應(yīng)力為基礎(chǔ)的有限元方法應(yīng)用于靈活的曲柄滑塊機(jī)構(gòu)外文文獻(xiàn)翻譯、中英文翻譯,應(yīng)力,基礎(chǔ),有限元,方法,法子,應(yīng)用于,靈活,靈便,曲柄,機(jī)構(gòu),外文,文獻(xiàn),翻譯,中英文 英文原稿 12thIFToMM World Congress,Besancon,June 18-21,2007 Application of Stress-based Finite Element Methodto a Flexible Slider Crank Mechanism Y.L.Kuo? W.L.Cleghorn University of Toronto University of Toronto Toronto,Canada Toronto,Canada Abstract—This 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 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 Zienkiewicz[1,2]were the first researchers introducing the stress-based finite element method.After that,the method was applied to a wide range of problems and its applications[3-5]In addition,there are various books providing details about the method[6,7]. 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______________________ ?Email:ylkuo@mie.utoronto.ca solutions are impossible to obtain.Cleghorn et al.[8-10]included 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 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 Young’s 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: 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 obtained[11-13]. 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 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 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 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;θ?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 letρbe 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 Young’s 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 by[14] 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: where[Me],[Ce]and[Ke]are mass,equivalent damping,and equivalent stiffness matrices of a element,respectively;{Fe}is a load vector of an element.When formulating the mass matrix of the coupler,the mass of the slider should be taken into account. 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{φ},the global equations can be obtained by directly summing up all of element equations,which can be expressed as where[M],[C],[K]are global mass,damping and stiffness matrices,respectively;{F}is 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 the system parameters of a flexible slider crank are as follows: R2=0.15(m),R3=0.30(m),ρA=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 whereαandβare two constants,which can be determined from two given damping ratio that correspond to two unequal frequencies of vibration[15]. In this paper,the values ofαandβare 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 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 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 follow[16]:R2=0.15(m),R3=0.30(m),ρA=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 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 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 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 [1]B.Fraeijs de Veubeke,“Displacement and equilibrium models in the finite element method”,Stress Analysis,edited by O.C.Zienkiewicz,Wiley,New York,1965. [2]B.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. [3]Z.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. [5]M.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.[6]O.C.Zienkiewicz and R.L.Taylor,The Finite Element Method,McGraw-Hill,London,2000. [7]R.H.Gallagher,Finite Element Fundamentals,Prentice-Hall,Englewood Cliffs,1975. [8]W.L.Cleghorn,1980,Analysis and design of high-speed flexible mechanism,Ph.D.Thesis,University of Toronto. [9]W.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. [10]W.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) [11]Y.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. [12]Y.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. [13]Y.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. [14]L.Meirovitch,1967,Analytical Methods in Vibrations Macmillan,New York,436-463. [15]K.J.Bathe,1996,Finite Element Procedures,Prentice Hall Englewood Cliffs,NJ,USA. [16]A.L.Schwab and J.P.Meijaard,2002,“Small vibrations superimposed on prescribed rigid body motion”,Multibody System Dynamics,8,29-49. [17]Y.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.  應(yīng)力為基礎(chǔ)的有限元方法應(yīng)用于靈活的曲柄滑塊機(jī)構(gòu) （多倫多大學(xué)：Y.L. Kuo .L. Cleghorn加拿大） 摘要:本文在歐拉一伯努利梁基礎(chǔ)上提出了一種新的適用于以應(yīng)力為基礎(chǔ)的有限元方法的程序。先選擇一個近似彎曲應(yīng)力的分布，然后通過一體化確定近似橫位移。該方法適用于解決靈活滑塊曲柄機(jī)構(gòu)問題，制定的依據(jù)是歐拉-拉格朗日方程，而拉格朗日包括與動能，應(yīng)變能有關(guān)的組件，并通過彈性橫向撓度構(gòu)成的軸向負(fù)荷的鏈接來工作。梁元模型以翻轉(zhuǎn)運(yùn)動為基礎(chǔ)，結(jié)果表明以應(yīng)力和位移為基礎(chǔ)的有限元方法。 關(guān)鍵詞：應(yīng)力為基礎(chǔ)的有限元方法，曲柄滑塊機(jī)構(gòu)，拉格-朗日方程 1.前言 以位移為基礎(chǔ)的有限元方法通過實(shí)行假定位移補(bǔ)充能量。這種方法可能由內(nèi)部因素產(chǎn)生不連續(xù)應(yīng)力場，同時由于采用了低階元素，邊界條件與壓力不能得到滿足。因此，另一種被成為以應(yīng)力為基礎(chǔ)采用假定應(yīng)力的有限元方法得到了應(yīng)用和發(fā)展。Veubeke和Zienkiewicz[1-2]首先對應(yīng)力有限元素進(jìn)行了研究。之后，這種方法被廣泛用于解決應(yīng)用程序中的問題[3-5]。此外，還有各種書籍提供更加詳細(xì)的方法[6，7]。 這一高速運(yùn)作機(jī)制采用振動，聲輻射，協(xié)同聯(lián)結(jié)，和撓度彈性鏈接的準(zhǔn)確定位。因此，有必要分析靈活的彈塑性動力學(xué)這一類的問題，而不是分析剛體動力學(xué)。 靈活的機(jī)制是一個由無限多個自由度組成的連續(xù)動力學(xué)系統(tǒng)，其運(yùn)動方程是由非線性偏微分方程建立的模型，但得不到分析解決方案。Cleghorn et al[8-10] 闡述了橫向振動上的軸向荷載對靈活四桿機(jī)構(gòu)的影響。并且通過能有效預(yù)測橫向振動和彎曲應(yīng)力的五次多項(xiàng)式建立了一個翻轉(zhuǎn)梁單元。 本文提出了一種新的方法來執(zhí)行建立在歐拉一伯努利基礎(chǔ)上的以應(yīng)力為基礎(chǔ)的有限元方法。改進(jìn)后的方法首先選定了假定應(yīng)力函數(shù)。然后通過整合假定應(yīng)力函數(shù)得到橫向位移函數(shù)。當(dāng)然，這種方法能解決沒有強(qiáng)制制約因素的應(yīng)力集中問題。我們可以通過這種方法解決靈活曲柄滑塊機(jī)構(gòu)體系中存在的問題。目的是通過這種方法提高準(zhǔn)確性，該系統(tǒng)存在的問題也可以通過取代基有限元方法來解決。結(jié)果可以證明偏差比較。 2.以應(yīng)力為基礎(chǔ)的歐拉一伯努利梁 歐拉一伯努利梁的彎曲應(yīng)力與橫向位移的二階導(dǎo)數(shù)相關(guān)，也就是曲率，可以近似的看做是形函數(shù)和交點(diǎn)變量： 這里[(i)N(c)]是連續(xù)載體的形函數(shù)；{(i)?e} 是列向量的交點(diǎn)函數(shù)，y是關(guān)于中性線的橫向定位，E是楊氏模量，（i）v是橫向位移，x軸向定位函數(shù)。 由方程（1）可以推導(dǎo)出橫向位移轉(zhuǎn)換方程： 橫向位移： 這里 (i)C1和(i)C2是兩個一體化常數(shù)，可以通過滿足兼容性來確定。 將方程（2）和（3）代入（1），可以得到有限元位移和回轉(zhuǎn)曲率，如下所示： 這里下標(biāo)（C），（R）和（D）分別代表曲率，自轉(zhuǎn)和位移。運(yùn)用變分原理，可以得到這些方程[11-13]。 表1 分別比較以位移和應(yīng)力為基礎(chǔ)的有限元方法的歐拉-伯努利梁元素 以位移為基礎(chǔ)的有限元方法 以應(yīng)力為基礎(chǔ)的有限元方法 近似橫向位移自由度 立方米 立方米 近似彎曲應(yīng)力 線性 線性 交點(diǎn)變量 兩端位移和回轉(zhuǎn) 兩端曲率 邊界應(yīng)力滿足條件 位移，回轉(zhuǎn) 位移，回轉(zhuǎn)，彎曲應(yīng)力 自由度數(shù)量 四 二 3.以位移和應(yīng)力為基礎(chǔ)的有限元方法的比較 主要區(qū)別在于以位移為基礎(chǔ)的有限元方法的應(yīng)力場存在不連續(xù)的內(nèi)部因素，同時具有低階形函數(shù)。主要是因?yàn)椴贿B續(xù)量的產(chǎn)生以及間離散分布。再者，它可能由于使用過多交點(diǎn)變量而產(chǎn)生剛度矩陣。 以應(yīng)力為基礎(chǔ)的方法與以位移為基礎(chǔ)的方法比較具有很多優(yōu)點(diǎn)。首先，以應(yīng)力為基礎(chǔ)的方法產(chǎn)生的交點(diǎn)變量較少（如表1）。第二，使用以應(yīng)力為基礎(chǔ)的方法時，彎曲應(yīng)力的邊