Electromechanical Problems Research Articles

Study on dynamic actuation in double microcantilever-based electrostatic microactuators with an in-house experimental set-up

This paper investigates the effect of dynamic excitation and its utility for enhancement of stable displacement range for a double cantilever-based electrostatic microactuator. A coupled electromechanical problem has been formulated using Galerkin method and solved considering dynamic actuation. The effect of excitation frequency is analyzed thoroughly to estimate the maximum stable displacement range of the actuator. Extensive studies illustrate that for suitable ac voltage–frequency combination, the maximum stable tip deflection for the cantilevers can be obtained even in the range of 50% to 90% of initial gap under specific damping. Such inherent dynamic characteristic of the actuator has been exploited here to achieve larger travel range. Furthermore, the theoretical observation has been experimentally demonstrated with a double microcantilever-based structure fabricated using silicon-on-insulator based process. The experimentation has been carried out using a developed in-house set-up. The major advantage in the present study is that unlike reported literature, the extended range has been achieved here without any additional complex circuits or design modifications. The proposed concept can be extended further to improve the displacement range of other microstructures toward different applications.

Mechanical behavior in superconducting composite wires

Numerical calculations of mechanical behavior in multi-filamentary Ag-alloy sheathed Bi-2212 superconducting round wires and cables are presented in this paper. Bi-2212 composite wires possess excellent current carrying capabilities even in magnetic field. The strain may lead to the degradation of critical current density. And the filament fracture caused by strain is a key issue to the application of superconductor. Due to the complex nonlinear electromagnetic characteristics of superconductor, we use the variational formulation based on the field-dependent critical state model to calculate the distributions of current and magnetic field in Bi-2212 composite superconducting wires and cables. The finite element method (FEM) is employed in the calculation of mechanical behavior. We have developed a numerical method by combining the variational formulation and FEM to calculate the two-dimensional electro-mechanical problem. The strain and stress distribution in infinity long Bi-2212 single wire, two wires, 6-around-1 cable and Rutherford cable carrying transport current in external magnetic field are calculated. The possible damage position in wires and cables is discussed.

Forced Vibrations and Dissipative Heating of a Hinged Bimorph Rectangular Plate with Open Electrodes

The problem of the forced vibrations and dissipative heating of piezoelectric rectangular plates with open electrodes and hinged ends under harmonic surface uniform pressure is stated. The electromechanical problem is reduced to an infinite system of algebraic equations. The exact analytical solution of this system is found and the exact analytical solution of the energy equation is obtained for vibrations at the first resonant frequency. A formula for the critical pressure reaching which the plate no longer performs its function due to loss of the piezoeffect is derived.

International Conference “Actual Problem of Electromechanics and Electrotechnology”

1. PrefaceInternational conference “Actual Problems of Electromechanics and Electrotechnology APEET-2017” was held November 13-16, 2017 in the Ural Federal University named after the first President of Russia B.N.Yeltsin.This conference presents papers that largely reflect problems that scientists and specialists in the field of electromechanics and modern electrotechnology solve, as well as in related fields of technology. This applies, first of all, to electrothermal equipment and electrotechnologies, power supplies and power supply systems of this equipment.Lists of committees/participants are available in this PDF.

Comprehensive Analytical Approximations of the Pull-In Characteristics of an Electrostatically Actuated Nanobeam under the Influences of Intermolecular Forces

In this paper, analytical closed-form expressions to accurately estimate the pull-in characteristics of an electrostatically-actuated doubly-clamped nanobeam are derived and examined. In this regard, a coupled electro-mechanical problem for the nano-actuator is first presented assuming a single mode approximation while taking into account all the possible structural, electrical and nanoscale effects: the fringing of the electrical actuating force, the geometric mid-plane stretching and intermolecular (van der Walls and Casimir) forces. The complicated nonlinear resultant equations are numerically approximated in order to derive the closed-form expressions for the important nano-actuator pull-in characteristics: i.e., the detachment length, the minimum reachable gap size before the collapse and the respective pull-in voltage. The resulting closed-form expressions are first quantitatively validated with other previously published results, and comparisons showed an acceptable agreement. Unlike the reported expressions in the literature, the proposed closed-form expressions in this work are proper approximations, fairly accurate and, more importantly, provide a quick estimate of the critical design pull-in parameters of the nano-actuator. In addition, the analysis of these expressions demonstrated that the consideration of the intermolecular forces together with the fringe effect tends to significantly reduce the threshold pull-in voltage, whereas the mid-plane stretching parameter tends to the contrary to increase the voltage at the pull-in collapse. The derived expressions of these analytical/approximate solutions could hopefully be appropriately used by NEMS engineers as simple/quick procedures for successful design and fabrication of electrostatically-actuated nano-devices.

Analysis of piezoelectric energy harvester under modulated and filtered white Gaussian noise

Abstract This paper proposes a comprehensive method for the electromechanical probabilistic analysis of piezoelectric energy harvesters subjected to modulated and filtered white Gaussian noise (WGN) at the base. Specifically, the dynamic excitation is simulated by means of an amplitude-modulated WGN, which is filtered through the Clough-Penzien filter. The considered piezoelectric harvester is a cantilever bimorph modeled as Euler-Bernoulli beam with a concentrated mass at the free-end, and its global behavior is approximated by the fundamental vibration mode (which is tuned with the dominant frequency of the dynamic input). A resistive electrical load is considered in the circuit. Once the Lyapunov equation of the coupled electromechanical problem has been formulated, an original and efficient semi-analytical procedure is proposed to estimate mean and standard deviation of the electrical energy extracted from the piezoelectric layers.

Piezoelectric TH-wave propagation in anisotropic solids subject to a bias

The transverse-horizontal wave propagating in a semi-infinite piezoelectric solid with hexagonal symmetry subject to initial electromechanical fields is investigated in this paper. The electromechanical boundary value problem is solved, and the phase velocity, the displacement, and the electric potential are obtained. For a metallized boundary surface, the dependency of the solution on the initial fields for several piezoelectric crystals is analyzed. These results may be proved useful to model the propagation of waves in anisotropic piezoelectric structures subject to a bias, serving as a benchmark for further numerical and experimental approaches.

Implicitly Coupled Electromechanical and Electromagnetic Transient Analysis Using a Frequency-Dependent Network Equivalent

This paper presents an improved solution of the implicitly coupled electromechanical and electromagnetic transient analysis problem using a frequency-dependent network equivalent (FDNE). In previous work, the fundamental frequency equivalent was used as the network equivalent of the external system. While simple, this fundamental frequency equivalent is only accurate for a single frequency, the fundamental frequency. This may lead to an inaccurate equivalent of the transient stability (TS) network. This paper extends previous work by using an FDNE derived from the vector-fitting technique. The FDNE is able to resolve the transient behavior over a wider frequency spectrum leading to a more accurate representation of the TS network. Results demonstrating the accuracy of the proposed scheme are presented on a 3-bus test system and the IEEE 118-bus system.

Analysis of laminated composite structures with embedded piezoelectric sheets by variable kinematic shell elements

In this article, the static analysis of multilayered shell structure embedding piezoelectric layers is performed using some advanced theories, obtained by expanding the unknown variables along the thickness direction using equivalent single-layer models, layer-wise models, and variable kinematic models. The variable kinematic models permit to reduce the computational cost of the analyses by grouping some layers of the multilayered structure with equivalent single-layer models and keeping the layer-wise models in other zones of the multilayer. This model is here extended to the static analysis of electro-mechanical problems. The used refined models are grouped in the Carrera Unified Formulation, and they accurately describe the displacement field, the stress distributions, and the electric potential along the thickness of the multilayered shell. The shell element has nine nodes, and the mixed interpolation of tensorial components method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement, and the finite element method is employed to solve them. Cross-ply plates and shells, with piezoelectric skins and simply supported edges, subjected to bi-sinusoidal mechanical or electrical load are analyzed. Various aspect ratios and radius-to-thickness ratios are considered. The results, obtained with different theories within Carrera Unified Formulation context, are compared with the elasticity solutions given in the literature. From the results, it is possible to conclude that the shell element based on Carrera Unified Formulation is very efficient in the study of electro-mechanical problems of composite structures. The variable kinematic models combining the equivalent single-layer with the layer-wise models permit to have a reduction of the computational costs, with respect to the full layer-wise theories, preserving the accuracy of the results in localized layers.

An efficient FE approach for attenuation of acoustic radiation of thin structures by using passive shunted piezoelectric systems

The present work concerns the numerical modeling of noise and vibration reduction of thin radiating structures in the low frequency range by using shunted piezoelectric elements. The aim is to propose an efficient approach able to predict the most radiating structural modes and to attenuate the corresponding resonances by using piezoelectric patches bonded on the structures and connected to resistive or resonant shunt circuits. The first step is to estimate the sound power radiated by the structure and to determine the vibration modes to be controlled. In a second step, an original finite element formulation, suitable for any elastic structure with surface-mounted piezoelectric patches, is proposed to solve the electromechanical problem. Finally, numerical examples are presented in order to validate and analyze the proposed approach.

Multiple cracks identification for piezoelectric structures

We propose a novel iterative procedure to solve the inverse problem of detecting multiple cracks in two-dimensional piezoelectric structures under dynamic electric loads. The effect of dynamic electric loads and a time-harmonic excitation with a single frequency is investigated. The crack is detected through the solution of an inverse time-dependent problem. The extended finite element method (XFEM) is employed for solving the forward problem as it allows the use of a single regular mesh for a large number of iterations with different crack geometries. The Newmark-beta method is employed for the time integration. In each iteration the forward problem is solved for various crack geometries, and at each iteration, the mechanical and electrical response of the piezoelectric structure is minimized at known specific points along the boundary to match the measured data. The minimization of the objective function is performed by multilevel coordinate search (MCS) method. The algorithm is an intermediate between purely heuristic methods and methods that allow an assessment of the quality of the minimum obtained and is in spirit similar to the direct method for global optimization. In this paper, the XFEM-MCS methodology under the dynamic electric load is applied to two-dimensional electromechanical problems where different number of straight cracks are considered. The results show that this methodology can be effectively employed for detecting multiple cracks in piezoelectric materials.

Monolithic modeling and finite element analysis of piezoelectric energy harvesters

This paper is devoted to monolithic modeling of piezoelectric energy harvesting devices. From a modeling perspective, piezoelectric energy harvesting is a strongly coupled phenomenon with two-way coupling between the electromechanical effect of the piezoelectric material and the harvesting circuit. Even in applications related to shunt damping, where the attached electrical circuit is passive, accurate modeling of the strong coupling is crucial for proper evaluation of the relevant parameters. The article proposes a monolithic mixed-hybrid finite element formulation for the predictive modeling and simulation of piezoelectric energy harvesting devices. The governing equations of the coupled electromechanical problem are converted into a single integral form with six independent unknown fields. Such a holistic approach provides consistent solution to the coupled field equations which involve structural dynamics, electromechanical effect of the piezoelectric patches and the dynamics of the attached harvesting circuit. This allows accurate computation of the eigenvalues and corresponding mode shapes of a harvester for any finite resistive load coupled to the harvester. The fully three-dimensional mixed-hybrid formulation is capable of analyzing structures with non-uniform geometry and varying material properties. The results of the finite element model are verified against the analytical results of a bimorph harvester with tip mass reported in the literature.

A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics

The paper presents aspects of implementation of a new high performance tensor contraction framework for the numerical analysis of coupled and multi-physics problems on streaming architectures. In addition to explicit SIMD instructions and smart expression templates, the framework introduces domain specific constructs for the tensor cross product and its associated algebra recently rediscovered by Bonet et al. (2015, 2016) in the context of solid mechanics. The two key ingredients of the presented expression template engine are as follows. First, the capability to mathematically transform complex chains of operations to simpler equivalent expressions, while potentially avoiding routes with higher levels of computational complexity and, second, to perform a compile time depth-first or breadth-first search to find the optimal contraction indices of a large tensor network in order to minimise the number of floating point operations. For optimisations of tensor contraction such as loop transformation, loop fusion and data locality optimisations, the framework relies heavily on compile time technologies rather than source-to-source translation or JIT techniques. Every aspect of the framework is examined through relevant performance benchmarks, including the impact of data parallelism on the performance of isomorphic and nonisomorphic tensor products, the FLOP and memory I/O optimality in the evaluation of tensor networks, the compilation cost and memory footprint of the framework and the performance of tensor cross product kernels. The framework is then applied to finite element analysis of coupled electro-mechanical problems to assess the speed-ups achieved in kernel-based numerical integration of complex electroelastic energy functionals. In this context, domain-aware expression templates combined with SIMD instructions are shown to provide a significant speed-up over the classical low-level style programming techniques.

An efficient multiscale computational formulation for geometric nonlinear analysis of heterogeneous piezoelectric composite

In this contribution, an efficient multiscale computational formulation is developed for the geometric nonlinear analysis of heterogeneous piezoelectric materials. In this formulation, the relations between the microscopic heterogeneous properties and macroscopic behaviors of the piezoelectric composite are established by the numerically constructed displacement and electric potential base functions. The heterogeneous and complex microstructure can be equivalent to a simple macroscopic piezoelectric coarse element through those constructed multiscale base functions. Then the equivalent tangent stiffness matrix and internal force vector of the macroscopic coarse element are deduced based on the co-rotational approach, which can describe the motion of macroscopic coarse element clearly and efficiently. Thus the original electro-mechanical coupling geometric nonlinear problem could be solved iteratively on the macroscopic scale, which will save a tremendous amount of computing time and cost. After all the macroscopic calculations, the microscopic mechanical and electrical responses could be retrieved and calculated from the macroscopic solutions by using the above-mentioned multiscale shape functions. To verify the validation and high-efficiency of the proposed multiscale computation formulation, several typical numerical examples are carried out. All the computation results indicate that the developed multiscale formulation not only could provide high precision solutions but also has high efficiency.

Analytical study of electromechanical buckling of a micro spherical elastic film on a compliant substrate part II: Postbuckling analysis

This work presents an analytical study of the electromechanical postbuckling of a spherical elastic thin film electrode that is bonded to a spherical dielectric compliant substrate. The compliant substrate is bonded to an inner rigid, fixed and electrically grounded spherical electrode. As will be shown, when the applied voltage reaches a critical value, the film buckles into a well ordered periodic surface patterns. The electromechanical postbuckling problem will analytically be solved for the one-dimensional, square checkerboard, hexagonal and herringbone periodic patterns. This work studies the effect of different geometry and material parameters on the selection and wavelength of the preferred/stable surface pattern of the buckled film. As will be deduced, the preferred buckling pattern can be determined to be either hexagonal or herringbone pattern depending on the geometry and material parameters of the film/substrate system. Furthermore, we study the effect of nonlinear stiffness of the compliant substrate on the electromechanical behavior of the film. The simplicity of generating and removing elastic surface wrinklings by On/Off voltage switching sufficiently increases the potential of the electromechanical buckling response to be employed in different MEMS application, such as, micro sensors, micro optical switches and deformable micro mirrors.

Cyclic steady states of nonlinear electro-mechanical devices excited at resonance

Author(s): Brandstetter, G; Govindjee, S | Abstract: We present an efficient numerical method to solve for cyclic steady states of nonlinear electro-mechanical devices excited at resonance. Many electro-mechanical systems are designed to operate at resonance, where the ramp-up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naive transient time-stepping from zero initial conditions. We use a recently developed high-order Eulerian–Lagrangian finite element method in combination with an energy-preserving dynamic contact algorithm in order to solve the coupled electro-mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro-electro-mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed-ups of the form Sn=n0.6·ξ−n0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley a Sons, Ltd.

Electromechanical decoupled model for cantilever-beam piezoelectric energy harvesters

Analysis of cantilever-based piezoelectric energy harvesting systems is usually performed using coupled equations that represent the mechanical displacement and the voltage output. These equations are then solved simultaneously. In contrast to this representation, we use analytical solutions of the governing equation to derive an algebraic equation of the power as a function of the beam displacement, electromechanical coefficients, and the load resistance. Such an equation can be more useful in the design of such harvesters. Particularly, the mechanical displacement is computed from a mechanical governing equation with modified natural frequency and damping ratio that account for the electromechanical coupling. The voltage and the harvested power are then obtained by relating them directly to the mechanical displacement. We validate the proposed analysis by comparing its solution including the tip displacement and harvested power with those of numerical simulations of the governing equations. To demonstrate the generality of the proposed approach, we consider the cases of base excitation, galloping, and autoparametric vibration. The model proposed in this study simplifies the electromechanical coupling problem for practical applications of cantilever-beam piezoelectric energy harvesting systems.

Theory for dielectrics considering the direct and converse flexoelectric effects and its finite element implementation

Flexoelectricity describes the linear energy coupling between the strain gradient and the electric polarization in a solid crystalline material. Based on strain-gradient theory, a new boundary-value model is developed for dielectrics when both the direct and converse flexoelectric effects are considered; here, the electric polarization is considered to be related to both the symmetrical and rotational strain gradients. Then, its finite element implementation is realized by establishing an equivalent-energy weak form of the problem. For this higher-order elastic problem, C1-continuous interpolations of the displacement and electric variables are required in the conventional displacement-based approach. In the present work, we construct an equivalent-energy weak form of the problem by introducing additional nodal degrees of freedom and enforcing the kinematic constraints between displacement and strain in the bulk and surface integration using Lagrange multipliers. The C1-continuous problem then reduces to a C0-continuous form. Using standard C0-continuous shape functions, some mixed-type finite elements are developed herein for dielectrics, in which the unknown variables, i.e., displacement, strain, rotational component and polarization intensity, are interpolated as independent nodal degrees of freedom. These elements are tested and applied to solve certain electromechanical problems, and their good convergence and accuracy are demonstrated via analysis of the mechanical and electrical properties of a dielectric.

Comparison of Nonlinear Domain Decomposition Schemes for Coupled Electromechanical Problems

The aim of this paper is to compare several domain decomposition schemes for nonlinear, coupled electromechanical problems. Both staggered and monolithic electrostatic/elastic formulations are considered for the multiphysics problem. The domain decomposition is applied either to a single multiphysics iteration, which corresponds to a linear problem, or to the full nonlinear multiphysics resolution. In the latter, the problem solved at each domain decomposition iteration is nonlinear. The influence of the elastic wave’s frequency and the electrostatic potential on the convergence rate and the computational cost of the algorithm is investigated on a 2-D model of a vibrating micromembrane array.

Transient thermoelectroelastic response of a piezoelectric material strip with two parallel cracks in arbitrary positions

ABSTRACTThis work is concerned with the thermoelectromechanical fracture behavior of two parallel cracks in arbitrary positions of a piezoelectric material strip under thermal shock loading. The crack faces are supposed to be insulated thermally and electrically. By using both the Laplace transform and the Fourier transform, the thermal and electromechanical problems are reduced to two systems of singular integral equations. The singular integral equations are solved numerically, and a numerical method is then employed to obtain the time-dependent solutions by way of a Laplace inversion technique. The intensity factors versus time for various geometric parameters are calculated and presented in graphical forms. The temperature, stress and electric displacement distributions in a transient state are also included.