行星架的數(shù)控加工與選用設(shè)計(jì)【NGW72-16二級(jí)行星圓柱齒輪減速器】,NGW72-16二級(jí)行星圓柱齒輪減速器,行星架的數(shù)控加工與選用設(shè)計(jì)【NGW72-16二級(jí)行星圓柱齒輪減速器】,行星,數(shù)控,加工,選用,設(shè)計(jì),ngw72,16,二級(jí),圓柱齒輪,減速器 常州工學(xué)院成人（繼續(xù)）教育學(xué)院 畢業(yè)設(shè)計(jì)（論文）任務(wù)書(shū) 系： 機(jī)電 專(zhuān)業(yè)：數(shù)控技術(shù)應(yīng)用 班級(jí)： 學(xué)生姓名 指導(dǎo)教師 職稱(chēng) 課題名稱(chēng) 行星架的數(shù)控加工與選用 課 題 工 作 內(nèi) 容 指 標(biāo) ︵ 目 標(biāo) ︶ 要 求 進(jìn) 程 安 排 主 要 參 與 文 獻(xiàn) [1]張國(guó)瑞、張展主編。行星傳動(dòng)技術(shù)。上海：上海交通大學(xué)出版社，1981 [2]毛賢德、李振清主編。袖珍機(jī)械傳動(dòng)設(shè)計(jì)手冊(cè)。北京：機(jī)械工業(yè)出版社。1994 [3]卞言主編。實(shí)用軸承技術(shù)。北京：機(jī)械工業(yè)出版社。2004 [4]林克主編。最新國(guó)內(nèi)外軸承代號(hào)對(duì)照手冊(cè)。北京：機(jī)械工業(yè)出版社。1998 [5]王之櫟、王大康主編。機(jī)械設(shè)計(jì)綜合課程設(shè)計(jì)。北京：機(jī)械工業(yè)出版社。2004 [6]江耕華、陳啟松主編。機(jī)械傳動(dòng)設(shè)計(jì)手冊(cè)。北京：煤炭工業(yè)出版社。1983 地 點(diǎn) 起止 日期 系主任： 指導(dǎo)教師： 年 月 日 年 月 日 常州工學(xué)院 （成人教育） 畢 業(yè) 設(shè) 計(jì)（論文） 題 目 行星架的數(shù)控加工與選用 副標(biāo)題 性 質(zhì)： √ 畢業(yè)設(shè)計(jì) 畢業(yè)論文 學(xué)生姓名 年 級(jí) 教 學(xué) 點(diǎn) 無(wú)錫技師學(xué)院 專(zhuān) 業(yè) 數(shù)控技術(shù)應(yīng)用 指導(dǎo)教師 評(píng)定成績(jī) 優(yōu) 良 中 及格 不及格 摘 要 國(guó)內(nèi)數(shù)控機(jī)械行業(yè)可以為制造業(yè)提供幾乎所有的裝備，可中國(guó)數(shù)控行業(yè)的水平和實(shí)力在國(guó)際市場(chǎng)上還不算是強(qiáng)者。數(shù)控機(jī)械行業(yè)在制造業(yè)中的地位，既取決于數(shù)控機(jī)械行業(yè)的技術(shù)進(jìn)步和結(jié)構(gòu)調(diào)整，也取決于制造業(yè)的發(fā)展與市場(chǎng)前景。 數(shù)控技術(shù)在數(shù)控機(jī)床加工中的應(yīng)用，成功地解決了某些形狀復(fù)雜、一致性要求較高的中、小批零件的加工自動(dòng)化問(wèn)題，不僅大大提高了生產(chǎn)效率和加工精度，而且減輕了工人的勞動(dòng)強(qiáng)度，縮短了勞動(dòng)周期，并推動(dòng)了航空、航天、國(guó)防、機(jī)電等工業(yè)的發(fā)展。目前，數(shù)控技術(shù)已逐步普及，數(shù)控機(jī)床在各個(gè)工業(yè)部門(mén)得到了廣泛應(yīng)用，已成為機(jī)床自動(dòng)化的一個(gè)重要發(fā)展方向。 而機(jī)床的發(fā)展，使得行星齒輪精度越來(lái)越高，也提高了行星齒輪減速器的精度。行星齒輪系在減速器中有非常重要的位置，而行星齒輪的裝配的重要部件行星架也得到了發(fā)展。行星架有其定位精度高的優(yōu)點(diǎn)，表面粗糙度高，孔的定位與尺寸精度高的要求，這為加工增加了不少的難度，隨著數(shù)控機(jī)床迅速的發(fā)展，為行星架的加工提供很大的便利。 關(guān)鍵詞：數(shù)控技術(shù)，編程，行星架，行星齒輪減速器 目 錄 第一章 緒論 ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 1 第二章 行星架的工藝分析 ¨¨¨¨¨¨¨¨¨ 2 2.1零件功能 ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨2 2.2零件結(jié)構(gòu)制造介紹¨¨¨¨¨¨¨¨¨¨¨¨¨¨2 2.3零件材料的選擇¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨3 2.4加工順序的安排¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨4 2.5工序集中和分散¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨4 2.6加工順序的安排¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨4 2.7工藝路線的擬訂¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨5 2.8加工方法說(shuō)明 ¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨5 2.9定位基準(zhǔn)的選擇¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨6 2.10機(jī)床及其工藝設(shè)備的選擇¨¨¨¨¨¨¨¨¨¨¨7 2.11工件的裝夾方式¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 7 第三章 數(shù)控加工 ¨¨¨¨¨¨¨¨¨¨¨¨¨ 8 3.1編寫(xiě)程序的步驟¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨8 3.2數(shù)控編程的方法 ¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 9 3.3程序代碼¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 9 第四章 行星架的選用¨¨¨¨¨¨¨¨¨¨¨ 12 第五章 總結(jié)¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 16 致謝¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 17 參考文獻(xiàn)¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨ 18 行星架的制造與應(yīng)用 第一章 緒論 數(shù)控技術(shù)及裝備是發(fā)展新興高新技術(shù)產(chǎn)業(yè)和尖端工業(yè)（如信息技術(shù)及其產(chǎn)業(yè)、生物技術(shù)及其航空、航天等國(guó)防工業(yè)產(chǎn)業(yè)）的使用技術(shù)和最基本的裝備。馬克思曾經(jīng)說(shuō)過(guò)“各種經(jīng)濟(jì)時(shí)代的區(qū)別，不在于生產(chǎn)什么，而在于怎么生產(chǎn)，用什么勞動(dòng)資料生產(chǎn)”。制造技術(shù)和裝備就是人類(lèi)生產(chǎn)活動(dòng)的最基本的生產(chǎn)資料，而數(shù)控技術(shù)又是當(dāng)今先進(jìn)制造技術(shù)和裝備最核心的技術(shù)。當(dāng)今世界各國(guó)制造業(yè)廣泛采用數(shù)控技術(shù)，以提高制作能力和水平，提高動(dòng)態(tài)多變市場(chǎng)的適應(yīng)能力和競(jìng)爭(zhēng)能力。此外，世界上各工業(yè)發(fā)達(dá)國(guó)家還將數(shù)控技術(shù)及數(shù)控裝備列為國(guó)家的戰(zhàn)略物資，不僅采取重大措施來(lái)發(fā)展自己的數(shù)控技術(shù)及其產(chǎn)業(yè)，而且在“高精尖”數(shù)控關(guān)鍵技術(shù)和裝備方面對(duì)我國(guó)實(shí)行封鎖和限制政策?？傊?，大力發(fā)展以數(shù)控技術(shù)為核心的先進(jìn)制造技術(shù)已成為世界各發(fā)達(dá)國(guó)家加速經(jīng)濟(jì)發(fā)展、提高綜合國(guó)力和國(guó)家地位的重要途徑。 數(shù)控技術(shù)是用數(shù)字信息對(duì)機(jī)械運(yùn)動(dòng)和工作過(guò)程進(jìn)行控制的技術(shù)，數(shù)控裝備是以數(shù)控技術(shù)為代表的新技術(shù)對(duì)傳統(tǒng)制造產(chǎn)業(yè)和新興制造業(yè)的滲透形成的機(jī)電一體化產(chǎn)品。 數(shù)控技術(shù)應(yīng)用不但給傳統(tǒng)制造業(yè)帶來(lái)了革命性的變化，使制造業(yè)成為工業(yè)蝦的象征，而且隨著數(shù)控技術(shù)的不斷發(fā)展和應(yīng)用領(lǐng)域的擴(kuò)大，它對(duì)國(guó)濟(jì)民生的一些重要行業(yè)（IT、汽車(chē)、輕工、醫(yī)療等）的發(fā)展起著越來(lái)越重要的作用，因?yàn)檫@些行業(yè)所需裝備數(shù)字化已是現(xiàn)代發(fā)展的大趨勢(shì)。從目前世界上數(shù)控技術(shù)及其裝備發(fā)展的趨勢(shì)來(lái)看，其主要研究熱點(diǎn)有以下幾個(gè)方面。 1） 高速、高精加工技術(shù)及裝備的新趨勢(shì) 效率、質(zhì)量是先進(jìn)制造技術(shù)的主體。高速、高精加工技術(shù)可大大地提高效率，提高產(chǎn)品的質(zhì)量和檔次，縮短生產(chǎn)周期和提高市場(chǎng)競(jìng)爭(zhēng)能力。 2）5軸聯(lián)動(dòng)加工和復(fù)合加工機(jī)床快速發(fā)展 采用5軸聯(lián)動(dòng)對(duì)三維曲面零件的加工，可用刀具最佳集合形狀進(jìn)行切削，不僅光潔度高，而且效率也大幅度提高。一般認(rèn)為，1臺(tái)5軸聯(lián)動(dòng)機(jī)床的效率可以等于2太3軸聯(lián)動(dòng)機(jī)場(chǎng)，特別是使用立方氮化硼等超硬材料銑刀進(jìn)行高速銑削淬硬鋼零件時(shí)，5軸聯(lián)動(dòng)加工可比3軸聯(lián)動(dòng)加工發(fā)揮更高的效益。但過(guò)去因5軸聯(lián)動(dòng)數(shù)控系統(tǒng)、主機(jī)結(jié)構(gòu)復(fù)雜等原因，其價(jià)格要比3軸聯(lián)動(dòng)數(shù)控機(jī)床高出數(shù)倍，加之編程技術(shù)難度較大，制約了5軸聯(lián)動(dòng)機(jī)床的發(fā)展 第五章 總結(jié) 畢業(yè)設(shè)計(jì)作為對(duì)大專(zhuān)教育所學(xué)知識(shí)的一次綜合檢驗(yàn)，起意義是不言而喻的。 在為期兩個(gè)多月的設(shè)計(jì)過(guò)程中，在指導(dǎo)老師的引導(dǎo)下對(duì)行星架典型的結(jié)構(gòu)和原理 進(jìn)行了研究。雖然這次設(shè)計(jì)已經(jīng)完成，但存在不少缺點(diǎn)，由于時(shí)間和個(gè)人水平的 限制，沒(méi)能將各種型號(hào)的行星齒輪一一介紹。在兩個(gè)多月的畢業(yè)設(shè)計(jì)過(guò)程中，有 過(guò)面對(duì)問(wèn)題的惶恐不安，也有解決問(wèn)題的信心滿懷，終于完成了這次設(shè)計(jì) 。 通過(guò)這次畢業(yè)設(shè)計(jì)使我掌握了做機(jī)械設(shè)計(jì)的基本方法和思路，為今后的工作打 下了基礎(chǔ)，而我的主要感受是在做任何事情前都應(yīng)做好相關(guān)數(shù)據(jù)資料的搜索。在 現(xiàn)今社會(huì)中，隨著計(jì)算機(jī)的普及以及網(wǎng)絡(luò)技術(shù)的發(fā)展，對(duì)于各種數(shù)據(jù)資料的搜索 已經(jīng)從圖書(shū)館的紙質(zhì)資料轉(zhuǎn)移到網(wǎng)絡(luò)平臺(tái)下的電子文檔。 本次設(shè)計(jì)不能算太完美，只可以說(shuō)基本完成了指導(dǎo)老師所提出的要求，因在設(shè)計(jì)過(guò)程對(duì)一些參數(shù)的選擇還存在一些不全面、不合理的選擇。因此在今后遇到類(lèi)似情況，將全面綜合考慮之后再選擇相關(guān)參數(shù)以及進(jìn)行有關(guān)計(jì)算。 致 謝 論文脫稿之際，特別感謝指導(dǎo)老師蔡昀老師為我講解、提供具有代表性的課題、材料，以及在我遇到難題時(shí)，能及時(shí)為我解答，使我順利完成這次設(shè)計(jì)。謹(jǐn)向蔡老師所授嚴(yán)謹(jǐn)、細(xì)致、認(rèn)真的指導(dǎo)和所付出的心血，致以崇高的敬意和衷心的感謝！ 曾經(jīng)教導(dǎo)過(guò)我的多位老師在我撰寫(xiě)畢業(yè)論文過(guò)程中以及在校學(xué)習(xí)期間，給予了我熱情幫助和大力支持。為此，向他們表示誠(chéng)摯的謝意！向所有關(guān)心、支持、幫助過(guò)我的各位老師、同學(xué)、朋友們表示謝意！ 最后，向熱心評(píng)審本論文和參加答辯的老師們致以衷心的感謝。 參考文獻(xiàn) [1]張國(guó)瑞、張展主編。行星傳動(dòng)技術(shù)。上海：上海交通大學(xué)出版社，1981 [2]毛賢德、李振清主編。袖珍機(jī)械傳動(dòng)設(shè)計(jì)手冊(cè)。北京：機(jī)械工業(yè)出版社。1994 [3]卞言主編。實(shí)用軸承技術(shù)。北京：機(jī)械工業(yè)出版社。2004 [4]林克主編。最新國(guó)內(nèi)外軸承代號(hào)對(duì)照手冊(cè)。北京：機(jī)械工業(yè)出版社。1998 [5]王之櫟、王大康主編。機(jī)械設(shè)計(jì)綜合課程設(shè)計(jì)。北京：機(jī)械工業(yè)出版社。2004 [6]江耕華、陳啟松主編。機(jī)械傳動(dòng)設(shè)計(jì)手冊(cè)。北京：煤炭工業(yè)出版社。1983 [7]顧京主編。數(shù)控加工編程及操作。北京：高等教育出版社。2003 17 MMK TRITA-MMK 2005:01 ISSN 1400-1179 ISRN/KTH/MMK/R-05/01-SE Relations between size and gear ratio in spur and planetary gear trains by Fredrik Roos planetary gears are commonly known to be compact and to have low inertia. Keywords Spur Gears, Planetary Gears, Gearhead, Servo Drive, Optimization Contents 1 INTRODUCTION / BACKGROUND. 5 2 EQUIVALENT LOAD. 6 3 SPUR GEAR ANALYSIS. 8 3.1 GEOMETRY, MASS AND INERTIA OF SPUR GEARS. 8 3.1.1 Geometrical relationships . 8 3.1.2 Gear pair mass . 9 3.1.3 Inertia . 9 3.2 NECESSARY GEAR SIZE . 10 3.2.1 Hertzian pressure on the teeth flanks . 10 3.2.2 Bending stress in the teeth roots. 12 3.2.3 Maximum allowed stress and pressure. 13 3.3 RESULTS AND SIZING EXAMPLES. 14 3.3.1 Necessary Size/Volume. 14 3.3.2 Gear pair mass, geometry and inertia. 15 4 ANALYSIS OF THREE-WHEEL PLANETARY GEAR TRAINS . 19 4.1 GEAR RATIO, RING RADIUS, MASS, INERTIA AND PERIPHERAL FORCE. 19 4.1.1 Gear ratio and geometry . 19 4.1.2 Weight/Mass . 20 4.1.3 Inertia . 20 4.1.4 Forces and torques . 22 4.2 PLANETARY GEAR SIZING MODELS BASED ON SS1863 AND SS1871. 23 4.2.1 Sun planet gear pair . 23 4.2.2 Planet and ring gear pair . 24 4.2.3 Maximum allowed stress and pressure. 26 4.3 RESULTS AND SIZING EXAMPLES. 27 4.3.1 Necessary size/volumes . 27 4.3.2 Weight, radius and inertia. 28 5 COMPARISON BETWEEN PLANETARY AND SPUR GEAR TRAINS . 32 6 CONCLUSIONS. 34 7 REFERENCES . 35 Relations between size and gear ratio in spur and planetary gear trains 4(35) Nomenclature Pressure angle rad Helix angel ( = 0 for spur gears) Transverse contact ratio Load angle rad Poissons number Mass density kg/m 3 F Root bending stress Pa Fmax Maximum allowed bending stress Pa H Hertzian flank pressure Pa Hmax Maximum allowed hertzian flank pressure Pa Angular velocity rad/s b Gear width m b c Carrier width m d Gear reference diameter m d a Gear tip diameter m d b Gear base diameter m E Module of elasticity Pa F Force N J Mass moment of inertia kgm 2 K F Factor describing the division of load between teeth K F Load distribution factor for bending K H Factor describing the division of load between teeth K H Load distribution factor for Hertzian pressure k ro Relation between outer and reference diameter of ring gear m Module m M Mass kg n Gear ratio of a complete gear train ( in / out ) p b Base pitch r Gear reference radius m S Safety factor T Torque Nm u Gear ratio of a single gear pair v Peripheral velocity m/s Y F Form factor for bending Y Helix angle factor for bending Y Contact ratio factor for bending z Number of teeth Z H Form factor for Hertzian pressure Z M Material factor for Hertzian pressure Z Contact ratio factor for Hertzian pressure Index 1 The small wheel (pinion) of a gear pair Index 2 The large wheel (gear wheel) of a gear pair Index r The ring gear in a planetary gear train Index s The sun gear in a planetary gear train Index p The planet gears in a planetary gear train Index c The planet carrier in a planetary gear train Relations between size and gear ratio in spur and planetary gear trains 5(35) 1 Introduction / Background This work was initiated within a research project about design and optimization methods for mechatronic systems. The goal with that research project is to derive methods for optimization of mechatronic actuation modules, with respect to weight, size and/or efficiency (Roos the heat generated in the motors winding is given by the RMS value of the motor current. Since the current is proportional to the motor torque, the RMS torque may be used for motor dimensioning. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1000 500 0 500 1000 time s Load profile, required torque and position Ang. Velocity rad/s Angle rad 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 40 20 0 20 40 60 RMS RMC Max time s torque Nm Figure 2. Example of an inertial load cycle, with RMS, RMC and max norms shown. Gear design is traditionally focused on strength of the gears. The load on the gear teeth is cyclic and therefore gear failure is most often a result of mechanical fatigue. The two classical limiting factors in gear design are surface fatigue and tooth root bending fatigue. The combination of cyclic loading of the gear teeth when in mesh and an applied load that varies with time makes it more difficult to find an expression for the equivalent load, than in the motor case. The exponent used in the torque norms used for gear sizing is not 2 as in the RMS norm, but ranges from 3 to 50 (Anthony 2003). These expressions for equivalent load are based on the so-called linear cumulative damage rule (the Palmgren-Minor rule). It assumes that the total life of a mechanical product can be estimated by adding up the percentage of life consumed by each stress cycle. The number of stress cycles on each tooth in a gear train can be huge during a lifetime. Anthony (2003) exemplifies this with a three-wheel planetary gear in which one sun gear tooth will be exposed to almost 3 million load cycles during an 8-hour period at 2000 rpm. Relations between size and gear ratio in spur and planetary gear trains 7(35) Area of unlimited load cycles Stres s l oad (l og S) Load Cycles (log N) E = 3 for bearings E = 8.7 for case hardening steel E = 17 for nitrided steel 10 6 10 9 10 3 10 12 E = 84 for carbo nitrided steel Figure 3. Whler curves for different steels. The exponent to use in the calculation of equivalent load depends on material type, heat treatments, and loading type (Antony 2003). It is however not obvious that the Palmgren-Minor rule can be used for infinite life design ( 10 6 load cycles), especially not in an application where the teeth will be subjected to the peak load more than 10 6 times. In fact only in applications where the total number of load cycles is below 210 6 is a higher load than the endurance limit load permissible (Antony 2003). This means that for infinite life dimensioning, the gears should be dimensioned with respect to the peak torque in the load cycle. Of course there are exceptions to this, for example load cycles where the peak load occurs while the gears are standing still. The calculated equivalent continuous torque, T cal is hence, for unlimited life design given by: max )(tTT cal = (1) This is the approach taken in this report; it is assumed that the teeth are subjected to the peak load more than 10 6 times, and therefore is the peak torque used for dimensioning. This research area is however very complex and are not investigated further in this report. By this approach, equation (1), is at least not a too low equivalent torque used. The sizing procedure gets even more complicated when the bearings are considered. For bearings, the Root Mean Cube (RMC) value of the load is often used as the equivalent continuous load (comp. Figure 3). This report will however only treat the actual dimensioning of the gears, not the bearings. But it should be noted that it may be the bearings that limit the maximum gear load. Relations between size and gear ratio in spur and planetary gear trains 8(35) 3 Spur gear analysis The analysis made here is mainly based on the formulas presented in the Swedish standard for calculation of load capacity of spur and helical gears, SS1871 and standard SS1863 for the spur gear geometry. Figure 4 shows a spur gear, to simplify the analysis only spur gears with no addendum modification are treated. Figure 4. Spur gear. 3.1 Geometry, mass and inertia of spur gears. 3.1.1 Geometrical relationships In order to simplify the rest of the analysis, it is useful to derive some simple geometrical relations. The gear ratio u is defined as: 2 1 1 2 1 2 = z z r r r r u in out (2) The center distance, a between the wheels is given by: 1221 rarrra =+= (3) Combining equations (2) and (3) gives: 1 1 r ra u = (4) Pinion Gear Wheel T in T out a r 2 r 1 Relations between size and gear ratio in spur and planetary gear trains 9(35) Finally, combining equations (3) and (4) gives the expressions for r 1 and r 2 1 1 + = u a r (5) 11 22 + = + = u au r u a ra (6) 3.1.2 Gear pair mass A gear wheel is here modeled as a cylinder, an approximation that is quite accurate. The mass, M of one gear is hence given by: 2 brM = (7) Where b is the face width, r is the reference radius and is the mass density of the wheel. The total mass of a gear pair can be expressed as: )( 2 2 2 121 rrbMMM tot +=+= (8) Finally by combining equation (2), (5) and (8) the following expression for the gear pair mass is obtained: 2 2 222 1 )1( 1 )1( u u baubrM tot + + =+= (9) 3.1.3 Inertia The inertia, J of a rotating cylinder is given by: 2 2 cyl cylcyl r MJ = (10) The inertia reflected on the pinion shaft (axis 1) of a gear pair is hence given by: 2 2 22 2 2 12 1 2 2 2 2 2 1 1 2 2 1 2 2 2 2 u r br r br u r M r M u J JJ tot +=+=+= (11) Figure 5. Gear mesh p F p F Relations between size and gear ratio in spur and planetary gear trains 10(35) Which, if combined with equations (5) and (6) result in the following expression of the gear pair inertia: 4 2 4 4 24 4 4 )1( 1 2 )1()1( 2 u u ba u ua u ab J tot + + = + + + = (12) 3.2 Necessary gear size According to SS 1871, the necessary gear size is determined from the teeth flank Hertzian pressure and the teeth root stress. Neglecting losses such as friction the peripheral force, see figure 5, acting on a gear tooth is given by: au uT r T F out out out p )1( + = (13) For a given load the necessary gear size is determined as function of gear ratio and number of pinion teeth. Depending on material properties, gear ratio, number of teeth, etc., either the flank pressure or the root stress sets the limit on the gear size, in general both stress levels must be checked. 3.2.1 Hertzian pressure on the teeth flanks The Hertzian pressure on a teeth flank is given by (SS1871): ubd uKKF ZZZ HHcal MHH 1 )1( + = (14) For gears with no addendum modification the form factor Z H is given by: t b H Z 2sin cos2 = (15) As shown below, the transverse section pressure angle t is, for spur gears, the same as the normal section pressure angle, n . The pressure angle will therefore only be noted as from now on. = nt n t 0 cos tan tan (16) Since the helix angle is zero on the pitch cylinder () it will be zero on the base cylinder too ( b ) : 1cos0 cos coscos cos = bb (17) This leads to the following expression for Z H : 2sin 2 = H Z (18) The material factor Z M is given by (SS1857): + = 2 2 2 1 2 1 11 2 EE Z M (19) Relations between size and gear ratio in spur and planetary gear trains 11(35) Where E is the module of elasticity of the respective gear and is poissons number. For spur gears the contact factor, Z is according to SS 1871 given by: 3 4 =Z (20) Where is the contact ratio. For an external spur gear pair it is, according to SS1863, given by: + = ww baba b a dddd p sin 22 1 2 2 2 2 2 1 2 1 (21) Where a w is the center distance between the wheels, for gears with no addendum modification, aa w = . The base pitch, p b is given by: cosmp b = (22) Where m is the module, which is defined as: 12 2 1 1 )1( 2 zu a z d z d m + = (23) The tip and base diameters, d a and d b, are, for external spur gears, given by (SS1863): mdd a 2+= cosdd b = ) 44 (sin) 44 cos1( cos44cos)2( 2 22 2 2222 222222222 z z d z z ddd z d mddmmddmddd ba ba +=+= =+=+= (24) From equation (5) and (6) the following expressions for the gears diameters can be derived: 1 2 , 1 2 21 + = + = u au d u a d (25) By combining equations (24) and (25), and inserting them into equation (21), the following expression for the contact ratio is obtained: + + + + + = sin2) 44 (sin )1( 4 ) 44 (sin )1( 4 2 1 cos2 )1( 22 1 1 2 2 22 2 1 1 2 2 2 1 a uz uz u ua z z u a a zu += sin)1( 44 sin 44 sin cos2 22 1 1 2 2 1 1 21 u uz uz u z z z (26) Inserting equation (13) and (25) into equation (14) results in: 22 3 2222 2 )1( uba uT KKZZZ cal HHMHH + = This equation can be rewritten as: 2 max 2 3 2222 2 )1( H cal HHMH u uT KKZZZba + = (27) Relations between size and gear ratio in spur and planetary gear trains 12(35) Where Z H , Z M , Z , are given by equation (18), (19) and (20). K H and K H are factors describing the division of load between teeth and the load distribution on each tooth respectively. Generally K H can be set to 1. K H is more complicated since it only can be 1 in theory (if the gears are perfect). Here, for simplicity, it is set to 1.3, but if more exact data is available it should be used instead, see SS1871 for more information and guidelines about how to select this constant. Equation (27) gives the minimum size of the gear pair (with respect to Hertzian pressure) given a material, Hmax , E 1 , E 2 , 1 , 1 , a gear ratio, u, the number of pinion teeth, z 1 , the pressure angle and the calculated torque T cal . 3.2.2 Bending stress in the teeth roots The bending stress in a tooth, F can be calculated according to SS1871 as follows: bm KKF YYY FFcal FF = (28) The calculation of the form factor, Y F is somewhat complicated the way it is done in SS1871, therefore Y F is approximated with the following expression (Maskinelement handbok 2003): 14/ 1.32.2 z F eY + (29) Y F will always be larger for the small wheel (pinion) since it decreases with z. The helix angle factor Y is 1 for spur gears. Y is the so called contact ratio factor, and it is according to SS1871 calculated as follows: 1 =Y (30) Where the contact ratio, is calculated as before with equation (26). By combining equation (13), (23) and (28) the following is obtained: bua uzT KKYY cal FFFF 2 2 1 2 )1( + = (31) The expression above can be rewritten as follows: max 2 12 2 )1( F cal FFF u uzT KKYYba + = (32) Where Y F and Y is given by equation (29) and (30). K F and K F are factors describing the load division between teeth and the load distribution on each tooth respectively. If no other data is available K F can be set to 1, and K F to the same value as K H (SS1871). Equation (32) can be used to calculate the minimum size of a gear pair, with respect to bending endurance. Relations between size and gear ratio in spur and planetary gear trains 13(35) 3.2.3 Maximum allowed stress and pressure The maximum allowed bending stress Fmax and the maximum allowed hertzian pressure on the teeth, Hmax is of course largely dependent on material choice and safety factor. There are a lot of factors that can be included into the calculation of the stress limits, including the number of load cycles. Here the number of load cycles is assumed to be larger than the endurance limit and therefore that factor is disregarded. The maximum allowed stress and pressure is here simply calculated as follows: H H H S lim max = F F F S lim max = Where Hlim and Flim are material properties and S H and S F are the safety factors. For more advanced calculations see SS1871. Values of Flim and Hlim can for example be retrieved from Maskinelement Handbok (2003) or from the standards. It should be noted that if the safety factor for bending is doubled, the necessary gear size is doubled. But if the safety factor for the flank stress is doubled, it requires a four time larger gear pair, see equations (27) and (32) respectively. Relations between size and gear ratio in spur and planetary gear trains 14(35) 3.3 Results and sizing examples 3.3.1 Necessary Size/Volume In this section, equations (27) and (32) are applied on a (equivalent) load of 20 Nm. First, material data from an induction hardened steel with a Hertzian fatigue limit of 1200 MPa and a bending fatigue limit of 300 MPa is used. The Pressure angle is 20 degrees, and all other constants are set to the standard value (Table 1). Material Properties Gear Properties E 206 GPa 20 deg. 0.3 K H 1 7800 kg/m 3 K H 1.3 Load K F 1 T cal 20 Nm K F 1.3 Table 1. Values of material and gear parameters used in all examples. 0 5 10 15 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 5 Gear ratio Gear pair volumes (a 2 b). Hmax : 1200 MPa. Fmax : 300 Mpa T cal = 20 Nm No of teeth, wheel 1 (z 1 ) a 2 b m 3 0 5 10 15 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 1.2 x 10 4 Gear ratio Gear pair volumes (a 2 b). Hmax : 1200 MPa. Fmax : 300 Mpa T cal = 20 Nm No of teeth, wheel 1 (z 1 ) a 2 b m 3 root stress F1 root stress F2 flank stress H Figure 6. Gear pair volume as function of gear ratio and number of teeth. The safety factors SH and SF is 1 in the graph to the left and 2 in the graph to the right. The following plots are obtained for a non-hardened steel with a Hertzian fatigue limit of 500 MPa and a bending fatigue limit of 200 MPa (e.g. SIS1550). The load and all other parameters are the same as in the example above. 0 5 10 15 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 x 10 4 Gear ratio Gear pair volumes (a 2 b). Hmax : 500 MPa. Fmax : 200 Mpa T cal = 20 Nm No of teeth, wheel 1 (z 1 ) a 2 b m 3 0 5 10 15 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 x 10 4 Gear ratio Gear pair volumes (a 2 b). Hmax : 500 MPa. Fmax : 200 Mpa T cal = 20 Nm No of teeth, wheel 1 (z 1 ) a 2 b m 3 root stress F1 root stress F2 flank stress H Figure 7. The same plots as in Figure 6, but for another steel. The safety factors are set to one in the plot to the left and 2 in the plot to the right. Relations between size and gear ratio in spur and planetary gear trains 15(35) From the plots above and a number of other examples not presented here, a couple of conclusions can be made. First of all, the Hertzian pressure is the limiting factor in the vast number of cases (it requires the largest gears). The root bending stress is the limiting factor only for steels with a large difference between the two stress limits, and only if the safety factor for Hertzian pressure is low. Of course it is possible to change a lot of constants to get a different result, but these are the results if the standard (SS1871) is applied with the constants set to the recommended values. Furthermore the surface representing the root stress of the pinion (wheel 1) is always higher than corresponding surface for the gear wheel (wheel 2). This is consistent with the previous conclusion that the root stress always is larger for the smaller wheel. Therefore the root stress is only calculated for the pinion in the rest of this report. The number of teeth has most influence on the root bending stress. Not surprisingly, the flank stress is in comparison almost independent of the number of teeth. By choosing a relatively small number of pinion teeth (wheel 1), the tooth flank stress will almost certainly be the limiting factor. 3.3.2 Gear pair mass, geometry and inertia By combining equation (9) with the results shown in the left part of Figure 6 (safety factors = 1, induction hardened steel) the following graph (Figure 8) of the gear pair mass is obtained: 0 5 10 15 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 Gear ratio Gearpair mass as function of number of teeth and gear ratio, T = 20 No of teeth z 1 Mass kg Figure 8. Gear pair weight as function of number of teeth of the pinion and gear ratio. To continue the analysis it will be necessary to lock one of the variables in Figure 8, at least if the results are to be visualized in 3D-plots. The number of teeth of wheel 1 (pinion) seems to be the most reasonable variable to lock. In the figure below (Figure 9) plots of two different pinion teeth numbers are shown, 28 and 17. The larger choice results in, depending on the gear ratio, that booth the flank and root stress are limiting. The choice of 17 teeth of the small wheel results in, as also seen earlier, that only the flank stress is limiting the volume of the gear pair, regardless of gear ratio. Relations between size and gear ratio in spur and planetary gear trains 16(35) 0 5 10 15 0.5 1 1.5 2 2.5 3 3.5 x 10 5 Cross section when z1 = const = 28, T = 20 Nm a 2 b m 3 Gear Ratio Flank Stress Root Stress limiting (max) 0 5 10 15 0.5 1 1.5 2 2.5 3 x 10 5 Cross section when z1 = const = 17, T = 20 Nm a 2 b m 3 Gear Ratio Flank Stress Root Stress limiting (max) Figure 9. Cross sections of the left part of Figure 6, at 28 and 17 teeth of wheel one. To eliminate unrealistic gear geometry t