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MATHEMATICAL COMPUTER PERGAMON Mathematical and Computer Modelling 29 (1999) 69-87 MODELLING Curvature Analysis of Roller-Follower Cam Mechanisms HONG-SEN YAN Department of Mechanical Engineering, National Cheng Kung University Tainan 70101, Taiwan, R.O.C. WEN-TENG CHENG Department of Mechanical Engineering, I-Shou University Ta-Shu, Kaohsiung Hsien 840, Taiwan, R.O.C. (Received January 1996; accepted January 1998) Abstract-The equations related to the curvature analysis of the roller-follower cam mechanisms are presented for roller surfaces being revolution surface, hyperboloidal surface, and globoidal surface. These equations give the expressions of the meshing function, the limit function of the first kind, and the limit function of the second kind. Once these functions are known, the principal curvatures of the cam surface, the relative normal curvatures of contacting surfaces, and the condition of undercutting can be derived. Three particular cam mechanisms with hyperboloidal roller are illustrated and the numerical comparison between 2-D and 3-D cam is given. 1999 Elsevier Science Ltd. All rights reserved. Keywords-” F ; 8”, , I 1 (3) 0 0 01 T23 = 0 cp -sp 0 (4 where we designate sine and cosine of the corresponding angle as symbols C and S, and the subscript ij in the designation Tij is the transformation matrix from coordinate systems Sj to s+ Transformation matrix Trs can be obtained by the successive matrix multiplication P13l = Pii GoI To21 P231. (5) Transformation matrix Trs is expressed in partition matrix as follows: P131 P13l fT131 = O 1 l where Rrs is a rotation matrix and drs is a translation column vector. Taking the derivatives of transformation matrix Tls, relative velocity matrix Wrs, and rela- tive angular velocity matrix firs are given by w131 = T131T i;3 = ;l3 $1 , (7) p131 = R131T , $1 , 1 (11) where wT31 = 1 Pl31 71311. (12) Expanding equation (1 l), we have where w, wy, and w, are the components of the relative angular velocity between the roller and the cam, and TV, rr, and rz are the components of the relative translational velocity between the roller and the cam at the origin of coordinate system S3. All the components of the relative velocities are expressed in coordinate system S3. For the roller-follower cam mechanism, the meshing function Cp is defined as qe,u,q E n(3) .vl) = nf W, q . (14) For the cam surface being conjugate to the roller surface at the point of contact, the equation of meshing is given by (e, 21, t = 0. (15) Simultaneous solution of equations (9) and (15) determines the contact line on the roller sur- face for any given time t, and simultaneous solution of equations (10) and (15) determines the corresponding contact line on the roller surface in the meantime. The limit function of the second kind at for mutually contacting surfaces Cr and C3 is expressed as (a,(e,u,t) = np T w; ?-a) * (16) Let KY and $) be the principal curvatures of the roller surface C3, and in and bn be the corresponding principal directions in coordinate system S3. Then, the limit function of the first kind E is defined as 7,12 Q=Jvnz+Iry+, E = K$nz + wn Y, (17) C=$)VnY-IIZ, where wnz, WQ, ynxr and vnV are the components of the relative angular and sliding velocities in the tangent plane of mutually contact surfaces C3 and Cr as follows: wnIs = wp T in 1 9 my = w3 1 (31) T bn, (18) v = g. Using equations (A2) and (A4), the components of the relative velocity matrix Wis becomes w, = WY = 0, W% = (4; - l)Wl, rz = -aSf - 1) Se) wi, vu = (-aC - 1) + c (46 - 1) CO) WI, u* = 0. From equation (41), the meshing function is given by = (-aS(0+2)+b(fC (e+42)+b+;se)f. From equations (48) to (50), the coefficients 5 and C, and the limit function of the first kind are given by =(ac(e+2)-b(:-1)Ce), c = 0, E = -a2 - b2 (4: - 1)2 + 2ab (4; - 1) CfSO) u tan y (Sa + tan ycace) +c2as2e(u set y tan r)2 - u tan y (s;sase + siC8) (u /Tsec y). Example 3. Concave Globoidal Cam with an Oscillating Hyperboloidal Follower The settings of the coordinate systems for the concave globoidal cam with a hyperboloidal follower is shown in Figure 7. The globoidal cam rotates about the input axis with rotation angle 41, while the follower oscillates about the output axis with rotation angle $2. Thus, let sr = 0 and 52 = 0. The shortest d is t ante between the input and output axes is a and the twisted 82 H.-S.YAN AND W.-T. CHENG Figure 7. Concave globoidal cam with an oscillating hyperboloidal follower. angle a is r/Z. For the relative location of the rotation axis of the roller and the output axis, the distance b = 0 and the twisted angle S = 7r/2. The roller has a distance d from the origin of the coordinate system Ss to its base circle. And, the relation between the input and the output displacements is given by 42 = ) , B = w1 (cdq5 - 21 (tan 7 (a + dS42) - c sec2 7959 - U2 tan 7 sec2 7S42) , c = 0. From equation (42), the equation of meshing is given by ysC2 + (234; + y32) (d + 215X” 7) = 0. Furthermore, the limit function of the second kind is given by !Bt = II II Nt3) -1(AtsinB+Btcosf3+Ct), a3 where Curvature Analysis At = W: (cdC) , Bt = wf (cd l(N(3)11-1 ( (z3$i + y3c42&) (d + ?JSeC2 7) . nom equations (48) to (SO), the coefficients c and C, and the limit function of the first kind are given by 542) - ya(d + 4h + 5542) . 150 2 (deg) I I r .L__ MS i __ 1 Dwell j 1 I I Dwell 120 Figure 8. Motion function. Example 4. Numerical Comparison Between 2-D and 3-D Cams The cam mechanisms of Example 1 and Example 3 are applied to offer the quantifiable com- parison between the 2-D and 3-D cams. They use the same roller radius, follower displacement, motion function, and distance between the input and output axes. The motion function cPs(&) shown in Figure 8 is divided into five intervals and that the second and the fourth intervals use modified sine motion. Table 1 shows the parameters and the functions which are used for the disk cam and the globoidal cam. Table 1. Parameters of disk cam and globoidal cam. a4 H.-S. YAN AND W.-T. CHENG Figure 9. Cam profile for disk cam. 50 0 Figure 10. Cam profile for globoidal cam. I I f I I I, I I I I I I I I I I 0 h (de Figure 11. Pressure angle for disk cam. Curvature Analysis 85 For the roller surface being a cylindrical surface, the pressure angles q&k and qs10 for the disk cam and the globoidal cam are derived as Cvdisk = IbSfJ WV (b2 + c2 + 2bcC6)“2 cqdO = (c2ce2 + u2)1/2. Figures 9-14 shows the cam profiles, the pressure angles, and the principal curvatures for the disk cam and the globoidal cam. As shown in Figure 10, the pressure angles for the Profiles 1 and 2 of the globoidal cam have the same value for the same 41 and u. CONCLUSION The rollers with cylindrical surface, conical surface, and globoidal surface are usually used in roller-follower cam mechanisms. The cylindrical surface and the conical surface are special cases of the hyperboloidal surface. For the rollers of revolution surface, hyperboloidal surface, and globoidal surface, the curvature analysis of the roller-follower cam mechanisms are presented in this paper. For the mutually contacting surfaces between the cam and the follower, the principal curvatures of the cam surface, the relative normal curvature, and the condition of undercutting are expressed in terms of the meshing function and the limit functions. And, these functions for the cam mechanisms with the three-roller surfaces are derived. The hyperboloidal surface and the globoidal surface are the particular cases of the axis-symmetric quadric surface while the later one is a particular case of the revolution surface. For the simplicity of programming, we just focus on the roller of revolution surface. Here, all the surface normals of the roller surfaces are directed outward the roller. Therefore, the limit function of the first kind must be minus in order to avoid the undercutting. APPENDIX The transformation matrix Trs is given by a1 CdJz + CaS41S42 a3(44lS42 + CaS41C4J2) - Sj3SaS1 Z3 = I -+%c42 + CaCd1S42 Pwiw42 + CffC41C2) - spsaclpl SffSdJ2 SPCa + C&9aC42 0 0 (AlI -SP(-ChS4Q + CaS&C95,) - cpsasq+l a% - szSc&h + b(C$IC& + Ca&s#) -ww1w2 + CaC41wJ2) - CphYCc#q -a%h - szSaC& + b(-ShW2 + c0rc4s4) -SPSaC& + CPCa -61 + s2Ca + bSaS& 1 I. 0 The relative velocity matrix Wrs is given by w131 = 0 -wz wy rz WZ 0 -% rrl -wy WI 0 72 0 0 0 0 with the components w, = -&s&pz, I I (4 wy = -&(SPCa + CPSaC42) + &sp, w, = -&(CPCa - Spsacqh) + 42cp, (A3) 86 H.-S. YAN AND W.-T. CHENG t I- u=S8 360 Figure 12. Pressure angle for globoidal cam. _._- 0 Figure 13. First principal curvature for disk cam. 360 0.04 , , , , ua58 / Figure 14. Principal curvatures for globoidal cam. Curvature Analysis 87 Tz = -&(aCoS+z + s2SaC&) - BlSdq2, Ty = $1 (-Ccc/3 (b + aCq52) + sosp (a + bC42) + s2SaC/w2) + cj2bCP - 81 (Cc&P + SaC/3C42) + B2SP, T= = $1 (CcxS (b + aC&) + SaCP (u + bCq&) - s2SaSPS42) - rj2bSP - B1 (Cc&p - S&3/%39) + B&p. The derivative of relative velocity matrix Wls is given by 0 -Ljz Lj, iz 1 w13 = WZ 0 -Ljz iv 1. I -&Jar iJz 0 i, 0 0 0 0 (A3)(cont.) (A4) with the components . . . . l& = -4142SaC42 - lSwJ2, Ljy = &2cpsasq52 - $1 (SPCa + C/wap2) + J,sp, Lj* = 4142spsas42 + $1 (-cpccu + S/mYCq52) + $2cp, i, = -sac42 &s2 + &Sl + &(-aCaCq52 + s2SaS42) ( - $1 (aCaSq52 + s2SaC42) - IlScYS42, iv = CSaS42 (qi 1S2 + 42Bl + &$2 (aCCcxS, - bSaS/W+2 + sCSCYC) (A5) + $1 a (SaSP - CPYC) + b (-CaCp + Sk164) + s2CPSaSqi2 + &bC/3 - 51 (CCYSP + SCYCPG#J) + i2Sp, iz = -S/3SaSqs2 (” 182 + $2.41 + $142 (-aSpCcuS& - bSaCPS& - s2S&SaC&) + $1 a (SaCP + SPCaC&) + b (CdV3 + CPSaC42) - sSM+ - &bS/3 + lil (-C&j3 + SCYS/C) + s2Cp. REFERENCES 1. M.L. 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Chakraborty, Curvature analysis of surfaces in higher pair, Part 1: An analytical investigation, ASME 2%ansactions, Journal of Engineering for Industry 98, 397-402, (1976). 9. S.G. Dhande and J. Chakraborty, Curvature analysis of surfaces in higher pair, Part 2: Application to spatial cam mechanisms, ASME Transactions, Journal of Engineering for Industry 98, 403-409, (1976). 10. J. Chakraborty and S.G. Dhande, Kinematics and Geometry of Planar and Spatial Mechanisms, Wiley, New York, (1977). 11. C.H. Chen, Formula of reduced curvature of two conjugate surfaces with conjugate motions of two degrees of freedom, In Proceedings of the flh World Congress on Theory of Machines and Mechanisms, pp. 842-845, (1983). 12. D.R. Wu and J.S. Luo, A Geometric Theory of Conjugate Tooth Surfaces, World Scientific, (1992). 湘潭大學(xué)興湘學(xué)院畢業(yè)設(shè)計工作中期檢查表 系 機電系 專業(yè) 機械設(shè)計制造及其自動化 班級 機械二班 姓 名 余啟良 學(xué) 號 2006183928 指導(dǎo)教師 胡自化 指導(dǎo)教師職稱 教授 題目名稱 弧面凸輪數(shù)控轉(zhuǎn)臺的設(shè)計—機械部分 題目來源 科研 ■ 企業(yè) 其它 課題名稱 8噸絞磨變速器設(shè)計 題目性質(zhì) ■ 工程設(shè)計 理論研究 科學(xué)實驗 軟件開發(fā) 綜合應(yīng)用 其它 資料情況 1、選題是否有變化 ■有 否 2、設(shè)計任務(wù)書 ■有 否 3、文獻綜述是否完成 完成 ■未完成 4、外文翻譯 完成 ■未完成 由 學(xué) 生 填 寫 目前研究設(shè)計到何階段、進度狀況： 了解了弧面凸輪在國內(nèi)外的發(fā)展現(xiàn)狀，弧面凸輪分度機構(gòu)的主要優(yōu)缺點及其應(yīng)用情況。在現(xiàn)有的研究基礎(chǔ)上深入了解了弧面凸輪的基本結(jié)構(gòu)類型，弧面凸輪的廓面方程、嚙合方程的推導(dǎo)過程，進行了弧面凸輪的造型設(shè)計。 由 老 師 填 寫 工作進度預(yù)測（按照任務(wù)書中時間計劃） 提前完成 按計劃完成 拖后完成 無法完成 工作態(tài)度（學(xué)生對畢業(yè)論文的認(rèn)真程度、紀(jì)律及出勤情況）： 認(rèn)真 較認(rèn)真 一般 不認(rèn)真 質(zhì)量評價（學(xué)生前期已完成的工作的質(zhì)量情況） 優(yōu) 良 中 差 存在的問題與建議： 指導(dǎo)教師（簽名）： 年 月 日 建議檢查結(jié)果： 通過 限期整改 緩答辯 系意見： 簽名： 年 月 日 注：1、該表由指導(dǎo)教師和學(xué)生填寫。 2、此表作為附件裝入畢業(yè)設(shè)計（論文）資料袋存檔。 湘 潭 大 學(xué) 興湘學(xué)院 本科畢業(yè)設(shè)計開題報告 題 目 弧面凸輪數(shù)控轉(zhuǎn)臺的設(shè)計—機械部分 姓 名 余啟良 學(xué)號 2006183928 專 業(yè) 機械設(shè)計制造及其自動化 班級 機械二班 指導(dǎo)教師 胡自化 職稱 教授 填寫時間 2010年4月22日 2010年4月  說 明 1．根據(jù)湘潭大學(xué)《畢業(yè)設(shè)計(論文)工作管理規(guī)定》，學(xué)生必須撰寫《畢業(yè)設(shè)計（論文）開題報告》，由指導(dǎo)教師簽署意見，系主任批準(zhǔn)后實施。 2．開題報告是畢業(yè)設(shè)計（論文）答辯委員會對學(xué)生答辯資格審查的依據(jù)材料之一。學(xué)生應(yīng)當(dāng)在畢業(yè)設(shè)計（論文）工作前期內(nèi)完成，開題報告不合格者不得參加答辯。 3．畢業(yè)設(shè)計(論文)開題報告各項內(nèi)容要實事求是，逐條認(rèn)真填寫。其中的文字表達要明確、嚴(yán)謹(jǐn)，語言通順，外來語要同時用原文和中文表達。第一次出現(xiàn)縮寫詞，須注出全稱。 4．本報告中，由學(xué)生本人撰寫的對課題和研究工作的分析及描述，應(yīng)不少于2000字。 5．開題報告檢查原則上在第2～4周完成，各系完成畢業(yè)設(shè)計開題檢查后，應(yīng)寫一份開題情況總結(jié)報告。 6. 填寫說明： (1) 課題性質(zhì)：可填寫A．工程設(shè)計；B．論文；C. 工程技術(shù)研究；E.其它。 (2) 課題來源：可填寫A．自然科學(xué)基金與部、省、市級以上科研課題；B．企、事業(yè)單位委托課題；C．校級基金課題；D．自擬課題。 (3) 除自擬課題外，其它課題必須要填寫課題的名稱。 (4) 參考文獻不能少于10篇。 (5) 填寫內(nèi)容的字體大小為小四，表格所留空不夠可增頁。 本科畢業(yè)設(shè)計(論文)開題報告 學(xué)生姓名 余啟良 學(xué) 號 2006183928 專 業(yè) 機械設(shè)計制造及其自動化 指導(dǎo)教師 胡自化 職 稱 教授 所在系 機電系 課題來源 導(dǎo)師發(fā)布 課題性質(zhì) 工程技術(shù)研究 課題名稱 弧面凸輪數(shù)控轉(zhuǎn)臺的設(shè)計—機械部分 一、選題的依據(jù)、課題的意義及國內(nèi)外基本研究情況 本設(shè)計是以新型傳動數(shù)控轉(zhuǎn)臺的的設(shè)計為研究平臺，針對弧面凸輪機構(gòu)的設(shè)計仿真分析。 由于生產(chǎn)工藝的要求，廣泛使用的各種自動機械中往往需要機構(gòu)來實現(xiàn)周期性的轉(zhuǎn)位、分度動作，實現(xiàn)這種運動的機構(gòu)稱為間歇機構(gòu)。隨著自動機械向高速化、精密化、輕量化的方向發(fā)展，對間歇機構(gòu)提出越來越高的要求。常用的間歇機構(gòu)主要包括棘輪機構(gòu)、槽輪機構(gòu)、針輪機構(gòu)、不完全齒輪機構(gòu)及各種凸輪型間歇機構(gòu)，其中前四種間歇機構(gòu)由于分度定位精度低，運動不夠穩(wěn)定，高速時有較大沖擊，只適用于低速、輕載的場合。凸輪型間歇機構(gòu)結(jié)構(gòu)簡單，能自動定位，動靜比可任意選擇，與傳統(tǒng)的幾種間歇機構(gòu)相比，更適用于要求高速、高分度精度的場合，因而成為現(xiàn)代間歇機構(gòu)發(fā)展的主要方向。采用弧面凸輪分度機構(gòu)的弧面凸輪分度箱，它已成為當(dāng)今世界上精密驅(qū)動的主流裝置。它具有高速性能好，運轉(zhuǎn)平穩(wěn)，傳遞扭矩大，定位時自鎖，結(jié)構(gòu)緊湊、體積小，噪音低、壽命長等顯著優(yōu)點，是代替槽輪機構(gòu)、棘輪機構(gòu)、不完全齒輪機構(gòu)等傳統(tǒng)間歇機構(gòu)的理想產(chǎn)品。 從參數(shù)化和可視化的虛擬設(shè)計技術(shù)出發(fā),基于UG軟件, 建立了弧面分度凸輪機構(gòu)的參數(shù)化設(shè)計、造型和運動仿真, 得出分度盤的轉(zhuǎn)速以及滾子與凸輪的嚙合力并進行分析,獲得比較直觀的結(jié)果.為弧面分度凸輪機構(gòu)的運動性能研究和企業(yè)的產(chǎn)品優(yōu)化設(shè)計提供研究參考。 研究現(xiàn)狀： 弧面分度凸輪機構(gòu)是二十世紀(jì)20年代美國工程師C.N.Neklutin發(fā)明的，當(dāng)時Neklutin稱此機構(gòu)為滾子齒形凸輪分度機構(gòu)。二十世紀(jì)50年代該機構(gòu)由C.N.Neklutin所創(chuàng)辦的Ferguson公司首先進行了標(biāo)準(zhǔn)化系列化生產(chǎn)。我國從二十世紀(jì)七十年代末對該機構(gòu)也開始了研制工作，在弧面分度凸輪機構(gòu)的理論研究、設(shè)計制造等方面做了大量的工作?；∶娣侄韧馆啓C構(gòu)從50年代開始投產(chǎn)以來，經(jīng)過不斷改進，已成為應(yīng)用最廣泛、產(chǎn)量最大的凸輪分度機構(gòu)產(chǎn)品。 二、研究內(nèi)容、預(yù)計達到的目標(biāo)、關(guān)鍵理論和技術(shù)、技術(shù)指標(biāo)、完成課題的方案和主要措施 本設(shè)計以新型傳動數(shù)控轉(zhuǎn)臺的的設(shè)計為研究平臺，針對弧面凸輪機構(gòu)的設(shè)計仿真分析是整個弧面凸輪數(shù)控轉(zhuǎn)臺項目中的一個重要環(huán)節(jié)。課題組在詳細了解國內(nèi)外在此方面的發(fā)展情況，并通過結(jié)合現(xiàn)在已開發(fā)的同類產(chǎn)品，在此基礎(chǔ)上進行優(yōu)化設(shè)計，使產(chǎn)品性能更加優(yōu)越，體積進一步減小。在項目研制過程中，我利用互聯(lián)網(wǎng)和學(xué)校圖書館詳細的了解了弧面凸輪的基本結(jié)構(gòu)類型，廓面方程，嚙合規(guī)律等方面的知識，對現(xiàn)有的弧面凸輪進行了了解，查閱了有關(guān)資料。本課題在設(shè)計造型和動態(tài)的模擬仿真方面采用計算機輔助設(shè)計的技術(shù)，利用UG軟件及基于UG二次開發(fā)模塊建模，UG的動態(tài)仿真，進一步縮短了設(shè)計周期，降低了設(shè)計成本，有助于促進了設(shè)計工作的規(guī)范化、系列化和標(biāo)準(zhǔn)化，從而提高該產(chǎn)品設(shè)計開發(fā)能力。 主要的工作內(nèi)容有以下幾個方面： 1）設(shè)計計算部分：在結(jié)合指導(dǎo)老師所給的數(shù)據(jù)的情況下，分析確定凸輪分度機構(gòu)傳動方案；在了解了弧面凸輪的廓面方程、嚙合方程的基礎(chǔ)上通過計算分析，確定弧面凸輪的參數(shù)，校核弧面凸輪強度；完成弧面凸輪的嚙合齒輪的設(shè)計計算；在傳動部分設(shè)計完成后，進行轉(zhuǎn)臺的聯(lián)接設(shè)計及轉(zhuǎn)臺自鎖問題的解決。 2）工程仿真分析部分：本論文利用三維軟件UG及基于UG二次開發(fā)模塊對弧面凸輪機構(gòu)進行三維建模，畫出零件三維圖形；利用UG軟件對弧面凸輪機構(gòu)模型進行模擬仿真；對內(nèi)嚙合齒輪傳動進行動力學(xué)分析。 三、主要特色及工作進度 主要特色： 利用計算機輔助設(shè)計技術(shù)，基于UG及其二次開發(fā)模塊等軟件對理論設(shè)計的進行參數(shù)化建模，動態(tài)仿真和結(jié)構(gòu)的優(yōu)化設(shè)計。 工作進度： 收集查閱了有關(guān)弧面凸輪的發(fā)展現(xiàn)狀、主要參數(shù)方程的推導(dǎo)等方面的資料，制定了設(shè)計提綱和計劃，完成了軟件的應(yīng)用學(xué)習(xí)。 四、主要參考文獻（按作者、文章名、刊物名、刊期及頁碼列出） ［1］濮良貴，紀(jì)名剛. 機械設(shè)計[M]. 北京:高等教育出版社，2002. ［2］胡宗武等. 非標(biāo)準(zhǔn)機械設(shè)備設(shè)計手冊[M]. 北京:機械工業(yè)出版社，2005. ［3］楊冬香,陽大志. 基于不同滾子從動件類型的弧面凸輪CAD 集成系統(tǒng)開發(fā)[J]. 機電工程技術(shù),2009. ［4］葛正浩,蔡小霞,王月華. 應(yīng)用包絡(luò)面理論建立弧面凸輪廓面方程［J］,2004. ［6］張高峰，楊世平，陳華章，周玉衡，譚援強.弧面分度凸輪的三維CAD[J].機械傳動,2003 ［7］王其超，我國弧面分度凸輪機構(gòu)研究的綜述及進展，機械設(shè)計，1997 ［8］胡自化,張平. 連續(xù)分度空間弧面凸輪的多軸數(shù)控加工工藝研究[J] . 中國機程,2006 ［9］張高峰,楊世平,陳華章,等. D-H 方法在弧面分度凸輪機構(gòu)設(shè)計中的應(yīng)用[J ] . 機械傳動,2003 ［10］張高峰　楊世平,陳華章,周玉衡,譚援強. 弧面分度凸輪機構(gòu)的研究與展望[J].機械傳動,2003 指導(dǎo)教師 意 見 指導(dǎo)教師簽名： 年 月 日 系意見 系主任簽名： 年 月 日 院意見 教學(xué)院長簽名： 年 月 日