Stresses distributions in a rotating functionally graded piezoelectric hollow cylinder

An analytic solution to the axisymmetric problem of a long, radially polarized, hollow cylinder composed of functionally graded piezoelectric material (FGPM) rotating about its axis at a constant angular velocity is presented. For the case that electric, thermal and mechanical properties of the material obey different power laws in the thickness direction, distributions for radial displacement, stresses and electric potential in the FGPM hollow cylinder are determined by using the theory of electrothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material nonhomogeneity, and adopting suitable graded indexes and applying suitable geometric size and rotating velocity ω may optimize the rotating FGPM hollow cylindrical structures. This will be of particular importance in modern engineering application.     

旋转功能梯度智能结构的刚-柔耦合动力学分析

基于一次耦合模型理论建立了中心刚体-压电层-功能梯度材料智能梁系统的刚柔耦合动力学模型．研究了开环状态下将压电材料作为传感器的压电效应和质量刚度效应对系统动力学特性的影响．通过仿真算例与另两种不同建模理论（传统零次近似耦合模型、一次近似耦合模型）作了对比．随着中心刚体外驱动力矩的增大,零次近似耦合模型和一次近似耦合模型计算结果逐渐发散,而本文的一次耦合模型的计算结果始终保持收敛,较其他近似耦合模型具有一定优势．对三种不同的结构的计算结果表明,压电材料的压电效应对系统的动力学特性影响显著,压电材料的质量刚度效应也会影响智能梁的动力学行为,前者比后者的影响大得多．此外,功能梯度材料功能梯度指数对系统动力学特性的影响也较大．     

A novel type of transverse surface wave propagating in a layered structure consisting of a piezoelectric layer attached to an elastic half-space

The existence and propagation of transverse surface waves in piezoelectric coupled solids is investigated, in which perfect bonding between a metal/dielectric substrate and a piezoelectric layer of finite-thickness is assumed. Dis- persion equations relating phase velocity to material con- stants for the existence of various modes are obtained in a simple mathematical form for a piezoelectric material of class 6mm. It is discovered and proved by numerical examples in this paper that a novel Bleustein-Gulyaev （B-G） type of transverse surface wave can exist in such piezoelectric cou- pled solid media when the bulk-shear-wave velocity in the substrate is less than that in the piezoelectric layer but greater than the corresponding B-G wave velocity in the same pie- zoelectric material with an electroded surface. Such a wave does not exist in such layered structures in the absence of pie- zoelectricity. The mode shapes for displacement and electric potential in the piezoelectric layer are obtained and discussed theoretically. The study extends the regime of transverse sur- face waves and may lead to potential applications to surface acoustic wave devices.     

Solution of the Reynolds Equation for Viscous-Fluid Flow in the Clearance between a Rotating Ellipsoid and a Surface

Slow viscous-fluid flows in the narrow clearance (i) between a moving ellipsoid and a straight tube of elliptic cross section and (ii) between a rotating ellipsoid and a toroidal tube, including the case of an ellipsoid near a plane, are considered. A solution of the boundary-value problem for the Reynolds equation describing the flow in the clearance is found. The similarity of the pressure profiles in the “ellipsoid-plane” and “ cylinder-plane” systems is indicated.     

Dynamically Equilibrium Shape of Intrusive Vortex Formations in the Ocean

The problem of finding the dynamically equilibrium shape of a rotating mass of liquid with homogeneous density (lens) submerged in a stratified ocean at rest on the rotating Earth is formulated. An equation for the shape of the interface between water masses is derived. An exact solution of the problem for an anticyclonically rotating lens in a linearly stratified ocean in the neighborhood of the lens depth shows that the dynamically equilibrium shape of the interface is a triaxial ellipsoid inclined with respect to the horizon and similar to an ellipsoid of revolution for real parameters of the phenomenon. The limiting values of the latitudes at which these formations can exist are determined. Degeneration of the shape with decrease in the intrinsic lens angular rate is investigated.     

Rotating sandwich cylindrical shells with an FGM core and two FGPM layers: free vibration analysis

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material(FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded(FG),the material properties vary along the thickness direction as one innovation of this study.Applying the first-order shear deformation theory(FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations(PDEs) using Hamilton's principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material(FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM.     

Problems of Hydrodynamics for a Triaxial Ellipsoid

Problems of motion of a triaxial ellipsoid in an ideal liquid and in a viscous liquid in the Stokes approximation and also equilibrium shapes of the rotating gravitating liquid mass are considered. Solutions of these problems expressed via four quadratures depending on four parameters are significantly simplified because they are expressed via the only function of two arguments. The efficiency of the proposed approach is demonstrated by means of analyzing the velocity and pressure fields in an ideal liquid, calculating the added mass of the ellipsoid, determining the viscous friction, and studying the equilibrium shapes and stability of the rotating gravitating capillary liquid. The pressure on the triaxial ellipsoid surface is expressed via the projection of the normal to the impinging flow velocity. The shape of an ellipsoid that ensures the minimum viscous drag at a constant volume is determined analytically. A simple equation in elementary functions is derived for determining the boundary of the domains of the secular stability of the Maclaurin ellipsoids. An approximate solution of the problem of equilibrium and stability of a rotating droplet is presented in elementary functions. A bifurcation point with non-axisymmetric equilibrium shapes branching from this point is found.     

Effect of rotation on Rayleigh waves in piezothermoelastic half space

The present paper deals with the study of rotation effect on the characteristics of Rayleigh waves propagating in a homogeneous, transversely isotropic piezothermoelastic half space in the framework of linear theory including Coriolis and Centrifugal forces. The medium is subjected to stress free, thermally insulated, electrically shorted/charge free boundary conditions and is rotating about an axis perpendicular to its plane. Characteristics of surface waves propagating in thermoelastic piezoelectric solids and their dependence upon various geometric and physical parameters are derived. After deriving secular equations in closed form and isolated mathematical conditions, the effect of rotation on dispersion curves and attenuation profiles is studied. The specific loss factor and relative frequency shift are also obtained in case of open and closed circuit electric surface conditions. Finally, in order to illustrate and verify the analytical results, numerical solution of various secular equations and other relevant relations are derived for cadmium selenide (6 mm) class material. The analysis shows that the rotation sensitivity at long wavelengths (in the vicinity of the surface) is substantially greater than those at short wavelengths (deep into the half space). The study is very helpful in the development of rotation sensors and other piezoelectric devices.     

主动控制压电旋转悬臂梁的参数振动稳定性分析

在工程实际中旋转机械由于制造和加工误差,装配的不均匀性等原因,往往会脉动运行,这将使得机械系统发生参数振动. 当脉动参数满足一定关系时,这种参数振动将会失稳,进而影响机械结构的正常运转. 本文针对这一问题,引入压电材料对 脉动旋转悬臂梁系统的振动进行控制,研究主动控制悬臂梁系统的参数振动优化设计问题,采用 Hamilton 变分原理与一阶 Galerkin 离散相结合的方法,建立了受速度反馈传感器主动控制的压电旋转悬臂梁的一阶近似线性控制方程. 运用多尺度方法,得到了压电旋转悬臂梁系统在发生1/2亚谐波参数共振时稳定性边界的控制方程,并利用直接分析方法验证了解析摄动解的正确性. 将摄动解中临界阻尼比和轮毂角速度脉动幅值的无量纲参数作为评价系统稳定性能的指标. 通过数值算例,分析了轮毂半径、轮毂角速度平均值和脉动幅值、梁长以及速度传感器的反馈增益系数对系统稳定性区域的影响. 研究结果表明,梁长、轮毂半径、脉动幅值会降低系统稳定性,反馈增益系数可以提高系统稳定性,而轮毂角速度平均值与系统稳定性之间有非单调的关系. 为进一步设计压电旋转机械结构提供了理论依据.      

Eigenvibrations in trapped-energy contoured piezoelectric resonators with a one-sided arbitrarily oriented elliptical convexity

A mathematical model of a piezoelectric plate is discussed with a one-sided convex surface of an arbitrary shape. Generic relations are given to calculate the frequency spectrum and distributions of the vibration amplitudes with normal drive levels. A particular case of the ellipsoidal convexity is studied, that is arbitrarily oriented with respect to the plate axes. A numerical example for such a curvature is also given. Based upon this, we show that the frequency spectrum is critically dependent on the angle between the main ellipsoid axis and the rotated coordinates of a piezoelectric plate and to the ratio of the main radii of an ellipsoid. We notice that such a surface allows for a proper placement of the anharmonics in the frequency spectrum, avoiding their interaction caused by environment. It may also be useful to model the manufacturing imperfection and its influence upon the vibration spectra.     

Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft

In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.     

压电/纤维复合材料耦合旋转驱动器力学设计模型

本文在经典层合板理论和三维压电材料本构方程的基础上,依据压电材料与纤维复合材料之间的相互作用、位移连续条件,建立了压电—纤维复合材料旋转耦合驱动器力学模型,并推导出电耦合方程。计算和初步实验说明,这种新型结构的压电伸缩变形—离轴纤维复合层扭转变形耦合旋转驱动器比同类型驱动器能量密度高、转矩大,且这种旋转电机结构简单紧凑、工作稳定、寿命长、预计随着研究的进一步深入,可望成为一种新型的微致驱动器而广泛应用于微型机械、精密测量和自动控制等领域。     

Propagation characteristics of thermoelastic waves in piezoelectric (6 mm class) rotating plate

The propagation of Lamb waves in a homogeneous, transversely isotropic (6 mm class), piezothermoelastic plate rotating with uniform angular velocity about normal to its boundary has been investigated. The generalized (non-classical) theories of thermoelasticity in contrast to Sharma and Pal [Sharma, J.N., Pal, M., 2004. Lamb wave propagation in transversely isotropic piezothermoelastic plate. J. Sound Vib. 270, 587–610] have been used to investigate the problem. The surfaces of the plate are subjected to stress free, thermally insulated/isothermal and electrically shorted boundary conditions. Secular equations for wave propagation modes in the plate are derived from a coupled system of governing partial differential equations of linear piezothermoelasticity. After obtaining the complex characteristic roots with the help of Descartes' algorithm, the transcendental secular equations have been solved by functional iteration numerical technique to compute phase velocity and attenuation coefficient. Finally, in order to illustrate the analytical development, numerical solution of secular equations is carried out for PZT-5A piezo-thermoelastic material. The corresponding simulated results of various physical quantities such as phase velocity, attenuation coefficients, specific loss factor of energy dissipation, thermo-mechanical coupling factor and relative frequency shifts have been presented graphically for both rotating and non-rotating plates for comparison purpose. There is a scope for extension of the present work to other classes of piezo/pyroelectric crystals. The study will be useful in design and construction of gyroscope, rotation sensors, temperature sensors and other pyro/piezoelectric surface acoustic wave (SAW) devices.     

旋转压电纳米梁的振动分析

本文基于非局部弹性理论，对旋转压电纳米梁模型的振动进行了分析．首先由哈密顿原理导出旋转压电纳米梁的动力学控制方程及相应的边界条件；再通过微分求积法对控制方程和两类边界条件进行离散；最后通过数值计算分析振动特性．通过改变旋转角速度、轮毂半径、非局部参数以及外部电压分析它们对压电纳米梁振动频率的影响关系．数值结果表明这些参数对压电纳米梁固有频率有不可忽略的影响，本文进一步讨论了旋转角速度对结构模态的影响．     

Control of stress response in a rotating infinite hollow multilayered piezoelectric cylinder

The theoretical study of the control of stress is developed for a rotating infinite hollow multilayered radially polarized piezoelectric cylinder. The exact solution is obtained by means of the state-space method. As an illustrative example, the distribution of the radial and tangential stresses in a rotating hollow internally pressurized five-layered piezoelectric cylinder subjected to different electric potential at the internal and external surfaces are performed. Numerical results show that the distribution of the stress can be controlled by applying appropriate electric potentials at the correct surfaces.     

Dynamic response of a rotating disk submerged and confined. Influence of the axial gap

In this paper, the natural frequencies and mode shapes of a rotating disk submerged and totally confined inside a rigid casing, have been obtained. These have been calculated analytically, numerically and experimentally for different axial gaps disk-casing. A simplified analytical model to analyse the dynamic response of a rotating disk submerged and confined, that has been used and validated in previous researches, is used in this case, generalised for arbitrary axial gaps disk-casing. To use this model, it is necessary to know the averaged rotating speed of the flow with respect to the disk. This parameter is obtained after an analytical discussion of the motion of the flow inside the casing where the disk rotates, and by means of CFD simulations for different axial positions of the disk. The natural frequencies of the rotating disk for the different axial confinements can be calculated following this method. A Finite Element Model has been built up to obtain the natural frequencies by means of computational simulation. The relative velocity of the flow with respect to the disk is also introduced in the simulation model in order to estimate the natural frequencies of the rotating disk. Experimental tests have been performed with a rotating disk test rig. A thin stainless steel disk (thickness of 8 mm, (h/r<5%) and mass of 7.6 kg) rotates inside a rigid casing. The position of the disk can be adjusted at several axial gaps disk-casing. A piezoelectric patch (PZT) attached on the rotating disk is used to excite the structure. Several miniature and submergible accelerometers have measured the response from the rotating frame. Excitation and measured signals are transmitted from the rotating to the stationary frame through a slip ring system. Experimental results are contrasted with the results obtained by the analytical and numerical model. Thereby, the influence of the axial gap disk-casing on the natural frequencies of a rotating disk totally confined and surrounded by a heavy fluid is determined.     

Dynamic analysis of a rotating rigid-flexible coupled smart structure with large deformations

Based on Hamilton's principle,a new kind of fully coupled nonlinear dynamic model for a rotating rigid-flexible smart structure with a tip mass is proposed.The geometrically nonlinear effects of the axial,transverse displacement and rotation angle are considered by means of the first-order approximation coupling(FOAC)model theory, in which large deformations and the centrifugal stiffening effects are considered.Three kinds of systems are established respectively,which are a structure without piezoelectric layer,with piezoelectric layer in open circuit and closed circuit.Several simulations based on simplified models are presented to show the differences in characteristics between structures with and without the tip mass,between smart beams in closed and open circuit, and between the centrifugal effects in high speed rotating state or not.The last simulation calculates the dynamic response of the structure subjected to external electrical loading.     

梁板结构埋入压电片的深度和厚度优化研究

根据梁的弯曲变形理论，以及压电材料的压电效应，对压电复合材料梁结构的应变和应力状态进行了分析。推导了压电材料埋入梁结构时其埋入深度和压电材料本身厚度的优化目标函数。根据优化目标函数，绘制出了在给定的基体材料和压电材料弹性模量比之下，压电材料的驱动力矩随埋入深度和压电材料厚度变化的三维曲面和等高线图，直观地表示出了埋入型压电材料智能结构获得最佳驱动力矩的配置方案。同时也分析了结构和压电材料两者的弹性模量之比对其结果的影响。结果表明，压电材料对于结构的作动力矩与其埋入基体材料的深度、本身厚度以及基体材料和压电材料的弹性模量之比都有着密切的联系。同时将结果也推广到了压电板结构，此时压电材料和基体材料的泊松比时其结果也有一定的影响。     

PMN-PT夹层弹性半空间结构中SH波的传播Propagation of SH waves in sandwich structures consisting of PMN-PT single crystal layer and elastic half-spaces

分析了弹性上下半空间和PMN‐PT单晶层组成的夹层结构中SH波的传播性质,PMN‐PT单晶沿[011]c方向极化,宏观上呈mm2对称,且晶体沿角度θ方向切割。基于正交各向异性压电材料和各向同性弹性材料的基本方程,得到了夹层结构中SH波传播时行列式形式的频散方程。通过对数值算例进行分析可以看出,PMN‐PT单晶的切割角度和弹性材料属性对结构中的相速度有很大影响,因此波的某些传播性能可以通过材料的设计以及晶体切割的方向来实现,这些结论为声表面波器件的开发和应用提供了理论依据。     

具有压电材料弹性环形板的屈曲

对在不同位置粘有任意多对压电片的弹性环形板在电压和径向压力作用下的轴对称屈曲进行了研究．根据压电效应的等效作用量，初参数法以及传递矩阵方法得到了环板屈曲的特征方程．对不同位置及不同宽度的压电片，计算了结构的屈曲载荷．并得到了稳定性边界曲线．