Forced vibration of composite cylindrical helical rods

The dynamic behavior of composite cylindrical helical rods subjected to time-dependent loads is theoretically investigated in the Laplace domain. The governing equations for naturally twisted and curved spatial laminated rods obtained using Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, the anisotropy of the rod material, effect of the rotary inertia, axial and shear deformations are considered in the formulations. The material of the rod is assumed to be homogeneous, linear elastic and anisotropic. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem accurately. The solutions obtained are transformed to the time domain using an appropriate numerical inverse Laplace transform method. The free vibration is then taken into account as a special case of forced vibration. The results obtained in this study are found to be in a good agreement with those available in the literature.     

Stress and electric potential fields in piezoelectric smart spheres

Piezoelectric materials produce an electric field by deformation, and deform when subjected to an electric field. The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices, including electric-mechanical actuators, sensors and structures. In this paper, a hollow sphere composed of a radially polarized spherically anisotropic piezoelectric material, e.g., PZT_5 or (Pb) (CoW) TiO₃ under internal or external uniform pressure and a constant potential difference between its inner and outer surfaces or combination of these loadings has been studied. Electrodes attached to the inner and outer surfaces of the sphere induce the potential difference. The governing equilibrium equations in radially polarized form are shown to reduce to a coupled system of second-order ordinary differential equations for the radial displacement and electric potential field. These differential equations are solved analytically for seven different sets of boundary conditions. The stress and the electric potential distributions in the sphere are discussed in detail for two piezoceramics, namely PZT_5 and (Pb) (CoW)TiO₃. It is shown that the hoop stresses in hollow sphere composed of these materials can be made virtually uniform across the thickness of the sphere by applying an appropriate set of boundary conditions.     

Transient response of circular plates and membranes: A numerical approach

A systematic discussion on the use of numerical Laplace transforms for the computation of response of circular plates and membranes is presented. Applying the Laplace transform to the governing differential equation of motion in polar coordinates reduces the dynamic problem to an equivalent static one. The resulting equations are discretized by using the finite difference technique and the solution is obtained in the Laplace plane. Time-domain solutions are obtained by using the numerical inversion technique proposed by Durbin based on sine—cosine transforms. The effects of internal viscoelastic damping and external viscous damping on the response are also studied. Several examples are tackled to illustrate the procedure.     

基于均匀化理论粘弹性复合材料松弛规律的研究

基于均匀化理论研究了粘弹性复合材料的松驰规律,推导出了均匀化方法计算等效松驰模量的一般形式。在Laplace域中计算等效松弛模量,并用最小二乘法拟合Prony级数表示的松弛函数,然后在时间域中进行数值的Laplace逆转换,得到时间域内等效复合模量。给出了算例来验证这个计算方法的可行性并讨论了松弛模量随夹杂体积分数变化的规律。     

Transient response of a permeable crack normal to a piezoelectric-elastic interface: Anti-plane problem

In this paper, the anti-plane transient response of a central crack normal to the interface between a piezoelectric ceramics and two same elastic materials is considered. The assumed crack surfaces are permeable. By virtue of integral transform methods, the electroelastic mixed boundary problems are formulated as two set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. Numerical values on the quasi-static stress intensity factor and the dynamic energy release rate are presented to show the dependences upon the geometry, material combination, electromechanical coupling coefficient and electric field.     

液体圆柱静压导轨设计参数对其性能的影响

为研究液体圆柱静压导轨的初始参数对导轨性能的影响,以内反馈节流形式的液体圆柱静压导轨为研究对象,列出力平衡方程、流量连续性方程,经推导和线性化处理得到液体圆柱静压导轨的线性化微分方程组,利用Laplace变换得到传递函数,推导出液体圆柱静压导轨的数学模型。从时域、频域内分别分析初始油膜厚度、油液黏度、供油压力对导轨性能的影响。研究表明在低频段减小初始油膜厚度、增大油液黏度和供油压力,在高频段增大初始油膜厚度,可增大导轨动态刚度,提高支承的稳定性,减小导轨间隙相对稳态位移值。在高频段,油液黏度、供油压力对液体圆柱静压导轨的动态性能影响不大。研究工作对液体圆柱静压导轨的设计提供参考价值。     

Transient free convection flow past an infinite vertical cylinder with thermal stratification

This paper presents an analytical solution of one-dimensional free convection flow past an infinite vertical circular cylinder in a stratified fluid medium. The dimensionless unsteady coupled linear governing equations are solved by Laplace transform technique for the case when the Prandtl number is unity. Due to the effect of thermal stratification, the velocity, temperature, skin-friction and Nusselt number shows oscillatory behavior at smaller times and then reaches steady state at larger times, while this behavior is not seen in the absence of stratification.     

Wave propagation in an elastic/viscoplastic hollow sphere

An analysis of the dynamic problem of a hollow sphere is presented. The medium is assumed elastic/viscoplastic, satisfying Mises condition, isotropic hardening and viscoplastic incompressibility. The cavity is subjected to a radially applied and continuously maintained impact load. The analysis for viscoplastic waves is based on the method of characteristics and a generalized form of Malvern's theory for strain-rate dependent materials. A bilinear shear stress-shear strain curve is considered and calculations are carried out on the IBM 7094 computer. The paper examines the influence of the important parameters governing spherical wave propagation on the response of the sphere; these parameters include the level of strain hardening, the viscosity coefficient, Poisson's ratio and the geometry of the sphere. It is shown that the degree of strain hardening has a significant effect on the amplitude of the oscillations with the period unaffected. The viscosity coefficient becomes more influential as the wall thickness of the sphere increases.     

Piezoelectric hollow cylinder with thermal gradient

The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, an analytic solution to the axisymmetric problem of a radially polarized, radially orthotropic piezoelectric hollow cylinder with thermal gradient is developed. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress and potential field distributions in the cylinder are obtained numerically for two piezoceramics. It is shown that the hoop stresses in a cylinder composed of these materials can be decreased throughout the cross-section by applying an appropriate set of boundary conditions. This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han Mahdi Saadatfar received a B. S. degree in Mechanical Engineering from University of Kashan 2006. He is currently a M.S student at the School of Mechanical Engineering at University of Tehran, Iran. He is currently researching about modeling of nanoindentation process in nanocomposites. Mr. Saadatfar’s research interests are in the area of piezoelectric Materials, Polymer/Clay nanocomposites and Finite element modeling. He has several published paper about piezoelectric materials and Finite element modeling of nanocomposites. Amin Shariat Razavi received a B.S degree in Mechanical Engineering from Kashan University in 2006. He is currently testing and examining an specific type of intelligent plasma cutting machine for process equipment that is designed by himself. Mr. Razavi’s research interests are smart materials and design of mechanical system.     

Numerical solution of two-dimensional or axi-symmetric incompressible flow using the vorticity equations

A method to integrate the vorticity-velocity form of two-dimensional or axi-symmetric incompressible Navier-Stokes equations is described. The method employs equi-potential lines and streamlines of an inviscid flow as coordinate lines and the velocity field is determined from the vorticity distribution with boundary conditions on the normal velocity only, while the tangential velocities are used as boundary conditions for the vorticity. The results of numerical experiments on time-dependent flow past an impulsively started circular cylinder and sphere are, then, presented to demonstrate the performance of the scheme. Numerical results show that the present method is very stable and accurate.     

Thermoelastic analysis of non-uniform pressurized functionally graded cylinder with variable thickness using first order shear deformation theory(FSDT) and perturbation method

Recently application of functionally graded materials(FGMs) have attracted a great deal of interest. These materials are composed of various materials with different micro-structures which can vary spatially in FGMs. Such composites with varying thickness and non-uniform pressure can be used in the aerospace engineering. Therefore, analysis of such composite is of high importance in engineering problems. Thermoelastic analysis of functionally graded cylinder with variable thickness under non-uniform pressure is considered. First order shear deformation theory and total potential energy approach is applied to obtain the governing equations of non-homogeneous cylinder. Considering the inner and outer solutions, perturbation series are applied to solve the governing equations. Outer solution for out of boundaries and more sensitive variable in inner solution at the boundaries are considered. Combining of inner and outer solution for near and far points from boundaries leads to high accurate displacement field distribution. The main aim of this paper is to show the capability of matched asymptotic solution for different non-homogeneous cylinders with different shapes and different non-uniform pressures. The results can be used to design the optimum thickness of the cylinder and also some properties such as high temperature residence by applying non-homogeneous material.     

Analytical aspects in boundary integral equation formulation for the generalized linear micropolar thermoelasticity

A formulation of the boundary integral equation method for generalized linear micropolar thermoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, Green–Naghdi, theory of type II without energy dissipation, Green–Naghdi, theory of type III and Chandrasekharaiah and Tzou theory with dual-phase lag, as well as to the dynamic coupled theory.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

Squeeze films in porous circular disks

The squeeze film behaviour between two circular disks is analysed. The governing equations are derived when one disk has a porous facing. The fluid in the film region satisfies the modified Reynolds equation and the flow in the porous facing. satisfies the Laplace equation. These equations are solved in closed form and expressions are derived for pressure distribution, load capacity and time of approach for the plates. It is found that with a suitable value of permeability for the porous facing, the pressure distribution becomes more even leading to uniform wear of the entire facing; however, increasing permeability has an adverse effect on the load capacity and time of approach. Numerical data, useful in the design of clutch plates with a porous lining, is given in tabular form.     

Large amplitude vibrations of thin elastic plates by the method of conformal transformation

A unified method for investigating large amplitude vibrations of thin elastic plates of any shape under clamped edge boundary conditions is presented, based on Von Karman governing equations generalised to the dynamical case. The conformal mapping technique is introduced and the domain is conformally transformed on to the unit circle. The deflection function is chosen beforehand in conformity with the prescribed boundary conditions and the stress function is solved taking only the first term of the mapping function. The transformed differential equations are solved by the Galerkin procedure to obtain the second order nonlinear differential equation for the unknown time function. The time equation is readily solved in terms of Jacobian elliptic functions. Frequency of linear and nonlinear oscillations as well as static nonlinear case are analysed for plates of circular, and regular polygonal shape. Results obtained are compared with other known results. From the comparative study of different results it is observed that the first term approximation of the mapping function yields fairly accurate results with less computational effort.     

Squeeze films between a rigid cylinder and an elastic layer bonded to a rigid foundation

A numerical solution to the elastohydrodynamic lubrication problem of a rigid cylinder and an elastic layer firmly bonded to a rigid substrate in normal approach and separated by a fluid of constant viscosity is presented. The rate of change of the deformation with time was neglected in the present investigation. The governing equations were solved via an iterative method in order to compute the pressure distribution and the corresponding film profile.Influences of the layer thickness, the layer compressibility and the central squeeze-film velocity on the results were investigated. In the case of the Hertzian contact, the present method was validated against the results reported in Herrebrugh [Elastohydrodynamic squeeze films between two cylinders in normal approach, Trans ASME, J Lubric Technol Ser F 1970; 92:292–302] where the results showed a very good agreement.     

Thermoelastic damping in cylindrical shells with application to tubular oscillator structures

Thermoelastic vibration and damping of cylindrical shell structures is studied in this paper. The general thermoelastic coupled equations for cylindrical thin shells are presented first. Since the general governing equations are quite complicated, they are then simplified with Donnell–Mushtari–Vlasov approaches for the cylindrical shells under transverse deflection-dominated vibrations. By solving the simplified thermoelastic equations with Galerkin method, the approximate solutions for thermoelastic damping in a cylindrical shell structure are obtained. The solutions are suitable for the predictions of thermoelastic damping of tubular oscillator structures. Some numerical examples for micro- or nano-tube resonators are provided to illustrate the results.     

厚壁圆筒安定问题的统一解析解

采用双剪统一强度准则对理想弹塑性材料的厚壁圆筒进行安定性分析,得出了厚壁圆筒加载应力、残余应力及安定极限压力的统一解析解。所得的解不仅能包含以往基于Tresca准则、Mohr—Coulomb准则和广义双剪准则的结果,而且给出一系列新的结果。这些结果能考虑材料的拉压强度差效应及中间主应力效应,因而适应于多种材料。最后,利用所得的解研究了材料的中间主应力效应和拉压强度差效应对厚壁圆筒安定极限压力的影响。结果表明：当考虑材料的拉压强度差效应及中间主应力效应时,厚壁圆筒安定极限压力将明显提高。     

Nonlinear transient heat transfer and thermoelastic analysis of thick-walled FGM cylinder with temperature-dependent material properties using Hermitian transfinite element

Nonlinear transient heat transfer and thermoelastic stress analyses of a thick-walled FGM cylinder with temperature-dependent materials are performed by using the Hermitian transfinite element method. Temperature-dependency of the material properties has not been taken into account in transient thermoelastic analysis, so far. Due to the mentioned dependency, the resulting governing FEM equations of transient heat transfer are highly nonlinear. Furthermore, in all finite element analysis performed so far in the field, Lagrangian elements have been used. To avoid an artificial local heat source at the mutual boundaries of the elements, Hermitian elements are used instead in the present research. Another novelty of the present paper is simultaneous use of the transfinite element method and updating technique. Time variations of the temperature, displacements, and stresses are obtained through a numerical Laplace inversion. Finally, results obtained considering the temperature-dependency of the material properties are compared with those derived based on temperature independency assumption. Furthermore, the temperature distribution and the radial and circumferential stresses are investigated versus time, geometrical parameters and index of power law. Results reveal that the temperature-dependency effect is significant.     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.