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（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(duì)齒輪傳動(dòng)有了更高的要求，影響齒輪的動(dòng)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(dòng)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(duì)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(duì)齒輪牙側(cè)面進(jìn)行了分析。通過對(duì)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(duì)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(dòng)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(dòng)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(duì)它的補(bǔ)充,那里被確定的安全相對(duì)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(dòng)系數(shù), 并且濾網(wǎng)傳輸表現(xiàn)成為相應(yīng)地幾乎滾動(dòng)的摩擦。因而這種新型的齒輪有許多好處，譬如更高的傳動(dòng)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(dòng)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α0。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(duì)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
3．LogiX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(duì)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(duì)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(dòng)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?jí)毫嵌取⑾鄬?duì)壓力角度δ,起始基圓半徑 G0,等等。這些參數(shù)的選擇對(duì)LogiX齒牙側(cè)面漸開線的影響非常大,它的結(jié)構(gòu)形式會(huì)直接影響力齒輪的傳動(dòng)能力。因此基本參數(shù)的選擇非常重要。
4.1起始?jí)毫堑倪x擇和影響
考慮到要設(shè)計(jì)較高傳動(dòng)性能的齒輪,起始?jí)毫菫?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?jí)毫嵌炔荒軐?duì)準(zhǔn)零位。比較相對(duì)于兩倍圓周-弧的齒輪，我們可以推出的起始?jí)毫窃叫?齒輪越大越容易產(chǎn)生根切。因此起始?jí)毫菓?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(duì)LogiX齒牙方面的影響可以用圖7來直接描述。顯然地，起始?jí)毫嵌葦?shù)的增加會(huì)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?jí)毫翘〉脑?。它的齒頂會(huì)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?jí)毫?當(dāng)LogiX 齒牙側(cè)面選較小時(shí)，選一個(gè)較大的起始?jí)毫恰?
通常,實(shí)踐計(jì)算經(jīng)驗(yàn)告訴我們，起始?jí)毫?取2度到12度，而且LogiX齒輪模型越大，起始?jí)毫窃叫 ?
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?jí)毫?。圖8所示的是當(dāng)給定參數(shù)α0和δ時(shí)G0對(duì)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(huì)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δ的選擇和影響
在圖 9中顯示參數(shù)δ的變化對(duì)齒牙的影響。根據(jù) LogiX齒牙的形成過程，參數(shù)δ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(duì)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)δ相應(yīng)的參數(shù)也比較大，選擇適當(dāng)?shù)钠鹗級(jí)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)δ，N-Ps的數(shù)字較大。當(dāng)δ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(huì)變的很小。也就是說，在整個(gè) LogiX齒輪 的運(yùn)動(dòng)過程中，兩個(gè)嚙合齒輪間在很短的時(shí)間內(nèi)會(huì)參數(shù)打滑和滾動(dòng)。N-Ps數(shù)目越多在兩齒輪間越長相反傳動(dòng)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：α0對(duì)LogiX齒輪側(cè)面的影響 圖8 ：G0對(duì)LogiX齒輪側(cè)面的影響
圖9 ：δ對(duì)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(duì)LogiX齒輪固有參數(shù)選擇的分析規(guī)則，對(duì)于不同的零件模型，當(dāng)它的相對(duì)壓力角為0.05度時(shí)，起始?jí)毫呛突鶊A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(dòng)規(guī)律。
2. 討論研究了齒輪自身基本參數(shù)譬如起始?jí)毫?，起始基圓半徑和相對(duì)壓力角以及參數(shù)的選擇，對(duì)LogiX 齒輪牙側(cè)和性能的影響。
3. 通過對(duì)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?jí)毫嵌?
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(dòng)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)
9
DOI 10.1007/s00170-003-1741-8
ORIGINAL ARTICLE
Int J Adv Manuf Technol (2004) 24: 789–793
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 productivity
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 gear’s transmission performance, a new type of gear
called the LogiX gear was developed in the early 1990s. However,
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 influence
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 forward;
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 mentioned
above, the new type of gear has some other excellent
F. Xianying (_) · W. Aiqun
School of Mechanical Engineering,
Shandong University,
P.R. China
E-mail: FXYing@sdu.edu.cn
Tel.: +86-531-8395852(0)
L. Lee
School of Mechanical & Manufacturing Engineering,
Singapore Polytechnic,
Singapore
characteristics. 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 zero
are assured at many points. Here, this kind of point is called the
null-point (N-P). The presence of many N-Ps during the mesh
process of LogiX gears can result in a smaller sliding coefficient,
and the mesh transmission performance becomes almost
rolling friction accordingly. Thus this new type of gear has many
advantages such as higher contact intensity, longer life and a
larger transmission-ratio power transfer than the standard involute
gear. Experimental results showed that, given a certain
number of N-Ps between two meshed LogiX gears, the contact
fatigue strength is 3 times and the bend fatigue strength 2.5 times
larger than those of the standard involute gear. Moreover, the
minimum tooth number can also be decreased to 3, much smaller
than that of the standard involute gear.
The LogiX gear, regarded as a new type of gear, still presents
some unsolved problems. The development of computer numerical
controlling (CNC) technology must also be taken into consideration
new high-efficiency methods to cut this new type of
gear. Therefore, further study of this new type of gear most
significantly impacts the acceleration of its broad and practical
application. This paper has the potential to usher in a new era in
the history of gear mesh theory and application.
2 Design principle of LogiX tooth profile
According to gear mesh and manufacturing theories, in order to
simplify problem analysis, generally a gear’s basic rack is begun
with some studies [2]. So here let us discuss the basic rack of
the LogiX gear first. Figure 1 shows the design principle of divided
involute curves of the LogiX rack. In Fig. 1, P.L represents
a pitch line of the LogiX rack. One point O1 is selected to form
the angle ?n0O1N1 =α0, P.L ? O1N1. The points of intersection
by two radials O1n0 and O1N1 and the pitch line P.L are N1
and n0. Let O1n0 = G1, extend O1n0 to O_
1 , and make two tangent
basic circles whose centres are O1, O_1 and radii are equal
to G1.. The point of intersection between circle O1 and pitch line
790
Fig. 1. Design principle of LogiX rack tooth profile
P.L is n0. The point of intersection between circle O2 and pitch
line P.L is n1. Make the common tangent g1s1 of basic circle O1
and O_1, then generate two minute involute curves m0s1 and s1m1
whose basic circle centres are O1 and O_1. The radii of curvature
at points m0 and m1 on the tooth profile should be: ρm0 = m0n0,
ρm1 = m1n1, and the centres are met on the pitch line.
Multiple different minute involutes consisting of a LogiX
profile should be arranged for a proper sequence. The pressure
angle of the next minute involute curve m1m2 should have an
increment comparable to its last segment m0m1. The centres of
curvature at extreme points m1, m2, etc. should be on the pitch
line, 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 rear
curves is that the radius of curvature at point m1 must be equal
to the radius of curvature just after point m1, and the radius of
curvature at point m2 must be equal to the radius of curvature
just after point m2. Figure 2 shows the connection and process of
generating minute involute curves. According to the above discussion,
the whole tooth profile can be formed.
Fig. 2. Connection of minute involute curves
3 Mathematicmodule of LogiX tooth profile
3.1 Mathematic module of the basic LogiX rack
According to the above-mentioned design principle, the curvature
centre of every finely divided profile curve should be located
at the rack pitch line, and the value of the relative curvature at
every point connecting different minute involute curves should
be zero. The design of the tooth profile is symmetrical with respect
to the pitch line, and the addendum is convex while the
dedendum is concave. Thus for the whole LogiX tooth profile, it
can 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 of
the coordinates O coincides with the point of intersection m0 between
rack pitch line P.L and the initial divided minute involute
curve.
According to the coordinates set up in Fig. 4, the formation
of initial minute involute curve m0m1 is shown in Fig. 5.
Fig. 3. LogiX rack tooth profile
Fig. 4. Set-up of coordinates
Fig. 5. Formation process of initial minute involute curve m0m1
791
Here: n0n_0 ? O1O_1 , n1n_1 ? O1O_1 , n1n1 ?n0n_0, and the parameters
α0, δ, G1 and ρm0 are given as initial conditions. The
curvature radius of the involute curve at point s1 is ρs1 = G1δ, or
ρs1 = ρm1+G1δ1. Thus the curvature radius and pressure angle
of the minute involute curve at point m1 are as follows:
ρm1 = ρs1?G1δ1 = G1(δ?δ1) (1)
α1 = α0+δ+δ1 . (2)
According to the geometrical relationship, we can deduce:
tg(α0+δ) =
2G1?G1 cos δ?G1 cos δ1
G1 sin δ?G1 sin δ1
=
2?(cos δ+cos δ1)
sin δ?sin δ1
. (3)
Based on Eqs. 1, 2 and 3 and the forming process of the LogiX
rack profile, the curvature radius formula of an arbitrary point on
the profile is deduced: ρmi =ρmi?1+Gi(δ?δi ). When i =k and
ρm0 = 0?, it is expressed as follows:
ρmk = G1(δ?δ1)+G2(δ?δ2)+· · ·+Gk(δ?δk)
=
k
_i=1
Gi(δ?δi) . (4)
Similarly, the pressure angle on an arbitrary k point of the tooth
profile can be deduced as follows:
αk = α0+(δ+δ1)+(δ+δ2)+· · · (δ+δk)
= α0+
k
_i=1
(δ+δi) = α0+kδ+
k
_i=1
δi . (5)
By ni?1ni = Gi(sin δ?sin δi)/ cos(αi?1 +δ), Eq. 5 can be
obtained:
n0nk =
k
_i=1
ni?1ni =
k
_i=1
Gi(sin δ?sin δi )
cos(αi?1 +δ)
. (6)
Thus the mathematical model of the No. 2 portion for the LogiX
rack profile is as follows:
_x1 = n0nk ?ρmk cos αk
y1 = ρmk sin αk
(No. 2) . (7)
Similarly, the mathematical models of the other three segments
can also be obtained as follows:
_x1 =?(n0nk ?ρmk cos αk)
y1 =?ρmk sin αk
(No.1) (8)
_x1 = s?(n0nk ?ρmk cos αk)
y1 = ρmk sin αk
(No.3) (9)
_x1 = s+n0nk ?ρmk cos αk
y1 =?ρmk sin αk
(No.4) . (10)
Fig. 6. Mesh coordinates
of LogiX gear and its basic
rack
3.2 Mathematical module of the LogiX gear
The coordinates O1X1Y1, O2X2Y2 and PXY are set up as shown
in Fig. 6 to express the mesh relationship between the LogiX
rack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, and
O1 is the point of intersection between the rack tooth profile and
its pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is the
gear’s centre. PXY is an absolute coordinate, and P is the point
of intersection of the rack’s pitch line and the gear’s pitch circle.
In accordance with gear meshing theories [3], if the above
model of the LogiX rack tooth profile is changed from coordinate
O1X1Y1 to OXY, and then again to O2X2Y2, a new type of gear
profile model can be deduced as follows:
_x2 =?ρmk cos αk cos ?2 ?(ρmk sin αk ?r2) sin ?2
y2 =?ρmk cos αk sin ?2 +(ρmk sin αk ?r2) cos ?2 .
(11)
Here the positive direction of ?2 is clockwise, and only the model
of the LogiX gear tooth profile in the first quadrant of the coordinates
is given.
4 Effect on the performance of the LogiX gear by its
inherent parameters and their reasonable selection
Besides the basic parameters of the standard involute rack, the
LogiX tooth profile has inherent basic parameters such as initial
pressure angle α0, relative pressure angle δ, initial basic circle
radius G0, etc. The selection of these parameters has a great influence
on the form of the LogiX tooth profile, and the form
directly influences gear transmission performance. Thus the reasonable
selection of these basic parameters is very important.
4.1 Influence and selection of initial pressure angle α0
Considering the higher transmission efficiency in practical design,
the initial pressure angle α0 should be selected as 0?. But
the final calculation result showed that the LogiX gear tooth profile
cut by the rack tool whose initial pressure angle was equal
to zero would be overcut on the pitch circle generally. Thus the
initial pressure angle α0 cannot be zero. Comparing the relative
double circle-arc gear [3], we can also deduce that the smaller
792
the initial pressure angle α0, the larger the gear number for producing
the overcut. Thus the initial pressure angle α0 should
not only not be zero, but should not be too small, either. From
Eqs. 3, 4 and 5, the influence of α0 on the LogiX tooth profile
can be directly described by Fig. 7. Obviously, increasing the initial
pressure angle will cause the curvature of the LogiX rack
tooth profile to become larger. If the rack selects a larger module
and too small an initial pressure angle α0, its addendum will
become too narrow or even overcut. Thus the LogiX tooth profile
that selects a larger module should select a smaller α0, and
the profile that selects a smaller module should select a larger
α0. Generally, by practical calculation experience, the selected
α0 should be located within a range of 2? ～ 12?, and the larger
the LogiX gear module, the smaller should be its initial pressure
angle α0.
4.2 Influence and selection of initial basic circle radius G0
According to the empirical formula Gi = G0{1?sin(0.6αi )} [1],
there are two parameters affecting the basic circle radius Gi of
the LogiX gear at different positions of tooth profile: one is the
G0 and the other is the initial pressure angle αi . Figure 8 shows
the influence of G0 on the LogiX tooth profile when certain
values of parameter α0 and δ are selected. Obviously, as G0 increases,
the curvature of the new type of gear tooth profile will
become smaller and smaller. Conversely, it will become increasingly
larger as G0 decreases. Thus the new type of rack with
a large module parameter should select a large G0 value, and
one having a small module parameter should select a small G0
value.
4.3 Influence and selection of relative pressure angle δ
Figure 9 shows the variable of the tooth profile affected by the δ
parameter. According to the forming process of the LogiX tooth
Fig. 7. Influence of α0 on
LogiX tooth profile
Fig. 8. Influence of G0 on
LogiX tooth profile
Fig. 9. Influence of δ on
LogiX tooth profile
profile, the smaller the selected parameter δ, the larger the number
of N-Ps meshing on the tooth profile of two LogiX gears.
From Sect. 2.1 the formula describing the relative pressure angle
δk of an arbitrary N-P mk can be deduced as follows:
sin(αk?1+δ)
cos(αk?1 +δ) =
2?(cos δ+cos δk)
sin δ?sin δk
. (12)
By Eqs. 5 and 12, the larger the δ parameter being selected, the
larger will be the δk parameter, and at certain selected values
of the initial pressure angle and maximum pressure angle, the
lower will be the number of N-Ps. By contrast, the smaller
the δ parameter, the larger the number of N-Ps. While δ is
0.0006?, the number of zero points can exceed 46,000. In this
case, selecting a gear module of m = 100, the length of the
micro-involute curve between two adjoining N-Ps will be only
a few microns. That is to say, during the whole meshing process
of the LogiX gear transmission, the sliding and rolling
motions happen alternately and last only a few micro-seconds
from one motion to another between two meshed gear tooth profiles.
The greater the number of N-Ps, the longer the relative
rolling time between two LogiX gears and the shorter the relative
sliding time between two LogiX gears. Thus abrasion of the
gear decreases and its loading capability and life span are improved.
But, considering the restriction of memory capability,
interpolation speed, angular resolution, etc. for the CNCmachine
tool used while cutting this type of gear, the relative pressure
angle selected should not be very small. δ _ 0.0006? is generally
satisfactory.
Table 1. Parameter values selected for LogiX rack at different modules
m(mm) α0 δ G0(mm)
1 10? 0.05? 6000
2 8.0? 0.05? 9500
4 6.0? 0.05? 10000
5 5.0? 0.05? 11000
6 4.0? 0.05? 12000
8 3.2? 0.05? 12024
10 2.8? 0.05? 14000
12 2.6? 0.05? 16500
15 2.5? 0.05? 20024
18 2.4? 0.05? 30036
20 2.4? 0.05? 35000
22 2.3? 0.05? 38000
793
4.4 Reasonable selection example
Based on the above analytical rules for LogiX gear inherent parameter
selection, a reasonable calculation and selection results
for the initial pressure angle and basic circle radius while selecting
different modules at the relative pressure angle δ = 0.05? are
listed in Table 1 for reference. In fact, the practical selections
should be reasonably modified by the concrete cutting conditions
and the special purpose requirement.
5 Conclusions
The following conclusions were made based on the findings presented
in this paper.
1. Two-dimensional meshing transmission models of LogiX
gears were deduced by further analysis of its forming
principle.
2. The influence on the LogiX gear tooth profile and its performance
by the gear’s own basic parameters such as initial
pressure angle, initial basic circle radius and relative pressure
angle was discussed and their reasonable selection was given.
3. The theoretical system of the LogiX gear was developed and
the mathematical basis for generating the LogiX tooth profile
by modern CNC technology was established. The characteristics
of the LogiX gear, which are different from those of the
ordinary standard involute gear, can have broad application
and most significantly impact the improvement of carrying
capacity, miniaturisation and longevity of kinetic transmission
products.
References
1. Komori T, Arga Y, Nagata S (1990) A new gear profile having
zero relative curvature at many contact points. Trans ASME 112(3):
430–436
2. Xutang W (1982) Gear meshing theory. Machinery Industry Press,
Beijing
3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing
6 Nomenclature
α0 initial pressure angle
αi pressure angle at contact point mi
δ parameter of pressure angle
ρs1 radius of curvature of gear tooth profile at contact point s1
ρmi radius of curvature of gear tooth profile at contact point mi
ρm1 radius of curvature of gear tooth profile at contact point
m1
G0 initial radius of basic circle in tooth profile
Gi radius of basic circle of mi point in gear tooth profile
?2 rotation angle of LogiX gear meshing with basic LogiX
rack
r2 radius of basic circle of LogiX gear meshing with basic
LogiX rack
m model of gear
z gear tooth number
s gear tooth thickness at pitch circle; here, i is an optional
number
DOI 10.1007/s00170-003-1741-8
ORIGINAL ARTICLE
Int J Adv Manuf Technol (2004) 24: 789–793
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 productivity
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 gear’s transmission performance, a new type of gear
called the LogiX gear was developed in the early 1990s. However,
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 influence
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 forward;
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 mentioned
above, the new type of gear has some other excellent
F. Xianying (_) · W. Aiqun
School of Mechanical Engineering,
Shandong University,
P.R. China
E-mail: FXYing@sdu.edu.cn
Tel.: +86-531-8395852(0)
L. Lee
School of Mechanical & Manufacturing Engineering,
Singapore Polytechnic,
Singapore
characteristics. 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 zero
are assured at many points. Here, this kind of point is called the
null-point (N-P). The presence of many N-Ps during the mesh
process of LogiX gears can result in a smaller sliding coefficient,
and the mesh transmission performance becomes almost
rolling friction accordingly. Thus this new type of gear has many
advantages such as higher contact intensity, longer life and a
larger transmission-ratio power transfer than the standard involute
gear. Experimental results showed that, given a certain
number of N-Ps between two meshed LogiX gears, the contact
fatigue strength is 3 times and the bend fatigue strength 2.5 times
larger than those of the standard involute gear. Moreover, the
minimum tooth number can also be decreased to 3, much smaller
than that of the standard involute gear.
The LogiX gear, rega