同功重比修形斜齒和直齒面齒輪性能對比研究

周鎮宇, 唐進元, 董建雄

(中南大學 高性能復雜制造國家重點實驗室, 湖南 長沙 410083)

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同功重比修形斜齒和直齒面齒輪性能對比研究

周鎮宇, 唐進元, 董建雄

(中南大學 高性能復雜制造國家重點實驗室, 湖南 長沙 410083)

為深入了解同功重比修形斜齒與直齒面齒輪的性能差異，選擇更適合于高速重載工況下的面齒輪傳動.基于嚙合原理推導了修形斜齒與直齒面齒輪齒面方程，基于CATIA建立了修形斜齒與直齒面齒輪三維模型，采用有限元接觸分析方法，以接觸應力、彎曲應力和重合度為面齒輪傳動性能指標展開研究.研究結果表明：修形斜齒面齒輪相比修形直齒面齒輪接觸應力大幅降低，算例最大接觸應力降低16.3%；修形斜齒面齒輪相比修形直齒面齒輪彎曲應力大幅降低，算例最大彎曲應力降低32.4%；修形斜齒面齒輪相比修形直齒面齒輪重合度大幅提高，算例重合度提高10.3%.所以同功重比情況下，修形斜齒面齒輪傳動性能優于修形直齒面齒輪，前者更適合于高速重載工況下的輕量化設計.

輕量化; 功重比; 接觸應力; 彎曲應力; 重合度; 修形

面齒輪傳動是一種相對較新的齒輪傳動方式.正交面齒輪傳動中,與直齒輪嚙合傳動的稱為直齒面齒輪,與斜齒輪嚙合傳動的稱為斜齒面齒輪.面齒輪目前主要應用于武裝直升機主傳動中,而輕量化設計已成為飛機、汽車設計的一大趨勢[1].對直升機主減速器而言,高功率密度是其主要性能指標,因此研究面齒輪功重比(單位質量所承載的功率)非常有必要.

斜齒面齒輪和直齒面齒輪均易出現邊緣接觸.為改善接觸軌跡的齒面分布,避免邊緣接觸,得到均勻的載荷分布,需對面齒輪進行修形.Litvin等[2-8]率先提出拋物線齒廓修形方法,并在斜齒面齒輪接觸分析中提出點接觸形成條件和安裝誤差坐標系,得到均勻的齒面接觸軌跡；初步給出面齒輪重合度的定義,但未給出具體計算方法.雷敦財[9]研究了有限元分析中輪轂結構對嚙合剛度的影響.

本文基于有限元分析方法,在相同功重比工況下對修形斜齒面齒輪和修形直齒面齒輪進行接觸應力、彎曲應力和重合度對比研究.

1　修形斜齒和直齒面齒輪齒面方程

齒輪嚙合傳動中主、被動齒輪的基節必須處處相等,從理論上講,精確的漸開線剛性齒輪是能夠實現上述目標的.但實際的齒輪副均為彈性體,在一定嚙合力作用下會產生相應彈性變形,使處于嚙合線位置的主動輪和被動輪基節出現變化,嚙合過程中產生嚙合沖擊.為了消除輪齒嚙合沖擊,通常采用齒廓修形方法,即沿齒高方向從齒面上去除部分材料,從而改變齒廓形狀,消除嚙合沖擊,減小嚙合過程中的最大應力,提高齒輪副的傳動平穩性和使用壽命.而拋物線齒廓是常用的修形齒廓.

考慮到標準的斜齒面齒輪和直齒面齒輪在嚙合傳動過程中均易產生邊緣接觸,故對這2種面齒輪均進行拋物線齒廓修形.修形面齒輪的齒面由修形產形輪齒面包絡得到[10],而修形產形輪齒面由修形齒條刀具包絡得到的,修形齒條齒廓直線被拋物線代替.

1.1修形斜齒面齒輪齒面方程

傳統加工斜齒輪的齒條刀刃∑c是直線,端面如圖1(a)所示,虛線所示∑cs和∑c1為修形拋物線齒廓,頂點Oa的位置由uo決定；空間齒廓如圖1(b)所示,∑i(i=cs,cl)為修形齒條齒面,li(i=cs,cl)為齒寬縱向變量參數；圖1(c)為拋物線修形齒條端面兩側齒廓,Sc(xc,yc,zc)為齒條動標系,Sa(xa,ya,za)和Sb(xb,yb,zb)為輔助坐標系,ui(i=s,l)為坐標系Sa(xa,ya,za)中橫向變量參數；圖1(d)所示為修形齒條加工修形產形輪的坐標系,Sm(xm,ym,zm)為靜標系,rpi(i=s,l)為節圓半徑,Si(xi,yi,zi)為

圖1　修形斜齒輪加工坐標系Fig.1　The processing coordinate system of the modified helical pinion

產形輪(i=s)和傳動輪(i=l)動標系.

圖1(c)中修形齒條刀具左右兩側齒面∑i和∑j(i,j=cs,cl)方程為(符號?和±均按左右順序表示)

(1)

動標系Sa到Sc的轉換矩陣[11]為Mca，得Sc中修形齒條刀具齒面∑i(i=cs,cl)方程為

rcl(ui,li)=Mcaral(ui,li)，

(2)

其齒面法向量為ncl；Sc到Ss的轉換矩陣為Msc(ψs)；得Ss中∑i(i=cs)包絡的修形產形輪齒面∑s1方程為

(3)

其齒面法向量為 ns2；圖2為修形斜齒面齒輪加工坐標系；Ss到S2轉化矩陣為M2s；φs和φ2滿足以下關系,其中Z2為面齒輪齒數:

(4)

圖2　修形斜齒面齒輪加工坐標系Fig.2　The processing coordinate system of the modified face gear with helical pinion

修形產形輪齒面∑s1包絡的修形斜齒面齒輪齒面∑1在動標系S2中的方程為

(5)

用同樣方法可由修形齒條右側刀刃包絡得到修形產形輪齒面,進而包絡求出修形斜齒面齒輪另一齒面的方程.

1.2修形直齒面齒輪齒面方程

與修形斜齒面齒輪不同的是,修形直齒面齒輪兩側齒面是對稱的,故只需推導一面的方程即可.此處所述修形直齒面齒輪的修形方法跟1.1節方法一致,推導過程也基本相同,不贅述其過程,在圖2所示加工坐標系中可求得動標系S2中修形面齒輪方程為

(6)

其中rs(us,ls)為修形直齒產形輪方程.

2　修形斜齒和直齒面齒輪高精度幾何建模

有限元仿真對幾何模型精度要求較高.表1和表2為修形斜齒和直齒面齒輪設計參數.

表1修形斜齒面齒輪設計參數

Table 1The design parameters of the modified face gear with helical pinion

設計參數傳動小輪產形輪修行斜齒面齒輪齒數Z2528160壓力角α/(°)2525—模數m/mm6.356.356.35旋向左旋左旋右旋螺旋角β/(°)151515外徑/mm——582內徑/mm——497軸交角γ/(°)909090修形系數as0.0020.003—齒頂系數1.01.251.0齒底系數1.251.251.25比值so/ωo0.90.9—

表2修形直齒面齒輪設計參數

Table 2The design parameters of the modified face gear with spur pinion

設計參數參數值小齒輪齒數Z119產形輪齒數Zs21面齒輪齒數Z273名義壓力角α/(°)25模數m/mm4.5面齒輪內徑/mm158面齒輪外徑/mm189拋物線位置參數Uo/mm3.035軸交角γ/(°)90產形輪齒頂高系數as1.25產形輪齒根高系數bs1小輪齒頂高系數a11小輪設計齒根高系數bm1

根據修形斜齒和直齒面齒輪方程,利用MATLAB計算功能計算得到齒面離散點,計算流程如圖3所示.

圖3　修形斜齒面齒輪齒面離散點計算流程圖Fig.3　The calculation flow chart of the tooth surface discrete points of the modified face gear with helical pinion

計算得到的齒面點數據,導入CATIA中進行擬合,經分割、陣列、去除等步驟可得到高精度修形斜齒面齒輪幾何模型裝配圖，如圖4所示[12-13].同樣方法可得修形直齒面齒輪幾何模型裝配圖，如圖5所示.

圖4　修形斜齒面齒輪Fig.4　The modified face gear with helical pinion

圖5　修形直齒面齒輪Fig.5　The modified face gear with spur pinion

3　修形斜齒和直齒面齒輪功重比對比分析

功重比為承載功率和質量的比值,因2組齒輪采用相同材料,質量等價于體積.本文采用有限元分析的方法,對比相同功重比的修形斜齒面齒輪和修形直齒面齒輪的承載能力.在CATIA中測得修形斜齒面齒輪大小輪體積總和為22 900 000 mm3，修形直齒面齒輪大小輪體積總和為31 320 000 mm3.

3.1修形斜齒和直齒面齒輪有限元建模

基于ABAQUS有限元軟件[14-17],對圖4和圖5的2組面齒輪進行有限元建模,其建模過程按以下步驟進行：1)幾何模型導入；2)網格劃分；3)裝配；4)設置材料屬性：E=2.06×105MPa,v=0.3；5)設置分析步；6)建立接觸與約束；7)定義邊界條件和載荷；8)設置網格類型以及創建并提交作業.

對修形斜齒面齒輪進行網格劃分時,需先在CATIA中將修形斜齒面齒輪的單齒進行剖分,過程如圖6所示.有些情況下,若齒輪內外端粗細不均,還需在齒寬方向進行剖分,把分塊同時導入ABAQUS進行網格劃分,再把劃分好網格的分塊進行接合.修形斜齒面齒輪有限元網格如圖7所示.

圖6　CATIA中剖分單齒Fig.6　The partition of single gear in CATIA

圖7　修形斜齒面齒輪有限元網格模型Fig.7　The finite element mesh model of the modified face gear with helical pinion

修形直齒面齒輪有限元建模過程相比修形斜齒面齒輪更加簡單,不具體介紹.

修形斜齒面齒輪轉速為5 000 r/min，功率為524 kW，計算得大輪扭矩為1 000 N·m.在相同轉速下,根據測得的體積,可求得同功重比的修形直齒面齒輪承載功率為716.7 kW，大輪扭矩為1 368 N·m.把已建好的2組齒輪有限元模型定義載荷后即可提交計算.

3.2接觸應力對比

2組齒輪經有限元分析后,得到同功重比情況下修形斜齒面齒輪最大接觸應力如圖8所示,修形直齒面齒輪最大接觸應力如圖9所示.

圖8　修形斜齒面齒輪最大接觸應力云圖Fig.8　The maximum contact stress cloud chart of the modified face gear with helical pinion

圖9　修形直齒面齒輪最大接觸應力云圖Fig.9　The maximum contact stress cloud chart of the modified face gear with spur pinion

由圖8和圖9可知修形斜齒面齒輪最大接觸應力為1 241 MPa，修形直齒面齒輪最大接觸應力為1 482 MPa.對比分析可知：同功重比工況下，修形斜齒面齒輪相比修形直齒面齒輪最大接觸應力降低16.3%((1482-1241)/1482×100%).

3.3彎曲應力對比

修形斜齒面齒輪最大彎曲應力如圖10所示,修形直齒面齒輪最大彎曲應力如圖11所示.

圖10　修形斜齒面齒輪最大彎曲應力云圖Fig.10　The maximum bending stress cloud chart of the modified face gear with helical pinion

圖11　修形直齒面齒輪最大彎曲應力云圖Fig.11　The maximum bending stress cloud chart of the modified face gear with spur pinion

由圖10可知修形斜齒面齒輪最大彎曲應力為579.9 MPa，由圖11可知修形直齒面齒輪最大彎曲應力為857.9 MPa.對比可知：相同功重比工況下,修形斜齒面齒輪相比修形直齒面齒輪最大彎曲應力降低32.4%((857.9-579.9)/857.9×100%).

3.4重合度對比

修形斜齒面齒輪有限元仿真結果的齒對接觸力隨時間分布如圖12所示.

圖12　修形斜齒面齒輪接觸力Fig.12　The contact force diagram of the modified face gear with helical pinion

修形直齒面齒輪齒對接觸力隨時間分布如圖13所示.

圖13　修形直齒面齒輪接觸力Fig.13　The contact force diagram of the modified face gear with spur pinion

同樣方法算得同功重比的修形直齒面齒輪傳動重合度為1.636.

對比分析可知：在上述同功重比工況下,修形斜齒面齒輪相比修形直齒面齒輪重合度提高10.3%((1.804-1.636)/1.636×100%).

4　結　論

同功重比工況下,修形斜齒面齒輪相比修形直齒面齒輪：最大接觸應力降低16.3%,最大彎曲應力降低32.4%,重合度提高10.3%.修形斜齒面齒輪比修形直齒面齒輪承載能力更強,重合度更高,更加符合高速重載工況下的輕量化設計要求.

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Comparative study on the performance of modified face gears withhelical pinion and spur pinion in the case of same power density

ZHOU Zhen-yu, TANG Jin-yuan, DONG Jian-xiong

(State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, China)

In order to know the performance difference of the modified face gears with helical pinion and spur pinion under the same power density better and choose a more suitable kind of face gear drive for high speed and heavy load conditions. Based on meshing theory, the tooth surface equations of the modified face gears with helical pinion and spur pinion were derived. The three-dimensional models of the modified face gears with helical pinion and spur pinion were built by CATIA. The finite element contact analysis method was used and the contact stress, bending stress and contact ratio were considered as performance indexes of the face gears to expand the research. Researches proved that the contact stress of modified face gear with helical pinion was much lower than that of modified face gear with spur pinion. The maximum contact stress was reduced by 16.3% in the case. The bending stress of modified face gear with helical pinion was much lower than that of modified face gear with spur pinion. The maximum bending stress was reduced by 32.4% in the case. The contact ratio of modified face gear with helical pinion was much higher than that of modified face gear with spur pinion. The contact ratio was increased by 10.3% in the case. So the transmission performance of the modified face gear with helical pinion is better than that of modified face gear with spur pinion under the same power density, the former is more suitable for the lightweight design under high speed and heavy load condition.

lightweight; power density; contact stress; bending stress; contact ratio; modification

2016-01-20.

本刊網址·在線期刊：http://www.journals.zju.edu.cn/gcsjxb

國家自然科學基金資助項目(51535012，51305462，51275530).

周鎮宇(1991—),男,江西宜春人,碩士生,從事齒輪傳動及數字化制造研究,E-mail:zhenyuzhou@csu.edu.cn.

10.3785/j.issn. 1006-754X.2016.04.007

TH 132.41

A

1006-754X(2016)04-0338-07

http://orcid.org//0000-0001-5888-2892

通信聯系人：唐進元(1962-),男,湖南長沙人,教授,從事齒輪傳動及數字化制造研究,E-mail:jytangcsu@163.com.http://orcid.org//0000-0001-7186-1316