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Aerospace Engineering

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#### Modeling of ultrasonic scattering experiments with applications to system and transducer characterization

The voltage signal output by the receiver electronics, which represents the observable quantity in an ultrasonic scattering experiment, is written as a product, in the frequency domain, of two factors: the system efficiency and the scattering coefficient. The system efficiency represents the combined electrical properties of both the generator and receiver electronics and is a function of frequency only. The scattering coefficient represents the acoustic nature of the experiment (the radiation, propagation, scattering and reception of ultrasonic waves) and depends on the distributed field properties of the transducers involved and their locations and orientations, on the number and type of scattering obstacles and their locations and orientations, on the acoustic properties of the media through which the waves travel, and on the nature and shape of any interfaces through which the waves pass. Based on a generalized principle of electroacoustic reciprocity, formulae are developed for the evaluation of the scattering coefficient. The most general of these involve an integration over either the volume or the surface of the scattering obstacle. More specific formulae are also developed which express the scattering coefficient in terms of either the spherical wave transition matrix or the plane wave scattering amplitude of the obstacle;In order to demonstrate the use of the formulae developed, the calculation of the scattering coefficient is considered for two common ultrasonic scattering experiments. The first experiment involves the pulse-echo scattering from an infinite, flat elastic plate immersed in water. This arrangement is often used for the measurement of the velocity and attenuation of elastic waves, and also as a reference experiment for the determination of the system efficiency. The second experiment involves the pulse-echo scattering from an elastic sphere immersed in water. Particular attention is given to the specular reflection component of the scattering, which is demonstrated to be approximately equivalent to a point measurement of the pressure field radiated by the transducer. This approximation is subsequently used as the basis for obtaining experimental data for transducer characterization. The characterization itself is based on expanding in a set of basis functions, each weighted by an unknown coefficient, the normal velocity profile across the plane flush with the face of the probe. Values for the coefficients are obtained by determining the best fit between the experimental pressure data and the pressure calculated from the assumed velocity profile. Results are presented for two commercially manufactured immersion transducers, one planar (unfocused) and the other focused.

#### Determination of secondary sources in noise cancellation with boundary element method

The direct boundary element method is proposed in this thesis to solve acoustic radiation problems as well as to achieve regional noise cancellation in half space with uniform finite impedance over the surface. The boundary integral equation and half space Green's function were derived to accomplish these goals. Those formulations were verified by comparing numerical simulations with theoretical solutions as well as experimental results. In addition, the above formulations were extended to achieve regional noise cancellation in half space by applying the boundary element method;Two methods were investigated to obtain noise cancellation in desired regions. They are the iterative control method and the coupled equation method. A set of Fortran programs including discretizing of geometries, incorporating boundary integral equations, and accommodating the noise cancellation technique were developed. Various problems concerning ill-conditioned matrices in numerical simulation and practical application of noise cancellation technique were discussed as well. Finally, data banks for various configurations of sound sources were set up for quick reference of the locations and driving functions of secondary sources. Thus, noise reduction in a designated area is shown to be feasible;A 6" speaker was used to simulate a noise source with uniform surface velocity. In addition, a ribbed aluminum plate with the dimension 71.12cm x 60cm was used to simulate a noise source with variable surface velocity. Four 10" speakers were used as secondary sources to achieve noise reduction in desired regions at certain frequencies. A multi-channel digital/analog converter was used in order to control desired driving functions for each individual secondary sources. The computer-controlled scanning system including a 2-channel controller, 2-D scanner, and stepping motors was used to place a quarter-inch microphone at certain locations. The acoustic pressure on a 120cm by 120cm plane at various distances above the source plane was measured. A Bruel and Kjaer model 2032 FFT analyzer was used to acquire and process signals from the microphone. The experimental results agreed well with numerical simulations. This indicated that the proposed noise cancellation technique attenuated the acoustic noise level successfully.

#### Vibrational and acoustic response of ribbed plates

A self-consistent mathematical formulation, using the Fourier transform method and a direct Gaussian numerical integration scheme, is developed and verified for analysis of both vibrational and acoustic responses of infinite submerged ribbed plates. Further steps developed from standard theories make structural intensity, acoustic intensity, and acoustic power calculations possible in the nearfield and farfield, and are demonstrated in this work;The direct numerical integration scheme adopted to obtain responses has proved to be straightforward and reliable. Although the double integration expression in some responses makes the technique infeasible, a practical way to overcome that difficulty is demonstrated using a standard branch-cut integration to eliminate one integration step analytically. The model and numerical scheme readily allow investigation of additional interesting topics, like the passband and stopband characteristic and the mode localization phenomenon that are observed in ribbed structures. Furthermore, an extension to comprehension of the mechanisms that generate the mode localization phenomenon on disordered structures has been realized;A secondary effort examines natural modes of vibration and acoustic radiation for finite stiffened multiple-span beams with the efficient transfer matrix method. This model shows that the mode localization phenomenon exists on disordered stiffened beams both under free-free and hinged-hinged end conditions. The sensitivity of the response to attachment disorder (perturbations in rib stiffness and location) has also been examined. An elaborate vibrational and acoustic experiment has been carried out on a baffled, stiffened, two-span, hinged beam to examine the existence of the localized modes and verify the predicted acoustic responses. Moreover, the radiation efficiency of finite beams has been investigated for comparison of the radiation behavior presented by the different stiffened beam arrangements;A thorough investigation of mode localization, frequency passbands and stopbands, structural and acoustic intensities and radiated acoustic power is presented for analysis of submerged infinite ribbed plates, with variable rib materials geometry and spacing (periodic and non-periodic). A second investigation of localized natural modes is demonstrated for analysis and experiment of finite stiffened beams in air.

#### Propagation of elastic waves through inhomogeneous, anisotropic materials

The recent advances in the field of ultrasonic nondestructive inspection have demanded a more precise knowledge of the propagation of sound beams within materials. Quantitative models for propagation of ultrasonic fields from a transducer can be used as tools to both analyze the inspection data and to predict inspection results. An approximate Gauss-Hermite model was developed a few years ago to predict the ultrasonic fields radiated into isotropic and anisotropic materials, through planar or simply curved interfaces by focused or unfocused transducers. The paraxial approximation used in the development of that model has allowed many functions to be evaluated analytically, thus, the model has the advantage of being computationally efficient. The Gauss-Hermite model has already been tested and verified for beam propagation through a single medium. However, there have been concerns about the performance of the model in multilayered media, and about the overall accuracy of the Gauss-Hermite solution. In the papers contained in this dissertation, more intense studies are presented evaluating the validity of the Gauss-Hermite model in a multilayered medium. First, a method of finding the elastic constants of anisotropic materials is presented. These elastic constants are important inputs to the Gauss-Hermite model. To validate the Gauss-Hermite beam model in multilayered media, comparisons have been made between its predictions and those found by the finite element method, which provides a more exact solution. The structures considered combine both isotropic and anisotropic layers. The comparison of the model with the finite element method showed good agreement around the central ray direction. Due to the use of the Fresnel approximation in the solution, the accuracy of the Gauss-Hermite model degrades as one moves away from the forward propagation direction. However, since most of the energy is concentrated in the vicinity of the central ray, which is the portion of the beam used in most NDE inspections, this problem does not appear to be too severe. To demonstrate the application of the Gauss-Hermite model to a real problem, it was used for sizing delaminations in a multilayered composite structure. This method, which utilizes the reflected signal from the delamination, showed considerable improvement over the traditional sizing method. Finally, the model in conjunction with a local ray tracing procedure, was also used to predict the ultrasonic field beyond an irregularly shaped interface (step discontinuity). The outcome of this study demonstrated a good agreement between the model predictions and experiment. Overall, this model has a good potential for being used in a variety of ultrasonic inspections.

#### Design, analysis, and modeling of giant magnetostrictuve transducers

The increased use of giant magnetostrictive, Terfenol-D transducers in a wide variety of applications has led to a need for greater understanding of the materials performance. This dissertation attempts to add to the Terfenol-D transducer body of knowledge by providing an in-depth analysis and modeling of an experimental transducer. A description of the magnetostriction process related to Terfenol-D includes a discussion of material properties, production methods, and the effect of mechanical stress, magnetization, and temperature on the material performance. The understanding of the Terfenol-D material performance provides the basis for an analysis of the performance of a Terfenol-D transducer. Issues related to the design and utilization of the Terfenol-D material in the transducers are considered, including the magnetic circuit, application of mechanical prestress, and tuning of the mechanical resonance. Experimental results from two broadband, Tonpilz design transducers show the effects of operating conditions (prestress, magnetic bias, AC magnetization amplitude, and frequency) on performance. In an effort to understand and utlilize the rich performance space described by the experimental results a variety of models are considered. An overview of models applicable to Terfenol-D and Terfenol-D transducers is provided, including a discussion of modeling criteria. The Jiles-Atherton model of ferromagnetic hysteresis is employed to describe the quasi-static transducer performance. This model requires the estimation of only six physically-based parameters to accurately simulate performance. The model is shown to be robust with respect to model parameters over a range of mechanical prestress, magnetic biases, and AC magnetic field amplitudes, allowing predictive capability within these ranges. An additional model, based on electroacoustics theory, explains trends in the frequency domain and facilitates an analysis of efficiency based on impedance and admittance analysis. Results and discussion explain the importance of the resonance of the electromechanical system, as distinct from the mechanical resonance. Conclusions are drawn based on the experimental work, transducer analysis, and modeling approaches.

#### Scattering investigation based on acoustical holography

The objective of this research is to investigate sound scattering by an object using a two-surface measurement technique that separates the incident field and the scattered field. The separation technique is developed in cartesian and cylindrical coordinates. The decomposition method in the cartesian coordinate system is based on the principle that any wave form can be decomposed into plane-wave components by using a two dimensional spatial Fourier transform. For the cylindrical coordinate system, a two plane separation technique is based on decomposing the sound field into cylindrical waves. Numerical simulations are performed and the effect of various parameters are investigated. Specifically, the distance between two measurement surfaces, the distance between measurement points, and the aperture size are investigated. In addition, experimental studies were conducted inside an anechoic chamber with a baffled loudspeaker as a source, illuminating four different scatterers. The decomposed scattered field is then used to estimate the far-field target strength. The experiments demonstrate the feasibility of the field separation technique.

#### Minimizing the radiated sound power with minimal structural modification

In cases where a structure's excitation cannot be altered, the sound radiation from the structure must be minimized by modifying the structure within given design constraints. This dissertation considers minimizing sound radiated by an existing structure with minimal changes to the structure itself. To accomplish this the sound power radiated by a structure was written as a function of the normal surface velocity using the boundary element method. The feasibility of reducing radiated sound power with small patches of constrained layer damping material is proved. Small patches of constrained layer damping material are placed on the structure to effectively reduce radiated sound using reactive structural shearing intensity. The size, shape, and number of patches is explored. Two gradient methods are then used to minimize sound power radiated by a structure as a function of the area covered by constrained layer damping. The method of simulated annealing was used to minimized sound power as a function of damping patch area in cases where gradient methods proved unsuccessful. Reductions in sound power radiated at a single frequency of over 10 dB were achieved by covering just 1.1 percent of the total structural surface area with constrained layer damping material.

#### The effect of rib attachment properties on structural acoustics

Sound Pressure measurements were taken of several 2.44m long thin beams which had 0.635cm wide t-ribs centered on them. The beams were excited by a chirp signal with the frequency range of 500Hz to 1500Hz. Two rib attachment methods were used on the beams. One group of beams were machined out of thick stock to the geometry of a rib welded to a beam to ensure that the rib and attachment were one continuous media. The other group of beams had ribs that were welded on. Fillets from the welding process were subjected to heat-treatment and machining to determine the effect of static stress in fillets. Farfield sound radiation from the beam and phase speeds of waves propagating through the beam were used to investigate the effects of the rib and its attachment properties on the beam response;The experimental results showed that the geometry and stress state of the attachment are the main parameters that alter wave propagation and sound radiation. Also, the maximum sound radiation from the rib was not centered directly over the rib, but rather ahead of the rib location. Furthermore, substantial phase speed increases were observed around the shaker location and the rib location;Attachment geometry and stress information were incorporated into a Euler-Bernoulli wave-based model. This wave-based model reproduced the static stress effect on the farfield sound radiation, but didn't reproduce the position of the sound radiation from the rib nor the experimental phase speed increases around the shaker or rib locations;An energy-based model was derived that included the geometry of the rib and attachment. The model used the Extended Hamilton's Principle. Cubic spline weighting functions were applied via Galerkin's method of weighted residuals. This energy-based model did reproduce both the position and the magnitude of the peak in the sound radiation field from the rib. It also reproduced the phase speed increases around the shaker and the rib locations.

#### Nonlinear and hysteretic magnetomechanical model for magnetostrictive transducers

The growing interest on magnetostrictive materials for generation of strains and forces in smart structure systems motivates the development of increasingly accurate models of the performance of these materials as used in transducers. The proposed magnetomechanical model provides a characterization of the material magnetization as well as the strain and force output by a transducer in response to quasistatic applied magnetic fields. The model is built in three steps. In the first, the mean field model for ferromagnetic hysteresis originally developed by Jiles and Atherton is used to compute the magnetization arising from the application of magnetic fields. While this model provides an accurate characterization of the field-induced magnetization at constant stress, it is insufficient in cases where the stress state of the magnetostrictive driver varies significantly during operation. To model the stress-induced magnetization changes, or magnetomechanical effect, a 'law of approach' to the anhysteretic magnetization is considered. The magnetization hysteresis model in combination with this law of approach provides a more realistic representation of the bidirectional energy transduction taking place in magnetostrictive transducers. In the second step, an even-term series expansion posed in terms of the magnetization is employed to calculate the magnetostriction associated with magnetic moment rotations within domains. While the magnetostriction provides a good description of the total material strain at the low field levels where elastic dynamics are of secondary significance, it is highly inaccurate at higher drive levels, in which the elastic response gains significance. This elastic or material response is considered in the third and last step, by means of force balancing in the form of a PDE system with magnetostrictive inputs and boundary conditions consistent with the transducer mechanical design. The solution to this PDE system provides the longitudinal displacements and corresponding strains and forces generated by the magnetostrictive driver. Since the formulation precludes analytic solution, a Galerkin discretization is employed to express the PDE in the form of a temporal system, which is subsequently solved using finite difference approximations. The ability of the model to accurately characterize the magnetomechanical behavior of magnetostrictive transducers is demonstrated via comparison of model simulations with experimental measurements collected from two Terfenol-D transducers.

#### Ultrasonic transducer characterization and transducer beam modeling for applications in nondestructive evaluation

In this work, analytic models have been developed for two different nondestructive evaluation (NDE) applications: the characterization of spherically focused ultrasonic immersion transducers, and the prediction of the radiated wave fields of a variety of commonly used ultrasonic transducers;When a spherically focused probe is correctly and completely characterized, its corresponding theoretical model can accurately predict the experimentally measured structure of its incident wave field. A new and efficient method for completely characterizing a spherically focused transducer and its accompanying measurement system is described. Predicted responses that make use of this method are shown to correspond very well to measured responses for a number of different commercial transducers;The ultrasonic beam modeling work is divided into three parts. First, the problem of a planar piston ultrasonic transducer radiating at oblique incidence through a planar fluid-solid interface is studied, and two new types of beam models representing this problem are developed--a surface integral model and a boundary diffraction wave (BDW) paraxial model. The less restrictive surface integral model is used to test the validity of the BDW paraxial model, particularly in the near field and at different angles of incidence. Second, a new edge element method has been developed for numerically evaluating a variety of ultrasonic transducer beam models. This edge element model, which uses a local Fraunhofer approximation, is used to evaluate the wave fields of a focused probe in water and of a planar probe radiating at oblique incidence to a plane fluid-solid interface. Third and finally, a complete elastodynamic model of an ultrasonic angle beam shear wave transducer is presented. This model is evaluated by the edge element method, and the various transmitted mode contributions are studied and compared to more approximate models.