几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程,运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比,验证了公式的正确性．基频近似解表达式简单明了,物理意义明确,清楚地说明了初始荷载及相关因素对板自由振动基频的影响,直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度,提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.     

圆薄板非对称大变形弯曲问题

本文首先导出圆薄板非轴对称大变形问题的位移基本方程及边界条件.利用变换和摄动法将非线性位移方程线性化,得到了近似边值问题.作为算例,文中研究了圆薄板在较复杂载荷作用下的非线性弯曲问题.     

Calibration of Eringen's small length scale coefficient for buckling circular and annular plates via Hencky bar-net model

This paper is focused on the modeling of circular and annular graphene sheets via Hencky bar-net model (HBM¹) and calibrating the Eringen's small length scale coefficient e₀ in Eringen's nonlocal theory. The buckling solutions of circular and annular graphene sheets based on Eringen's nonlocal continuum plate theory are first obtained. On the other hand, HBM is developed to model the same structure from the discrete view. HBM is a grid system comprising rigid bars and arcs connected by frictionless hinges with elastic rotational and torsional springs. By regarding the length of straight segments in HBM equal to the characteristic length of Eringen's nonlocal model (ENM²) and matching their solutions, the Eringen's small length scale coefficient e₀ is calibrated. It is found that for circular graphene sheet, e₀ = 0.258 for clamped edge and e₀ = 0.300 for simply supported edge. For annular graphene sheet, e₀ is dependent on the inner to outer radius ratio χ and boundary conditions. The scale coefficient e₀ takes 0.307–0.367 for clamped edges while 0.219–0.290 for simply supported edges with χ varying from 0.2 to 0.8. Another finding is that the graphene sheet will buckle with a very small load when its dimension is large, regardless of models adopted. However for small dimensions, ENM and HBM predict lower buckling loads than the classical local model because the scale effect is more obvious.     

An Analogue-difference Method and Application to Induction Motor Models

The paper established a so-called analogue-difference method (ADM) to compute the numerical solutions for boundary value problems of higher-order differential equations, which can be a fundamental method and performs much better than the finite difference method (FDM), even for second-order boundary value problems. Numerical examples and results illustrate the simplicity, efficiency and applicability of the method, which also show that the proposed method has obvious advantages over the methods presented by recent state-of-the-art work for induction motor models.     

极正交各向异性圆板非线性弯曲的定性分析及单调迭代解

本文对极正交各向异性圆板在任意轴对称载荷和边界条件下的非线性弯曲问题进行了较为系统的研究.首先,将边值问题归结为等价的积分方程,并且借助于广义函数得到了线性问题的一般解答.其次,对导出的非线性积分方程解的性质作了较为细致的讨论,例如边缘皱褶,非负性和奇性等.然后,构造了解的双边单调迭代格式,并给出了迭代格式的收敛性判据和误差估计,同时还讨论了解的全局存在唯一性.最后,给出了一个数值例子来说明本文方法和结论的应用.本文某些结果是由作者新得到的.     

A symplectic analytical wave propagation model for damping and steady state forced vibration of orthotropic composite plate structure

An analytical wave propagation model is proposed in this paper for damping and steady state forced vibration of orthotropic composite plate structure by using the symplectic method. By solving an eigen-problem derived in the symplectic dual system of free bending vibration of orthotropic rectangular thin plates, the wave shape of plate is obtained in symplectic analytical form for any combination of simple boundary conditions along the plate edges. And then the specific damping capacity of wave mode is obtained symplectic analytically by using the strain energy theory. The steady state forced vibration of built-up plates structure is calculated by combining the wave propagation model and the finite element method. The vibration of the uniform plate domain of the built-up plates structure is described using symplectic analytical waves and the connector with discontinuous geometry or material is modeled using finite elements. In the numerical examples, the specific damping capacity of orthotropic rectangular thin plate with three different combinations of boundary condition is first calculated and analyzed. Comparisons of the present method results with respect to the results from the finite element method and from the Rayleigh–Ritz method validate the effectiveness of the present method. The relationship between the specific damping capacity of wave mode and that of modal mode is expounded. At last, the damped steady state forced vibration of a two plates system with a connector is calculated using the hybrid solution technique. The availability of the symplectic analytical wave propagation model is further validated by comparing the forced response from the present method with the results obtained using the finite element method.     

变厚度簿板弯曲问题的任意网格差分解法

本文提出一种求解变厚度薄板弯曲问题的任意网格差分格式,可适应不同边界,各种荷载和复杂形状板.计算实例表明,该方法具有格式简单、通用性强、计算精度高,计算量少等特点.     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates

Exact bending solutions of fully clamped orthotropic rectangular thin plates subjected to arbitrary loads are derived using the finite integral transform method. In the proposed mathematical method one does not need to predetermine the deformation function because only the basic governing equations of the classical plate theory for orthotropic plates are used in the procedure. Therefore, unlike conventional semi-inverse methods, it serves as a completely rational and accurate model in plate bending analysis. The applicability of the method is extensive, and it can handle plates with different loadings in a uniform procedure, which is simpler than previous methods. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the bibliography.     

双参数地基上Reisner板弯曲问题的边界积分方程

本文应用广义函数的Ｆｏｕｒｉｅｒ积分变换，导出了双参数地基上Ｒｅｉｓｓｎｅｒ板弯曲问题的两个基本解·在此基础上，从虚功原理出发，依据胡海昌导出的Ｒｅｉｓｓｎｅｒ板弯曲理论，推导出适用于任意形状、任意荷载、任意边界条件情形的三个边界积分方程，为边界元法在这一问题中的应用提供了理论基础·文中给出了固支、简支、自由三类边界的算例，并与解析解比较，均得到满意的结果·     

3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates

Within the framework of three-dimensional elasticity theory, this paper investigates the axisymmetric bending of novel functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite circular and annular plates under a combined action of a uniformly distributed transverse load and a through-thickness steady temperature field. Three different distribution patterns of GPLs within the polymer matrix are considered. The present analytical solutions are validated through comparisons against those available in open literature for the reduced cases. A parametric study is conducted to examine the effects of GPL weight fraction, distribution pattern, plate thickness to radius ratio, and boundary conditions on the stress and deformation fields of the plate. The results show that GPL nanofillers with a low content can have a significant reinforcing effect on the bending behavior of the thermo-mechanically loaded plate.     

Approximation of boundary values in the boundary element method

Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions.Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined.Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time.Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

夹层圆板的非线性弯曲

本文导出了具有软夹心的夹层圆板的非线性轴对称弯曲理论的基本方程和边界条件,并给出了表板很薄情况下的这些方程和边界条件的简化形式.作为算例,研究了在均布横向载荷作用下具有滑动固定边界条件的夹层圆板,使用修正迭代法,得到了相当精确的解析解.     

Dynamic analysis of a non-conservative band–wheel system with a moving boundary part 2: Numerical results

The dynamic formulations of a non-conservative band/wheel system with a moving boundary are derived in the companion paper by use of Hamilton’s principle and calculus of variation. In this paper, numerical simulations for the system are obtained in transient amplitudes of the string and positions of the moving boundary by a modified finite difference method (FDM). Since the moving boundary position may not locate exactly at a grid point for any computational time, a special technique of the FDM is proposed to approximate both the transversality condition of a moving boundary and the partial differential equation of the neighboring grid points. The effects of parameters such as radius of the wheel, tension of the string, propagation speeds of the longitudinal and transverse wave and various initial conditions on the transient responses are investigated and compared with those of the fixed boundary problem.     

非均匀弹性地基圆板轴对称弯曲的一般解

本文在阶梯折算法的基础上,提出一个新的方法——精确解析法,得到了非均匀弹性地基圆板弯曲的一般解.文中导出了在任意轴对称载荷和边界条件下求解非均匀弹性地基圆板和中心带孔圆板弯曲的一般公式,并给出一致收敛于精确解的证明.文中得到的一般解可直接计算无弹性地基圆板的弯曲问题.问题最后归结为求解一个二元一次代数方程.文末给出算例,算例表明无论内力和位移均可得到满意的结果.     

An artificial boundary method for the Hull-White model of American interest rate derivatives

The Hull-White (HW) model is a widely used one-factor interest rate model because of its analytical tractability on liquidly traded derivatives, super-calibration ability to the initial term structure and elegant tree-building procedure. As an explicit finite difference scheme, lattice method is subject to some stability criteria, which may deteriorate the computational efficiency for early exercisable derivatives. This paper proposes an artificial boundary method based on the partial differential equations (PDEs) to price interest rate derivatives with early exercise (American) feature under the HW model. We construct conversion factors to extract the market information from the zero-coupon curve and then reduce the infinite computational domain into a finite one by using an artificial boundary on which an exact boundary condition is derived. We then develop an implicit θ-scheme with unconditional stability to solve the PDE in the reduced bounded domain. With a finite computational domain, the optimal exercise strategy can be determined efficiently. Our numerical examples show that the proposed scheme is accurate, robust to the truncation size, and more efficient than the popular lattice method for accurate derivative prices. In addition, the singularity-separating technique is incorporated into the artificial boundary method to enhance accuracy and flexibility of the numerical scheme.     

Thermal transpiration for the linearized Boltzmann equation

The phenomena of thermal transpiration due to the boundary temperature gradient is studied on the level of the linearized Boltzmann equation for the hard‐sphere model. We construct such a flow for a highly rarefied gas between two plates and also in a circular pipe. It is shown that the flow velocity parallel to the plates is proportional to the boundary temperature gradient. For a highly rarefied gas, that is, for a sufficiently large Knudsen number κ, the flow velocity between two plates is of the order of log κ, and the flow velocity in a pipe is of finite order. Our analysis is based on certain pointwise estimates of the solutions of the linearized Boltzmann equation. © 2006 Wiley Periodicals, Inc.     

Analysis of notches and cracks in circular Kirchhoff plates using the scaled boundary finite element method

A new formulation of the scaled boundary finite element method (SBFEM) is presented for the analysis of circular plates in the framework of Kirchhoff's plate theory. Essential for the SBFEM is, that a domain is described by the mapping of its boundary with respect to a scaling centre. The governing partial differential equations are transformed into scaled boundary coordinates and are reduced to a set of ordinary differential equations, which can be solved in a closed-form analytical manner. If the scaling centre is selected at the root of an existent crack or notch, the SBFEM enables the effective and precise calculation of singularity orders of cracked and notched structures. The element stiffness matrices for bounded and unbounded media are derived. Numerical examples show the performance and efficiency of the method, applied to plate bending problems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

复合材料层合板的弯曲与振动

本文应用应力杂交有限元方法分析了复合材料层合板的弯曲与振动.在本文中,首先根据修正的余能变分原理,构造了一个适合于复合材料层合板特点的矩形应力杂交板弯曲单元.在单元内,分层假设应力参数,在单元的边界上,根据YNS理论的假设确定边界位移场.这样使得构造出来的单元不仅能够考虑横向剪切变形的影响和局部扭曲效应,而且具有较少的自由度数.其次,用此单元求解了层合板的弯曲与振动问题,并将计算结果与精确解进行了比较,比较表明二者非常接近.这说明了在计算方面本文单元具有较高的精确度.