通过控制接触的路径和传输误差来为螺旋锥轮设计齿轮机床设置外文文献翻译、中英文翻译

华北科技学院毕业设计 第 1 页 共 33 页 附录 外文文献 Design of Pinion Machine Tool-settings foSpiral Bevel Gearsby Controlling Contact Path and Transmission Errors Cao Xuemeia,*,Fang Zongdea,Xu Haob,Su Jinzhana a:School of Mechatronics,Northwestern Polytechnic University,Xian 710072,China b:Zhongnan Transmission Machinery Works of Changsha Aviation Industries, Changsha 410200,China Received 16 September 2007；accepted 20 February 2008 Abstract： This paper proposes a new approach to design pinion machine tool-settings for spiral bevel gears by controlling contact path and transmission errors.It is based on the satisfaction of contact condition of three given control points on the tooth surface.The three meshing points are controlled to be on a predesigned straight contact path that meets the pre-designed parabolic function of transmission errors.Designed separately,the magnitude of transmission errors and the orientation of the contact path are subjected to precise control.In addition,in order to meet the manufacturing requirements,we suggest to modify the values of blank offset,one of the pinion machine tool-settings,and redesign pinion machine tool-settings to ensure that the magnitude and the geometry of transmission errors should not be influenced apart from minor effects on the predesigned straight contact path.The proposed approach together with its ideas has been proven by a numerical example and the manufacturing practice of a pair of spiral bevel gears. Keywords:spiral bevel gear；contact path；transmission error；blank offset；tooth contact analysis 1 Introduction Spiral bevel gears are among the key components of aerospace power plants in general,helicopter gear drives in particular,which regard the meshing performance,endurance and reliability as critical safety factors.Therefore,designing spiral bevel gears has all the time 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 2 页 共 33 页 been drawing close attention of researchers in many companies and institutions.The requirements of reducing noise level and increasing endurance of spiral bevel gears have raise formidable challenge to designers too. The bases for designing low-noise spiral bevel gears with localized bearing contacts were presented in related Refs.In order to absorb larger errors in alignment and have better stability,the contact path should be designed to be a straight line.As transmission errors are mostly blamed for noise and vibration in gearing systems,the transmission errors of parabolic type are considered to be able to absorb linear discontinuous effects caused by misalignment referred to as the main source of noise. In some cases,the machine tool-settings designed by way of the existing local synthetic method are well beyond the appropriate applicable range of the machine. In this paper,a new integrated approach is proposed on the base of meeting the contact conditions inclusive of the predesigned parabolic function of transmission errors and the specifically oriented straight contact path through three given control points on the tooth surface.As a result,as early as in the designing stage,the operating performance could be controlled.In addition,the values of blank offset can be so modified as to be within the appropriate range of the machine without any influence on the magnitude of transmission errors apart from there being minor effects on the predesigned linear contact path. The proposed approach is based on the following assumptions: (1)The gear machine tool-settings are predetermined and can be adopted. (2)The main input parameters are 2 and m21 .Of them, 2 determines the orientation angle of the predesigned straight contact path,and m21 the sec- ond derivative of the transmission function,which determines the predesigned magnitude of the parabolic function of transmission error.In this paper,the values of 2 and m21 are given in advance,but,in practices,they can be optimized depending on the applied loads to obtain the favorable meshing performance through the loaded tooth contact analysis(LTCA). 2 Active Tooth Surface Design by Three Given Meshing Points After the three control meshing points have been determined on the gear surface by the predesigned straight contact path and the parabolic function of transmission errors,the pinion 华北科技学院毕业设计 第 3 页 共 33 页 machine tool-settings can be determined. 2.1 Determination of three contact points Fig.1 shows the three contact points on the gear tooth surface.2 is the pitch angle of the gear.The contact path is designed to be a straight line and 2 is the orientation of contact path. Fig.1 Three control meshing points and contact path. When the gear surface 2 rotates by 02, 12 and22,the pinion surface 1 contacts with it at meshing points M0,M1 and M2 with rotational angles 02, 12 and 22 respectively.Let the cycle of pinion meshing be 2/Z1,where Z1 is the tooth number of pinion,and then 11=/Z1+01 and 21=/Z1+01 can be determined. At the mean contact point M0,the instantaneous transmission ratio is equal to the gear ratio.Usually this point is chosen to be in the middle of the tooth surface,and its location can be adjusted according to design requirements.The rotation angles of the gear 02 and the pinion at this point 01,can be determined from the location and the meshing equation at M0. 2.2 Determination of transmission errors The transmission error function is represented by 2=(202) (101) Z1/ Z2 (1) Where i(i=1,2)is the rotation angle of the pinion(i=1)or the gear(i=2)in the process of meshing,and Zi the tooth number of the pinion(i=1)or the gear(i=2). The parabolic function of transmission error is represented by 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 4 页 共 33 页 2= m21(101)2 (2) where m21 is the derivative of the transmission ratio. 2 is determined as follows 2=Z1(101)/Z2+02+ m21(101)2 (3) The rotation angles of gear 12 and 22,at the meshing points M1 and M2,can be determined from Eq.(3). 2.3 Determination of orientation of contact path As the contact path is designed to be a straight line,the three meshing points M0,M1 and M2 are on it.The meshing equation between the gear surface 2 and the pinion surface 1 at the contact point M1 should observe n12=f(g,g, 12)=0 (4) where n and 12 are the unit normal and the relative velocity at the meshing point, g and g represent the surface coordinates of the gear tooth surface. Based on the location of the contact point M0 and the given orientation of contact path 2,the location of the contact point M1 can be determined.The location equation of contact point M1 is g(g,g, 12)=0 (5) By solving Eqs.(3)-(5),the position vector and unit normal of the point M1 at the gear surface 2 can be obtained.The position vector and the unit normal of the point M2 at the surface 2 can also be determined the same way. 2.4 Determination of 11 and 21 Fig.2 shows the coordinate systems for pinion generation.The coordinate systems Sm1,Sc and Sb are fixed on the machine.The pinion machine root angle 1 determines the orientation of Sb with respect to Sm1.XG1 is the machine center to back for generation of pinion；Em1 the blank offset and XB1 the sliding base.The cradle coordinate system Sp rotates about the Zm1-axis.The angle p is the current rotation angle of the cradle in the process of generation.The coordinate system Sf is connected rigid to the pinion head-cutter,which in the process of generation performs rotation with the cradle (transfer motion)and relative motion with respect to 华北科技学院毕业设计 第 5 页 共 33 页 Fig.2 Coordinate systems for pinion generation. the cradle about the Zf-axis.The movable coordinate system S1 is connected rigid to the generated pinion and rotates about the Xb-axis.The angle 1 is the current rotation angle of pinion in the process of generation. 11 and 21 are the rotation angles of pinion in the process of generation at points M1 and M2.By transforming the coordinate from S2 to Sc,the unit normal nc of contact points M1 and M2 in system Sc can be determined.Then by transforming the coordinate from Sf to Sm1,the unit normal nm1 of pinion-cutter-generating surface in Sm1 can be determined.Since the axis of Sm1 is parallel to the axis of Sc,the unit normal of contact point in the system Sc is equal to that in the system Sm1.The following equation holds true. nc (1) =nm1(p+ p) (6) from which 11 and 21 can be determined. 2.5 Determination of XG1,Em1, 1p and 2p The position vectors rcf of points M1 and M2 in the system Sc can be derived from their position vectors on pinion-cutter-generating surfaces by transforming the coordinate from Sf to Sc.The position vectors rc can be derived from the position vectors of contact points M1 and M2 on the surface 2 by transforming the coordinate from S2 to Sc.rcf and rc coincide with each other at the instantaneous point of contact M1(M2).The position vector equation is rcf=(XG1,Em1, p,sp)=rc (7) As for the points M1 and M2,the position vector equations,which are equivalent to six independent scalar equations with six unknowns,can be 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 6 页 共 33 页 used to determine the six unknowns:XG1,Em1, 1p, 2p,s1p and s2p.Here 1p and 2p are the rotation angles of cradle in the process of generating contact points M1 and M2. 2.6 Determination of rp,sr1,q1,XB1 and mp1 From the given angle 2 on the contact path 2,and the length of the major axis of the instantaneous contact ellipse,the principal curvatures and directions at mean contact point M0 for the pinion head-cutter can be determined with the local synthesis.Then the cutter point radius rp can be determined.Based on the position vectors for the point M0 in the systems Sc and Sm1,the machine tool-settings sr1,q1 and XB1 can be determined.Likewise,based on the unit normal for the point M0 in the system Sc and the equation of meshing between the pinion head-cutter and the pinion to be generated,the cutting ratio mp1 can be determined. 2.7 Determination of modified roll The modified roll means the cutting ratio not being constant in the process of generation.The rotation angle of the cradle p and the rotation angle of the pinion to be generated 1 are related by a polynomial function,usually,of the third order 1=pC(p)2D(p)3 (8) where C and D are modified roll coefficients.As the rotation angles 1p and 2p of the cradle and the rotation angles 11 and 21 of the pinion in the process of generating points M1 and M2 are all known.It is possible to use Eq.(8)to determine the modified roll coefficients. 3 Redesigning Pinion Machine Tool-settings Based on Blank Offset If the blank offset,calculated according to Section 2,is well beyond the appropriate applicable range of the machine,it should be modified,and the pinion machine tool-settings be redesigned on the result acquired in Section 2. 3.1 Determination of XG1,rp and mp1 Principal directions(ef, eh)and principal curvatures(kf,kh)of a pinion tooth surface 1 at the mean contact point M0 can be determined by local synthesis between the gear tooth surface 2 and the pinion tooth surface 1.Then using the meshing equation of pinion and head-cutter on the mean contact point M0,the machine center to back XG1 can be determined. Based on the Rodrigues formula and the condition of continuous tangency of head-cutter surface p and pinion tooth surface 1 along the line,the following two equations can be 华北科技学院毕业设计 第 7 页 共 33 页 obtained 212=1122 (9) 1123=1213 (10) From Eq.(9),the principal curvatures ks of point M0 on the cone surface of the head-cutter can be determined while the other principal curvature being zero.Then the cutter point radius rp can be determined.From Eq.(10),the ratio of pinion roll mp1 can be determined. 3.2 Determination of modified roll coefficients r(1)h(p,p,1)= r(2)h(g,g,1) (11) n(1)h(p,p,1)= n(2)h(g,g,1) (12) Eqs.(11)-(12)describe the continuous tangency of the pinion and the gear tooth surfaces 1 and 2, the subscripts 1 and 2 denote the pinion tooth surface and the gear tooth surface,respectively.Eq.(11) indicates that the position vectors of the point on 1 and the point on 2 coincide at the instantaneous contact point in the fixed coordinate system Sh,and Eq.(12)that the surface unit normals do at the contact point.Eqs.(11)-(12)are equivalent to five independent scaler equations with five unknowns.The parameters p and p represent the surface coordinates of 1,and g and g of 2.The parameter 1 denotes the rotation angle of pinion in the process of generation at the contact point. The rotation angles of gear 12 and 22 at the contact points M1 and M2 can be determined byEq.(3),and are chosen to be the input in solving Eqs.(11)-(12).Then 11, 21and 1p, 2p can be determined. By Eq.(8),the modified roll coefficients C and D can be determined. The solutions of Eqs.(11)-(12)are not unique.Rather,the different solution determines the different contact point under the same rotation angle of gear 2.In order to keep the contact point as close to the predesigned contact point as possible,the corresponding parameters of the point M1(M2)from Section 2 should be chosen as the initial values of the five unknowns to solve Eq.(11)and Eq.(12). An example is taken to show the influences of modification of blank offset on the contact path,of which the design parameters are listed in Table 1.Fig.3(a)shows the contact pattern designed according to Section 2.The contact path is a straight line and the blank offset calculated is27.604 mm.Then by assuming the blank offset to be 0 mm,the redesign 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 8 页 共 33 页 according to Section 3 is accomplished on the base of the first design.Fig.3(b)shows the contact pattern. Table 1 Blank data 华北科技学院毕业设计 第 9 页 共 33 页 Fig.3 Influences of blank offset on contact path. The redesign aims at making the values of blank offset within the appropriate applicable range of the machine.Since the rotation angles of pinion at control meshing points are known,and the corresponding rotation angles of gear can be determined from Eq.(3),the modification of blank offset does not change the magnitude of transmission errors. The nonlinear Eqs.(11)and(12)have multiple solutions.The parameters of control meshing points M1 and M2 calculated in active tooth surface design(see Sections 2.3 and 2.5)are used to be the initial values to solve equations,so the control meshing points redesigned can be as close to the corresponding points M1 and M2 as possible.Although the redesigned contact path has a very small curvature,it still comes extremely close to the straightline shape resulting from the function-oriented active tooth surface design. In order to absorb larger errors of misalignment, the contact path should be designed to be a straigh line.On the other hand,the values of blank offse depend on the appropriate applicable range of the machine.Therefore,the modification of blank offse should be within the appropriate applicable range of the machine and made to reduce the curvature of the contact path as much as possible. 4 Example of Designing Spiral Bevel Gear An example of designing a spiral bevel gear has been accomplished to illustrate the proposed approach.The design parameters are listed in Table 1. 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 10 页 共 33 页 The concave side of the pinion tooth surface and the convex side of the gear tooth surface are considered the driving and driven surfaces,respectively.On the working flank,the geometry of the transmission error is designed to be of a parabolic function；the magnitude of the transmission error is 8.25,and the predesigned contact path orientation is 22o from the root cone.On the non-working flank,the magnitude of the transmission error is designed to be 11.25and the predesigned contact path orientation 14o from the root cone.Blank offset is assumed to be 0 mm. Table 2 and Table 3 show the machine tool-settings of the gear and the pinion. Table 2 Gear machine tool-settings Table 3 Pinion machine tool-settings 华北科技学院毕业设计 第 11 页 共 33 页 The results from tooth contact analysis(TCA) are shown in Fig.4,which include the adjusted contact pattern and the obtained function of transmission errors. 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 12 页 共 33 页 Fig.4 Contact patterns and transmission errors. 5 Experimental The spiral bevel gear pair designed according to Section 4 is processed by the Phoenix 800PG grinding machine.The actual tooth surfaces are measured on the Mahrs measurement device.The theoretically calculated tooth surface is used as a baseline for comparison.Fig.5 compares the tooth 华北科技学院毕业设计 第 13 页 共 33 页 Fig.5 Deviations of actual tooth surface from theoretical tooth surfaces for gear and pinion. topographies,obtained from the mathematical model and the data measured on the real manufactured gears.For the gear,the maximum surface deviations are 0.015 mm on the convex side and 0.010 mm on the concave side,and for the pinion, the maximum surface deviations are 0.004 mm on the convex side and 0.004 mm on the concave side.Moreover,the maximum deviations are all located far away from the contact area while the deviations on the contact area are near zero.Fig.6 and Fig.7 show the real contact patterns on the working flank and the non-working flank,which are quite consistent with the theoretically calculated results. 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 14 页 共 33 页 Fig.6 Contact patterns on working flank 华北科技学院毕业设计 第 15 页 共 33 页 Fig.7 Contact patterns on non-working flank. 6 Conclusions From the computer calculation,simulation and experiment,some conclusions can be made as follows: (1)The proposed approach to design pinion surfaces is based on controlling three meshing points.The geometry of transmission errors is designed to be a parabolic function and the magnitude can be calculated by derivation of transmission ratio m21,an input variable.The contact path is designed to be a straight line and its orientation can be adjusted.The magnitude of transmission errors and the contact path are designed separately.This provides a better ground for the further design of the transmission errors under loads. (2)The values of blank offset can be so modified as to have no influences on the 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 16 页 共 33 页 magnitude of transmission error apart from there being few of effects on the predesigned straight contact path. (3)On the Phoenix grinding machine,a spiral bevel gear pair is produced,whose meshing mark verifies the computer-calculated and simulation results. References 1Litvin F L.Local synthesis and tooth analysis of face-milled of spiral bevel gears.NASA-CR-4342,1990. 2Litvin F L.Gear geometry and applied theory.Englewood Cliffs:Prentice Hall,1994. 3Lewicki D G,Handschuh R F,Henry Z S.Low-noise,high strength spiral bevel gears for helicopter transmission.Journal of Propulsion and Power 1994；10(3):356-361. 4Zhang Y.Computering analysis of meshing and contact of gear real tooth surfaces.ASME Journal of Mechanical Design 1994；116(6):677-671. 5Zhang Y,Litvin F L,Handschuh R F.Computerized design of low-noise face-milled spiral bevel gears.Mechanism and Machine Theory 1995；30(8):1171-1178. 6Lin C Y,Tsay C B,Fong Z H.Mathematical model of spiral bevel and hypoid gears manufactured by the modified roll method. Mechanism and Machine Theory 1997；32(1):121-136. 7Litvin F L,Wang A G,Handschuh R F.Computerized generation and simulation of meshing and contact of spiral bevel gears with improved geometry.Journal Computer Methods in Applied Me-chanics and Engineering 1998；158(1):35-64. 8Lin C Y,Tsay C B,Fong Z H.Computer-aided manufacturing of spiral bevel and hypoid 华北科技学院毕业设计 第 17 页 共 33 页 gears by applying optimization tech-niques.Journal of Materials Processing Technology 2001；114(1): 22-35. 9Simon V.Optimal machine tool settings for hypoid gears improving load distribution.ASME Journal of Mechanical Design 2001；123(12):557-582. 10Fuentes A,Litvin F L,Woods B R,et al.Design and stress analysis of low-noise adjusted bearing contact spiral bevel gears.ASME Journal of Mechanical Design 2002；124(4):524-532. 11Litvin F L.Computerized design,simulation of meshing,and contact and stress analysis of face-milled formate generated spiral bevel gears.Mechanism and Machine Theory 2002；37(3): 441-459. 12Argyris J.Computerized integrated approach for design and stress of spiral bevel gears.Comput Methods Appl Mech Engrg 2002；191(8):1057-1095. 13Fang Z D.Tooth contact analysis of spiral bevel gears based on the design of transmission error.Acta Aeronautica et Astronautica Sinica 2002；23(3):226-230.in Chinese 14Cao X M,Fang Z D,Zhang J L.Function-oriented active tooth surface design of spiral bevel gears.Chinese Journal of Mechanical Engineering 2007；43(8):155-158.in Chinese 15Cao X M,Fang Z D,Zhang J L.Analysis and design of the pinion machine settings for spiral bevel gears.China Mechanical Engineering 2007；18(13):1584-1587.in Chinese 16Wang P Y,Fong Z H.Fourth-order kinematic synthesis for face milling spiral bevel gears with modified radial motion(MRM) correction.ASME Journal of Mechanical Design 2006；128(2): 457-467. 17Medvedev V I,Volkov A E.Synthesis of spiral bevel gear transmissions with a small shaft angle.Journal of Mechanical Design 2007；129(9):949-959. 18Fang Z D,Cao X M,Zhang J L.Measuring date processing of aviation spiral bevel gears by using coordinate measurement.Acta Aeronautica et Astronautica Sinica 2007；28(2):456-459.in Chinese Biographies: Cao Xuemei Born in 1970,her main research interests include design,manufacture and measurement of spiral bevel and hypoid gears. E-mail:2004 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 18 页 共 33 页 Fang Zongde Born in 1948,Ph.D.,professor.His main research interests include dynamics of structure,micro-air- craft design and CAE. E-mail: 华北科技学院毕业设计 第 19 页 共 33 页 译文： 通过控制接触的路径和传输误差来为螺旋锥 轮设计齿轮机床设置 曹雪梅 a 方宗德 a 徐昊 b 苏金展 a a:西北理工大学 机电一体化 中国 西安 邮编 710072 b:中南航空工业学院 中国 长沙 邮编 410200 2007 年 9月 16 日收到 2008 年 2 月审核通过 摘要 :本文提出了一种通过控制接触的路径和传输 误差来为 螺旋锥齿轮 设计齿轮机床设置 的 新方法 。此方法是基于齿面上三个给定的控制点接触情况的符合度。这 三个啮合点是要控制在一个预先直接联系的 路径内 ， 且此路径 符合预先的抛物线功能传输 误差 设计 。 另外，设计 、 规模传输 误差 和方向 与 联络道路受到精确的 控制。 此外，为了满足生产要求，我们建议修改空白补偿 的价值、其中一种 的插齿机床设置，并重新设计齿轮机床设置，以确保规模和几何形状，传输 误差 不 会 受到影响，除了对预 先直接 路径 的影响。对于 一对螺旋锥齿轮 来说 ,此 方法连同其理念已被一个数值 模型所 证明 ,并且是一个制造业的实践 。 关键词 : 螺旋锥齿轮 ； 联络 路 径；传输误差；空白补偿； 齿面接触分析 1 前言 一般说来， 螺旋锥齿轮是航天发电厂其中的关键组成部 分 ，尤其是直升机齿轮传动 。 啮合性能，耐力和可靠性 是 这方面重要的安全 参数。因而 ，许多公司和机构 的 研究人员 一直都在 密切关注螺旋锥齿轮 的 设计。 同时，也在设计 减少噪音水平和增加耐力的螺旋锥齿轮 方面给设计师带来了 艰巨的挑战。 低噪声局部轴承接触螺旋锥齿轮 的基础设计 ， 参见相关参考文献上。 为了 吸收较大的 误差 路线和有较好的稳定性，联络 路径 的设计应是一 条 直 线。因为 传输 误差 ，大多归咎于噪音 与 齿轮系统 振动 ，抛物型 的 传输 误差 被认 为能够吸收线性间断的影响所造成的偏心 ，而成 为主要噪音来源。 在某些情况下， 运用 现有的合成方法 来 机床设置设计的， 将 远远 超过机器的 适当的适用范围 。 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 20 页 共 33 页 在 本文中 ，在满足接触条件包容性的预抛物功能的传输 误差 和通过 在齿面上的 三个给定控制点的 直 接 接触路径 的 基础上 ，提 出了一种新的综合 方 法 。 因此，早在设计阶段，经营业绩可 以得到预先控制。 此外，空白抵销 的价值的 修改 可以 在 机器的 适当范围内 ，从而 对规模传输错误没有任何影响，除了对预线性联系的路径有轻微的 影响。 此 方法是基于以下的假设 ： （ 1） 齿轮机床设置 可以 预 先设定且能被采纳。 （ 2）主要的输入参数是 2 和 m21。 其中， 2决定了 预 先直接 接触 路径 的 方位角 ，也决定了 传输功能 的 二阶导数 m21，而 m21决定了传输 误差 预先设计的规模抛物线的功能 。在本文中，参数 2 和 m21事先给定的。但在实际中， 通过加载齿面接触分析 ，它们 可以依赖于应用负载以取得良好的啮合性能， 从而得到优化。 2 通过三个给定啮合点进行活动齿面设计 由预 先直接 接触路径和抛物功能的传输 误差确定 三个 齿面上的 控制啮合点 后 ，齿轮机床设置才能 得到 确定 。 2.1 三个接触点的确定 图 1 展示出了齿面上的三个接触点。 2是齿轮的俯仰角。接触路径被设计成一条直径，同时， 2是接触路径的方位。 图 1 三个控制啮合点与接触路径 当齿轮表面 2由 02, 12 和 22决定时 ,齿轮表面 1将分别以角度 02, 12 和 22在点 M0,M1 和 M2 与其接触。如果齿轮啮合的周长为 2 /Z1， Z1 是齿轮的齿数，那么华北科技学院毕业设计 第 21 页 共 33 页 11= /Z1+ 01 和 21= /Z1+ 01得到确定。 对于接触点 M0， 瞬时传动比等于齿轮比率 。通常这个点被选在齿轮表面的中央，而且它的位置可以根据设计要求进行调整。 齿轮 02和在 01这一点上 的 插齿 的旋转 角度，可以通过 M0点的 位置 与 啮合方程 来确定。 2.2 确定传输误差 传输误差函数由下式表示： 2=( 2 02) ( 1 01) Z1/ Z2 (1) i(i=1,2)是啮合过程中插齿 (i=1)或者齿轮 (i=2)的方位角， Zi是插齿 (i=1)或者齿轮 (i=2)的齿轮数目。 传输误差的抛物线函数由下式表示： 2= m21( 1 01)2 (2) m21是传动比率的导数。 2 又由下式表示： 2=Z1( 1 01)/Z2+ 02+ m21( 1 01)2 (3) 在点 M1和 M2的齿轮 12 和 22的方位角可以由公式（ 3）得出。 2.3 接触路径方位的确定 因为接触路径要被设计成一条直线 ,同时三个啮合点 M0,M1和 M2在其之上，所以齿轮表面 2与插齿表面 1在点 M1的啮合方程必须满足： n 12=f( g,g, 12)=0 (4) n and 12是在啮合点上的普通单元与相当速度， g and g表示齿轮表面的 表面坐标。 在接触点 M0的位置与接触路径 2给定的方位基础上 ,接触点 M1的位置才能得到确定。接触点 M1的位置方程是： g( g,g, 12)=0 (5) 通过解出方程 (3)-(5), 在齿轮表面 2上的 接触点 M1的位置矢量和普通单元可以得出。同理可以得出 齿轮表面 2上的 接触点 M2的位置矢量和普通单元。 2.4 确定 11 和 21 图 2 表示出了齿轮的坐标系。坐标系 Sm1,Sc 和 Sb是固定在机床上的。插齿机器的普通中型车床主轴箱与齿轮加工工艺及工装设计 第 22 页 共 33 页 基准角度 1和 Sm1共同决定了 Sb的方位。 XG1是机床的中心，它支持着齿轮的运转、空白补偿 Em1和滑动地点 XB1。基准坐标系 Sp围绕着 Zm1轴旋转。角度 p是基准坐标系旋转过程中的即时方位角。坐标系 Sf牢固地连接在齿轮刀头上 ,而它是在执行基准坐标系 (运动转变 )的旋转过程和围绕着 Zf轴的相对转动中的。可移动坐标系 S1牢固地连接在运动的插齿上并围绕着 Xb旋转。角度 1是插齿旋转过程中的即时方位角。 11和 21分别是插齿运动到点 M1和 M2处的方位角。通过将坐标从 S2改变到 Sc，坐标系 Sc里接触点 M1和 M2的普通单元 nc就能得到确定。然后通过将坐 标从 Sf改变到 Sm1，坐标系 Sm1里齿轮运转机表面上的普通单元 nm1就能得到确定。由于 Sm1轴和 Sc轴是平行的，那么两个坐标系中接触点上的普通单元是等同的，有下面的方程成立： nc (1) =nm1( p+ p) (6) 从中可以得出 11和 21的值。 图 2 插齿的坐标系 2.5 确定 XG1,Em1, 1p 和 2p 坐标系 Sc 中接触点 M1和 M2的方位向量 rcf，通过将坐标从 Sf改变到 Sc，可以在齿轮运转机表面上得到。方位向量 rc，通过将坐标从 S2改 变到 Sc，可以从齿轮表面 2上接触点 M1和 M2上得出。 rcf和 rc在瞬时接触点 M1（ M2）是相互关联的。其方位向量方程是： rcf=(XG1,Em1, p,sp)=rc (7) 对于点 M1和 M2来说，方位向量方程 相当于 6 个独立的标量方程与 6 个 未知数，可用来确定 6个 未知数 ： XG1,Em1, 1p, 2p,s1p和 s2p。这里的 1p和 2p是基准坐标系旋转过程中华北科技学院毕业设计 第 23 页 共 33 页 接触点 M1和 M2的方位角。 2.6 确定 rp,sr1,q1,XB1和 mp1 从接触路径 2 上给定的角度 2，和 主要轴线的瞬时接触椭圆形 的长度以及齿轮头到的主要接触点 M0主曲率和方向 可以由局部的合成得到。之后刀头半径 rp就可以得出。基于坐标系 Sc和 Sm1中接触点 M0的方位矢量，机床设置值 sr1,q1 和 XB1即可得出。同理，基于坐标系 Sc中接触点 M0上的普通单元和刀头齿轮与旋转齿轮间的啮合方程，可以得出切割比率 mp1 2.7 确定可调节 轧辊 可调节 轧辊 指的是在齿轮旋转过程中切割比率并不是恒量。起始 p的方位角度和启动齿轮的 1的方位角度，通常来说与一个三次多项式函数有关。 1=pC(p)2D(p)3 (8) 其中 C 和 D 是 可调节 轧辊 系数。由于起始坐标系的方位角 1p与 2p和在旋转过程中的接触点 M1与 M2的方位角 11与 21都是已知的，所以可以运用公式（ 8）来确定可调节轧辊 系数。 3 基于空白补偿的齿轮机床设置的再设计 如果空白补偿根据章节 2来计算，大大超过了机床的合适的适用范围，那么就应该对它进行调整。同时齿轮机床设置也应该根据从章节 2 得到的结果进行再设计。 3.1 确定 XG1,rp和 mp1 通过对齿轮表面 2 与插齿表面 1 间的局部合成，在主要 接触点 M0 上齿轮表面 1的主方向 (ef, eh) 与主曲率 (kf,kh)即可以得出。然后在主要接触点 M0上运用齿轮与刀头间的啮合方程，支撑 XG1的机床中心可以得到确定。 基于 Rodrigues 公式和刀头沿直线连续切削齿轮表面 p与齿轮齿面 1的情况，可以得出以下两个公式： 212= 11 22 (9) 11 23= 12 13 (10) 根据公式（ 9），当另一个主曲率为零时，可以得出刀头的锥面上的接触点 M0 的主曲率 ks。之后切削点的半径 rp 也可以得出。根据公式（ 10），可以得出 插齿辊 mp1的比例。 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 24 页 共 33 页 3.2 确定可调节 轧辊系数 r(1)h( p,p,1)= r(2)h( g,g,1) (11) n(1)h( p,p,1)= n(2)h( g,g,1) (12) 公式 （ 11） -（ 12） 描述了齿轮的连续切削 ,和插齿齿面 1 与齿轮齿面 2,下标 1与 2 分别代表表示插齿齿面 与齿轮齿面。公式（ 11）表明表面 1与表面 2上的点的方位矢量在 固定坐标系 Sh的 瞬时接触点 上相合。同时根据公式（ 12），齿面上的普通单元在接触点上。公式（ 11） -（ 12）相当于拥有 5 个未知数的 5 个独立的标量方程。参数 p和 p表示表面 1上的坐标 ,而 g和 g则表示表面 2上的坐标。参数 1表示插齿在转动过程中在接触点上的旋转角度。 通过方程（ 3），齿轮 12和 22在接触点 M1和 M2的旋转角度即可以得出，然后被选出，作为输入量来解方程（ 11） -（ 12）。然后可以得出 11, 21和 1p, 2p。 通 过方程（ 8），可以得出 可调节 轧辊 系数 C和 D。 方程（ 11） -（ 12）的解并不是惟一的。当齿轮 2的旋转角度一定时，不同的解将得出不同的接触点。为了使这个得出的接触点与预设计时的接触点尽可能地接近，来自章节 2的相应的参数 M1(M2)将被选出，作为方程（ 11）和（ 12） 5个未知数的初始值来解这两个方程。 有一个例子可以看出改变接触路径上的空白补偿所带来的影响，其设计参数见表 1。图 3（ a）表示出了根据章节所得出的接触方式设计。其中接触路径是一条直线，同时计算得出空白补偿为 27.604 mm。然后假设空白补偿为 0mm，那么在第一次设计的基础上，可以根据章节 3完成再设计。 表 1 空白补偿数据 华北科技学院毕业设计 第 25 页 共 33 页 图 3 在接触路径上空白补偿的影响 再设计的目的是在机床合适的适用范围内是空白补偿产生价值。因为在一个可控制啮合点上的插齿的旋转角度是已知的，同时可以从方程（ 3）中得出齿轮的旋转角度，普通中型车床主轴箱与齿轮加工工艺及工装设计 第 26 页 共 33 页 那么空白补偿的变化不会改变传输误差的大小。方程（ 11） -（ 12）有多组解。在运动齿轮齿面（参见章节 2.3 和 2.5）中计算得出的可控制啮合点 M1和 M2的参数，被作为初始值来解方程，那么再设计的可控制啮合点可以做到尽可能地接近相应的接触点 M1 和M2。尽管再设计的路径会有一个很小的曲率，但是它还是能十分接近面向功能的运动齿轮齿面的设计出的直线路径。 为了能抵消偏心所造成的误差，接触路径必须设计成一条直线。另一方面，空白补偿的作用取决于机床合适的适用范围的大小。因而，空白补偿的变化应该在机床合适的适用范围之内，同时它的变化要尽可能地减小接触路径的曲率。 4 螺旋锥齿轮的设计举例 现在已经完成了一个 螺旋锥齿轮 设计的例子，以此来展现本文中的再设计方法。其设计参数见表 1。 凹侧的插齿齿面 与 凸侧的齿轮齿面 分别是主动切削表面与被切削表面。在工作侧翼，传输误差 的几何形状被设计成包含一个抛物线函数；传输误差的大小为 8.25，从齿轮根锥到预设计的接触路径的方位角为 22o。在非工作侧翼，将传输误差的大小设计为 11.25，从齿轮根锥到预设计的接触路径的方位角为 14o。空白补偿假设为 0mm。 表 2和表 3表示了齿轮与插齿的机床设置。 表 2 齿轮的机床设置 表 3 插齿的机床设置 华北科技学院毕业设计 第 27 页 共 33 页 图 4 中表示出了齿面接触分析 (TCA)的结果，它包括调整了的接触方式和来自传输误差的功能。 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 28 页 共 33 页 图 4 接触方式与传输误差 5 实验 根据章节 4得出的 螺旋锥齿轮对设计 将在凤凰 800PG 磨床上 进行实验。在 马尔的测量装置 对实际齿面进行测量。 该理论计算的齿面是用来作为 比较基准的。图 5比较了齿轮的 拓朴图 ，此图由数学模型和实际制造的齿轮的测量数据所得来。齿轮的凸面最高表面偏差为 0.015mm，凹面最高最高表面偏差为 0.010mm。插齿的 凸面最高表面偏差为0.004mm，凹面最高最高表面偏差也为 0.004mm。最高的偏差都是远离接触面的区域，而接触面的偏差几乎为零。 华北科技学院毕业设计 第 29 页 共 33 页 图 5 齿轮与插齿实际齿面与理论齿面的偏差 图 6 与图 7 表示出了齿轮工作侧翼和非工作侧翼的实际接触路径，这与理论计算结果相当一致。 普通中型车床主轴箱与齿轮加工工艺及工装设计 第 30 页 共 33 页 图 6 工作侧翼的接触路径 华北科技学院毕业设计 第 31 页 共 33 页 图 7 非工作侧翼的接触路径 6 结论 经过计算机的计算、模拟与实验，可以得出如下一些结论： （ 1） 本文中 的设计插齿表面 的 方法是基于控制三个啮合点 的。 传输误差的几何形状被设计成包含一个抛物线函数，其大小可由传动比率（一个输入变量）的偏差所计算出。接触路径被设计成一条直线且其方向可以调整。传输误差大小与接触路径是分开进行设计的，这为在负载下对传输误差做进一步设计打下了一个更好的基础。 （ 2）空白补偿具有很大的可变性，以致于除了对预设计的直线接触路径有微弱影响外，其对于传输误差的大小没 有影响。 （ 3）在凤凰磨床上制造出一对 螺旋锥齿轮 ，而齿轮的啮合路径验证了计算机计算与模拟的结果。 普通中型车床主轴箱