应力为基础的有限元方法应用于灵活的曲柄滑块机构外文文献翻译@中英文翻译@外文翻译

1 英文原稿 Application of Stress-based Finite Element Method to a Flexible Slider Crank Mechanism （ Y.L.Kuo University of Toronto W.L.Cleghorn University of Canada） AbstractThis paper presents a new procedure to apply the stress-based finite element method on Euler-Bernoulli beams.An approximated bending stress distribution is selected,and then the approximated transverse displacement is determined by integration.The proposed approach is applied to solve a flexible slider crank mechanism.The formulation is based on the Euler-Lagrange equation,for which the Lagrangian includes the components related to the kinetic energy,the strain energy,and the work done by axial loads in a link that undergoes elastic transverse deflection.A beam element is modeled based on a translating and rotating motion.The results demonstrate the error comparison obtained from the stress-and displacement-based finite element methods. Keywords:stress-based finite element method；slider crank mechanism；Euler-Lagrange equation. 1.Introduction The displacement-based finite element method employs complementary energy by imposing assumed displacements.This method may yield the 2 discontinuities of stress fields on the inter-element boundary while employing low-order elements,and the boundary conditions associated with stress could not be satisfied.Hence,an alternative approach was developed and called the stress-based finite element method,which utilizes assumed stress functions.Veubeke and Zienkiewicz1,2were the first researchers introducing the stress-based finite element method.After that,the method was applied to a wide range of problems and its applications3-5In addition,there are various books providing details about the method6,7. The operation of high-speed mechanisms introduces vibration,acoustic radiation,wearing of joints,and inaccurate positioning due to deflections of elastic links.Thus,it is necessary to perform an analysis of flexible elasto-dynamics of this class of problems rather than the analysis of rigid body dynamics.Flexible mechanisms are continuous dynamic systems with an infinite number of degrees of freedom,and their governing equations of motion are modeled bynonlinear partial differential equations,but their analytical solutions are impossible to obtain.Cleghorn et al.8-10included the effect of axial loads on transverse vibrations of a flexible four-bar mechanism.Also,they constructed a translating and rotating beam element with a quintic polynomial,which can effectively predict the transverse vibration and the bending stress. This paper presents a new approach for the implementation of the 3 stress-based finite element method on the Euler-Bernoulli beams.The developed approach first selects an assumed stress function.Then,the approximated transverse displacement function is obtained by integrating the assumed stress function.Thus,this approach can satisfy the stress boundary conditions without imposing a constraint.We apply this approach to solve a flexible slider crank mechanism.In order to show the accuracy enhancement by this approach,the mechanism is also solved by the displace-based finite element method.The results demonstrate the error comparison. II.Stress-based Method for Euler-Bernoulli Beams The bending stress of Euler-Bernoulli beams is associated with the second derivative of the transverse displacement,namely curvature,which can be approximated as the product of shape functions and nodal variables: Where is a row vector of shape functions for the ith element； is a column vector of nodal curvatures,y is the lateral position with respect to the neutral line of the beam,E is the Youngs modulus,and is the transverse displacement,which is a function of axial position x. Integrating Eq.(1)leads to the expressions of the rotation and the transverse displacement as Rotation： Transverse displacement: 4 Where and are two integration constants for the ith element,which can be determined by satisfying the compatibility. Substituting Eqs.(2)and(3)into(1),the finite element displacement,rotation and curvature can be expressed as: where the subscripts(C),(R)and(D)refer to curvature,rotation and displacement,respectively.By applying the variational principle,the element and global equations can be obtained11-13. Table 1:Comparison of the displacement-and the stress-based finite element methods for an Euler-Bernoulli beam element III.Comparisons of the Displacement-and Stress-based Finite Element Methods The major disadvantage of the displacement-based finite element method 5 is that the stress fields at the inter-element nodes are discontinuous while employing low-degree shape functions.This discontinuity yields one of the major concerns behind the discretization errors.In addition,it might use excessive nodal variables while formulating stiffness matrices. The stress-based method has several advantages over the displacement-based finite method.First of all,the stress-based method produces fewer nodal variables (Table 1).Secondly,when employing the stress-basedfinite method,the boundary conditions of bending stress can be satisfied,and the stress is continuous at theinter-element nodes.Finally,the stress is calculated directly from the solution of the global system equations.However,the only disadvantage of the stress-based finite method is that the integration constants are different for each element. IV.Generation of Governing Equation The slider crank mechanism shown in Fig.1 is operated with a prescribed rigid body motion of the crank,and the governing equations are derived using a finite element formulation.The derivation procedure of the finite element equations involves:(1)deriving the kinematics of a rigid body slider crank mechanism；(2) constructing a translating and rotating beam element based on the rigid body motion of the mechanism；(3)defining a set of global variables to describe the motion of a flexible slider crank mechanism；(4)assembling all beam elements.Finally,the global finite 6 element equations can be obtained,and the time response of a flexible slider crank mechanism can be obtained by time integration. A.Element equation of a translating and rotating beam Consider a flexible beam element subjected to prescribed rigid body translations and rotations.Superimposed on the rigid body trajectory,a finite number of deflection variables in the longitudinal and transverse directions is allowed.The Euler-Lagrange equation is used to derive the governing differential equations for an arbitrarily translating and rotating flexible member.Since elastic deflections are considered small,and there is a finite number of degrees of freedom,the governing equations are linear and are conveniently written in matrix form.The derivation of the element equations has been precisely presented in 8-10,and this section provides a brief summary. In view of high axial stiffness of a beam,it is reasonable to consider the beam as being rigid in its longitudinal direction.Hence,the longitudinal deflection is given as where u1 is a nodal variable,which is constant with respect to the x 7 direction shown in Fig.2.The transverse deflection can be represented as The velocity of an arbitrary point on the beam element with a translating and rotating motion is given as where is the absolute velocity of point O of the beam element shown in Fig.2；?is the angular velocity of the beam element； are the longitudinal and transverse displacements of an arbitrary point on the beam element,respectively；x is a longitudinal position on the beam element shown in Fig. 2. If we letbe the mass per unit volume of element material；A,the element cross-sectional area,and L the element length,then the kinetic energy of an element is expressed as The flexural strain energy of uniform axially rigid element with the Youngs modulus,E,and second moment of area,I,is given as The work done by a tensile longitudinal load,(i)P,in an element that undergoes an elastic transverse deflection is given by14 8 Longitudinal loads in a moving mechanism element are not constant,and depend both on the position in the element and on time.With the longitudinal elastic motions neglected,the longitudinal loads may be derived from the rigid body inertia forces,and can be expressed as where PR is an external longitudinal load acting at theright hand end of an element,andox (i )ais the absolute eacceleration of the point O in the x direction shown in Fig.2. The Lagrangian takes the form Substituting Eqs.(5-10)into(12),and employing the Euler-Lagrange equations,the governing equations of motion for a rotating and translating elastic beam can be expressed in the following matrix form: whereMe,CeandKeare mass,equivalent damping,and equivalent stiffness matrices of a element,respectively；Feis a load vector of an element.When formulating the mass matrix of the coupler,the mass of the slider should be taken into account. 9 B.Global equations of slider crank mechanism For the proposed approach to solve a flexible slider crank mechanism,the global variables are the curvatures on the nodes.For assembling all elements,it is necessary to consider the boundary conditions applied to the mechanism.Since a prescribed motion applied to the base of the crank,there is a bending moment at point O shown in Fig.1,i.e.,the curvature at point O exists.For points A and B shown in Fig.1,we presume that both points refer to pin joints.Thus,the bendingmoments and the curvatures at both points are zeros. Since Eq.(13)is a matrix-form expression in terms of the vector of global variables,the global equations can be obtained by directly summing up all of element equations,which can be expressed as whereM,C,Kare global mass,damping and stiffness matrices,respectively；Fis a global load vector. V.Numerical simulation based on steady state The rotating speed of the crank is operating at 150rad/s(1432 rpm),and 10 the system parameters of a flexible slider crank are as follows: R2=0.15(m),R3=0.30(m),A=0.225(kg/m),EI=12.72(N-m2),mB=0.03375(kg) where R2 and R3 are the lengths of the crank and coupler,respectively；mB is the mass of the slider. The analytical results of this paper are presented by plotting steady state transverse displacements and bending strains of midpoints on crank and coupler throughout a cycle of motion.The steady state can be obtained by adding a physical damping matrix,namely Rayleigh damping whereandare two constants,which can be determined from two given damping ratio that correspond to two unequal frequencies of vibration15. In this paper,the values ofandare determined based on the first two natural frequencies. By adding physical damping to the equations of motion,the analytical solution is obtained by performing the constant time-step Newmark method over twenty cycles of motion.The initial conditions are set to zeros when performing numerical time integration. The error indicator is defined as where QFE and QRef are two quantities based on a finite element solution and a reference solution,respectively.Generally,they are functions 11 of time,and they can be arbitrarily selected,such as energy,displacement,bending strain,etc.t1 and t2 refer to the interval of timeintegration,which are usually one cycle after steady-state condition has been reached.Since an exact solution is not available,a reference solution is obtained by the displacement-based finite element method based on twenty elements per link with quintic polynomials in this paper. Fig.3.Time responses of the total energy,mensionless midpoint deflection of the coupler,and he midpoint strain of the coupler at the steady state condition VI.Numerical Simulations In the section,we consider the mechanism with a rigid crank.The coupler is the only flexible link.Based on the beam element constructed in Section IV.,the beam element has a rigid axial motion,but it has a transverse deflection. When we implement the stress-based finite elementmethod proposed in Section III.,it is necessary to consider the boundary conditions of the modeled links and the approximated degree of shape functions.In this 12 example,we select a linear function along the axial axis to approximate the strain distribution of the coupler,and the boundary conditions of the coupler are considered without zero bending moment.Thus,it is impossible to model the coupler with one element. In the example,we consider the coupler discretized as two,three,four,and five elements,and its curvature distribution is approximated by a linear function as And then,the time responses and the errors of the total energy,the midpoint deflection of the coupler,the midpoint strain of the coupler is obtained by the stress-based finite element method.Also,the first natural frequency is evaluated. The rotating speed of the crank is operating at 150rad/s(1432 rpm),and the system parameters of a flexible slider-crank are as follow16:R2=0.15(m),R3=0.30(m),A=0.225(kg/m),EI=12.72(N-m 2),mB=0.03375(kg)where R2 and R3 are the lengths of the crank and coupler,respectively；mB is the mass of the slider. In order to compare the errors obtained by the displacement-based finite element method,we also use it to solve the mechanism,and its results are based on Ref.17. Table 2.Errors of the first natural frequency by both finite element methods 13 Fig.3.shows the time responses of the total energy,the dimensionless midpoint deflection of coupler,and the midpoint strain of the coupler on the steady state condition.Tables 2 to 5 show the error comparisons of the first natural frequency,the total energy,the midpoint deflection of the coupler,and the midpoint strain of the coupler by the stress-and the displacement-based finite element methods.The error calculation is based on Eq. (16).The results show that the errors from the stress-based finite 14 element method are greater than the errors from the displacement-based finite element method,when we consider the same number of elements for both methods.However,when the number of degrees of freedom is the same,the errors from the stress-based finite element method is much smaller than the errors from the displacement-based finite element method.Also,we notice that except for the errors of the first natural frequency,the errors from the stress-based finite element method are smaller than the errors from the displacement-based finite element method under the same number of elements.It illustrates that the stress-based finite element method can provide much accurate approximated solutions for kineto-elasto-dynamic problems. VII.Conclusions This paper proposed a new approach to implement the stress-based finite element method to Euler-Bernoulli beam problems.Especially,this 15 approach can be applied to kineto-elasto-dynamic problems.The proposed approach is to approximate the curvature of a beam. Then,we can obtain the transverse deflection and the stress distribution by integrating the approximate curvature distribution.During the integration procedure, it is necessary to make the boundary conditions of a beam element satisfied,which can derive the integration constant.In this paper,we apply the proposed approach to solve a flexible slider crank mechanism operating a high-speed motion.The results illustrate that the errors from the stress-based finite element method are much smaller than the errors from the conventional approach, the displacement-based finite element method,when we compare the errors under the same degrees of freedom. Also,some errors show that the stress-based finite element method can provide more accurate solutions under the same number of elements. References 1B.Fraeijs de Veubeke,“Displacement and equilibrium models in the finite element method”,Stress Analysis,edited by 16 O.C.Zienkiewicz,Wiley,New York,1965. 2B.Fraeijs de Veubekd and O.C.Zienkiewicz,“Strain-energy bounds in finite-element analysis by slab analogy”,J.Strain Analysis,Vol.2,pp.265-271,1967. 3Z.Wieckowski,S.K.Youn,and B.S.Moon,“Stressed-based finite element analysis of plane plasticity problems”,Int.J.Numer.Meth.Engng.,Vol.44,pp.1505-1525,1999. 4 H.Chanda and K.K.Tamma,“Developments encompassing stress based finite element formulations for materially nonlinear static dynamic problems”,Comp.Struct.,Vol.59, No.3,pp.583-592,1996. 5M.Kaminski,“Stochastic second-order perturbation approach to the stress-based finite element method”,Int.J.Solids and Struct.,Vol.38,No.21,pp.3831-3852,2001.6O.C.Zienkiewicz and R.L.Taylor,The Finite Element Method,McGraw-Hill,London,2000. 7R.H.Gallagher,Finite Element Fundamentals,Prentice-Hall,Englewood Cliffs,1975. 8W.L.Cleghorn,1980,Analysis and design of high-speed flexible mechanism,Ph.D.Thesis,University of Toronto. 9W.L.Cleghorn,R.G.Fenton,and B.Tabarrok,1981,“Finite element analysis of high-speed flexible mechanisms”,Mechanism and Machine Theory,16(4),407-424. 17 10W.L.Cleghorn,R.G.Fenton,and B.Tabarrok,1984,“ Steady-state vibrational response of high-speed flexible mechanisms”,Mechanism and Machine Theory,19(4/5) 11Y.L.Kuo,W.L.Cleghorn and K.Behdinan,“Stress-based Finite Element Method for Euler-Bernoulli Beams”,Transactions of the Canadian Society for Mechanical Engineering,Vol.30(1),pp.1-6,2006. 12Y.L.Kuo,W.L.Cleghorn,and K.Behdinan“Applications of Stress-based Finite Element Method on Euler-Bernoulli Beams”,Proceedings of the 20th Canadian Congress of Applied Mechanics,Montreal,Quebec,Canada,May 30-Jun2,2005. 13Y.L.Kuo,Applications of the h-,p-,and r-refinements of the Finite Element Method on Elasto-dynamic Problems,Ph.D.Thesis,University of Toronto,2005. 14L.Meirovitch,1967,Analytical Methods in Vibrations Macmillan,New York,436-463. 15K.J.Bathe,1996,Finite Element Procedures,Prentice Hall Englewood Cliffs,NJ,USA. 16A.L.Schwab and J.P.Meijaard,2002,“Small vibrations superimposed on prescribed rigid body motion”,Multibody System Dynamics,8,29-49. 17Y.L.Kuo and W.L.Cleghorn,“The h-p-r-refinement FiniteElement Analysis of a High-speed Flexible Slider Crank Mechanism”,Journal of Sound and Vibration,in press. 18 英文翻译 应力为基础的有限元方法应用于灵活的曲柄滑块机构 （多伦多大学： Y.L. Kuo .L. Cleghorn 加拿大） 摘要 :本文 在 欧拉一伯努利梁 基础上 提出了一种新的适用 于以 应力为基础的有限元方法 的程序。 先选择一个 近似弯曲应力 的分布 ，然后 通过一体化确定近似横位移 。该方法适用于解决灵活滑块曲柄机构 问题， 制定的依据是欧拉 -拉格朗日方程，而拉格朗日包括 与 动能，应变能 有关的组件 ， 并通过 弹性横向 挠度构成的 轴向负荷的链接 来工作 。梁元模型 以翻转运动为基础， 结果表明 以 应力和位移为基础的有限元方法。 关键词 ： 应力为基础的有限元方法，曲柄滑块机构，拉格 -朗日方程 1.前言 以位移为基础的有限元方法通过实行假定位移补充能量。 这种方法可能由内部因素产生不连续应力场，同时由于采用了低阶元素，边界条件与压力不能得到满足。因此，另一种被成为以应力为基础采用假定应力的有限元方法得到了应用和发展。 Veubeke 和 Zienkiewicz1-2首先对应力有限元素进行了研究。之后，这种方法被广泛用于解决应用程序中的问题 3-5。此外，还有各种书籍提供更加详细的方法 6.7。 这一高速运作机制采用振动，声辐射，协同联结，和挠度弹性链接的准确定位。因此，有必要分析灵活的弹塑性动力学这一类的问题，而不是分析刚体动力学。 灵活的机制是一个由无限 多个自由度组成的连续动力学系统，其运动方程是由非线性偏微分方程建立的模型，但得不到分析解决方案。 Cleghorn et al8-10 阐述了横向振动上的轴向荷载对灵活四杆机构的影响。并且通过能有效预测横向振动和弯曲应力的五次多项式建立了一个翻转梁单元。 本文提出了一种新的方法来执行建立在欧拉一伯努利基础上的以应力为基础的有限元方法。改进后的方法首先选定了假定应力函数。然后通过整合假定应力函数得到横向位移函数。当然，这种方法能解决没有强制制约因素的应力集中问题。我们可以通过这种方法解决灵活曲柄滑块机构体系 中存在的问题。目的是通过这种方法提高准确性，该系统存在的问题也可以通过取代基有限元方法来解决。结果可以证明偏差比较。 2.以应力为基础的欧拉一伯努利梁 欧拉一伯努利梁的弯曲应力与横向位移的二阶导数相关，也就是曲率，可以近似的看做是形函数和交点变量： 19 这里 (i)N(c)是连续载体的形函数； (i)e 是列向量的交点函数， y 是关于中性线的横向定位， E是杨氏模量，（ i） v是横向位移， x 轴向定位函数。 由方程（ 1）可以推导出横向位移转换方程： 横向位移： 这里 (i)C1 和 (i)C2 是两个一 体化常数，可以通过满足兼容性来确定。 将方程（ 2）和（ 3）代入（ 1），可以得到有限元位移和回转曲率，如下所示： 这里下标（ C），（ R）和（ D）分别代表曲率，自转和位移。运用变分原理，可以得到这些方程 11-13。 表 1 分别比较以位移和应力为基础的有限元方法的欧拉 -伯努利梁元素 以位移为基础的有限元方法 以应力为基础的有限元方法 近似横向位移自由度 立方米 立方米 近似弯曲应力 线性 线性 交点变量 两端位移和回转 两端曲率 边界应力满足条件 位移，回转 位移，回转，弯曲应力 自由度数量 四 二 3.以位移和应力为基础的有限元方法的比较 主要区别在于以位移为基础的有限元方法的应力场存在不连续的内部因素，同时具有低阶形函数。主要是因为不连续量的产生以及 间离散分布。再者， 它可能 由于 使用过多交点变量而 产生 刚度矩阵 。 20 以应力为基础的方法与以位移为基础的方法比较具有很多优点。首先，以应力为基础的方法产生的交点变量较少（如表 1）。第二，使用以应力为基础的方法时，弯曲应力的边界条件可以得到满足。最后，应力由体系方程直接计算得到。 4.方程推导 曲柄滑块机构如图 1所示，由做刚体运动的曲柄来运作 ，该方程由有限元公式推导而得。有限元方程的推导过程如下：（ 1）建立刚体运动学曲柄滑块机构；（ 2）构建基于刚体运动学机构的翻转梁单元；（ 3）确定一套变量用来描述灵活曲柄滑块机构的运动；（ 4）装配所有梁单元。最后，就可以得到有限元方程，同时该灵活曲柄滑块机构的时间响应可以通过时间一体化确定。 图 1 灵活曲柄滑块机构 A翻转梁的元方程 考虑灵活的梁单元受到刚体翻转和回转运动。叠加在刚体运动轨迹时，纵向和横向方向上允许一些挠度变量。通过拉格 -朗日方程可以得到任意灵活翻转的组件的微分方程。由于弹性变形认为是很 小的，而且自由度是有限的，这个方程是线性的并且很容易画出来。推导公式的元素也被很明确的列出来 8-10，并且做了简要的介绍。 鉴于在轴向有很强的刚度，因此很有必要在纵向方向上合理考虑为刚性梁。所以，纵向方向如一下所示： （ 5） 这里 u1 是交点变量，是关于 x 轴方向的常数，如图 2 所示。横向可以表示为： 21 翻转梁单元上任意点的速度可以表示如下： 这里（ i） Vax（ i） Vay）是梁单元在 O点的绝对速度，如图 2所示； 是梁单元的 角速度；（ i） u（ i） v）分别是梁单元上任意点纵向和横向的位移， x是梁单元纵向的定位，如图 2所示。 图 2 旋转梁 如果我们把 当作组件材料的单位体积； A 是组件的横截面积， L是组件的长度，组件的动能可以表示如下： 均匀刚性组件的轴向弯曲应变能量与杨氏模量 E 有关，得到二阶矩阵 I，如下所示： 22 由纵向拉伸负荷工作，（ i） P,组件的横向挠度表示如下： 运功机制的纵向负荷不是一成不变的，与位置和时间有关。在忽略纵向弹性形变的前提下，纵向负荷可能来自于刚性惯性力，可以表示如下： 这里 PR 是元件右侧的外部纵向负载， 是 x轴方向上 O点的绝对加速度。如图 2所示。 拉格 -朗日形式表示如下： 将公式（ 5-100）代入（ 12），并且运用欧拉 -拉格朗日方程，旋转梁的运动方程可以表示为一下形式： 这里 Me、 Ce和 Ke分别是元件的质量、等效阻尼和等效刚度矩阵； Fe是元件的载荷向量。当建立质量耦合矩阵时，应主要考虑滑块机构。 B.曲柄滑块机构方程 提出解决曲柄滑块机构问题的方法，变量是曲率的节点。装配所有元件时，考虑机构的边界条件是很有必要的。因为该动力适用于基础曲 柄结构，在 O点 存在弯矩，如图 1 所示，在 O点也存在曲率。如图 1所示的 A 点和 B 点，我们假定它们是很小的点。然而，实际上，弯矩和曲率在这两个点上都为零。 因为公式（ 13）是变量的矩阵表示方式 ，这个公式可以通过总结所有的方程来得到，可以表示如下： 这里 M、 C、 K分别是质量、阻尼和刚度矩阵， F是负载向量。 5.稳定状态基础上的数值模拟 曲柄的转速是 150rad/s (1432rpm),该灵活曲柄滑块机构的各项数值表示如下：R2=0.15(m)， R3=0.30(m), =0.225(kg/m), EI=12.72(N-m2), 23 mB=0.03375(kg)。 这里 R2 和 R3 分别是曲柄和耦合器的长度， mB 是滑块的质量。 通过曲柄和耦合器的一个运动周期，可以看出稳态横向位移和中点弯曲应力的变化情况，以及分析本课题的结果。可以通过增加物理阻尼矩阵提高稳定性，被称作瑞利阻尼： 这里和是两个常数，可以从 15中对应于两个不同频率的振动的阻尼比得到。本文中和的值取决于自然频率。 通过在运动方程中增加物理阻尼，也可以通过 Newmark 时间步骤观测超过 20