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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(lèi)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(Z1)siny1=mZ12+hcos(Z1)cos (1)x2=mZ12+hcosZ1sin-1+1i1+asin1iy2=mZ12+hcosZ1cos-1+1i1-asin1i (2)x=mZ12+hcosZ1sin-1y=mZ12+hcosZ1cos-1-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，是相對(duì)于1O1,x1,y1 坐標(biāo)系的旋轉(zhuǎn)角如圖.2所示，是余弦曲線上任意點(diǎn)處的切線與x1 軸的交角，1是齒輪1的旋轉(zhuǎn)角，可以通過(guò)如下公式得到1=arcsinmZ12+hcosZ1sin+mZ12-=arctan-mZ12+hcosZ1tan-hZ1sinZ1mZ12+hcosZ1-hZ1tansinZ1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：=e-f2Z1 (4)式中：e 和 f 分別表示當(dāng)x=xe及x=xf 時(shí)的旋轉(zhuǎn)角1，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)Z1Z2mmm153231.2641.575174031.2431.614216031.2401.6772.2 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?；瑒?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+hcosZ11-1cos-1-AmZ12+hcosZ11-1sin-1+B (7)式中：1 和 分別是 1 和 與 的差，可以表示成如下公式:1=mZ12+hcosZ11+cos+-Cm2Z12-mZ12+hcosZ12sin2+-=D+EmZ12+hcosZ1-hZ1tansinZ12+hZ12mZ12+hcosZ1sinZ1+tan2cosZ1mZ12+hcosZ1-hZ1tansinZ12式中：A=hZ1sin-1sinZ1 B=hZ1cos-1sinZ1 C=2hZ1sinZ1sin+ D=-mZ12+hcosZ1sec2-2h2Z12sin2Z1sec2 E=h2Z13tansin2Z1-sinZ1cosZ1 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+hcosZ11-1sin-1+hZ1cos-1sinZ1+12mZ1+hcosZ1cos-1-12mZ1 (9)式中： G=mZ12+hcosZ121-1sin-1cos-1 F=hZ1mZ12+hcosZ1sin2-1sinZ1而rk1 ，rk2 和 可以由下列公式得到：rk1=mZ12+hcosZ1rk2=rk12+a2-2rk1acos將 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，=220 ，寬度b=75 mm?；诹W(xué)性能的彈性模量E=210 Gpa。泊松比=0.29。扭矩為98790 Nmm。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c（張力）bt（壓力）bc余弦齒輪498.9886.0495.59漸開(kāi)線齒輪641.58115.24134.00 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=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 i1201.62251.63301.6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Z1c（拉伸）bt（壓縮）bc20569.76117.51124.9825498.9886.0495.5930410.6164.5274.41 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 /(0)傳動(dòng)比 i1221.62231.63241.6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（=230）（MPa）圖.12 余弦齒輪傳動(dòng)的應(yīng)力分布（=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/(0)c（拉伸）bt（壓縮）bc22498.9886.0495.5923448.9680.8991.0224395.4371.8186.32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=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=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?；跀?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。 余弦齒輪傳動(dòng)的傳動(dòng)特性分析Wang jianLuo shanmingChen lifengHu huaringSchool of electromechanical engineeringHunan 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 design1 . 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 al2 , 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 al3, 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 al4, 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 al5, introduced the generation process of a micro-segment gear. ZHANG, et al6, presented a double involute curves, which are linked by a transition curve and form the ladder shape of tooth. LUO, et al7, 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 investigated8. 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 . 11 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, is the rotation angle relative to system 1O1,x1,y1 as shown in Fig.2, is the angle between x1-axis and the tangent of any point on the cosine profile, 1 is the rotational angle of gear 1 which can be given as follows 公式 Fig.22 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 gears2.1 Contact ratioThe 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 teeth9 . 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 driveThree examples as shown in Table 1 have been carried out by using program MatlabThe 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 Refs10-11, 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 12.2 Sliding coefficientSliding 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 follows12 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 Fig4 FIG.4Therefore，slope k of the straight line PH can be expressed as follows 公式6Substituting Eq(3) into Eq(6) gives 公式7where 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8Where (x0，Y0，) denotes the coordinate of the contact point in coordinate system Substituting Eq(3)and Eq(7) into Eq(8) gives 公式9Substituting 0 and Eq(9) into Eq(5)，the sliding coefficients can be obtainedThe gears are designed to have a module of m=3 mma number of teeth of Z1=35，and a transmission ratio of i=2The pressure angle of the involute gear is 20owhile it is 22。 for the cosine gearAccording 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 Fig5The 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52.3 Contact and bending stressesIn 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 timeThis 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 mmThe basic mechanical properties are modulus of elasticity E = 210 GPaand Poissons ratio = 029 The torque is 98790 N mmTwo sides of each model sufficiently far from the fillet are chosen to justify the rigid constraints applied along the boundariesA large enough part of the wheel below the teeth is chosen for the fixed boundaryAreas are meshed by using plane-82 elementsThe finite element models are shown in Fig.6, and there are 3373 elements and 10053 nodesTwo options related to the contact problem. Small sliding and no friction have been selected Fig7 shows the contour plot of Von-Mises stressThe numerical results are listed in Table 2圖6 Tu7Table 2Under the same parameters，stress distribution of an involute gear drive shown in Fig8 is also analyzed for the purpose of comparisonThe 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.Tu8From the obtained numerical results， the following conclusions can be made：the maximum contact stress of the cosine Rear is reduced by about 2223 in comparison with the involute gearThe tension bending stress of the cosine gear is 2534 less than that of the involute gear, and the compression bending stress is reduced by about 2867 in comparison with the involute gearAn application of a cosine tooth profile allows reducing both，contact and bending stresses24 Influences of design parameters on stressesBased 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 stressesExample 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 mmThe other main parameters are shown in Table 3Table3With the same material parameters as aforementioned，the contact and bending stresses of three sets of cosine gears are analyzed by using program AnsysResults are shown in Fig9，F(xiàn)ig7 and Fig10，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 teethFor instance，the contact stress，tension and compression bending stresses are 56976 MPa11 75 1 MPa and 12498 MPa，respectively，as the number of teeth Z1=20，while 41061 Mpa6452Mpa and 7441 MPa as the number of teethZ1=30Tu9Tu10Table 4Example 2：the gears are designed to have a module of m=3mm，number of teeth Zt=25，a width of b=75mmThe other main parameters are shown in Table 5Table5 With the same material parameters as aforementioned, the contact and bending stresses are also computed by using program AnsysResults are shown in Fig7，F(xiàn)ig11 an d Fig12，and the values of the contact and bending stresses are shown in Table 6Tu11Tu12Table6According to Table 6，the contact and bending stresses decrease with the growth of the pressure angleFor instance，the contact stress，tension and compression bending stresses are49898 M Pa8604 MPa and 9559 MPa，respectively，as the pressure angle of =22。while 39543 MPa，7 18 1 MPa，and 86.32 MPa as the pressure angle of =24。3 CONCLUSIONSA new type of gear drivesa cosine gear drive is investigatedwhich takes a cosine curve as the tooth profileBased on the mathematical model, the characteristics including the contact ratiothe sliding coefficient and stresses are studiedThe effects of gear design parameterssuch as the number of teeth，pressure angle at pitch circle，on stresses of cosine gears have also been analyzedThe results of performed research allow the following conclusions to be drawn(1) The contact ratio of the cosine gear drive is about 12 to13which 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 Fig5(3)The contact and the bending stresses of the cosine gear drive are lower than that of the involute gear driveFor instance，under the given parameters as shown in section 2, the maximum contact stress of the cosine gear is reduced by about 2223 in comparison with the involute gear, and the compression bending stress is 2867 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 drivesThereforeother characteristics such as inspection，sensitivity of center distance error of this drive and its manufacturing should be researched further.
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