Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching-bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

一种基于K-means与Close-Form融合的树木图像提取方法

基于图像的树木自动提取是其种类、长势、形态等信息智能化判别的基础,如何实现树木的自动、准确、快速提取是具有实用性的科学问题。在自然场景中,由于图像元素多样、颜色差异大,树木自身存在不规则性,树木提取难度非常大。针对现有的图像分割与图像抠图法在树木提取过程中分别存在的误分割与过程复杂所导致的计算量大的问题,提出了一种基于K-means聚类算法优化Close-Form图像抠图的树木提取方法。在少量的标记下,依据颜色线性假设进行最小化代价函数计算,得到图像透明度;对透明度图像依次进行中值滤波、高斯滤波,得到透明度的去噪图像;对滤波后透明度与标准化的图像绿色分量组成二维空间进行K-means二聚类,实现背景与前景对象的准确判定,进而完成自然背景的树木图像提取。为了验证所提方法在不同场景和不同标记下的树木提取有效性,设计了基于K-means图像分割和传统Close-Form抠图方法的比较性试验。结果表明,基于K-means优化Close-Form的树木提取方法解决了传统Close-Form算法在少量标记下图像前景、背景估计不准确问题,克服了图像分割存在的误分类情况,实现了不同自然环境和多目标树木对象的提取。此方法具有对象提取稳定、计算时间快的优点,相对原Close-Form算法用时减少49.98%。

Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.

Closed-form solution to thin-walled box girders considering effects of shear deformation and shear lag

Considering three longitudinal displacement functions and uniform axial displacement functions for shear lag effect and uniform axial deformation of thin-walled box girder with varying depths,a simple and efficient method with high precision to analyze the shear lag effect of thin-walled box girders was proposed.The governing differential equations and boundary conditions of the box girder under lateral loading were derived based on the energy-variational method,and closed-form solutions to stress and deflection corresponding to lateral loading were obtained.Analysis and calculations were carried out with respect to a trapezoidal box girder under concentrated loading or uniform loading and a rectangular box girder under concentrated loading.The analytical results were compared with numerical solutions derived according to the high order finite strip element method and the experimental results.The investigation shows that the closed-form solution is in good agreement with the numerical solutions derived according to the high order finite strip method and the experimental results,and has good stability.Because of the shear lag effect,the stress in cross-section centroid is no longer zero,thus it is not reasonable enough to assume that the strain in cross-section centroid is zero without considering uniform axial deformation.

Closed-form solution for shear lag effects of steel-concrete composite box beams considering shear deformation and slip

Based on the consideration of longitudinal warp caused by shear lag effects on concrete slabs and bottom plates of steel beams,shear deformation of steel beams and interface slip between steel beams and concrete slabs,the governing differential equations and boundary conditions of the steel-concrete composite box beams under lateral loading were derived using energy-variational method.The closed-form solutions for stress,deflection and slip of box beams under lateral loading were obtained,and the comparison of the analytical results and the experimental results for steel-concrete composite box beams under concentrated loading or uniform loading verifies the closed-form solution.The investigation of the parameters of load effects on composite box beams shows that:1) Slip stiffness has considerable impact on mid-span deflection and end slip when it is comparatively small;the mid-span deflection and end slip decrease significantly with the increase of slip stiffness,but when the slip stiffness reaches a certain value,its impact on mid-span deflection and end slip decreases to be negligible.2) The shear deformation has certain influence on mid-span deflection,and the larger the load is,the greater the influence is.3) The impact of shear deformation on end slip can be neglected.4) The strain of bottom plate of steel beam decreases with the increase of slip stiffness,while the shear lag effect becomes more significant.

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

Closed-form solution of mid-potential between two parallel charged plates with more extensive application

Efficient calculation of the electrostatic interactions including repulsive force between charged molecules in a biomolecule system or charged particles in a colloidal system is necessary for the molecular scale or particle scale mechanical analyses of these systems. The electrostatic repulsive force depends on the mid-plane potential between two charged particles. Previous analytical solutions of the mid-plane potential, including those based on simplified assumptions and modern mathematic methods, are reviewed. It is shown that none of these solutions applies to wide ranges of interparticle distance from 0 to 10 and surface potential from 1 to 10. Three previous analytical solutions are chosen to develop a semi-analytical solution which is proven to have more extensive applications. Furthermore, an empirical closed-form expression of mid-plane potential is proposed based on plenty of numerical solutions. This empirical solution has extensive applications, as well as high computational efficiency.

MODIFIED CLOSED-FORM NUMERICAL ALGORITHM FOR PERIODIC VIBRATION RESPONSE OF MECHANICAL TRANSMISSIONS

A closed-form numerical algorithm (CFNA) is analyzed in detail. CFNA iswidely used in mechanical dynamics for periodic solution of second-order original differentialequations (SODE) with periodic time-variant coefficients. The principle of the algorithm is todiscretize the motion period into many short time intervals, so the coefficient matrices of theequation set are regarded as constant in a time interval. Defects are found in the originalalgorithm in treating the modal coordinates at the two end-nodes and important modifications to thedefects is made for the algorithm. The modified algorithm is finally used to solve the dynamicproblem of a three-ring planetary gear transmission.

Closed form solution of stress intensity factors for cracks emanating from surface semi-spherical cavity in finite body with energy release rate method

In this paper, a new semi-analytical and semi-engineering method of the closed form solution of stress intensity factors （SIFs） of cracks emanating from a surface semi-spherical cavity in a finite body is derived using the energy release rate theory. A mode of crack opening displacements of a normal slice is established, and the normal slice relevant functions are introduced. The proposed method is both effective and accurate for the problem of three-dimensional cracks emanating from a surface cavity. A series of useful results of SIFs are obtained.

Novel Closed-Form Solution for Analyzing Mutual Coupling Between Cylindrical Comformal Rectangular Microstrip Patch Antennas

Based on the integral equation formulations and the moment method, a novel closed form solution for analyzing the mutual coupling effect between the cylindrical comformal rectangular microstrip patch antennas is presented. By using this algorithm, the elements of the impedance matrix and exciting vector are cast into closed forms, thus the computational efficiency is improved dramatically. Numerical results are presented to verify the validity and reliability of the algorithm.

Closed form solution and numerical analysis for Eshelby's elliptic inclusion in plane elasticity

This paper presents a closed form solution and numerical analysis for Es- helby＇s elliptic inclusion in an infinite plate. The complex variable method and the confor- real mapping technique are used. The continuity conditions for the traction and displace- ment along the interface in the physical plane are reduced to the similar conditions along the unit circle of the mapping plane. The properties of the complex potentials defined in the finite elliptic region are analyzed. From the continuity conditions, one can separate and obtain the relevant complex potentials defined in the inclusion and the matrix. From the obtained complex potentials, the dependence of the real strains and stresses in the inclusion from the assumed eigenstrains is evaluated. In addition, the stress distribution on the interface along the matrix side is evaluated. The results are obtained in the paper for the first time.

Sample geometric mean versus sample median in closed form framework of seismic reliability evaluation: a case study comparison

Earthquake engineers have made a lot of efforts to derive a comprehensive set of closed form expressions for performance evaluation of frames, which are already presented in guidelines such as SAC/FEMA. These analytical expressions have been developed to estimate the annual probability of exceeding a limit state. In the process of such seismic assessments, some essential assumptions are adopted to simplify the process. One of these fundamental assumptions declares that drift demand at any seismic intensity level follows a lognormal distribution around its median. To investigate the validity of this assumption, this paper describes a case study of the types of errors that could be produced by using the sample median as the central tendency. Based on the Maximum Likelihood Estimation method as well as other statistical evidence, this paper proposes the use of the sample geometric mean instead of the sample median for the central tendency. Further, the results of seismic reliability evaluations of 4 sample frames are compared based on utilizing both the geometric mean and the sample median. In this process, both first and second order power law fits of the hazard curve are implemented to compare the effects of hazard estimation and the selection of the central tendency on the final results. It is observed in the application example that the sample geometric mean could lead to more accurate results.

Closed-form Solutions to the Matrix Equation AX-EXF=BY with F in Companion Form

A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .

Method of closed loop springback compensation for incremental sheet forming process

The closed loop control model was built up for compensating the springback and enhancing the work piece precision.A coupled closed loop algorithm and a finite element method were developed to simulate and correct the springback of incremental sheet forming.A three-dimensional finite element model was established for simulation of springback in incremental sheet forming process.The closed loop algorithm of trajectory profile for the incremental sheet forming based on the wavelet transform combined with fast Fourier transform was constructed.The profile of processing tool path of shallow dishing with spherical surface was designed on the basis of the profile correction algorithm.The result shows that the algorithm can predict an ideal profile of processing track,and the springback error of incremental sheet forming is eliminated effectively.It has good convergence efficiency,and can improve the workpiece dimensional accuracy greatly.

Closed-form internal impedance model and characterization of mixed carbon nanotube bundles for three-dimensional integrated circuits

Based on the complex effective conductivity method, a closed-form expression for the internal impedance of mixed carbon nanotube （CNT） bundles, in which the number of CNTs for a given diameter follows a Gaussian distribution, is proposed in this paper. It can appropriately capture the skin effect as well as the temperature effect of mixed CNT bundles. The results of the closed-form expression and the numerical calculation are compared with various mean diameters, standard deviations, and temperatures. It is shown that the proposed model has very high accuracy in the whole frequency range considered, with maximum errors of 1% and 2.3% for the resistance and the internal inductance, respectively. Moreover, by using the proposed model, the high-frequency electrical characteristics of mixed CNT bundles are deeply analyzed to provide helpful design guidelines for their application in future high-performance three-dimensional integrated circuits.

Shape formation of closed-cell aluminum foam in solid–liquid–gas coexisting state

The mold pressing process was applied to investigate the formability of closed-cell aluminum foam in solid–liquid–gas coexisting state.Results show that the shape formation of closed-cell aluminum foam in the solid–liquid–gas coexisting state was realized through cell wall deformation and cell movement caused by primary α-Al grains that slid,rotated,deformed,and ripened within cell walls.During formation,characteristic parameters of closed-cell aluminum foam were almost unchanged.Under proper forming conditions,shaped products of closed-cell aluminum foam could be fabricated through mold pressing.

设置侧向限位阻尼器的层间隔震结构减震性能研究

对在隔震层设置侧向限位黏弹性阻尼器组成混合耗能结构随机激励下的减震性能进行了系统研究.首先,建立混合耗能结构的地震动方程,利用双过滤白噪声激励地震动的滤波方程,将该类激励下混合耗能结构的动力学问题精确转化为基于易于获得简明封闭解的白噪声激励问题.其次,基于复模态法和白噪声激励的Dirac函数性质推导了混合耗能结构系列响应(结构位移、层间位移和阻尼器阻尼力)的方差和0~2阶谱矩的简明封闭解.最后,通过算例在验证所提封闭解正确的基础上研究了混合耗能结构减震性能的影响因素.研究表明:层间隔震层以上楼层的结构响应量随着隔震层刚度的增大而增大;黏弹性阻尼器阻尼参数对隔震层以上楼层的影响特征为,结构位移在一定阻尼参数下可达到最小,而层间位移则随着阻尼参数的增大而增大.该研究可为层间隔震混合耗能结构的设计提供参考.

基于全概率公式的M-APSK误码率分析模型

为有效评估加性白高斯噪声信道中M-APSK的性能,提出了一种基于全概率公式的理论误码率分析模型。首先根据星座点分布及其编码特点对星座平面进行区域划分,然后由接收符号落在相应区域的概率推导出各比特的判决错误概率表达式,最后得到理论误码率的精确闭式解。以16APSK和32APSK为例,推导出了其各类星座图的理论误码率计算公式,与Monte Carlo仿真结果非常吻合。模型具有通用性,PSK、QAM、PAM、APSK等调制方式均可利用该模型得到理论误码率的精确闭式解。相较于依赖大量数值计算的复杂二维概率积分方法,所提模型能够更方便地给出M-APSK的误码率性能。

地震作用下调谐黏滞质量阻尼器阻尼比增效效应与优化设计研究

调谐黏滞质量阻尼器(Tuned Viscous Mass Damper,TVMD)是一种有效的被动惯容减震装置,本文针对地震作用下建筑结构TVMD阻尼比增效效应与优化设计展开研究。将TVMD对结构自身阻尼耗能功率的控制效果归纳为TVMD等效附加阻尼比,并基于随机振动理论推导了等效附加阻尼比的理论表达式。为了使TVMD更具实际应用价值,TVMD理论上应取得比同阻尼系数的黏滞阻尼器(VD)更大的等效附加阻尼比,这一现象定义为TVMD阻尼比增效效应,并定义了阻尼比增效系数来量化评估阻尼比增效效应。将等效附加阻尼比和阻尼比增效系数均作为优化目标,提出了TVMD最优设计参数理论解。参数分析结果表明,本文解具有良好的稳定性和适用性,为了更高效地发挥阻尼比增效效应,推荐TVMD质量比不超过0.3或阻尼比不超过0.1。以某七层标准钢框架结构作为工程算例展示了TVMD设计流程,并验证了本文解的有效性和优越性。算例分析结果表明,使用本文解设计TVMD能显著放大其阻尼元件变形,表现出了理想的阻尼比增效效应。与传统解相比,本文解还具有另一个明显优势,即保证TVMD的减震效果优于同阻尼系数的VD,不存在减震效率问题。

内界膜剥除术与内界膜翻转遮盖术对特发性黄斑裂孔术后视力及视网膜结构的影响

目的评估内界膜(ILM)翻转遮盖术与ILM剥除术在特发性黄斑裂孔(IMH)手术中的效果,观察2种手术方法对黄斑裂孔闭合方式及术后视力的影响。方法回顾性分析2021年6月—2022年3月于我院就诊的IMH患者54例(54眼)的临床资料。根据不同手术方式将患者分为2组,采用ILM剥除术28例(28眼),采用ILM翻转遮盖术26例(26眼),比较2组患者术后6个月时最佳矫正视力(BCVA)、黄斑孔闭合形态及视网膜外层结构。结果术后6个月时,剥除组和翻转组平均视力(LogMAR)分别为0.85±0.25和0.71±0.19,2组视力较术前均有所提高,差异有统计学意义(t_(1)=12.67,P1<0.001;t_(2)=11.47,P_(2)<0.001)。术后视力翻转组较剥除组更好(t=2.20,P=0.033)。术后6个月时,剥除组黄斑裂孔(MH)成功闭合24例(85.7%),其中U型闭合11例(45.8%),EZ完整10例(41.7%),外界膜(ELM)完整17例(70.8%);翻转组MH成功闭合24例(92.3%),U型闭合18例(75.05%),EZ完整16例(66.7%),EIM完整20例(83.3%),翻转遮盖组术后U型闭合及EZ、ELM恢复均高于剥除组。结论2种手术方式在提高MH闭合率方面均有有效性,ILM翻转遮盖术更有利于裂孔沿着原有解剖形态愈合,有效恢复术后视网膜外层结构,提高患者预后视力。