高空作业车的转台结构设计及有限元分析【含CAD图纸和说明书】
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（译文）选择固定参数研究齿轮牙侧面的设计规则摘要：科学技术和生产的发展对齿轮传动有了更高的要求，影响齿轮的动态性能的关键因素是齿轮牙的侧面啮合形式。为了提高齿轮的传动性能，一种新的被称着为LogiX的齿轮在19世纪发明出来。然而，对于这种特殊的齿轮还有很多理论和实践上的未知问题等待解决。 本文进一步研究这种新型齿轮的设计准则并运用数学推理的方法对齿轮牙侧面进行了分析。通过对齿轮牙侧面啮合形式的影响参数的讨论，说明了这种LogiX齿轮的参数选择的合理性，发展和加强了LogiX齿轮的理论体系。这种对参数等的研究最终是为了能够发明出现代耐久的传动产品。关键词：基本参数 设计原理 LogiX齿轮 详细概括 齿轮牙的侧面1介绍为了提高齿轮的传动性能和满足一些特殊的要求，一种新的齿轮应运而生；他被命名为“LogiX”为增加一些优异的性能和渐开线齿轮另外拥有以上两种类型齿轮的优点的新型齿轮还有一些别的特殊的优点.在这种新的齿轮牙，连续的凸面联络被执行从它的齿根高对它的补充,那里被确定的安全相对弯曲有很多点。这里这种点被叫着零位点（N-P）许多的出现在（N-Ps）点在LogiX 齿轮期间的滤网过程可能导致一个滑动系数, 并且滤网传输表现成为相应地几乎滚动的摩擦。因而这种新型的齿轮有许多好处，譬如更高的传动强度、寿命长而且比标准断开线齿轮有更大的传输比率。实验性结果表示, 给一定数量的（N-Ps） 在二个捕捉的LogiX 齿轮之间, 3倍的传动疲劳强度和2.5倍的抗弯疲劳强度远远比那些标准断开线齿轮承受能力大。而且, 最小的牙数可达到3个这是那些标准断开线齿轮没法比的。这种被认为新型的LogiX 齿轮还有很多未被解决的问题。计算机数字控制(CNC) 技术的发展必然被用来研究更高效率的方法来发展这种新型齿轮。因此进一步提高研究这种齿轮的宽度和实际应用的加速度显的很重要。本文有信心在这个新时代里把齿轮滤网理论和应用取得历史性的突破。 2齿牙侧面的设计原则 根据齿轮滤网和制造业理论, 为了简化问题分析, 从齿轮的基本的机架开始入手来研究。让我们首先从讨论LogiX 齿轮的基本机架开始。图1 显示LogiX 机架的设计原则的划分和断开线曲线。在图一中P.L代表 LogiX 机架的节线。选择点是为了形成角0。P.L O1N1.两个径向的交叉点O1n0 和O1N1，节线P.L和N1，n0。使得延长到使得两个基圆的切线相交到，和幅线相交于两个圆的交点和节线P.L交点，两个圆的交点和节线P.L交点。使得公切线和基圆相交和，在有关齿侧面点m0和m1方面曲率半径应该是: ，在节线上相交于中心。不同倍数的回旋包括LogiX 外形应该被安排为一个适当的顺序。下个渐开弯曲线m1m2的压力角度应该比前段m0m1的有所增加。中心曲度在极端点m1 、m2, 等应该是在节线上, 并且基本的圈子压力半径的作用变化应该是从G1 到G2 。形成的条件为前曲线和后曲线的半径曲度必须是在和点m1相等的对半径曲度在点m1 之后同时半径曲度在点m2处 必须是与半径曲度相等的在点m2 之后。图2 显示渐开线曲线的准确连接的过程。根据上述讨论,整体外牙形成。 图1：LogiX 机架外形牙的设计原则 图2.渐开线曲线的准确连接3LogiX的 侧牙数学模块3.1 基本的LogiX 机架的数学模块根据上述设计原则, 每个精确曲线外形的曲率中心应该在机架节线上找出, 和每点在相对曲度连接的价值不同，在渐开线曲线上的应该是零点。外形牙的设计是关于节线对称的,齿根高凹面和凸面是互补的。因而作为LogiX外形牙的整体, 它可以被确切的划分成为四份, 如图3所示 。座标设定如图4中所示, 节线P.L在座标起点O 与交点m0之间和形成最初的渐开线曲线。 根据图4中的座标设定, 形成最初的渐开线曲线m0m1 如图5所示。 图3. LogiX 机架外形牙 图4. 座标设定 图5. 最初的渐开线曲线m0m1的形成过程 图6：LogiX齿轮啮合和基圆 现在，和参数量，和作为已知条件。在曲线和渐开线的交点是，或者。因此曲率半径和压力角在渐开线的交点处的关系如下： （1） （2）根据几何关系我们可以得出以下结论： （3）根据式1，2和3和有关LogiX齿的形成材料，根据曲率半径的形成规则可得到如下关系：。当且，可得出特殊关系式如下： （4）显然，在任一齿牙侧面的k 点处压力角的关系如下： （5）当是关系式5可变为如下： （6）根据数学几何关系可得到No.2的关系式： （No2） （7）显然根据几何关系可得别的式子如下：（No1） （8）（No3） （9）（No4） （10）3.2 LogiX齿轮的数学模块 配合角，和PXY如图6所示，在LogiX齿轮架和LogiX齿轮间的精确的啮合关系。这里被定位在齿轮架上是齿轮侧面和节线的交点被定位在齿轮的啮合处是齿轮的中心。PXY是完全的中空角，P点是齿轮的切线和基圆的交点。 为了和理论建立起一致的关系。假如上面的例子中LogiX齿轮牙的侧面从 改动到OXY同时再改变到，一种新型的齿轮模型的关系就产生如下： （11）在这里理想的正角，只在LogiX齿轮模型的第一象限中给出。4自身固有参数和它们选择的原因对LogiX齿轮的影响除标准渐开线的基本参数外, LogiX齿牙侧面有自己固有的基本参数，譬如起始压力角度、相对压力角度,起始基圆半径 G0,等等。这些参数的选择对LogiX齿牙侧面渐开线的影响非常大,它的结构形式会直接影响力齿轮的传动能力。因此基本参数的选择非常重要。4.1起始压力角的选择和影响考虑到要设计较高传动性能的齿轮,起始压力角为0度。但是最后的计算结果表示 LogiX 齿牙侧面加工工具的角度和起始的压力角度是相等的。这样起始压力角度不能对准零位。比较相对于两倍圆周-弧的齿轮，我们可以推出的起始压力角越小,齿轮越大越容易产生根切。因此起始压力角应该不仅仅是零,但也不能太小，同时，从例 3,4 和5,可以看出对LogiX齿牙方面的影响可以用图7来直接描述。显然地，起始压力角度数的增加会引起 LogiX 齿架的曲率的增大。如果选择一个较大的齿架，而起始压力角太小的话。它的齿顶会变的很窄或产生根切现象。因此 LogiX 齿牙侧面选较大时，应该选一个较小的起始压力角,当LogiX 齿牙侧面选较小时，选一个较大的起始压力角。通常,实践计算经验告诉我们，起始压力角 取2度到12度，而且LogiX齿轮模型越大，起始压力角越小。4.2起始基圆半径 G0的选择和影响根据公式在LogiX 齿轮牙的侧面的不同位置有两个参数影响基圆半径 Gi：在牙齿描绘的不同位置的 LogiX 齿轮: 一个是G0 另一个是起始压力角。图8所示的是当给定参数0和时G0对LogiX齿侧面的影响。显然地,如果G0增加，新型齿轮牙的侧面曲率将变得越来越小。显而易见，它会随着G0的减小而逐渐增大。因此新型齿架的参数大时G0也应选大的，同时当齿架的参数小时G0也选小。4.3压力角的选择和影响在图 9中显示参数的变化对齿牙的影响。根据 LogiX齿牙的形成过程，参数越小在LogiX齿轮的两齿之间形成的N-Ps越大。根据2.1中的描述相对压力角在N-P mk中的关系如下： 如式5和12,选择比较大的参数相应的参数也比较大，选择适当的起始压力角和最大压力角，压力角越小N-Ps越多，相反地,比较小的参数，N-Ps的数字较大。当取0.0006度时，零点的数字将超过 46,000 。在这情形,选择一个齿轮模数m =100, 两个N-Ps点之间会变的很小。也就是说，在整个 LogiX齿轮 的运动过程中，两个啮合齿轮间在很短的时间内会参数打滑和滚动。N-Ps数目越多在两齿轮间越长相反传动时间越短。因此它的磨损减少了使用寿命就增长了。但是, 考虑到承载能力的限制，速度的改变、角度的因素等等当切割这种类型的齿轮时必须用CNC机床刀具，相关压力角的选择非常小，一般必须满足度。 图7：0对LogiX齿轮侧面的影响 图8 ：G0对LogiX齿轮侧面的影响图9 ：对LogiX齿轮侧面的影响4.4 选择合理参数举例 基于上述对LogiX齿轮固有参数选择的分析规则，对于不同的零件模型，当它的相对压力角为0.05度时，起始压力角和基圆半径合理的计算结果如下表1所作的参考。事实上, 实际的选择应该根据具体切断情况和特殊需求而定。5 结论下面是根据调查结果得出的结论：1. 通过进一步的深入研究可推出LogiX齿轮的二维啮合传动规律。2. 讨论研究了齿轮自身基本参数譬如起始压力角，起始基圆半径和相对压力角以及参数的选择，对LogiX 齿轮牙侧和性能的影响。 3. 通过对LogiX 齿轮理论系统和数学基础的进一步研究建立了现代 CNC 技术。LogiX 齿牙的特性：它不如常规渐开线齿轮应用广但是它是一种承载能力大，体积小，寿命长的产品。6 命名法 起始压力角度 交点 mi 处的压力角 压力角度参数 在交点 s 1 处的 齿轮牙齿侧面的曲率半径 在交点 mi处的 齿轮牙齿侧面的曲率半径 在交点 m1处的 齿轮牙齿侧面的曲率半径 起始基圆半径 齿轮牙侧面上点mi初基圆的半径 LogiX齿轮啮合转动时和基架LogiX的夹角r2 LogiX 齿轮啮合时的基圆和基架LogiX的半径m 齿轮的模型z 齿数s 齿厚，这里， i 是任意数 9DOI 10.1007/s00170-003-1741-8ORIGINAL ARTICLEInt J Adv Manuf Technol (2004) 24: 789793Feng Xianying Wang Aiqun Linda LeeStudy on the design principle of the LogiX gear tooth profileand the selection of its inherent basic parametersReceived: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004Abstract The development of scientific technology and productivityhas called for increasingly higher requirements of geartransmission performance. The key factor influencing dynamicgear performance is the form of the meshed gear tooth profile. Toimprove a gears transmission performance, a new type of gearcalled the LogiX gear was developed in the early 1990s. However,for this special kind of gear there remain many unknowntheoretical and practical problems to be solved. In this paper, thedesign principle of this new type of gear is further studied andthe mathematical module of its tooth profile deduced. The influenceon the form of this type of tooth profile and its meshperformance by its inherent basic parameters is discussed, andreasonable selections for LogiX gear parameters are provided.Thus the theoretical system information about the LogiX gear aredeveloped and enriched. This study impacts most significantlythe improvement of load capacity, miniaturisation and durabilityof modern kinetic transmission products.Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile1 IntroductionIn order to improve gear transmission performance and satisfysome special requirements, a new type of gear 1 was put forward;it was named “LogiX” in order to improve some demeritsof W-N (Wildhaver-Novikov) and involute gears.Besides having the advantages of both kinds of gears mentionedabove, the new type of gear has some other excellentF. Xianying (_) W. AiqunSchool of Mechanical Engineering,Shandong University,P.R. ChinaE-mail: FXYTel.: +86-531-8395852(0)L. LeeSchool of Mechanical & Manufacturing Engineering,Singapore Polytechnic,Singaporecharacteristics. On this new tooth profile, the continuous concave/convex contact is carried out from its dedendum to its addendum,where the engagements with a relative curvature of zeroare assured at many points. Here, this kind of point is called thenull-point (N-P). The presence of many N-Ps during the meshprocess of LogiX gears can result in a smaller sliding coefficient,and the mesh transmission performance becomes almostrolling friction accordingly. Thus this new type of gear has manyadvantages such as higher contact intensity, longer life and alarger transmission-ratio power transfer than the standard involutegear. Experimental results showed that, given a certainnumber of N-Ps between two meshed LogiX gears, the contactfatigue strength is 3 times and the bend fatigue strength 2.5 timeslarger than those of the standard involute gear. Moreover, theminimum tooth number can also be decreased to 3, much smallerthan that of the standard involute gear.The LogiX gear, regarded as a new type of gear, still presentssome unsolved problems. The development of computer numericalcontrolling (CNC) technology must also be taken into considerationnew high-efficiency methods to cut this new type ofgear. Therefore, further study of this new type of gear mostsignificantly impacts the acceleration of its broad and practicalapplication. This paper has the potential to usher in a new era inthe history of gear mesh theory and application.2 Design principle of LogiX tooth profileAccording to gear mesh and manufacturing theories, in order tosimplify problem analysis, generally a gears basic rack is begunwith some studies 2. So here let us discuss the basic rack ofthe LogiX gear first. Figure 1 shows the design principle of dividedinvolute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1 is selected to formthe angle n0O1N1 =0, P.L O1N1. The points of intersectionby two radials O1n0 and O1N1 and the pitch line P.L are N1and n0. Let O1n0 = G1, extend O1n0 to O_1 , and make two tangentbasic circles whose centres are O1, O_1 and radii are equalto G1. The point of intersection between circle O1 and pitch line790Fig. 1. Design principle of LogiX rack tooth profileP.L is n0. The point of intersection between circle O2 and pitchline P.L is n1. Make the common tangent g1s1 of basic circle O1and O_1, then generate two minute involute curves m0s1 and s1m1whose basic circle centres are O1 and O_1. The radii of curvatureat points m0 and m1 on the tooth profile should be: m0 = m0n0,m1 = m1n1, and the centres are met on the pitch line.Multiple different minute involutes consisting of a LogiXprofile should be arranged for a proper sequence. The pressureangle of the next minute involute curve m1m2 should have anincrement comparable to its last segment m0m1. The centres ofcurvature at extreme points m1, m2, etc. should be on the pitchline, and the radius of the basic circle is a function of pressure 1 it varies from G1 to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1 must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2 must be equal to the radius of curvaturejust after point m2. Figure 2 shows the connection and process ofgenerating minute involute curves. According to the above discussion,the whole tooth profile can be formed.Fig. 2. Connection of minute involute curves3 Mathematicmodule of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curvaturecentre of every finely divided profile curve should be locatedat the rack pitch line, and the value of the relative curvature atevery point connecting different minute involute curves shouldbe zero. The design of the tooth profile is symmetrical with respectto the pitch line, and the addendum is convex while thededendum is concave. Thus for the whole LogiX tooth profile, itcan be dealt with by dividing it into four parts, as shown in Fig. 3.Set up the coordinates as shown in Fig. 4, where the origin ofthe coordinates O coincides with the point of intersection m0 betweenrack pitch line P.L and the initial divided minute involutecurve.According to the coordinates set up in Fig. 4, the formationof initial minute involute curve m0m1 is shown in Fig. 5.Fig. 3. LogiX rack tooth profileFig. 4. Set-up of coordinatesFig. 5. Formation process of initial minute involute curve m0m1791Here: n0n_0 O1O_1 , n1n_1 O1O_1 , n1n1 n0n_0, and the parameters0, , G1 and m0 are given as initial conditions. Thecurvature radius of the involute curve at point s1 is s1 = G1, ors1 = m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1 are as follows:m1 = s1G11 = G1(1) (1)1 = 0+1 . (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1 cos G1 cos 1G1 sin G1 sin 1=2(cos +cos 1)sin sin 1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point onthe profile is deduced: mi =mi1+Gi(i ). When i =k andm0 = 0, it is expressed as follows:mk = G1(1)+G2(2)+ +Gk(k)=k_i=1Gi(i) . (4)Similarly, the pressure angle on an arbitrary k point of the toothprofile can be deduced as follows:k = 0+(+1)+(+2)+ (+k)= 0+k_i=1(+i) = 0+k+k_i=1i . (5)By ni1ni = Gi(sin sin i)/ cos(i1 +), Eq. 5 can beobtained:n0nk =k_i=1ni1ni =k_i=1Gi(sin sin i )cos(i1 +). (6)Thus the mathematical model of the No. 2 portion for the LogiXrack profile is as follows:_x1 = n0nk mk cos ky1 = mk sin k(No. 2) . (7)Similarly, the mathematical models of the other three segmentscan also be obtained as follows:_x1 =(n0nk mk cos k)y1 =mk sin k(No.1) (8)_x1 = s(n0nk mk cos k)y1 = mk sin k(No.3) (9)_x1 = s+n0nk mk cos ky1 =mk sin k(No.4) . (10)Fig. 6. Mesh coordinatesof LogiX gear and its basicrack3.2 Mathematical module of the LogiX gearThe coordinates O1X1Y1, O2X2Y2 and PXY are set up as shownin Fig. 6 to express the mesh relationship between the LogiXrack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, andO1 is the point of intersection between the rack tooth profile andits pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is thegears centre. PXY is an absolute coordinate, and P is the pointof intersection of the racks pitch line and the gears pitch circle.In accordance with gear meshing theories 3, if the abovemodel of the LogiX rack tooth profile is changed from coordinateO1X1Y1 to OXY, and then again to O2X2Y2, a new type of gearprofile model can be deduced as follows:_x2 =mk cos k cos 2 (mk sin k r2) sin 2y2 =mk cos k sin 2 +(mk sin k r2) cos 2 .(11)Here the positive direction of 2 is clockwise, and only the modelof the LogiX gear tooth profile in the first quadrant of the coordinatesis given.4 Effect on the performance of the LogiX gear by itsinherent parameters and their reasonable selectionBesides the basic parameters of the standard involute rack, theLogiX tooth profile has inherent basic parameters such as initialpressure angle 0, relative pressure angle , initial basic circleradius G0, etc. The selection of these parameters has a great influenceon the form of the LogiX tooth profile, and the formdirectly influences gear transmission performance. Thus the reasonableselection of these basic parameters is very important.4.1 Influence and selection of initial pressure angle 0Considering the higher transmission efficiency in practical design,the initial pressure angle 0 should be selected as 0. Butthe final calculation result showed that the LogiX gear tooth profilecut by the rack tool whose initial pressure angle was equalto zero would be overcut on the pitch circle generally. Thus theinitial pressure angle 0 cannot be zero. Comparing the relativedouble circle-arc gear 3, we can also deduce that the smaller792the initial pressure angle 0, the larger the gear number for producingthe overcut. Thus the initial pressure angle 0 shouldnot only not be zero, but should not be too small, either. FromEqs. 3, 4 and 5, the influence of 0 on the LogiX tooth profilecan be directly described by Fig. 7. Obviously, increasing the initialpressure angle will cause the curvature of the LogiX racktooth profile to become larger. If the rack selects a larger moduleand too small an initial pressure angle 0, its addendum willbecome too narrow or even overcut. Thus the LogiX tooth profilethat selects a larger module should select a smaller 0, andthe profile that selects a smaller module should select a larger0. Generally, by practical calculation experience, the selected0 should be located within a range of 2 12, and the largerthe LogiX gear module, the smaller should be its initial pressureangle 0.4.2 Influence and selection of initial basic circle radius G0According to the empirical formula Gi = G01sin(0.6i ) 1,there are two parameters affecting the basic circle radius Gi ofthe LogiX gear at different positions of tooth profile: one is theG0 and the other is the initial pressure angle i . Figure 8 showsthe influence of G0 on the LogiX tooth profile when certainvalues of parameter 0 and are selected. Obviously, as G0 increases,the curvature of the new type of gear tooth profile willbecome smaller and smaller. Conversely, it will become increasinglylarger as G0 decreases. Thus the new type of rack witha large module parameter should select a large G0 value, andone having a small module parameter should select a small G0value.4.3 Influence and selection of relative pressure angle Figure 9 shows the variable of the tooth profile affected by the parameter. According to the forming process of the LogiX toothFig. 7. Influence of 0 onLogiX tooth profileFig. 8. Influence of G0 onLogiX tooth profileFig. 9. Influence of onLogiX tooth profileprofile, the smaller the selected parameter , the larger the numberof N-Ps meshing on the tooth profile of two LogiX gears.From Sect. 2.1 the formula describing the relative pressure anglek of an arbitrary N-P mk can be deduced as follows:sin(k1+)cos(k1 +) =2(cos +cos k)sin sin k. (12)By Eqs. 5 and 12, the larger the parameter being selected, thelarger will be the k parameter, and at certain selected valuesof the initial pressure angle and maximum pressure angle, thelower will be the number of N-Ps. By contrast, the smallerthe parameter, the larger the number of N-Ps. While is0.0006, the number of zero points can exceed 46,000. In thiscase, selecting a gear module of m = 100, the length of themicro-involute curve between two adjoining N-Ps will be onlya few microns. That is to say, during the whole meshing processof the LogiX gear transmission, the sliding and rollingmotions happen alternately and last only a few micro-secondsfrom one motion to another between two meshed gear tooth profiles.The greater the number of N-Ps, the longer the relativerolling time between two LogiX gears and the shorter the relativesliding time between two LogiX gears. Thus abrasion of thegear decreases and its loading capability and life span are improved.But, considering the restriction of memory capability,interpolation speed, angular resolution, etc. for the CNCmachinetool used while cutting this type of gear, the relative pressureangle selected should not be very small. _ 0.0006 is generallysatisfactory.Table 1. Parameter values selected for LogiX rack at different modulesm(mm) 0 G0(mm)1 10 0.05 60002 8.0 0.05 95004 6.0 0.05 100005 5.0 0.05 110006 4.0 0.05 120008 3.2 0.05 1202410 2.8 0.05 1400012 2.6 0.05 1650015 2.5 0.05 2002418 2.4 0.05 3003620 2.4 0.05 3500022 2.3 0.05 380007934.4 Reasonable selection exampleBased on the above analytical rules for LogiX gear inherent parameterselection, a reasonable calculation and selection resultsfor the initial pressure angle and basic circle radius while selectingdifferent modules at the relative pressure angle = 0.05 arelisted in Table 1 for reference. In fact, the practical selectionsshould be reasonably modified by the concrete cutting conditionsand the special purpose requirement.5 ConclusionsThe following conclusions were made based on the findings presentedin this paper.1. Two-dimensional meshing transmission models of LogiXgears were deduced by further analysis of its formingprinciple.2. The influence on the LogiX gear tooth profile and its performanceby the gears own basic parameters such as initialpressure angle, initial basic circle radius and relative pressureangle was discussed and their reasonable selection was given.3. The theoretical system of the LogiX gear was developed andthe mathematical basis for generating the LogiX tooth profileby modern CNC technology was established. The characteristicsof the LogiX gear, which are different from those of theordinary standard involute gear, can have broad applicationand most significantly impact the improvement of carryingcapacity, miniaturisation and longevity of kinetic transmissionproducts.References1. Komori T, Arga Y, Nagata S (1990) A new gear profile havingzero relative curvature at many contact points. Trans ASME 112(3):4304362. Xutang W (1982) Gear meshing theory. Machinery Industry Press,Beijing3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing6 Nomenclature0 initial pressure anglei pressure angle at contact point mi parameter of pressure angles1 radius of curvature of gear tooth profile at contact point s1mi radius of curvature of gear tooth profile at contact point mim1 radius of curvature of gear tooth profile at contact pointm1G0 initial radius of basic circle in tooth profileGi radius of basic circle of mi point in gear tooth profile2 rotation angle of LogiX gear meshing with basic LogiXrackr2 radius of basic circle of LogiX gear meshing with basicLogiX rackm model of gearz gear tooth numbers gear tooth thickness at pitch circle; here, i is an optionalnumberDOI 10.1007/s00170-003-1741-8ORIGINAL ARTICLEInt J Adv Manuf Technol (2004) 24: 789793Feng Xianying Wang Aiqun Linda LeeStudy on the design principle of the LogiX gear tooth profileand the selection of its inherent basic parametersReceived: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004Abstract The development of scientific technology and productivityhas called for increasingly higher requirements of geartransmission performance. The key factor influencing dynamicgear performance is the form of the meshed gear tooth profile. Toimprove a gears transmission performance, a new type of gearcalled the LogiX gear was developed in the early 1990s. However,for this special kind of gear there remain many unknowntheoretical and practical problems to be solved. In this paper, thedesign principle of this new type of gear is further studied andthe mathematical module of its tooth profile deduced. The influenceon the form of this type of tooth profile and its meshperformance by its inherent basic parameters is discussed, andreasonable selections for LogiX gear parameters are provided.Thus the theoretical system information about the LogiX gear aredeveloped and enriched. This study impacts most significantlythe improvement of load capacity, miniaturisation and durabilityof modern kinetic transmission products.Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile1 IntroductionIn order to improve gear transmission performance and satisfysome special requirements, a new type of gear 1 was put forward;it was named “LogiX” in order to improve some demeritsof W-N (Wildhaver-Novikov) and involute gears.Besides having the advantages of both kinds of gears mentionedabove, the new type of gear has some other excellentF. Xianying (_) W. AiqunSchool of Mechanical Engineering,Shandong University,P.R. ChinaE-mail: FXYTel.: +86-531-8395852(0)L. LeeSchool of Mechanical & Manufacturing Engineering,Singapore Polytechnic,Singaporecharacteristics. On this new tooth profile, the continuous concave/convex contact is carried out from its dedendum to its addendum,where the engagements with a relative curvature of zeroare assured at many points. Here, this kind of point is called thenull-point (N-P). The presence of many N-Ps during the meshprocess of LogiX gears can result in a smaller sliding coefficient,and the mesh transmission performance becomes almostrolling friction accordingly. Thus this new type of gear has manyadvantages such as higher contact intensity, longer life and alarger transmission-ratio power transfer than the standard involutegear. Experimental results showed that, given a certainnumber of N-Ps between two meshed LogiX gears, the contactfatigue strength is 3 times and the bend fatigue strength 2.5 timeslarger than those of the standard involute gear. Moreover, theminimum tooth number can also be decreased to 3, much smallerthan that of the standard involute gear.The LogiX gear, regarded as a new type of gear, still presentssome unsolved problems. The development of computer numericalcontrolling (CNC) technology must also be taken into considerationnew high-efficiency methods to cut this new type ofgear. Therefore, further study of this new type of gear mostsignificantly impacts the acceleration of its broad and practicalapplication. This paper has the potential to usher in a new era inthe history of gear mesh theory and application.2 Design principle of LogiX tooth profileAccording to gear mesh and manufacturing theories, in order tosimplify problem analysis, generally a gears basic rack is begunwith some studies 2. So here let us discuss the basic rack ofthe LogiX gear first. Figure 1 shows the design principle of dividedinvolute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1 is selected to formthe angle n0O1N1 =0, P.L O1N1. The points of intersectionby two radials O1n0 and O1N1 and the pitch line P.L are N1and n0. Let O1n0 = G1, extend O1n0 to O_1 , and make two tangentbasic circles whose centres are O1, O_1 and radii are equalto G1. The point of intersection between circle O1 and pitch line790Fig. 1. Design principle of LogiX rack tooth profileP.L is n0. The point of intersection between circle O2 and pitchline P.L is n1. Make the common tangent g1s1 of basic circle O1and O_1, then generate two minute involute curves m0s1 and s1m1whose basic circle centres are O1 and O_1. The radii of curvatureat points m0 and m1 on the tooth profile should be: m0 = m0n0,m1 = m1n1, and the centres are met on the pitch line.Multiple different minute involutes consisting of a LogiXprofile should be arranged for a proper sequence. The pressureangle of the next minute involute curve m1m2 should have anincrement comparable to its last segment m0m1. The centres ofcurvature at extreme points m1, m2, etc. should be on the pitchline, and the radius of the basic circle is a function of pressure 1 it varies from G1 to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1 must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2 must be equal to the radius of curvaturejust after point m2. Figure 2 shows the connection and process ofgenerating minute involute curves. According to the above discussion,the whole tooth profile can be formed.Fig. 2. Connection of minute involute curves3 Mathematicmodule of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curvaturecentre of every finely divided profile curve should be locatedat the rack pitch line, and the value of the relative curvature atevery point connecting different minute involute curves shouldbe zero. The design of the tooth profile is symmetrical with respectto the pitch line, and the addendum is convex while thededendum is concave. Thus for the whole LogiX tooth profile, itcan be dealt with by dividing it into four parts, as shown in Fig. 3.Set up the coordinates as shown in Fig. 4, where the origin ofthe coordinates O coincides with the point of intersection m0 betweenrack pitch line P.L and the initial divided minute involutecurve.According to the coordinates set up in Fig. 4, the formationof initial minute involute curve m0m1 is shown in Fig. 5.Fig. 3. LogiX rack tooth profileFig. 4. Set-up of coordinatesFig. 5. Formation process of initial minute involute curve m0m1791Here: n0n_0 O1O_1 , n1n_1 O1O_1 , n1n1 n0n_0, and the parameters0, , G1 and m0 are given as initial conditions. Thecurvature radius of the involute curve at point s1 is s1 = G1, ors1 = m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1 are as follows:m1 = s1G11 = G1(1) (1)1 = 0+1 . (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1 cos G1 cos 1G1 sin G1 sin 1=2(cos +cos 1)sin sin 1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point onthe profile is deduced: mi =mi1+Gi(i ). When i =k andm0 = 0, it is expressed as follows:mk = G1(1)+G2(2)+ +Gk(k)=k_i=1Gi(i) . (4)Similarly, the pressure angle on an arbitrary k point of the toothprofile can be deduced as follows:k = 0+(+1)+(+2)+ (+k)= 0+k_i=1(+i) = 0+k+k_i=1i . (5)By ni1ni = Gi(sin sin i)/ cos(i1 +), Eq. 5 can beobtained:n0nk =k_i=1ni1ni =k_i=1Gi(sin sin i )cos(i1 +). (6)Thus the mathematical model of the No. 2 portion for the LogiXrack profile is as follows:_x1 = n0nk mk cos ky1 = mk sin k(No. 2) . (7)Similarly, the mathematical models of the other three segmentscan also be obtained as follows:_x1 =(n0nk mk cos k)y1 =mk sin k(No.1) (8)_x1 = s(n0nk mk cos k)y1 = mk sin k(No.3) (9)_x1 = s+n0nk mk cos ky1 =mk sin k(No.4) . (10)Fig. 6. Mesh coordinatesof LogiX gear and its basicrack3.2 Mathematical module of the LogiX gearThe coordinates O1X1Y1, O2X2Y2 and PXY are set up as shownin Fig. 6 to express the mesh relationship between the LogiXrack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, andO1 is the point of intersection between the rack tooth profile andits pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is thegears centre. PXY is an absolute coordinate, and P is the pointof intersection of the racks pitch line and the gears pitch circle.In accordance with gear meshing theories 3, if the abovemodel of the LogiX rack tooth profile is changed from coordinateO1X1Y1 to OXY, and then again to O2X2Y2, a new type of gearprofile model can be deduced as follows:_x2 =mk cos k cos 2 (mk sin k r2) sin 2y2 =mk cos k sin 2 +(mk sin k r2) cos 2 .(11)Here the positive direction of 2 is clockwise, and only the modelof the LogiX gear tooth profile in the first quadrant of the coordinatesis given.4 Effect on the performance of the LogiX gear by itsinherent parameters and their reasonable selectionBesides the basic parameters of the standard involute rack, theLogiX tooth profile has inherent basic parameters such as initialpressure angle 0, relative pressure angle , initial basic circleradius G0, etc. The selection of these parameters has a great influenceon the form of the LogiX tooth profile, and the formdirectly influences gear transmission performance. Thus the reasonableselection of these basic parameters is very important.4.1 Influence and selection of initial pressure angle 0Considering the higher transmission efficiency in practical design,the initial pressure angle 0 should be selected as 0. Butthe final calculation result showed that the LogiX gear tooth profilecut by the rack tool whose initial pressure angle was equalto zero would be overcut on the pitch circle generally. Thus theinitial pressure angle 0 cannot be zero. Comparing the relativedouble circle-arc gear 3, we can also deduce that the smaller792the initial pressure angle 0, the larger the gear number for producingthe overcut. Thus the initial pressure angle 0 shouldnot only not be zero, but should not be too small, either. FromEqs. 3, 4 and 5, the influence of 0 on the LogiX tooth profilecan be directly described by Fig. 7. Obviously, increasing the initialpressure angle will cause the curvature of the LogiX racktooth profile to become larger. If the rack selects a larger moduleand too small an initial pressure angle 0, its addendum willbecome too narrow or even overcut. Thus the LogiX tooth profilethat selects a larger module should select a smaller 0, andthe profile that selects a smaller module should select a larger0. Generally, by practical calculation experience, the selected0 should be located within a range of 2 12, and the largerthe LogiX gear module, the smaller should be its initial pressureangle 0.4.2 Influence and selection of initial basic circle radius G0According to the empirical formula Gi = G01sin(0.6i ) 1,there are two parameters affecting the basic circle radius Gi ofthe LogiX gear at different positions of tooth profile: one is theG0 and the other is the initial pressure angle i . Figure 8 showsthe influence of G0 on the LogiX tooth profile when certainvalues of parameter 0 and are selected. Obviously, as G0 increases,the curvature of the new type of gear tooth profile willbecome smaller and smaller. Conversely, it will become increasinglylarger as G0 decreases. Thus the new type of rack witha large module parameter should select a large G0 value, andone having a small module parameter should select a small G0value.4.3 Influence and selection of relative pressure angle Figure 9 shows the variable of the tooth profile affected by the parameter. According to the forming process of the LogiX toothFig. 7. Influence of 0 onLogiX tooth profileFig. 8. Influence of G0 onLogiX tooth profileFig. 9. Influence of onLogiX tooth profileprofile, the smaller the selected parameter , the larger the numberof N-Ps meshing on the tooth profile of two LogiX gears.From Sect. 2.1 the formula describing the relative pressure anglek of an arbitrary N-P mk can be deduced as follows:sin(k1+)cos(k1 +) =2(cos +cos k)sin sin k. (12)By Eqs. 5 and 12, the larger the parameter being selected, thelarger will be the k parameter, and at certain selected valuesof the initial pressure angle and maximum pressure angle, thelower will be the number of N-Ps. By contrast, the smallerthe parameter, the larger the number of N-Ps. While is0.0006, the number of zero points can exceed 46,000. In thiscase, selecting a gear module of m = 100, the length of themicro-involute curve between two adjoining N-Ps will be onlya few microns. That is to say, during the whole meshing processof the LogiX gear transmission, the sliding and rollingmotions happen alternately and last only a few micro-secondsfrom one motion to another between two meshed gear tooth profiles.The greater the number of N-Ps, the longer the relativerolling time between two LogiX gears and the shorter the relativesliding time between two LogiX gears. Thus abrasion of thegear decreases and its loading capability and life span are improved.But, considering the restriction of memory capability,interpolation speed, angular resolution, etc. for the CNCmachinetool used while cutting this type of gear, the relative pressureangle selected should not be very small. _ 0.0006 is generallysatisfactory.Table 1. Parameter values selected for LogiX rack at different modulesm(mm) 0 G0(mm)1 10 0.05 60002 8.0 0.05 95004 6.0 0.05 100005 5.0 0.05 110006 4.0 0.05 120008 3.2 0.05 1202410 2.8 0.05 1400012 2.6 0.05 1650015 2.5 0.05 2002418 2.4 0.05 3003620 2.4 0.05 3500022 2.3 0.05 380007934.4 Reasonable selection exampleBased on the above analytical rules for LogiX gear inherent parameterselection, a reasonable calculation and selection resultsfor the initial pressure angle and basic circle radius while selectingdifferent modules at the relative pressure angle = 0.05 arelisted in Table 1 for reference. In fact, the practical selectionsshould be reasonably modified by the concrete cutting conditionsand the special purpose requirement.5 ConclusionsThe following conclusions were made based on the findings presentedin this paper.1. Two-dimensional meshing transmission models of LogiXgears were deduced by further analysis of its formingprinciple.2. The influence on the LogiX gear tooth profile and its performanceby the gears own basic parameters such as initialpressure angle, initial basic circle radius and relative pressureangle was discussed and their reasonable selection was given.3. The theoretical system of the LogiX gear was developed andthe mathematical basis for generating the LogiX tooth profileby modern CNC technology was established. The characteristicsof the LogiX gear, which are different from those of theordinary standard involute gear, can have broad applicationand most significantly impact the improvement of carryingcapacity, miniaturisation and longevity of kinetic transmissionproducts.References1. Komori T, Arga Y, Nagata S (1990) A new gear profile havingzero relative curvature at many contact points. Trans ASME 112(3):4304362. Xutang W (1982) Gear meshing theory. Machinery Industry Press,Beijing3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing6 Nomenclature0 initial pressure anglei pressure angle at contact point mi parameter of pressure angles1 radius of curvature of gear tooth profile at contact point s1mi radius of curvature of gear tooth profile at contact point mim1 radius of curvature of gear tooth profile at contact pointm1G0 initial radius of basic circle in tooth profileGi radius of basic circle of mi point in gear tooth profile2 rotation angle of LogiX gear meshing with basic LogiXrackr2 radius of basic circle of LogiX gear meshing with basicLogiX rackm model of gearz gear tooth numbers gear tooth thickness at pitch circle; here, i is an optionalnumberDOI 10.1007/s00170-003-1741-8ORIGINAL ARTICLEInt J Adv Manuf Technol (2004) 24: 789793Feng Xianying Wang Aiqun Linda LeeStudy on the design principle of the LogiX gear tooth profileand the selection of its inherent basic parametersReceived: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004Abstract The development of scientific technology and productivityhas called for increasingly higher requirements of geartransmission performance. The key factor influencing dynamicgear performance is the form of the meshed gear tooth profile. Toimprove a gears transmission performance, a new type of gearcalled the LogiX gear was developed in the early 1990s. However,for this special kind of gear there remain many unknowntheoretical and practical problems to be solved. In this paper, thedesign principle of this new type of gear is further studied andthe mathematical module of its tooth profile deduced. The influenceon the form of this type of tooth profile and its meshperformance by its inherent basic parameters is discussed, andreasonable selections for LogiX gear parameters are provided.Thus the theoretical system information about the LogiX gear aredeveloped and enriched. This study impacts most significantlythe improvement of load capacity, miniaturisation and durabilityof modern kinetic transmission products.Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile1 IntroductionIn order to improve gear transmission performance and satisfysome special requirements, a new type of gear 1 was put forward;it was named “LogiX” in order to improve some demeritsof W-N (Wildhaver-Novikov) and involute gears.Besides having the advantages of both kinds of gears mentionedabove, the new type of gear has some other excellentF. Xianying (_) W. AiqunSchool of Mechanical Engineering,Shandong University,P.R. ChinaE-mail: FXYTel.: +86-531-8395852(0)L. LeeSchool of Mechanical & Manufacturing Engineering,Singapore Polytechnic,Singaporecharacteristics. On this new tooth profile, the continuous concave/convex contact is carried out from its dedendum to its addendum,where the engagements with a relative curvature of zeroare assured at many points. Here, this kind of point is called thenull-point (N-P). The presence of many N-Ps during the meshprocess of LogiX gears can result in a smaller sliding coefficient,and the mesh transmission performance becomes almostrolling friction accordingly. Thus this new type of gear has manyadvantages such as higher contact intensity, longer life and alarger transmission-ratio power transfer than the standard involutegear. Experimental results showed that, given a certainnumber of N-Ps between two meshed LogiX gears, the contactfatigue strength is 3 times and the bend fatigue strength 2.5 timeslarger than those of the standard involute gear. Moreover, theminimum tooth number can also be decreased to 3, much smallerthan that of the standard involute gear.The LogiX gear, regarded as a new type of gear, still presentssome unsolved problems. The development of computer numericalcontrolling (CNC) technology must also be taken into considerationnew high-efficiency methods to cut this new type ofgear. Therefore, further study of this new type of gear mostsignificantly impacts the acceleration of its broad and practicalapplication. This paper has the potential to usher in a new era inthe history of gear mesh theory and application.2 Design principle of LogiX tooth profileAccording to gear mesh and manufacturing theories, in order tosimplify problem analysis, generally a gears basic rack is begunwith some studies 2. So here let us discuss the basic rack ofthe LogiX gear first. Figure 1 shows the design principle of dividedinvolute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1 is selected to formthe angle n0O1N1 =0, P.L O1N1. The points of intersectionby two radials O1n0 and O1N1 and the pitch line P.L are N1and n0. Let O1n0 = G1, extend O1n0 to O_1 , and make two tangentbasic circles whose centres are O1, O_1 and radii are equalto G1. The point of intersection between circle O1 and pitch line790Fig. 1. Design principle of LogiX rack tooth profileP.L is n0. The point of intersection between circle O2 and pitchline P.L is n1. Make the common tangent g1s1 of basic circle O1and O_1, then generate two minute involute curves m0s1 and s1m1whose basic circle centres are O1 and O_1. The radii of curvatureat points m0 and m1 on the tooth profile should be: m0 = m0n0,m1 = m1n1, and the centres are met on the pitch line.Multiple different minute involutes consisting of a LogiXprofile should be arranged for a proper sequence. The pressureangle of the next minute involute curve m1m2 should have anincrement comparable to its last segment m0m1. The centres ofcurvature at extreme points m1, m2, etc. should be on the pitchline, and the radius of the basic circle is a function of pressure 1 it varies from G1 to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1 must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2 must be equal to the radius of curvaturejust after point m2. Figure 2 shows the connection and process ofgenerating minute involute curves. According to the above discussion,the whole tooth profile can be formed.Fig. 2. Connection of minute involute curves3 Mathematicmodule of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curvaturecentre of every finely divided profile curve should be locatedat the rack pitch line, and the value of the relative curvature atevery point connecting different minute involute curves shouldbe zero. The design of the tooth profile is symmetrical with respectto the pitch line, and the addendum is convex while thededendum is concave. Thus for the whole LogiX tooth profile, itcan be dealt with by dividing it into four parts, as shown in Fig. 3.Set up the coordinates as shown in Fig. 4, where the origin ofthe coordinates O coincides with the point of intersection m0 betweenrack pitch line P.L and the initial divided minute involutecurve.According to the coordinates set up in Fig. 4, the formationof initial minute involute curve m0m1 is shown in Fig. 5.Fig. 3. LogiX rack tooth profileFig. 4. Set-up of coordinatesFig. 5. Formation process of initial minute involute curve m0m1791Here: n0n_0 O1O_1 , n1n_1 O1O_1 , n1n1 n0n_0, and the parameters0, , G1 and m0 are given as initial conditions. Thecurvature radius of the involute curve at point s1 is s1 = G1, ors1 = m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1 are as follows:m1 = s1G11 = G1(1) (1)1 = 0+1 . (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1 cos G1 cos 1G1 sin G1 sin 1=2(cos +cos 1)sin sin 1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point onthe profile is deduced: mi =mi1+Gi(i ). When i =k andm0 = 0, it is expressed as follows:mk = G1(1)+G2(2)+ +Gk(k)=k_i=1Gi(i) . (4)Similarly, the pressure angle on an arbitrary k point of the toothprofile can be deduced as follows:k = 0+(+1)+(+2)+ (+k)= 0+k_i=1(+i) = 0+k+k_i=1i . (5)By ni1ni = Gi(sin sin i)/ cos(i1 +), Eq. 5 can beobtained:n0nk =k_i=1ni1ni =k_i=1Gi(sin sin i )cos(i1 +). (6)Thus the mathematical model of the No. 2 portion for the LogiXrack profile is as follows:_x1 = n0nk mk cos ky1 = mk sin k(No. 2) . (7)Similarly, the mathematical models of the other three segmentscan also be obtained as follows:_x1 =(n0nk mk cos k)y1 =mk sin k(No.1) (8)_x1 = s(n0nk mk cos k)y1 = mk sin k(No.3) (9)_x1 = s+n0nk mk cos ky1 =mk sin k(No.4) . (10)Fig. 6. Mesh coordinatesof LogiX gear and its basicrack3.2 Mathematical module of the LogiX gearThe coordinates O1X1Y1, O2X2Y2 and PXY are set up as shownin Fig. 6 to express the mesh relationship between the LogiXrack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, andO1 is the point of intersection between the rack tooth profile andits pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is thegears centre. PXY is an absolute coordinate, and P is the pointof intersection of the racks pitch line and the gears pitch circle.In accordance with gear meshing theories 3, if the abovemodel of the LogiX rack tooth profile is changed from coordinateO1X1Y1 to OXY, and then again to O2X2Y2, a new type of gearprofile model can be deduced as follows:_x2 =mk cos k cos 2 (mk sin k r2) sin 2y2 =mk cos k sin 2 +(mk sin k r2) cos 2 .(11)Here the positive direction of 2 is clockwise, and only the modelof the LogiX gear tooth profile in the first quadrant of the coordinatesis given.4 Effect on the performance of the LogiX gear by itsinherent parameters and their reasonable selectionBesides the basic parameters of the standard involute rack, theLogiX tooth profile has inherent basic parameters such as initialpressure angle 0, relative pressure angle , initial basic circleradius G0, etc. The selection of these parameters has a great influenceon the form of the LogiX tooth profile, and the formdirectly influences gear transmission performance. Thus the reasonableselection of these basic parameters is very important.4.1 Influence and selection of initial pressure angle 0Considering the higher transmission efficiency in practical design,the initial pressure angle 0 should be selected as 0. Butthe final calculation result showed that the LogiX gear tooth profilecut by the rack tool whose initial pressure angle was equalto zero would be overcut on the pitch circle generally. Thus theinitial pressure angle 0 cannot be zero. Comparing the relativedouble circle-arc gear 3, we can also deduce that the smaller792the initial pressure angle 0, the larger the gear number for producingthe overcut. Thus the initial pressure angle 0 shouldnot only not be zero, but should not be too small, either. FromEqs. 3, 4 and 5, the influence of 0 on the LogiX tooth profilecan be directly described by Fig. 7. Obviously, increasing the initialpressure angle will cause the curvature of the LogiX racktooth profile to become larger. If the rack selects a larger moduleand too small an initial pressure angle 0, its addendum willbecome too narrow or even overcut. Thus the LogiX tooth profilethat selects a larger module should select a smaller 0, andthe profile that selects a smaller module should select a larger0. Generally, by practical calculation experience, the selected0 should be located within a range of 2 12, and the largerthe LogiX gear module, the smaller should be its initial pressureangle 0.4.2 Influence and selection of initial basic circle radius G0According to the empirical formula Gi = G01sin(0.6i ) 1,there are two parameters affecting the basic circle radius Gi ofthe LogiX gear at different positions of tooth profile: one is theG0 and the other is the initial pressure angle i . Figure 8 showsthe influence of G0 on the LogiX tooth profile when certainvalues of parameter 0 and are selected. Obviously, as G0 increases,the curvature of the new type of gear tooth profile willbecome smaller and smaller. Conversely, it will become increasinglylarger as G0 decreases. Thus the new type of rack witha large module parameter should select a large G0 value, andone having a small module parameter should select a small G0value.4.3 Influence and selection of relative pressure angle Figure 9 shows the variable of the tooth profile affected by the parameter. According to the forming process of the LogiX toothFig. 7. Influence of 0 onLogiX tooth profileFig. 8. Influence of G0 onLogiX tooth profileFig. 9. Influence of onLogiX tooth profileprofile, the smaller the selected parameter , the larger the numberof N-Ps meshing on the tooth profile of two LogiX gears.From Sect. 2.1 the formula describing the relative pressure anglek of an arbitrary N-P mk can be deduced as follows:sin(k1+)cos(k1 +) =2(cos +cos k)sin sin k. (12)By Eqs. 5 and 12, the larger the parameter being selected, thelarger will be the k parameter, and at certain selected valuesof the initial pressure angle and maximum pressure angle, thelower will be the number of N-Ps. By contrast, the smallerthe parameter, the larger the number of N-Ps. While is0.0006, the number of zero points can exceed 46,000. In thiscase, selecting a gear module of m = 100, the length of themicro-involute curve between two adjoining N-Ps will be onlya few microns. That is to say, during the whole meshing processof the LogiX gear transmission, the sliding and rollingmotions happen alternately and last only a few micro-secondsfrom one motion to another between two meshed gear tooth profiles.The greater the number of N-Ps, the longer the relativerolling time between two LogiX gears and the shorter the relativesliding time between two LogiX gears. Thus abrasion of thegear decreases and its loading capability and life span are improved.But, considering the restriction of memory capability,interpolation speed, angular resolution, etc. for the CNCmachinetool used while cutting this type of gear, the relative pressureangle selected should not be very small. _ 0.0006 is generallysatisfactory.Table 1. Parameter values selected for LogiX rack at different modulesm(mm) 0 G0(mm)1 10 0.05 60002 8.0 0.05 95004 6.0 0.05 100005 5.0 0.05 110006 4.0 0.05 120008 3.2 0.05 1202410 2.8 0.05 1400012 2.6 0.05 1650015 2.5 0.05 2002418 2.4 0.05 3003620 2.4 0.05 3500022 2.3 0.05 380007934.4 Reasonable selection exampleBased on the above analytical rules for LogiX gear inherent parameterselection, a reasonable calculation and selection resultsfor the initial pressure angle and basic circle radius while selectingdifferent modules at the relative pressure angle = 0.05 arelisted in Table 1 for reference. In fact, the practical selectionsshould be reasonably modified by the concrete cutting conditionsand the special purpose requirement.5 ConclusionsThe following conclusions were made based on the findings presentedin this paper.1. Two-dimensional meshing transmission models of LogiXgears were deduced by further analysis of its formingprinciple.2. The influence on the LogiX gear tooth profile and its performanceby the gears own basic parameters such as initialpressure angle, initial basic circle radius and relative pressureangle was discussed and their reasonable selection was given.3. The theoretical system of the LogiX gear was developed andthe mathematical basis for generating the LogiX tooth profileby modern CNC technology was established. The characteristicsof the LogiX gear, which are different from those of theordinary standard involute gear, can have broad applicationand most significantly impact the improvement of carryingcapacity, miniaturisation and longevity of kinetic transmissionproducts.References1. Komori T, Arga Y, Nagata S (1990) A new gear profile havingzero relative curvature at many contact points. Trans ASME 112(3):4304362. Xutang W (1982) Gear meshing theory. Machinery Industry Press,Beijing3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing6 Nomenclature0 initial pressure anglei pressure angle at contact point mi parameter of pressure angles1 radius of curvature of gear tooth profile at contact point s1mi radius of curvature of gear tooth profile at contact point mim1 radius of curvature of gear tooth profile at contact pointm1G0 initial radius of basic circle in tooth profileGi radius of basic circle of mi point in gear tooth profile2 rotation angle of LogiX gear meshing with basic LogiXrackr2 radius of basic circle of LogiX gear meshing with basicLogiX rackm model of gearz gear tooth numbers gear tooth thickness at pitch circle; here, i is an optionalnumberDOI 10.1007/s00170-003-1741-8ORIGINAL ARTICLEInt J Adv Manuf Technol (2004) 24: 789793Feng Xianying Wang Aiqun Linda LeeStudy on the design principle of the LogiX gear tooth profileand the selection of its inherent basic parametersReceived: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004Abstract The development of scientific technology and productivityhas called for increasingly higher requirements of geartransmission performance. The key factor influencing dynamicgear performance is the form of the meshed gear tooth profile. Toimprove a gears transmission performance, a new type of gearcalled the LogiX gear was developed in the early 1990s. However,for this special kind of gear there remain many unknowntheoretical and practical problems to be solved. In this paper, thedesign principle of this new type of gear is further studied andthe mathematical module of its tooth profile deduced. The influenceon the form of this type of tooth profile and its meshperformance by its inherent basic parameters is discussed, andreasonable selections for LogiX gear parameters are provided.Thus the theoretical system information about the LogiX gear aredeveloped and enriched. This study impacts most significantlythe improvement of load capacity, miniaturisation and durabilityof modern kinetic transmission products.Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile1 IntroductionIn order to improve gear transmission performance and satisfysome special requirements, a new type of gear 1 was put forward;it was named “LogiX” in order to improve some demeritsof W-N (Wildhaver-Novikov) and involute gears.Besides having the advantages of both kinds of gears mentionedabove, the new type of gear has some other excellentF. Xianying (_) W. AiqunSchool of Mechanical Engineering,Shandong University,P.R. ChinaE-mail: FXYTel.: +86-531-8395852(0)L. LeeSchool of Mechanical & Manufacturing Engineering,Singapore Polytechnic,Singaporecharacteristics. On this new tooth profile, the continuous concave/convex contact is carried out from its dedendum to its addendum,where the engagements with a relative curvature of zeroare assured at many points. Here, this kind of point is called thenull-point (N-P). The presence of many N-Ps during the meshprocess of LogiX gears can result in a smaller sliding coefficient,and the mesh transmission performance becomes almostrolling friction accordingly. Thus this new type of gear has manyadvantages such as higher contact intensity, longer life and alarger transmission-ratio power transfer than the standard involutegear. Experimental results showed that, given a certainnumber of N-Ps between two meshed LogiX gears, the contactfatigue strength is 3 times and the bend fatigue strength 2.5 timeslarger than those of the standard involute gear. Moreover, theminimum tooth number can also be decreased to 3, much smallerthan that of the standard involute gear.The LogiX gear, regarded as a new type of gear, still presentssome unsolved problems. The development of computer numericalcontrolling (CNC) technology must also be taken into considerationnew high-efficiency methods to cut this new type ofgear. Therefore, further study of this new type of gear mostsignificantly impacts the acceleration of its broad and practicalapplication. This paper has the potential to usher in a new era inthe history of gear mesh theory and application.2 Design principle of LogiX tooth profileAccording to gear mesh and manufacturing theories, in order tosimplify problem analysis, generally a gears basic rack is begunwith some studies 2. So here let us discuss the basic rack ofthe LogiX gear first. Figure 1 shows the design principle of dividedinvolute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1 is selected to formthe angle n0O1N1 =0, P.L O1N1. The points of intersectionby two radials O1n0 and O1N1 and the pitch line P.L are N1and n0. Let O1n0 = G1, extend O1n0 to O_1 , and make two tangentbasic circles whose centres are O1, O_1 and radii are equalto G1. The point of intersection between circle O1 and pitch line790Fig. 1. Design principle of LogiX rack tooth profileP.L is n0. The point of intersection between circle O2 and pitchline P.L is n1. Make the common tangent g1s1 of basic circle O1and O_1, then generate two minute involute curves m0s1 and s1m1whose basic circle centres are O1 and O_1. The radii of curvatureat points m0 and m1 on the tooth profile should be: m0 = m0n0,m1 = m1n1, and the centres are met on the pitch line.Multiple different minute involutes consisting of a LogiXprofile should be arranged for a proper sequence. The pressureangle of the next minute involute curve m1m2 should have anincrement comparable to its last segment m0m1. The centres ofcurvature at extreme points m1, m2, etc. should be on the pitchline, and the radius of the basic circle is a function of pressure 1 it varies from G1 to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1 must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2 must be equal to the radius of curvaturejust after point m2. Figure 2 shows the connection and process ofgenerating minute involute curves. According to the above discussion,the whole tooth profile can be formed.Fig. 2. Connection of minute involute curves3 Mathematicmodule of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curvaturecentre of every finely divided profile curve should be locatedat the rack pitch line, and the value of the relative curvature atevery point connecting different minute involute curves shouldbe zero. The design of the tooth profile is symmetrical with respectto the pitch line, and the addendum is convex while thededendum is concave. Thus for the whole LogiX tooth profile, itcan be dealt with by dividing it into four parts, as shown in Fig. 3.Set up the coordinates as shown in Fig. 4, where the origin ofthe coordinates O coincides with the point of intersection m0 betweenrack pitch line P.L and the initial divided minute involutecurve.According to the coordinates set up in Fig. 4, the formationof initial minute involute curve m0m1 is shown in Fig. 5.Fig. 3. LogiX rack tooth profileFig. 4. Set-up of coordinatesFig. 5. Formation process of initial minute involute curve m0m1791Here: n0n_0 O1O_1 , n1n_1 O1O_1 , n1n1 n0n_0, and the parameters0, , G1 and m0 are given as initial conditions. Thecurvature radius of the involute curve at point s1 is s1 = G1, ors1 = m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1 are as follows:m1 = s1G11 = G1(1) (1)1 = 0+1 . (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1 cos G1 cos 1G1 sin G1 sin 1=2(cos +cos 1)sin sin 1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point onthe profile is deduced: mi =mi1+Gi(i ). When i =k andm0 = 0, it is expressed as follows:mk = G1(1)+G2(2)+ +Gk(k)=k_i=1Gi(i) . (4)Similarly, the pressure angle on an arbitrary k point of the toothprofile can be deduced as follows:k = 0+(+1)+(+2)+ (+k)= 0+k_i=1(+i) = 0+k+k_i=1i . (5)By ni1ni = Gi(sin sin i)/ cos(i1 +), Eq. 5 can beobtained:n0nk =k_i=1ni1ni =k_i=1Gi(sin sin i )cos(i1 +). (6)Thus the mathematical model of the No. 2 portion for the LogiXrack profile is as follows:_x1 = n0nk mk cos ky1 = mk sin k(No. 2) . (7)Similarly, the mathematical models of the other three segmentscan also be obtained as follows:_x1 =(n0nk mk cos k)y1 =mk sin k(No.1) (8)_x1 = s(n0nk mk cos k)y1 = mk sin k(No.3) (9)_x1 = s+n0nk mk cos ky1 =mk sin k(No.4) . (10)Fig. 6. Mesh coordinatesof LogiX gear and its basicrack3.2 Mathematical module of the LogiX gearThe coordinates O1X1Y1, O2X2Y2 and PXY are set up as shownin Fig. 6 to express the mesh relationship between the LogiXrack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, andO1 is the point of intersection between the rack tooth profile andits pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is thegears centre. PXY is an absolute coordinate, and P is the pointof intersection of the racks pitch line and the gears pitch circle.In accordance with gear meshing theories 3, if the abovemodel of the LogiX rack tooth profile is changed from coordinateO1X1Y1 to OXY, and then again to O2X2Y2, a new type of gearprofile model can be deduced as follows:_x2 =mk cos k cos 2 (mk sin k r2) sin 2y2 =mk cos k sin 2 +(mk sin k r2) cos 2 .(11)Here the positive direction of 2 is clockwise, and only the modelof the LogiX gear tooth profile in the first quadrant of the coordinatesis given.4 Effect on the performance of the LogiX gear by itsinherent parameters and their reasonable selectionBesides the basic parameters of the standard involute rack, theLogiX tooth profile has inherent basic parameters such as initialpressure angle 0, relative pressure angle , initial basic circleradius G0, etc. The selection of these parameters has a great influenceon the form of the LogiX tooth profile, and the formdirectly influences gear transmission performance. Thus the reasonableselection of these basic parameters is very important.4.1 Influence and selection of initial pressure angle 0Considering the higher transmission efficiency in practical design,the initial pressure angle 0 should be selected as 0. Butthe final calculation result showed that the LogiX gear tooth profilecut by the rack tool whose initial pressure angle was equalto zero would be overcut on the pitch circle generally. Thus theinitial pressure angle 0 cannot be zero. Comparing the relativedouble circle-arc gear 3, we can also deduce that the smaller792the initial pressure angle 0, the larger the gear number for producingthe overcut. Thus the initial pressure angle 0 shouldnot only not be zero, but should not be too small, either. FromEqs. 3, 4 and 5, the influence of 0 on the LogiX tooth profilecan be directly described by Fig. 7. Obviously, increasing the initialpressure angle will cause the curvature of the LogiX racktooth profile to become larger. If the rack selects a larger moduleand too small an initial pressure angle 0, its addendum willbecome too narrow or even overcut. Thus the LogiX tooth profilethat selects a larger module should select a smaller 0, andthe profile that selects a smaller module should select a larger0. Generally, by practical calculation experience, the selected0 should be located within a range of 2 12, and the largerthe LogiX gear module, the smaller should be its initial pressureangle 0.4.2 Influence and selection of initial basic circle radius G0According to the empirical formula Gi = G01sin(0.6i ) 1,there are two parameters affecting the basic circle radius Gi ofthe LogiX gear at different positions of tooth profile: one is theG0 and the other is the initial pressure angle i . Figure 8 showsthe influence of G0 on the LogiX tooth profile when certainvalues of parameter 0 and are selected. Obviously, as G0 increases,the curvature of the new type of gear tooth profile willbecome smaller and smaller. Conversely, it will become increasinglylarger as G0 decreases. Thus the new type of rack witha large module parameter should select a large G0 value, andone having a small module parameter should select a small G0value.4.3 Influence and selection of relative pressure angle Figure 9 shows the variable of the tooth profile affected by the parameter. According to the forming process of the LogiX toothFig. 7. Influence of 0 onLogiX tooth profileFig. 8. Influence of G0 onLogiX tooth profileFig. 9. Influence of onLogiX tooth profileprofile, the smaller the selected parameter , the larger the numberof N-Ps meshing on the tooth profile of two LogiX gears.From Sect. 2.1 the formula describing the relative pressure anglek of an arbitrary N-P mk can be deduced as follows:sin(k1+)cos(k1 +) =2(cos +cos k)sin sin k. (12)By Eqs. 5 and 12, the larger the parameter being selected, thelarger will be the k parameter, and at certain selected valuesof the initial pressure angle and maximum pressure angle, thelower will be the number of N-Ps. By contrast, the smallerthe parameter, the larger the number of N-Ps. While is0.0006, the number of zero points can exceed 46,000. In thiscase, selecting a gear module of m = 100, the length of themicro-involute curve between two adjoining N-Ps will be onlya few microns. That is to say, during the whole meshing processof the LogiX gear transmission, the sliding and rollingmotions happen alternately and last only a few micro-secondsfrom one motion to another between two meshed gear tooth profiles.The greater the number of N-Ps, the longer the relativerolling time between two LogiX gears and the shorter the relativesliding time between two LogiX gears. Thus abrasion of thegear decreases and its loading capability and life span are improved.But, considering the restriction of memory capability,interpolation speed, angular resolution, etc. for the CNCmachinetool used while cutting this type of gear, the relative pressureangle selected should not be very small. _ 0.0006 is generallysatisfactory.Table 1. Parameter values selected for LogiX rack at different modulesm(mm) 0 G0(mm)1 10 0.05 60002 8.0 0.05 95004 6.0 0.05 100005 5.0 0.05 110006 4.0 0.05 120008 3.2 0.05 1202410 2.8 0.05 1400012 2.6 0.05 1650015 2.5 0.05 2002418 2.4 0.05 3003620 2.4 0.05 3500022 2.3 0.05 380007934.4 Reasonable selection exampleBased on the above analytical rules for LogiX gear inherent parameterselection, a reasonable calculation and selection resultsfor the initial pressure angle and basic circle radius while selectingdifferent modules at the relative pressure angle = 0.05 arelisted in Table 1 for reference. In fact, the practical selectionsshould be reasonably modified by the concrete cutting conditionsand the special purpose requirement.5 ConclusionsThe following conclusions were made based on the findings presentedin this paper.1. Two-dimensional meshing transmission models of LogiXgears were deduced by further analysis of its formingprinciple.2. The influence on the LogiX gear tooth profile and its performanceby the gears own basic parameters such as initialpressure angle, initial basic circle radius and relative pressureangle was discussed and their reasonable selection was given.3. The theoretical system of the LogiX gear was developed andthe mathematical basis for generating the LogiX tooth profileby modern CNC technology was established. The characteristicsof the LogiX gear, which are different from those of theordinary standard involute gear, can have broad applicationand most significantly impact the improvement of carryingcapacity, miniaturisation and longevity of kinetic transmissionproducts.References1. Komori T, Arga Y, Nagata S (1990) A new gear profile havingzero relative curvature at many contact points. Trans ASME 112(3):4304362. Xutang W (1982) Gear meshing theory. Machinery Industry Press,Beijing3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing6 Nomenclature0 initial pressure anglei pressure angle at contact point mi parameter of pressure angles1 radius of curvature of gear tooth profile at contact point s1mi radius of curvature of gear tooth profile at contact point mim1 radius of curvature of gear tooth profile at contact pointm1G0 initial radius of basic circle in tooth profileGi radius of basic circle of mi point in gear tooth profile2 rotation angle of LogiX gear meshing with basic LogiXrackr2 radius of basic circle of LogiX gear meshing with basicLogiX rackm model of gearz gear tooth numbers gear tooth thickness at pitch circle; here, i is an optionalnumberDOI 10.1007/s00170-003-1741-8ORIGINAL ARTICLEInt J Adv Manuf Technol (2004) 24: 789793Feng Xianying Wang Aiqun Linda LeeStudy on the design principle of the LogiX gear tooth profileand the selection of its inherent basic parametersReceived: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004Abstract The development of scientific technology and productivityhas called for increasingly higher requirements of geartransmission performance. The key factor influencing dynamicgear performance is the form of the meshed gear tooth profile. Toimprove a gears transmission performance, a new type of gearcalled the LogiX gear was developed in the early 1990s. However,for this special kind of gear there remain many unknowntheoretical and practical problems to be solved. In this paper, thedesign principle of this new type of gear is further studied andthe mathematical module of its tooth profile deduced. The influenceon the form of this type of tooth profile and its meshperformance by its inherent basic parameters is discussed, andreasonable selections for LogiX gear parameters are provided.Thus the theoretical system information about the LogiX gear aredeveloped and enriched. This study impacts most significantlythe improvement of load capacity, miniaturisation and durabilityof modern kinetic transmission products.Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile1 IntroductionIn order to improve gear transmission performance and satisfysome special requirements, a new type of gear 1 was put forward;it was named “LogiX” in order to improve some demeritsof W-N (Wildhaver-Novikov) and involute gears.Besides having the advantages of both kinds of gears mentionedabove, the new type of gear has some other excellentF. Xianying (_) W. AiqunSchool of Mechanical Engineering,Shandong University,P.R. ChinaE-mail: FXYTel.: +86-531-8395852(0)L. LeeSchool of Mechanical & Manufacturing Engineering,Singapore Polytechnic,Singaporecharacteristics. On this new tooth profile, the continuous concave/convex contact is carried out from its dedendum to its addendum,where the engagements with a relative curvature of zeroare assured at many points. Here, this kind of point is called thenull-point (N-P). The presence of many N-Ps during the meshprocess of LogiX gears can result in a smaller sliding coefficient,and the mesh transmission performance becomes almostrolling friction accordingly. Thus this new type of gear has manyadvantages such as higher contact intensity, longer life and alarger transmission-ratio power transfer than the standard involutegear. Experimental results showed that, given a certainnumber of N-Ps between two meshed LogiX gears, the contactfatigue strength is 3 times and the bend fatigue strength 2.5 timeslarger than those of the standard involute gear. Moreover, theminimum tooth number can also be decreased to 3, much smallerthan that of the standard involute gear.The LogiX gear, regarded as a new type of gear, still presentssome unsolved problems. The development of computer numericalcontrolling (CNC) technology must also be taken into considerationnew high-efficiency methods to cut this new type ofgear. Therefore, further study of this new type of gear mostsignificantly impacts the acceleration of its broad and practicalapplication. This paper has the potential to usher in a new era inthe history of gear mesh theory and application.2 Design principle of LogiX tooth profileAccording to gear mesh and manufacturing theories, in order tosimplify problem analysis, generally a gears basic rack is begunwith some studies 2. So here let us discuss the basic rack ofthe LogiX gear first. Figure 1 shows the design principle of dividedinvolute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1 is selected to formthe angle n0O1N1 =0, P.L O1N1. The points of intersectionby two radials O1n0 and O1N1 and the pitch line P.L are N1and n0. Let O1n0 = G1, extend O1n0 to O_1 , and make two tangentbasic circles whose centres are O1, O_1 and radii are equalto G1. The point of intersection between circle O1 and pitch line790Fig. 1. Design principle of LogiX rack tooth profileP.L is n0. The point of intersection between circle O2 and pitchline P.L is n1. Make the common tangent g1s1 of basic circle O1and O_1, then generate two minute involute curves m0s1 and s1m1whose basic circle centres are O1 and O_1. The radii of curvatureat points m0 and m1 on the tooth profile should be: m0 = m0n0,m1 = m1n1, and the centres are met on the pitch line.Multiple different minute involutes consisting of a LogiXprofile should be arranged for a proper sequence. The pressureangle of the next minute involute curve m1m2 should have anincrement comparable to its last segment m0m1. The centres ofcurvature at extreme points m1, m2, etc. should be on the pitchline, and the radius of the basic circle is a function of pressure 1 it varies from G1 to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1 must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2 must be equal to the radius of curvaturejust after point m2. Figure 2 shows the connection and process ofgenerating minute involute curves. According to the above discussion,the whole tooth profile can be formed.Fig. 2. Connection of minute involute curves3 Mathematicmodule of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curvaturecentre of every finely divided profile curve should be locatedat the rack pitch line, and the value of the relative curvature atevery point connecting different minute involute curves shouldbe zero. The design of the tooth profile is symmetrical with respectto the pitch line, and the addendum is convex while thededendum is concave. Thus for the whole LogiX tooth profile, itcan be dealt with by dividing it into four parts, as shown in Fig. 3.Set up the coordinates as shown in Fig. 4, where the origin ofthe coordinates O coincides with the point of intersection m0 betweenrack pitch line P.L and the initial divided minute involutecurve.According to the coordinates set up in Fig. 4, the formationof initial minute involute curve m0m1 is shown in Fig. 5.Fig. 3. LogiX rack tooth profileFig. 4. Set-up of coordinatesFig. 5. Formation process of initial minute involute curve m0m1791Here: n0n_0 O1O_1 , n1n_1 O1O_1 , n1n1 n0n_0, and the parameters0, , G1 and m0 are given as initial conditions. Thecurvature radius of the involute curve at point s1 is s1 = G1, ors1 = m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1 are as follows:m1 = s1G11 = G1(1) (1)1 = 0+1 . (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1 cos G1 cos 1G1 sin G1 sin 1=2(cos +cos 1)sin sin 1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point onthe profile is deduced: mi =mi1+Gi(i ). When i =k andm0 = 0, it is expressed as follows:mk = G1(1)+G2(2)+ +Gk(k)=k_i=1Gi(i) . (4)Similarly, the pressure angle on an arbitrary k point of the toothprofile can be deduced as follows:k = 0+(+1)+(+2)+ (+k)= 0+k_i=1(+i) = 0+k+k_i=1i . (5)By ni1ni = Gi(sin sin i)/ cos(i1 +), Eq. 5 can beobtained:n0nk =k_i=1ni1ni =k_i=1Gi(sin sin i )cos(i1 +). (6)Thus the mathematical model of the No. 2 portion for the LogiXr
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