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DOI 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 pro-ductivity has 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. How-ever, 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 in-fluence on 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 for-ward; 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 men-tioned above, the new type of gear has some other excellentF. Xianying (a117) 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 con-cave/convex contact is carried out from its dedendum to its ad-dendum, 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 coeffi-cient, 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 in-volute gear. 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 numer-ical controlling (CNC) technology must also be taken into con-sideration new 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 di-vided involute curves of the LogiX rack. In Fig. 1, P.L representsa pitch line of the LogiX rack. One point O1is selected to formthe angle n0O1N1= 0, P.L O1N1. The points of intersectionby two radials O1n0and O1N1and the pitch line P.L are N1and n0.LetO1n0= G1,extendO1n0to Oprime1,andmaketwotan-gent basic circles whose centres are O1, Oprime1and radii are equalto G1. The point of intersection between circle O1and pitch line790Fig.1. Design principle of LogiX rack tooth profileP.L is n0.The point of intersection between circle O2and pitchline P.L is n1. Make the common tangent g1s1of basic circle O1and Oprime1, then generate two minute involute curves m0s1and s1m1whose basic circle centres are O1and Oprime1. The radii of curvatureat points m0and m1on 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 m1m2should 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 G1to G2. The condition for joining front and rearcurves is that the radius of curvature at point m1must be equalto the radius of curvature just after point m1, and the radius ofcurvature at point m2must 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 dis-cussion, the whole tooth profile can be formed.Fig.2. Connection of minute involute curves3 Mathematic module of LogiX tooth profile3.1 Mathematic module of the basic LogiX rackAccording to the above-mentioned design principle, the curva-ture centre 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 re-spect to 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 m0be-tween rack 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 m0m1is shown in Fig. 5.Fig.3. LogiX rack tooth profileFig.4. Set-up of coordinatesFig.5. Formation process of initial minute involute curve m0m1791Here: n0nprime0 O1Oprime1, n1nprime1 O1Oprime1, n1n1n0nprime0,andthepa-rameters 0, , G1and m0are given as initial conditions. Thecurvature radius of the involute curve at point s1is s1= G1,ors1= m1+G11. Thus the curvature radius and pressure angleof the minute involute curve at point m1are as follows:m1= s1G11= G1(1) (1)1= 0+1. (2)According to the geometrical relationship, we can deduce:tg(0+) =2G1G1cos G1cos 1G1sin G1sin 1=2(cos +cos 1)sin sin1. (3)Based on Eqs. 1, 2 and 3 and the forming process of the LogiXrack profile, the curvature radius formula of an arbitrary point ontheprofileisdeduced:mi= mi1+Gi(i).Wheni = k andm0= 0, it is expressed as follows:mk= G1(1)+G2(2)+Gk(k)=ksummationdisplayi=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+ksummationdisplayi=1(+i) = 0+k+ksummationdisplayi=1i. (5)By ni1ni= Gi(sin sin i)/ cos(i1+ ), Eq. 5 can beobtained:n0nk=ksummationdisplayi=1ni1ni=ksummationdisplayi=1Gi(sin sini)cos(i1+). (6)Thus the mathematical model of the No. 2 portion for the LogiXrack profile is as follows:braceleftbiggx1= n0nkmkcos ky1= mksin k(No. 2) . (7)Similarly, the mathematical models of the other three segmentscan also be obtained as follows:braceleftbiggx1=(n0nkmkcos k)y1=mksin k(No.1) (8)braceleftbiggx1= s(n0nkmkcos k)y1= mksin k(No.3) (9)braceleftbiggx1= s+n0nkmkcos ky1=mksin k(No.4) . (10)Fig.6. Mesh coordinatesof LogiX gear and its ba-sic rack3.2 Mathematical module of the LogiX gearThe coordinates O1X1Y1, O2X2Y2and PXY are set up as shownin Fig. 6 to express the mesh relationship between the LogiXrack and the LogiX gear. Here, O1X1Y1is fixed on the rack, andO1is the point of intersection between the rack tooth profile andits pitch line. O2X2Y2is fixed on the meshed gear, and O2is 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 coordinateO1X1Y1to OXY, and then again to O2X2Y2, a new type of gearprofile model can be deduced as follows:braceleftbiggx2=mkcos kcos 2(mksin kr2) sin 2y2=mkcos ksin 2+(mksin kr2) cos 2.(11)Here the positive direction of 2is clockwise, and only the modelof the LogiX gear tooth profile in the first quadrant of the coordi-nates is 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 in-fluence on the form of the LogiX tooth profile, and the formdirectly influences gear transmission performance. Thus the rea-sonable selection of these basic parameters is very important.4.1 Influence and selection of initial pressure angle 0Considering the higher transmission efficiency in practical de-sign, the initial pressure angle 0should be selected as 0.Butthe final calculation result showed that the LogiX gear tooth pro-file cut by the rack tool whose initial pressure angle was equalto zero would be overcut on the pitch circle generally. Thus theinitial pressure angle 0cannot 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 pro-ducing the overcut. Thus the initial pressure angle 0shouldnot only not be zero, but should not be too small, either. FromEqs. 3, 4 and 5, the influence of 0on the LogiX tooth profilecan be directly described by Fig. 7. Obviously, increasing the ini-tial pressure angle will cause the curvature of the LogiX racktooth profile to become larger. If the rack selects a larger mod-ule and too small an initial pressure angle 0, its addendum willbecome too narrow or even overcut. Thus the LogiX tooth pro-file that 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 selected0should 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 Giofthe LogiX gear at different positions of tooth profile: one is theG0and the other is the initial pressure angle i. Figure 8 showsthe influence of G0on the LogiX tooth profile when certainvalues of parameter 0and are selected. Obviously, as G0in-creases, the curvature of the new type of gear tooth profile willbecome smaller and smaller. Conversely, it will become increas-ingly larger as G0decreases. Thus the new type of rack witha large module parameter should select a large G0value, 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 0onLogiX tooth profileFig.8. Influence of G0onLogiX tooth profileFig.9. Influence of onLogiX tooth profileprofile, the smaller the selected parameter , the larger the num-ber of N-Ps meshing on the tooth profile of two LogiX gears.From Sect. 2.1 the formula describing the relative pressure anglekof an arbitrary N-P mkcan 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 kparameter, 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 pro-cess of 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 pro-files. The greater the number of N-Ps, the longer the relativerolling time between two LogiX gears and the shorter the rela-tive sliding time between two LogiX gears. Thus abrasion of thegear decreases and its loading capability and life span are im-proved. But, considering the restriction of memory capability,interpolation speed, angular resolution, etc. for the CNC machinetool used while cutting this type of gear, the relative pressureangle selected should not be very small. greaterorequalslant 0.0006is generallysatisfactory.Table 1. Parameter values selected for LogiX rack at different modulesm(mm) 0 G0(mm)1100.05600028.0.059500460.05100005500.051100064.0.05120008320.051202410 2.80.051400012 2.60.051650015 2.50.052002418 2.40.053003620 2.40.053500022 2.30.05380007934.4 Reasonable selection exampleBased on the above analytical rules for LogiX gear inherent pa-rameter selection, a reasonable calculation and selection resultsfor the initial pressure angle and basic circle radius while select-ing different modules at the relative pressure angle = 0.05arelisted 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 pre-sented in this paper.1. Two-dimensional meshing transmission models of LogiXgears were deduced by further analysis of its forming