王亞珍，汪安明，趙 坤，宋 麗

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基于薄壁圓環(huán)理論的機(jī)器人用柔性軸承變形特征快速求解

王亞珍1，汪安明1，趙 坤2，宋 麗2

（1. 上海大學(xué)機(jī)電工程與自動(dòng)化學(xué)院，上海 201900；2. 寧波慈興軸承有限公司，寧波 315301）

諧波減速器內(nèi)部柔性軸承是機(jī)器人關(guān)節(jié)的重要傳動(dòng)部件，因在工作中會(huì)產(chǎn)生較大的預(yù)變形而與普通軸承不同，導(dǎo)致傳統(tǒng)軸承理論不適用，所以建立新的研究方法對(duì)其各項(xiàng)性能進(jìn)行分析非常必要。該文通過(guò)建立柔性軸承的理論計(jì)算模型，求解計(jì)算柔性軸承工作時(shí)內(nèi)部應(yīng)力與變形特征，具體步驟包括：1）建立變形協(xié)調(diào)方程，并通過(guò)莫爾積分定理求解方程，得到變形過(guò)程中柔性軸承外圈的彎曲力矩；2）根據(jù)薄壁圓環(huán)理論，求得外圈的變形特征與內(nèi)圈彎曲力矩；3）建立三彎矩方程，計(jì)算外載荷作用下柔性軸承外圈所承受的最大彎曲力矩。最后，建立柔性軸承有限元仿真模型對(duì)比驗(yàn)證理論模型，兩者最大誤差為7%，其中理論求解時(shí)間為5～8 min，有限元計(jì)算需4～5 h，通過(guò)理論模型可以快速求得柔性軸承內(nèi)部變性特征與受力情況。計(jì)算結(jié)果表明，對(duì)于CSF-25-80型柔性軸承，外圈厚度設(shè)計(jì)在1.3～1.6 mm，寬度設(shè)計(jì)在9 mm左右，都可以有效改善外圈的應(yīng)力狀況。該研究可為對(duì)柔性軸承的設(shè)計(jì)和優(yōu)化提供理論參考。

機(jī)器人；軸承；模型；薄壁圓環(huán)理論；預(yù)變形

0 引 言

農(nóng)業(yè)產(chǎn)業(yè)一直是中國(guó)經(jīng)濟(jì)發(fā)展的重要支柱，隨著人口老齡化問(wèn)題的日趨嚴(yán)重，勞動(dòng)力資源嚴(yán)重不足，生產(chǎn)成本不斷增高，機(jī)械化作業(yè)將作為未來(lái)國(guó)內(nèi)農(nóng)業(yè)發(fā)展的重要方向，在降低勞動(dòng)成本或提高產(chǎn)量等方面有著巨大優(yōu)勢(shì)[1-2]。雖然中國(guó)對(duì)農(nóng)業(yè)機(jī)器人的研究已取得了較大的發(fā)展，但相對(duì)于國(guó)外來(lái)說(shuō)，仍然存在較大差距。

諧波減速器內(nèi)部柔性軸承是機(jī)器人[3]關(guān)節(jié)的重要傳動(dòng)部件，其性能會(huì)嚴(yán)重影響到整個(gè)機(jī)構(gòu)的精度。因此，對(duì)柔性軸承在工作中各項(xiàng)性能進(jìn)行分析計(jì)算是非常重要 的[4-5]。雖然，目前通過(guò)有限元的仿真計(jì)算可以得到較為精準(zhǔn)的計(jì)算模型，但是由于柔性軸承滾動(dòng)體數(shù)目多（一般23個(gè)球），且每個(gè)滾動(dòng)體與內(nèi)外圈的接觸狀態(tài)各不相同，導(dǎo)致有限元計(jì)算中含有大量接觸對(duì)，計(jì)算時(shí)間長(zhǎng)（一般高性能工作站計(jì)算時(shí)間大于4 h），且計(jì)算過(guò)程難以收斂、頻繁中斷，很難得到正確的計(jì)算結(jié)果。

在理論研究方面，普通軸承在工作中，一般是作為支撐件存在[6]，傳統(tǒng)的軸承理論主要對(duì)其在徑向或軸向載荷的作用下，滾動(dòng)體與內(nèi)外圈之間的接觸載荷計(jì)算[7]。與普通軸承工作原理不同，柔性軸承作為傳動(dòng)部件,預(yù)先裝配在一個(gè)橢圓形的波發(fā)生器凸輪上[8]，軸承會(huì)發(fā)生較大的預(yù)變形。在傳統(tǒng)軸承的計(jì)算理論中并不包含對(duì)這一變形階段的分析，目前國(guó)內(nèi)在此方面的研究較少。因此建立一個(gè)合理的柔性軸承理論計(jì)算模型，求出內(nèi)外圈的變形特征與內(nèi)部受力情況，是柔性軸承力學(xué)分析與負(fù)荷能力計(jì)算的前提。對(duì)于柔性軸承及整體減速器的設(shè)計(jì)，完善其系列化、標(biāo)準(zhǔn)化都具有重要意義。

柔性軸承被強(qiáng)制變形之后，內(nèi)圈根據(jù)中性面不伸長(zhǎng)原理，完全貼合在波發(fā)生器凸輪之上，而外圈則由于在多個(gè)離散滾動(dòng)體支撐下，變形形狀比較復(fù)雜且與內(nèi)圈不完全相同。因此，在進(jìn)行柔性軸承的分析計(jì)算時(shí)，需要對(duì)內(nèi)外圈分別進(jìn)行單獨(dú)的分析[9]。本文提出了一種基于變形協(xié)調(diào)方程、莫爾積分定理、薄壁圓環(huán)理論與三彎矩方程對(duì)柔性軸承內(nèi)外圈變形特征與彎曲應(yīng)力進(jìn)行理論求解的方法。最后建立了CSF-25-80型號(hào)柔性軸承的有限元模型，并將仿真計(jì)算結(jié)果與理論模型計(jì)算結(jié)果進(jìn)行對(duì)比，驗(yàn)證求解的準(zhǔn)確性。

1 外圈變形彎矩計(jì)算

與普通軸承最大的不同之處在于柔性軸承工作前，被預(yù)先裝配在非圓形波發(fā)生器凸輪之上，產(chǎn)生強(qiáng)制性的變形，變形過(guò)程如圖1所示。

外圈在波發(fā)生器凸輪的作用下，長(zhǎng)軸處形成的最大變形量為max。圖2給出了外圈中性層半徑與其橫截面尺寸：×（寬度與厚度），由于和都遠(yuǎn)小于，因此可將外圈等效為一薄壁圓環(huán)進(jìn)行求解。

注：δmax為軸承最大變形量，mm；[Q1, Q2,…, Qi]為第[1, 2, …, i]個(gè)滾動(dòng)體的支撐反力，N；F為凸輪與軸承之間相互作用力，N。

典型波發(fā)生器凸輪所形成的徑向位移為[10]

由圖1可知，軸承受到波發(fā)生器凸輪作用變形后，其外圈是在多個(gè)離散滾動(dòng)體產(chǎn)生的徑向力[1,2,…,Q]共同作用下發(fā)生變形，其受力狀態(tài)沿變形長(zhǎng)軸對(duì)稱，因此只需對(duì)半圓環(huán)進(jìn)行受力分析。將圖1外圈等效為薄壁圓環(huán)后可得圖2，外圈在[1,2,…,Q]作用下變形，由于其變形長(zhǎng)軸與短軸所在截面和的轉(zhuǎn)角都是零，故可以將面作為固定端，解除面約束得到基準(zhǔn)靜定基，以面相對(duì)于面的轉(zhuǎn)角等于0作為變形協(xié)調(diào)條件，對(duì)圓環(huán)進(jìn)行分析。

注：是外圈寬度，mm；是外圈厚度，mm；是外圈半徑，mm；、是水平軸與垂直軸；M解除面約束后的彎矩，N·mm；P是解除面約束后的拉力，N；γ是Q與的夾角，rad；是積分角變量，rad。

Note:is outer ring width, mm;is outer ring thickness, mm;is outer ring radius, mm;andare horizontal axis and vertical axis;Mis bending moment after releasing constraint of, N·mm;Pis force after releasing constraint of, N;γis angle betweenQand, rad;is integral angle variable, rad.

圖2 外圈變形受力簡(jiǎn)圖

Fig.2 Forces state of outer ring

由于對(duì)稱性可知面解除約束后會(huì)產(chǎn)生拉力與彎矩，以面轉(zhuǎn)角為零作為條件，建立變形協(xié)調(diào)方程[11]

將式（2）轉(zhuǎn)化為正則方程

因?yàn)槊娼獬s束后變?yōu)樽杂啥?，?duì)于基準(zhǔn)靜定基來(lái)說(shuō)，支撐力 Q在QD段不再產(chǎn)生彎矩，QD彎矩由解除約束后的MP產(chǎn)生。則由Q引起的力矩為[12]

根據(jù)莫爾積分定理，可求出

式中為材料彈性模量，MPa；為截面慣性矩，mm4。

對(duì)于> 1，根據(jù)靜力平衡條件

代入式（3），得到基準(zhǔn)靜定基在MP與Q共同作用下，其任意角度處的彎矩為

其中

即在單位力作用下圓環(huán)內(nèi)的彎矩為

再次使用莫爾積分，得到在Q作用下圓環(huán)的變形量

則[1,2,…,Q]的共同作用下，外圈長(zhǎng)軸最大撓度變形max的計(jì)算方程

根據(jù)軸承滾動(dòng)體受力分布理論[13]，結(jié)合典型波發(fā)生器凸輪所形成的徑向位移，各軸承力之間的關(guān)系為

式中= 3/2（對(duì)于球軸承）。

聯(lián)立方程求解，可得到支撐力Q的大小，再帶入式（9）中就可以解得任意處的彎矩

2 外圈變形計(jì)算與內(nèi)圈彎矩計(jì)算

2.1 外圈變形計(jì)算

2.1.1 徑向位移計(jì)算

根據(jù)薄壁圓環(huán)理論[14]，圓環(huán)徑向位移/mm通過(guò)式（18）求解

為了便于微分方程的求解，將第一節(jié)求得的彎矩方程進(jìn)行修改，設(shè)

結(jié)合式（9）、（10）、（11），圓環(huán)彎矩方程的簡(jiǎn)化 如下

將式（22）代入式（18），解得

根據(jù)對(duì)稱性得到2個(gè)邊界條件與2個(gè)連續(xù)性條件：1）= 0時(shí)，d/d= 0；2）= π/2時(shí)，d/d= 0；3）=時(shí)，(d1/d) = (d2/d)；4）=時(shí)，12。式中1、2、3、4為待定解，由上面4個(gè)邊界條件求得，由于表達(dá)式過(guò)于繁瑣，這里不予給出。

2.1.2 周向位移計(jì)算

根據(jù)薄壁圓環(huán)理論，周向位移/mm與徑向位移的關(guān)系由式d/d=-確定[15]，將1、2代入可得周向位移方程

2.2 內(nèi)圈彎矩計(jì)算

根據(jù)柔性軸承設(shè)計(jì)中的中性面不伸長(zhǎng)原理，軸承內(nèi)圈變形后，應(yīng)與波發(fā)生器凸輪完全貼合，即徑向位移為

將式（27）代入式（18），得到內(nèi)圈在變形作用下產(chǎn)生的內(nèi)部彎矩（R為內(nèi)圈半徑）

3 外部載荷作用

3.1 外部載荷分析

在諧波減速器中，柔性軸承受到減速器內(nèi)部柔輪與剛輪之間嚙合反力的作用，承受較大的外部載荷[16]。剛輪與柔輪之間的嚙合力與輪齒的嚙合區(qū)域、傳遞的扭矩有關(guān)，其受載如圖3所示。

注；qr是徑向均布載荷，MPa；qφ是周向均布載荷，MPa；φ1是載荷偏角，rad；φ2是載荷左分布角，rad；φ3是載荷右分布角，rad。

根據(jù)Ivanov的試驗(yàn)所測(cè)，減速器內(nèi)柔輪所承受的載荷分布為[17]。

式中表示諧波減速器輸入扭矩，N·mm，d、b分別表示及其內(nèi)部柔輪的節(jié)圓直徑與寬度，mm，為柔輪輪齒壓力角，rad。

柔輪將載荷傳遞至柔性軸承[18-19]，不考慮偏載的影響，即1= 0，切向力q帶動(dòng)軸承轉(zhuǎn)動(dòng)，徑向力q引起柔性軸承產(chǎn)生附加的彎曲力矩與應(yīng)力。載荷的分布角為

3.2 載荷作用下的外圈最大彎矩求解

與內(nèi)圈不同，柔性軸承受載時(shí)，其外圈支承在多個(gè)離散的滾動(dòng)體上，可以簡(jiǎn)化為材料力學(xué)里的多跨度梁結(jié)構(gòu)。連續(xù)梁的三彎矩方程可用于求解這種多跨梁[20]。將每個(gè)支撐點(diǎn)的約束分解成一對(duì)力矩，它們大小相等且方向相反。如圖4所示，取任意相鄰的3個(gè)支撐點(diǎn)的彎曲力矩M-1、M、M+1可建立方程。

式中l、l+1為相鄰支撐點(diǎn)之間的跨距，ω、ω+1是跨距內(nèi)載荷q()、q+1()單獨(dú)作用下的彎矩圖面積，a、b+1表示彎矩圖的形心到左右兩端的距離。

對(duì)每個(gè)支點(diǎn)建立三彎矩方程，聯(lián)立求解，可得到外部載荷作用時(shí)，各支撐點(diǎn)處所產(chǎn)生的額外彎矩。

注：Mn是支撐點(diǎn)彎矩，N·mm；ln是相鄰支撐點(diǎn)之間的跨距，mm；qn(θ)是跨距內(nèi)載荷，MPa；ωn是跨距內(nèi)彎矩圖面積，mm2；an是跨距內(nèi)彎矩圖形心與左端的距離，mm；bn+1是跨距內(nèi)彎矩圖形心與右左端的距離，mm。

4 結(jié)果與分析

本文以CSF-25-80型號(hào)柔性軸承為例，將理論模型編制為MATLAB程序進(jìn)行計(jì)算，計(jì)算流程如圖5所示，計(jì)算時(shí)間約為5～8 min。同時(shí)，建立了基于ANSYS Workbench的柔性軸承的有限元仿真模型[21-23]與其對(duì)比，模型使用HPZ840工作站進(jìn)行求解，計(jì)算時(shí)間約為4～5 h。

首先，對(duì)于軸承的變形過(guò)程，計(jì)算最大變形量max= 0.4 mm時(shí)內(nèi)外圈的彎矩，代入式= (())/中（式中為抗彎截面系數(shù)，mm4），得到內(nèi)外圈變形長(zhǎng)軸處彎曲應(yīng)力（圖6）；同時(shí)，計(jì)算外圈由于滾動(dòng)體支撐的作用軸承外圈所產(chǎn)生的實(shí)際變形量（圖7）。

圖5 計(jì)算流程圖

圖6 內(nèi)外圈彎曲應(yīng)力對(duì)比

圖7 外圈變形量對(duì)比

從圖6和圖7中可以看出，理論值與仿真值相差不大。表1給出圖6、圖7中重要計(jì)算結(jié)果的對(duì)比。結(jié)果表明，理論計(jì)算值和仿真計(jì)算結(jié)果的誤差值在7%以內(nèi)，進(jìn)一步驗(yàn)證了理論計(jì)算的合理性和準(zhǔn)確性。

表1 內(nèi)外圈彎曲應(yīng)力與變形特征關(guān)鍵值對(duì)比

內(nèi)外圈的長(zhǎng)軸處最大應(yīng)力反映了其可承受的變形量的大小。而短軸處的應(yīng)力值為負(fù)，則為壓應(yīng)力，其大小會(huì)影響軸承的壽命[24]。外圈在短軸處位移量小于max，因此軸承變形短軸處內(nèi)外圈與滾動(dòng)體之間會(huì)形成一定間隙，約為0.016 mm，間隙的大小將影響諧波減速器輪齒的嚙合度[25]。最大周向位移反映了外圈在切向方向的變形程度[26-27]。外圈受載后的最大應(yīng)力由三彎矩方程求得，其決定了軸承的承載能力。

由于柔性軸承在運(yùn)轉(zhuǎn)過(guò)程中，外圈會(huì)承受載荷帶來(lái)的額外彎曲應(yīng)力，而外圈的主要參數(shù)包括其寬度與厚度。因此在這里通過(guò)上述理論分析了這2個(gè)參數(shù)對(duì)外圈強(qiáng)度影響[28-29]，如圖8所示。

圖8 外圈參數(shù)對(duì)其應(yīng)力的影響

由圖8a可知，厚度較小時(shí)，柔性軸承的外圈在載荷作用下的最大彎曲應(yīng)力較大，隨著厚度的增加，載荷作用產(chǎn)生的彎曲應(yīng)力減小，而變形產(chǎn)生的彎曲應(yīng)力隨之增加，所以外圈的厚度存在一定的最優(yōu)值。

圖8b顯示了寬度的增加對(duì)于變形產(chǎn)生的彎曲應(yīng)力沒(méi)有影響，但可以減小載荷產(chǎn)生的最大應(yīng)力。因此，在設(shè)計(jì)柔性軸承時(shí)需要選擇合理的外圈厚度并在允許的范圍內(nèi)[30]適當(dāng)增加其寬度。對(duì)于CSF-25-80型柔性軸承，外圈厚度設(shè)計(jì)在1.3～1.6 mm，寬度設(shè)計(jì)在9 mm左右，都可以有效改善外圈的應(yīng)力狀況，提高軸承的使用壽命。

5 結(jié) 論

1）本文提出了一種機(jī)械人關(guān)節(jié)諧波減速器內(nèi)柔性軸承變形特征與內(nèi)部受力的理論求解方法，分析了柔性軸承變形后內(nèi)外圈彎曲應(yīng)力大小及外圈的變形量，并通過(guò)三彎矩方程求解載荷作用下由于滾動(dòng)體支撐作用產(chǎn)生的最大彎曲應(yīng)力。計(jì)算結(jié)果對(duì)于柔性軸承的設(shè)計(jì)與優(yōu)化具有重要的指導(dǎo)意義。

2）柔性軸承內(nèi)圈的變形與橢圓波發(fā)生器一致；外圈變形由多個(gè)滾動(dòng)體的離散支撐產(chǎn)生，長(zhǎng)軸處的變形量與內(nèi)圈相同，均為軸承最大變形量，短軸處變形較小，因此在短軸處將產(chǎn)生一定的徑向間隙。間隙的大小將影響諧波減速器輪齒的嚙合度。

3）將有限元仿真分析結(jié)果與理論計(jì)算進(jìn)行比較。本次仿真計(jì)算使用HP Z840工作站進(jìn)行求解，計(jì)算時(shí)間約為4～5 h，而通過(guò)理論對(duì)方程進(jìn)行求解僅需5～8 min。二者之間的結(jié)果非常接近，對(duì)幾個(gè)重要的計(jì)算結(jié)果進(jìn)行了對(duì)比后，最大誤差控制在7%以內(nèi)，驗(yàn)證了理論計(jì)算的合理性和準(zhǔn)確性。

4）理論分析了柔性軸承外圈彎曲應(yīng)力隨厚度與寬度的變化。計(jì)算結(jié)果表明，對(duì)于CSF-25-80型柔性軸承，外圈厚度設(shè)計(jì)在1.3～1.6 mm，寬度設(shè)計(jì)在9 mm左右，都可以有效改善外圈的應(yīng)力狀況，提高軸承的使用壽命。

[1] 王儒敬，孫丙宇. 農(nóng)業(yè)機(jī)器人的發(fā)展現(xiàn)狀及展望[J]. 中國(guó)科學(xué)院院刊，2015(6)：803－809. Wang Rujing, Sun Bingyu. Development status and expectation of agricultural robot[J]. Bulletin of Chinese Academy of Sciences, 2015(6): 803－809. (in Chinese with English abstract)

[2] 李玉林，崔振德，張園，等. 中國(guó)農(nóng)業(yè)機(jī)器人的應(yīng)用及發(fā)展現(xiàn)狀[J]. 熱帶農(nóng)業(yè)工程，2014，38(4)：30－33. Li Yulin, Cui Zhende, Zhang Yuan, et al. Application and development status of China's agricultural robot[J]. Tropical Agricultural Engineering, 2014, 38(4): 30－33. (in Chinese with English abstract)

[3] 夏田，楊世勇，何乃如，等. 工業(yè)機(jī)器人用諧波減速器傳動(dòng)性能正交試驗(yàn)分析[J]. 科學(xué)技術(shù)與工程，2017(19)： 138－141.

Xia Tian, Yang Shiyong, He Nairu, et al. Orthogonal experiment analysis on transmission performance of harmonic drive for industrial robots[J]. Science Technology and Engineering, 2017(19): 138－141. (in Chinese with English abstract)

[4] Taghirad H D, Belanger P R. Modeling and parameter identification of harmonic drive systems[J]. Journal of Dynamic Systems Measurement & Control, 1997, 120(4): 439－444.

[5] Timothy D, Tuttle T, Seering W P, et al. A nonlinear model of a harmonic drive gear transmission[J]. IEEE Transactions on Robotics and Automation, 1996, 12(3): 368－374.

[6] Shah D B, Patel K M, Trivedi R D. Analyzing hertzian contact stress developed in a double row spherical roller bearing and its effect on fatigue life[J]. Industrial Lubrication & Tribology, 2016, 68(3): 361－368.

[7] Gunia D, Smolnicki T. The analysis of the stress distribution in contact pairs ball-wire and wire-ring in wire raceway slewing bearing[C]//International Conference on Renewable Energy Sources-Research & Business. Springer, Cham, 2016.

[8] You Bindi, Wen Jianmin. Load distribution calculation of fexible ball bearing with elliptical cam wave generator[C]// International Conference on Mechanics, 2016.

[9] 張立勇，劉新猛，王長(zhǎng)路，等. 徑向變形量對(duì)諧波減速器嚙合特性及柔輪應(yīng)力的影響分析[J]. 機(jī)械傳動(dòng)，2017(9):172－175.

Zhang Liyong, Liu Xinmeng, Wang Changlu, et al. Influence of radial deformation on stress of flexspline and meshing characteristic of harmonic reducer[J]. Journal of Mechanical Transmission, 2017(9):172－175. (in Chinese with English abstract)

[10] 關(guān)崇復(fù). 柔性軸承動(dòng)載荷下的接觸應(yīng)力計(jì)算[J]. 燕山大學(xué)學(xué)報(bào)，1994(3)：220－226. Guan Chongfu. Calculation for the contact pressure of flexible bearings under dynamic load[J]. Journal of Yanshan University, 1994(3): 220－226. (in Chinese with English abstract)

[11] 劉鴻文. 材料力學(xué)[M]. 北京：高等教育出版社，1985.

[12] 陳曉霞，劉玉生，邢靜忠，等. 諧波齒輪中柔輪中性層的伸縮變形規(guī)律[J]. 機(jī)械工程學(xué)報(bào)，2014，50(21)：189－196. Cheng Xiaoxia, Liu Yusheng, Xing Jingzhong, et al. Neutral line stretch of flexspline in harmonic driver[J]. Journal of Mechnical Engineering, 2014, 50(21): 189－196. (in Chinese with English abstract)

[13] Harris T A, Kotzalas M N. Advanced Concepts of Bearing Technology: Rolling Bearing Analysis[M].Boca Raton: CRC Press, 2007.

[14] 王靜靜. 某諧波減速機(jī)波發(fā)生器柔性軸承疲勞壽命研究與結(jié)構(gòu)優(yōu)化[D]. 長(zhǎng)沙：湖南大學(xué)，2016. Wang Jingjing. The Research on Fatigue Life and Structure Optimization of the Flexible Bearing in a Harmonic Rreducer Wave Generator[D]. Changsha: Hunan University, 2016. (in Chinese with English abstract)

[15] 伊萬(wàn)諾夫，沈允文，李克美. 諧波齒輪傳動(dòng)[M]. 北京：國(guó)防工業(yè)出版社，1987.

[16] 柏德恩，全齊全，李賀，等. 伺服加載的諧波減速器啟動(dòng)力矩測(cè)試系統(tǒng)[J]. 吉林大學(xué)學(xué)報(bào)：工學(xué)版，2017(6)：141－147.

Bai Deen, Quan Qiquan, Li He, et al. Starting torque test system for harmonic driver based on servo loading[J]. Journal of Jilin University: Engineering and Technology Edition, 2017(6):141－147. (in Chinese with English abstract)

[17] Kayabasi O, Erzincanli F. Shape optimization of tooth profile of a flexspline for a harmonic drive by finite element modelling[J]. Materials & Design, 2007, 28(2): 441－447.

[18] 唐小歡，王湘江，馮棟彥. 諧波減速器輸入軸遲滯曲線的特性研究[J]. 機(jī)械傳動(dòng)，2016(8)：29－32. Tang Xiaohuan, Wang Xiangjiang, Feng Dongyan. Study on hysteresis curve characteristic of input shaft of harmonic reducer[J]. Journal of Mechanical Transmission, 2016(8): 29－32. (in Chinese with English abstract)

[19] Tian Lin, Jiang Yi, Wang Yazhen, et al. Load distribution research on flexible bearing in harmonic drive[C]//International Conference on Mechanical Design. Springer, Singapore, 2017: 165－176.

[20] 劉正寧. 諧波齒輪傳動(dòng)柔性軸承受力分析[J]. 大連大學(xué)學(xué)報(bào)，1995(4)：512－517. Liu Zhengning. Analysing stress of fIexible bearing under harmonic gear drive[J]. Journal of Dalian University, 1995(4): 512－517. (in Chinese with English abstract)

[21] Ostapski W. Analysis of the stress state in the harmonic drive generator-flexspline system in relation to selected structural parameters and manufacturing deviations[J]. Bulletin of the Polish Academy of Sciences: Technical Sciences, 2010, 58(4): 683－698.

[22] Ostapski W, Mukha I. Stress state analysis of harmonic drive elements by FEM[J]. Bulletin of the Polish Academy of Sciences: Technical Sciences, 2007, 55(1): 115－123.

[23] Pacana J, Witkowski W, Mucha J. FEM analysis of stress distribution in the hermetic harmonic drive flexspline[J]. Strength of Materials, 2017, 49(1):1－11.

[24] Li Junyang, Wang Jiaxun, Fan Kaijie, et al. Accelerated life model for harmonic drive under adhesive wear[J]. Tribology, 2016(3): 297－303.

[25] Liang Yong, Fan Yuanxun. Analysis of tooth deflection of flexspline and its influence on harmonic drive[J]. Journal of Mechanical Transmission, 2018(2): 31－35.

[26] Gravagno F, Mucino V H, Pennestrì, et al. Influence of wave generator profile on the pure kinematic error and centrodes of harmonic drive[J]. Mechanism and Machine Theory, 2016, 104: 100－117.

[27] Dennis León, Arzola N, Andrés Tovar. Statistical analysis of the influence of tooth geometry in the performance of a harmonic drive[J]. Journal of the Brazilian Society of Mechanical Sciences & Engineering, 2015, 37(2):723－735.

[28] 沙曉晨，范元?jiǎng)? 諧波減速器傳動(dòng)誤差的研究[J]. 機(jī)械制造與自動(dòng)化，2015(5)：50－54. Sha Xiaochen, Fan Yuanxun. Study of transmission error of harmonic drive reducer[J]. Machine Building & Automation, 2015(5): 50－54. (in Chinese with English abstract)

[29] Ye Zhenhuan, Liu Zhansheng, Wang Liqin. Optimization analysis of structure parameter of angular-contact ball bearings based on fatigue life[J]. Journal of Mechanical Transmission, 2016(1): 59－63.

[30] Kim S W, Kang K , Yoon K, et al. Design optimization of an angular contact ball bearing for the main shaft of a grinder[J]. Mechanism & Machine Theory, 2016, 104: 287－302.

Fast solution for deformation characteristics of flexible bearing of robot based on thin-walled ring theory

Wang Yazhen1, Wang Anming1, Zhao Kun2, Song Li2

(1.201900,; 2.315301,)

Agricultural robots are an important part of the modern agriculture, and its function execution is mainly accomplished by robotic arm. Flexible bearing of the harmonic driver is a key part of the joint of the robots' arm. At present, there is little research on the design of flexible bearings. Different from most rolling bearings which are used as supporting components in ordinary, the flexible bearings are used as transmission components. A non-circular cam which is called wave generator is assembled into the inner ring of the bearing before working and causes a greatly pre-deformation of the flexible bearing. Previous theories of rolling bearing will not applicable because of this deformation. Meanwhile, a pair of radial force in the opposite direction is applied to both ends of the long axis of deformation of the flexible bearing in the transmission process. Therefore, more complicated shape state and bending stresses are generated. Those unusual working conditions will lead the flexible bearing more prone to damage than ordinary bearing. Although FEA simulation can get relatively accurate results, it usually takes several hours to solve finite element model, and the calculation process is difficult to converge. Therefore, it is necessary to establish a new calculation method to obtain the performance of flexible bearings. In this paper, the stress and deformation characteristics of the inner and outer rings of the flexible bearing were solved separately by the following methods: 1) First, the outer ring was equivalent to a statically indeterminate structure. The deformation coordination equation of the outer ring was established according to the equivalent model and solved by Mohr's integral theory. Overall bending moment of the outer ring of the flexible bearing formed by deformation was obtained. 2) Combined the theory of thin-walled ring with the bending moment equation which was obtained above, the radial and circumferential deformation characteristics of the outer ring of the flexible bearing were obtained. 3) According to the theory of multi span beam which was introduced in mechanics of materials, the loading model of the outer ring of the flexible bearing was built into three moment equations. The maximum stress was obtained by combining the three moment equation and the loading formula which was summarized by Ivanov’s experiment. Above theoretical equations were compiled by MATLAB programs. Finally, a finite element simulation model of flexible bearing was established by ANSYS Workbench. The time consumed by simulation was about 4 - 5 hours while the calculation of theoretical equations only needed 5-8 min. By comparison, the maximum error between simulation value and theoretical value was only within 7%, it proved the correctness of the theoretical model calculation. Through analysis of bending stress and deformation of the flexible bearing, conclusions can be drawn as the following: 1) the force state of the outer ring was different from the inner one during rotation of flexible bearings in harmonic drive, cyclic deformation of the outer ring caused a large alternating bending stress and prone to fatigue failure; 2) stress of the outer ring formed by deformation increased sharply with the thickness while stress caused by external load would decline. And width only affected stress caused by external load. Increase of thickness was beneficial to carrying capacity of flexible bearing. However, increase of thickness lead to increasing of bending stress formed by deformation. The total bending stress had a minimum value in optimal thickness. 3) Width was an important parameter which had a greater effect on the carrying capacity of the bearing, but it was constrained by external structural and cannot be too large. Calculation results would provide a theoretical reference for the design and optimization of flexible bearings.

robots; bearings; models; thin-walled ring theory; pre-deformation

王亞珍，汪安明，趙 坤，宋 麗. 基于薄壁圓環(huán)理論的機(jī)器人用柔性軸承變形特征快速求解[J]. 農(nóng)業(yè)工程學(xué)報(bào)，2019，35(3)：60－66.doi：10.11975/j.issn.1002-6819.2019.03.008 http://www.tcsae.org

Wang Yazhen, Wang Anming, Zhao Kun, Song Li. Fast solution for deformation characteristics of flexible bearing of robot based on thin-walled ring theory[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2019, 35(3): 60－66. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2019.03.008 http://www.tcsae.org

2018-10-08

2018-12-30

國(guó)家“863”計(jì)劃（2015AA043005）；寧波市科技攻關(guān)項(xiàng)目（2014B1006）；寧波市科技創(chuàng)新團(tuán)隊(duì)項(xiàng)目（2015B11012）

王亞珍，副研究員，研究方向?yàn)檩S承摩擦學(xué)及優(yōu)化設(shè)計(jì)。 Email：meyzwang@shu.edu.cn

10.11975/j.issn.1002-6819.2019.03.008

TH133.33; TH113

A

1002-6819(2019)-03-0060-07