法蘭成型機傳動系統(tǒng)設(shè)計【型鋼卷圓機傳動箱的傳動系統(tǒng)】

法蘭成型機傳動系統(tǒng)設(shè)計【型鋼卷圓機傳動箱的傳動系統(tǒng)】,型鋼卷圓機傳動箱的傳動系統(tǒng),法蘭成型機傳動系統(tǒng)設(shè)計【型鋼卷圓機傳動箱的傳動系統(tǒng)】,法蘭,成型,傳動系統(tǒng),設(shè)計,型鋼,卷圓機,傳動 江西農(nóng)業(yè)大學(xué)畢業(yè)設(shè)計(論文)任務(wù)書 設(shè)計（論文） 課題名稱 法蘭成型機傳動系統(tǒng)設(shè)計 學(xué)生姓名 院（系） 工學(xué)院 專 業(yè) 機械設(shè)計制造及其自動化 指導(dǎo)教師 職 稱 副教授 學(xué) 歷 畢業(yè)設(shè)計（論文）要求： 1、 能獨立擬定設(shè)計方案，提出方案的構(gòu)思以及技術(shù)、經(jīng)濟條件等方面的可行性論證報告。 2、 能熟練應(yīng)用已學(xué)過的理論知識，采用工程分析計算方法或數(shù)值計算方法，正確完成設(shè)計中的計算工作。 3、 能熟練掌握機械制圖的方法和技巧，并運用計算機繪圖、計算機輔助設(shè)計等，按國家標準正確地完成繪圖工作。 4、 能按設(shè)計任務(wù)書的要求，編寫出設(shè)計說明書。 畢業(yè)設(shè)計（論文）內(nèi)容與技術(shù)參數(shù)： 1、 完成傳動系統(tǒng)設(shè)計，確定傳動系統(tǒng)各部分尺寸大小。 2、 設(shè)計傳動系統(tǒng)的方案。 3、 畫出法蘭成型機傳動系統(tǒng)的裝配圖和主要零件圖。 4、 編寫設(shè)計說明書。 畢業(yè)設(shè)計（論文）工作計劃： 1、 調(diào)查實習(xí)、查閱文獻、收集資料：2008.12～2009.1 2、 方案選擇設(shè)計：2009.2 3、 總體設(shè)計：2009.2 4、 詳細計算、結(jié)構(gòu)設(shè)計：2009.3 5、 工程圖的繪制：2009.4 6、 編寫設(shè)計說明書：2009.4 7、 修改設(shè)計、準備答辯：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（主任）簽名 年 月 日 余弦齒輪傳動的傳動特性分析 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的幾個特性，比如重合度、滑動系數(shù)、接觸應(yīng)力和彎曲應(yīng)力等。同時還與漸開線齒輪傳動的這些特性進行了對比研究。分析了一些設(shè)計參數(shù)對傳動的影響，包括輪齒的數(shù)目、壓力角、接觸應(yīng)力及彎曲應(yīng)力等。并且驗證了以下結(jié)論：余弦齒輪傳動的重合度大約為1.2到1.3左右，與漸開線齒輪傳動相比縮減了20%；余弦齒輪傳動的滑動系數(shù)小于漸開線齒輪傳動；余弦齒輪傳動的接觸應(yīng)力和彎曲應(yīng)力比漸開線齒輪傳動低；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)力會逐漸降低。 關(guān)鍵詞：齒輪傳動 余弦齒形 重合度 滑動系數(shù) 應(yīng)力 引言 目前，在齒輪的設(shè)計中，漸開線齒輪、圓形齒輪及擺線針輪行星傳動這三種類型被廣泛應(yīng)用。由于其不同的優(yōu)缺點，它們被應(yīng)用于各種不同的場合。隨著計算機數(shù)字控制技術(shù)（數(shù)控）的發(fā)展，大量文獻提出了有關(guān)齒輪成形的結(jié)構(gòu)和方法等方面的研究報告。ARIGA等人[2]利用一種結(jié)合了圓弧和漸開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齒輪對中心距變化敏感的問題。TSAY等人[3]研究了一種由漸開線及圓弧夠成的螺旋齒輪，這種齒輪在任何時刻的齒面接觸都是一個點而不是一條直線。KOMORI等[4]開發(fā)了一種邏輯齒輪，其在各接觸點的相對曲率為零。這種齒輪與漸開線齒輪相比具有更高的耐久性和強度。ZHAO等人[5]提出了微線段齒輪的生成過程。ZHANG等人[6]提出了雙漸開線曲線的概念，這是一種聯(lián)系在一起的過度曲線，并最終形成階梯形的齒牙。 LUO等人[7]提出了余弦齒輪傳動，它采用了余弦曲線的零線作為分度圓，余弦曲線的波長作為齒間距，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線齒輪非常接近。但在齒根區(qū)域，余弦齒輪的齒厚比漸開線齒輪的齒厚更大。 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、共軛齒廓方程及運動路線方程等都已建立。同時還建立了余弦齒輪的實體模型，并對齒輪傳動的嚙合進行了仿真分析[8]。這項工作的目的就是在于分析余弦齒輪傳動的特性。接下來的文章將分為三節(jié)。第一節(jié)，主要是對余弦齒輪傳動數(shù)學(xué)模型的介紹。第二節(jié)，主要對余弦齒輪傳動的幾個特性進行了分析，包括重合度、滑動系數(shù)、接觸應(yīng)力及彎曲應(yīng)力等。并與漸開線齒輪傳動的這些特性進行了對比研究。分析一些設(shè)計參數(shù)對齒輪傳動的影響，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)對研究進行總結(jié)。 圖.1 余弦齒輪與漸開線齒輪 1 余弦齒輪傳動的數(shù)學(xué)模型 根據(jù)參考文獻[8]，余弦齒廓、共軛齒廓及運動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是相對于1O1,x1,y1 坐標系的旋轉(zhuǎn)角如圖.2所示， β是余弦曲線上任意點處的切線與x1 軸的交角， φ1是齒輪1的旋轉(zhuǎn)角，可以通過如下公式得到 φ1=arcsinmZ12+hcosZ1θsinθ+βmZ12-ββ=arctan-mZ12+hcosZ1θtanθ-hZ1sinZ1θmZ12+hcosZ1θ-hZ1tanθsinZ1θ 圖.2 余弦齒輪傳動的原理 2 余弦齒輪傳動的特性 基于數(shù)學(xué)模型，分析余弦齒輪傳動的三個重要特性：重合度、滑動系數(shù)和應(yīng)力。包括將這些特性與漸開線齒輪傳動進行對比研究。 2.1 重合度 重合度可以表示一對齒輪在嚙合時的平均輪齒對數(shù)，其定義為一對輪齒從剛開始嚙合到分離時齒輪所旋轉(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-φf2πZ1 (4) 式中：φe 和 φf 分別表示當x=xe及x=xf 時的旋轉(zhuǎn)角φ1，它們可以通過公式(3)計算得到。 圖.3 余弦齒輪傳動的重合度 通過使用數(shù)學(xué)軟件Matlab，列舉了三個例子如數(shù)表1所示。同時在表1中還列出了漸開線齒輪傳動的參數(shù)，以方便進行對比。根據(jù)表1可知，余弦齒輪傳動的重合度為1.2到1.3左右，這比漸開線齒輪傳動的重合度縮減了20%。根據(jù)參考文獻[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，余弦齒輪傳動可以應(yīng)用于齒輪泵領(lǐng)域。 表1 余弦齒輪傳動的重合度 齒數(shù) 齒數(shù) 模量 余弦 齒輪傳動 漸開線 齒輪傳動 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 滑動系數(shù) 滑動系數(shù)是指齒輪在一個嚙合周期的滑移量。由于摩擦變小，較低的滑動系數(shù)將會有更大的動力傳動效率?；瑒酉禂?shù)被定義為其滑動弧長的比例相當于平面嚙合時的弧長比例?；瑒酉禂?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H在P，x，y坐標系的縱坐標； H是接觸點法線與O1O2 線的交點，如圖.4所示。 圖.4 余弦齒輪傳動的相當滑動 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H在坐標系P，x，y上的縱坐標可以表示為： L=-kx0+y0 (8) 式中：（x0，y0）表示接觸點在坐標系P，x，y上的坐標。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）就可得到滑動系數(shù)。 這種齒輪被設(shè)計成模數(shù)m=3 mm，齒數(shù)Z1=35，傳動比i=2 。漸開線齒輪的壓力角為200，余弦齒輪的壓力角為220。根據(jù)公式（5）-（9），建立余弦齒輪傳動的主動輪及從動輪的滑動系數(shù)曲線圖，如圖.5所示。同時，為了方便進行對比，在圖.5上還畫出了漸開線齒輪傳動的滑動系數(shù)[13]。根據(jù)圖.5可知余弦齒輪傳動的滑動系數(shù)小于漸開線齒輪傳動，這可以幫助改善其傳動性能。 (a) 主動輪 (b) 從動輪 圖.5 余弦齒輪傳動的滑動系數(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，整個齒輪傳動的有限元模型是首選地，特別是考慮到計算機的內(nèi)存限制和節(jié)約計算時間的需要。本文建立了余弦齒輪傳動的三種接觸齒形的有限元模型。其中兩個模型是基于真實的齒輪幾乎尺寸，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軟件進行應(yīng)力分析。 使用下列設(shè)計參數(shù)對余弦齒輪傳動進行數(shù)值計算：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。每個模型的兩面應(yīng)盡量的遠，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個單元和10053個節(jié)點。考慮了有關(guān)接觸的兩個問題：微小滑動和無摩擦。圖.7展示了馮-米塞斯應(yīng)力的等高線圖。計算結(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 漸開線齒輪 641.58 115.24 134.00 圖.8 為在相同參數(shù)下的漸開線齒輪傳動的應(yīng)力分布圖，為了方便進行對比。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 漸開線齒輪傳動的應(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)力比漸開線齒輪減少了25.34%，而壓縮應(yīng)力比漸開線齒輪減少了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)計參數(shù)對應(yīng)力的影響 用兩個例子，在有限元模型的基礎(chǔ)上對設(shè)計參數(shù)的影響進行說明，設(shè)計參數(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)計參數(shù)（例子1） 序號 齒數(shù) Z1 傳動比 i 1 20 1.6 2 25 1.6 3 30 1.6 使用上述材料參數(shù)，通過ANSYS軟件同時對三組余弦齒輪的接觸和彎曲應(yīng)力進行分析。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)力會逐漸減小。此例子中，當齒數(shù)Z1=20時，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，當齒數(shù)Z1=30時，它們分別為410.61MPa、64.52MPa和74.41MPa。 圖.9 余弦齒輪傳動的應(yīng)力分析（Z1=20）(MPa) 圖.10 余弦齒輪傳動的應(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 齒輪的主要計算參數(shù)（例子2） 序號 壓力角 α/(0) 傳動比 i 1 22 1.6 2 23 1.6 3 24 1.6 使用上述材料參數(shù)，通過ANSYS軟件對其接觸應(yīng)力和彎曲應(yīng)力進行分析。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 余弦齒輪傳動的應(yīng)力分析（α=230）（MPa） 圖.12 余弦齒輪傳動的應(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，當壓力角α=220時，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，當壓力角α=240時，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——余弦齒輪傳動。這種齒輪以余弦曲線作為齒廓。基于數(shù)學(xué)模型對余弦齒輪的特性進行了研究，包括重合度、滑動系數(shù)和應(yīng)力。分析了設(shè)計參數(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，余弦齒輪傳動的重合度約為1.2到1.3，比漸開線齒輪傳動縮減了20%。 （2）根據(jù)圖.5 余弦齒輪傳動的滑動系數(shù)略低于漸開線齒輪傳動。 （3）余弦齒輪傳動的接觸和彎曲應(yīng)力比漸開線齒輪傳動低。研究顯示，在第2節(jié)所給出的參數(shù)下，余弦齒輪傳動的最大接觸應(yīng)力與漸開線齒輪傳動相比減小了22.23%，其壓縮彎曲應(yīng)力與漸開線齒輪傳動相比減小了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）余弦齒輪傳動是一種新型的齒輪傳動，因此，其他的一些特性，如檢測、對中心距變化的敏感度以及其制造過程等都應(yīng)在將來進行仔細的研究分。 http:/ http:/ http:/ http:/ http:/ 余弦齒輪傳動的傳動特性分析 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.