Guided waves can propagate along thin-walled structures, such as pipes and plates, with limited energy loss, thus enabling the efficient inspection and monitoring of large structures from a limited number of sensor locations. This allows Structural Health Monitoring (SHM) using permanently installed monitoring systems with limited access and sensor requirements. However, guided wave propagation is complicated due to multiple propagating wave modes and dispersion, potentially causing the signals to become distorted. Both analytical and numerical analysis and experimental measurements are essential for guided wave research. Specialized laboratory equipment such as non-contact laser vibrometers can be expensive and unaffordable. This contribution presents a preliminary investigation on what can be achieved experimentally using low-cost sensors, suitable for research and teaching in circumstances with limited budgets. For low frequency guided wave propagation in an isotropic plate (A0 Lamb wave mode), the influence of different measurement configurations on the accuracy of group velocity quantification was investigated. Results from experimental setups employing different receive transducers were evaluated and quantified. Accuracy of sensor placement and coupling as well as measurement repeatability and Signal-to-Noise Ratio (SNR) were investigated. Accurate experimental quantification of the group velocity, using the different movable sensors, was demonstrated by comparison to theoretical predictions based on the nominal aluminum material properties.
KEYWORDS: Waveguides, Wave propagation, Temperature metrology, Phase velocity, Dispersion, Finite element methods, 3D modeling, Phase shifts, Phase measurement, Biological research
Guided wave propagation has been proposed as a means to monitor the axial loads in continuously welded railway rails
although no practical system has been developed. In this paper, the influence of axial load on the guided wave
propagation characteristics was analyzed using the semi-analytical finite element method, extended to include axial
loads. Forty modes of propagation were analyzed up to a maximum frequency of 100 kHz. The sensitivity of the modes
to axial load or changes in elastic modulus was formulated analytically and computed. In practice, by using separation
of signals in time it would only be possible to separate the mode with the greatest group velocity over a reasonable
distance. It was found that the influence of axial load on the wavelength of such a mode should be measureable.
However, the influence of changes in the elastic modulus due to temperature is expected to be an order of magnitude larger. In order to develop a practical measurement technique it would be necessary to eliminate or compensate for this and other influences.
There are many applications in which a laser beam with a flat-top intensity profile would be ideal, as compared to a laser
beam with a non-uniform energy distribution. Standard stable optical resonators will unfortunately not generate such a
laser beam as the oscillating mode. Single-mode oscillation would typically be Gaussian in profile, while multimode
oscillation might deliver a beam with an averaged flat-like profile in the near field, but would diverge very quickly due
to the higher order modes. In addition, if the modes are coherently coupled, then large intensity oscillations could be
expected across the beam. Techniques exist to generate flat-top beams external to the cavity, but this is usually at the
expense of energy, and almost always requires very precise input beam parameters. In this paper we present the design
of an optical resonator that produces as the stable transverse mode a flat-top laser beam, by making use of an intra-cavity
diffractive mirror. We consider the modal build-up in such a resonator and compare the mode competition between
flat-top like beams, including Flattened Gaussian beams, Fermi-Dirac beams, and super-Gaussian beams. Finally, we
remark on the use of an intra-cavity piezoelectric unimorph mirror for selecting a particular class of flat-top beam as the
fundamental mode of the resonator.
In this paper we present the design of an optical resonator that produces as the stable transverse mode a flattened
Gaussian laser beam by making use of an intra-cavity diffractive mirror. We consider the modal build-up in such a
resonator and propose the required dynamic changes to an intra-cavity piezoelectric unimorph mirror for selecting the
flattened Gaussian beam order of the stable mode. The feasibility of using a deformable diffractive mirror is
demonstrated numerically. An optimization approach is employed to determine the optimal voltage distribution required
to deform the mirror into a prescribed shape for the selection of the flattened Gaussian beam order. Good agreement
between an ideal static diffractive mirror and the proposed adaptive mirror is achieved.
One important application of guided wave ultrasound is that of rail condition monitoring where long lengths of rail can
be monitored from permanently attached transducer locations. During the development of transducers for such a system
it is advantageous to be able to measure the amplitude of the individual modes of propagation on a short length of rail in
the laboratory. This paper describes a method of extracting modal amplitudes from measured time domain signals
performed at a limited set of points on the waveguide. The method uses the wave propagation characteristics of the
waveguide, predicted by a semi-analytical finite element model, to extract the modal amplitudes from experimental
measurements. The frequency response at a set of measurement locations is described by a superposition (with
unknown amplitude coefficients) of the frequency response of the modes that propagate in the frequency range of
interest. Experimental time domain responses are measured and transformed to frequency responses. The amplitude of
each mode is estimated using the pseudo-inverse to provide a minimum norm least-squares estimate. The technique is
demonstrated on a rail excited by a piezoelectric patch transducer. A laser vibrometer was used to measure
displacements at five points around the rail circumference at three distances giving a total of 15 measurements. Eight
propagating modes were extracted from these measurements. The extracted modes were then used to predict the
response at points further along the waveguide and these predictions were verified by further measurements indicating
that the modes of propagation were accurately estimated. The technique requires that the distance between the
measurement points be known but does not require that the distance from the transducer be known. This feature and the
fact that only a few measurements are required make the method suitable for measuring the propagation of individual
modes over long distances in the field.
This paper describes the development of a deformable mirror to be used in conjunction with diffractive optical elements
inside a laser cavity. A prototype piezoelectric unimorph adaptive mirror was developed to correct for time dependent
phase aberrations to the laser beam, such as those caused by thermal expansion of materials. The unimorph consists of a
piezoelectric disc bonded to the back surface of a copper reflective mirror. The rear electrode of the piezoelectric
ceramic disc is divided into segments so that a number of different control voltages can be applied to deform the mirror
in a desired displacement distribution. The mirror is required to be able to deform in the shape of each of the lower order
Zernike polynomials, which describe aberrations in optical systems. A numerical model of the device was used to
determine a suitable electrode configuration. Finally, the device was constructed and the deformed shapes measured
using a laser vibrometer.
An actuation system, making use of paraffin wax as a smart material, has been developed for high force, large
displacement applications. Wax actuators exploit the significant volumetric expansion (typically between 10 and 15%)
experienced during the solid to liquid phase change of paraffin wax. When contained, this expansion results in
considerable hydrostatic pressure. Traditionally, wax actuators are designed such that the wax acts directly, via a
compliant seal, on an output device such as a piston. We propose using an additional intermediate (passive) fluid to
transmit pressure to a separate remote actuator. In essence, we propose a solid-state 'pump' for hydraulic actuation, with
no moving parts and which requires no maintenance. The pump makes use of paraffin wax pellets, submerged in
hydraulic fluid. The pellets are encapsulated in silicone rubber to prevent contamination of the hydraulic fluid. Upon
melting, the volumetric expansion is used to displace the hydraulic working fluid, which is in turn used to drive a
conventional hydraulic actuator. Making use of only 65g of paraffin wax, heated from room temperature to 80ºC, the
pump generated a blocked pressure of 45MPa and displaced 15.7ml of hydraulic fluid. The pump was used to drive a
commercial actuator, and achieved a free stroke of 24.4mm and a blocked force of approximately 29kN.
Piezoelectric transducers are commonly used to excite waves in elastic waveguides such as pipes, rock bolts and rails.
While it is possible to simulate the operation of these transducers attached to the waveguide, in the time domain, using
conventional finite element methods available in commercial software, these models tend to be very large. An
alternative method is to use specially formulated waveguide finite elements (sometimes called Semi-Analytical Finite
Elements). Models using these elements require only a two-dimensional finite element mesh of the cross-section of the
waveguide. The waveguide finite element model was combined with a conventional 3-D finite element model of the
piezoelectric transducer to compute the frequency response of the waveguide. However, it is difficult to experimentally
verify such a frequency domain model. Experiments are usually conducted by exciting a transducer, attached to the
waveguide, with a short time signal such as a tone-burst and measuring the response at a position along the waveguide
before reflections from the ends of the waveguide are encountered. The measured signals are a combination of all the
modes that are excited in the waveguide and separating the individual modes of wave propagation is difficult if there are
numerous modes present. Instead of converting the measured signals to the frequency domain we transform the modeled
frequency responses to time domain signals in order to verify the models against experiment. The frequency response
was computed at many frequency points and multiplied by the frequency spectrum of the excitation signal, before an
inverse Fourier transform was used to transform from the frequency domain to the time domain. The time response of a
rail, excited by a rectangular piezoelectric ceramic patch, was computed and found to compare favorably with
measurements performed using a laser vibrometer. By using this approach it is possible to determine which modes of
propagation dominate the response and to predict the signals that would be obtained at large distances, which cannot be
measured in the lab, and would be computationally infeasible using conventional finite element modeling.
Guided wave ultrasonic inspection is becoming an important method of non-destructive testing for long, slender structures such as pipes and rails. Often it is desirable to use transducers that can strongly excite a specific mode of wave propagation in the waveguide. Piezoelectric patch transducers are frequently employed, by researchers, for exciting waves in beam like structures. Sonar systems frequently make use of resonant transducers, such as sandwich
transducers, for acoustic wave generation and this principle has been used to excite waves in a rail. This paper compares the two transduction approaches, for launching bending waves in rectangular waveguides, with numerical modeling. The numerical modeling combined a waveguide finite element model, of the waveguide, with conventional
three-dimensional piezoelectric finite element models of the transducers. The waveguide finite elements were formulated using a complex exponential to describe the wave propagation along the structure and conventional finite element interpolation over the area of the element. Consequently, only a two-dimensional finite element mesh covering the cross-section of the waveguide is required. The harmonic forced response of the waveguide was used to compute a complex dynamic stiffness matrix which represented the waveguide in the transducer model. The effects of geometrical
parameters of patch and sandwich transducers were considered before the comparison was made. It appears that piezoelectric patch transducers offer advantages at low frequencies while sandwich transducers are superior at high frequencies, where resonance can be exploited, at the cost of more complex design.
Wave propagation characteristics can be computed analytically for simple geometries such as plates and cylinders but geometries that are more complex and waveguides comprising combinations of different materials require numerical analysis. Piezoelectric waveguide finite elements, which can model wave propagation in piezoelectric waveguides of arbitrary cross-section, were formulated and implemented. In these elements wave functions are used to describe the displacement variation along the waveguide with conventional finite element interpolation functions over the crosssection of the waveguide. The resulting two-dimensional element is very efficient for computing wave propagation in waveguides. The accuracy of the elements was verified by comparison with a three-dimensional finite element model with appropriate boundary conditions to represent a waveguide. An analytical expression was derived to compute the group velocities of waves in piezoelectric waveguides. Wavenumber and group velocity versus frequency curves were plotted for a piezoelectric waveguide with square cross-section. The elements were used to model 1-3 piezoelectric composite material. A unit cell, comprising one-quarter of the cross-section of the piezoelectric pillar and half the neighboring (non-piezoelectric) polymer was modeled with appropriate boundary conditions to represent the periodicity of the material. The results were verified with three-dimensional finite element modeling and the waveguide element model was found to be very accurate and efficient.
Variable stiffness springs allow vibration absorbers and isolators to adapt to changing operating conditions. This paper describes the development of such a spring. The spring was a compound leaf spring and variable stiffness was achieved by separating the two leaf springs using a wax actuator. In the selected design, each spring consisted of an outer (220mm in diameter) and an inner ring connected by three radial beams. A paraffin wax actuator was chosen to affect the separation of the leaf springs. This actuator consisted of a small copper cup containing paraffin wax. When the wax is heated, it changes from a solid to a liquid with an associated volume change that is used to drive an output shaft. A hot-air gun was used to heat and cool the wax actuator. It was found that the wax actuator could produce an 8mm separation of the springs, which increased the stiffness of the spring by 2.7 times, exceeding the typical requirement for adaptive absorbers and isolators. The loss factor, of the variable stiffness spring, was less than 0.12. The measured response times for the open-loop system were 82s and 109s for heating and cooling respectively. A linear sliding potentiometer was used to measure the spring separation and proportional and derivative feedback control was used to control the current supplied to the heating element thus reducing the response time to less than 30s for small step changes. Further improvement in response time could be achieved by more directly heating and cooling of the paraffin wax in the actuator.
Many ultrasonic motors are based on resonators that have two modes of vibration with equal resonant frequencies. Excitation of the two modes with a 90°-phase shift is used to generate elliptical motion at the contact points between resonator and slider. An alternative motor using two modes with a resonant frequency ratio of 1:2 has been demonstrated in the literature. The contact point in this stator describes a Lissajous figure ('8') instead of an ellipse. The motor requires only a single power amplifier and piezoelectric ceramic element. The challenge of designing such a resonator is to achieve the predetermined resonant frequency ratio with corresponding modes having usable motion at the contact points. In this paper, topology optimization was applied to design a resonator that has two modes satisfying these conditions. The topology optimization problem was formulated in an unusual manner with direct minimization of the intermediate density and a constraint on the natural frequency ratio. The design domain was discretized using four-noded membrane elements with drilling degrees of freedom due to their superior efficiency. Manual shape optimization, considering 3-D effects, was performed on the design produced by the topology optimization step. A prototype resonator was constructed and tested to verify the numerical modeling.
A method for designing practical displacement amplification mechanisms for piezoelectric stack actuators was developed. The amplification mechanisms and the piezoelectric stack actuators were modeled using plane-strain finite elements. Optimal sizing and topology optimization were performed simultaneously to maximize the first natural frequency while satisfying free stroke and stress constraints. Optimal sizing variables were selected to control the kinematic behavior of the mechanism while a restricted variable thickness sheet topology optimization method was used to remove unnecessary material from stiff regions of the structure. Calculation of sensitivities was very efficient for the topology optimization variables but required the major portion of computational time for the optimal sizing variables. The method was applied to beam-type lever amplification mechanisms and two devices that included pre-stressing of the piezoelectric ceramics and pure translation of the output point were optimized, manufactured and tested. The results demonstrate that the method presented can be used to design amplified piezoelectric actuators that can be manufactured without interpretation by the designer.
KEYWORDS: Actuators, Amplifiers, Systems modeling, Optimization (mathematics), Chemical elements, Solid state electronics, Ceramics, Data modeling, Sensing systems, Active vibration control
Various solid-state mechanisms for the amplification of the small stroke, produced by piezoelectric materials, have been presented in the literature. A designer tasked with designing a device such as a micro-positioner must choose between these mechanisms. In this paper, the use of topological optimization to produce characteristic functions for amplification mechanisms, forming a basis for comparison of different designs, is investigated. The optimization problem was formulated as a variable thickness sheet problem where the stiffness was maximized subject to a constraint on the free stoke. Apart from specifying the design domain, no volume constraints were imposed. The design domain, comprising a piezoelectric and a metallic region, was discretized with eight-noded, plane-strain finite elements. This formulation was found to produce designs with negligible intermediate thickness. These designs are non-unique and repeatedly solving the problem from different starting material distributions results in slightly different 'optimal' stiffness values. The resulting maximum stiffness can be plotted as a function of free stroke producing a curve that is characteristic of the amplification mechanism. Irregularities in this curve would indicate that a local maximum with poor performance has been found. The method is demonstrated by computing the characteristic curve for two amplifier mechanisms.
KEYWORDS: Transducers, Finite element methods, Wave propagation, 3D modeling, Telecommunications, Signal detection, Head, Receivers, Transmitters, Signal to noise ratio
A system has been developed and installed to continuously monitor the integrity of a railway track. In this system elastic waves are transmitted, along the rails, between transmit and receive stations spaced at 2.5 km intervals along the length of the track. The development of piezoelectric transducers, which transmit and receive the elastic waves is described. The requirements and conceptual design of the transducer are described. Results of measurements conducted in the laboratory and in the field are presented. The transducer achieved transmission over the required 2.5 km with a signal to noise ratio of approximately 30 dB at the receiver. Finite element modeling was used to obtain a better understanding of the transducer operation. Improved modeling of the wave propagation and energy loss mechanisms in the effectively infinite rail is required before the model can be used to predict optimal frequencies and methods of excitation. The present system was designed to detect complete breaks in the rail. A 'smarter' system with communication between the transmit and receive stations, more sophisticated signal processing, and wave propagation confined to the critical regions of the rail cross-section could indicate the growth of cracks before the rail breaks thus adding substantial value to the system.
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