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應力為基礎的有限元方法應用于靈活的曲柄滑塊機構
（多倫多大學：Y.L. Kuo .L. Cleghorn加拿大）
摘要:本文在歐拉一伯努利梁基礎上提出了一種新的適用于以應力為基礎的有限元方法的程序。先選擇一個近似彎曲應力的分布，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構問題，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，應變能有關的組件，并通過彈性橫向撓度構成的軸向負荷的鏈接來工作。梁元模型以翻轉運動為基礎，結果表明以應力和位移為基礎的有限元方法。
關鍵詞：應力為基礎的有限元方法，曲柄滑塊機構，拉格-朗日方程
1.前言
以位移為基礎的有限元方法通過實行假定位移補充能量。這種方法可能由內部因素產生不連續(xù)應力場，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，另一種被成為以應力為基礎采用假定應力的有限元方法得到了應用和發(fā)展。Veubeke和Zienkiewicz[1-2]首先對應力有限元素進行了研究。之后，這種方法被廣泛用于解決應用程序中的問題[3-5]。此外，還有各種書籍提供更加詳細的方法[6，7]。
這一高速運作機制采用振動，聲輻射，協(xié)同聯(lián)結，和撓度彈性鏈接的準確定位。因此，有必要分析靈活的彈塑性動力學這一類的問題，而不是分析剛體動力學。 靈活的機制是一個由無限多個自由度組成的連續(xù)動力學系統(tǒng)，其運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構的影響。并且通過能有效預測橫向振動和彎曲應力的五次多項式建立了一個翻轉梁單元。
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上的以應力為基礎的有限元方法。改進后的方法首先選定了假定應力函數(shù)。然后通過整合假定應力函數(shù)得到橫向位移函數(shù)。當然，這種方法能解決沒有強制制約因素的應力集中問題。我們可以通過這種方法解決靈活曲柄滑塊機構體系中存在的問題。目的是通過這種方法提高準確性，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。結果可以證明偏差比較。
2.以應力為基礎的歐拉一伯努利梁
歐拉一伯努利梁的彎曲應力與橫向位移的二階導數(shù)相關，也就是曲率，可以近似的看做是形函數(shù)和交點變量：
這里[(i)N(c)]是連續(xù)載體的形函數(shù)；{(i)?e} 是列向量的交點函數(shù)，y是關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）可以推導出橫向位移轉換方程：
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曲率，如下所示：
這里下標（C），（R）和（D）分別代表曲率，自轉和位移。運用變分原理，可以得到這些方程[11-13]。
表1 分別比較以位移和應力為基礎的有限元方法的歐拉-伯努利梁元素
以位移為基礎的有限元方法
以應力為基礎的有限元方法
近似橫向位移自由度
立方米
立方米
近似彎曲應力
線性
線性
交點變量
兩端位移和回轉
兩端曲率
邊界應力滿足條件
位移，回轉
位移，回轉，彎曲應力
自由度數(shù)量
四
二
3.以位移和應力為基礎的有限元方法的比較
主要區(qū)別在于以位移為基礎的有限元方法的應力場存在不連續(xù)的內部因素，同時具有低階形函數(shù)。主要是因為不連續(xù)量的產生以及間離散分布。再者，它可能由于使用過多交點變量而產生剛度矩陣。
以應力為基礎的方法與以位移為基礎的方法比較具有很多優(yōu)點。首先，以應力為基礎的方法產生的交點變量較少（如表1）。第二，使用以應力為基礎的方法時，彎曲應力的邊界條件可以得到滿足。最后，應力由體系方程直接計算得到。
4.方程推導
曲柄滑塊機構如圖1所示，由做剛體運動的曲柄來運作，該方程由有限元公式推導而得。有限元方程的推導過程如下：（1）建立剛體運動學曲柄滑塊機構；（2）構建基于剛體運動學機構的翻轉梁單元；（3）確定一套變量用來描述靈活曲柄滑塊機構的運動；（4）裝配所有梁單元。最后，就可以得到有限元方程，同時該靈活曲柄滑塊機構的時間響應可以通過時間一體化確定。
圖1 靈活曲柄滑塊機構
A．翻轉梁的元方程
考慮靈活的梁單元受到剛體翻轉和回轉運動。疊加在剛體運動軌跡時，縱向和橫向方向上允許一些撓度變量。通過拉格-朗日方程可以得到任意靈活翻轉的組件的微分方程。由于彈性變形認為是很小的，而且自由度是有限的，這個方程是線性的并且很容易畫出來。推導公式的元素也被很明確的列出來[8-10]，并且做了簡要的介紹。
鑒于在軸向有很強的剛度，因此很有必要在縱向方向上合理考慮為剛性梁。所以，縱向方向如一下所示：
（5）
這里u1是交點變量，是關于x軸方向的常數(shù)，如圖2所示。橫向可以表示為：
翻轉梁單元上任意點的速度可以表示如下：
這里（（i）Vax（i）Vay）是梁單元在O點的絕對速度，如圖2所示； 是梁單元的角速度；（（i）u（i）v）分別是梁單元上任意點縱向和橫向的位移，x是梁單元縱向的定位，如圖2所示。
圖2 旋轉梁
如果我們把 當作組件材料的單位體積；A是組件的橫截面積，L是組件的長度，組件的動能可以表示如下：
均勻剛性組件的軸向彎曲應變能量與楊氏模量E有關，得到二階矩陣I，如下所示：
由縱向拉伸負荷工作，（i）P,組件的橫向撓度表示如下：
運功機制的縱向負荷不是一成不變的，與位置和時間有關。在忽略縱向彈性形變的前提下，縱向負荷可能來自于剛性慣性力，可以表示如下：
這里PR是元件右側的外部縱向負載， 是x軸方向上O點的絕對加速度。如圖2所示。 拉格-朗日形式表示如下：
將公式（5-100）代入（12），并且運用歐拉-拉格朗日方程，旋轉梁的運動方程可以表示為一下形式：
這里[Me]、[Ce]和[Ke]分別是元件的質量、等效阻尼和等效剛度矩陣；{Fe}是元件的載荷向量。當建立質量耦合矩陣時，應主要考慮滑塊機構。
B.曲柄滑塊機構方程
提出解決曲柄滑塊機構問題的方法，變量是曲率的節(jié)點。裝配所有元件時，考慮機構的邊界條件是很有必要的。因為該動力適用于基礎曲柄結構，在O點存在彎矩，如圖1所示，在O點也存在曲率。如圖1所示的A點和B點，我們假定它們是很小的點。然而，實際上，彎矩和曲率在這兩個點上都為零。
因為公式（13）是變量的矩陣表示方式{?} ，這個公式可以通過總結所有的方程來得到，可以表示如下：
這里[M]、[C]、[K]分別是質量、阻尼和剛度矩陣，{F}是負載向量。
5.穩(wěn)定狀態(tài)基礎上的數(shù)值模擬
曲柄的轉速是150rad/s (1432rpm),該靈活曲柄滑塊機構的各項數(shù)值表示如下：R2=0.15(m)，R3=0.30(m), =0.225(kg/m), EI=12.72(N-m2), mB=0.03375(kg)。
這里R2 和R3分別是曲柄和耦合器的長度，mB是滑塊的質量。
通過曲柄和耦合器的一個運動周期，可以看出穩(wěn)態(tài)橫向位移和中點彎曲應力的變化情況，以及分析本課題的結果。可以通過增加物理阻尼矩陣提高穩(wěn)定性，被稱作瑞利阻尼：
這里α和β是兩個常數(shù)，可以從[15]中對應于兩個不同頻率的振動的阻尼比得到。本文中α和β的值取決于自然頻率。
通過在運動方程中增加物理阻尼，也可以通過Newmark時間步驟觀測超過20個周期的運動，從而得到分析結果。當采用數(shù)值時間積分是出示條件從零開始。
誤差可以表示為：
這里QFEQRef 和分別表示以有限元方法和參考方法為基礎的兩個值，總的來說，可以建立時間方程，而且很容易被接受，比如能量、位移、彎矩等等。t1 和t2指的是時間積分的間隔，通常指的是穩(wěn)態(tài)條件下的以個周期。因為沒有一個合適的準確的方法，在本文中可以通過一個五次多項式表示20個元件鏈接為基礎的位移有限元方法得到參考值。
Fig. 3. Time responses of the total energy, dimensionless midpoint deflection of the coupler, and the midpoint strain of the coupler at the steady state
condition
圖3 總能量的時間響應，耦合器的量綱中點撓度，耦合器在穩(wěn)態(tài)條件下的中點應變。
6.數(shù)值模擬
在這一節(jié)中，我們討論剛性曲柄機構。耦合器是唯一的一個靈活的連桿。在第六節(jié)中以以梁單元為基礎，該梁單元可以做剛性軸運動，但是存在橫向撓度。
在第三節(jié)中討論以有限元為基礎的方法時，很有必要考慮模型的邊界條件和形函數(shù)的相近程度，我們粗略的建立了耦合器應變線性分布方程，而且在彎矩不為零的條件下考慮耦合器的邊界條件。
在下面這個例子中，我們認為耦合器是由兩個、三個、四個或者五個元件組成的，同時它的曲率分布可以表示為線性方程：
于是，時間響應和總能量誤差，耦合器的中點撓度和應變都可以通過以應力為基礎的有限元方法得到。同時，也評估了第一自然頻率。
曲柄的轉速為150rad/s (1432rpm) ，該靈活的曲柄滑塊機構中各個部件的值可以表示如下[16]：
R2=0.15(m)，R3=0.30(m), =0.225(kg/m), EI=12.72(N-m2), mB=0.03375(kg)。
這里R2 和R3分別是曲柄和耦合器的長度，mB是滑塊的質量。
為了通過以位移為基礎的有限元方法比較誤差，我們同樣要用它建立一個機構，結果可以參考文獻[17]。
表2 兩種有限元方法的第一自然頻率誤差
元件數(shù)目
第一自然頻率
以位移為基礎的有限元方法
以應力為基礎的有限元方法
1
1.10E-1(DOF=2)
2
3.91E-3(DOF=4)
7.21E-3(DOF=1)
3
8.10E-4(DOF=6)
1.12E-3(DOF=2)
4
2.60E-4(DOF=8)
3.05E-4(DOF=3)
5
1.07E-4(DOF=10)
1.19E-4(DOF=4)
DOF：自由度數(shù)目
表3 兩種有限元方法的總能量誤差
元件數(shù)目
第一自然頻率
以位移為基礎的有限元方法
以應力為基礎的有限元方法
1
1.94E-2(DOF=2)
2
3.21E-3(DOF=4)
8.85E-4(DOF=1)
3
1.92E-3(DOF=6)
3.77E-4(DOF=2)
4
1.20E-3(DOF=8)
2.82E-4(DOF=3)
5
9.00E-4(DOF=10)
2.17E-4(DOF=4)
DOF：自由度數(shù)目
圖3顯示了總能量的時間響應，耦合器的量綱中點撓度，耦合器在穩(wěn)態(tài)條件下的中點應變。表2-5分別比較了以位移為基礎和以應力為基礎的有限元方法的第一自然頻率誤差、總能量、耦合器的中點撓度量綱、以及耦合器的中點應變。誤差可以由公式（16）得到。結果表明，當兩種方法中的元件數(shù)目相同時，以應力為基礎的方法誤差較以位移為基礎的誤差大。但是，當自由度的數(shù)目相同時，以應力為基礎的有限元方法的誤差比以位移為基礎的有限元方法的誤差小很多。同時，我們注意到當元件相同，除去第一自然頻率誤差時，以應力為基礎的有限元方法的誤差也比以位移為基礎的有限元方法的小很多。這說明以應力為基礎的有限元方法可以提供大量精確的解決動態(tài)彈塑性問題的方法。
表4 兩種有限元方法的耦合器中點撓度誤差
元件數(shù)目
第一自然頻率
以位移為基礎的有限元方法
以應力為基礎的有限元方法
1
3.60E-1(DOF=2)
2
5.27E-2(DOF=4)
1.29E-2(DOF=1)
3
3.26E-2(DOF=6)
8.41E-3(DOF=2)
4
2.14E-2(DOF=8)
6.12E-3(DOF=3)
5
1.57E-2(DOF=10)
4.90E-3(DOF=4)
DOF：自由度數(shù)目
表5 兩種有限元方法的耦合器中點應變誤差
元件數(shù)目
第一自然頻率
以位移為基礎的有限元方法
以應力為基礎的有限元方法
1
4.25E-1(DOF=2)
2
1.65E-1(DOF=4)
2.38E-1(DOF=1)
3
5.35E-2(DOF=6)
2.89E-2(DOF=2)
4
4.38E-2(DOF=8)
2.28E-2(DOF=3)
5
2.27E-2(DOF=10)
1.80E-2(DOF=4)
DOF：自由度數(shù)目
7.結論
本文提出了一種新的以應力為基礎的有限元方法來解決歐拉-拉格朗日梁問題。該方法尤其適用于解決動態(tài)彈塑性問題。并且提出了梁的近似曲率。然后我們可以通過整合近似曲率得到橫向撓度和應力分布。在整合過程中，有必要使梁單元的邊界條件得到滿足，從而可以得到整合常數(shù)。本文中，我們提出了在高速運作下解決靈活曲柄滑塊機構問題。結果表明，在同樣的自由度下，以應力為基礎的有限元方法的誤差小于常規(guī)方法的誤差，常規(guī)方法也就是以位移為基礎的有限元方法。同樣，在元件數(shù)目相同的條件下，以應力為基礎的有限元方法可以提供更多準確的解決方法。
參考文獻
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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 displace
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