高空作業(yè)車的轉(zhuǎn)臺結(jié)構(gòu)設(shè)計(jì)及有限元分析【含CAD高清圖紙和說明書】

【溫馨提示】 dwg后綴的文件為CAD圖，可編輯，無水印，高清圖，壓縮包內(nèi)文檔可直接點(diǎn)開預(yù)覽，需要原稿請自助充值下載，請見壓縮包內(nèi)的文件及預(yù)覽，所見才能所得，請細(xì)心查看有疑問可以咨詢QQ：414951605或1304139763 DOI 10.1007/s00170-003-1741-8 ORIGINAL ARTICLE Int J Adv Manuf Technol (2004) 24: 789793 Feng Xianying Wang Aiqun Linda Lee Study on the design principle of the LogiX gear tooth profile and the selection of its inherent basic parameters Received: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004 Springer-Verlag London Limited 2004 Abstract The development of scientific technology and pro- ductivity has called for increasingly higher requirements of gear transmission performance. The key factor influencing dynamic gear performance is the form of the meshed gear tooth profile. To improve a gears transmission performance, a new type of gear called the LogiX gear was developed in the early 1990s. How- ever, for this special kind of gear there remain many unknown theoretical and practical problems to be solved. In this paper, the design principle of this new type of gear is further studied and the mathematical module of its tooth profile deduced. The in- fluence on the form of this type of tooth profile and its mesh performance by its inherent basic parameters is discussed, and reasonable selections for LogiX gear parameters are provided. Thus the theoretical system information about the LogiX gear are developed and enriched. This study impacts most significantly the improvement of load capacity, miniaturisation and durability of modern kinetic transmission products. Keywords Basic parameter Design principle LogiX gear Minute involute Tooth profile 1 Introduction In order to improve gear transmission performance and satisfy some special requirements, a new type of gear 1 was put for- ward; it was named “LogiX” in order to improve some demerits of 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 excellent F. Xianying (a117) W. Aiqun School of Mechanical Engineering, Shandong University, P.R. China E-mail: FXYing Tel.: +86-531-8395852(0) L. Lee School of Mechanical here, i is an optional number （譯文）選擇固定參數(shù)研究齒輪牙側(cè)面的設(shè)計(jì)規(guī)則摘要：科學(xué)技術(shù)和生產(chǎn)的發(fā)展對齒輪傳動有了更高的要求，影響齒輪的動態(tài)性能的關(guān)鍵因素是齒輪牙的側(cè)面嚙合形式。為了提高齒輪的傳動性能，一種新的被稱著為LogiX的齒輪在19世紀(jì)發(fā)明出來。然而，對于這種特殊的齒輪還有很多理論和實(shí)踐上的未知問題等待解決。 本文進(jìn)一步研究這種新型齒輪的設(shè)計(jì)準(zhǔn)則并運(yùn)用數(shù)學(xué)推理的方法對齒輪牙側(cè)面進(jìn)行了分析。通過對齒輪牙側(cè)面嚙合形式的影響參數(shù)的討論，說明了這種LogiX齒輪的參數(shù)選擇的合理性，發(fā)展和加強(qiáng)了LogiX齒輪的理論體系。這種對參數(shù)等的研究最終是為了能夠發(fā)明出現(xiàn)代耐久的傳動產(chǎn)品。關(guān)鍵詞：基本參數(shù) 設(shè)計(jì)原理 LogiX齒輪 詳細(xì)概括 齒輪牙的側(cè)面1介紹為了提高齒輪的傳動性能和滿足一些特殊的要求，一種新的齒輪應(yīng)運(yùn)而生；他被命名為“LogiX”為增加一些優(yōu)異的性能和漸開線齒輪另外擁有以上兩種類型齒輪的優(yōu)點(diǎn)的新型齒輪還有一些別的特殊的優(yōu)點(diǎn).在這種新的齒輪牙，連續(xù)的凸面聯(lián)絡(luò)被執(zhí)行從它的齒根高對它的補(bǔ)充,那里被確定的安全相對彎曲有很多點(diǎn)。這里這種點(diǎn)被叫著零位點(diǎn)（N-P）許多的出現(xiàn)在（N-Ps）點(diǎn)在LogiX 齒輪期間的濾網(wǎng)過程可能導(dǎo)致一個(gè)滑動系數(shù), 并且濾網(wǎng)傳輸表現(xiàn)成為相應(yīng)地幾乎滾動的摩擦。因而這種新型的齒輪有許多好處，譬如更高的傳動強(qiáng)度、壽命長而且比標(biāo)準(zhǔn)斷開線齒輪有更大的傳輸比率。實(shí)驗(yàn)性結(jié)果表示, 給一定數(shù)量的（N-Ps） 在二個(gè)捕捉的LogiX 齒輪之間, 3倍的傳動疲勞強(qiáng)度和2.5倍的抗彎疲勞強(qiáng)度遠(yuǎn)遠(yuǎn)比那些標(biāo)準(zhǔn)斷開線齒輪承受能力大。而且, 最小的牙數(shù)可達(dá)到3個(gè)這是那些標(biāo)準(zhǔn)斷開線齒輪沒法比的。這種被認(rèn)為新型的LogiX 齒輪還有很多未被解決的問題。計(jì)算機(jī)數(shù)字控制(CNC) 技術(shù)的發(fā)展必然被用來研究更高效率的方法來發(fā)展這種新型齒輪。因此進(jìn)一步提高研究這種齒輪的寬度和實(shí)際應(yīng)用的加速度顯的很重要。本文有信心在這個(gè)新時(shí)代里把齒輪濾網(wǎng)理論和應(yīng)用取得歷史性的突破。 2齒牙側(cè)面的設(shè)計(jì)原則 根據(jù)齒輪濾網(wǎng)和制造業(yè)理論, 為了簡化問題分析, 從齒輪的基本的機(jī)架開始入手來研究。讓我們首先從討論LogiX 齒輪的基本機(jī)架開始。圖1 顯示LogiX 機(jī)架的設(shè)計(jì)原則的劃分和斷開線曲線。在圖一中P.L代表 LogiX 機(jī)架的節(jié)線。選擇點(diǎn)是為了形成角0。P.L O1N1.兩個(gè)徑向的交叉點(diǎn)O1n0 和O1N1，節(jié)線P.L和N1，n0。使得延長到使得兩個(gè)基圓的切線相交到，和幅線相交于兩個(gè)圓的交點(diǎn)和節(jié)線P.L交點(diǎn)，兩個(gè)圓的交點(diǎn)和節(jié)線P.L交點(diǎn)。使得公切線和基圓相交和，在有關(guān)齒側(cè)面點(diǎn)m0和m1方面曲率半徑應(yīng)該是: ，在節(jié)線上相交于中心。不同倍數(shù)的回旋包括LogiX 外形應(yīng)該被安排為一個(gè)適當(dāng)?shù)捻樞?。下個(gè)漸開彎曲線m1m2的壓力角度應(yīng)該比前段m0m1的有所增加。中心曲度在極端點(diǎn)m1 、m2, 等應(yīng)該是在節(jié)線上, 并且基本的圈子壓力半徑的作用變化應(yīng)該是從G1 到G2 。形成的條件為前曲線和后曲線的半徑曲度必須是在和點(diǎn)m1相等的對半徑曲度在點(diǎn)m1 之后同時(shí)半徑曲度在點(diǎn)m2處 必須是與半徑曲度相等的在點(diǎn)m2 之后。圖2 顯示漸開線曲線的準(zhǔn)確連接的過程。根據(jù)上述討論,整體外牙形成。 圖1：LogiX 機(jī)架外形牙的設(shè)計(jì)原則 圖2.漸開線曲線的準(zhǔn)確連接3LogiX的 側(cè)牙數(shù)學(xué)模塊3.1 基本的LogiX 機(jī)架的數(shù)學(xué)模塊根據(jù)上述設(shè)計(jì)原則, 每個(gè)精確曲線外形的曲率中心應(yīng)該在機(jī)架節(jié)線上找出, 和每點(diǎn)在相對曲度連接的價(jià)值不同，在漸開線曲線上的應(yīng)該是零點(diǎn)。外形牙的設(shè)計(jì)是關(guān)于節(jié)線對稱的,齒根高凹面和凸面是互補(bǔ)的。因而作為LogiX外形牙的整體, 它可以被確切的劃分成為四份, 如圖3所示 。座標(biāo)設(shè)定如圖4中所示, 節(jié)線P.L在座標(biāo)起點(diǎn)O 與交點(diǎn)m0之間和形成最初的漸開線曲線。 根據(jù)圖4中的座標(biāo)設(shè)定, 形成最初的漸開線曲線m0m1 如圖5所示。 圖3. LogiX 機(jī)架外形牙 圖4. 座標(biāo)設(shè)定 圖5. 最初的漸開線曲線m0m1的形成過程 圖6：LogiX齒輪嚙合和基圓 現(xiàn)在，和參數(shù)量，和作為已知條件。在曲線和漸開線的交點(diǎn)是，或者。因此曲率半徑和壓力角在漸開線的交點(diǎn)處的關(guān)系如下： （1） （2）根據(jù)幾何關(guān)系我們可以得出以下結(jié)論： （3）根據(jù)式1，2和3和有關(guān)LogiX齒的形成材料，根據(jù)曲率半徑的形成規(guī)則可得到如下關(guān)系：。當(dāng)且，可得出特殊關(guān)系式如下： （4）顯然，在任一齒牙側(cè)面的k 點(diǎn)處壓力角的關(guān)系如下： （5）當(dāng)是關(guān)系式5可變?yōu)槿缦拢?（6）根據(jù)數(shù)學(xué)幾何關(guān)系可得到No.2的關(guān)系式： （No2） （7）顯然根據(jù)幾何關(guān)系可得別的式子如下：（No1） （8）（No3） （9）（No4） （10）3.2 LogiX齒輪的數(shù)學(xué)模塊 配合角，和PXY如圖6所示，在LogiX齒輪架和LogiX齒輪間的精確的嚙合關(guān)系。這里被定位在齒輪架上是齒輪側(cè)面和節(jié)線的交點(diǎn)被定位在齒輪的嚙合處是齒輪的中心。PXY是完全的中空角，P點(diǎn)是齒輪的切線和基圓的交點(diǎn)。 為了和理論建立起一致的關(guān)系。假如上面的例子中LogiX齒輪牙的側(cè)面從 改動到OXY同時(shí)再改變到，一種新型的齒輪模型的關(guān)系就產(chǎn)生如下： （11）在這里理想的正角，只在LogiX齒輪模型的第一象限中給出。4自身固有參數(shù)和它們選擇的原因?qū)ogiX齒輪的影響除標(biāo)準(zhǔn)漸開線的基本參數(shù)外, LogiX齒牙側(cè)面有自己固有的基本參數(shù)，譬如起始壓力角度、相對壓力角度,起始基圓半徑 G0,等等。這些參數(shù)的選擇對LogiX齒牙側(cè)面漸開線的影響非常大,它的結(jié)構(gòu)形式會直接影響力齒輪的傳動能力。因此基本參數(shù)的選擇非常重要。4.1起始壓力角的選擇和影響考慮到要設(shè)計(jì)較高傳動性能的齒輪,起始壓力角為0度。但是最后的計(jì)算結(jié)果表示 LogiX 齒牙側(cè)面加工工具的角度和起始的壓力角度是相等的。這樣起始壓力角度不能對準(zhǔn)零位。比較相對于兩倍圓周-弧的齒輪，我們可以推出的起始壓力角越小,齒輪越大越容易產(chǎn)生根切。因此起始壓力角應(yīng)該不僅僅是零,但也不能太小，同時(shí)，從例 3,4 和5,可以看出對LogiX齒牙方面的影響可以用圖7來直接描述。顯然地，起始壓力角度數(shù)的增加會引起 LogiX 齒架的曲率的增大。如果選擇一個(gè)較大的齒架，而起始壓力角太小的話。它的齒頂會變的很窄或產(chǎn)生根切現(xiàn)象。因此 LogiX 齒牙側(cè)面選較大時(shí)，應(yīng)該選一個(gè)較小的起始壓力角,當(dāng)LogiX 齒牙側(cè)面選較小時(shí)，選一個(gè)較大的起始壓力角。通常,實(shí)踐計(jì)算經(jīng)驗(yàn)告訴我們，起始壓力角 取2度到12度，而且LogiX齒輪模型越大，起始壓力角越小。4.2起始基圓半徑 G0的選擇和影響根據(jù)公式在LogiX 齒輪牙的側(cè)面的不同位置有兩個(gè)參數(shù)影響基圓半徑 Gi：在牙齒描繪的不同位置的 LogiX 齒輪: 一個(gè)是G0 另一個(gè)是起始壓力角。圖8所示的是當(dāng)給定參數(shù)0和時(shí)G0對LogiX齒側(cè)面的影響。顯然地,如果G0增加，新型齒輪牙的側(cè)面曲率將變得越來越小。顯而易見，它會隨著G0的減小而逐漸增大。因此新型齒架的參數(shù)大時(shí)G0也應(yīng)選大的，同時(shí)當(dāng)齒架的參數(shù)小時(shí)G0也選小。4.3壓力角的選擇和影響在圖 9中顯示參數(shù)的變化對齒牙的影響。根據(jù) LogiX齒牙的形成過程，參數(shù)越小在LogiX齒輪的兩齒之間形成的N-Ps越大。根據(jù)2.1中的描述相對壓力角在N-P mk中的關(guān)系如下： 如式5和12,選擇比較大的參數(shù)相應(yīng)的參數(shù)也比較大，選擇適當(dāng)?shù)钠鹗級毫呛妥畲髩毫?，壓力角越小N-Ps越多，相反地,比較小的參數(shù)，N-Ps的數(shù)字較大。當(dāng)取0.0006度時(shí)，零點(diǎn)的數(shù)字將超過 46,000 。在這情形,選擇一個(gè)齒輪模數(shù)m =100, 兩個(gè)N-Ps點(diǎn)之間會變的很小。也就是說，在整個(gè) LogiX齒輪 的運(yùn)動過程中，兩個(gè)嚙合齒輪間在很短的時(shí)間內(nèi)會參數(shù)打滑和滾動。N-Ps數(shù)目越多在兩齒輪間越長相反傳動時(shí)間越短。因此它的磨損減少了使用壽命就增長了。但是, 考慮到承載能力的限制，速度的改變、角度的因素等等當(dāng)切割這種類型的齒輪時(shí)必須用CNC機(jī)床刀具，相關(guān)壓力角的選擇非常小，一般必須滿足度。 圖7：0對LogiX齒輪側(cè)面的影響 圖8 ：G0對LogiX齒輪側(cè)面的影響圖9 ：對LogiX齒輪側(cè)面的影響4.4 選擇合理參數(shù)舉例 基于上述對LogiX齒輪固有參數(shù)選擇的分析規(guī)則，對于不同的零件模型，當(dāng)它的相對壓力角為0.05度時(shí)，起始壓力角和基圓半徑合理的計(jì)算結(jié)果如下表1所作的參考。事實(shí)上, 實(shí)際的選擇應(yīng)該根據(jù)具體切斷情況和特殊需求而定。5 結(jié)論下面是根據(jù)調(diào)查結(jié)果得出的結(jié)論：1. 通過進(jìn)一步的深入研究可推出LogiX齒輪的二維嚙合傳動規(guī)律。2. 討論研究了齒輪自身基本參數(shù)譬如起始壓力角，起始基圓半徑和相對壓力角以及參數(shù)的選擇，對LogiX 齒輪牙側(cè)和性能的影響。 3. 通過對LogiX 齒輪理論系統(tǒng)和數(shù)學(xué)基礎(chǔ)的進(jìn)一步研究建立了現(xiàn)代 CNC 技術(shù)。LogiX 齒牙的特性：它不如常規(guī)漸開線齒輪應(yīng)用廣但是它是一種承載能力大，體積小，壽命長的產(chǎn)品。6 命名法 起始壓力角度 交點(diǎn) mi 處的壓力角 壓力角度參數(shù) 在交點(diǎn) s 1 處的 齒輪牙齒側(cè)面的曲率半徑 在交點(diǎn) mi處的 齒輪牙齒側(cè)面的曲率半徑 在交點(diǎn) m1處的 齒輪牙齒側(cè)面的曲率半徑 起始基圓半徑 齒輪牙側(cè)面上點(diǎn)mi初基圓的半徑 LogiX齒輪嚙合轉(zhuǎn)動時(shí)和基架LogiX的夾角r2 LogiX 齒輪嚙合時(shí)的基圓和基架LogiX的半徑m 齒輪的模型z 齒數(shù)s 齒厚，這里， i 是任意數(shù) 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: FXYingsdu.edu.cnTel.: +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: FXYingsdu.edu.cnTel.: +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, rega