有限元分析英文文献

毕业设计（论文）中文题目：京沪高铁腕臂结构及其定位装置静力学强度有限元分析英文题目：The Structural Static Strength Analysis Of Cantilever and Positioning Device OfBeijing-Shanghai High-Speed Rail By FEM学院：机械与电子控制工程学院专业：机械工程及自动化学生姓名：陈奕舟学号：08221003指导教师：冯超2012年 6 月 5 日学号：学士论文版权使用授权书本学士论文作者完全了解北京交通大学有关保留、使用学士论文的规定。

特授权北京交通大学可以将学士论文的全部或部分内容编入有关数据库进行检索，提供阅览服务，并采用影印、缩印或扫描等复制手段保存、汇编以供查阅和借阅。

（保密的学士论文在解密后适用本授权说明）学士论文作者签名：指导教师签名：签字日期：年月日签字日期：年月日中文摘要本文以京沪高铁的正定位腕臂为研究对象，研究腕臂结构的静力学强度。

通过对京沪高铁所用正定位腕臂结构分别采用梁单元、壳和实体单元两种网格划分方案进行了腕臂结构的有限元模拟仿真，论文的主要工作以及研究成果表现为：1.通过对京沪高铁某直线段正定位腕臂的受力进行了分析，给出了200km/h和350km/h下的腕臂载荷表。

2.对京沪高铁正定位腕臂结构进行了梁单元、壳和实体单元的有限元模型构建。

3.实体和壳单元分网策略下能够较真实地反映出定位钩和定位环之间的接触状态，能够较清楚地显示结构上的应力分布彩色云图。

4.通过对腕臂结构有限元仿真结果的分析，腕臂结构200km/h工况下的静力学强度分析中应力最大值为74MPa，小于材料许用应力150MPa，结构安全。

5.腕臂结构在350km/h工况下，其应力为162MPa，大于结构材料的许用应力（即铝合金材料的屈服极限），会导致腕臂结构塑性变形。

但是由于铝合金材料的断裂应力为300MPa，所以静力情况下，腕臂结构基本安全。

The Basics of FEA Procedure有限元分析程序的基本知识2.1IntroductionThis chapter discusses the spring element,especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures;however,it represents a structure in an ideal form for an FEA analysis.Spring element doesn’t require discretization(division into smaller elements)and follows the basic equation F=ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。

弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F=ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。

2.2Overview概述Finite Element Analysis(FEA),also known as finite element method(FEM)is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F=ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F=ku切合的关系。

南京林业大学本科毕业设计（论文）外文资料翻译翻译资料名称(外文)Stress analysis of heavy duty truck chassis as apreliminary data for its fatigue life predictionusing FEM翻译资料名称(中文)利用重型载货汽车的有限元应力分析的初步数据预测其疲劳寿命院（系）：汽车与交通工程学院专业：机械制造及其自动化（汽车设计方向）姓名：学号：指导教师：完成日期： 2012/5/31利用重型载货汽车的有限元应力分析的初步数据预测其疲劳寿命Roslan Abd Rahman, Mohd Nasir Tamin, Ojo Kurdi马来西亚工程大学机械工程系81310 UTM, Skudai,Johor Bahru摘要本文对一重型货车底盘做了应力分析。

应力分析能够确定零件的最大受力点，是分析零部件疲劳研究和寿命预测的重要手段。

前人已有用商用有限元软件ABAQUS软件对底盘模型进行分析的。

本次研究的底盘长12.35米，宽2.45米，材料是ASTM低合金钢710（3级），屈服极限552MPa，抗拉强度620MPa。

分析结果显示，最大应力点出现在底盘与螺栓连接的空缺处，最大应力为386.9MPa，底盘的疲劳破坏将会从最大应力点开始向车架各部位蔓延。

关键字：应力分析，疲劳寿命预测，货车底盘1.0简介在马来西亚，很多货车的车架寿命都有20多年，20多年架就会有使用安全的问题。

因此，为了确保底盘在工作期间的安全性能，就有必要对底盘作疲劳研究和寿命预测。

利用有限元法作应力分析能够确定受最大应力的关键点，这个关键点是导致底盘疲劳损伤的因素之一。

应力的大小能够预测底盘的寿命，所以可以根据应力分析的结果精确地预测底盘的寿命，应力分析越精确，底盘寿命预测的越合理。

本文是用商用有限元软件ABAQUS 软件完成底盘应力分析的。

汽车工业（汽车总成及各部件）在马来西亚的工业中占据非常重要的地位。

有限元分析中英文对照资料The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditions Finite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kindof problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method. Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options fordifferent weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sumof the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, sochoose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use thecoordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation.(5) the overall synthesis: after the finite element equation, the area of all elements inthe finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.有限元分析有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂有限元模型：它是真实系统的理想化的数学抽象。

附录1：外文翻译CAE的技术种类有很多，其中包括有限元法,边界元法,有限差法等。

每一种方法各有其应用的领域，而其中有限元法应用的领域越来越广，现已应用于结构力学、结构动力学、热力学、流体力学、电路学、电磁学等。

ANSYS软件是融结构、流体、电场、磁场、声场分析于一体的大型通用有限元分析软件。

由世界上最大的有限元分析软件公司之一的美国ANSYS开发，它能与多数CAD软件接口，实现数据的共享和交换，如Pro/Engineer, NASTRAN, Alogor, I－DEAS, AutoCAD等，是现代产品设计中的高级CAE工具之一。

ANSYS有限元软件包是一个多用途的有限元法计算机设计程序，可以用来求解结构、流体、电力、电磁场及碰撞等问题。

因此它可应用于以下工业领域：航空航天、汽车工业、生物医学、桥梁、建筑、电子产品、重型机械、微机电系统、运动器械等。

有限元分析（FEA，Finite Element Analysis）的基本概念是用较简单的问题代替复杂问题后再求解。

它将求解域看成是由许多称为有限元的小的互连子域组成，对每一单元假定一个合适的(较简单的）近似解，然后推导求解这个域总的满足条件(如结构的平衡条件），从而得到问题的解。

这个解不是准确解，而是近似解，因为实际问题被较简单的问题所代替。

由于大多数实际问题难以得到准确解，而有限元不仅计算精度高，而且能适应各种复杂形状，因而成为行之有效的工程分析手段。

有限元是那些集合在一起能够表示实际连续域的离散单元。

有限元的概念早在几个世纪前就已产生并得到了应用，例如用多边形（有限个直线单元）逼近圆来求得圆的周长，但作为一种方法而被提出，则是最近的事。

有限元法最初被称为矩阵近似方法，应用于航空器的结构强度计算，并由于其方便性、实用性和有效性而引起从事力学研究的科学家的浓厚兴趣。

经过短短数十年的努力，随着计算机技术的快速发展和普及，有限元方法迅速从结构工程强度分析计算扩展到几乎所有的科学技术领域，成为一种丰富多彩、应用广泛并且实用高效的数值分析方法。

The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditionsFinite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system. Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finiteelement analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary. At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitutedifferent finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method. Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functionscan be used as weight function. According to the weight function of the different options for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force of the sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinearelastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy. The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of thefinite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs will drive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell.For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximatestructure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. T ake an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integratedtotal stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell. For the finite element method, the basic ideas and steps can be summarized as: (1) to establish integral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value. (3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the实用标准文案unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation. (5) the overall synthesis: after the finite element equation, the area of all elements in the finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.精彩文档。

The Basics of FEA Procedure有限元分析程序的基本知识2.1 IntroductionThis chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique.本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的A spring element is not very useful in the analysis of real engineering structures; however, it represents a structure in an ideal form for an FEA analysis. Spring element doesn’t require discretization (division into smaller elements) and follows the basic equation F = ku.在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一个理想的形式分析。

弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程F = ku We will use it solely for the purpose of developing an understanding of FEA concepts and procedure.我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。

2.2 Overview概述Finite Element Analysis (FEA), also known as finite element method (FEM) is based on the concept that a structure can be simulated by the mechanical behavior of a spring in which the applied force is proportional to the displacement of the spring and the relationship F = ku is satisfied.有限元分析(FEA),也称为有限元法(FEM)，是基于一个结构可以由一个弹簧的力学行为模拟的应用力弹簧的位移成正比,F = ku切合的关系。

CHAPTER 3Truss Element3.1 IntroductionThe single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for representing a real engineering structure, and therefore, their selection must be a true representation of geometry and mechanical properties of the structure. Any deviation from either the geometry or the mechanical properties would yield erroneous results.The elements used in commercial codes can be classified in two basic categories:1.Discrete elements: These elements have a well defined deflection equation that canbe found in an engineering handbook, such as, Truss and Beam/Frame elements. The geometry of these elements is simple, and in general, mesh refinement does not affect the results. Discrete elements have a very limited application; bulk of the FEAapplication relies on the Continuous-structure elements.2.Continuous-structure Elements: Continuous-structure elements do not have a welldefine deflection or interpolation function, it is developed and approximated by using the theory of elasticity. In general, a continuous-structure element can have anygeometric shape, unlike a truss or beam element. The geometry is represented by 1-D, 2-D, or a 3-D solid element. Since elements in this category can have any shape, it is very effective in calculation of stresses at a sharp curve or geometry, i.e., evaluation of stress concentrations. Since discrete elements cannot be used for this purpose,continuous structural elements are extremely useful for finding stress concentration points in structures.NodesAs explained earlier, for analyzing an engineering structure, we divide the structure into small sections and represent them by appropriate elements. Nodes define geometry of the structure and elements are generated when the applicable nodes are connected. Resultsare always obtained for node points – and not for elements - which are then interpolated to provide values for the corresponding elements.For a static structure, all nodes must satisfy the equilibrium conditions and the continuity of displacement, translation and rotation. However, the equilibrium conditions may not be satisfied in the elements.In the following sections, we will get familiar with characteristics of the basic finite elements.3.2 Structures & ElementsMost 3-D structures can be analyzed using 2-D elements (idealization), which require relatively less computing time than the 3-D solid elements. Therefore, in FEA, 2-D elements are the most widely used elements. However, there are cases where we must use 3-D solid elements. In general, elements used in FEA can be classified as: - Trusses-Beams-Plates-Shells-Plane solids-Axisymmetric solids-3-D solidsSince Truss element is a very simple and discrete element, let us look at its properties and application first.3.3Truss ElementsThe characteristics of a truss element can be summarized as follows:Truss is a slender member (length is much larger than the cross-section).It is a two-force member i.e. it can only support an axial load and cannot support a bending load. Members are joined by pins (no translation at the constrained node, but free to rotate in any direction).The cross-sectional dimensions and elastic properties of each member are constant along its length.The element may interconnect in a 2-D or 3-D configuration in space.The element is mechanically equivalent to a spring, since it has no stiffness against applied loads except those acting along the axis of the member.However, unlike a spring element, discussed in previous chapters, a truss element can be oriented in any direction in a plane, and the same element is capable of tension as well as compression.jiFigure 3.2 A Truss Element3.3.1 Stress – Strain relation :As stated earlier, all deflections in FEA are evaluated at the nodes. The stress and strainin an element is calculated by interpolation of deflection values shared by nodes of the element. Since the deflection equation of the element is clearly defined, calculation of stress and strain is rather simple matter. When a load F is applied on a truss member, the strain at a point is found by the following relationship. xor, ε = δL/LL + δLFigure 3.3 Truss member in Tensionwhere, ε = strain at a pointu = axial displacement of any point along the length LBy hook’s law,Where, E = young’s modulus or modulus of elasticity.From the above relationship, and the relation,F = A σdxdu =εεσE =the deflection, δL, can be found asδL = FL/AE (3.1) Where, F = Applied loadA = Cross-section areaL = Length of the element3.3.2 Treatment of Loads in FEAFor a truss element, loads can be applied on a node only. If loads are distributed on a structure, they must be converted to the equivalent loads that can be applied at nodes. Loads can be applied in any direction at the node, however, the element can resist only the axial component, and the component perpendicular to the axis merely causes free rotation at the joint.3.3.3 Finite Element Equation of a Truss StructureIn this section, we will derive the finite element equation of a truss structure. The procedure presented here is the basis for all FEA analyses formulations, wherever h-element are used.Analogues to the previous chapter, we will use the direct or equilibrium method for generating the finite element equations. Assembly procedure for obtaining the global matrix will remain the same.In FEA, when we find deflections at nodes, the deflections are measured with respect to a global coordinate system, which is a fixed frame of reference. Displacements of individual nodes with respect to a fixed coordinate system are desirable in order to see the overall deformed structural shape. However, these deflection values are not convenient in the calculation of stress and strain in an element. Global coordinate system is good for predicting the overall deflections in the structure, but not for finding deflection, strain, and stress in an element. For this, it’s much easier to use a local coordinate system. We will derive a general equation, which relates local and global coordinates.In Figure 3.4, the global coordinates x-y can give us the overall deflections measured with respect to the fixed coordinate system. These deflections are useful for finding the final shape or clearance with the surroundings of the structure. However, if we wish to find the strain in some element, say, member 2-7 in figure 3.4, it will be easier if we know the deflections of node 2 and 3, in the y’ direction. Thus, calculation of strain value is much easier when the local deflection values are known, and will be time- consuming if we have to work with the x and y values of deflection at these nodes.Therefore, we need to establish a trigonometric relationship between the local and global coordinate systems. In Figure 3.4, xy coordinates are global, where as, x’y’ are local coordinates for element 4-7Nodes2 3 4 5 xFigure 3.4. Local and Global Coordinates3.3.4 Relationship Between Local and Global DeflectionsLet us consider the truss member, shown in Figure 3.5. The element is inclined at an angle θ, in a counter clockwise direction. The local deflections are δ1 and δ2. The global deflections are: u1, u2, u3, and u4. We wish to establish a relationship between these deflections in terms of the given trigonometric relations.δ2, R2R1, δ1Figure 3.5 Local and Global DeflectionsBy trigonometric relations, we have,δ1 = u1x cosθ + u2 sinθ = c u1x + s u1yδ2 = u2x cosθ + u2y sinθ = c u2x + s u2ywhere, cosθ = c, and sinθ = sWriting the above equations in a matrix form, we get,u1xδ1 c s 0 0 u1y= u2x (3.2) δ10 0 c s u2yOr, in short form, δ = T uWhere T is called Transformation matrix.Along with equation (3.2), we also need an equation that relates the local and global forces.3.3.5 Relationship Between Local and Global ForcesBy using trigonometric relations similar to the previous section, we can derive the desired relationship between local and global forces. However, it will be easier to use the work-energy concept for this purpose. The forces in local coordinates are: R1 and R2, and in global coordinates: f1, f2, f3, and f4, see Figure 3.6 for their directions.Since work done is independent of a coordinate system, it will be the same whether we use a local coordinate system or a global one. Thus, work done in the two systems is equal and given as,W = δT R = u T f, or in an expanded form,R1 f1W = δ1 δ2 = u1x u1y u2x u2y f2R2 f3f4= {δ}T {R} = {u}T{f}Substituting δ = T u in the above equation, we get,[[T] {u}]T {R} = {u}T {f}, or{u}T [T]T {R} = {u}T {f}, dividing by {u}T on both sides, we get,[T]T {R} = {f}(3.3)Equation (3.3) can be used to convert local forces into global forces and vice versa.2, δ2 R 1, δ1 11x Figure 3.6 Local and Global Forces3.3.6 Finite Element Equation in Local Coordinate SystemNow we will derive the finite element equation in local coordinate system. This equation will be converted to global coordinate system, which can be used to generate a global structural equation for the given structure. Note that, we can not use the elementequations in their local coordinate form, they must be converted to a common coordinate system, the global coordinate system.Consider the element shown below, with nodes 1 and 2, spring constant k, deflections δ1, and δ2, and forces R 1 and R 2. As established earlier, the finite element equation in local coordinates is given as,R 1 k -k δ1 = δ1, R 1 R 2 -k k δ1 22Figure 3.7 A Truss Element Recall that, for a truss element, k = AE/LLet k e = stiffness matrix in local coordinates, then,AE/L -AE/Lk e = Stiffness matrix in local coordinates-AE/L AE/L3.3.7 Finite Element Equation in Global CoordinatesUsing the relationships between local and global deflections and forces, we can convert an element equation from a local coordinate system to a global system.Let k g = Stiffness matrix in global coordinates.In local system, the equation is: R = [k e]{ δ} (A)We want a similar equation, but in global coordinates. We can replace the local force R with the global force f derived earlier and given by the relation:{f} = [T T]{R}Replacing R by using equation (A), we get,{f} = [T T] [[k e]{ δ}],and δ can be replaced by u, using the relation δ = [T]{u}, therefore,{f} = [T T] [k e] [T]{u}{f} = [k g] { u}Where, [k g] = [T T] [k e] [T]Substituting the values of [T]T, [T], and [k e], we get,c 0[k g] = s 0 AE/L -AE/L c s 0 00 c -AE/L AE/L 0 0 c s0 sSimplifying the above equation, we get,c2cs -c2-cscs s2-cs -s2[k g] = -c2-cs c2cs (AE/L)-cs -s2cs s2This is the global stiffness matrix of a truss element. This matrix has several noteworthy characteristics:The matrix is symmetricSince there are 4 unknown deflections (DOF), the matrix size is a 4 x 4.The matrix represents the stiffness of a single element.The terms c and s represent the sine and cosine values of the orientation of element with the horizontal plane, rotated in a counter clockwise direction(positive direction).The following example will illustrate its application.ExamplesFind displacements of joints 2 and 3Find stress, strain, & internal forcesin each member.A AL = 200 mm2 , A ST = 100 mm2All other dimensions are in mm.SolutionLet the following node pairs form the elements:Element Node Pair(1) 1-3(2) 2-1(3) 2-3Using Shigley’s Machine Design book for yield strength values, we have, S y(AL) = 0.0375kN/mm2 (375 Mpa)S y (ST) = 0.0586kN/mm2 (586 Mpa)E (AL) = 69kN/mm2 , E (ST) = 207kN/mm2A(1) = A(2) = 200mm2 , A(3) =100mm2Find the stiffness matrix for each elementu1y u3y Element (1)L(1 ) = 260 mm, u1x u3x E(1) = 69kN/mm2A(1) = 200mm2θ = 0c = cosθ = 1, c2 = 1s = sinθ = 0, s2 = 0cs = 0EA/L = 69 kN/mm2 x 200 mm2 x 1/(260mm) = 53. 1 kN/mmc2cs - c2-cs[K g](1) = (AE/L) x cs s2-cs - s2-c2-cs c2cs-cs -s2cs s21 0 -1 0[K g](1) = (53.1) x 0 0 0 0-1 0 1 00 0 0 0u1y Element 2θ = 900c = cos 900 = 0, c2 = 0 u1x s = sin 900 = cos 00 = 1, s2 = 1cs = 0EA/L = 69 x 200 x (1/150) = 92 kN/mmu2xu2x u2y u1x u1y0 0 0 0 u2x0 1 0 -1 u2y[k g](2) = (92) 0 0 0 0 u1x0 -1 0 1 u1yElement 3θ = 300c = cos 300 = 0.866, c2 = 0.753x s = cos 600 = .5, s2 = 0.25cs = 0.433 uEA/L = 207 x 100 x (1/300) = 69 kN/mmu2x u2y u3x u3yu2x .75 .433 -.75 -.433u2y -.433 .25 -.433 -.25 (69)[k g](3) = u3x-.75 -.433 .75 .433u3y-.433 -.25 .433 .25Assembling the stiffness matricesSince there are 6 deflections (or DOF), u1 through u6, the matrix is 6 x 6. Now, we will place the individual matrix element from the element stiffness matrices into the global matrix according to their position of row and column members.Element [1]u1x u1y u2x u2y u3x u3yu1x 53.1 -53.1u1yu2xu2yu3x -53.1 53.1u3yThe blank spaces in the matrix have a zero value.Element [2]u1x u1y u2x u2y u3x u3yu1x92 -92u1yu2xu2y -92 92u3xu3yElement [3]u1x u1y u2x u2y u3x u3yu1xu1yu2x51.7 29.9 -51.7 -29.9u2y29.9 17.2 -29.9 -17.2u3x-51.7 -29.9 51.7 29.9u3y-29.9 -17.2 29.9 17.2Assembling all the terms for elements [1] , [2] and [3], we get the complete matrix equation of the structure.u1x u 1yu 2x u 2y u 3x u 3y 53.1 0 0 0 -53.1 0 u 1x F 1 0 92 0 -92 0 0 u 1y F 1 0 0 51.7 29.9 -51.7 -29.9 u 2x = F 1 0 -92 29.9 109.2 -29.9 -17.2 u 2y F 1 -53.1 0 -51.7 -29.9 104.8 29.9 u 3x F 1 0 0 -29.9 -17.2 29.9 17.2u 3y F 1Boundary conditionsNode 1 is fixed in both x and y directions, where as, node 2 is fixed in x-direction only and free to move in the y-direction. Thus,u 1x = u 1y = u 2x = 0.Therefore, all the columns and rows containing these elements should be set to zero. The reduced matrix is:Writing the matrix equation into algebraic linear equations, we get,29.9u 2y - 29.9 u 3x - 17.2u 3y =0 -29.9u 2y + 104 u 3x + 29.9u 3y = 0 -17.2u 2y + 29.9u 3x + 17.2u 3y = -0.4solving, we get u 2y = -0.0043 u 3x = 0.0131 u 3y = -0.0502Sress, Strain and deflectionsElement (1)Note that u 1x , u 1y , u 2x , etc. are not coordinates, they are actual displacements.−= −−−−4.0002.179.292.179.298.1049.292.179.292.109332y x y u u uL = u3x = 0.0131= L/L = 0.0131/260 = 5.02 x 10-5 mm/mm= E = 69 x 5.02 x 10-5 = 0.00347 kN/mm2Reaction R = A = 0.00347 kNElement (2)L = u2y = 0.0043= L/L = 0.0043/150 = 2.87 x 10-5 mm/mm= E = 69 x 2.87 x 10-5 = 1.9803 kN/mm2Reaction R = A = (1.9803 x 10-3) (200) = 0.396 kNElement (3)Since element (3) is at an angle 300, the change in the length is found by adding the displacement components of nodes 2 and 3 along the element (at 300). Thus,L = u3x cos 300 + u3y sin 300 – u2y cos600= 0.0131 cos300 -0.0502 sin300 + 0.0043 cos600= -0.0116= L/L = -0.0116/300 = -3.87 x 10-5 = 3.87 x 10-5 mm/mm= E = 207 x -3.87.87 x 10-5 = -.0080 kN/mm2Reaction R = A = (-0.0087) (100) = 0.-0.800 kNFactor of SafetyFactor of safety ‘n’ is the ratio of yield strength to the actual stress found in the part.The lowest factor of safety is found in element (3), and therefore, the steel bar is the most likely to fail before the aluminum bar does.Final Notes- The example presented gives an insight into how the element analysis works. The example problem is too simple to need a computer based solution; however, it gives the insight into the actual FEA procedure. In a commercial FEA package, solution of a typical problem generates a very large stiffness matrix, which will require a computer assisted solution.- In an FEA software, the node and element numbers will have variable subscriptsso that they will be compatible with a computer-solution- Direct or equilibrium method is the earliest FEA method.0.0375(1)10.80.003470.0375(2)18.90.001980.0586(3)7.3250.0080yyyS Element n S Element n S Element n σσσ=========Example 2Given:Elements 1 and 2: Aluminum Element 3: steel A (1) = 1.5in 2 A (2) = 1.0in 2 A (3) = 1.0in 2Required:Find stresses and displacements using hand calculationsy.SolutionCalculate the stiffness constants:Calculate the Element matrix equations.inlb L AE K in lb L AE K in lb L AE K 563562561100.650101030105.24010101105.72010105.1×=××==×=××==×=××==Element (1)u2x Denoting the Spring constant for element (1) by k1, and the stiffness matrix by K(1), the stiffness matrix in global coordinates is given as,u1x u1y u2x u2yc2cs -c2-cs u1xcs s2-cs -s2 u1y[K g](1) = K1c2-cs c2cs u2x-cs -s2cs s2 u2yFor element (1), θ = 00, thereforec =1, c2 = 1s = 0, s2 =0, and cs = 0u1x u1y u2x u2y1 0 -1 0 u1x0 0 0 0u1y[k(1)] = k1 1 0 1 0 u2x0 0 0 0u2yElement (2) 2xFor this element, θ = 900, Therefore,c= cosθ = 0, c2 = 0 0s = sinθ = 1, s2 = 1 u3xcs= 0The stiffness matrix is, u3yu3x u3y u2x u2yc2cs -c2-cs u3xcs s2-cs -s2 u3y[k g](2) = k2c2-cs c2cs u2x-cs -s2cs s2 u2yu3x u3y u2x u2y0 0 0 0 u3x0 1 0 -1u3y[k g](2) = k20 0 0 0 u2x0 -1 0 1u2yElement 3 uFor element (3), θ = 126.90.c= cos(126.90) = -0.6, c20 s = sin(126.90, s2 = .64cs = -0.484yu4x u4y u2x u2y.36 -.48 -.36 .48 u4x-.48 .64 .48 -.64u4y[k g](3) = k3-.36 .48 .36 -.48 u2x.48 -.64 -.48 .64u2yAssembling the global MatrixFollowing the procedure for assembly described earlier, the assembled matrix is,[K g] =111133333232332233333333 00000010000000020.36.4800.36.48300.48.640.48.644000000005000000600.36.4800.36.48700.48.6400.48.648K KK K K K K KK K K K K KK KK K K KK K K K−−+−−+−−−−−−−The boundary conditions are:u1x = u1y = u3x = u3y = u4x = u4y = 0We will suppress the corresponding rows and columns. The reduced matrix is a 2 x2, given below,u2x u2yK1-.48K3 u2x[K g] =-.48K3 K2 + .64 K3 u2yThe final equation is,K1 + .36 K3 -.48K3 u2x -4000=-.48K3 K2 + .64 K3 u2x 8000Substituting values for k1, k2, and k3, we getChapter 3 Truss ElementFEA Lecture Notes © by R. B. Agarwal 3-21 9.66 -2.88 u 2x -4000 105 = -2.88 6.34 u 2y 8000 u 2x = - 0.0000438 in. u 2y = - 0.01241 in.1 = P/A = K 1 u/A 1 = [(7.5 x 105)(- 0.0000438)] / (1.5) = 214 psi2= P/A = K 2 u/A 2 = [(2.5 x 105)(- 0.012414)] / (1.0) = 3015 psi3 = P/A = K 3 u/A 3 = [(6 x 105)(- 0.0000438 cos 53.10 + 0.012414 sin 53.10)] / (1.0) = 6119 psi。

Studies in dynamic design of drilling machine using updated finite element modelsAbstractThe aim of the present work is to develop updated FE models of a drilling machine using analytical and experimental results. These updated FE models have been used to predict the effect of structural dynamic modifications on vibration characteristics of the drilling machine. Two studies have been carried out on the machine. In the first study, modal tests have been carried out on a drilling machine using instrumented impact hammer. Modal identification has been done using global method of modal identification. For analytical FE modeling of the machine, a computer program has been developed. The results obtained using FEM, have been correlated with the experimental ones using mode shape comparison and MAC values. Analytical FE model has been updated, with the help of a program, which has been developed using direct methods of model updating. In the second study, modal testing has been carried out using random noise generator and modal exciter. Global method has been used for modal identification. Analytical FE modeling has been done using I-DEAS software. Correlation of FE results with the experimental ones has been carried out using FEMtools software. Updating of the analytical FE model has also been done using the above software, based on an indirect technique viz. sensitivity based parameter estimation technique. The updated FE models, obtained from both the studies have been used for structural dynamic modifications (SDM), for the purpose of dynamic design and the results of SDM predictions are seen to be reasonably satisfactory.Article Outline1. Introduction2. Modal testing and identification3. Finite element formulation of drilling machine4. Comparison of analytical FE and experimental results (model correlation)5. Finite element model updating6. SDM studies using updated models for dynamic design7. ConclusionsReferences1. IntroductionDynamic design aims at obtaining desired dynamic characteristics in machines and structures, which may include shifting of natural frequencies, desired mode shapes and vibratory response. The ultimate objectives are to have a quieter and more comfortable environment, higher reliability and better quality of product. The conventional dynamic design is basically hit and trial method in which we try to achieve desired dynamic characteristics by making several prototypes. The disadvantage of this technique is that actual design cycle takes a lot of time and therefore it is not cost effective. However, model updating based dynamic design saves design cycle time as well as reduces the cost involved. Various tools used for updating based dynamic design are: experimental modal analysis (EMA) including modal testing and modal identification, model updating and structural dynamic modification.Ewins [1]and Maia and Silva [2]have explained the basic concepts of modal testing, which is an experimental approach to obtain mathematical model of a structure. In a modal test, the structure under test is excited either by an impact hammer or by a modal exciter, and the response of the structure is recorded at several experimental points, in the form of frequency response functions (FRFs), using a dual channel FFT analyzer. The experimental modal model gives information about the natural frequencies, corresponding mode shapes and modal damping factor and is useful for model updating. The model updating techniques helps us to bring analytical finite element models closer to real systems. In model updating an initial analytical FE model constructed for analyzing the dynamics of a structure is refined or updated using test data measured on actual structure such that the updated model describes the dynamic properties of the structure more correctly. The inaccuracies in FEM, when applied to dynamic problems are due to uncertainties in boundary conditions and structural damping etc.Friswell and Mottershead [3] have discussed the finite element model updating in structural dynamics. Baruch and Bar-Itzhack and Baruch [4] and [5] considered analytical mass matrix to be exact and developed a direct method for updating using test data. Berman and Nagy [6]developed a method of model updating, which uses measured modes and natural frequencies to improve analytical mass and stiffness matrices. Structural dynamic modification (SDM) techniques [7]and [8]are the methods by whichdynamic behaviour of the structure is improved by predicting the modified behaviour brought about by adding modifications like those of lumped masses, rigid links, dampers etc. Thus the dynamic design using updated model is expected to be helpful in order to predict accurately and quickly, the effect of possible modifications on the dynamic characteristics of the structure at computer level itself, thus saving time and cost.Sestieri [7]has discussed SDM application to machine tools and engines. Kundra [8]gave the method of structural dynamic modification via models. Modak [9] has discussed SDM predictions using updated FE model for an F-structure. He used constrained nonlinear optimization method for updating of a machine tool using stiffness parameters at the boundary [10]. The present paper deals with the FE model updating using direct as well as indirect method, and to use this updated FE model for dynamic design based on SDM predictions of a machine tool viz. a drilling machine. Two different studies are reported using different techniques for analytical and experimental analysis and for updating. Various objectives with which the present research work has been carried out are• To develop updated FE models of a complex structure like that of a drilling machine and to use these updated models to predict the effect of various modifications on modal properties of the machine.• To see whether hammer excitation yields good results for fairly complex structures like drilling machine or not, and to compare these results with those obtained from modal exciter.• To analyze the results of SDM predictions obtained using the updated models derived in the studies.2. Modal testing and identificationIn the two studies mentioned earlier, different techniques have been used, for modal testing and identification. In the first study, impact hammer is used to excite the drilling machine structure, at various points as shown in Fig. 1 and Fig. 2. Response is taken at a fixed point with the help of an accelerometer.(12K)Fig. 1. Experimental setup (Study 1).(4K)Fig. 2. Hammer excitation locations.In the present study, the drilling machine is excited at 30 locations and therefore, 30 FRFs are obtained. These FRFs are recorded in the form of inertance. The experimental FRFs, thus obtained are transferred to computer. Modal identification or modal parameter extraction consists of curve fitting a theoretical expression for an individual FRF to the actual measured data obtained. The experimental FRFs are analyzed by GRF-M method using modal analysis software ICATS [11] to obtain modal parameters of the drilling machine. In the second study, the machine tool structure has been excited at the base at point 28, referring to Fig. 2, using modal exciter and response has been measured at various points using piezoelectric accelerometers. The modal identification of the FRFs, thus measured has been carried out using global method GRF-M method in ICATS software.Table 1compares the experimental natural frequencies obtained from both the methods, which shows minor differences in the two modal frequenciesTable 1.Mode110.29 Hz8.67 HZ0.95311.20 Hz8.79 Hz0.946Mode 265.14 Hz47.34 HZ0.90163.37 Hz44.40 Hz0.9063. Finite element formulation of drilling machineSeveral books have given the basic concepts of finite element analysis, some of them are: Zienkiewicz [12] and Bathe [13].The drilling machine structure is very complicated with different mountings and accessories. Therefore exact modeling and analysis of the actual structure is difficult and it takes more computational effort. However for analytical FE analysis, simplified model of drilling machines has been considered. In study 1, the finite element modelling has been done using a program developed in MATLAB. Beam elements have been used for the analysis. The joints and boundary conditions are considered to be rigid and influence of structural damping on modal model parameters, is ignored. The relevant data used for the drilling machine is given below:25 mm pillar type, height = 1.655 m, mass density = 7800 kg/m3, Young’s modulus= 200 Gpa, number of nodes = 30, number of elements = 29, number of nodes per element = 2, degrees of freedom per node = 3.Fig. 3shows the structure of the drilling machine with the node numbers given for study 1.(7K)Fig. 3. Drilling machine structure for FE analysis.The eigenvalues and eigenvectors have been calculated. The analytical FE model of the structure consists of 90 ×90-size mass and stiffness matrices (30 ×3, 30 nodes and 3 d.o.f. per node). But by experiment only 30 coordinates can be measured. Therefore FE model has been reduced using Guyan [14] reduction method with the help of a program developed in MATLAB.In study 2, the finite element modelling has been done using I-DEAS software. The model has been made using beam mesh. Although the FE model has been simplified but the beam elements has rotational degree of freedom, which cannot be measured experimentally. Therefore the FE model needs to be reduced. The FE model has been reduced using model reduction utilityin FEMtools software. Fig. 4 and Fig. 5 shows the mode shape animation for the first and second mode respectively, using I-DEAS software.(31K)Fig. 4. Mode shape animation (first mode).(23K)Fig. 5. Mode shape animation (second mode).4. Comparison of analytical FE and experimental results (model correlation)The first stage of any reconciliation exercise is to determine how closely the experimental and analytical models correspond. If we are unable to obtain a satisfactory degree of correlation between the initial analytical FE model and the test data, then it is extremely unlikely that any form of model updating will succeed. Thus, a successful correlation is crucial for the success of model updating. Table 1gives the comparison between experimental and analytical natural frequencies. There are differences between analytical FE model predictions and experimental results. Thus the FE models need to be updated. However, the differences between the corresponding results of both studies are minor.Apart from natural frequency comparison (as given in Table 1), another method of model correlation is mode shape comparison. To compare the mode shapes, we plot the deformed shapes of the structure for a particular mode, using experimental as well as analytical model. These mode shapes are plotted side-by-side for quick comparison. Mode shape corresponding to second mode is shown in Fig. 6, using ICATS software. It shows a fairlygood level of correlation between the experimental and analytical FE model.(87K)Fig. 6. Mode shape comparison.Several researchers have developed techniques for quantifying the comparison between measured and predicted mode shapes. As an alternative to the graphical approach, Model Assurance Criterion i.e. MAC, [15]) is a widely used technique to estimate the degree of correlation between mode shape vectors. This provides a measure of the least squares deviation or ‘scatter’ of the points from the straight-line correlation. The MAC between a measured and analytical mode is:(1)wherem andarepresents measured or experimental and analytical modeshapes respectively. MAC is a scalar quantity whose value is between 0 and 1. A value of MAC close to 1 shows a good degree of correlation between experimental and analytical FE model. We can see in Table 1 that the MAC numbers are close to 1, though somewhat lower for the second mode. Table 1 also shows that the results obtained from both the studies are quite close to each other.5. Finite element model updatingModel updating can be defined as “the process of correcting the numerical values of individual parameters in a analytical FE model using data obtained from an associated experimental model such that the updated model correctly describes the dynamic properties of the subject structure”.Various model updating methods can be classified into two major groups: • Direct matrix methods• Indirect or iterative methodsDirect methods are capable of reproducing measured data exactly, but they provide no opportunity for the user to select parameters for updating. Here parameter means any physically realizable quantity like Young’s modulus, Poisson’s ratio, mass density etc. When using the direct methods, the entire stiffness and mass matrices are updated in a single(non-iterative) solution step. Consequently any physical meaning, which the initial finite element model might have possessed, is lost in the updating process.Techniques like the indirect methods, allow the updating parameters to be selected. So, considerable physical insight is required if the model is to be improved, not only in its ability to reproduce test results, but also in interpreting the parameters physically. Methods in this second group are iterative and, as such, considerably more expensive of computer effort.Two studies have been carried out for model updating and computer programs for the same were developed in the present work using MATLAB. In the first study, two direct methods are applied to update the analytical FE model of the drilling machine structure.Baruch and Bar-ltzhack [4] and Baruch [5] considered the mass matrix of the analytical model to be exact. The measured eigenvectors are corrected by using the relation:(2)The stiffness matrices of the analytical FE model after updating is given as:K=K a-K a T M a+M a T K a+M a T K a T M a+M aΛT M a(3)uBerman and Nagy [6] used a method similar to that of Baruch. They update mass and stiffness matrices while the mass matrix is updated to ensure the orthogonality of the exact FE model modes. The mass matrix is updated as:(4)(5) The stiffness matrix is updated using following equation:(6) The updated mass matrix obtained will be symmetric and stiffness matrix will be close to that of exact stiffness matrix. Results obtained fromBaruch and Berman–Nagy model updating methods are tabulatedin Table 2. It is clear from Table 2, that the updated model reproduces the measured frequencies. After updating, MAC values have been calculated again using Eq. It can be observed that updated MAC values show some improvement over the initial MAC values.Table 2.Comparison of experimental and updated FE frequencies and MAC valuesModenumberStudy 1Study 2Measured frequency Baruch method Berman method MeasuredfrequencySensitivitymethodUpdatedfrequencyMACUpdatedfrequencyMACUpdatedfrequencyMACMod e 18.67 Hz8.67 Hz0.9258.67 Hz0.9348.79 Hz8.79 Hz0.947Mod e 247.34 Hz47.34 Hz0.96647.34 Hz0.94744.40 Hz44.43 Hz0.913The results after updating have been tabulated in Table 2. It clearly shows that after updating, the updated FE model closely represents the actual machine tool structure. It also shows that MAC values have also been improved after updating.6. SDM studies using updated models for dynamic designStructural dynamic modification (SDM) techniques are methods by which dynamic characteristics of the structure can be improved by adding the modifications like changing mass, spring, damping etc. The mass modification has been considered here for predicting dynamic characteristics using updated FE model.A mass modification on drilling machine is introduced in the form of a lumped mass of 14.3 kg at the top of the vertical pillar, i.e. node 20 as in Fig. 3. The modal test for the mass modified machine is carried out by Modak [10], using impact hammer for excitation. The FRFs are analyzed in ICATS in order to obtain an experimental estimate of the altered dynamic characteristics of the drilling machine, as given in Table 3.Table 3.Comparison of measured and predicted frequencies, after mass modification, using updated FE modelMode no.MeasuredfrequenciesSDM prediction using updated FE modelBaruch’s method Berman and Nagy’s method Sensitivity method18.37 Hz8.34 Hz8.17 Hz8.40 Hz246.05 Hz47.15 Hz46.96 Hz43.20 HzThe effect of the same mass modification on the dynamic characteristics of the drilling machine has also been predicted by updated FE model. Table 3 gives a comparison of the predictions based on the updated FE models obtained by direct methods of Baruch, Berman and Nagy and by indirectmethod based on sensitivity analysis, with that of the measured modified characteristics.It is seen from the Table 3 that the updated FE model predictions of the natural frequencies are quite close to the measured value of natural frequencies. This shows the capability of the updated FE model to accurately predict the effect of structural modifications on the dynamic properties of the structure. Now, this updated FE model has been further used to predict, at computer level, the effect of various mass modifications on the structural dynamics of the drilling machine, for the purpose of dynamic design.The mass modification can bring about significant changes in the natural frequencies of a structure. For predicting the effect of modifications using updated FE model, modification at node 20 and node 25, on the drilling machine has been considered. The effects on FRFs due to these modifications have been shown in Fig. 8and Fig. 9respectively. The values of the first two natural frequencies have been predicted using updated FE models and MODIFY module of ICATS software and are shown in Table 4and Table 5.(33K)Fig. 8. Regenerated FRF due to mass modification at node 20.(29K)Fig. 9. Regenerated FRF due to mass modification at node 25.Table 4.Predicted natural frequencies after mass modification using 20 kg at node 20 (PanelA), predicted natural frequencies after mass modification using 40 kg at node 20 (PanelB)Comparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured Frequencies Panel AMass modification of 20 kg8.22 Hz7.99 Hz8.3 Hz8.2 Hz8.3 Hz47.08 Hz46.83 Hz42.87 Hz47.1 Hz42.8 HzPanel BMass modification of 40 kg7.83 Hz7.45 Hz7.76 Hz7.75 Hz7.8 Hz46.86 Hz46.44 Hz41.27 Hz46.6 Hz41.3 HzTable 5.Predicted natural frequencies after mass modification using 20 kg at node 25 (PanelA), predicted natural frequencies after mass modification using 40 kg at node 25 (PanelB)Comparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured frequenciesPanel AMass modification of 20kg8.34 Hz8.44 Hz8.41 Hz8.35 Hz8.4 Hz 42.24 Hz43.74 Hz41.67 Hz42.2 Hz41.6 HzComparisons of SDM predictions using updated FE modelBaruch method Berman method Sensitivity method ICATS MODIFY Measured frequenciesPanel BMass modification of 40kg8.03 Hz8.21 Hz8.04 Hz8.1 Hz8.1 Hz38.89 Hz41.02 Hz39.67 Hz39.15 Hz39.5 Hz7. ConclusionsComparison of results obtained from experimental modal analysis and FE models of a drilling machine, indicate that its finite element models need to be updated. This is necessary in order to predict dynamic behavior of the complex structure with an acceptable accuracy.An experimentation involving modal testing has been carried out on the drilling machine using impact hammer as well as modal exciter. It has been observed that both impact hammer and modal exciter yield good results for fairly complex structures like drilling machine.Analytical FE model has been updated in the light of experimental data using direct as well as indirect methods. Both the methods give results, which are fairly close when used for predicting results of SDM attaching additional mass on the machine at different locations. The predicted results have been validated by comparison with measured results and are seen to be fairly accurate in particular for the indirect method using sensitivity analysis.References[1]D.J. Ewins, Modal Testing: Theory and Practice, John Willey and Sons, New York (2000).[2] N.M.M. Maia and J.M.M. Silva, Theoretical and Experimental Modal Analysis, John Willey and Sons, New York (1997).[3]M.I. Friswell and J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht (1995).[4] M. Baruch and I.Y. Bar-ltzhack, Optimal weighted orthogonalization of measured modes, AIAA Journal16 (1978), pp. 346–351.[5] M. Baruch, Optimisation procedure to correct stiffness and flexibility matrices using vibration test data, AIAA Journal16 (1978), pp. 1208–1210.[6]A. Berman and E.J. Nagy, Improvement of a large analytical model using test data, AIAA Journal21 (1983), pp. 1168–1173.[7] A. Sestieri, SDM application to machine tools and engines, Sadhana 25 (2000), pp. 305–317.[8] T.K. Kundra, Structural dynamic modification via model, Sadhana25 (2000), pp. 261–276.[9]S.V. Modak, Studies in Finite Element Model Updating and Application to Dynamic Design, Ph.D. Thesis, Department of Mechanical Engineering, IIT Delhi, 2001.[10]S.V. Modak, T.K. Kundra, B.C. Nakra, Dynamic design of machine tool structures using an updated model Pro. IMAC- XX, 2002, pp. 1489–1494.[11] ICATS Reference Manual, Imperial College, London, 1996.[12]O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill Publishing Company, London (1977).[13] K.J. Bathe and E.J. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ (1982).[14] R.J. Guyan, Reduction of stiffness and mass matrices, AIAA Journal 3 (1965), p. 380.[15]R.L. Allemang, BD.L. Rown. A correlation coefficient for modal vector analysis, Proc. 1st IMAC, 1982, pp. 110–116.[16] FEMtools Manual Version 2.0, Dynamic design solutions, 2000.Corresponding author. Steam Turbine Engineering, TCGT Division, Bharat Heavy Electricals Limited, Hyderabad-502032, India译文利用有限元模型对钻机进行动态分析的研究摘要现在的工作的目的是使用分析和实验的结果发展和改进钻机的有限元分析模型，这种有限元分析已经用在监测将钻机的工作特性修改后对钻机动态的结构的影响；对钻机将有两个研究就进行。

The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditionsFinite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary.At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method.Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force ofthe sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs willdrive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establishintegral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value.(3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation.(5) the overall synthesis: after the finite element equation, the area of all elements in the finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.有限元分析有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂有限元模型：它是真实系统的理想化的数学抽象。

基于有限元方法的陀螺仪的盘型制动系统的尖叫分析作者：Jaeyoung Kang【摘要】本文对一辆车的制动系统中旋转阀瓣接触两个固定垫的动力失稳性进行了研究。

在现行的近似几何中，盘被有限元分析法以帽盘型结构为模型。

从参考坐标系和移动坐标系见的坐标变换，对盘和垫之间的接触运动学进行了阐述。

通过引入统一的二维网的方法来构造阀瓣相应的陀螺矩阵。

陀螺仪的非保守性制动系统的动力不稳定性是对系统参数的数值预测。

结果表明, 尖叫声倾向于转速，转速取决于参与尖叫声模式下的振动模式。

而且,它强调摩擦系数的负斜率对在盘的面内扭转模式下产生尖叫声起着至关重要的作用。

【关键词】陀螺仪；盘型制动；制动尖叫；耦合模式1.介绍盘式制动尖叫已经被许多学者研究了数十年。

通过对尖叫机械的研究积累了许多有价值的信息。

Kinkaid等[1]提供了关于各种盘型制动尖叫研究的概述。

Ouyang 等[2]发行了以汽车盘型制动尖叫的数值分析为集中研究的评论性文章。

他们显示一个主要研究制动尖叫的方法，是线性稳定分析。

从线性化的运动微分方程来看,真正的部分特征值被计算出来，用于决定均衡的稳定性。

在文献中，有两个关于线性尖叫分析的主要方向：静态平稳的复杂特征值分析——滑动平稳[3–8]和旋转制动系统的稳定性分析[9–12,14]。

固定盘和垫的静态的滑动稳定的稳定性分析提供尖叫原理作为频繁摩擦领域里的合并模式的特性。

Huang等[6]使用本征值摄动法发展必要的条件没有直接的本征结果。

Kang等[7]推导了盘对之间的合并模式的封闭解。

由于固定盘假设,有限元(FE)方法被容易地应用于上面提到的评论性文章[2]. 同样的，Cao 等[13]从一个有移动垫和固定盘的FE盘型制动模型模型研究了移动荷载效应，因此,陀螺仪的影响被忽视了。

Giannini等[15,16]验证其合并模式行为,通过使用实验尖叫频率作为尖叫开始。

另一方面，旋转盘型制动的稳定性已经调查了分析的方法。

Finite element analysis system development present situation and forecastAlong with modern science and technology development, the people unceasingly are making the faster transportation vehicle, the large-scale building, the greater span bridge, the high efficiency power set and the preciser mechanical device. All these request engineer to be able precisely to forecast in the design stage the product and the project technical performance, needs to be static, technical parameter and so on dynamic strength to the structure as well as temperature field, flow field, electromagnetic field and transfusion carries on the analysis computation. For example analysis computation high-rise construction and great span bridge when earthquake receives the influence, has a look whether can have the destructive accident; The analysis calculates the nuclear reactor the temperature field, the determination heat transfer and the cooling system are whether reasonable; Analyzes in the new leaf blade the hydrodynamics parameter, enhances its operating efficiency. The sell may sum up as the solution physics question control partial differential equations often is not impossible.In recent years the finite element analysis which develops in the computer technology and under the numerical analysis method support(FEA, Finite Element Analysis) the side principle for solves these complex project analysis estimation problems to provide the effective way. Our country in " 95 " Plan period vigorously promotes the CAD technology, mechanical profession large and middle scalene terries CAD popular rate from " 85 " End 20% enhances that present 70%.With enterprise application of CAD, engineering and technical personnel has gradually get rid drawing board, and will join the main energy how to optimize the design, engineering and improving the quality of products, computer-aided engineering analysis (CAE. Computer Aided Engineering) method and software will be the key technical elements .ln engineering practice, finite element analysis software and CAD system integration design standards should be a qualitative leap, mainly in the following aspects :The increase design function, reduces the design cost;Reduces design and the analysis cycle period;Increase product and project reliability;Uses the optimized design, reduces the material the consumption or the cost;Discovers the latent question in advance before the product facture or the project construction;Simulates each kind of experimental plan, reduces the experimental time and the funds;Carries on the machine accident to analyze, search accident cause factor.In vigorously promotes the CAD technology today, from the bicycle to the aerospace craft, all designs manufacture all will not be able to leave the finite element analysis computation, FEA obtains the more and more widespread value in the engineering design and the analysis. The next chart is the American San Francisco bay bridge earthquake responds the computation the finite element analysis model. The development direction and on significant progress international early20th century in the end of the 50's, at the beginning of the 60's has the formidable function on the investment massive manpower and the physical resource development the finite element analysis procedure. What is most famous was (NASA) entrusts US by American country Space Agency in 1965 to calculate the NASTRAN finite element analysis system which the scientific company and the Bell aerospace system company developed.This system development until now had several dozens editions, is in the present world the scale is biggest, the function strongest finite element analysis system. From at that time to the present, the world each place development facility and the university also developed one batch of scales smaller but to use nimble, the price is lower special-purpose or general finite element analysis software, mainly had German ASKA, England's PAFEC, France's company's the and so on SYSTUS, US'S ABQUS, ADINA, ANSYS, BERSAFE, BOSOR, COSMOS, ELAS, MARC and STARDYNE product. Now on the international FEA method and the software development presents below some tendency characteristics:1.develops from the pure structure mechanics computation to solves many physicalfields question finite element analysis method most early is comes from the structure matrix analysis development, gradually promotes to the board, the shell and the entity and so on the continual body solid mechanics analysis, the practice proved this is one extremely effective numerical analysis method. Moreover the or etically also already proved, so long as uses in to be separated solution object theunit enough to be small, the obtained solution may enough approach to the precise value.Therefore the recent years finite element method has developed question the and so on hydromechanics, temperature field, electricity conduction, magnetic field, transfusion and sound field solution computation, recently developed solves several inter disciplinary studies. the questions. For example when the air current flows a very high iron tower to have the distortion, but the tower distortion in turn affects to the air current flows. This needs to use the solid mechanics and the hydrodynamics finite element analysis finally overlapping iteration solves, namely so-called " Flows solid coupling " Question.2 Progresses from the solution linear project question to the analysis non-linear problem along with the science and technology development, the linear theory already by far could not satisfy the design there quest. For example in construction profession high-rise construction and great span hanging bridge appearance, request consideration geometry non-linear problem and so on structure big displacement and big strain; Astronautics and power engineering high temperature part existence thermal deformation and thermal load, also must consider the material the non-linear problem; Such as the plastic, the rubber and the compound material and so on each kind of new material appearance ,only depends on the question which the linear computation theory is insufficient to solve meets, only has uses the non-linear finite element algorithm to be able to solve. It is well known, the non-linear value computation is very complex, it involves to the very many special mathematics question and the operation skill, thanks the general engineering technical personnel to grasp very much. Has spent massive specialty and so on manpower and investment development such as MARC, ABQUS and ADINA for this recent years overseas some companies to solution non-linear problem finite element analysis software, and widely applies to the project practice. These software common characteristics are have the highly effective non-linear solution as well as rich and the practical non-linear material. Definitely.3 The enhancement visible pretag e modelling and the post positioned data processing function early time finite element analysis software research key lies in infers new high efficiency solution method and the high accuracy unit. Along with the numerical analysis method gradually consummation, the computer operating speed rapid development, the entire computing system uses in to solve the operation inparticular the time more and more few, but the data preparation and the operation result performance question day by day is actually prominent. On the present project workstation, solves to contain 100,000 equations the finite lement models only to need to use several dozens minutes. But if establishes this model with the manual way, then the rehandling massive computed result must use several week-long time. May not exaggerate said, engineer calculates a project question when the analysis has 80% above the energy all to spend in the data preparation and the result analysis. Therefore the present nearly all shangye finite element software all has the function very strong pretage modeling and the post positioned data processing module. Is emphasizing " Visible " Today ,very many procedures have all established to user extremely friendly GUI (Graphics User Interface), enables the user direct-viewing fast to carry on the grid automatic division by the visible graphic mode, the production finite element analysis needs the data, and the massive computed result will reorganize the distortion chart, the equivalentd is tribution cloud chart according to the request, will be advantageous for the extreme value search and needs the data to tabulate the output.4 With the CAD software seamless integration now the finite element analysis system another characteristic is and the general CAD software integrated use namely, after uses CAD software to complete the part and the components modeling design, automatically produces the finite element grid and carries on the computation, if analyzes the result does not conform to the design requirement then recarries on the modeling and the computation, until satisfaction, thus enormously raised the design level and the efficiency.The complex project analysis question today, which engineer may quickly solves in integrated CAD and in the FEA software environment in before is unable to deal with. Therefore now all shangye finite element system business all has developed and famous CAD software (for example Pro/ENGINEER, Unigraphics, SolidEdge, SolidWorks, IDEAS, Bentley and AutoCAD and so on) connection.5 Basically in Wintel platform development early time finite element analysis software all is in the large and middle scale computers(mainly is Mainframe) on the development and the movement, after wards also developed take the project workstation (EWS, Engineering WorkStation) as the platform, their common characteristic all uses the UNIX operating system.. The PC machine appearance caused the computer the application to have he fundamental change, engineer hadlonged for completed the complex project analysis on the desk the dream to become the reality. But early PC machine uses 16 CPU and the DOS operating system, in the memory public block data is restricted, at that time there fore calculated the model the scale not to be able to surpass 10,000 steps equations. Microsoft the Windows operating system and 32 Intel Pentium processor promoting for used in the finite element analysis PC machine to provide essential software and the hardware supports the platform. Therefore on the current international famous finite element procedure research and the development organization all in abundance move theirs software the value to the Wintel platform in. Next cava logged version environment and the SGI workstation simultaneously calculates the solution time with ADINA the V7.3 in PC machine Windows on the NT which 4 projects examples needed. May see the newest upscale PC machine solution ability already and center low-grade EWS is equally matched.In order to will develop on the large and middle scale computers and EWS the finite element procedure will move the value to PE machine on, frequently will need to use Hummingbird Corporation simulation software Exceed. Does this the result quite is troublesome, moreover cannot fully use the PC machine software and hardware resources. Therefore recently some companies, for example IDEAS, ADINA and R&D starts in the Windows platform to develop the finite element procedure, is called as " Native Windows" The edition, meanwhile has the finite element routine package which in on Linux operating system environment develops in PC machine.The domestic development situation and the prospect in 1979 US'S SAP5linear structure was static, the mechanical analysis procedure transplants successfully to the domestic introduction, raised the application general finite element procedure to analyze the computation project question the high tide. This high tide continuously continued to 1981 the ADINA non-linear structure analysis procedure introduction, many continuously was unable the project difficult problem which solved all to be easily solved for a while. Everybody also starts to realize indeed is engineer carries on then is computation to the finite element analysis procedure using the computer the important tool。

有限元分析与应用上机实验报告学院：机电工程学院专业：机械工程班级：硕士1606班*名：***学号： *********指导教师：***日期： 2016.12.021.QuestionFig.1. Schematic diagram of herringbone roof truss.Question: The geometric dimensions of the chevron roof is shown in Fig.1,you should analyze it by statics，as a result you should give the displacement and axial force and axial force diagram of the deformation diagram.Conditions: The ends of the roof truss were fixed, the sectional area of the truss is 0.01m2, elastic modulus is2.0×1011 N/m2, poisson's ratio is 0.3.2.The software usedANSYS Finite element software (APDL 15.0)3.Solving processPoint 1 was choosed as the Coordinate point, horizontal to the right was the X axis，the upright direction is choosed as the Y axis to create a coordinate system. The nodes was numbered as shown in Figure 1，node 1 and node 5 was fixed，and the force on node 6,7,8 was is 1k N，the direction is opposite to the Y-axis3.1 The preparatory work before analysis（1）Specify the new file name. Select Utility>Menu> File>Change Jobname, then pop-up the dialog box Change Jobname, input the the working file name ‘2D-sp’ in the Enter New Jobname, click OK to finish the difinition, as shown in Fig.2.Fig.2. The difinition of working file name.（2）Specify a new title. Select Utility>Menu>File>Change Title，then pop-up the dialog box Change Title, input the the file name ‘2D-sp pro’ in the Enter New Title, click OK to finish the difinition, as shown in Fig.3.Fig.3. The difinition of file name.（3）Re-refresh the graphics window.Select Utility>Menu>Plot>Replot, the defined information was displayed in the graphics window.（4）Define the structural analysis. Run the main menu Main Menu>Preferences，then choose the Structural, click OK to complete the definition of the analysis type , as shown in Fig.4.Fig.4. The definition of the structural analysis.3.2 Define the element typeRun the main menu Main Menu>Preprocessor>Element Type>Add, then pop-up the dialog box Element Types, click the button Add to build a new element type, then pop-up the dialog box Library of Element Types, choose Link first, and then select 3D finit stn 180（Link1），click the button OK to finish the definition of the element type, click the button Close to finish the settings, as shown in Fig.5.Fig.5. The definition of the element type.3.3 Define the real constantsRun the main menu Main Menu>Preprocessor>Real Constants Add, then pop-up the dialog box of real constants, click the button Add to come into the constant input dialog box, as shown in Fig.6. Input the sectional area of the truss (0.01m2) in AREA, click the button OK to finish the input of the real constants, as shown in Fig.7.Fig.6. Get into the instance constant dialog box.Fig.7. The definition of the real constants.3.4 Define the material propertiesRun the main menu Main Menu>Preprocessor>Material Props>Material Models,then pop-up the dialog box of material properties, and select Structure、Linear、Elastic、Isotropic, as shown in Fig.8.When the selection was completed, appeared the MaterialProperties input dialog box appears，then input elastic modulus 2e11 and the poisson's ratio 0.3, as shown in Fig.9, click OK to finish the input of the material properties.Fig.8. Get into the setting of material properties.Fig.9. The difinition of the material properties.3.5 Establish the analytical model（1）Create the nodes. The coordinates of the 1—8 nodes as shown in Table.1. Run the main menu Main Menu>Preprocessor> Modeling>Create>Nodes>In Active CS. The create node entry dialog box appears as shown in Fig.10. I nput the first node 1 and its’ x, y, z coordinate, then click Apply to finish the creating of the node. Similarly, create the nodes 2—8, click OK to finish the creating of the nodes, as shown in Fig.10.Table.1.The coordinates of geometry model nodes.Fig.10. The dialog box of nodes input.Fig.11. The creation of the nodes.The created nodes as shown below：（2）Created the bar unitRun the main menu MainMenu>Preprocessor>Modeling>Create >Elements>Auto Numbered>Thru Nodes, then pop-up the dialog box Element from Nodes, pick the nodes 1 and 2, click Apply to complete the first lever unit . Similarly, pick both ends of the rod in turn, click Apply to complete the lever unit, lastly click OK, as shown in Fig.12.Fig.12. Model establishment of the herringbone truss.3.6 Apply constraints and loads（1）Impose constraintsRun the main menu Main Menu>Solution>Define Loads>Apply>Structural> Displacement>On Nodes, the pick menu appears, select nodes 1 and 5 in turn, then click OK, the constraint definition dialog box appears，as shown in Fig.13. Choose All DOF to constrain all degrees of freedom，other items by default，then click the button OK to complete the constraint definition.Fig.13. Impose the constraints.（2）Apply the loadRun the main menu Main Menu>Solution>Define Loads>Apply>Structural> Force/Moment>On Nodes, the pick menu appears, select nodes 6, 7 and 8 in turn, click OK , then the load definitions dialog box appears，as shown in Fig.14. The load type is concentrated force FY. The value is -1000 , Then clickthe OK button to complete the application of the load.Fig.14. The application of the load.Figure after load application as shown below:3.7 SolutionRun the main menu Main Menu>Solution>Solve>Current LS, then pop-up the dialog box Solve Current Load Step, click STAT Command, then click STAT Command>File>Close toclose the STAT Command window, then click OK to finish the steps.Then the computer started the solution. The promption of "Solution is done" indicated that the solution was completed, click the button Close to finish the solution, as shown in Fig.15.Fig.15. The process of the solution.3.8 View the analysis results（1）Define the cell table. In ANSYS, some data can not be directly accessed, so we need to complete the definition of cell to access the results.Run the main menu Main Menu>General Postproc>Element Table>Define Table, then pop-up the dialog box, click the button Add. The unit table definition dialog box appeared, as shown in Fig.16.Fig.16a. The definition of the cell table.Fig.16b. The definition of the cell table.（2）Display the axial force (axial stress) diagram. Run the main menu Main Menu>General Postproc>Plot Results> Contour Plot>Elem Table. Then pop-up the cell table result selection dialog box , as shown in Fig.17. Choose A-STR and click OK to view the axial force (axial stress) diagram , as shown in Fig.18.Fig.17. Contor plot of Element Table Data.（3）Display the result. The result as shown in Fig.18.Fig.18. The final result.。

The finite element analysisFinite element method, the solving area is regarded as made up of many small in the node connected unit (a domain), the model gives the fundamental equation of sharding (sub-domain) approximation solution, due to the unit (a domain) can be divided into various shapes and sizes of different size, so it can well adapt to the complex geometry, complex material properties and complicated boundary conditionsFinite element model: is it real system idealized mathematical abstractions. Is composed of some simple shapes of unit, unit connection through the node, and under a certain load.Finite element analysis: is the use of mathematical approximation method for real physical systems (geometry and loading conditions were simulated. And by using simple and interacting elements, namely unit, can use a limited number of unknown variables to approaching infinite unknown quantity of the real system.Linear elastic finite element method is a ideal elastic body as the research object, considering the deformation based on small deformation assumption of. In this kind of problem, the stress and strain of the material is linear relationship, meet the generalized hooke's law; Stress and strain is linear, linear elastic problem boils down to solving linear equations, so only need less computation time. If the efficient method of solving algebraic equations can also help reduce the duration of finite element analysis.Linear elastic finite element generally includes linear elastic statics analysis and linear elastic dynamics analysis from two aspects. The difference between the nonlinear problem and linear elastic problems:1) nonlinear equation is nonlinear, and iteratively solving of general;2) the nonlinear problem can't use superposition principle;3) nonlinear problem is not there is always solution, sometimes even no solution. Finite element to solve the nonlinear problem can be divided into the following three categories:1) material nonlinear problems of stress and strain is nonlinear, but the stress and strain is very small, a linear relationship between strain and displacement at this time, this kind of problem belongs to the material nonlinear problems. Due to theoretically also cannot provide the constitutive relation can be accepted, so, general nonlinear relations between stress and strain of the material based on the test data, sometimes, to simulate the nonlinear material properties available mathematical model though these models always have their limitations. More important material nonlinear problems in engineering practice are: nonlinear elastic (including piecewise linear elastic, elastic-plastic and viscoplastic, creep, etc.2) geometric nonlinear geometric nonlinear problems are caused due to the nonlinear relationship between displacement. When the object the displacement is larger, the strain and displacement relationship is nonlinear relationship. Research on this kind of problemIs assumes that the material of stress and strain is linear relationship. It consists of a large displacement problem of large strain and large displacement little strain. Such as the structure of the elastic buckling problem belongs to the large displacement little strain, rubber parts forming process for large strain.3) nonlinear boundary problem in the processing, problems such as sealing, the impact of the role of contact and friction can not be ignored, belongs to the highly nonlinear contact boundary.At ordinary times some contact problems, such as gear, stamping forming, rolling, rubber shock absorber, interference fit assembly, etc., when a structure and another structure or external boundary contact usually want to consider nonlinear boundary conditions. The actual nonlinear may appear at the same time these two or three kinds of nonlinear problems.Finite element theoretical basisFinite element method is based on variational principle and the weighted residual method, and the basic solving thought is the computational domain is divided into a finite number of non-overlapping unit, within each cell, select some appropriate nodes as solving the interpolation function, the differential equation of the variables in the rewritten by the variable or its derivative selected interpolation node value and the function of linear expression, with the aid of variational principle or weighted residual method, the discrete solution of differential equation. Using different forms of weight function and interpolation function, constitute different finite element methods. 1. The weighted residual method and the weighted residual method of weighted residual method of weighted residual method: refers to the weighted function is zero using make allowance for approximate solution of the differential equation method is called the weighted residual method. Is a kind of directly from the solution of differential equation and boundary conditions, to seek the approximate solution of boundary value problems of mathematical methods. Weighted residual method is to solve the differential equation of the approximate solution of a kind of effective method.Hybrid method for the trial function selected is the most convenient, but under the condition of the same precision, the workload is the largest. For internal method and the boundary method basis function must be made in advance to meet certain conditions, the analysis of complex structures tend to have certain difficulty, but the trial function is established, the workload is small. No matter what method is used, when set up trial function should be paid attention to are the following:(1) trial function should be composed of a subset of the complete function set. Have been using the trial function has the power series and trigonometric series, spline functions, beisaier, chebyshev, Legendre polynomial, and so on.(2) the trial function should have until than to eliminate surplus weighted integral expression of the highest derivative low first order derivative continuity.(3) the trial function should be special solution with analytical solution of the problem or problems associated with it. If computing problems with symmetry, should make full use of it. Obviously, any independent complete set of functions can be used as weight function. According to the weight function of the different options for different weighted allowance calculation method, mainly include: collocation method, subdomain method, least square method, moment method and galerkin method. The galerkin method has the highest accuracy.Principle of virtual work: balance equations and geometric equations of the equivalent integral form of "weak" virtual work principles include principle of virtual displacement and virtual stress principle, is the floorboard of the principle of virtual displacement and virtual stress theory. They can be considered with some control equation of equivalent integral "weak" form. Principle of virtual work: get form any balanced force system in any state of deformation coordinate condition on the virtual work is equal to zero, namely the system of virtual work force and internal force ofthe sum of virtual work is equal to zero. The virtual displacement principle is the equilibrium equation and force boundary conditions of the equivalent integral form of "weak"; Virtual stress principle is geometric equation and displacement boundary condition of the equivalent integral form of "weak". Mechanical meaning of the virtual displacement principle: if the force system is balanced, they on the virtual displacement and virtual strain by the sum of the work is zero. On the other hand, if the force system in the virtual displacement (strain) and virtual and is equal to zero for the work, they must balance equation. Virtual displacement principle formulated the system of force balance, therefore, necessary and sufficient conditions. In general, the virtual displacement principle can not only suitable for linear elastic problems, and can be used in the nonlinear elastic and elastic-plastic nonlinear problem.Virtual mechanical meaning of stress principle: if the displacement is coordinated, the virtual stress and virtual boundary constraint counterforce in which they are the sum of the work is zero. On the other hand, if the virtual force system in which they are and is zero for the work, they must be meet the coordination. Virtual stress in principle, therefore, necessary and sufficient condition for the expression of displacement coordination. Virtual stress principle can be applied to different linear elastic and nonlinear elastic mechanics problem. But it must be pointed out that both principle of virtual displacement and virtual stress principle, rely on their geometric equation and equilibrium equation is based on the theory of small deformation, they cannot be directly applied to mechanical problems based on large deformation theory. 3,,,,, the minimum total potential energy method of minimum total potential energy method, the minimum strain energy method of minimum total potential energy method, the potential energy function in the object on the external load will cause deformation, the deformation force during the work done in the form of elastic energy stored in the object, is the strain energy.The convergence of the finite element method, the convergence of the finite element method refers to when the grid gradually encryption, the finite element solution sequence converges to the exact solution; Or when the cell size is fixed, the more freedom degree each unit, the finite element solutions tend to be more precise solution. Convergence condition of the convergence condition of the finite element finite element convergence condition of the convergence condition of the finite element finite element includes the following four aspects: 1) within the unit, the displacement function must be continuous. Polynomial is single-valued continuous function, so choose polynomial as displacement function, to ensure continuity within the unit. 2) within the unit, the displacement function must include often strain. Total can be broken down into each unit of the state of strain does not depend on different locations within the cell strain and strain is decided by the point location of variables. When the size of the units is enough hours, unit of each point in the strain tend to be equal, unit deformation is uniform, so often strain becomes the main part of the strain. To reflect the state of strain unit, the unit must include the displacement functions often strain. 3) within the unit, the displacement function must include the rigid body displacement. Under normal circumstances, the cell for a bit of deformation displacement and displacement of rigid body displacement including two parts. Deformation displacement is associated with the changes in the object shape and volume, thus producing strain; The rigid body displacement changing the object position, don't change the shape and volume of the object, namely the rigid body displacement is not deformation displacement. Spatial displacement of an object includes three translational and three rotational displacement, a total of six rigid body displacements. Due to a unit involved in the other unit, other units do rigid body displacement deformation occurs willdrive unit, thus, to simulate real displacement of a unit, assume that the element displacement function must include the rigid body displacement. 4) the displacement function must be coordinated in public boundary of the adjacent cell. For general unit of coordination is refers to the adjacent cell in public node have the same displacement, but also have the same displacement along the edge of the unit, that is to say, to ensure that the unit does not occur from cracking and invade the overlap each other. To do this requires the function on the common boundary can be determined by the public node function value only. For general unit and coordination to ensure the continuity of the displacement of adjacent cell boundaries. However, between the plate and shell of the adjacent cell, also requires a displacement of the first derivative continuous, only in this way, to guarantee the strain energy of the structure is bounded. On the whole, coordination refers to the public on the border between neighboring units satisfy the continuity conditions. The first three, also called completeness conditions, meet the conditions of complete unit is complete unit; Article 4 is coordination requirements, meet the coordination unit coordination unit; Otherwise known as the coordinating units. Completeness requirement is necessary for convergence, all four meet, constitutes a necessary and sufficient condition for convergence. In practical application, to make the selected displacement functions all meet the requirements of completeness and harmony, it is difficult in some cases can relax the requirement for coordination. It should be pointed out that, sometimes the coordination unit than its corresponding coordination unit, its reason lies in the nature of the approximate solution. Assumed displacement function is equivalent to put the unit under constraint conditions, the unit deformation subject to the constraints, this just some alternative structure compared to the real structure. But the approximate structure due to allow cell separation, overlap, become soft, the stiffness of the unit or formed (such as round degree between continuous plate unit in the unit, and corner is discontinuous, just to pin point) for the coordination unit, the error of these two effects have the possibility of cancellation, so sometimes use the coordination unit will get very good results. In engineering practice, the coordination of yuan must pass to use "small pieces after test". Average units or nodes average processing method of stress stress average units or nodes average processing method of stress average units or nodes average processing method of stress of the unit average or node average treatment method is the simplest method is to take stress results adjacent cell or surrounding nodes, the average value of stress.1. Take an average of 2 adjacent unit stress. Take around nodes, the average value of stressThe basic steps of finite element method to solve the problemThe structural discretization structure discretization structure discretization structure discretization to discretization of the whole structure, will be divided into several units, through the node connected to each other between the units; 2. The stiffness matrix of each unit and each element stiffness matrix and the element stiffness matrix and the stiffness matrix of each unit (3) integrated global stiffness matrix integrated total stiffness matrix integrated overall stiffness matrix integrated total stiffness matrix and write out the general balance equations and write out the general balance equations and write out the general balance equations and write a general equation 4. Introduction of supporting conditions, the displacement of each node 5. Calculate the stress and strain in the unit to get the stress and strain of each cell and the cell of the stress and strain and the stress and strain of each cell.For the finite element method, the basic ideas and steps can be summarized as: (1) to establishintegral equation, according to the principle of variational allowance and the weight function or equation principle of orthogonalization, establishment and integral expression of differential equations is equivalent to the initial-boundary value problem, this is the starting point of the finite element method. Unit (2) the area subdivision, according to the solution of the shape of the area and the physical characteristics of practical problems, cut area is divided into a number of mutual connection, overlap of unit. Regional unit is divided into finite element method of the preparation, this part of the workload is bigger, in addition to the cell and node number and determine the relationship between each other, also said the node coordinates, at the same time also need to list the natural boundary and essential boundary node number and the corresponding boundary value.(3) determine the unit basis function, according to the unit and the approximate solution of node number in precision requirement, choose meet certain interpolation condition basis function interpolation function as a unit. Basis function in the finite element method is selected in the unit, due to the geometry of each unit has a rule in the selection of basis function can follow certain rules. (4) the unit will be analysis: to solve the function of each unit with unit basis functions to approximate the linear combination of expression; Then approximate function generation into the integral equation, and the unit area integral, can be obtained with undetermined coefficient (i.e., cell parameter value) of each node in the algebraic equations, known as the finite element equation.(5) the overall synthesis: after the finite element equation, the area of all elements in the finite element equation according to certain principles of accumulation, the formation of general finite element equations. (6) boundary condition processing: general boundary conditions there are three kinds of form, divided into the essential boundary conditions (dirichlet boundary condition) and natural boundary conditions (Riemann boundary conditions) and mixed boundary conditions (cauchy boundary conditions). Often in the integral expression for natural boundary conditions, can be automatically satisfied. For essential boundary conditions and mixed boundary conditions, should be in a certain method to modify general finite element equations satisfies. Solving finite element equations (7) : based on the general finite element equations of boundary conditions are fixed, are all closed equations of the unknown quantity, and adopt appropriate numerical calculation method, the function value of each node can be obtained.有限元分析有限元法求解区域是由许多小的节点连接单元（域），该模型给出了切分的基本方程（子域名）的近似解，由于单位（域）可以分为不同的形状和大小不同的尺寸，所以它能很好的适应复杂的几何形状、材料特性和边界条件复杂，复杂有限元模型：它是真实系统的理想化的数学抽象。

7. INTRODUCTION TO THE FINITE ELEMENT METHODEngineers use a wide range of tools and techniques to ensure that the designs they create are safe. However, accidents sometimes happen and when they do, companies need to know if a product failed because the design was inadequate or if there is another cause, such as an user error. But they have to ensure that the product works well under a wide range of conditions, and try to avoid to the maximum a failure produced by any cause. One important tool to achieve this is the finite element method.“The finite element method is one of the most powerful numerical techniques ever devised for solving differential (and integral) equations of initial and boundary-value problems in geometrically complicated regions.” (Red dy, 1988). There is some data that can not be ignored when analyzing an element by the finite element method. This input data is to define the domain, the boundary and initial conditions and also the physical properties. After knowing this data, if the analysis is done carefully, it will give satisfactory results. It can be said that the process to do this analysis is very methodical, and that it is why it is so popular, because that makes it easier to apply. “The finite element analysis of a problem is so systematic that it can be divided into a set of logical steps that can be implemented on a digital computer and can be utilized to solve a wide range of problems by merely changing the data input to the computer program.” (Reddy,1988).The finite element analysis can be done for one, two and three-dimensional problems. But generally, the easier problems are those including one and two dimensions, and those can be solved without the aid of a computer, because even if they give a lot of equations, if they are handled with care, an exact result can be achieved. But if the analysis requires three-dimensional tools, then it would be a lot more complicated, because it will involve a lot of equations that are very difficult to solve without having an error. That is why engineers have developed softwares that can perform these analyses by computer, making everything easier. These softwares can make analysis of one, two and threedimensional problems with a very good accuracy.A basic thing to understand how finite element works is to know that it divides the whole element into a finite number of small elements. “The domain of the problem is viewed as a collection of nonintersecting simple subdomains, called finite elements… The subdivision of a domain into elements is termed finite element discretization. The collection of the elements is called the finite element mesh of the domain.” (Reddy,1988). The advantage of dividing a big element into small ones is that it allows that every small element has a simpler shape, which leads to a good approximation for the analysis. Another advantage is that at every node (the intersection of the boundaries) arises an interpolant polynomial, which allows an accurate result at a specific point. Before the finite element method, engineers and physicians used a method that involved the use of differential equations, which is known as the finite difference method.The method of the finite element is a numerical technique that solves or at least approximates enough to a solution of a system of differential equations related with a physics or engineering problem. As explained before, this method requires a completely defined geometrical space, and then it would be subdivided into small portions, which together will form a mesh. The difference between the method of the finite element and the method of the finite difference is that in the second one, the mesh consists of lines and rows of orthogonal lines, while in the method of the finite element the division does not necessarily involves orthogonal lines, and this results in a more accurate analysis (Figure 38).Figure 38 (taken from Algor 13 ®)The equations used for the finite element method are a lot, but they have the basison some single equations that describe a particular phenomenon. Those equations are:The elliptic equation is described by22220 dx dy∂Φ∂Φ+=The parabolic equation is described by220 dx t∂Φ∂Φ-=∂The hyperbolic equation is described by221 220 t x∂Φ∂Φ-=∂∂No matter which is the cause of the internal forces and the deformation that they cause, there are three basic conditions that allow the finite element analysis: the equilibrium of forces, the compatibility of displacements and the laws of material behavior. “The first condition merely requires that the internal forces balance the external applied loads.” (Rocke t et. al., 1983). That is the most important condition, but the other two assure that the system will be a statically determinate problem. Another condition that must be taken into account is that there exists a relationship between the load applied and the deformation, and this is given by Hooke’s law, as explained in past chapters, but only in the elastic range.In order to achieve a structural analysis by matrix methods, there might be threeways: stiffness (displacement) method, flexibility (force) method and mixed method. In the first two methods, two basic conditions of nodal equilibrium and compatibility must be reached. In the first method, the once the displacement compatibility conditions are reached, then an answer can be given. In the second method, once the conditions of nodal equilibrium are satisfied and then the compatibility of nodal displacement, forces are known in the members.One of the principles that is the basis of the finite element method is the one known as principle of virtual work. “This principle is conc erned with the relationship which exists between a set of external loads and the corresponding internal forces which together satisfy the equilibrium condition, and also with sets of joint (node) displacements and the corresponding member deformations which satisfy the conditions of compatibility.” This principle can be stated in terms of an equation of equilibrium of loads, where the work done by the external loads is equal to the internal virtual work absorbed by the element. Or expressed as an equation:v F d δσε⋅ =⋅⋅∑⎰where F are the external loads, the deflection, the system of internal forces, and the internal deformations.A pin ended tie has similar characteristics to those of an elastic spring that is subjected on one end, looking downwards, suffering the effects of gravity and the effect of an external load. The direct relationship between the force and the displacement of the free end is:F k δ=⋅the vale k is known as the stiffness of the spring. Once the value of the applied force and the value of the stiffness, the equation can be inverted to find the displacement: 1F kδ= This is a simple example for systems that imply only a few data. But when the problem implies more complex systems, then the equations become a little more complicated. When a number of simple members are interconnected at a number of nodes, the displacement caused by the load can only be described by simultaneous equations. Then, the simpler equation seen before becomes:{}[]{}F K δ=where K is the stiffness of the whole structure.For example, for a spring that has two pins, it generates two forces and two displacements. Therefore, the stiffness matrix would be of order 2 x 2:111121221222F k k u F k k u ⎧⎫⎡⎤⎧⎫=⎨⎬⎨⎬⎢⎥⎩⎭⎣⎦⎩⎭where u represents the displacement.Boundary conditions are the limitations set for the problem. These limitations are necessary in order to solve it, because otherwise, the system would be taken as a rigid body. The limitations stated by these boundary conditions are like where does the element is likely to move, and were it is restricted. If there were not boundary conditions, the body would be floating in the space, and under the action of any load, it would not suffer any deformation, but it would move around the space as a rigid body. So, when assuming boundary conditions, it has to be assured that the element has enough of them in order to prevent moving as a rigid body. Once this is done, the values of the displacements are obtained and can be substituted in the last equation seen, which will give that the displacement is equal to zero, because the element can not move in any direction. Then, algebra is applied, and the values of the forces or of the stiffness are known.7.1 Applications of the finite element method in engineeringEvery designed made has to fulfill certain specifications, and among them is working under a variety of conditions: temperature, humidity, vibrations, etc. The job of the designer is to achieve this, and to assure that the product will work effectively, taking care of the user and of the element. An engineer has to follow certain steps in order to create a good product with high quality. First, the steps of the design flow chart must be followed:1. recognize a need2. specifications and requirements3. feasibility sturdy4. creative design and development5. detailed drawings6. prototype building and testing7. design for manufactureThese are the basic concepts of the design, but it involves a lot of other things to be able to assure a good performance of the product. After the designer has the calculations of the dimensions, tolerances, manufacturing parameters, etc, some other tests should be done. For example, an engineer has to know whether the product is going to support certain loads, or how it is going to behave with temperature variations, or what could happen if vibrations are present. This is where the finite element method enters in action in engineering.In the last pages, it has been explained how the finite element works and which are the basis of it. Now it is time to explain what is it good for, the applications, the benefits, etc. For example, the first application of the method was introduced by Richard Courant to solve torsion on a cylinder. Then, in the middle 50’s, the method started to be applied for airframe and structural analysis, and then used in civil engineering. In general, finite element methods are used in a wide variety of engineering applications, like in computer graphics, heat transfer, electrical and magnetic fields, among others.Use of finite element method in mechanical engineering is very wide. For example, it is used in mechanics of materials, for structures and trusses. It is used to understand and to prevent how some structures are going to behave under the action of some loads. For example, for a bridge, how is it going to behave with the vibrations, or with the effect of the air, or with the variations of temperature. The aircraft industry uses this method to determine the static and dynamic answer of planes and space crafts to the great variety of environments and conditions that can be found during their operation.In the case of mechanics of fluids, the method is used to know how a wing of a plane is going to behave with the air flux passing through it, if it is going to resist, how much vibration is this going to cause, in order to avoid resonance (because of the vibrations caused by the turbines too). The finite element method allows calculating the drag and the flotation forces caused during operation.For heat transfer, it allows to know how a turbine is going to behave, and how the material is going to be affected for the effect of the heat. It is known that heat also creates stresses, and this is a fundamental concept when talking about design of turbines. Another important thing to mention here is that when turbines are working, they reach very high temperatures, and they use a coolant to maintain the temperature under some point. This coolant, when touching the hot turbine, creates a thermal shock, which also producesstresses. These stresses can be measured through the finite element method. Something else that it is important is that some components of the turbine (like blades) have holes to let the air in to act as a coolant to avoid overheating. These holes can act as stress concentrators, and by this method, it can be analyzed which is the way they affect less the stress distribution.One important area where this method is used is in design of mechanical elements. Generally, these elements have to support high loads, whether they are radial, tension, compression, tangential, etc. Sometimes these loads are combined, which makes the element designed more prone to failure. Most of the time, mechanical parts also have holes for assembly, for flow of a certain fluid, etc, and these holes create stress concentrations. A very specific and careful analysis has to be made. Here is where the finite element analysis plays an important role. For example, in the design of a crank, it is subjected to different loads, and it has specific boundary conditions. It is tested with total restriction on one side, and with a load applied tangentially on the other side. The results are like in Figure 39, where the areas that have bigger stresses can be identified according to the color specified on the right of the screen.Figure 39Other application in design is in the design of a control arm. This is for a combination of mechanics with electronics, but it still has forces applied, and holes manufactured for assembly. The boundary conditions have to be established according to the allowable movements of the robot, and then the loads applied in the points where it carries something. The results are those in Figure 40Figure 40The automotive industry uses this method to see how the cars behave under some conditions of load, and for crash tests. For this industry, it has a lot of advantages. After the car is modeled in a CAD software, it is exported to a finite element analysis software, and after all the conditions are set, the test begins. The method has the advantage of giving very accurate information. For example, the deformation suffered point by point (node by node) at a specific time. The fact of they doing these type of analyses represents a great advantage for them, because they save money, they save time, and they also save material, and yet they have very reliable information of the behavior of the car. For example, Figure 41, which is a model of a thesis done the last semester, where the student was analyzing the effect of the impact on the car, and also on a dummy. He could calculate the stress at any point he wanted to, and at any time of the collision, and he was able to determine which areas of the car and the dummy were the most affected with the impact. Another example is Figure 42.Figure 41 (from Alfredo Pérez Mitre)Figure 427.2 AdvantagesThe use of the finite element method has a lot of advantages, and most of them have already been commented in this thesis, but here it is going to be made a short condensation of them.First, it is a very important tool for stress and strain analysis, not only because it provides accurate information, but because it also saves a lot of money and time by simulating the events in computer and not in real life. It is a really easy to use tool. Once some tutorials are followed, the only thing left to do is to explore a little bit of the software, and then to apply all the knowledge acquired. The principles of the method are easy to understand, even if when the model is complex, the analyses are also complex.For every engineer it is a very reliable tool, because it is very specific for each occasion, and it is able to perform different analyses of the same model under different circumstances with simple changes in the boundary conditions, in the loads, in the material, or whatever the problem demands.The method can help to modify each design in order to increase the service life of it as much as possible. That is the case of study of this thesis, where some analyses are run in the computer to see how the presence of a stress concentrator affects the behavior of the element. If the stress concentration is high, the element can be modified with ease and thensubjected to analysis again, and depending on the results, a decision has to be taken to see if it needs more changes or if it has reached or if it is even close to its maximum service life condition.The combination of this as a software with other types of softwares is a very useful tool, because the program of finite element analysis allows the designer to import models from other CAD softwares, and that way it does not forces the engineer to make the solid in the FEA CAD. That way, complex models can be created and analyzed with the combination of those two powerful tools.7.有限元方法的简介工程师需要使用范围更广的工具和技术来保证他们创建的设计是安全的。

机器人机构优化设计有限元分析中英文资料对照外文翻译文献综述FEM Optimization for Robot StructureAbstractIn optimal design for robot structures, design models need to he modified and computed repeatedly. Because modifying usually can not automatically be run, it consumes a lot of time. This paper gives a method that uses APDL language of ANSYS 5.5 software to generate an optimal control program, which mike optimal procedure run automatically and optimal efficiency be improved.1）IntroductionIndustrial robot is a kind of machine, which is controlled by computers. Because efficiency and maneuverability are higher than traditional machines, industrial robot is used extensively in industry. For the sake of efficiency and maneuverability, reducing mass and increasing stiffness is more important than traditional machines, in structure design of industrial robot.A lot of methods are used in optimization design of structure. Finite element method is a much effective method. In general, modeling and modifying are manual, which is feasible when model is simple. When model is complicated, optimization time is longer. In the longer optimization time, calculation time is usually very little, a majority of time is used for modeling and modifying. It is key of improving efficiency of structure optimization how to reduce modeling and modifying time.APDL language is an interactive development tool, which is based on ANSYS and is offered to program users. APDL language has typical function of some large computer languages. For example, parameter definition similar to constant and variable definition, branch and loop control, and macro call similar to function and subroutine call, etc. Besides these, it possesses powerful capability of mathematical calculation. The capability of mathematical calculation includes arithmetic calculation, comparison, rounding, and trigonometric function, exponential function and hyperbola function of standard FORTRAN language, etc. By means of APDL language, the data can be read and then calculated, which is in database of ANSYS program, and running process of ANSYS program can be controlled.Fig. 1 shows the main framework of a parallel robot with three bars. When the length of three bars are changed, conjunct end of three bars can follow a given track, where robot hand is installed. Core of top beam is triangle, owing to three bars used in the design, which is showed in Fig.2. Use of three bars makes top beam nonsymmetrical along the plane that is defined by two columns. According to a qualitative analysis from Fig.1, Stiffness values along z-axis are different at three joint locations on the top beam and stiffness at the location between bar 1 and top beam is lowest, which is confirmed by computing results of finite element, too. According to design goal, stiffness difference at three joint locations must he within a given tolerance. In consistent of stiffness will have influence on the motion accuracy of the manipulator under high load, so it is necessary to find the accurate location of top beam along x-axis.To the questions presented above, the general solution is to change the location of the top beam many times, compare the results and eventually find a proper position, The model will be modified according to the last calculating result each time. It is difficult to avoid mistakes if the iterative process is controlled manually and the iterative time is too long. The outer wall and inner rib shapes of the top beam will be changed after the model is modified. To find the appropriate location of top beam, the model needs to be modified repetitiously.Fig. 1 Solution of Original DesignThis paper gives an optimization solution to the position optimization question of the top beam by APDL language of ANSYS program. After the analysis model first founded, the optimization control program can be formed by means of modeling instruction in the log file. The later iterative optimization process can be finished by the optimization control program and do not need manual control. The time spent in modifying the model can be decreased to the ignorable extent. The efficiency of the optimization process is greatly improved.2）Construction of model for analysisThe structure shown in Fig. 1 consists of three parts: two columns, one beam and three driving bars. The columns and beam are joined by the bolts on the first horizontal rib located on top of the columns as shown in Fig.1. Because the driving bars are substituted by equivalentforces on the joint positions, their structure is ignored in the model.The core of the top beam is three joints and a hole with special purpose, which can not be changed. The other parts of the beam may be changed if needed. For the convenience of modeling, the core of the beam is formed into one component. In the process of optimization, only the core position of beam along x axis is changed, that is to say, shape of beam core is not changed. It should be noticed that, in the rest of beam, only shape is changed but the topology is not changed and which can automatically be performed by the control program.Fig.1, six bolts join the beam and two columns. The joint surface can not bear the pull stress in the non-bolt joint positions, in which it is better to set contact elements. When the model includes contact elements, nonlinear iterative calculation will be needed in the process of solution and the computing time will quickly increase. The trial computing result not including contact element shows that the outside of beam bears pulling stress and the inner of beam bears the press stress. Considering the primary analysis object is the joint position stiffness between the top beam and the three driving bars, contact elements may not used, hut constructs the geometry model of joint surface as Fig.2 showing. The upper surface and the undersurface share one key point in bolt-joint positions and the upper surface and the under surface separately possess own key points in no bolt positions. When meshed, one node will be created at shared key point, where columns and beam are joined, and two nodes will be created at non shared key point, where column and beam are separated. On right surface of left column and left surface of right column, according to trial computing result, the structure bears press stress. Therefore, the columns and beam will share all key points, not but at bolts. This can not only omit contact element but also show the characteristic of bolt joining. The joining between the bottoms of the columns and the base are treated as full constraint. Because the main aim of analysis is the stiffness of the top beam, it can be assumed that the joint positions hear the same as load between beam and the three driving bars. The structure is the thin wall cast and simulated by shell element . The thickness of the outside wall of the structure and the rib are not equal, so two groups of real constant should he set. For the convenience of modeling, the two columns are alsoset into another component. The components can create an assembly. In this way, the joint positions between the beam core and columns could he easily selected, in the modifying the model and modifying process can automatically be performed. Analysis model is showed Fig.1. Because model and load are symmetric, computing model is only half. So the total of elements is decreased to 8927 and the total of nodes is decreased to 4341. All elements are triangle.3.）Optimization solutionThe optimization process is essentially a computing and modifying process. The original design is used as initial condition of the iterative process. The ending condition of the process is that stiffness differences of the joint locations between three driving bars and top beam are less than given tolerance or iterative times exceed expected value. Considering the speciality of the question, it is foreseen that the location is existent where stiffness values are equal. If iterative is not convergent, the cause cannot be otherwise than inappropriate displacement increment or deficient iterative times. In order to make the iterative process convergent quickly and efficiently, this paper uses the bisection searching method changing step length to modify the top beam displacement. This method is a little complex but the requirement on the initial condition is relatively mild.The flow chart of optimization as follows:1. Read the beam model data in initial position from backup file;2. Modify the position of beam;3. Solve;4. Read the deform of nodes where beam and three bars are joined;5. Check whether the convergent conditions are satisfied, if not, then continue to modify the beam displacement and return to 3, otherwise, exit the iteration procedure.6. Save the results and then exit.The program's primary control codes and their function commentaries are given in it, of which the detailed modeling instructions are omitted. For the convenience of comparing with the control flow, the necessary notes are added.the flag of the batch file in ANSYSBATCH RESUME, robbak.db, 0read original data from the backupfile robbak,.db/PREP7 enter preprocessordelete the joint part between beam core and columnsmove the core of the beam by one :step lengthapply load and constraint on the geometry meshing thejoint position between beam core and columns FINISH exit the preprocessorISOLU enter solverSOLVE solveFINISH exit the solverPOST1 enter the postprocessor*GET ,front,NODE,2013,U,Z read the deformation of first joint node on beam*GET,back,NODE, 1441 ,U,Z read the deformation of second joint node on beam intoparameter hacklastdif-1 the absolute of initial difference between front and hacklast timeflag=- 1 the feasibility flag of the optimizationstep=0.05 the initial displacement from initial position to the currentposition*D0,1,1,10,1 the iteration procedure begin, the cycle variable is I andits value range is 1-10 and step length is 1dif=abs(front-back) the absolute of the difference between front and hack inthe current result*IF,dif,LE,l .OE-6,THEN check whether the absolute difference dif satisfies therequest or noflag=l yes, set flag equal to 1*EXIT exit the iterative calculation*ELSEIF,dif,GE,lastdif,THEN check whether the dif value becomes great or not flag=2yes, set flag 2 modify step length by bisection methodperform the next iterative calculation, use the lastposition as the current position and modified last steplength as the current step lengthELSE if the absolute of difference value is not less thanexpected value and become small gradually, continue tomove top beam read the initial condition from back upfile enter the preprocessorMEN, ,P51X, , , step,, , ,1 move the core of the beam by one step length modify thejoint positions between beam core and column applyload and constraint meshingFINISH exit preprocessorISOLU enter solverSOLVE solveFINISH exit the solver/POST1 exit the postprocessor*GET,front,NODE,201 3,U,Z read the deformation of first joint node to parameter front *GET,back,NODE, 144 1,U,Z read the deformation of second joint node to parameter back lastdif-dif update the value of last dif*ENDIF the end of the if-else*ENDDO the end of the DO cycleMost of the control program above is copied from log file, which is long. The total of lines is up to about 1000 lines. Many codes such as modeling and post-process codes are used repeatedly. To make the program construct clear, these instructions can he made into macros, which are called by main program. This can efficiently reduce the length of the main program. In addition, modeling instructions from log file includes lots of special instructions that are only used under graphic mode but useless under hatch mode. Deleting and modifying these instructions when under batch mode in ANSYS can reduce the length of the file, too.In the program above, the deformation at given position is read from node deformation. In meshing, in order to avoid generating had elements, triangle mesh is used. In optimization, the shape of joint position between columns and beam continually is changed. This makes total of elements different after meshing each time and then element numbering different, too. Data read from database according to node numbering might not he data to want. Therefore, beam core first needs to he meshed, then saved. When read next time, its numbering is the same as last time.Evaluating whether the final result is a feasible result or not needs to check the flag value. If only the flag value is I, the result is feasible, otherwise the most proper position is not found. The total displacement of top beam is saved in parameter step. If the result is feasible, the step value is the distance from initial position to the most proper position. The sum of iterative is saved in parameter 1. According to the final value of I, feasibility of analysis result and correctness of initial condition can he evaluated.4）Optimization resultsThe sum of iterative in optimization is seven, and it takes about 2 hour and 37 minutes to find optimal position. Fig.3 shows the deformation contour of the half-construct. In Fig.3, the deformations in three joints between beam and the three driving bars is the same as level, and the corresponding deformation range is between -0.133E-04 and -0.1 15E-O4m, the requirement of the same stiffness is reached. At this time, the position of beam core along x-axis as shown in Fig. 1 has moved -0.71E-01m compared with the original designed positionBecause the speed of computer reading instruction is much faster than modifying model manually, the time modifying model can be ignored. The time necessary foroptimization mostly depends on the time of solution. Compared with the optimization procedure manually modifying model, the efficiency is improved and mistake operating in modeling is avoided.5）ConclusionThe analyzing result reveals that the optimization method given in this paper is effective and reaches the expected goal. The first advantage of this method is that manual mistakes do not easily occur in optimization procedure. Secondly, it is pretty universal and the control codes given in this paper may he transplanted to use in similar structure optimization design without large modification. The disadvantage is that the topology structure of the optimization object can not be changed. The more the workload of modifying the model, the more the advantages of this method are shown. In addition, the topology optimization function provided in ANSYS is usedto solve the optimization problem that needs to change the topology structure.The better optimization results can he achieved if the method in this paper combined with it.中文译文：机器人机构优化设计有限元分析摘要机器人结构最优化设计，设计模型需要反复的修正和计算。