Analysis of a disk-type stator for the piezoelectric ultrasonic motor using impedance matrix

Impedance and admittance matrices of a piezoelectric annular actuator with segmented electrodes are presented for the analysis of the disk-type piezoelectric ultrasonic motors (USM). Equations of motion and the conjugate parameters for the impedance and admittance matrices are derived using the variational principle. In the derivation, the electric field in the piezoelectric layer is assumed to be constant over the area covered by a particular electrode, and the effects of both shear deformation and rotary inertia are taken into account. The resonance and antiresonance frequencies and the vibrating modes are calculated for the various resonance modes and boundary conditions, and the results are compared with those by the three-dimensional finite element methods. They are in excellent agreement with each other. It is expected that the derived impedance matrix can be effectively applied to the analysis and the design of the USM.     

MODELLING DYNAMICS OF A CONTINUOUS STRUCTURE WITH A PIEZOELECTRIC SENSORACTUATOR FOR PASSIVE STRUCTURAL CONTROL

This paper presents the concept of a vibration control system in which motions of a continuous structure with piezoelectric sensors/actuators can be suppressed (or activated) through transforming mechanical energy to electrical one and vice versa. The study is focused on distributed parameter structures, in which electromechanical variables are spatially dependent, and therefore traditional methods of design of piezoelectric transformers do not apply. In this case, a different approach is necessary to account for the spatial dependency of the variables. To examine the feasibility of the proposed vibration control system, we have performed the vibration suppression analysis of the cantilevered beam with piezoelectric sensors/actuators subjected to an exciting force/moment(s). The experimental results indicate that the damping of the composite system increases by 8-10 times in comparison with the mechanical system.As a result, the paper significantly expands the concept of passive damping mechanism for structural systems to take into account the dynamics of a continuous elastic structure piezoelectrically coupled to electrical network.     

Input-output matching in linear mechanical systems

The steady state forced harmonic motion of a linear mechanical system is considered. It is shown that the output q of the system can be matched with its input p, in the sense that the condition p = λVq is satisfied, where V is a given weighting matrix. The quantities λ and q, satisfying the above matching equation, are called the matching coefficient and mode, respectively. The application of the matching method to the identification of unknown system matrices, and hence to shifting a system resonance frequency, is presented. Three practical methods of achieving input-output matching are proposed.     

An alternative derivation of dynamic admittance matrix of piezoelectric cantilever bimorph

In this paper, the derivation method used in (J. Microelectromech. Systems 3 (1994) 105) and the solutions of dynamic admittance matrix of a piezoelectric device derived from the method are reviewed. By solving the problem of dynamic responses of a piezoelectric cantilever bimorph with mode analysis method, an alternative approach in the derivation of the dynamic admittance matrix and other related parameters of a piezoelectric system, which can be expressed explicitly in terms of series resonance characteristics of the structure, is presented. It is shown that this form of solutions may offer some conveniences in studying mechanical and electrical properties of the system in the vicinity of resonance frequencies.     

Damping of flexural vibrations in rectangular plates using the acoustic black hole effect

The reduction of flexural vibration in plate structures has been investigated using the recently reported acoustic black hole effect for flexural wave reflection in plates with the local thickness varying according to h(x)=εx^m and m≥2. Since sharp edges of such plates (wedges) are always truncated before x=0, the real reflection coefficients are relatively high, therefore the application of a small amount of damping is required to achieve large reductions in vibration amplitude. This paper presents a numerical model of a plate incorporating an acoustic black hole wedge, with predictions for vibration amplitudes. These are compared to equivalent experimental measurements for a range of applied damping layers. It is concluded that the above-mentioned power-law wedges can be used as effective vibration dampers in plate structures over a wide frequency range of interest.     

Absorption of flexural waves by a pair of mechanical resonator chains mounted on a plate

The scattering of flexural waves by two chains of mechanical resonators characterized by effective admittances is considered. In the first (lower) chain, the effective admittance is represented by a susceptance, whereas, in the second (upper) chain, the resonators are characterized by a complex admittance. The spatial periods of the chains are identical. A plane harmonic flexural wave is obliquely incident on the upper chain, and the scattered field formed by the chains is expressed as a superposition of homogeneous and inhomogeneous Bragg spectra. Intense scattering of the incident wave only occurs in the case of mutual compensation of the resonator susceptance and the radiation admittance. A pair of chains with periods not exceeding the half-wavelength of the flexural wave represents an effective insulator for this wave. In the half-space behind the first chain, the zeroth spectral component of the scattered field completely cancels the resonance-frequency incident flexural wave. Let the second chain be positioned at one of the displacement antinodes of the total field formed by the incident field and the zeroth scattered spectrum. Then, if the active components of the effective admittance of resonators belonging to the second chain are identical to the radiation admittance, the incident flexural wave is completely absorbed by the resonators.     

Vibration of prolate spheroidal shells with shear deformation and rotatory inertia: axisymmetric case

This paper presents the derivation of the equations for nonaxisymmetric motion of prolate spheroidal shells of constant thickness. The equations include the effect of distributed mechanical surface forces and moments. The shell theory used in this derivation includes three displacements and two thickness shear rotations. Thus, the effects of membrane, bending, shear deformation, and rotatory inertia are included in this theory. The resulting five coupled partial differential equations are self-adjoint and positive definite. The frequency-wave-number spectrum has five branches, two acoustic and three optical branches representing flexural, extensional, torsional, and two thickness shear. For the case of axisymmetric motion, these were computed for various spheroidal shell eccentricities and thickness-to-length ratios for a large number of modes. The axisymmetric dynamic response for damped shells of various eccentricities and thicknesses under point and ring surface forces are presented.     

A low frequency piezoelectric power harvester using a spiral-shaped bimorph

We propose a spiral-shaped piezoelectric bimorph power harvester operating with coupled flexural and extensional vibration modes for applications to low frequency energy sources. A theoretical analysis is performed and the computational results show that the spiral structure has relatively low operating frequency compared to beam power harvesters of the same size. It is found that to optimize the performance of a piezoelectric spiral-shaped harvester careful design is needed.     

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0, n), torsional T(0, n) and flexural F(m, n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method.     

Piezoelastic vibrations of composite Reissner-Mindlin-type plates

Linear vibrations of Reissner-Mindlin-type composite plates in the presence of piezoelectric eigenstrains are studied. Piezoelectric eigenstrains are produced by applying electrical loads to piezoelectric layers embedded in or attached to substrate layers. The influence of the mechanical field upon the electric field is taken into account in the modelling, ending up with electro-mechanically coupled field equations and boundary conditions, which describe the mechanical and the electrical dynamic response of the plate.The mechanical displacements are approximated by means of the kinematic hypothesis of Hencky. The electric potential distribution is assumed to be composed of a superposition of a linear and a parabolic distribution in the thickness direction. The linear part accounts for the electric potential difference between the electrodes of the totally electroded piezoelectric layers. The parabolic part is considered in order to take into account the influence of the mechanical field upon the electric potential by means of the direct piezoelectric effect. A weak two-dimensional formulation of the three-dimensional field equations is obtained by utilizing mechanical and electrical variational principles. This formulation is characterized by resultants of stress and electric displacement. The electro-mechanically coupled behaviour comes into play by means of the constitutive relations. In case the electric potential difference is not prescribed, it can be calculated from a relation, which connects the total electric charge and the electric potential difference to each other. Because this relation is obtained from the Gauss law of electrostatics, requiring integration with respect to the area of the electrode, non-local constitutive relations for the plate are found. The non-local constitutive relations bring a new aspect into the theory of plates. An analysis for the practically interesting one-dimensional case of composite, piezoelectric plates in cylindrical motion completes the paper.     

Generalized thermoelastic extensional and flexural wave motions in homogenous isotropic plate by using asymptotic method

In this paper the asymptotic method has been applied to investigate propagation of generalized thermoelastic waves in an infinite homogenous isotropic plate. The governing equations for the extensional, transversal and flexural motions are derived from the system of three-dimensional dynamical equations of linear theories of generalized thermoelasticity. The asymptotic operator plate model for extensional and flexural free vibrations in a homogenous thermoelastic plate leads to sixth and fifth degree polynomial secular equations, respectively. These secular equations govern frequency and phase velocity of various possible modes of wave propagation at all wavelengths. The velocity dispersion equations for extensional and flexural wave motion are deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate. The approximation for long and short waves along with expression for group velocity has also been obtained. The Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations and its equivalence established with that of asymptotic method. The numeric values for phase velocity, group velocity and attenuation coefficients has also been obtained using MATHCAD software and are shown graphically for extensional and flexural waves in generalized theories of thermoelastic plate for solid helium material.     

Elastic guided wave propagation in a periodic array of multi-layered piezoelectric plates with finite cross-sections

In this study, we present a model study of guided wave dispersion and resonance behavior of an array of piezoelectric plates with arbitrary cross-sections. The objective of this work is to analyze the influence of the geometry of an element of a 1D-array ultrasound transducer on generating multi-resonance frequency so as to increase the frequency bandwidth of the transducers. A semi-analytical finite-element (SAFE) method is used to model guided wave propagation in multi-layered 1D-array ultrasound transducers. Each element of the array is composed of LiNbO₃ piezoelectric material with rectangular or subdiced cross-section. Four-node bilinear finite-elements have been used to discretize the cross-section of the transducer. Dispersion curves showing the dependence of phase and group velocities on the frequency, and mode shapes of propagating modes were obtained for different geometry consurations. A parametric analysis was carried out to determine the effect of the aspect ratio, subdicing, inversion layer and matching layers on the vibrational behavior of 1D-array ultrasound transducers. It was found that the geometry with subdiced cross-section causes more vibration modes compared with the rectangular section. Modal analysis showed that the additional modes correspond to lateral modes of the piezoelectric subdiced section. In addition, some modes have strong normal displacements, which may influence the bandwidth and the pressure field in front of the transducer. In addition, the dispersion curves reveal strong coupling between waveguide modes due to the anisotropy of the piezoelectric crystal. The effect of the matching layers was to cluster extensional and flexural modes within a certain frequency range. Finally, inversion layer is found to have a minor effect on the dispersion curves. This analysis may provide a means to analyze and understand the dynamic response of 1D-array ultrasound transducers.     

Modeling and analysis of laminate composite plates with embedded active-passive piezoelectric networks

The objective of this work is to present the finite element modeling of laminate composite plates with embedded piezoelectric patches or layers that are then connected to active-passive resonant shunt circuits, composed of resistance, inductance and voltage source. Applications to passive vibration control and active control authority enhancement are also presented and discussed. The finite element model is based on an equivalent single layer theory combined with a third-order shear deformation theory. A stress-voltage electromechanical model is considered for the piezoelectric materials fully coupled to the electrical circuits. To this end, the electrical circuit equations are also included in the variational formulation. Hence, conservation of charge and full electromechanical coupling are guaranteed. The formulation results in a coupled finite element model with mechanical (displacements) and electrical (charges at electrodes) degrees of freedom. For a Graphite-Epoxy (Carbon-Fibre Reinforced) laminate composite plate, a parametric analysis is performed to evaluate optimal locations along the plate plane (xy) and thickness (z) that maximize the effective modal electromechanical coupling coefficient. Then, the passive vibration control performance is evaluated for a network of optimally located shunted piezoelectric patches embedded in the plate, through the design of resistance and inductance values of each circuit, to reduce the vibration amplitude of the first four vibration modes. A vibration amplitude reduction of at least 10 dB for all vibration modes was observed. Then, an analysis of the control authority enhancement due to the resonant shunt circuit, when the piezoelectric patches are used as actuators, is performed. It is shown that the control authority can indeed be improved near a selected resonance even with multiple pairs of piezoelectric patches and active-passive circuits acting simultaneously.     

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2ⁿ phases, each constant value in the present method denotes one material phase and 2ⁿ phases can be represented by 2ⁿ pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.     

Non-linear flexural waves in thin layers

Non-linear flexural waves in thin plates or layers have been analyzed in this paper. The equation of motion of the plate is derived assuming that the motion is antisymmetric about the mid-plane of the plate and that the plate is thin. The plate is considered to be elastic. The Von Karman non-linear strains and Landau elastic constants have been used to model geometric and material non-linearities, respectively. An asymptotic analysis of wave motion is presented using the method of multiple scales. Evolution equations are derived for small amplitude traveling flexural elastic waves. Numerical results show waveform distortion, amplitude amplification, and harmonic generation.     

Multipulse sequences for explosives detection by NQR under conditions of magnetoacoustic and piezoelectric ringing

A number of methods for cancelling magnetoacoustic and piezoelectric ringing signals in the spectroscopy of the nuclear quadrupole resonance are presented. The suggested methods include using the sequence (?₀)_?-(τ-?_x-2τ-? _y -2τ-?_?x -2τ-?_?y -τ) _n and a multipulse analog of the two-pulse Hahn sequence with the first pulse replaced by a short steady-state sequence. Another method presented is the method of orthogonal effective fields for a fast saturation of the quadrupole spin system which can be used for subtracting the magnetoacoustic and piezoelectric components from the signal. The suggested methods can be used for the practical purposes of detecting explosive substances and narcotics.     

The theoretical simulation of magnetized electron beam effects on radially polarized of an annular cylindrical piezoelectric crystal

By using the equilibrium equations for a hollow cylindrical piezoelectric layer in absence of body forces and taking into the account a magnetized electron beam in hollow region of this system the effects of thermal pressure, electrostatic self field and the strength of external magnetic field of a non-relativistic magnetized electron beam column on radially polarized of an annular cylindrical piezoelectric crystal in the steady state are simulated. The electrostatic potential profile in piezoelectric layer due to the mechanical pressure and electrostatic self field of electron beam are studied. Furthermore the graphs of the difference of potential Δϕ between inner and outer surfaces of piezoelectric versus to electron temperature, density of beam and the strength of external magnetic field are presented.     

Explicit time-domain approaches based on numerical Green’s functions computed by finite differences – The ExGA family

The present paper describes a new family of time stepping methods to integrate dynamic equations of motion. The scalar wave equation is considered here; however, the method can be applied to time-domain analyses of other hyperbolic (e.g., elastodynamics) or parabolic (e.g., transient diffusion) problems. The algorithms presented require the knowledge of the Green’s function of mechanical systems in nodal coordinates. The finite difference method is used here to compute numerically the problem Green’s function; however, any other numerical method can be employed, e.g., finite elements, finite volumes, etc. The Green’s matrix and its time derivative are computed explicitly through the range [0, Δt] with either the fourth-order Runge–Kutta algorithm or the central difference scheme. In order to improve the stability of the algorithm based on central differences, an additional matrix called step response is also calculated. The new methods become more stable and accurate when a sub-stepping procedure is adopted to obtain the Green’s and step response matrices and their time derivatives at the end of the time step. Three numerical examples are presented to illustrate the high precision of the present approach.     

Transfer constant of a multilayer acoustic transducer at the direct and inverse transformation

The algorithms for calculating the direct and inverse transfer constants of an acoustic transducer with an arbitrary number of intermediate layers between the piezoelectric layer, the acoustic duct, and the rear acoustic load are described. The results of a numerical analysis are presented and discussed. As an illustration, a 100-MHz transducer formed by a (Y+36°)-cut LiNbO₃ plate fixed on a fused-quartz acoustic duct with the help of five metal layers is considered. The other side of the plate carries two metal layers and a rear load. The phase-frequency characteristics and the transformation loss as a function of frequency are analyzed for the cases of direct and inverse transformation under the assumption that the signal is supplied and retrieved by a two-wire line.     

The effects of wall discontinuities on the propagation of flexural waves in cylindrical shells

The transmission of flexural type waves through various discontinuities in the walls of cylindrical shells is investigated. Theoretical curves of transmission loss are obtained for different circumferential wavenumbers and wave types, as functions of frequency. Material stiffness and extensional phase speed, together with the relationship between radial vibration amplitude and total wave power of propagation, are important factors which are found to strongly influence wave transmission through discontinuities. Some practical results useful for predicting the performance of typical pipe isolators (in vacuo) are obtained.