摘要
双圆弧齿轮传动比渐开线齿轮传动具有承载能力高、寿命长等特点，在石油、化工、矿山、冶金等行业得到了广泛应用。在新型井下驱动双螺杆泵中，实现两个螺杆同步运转和功率分流的双圆弧齿轮是决定泵工作能力的关键部件。泵挂深度超过1200m、沉没度超过500m(双螺杆泵整体浸泡在原油中)、井底温度超过80℃、设备不间断工作30个月、同步齿轮箱内充满润滑油、动力系统运转速度3000r／min左右、传递功率45kW，同时由于井筒空间限制，泵体直径≤140mm，且润滑油得不到更换和补充，狭小的空间又不利于散热等非常规工况条件又对双圆弧齿轮的工作能力提出了更高的设计要求。
本文的主要研究内容为：
(1)分析了双圆弧齿轮跑合前的齿廓方程，建立了有限元接触计算在静止状态和运动状态下的分析模型。并计算讨论了双圆弧齿轮的接触、弯曲强度；测试了齿轮弯曲应力以验证接触模型的正确性。
(2)研究了双圆弧齿轮跑合仿真计算的基本原理，以ANSYS接触算法为子程序建立了双圆弧齿轮跑合仿真模型，对一对双圆弧齿轮进行了跑合仿真以预测所采用的跑合措施的效果，并与跑合前的计算结果做了比较。
(3)针对双圆弧齿轮传动系统，通过划分热网络节点，建立了热网络方程；
研究了有润滑条件下的功率损失、热阻及对流换热计算模型，确定了不同部件的对流换热系数；计算了各节点在不同润滑条件下的温度值及轴承、齿轮等摩擦副零件的热变形量。
(4)建立了无油润滑条件下轴承、齿轮等关键部件的功率损失和对流换热计算模型，确定了传动系统瞬态计算模型；研究了系统温度分布和变化规律并计算了摩擦副零件的热变形量。
5)在热弹耦合理论的基础上，应用ANSYS软件，计算了双圆弧齿轮在热弹耦合场环境中跑合前和跑合后的耦合变形、耦合应力，并利用离散干涉检查算法实现了变形后双圆弧齿轮的运动干涉检查，提出了最佳侧隙。
(6)根据热弹耦合计算结果，为反映双螺杆泵的实际运转工况，编制了全寿命历程载荷谱，完成了双圆弧齿轮疲劳寿命预测，并与同尺寸渐开线齿轮寿命进行了比较。
主要研究成果及创新点如下：
1．以热弹耦合状态下双圆弧齿轮跑合前后接触强度、变形及跑合后疲劳寿命为工作能力评价参数，基于齿轮啮合原理、轮齿接触分析、摩擦学、传热学和疲劳寿命分析等理论，将有限元、热网络、碰撞干涉检查和雨流计数等方法相结合，提出了一种非常规工况条件下双圆弧齿轮传动工作能力研究新方法。
2．分析了影响建模的因素和边界条件、载荷处理方法，基于柔性轮缘和ANSYS／LS-DYNA建立了双圆弧齿轮动态接触应力分析用柔性体齿轮接触模型。
3．基于数值齿面思想，利用ANSYS软件中的参数化语言APDL建立了双圆弧齿轮跑合接触仿真模型，模拟其跑合过程并得到了跑合后接触应力分布。
4．基于热弹性力学和离散碰撞干涉算法，提出了双圆弧齿轮热弹耦合状态下运动干涉检查模型，完成了运动干涉检查，为齿轮动态设计奠定了基础。
5．变潜油电机额定工况为双螺杆执行机构实际运转工况，利用雨流计数法对双圆弧齿轮的随机载荷数据进行了处理，编制全寿命历程载荷谱，预测了非常规工况条件下双圆弧齿轮的疲劳寿命。
A double circular arc gear drive has the characteristics of higher carrying capacity and longer life than an involute gear drive, which is widely used in the petroleum, chemistry, mining, metallurgy industries. In a newly down-hole driven twin-screw pump, the double circular arc gears, which make the twin-screws run simultaneously and realize power split, are the key components in determining the working capacity of a pump. Meanwhile, better design requirements about higher working capacity of a double circular arc gear are needed under such abnormal working conditions as the setting depth of the pump beyond 1200m, the submergence depth over 500m that immersed a twin-screw pump in the crude oil, the temperature of the bottom of a well above 80℃, 30 months of uninterrupted work, full of lubricant in the synchronous gear box, a rapid power system at a speed of 3000r/min, 45 kW of transfer power, a body diameter of the pump less than 140mm, the lubricant unable to replace or renew, the space too small to disperse the heat, etc.
This thesis lays emphasis on the following points:
(1) The tooth profile equations of a double circular arc gear before the running-in are analyzed, and the contact computation models with finite elements under the static status and the dynamic status are set up. Moreover, the contact strength and the bending strength of the double circular arc gears are calculated, and the bending stress is tested to verify the contact model.
(2) The fundamental about the running-in simulation of a double circular arc gear is studied, and its model is set up on the basis of ANSYS contact arithmetic. Furthermore, the running-in simulation of a pair of double circular arc gears has been realized to predict the effect of the running-in method, and the simulation results have been compared with the computation results before running-in.
(3) By dividing different heat nodes and meshes, the heat net equation is determined. The computation models of power loss, heat resistance and heat convection with the lubrication are discussed, and heat convection coefficients of different parts are presented. Furthermore, the temperature of all nodes under the different lubricating conditions and the thermal deformation of the friction parts, such as bearings and gears, are calculated.
(4) The computation models of power loss and heat convection of key parts, such as bearings and gears, without any lubrication, are established, and the transient computation model of the drive is determined. In addition, the distribution and changing rule of the temperature of the system are investigated, and the thermal deformations of a friction pair of parts are calculated.
(5) Based on the thermo-elastic coupling theory and ANSYS, the coupling deformation and the stress of a double circular arc gear in the thermo-elastic coupling field, before and after the running-in, are computed. By the discrete interference inspection arithmetic, the moving interference inspection of the double circular arc gears after deformation is carried out, and the best backlash is presented.
(6) To demonstrate the real working condition of the twin-screw pump, and on the basis of the computation results of thermo-elastic coupling, the full life history of loading spectrum is compiled, the prediction of the fatigue life of the double circular arc gear is implemented, and its life is compared with that of the involute gear drive at the same size.
The major innovative points and results are as follows:
1. Based on the theories of gear geometry, gear tooth contact analysis, tribology, heat transmission and fatigue life analysis, and combined with the methods of finite elements, heat net, collision interference inspection and rain flow counting, a new method on studying the working capacity of a double circular arc gear under abnormal working condition is put forward by taking contact strengths and deformations before and after running-in, and fatigue life after running-in, of a double circular arc gear under thermo-elastic coupling status as the evaluation parameters of working capacity.
2. The factors of influenced modeling, boundary conditions and the method of load processing are analyzed, and the contact model of a flexible gear to analyze the dynamic contact stress of a double circular arc gear is established on the basis of the flexible rim and ANSYS/LS-DYNA.
3. On the idea of a numerical tooth flank and utilized APDL of ANSYS, the running-in contact simulation model of a double circular arc gear is constructed, the contact stress distribution after running-in is obtained by the simulation of running-in course.
4. Based on the thermo-elastic mechanics and the discrete collision interference arithmetic, the moving interference inspection model of a double circular arc gears under thermo-elastic coupling status is presented, and a moving interference inspection is carried out, which is served as a theoretical basis to the dynamic design of a gear.
5. To change the rated condition of a submersible electric motor into the working condition of a twin-screw and utilize the rain flow counting method, the random load data of double circular arc gear are processed The full life history of loading spectrum is compiled, and the fatigue life of double circular arc gear drive under abnormal working condition has been predicted.
引文
[1] 林德旺．齿轮传动装置[M]．上海：上海科技文献出版社，1989．
[2] Hellmann D H. Pumps for Multiphase Boosting. Proceedings of the Second International Conference on Pumps and Fans. Beijing, 1995, (1): 43～61.
[3] 李增亮，张来斌，何元君，李华林，徐鸿伟．双螺杆油气混输泵设计使用中有关问题的讨论．石油机械，2003，30(10)：48～50．
[4] Chiesa G, Donno S D, Aggradi G F. Subsea Testing of a Multiphase Pumping Unit Prototype.The 26th Annual Offshore Technology Conference in Houston, Texas, USA, 1994, 5: 77～84.
[5] 孙桓，陈作模．机械原理(第六版)．高等教育出版社，2000．
[6] Ebubekir Atan. On the prediction of the design criteria for modification of contact stresses due to thermal stresses in the gear mesh. Tribology International, 2005, 38: 227～233.
[7] 邵家辉．圆弧齿轮．机械工业出版社，1994．
[8] Wildhaber B. Helical gearing, US Patent No. 1, 601, 750, 1926.
[9] Novikov M L. USSR Patent No. 109, 750, 1956.
[10] [日]保延诚，特许公报，昭45-4569．
[11] Studer R M. US. Pat. NO. 3553300. 1970.
[12] 成大先．机械设计手册第四版第3卷．化学工业出版社．2002．1．
[13] Litvin F L．齿轮啮合原理(第二版)，上海：上海科学技术出版社，1984．
[14] Litvin F L, Tsay C B. Helical gears with circular arc teeth: simulation of conditions of meshing and bearing contact, ASME J. Mech. Trans. Auto. Des. 1985, 107: 556～564.
[15] Litvin F L. Theory of Gearing. NASA RP-1212(AVSCOM88-C-C035), Washington DC, 1989.
[16] Litvin F L, zhang Yi. Local synthesis and tooth contact analysis of fat-milled spiral bevel gears. NASA CP-4342, 1991, 1.
[17] Litvin F L. Gear Geometry and Applied Theory. Prentice-Hall, Englewood Cliffs, NJ, 1994.
[18] Litvin F L, Chen N X and Chen J S. Computerized determination of curvature relations and contact ellipse for conjugate surfaces. Computer Methods in Applied Mechanics and Engineering, 1995, 125(4): 151～170.
[19] Lu J, Litvin F L and Chen J S. Load Share and Finite Element Stress Analysis for Double Circular-Arc Helical Gears. Mathematical and Computer Modeling, 1995, 21(10): 31～47.
[20] Litvin F L, Lu J. Computerized design and generation of double circular-arc helical gears with low transmission errors. Comput Methods Appl. Mech. Engrg. 1995, 127: 57～86.
[21] Litvin F L, Feng P H, Lagutin S A. Computerized generation and simulation of meshing and contact of new type of Novikov-Wildhaber helical gears. NASA ARL-CR-428, 2000.
[22] Litvin F L,Fuentes A, Zanzi C, Pontiggia M, Handschuh R F. Face gear drive with spur involute pinion: geometry, generation by a worm, stress analysis.Computer Methods in Applied Mechanics and Engineering, 2002, 191: 2785～2813.
[23] Litvin F Let al. Helical and spur gear drive with double crowned pinion tooth surfaces and conjugated gear tooth surfaces. US Patent No. 6, 205, 879, 2001.
[24] Lit-vin F L and Lu J.Computerized simulation of generation, meshing and contact of double circular-arc helical gears. Mathematical and Computer Modeling, 1993, 18(5): 31～47.
[25] Litvin F Let al. New version of Novikov-Wildhaber helical gears:computerized design, simulation of meshing and stress analysis, Comput. Methods Appl. Mech. Engrg. 2002, 191: 5707～5740.
[26] Litvin F L, Chen J S. Application of Finite Element Analysis for Determination of Load Share, Real Contact Ratio, Precision of Motion and Stress Analys. ASME Journal of mechanical design, 1996, 118: 561～567.
[27] Argyris J, Fuentes A, Litvin F L. Computerized integrated approach for design and stress analysis of spiral bevel gears. Comput. Methods Appl. Mech. Engrg.2002, 191: 1057～1095.
[28] 武宝林，孟惠荣，邵家辉．双圆弧齿轮传动的动力学模型．太原工业大学学报，1997，3：12～16．
[29] 武宝林，孟惠荣，邵家辉．齿宽、重合度对双圆弧圆柱齿轮动态行为的影响分析．机械科学与技术，1999，18(1)：49～50．
[30] 武宝林，邵家辉，孟惠荣．螺旋角对双圆弧圆柱齿轮动态行为的影响分析．兵工学报·坦克装甲车与发动机分册，1999，1：15～19．
[31] 武宝林，邵家辉，孟惠荣．双圆弧齿轮的螺旋角误差对其传动误差的影响分析．兵工学报·坦克装甲车与发动机分册，1999，1：20～24．
[32] 武宝林，邵家辉．轴线平行度误差对双圆弧齿轮传动误差的影响分析．机械传动，2001，25，1：31～33．
[33] 武宝林，邵家辉，孟惠荣．双圆弧齿轮接触迹间载荷分配状况分析．航空动力学报，1999，14(1)：297～300，333．
[34] 武宝林，邵家辉，孟惠荣．双圆弧齿轮传动的动态性能试验分析．机械传动，1999，23，2：35～37
[35] 张静，王铁．双圆弧齿轮齿形参数的模糊优化．太原理工大学学报，2002，2：6～8．
[36] 李阳星，周广林，李光煜．双圆弧齿轮传动的模糊可靠性优化设计．机械科学与技术，2003，22(1)：75～77．
[37] 张秀亲，李瑞斌，张应顺．双圆弧齿轮齿根弯曲应力的研究．机械科学与技术，2003，22(5)：764～766．
[38] 陈胡兴．双圆弧齿轮传动的啮合特性及应力的有限元分析．南京航空航天大学硕士论文，2005．7．
[39] 郭辉．双圆弧齿轮的接触分析研究及应用．西北工业大学硕士论文，2005．6．
[40] 孙大乐，杨文通，蔡春源．双圆弧齿轮跑合性能及计算机仿真．机械工程学报，1997，(1)：9～13．
[41] 孙大乐，杨文通，蔡春源．双圆弧齿轮磨合磨损的试验研究．摩擦学学报，1997，17(4)：371～374．
[42] 孙大乐，杨文通，蔡春源，张宝忠．齿形参数对双圆弧齿轮接触跑合性能的影响．机械，1998，25(5)：17～19．
[43] 吴洁，王善政，蔡春源．提高圆弧齿轮跑合性能的滚刀设计方法．机械设计与制造， 2000，1：58～59．
[44] 吴洁，蔡春源．基于跑合性能的双圆弧齿轮齿形参数优化设计．煤矿机械，2001，2：6～7．
[45] 罗立凤，陈谌闻．圆弧齿轮跑合过程计算和加载啮合静态接触迹测试．矿山机械，1993，12：2-5．
[46] 罗立凤，陈谌闻．计入跑合效果的JB929—67型圆弧齿轮的接触应力．机械传动，1994，3：16-17，56．
[47] Rumbarger J H, Filetti E G, Gubemick D. Gas turbine engine mainshaft roller bearing system analysis. ASME Journal of Lubdcation Technology. 1973, 95(4): 401～416.
[48] 1978, 171.
[49] Anderson N E, Loewenthal S H. Effect of geometry and operating condition on Spur gear system power loss. Journal of Mechanical Design, 1981, 103(4): 151～160.
[50] Anderson N E, Loewenthal S H. Design of Spur gears for improved efficiency. 1982, 104(4): 767～774.
[51] Anderson N E, Loewenthal S H. An analytical method to predict of aircraft gearboxes. NASA-TM-83716(N84-25606), 1984.
[52] Anderson N E, Loewenthal S H. Efficiency of nonstandard and high contact ratio Spur gears. NASA-TM-83725(N84-29223), 1984, 35.
[53] Masatoshi Yoshizaki,et al.Study on frictional loss of spur gears (concerning the influence of tooth form, load, tooth surface roughness, and lubricating oil). Tfib. Trans, 1991, 34(1): 138～146.
[54] Michaelis K, Hohn B R. Influence of lubricants on power loss of cylindrical gears. Trib.Trans, 1994, 37(1): 161～167.
[55] HalTiS T A. Rolling bearing analysis. 3rd John Wiley&Sons, Inc. 1990.
[56] Crecelius W J, Pirvics J. Computer Program Operation Manual on SHABERTH: A Computer Program for the Analysis of the Steady State and Transient Thermal Performance of Shaft Bearing Systems. SKF-AL76P030, AD-AO42981, 1976. 246p.
[57] Coe H H, Schuler F T. Calculated and experimental data for a 118mm bore roller bearing to 3 million DN. NASA-TM-81427(N80-19496), 1980, 37p.
[58] Coe H H. Thermal analysis of planetary transmission with spherical roller bearings operating after complete loss of oil. NASA-TP-2367(N84-32824), 1984, 14p.
[59] Nadir Patir, Cheng H S. Prediction of the bulk temperature in spur gears based on f'mite temperature analysis.ASLE Transaction, 1979, 22(1): 25～36.
[60] Handschuh R F. Thermal Behavior of Spiral Bevel Gears. Ph, D. Thesis, Case Western Reserve University, Ang. 1993, NASA-TM-106518 (N95-18935), 1995.
[61] Long H, Lord A A, Gethin D T, et al. Operating temperatures of oil-iubricated medium-speed gears: numerical models and experimental results. Jounal of Aerospace Engineering, 2003, 217(2): 87～106.
[62] Coe H H. Predicted and experimental performance of large-bore high-speed ball and roller beatings. NASA-CP-2210(AD-A 126186, N83-20119)AD-p000710, 1983, 18p.
[63] Parker R J. Comparison of predicted and experimental thermal performance of angular-contact ball bearings. NASA-TP-2275(N84-18654), 1984, 19p.
[64] Hadden G B, Kleckner R J, Ragen M A, et al. Research Report: user'smanual for computer program AT81Y003 SHABERTHteady state and transient thermal analysis of a shaft beating system including ball, cylindrical and tapered roller beatings. NASA- CR-165365(N82-31969), 1982, 254p.
[65] 刘志全，沈允文，张鹏顺，力华东．高速圆柱滚子轴承温度分布的计算与测试．机械科学与技术，1997，16(5)：805～811
[66] 李健，刘志全，袁培益．航空发动机轴承腔热状态分析模型及温度场计算．航空动力学报，1999，14(3)：242～246，328．
[67] 韩凤琴，桂中华，久保田乔．基于奇异谱分析的轴承温度趋势预测及其应用(英文)．华南理工大学学报(自然科学版)，2005，33(9)：51～54，72．
[68] Blok H. Surface temperature under extreme pressure lubrication conditions. Proceedings of 2nd World Petroleum Congress, Section Ⅳ, 1937, 3: 471～486.
[69] Blok H. Theoretical study of temperature rise at surfaces actual contact under oiliness. Proceedings of Gen. Disc on lubrication. Instn. Mech. Engs., London, 1937, 2: 222～235.
[70] Blok H. Measurement of temperature on gear teeth under extreme conditions. Proceedings of Gen. Disc on lubrication. 1937, 2: 14～20.
[71] Terauchi Y, Mori H. Comparison of theories and experimental results for surface temperature of pur gear teeth. Transanctions of ASME, Journal of Engineering for Industry, 1974, 2.
[72] Wang K L, Cheng H S. A numerical solution to the dynamic load, film thickness and surface temperatures in spur gears. part Ⅰ analysis. Transanctions of the ASME, Journal of Mechanical Design. 1981, 103.
[73] Wang K L, Cheng H S. A numerical solution to the dynamic load, film thickness and surface temperatures in spur gears, partⅡresults. Transanctions of the ASME, Journal of Mechanical Design. 1981, 103.
[74] 沈允文，王彤，王三民，张永红，刘志全．弧齿锥齿轮传动的稳态本体温度场分析．机械传动，2001，25(3)：1～4．
[75] 张永红，苏华，刘志全，沈允文．行星齿轮传动系统的瞬态热分析．航空学报，2000，21(6)：542～544．
[76] 龙慧，张光辉，罗文军．旋转齿轮瞬时接触应力和温度的分析模拟，机械工程学报，2004，40(8)：24～29．
[77] 李桂华，费业泰．温度变化对啮合齿轮侧隙的影响．合肥工业大学学报(自然科学版)，2004，27(10)：1147～1150．
[78] 李桂华，费业泰．温度变化对圆柱齿轮齿形的影响．机械设计，2005，22(2)：22～23，47．
[79] Adelman H M, Sawyer P L. Inclusion of explicit thermal requirements in optimum structural design. NASA TM X-74017, 1977, 3.
[80] Haftka R T, Shore C P. Approximation methods for combined thermal/structural design. NASA TP 1428, 1978, 6.
[81] Thomas H L, Shyy Y K. Optimization of structures with static and thermal loads. Proceedings of the AIAA/ASME/ASCE/AHS/ASC 34th Structure, Structural Dynamics, and Materials Conference, Lajolla, GA, 1993: 1438～1442.
[82] Hattori T, Ohnishi H. Optimum design of rotational loading. JSME, International Journal, Series Ⅲ, 1989, 32(4): 597～605.
[83] Behzad Bigdeli. A finite element thermal-structural analysis of the mast of the international space station.41~(st) AIAA/ASME/ASCE/AHS/ASC 34th Structure, Structural Dynamics, and Materials Conference and Exhibit, Lajolla, GA, 2000, 4: 1～11.
[84] Barut A, Guven I, Madenci E. Analysis of singular stress fields at junctions of multiple dissimilar materials under mechanical and thermal loading.International Journal of Solids and Structures, 2001, 38: 9077～9109.
[85] Srivastava A K, Veldhuis S C, Elbestawit M A. Modeling geometric and thermal errors in a five-axis CNC machine tool. International Journal of Machine Tools & Manufacture. 1995, 34(9): 1321～1337.
[86] Ramesh R, Mannan M A, Poo A N, Keerthi S S. Thermal errors measurement and modeling in machine tools. International Journal of Machine Tools & Manufacture, 2003, 43: 405～419.
[87] 顾元宪，刘涛，亢战，赵国忠．热结构稳态响应的耦合灵敏度分析方法．计算力学学报，2004，21(5)：535～538．
[88] 顾元宪，刘涛，亢战，赵国忠．热结构瞬态响应的耦合灵敏度分析方法与优化设计．力学学报，2004，36(1)：37～42．
[89] 张淑杰．空间可展桁架结构的设计与热分析．浙江大学博士学位论文，2001．6．
[90] 李松年．航天结构热力力学的任务和应用．力学进展，1994，24(1)：1～23．
[91] 程乐锦，薛明德．大型空间结构热动力学耦合有限元分析．清华大学学报(自然科学版)，2004，44(5)：681～684，688
[92] 马迅，秦剑．制动鼓的热-结构耦合分析．湖北汽车工业学院学报，2004，18(3)：5～9．
[93] 马思群，兆文忠，谢素明，万朝燕．列车制动盘热—机耦合过程的数值仿真．机械设计与制造，2003，4：73～74．
[94] 漆文凯，陈伟，高德平．航空发动机涡轮盘热—机械载荷耦合响应分析研究．机械设计与制造，2004，1：89～91．
[95] 张耀满，王旭东，李明达．大型回转零件有限元热与瞬态动力学耦合分析．机械设计与制造，2005，1：56～58．
[96] 钟明，张永元．三维耦合热裂纹体的时域边界元解及其收敛性．上海交通大学学报，2001，35(4)：621～624．
[97] 李润方，汤庆平．运转过程中轮齿耦合热弹性接触有限元分析．齿轮，1989，13(1)：5～8．
[98] 李绍彬，李润方，林腾蛟．基于热弹变形的圆柱齿轮理想修形曲线．中国机械工程，2003，14(14)：1175～1179．
[99] Avinash Singh. An Experimental investigation of Bending Fatigue initiation and Propagation Lives. Transactions of the ASME, Journal of Mechanical Design, September 2001, 123: 431～435.
[100] Kramberger J, Sraml M, Glodez S, Flasker J, Potrc. I. Computational model for the analysis of bending fatigue in gears. Computers and Structures, 2004, 82: 2261～2269.
[101] Kramberger J, Sraml M, Flasker J, Potrc I. Numerical calculation of bending fatigue life of thin-rim spur dears. Engineering Fracture Mechanics,2004,71:647～656.
[102] Glodez S,Sraml M,Kramberger J.A computational model for determination of service life of gears. International Journal of Fatigue, 2002,24:1013～1020.
[103] David G L,Robert F H,et al.Consideration of Moving Tooth Load in Gear Crack Propagation Predictions.Transactions of the ASME,Journal of Mechanical Design, September 2001,123:118～124.
[104] Aslantas K,Tasgetiren S. A study of spur gear pitting formation and life prediction. Wear,2004,257:1167～1175.
[105] James Li,Hyungdae Lee. Gear fatigue crack prognosis using embedded model, gear dynamic model and fracture mechanics. Mechanical Systems and Signal Processing, 2005, 19: 836～846.
[106] Timothy Krantz,Clark Cooper,Dennis Townsend,Bruce Hansen.Increased Surface Fatigue Lives of Spur Gears by Application of a Coating. Transactions of the ASME,Journal of Mechanical Design, 11 2004,126:1047～1054.
[107] Cedergren J,Melin S,Lidstrom P.Numerical investigation of Powder Metallurgy manufactured gear wheels subjected to fatigue loading. Powder Technology,2005,160: 161～169.
[108] Deng Gang,Tsutomu Nakanishi,Masana Kato. Surface Temperature Calculation and its Application to Surface Fatigue Strength Evaluation. Transactions of the ASME,Journal of Mechanical Design,2002,124:805～812.
[109] 李艳，张祖明．齿轮接触疲劳寿命、强度及应力最佳分布类型的确定．北京印刷学院学报，1999，7(3)：3～9．
[110] 亓秀梅，周谋，高创宽．油膜比厚对齿轮接触疲劳寿命的影响．机械科学与技术，2003，S1：84～86．
[111] 陈举华，徐楠，安艳秋．42CrMo材料硬齿面齿轮全寿命试验及数据分析．机械传动，2005，29(5)：63～65．
[112] 李玉春，姚卫星，温卫东．应力场强法在多轴疲劳寿命估算中的应用．机械强度，2002，24(2)：258～261．
[113] 童小燕，姚磊江，吕胜利．疲劳能量方法研究回顾．机械强度，2004，26(S)：216～221．
[114] 李洪双，吕震宙．预测复杂结构系统安全疲劳寿命的一种新数值方法．西北工业大学学报，2005，23(4)：435～439．
[115] 胡来瑢．空间啮合原理及应用，北京：煤炭工业出版社，1996．
[116] 卢贤瓒，尚俊开．圆弧齿轮啮合原理，北京：机械工业出版社，2003．
[117] 武昭晖．双圆弧齿轮传动的齿形参数对轮齿弯曲强度影响的研究．太原理工大学硕士学位论文，2000．4．
[118] 邓季贤，李进宝．双圆弧齿轮端面齿廓方程及其啮合过程．煤矿机械，2003，(1)：19～21．
[119] 贾毅．双圆弧齿轮变形及修形降低噪音的研究．太原工业大学硕士学位论文，1988．
[120] 何晓庚，朱景梓．圆弧齿轮应用分析的理论基础及试验研究．太原工学院齿轮强度研究室，1982．
[121] 佟景伟，徐步青．冲击载荷作用下齿轮动应力的研究．机械强度，2002，24(1)：125～127．
[122] Vijayarangan, S.Ganesan. A study of dynamic stresses in a spur gear under a moving line load and impact load conditions by a three dimensional finite element method. Joural of sound and vibration, 1993,162(1): 185～189.
[123] [苏联]耿坎．М．П，雷约夫．М．А，雷约夫．Н．М．提高重载齿轮传动的可靠性．北京：机械工业出版社，1986．
[124] Archard J F. Wear Control Handbook. ASME,New York,1980:35～80.
[125] 张秀亲，张应顺，邵家辉．双圆弧齿轮齿腰应力的研究．太原工业大学学报，1996，27(1)：91～96．
[126] C J Hooke,K Mao. Measurement and prediction the surface temperature in gears and its relationship to gear Wear. Transactions of the ASME Journal of Tribology, 1993.
[127] 机械工程手册编辑委员会，机械工程手册第七卷，北京：机械工业出版社，1982．
[128] 杨世铭，陶文铨．传热学(第三版)，北京：高等教育出版社，1998．
[129] 胡鹏浩，非均匀温度场中的机械零部件热变形的理论及应用研究．合肥工业大学博士论文，2001．7．
[130] 刘志全等．某直升机齿轮传动系统的瞬态热分析．航空动力学报，1999，14(3)：309～312，334．
[131] 刘志全．武装直升机生存能力的研究．传动系统的稳态及瞬态热分析．西北工业大学博士后研究工作报告，1998．
[132] 吴林峰．转炉托圈机械热应力耦合三维非线性分析．南京理工大学硕士论文，2005．6．
[133] 程昌钧，朱媛媛．弹性力学．上海大学出版社，2005．9．
[134] 李维特，黄保海，毕仲波．热应力理论分析及应用．中国电力出版社，2004．
[135] 李绍彬．高速重载齿轮传动热弹变形及非线性耦合动力学研究．重庆大学博士论文，2003．11．
[136] 罗枫．三维网格模型的快速碰撞检测及相交体计算．浙江大学硕士论文，2005．3．
[137] 李碧浩．金属双螺杆泵螺杆转子性能分析研究．西北工业大学硕士论文，2006．3．
[138] 赵少沛．抗疲劳设计．北京：机械工业出版社，1994．
[139] 姚卫星．结构疲劳寿命分析．北京：国防工业出版社，2003．
[140] 刘维信．机械可靠性设计．北京：清华大学出版社，1996．
[141] 袁菲．某火炮传动系统随机载荷下齿轮疲劳寿命预测方法研究．西北工业大学硕士学位论文．2006．5．
[142] 吴富民．结构疲劳强度．西安：西北工业大学出版社，1985．
[143] Yao W X. Stress field intensity approach for prediction fatigue life. International Journal of Fatigue, 1993,15(3): 233～245.