Dynamic behavior of the interface between two solid phases in an elastic bar

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a semi-infinite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail, and an exact solution of the dynamical problem, which is governed by a nonlinear resursive formula, is obtained. It is shown that the phase boundary reaches a new equilibrium state for large time. Numerical calculations based on the recursive formula are carried out to illustrate analytical results.Address after August 15, 1995: Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA24061, USA.     

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is studied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special case is also given.     

Justification of equivalent substitution circuits used to optimize the dissipative properties of electroelastic bodies with external electric circuits

We consider elastoplastic systems which are piecewise homogeneous bodies composed of piezoelectric elements some of which have piezoelectrical properties. Electric series circuits consisting of resistors, capacitors, and inductance coils are applied to piezoelectric elements through the electrode coating on the body surface. The goal of the study is to develop efficient methods of mathematical modelling for determining the parameters of elements of the external electric circuit, which ensure, at prescribed resonance frequencies, the maximum damping properties of electroelastic bodies with external electric circuits. To choose effective circuits for solving the problem posed above, we suggest to pose the problem of natural vibrations of elastic bodies whose elements exhibit piezoeffect and have external electric circuits.As the most efficient approaches for calculating the electric circuit parameters necessary for the maximal damping, we propose some versions of equivalent circuits, which can be used to substitute elastic systems with piezoelectric elements. The most reliable equivalent substitution circuits are justified on the basis of the proposed problem of natural vibrations. Numerical results are obtained for a cantilever plate with a piezoelement connected through the electrode coated surface with a series electric circuit consisting of resistors, capacitors and inductance coils.     

Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading

The forced monoharmonic bending vibrations and dissipative heating of a piezoelectric circular sandwich plate under monoharmonic mechanical and electrical loading are studied. The core layer is passive and viscoelastic. The face layers (actuators) are piezoelectric and oppositely polarized over the thickness. The plate is subjected to harmonic pressure and electrical potential. The viscoelastic behavior of the materials is described by complex moduli dependent on the temperature of heating. The coupled nonlinear problem is solved numerically. A numerical analysis demonstrates that the natural frequency, amplitude of vibrations, mechanical stresses, and temperature of dissipative heating can be controlled by changing the area and thickness of the actuator. It is shown that the temperature dependence of the complex moduli do not affect the electric potential applied to the actuator to compensate for the mechanical stress __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 79–89, January 2008.     

Finite Oscillations of a Cylinder in a Coaxial Duct Subjected to Annular Compressible Flow

As a generalization considering small fluid-structural vibrations, the present paper examines the finite magnitude oscillatory motion of an elastically supported rigid cylinder in a cylindrical rigid duct conveying a compressible flow. The fluid is assumed to be inviscid and irrotational and free purely transverse vibrations of the body are dealt with. The governing equations of motion are the fully nonlinear Euler equations together with the continuity equation and a state equation (here for an ideal gas), the ordinary differential equation for the vibrating cylinder, and the kinematical transition and boundary conditions at the moving contact interface between fluid and body and the outside fluid border, respectively. A pertubation analysis is performed to calculate not only the dynamic characteristics for small coupled oscillations but also the corrections due to the inherent nonlinearities of the vibroacoustic problem. To make the calculation steps more transparent, the simpler problem of a two-dimensional channel flow between a rigid wall and an elastically supported rigid plate is also included in the present study. As an outlook, the influence of flexibility of the cylinder (or the plate) is addressed and the problem of forced vibrations is touched. This revised version was published online in July 2006 with corrections to the Cover Date.     

Influence of shear strains on damping the vibrations of a rectangular plate with piezoelectric actuators

The problem of active damping of the nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators is solved using Timoshenko’s hypotheses. To this end, two methods are employed: (i) a classical method of balancing the principal vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamical-programming method that reduces the problem to the matrix algebraic Riccati equation. The results obtained by both approaches are compared. The effect of the shear modulus on the amplitude of the damping potential difference and the deflection of the plate is analyzed     

Damping the vibrations of a rectangular plate with piezoelectric actuators

The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

Resonant vibrations of a clamped thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a clamped thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of the mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of self-heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of the resonance vibrations of the plate is studied     

Resonant vibrations of a hinged thermoviscoelastic rectangular plate with sensors and actuators

The paper discusses the active damping of the resonant flexural vibrations of a hinged thermoviscoelastic rectangular plate with distributed piezoelectric sensors and actuators. The thermoviscoelastic behavior of the passive and active materials is described using the concept of complex characteristics. The interaction of mechanical and thermal fields is taken into account. The Bubnov–Galerkin method is used. The effect of dissipative heating, the dimensions of the piezoelectric inclusions, and the feedback factor on the effectiveness of active damping of resonance vibrations of the plate is studied     

On solution of problems of dynamics of plates and shells with local structure heterogeneities

A general approach to the solution of nonstationary problems of the theory of shells with local structure heterogeneities is proposed. The approach is based on representation of solutions for the systems of differential motion equations of shell theory in the form of expansions in terms of free forms of vibrations (FFV). Two implementations of the approach, involving FFV expansions of certain members and FFV expansions of the system as a whole, are considered. The capabilities of the approach are illustrated by results for specific problems. In the first implementation, we determine the dynamic stress-strain state of a square plate with three rod supports. A round plate connected to a coaxial circular cylindrical shell is considered as an example of FFV expansion of the system as a whole. Special Design and Engineering Office, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 46–53, October, 1999     

压电功能梯度板自由振动的三维解

基于三维弹性理论和压电理论，导出了有限长矩形压电功率梯度板的动力学方程及相应的边界条件，并用幂级数展开法进行了求解，得到了压电功能梯度板自由振动的三维精确解公式，求解了自由振动的固有频率，并分析了压电系数的梯度变化对不同电学边界条件下压电板的自由振动频率的影响，结果可用于校核不同的近似理论及理解压电结构的动态行为。     

VIBRATION CONFINEMENT OF THICKNESS-SHEAR AND THICKNESS-TWIST MODES IN A FUNCTIONALLY GRADED PIEZOELECTRIC PLATE

We analyze shear-horizontal vibrations of a functionally graded piezoelectric plate of polarized ceramics or crystals of 6mm symmetry.The material of the central portion of the plate is different from that of the rest of the plate.It is shown that when the material properties of the central portion and the rest of the plate satisfy certain conditions,there exist trapped thicknessshear and thickness-twist modes whose vibrations are mainly confined within the central portion and decay quickly outside the central portion.The effects of the functionally graded material properties on the behaviors of the trapped modes are examined.     

Analytical solution for the cylindrical bending vibration of piezoelectric composite plates

An analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezoelectric materials. The laminated plate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. Fourier basis functions for the mechanical displacements and electric potential that identically satisfy the equations of motion and the charge equation of electrostatics are used to solve boundary value problems via the superposition principle. The coefficients in the infinite series solution are determined from the boundary conditions at the edges and continuity conditions at the interfaces between laminae, which are satisfied in the sense of Fourier series. The formulation admits different boundary conditions at the edges of the laminated piezoelectric composite plate. Results for laminated elastic plates with either distributed or segmented piezoelectric actuators are presented for different sets of boundary conditions at the edges.     

Transverse vibrations and vibrational heating of a rectangular bimorphous plate of dissipative piezoelectric material

The problem of induced resonance vibrations and dissipation heating in a rectangular bimorphous plate made of a dissipative piezoelectric material under a harmonic potential difference is tackled. The edges of the plate are considered to be hinged and ideally thermally insulated. The dissipation properties of the material are taken into account on the basis of the concept of complex characteristics, which are assumed to be temperature-independent. An exact solution is found for the problem. The critical value of the load parameter is determined when the maximum temperature reaches the Curie point. A finite-element method has been developed for investigating the dynamic behavior and temperature of vibrational heating that bimorphous plates made of a viscoelastic material undergo under a harmonic load. The results obtained for the electromechanical vibrations of plates by finite-element calculations and by an analytical solution are compared. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 85–93, September, 1999.     

Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic plate of variable thickness

An analysis and numerical results are presented for free transverse vibrations of non-homogeneous visco-elastic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through plate centre. The variation in density is assumed as parabolic along a line through plate centre, which raise non-homogeneity of the plate materials and make problem interesting as introducing variation in non-homogeneity of the material mass density reduce the problem practical importance in comparison to homogenous plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The governing differential equation of motion has been solved by Galerkin’s technique. Deflection, time period and logarithmic decrement corresponding to the first two modes of vibrations of a clamped non-homogeneous visco-elastic elliptic plate for various values of taper constant, thermal constants, non-homogeneity constant and aspect ratio are obtained and shown graphically.     

Basic relations of the theory of thermoviscoelastic plates with distributed sensors

The basic relations of the thermomechanics of thin-walled viscoelastic plates with distributed piezoelectric sensors under monogarmonic mechanical loading are presented. To describe the thermomechanical behavior of materials, the concept of complex characteristics is used. Numerical and variational methods are used to study the forced resonant vibrations of viscoelastic plates with distributed piezoelectric sensors. The effect of dissipative heating on the readings of the sensors of a circular viscoelastic plate undergoing axisymmetric resonant bending vibrations is analyzed as an example     

Forced axisymmetric vibrations and self-heating of a circular viscoelastic plate,controlled with piezoelectric sensors and actuators

The forced monoharmonic vibrations and self-heating of a circular thermoviscoelastic plate with piezoelectric sensor and actuator are studied. The viscoelastic behavior of the passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli. The problems of electroelasticity and heat conduction are solved numerically, assuming that the mechanical load is unknown. The effect of self-heating on the active damping of the vibrations of the plate is analyzed     

Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory

A finite element formulation of the piezoelectric vibrations of quartz resonators based on Mindlin plate theory is derived. The higher-order plate theory is employed for the development of a collection of successively higher-order plate elements which can be effective for a broad frequency range including the fundamental and overtone modes of thickness-shear vibrations. The presence of electrodes is also considered for their mechanical effects.The mechanical displacements and electric potential are combined into a generalized displacement field, and the subsequent derivations are carried out with all the generalized equations. Through the standard finite element procedure, the vibration frequency, the vibration mode shapes and the electric potential distribution are obtained. The frequency spectra are compared with some well-known experimental results with good agreement.Our previous experience with finite element analysis of high-frequency quartz plate vibrations leads us to believe that memory and computing time will always remain as key issues despite the advances in computers. Hence, the use of sparse matrix techniques, efficient eigenvalue solvers, and other reduction procedures are explored.     

Flexural Vibrations of a Piezoelectric Bimorph with a Cut Internal Electrode

Stationary vibrations of a bimorph plate composed of two piezoelectric layers of equal thickness are studied. There is an infinitely thin cut electrode between the layers. A model of flexural vibrations of the bimorph that is based on the variational equation generalizing the Hamilton principle in electroelasticity is proposed. For the plane problem, a system of equations of motion is derived and the boundary conditions and the conjugate conditions at the interface of the regions of the cut electrode are formulated. For the TsTS–19 piezoceramics, resonance and antiresonance frequencies are calculated. The values obtained are compared with the calculation results obtained with the use of the Kirchhoff model and the finite–element method. It is shown that the use of a plate with a cut electrode allows one to increase the efficiency of vibration excitation compared to the case of a continuous internal electrode.     

Viscoelastic vibrations of a triangular plates

The problem of stationary vibrations of a viscoelastic equilateral triangular plate is treated. The vibrations are caused by the action of a uniformly distributed load which varies harmonically and by the motion of the simply supported boundary of the plate as a rigid body which vibrates at the same frequency. The level lines of the vibration amplitude are studied and the graphs of the amplitude distribution along the height of the triangle are given.