關(guān)節(jié)軸承搖擺試驗機設(shè)計

關(guān)節(jié)軸承搖擺試驗機設(shè)計,關(guān)節(jié)軸承搖擺試驗機設(shè)計,關(guān)節(jié)軸承,搖擺,扭捏,搖曳,試驗,實驗,設(shè)計 關(guān)節(jié)軸承搖擺試驗機設(shè)計說明書 摘 要 關(guān)節(jié)軸承作為通用機械零件能滿足重載荷、長壽命要求，且具有轉(zhuǎn)動靈活、少維護、結(jié)構(gòu)緊湊、易于裝拆，在工作過程中可以免維修和無需添加潤滑劑等優(yōu)異特性，廣泛應(yīng)用于工程機械、載重汽車、水利設(shè)施、軍工機械等方面。關(guān)節(jié)軸承的主要失效形式是磨損，磨損使軸承內(nèi)部的游隙明顯增大，從而引起軸承支承部位的振動和噪聲增加，使機械的運行狀態(tài)變差，導(dǎo)致軸承不能正常工作。因此研究關(guān)節(jié)軸承的摩擦磨損壽命性能是一項基礎(chǔ)而又重要的工作。 關(guān)節(jié)軸承搖擺試驗機是針對關(guān)節(jié)軸承力學(xué)性能試驗而開發(fā)的特殊試驗機，該試驗機采用成熟的偏心輪擺動技術(shù)，絲杠加載等技術(shù)，大大提高了系統(tǒng)的穩(wěn)定性和可靠性。 關(guān)節(jié)軸承搖擺試驗機模擬關(guān)節(jié)軸承實際載荷譜、轉(zhuǎn)速譜工況進行試驗并考核其各種性能。對試樣采用“兩端支撐式”結(jié)構(gòu)，通過對旋轉(zhuǎn)軸承施加靜態(tài)的徑向力，試驗軸承所受到的扭矩的疲勞壽命。旋轉(zhuǎn)和徑向力的加載均采用公司成熟的技術(shù)，能夠自動記錄并處理試驗數(shù)據(jù)，在試驗中出現(xiàn)故障能自動判別并停機。該機主要由主機、控制系統(tǒng)、夾具和其他必要的附件等組成。 ABSTRACT The design of hydraulic drive manipulator movements under theprovisions of the order ， use the basic theory, basic knowledge and related mechanical design expertise comprehensively to complete the design，and drawing the necessary assembly, hydraulic system map, PLC control system diagram . Manipulator mechanical structure using tanks, screw ,guide tubes and other mechanical device component ；In the hydraulic drive bodies ，manipulator arm stretching using telescopic tank ，rotating column of tanks used rack ，manipulator movements using tank movements ，the column takes the horizontal movement of tanks This text introduce upper and lower material spend design process of manipulator , it include to whole job requirements and analysis of situation of system mainly, confirm the whole structural design systematic in hydraulic pressure through the working course of the system. Analyse whole cyclical process , confirm systematic operation principle picture , require the hydraulic pressure component of the standard for selection according to systematic parameter, finish the installation diagram systematic in hydraulic pressure. Hydraulic pressure integrate piece as now main part , hydraulic pressure of system , hydraulic pressure at present integrate pieces of application and development receive domestic and international hydraulic pressure extensive attention of circle, hydraulic pressure integrate research and development of CAD of piece already offer effective support for engineering design of the hydraulic pressure, while design the system integration one of hydraulic pressure of the manipulator, can combine with real processing technology . And CAD integrating one to the present hydraulic pressure has very good understanding. Keywords: manipulator ,Drive , Hydraulic manifold block , Elements 目錄 一、前言-----------------------------------------------------------3 1.1關(guān)節(jié)軸承的研究與發(fā)展----------------------------------------3 1.2 設(shè)計意義----------------------------------------------------4 二、產(chǎn)品設(shè)計過程----------------------------------------------------5 2.1徑向加載部分電機的設(shè)計計算--------------------------------5 2.2軸動加載部分電機的設(shè)計計算-------------------------------6 2.3扭矩傳感器的設(shè)計------------------------------------------7 三、試驗機強度的校核-----------------------------------------8 3.1軸承強度的校核--------------------------------------9 3.2聯(lián)軸器強度的校核-----------------------------------10 四、關(guān)節(jié)軸承搖擺試驗機主機的三維建模-------------------------16 4.1 徑向加載部分的三維建模------------------------------------17 4.2 軸向加載部分的三維建模-----------------------------------18 4.3 夾具部分的三維建模---------------------------------------19 四、設(shè)計總結(jié)--------------------------------------------------21 參考文獻---------------------------------------------------22 致謝---------------------------------------------------------------23 前言 關(guān)節(jié)軸承的旋轉(zhuǎn)擺動因幅度大而不同于微動，也不同于普通的單向直線和旋轉(zhuǎn)運動，因此關(guān)節(jié)軸承在擺動運動下的摩擦磨損性能研究逐漸受到人們的關(guān)注。 目前評價關(guān)節(jié)軸承摩擦學(xué)性能的試驗機主要有2種：一種是以軸承為試驗對象，把主軸設(shè)計成圓周方向旋轉(zhuǎn)擺動，進行軸承旋轉(zhuǎn)摩擦試驗；另一種則是以軸承材料的標(biāo)準(zhǔn)試塊為試驗對象的直線往復(fù)式摩擦磨損試驗機［5］。文獻［4，－8］研制了關(guān)節(jié)軸承隨主軸做圓周方向旋轉(zhuǎn)擺動的試驗機，并能對一定尺寸的關(guān)節(jié)軸承做磨損試驗。文獻［4］研制了大中型關(guān)節(jié)軸承磨損壽命試驗機，該試驗機加載系統(tǒng)采用液壓加載；擺動系統(tǒng)采用液壓推動齒條，帶動主軸齒輪擺動；測量系統(tǒng)采用測長傳感器測量徑向磨位移量，采用鉑電阻通過顯示儀表監(jiān)測軸承外圈端面溫度。該試驗機的參數(shù)為：被試關(guān)節(jié)軸承內(nèi)徑50～150mm，擺動角度+40°～－40°，擺動頻率0～15次/min，最大加載值1000kN。向心關(guān)節(jié)軸承的運動形式一般是擺動，所受的載荷主要是徑向載荷。文獻［6］按關(guān)節(jié)軸承在徑向載荷作用下做擺動運動來設(shè)計，研制了新型關(guān)節(jié)軸承壽命試驗機，如圖1所示。該試驗機以整個關(guān)節(jié)軸承為試驗對象，把關(guān)節(jié)軸承安裝在主軸上，電動機通過皮帶帶動減速器，再由曲柄、連桿和搖桿機構(gòu)實現(xiàn)主軸的擺動運動，并利用螺紋的自鎖性實現(xiàn)無級加載。通過測量軸承外圈的徑向位移和摩擦系數(shù)判斷軸承是否失效，并監(jiān)測軸承外圈的溫度以了解軸承內(nèi)部摩擦力的變化情況。該試驗機的技術(shù)參數(shù)為：被試關(guān)節(jié)軸承內(nèi)徑為15～50mm，徑向載荷為0～98kN，擺動頻率為02．5Hz，擺動角度為20°～100°。圖1新型關(guān)節(jié)軸承壽命試驗機工作原理圖文獻［7］研制了E06－12型自潤滑桿端關(guān)節(jié)軸承試驗機，該試驗機采用曲柄搖桿機構(gòu)驅(qū)動主軸擺動，主軸帶動裝配在主軸一側(cè)的2個試驗頭在一定的空間范圍內(nèi)同步擺動。試驗軸承采用懸掛式，杠桿加載裝置施加的力通過軸承殼體螺紋部分傳遞到軸承內(nèi)圈，以使軸承試驗條件與實際使用狀況相吻合的方法進行磨損試驗。該試驗機的技術(shù)參數(shù)為：試驗軸承的內(nèi)徑尺寸為6～12mm，單套試驗軸承的最大載荷為9．8kN，擺頻20～40次/min，擺幅0°～±40°。文獻［8］研制了可滿足各種載荷的大、中、小關(guān)節(jié)軸承測試需求的磨損試驗機。該軸承磨損試驗機主要由主機、液壓源、測量控制系統(tǒng)3部分組成，軸承徑向載荷采用伺服油缸加載。利用電液伺服閉環(huán)控制原理，進行載荷和位移控制，扭轉(zhuǎn)部分通過伺服電動機帶動擺線針輪減速機輸出扭矩，由扭矩傳感器、光電編碼器分別測量扭矩、扭角，并組成閉環(huán)系統(tǒng)進行控制。磨損測量采用差動壓器式位移傳感器，測量試驗過程中外圈相對于內(nèi)圈的徑向位移。計算機可實時采集試驗數(shù)據(jù)繪制曲線，并可存儲試驗數(shù)據(jù)進行后處理。該試驗機的參數(shù)為：徑向載荷最大試驗力分別為00，300和100kN，擺動角度±25°，擺動頻率為1～15次/min。與關(guān)節(jié)軸承繞單軸做圓周方向旋轉(zhuǎn)擺不同，文獻［9］研制了直升機自動傾斜器球鉸自潤滑關(guān)節(jié)軸承試驗機，該試驗機采用2套空間四桿機構(gòu)實現(xiàn)了關(guān)節(jié)軸承在2個坐標(biāo)軸方向的任意擺動；通過1套凸輪機構(gòu)實現(xiàn)主軸的上、下往復(fù)運動。以運動分解的方式獲得了關(guān)節(jié)軸承3個自由度的運動目標(biāo)，采用可編程控制器及觸摸屏實時控制關(guān)節(jié)軸承在試驗過程中的擺動速度、頻率和載荷的大小，可以監(jiān)測、顯示襯墊磨損量、摩擦力、溫度和振動信號。該試驗機的參數(shù)為：最大載荷8kN，最大頻率3Hz，擺動角度±9°。此外，還有學(xué)者通過研制直線往復(fù)式摩擦磨損試驗機來研究自潤滑材料的性能，如文獻［10］中的試驗機能夠通過六桿機構(gòu)實現(xiàn)直線往復(fù)運動，并用液壓機實現(xiàn)加載對自潤滑材料進行試驗，從而得出其摩擦、磨損性能。這為研制關(guān)節(jié)軸承試驗機提供了參考。2關(guān)軸承磨損性能的影響因素關(guān)節(jié)軸承磨損性能受到載荷、溫度、軸承摩擦副材料及自潤滑材料等因素的影響。2．1載荷文獻［1113］研究了關(guān)節(jié)軸承受重載情況下的磨損性能，為研究關(guān)節(jié)軸承磨損性能提供了重要參考。文獻［11］采用上海市軸承技術(shù)研究所研制的ZMS1500磨損試驗機對織物類襯墊鋁合金自潤滑關(guān)節(jié)軸承進行磨損試驗，該試驗對內(nèi)徑尺寸為110和130mm的關(guān)節(jié)軸承進行了擺角為25°、擺頻為10次/min、載荷為200～600kN的擺動磨損試驗。試驗中采用測長法測量磨損量，即在試件內(nèi)、外套圈相對擺動后，以徑向尺寸發(fā)生的變化測定磨損量，并采用徑向位移量與電測量相互轉(zhuǎn)換的方法，對磨損量進行連續(xù)測量。試驗結(jié)果表明：凡是內(nèi)、外球面滾道接觸良好者，其磨損性能優(yōu)良；若磨痕區(qū)域呈黑褐色狹長狀，則其磨損性能較差。所謂內(nèi)、外球面滾道接觸良好，即吻合度好， 接觸面積相對較大，因此，在相同載荷條件下單位面上的壓強小，磨損均勻，故其磨損性能較好，反之磨損性能則差。文獻［12］研究了PTFE編織復(fù)合材料重載關(guān)節(jié)軸承的旋轉(zhuǎn)摩擦特性，采用自制的重載摩擦試驗機，在轉(zhuǎn)速為2．5r/min時測試了PTFE編織復(fù)合材料關(guān)節(jié)軸承承載能力與摩擦系數(shù)的關(guān)系，以及摩擦的時間效應(yīng)。在承載為135MPa時，測試了軸承的磨損曲線、軸承的溫升以及摩擦系數(shù)隨連續(xù)擺動時間變化的關(guān)系曲線。試驗結(jié)果表明，隨著承載由25MPa增加到135MPa，干摩擦系數(shù)由0．061降低到0．038。通過掃描電鏡分析了軸承失效機理，在擺動的過程中PTFE不斷被擠出，軸承自潤滑功能下降，導(dǎo)致編織基體材料發(fā)生磨損。文獻［13］利用自制的高頻重載關(guān)節(jié)軸承摩擦磨損試驗機，研究了不同擺動頻率和接觸壓強條件下PTFE/銅網(wǎng)復(fù)合材料襯關(guān)節(jié)軸承摩擦系數(shù)、線磨損量和摩擦溫度變化規(guī)律。借助掃描電子顯微鏡（SEM）和能譜分析儀（EDS）分析了關(guān)節(jié)軸承襯墊的磨損表面并探討了其磨損機理。結(jié)果表明：在接觸應(yīng)力≤47．6MPa時，擺動頻率的升高對軸承的摩擦系數(shù)、磨損量和摩擦溫度影響較小；在接觸應(yīng)力＞47．6MPa，隨著擺動頻率的變化，軸承的摩擦系數(shù)、磨損量波動范圍較大，摩擦溫度上升較快。由圖2a可以看出，襯墊表層保存完好，基體材料銅尚未顯露出，襯墊材料幾乎沒有遭到破壞，自潤滑性能依然良好；隨著擺動頻率升高．8Hz，摩擦過程產(chǎn)生的熱應(yīng)力以及摩擦面上的剪切拉引起聚合物表面龜裂，并且發(fā)生塑性變形，甚至出現(xiàn)了剝落說明此狀態(tài)下襯墊材料發(fā)生剝落磨損（圖2b）。當(dāng)接觸應(yīng)力為95．2MPa，擺動頻率為1．2Hz時，軸承襯墊表層材料磨損較為嚴(yán)重，襯墊表層材料已幾乎被磨完，襯墊基體材 料也遭到了破壞，剝落面積增大（圖2c）；當(dāng)擺動頻率為8Hz時，襯墊表層材料已經(jīng)磨完，表面有大量的附著顆粒（圖2d）。載荷發(fā)生變化時會影響自潤滑關(guān)節(jié)軸承的力學(xué)性能。在重載條件下，軸承內(nèi)外圈擺動將在內(nèi)、外圈接觸面產(chǎn)生摩擦熱，襯墊材料發(fā)生塑性變形、擠壓變形和剝落，致關(guān)節(jié)軸承自潤滑能力下降，進而發(fā)生粘著磨損和磨粒磨損，導(dǎo)致軸承失效。圖2PTFE編織物磨損表面形貌SEM照片［3］2．2溫度當(dāng)關(guān)節(jié)軸承應(yīng)用于航空航天等領(lǐng)域時，溫度對自潤滑材料摩擦磨損特性的影響尤為明顯，因而溫度對關(guān)節(jié)軸承性能的響也受到了廣泛的關(guān)注。文獻［14］為研究低溫環(huán)境下PTFE的摩擦磨損屬性，進行了環(huán)境溫度為～77K熱力學(xué)溫度的盤銷磨損試驗，果表明，TFE在低溫環(huán)境下的摩擦性能提高。這是由于在低溫環(huán)境下PT-FE的硬度和機械強度提高。文獻［15］研究了速度、載荷、溫度對35%填充TFE復(fù)合材料摩擦磨損特性的響。試驗結(jié)果表明，通過使軸承冷卻，載荷和速度對磨損的影響減少。因此，可以通過降低關(guān)節(jié)軸承的工作溫度來提高PTFE的磨損性能，進而提高關(guān)節(jié)軸承的磨損性能。2．3軸承材料和自潤滑材料軸承材料的選擇直接影響著關(guān)節(jié)軸承的使用性能和壽命，因此，為了得到關(guān)節(jié)軸承優(yōu)良的使用性能，選擇合理的軸承材料非常重要。文獻［16］為了避免軸承鋼關(guān)節(jié)軸承發(fā)生銹蝕卡死，采用洛陽軸承研究所研制的能同時監(jiān)視軸承的摩擦、溫度以及磨損情況的SPBTM－Ⅱ型關(guān)節(jié)軸承試驗機對UG20和UC20X進行了摩擦性能試驗對比分析。試驗結(jié)果表明：不銹鋼關(guān)節(jié)軸承UC20X的壽命高于軸承鋼關(guān)節(jié)軸承UG20，這是因為不銹鋼材料的自適應(yīng)性和耐磨性優(yōu)于軸承鋼的緣故。自潤滑材料的性能直接影響關(guān)節(jié)軸承的磨損性能和使用壽命。陶瓷基復(fù)合材料是以陶瓷作為黏接劑的自潤滑材料，這類材料具有高硬度、高強度、高剛度、低密度和優(yōu)異的化學(xué)穩(wěn)定特性，也具有良好的減摩耐磨特性。文獻［17］指出，CaF2在高溫下由脆性向塑性轉(zhuǎn)變而具有潤滑性。在摩擦過程中，CaF2中的氟與磨損表面所起的化學(xué)作用也是具有潤滑性的重要原因。但是材料的力學(xué)、摩擦學(xué)性能并不一定隨固體潤滑劑含量的增加呈線性變化。文獻［18］研究了通過熱壓成形工藝制的Al2O3TiC/CaF2自潤滑陶瓷材料在室溫下的摩擦磨損性能，結(jié)果表明：當(dāng)CaF2含量為10%時，該材料具有較好的力學(xué)性能，其摩擦系數(shù)隨aF2含量、載荷和速度的增加而降低。Al23TiC/CaF2材料在高速摩擦條件下能夠在磨損表面形成一層固體潤滑膜，正是由于這層膜的存在使得其在高速、高載荷下具有較低的摩擦系數(shù)；而低速下其磨損機理主要是磨粒磨損。 二、產(chǎn)品設(shè)計過程 2.1徑向加載部分電機的設(shè)計計算 由于徑向加載部分的電機需要可調(diào)速的，則從電機的運動路線為定比傳動，其總的傳動比可利用自身攜帶的減速器來得到。 電機的轉(zhuǎn)速不易太高，因為絲桿的移動能力并不是隨轉(zhuǎn)速增加而增加。當(dāng)速度達到一定值以后，效率反而下降，因此電機的轉(zhuǎn)速一般在200一400r/min比較適宜。在本機選用326r/min。 由傳動比標(biāo)準(zhǔn)系列查B2表2－1 初步取1.76 2.5 根據(jù)選用的電機和絞籠轉(zhuǎn)速要求設(shè)計傳動路線如下： 2.1電機的選擇 N=＝4(KW) G－絲桿的轉(zhuǎn)動慣量，1000kg/h W－切割1kg物料耗用能量，其值與孔眼直徑有關(guān)，d小則w大，當(dāng)d＝3mm， 取w＝0.0030kw.h/kg。（查B5p） －傳動效率，取0.75 所以根據(jù)N＝4kw，n＝1500r/min，查B1表10-4-1選用Y112M-4，再查B1表10-4-2得Y112M-4電機的結(jié)構(gòu)。 圖4-1 Y112M-4電動機的外觀圖 2.2軸向加載部分電機的設(shè)計計算 由于軸向加載部分的電機需要可調(diào)速的，則從電機的運動路線為定比傳動，其總的傳動比可利用自身攜帶的減速器來得到。 電機的轉(zhuǎn)速不易太高，因為絲桿的移動能力并不是隨轉(zhuǎn)速增加而增加。當(dāng)速度達到一定值以后，效率反而下降，因此電機的轉(zhuǎn)速一般在200一400r/min比較適宜。在本機選用326r/min。 由傳動比標(biāo)準(zhǔn)系列查B2表2－1 初步取1.76 2.5 根據(jù)選用的電機和絞籠轉(zhuǎn)速要求設(shè)計傳動路線如下： 2.2．1電機的選擇 N=＝4(KW) G－絲桿的轉(zhuǎn)動慣量，1000kg/h W－切割1kg物料耗用能量，其值與孔眼直徑有關(guān)，d小則w大，當(dāng)d＝3mm， 取w＝0.0030kw.h/kg。（查B5p） －傳動效率，取0.75 所以根據(jù)N＝4kw，n＝1500r/min，查B1表10-4-1選用Y112M-4，再查B1表10-4-2得Y112M-4電機的結(jié)構(gòu)。 2.3扭力傳感器的選型計算 彈簧機構(gòu)扭矩測量傳感器選型 主要查找了力傳感器和扭矩傳感器力傳感器有國內(nèi)和國外兩種傳感器，均為壓電式傳感器，尺寸和安裝方面比較合適（后面1，2）；扭矩傳感器都比較大，沒有找到合適安裝的傳感器（后面3，4）力測量方法接測量直接測量力需要將結(jié)構(gòu)沿垂直力傳遞垂直分成兩部分, 以便安裝已校準(zhǔn)的力傳感器。這樣會給試驗結(jié)構(gòu)帶來一定的影響, 安裝力傳感器必須滿足試驗結(jié)構(gòu)的強度和剛度要求。傳感器測量范圍必須大于被測量的過程力。這種安裝方法的一個主要優(yōu)點是不需考慮力作用點, 總是可以準(zhǔn)確、線性完美地測量力。應(yīng)用:試驗室單分量與多分量力測量, 微小力測量, 絕對力測量適用的傳感器類型:已校準(zhǔn)的單分量或多分量力傳感器。奇石樂力測量技術(shù)主要用于生產(chǎn)、車輛工程和生物力學(xué)。我們的實踐經(jīng)驗、試驗臺以及與客戶的緊密聯(lián)系使得我們成為全球這一高要求的市場中理想的合作伙伴。除了傳統(tǒng)應(yīng)用領(lǐng)域, 奇石樂還在眾多不同的挑戰(zhàn)性的測力應(yīng)用領(lǐng)域發(fā)揮著重要作用。無論很大的力還是微小力, 單向力或是多向力, 只要能夠在不影響部件性能的前提下安裝力傳感器, 就可以使用奇石樂力傳感器測量。壓電式測量是唯一滿足高固有頻率要求的測量技術(shù)。幾乎無限的使用壽命是選擇奇石樂力傳感器解決測量難題的另一個充分理由。間接測量如需測量很大的力或結(jié)構(gòu)不能分解成兩部分, 必須測量部分力。傳感器安裝在力的分流道中的合適位置, 并與測試結(jié)構(gòu)堅固結(jié)合, 因此它只能測量部分力。部分力的大小取決于傳感器安裝的方式。這種安裝方法的優(yōu)點在于對已有結(jié)構(gòu)的改動較小。只需要量程較小的傳感器。一旦傳感器安裝后, 需要對力進行現(xiàn)場校準(zhǔn), 三、試驗機強度的校核 3.1軸承強度的校核 軸承的壽命與所受負(fù)荷的大小有關(guān)，工作負(fù)荷愈大，軸承的壽命就愈短。國家標(biāo)準(zhǔn)規(guī)定，基本額定壽命為一百萬轉(zhuǎn)（=106轉(zhuǎn)）時，軸承所能承受的負(fù)荷稱為基本額定動負(fù)荷，單位為牛頓（）。對于徑向接觸軸承，這一負(fù)荷是指純徑向負(fù)荷，對于角接觸軸承和圓錐滾子軸承，是使軸承套圈之間只產(chǎn)生徑向位移的負(fù)荷的徑向分量，對這些軸承，就具體稱為徑向基本額定動負(fù)荷，用符號r表示；對于推力軸承，是指作用于軸承中心的純軸向負(fù)荷，具體稱為軸向基本額定動負(fù)荷a。 3.1.1 壽命計算公式 根據(jù)大量試驗和理論分析結(jié)果推導(dǎo)出軸承疲勞壽命計算公式如下： 公式一 式中 ——基本額定動負(fù)荷。對向心軸承為Cr，推力軸承為Ca，N； ——當(dāng)量動負(fù)荷，Ｎ； ——溫度系數(shù)； ——負(fù)荷系數(shù)； ——壽命指數(shù)，球軸承ε=3，滾子軸承ε=10/3； ——軸承的工作轉(zhuǎn)速，r/min。 溫度系數(shù)ft 工作溫度，/℃ <120 125 150 175 200 225 250 300 ft 1.0 0.95 0.90 0.85 0.80 0.75 0.70 0.6 沖擊負(fù)荷系數(shù)fp 負(fù)荷性質(zhì) fp 舉例 無沖擊或輕微沖擊 1.0～1.2 電機、汽輪機、通風(fēng)機、水泵 中等沖擊 1.2～1.8 車輛、機床、起重機、冶金設(shè)備、內(nèi)燃機 強大沖擊 1.8～3.0 破碎機、軋鋼機、石油鉆機、振動篩 如果設(shè)計時要求軸承達到規(guī)定的預(yù)期壽命， 則在已知當(dāng)量動負(fù)荷P和轉(zhuǎn)速n的條件下，可按下式算得軸承應(yīng)當(dāng)具有的基本額定動負(fù)荷CC，但使CC小于所選軸承的C值：　 公式二 式中 ——軸承預(yù)期壽命，（ｈ），推薦的軸承使用壽命見下表。 軸承預(yù)期壽命薦用值 使用條件 使用壽命Lh/h 不經(jīng)常使用的儀器和設(shè)備 300~3000 短期或間斷使用的機械，中斷使用不致引起嚴(yán)重后果，如手動機械、農(nóng)業(yè)機械、裝配吊車、回柱絞車等 3000~8000 間斷使用的機械，中斷使用將引起嚴(yán)重后果，如發(fā)電站輔助設(shè)備、流水線傳動裝置、升降機、膠帶輸送機等 8000~12000 每天工作八小時的機械（利用率不高），如電機、一般齒輪裝置、破碎機、起重機等 10000~25000 每天工作八小時的機械（利用率較高），如機床、工程機械、印刷機械、木材加工機械等 20000~30000 24小時連續(xù)運轉(zhuǎn)的機械，如壓縮機、泵、電機、軋機齒輪裝置、礦井提升機等 40000~50000 24小時連續(xù)工作的機械，中斷使用將引起嚴(yán)重后果，如造紙機械、電站主要設(shè)備、礦用水泵、通風(fēng)機等 約100000 公式一和公式二分別用于不同情況。當(dāng)軸承型號已定時，用式一校核軸承的壽命，要求≥；型號未定時，用式二選軸承型號，要求CC≤Ｃ。 2.4.2. 當(dāng)量動負(fù)荷的計算 滾動軸承的基本額定動負(fù)荷是在向心軸承只受徑向負(fù)荷，推力軸承只受軸向負(fù)荷的特定條件下確定的。實際上，軸承往往承受著徑向負(fù)荷和軸向負(fù)荷的聯(lián)合作用，因此，須將該實際聯(lián)合負(fù)荷等效為一假想的當(dāng)量動負(fù)荷P來處理，在此載荷作用下，軸承的工作壽命與軸承在實際工作負(fù)荷下的壽命相同。 （1)只承受徑向負(fù)荷P的徑向接觸軸承 （2）對于只承受軸向負(fù)荷P的軸向接觸軸承 （3）對于同時承受徑向負(fù)荷和軸向負(fù)荷的深溝球軸承和角接觸軸承 式中，X、Y分別為徑向負(fù)荷系數(shù)和軸向負(fù)荷系數(shù)，可查表得其數(shù)值。 2．4.3. 向心角接觸軸承軸向負(fù)荷的計算 向心角接觸軸承（3類、7類）在受到徑向載荷作用時，將產(chǎn)生使軸承內(nèi)外圈分離的附加的內(nèi)部軸向力（見右圖），其值按表所列公式計算，其方向由軸承外圈寬邊所在端面（背面），指向外圈窄邊所在端面（前面）。 為了保證軸承正常工作，向心角接觸軸承通常成對使用。成對布置的方式有兩種：前面對前面的安裝稱為正裝，背面對背面的安裝稱為反裝。 由于向心角接觸軸承產(chǎn)生內(nèi)部軸向力，故在計算其當(dāng)量動負(fù)荷時，式中的軸向負(fù)荷并不等于軸向外力，而是應(yīng)根據(jù) 整個軸上所有軸向受力（軸向外力、內(nèi)部軸向力，）之間的平衡關(guān)系確定兩個軸承最終受到的軸向負(fù)，。下面以正裝情況為例進行分析。 (1)當(dāng)時，軸有向右移動的趨勢，使右端軸承壓緊，左端軸承松由力平衡條件可知 “壓緊”端軸承所受的軸向負(fù)荷= “放松”端軸承所受的軸向負(fù)荷= (2) 當(dāng)時，軸有向左移動的趨勢，使左端軸承壓緊，右端軸松由力平衡條件可知 “壓緊”端軸承所受的軸向負(fù)荷=- “放松”端軸承所受的軸向負(fù)荷= 由此可總結(jié)出計算向心角接觸軸承軸向負(fù)荷Fa的步驟如下： （1）確定軸承內(nèi)部軸向力，的方向（由外圈寬邊指向窄邊，即正裝時相向，反裝時背向），并按表所列公式計算內(nèi)部軸向力的值； 2） 判斷軸向合力++（計算時各帶正負(fù)號）的指向，確定被“壓緊”和被“放松”的軸承。正裝時，軸向合力指向的一端為緊端；反裝時，軸向合力指向的一端為松端； （3）松端軸承的軸向負(fù)荷僅為其本身的內(nèi)部軸向力；緊端軸承的軸向負(fù)荷則為除去本身的內(nèi)部軸向力后其余各軸向力的代數(shù)和。即 式中，下標(biāo)“緊”和“松”，分別代表緊端軸承和松邊軸承的受力；為代數(shù)和，即與Ka同向時加，反向時減，取絕對值。 上兩式對正裝與反裝的各種情況都適用，使用時只需將“應(yīng)的軸承編號即可。 3.2聯(lián)軸器強度的校核 聯(lián)軸器如圖4-6，為HL5型彈性柱銷聯(lián)軸器，此聯(lián)軸器許用轉(zhuǎn)矩為2000 N.m；許用轉(zhuǎn)速為3150r/min。滿足能量補償傳遞扭矩的要求，而且此型號聯(lián)軸器結(jié)構(gòu)簡單，制造容易，更換方便。由于柱銷為聯(lián)軸器強度的關(guān)鍵，采用MC尼龍6材質(zhì)，其具有彈性強大，抗剪切性強的特點。為了確保長期運行平穩(wěn)，對尼龍材料的機械性能進行了校核。材料手冊中表明，普通的MC尼龍6其性能指標(biāo)如下：抗拉強度≥54MPa 抗壓強度≥70MPa 抗剪強度≥52MPa 柱銷轉(zhuǎn)矩值Tz通過近似公式（4-10）進行計算：Tz=(D1/13)3 （4-10）式中D1為柱銷中心分布園直徑，mm 通過公式（4-11）進行計算：D1=d3/0.14 （-11）式中d3為尼龍柱銷直徑,mm 得出Tz為4460 N.m，大于聯(lián)軸器的許用轉(zhuǎn)矩Tn=2000 N.m，滿足使用需求。 二、聯(lián)軸器 1、國家標(biāo)準(zhǔn)規(guī)定參數(shù)情況現(xiàn)在井上使用的為GB5014-1985 HL5彈性柱銷聯(lián)軸器，其主要參數(shù)參見機械工業(yè)出版社出版的《機械設(shè)計手冊》第4卷第41篇第5章第41-116頁，主要性能參數(shù)如下：許用轉(zhuǎn)矩2000N.m 許用轉(zhuǎn)速3500r/min 而我們聯(lián)軸器與電機主軸相聯(lián)，電機為YB225S-8，轉(zhuǎn)速只有710 r/min。單級補償器的工作扭矩為1000 N.m 對于聯(lián)軸器的強度可以計算尼龍柱銷的抗剪強度，尼龍材料的許用切應(yīng)力：τ=110Kgf/cm2。那么根據(jù)φ30柱銷的許用剪切強度可計算出其許用剪切力為F=S柱XτXn=1.52X3.14X110X8=6217.2Kgf 其中：S柱為柱銷截面積τ為許用切應(yīng)力n為柱銷個數(shù)單極補償裝置需要傳遞的力為4000 Kgf. 四、關(guān)節(jié)軸承搖擺試驗機主機的三維建模 4.1徑向加載部分的三維建模 徑向加載部分安裝在主機橫梁上部，加載裝置中的減速電機帶動傳動絲杠通過杠桿、連接部件對安裝在夾具內(nèi)上的軸承進行徑向加載，加載的徑向力通過傳感器測量。機構(gòu)圖如下： 4.2徑向加載部分的三維建模 4.3夾具部分的三維建模 設(shè)計總結(jié) 本設(shè)計主要是基于國內(nèi)關(guān)節(jié)軸承搖擺試驗機的發(fā)展，節(jié)水和環(huán)境保護以及車業(yè)投資和成本控制的要求，設(shè)計具有一定經(jīng)濟和好實用性。 本設(shè)計主要完成的工作包括以下幾個方面： 1、通過對關(guān)節(jié)軸承搖擺試驗機現(xiàn)狀分析，找到設(shè)計出發(fā)點，確定設(shè)計的方向。 2、完成了關(guān)節(jié)軸承搖擺是樣機的結(jié)構(gòu)設(shè)計、系統(tǒng)設(shè)計，同時進行創(chuàng)新設(shè)計，突出質(zhì)量的優(yōu)勢，實現(xiàn)自己動手洗車簡潔化。通過三維軟件對各部分零件建模，在建模期間一般應(yīng)進行深入的特征分析，搞清零件是由那幾個特征組成，明確各個特征的形狀，他們之間的相對位置和表面連接關(guān)系，然后按照特征的主次關(guān)系，按一定的順序進行建模。一個復(fù)雜的零件，可能是許多個簡單特征經(jīng)過相互之間的疊加、切除或相交組成。所以零件建模時，特征的生成順序十分重要，不同的建模過程雖然可以構(gòu)造出同樣的實體零件，但其造型過程及實體的構(gòu)型結(jié)構(gòu)卻直接影響到實體模型的穩(wěn)定性、可修改性、可理解性及實體模型的應(yīng)用。 尤其在二維圖紙上，我們能看到的只是零件的平面圖，而內(nèi)部特征則以虛線給予表示，另外還有零件的相貫線，這表示了各個特征相交時出現(xiàn)線段。在零件的草圖繪制過程中，必須要選好第一個草繪平面，這很關(guān)鍵，這個平面決定了往后建模的所用到的命令，簡單的說，一個圓柱可以作一個圓形然后拉伸，也可以作一個長方體旋轉(zhuǎn)，雖然他們的結(jié)果都一樣，但所用的草繪平面和命令就截然不同。如果我們要的是一條軸，那我們就應(yīng)該選擇第二種方法為好了。 致 謝 時光飛逝，歲月如梭，轉(zhuǎn)眼之間大學(xué)生活已經(jīng)接近尾聲，回首大一剛?cè)雽W(xué)的場景依然歷歷在目，仿佛還是昨天的事情。再從頭到尾看一看這篇畢業(yè)設(shè)計，每一個環(huán)節(jié)都離不開xxx老師的幫助，從題目的擬定，到結(jié)構(gòu)框架，資料搜集，整理提煉，以及最后的反復(fù)斟酌。在具體的寫作過程中，我遇到了許多這樣或那樣的預(yù)料之外的困難，讓我感到困惑和焦慮，但最終在xxx老師的指導(dǎo)和幫助下，還是獨立完成了單拐曲軸的機械加工工藝規(guī)程及夾具設(shè)計。論文的最終完成，是一波三折的過程，在不斷完善和修改的過程中讓我更加懂得“一分耕耘一分收獲”的道理。 除了要衷心的感謝我的指導(dǎo)老師以外，我還要感謝在校期間所有傳授過我知識的老師們，他們孜孜不倦的教誨是我完成這篇論文的基礎(chǔ)。作為一個機械工程及自動化專業(yè)的學(xué)生，在一些其他課程上遇到過許多問題，多虧老師們在百忙之中抽出時間為我答疑解難，給予我耐心的指點，讓我少走了許多彎路。在此我對他們無私的愛心表示由衷的感謝。同時也感謝大學(xué)生活中與我朝夕相處的同學(xué)們，他們給我留下了最難忘的回憶。在一起走過的日子里，我們一同歡笑，一起悲傷，相互鼓勵，共同進步。我要感謝在這漫長而短暫的四年里陪我一同走過的同學(xué)們，因為有了你們，我的生命中多了很多歡樂的瞬間和美好的回憶，愿我們的情義地久天長。 最后感謝我的家人，是他們一直站在我身后，做我堅強的后盾，無論我成功與否，他們都默默地給予我支持與鼓勵，讓我感到我不是一個人在戰(zhàn)斗。謝謝你們，我一定會更加努力，不辜負(fù)你們對我的期望。 四年時光，說長不長說短不短，不知不覺中我們已經(jīng)共同經(jīng)歷了許多，我們手拉著手一同走過了快樂與悲傷，收貨了屬于我們的成長與堅強。也許在今后的日子里還有更多的困難等待著我們，但是我不會害怕，我會勇敢的迎接挑戰(zhàn)，我要用自己的努力搏一個美好的未來。最后我想說：畢業(yè)，是終點，亦是起點。 本文是在導(dǎo)師xx老師的悉心指導(dǎo)下完成的，字里行間都凝聚者導(dǎo)師的智慧和心血。半年來，導(dǎo)師不僅在學(xué)術(shù)上循循善誘，引導(dǎo)學(xué)生不斷進取、精益求精，而且在思想方法上諄諄教誨，傳授學(xué)生生活和做人的道理。導(dǎo)師活躍的學(xué)術(shù)思想、淵博的學(xué)識和對工作一絲不茍的工作作風(fēng)將對我的一生產(chǎn)生重要的影響。在畢業(yè)之際，謹(jǐn)向?qū)熤乱陨钌畹闹x意。 感謝導(dǎo)師xxx老師在畢業(yè)設(shè)計過程中的關(guān)心和支持。 也感謝各位同學(xué)在設(shè)計過程中的鼎立相助。 原此次設(shè)計順利完成，以答謝各位老師和同學(xué)的支持！ 最后，向在百忙之中評閱本文的各位老師表示衷心的感謝！ 參考文獻 [1] 張煉. 以政策和市場為導(dǎo)向、推動產(chǎn)學(xué)研合作教育深入開展. 中國高等教育： 2000（21）. 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[13] 陳友玲，劉乘． 基于工業(yè)設(shè)計的微水洗車產(chǎn)品設(shè)計分析研究［J］． 包裝工程，2011，32( 16) : 55 －58． [14] 孫桓.機械原理.北京:高等教育出版社，2000年 [15] 張政宏． 便攜微型洗車器: 中國，CN201472335U［P］． 2010 －05 －19． [16］王魯佳． 車用便攜洗車器: 中國，CN201670209U［P］． 2010 －12 －05． [17］陳霞，白小軍． 基于單片機的液位監(jiān)控系統(tǒng)［J］． 武漢理工大學(xué)學(xué)報，2007，29( 3) : 38 －41． [18］李偉光，朱金華，彭永紅． 串行模數(shù)裝換芯片TLC1549 及其應(yīng)用［J］． 機械工程師，2003( 11) : 36 － 38． [19] 曹清、李建昌 一種家用循環(huán)用水除塵洗車器的設(shè)計 《節(jié)能技術(shù)》2013年1月 第1期 Optimal design of a 1-rotational DOF flexure joint for a 3-DOF H-type stage Abstract A 3-DOF H-type stage using a flexure joint to accomplish rotational motion about the Z axis of a gantry stage is presented. To employ the rotational motion about the Z axis in an H-type stage, a 1-rotational DOF flexure joint is proposed. The proposed flexure joint in the H-type stage has high off-axis stiffness and adequate durability against high thrust force. A 6-DOF stiffness equation of the proposed flexure joint is obtained by analysis of leaf spring stiffness. To satisfy the required stage dynamic performance, the optimal design is executed on the geometric parameters of the proposed flexure joint using Sequential Quadratic Programing. The results of the optimal design are verified by experiment on the actual flexure joint. Keywords · H-type stage; · Flexure joint; · DOF analytical stiffness equation; · 6-Optimal design; · System mode analysis 1. Introduction Information technology has developed dramatically in the world since the invention of the PC. The development of information technology requires large and high resolution displays. The long range precision positioning stage is the essential system for implementing large and high resolution displays like LCDs, OLEDs and PDPs. The long range stage requires high thrust force to satisfy the high throughout. Consequently, to support the development of the display industry or semiconductor industry, the required performances are the high thrust, precise accuracy, and long range motion. The H-type stage has generally been used as the precision positioning system. The H-type stage has been developed to achieve larger range, more precision accuracy, and higher thrust force. In particular, accomplishing long range in the H-type stage conflicts with high accuracy due to manufacturing errors and assembly errors. A rotation error about the Z axis in the gantry stage, which is the guide for the X axis drive, is the dominant position error in the H-type stage. Recently, the monolithic flexure hinge has been used to guide a high precision motion system. There have been many efforts to compensate the Z rotational error. In et al. [1] developed a planar-type redundantly actuated parallel mechanism that rotated the moving platform about the Z axis. It had poor angular accuracy because it used the bearing at the revolute joint. Shinno et al. [2] proposed the VCM (Voice Moil Motor) stage with aerostatic levitation. It was used for positioning the nano-machining process. The Z rotational motion was arose from 8?VCM modules. It had 0.1?μrad accuracy. In this paper, we propose an H-type stage using flexure joints to compensate for the Z rotation error. The flexure guide mechanism has many advantages: negligible backlash and stick–slip friction; smooth and continuous displacement; adequate amplification for the output displacement of actuation; and inherently infinite resolution. Therefore, the flexure joint has been used for various applications, such as a micromanipulation system [3], an atomic force microscope (AFM) [4], and a dual actuation system for the flat panel display process [5]. The flexure joint, which is placed between the gantry stage and the tandem linear motors, makes the gantry stage rotate in the θz axis. The flexure hinge mechanism needs to have high stiffness except for the Z rotational DOF to guarantee the motion range required for the H-type stage. High off-axis stiffness in the flexure joint is important, because unwanted motions in the H-type stage appear by low stiffness components in the stage. There are many flexure joints for a 1-rotational DOF. The most basic flexure joint is the notch-type flexure joint [6]. A notch-type flexure joint has a low rotation angle due to the stress concentration around the central pivot point. A cross strip flexure joint, axial strip flexure [6], and cartwheel flexure [7] have been proposed to increase the motion of the flexure joint. These flexure joints are not enough for the off-axis stiffness, and are against a high thrust force. However, the proposed flexure joint, which is an over-constrained structure, can give high structural stiffness. There have been many efforts to derive stiffness modeling of the flexure joint. Paros and Weisbord’s model [8] was developed to calculate the spring rates of a single-axis flexure hinge mechanism. Ryu et al. [9] developed a stiffness modeling process for a whole flexure system using a compliance equation of the flexure joint in 6-DOF. This method can be used in a complex flexure system. However, this method is complex and it is difficult to find modeling error when calculation errors arise in the coordinate change and modeling errors emerge in the stiffness modeling of the flexure mechanism. Recently, a computer-based FEM method has been used to analyze the elasticity, natural frequency, and dynamic characteristics of a whole flexure mechanism automatically [10]. In this paper, we analyzed the stiffness of the flexure joint in 6-DOF using a simple stiffness calculation method. We present the optimal design of the proposed flexure joint to satisfy the desired specifications of the H-type stage. Using MATLAB’s Sequential Quadratic Programing (SQP), the size of the flexure joint is optimally designed for the required rotation motion about the Z axis and high off-axis stiffness. The stiffness equation of the proposed flexure joint is verified by the FEM results and experiment. 2. System configuration of 3-DOF H-type stage Fig. 1 shows the configuration of the proposed stage, which consists of the gantry stage, tandem Y axis motors, and slider. The proposed flexure joints are placed between the gantry stage and tandem Y axis motors to allow rotational motion of the gantry stage about the Z axis. The slider and tandem Y axis motors use a linear motor as an actuator with an air bearing guide, which is generally used in precision positioning systems. The guide mechanism uses the magnet preload mechanism to strengthen the stiffness of air bearing used in the slider and tandem Y axis motors. The sensors for position feedback in the X, Y1 and Y2 axes are an optical linear encoder with 1-nm resolution by 12-bit interpolation. Fig. 1.?3-D modeling of the H-type stage. The tandem Y axis motors are separated into master axis and slave axis as the reference for accuracy. The master axis (Y1) motor supports the reference of the Y axis motion to determine the accuracy and motion error in the Y axis. When the gantry stage rotates about the Z axis, the translational motion in the X axis is required to release a large rotation of the gantry stage. Fig. 2 shows the required freedoms in the H-type stage for Z rotational DOF. In the master axis (Y1), rotational joint implements θZ DOF between the gantry stage and the master axis motor. In the slave axis (Y2), the translational guide is equipped with a rotational flexure joint to allow X translation motion. There is no translational motion between the master axis (Y1) and the gantry stage. Fig. 2.?Schematic of the H-type stage for analysis of DOF. Fig. 2.?Schematic of the H-type stage for analysis of DOF. View thumbnail images The translational mechanism is implemented using a linear motion (LM) guide. The LM guide makes the gantry stage decouple from the slave axis motor (Y2). The LM guide prevents contact with the air bearing stage when the gantry stage is rotating. 3. 1R-DOF flexure joint for a 3-DOF H-type stage 3.1. Introduction to the proposed flexure joint The proposed flexure joint has 1R-DOF to implement rotational motion of the gantry stage about the Z axis. Fig. 3 shows the flexure joint which has a wheel shape. Even though the flexure joint is an over-constraint structure, the rotational motion of the gantry stage about the Z axis occurs with the elastic deformation of a leaf spring. Fig. 3.?3-D modeling of the flexure joint. The stiffness of the flexure joint influences the structure stiffness of the stage, which determines the whole dynamics of the H-type stage. It is important that the off-axis stiffness of the flexure joint is maximized to obtain high precision motion. Thus, the proposed flexure joint is adequate for implementation in the H-type stage due to the high off-axis stiffness. To satisfy the desired specifications of the H-type stage, 6-DOF stiffness modeling of the proposed flexure joint is required to perform an optimal design. 3.2. 6-DOF stiffness modeling of the proposed flexure joint The proposed flexure joint is symmetric in the X and Y axes, so the compliance matrix of the flexure joint has a diagonal matrix form as in Eq. (1), and the 6-DOF stiffness equations of the flexure joint reduce to four equations. (1) The proposed flexure joint is composed of the same eight leaf springs. The stiffness modeling of the flexure joint can be derived from the stiffness of each leaf spring. To derive the stiffness modeling, it is necessary to analyze the 6-DOF stiffness of one leaf spring. There have been many studies to determine precise stiffness modeling of a leaf spring, but it is hard to derive the stiffness modeling of the flexure joint whole motion range due to the complexity of finite element behavior. Smith [4] analyzed motion stiffness of a leaf spring in desired motion. Kang [7] derived a 6-DOF stiffness equation for a clamped leaf spring.Fig. 4 is a clamped leaf spring. The compliance matrix for a leaf spring is as [7]. (2) where b, l, and t are the height, length, and thickness of the leaf spring respectively in Fig. 3, E is Young’s modulus, G is the shear modulus, k2 is the modeling coefficient determined by b/t. 6-DOF stiffness equations of the flexure joint were derived by analyzing the deformation of all the leaf springs. In the next section, we present methods to determine 6-DOF stiffness equations for the flexure joint. Fig. 4.?Parameters and axis definition of a leaf spring [7]. View thumbnail images 3.3.1. Translational stiffness equation in the X axis For convenience, the flexure joint can be separated into?+?shaped leaf springs and an × shaped leaf spring as shown in Fig. 5. Eq. (3) represents the translational stiffness of a?+?shaped leaf spring and Eq. (4) represents the translational stiffness of an × shaped leaf spring. (3) (4) where E is Young’s modulus, G is the shear modulus, and φ is the angle between the leaf spring and the X axis. Using Eqs. (3)?and?(4), the translational stiffness in the X direction (dFx/dx) of the proposed flexure joint is shown in the following equation: (5) Fig. 5.?(a) a?+?shaped leaf springs and (b) an × shaped leaf springs. View thumbnail images 3.3.2. Rotational stiffness about the Z axis When the flexure joint rotates about the Z axis, the leaf springs encounter the axial force, normal force and moment about the Z axis as shown in Fig. 6. The sum of the total axial force in the flexure joint is zero due to the cancelation of all the force components, and the normal force is also zero. Therefore, the rotational stiffness about the Z axis is derived from the sum of the moment about the Z axis of all the leaf springs. Eq. (6) is the rotational stiffness of the flexure joint about the Z axis. (6) where ri is the radius of the inner body. Fig. 6.?Deformation of the flexure joint by Mz. View thumbnail images 3.3.3. Translational stiffness in the Z axis The Z translational stiffness of eight leaf springs makes the Z translational motion of the flexure joint. Thus, it can be modeled easily in the following equation: (7) 3.3.4. Rotational stiffness about the X axis The X rotational motion, which is out-of-plane motion has a complex deformation in leaf springs. Fig. 7 shows the deformation of each leaf spring. Fig. 7.?Deformation of flexure joint under Mx(a) Flexure joint under the moment Mx, (b) Free body diagram of 3, 7 leaf spring, (c) Free body diagram of 1, 5. leaf spring, (d) Free body diagram of 2, 4, 6, and 8 leaf spring. View thumbnail images The X rotational stiffness of the flexure joint is derived from three types of deformation of a leaf spring. The first deformation of the leaf spring is like 3, 7 leaf springs which undergo torsional moment about the X axis as shown in Fig. 7b. The second deformation is like 1, 5 leaf springs which are deformed by moment Myl1 and Myl2. The third deformation is like 2, 4, 6, 8 leaf springs, which are influenced by the previous two deformations equally. Eq. (8) indicates the rotational stiffness of the flexure joint about the X axis. where le is the effective length of the leaf spring. 3.3.5. Maximum stress The maximum stress occurs at the end position of the leaf spring when the flexure joint undergoes Z rotational deformation. Eq. (9) shows the maximum stress of the flexure joint. (9) where σmax is the maximum stress of the flexure joint and Kt is the stress concentration factor given by Peterson and co-workers [11]. 3.4. Verification of the flexure joint modeling To check the effectiveness of the stiffness equations of the flexure joint, it is verified by a FEM program named Pro Engineering/MECHANICA. Table 1 gives the results of the verification. The parameters of the flexure joint are ri?=?60?mm, l?=?40?mm, b?=?30?mm and t?=?2.5?mm. Table 1. Verification results of the stiffness modeling of the flexure joint. Unit Analytic model FEM simulation Error (%) kx N/μm 441.630 416.899 5.6 ky N/μm 441.630 416.899 5.6 kz N/μm 168.013 148.691 11.5 kθx Nm/μrad 0.69294 0.60492 12.7 kθy Nm/μrad 0.69294 0.60492 12.7 kθz Nm/μrad 0.02497 0.02282 8.6 σmax MPa 202.05 221.851 9.8 Full-size table The model of the flexure joint shows a reasonable prediction of the stiffness modeling with less than 13% errors. 4. Parametric analysis To create the optimal design, a parametric analysis on the flexure design is required to investigate how the design parameters of the flexure joint affect the 6-DOF stiffness and the maximum stress in the flexure joint. The results of the parametric analysis will ensure that the result of the optimal design is reasonable The design parameters of the flexure joint are as follows: Height of the leaf spring: b. Length of the leaf spring : l. Thickness of the leaf spring: t. Radius of the inner body: ri. We can derive the sensitivity analysis on the flexure joint with respect to variations of the design parameters. The parametric analysis results are shown in Fig. 8. Fig. 8.?Parametric analysis results of the flexure joint. View thumbnail images The length of the flexure joint(l) is the most sensitive design parameter in designing the flexure joint as shown in Fig. 8. The most sensitive property of the flexure joint is the rotational stiffness about the Z axis and the maximum stress as shown in Fig. 8. The design parameter b does not affect the maximum stress from Fig. 8b. The design parameter ri affects only the rotational stiffness about the Z axis and the maximum stress. 5. Optimal design of flexure joint The flexure joint is used to accomplish the yaw motion in the H-type stage. The rotational angle is important in the H-type stage to compensate for yaw errors. Thus, the flexure joint must have a low enough rotational stiffness about the Z axis. When driving the H-type stage, the settling time is determined by the stiffness of the system structure. The flexure joint in the H-type stage plays the role of a bottleneck in designing the H-type stage. To obtain a desired stiffness of the flexure joint that does not disturb the required specifications of the H-type stage, the optimal design is required to obtain enough rotational stiffness about the Z axis and the high off-axis stiffness. The design variables for the flexure joint are l, b, t, and ri. The influence of each design variable was discussed in the previous section. The cost function of the optimal design is determined to minimize the rotational stiffness about the Z axis and maximize the off-axis stiffness. Eq. (10) shows the cost function of the optimal design for the flexure joint. (10) where c1, c2, c3, and c4 are coefficients to make the design variables unity. The optimal design minimizes the cost function. The problem includes several constraints. For example, the maximum stress in theflexure joint should be less than the yield stress. There are also constraints for the system size and the off-axis stiffness. (11) where Sf1 is the safety factor of the maximum stress constraint, σyield is the yield strength of the flexure joint, and θzd is the desired Z rotation angle. Second, it is necessary that the rotation angle(θz) about the Z axis be larger than the desired specification, when the normal force of the linear motor is applied to the stage. θz should be as (12) where Sf2 is the safety factor for the rotational angle constraint and Mz is the drive moment about the Z axis by the thrust of the Y axis linear motor. Third, the flexure joint dominantly affects the dynamics of the H-type stage, because the flexure joint has the smallest stiffness in the H-type stage system. The settling time influenced by the system dynamics is an important specification in determining how fast the stage drives. Thus, to achieve the required settling time, the flexure joint must have enough stiffness in all degrees of freedom. When applying the X motion to the slider, the flexure joint deformed to the X translation and rotational motion about the Y axis. When applying the Y motion, deformation about the X rotational motion occurs in the flexure joint. The constraints for the settling time of the stage are as follows by the dynamic modeling of the H-type stage [12]. (14) where kx and kθx are the desired stiffness values for the X axis and the rotational X axis, respectively. There are the constraints for geometry of the flexure joint. The geometric constraints are listed as follows: (15) g5=ri+2l-C5?0 In summary, the described optimization problem can be rewritten as and The flexure joint is optimally designed using above optimal design work. Table 2 shows the constants for the optimal design of the flexure joint. Table 2. Constants for the optimal design of the flexure joint. Constant Unit Value σyield Mpa 503 E Gpa 68.95 θz Degree 0.3 kxd N/m 200 kθxd Nm/μrad 0.45 Kt – 15.7 Sf1 – 2.0 Sf2 – 2.0 Sf3 – 2.0 Sf4 – 2.0 Full-size table 6. Design results To accomplish the optimal design, we adopted SQP which uses a positive define quasi-Newton approximation of the Hessian of the Lagrange function implemented on the MATLAB program. This method generally guarantees the local minimum [13]. Fig. 9 shows the convergence profile of the cost function. The cost function value gradually converged to certain value. Fig. 10 shows that the final cost function values with various initial points converge to the same value. This proves that the optimal design result reaches sub minimum. Fig. 9.?Convergence profile of cost function. View thumbnail images Thus, the optimal design results with diffident initial points were checked. Table 3 shows the design variable sets for the optimal design. However, because of manufacturing cost effect of Electrical Discharge Machining (EDM), the design variables were chosen as in Table 4. Table 5 shows the calculated characteristics from modeling of the flexure joint. Table 3. Design variable sets. Design variables Start points Optimum results Unit S1 S2 S3 S4 Sopt l mm 25 30 40 50 42.10 b mm 20 25 30 40 40.00 t mm 1.5 1.8 2.0 2.5 1.92 ri mm 55 60 70 75 61.56 Full-size table Table 4. Final dimension of the design variables. Final dimension l (mm) b (mm) t (mm) ri (mm) Value 42.0 40.00 1.90 61.60 Full-size table Table 5. Simulated characteristics of the flexure joint. Start poi