Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators

The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.     

Antiplane Problem of Electroelasticity for a Hollow Piezoceramic Cylinder Excited by a System of Surface Electrodes

An antiplane mixed boundary-value problem of electroelasticity is considered for a hollow piezoceramic cylinder with an arbitrary system of active surface electrodes generating its harmonic vibrations. The problem is solved using a method elaborated earlier for investigating vibrations of a solid piezoceramic cylinder with a system of active surface electrodes. The scheme of numerical solution of the obtained singular integro-differential equations of the boundary-value problem is based on the quadrature method. Calculation results are presented that describe the amplitude-frequency characteristics of a hollow cylinder and the behavior of some mechanical and electric quantities both within the cylinder and on its boundary.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.     

Numerical modeling of the active damping of forced thermomechanical resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric inclusions

We consider the problem of active damping of forced resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric sensors and actuators. Here, the interaction of electromechanical and thermal fields is taken into account. For modeling of vibrations, we use the Kirchhoff–Love hypotheses as well as hypotheses adequate to them and describing the distribution of temperature and electric field quantities. The shell temperature increases as a result of dissipative heating. For the active damping of vibrations, piezoelectric sensors and actuators are used. It is supposed that the electromechanical characteristics of materials depend on the temperature. The solution of this complex nonlinear problem has been obtained by the iterative method and finite element method. We have investigated the influence of temperature of dissipative heating on the efficiency of active damping of vibrations of a viscoelastic cylindrical panel with rigid restraint of its edges.     

Mechatronic Subsystem Studied of Approximate Method Modeled by Hypergraph

The transverse vibrating mechatronic subsystem is considered. Integral parts of this system are: a continuous beam with known boundary conditions and a transducer, extorted by harmonic voltage excitation, to be perfectly bonded to the beam surface. Findings this article are dynamical characteristics of the discussed mechatronic and mechanical system to model them by hypergraphs. Research limitation is that the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered, however for this kind of systems the approach is sufficient. Practical implications of this researches was that global approach is presented, that means in the domain of frequency spectrum analysis. The methods of analysis and obtained results can be base of design and investigation for this type of mechatronic systems. Originality of this paper is that the mechatronic system created from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far because is it relies on application approximate methods of analysis of mechatronic subsystem and modeling the one by hypergraph [1-7]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced Vibration of Three-Layered Spherical and Ellipsoidal Shells under Axisymmetric Loads

A variant of vibration theory for three-layered shells of revolution under axisymmetric loads is elaborated by applying independent kinematic and static hypotheses to each layer, with account of transverse normal and shear strains in the core. Based on the Reissner variational principle for dynamic processes, equations of nonlinear vibrations and natural boundary conditions are obtained. The numerical method proposed for solving initial boundary-value problems is based on the use of integrodifferential approach for constructing finite-difference schemes with respect to spatial and time coordinates. Numerical solutions are obtained for dynamic deformations of open three-layered spherical and ellipsoidal shells, over a wide range of geometric and physical parameters of the core, for different types of boundary conditions. A comparative analysis is given for the results of investigating the dynamic behavior of three-layered shells of revolution by the equations proposed and the shell equations of Timoshenko and Kirhhoff-Love type, with the use of unified hypotheses across the heterogeneous structure of shells.     

Dynamic interaction of an elastic medium with a thin-walled curvilinear piezoelectric inclusion under longitudinal vibrations of a composite

Using the method of matched asymptotic expansions, we obtain models of dynamic interaction of a thin-walled curvilinear piezoelectric inclusion of variable thickness with an elastic isotropic matrix under stationary vibrations of the composite. The elastic system is under conditions of longitudinal shear. Different cases of electric boundary conditions on the surface of the heterogeneity are considered. We propose an algorithm for the construction of boundary layer corrections for refining the behavior of displacements and stresses in the vicinity of the edge of the inclusion for its different shapes.     

热-压电效应对轴压混合层合圆柱曲板后屈曲的影响

基于Karman-Donnell型非线性壳体方程，给出带压电作动器混合层合圆柱曲板在机械荷载、电荷载和热荷载作用下的后屈曲分析．假定温度场为均匀分布，电场仅有沿板厚方向的分量Ez，且假定材料性能常数与温度和电场的变化无关。将壳体屈曲的边界层理论推广到混合层合圆柱曲板受复合荷载作用的情况.相应的奇异摄动法用于确定圆柱曲板的屈曲荷载和后屈曲平衡路径.分析中同时考虑非线性前屈曲变形和初始几何缺陷的影响．数值算例给出完善和非完善，含整体覆盖或内埋压电作动器正交铺设层合圆柱曲板的后屈曲平衡路径。讨论了温度变化、控制电压、铺层方式、面内边界条件和初始几何缺陷等各种参数变化的影响。     

On the investigation of free vibrations of nonthin cylindrical shells of variable thickness by the spline-collocation method

For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization. The problem has been solved within the framework of the refined Timoshenko–Mindlin theory. We have also investigated the influence of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions.     

Deflection analysis of micro-scale microstretch thermoelastic beam resonators under harmonic loading

In this paper, the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler–Bernoulli beam resonators, due to time harmonic load have been investigated. The axial ends of the beam are assumed to be at either clamped-clamped, simply supported-simply supported or clamped-free conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection. The numerical simulation has been carried out with the help of MATLAB software for magnesium like material. The graphical representations and observations have been discussed for deflection of beam under various boundary conditions and for distinct considered values of time and space as well.     

A parametric investigation of the harmonic vibrations of a poroelastic rod

The problems of calculating the forced harmonic vibrations of porous structures saturated with a fluid are considered. The equations of motion are obtained by starting from general relations of the Biot theory of poroelasticity and continuous mechanics after taking into account the anisotropy of the elastic and hydraulic properties of the medium. The influence of the mechanical constants of the material on its dynamical characteristics is investigated using the example of the flexural vibrations of a simple rod-shaped structure.     

On 2D structured mesh generation by using mappings

A grid generation problem in two‐dimensional domains is considered by using a quasi‐conformal mapping of the parametric domain with a given square mesh onto the physical domain where the grid is required. To this end, a harmonic mapping is first applied, which, by the Radó theorem, is a diffeomorphism subject to some known conditions. However, the discrete harmonic mapping may produce folded meshes on a nonconvex domain with a strongly bent boundary. We demonstrate that it is caused by the truncation error. With the aim of controlling grid node location, an additional mapping is used. The Dirichlet problem for the universal elliptic partial differential equations is solved to construct the mapping. This allows to produce unfolded grids with a prescribed cell shape. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1072–1091, 2011     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.     

Vibrations of Piezoceramic Rods

Presently, most of the research on vibrations of monolithic piezoelectric rods at weak electric fields is restricted to longitudinal oscillations of such structural members where free and forced vibrations have been dealt with and in the case of resonance conditions not only linear but also nonlinear effects within the constitutive relations have been incorporated. On the other hand, bending and torsional vibrations of piezoceramic one‐parametric rods have not been examined yet. The present contribution develops a linear vibration theory of rods with a focus to bending vibrations taking into account rotatory inertia and shear deformation. The governing boundary value problem for beams with both longitudinal and as well transversal polarization is derived, in particular free vibrations are analyzed. Also nonlinear extensions not only of physical nature but also geometrical ones are addressed. A possible technical application is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenfrequencies of radial vibrations of a spherical sandwich shell

Free across-the-thickness vibrations of a closed spherical shell consisting of three rigidly connected layers with arbitrary physical constants and thicknesses are studied. A closed-form solution in displacements to a one-dimensional (along the radius) vibration problem for a homogeneous spherical shell is derived and then used in posing a boundary-value problem on free vibrations of a heterogeneous sphere. Based on the degeneration of the sixth-order determinant of a system of homogeneous equations satisfying the corresponding boundary conditions, a transcendental equation for eigenfrequencies is found. Transformation variants for the equation of eigenfrequencies in the cases of degeneration of physical and geometric parameters of the compound shell are considered. The main attention in investigating the lowest frequency is given to its dependence on the structure of shell wall, whose parameters greatly affect the calculated values of the high-frequency vibration spectrum of the shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 839–852, November–December, 2008.     

Generation of structured difference grids in two-dimensional nonconvex domains using mappings

Generation of structured difference grids in two-dimensional nonconvex domains is considered using a mapping of a parametric domain with a given nondegenerate grid onto a physical domain. For that purpose, a harmonic mapping is first used, which is a diffeomorphism under certain conditions due to Rado’s theorem. Although the harmonic mapping is a diffeomorphism, its discrete implementation can produce degenerate grids in nonconvex domains with highly curved boundaries. It is shown that the degeneration occurs due to approximation errors. To control the coordinate lines of the grid, an additional mapping is used and universal elliptic differential equations are solved. This makes it possible to generate a nondegenerate grid with cells of a prescribed shape.     

Magneto-Structural analysis of rotating electrical machines

The vibrations of the stator core of a rotating electrical machine induce acoustic noise. These oscillations of the stator yoke are excited of the force density due to the magnetic field in the air gap. This requires a transient magnetic field analysis coupled with a dynamic mechanical analysis. Coupling these two different physical fields results in a high numerical effort and usually one direction of the interaction is disregarded. This paper presents a method to calculate the vibrations of a stator core under design operating conditions. For this purpose, harmonic electromagnetic excitation forces have been calculated in a linear magnetic field analyses using the finite element method. The resulting forces have been applied to a linear structural dynamic FE model in the frequency domain. The results of the calculations are harmonic velocities specified by amplitude and phase from the structural surface of the stator core. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于扭转模态的角振动压电俘能器研究

以线性压电理论为基础,分析了压电陶瓷圆柱壳的扭转振动问题.由于压电陶瓷柱壳的扭转振动使得覆盖于柱壳两端的电极之间出现电势差,通过导线连接两电极与外负载,则圆柱壳在振动过程中所捕获的能量能有效地供给微电子器件工作.得到了输出电压、电流、能量、效率以及输出功率密度的解析表达式,并数值分析了这类压电俘能器的基本性能.结果表明,这种结构能够用作压电俘能器,将旋转机械能转化成为电能.