Dynamic response of elastic rectangular plates by spline state variable method

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｅｎｔｈｅｓｔｒｕｃｔｕｒｅｓａｒｅｓｕｂｊｅｃｔｅｄｔｏｔｒａｎｓｉｅｎｔｌｏａｄｓｓｕｃｈａｓｂｏｍｂ＿ｌｏａｄ,ｅａｒｔｈｑｕａｋｅａｎｄｓｏｏｎ,ｅｎｇｉｎｅｅｒｓｐａｙｓｐｅｃｉａｌａｔｔｅｎｔｉｏｎｔｏｔｈｅｓａｆｅｔｙｂｅｈａｖｉｏｒｏｆｓｔｒｕｃｔｕｒｅｓ.Ｔｈｅｒｅｉｓｌｏｔｓｏｆｒｅｓｅａｒｃｈｗｏｒｋｏｎｔｈｅｃｏｎｖｅｎｔｉｏｎａｌｍｅｔｈｏｄｏｆｄｙｎａｍｉｃｒｅｓｐｏｎｓｅａｎａｌｙｓｉｓｆｏｒｓｔｒｕｃｔｕｒａｌｅｎｇｉｎｅｅｒｉｎｇｂｏｔ…     

Analytical modeling of sound transmission through clamped triple-panel partition separated by enclosed air cavities

An analytical model for sound transmission through a clamped triple-panel partition of finite extent and separated by two impervious air cavities is formulated. The solution derived from the model takes the form of that for a clamp supported rectangular plate. A set of modal functions (or more strictly speaking, the basic functions) are employed to account for the clamped boundary conditions, and the application of the virtual work principle leads to a set of simultaneous algebraic equations for determining the unknown modal coefficients. The sound transmission loss (STL) of the triple-panel partition as a function of excitation frequency is calculated and compared with that of a double-panel partition. The model predictions are then used to explore the physical mechanisms associated with the various dips on the STL versus frequency curve, including the equivalent ‘mass-spring’ resonance, the standing-wave resonance and the panel modal resonance. The asymptotic variation of the solution from a finite-sized partition to an infinitely large partition is illustrated in such a way as to demonstrate the influence of the boundary conditions on the soundproofing capability of the partition. In general, a triple-panel partition outperforms a double-panel partition in insulating the incident sound, and the relatively large number of system parameters pertinent to the triple-panel partition in comparison with that of the double-panel partition offers more design space for the former to tailor its noise reduction performance.     

The solution of rectangular plates with large deflection by spline functions

In this paper, Von Karman ’s set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.     

冰与海洋结构物作用的极限承载力

为进一步研究冰与海洋结构物作用时的变形和影响,本文将岩土力学中的Smith屈服准则引入到冰材料的弹塑性分析中,从细观力学的角度分析了孔洞存在以及拉、压破坏强度不同的条件下冰材料的力学性能,构造了冰的本构方程;采用分区构造方法分析给出冰与矩形直立结构物作用时应力场的解析解;最后用数值分析的方法分析给出不同压力敏感性参数和拉压强度比下冰的极限承载力。分析结果表明Smith屈服准则能够准确反应孔洞的存在及拉压屈服强度不同对冰材料力学性能的影响。     

弹性动力学的多变量响应问题的解法

基于瞬时混合变分原理与乘积型二元三次 B样条函数 ,以板壳为例 ,建立样条动力方程。引入样条参数及其对时间的导数作为状态变量 ,导出状态方程。对空间域采用混合样条元法 ,对时间域采用现代控制论中的状态空间法。文末数值算例表明 ,计算精度与效率是令人满意的。本文方法对计算多输入与多输出 ,时不变与时变系统和线性与非线性系统等多变量动力响应问题 ,有广阔的应用与发展前景     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

Reconciliation of size-density bivariate distributions over a separating node

Data reconciliation considers the restoration of mass balance among the noise prone measured data by way of component adjustments for the various particle size or particle density classes or assays over the separating node. In this paper, the method of Lagrange multipliers has been extended to balance bivariate feed and product size-density distributions of coal particles split from a settling column. The settling suspension in the column was split into two product fractions at 40% height from the bottom after a minute settling of homogenized suspension at start. Reconciliation of data assists to estimate solid flow split of particles to the settled stream as well as helps to calculate the profiles of partition curves of the marginal particle size or particle density distributions. In general, Lagrange multiplier method with uniform weighting of its components may not guarantee a smooth partition surface and thus the reconciled data needs further refinement to establish the nature of the surface. In order to overcome this difficulty, a simple alternative method of reconciling bivariate size-density data using partition surface concept is explored in this paper.     

Reconciliation of size–density bivariate distributions over a separating node

Data reconciliation considers the restoration of mass balance among the noise prone measured data by way of component adjustments for the various particle size or particle density classes or assays over the separating node. In this paper, the method of Lagrange multipliers has been extended to balance bivariate feed and product size-density distributions of coal particles split from a settling column. The settling suspension in the column was split into two product fractions at 40% height from the bottom after a minute settling of homogenized suspension at start. Reconciliation of data assists to estimate solid flow split of particles to the settled stream as well as helps to calculate the profiles of partition curves of the marginal particle size or particle density distributions. In general, Lagrange multiplier method with uniform weighting of its components may not guarantee a smooth partition surface and thus the reconciled data needs further refinement to establish the nature of the surface. In order to overcome this difficulty, a simple alternative method of reconciling bivariate size-density data using partition surface concept is explored in this paper.     

计算力学中的样条有限元法的进展

主要评述基于变分原理、样条函数理论与状态空间理论的样条有限元法在近２０多年来的进展以及进一步发展的趋势．首先评述了国外的早期工作──截断式样条函数在力学中的应用情况．接着评述国内工作──样条有限元、样条有限点、样条单元法等优缺点,最后还介绍了目前国内外尚未报导过的,即多变量样条有限元法和样条状态变量法在动力响应中的应用．关键词变分原理,样条函数,状态空间理论,样条有限元法      

花键轴冷挤压成形后的三维光塑性模拟研究

本文利用光塑性方法,以聚碳酸酯（Ｐｏｌｙｃａｒｂｏｎａｔｅ）为材料研究了矩形花键轴冷挤压成形过程,获得了花键挤压成形的三维塑性应变场分布,提出其变形的应变分布特征,并对典型截面的应变进行了计算分析;文中还通过对聚碳酸酯材料的实验,得到了使花键齿形充满的最佳坯料尺寸,为实际工艺生产优选出合理的工艺参数。     

Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate

An exact three-dimensional analysis is presented for a functionally gradient piezoelectric material rectangular plate that is simply supported and grounded along its four edges. The state equations of the functionally gradient piezoelectric material are developed based on the state space approach. Assuming that the mechanical and electric properties of the material have the same exponent-law dependence on the thickness-coordinate, we obtain an exact three-dimensional solution of the coupling electroelastic fields in the plate under mechanical, and electric loading on the upper and lower surfaces of the plate. The influences of the different functionally gradient material properties on the structural response of the plate to the mechanical and electric stimuli are then studied through examples.     

矩形箱梁力学性能分析的修正计算方法

本文对矩形箱梁翼板设置了不同的剪滞翘曲位移差函数,继而综合考虑剪力滞效应、剪切变形以及剪滞翘曲应力和弯矩自平衡条件等因素,且以能量变分原理为基础建立了矩形箱梁的弹性控制微分方程和自然边界条件,基于此修正了现行薄壁结构分析方法。与传统剪滞理论相比,本文方法深刻反映了矩形箱梁的力学特性。研究表明,(1)由于剪滞翘曲应力和弯矩自平衡条件的引入,矩形箱梁力学性能分解为独立的初等梁理论和剪滞理论体系,且箱梁力学性能为两者的叠加效应;(2)矩形箱梁断面尺寸确定,剪滞效应对其正应力的影响值不变,即剪滞效应的竖向力学行为与箱梁跨径无关;(3)尽管矩形箱梁的梁高对箱形梁剪滞翘曲应力和初等梁理论的应力值皆有一定影响,但其剪力滞系数不变,因此剪力滞效应与梁高无关;(4)剪力滞效应不仅影响箱梁翼板力学性能,而且对其腹板力学行为的影响不可忽视。因而,与传统剪滞理论相比,本文修正法不仅计算精度明显提高,而且更能真实反映矩形箱梁的力学性能。     

Free vibrations of shallow nonthin shells with variable thickness and rectangular planform

The free vibrations of shallow orthotropic shells with variable thickness and rectangular planform are studied. The shear strains are taken into account. The spline approximation of unknown functions is used. The natural frequencies are calculated for different boundary conditions. The dependence of the natural frequencies on the curvature of the midsurface is examined. The natural frequencies of shells with constant and variable thickness are compared     

Effect of a permeable partition on convective instability in a rectangular cavity

The equilibrium stability of a fluid, heated from below, in a rectangular cavity with a vertical permeable partition is investigated. The small perturbation problem is solved by the Galerkin-Kantorovich method. The relations obtained for the dependence of the critical Rayleigh numbers on the partition parameters and the cavity dimensions make it possible to identify regions in which either even or odd perturbations, sensitive to only the normal or only the tangential resistance of the partition, respectively, are responsible for equilibrium crisis. The effect of a permeable partition on the convective instability of a horizontal layer of fluid under various heating conditions was considered in [1–3], where a significant dependence of the critical Rayleigh numbers on the properties of the partition was established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 6–10, May–June, 1989.     

样条状态变量法分析扁球壳的动力响应

基于瞬时变分原理与样条函数理论,建立扁球壳的动力方程,引入样条参数及其对时间的导数作为状态变量,导出状态方程,对空间域,采用样条有限元法,对时间域,采用现代控制论中的状态空间法,此外,建立了一种状态变量的递推计算格式,在计算过程中,不解线性方程组,也不需采用解耦技术,只要作矩阵运算,可直接算出动力响应量,文末给出的数值算例表明本文方法是可靠 与有效的。     

NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE$^{\bf 1)}$

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.     

饱和多孔地基与矩形板动力相互作用的非轴对称混合边值问题

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.      

THREE-DIMENSIONAL THERMOELASTIC ANALYSIS OF FUNCTIONALLY GRADED PLATE

Three-dimensional thermoelastic analysis is presented for an orthotropic functionally graded rectangular plate,which is simply supported and isothermal on its four lateral edges.With the assumption that material properties have arbitrary dependence on the thickness-coordinate,a Peano-Baker series solution is obtained for the thermoelastic fields of the functionally graded plate subjected to mechanical and thermal loads on its upper and lower surfaces by means of state space method.The correctness of the obtained series solution is validated through numerical examples.The influence of different material properties distributions on the structural response of the plate is also studied.     

Numerical simulations of flow-field interactions between moving and stationary objects in idealized street canyon settings

Numerical simulations of an ideal model of street canyons with moving objects in the horizontal plane were conducted. The simulations were based on the unsteady two-dimensional incompressible Navier–Stokes equations, discretized on an overlapping grid with a numerical scheme that is second-order accurate in both space and time. The computational domain consists of a rectangular background with eight fixed objects arranged in two parallel columns representing the street canyon. Four identical objects were put in each column equidistantly. One or two identical objects were moving along the symmetry line of the computational domain. The objects were either circular or rectangular with rounded corners in shape. The numerical method was first validated by comparing with existing experimental and simulation data. A parametric study was carried out to investigate the influence of the characteristic parameters (such as canyon width, velocity of the moving objects, and separation distance between them) on the wake of the moving objects.     

弹性半空间地基上预应力中厚矩形板的横向振动

基于Reissner-Mindlin一阶剪切变形理论,讨论在预加面内机械荷载作用下,弹性半空间地基上四边自由中厚矩形板的横向振动问题。把地基看作三维弹性半空间体,考虑地基变形的衰减。用一组数学上完备的二元多项式作为位形函数,采用pb-2 Rayleigh-Ritz法求得四边自由中厚矩形板的自振频率和在横向简谐荷载作用下的动力响应。讨论了板的长宽比、宽厚比及弹性地基和板的相对刚度对板的自振频率的影响。