Reformulation of Navier equations for solving three-dimensional elasticity problems with applications to thick plate analysis

In this paper, a new reformulation of the Navier equations of motion is introduced for solving the known three-dimensional elastostatics and elastodynamics problems. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. These equations are also invariant with respect to the choice of the coordinate system. In order to solve a three-dimensional elasticity problem based on the presented formulation, one of the three equations in terms of displacement components and the corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. In order to verify the relations, the closed-form solutions are obtained for deflection and natural frequencies of the thick rectangular plate. The numerical results are compared with available results in the literature and it can be seen that the results of the present study are identical to those of the previous works.     

黏弹性三参数地基上Timoshenko梁横向自由振动

运用复模态分析方法研究了黏弹性三参数地基上Timoshenko梁的横向振动特征,得到简支边界条件下的频率方程以及模态函数表达式。通过具体算例，分析了各项地基参数对固有频率和模态函数的影响，比较了相同地基上作用的Timoshenko梁和Euler-Bernoulli梁的振动特征。结果表明，随着地基刚度、剪切参数的增大以及黏性系数的减小，各阶固有频率值均增大；Timoshenko梁的固有频率略低于Euler-Bernoulli梁。     

大型发电机定子端部绕组振动的叠层加筋圆锥壳模型

采用正交各向异性叠层复合材料加筋圆锥壳模型,对大型汽轮发电机定子端部绕组整体结构的固有振动问题进行了研究。首先,建立了复合材料加筋圆锥壳体的基本运动方程,以及用位移形式表示的定子端部绕组固有振动控制方程,并采用复合材料理论给出了确定单根绕组及整体结构弹性常数的计算表达式。然后,采用伽辽金积分法推得了确定端部绕组固有振动频率的广义特征值方程。最后,通过算例分析,得到了在不同支架支承条件下发电机端部绕组整体固有振动的频率值及变化规律,并与实测结果进行了比较。     

Vibration and stability of an initially stressed laminated plate based on a higher-order deformation theory

Equations of motion for a laminated plate in a general state of nonuniform initial stress where the higher-order shear deformation terms are included are derived. The equations are derived by using the average stress method. The natural frequencies and buckling loads of cross-ply laminated plates subjected to initial stress are investigated. The initial stress is taken to be a combination of uniaxial pure bending and extensional stresses in the plane of the plate. The results obtained from the present higher-order theory are compared with first-order theory and other higher-order theory results. In addition, the effects of various parameters on the natural frequencies and buckling loads are studied.     

Dynamic and contact analysis of a bimodal ultrasonic motor

A bimodal ultrasonic motor, which operates with only one power amplifier, uses two simultaneously excited modes to drive the rotor; a longitudinal mode and a flexural mode. The equations of motion describing the vibrations and contact behavior are derived by Hamilton's principle and the geometry constraint. The Lagrange multiplier method is used to treat the frictional contact problem. The finite element method and numerical integration scheme are used to simulate the dynamic responses of this system with and without contact. Some important factors are studied for the bimodal ultrasonic motor design. The factors include structure design, amplitude of input voltage, phase displacement, exciting frequency, and contact behavior.     

圆弧曲梁面内自由振动的微分容积解法

用微分容积法求解圆弧曲梁在面内的自由振动问题。通过微分容积法将曲梁自由振动的控制微分方程和边界约束方程离散成为一组线性齐次代数方程组，这是一典型的特征值问题，求解这一特征值问题可以求得其自由振动的圆频率。中采用了考虑轴向变形、剪切变形和转动效应的理论，并采用子空间迭代法求解频率方程。数值算例表明，本方法稳定收敛、精度较高，对圆弧曲梁问题简单、有效。     

固定边界超临界轴向运动梁平面振动频率

通过数值方法研究超临界速度下,两端固定边界的轴向运动梁平面耦合非线性振动固有频率。发展有限差分法,确定在超临界范围轴向运动梁的径向与横向耦合平面内非平凡静平衡位形。基于非平凡静平衡位形,经坐标变换,建立超临界轴向运动梁连续陀螺系统的标准控制方程。运用高阶Galerkin截断,研究超临界运动状态下梁平面振动的固有频率;并研究Galerkin截断阶数对计算结果的影响。     

初应力位形上小的变形问题

论文研究初应力杆的小变形问题。用初应力作为参数,提出初应力位形上小变形问题的位移变分原理;针对直杆的几何特点,提出了用数值参数描写直杆初应力,用所提出的变分原理建立了初应力杆的小变形位移分量的定解问题。所得的方程中包含了初应力的数值参数。分析表明,初应力的不同的数值参数可以对相关的刚度、压杆的临界力以及基本变形的耦合与固有频率产生显著的影响。     

DQE方法及其在平板振动分析中的应用

介绍一种新的数值方法－DQE方法用以分析具有不连续几何形状和具有集中质量平板的自由振动,其基本思想是根据平板的具体情况将求解区域分割为几个单元,在每个单元内部用DQ方法将其控制方程离散,各单元之间的连接点上应用平衡条件和连续条件,再加上边界约束条件可得到问题的特征方程。作为数值算例,计算了具有集中质量平板和L型平板的固有频率,并与已有数值结果作了比较。     

Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates

In this paper, a three dimensional vibration analysis of nano-plates is studied by decoupling the field equations of Eringen theory. Considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. In order to find the solution for a nano-plate based on the presented formulation, one of the three equations in terms of displacement components and corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. A Navier-type method for finding the exact three dimensional solution of a nano-plate is presented using the Fourier series technique. Exact natural frequencies of nano-plates are presented and compared with the results of nonlocal first order and third order shear deformation theories.     

Micro-slot injection into a boundary layer driven by a favourable pressure gradient

This study aims to investigate the nonlinear forced vibration of functionally graded (FG) nanobeams. It is assumed that material properties are gradually graded in the direction of thickness. Nonlocal nonlinear Euler–Bernoulli beam theory is used to derive nonlocal governing equations of motion. The linear eigenmodes of FG nanobeams are used to transform a partial differential equation of motion into a system of ordinary differential equations via the Galerkin method. The multiple scale method is used to find the governing equations of the steady-state responses of FG nanobeams excited by a distributed harmonic force with constant intensity. It is also assumed that the working frequency is close to three times greater than the lowest natural frequency. Based on the equation governing the linear natural frequencies of FG nanobeams, the influence of the small scale parameter, material composition, and stiffness of the foundation on the linear relationship among natural frequencies is studied. Results show that superharmonic response or a combination of resonances may occur as well as a subharmonic response depending on the power-law index and stiffness of the foundation. Then the governing equations of a steady-state response of FG nanobeams for four possible solutions are obtained depending on the value of the small scale parameter. It is shown that the simplest response of FG nanobeams is a subharmonic response or superharmonic response. The equations governing the frequency–response curves are obtained and the effects of the power-law index and small scale parameter on them are discussed.     

电动机-弹性连杆机构系统谐振机理研究

以三相交流电动机-弹性连杆机构系统为研究对象，考虑三相交流电机转子振动偏心时不均匀气隙的气隙磁场对系统的影响，用有限单元法建立了系统的耦合动态方程，并将驱动电机和机构作为一个完整的系统对其谐振机理进行了研究。研究表明：该系统的谐振现象不仅与系统转速有关，而且还与电动机的同步转速有关；该系统的低阶临界转速不仅与系统的固有频率有关，也与电动机的同步转速有关。     

基于有限元与宽频快速多极边界元的二维流固耦合声场分析

采用有限元/快速多极边界元法进行水下弹性结构的辐射和散射声场分析。Burton-Miller法用于解决传统单Helmholtz边界积分方程在求解外边界值问题时出现的非唯一解的问题。该文采用GMRES和快速多极算法加速求解系统方程。针对传统快速算法在高频处效率低和对角式快速算法在低频处不稳定这一问题,该文通过结合这两种快速算法形成宽频快速算法来克服。同时该文通过观察不同参数条件设置下,宽频快速多极法得到的数值结果在计算精度和计算时间上的变化,得到最优的参数组合值。最后通过数值算例验证该文算法的正确性和有效性。     

Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach

A boundary continuous displacement based Fourier series solution to the boundary-value problem of free vibration of an arbitrarily laminated thin rectangular plate is presented. This powerful approach is employed to solve a system of three highly coupled partial differential equations arising from the Kirchhoff hypothesis as applied to an anisotropic laminate, with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the computed eigenvalues (natural frequencies) is ascertained by studying the convergence characteristics of the lowest seven natural frequencies, and also by comparison with the computed degenerate FEM (finite element methods) results. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as fiber orientation angle and longitudinal-to-transverse modulus ratio.     

Experimental study of construction mechanism of V(z) curves obtained by line-focus-beam acoustic microscopy

The propagation characteristics, viz., phase velocity and attenuation, of leaky surface acoustic waves (LSAWs), excited on the water/sample boundary are obtained through analyzing the V(z) curves measured by line-focus-beam acoustic microscopy. However, different values of these characteristics are obtained, depending upon different ultrasonic devices and operating frequencies employed. The construction mechanism of V(z) curves was investigated experimentally by measuring the amplitude and phase for Teflon to provide an understanding of the device performance for velocity measurements. A V(z) curve measured for Teflon, on which no leaky waves are excited when water is the coupling medium, can be used for the characteristic device response, depending only upon the device parameters and the operating frequencies. From the investigation of the ultrasonic device and the frequency dependences of the characteristic device responses, the phase gradient was found to be directly related to values of measured LSAW velocities. From this result, apparent frequency dependences in LSAW velocity measurements are explained quantitatively for a specimen of gadolinium gallium garnet.     

Free vibration analysis of orthogonal-woven fabric composites

This paper presents a study on the free vibration analysis of orthogonal-woven fabric composites. The fabrics are composed of two sets of mutually orthogonal yarns of either the same material (nonhybrid fabrics) or different materials (hybrid fabrics). We focus on the repeating unit of the woven structure, known as a `unit cell'. Based upon the one-dimensional (1D) elasto-dynamic analysis developed by the authors for such a woven fabric composite, the free vibration problem is formulated and solved for four basic boundary conditions. Natural frequency equations and natural modes are obtained analytically. For the special case of two bonded isotropic layers, closed-form solutions of natural frequencies and natural modes are given. It has been shown that numerical solutions can readily be carried out for woven fabric composites with arbitrary combinations of material and geometrical parameters. As an example, the first ten natural frequencies for plain weave composites based upon graphite/epoxy and glass/polyester are presented and discussed.     

Numerical model of longitudinal wave scattering in polycrystals

The scattering of elastic waves in polycrystalline materials is relevant for ultrasonic materials characterization and nondestructive evaluation (NDE). Ultrasonic attenuation is used widely to extract microstructural parameters such as grain size. Accurate interpretation of experimental data requires robust scattering models. Such models typically assume constant density, uniform grain size, and ergodicity hypotheses. The accuracy and limits of applicability of these models cannot be fully tested with experiments due to practical limits of real materials processing. Here, this problem is examined in terms of numerical simulations using Voronoi polycrystals that are discretized using finite elements. Wave propagation is studied by integrating the system directly in time using a planestrain formulation. Voronoi polycrystals with cubic symmetry and random orientations are used making the bulk material statistically isotropic. Example numerical results for materials with various degrees of scattering that are of common interest are presented. The numerical results are presented and compared with scattering theory for a wide range of frequencies. The numerical results show good agreement with the theory for the examples examined with evidence that the correlation function is frequency dependent. These results are anticipated to impact ultrasonic NDE of polycrystalline media.     

Dynamic modeling and analysis of a bimodal ultrasonic motor

A dynamic model that includes four subsystems is developed to analyze the fundamental characteristics of a bimodal ultrasonic motor. The first subsystem is the driving circuit designed for the motor to achieve bidirectional motion. The stator is modeled as a Timoshenko beam, and the assumed mode energy method is used to obtain the dynamic equations. The normal interface force is represented by an elastic spring existing in between the tip of the stator and the moving platform. The interface forces are coupled into the dynamic formulations of the stator and the moving platform. The behavior of the force transmission between the stator and the moving platform are analyzed using the developed model. Transient and steady-state responses of the system are obtained by numerical simulation, and the results are validated by experiments. Furthermore, the existing of a nonlinear deadzone is predicted analytically, and the causes of this nonlinearity are clarified.     

负载谐波诱发轧机主传动机电耦合扭振仿真研究

现场测试发现:轧机在轧制过程中由于负载谐波频率与轧机机械传动系统固有频率接近时使得主传动系统发生强烈扭振。该文结合现场参数,通过计算机仿真,得出负载谐波力矩会引发电机转速振荡,而在机电耦合作用下电机转速振荡会激发电机输出相同频率的电磁力矩,当含有与机械系统固有频率相接近频率的谐波力矩和电磁力矩作用在轧机主传动系统两端时,主传动系统将会出现强烈的扭振。仿真结果验证了存在这种振动,为抑振措施提供理论基础。     

Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

Natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations and rotatory inertia. The modulus of elasticity of the plates is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded (FG) plates is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Critical buckling stresses of FG plates subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG plates without in-plane stresses is presented. The distributions of modal displacements and modal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2-D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG plates.