomography imaging of submerged objects

In a continuing study of acoustic reflection tomography [C. F. Gaumond and P. B. Abraham, J. Acous. Soc. Am. Suppl. 1 78, S80 (1985)], results obtained from synthetic data for prolate spheroids will be presented. These will be compared with similar results for various spherical scatterers. This novel method of image reconstruction of strong scatterers will be briefly reviewed.