作者：Chunhui Yang Shimin Luo國籍：China出處：2010 Intemational Conference on System,Engineering Drdign and Manufacturing InformatizationCircular are flexible hinge stiffness character analysisAbstract-Flexible hinges are widely used in micro robotic. Its tigidity directly influences an organizations terminal localization. Its actual structure geometry size cannot satisfy the theoretical analysis completely in a theoretical supposition condition. In this paper. We analayzed the rotation rigidity of a ellliptical flexible hinge in different parameters using finite element software ANSYS. The errors are discovered and compared with theoretical result. Though the graph of the flexible hinge parameters on the performance of a elliptical flexible hinge was carried out. The key manufacture parameters that affect the performance of an elliptical flexible hinge the most and rules of design are given, which can provide directions of design precision for the flexible hinge.Keywords-flexible hinge; elllipse; finite element analysisl;rigidity.INTRODUCTIONFlexible hinge have some characteristics, such as small volume, without tubs, ceaseless, good rigidity and high sensitivity. With micocomputer electrical system series (MEMS) technical rapidly expanding, flexible hinge are widely applied in the displacement which requests small angular and high-precision rotation, such as gyroscopes,accelerometers, percision instruments and so on. It has broad application prospects in the micron level domain.The common flexible hinge is in two kinds:beam-shape flexible hinge and arc-shaped flexible hinge. The beam-shaped flexible hinge has a big slewing area, but the movement precision is bad. The arc-shaped flexible hinges movement percision is bad. The arc-shaped flexible hinge is relatively small. In order to take into account the movement ptecision and scope, the following several rotation flexible hinges have been generated: parabolic flexure hinge, an elliptical flexure hinge and a hyperbola-shaped hinge, etc. The properties of flexible hinges are rigidity precision and stress characteristic etc. the rigidity performance reflects the stress ability and also manifests movement to a vice-flexible degree. In 1965, Parosetal announced his design development of the circular flexible hinge for the first time, and gave the rigidity formula.Smithetal used the similar method to obtain an elliptic flexible hinge mechanics expression. Nicolae Lodonitu inferred the parabola and the hyperbolic flexible hinges rigidity formula. Wei Xu and Tim king analyzed the tectangular and elllipse flexible hinges rigidity and rotarion precision using the finite element method.In this paper the elliptical flexible hinge stiffness to different geometrical parameters is analyzed with software ANSYs10.0. Compared with ressults of theoretical analysis and finite element analysis(FEA), the errors are analyzed. Theough the graph of the flexible hinge parameters and its performanve, an analysis of changes of parameters on the performance of the elliptical fexible hinge was carried out. The key manufacture parameters that affect the performances of an elliptical flexible hinge the most and rules of design are given, which can guve direcrions of design precision for the flexible hinge.RIGIDITY FORMULA OF THE ELLIPTICAL FLEXIBLE HINGE An elliptical flexible hing,as shown in Figure 1, is a particular type of flexure that consists of a neckde down section. Parameters t,h,b are flexible hinges smallest thickness,height and width ,resoectively, Parameter is the semimajor axis of ellipse, and is the semimajor axis of elllipse.As shown in Figure 1(a), the infinitesimal is intercepted in the abscissa axis. To begin, the infinitesimal section is vertical to the abscissa axis. The flexible hinges angular deformation is generated under torque as given in Equation(1).（1）（2）WhereThe rotation rigidity fotmula is given by Euation(3)(3)When , the flexible hinge is the circular flexible hinge. Its rotantion rigidity formula is given by Equatuin(4)(4)The rorarion rigidity formula of circular flexible hinge is consistent with the reference2.(a)(b)Figure 1 Model of elliptical flexible hinge . FEA MODEL OF ELLIPTOCAL FLEXIBLE HINGE Ansys has some characteristics that the general finite element analysis technologt, powerful computing, and reliable result. The elliptical flexible hinges basic srtuture size is . The materical is bertllium copper alloy, ,. The FEA model is shown in Figure2(a). The model left end surface is restrained completely, the right end surface exerts bending moment . The special node 1 of the grid model right end surface represents the output displacement. The unit type chooses 3_D the entity SOLID92 unit model, the entire model usrs Smartsize to free mesh. The deformation of the FEA model is shown in Figure 2(b). The FEA model and the output displacement have been obtained with changing elliptical flexible hinges paramters E,b,t and as well as (a)(b)Figure 2 . FEA model of elliptical flexure hingeThe theoretical calculation and FEA rotational stiffness are obtaoned through changing elliptical flexure hinges parameters E,b,t and as well as , as shown in Figure 3 to 7.Figure 3. Comparison of FEA value and the theoretical value of flexure hinge with changing EFigure 3. Comparison of FEA value and the theoretical value of flexure hinge with changing width bFigure 5. Comparison of FEA value and the theoretical value of flexure hinge with chinging thickness tFigure 6. Comparison of FEA value and the theoretical value of flexure hinge with chinging semimajor axis Figure 7. Comparison of FEA value and the theoretical value of flexure hinge with chinging semimajor axis From Equation (3) and Figures 3 to 7 the following conclusions can be observed.(1) From Figures 3and 4 it can be observed that the rotation stiffness is a linearly increasing realtion with material youngs modulus E and width b. The FEA value is bigger than the theoretical value. When E and b are smallerm the FEA value and the theoretical value are closer .(2) From Figure 5 it can observed that the rotantion of the stiffness FEA value and the theoretical value is a vurve increasing with thickness t, end the speed-up is getting quicker and quicker. When t is bigger, the difference of the theoretical value and the FEA value is bigger. When t<2mm, the FEA value and the theorerical value are close.(3) From Figure 6 it can observed that the rotantion of rotation rigidity and semimajor axis is a decreasing curve, and the rate of reduced scope is gradually decreased. When >3mm, the FEA value and the theoretical value are closer.(4) From Figure 7 it can observed that the rotantion of rotation ridity and semiminor axis is a linearly increasing, but the increasing scope is small. The FEA value is bigger than the theoretical value.From Figure 3 to 7 it can observed that the influence of flexure hinges parameter to its rotation stiffness is:the influence of thickness t is biggest, followed by semimajor axis , semimajor axis width b and E.The theoretical value and the FEA value of an elliptical flexible hinge rotation rigidity is not equal, even if has a big differential value. The reasons are:(1) The flexible hinge theoretical model that is eastablished using materials mechanics bending strain theory is built on the basis of certain assumptions.From Figure 2(b) it can be observed that the FEA model not only has the displancement in the y axis direction, but also has the displacement change on the z axis direcrion, when the torque exerted on the z axis for the model. In other words, the flexible hinge not only has the bending strain,but also will have the shearing force to cause upward deformation. Under certain design parameters, the flexible hinges theory and FEA solution achieve a good match.CONCLUSIONSThe different design parameters to the flexible hinge rotation rigidity influence and the linear relationship are obtained by comparing the rotation rigidity theory solution stiffness is:the influence of thickness is biggest,followed by semimajor axis R,width b and E. The teasons that the theoretical value and the FEA value of an elliptical fexible hinge not only has the bending strain,but also will have the shearing force to cause upward deformation. It is helpful to further analyze the movement of the mechanical deformation mechanism, using the finite element technology to simulate the flexible hinge performance.圓弧型柔性鉸鏈剛度特性分析摘 要： 柔性鉸鏈?zhǔn)悄壳氨粡V泛用于微動(dòng)機(jī)器人的主要部件之一，其剛度性能直接影響微動(dòng)機(jī)器人的終端定位。 由于實(shí)際需要的多樣性和復(fù)雜性，使得其實(shí)際結(jié)構(gòu)的幾何尺寸不能完全滿足傳統(tǒng)理論分析的假設(shè)條件，因此影響到對(duì)其性能的準(zhǔn)確分析。利用有限元軟件ANSYS對(duì)雙邊圓弧形柔性鉸鏈和單邊圓弧形柔性鉸鏈在不同結(jié)構(gòu)參數(shù)下的轉(zhuǎn)動(dòng)剛度進(jìn)行分析，并與其解析計(jì)算結(jié)果進(jìn)行了比較，分析了其間產(chǎn)生誤差的原因。為MEMS中柔性鉸鏈的設(shè)計(jì)與應(yīng)用提供一定的依據(jù)。關(guān)鍵詞: 雙邊圓弧形柔性鉸鏈，單邊圓弧形柔性鉸鏈，有限元分析，剛度引言隨著微機(jī)電系統(tǒng)（MEMS）技術(shù)的迅速發(fā)展，柔性鉸鏈因具有體積小、無機(jī)械摩擦、無間隙、剛性好和高靈敏度的特點(diǎn)，柔性鉸鏈被廣泛地應(yīng)用于各種要求小角位移、高精度轉(zhuǎn)動(dòng)的場(chǎng)合，如陀螺儀、加速度計(jì)、精密天平等儀器儀表。柔性鉸鏈在微米級(jí)領(lǐng)域內(nèi)有著廣闊的應(yīng)用前景。常見的柔性鉸鏈有兩種：直梁形柔性鉸鏈和圓弧形柔性鉸鏈。直梁形柔性鉸鏈有較大的轉(zhuǎn)動(dòng)范圍，但運(yùn)動(dòng)精度較差；而圓弧形柔性鉸鏈的運(yùn)動(dòng)精度較高，但轉(zhuǎn)動(dòng)范圍相對(duì)小。為了兼顧運(yùn)動(dòng)精度和運(yùn)動(dòng)范圍，又衍生出下面幾種轉(zhuǎn)動(dòng)柔性鉸鏈：拋物線形柔性鉸鏈、橢圓形柔性鉸鏈、雙曲線形柔性鉸鏈等。柔性鉸鏈的基本性能主要包括剛度、精度及應(yīng)力特性等幾方面，其中剛度性能直接反映了柔性鉸鏈抵抗外載的能力，也體現(xiàn)了運(yùn)動(dòng)副的柔性程度，是主要的研究因素。1965年，Paros 等人首次公布其設(shè)計(jì)開發(fā)的圓形柔性鉸鏈，并給出了剛度計(jì)算公式。本文應(yīng)用有限元軟件ANSYS10.0 對(duì)雙邊圓弧形柔性鉸鏈和單邊圓弧形柔性鉸鏈在不同結(jié)構(gòu)參數(shù)下的轉(zhuǎn)動(dòng)剛度進(jìn)行分析，并與其解析計(jì)算結(jié)果進(jìn)行了比較， 分析了其間產(chǎn)生誤差的原因。通過建立的雙邊、單邊圓弧形柔性鉸鏈參數(shù)與其剛度性能關(guān)系圖，分析了圓弧形柔性鉸鏈各參數(shù)變化對(duì)其剛度性能的影響；通過分析比較雙邊、單邊圓弧形鉸鏈的剛度特性，結(jié)果顯示單邊圓弧形柔性鉸鏈具有更大的轉(zhuǎn)動(dòng)能力，但其對(duì)軸向載荷的影響更為敏感，設(shè)計(jì)的單邊圓弧形柔性鉸鏈更適用于結(jié)構(gòu)緊湊、需要大位移應(yīng)用場(chǎng)合。給出了對(duì)柔性鉸鏈影響最大的關(guān)鍵加工參數(shù)，為圓弧形鉸鏈的選擇和工程設(shè)計(jì)提供了理論依據(jù)。圓弧形柔性鉸鏈剛度計(jì)算公式雙邊圓弧形柔性鉸鏈的主視圖如圖1（a）所示，其橫截面為矩形。立體圖如圖1（b）所示。t、h、b 分別為圓弧型柔性鉸的最小厚度、高度和寬度，R 為圓弧的圓半徑，為圓弧的圓心角。如圖1（a）所示，在橫軸上截取微元，在受力作用前微元截面垂直于橫軸。微元高度為a，長度為 ，寬度等于圓弧的最小寬度b，在力矩Mz 的作用下柔性鉸鏈的角變形為（1）（2）式中：圓弧型鉸鏈的轉(zhuǎn)動(dòng)剛度計(jì)算公式為：（3）當(dāng)（4）（a）(b)圖1 雙邊圓弧型柔性鉸鏈模型圓弧型柔性鉸鏈的有限元模型ANSYS 是一種大型通用的有限元分析技術(shù)，計(jì)算功能強(qiáng)大，計(jì)算結(jié)果可靠的軟件。圓弧型柔性鉸鏈的基本結(jié)構(gòu)尺寸為：；材料選用鈹銅合金，其楊氏模量，泊松比，材料特性為各向同性。雙邊圓弧形柔性鉸鏈有限元模型如圖2（a）所示，對(duì)該模型左端面進(jìn)行全約束，網(wǎng)格模型右端面的自由端為載荷的施加位置，施加彎矩；網(wǎng)格模型右端面的特殊節(jié)點(diǎn)1 為節(jié)點(diǎn)位移代表輸出位移；單元類型選擇3-D 實(shí)體SOLID92 單元模型，對(duì)整個(gè)模型使用Smartsize進(jìn)行自由網(wǎng)格劃分。單邊圓弧形柔性鉸鏈的有限元模型如圖2（b）所示。通過分別改變圓弧型柔性鉸鏈的結(jié)構(gòu)尺寸參數(shù)b、t、以及R，得到各類結(jié)構(gòu)參數(shù)改變的有限元模型。（a）（b）圖2 有限元模型剛度性能分析改變圓弧形柔性鉸鏈的結(jié)構(gòu)尺寸參數(shù)b、t、以及R，分別計(jì)算出在不同結(jié)構(gòu)參數(shù)下圓弧型柔性鉸鏈的理論計(jì)算轉(zhuǎn)動(dòng)剛度和有限元分析結(jié)果轉(zhuǎn)動(dòng)剛度，如圖3 7所示。圖3 改變材料的楊氏模量E的圓弧型柔性鉸鏈轉(zhuǎn)動(dòng)剛度有限元值與理論值比較圖4 改變寬度b的柔性鉸鏈轉(zhuǎn)動(dòng)剛度有限元值與理論值比較圖5 改變厚度t的圓弧型柔性鉸鏈轉(zhuǎn)動(dòng)剛度有限元值與理論值比較圖6 改變圓弧半徑的圓弧型柔性鉸鏈轉(zhuǎn)動(dòng)剛度有限元值與理論值比較圖7改變圓弧半徑的圓弧型柔性鉸鏈轉(zhuǎn)動(dòng)剛度有限元值與理論值比較由式（3）、（4）和圖3-7分析得：(1) 由圖3、4 可見，轉(zhuǎn)動(dòng)剛度與楊氏模量E、寬度b 呈線性遞增關(guān)系，且理論值比有限元值大，楊氏模量E 和寬度b 越小，有限元值與理論值越接近。(2) 由圖5 可知，轉(zhuǎn)動(dòng)剛度的有限元值和理論值都與厚度t 呈曲線遞增關(guān)系，且增速越來越快。當(dāng)t 越大，有限元值與理論值相差越大；當(dāng)t 2mm，有限元值與理論值越接近。(3）由圖6-7 可知，轉(zhuǎn)動(dòng)剛度的有限元和理論值都與半徑R 呈曲線遞減關(guān)系，減幅較??；轉(zhuǎn)動(dòng)剛度的有限元值與理論值接近。(4）分析圖3 - 7可知，雙邊圓弧形鉸鏈的剛度比單邊圓弧形鉸鏈的剛度大。分析圖3-7可得圓弧型柔性鉸鏈各設(shè)計(jì)參數(shù)對(duì)其轉(zhuǎn)動(dòng)剛度的影響程度依次為：厚度t 影響最大，其次為圓弧半徑R，再次為圓心角，最后為楊氏模量E、寬度b。圓弧型柔性鉸鏈的轉(zhuǎn)動(dòng)剛度理論值與有限元值不相等，甚至有較大差值的可能原因是：（1）利用材料力學(xué)的梁彎曲理論建立的柔性鉸鏈理論模型，是建立在一定假設(shè)條件基礎(chǔ)之上。（2）從有限元模型圖3可見，當(dāng)給模型自由端施加Z 軸的轉(zhuǎn)矩MZ 時(shí)，有限元模型不僅在Y 軸方向產(chǎn)生位移，同時(shí)在Z 軸上也有位移變化。也就是說，柔性鉸鏈不僅只發(fā)生彎曲變形，同時(shí)有剪力將引起柔性鉸鏈截面曲翹變形。在一定的結(jié)構(gòu)參數(shù)下，柔性鉸鏈的理論解與有限元分析吻合的很好。 結(jié)論通過分析比較雙邊、單邊圓弧形柔性鉸鏈的轉(zhuǎn)動(dòng)剛度理論解與有限元分析，得出不同結(jié)構(gòu)參數(shù)對(duì)柔性鉸鏈轉(zhuǎn)動(dòng)剛度的影響程度及線性關(guān)系。其中圓弧型柔性鉸鏈各設(shè)計(jì)參數(shù)對(duì)其轉(zhuǎn)動(dòng)剛度的影響程度依次為：厚度t 影響最大，其次為圓弧半徑R，最后為楊氏模量E、寬度b。并闡述了圓弧型柔性鉸鏈的轉(zhuǎn)動(dòng)剛度理論值與有限元值不相等的主要原因是：柔性鉸鏈不僅只發(fā)生彎曲變形，同時(shí)有剪力將引起柔性鉸鏈截面曲翹變形。通過分析比較雙邊、單邊圓弧形鉸鏈的剛度特性，結(jié)果顯示單邊圓弧形柔性鉸鏈具有更大的轉(zhuǎn)動(dòng)能力，但其對(duì)軸向載荷的影響更為敏感，設(shè)計(jì)的單邊圓弧形柔性鉸鏈更適用于結(jié)構(gòu)緊湊、需要大位移應(yīng)用場(chǎng)合。為圓弧形鉸鏈的選擇和工程設(shè)計(jì)提供了理論依據(jù)。