法蘭成型機(jī)傳動(dòng)系統(tǒng)設(shè)計(jì)【型鋼卷圓機(jī)傳動(dòng)箱的傳動(dòng)系統(tǒng)】,型鋼卷圓機(jī)傳動(dòng)箱的傳動(dòng)系統(tǒng),法蘭成型機(jī)傳動(dòng)系統(tǒng)設(shè)計(jì)【型鋼卷圓機(jī)傳動(dòng)箱的傳動(dòng)系統(tǒng)】,法蘭,成型,傳動(dòng)系統(tǒng),設(shè)計(jì),型鋼,卷圓機(jī),傳動(dòng) 江西農(nóng)業(yè)大學(xué)畢業(yè)設(shè)計(jì)(論文)任務(wù)書(shū) 設(shè)計(jì)（論文） 課題名稱 法蘭成型機(jī)傳動(dòng)系統(tǒng)設(shè)計(jì) 學(xué)生姓名 院（系） 工學(xué)院 專 業(yè) 機(jī)械設(shè)計(jì)制造及其自動(dòng)化 指導(dǎo)教師 職 稱 副教授 學(xué) 歷 畢業(yè)設(shè)計(jì)（論文）要求： 1、 能獨(dú)立擬定設(shè)計(jì)方案，提出方案的構(gòu)思以及技術(shù)、經(jīng)濟(jì)條件等方面的可行性論證報(bào)告。 2、 能熟練應(yīng)用已學(xué)過(guò)的理論知識(shí)，采用工程分析計(jì)算方法或數(shù)值計(jì)算方法，正確完成設(shè)計(jì)中的計(jì)算工作。 3、 能熟練掌握機(jī)械制圖的方法和技巧，并運(yùn)用計(jì)算機(jī)繪圖、計(jì)算機(jī)輔助設(shè)計(jì)等，按國(guó)家標(biāo)準(zhǔn)正確地完成繪圖工作。 4、 能按設(shè)計(jì)任務(wù)書(shū)的要求，編寫出設(shè)計(jì)說(shuō)明書(shū)。 畢業(yè)設(shè)計(jì)（論文）內(nèi)容與技術(shù)參數(shù)： 1、 完成傳動(dòng)系統(tǒng)設(shè)計(jì)，確定傳動(dòng)系統(tǒng)各部分尺寸大小。 2、 設(shè)計(jì)傳動(dòng)系統(tǒng)的方案。 3、 畫(huà)出法蘭成型機(jī)傳動(dòng)系統(tǒng)的裝配圖和主要零件圖。 4、 編寫設(shè)計(jì)說(shuō)明書(shū)。 畢業(yè)設(shè)計(jì)（論文）工作計(jì)劃： 1、 調(diào)查實(shí)習(xí)、查閱文獻(xiàn)、收集資料：2008.12～2009.1 2、 方案選擇設(shè)計(jì)：2009.2 3、 總體設(shè)計(jì)：2009.2 4、 詳細(xì)計(jì)算、結(jié)構(gòu)設(shè)計(jì)：2009.3 5、 工程圖的繪制：2009.4 6、 編寫設(shè)計(jì)說(shuō)明書(shū)：2009.4 7、 修改設(shè)計(jì)、準(zhǔn)備答辯：2009.5 接受任務(wù)日期 2008 年 12 月 1 日 要求完成日期 2009 年 5 月 10 日 學(xué) 生 簽 名 年 月 日 指導(dǎo)教師簽名 年 月 日 院長(zhǎng)（主任）簽名 年 月 日 余弦齒輪傳動(dòng)的傳動(dòng)特性分析 摘要：本文將基于數(shù)學(xué)模型分析一種的新型余弦齒輪傳動(dòng)的幾個(gè)特性，比如重合度、滑動(dòng)系數(shù)、接觸應(yīng)力和彎曲應(yīng)力等。同時(shí)還與漸開(kāi)線齒輪傳動(dòng)的這些特性進(jìn)行了對(duì)比研究。分析了一些設(shè)計(jì)參數(shù)對(duì)傳動(dòng)的影響，包括輪齒的數(shù)目、壓力角、接觸應(yīng)力及彎曲應(yīng)力等。并且驗(yàn)證了以下結(jié)論：余弦齒輪傳動(dòng)的重合度大約為1.2到1.3左右，與漸開(kāi)線齒輪傳動(dòng)相比縮減了20%；余弦齒輪傳動(dòng)的滑動(dòng)系數(shù)小于漸開(kāi)線齒輪傳動(dòng)；余弦齒輪傳動(dòng)的接觸應(yīng)力和彎曲應(yīng)力比漸開(kāi)線齒輪傳動(dòng)低；隨著輪齒數(shù)目的增加以及壓力角的增大，其接觸應(yīng)力和彎曲應(yīng)力會(huì)逐漸降低。 關(guān)鍵詞：齒輪傳動(dòng) 余弦齒形 重合度 滑動(dòng)系數(shù) 應(yīng)力 引言 目前，在齒輪的設(shè)計(jì)中，漸開(kāi)線齒輪、圓形齒輪及擺線針輪行星傳動(dòng)這三種類型被廣泛應(yīng)用。由于其不同的優(yōu)缺點(diǎn)，它們被應(yīng)用于各種不同的場(chǎng)合。隨著計(jì)算機(jī)數(shù)字控制技術(shù)（數(shù)控）的發(fā)展，大量文獻(xiàn)提出了有關(guān)齒輪成形的結(jié)構(gòu)和方法等方面的研究報(bào)告。ARIGA等人[2]利用一種結(jié)合了圓弧和漸開(kāi)線的齒輪銑刀制造出新型的“維爾德哈貝爾-諾維科夫”齒輪。這種特殊的齒形可以解決常規(guī)W-N齒輪對(duì)中心距變化敏感的問(wèn)題。TSAY等人[3]研究了一種由漸開(kāi)線及圓弧夠成的螺旋齒輪，這種齒輪在任何時(shí)刻的齒面接觸都是一個(gè)點(diǎn)而不是一條直線。KOMORI等[4]開(kāi)發(fā)了一種邏輯齒輪，其在各接觸點(diǎn)的相對(duì)曲率為零。這種齒輪與漸開(kāi)線齒輪相比具有更高的耐久性和強(qiáng)度。ZHAO等人[5]提出了微線段齒輪的生成過(guò)程。ZHANG等人[6]提出了雙漸開(kāi)線曲線的概念，這是一種聯(lián)系在一起的過(guò)度曲線，并最終形成階梯形的齒牙。 LUO等人[7]提出了余弦齒輪傳動(dòng)，它采用了余弦曲線的零線作為分度圓，余弦曲線的波長(zhǎng)作為齒間距，而齒頂高就是余弦曲線的振幅。如圖.1所示，在分度圓附近或以上的區(qū)域即齒頂高部分，余弦齒輪的齒廓與漸開(kāi)線齒輪非常接近。但在齒根區(qū)域，余弦齒輪的齒厚比漸開(kāi)線齒輪的齒厚更大。 在數(shù)學(xué)模型中，基于齒輪嚙合理論，很多方程式包括余弦齒輪齒廓方程、共軛齒廓方程及運(yùn)動(dòng)路線方程等都已建立。同時(shí)還建立了余弦齒輪的實(shí)體模型，并對(duì)齒輪傳動(dòng)的嚙合進(jìn)行了仿真分析[8]。這項(xiàng)工作的目的就是在于分析余弦齒輪傳動(dòng)的特性。接下來(lái)的文章將分為三節(jié)。第一節(jié)，主要是對(duì)余弦齒輪傳動(dòng)數(shù)學(xué)模型的介紹。第二節(jié)，主要對(duì)余弦齒輪傳動(dòng)的幾個(gè)特性進(jìn)行了分析，包括重合度、滑動(dòng)系數(shù)、接觸應(yīng)力及彎曲應(yīng)力等。并與漸開(kāi)線齒輪傳動(dòng)的這些特性進(jìn)行了對(duì)比研究。分析一些設(shè)計(jì)參數(shù)對(duì)齒輪傳動(dòng)的影響，包括輪齒的數(shù)目、壓力角、接觸應(yīng)力及彎曲應(yīng)力等。最后將在第三節(jié)對(duì)研究進(jìn)行總結(jié)。 圖.1 余弦齒輪與漸開(kāi)線齒輪 1 余弦齒輪傳動(dòng)的數(shù)學(xué)模型 根據(jù)參考文獻(xiàn)[8]，余弦齒廓、共軛齒廓及運(yùn)動(dòng)路線方程可以表示成如下方程式 x1=mZ12+hcos(Z1θ)sinθy1=mZ12+hcos(Z1θ)cosθ (1) x2=mZ12+hcosZ1θsinθ-1+1iφ1+asinφ1iy2=mZ12+hcosZ1θcosθ-1+1iφ1-asinφ1i (2) x=mZ12+hcosZ1θsinθ-φ1y=mZ12+hcosZ1θcosθ-φ1-mZ12 (3) 式中：m和Z1 代表模量和齒數(shù)， h、I 和 a 分別表示齒頂高、重合度和中心距， θ是相對(duì)于1O1,x1,y1 坐標(biāo)系的旋轉(zhuǎn)角如圖.2所示， β是余弦曲線上任意點(diǎn)處的切線與x1 軸的交角， φ1是齒輪1的旋轉(zhuǎn)角，可以通過(guò)如下公式得到 φ1=arcsinmZ12+hcosZ1θsinθ+βmZ12-ββ=arctan-mZ12+hcosZ1θtanθ-hZ1sinZ1θmZ12+hcosZ1θ-hZ1tanθsinZ1θ 圖.2 余弦齒輪傳動(dòng)的原理 2 余弦齒輪傳動(dòng)的特性 基于數(shù)學(xué)模型，分析余弦齒輪傳動(dòng)的三個(gè)重要特性：重合度、滑動(dòng)系數(shù)和應(yīng)力。包括將這些特性與漸開(kāi)線齒輪傳動(dòng)進(jìn)行對(duì)比研究。 2.1 重合度 重合度可以表示一對(duì)齒輪在嚙合時(shí)的平均輪齒對(duì)數(shù)，其定義為一對(duì)輪齒從剛開(kāi)始嚙合到分離時(shí)齒輪所旋轉(zhuǎn)的角度[9]。如圖.3所示，余弦齒輪的重合度可以如下表示： ε=φe-φf(shuō)2πZ1 (4) 式中：φe 和 φf(shuō) 分別表示當(dāng)x=xe及x=xf 時(shí)的旋轉(zhuǎn)角φ1，它們可以通過(guò)公式(3)計(jì)算得到。 圖.3 余弦齒輪傳動(dòng)的重合度 通過(guò)使用數(shù)學(xué)軟件Matlab，列舉了三個(gè)例子如數(shù)表1所示。同時(shí)在表1中還列出了漸開(kāi)線齒輪傳動(dòng)的參數(shù)，以方便進(jìn)行對(duì)比。根據(jù)表1可知，余弦齒輪傳動(dòng)的重合度為1.2到1.3左右，這比漸開(kāi)線齒輪傳動(dòng)的重合度縮減了20%。根據(jù)參考文獻(xiàn)[10-11]，在齒輪泵的應(yīng)用中，齒輪的重合度約為1.1到1.3，因此，余弦齒輪傳動(dòng)可以應(yīng)用于齒輪泵領(lǐng)域。 表1 余弦齒輪傳動(dòng)的重合度 齒數(shù) 齒數(shù) 模量 余弦 齒輪傳動(dòng) 漸開(kāi)線 齒輪傳動(dòng) Z1 Z2 mmm 15 32 3 1.264 1.575 17 40 3 1.243 1.614 21 60 3 1.240 1.677 2.2 滑動(dòng)系數(shù) 滑動(dòng)系數(shù)是指齒輪在一個(gè)嚙合周期的滑移量。由于摩擦變小，較低的滑動(dòng)系數(shù)將會(huì)有更大的動(dòng)力傳動(dòng)效率?；瑒?dòng)系數(shù)被定義為其滑動(dòng)弧長(zhǎng)的比例相當(dāng)于平面嚙合時(shí)的弧長(zhǎng)比例。滑動(dòng)系數(shù)U1和U2可以由如下公式表示[12]： U1=1-r2-Lr1+Li21U2=1-r1+Lr2-Li12 (5) 式中：r1和r2分別表示兩齒輪分度圓的半徑； L表示點(diǎn)H在P，x，y坐標(biāo)系的縱坐標(biāo)； H是接觸點(diǎn)法線與O1O2 線的交點(diǎn)，如圖.4所示。 圖.4 余弦齒輪傳動(dòng)的相當(dāng)滑動(dòng) i12=1i21=r2r1 因此，直線PH的斜率k可以由如下公式表示 k=-dxdy (6) 帶入公式（3）代人公式（6）可得： k=mZ12+hcosZ1θ1-φ1'cosθ-φ1-AmZ12+hcosZ1θ1-φ1'sinθ-φ1+B (7) 式中：φ1' 和 β' 分別是 φ1 和 β 與 θ 的差，可以表示成如下公式: φ1'=mZ12+hcosZ1θ1+β'cosθ+β-Cm2Z12-mZ12+hcosZ1θ2sin2θ+β-β' β'=D+EmZ12+hcosZ1θ-hZ1tanθsinZ1θ2+hZ12mZ12+hcosZ1θsinZ1θ+tan2θcosZ1θmZ12+hcosZ1θ-hZ1tanθsinZ1θ2 式中：A=hZ1sinθ-φ1sinZ1θ B=hZ1cosθ-φ1sinZ1θ C=2hZ1sinZ1θsinθ+β D=-mZ12+hcosZ1θsec2θ-2h2Z12sin2Z1θsec2θ E=h2Z13tanθsin2Z1θ-sinZ1θcosZ1θ 因此，點(diǎn)H在坐標(biāo)系P，x，y上的縱坐標(biāo)可以表示為： L=-kx0+y0 (8) 式中：（x0，y0）表示接觸點(diǎn)在坐標(biāo)系P，x，y上的坐標(biāo)。將公式（3）和公式（7）代人公式（8）可得： L=F-GmZ12+hcosZ1θ1-φ1'sinθ-φ1+hZ1cosθ-φ1sinZ1θ+12mZ1+hcosZ1θcosθ-φ1-12mZ1 (9) 式中： G=mZ12+hcosZ1θ21-φ1'sinθ-φ1cosθ-φ1 F=hZ1mZ12+hcosZ1θsin2θ-φ1sinZ1θ 而rk1 ，rk2 和 θ 可以由下列公式得到： rk1=mZ12+hcosZ1θrk2=rk12+a2-2rk1acosθ 將θ 和公式（9）代人公式（5）就可得到滑動(dòng)系數(shù)。 這種齒輪被設(shè)計(jì)成模數(shù)m=3 mm，齒數(shù)Z1=35，傳動(dòng)比i=2 。漸開(kāi)線齒輪的壓力角為200，余弦齒輪的壓力角為220。根據(jù)公式（5）-（9），建立余弦齒輪傳動(dòng)的主動(dòng)輪及從動(dòng)輪的滑動(dòng)系數(shù)曲線圖，如圖.5所示。同時(shí)，為了方便進(jìn)行對(duì)比，在圖.5上還畫(huà)出了漸開(kāi)線齒輪傳動(dòng)的滑動(dòng)系數(shù)[13]。根據(jù)圖.5可知余弦齒輪傳動(dòng)的滑動(dòng)系數(shù)小于漸開(kāi)線齒輪傳動(dòng)，這可以幫助改善其傳動(dòng)性能。 (a) 主動(dòng)輪 (b) 從動(dòng)輪 圖.5 余弦齒輪傳動(dòng)的滑動(dòng)系數(shù) 2.3 接觸應(yīng)力和彎曲應(yīng)力 一般情況下，組成一個(gè)有限元模型的有限單元越多，其分析的結(jié)果越精確。然而，整個(gè)齒輪傳動(dòng)的有限元模型是首選地，特別是考慮到計(jì)算機(jī)的內(nèi)存限制和節(jié)約計(jì)算時(shí)間的需要。本文建立了余弦齒輪傳動(dòng)的三種接觸齒形的有限元模型。其中兩個(gè)模型是基于真實(shí)的齒輪幾乎尺寸，使用Pro/E軟件建立齒輪的齒形，并輸出IGES格式文件 ，然后輸入ANSYS軟件進(jìn)行應(yīng)力分析。 使用下列設(shè)計(jì)參數(shù)對(duì)余弦齒輪傳動(dòng)進(jìn)行數(shù)值計(jì)算：Z1=25，Z2=40，m=3 mm，α=220 ，寬度b=75 mm?；诹W(xué)性能的彈性模量E=210 Gpa。泊松比μ=0.29。 扭矩為98790 N?mm。每個(gè)模型的兩面應(yīng)盡量的遠(yuǎn)，圓角的選擇應(yīng)足以適用沿邊界的剛性約束。選擇輪齒下面足夠大的部分作為固定邊界。網(wǎng)狀區(qū)域使用平面-82單元。有限元模型如圖.6所示，共有3373個(gè)單元和10053個(gè)節(jié)點(diǎn)?？紤]了有關(guān)接觸的兩個(gè)問(wèn)題：微小滑動(dòng)和無(wú)摩擦。圖.7展示了馮-米塞斯應(yīng)力的等高線圖。計(jì)算結(jié)果在填入表2。 圖.6 有限元分析的應(yīng)用模型 圖.7 余弦齒輪傳動(dòng)的應(yīng)力分布（MPa） 表2 最大彎曲應(yīng)力和接觸應(yīng)力 MPa 齒輪 接觸應(yīng)力 彎曲應(yīng)力 彎曲應(yīng)力 σc （張力）σbt （壓力）σbc 余弦齒輪 498.98 86.04 95.59 漸開(kāi)線齒輪 641.58 115.24 134.00 圖.8 為在相同參數(shù)下的漸開(kāi)線齒輪傳動(dòng)的應(yīng)力分布圖，為了方便進(jìn)行對(duì)比。在輪齒圓角接觸面獲得的彎曲應(yīng)力視為拉伸應(yīng)力，而在輪齒背面的視為壓縮應(yīng)力。 圖.8 漸開(kāi)線齒輪傳動(dòng)的應(yīng)力分布（MPa） 從獲得的數(shù)值結(jié)果中可以得到以下結(jié)論：與漸開(kāi)線齒輪相比，改成余弦后期最大接觸應(yīng)力減速了約22.23%；余弦齒輪彎曲應(yīng)力中的拉伸應(yīng)力比漸開(kāi)線齒輪減少了25.34%，而壓縮應(yīng)力比漸開(kāi)線齒輪減少了28.67%；余弦齒輪在應(yīng)用中允許減少其接觸和彎曲應(yīng)力。 2.4 設(shè)計(jì)參數(shù)對(duì)應(yīng)力的影響 用兩個(gè)例子，在有限元模型的基礎(chǔ)上對(duì)設(shè)計(jì)參數(shù)的影響進(jìn)行說(shuō)明，設(shè)計(jì)參數(shù)包括輪齒數(shù)目、壓力角、接觸和彎曲應(yīng)力等 例子1：齒輪的壓力角α=220，在分度圓上，模量m=3 mm，寬b=75 mm。其他主要參數(shù)在表.3中顯示 表3 齒輪的主要設(shè)計(jì)參數(shù)（例子1） 序號(hào) 齒數(shù) Z1 傳動(dòng)比 i 1 20 1.6 2 25 1.6 3 30 1.6 使用上述材料參數(shù)，通過(guò)ANSYS軟件同時(shí)對(duì)三組余弦齒輪的接觸和彎曲應(yīng)力進(jìn)行分析。結(jié)果如圖.9，圖.7及圖.10所示，接觸與彎曲應(yīng)力的數(shù)值如表4所示。根據(jù)表4可知隨著輪齒數(shù)目的增加，接觸應(yīng)力和彎曲應(yīng)力會(huì)逐漸減小。此例子中，當(dāng)齒數(shù)Z1=20時(shí)，其接觸應(yīng)力、拉伸和壓縮彎曲應(yīng)力分別為569.76MPa、117.51MPa和124.98MPa，當(dāng)齒數(shù)Z1=30時(shí)，它們分別為410.61MPa、64.52MPa和74.41MPa。 圖.9 余弦齒輪傳動(dòng)的應(yīng)力分析（Z1=20）(MPa) 圖.10 余弦齒輪傳動(dòng)的應(yīng)力分布（Z1=30）(MPa) 表4 余弦齒輪在不同齒數(shù)下的應(yīng)力 MPa 齒數(shù) 接觸應(yīng)力 彎曲應(yīng)力 彎曲應(yīng)力 Z1 σc （拉伸）σbt （壓縮）σbc 20 569.76 117.51 124.98 25 498.98 86.04 95.59 30 410.61 64.52 74.41 例子2：齒輪的模量m=3 mm，齒數(shù)Z1=25，寬b=75 mm。其他主要參數(shù)如表5所示。 表5 齒輪的主要計(jì)算參數(shù)（例子2） 序號(hào) 壓力角 α/(0) 傳動(dòng)比 i 1 22 1.6 2 23 1.6 3 24 1.6 使用上述材料參數(shù)，通過(guò)ANSYS軟件對(duì)其接觸應(yīng)力和彎曲應(yīng)力進(jìn)行分析。結(jié)果如圖.7、圖.11和圖.12所示，接觸應(yīng)力和彎曲應(yīng)力的數(shù)值如表6所示。 圖.11 余弦齒輪傳動(dòng)的應(yīng)力分析（α=230）（MPa） 圖.12 余弦齒輪傳動(dòng)的應(yīng)力分布（α=240）（MPa） 表6 不同壓力角下余弦齒輪的應(yīng)力 壓力角 接觸應(yīng)力 彎曲應(yīng)力 彎曲應(yīng)力 α/(0) σc （拉伸）σbt （壓縮）σbc 22 498.98 86.04 95.59 23 448.96 80.89 91.02 24 395.43 71.81 86.32 根據(jù)表6，接觸應(yīng)力和彎曲應(yīng)力的大小隨著壓力角的增大而減小。此例子中，當(dāng)壓力角α=220時(shí)，其接觸應(yīng)力、拉伸和壓縮彎曲應(yīng)力分布為498.98MPa、86.04MPa和95.59MPa，當(dāng)壓力角α=240時(shí)，它們分別為395.43MPa、71.84MPa和86.32MPa。 3 總結(jié) 研究了一種新型的齒輪傳動(dòng)——余弦齒輪傳動(dòng)。這種齒輪以余弦曲線作為齒廓。基于數(shù)學(xué)模型對(duì)余弦齒輪的特性進(jìn)行了研究，包括重合度、滑動(dòng)系數(shù)和應(yīng)力。分析了設(shè)計(jì)參數(shù)的影響，包括輪齒數(shù)目、分度圓上的壓力角及應(yīng)力等。研究所得到的結(jié)果得出了以下結(jié)論。 （1）根據(jù)表1，余弦齒輪傳動(dòng)的重合度約為1.2到1.3，比漸開(kāi)線齒輪傳動(dòng)縮減了20%。 （2）根據(jù)圖.5 余弦齒輪傳動(dòng)的滑動(dòng)系數(shù)略低于漸開(kāi)線齒輪傳動(dòng)。 （3）余弦齒輪傳動(dòng)的接觸和彎曲應(yīng)力比漸開(kāi)線齒輪傳動(dòng)低。研究顯示，在第2節(jié)所給出的參數(shù)下，余弦齒輪傳動(dòng)的最大接觸應(yīng)力與漸開(kāi)線齒輪傳動(dòng)相比減小了22.23%，其壓縮彎曲應(yīng)力與漸開(kāi)線齒輪傳動(dòng)相比減小了28.67%。 （4）根據(jù)有限元模型例子可得，接觸應(yīng)力和彎曲應(yīng)力都隨著齒數(shù)和壓力角的增大而減小。 （5）余弦齒輪傳動(dòng)是一種新型的齒輪傳動(dòng)，因此，其他的一些特性，如檢測(cè)、對(duì)中心距變化的敏感度以及其制造過(guò)程等都應(yīng)在將來(lái)進(jìn)行仔細(xì)的研究分。 http:/ http:/ http:/ http:/ http:/ 余弦齒輪傳動(dòng)的傳動(dòng)特性分析 Wang jian Luo shanming Chen lifeng Hu huaring School of electromechanical engineering Hunan university of science, and technology, Xiangtan 411201,china Abstract: Based on the mathematical model of a novel cosine gear drive, a few characteristics, such as the contact ratio, the sliding coefficient, and the contact and bending stresses, of this drive are analyzed. A comparison study of these characteristics with the involute gear drive is also carried out. The influences of design parameters including the number of teeth and the pressure angle on the contact and bending stresses are studied. The following conclusions are achieved: the contact ratio of the cosine gear drive is about 1.2 to 1.3, which is reduced by about 20% in comparison with that of the involute gear drive. The sliding coefficient of the cosine gear drive is smaller than that of the involute gear drive. The contact and bending stresses of the cosine gear drive are lower then those of the involute gear drive. The contact and bending stresses decrease with the growth of the number of teeth and the pressure angle. Key words: Gear drive Cosine profile Contact ratio Sliding coefficient Stress 0 introduction Currently, the involute, the circular are and the cycloid profiles are three types of tooth profiles that are widely used in the gear design[1] . All of these gears used in different fields due to their different advantages and disadvantages. With the development of computerized numerical control (CNC) technology, a large amount of literature is presented in investigations on mechanisms and methods for tooth profile generation. ARIGA, et al[2] , used a cutter with combined circular-arc and involute tooth profiles to generate a new type of Wildhaber-Novikov gear. This particular tooth profile can solve the problem of conventional W-N gear profile, that is, the profile sensitivity to center distance variations. TSAY, et al[3], studied a helical gear drive whose profiles consist of involute and circular-arc. The tooth surfaces of this gearing contact with each other at every instant at a point instead of a line. KOMORI, et al[4], developed a gear with logic tooth profiles which have zero relative curvature at many contact points. The gear has higher durability and strength then involute gear. ZHAO, et al[5], introduced the generation process of a micro-segment gear. ZHANG, et al[6], presented a double involute curves, which are linked by a transition curve and form the ladder shape of tooth. LUO, et al[7], presented a cosine gear drive, which takes the zero line of cosine curve as the pitch circle, a period of the curve as a tooth space, and the amplitude of the curve as tooth addendum. As shown in Fig. 1, the cosine tooth profile appears very close to the involute tooth profile in the area near or above the pitch circle, i.e., the part of addendum. However, in area of dedendum, the tooth thickness of cosine gear is greater then that of involute gear. The mathematical models, including the equation of the cosine tooth profile, the equation of the conjugate tooth profile and the equation of the line of action, have been established based on the meshing theory. The solid model of cosine gear has been built , and the meshing simulation of this drive has also been investigated[8]. The aim of this work is to analyze the characteristics of the cosine gear drive. The remainder is organized in three sections 1, the mathematical models of the cosine gear drive are introduced. In section 2, the characteristics, including contact ratio, sliding coefficient, contact and bending stresses, of the cosine gear drive are analyzed, and a comparison study of these characteristics with the involute gear drive is also carried out. The influences of design parameters, including the number of teeth and the pressure angle, on contact and bending stresses are studied. Finally, a conclusion summary of this study is given in section 3. Fig . 1 1 Mathematical Model of the cosine gear drive According to Ref.[8], the equation of the cosine tooth profile, the conjugate tooth profile and the line of action can be expressed as follows 公式 Where m and Z1 represent the modulus and the number of teeth, respectively, h is the addendum, I and a denote the contact ratio and the center distance, respectively, θ is the rotation angle relative to system 1O1,x1,y1 as shown in Fig.2, β is the angle between x1-axis and the tangent of any point on the cosine profile, φ1 is the rotational angle of gear 1 which can be given as follows 公式 Fig.2 2 CHARACTERISTICS OF THE COSINE GEAR DRIVE Based on the mathematical model of the cosine gear drive, three characteristics, contact ratio, sliding coefficient, and stresses, are analyzed. In addition, all these characteristics are compared with those of the involute gears． 2.1 Contact ratio The contact ratio could be considered as an indication of average teeth-pairs in mesh of a gear-pair and naturally is ought to be defined according to the rotation angle of a gear from gear-in to gear-out of a pair of teeth[9] . As shown in Fig.3, the contact ratio of the cosine gear can be expressed as follows 公式 where and are the values of rotation angle as = and = ， respectively, which can be calculated by Eq．(3)． Fig.3 Contact ratio of the cosine gear drive Three examples as shown in Table 1 have been carried out by using program Matlab．The contact ratios of the involute gear drives with the same parameters are also shown in Table 1 for the purpose of comparison. According to Table 1, the contact ratio of the cosine gear drive is about 1.2 to 1.3, which is about 20% less than that of the involute gear drive. According to Refs．[10-11], the contact ratio of gears applied in gear pump is about 1.1 to 1,3, therefore, such cosine gear drive can be applied in the field of gear pump. Table 1 2.2 Sliding coefficient Sliding coefficient is a measure of the sliding action during the meshing cycle. A lower coefficient will have greater power transmission efficiency because of the less friction. The sliding coefficient is defined as the limit of the ratio of the sliding arc length to the corresponding arc length in plane meshing. The sliding coefficients U1 and U2 can be expressed as follows[12] 公式 Where and denote the radius of the pitch circle，respectively，L represents the vertical coordinate of point H in coordinate system ，H is the intersection point of the normal line of the contact point and the line ，as shown in Fig．4． FIG.4 Therefore，slope k of the straight line PH can be expressed as follows 公式6 Substituting Eq．(3) into Eq．(6) gives 公式7 where and are the differential coefficients of and to , respectively, which can be expressed as 公式 Therefore， the vertical coordinate of the point H in coordinate system can be expressed as follows 公式8 Where (x0，Y0，) denotes the coordinate of the contact point in coordinate system ．Substituting Eq．(3)and Eq．(7) into Eq．(8) gives 公式9 Substituting 0 and Eq．(9) into Eq．(5)，the sliding coefficients can be obtained． The gears are designed to have a module of m=3 mm．a(chǎn) number of teeth of Z1=35，and a transmission ratio of i=2．The pressure angle of the involute gear is 20o．while it is 22。 for the cosine gear．According to Eqs．(5)-(9)，a computer simulation to plot the graphs of sliding coefficients for the driving and the driven gears of the cosine gear drive is developed as shown in Fig．5．The sliding coefficients of the involute gear drive [13] are also listed in Fig.5 for the purpose of comparison. According to Fig.5 the sliding coefficients of the cosine gear drive is smaller than that of the involute gear drive. which can help to improve the transmission performance. 圖5 2.3 Contact and bending stresses In general, an FEA model with a larger number of elements for finite element stress analysis may lead to more accurate results. However, an FEA model of the whole gear drive is not preferred, especially considering the limit of computer memories and the need for saving computational time．This paper establishes an FEA model of three pairs of contact teeth for the cosine gear drive. Two models of contacting teeth based on the real geometry of the pinion and the gear teeth surfaces created in Pro/Engineer are exported as a IGES file which is then imported into the software Ansys for stress an analysis. The numerical computations have been performed for the cosine drive with the following design parameters：Z1=25，Z2=40。 m=3 mm，a=22。，a width of b=75 mm．The basic mechanical properties are modulus of elasticity E = 210 GPa．a(chǎn)nd Poisson’s ratio = 0．29． The torque is 98790 N ·mm．Two sides of each model sufficiently far from the fillet are chosen to justify the rigid constraints applied along the boundaries．A large enough part of the wheel below the teeth is chosen for the fixed boundary．Areas are meshed by using plane-82 elements．The finite element models are shown in Fig.6, and there are 3373 elements and 10053 nodes．Two options related to the contact problem. Small sliding and no friction have been selected ．Fig．7 shows the contour plot of Von-Mises stress．The numerical results are listed in Table 2． 圖6 Tu7 Table 2 Under the same parameters，stress distribution of an involute gear drive shown in Fig．8 is also analyzed for the purpose of comparison．The bending stress obtained in the fillet of the contacting tooth side are considered as tension stresses，and those in the fillet of the opposite tooth side are considered as compression stresses. Tu8 From the obtained numerical results， the following conclusions can be made：the maximum contact stress of the cosine Rear is reduced by about 22．23％ in comparison with the involute gear．The tension bending stress of the cosine gear is 25．34％ less than that of the involute gear, and the compression bending stress is reduced by about 28．67％ in comparison with the involute gear．An application of a cosine tooth profile allows reducing both，contact and bending stresses． 2．4 Influences of design parameters on stresses Based on the finite element models，two examples are used to clarify the influences of design parameters including the number of teeth and the pressure angle on contact and bending stresses． Example l：the gears are designed to have a pressure angle of a=22o. at the pitch circle，a module of m =3 mm。a width of b=75 mm．The other main parameters are shown in Table 3． Table3 With the same material parameters as aforementioned，the contact and bending stresses of three sets of cosine gears are analyzed by using program Ansys．Results are shown in Fig．9，F(xiàn)ig．7 and Fig．10，and the values of the contact and bending stresses are shown in Table.4 According to Table 4. both the contact and bending stresses decrease with the growth of the number of teeth．For instance，the contact stress，tension and compression bending stresses are 569．76 MPa．11 7．5 1 MPa and 124．98 MPa，respectively，as the number of teeth Z1=20，while 410．61 Mpa．64．52Mpa and 74．41 MPa as the number of teeth Z1=30． Tu9 Tu10 Table 4 Example 2：the gears are designed to have a module of m=3mm，number of teeth Zt=25，a width of b=75mm．The other main parameters are shown in Table 5． Table5 With the same material parameters as aforementioned, the contact and bending stresses are also computed by using program Ansys．Results are shown in Fig．7，F(xiàn)ig．11 an d Fig．12，and the values of the contact and bending stresses are shown in Table 6． Tu11 Tu12 Table6 According to Table 6，the contact and bending stresses decrease with the growth of the pressure angle．For instance，the contact stress，tension and compression bending stresses are498．98 M Pa．86．04 MPa and 95．59 MPa，respectively，as the pressure angle of =22。．while 395．43 MPa，7 1．8 1 MPa，and 86.32 MPa as the pressure angle of =24。． 3 CONCLUSIONS A new type of gear drives—a cosine gear drive is investigated．which takes a cosine curve as the tooth profile．Based on the mathematical model, the characteristics including the contact ratio．the sliding coefficient and stresses are studied．The effects of gear design parameters．such as the number of teeth，pressure angle at pitch circle，on stresses of cosine gears have also been analyzed．The results of performed research allow the following conclusions to be drawn． (1) The contact ratio of the cosine gear drive is about 1．2 to1．3．which is about 20％ less than that of the involute gear drive according to Table 1． (2)The sliding coefficient of the cosine gear drive is smaller than that of the involute gear drive according to Fig．5． (3)The contact and the bending stresses of the cosine gear drive are lower than that of the involute gear drive．For instance，under the given parameters as shown in section 2, the maximum contact stress of the cosine gear is reduced by about 22．23％ in comparison with the involute gear, and the compression bending stress is 28．67％ less than that of the involute gear． (4) Both the contact and bending stresses decrease with the growth of the number of teeth and the pressure angle according to simulation results of the example FE mode1． (5)The cosine gear drive is a new type of gear drives．Therefore．other characteristics such as inspection，sensitivity of center distance error of this drive and its manufacturing should be researched further.