A Finite Element Approach to Modelling Fractal Ultrasonic Transducers

Piezoelectric ultrasonic transducers usually employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwith device it would seem natural therefore to utilize resonators that span a range of length scales. In this article we derive a mathematical model to predict the dynamics of a fractal ultrasound transducer; the fractal in this case being the Sierpinski gasket. Expressions for the electrical and mechanical fields that are contained within this structure are expressed in terms of a finite element basis. The propagation of an ultrasonic wave in this transducer is then analyzed and used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities as a function of the driving frequency. Comparing these key performance measures to an equivalent standard (Euclidean) design shows some benefits of these fractal designs

Ultrasonic wave propagation in heterogenous media

The non-destructive testing of austenitic steel welds using ultrasound is of vital importance for assessing safety critical structures such as those found in the nuclear industry. The internal geometry of these welds is heterogeneous and highly scattering and this makes it dicult to detect and characterise any defects within them. To help overcome these diculties the use of ultrasound transducer arrays and the associated Full Matrix Capture is becoming more widespread. There is a need therefore to develop post-processing algorithms that best utilise the data from such devices. This paper considers the use of a time-frequency domain method known as the Decomposition of the Time Reversal Operator (DORT) method. To develop this method and to demonstrate its ecacy in tackling this problem a series of simulated data sets are used. The simulated data is generated using a finite element method (PZFLEX) with the heterogeneous internal microstructure of the weld being given by previous Electron Backscatter Diraction measurements. A range of artificial flaws are then inserted into this geometry. By varying the flaw size and type a comparison is conducted between the DORT method and the Total Focusing Method (TFM) and their relative ability to perform flaw detection assessed. Importantly, however, the DORT method relies on a Singular Value Decomposition in time and frequency space and this spectral information contains information about the flaw size and shape

Modelling of artefacts in estimations of particle size of needle-like particles from laser diffraction measurements

Manufacturing of particulate products across many industries relies on accurate measurements of particle size distributions in dispersions or powders. Laser diffraction (or small angle light scattering) is commonly used, usually off-line, for particle size measurements. The estimation of particle sizes by this method requires the solution of an inverse problem using a suitable scattering model that takes into account size, shape and optical properties of the particles. However, laser diffraction instruments are usually accompanied by software that employs a default scattering model for spherical particles, which is then used to solve the inverse problem even though a significant number of particulate products occur in strongly non-spherical shapes such as needles. In this work, we demonstrate that using the spherical model for the estimation of sizes of needle-like particles can lead to the appearance of artefacts in the form of multimodal populations of particles with size modes much smaller than those actually present in the sample. This effect can result in a significant under-estimation of the mean particle size and in false modes in estimated particles size distributions.Comment: 28 pages 8 figures accepted in the journal of Chemical Engineering Scienc

Vibration analysis of a circular plate in interaction with an acoustic cavity leading to extraction of structural modal parameters

When carrying out vibration health monitoring (VHM) of a structure it is usually assumed that the structure is in the absence of fluid interaction and that any environmental effects which can cause changes in natural frequency either remain constant or are negligible. In certain cases, this condition cannot be assumed and therefore it is necessary to extract values of natural frequencies of the structure for the condition with no fluid interaction from those values measured. This paper considers the case of a thin circular plate in contact with a fluid cavity giving rise to strong structural/fluid vibration interaction. The paper details the free vibration analysis of the coupled system and through consideration of modal energy, illustrates how the affined modes of vibration of the plate and the fluid can be qualitatively described. The paper then introduces a method by which the natural frequencies of the plate in the absence of fluid interaction can be obtained from those of the plate in interaction with the fluid

Harmonic analysis of lossy piezoelectric composite transducers using the plane wave expansion method

Periodic composite ultrasonic transducers oer many advantages but the periodic pillar architecture can give rise to unwanted modes of vibration which interfere with the piston like motion of the fundamental thickness mode. In this paper, viscoelastic loss is incorporated into a three dimensional plane wave expansion model (PWE) of these transducers. A comparison with experimental and nite element data is conducted and a design to damp out these lateral modes is investigated. Scaling and regularisation techniques are introduced to the PWE method to reduceill-conditioning in the large matrices which can arise. The identication of the modes of vibration is aided by examining proles of the displacements, electrical potentialand Poynting vector. The dispersive behaviour of a 2-2 composite transducer with high shear attenuation in the passive phase is examined. The model shows thatthe use of a high shear attenuation ller material improves the frequency band gap surrounding the fundamental thickness mode

Renormalisation analysis of a composite ultrasonic transducer with a fractal architecture

To ensure the safe operation of many safety critical structures such as nuclear plants, aircraft and oil pipelines, non-destructive imaging is employed using piezoelectric ultrasonic transducers. These sensors typically operate at a single frequency due to the restrictions imposed on their resonant behaviour by the use of a single length scale in the design. To allow these transducers to transmit and receive more complex signals it would seem logical to use a range of length scales in the design so that a wide range of resonating frequencies will result. In this article we derive a mathematical model to predict the dynamics of an ultrasound transducer that achieves this range of length scales by adopting a fractal architecture. In fact, the device is modelled as a graph where the nodes represent segments of the piezoelectric and polymer materials. The electrical and mechanical fields that are contained within this graph are then expressed in terms of a finite element basis. The structure of the resulting discretised equations yields to a renormalisation methodology which is used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities. A comparison with a standard design shows some benefits of these fractal designs

Reconstruction of the spatial dependency of dielectric and geometrical properties of adhesively bonded structures

An inverse problem motivated by the nondestructive testing of adhesively bonded structures used in the aircraft industry is studied. Using transmission line theory, a model is developed which, when supplied with electrical and geometrical parameters, accurately predicts the reflection coefficient associated with such structures. Particular attention is paid to modelling the connection between the structures and the equipment used to measure the reflection coefficient. The inverse problem is then studied and an optimization approach employed to recover these electrical and geometrical parameters from experimentally obtained data. In particular the approach focuses on the recovery of spatially varying geometrical parameters as this is paramount to the successful reconstruction of electrical parameters. Reconstructions of structure geometry using this method are found to be in close agreement with experimental observations

A theoretical model of an ultrasonic transducer incorporating spherical resonators

This article considers the theoretical modelling of a novel electrostatic transducer in which the backplate consists of many spherical resonators. Three analytical models are considered, each of which produce impedance profiles of the device, in addition to transmission voltage responses and reception force responses, all of which closely agree. Design parameters are then varied to investigate their influence on the resonant frequencies and other model outputs

A Composite Ultrasonic Transducer with a Fractal Architecture

To ensure the safe operation of many safety critical structures such as nuclear plants, aircraft and oil pipelines, non-destructive imaging is employed using piezoelectric ultrasonic transducers. These sensors typically operate at a single frequency due to the restrictions imposed on its resonant behaviour by the use of a single length scale in its design. To allow these transducers to transmit and receive more complex signals it would seem logical to use a range of length scales in the design so that a wide range of resonating frequencies will result. In this article we derive a mathematical model to predict the dynamics of an ultrasound transducer that achieves this range of length scales by adopting a fractal architecture. In fact, the device is modelled as a graph where the nodes represent segments of the piezoelectric and polymer materials. The electrical and mechanical fields that are contained within this graph are then expressed in terms of a finite element basis. The structure of the resulting discretised equations yields to a renormalisation methodology which is used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities. A comparison with a homogenised (standard) design shows some benefits of these fractal designs