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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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