Jean C. Piquette

A nonlinear material model of lead magnesium niobate (PMN)

A one-dimensional material model of lead magnesium niobate (PMN) is presented. The model includes saturation phenomenology, but excludes hysteresis and dispersion. (Constant temperature is assumed.) It is shown that the strain can be taken to respond as an exactly quadratic function in the electric displacement D throughout the saturation region and yet still deduce the full observed response of the material, including flattening of the curve, when the strain is expressed as a function of electric field E. The model developed here is shown to be compatible with experimental measurements previously acquired by other researchers.

Generalized material model for lead magnesium niobate (PMN) and an associated electromechanical equivalent circuit

An existing one-dimensional nonlinear material model of lead magnesium niobate [J. C. Piquette and S. E. Forsythe, “A nonlinear model of lead magnesium niobate (PMN),” J Acoust. Soc. Am. 101, 289–296 (1997)] is generalized to three dimensions. The resulting theory is applied to two practical systems: the “thickness expander plate” and the “length expander bar.” Linearizing the theory results in an electromechanical equivalent circuit that is applicable to predicting the first-order behavior of transducers based on either of these practical systems. The methods used are sufficiently general that the circuit is also appropriate for piezoelectric, and even for electrostatic, transducers. Preliminary experimental data that confirm the validity of the circuit are presented. Connections between the constants of the theory and those of piezoelectricity are derived, and a general expression for the coupling coefficient is obtained. Known theoretical coupling coefficients for piezoelectric and electrostatic transd...

Nonlinear scattering of acoustic waves by vibrating obstacles

The problem of scattering of an acoustic wave (at angular frequency ω) by an obstacle whose surface vibrates harmonically (at angular frequency Ω) was studied both theoretically and experimentally. The theoretical approach involved solving the nonlinear wave equation, subject to appropriate boundary conditions, by use of a perturbation expansion of the fields and a Greens function method. This problem was previously studied theoretically by D. Censor [(J. Sound Vib. 25, 101–110 (1972)], who used the linear wave equation together with nonlinear boundary conditions to obtain his solution. In addition to ordinary rigid‐body scattering, Censor predicted nongrowing waves at the sum and difference frequencies ω± = ω ± Ω. The solution to the nonlinear wave equation also yields scattered waves at frequencies ω±. However, the amplitudes of these waves tend to grow with increasing distance from the scatterers surface and after a very small distance dominate those predicted by Censor. Preliminary experimental resu...

Censor's acoustical Doppler effect analysis--Is it a valid method?

In the article, ‘‘Acoustical Doppler effect analysis—Is it a valid method?’’ [J. Acoust. Soc. Am. 83, XXX–XXX (1988)], Censor considers the problem of scattering in the presence of moving objects under conditions of space‐ and time‐dependent moving media. He considers this situation in the context of a generalized linear wave equation. This equation predicts the generation of Doppler‐type spectral components at frequencies that are equal to (i) the sum of the frequencies of the primary waves, (ii) the difference of the frequencies of the primary waves, and (iii) harmonics of these sum‐and‐difference frequencies. However, since the nonlinear wave equation also predicts scattered spectral components at these same frequencies, and since those predicted by the nonlinear theory are usually much stronger than those predicted by Censor’s theory, the generalized linear wave equation used by Censor is generally inadequate for accurately predicting the amplitudes of the spectral components of interest. However, a l...

Spherical wave scattering by an elastic solid cylinder of infinite length

The problem of the scattering of a spherical acoustic wave by an elastic (lossless) solid cylinder of infinite length immersed in an infinite, inviscid fluid medium is investigated theoretically. The solution is obtained by imposing appropriate boundary conditions (involving stress and normal displacement) at the fluid–solid interface on the relevant differential equations. In order to be able to solve the differential equations, an approximation is made that is equivalent to assuming that the most significant additional contributions to the scattered wave appearing in the fluid (compared with the contributions to the scattered wave arising in the incident plane‐wave case) are those associated with the waves propagating along the z axis within the solid. Numerical results are presented for a 1000‐Hz wave incident on a 2‐cm‐diam metallic cylinder in water. This is a low ka calculation (where k is the wavenumber in the fluid and a is the radius of the scatterer). Several different metals are considered. The...

Method for transducer transient suppression. I: Theory

The problem of driving a transducer in such a way as to produce a tone burst of steady‐state sound radiation in the surrounding fluid medium is considered. The goal is to determine the driving voltage waveform to apply to a transducer to produce an acoustic pressure waveform in the fluid that is a segment of a steady‐state sine wave, beginning and ending at zero crossings of the sine, i.e., the usual turnon and turnoff transients are suppressed. The theoretical driving voltage waveform for a spherical transducer is shown to consist of a sum of a pedestal voltage, a ramp voltage, and a sinusoidal voltage that is phase shifted with respect to the sinusoid appearing in the fluid. Both theoretical and numerical calculations are given here. The following paper presents results of experimental measurements. The measurements were carried out on several spherical transducers (one of which was selected for presentation) and on an array of piezoelectric tubes. These experiments confirm the validity of the theory.

Low-frequency echo-reduction and insertion-loss measurements from small passive-material samples under ocean environmental temperatures and hydrostatic pressures.

System L is a horizontal tube designed for acoustical testing of underwater materials and devices, and is part of the Low Frequency Facility of the Naval Undersea Warfare Center in Newport, Rhode Island. The tube contains a fill fluid that is composed of a propylene glycol/water mixture. This system is capable of achieving test temperatures in the range of -3 to 40 deg Centigrade, and hydrostatic test pressures in the range 40 to 68,950 kPa. A unidirectional traveling wave can be established within the tube over frequencies of 100 to 1750 Hz. Described here is a technique for measuring the (normal-incidence) echo reduction and insertion loss of small passive-material samples that approximately fill the tube diameter of 38 cm. (Presented also is a waveguide model that corrects the measurements when the sample fills the tube diameter incompletely.) The validity of the system L measurements was established by comparison with measurements acquired in a large acoustic pressure-test vessel using a relatively large panel of a candidate material, a subsample of which was subsequently evaluated in system L. The first step in effecting the comparison was to least-squares fit the data acquired from the large panel to a causal material model. The material model was used to extrapolate the panel measurements into the frequency range of system L. The extrapolations show good agreement with the direct measurements acquired in system L.

Technique for Evaluating Indefinite Integrals Involving Products of Certain Special Functions

A new technique is described for evaluating a general class of indefinite integrals involving products of many of the special functions of physics such as Bessel functions, Legendre functions, Hermite functions, etc. The technique is a generalization of the method used by Sonine to evaluate certain indefinite integrals of Bessel functions. It involves replacing the integral to be evaluated by a coupled set of linear, inhomogeneous differential equations. A particular solution of the set of differential equations is then sufficient to express the result of integration. Several examples are given to illustrate the technique.

Quasistatic coupling coefficients for electrostrictive ceramics

A generalized definition of the coupling coefficient, useful for the evaluation of transducers that incorporate an electrostrictive active element, is introduced. The definition is expressed under quasistatic conditions and becomes zero when no bias is applied (assuming that the effects of remanence are negligible), and remains zero under zero bias even when a significant prestress is present. This reflects a property of the piezoelectric coupling coefficient, which vanishes when the ceramic becomes depoled. The behavior of this definition thus differs from that of another definition, introduced elsewhere, which produces a significant nonzero result at zero bias. [See C. Hom et al., “Calculation of quasi-static electromechanical coupling coefficients for electrostrictive ceramic materials,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 542–551 (1994).] The present definition also leads in a natural way to a coupling coefficient for biased piezoelectric ceramics, and an equation is given for that ca...

One-dimensional phenomenological model of hysteresis. I. Development of the model

A one-dimensional phenomenological model of hysteresis is presented. The model is suitable for application both to electrostrictive materials, such as lead magnesium niobate (PMN), and to magnetostrictive materials, such as Terfenol D. The concepts of “inflation,” “field space,” and the “reference ellipse” are introduced as suitable mechanisms for transforming measured hysteretic data into corresponding anhysteretic versions. An anhysteretic model is then fitted (in the least-squares sense) to the transformed data. By applying the inverse transforms to the fitted anhysteretic model, a hysteretic model is deduced. Good agreement with the original (hysteretic) data is seen. It is shown that, when a sample is driven by a monofrequency electric field, the area of the polarization vs electric-field hysteresis loop is independent of all harmonics in the polarization but the first. A principle useful for understanding the shapes assumed by P–E and M–H hysteresis loops in general is described.