Nonlinear Transducer Research Articles

Modeling piezoceramic transducer hysteresis in the structural vibration control problem

To appropriately assess, compensate for and/or potentially utilize piezoceramic transducer nonlinearities, reversible and irreversible, in the structural vibration control problem, a suitable theoretical framework is needed. In this study such a framework is developed and experimentally evaluated for a basic pedagogical structural vibration control system, the simply supported beam with a monolithic piezoceramic wafer bonded to it that can be both electrically shunted and/or driven by an external electric source. A constitutive model for the piezoceramic (PZT) wafer by itself is formulated that incorporates reversible (higher order polynomial) and irreversible (hysteretic) dielectric nonlinearity. An identification scheme for the model is validated experimentally. The nonlinear PZT model is then integrated into the coupled dynamic equations of the overall system consisting of the simply supported beam and the electrically shunted PZT wafer bonded to the beam. The theoretical system model is then evaluated by comparing its predictions to experimental results. Energy loss and transduction mechanisms in the integrated structural system are investigated.

Principal dynamic mode analysis of nonlinear transduction in a spider mechanoreceptor.

The nonlinear dynamics of the mechanoelectrical transduction in an arthropod mechanoreceptor (cuticular slit sense organ of the spider Cupiennius salei) were studied using Volterra kernel measurements and the recently proposed method of principal dynamic modes. Since mechanoreceptors must operate with sufficient gain sensitivity to rapidly varying displacement stimuli over a broad bandwidth and for a wide range of amplitudes, the experimental data were generated by applying pseudorandom broadband mechanical displacements of various mean levels to the cuticular slits. The recorded response data were intracellular current and potential. The purpose of this paper is to demonstrate the use of the principal dynamic mode (PDM) methodology in elucidating the nonlinear dynamics of a spider mechanoreceptor. The results obtained demonstrate that two PDMs suffice to provide a complete nonlinear dynamic model of this insect mechanoreceptor. The first PDM resembles the first-order kernel and has a low pass characteristic (position dependent), while the second PDM has a high-pass characteristic (velocity-dependent) and resides entirely in the second-order kernel (nonlinear adaptation). This study may serve as an example of the proper use of this new methodology for the analysis of nonlinear physiological systems.

Characterizing human perceptual inefficiencies with equivalent internal noise.

A widely used method for characterizing and comparing inefficiencies in perceptual processes is the method of equivalent internal noise--the amount of random internal noise necessary to produce the degree of inefficiency exhibited by the perceptual system in processing [J. Opt. Soc. Am. 46, 634 (1956)]. One normally estimates the amount of equivalent internal noise by systematically increasing the amount of external noise added to the signal stimulus and observing how threshold--signal stimulus energy required for an observer to maintain a given performance level--depends on the amount of external noise. In a variety of perceptual tasks, a simple noisy linear amplifier model [D. Pelli, Ph.D. dissertation (University of Cambridge, Cambridge, UK 1981)] has been utilized to estimate the equivalent internal noise Ninternal by fitting of the relation between threshold contrast c tau and external noise N(ext) at a single (d') performance level: c tau 2 =(d'/beta)2(N(ext)2 + Ninternal2). This model makes a strong prediction: Independent of observer and external noise contrast, the ratio between two thresholds at each external noise level is equal to the ratio of the two corresponding d' values. To our knowledge, this potential test for the internal consistency of the model had never been examined previously. In this study we estimated threshold ratios between multiple performance levels at various external noise contrasts in two different experiments: Gabor orientation identification, and Gabor detection. We found that, in both identification and detection, the observed threshold ratios between different performance levels departed substantially from the d' ratio predicted by the simple noisy linear amplifier model. An elaborated perceptual template model [Vision Res. 38, 1183 (1998)] with nonlinear transducer functions and multiplicative noise in addition to the additive noise in the simple linear amplifier model leads to a substantially better description of the data and suggests a reinterpretation of earlier results that relied on the simple noisy linear amplifier model. The relationship of our model and method to other recent parallel and independent developments [J. Opt. Soc. Am. A 14, 2406 (1997)] is discussed.

Terfenol-D Nonlinear Vibration Absorber

An active nonlinear vibration absorber scheme for flexible structures is demonstrated. The absorber exploits the nonlinear transduction of nonbiased Terfenol-D actuators. It uses a quadratic feedback control law and the "quadratic" strain-field characteristic of a nonbiased Terfenol-D actuator to achieve a nonlinear coupling between the absorber and the forced structure. A saturation phenomenon is induced by tuning the frequency of the underdamped second-order absorber to one-half that of the primary structure. As a result, the structural vibrations are greatly reduced. This type of control has been demonstrated previously using linear actuators; the nonlinearities have been introduced via the control algorithm. Here, we show that nonbiased Terfenol-D actuators can perform effectively the "squaring" operation, instead of the algorithm, and achieve similar results. We develop the theory and present experimental results for the control of the first and second modes of a cantilever beam. We also consider application of the strategy experimentally when the forcing is due to a rotating imbalance. In this case, the excitation source is nonideal. Our results indicate that the saturation-based control technique implemented with a Terfenol-D actuator constitutes an effective nonlinear vibration absorber.

Terfenol-D Nonlinear Vibration Absorber

An active nonlinear vibration absorber scheme for flexible structures is demonstrated. The absorber exploits the nonlinear transduction of nonbiased Terfenol-D actuators. It uses a quadratic feedback control law and the "quadratic" strain-field characteristic of a nonbiased Terfenol-D actuator to achieve a nonlinear coupling between the absorber and the forced structure. A saturation phenomenon is induced by tuning the frequency of the underdamped second-order absorber to one-half that of the primary structure. As a result, the structural vibrations are greatly reduced. This type of control has been demonstrated previously using linear actuators; the nonlinearities have been introduced via the control algorithm. Here, we show that nonbiased Terfenol-D actuators can perform effectively the "squaring" operation, instead of the algorithm, and achieve similar results. We develop the theory and present experimental results for the control of the first and second modes of a cantilever beam. We also consider application of the strategy experimentally when the forcing is due to a rotating imbalance. In this case, the excitation source is nonideal. Our results indicate that the saturation-based control technique implemented with a Terfenol-D actuator constitutes an effective nonlinear vibration absorber.

Electroelastic nonlinearities in piezoelectric transducers and their effect on vibratory energy transmission, transduction, and dissipation in structural systems

Though all structural dynamic systems are inherently nonlinear, the majority of analyses are typically based on the assumption of linearity. While vibration of passive systems is often reasonably described by linear system theory, particularly in the audio and ultrasonic frequency ranges, the vibrations of systems employing integrated actuators fabricated from smart materials often are not, whether they be based on piezoelectric, electrostrictive, or magnetostrictive technologies, etc. Nonlinearities can affect the system performance through a loss of actuation authority, stability, and/or system functionality. Recognizing this, some researchers have included nonlinearities in their formulations for the constitutive equations of the smart material. Typical nonlinear operators are either in the form of higher-order polynomial expressions or various hysteresis operators involving the electrical (magnetic), elastic and/or electroelastic (magnetoelastic) relations. An important fundamental issue is how the different types of nonlinearities affect the smart material in its role as source, conduit, transformer and/or receiver/dissipator of vibroacoustic energy in a smart structure. This is the principle focus of this study in which a generic active vibration isolation application is considered. The effect of different types of nonlinearities with respect to how vibratory energy flows throughout the smart structural system is investigated. [Work supported by NSF.]

On the equivalence problem for letter-to-letter top-down tree transducers

Letter to letter top-down tree transducers are investigated in this paper. Informally, trees which appear in the rules of such transducers are reduced to one letter in the right-hand side as in the left one. With an encoding of the tree transformations induced by such transducers into recognizable forests, we recently established the decidability of equivalence for linear top-down transducers. Here, in order to capture the non-linearity of top-down transducers, we introduce new classes of tree automata with equivalence constraints between direct subterms for which equivalence is decidable. We then show that the equivalence problem for non-linear top-down transducers can be reduced to the equivalence problem of automata with equivalence constraints.

Time-domain analysis of nonlinear transducer networks

The existence of computer methods for the construction of network system equations from a netlist description and numerical methods for the solution of initial-value problems in mixed differential-algebraic equations (DAE’s) allow the time-domain analysis of networks containing nonlinear elements. This paper will demonstrate these techniques in application to electromechanical transduction systems exhibiting small- and large-signal nonlinear behavior. Nonlinear constitutive equations defining electromechanical elements in transducer networks will be developed in the proper form for inclusion in a modified nodal formulation. It will be shown that consistent sets of initial conditions can be determined from a modified netlist under limited circumstances, but these limitations do not significantly constrain the utility of method in this application. In fact, this approach opens the method up to elements with static behavior, such as permanent magnets or electrets. Time-domain behavior of the system is numerically derived from the descriptor equations by a backward difference formula (BDF) DAE integrator appropriate for stiff systems.

Identification of the nonlinearity governing even-order distortion products in cochlear potentials

In order to characterize the cochlear transducer nonlinearities which are involved in the generation of the summating potential (SP), we investigated the effect of a change in the electrical operating point of the cochlear transducer on the SP. The electrical operating point of the cochlear transducer was affected by suppressing reversibly the endocochlear potential (EP). This was realized by intravenous injection of furosemide in guinea pig. A differential recording technique was used in the basal turn of the cochlea to measure locally generated even-order distortion products: the SP and the second harmonic component (2F 0) of the cochlear microphonics (CM). These potentials were evoked by 2 and 8 kHz stimuli presented at 60 dB SPL. Following furosemide injection, the SP changed polarity twice over time. The zero crossings of the SP coincided with a minimum in the amplitude of 2F 0. Concomitantly, the phase of 2F 0 shifted about 120°. The changes in the electrical even-order products were comparable to the changes that occurred in a mechanical even-order intermodulation distortion product (the difference tone F 2−F 1 otoacoustic emission) after furosemide application (Mills et al., J. Acoust. Soc. Am. 94 (1993) 2108–2122). The combined results suggest that only one sigmoidal transfer function may account for the SP, 2F 0, and the emission of the difference tone F 2−F 1, and that shifts in the operating point of the transfer function would be the major cause behind the furosemide-induced changes in the even-order distortion products. The sigmoidal transfer function is likely associated with the mechano-electrical transducer channel at the apical pole of the outer hair cell.

Contrast discrimination function: spatial cuing effects.

The effects of spatial cuing were measured for discrimination between an increment and a decrement on a target's pedestal contrast. Discrimination thresholds measured in the absence of a spatial cue were always higher than corresponding thresholds measured in the presence of a spatial cue, except when pedestal contrast was near zero. Uncued discrimination thresholds rose monotonically with pedestal contrast; cued discrimination thresholds formed a dipper function of pedestal contrast. A spatial-uncertainty model incorporating a nonlinear transducer produced similar results.

Entropy Masking

Current vision theory admits two principal types of visual masking: contrast masking and noise masking. The former is thought to operate via mechanisms such as a nonlinear transducer, contrast gain control, and lateral inhibitory influences. The latter is understood to be due to the randomness of the target + mask combination. Here I propose a new category: entropy masking. Entropy masking is evident when the mask is a complex but deterministic signal. Because the mask is deterministic, and because experiments demonstrate that the masking power of such signals is much greater than predicted by contrast gain control, entropy masking cannot be due to either contrast masking or noise masking. Entropy masking occurs because the observer cannot discount the background, as supposed by ideal-observer theory, and hence must learn the target + mask signal. The ease with which this is done is a function of the entropy of both target and mask. Experiments that illustrate these general principles are described.

A Method of Identification of Analyser Characteristics That is Free of Probability Summation Effect

A detection model (originally proposed by Quick) comprising, in a sequence of linear analysers, varphi1, …, varphi n, nonlinear transducer functions, and the Minkowski decision rule, is widely used, especially when it is necessary to take into account the effect of probability summation. However, there is a general belief that the analyser characteristics cannot be determined in detection experiments since there is a trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, ie how to identify the analysers varphi1, …, varphi n despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equidetection surface (EDS). Each analyser varphi i is expressed as a weighted sum of linear (coordinate) analysers {phi j}: varphi i=sum j=1 n a ijphi j, so that an identification of the analysers {phi i} is then reduced to evaluating the weight matrix A={ a ij}. It has been proven that A can be uniquely recovered from an ellipsoidal approximation of EDS in the neighbourhood of at least two points. More specifically, the following equation holds true: A−1 DA= H1−1 H2, where D is a diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the n-dimensional ellipsoids approximating EDS. Thus, the matrix H1−1 H2 known from experiment is a similarity transform of the diagonal matrix, the columns of A being the eigenvectors of H1−1 H2. Hence, any eigensystem routine can be used to derive A from H1−1 H2.

Analysis of harmonic distortion in electroacoustic transducers under indirect drive conditions

Nonlinear transducer equations for the six major types of electroacoustic transducers (electrostatic, variable reluctance, moving coil, piezoelectric, electrostrictive, and magnetostrictive) are formulated in a common way and solved approximately by perturbation analysis for indirect drive, i.e., current drive of electric field transducers and voltage drive of magnetic field transducers. The direct drive analysis, i.e., voltage drive of electric field transducers and current drive of magnetic field transducers, was given in J. Acoust. Soc. Am. 98, 1596–1611 (1995). Expressions and graphs are given for some of the indirect drive harmonics which apply to all six transducer types. The solutions for direct and indirect drive are compared, and some of the differences are discussed with emphasis on the question of which drive gives lower harmonic distortion. No answer to this question has been found that is applicable to all nonlinear mechanisms in all transducers at all frequencies. However, it is shown that for transducers of all types with low electrical loss operated near resonance direct drive usually gives lower second harmonics. The exceptions occur when mechanical springs are the dominant nonlinearity and the electromechanical coupling is very high; then indirect drive gives lower second harmonics. It is shown how the results, although based on simple lumped parameter models, can be applied in some cases to more complex transducer structures.

Stochastic resonance: A chaotic dynamics approach.

For a class of multistable systems it follows from qualitative results of Melnikov theory that deterministic and stochastic excitations play equivalent roles in the promotion of chaos. We use such results to suggest: (1) a method for assessing the role of the noise spectrum in enhancing the signal-to-noise ratio (SNR), the most effective spectral shape being that for which the power is distributed closest to the frequency of the Melnikov scale factor's peak; (2) a method for more effective SNR enhancement than can be achieved by increasing the noise, wherein the noise is left unchanged and a harmonic excitation with frequency based on the system's Melnikov scale factor is added to the system. The effectiveness of our Melnikov-based methods is confirmed by numerical simulations. The principle of a practical and effective nonlinear transduction device for enhancing SNR is proposed and demonstrated numerically. \textcopyright{} 1996 The American Physical Society.

An electroacoustic method for measuring the intensity of high-power microwave pulses

A microwave electric field intensity meter for single high-power radiowave pulses is described. A surface acoustic wave (SAW) dielectric acoustic line is used which has nonlinear electroacoustic properties and is comprised of two transducers. The first (nonlinear) transducer is affected by a microwave electric field and provides the SAW excitation due to nonlinear effects. A delayed SAW signal, carrying information about the electric field strength, is received by a linear interdigital transducer.

Multiplexed Low-Frequency Electric and Magnetic Field Fiber Optic Sensors

A frequency-multiplexed fiber optic interferometric sensor designed for the simultaneous measurement of low-frequency electric and magnetic fields is demonstrated. The sensor relies on the nonlinear transduction mechanism in electrostrictive and magnetostrictive materials for frequency-division multiplexing in a single interferometer with one unmodulated laser. The sensor exhibits extremely low cross talk (< −75 dB) between channels. Effects resulting from the nonlinear response of the fiber optic interferometer are discussed with regard to sensor multiplexing. Finally, low-frequency electromagnetic signals from remotely located electric and magnetic dipoles are detected with the fiber optic sensor.

Analysis of harmonic distortion in electroacoustic transducers

Nonlinear transducer equations are formulated for the six major types of electroacoustic transducers (electrostatic, variable reluctance, moving coil, piezoelectric, electrostrictive, and magnetostrictive) in a common form using the various physical and mathematical similarities among the transduction mechanisms. The common nonlinear equation is solved approximately by perturbation analysis giving simple expressions and graphs for harmonic amplitudes caused by all the frequently occurring nonlinear mechanisms. The expressions and graphs can be applied to all the transducer types by using nonlinear parameters tabulated for each type. These results can facilitate diagnosis of the cause of measured harmonics, prediction of harmonics during design, and development of electrical drives to suppress harmonics.

Transient response of arrays of linear and nonlinear transducers

In this paper a method is presented for the time-domain analysis of the mutual interaction effects in transducer arrays. This method is applicable to a large class of linear and nonlinear devices. Most analyses of array acoustic interactions to date have utilized a frequency-domain approach. However, a time-domain approach is more natural when the transducer elements are nonlinear or when it is desired to study transient effects. A method for calculating array acoustic interaction effects in the time domain is presented that can be easily coupled to models for many linear and nonlinear transducer elements. The boundary integral equation program CHIEF is used to generate acoustic interaction coefficients in the frequency domain which are then Fourier transformed to the time domain. When coupled with the transducer equations the result is a coupled set of integral-differential equations that can be solved by an explicit time-stepping procedure. Numerical results are presented for a small closely packed array of simple linear transducer elements and for an array of nonlinear hydraulically driven transducers. [Work supported by SPAWAR, PMW 183.]

Three-channel low-frequency fiber-optic voltage sensor

We demonstrate a frequency-multiplexed three-channel fiber-optic interferometric dc and low-frequency voltage sensor based on the electrostrictive effect in lead magnesium niobate transducers. The sensor relies on the nonlinear transduction mechanism in electrostrictive materials for frequency-division multiplexing by use of a single interferometer and a single unmodulated laser. To our knowledge this is the first direct measurement of three low-frequency voltage signals with a single fiber-optic interferometer. Demonstrated cross talk between channels is less than −75 dB. The cross talk is shown to be caused by the nonlinear output of the interferometer at large applied strains.

Geometric nonlinearities in the design of force transducers

In order to design highly accurate force transducers, the nonlinear relationship of electrical output versus an incrementally applied force must be extremely small. This error is primarily due to geometric nonlinear changes in the transducer structure. A finite element analysis was carried out to accurately predict the geometric nonlinear response of a force transducer due to an incrementally applied load. The finite element results were verified through experimentation. This work demonstrates that a finite element analysis is very useful in reducing or eliminating geometric nonlinearities in force transducers.