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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-&m mechanism, Roller-follower, Principal curvature, Undercutting, Limit function. NOMENCLATURE a A,B,C &,&act b c, d I-h1 E h-II, linl *(3) shortest distance between axes zi and z, coefficients of equation of meshing derivatives of the coefficients A, B, C with respect to time shortest distance between axes zs and z3 offset coordinates of center of base circle for roller surface 3 x 1 translational displacement of the roller relative to the cam at the origin one of fundamental magnitudes of the first order for roller surface principal directions of C3 roller surface in coordinate system Ss unit normal of roller surface C3 N(3) b$)l T TV) t R P131 $1 92 t Tijl surface normal of the roller unit normal of roller surface C3 in coordinate system 53 radius of generating circular arc of globoidal roller position of points on roller sur- face Ci in coordinate system Si distance from points on roller surface to z-axis rotation matrix from coordinate systems S3 to Sr linear displacement of the cam along axis .Zi linear displacement of the follower along axis to time transformation matrix from coordinate systems Sj to Si 0895-7177/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: SO895-7177(98)00179-4 Typeset by &G-TkX H.-S. YAN AND W.-T. CHENG relative sliding velocity at contact point between the roller and the cam relative sliding velocity between the roller and the cam in coordinate system S3 components of relative sliding velocity uf31) in coordinate system S3 *t $), ,w 2 e,u components of relative sliding velocity uf31) in coordinate system Sn (T 4 x 4 relative velocity matrix of the roller with respect to the cam 7131 3 x 4 relative velocity matrix of the roller with respect to the cam Wl coordinates of points on roller surface C3 in coordinate sys- tem S3 “ coordinates of points on roller surface C3 in coordinate sys- wz,wy,wz tem S3 twisted angle from axis zi to axis zo about axis xi wnz I 4ly twisted angle from axis z2 to axis z3 about axis 12 twisted angle of generating line hyperboloidal surface Pl31 angular displacement of the cam E?C rotating about axis .zi 9 angular displacement of the follower rotating about axis ze meshing function limit function of the second kind principal normal curvatures of roller surface Ci curvilinear coordinates of roller surfaces angle between principal direc- tions of mutually contacting surfaces Cl and C3 relative velocity between the roller and the cam at the origin in coordinate system S3 components of relative velocity t7131 angular velocity of the cam relative angular velocity between the roller and the cam in coordinate system S3 components of relative angular velocity w() in coordinate system S3 components of relative angular velocity w() in coordinate system Sn 3 x 3 relative angular velocity matrix of the roller with respect to the cam coefficients limit function of the first kind INTRODUCTION For designing of a transmission mechanism imparted by a higher pair, the relative location be- tween the input axis and the output axis, and the input-output relation are generally specified. Based on conjugate surfaces, the input link and the output link contact with each other. After the selection of a generating surface being one of mutually contacting surfaces, the generated surface, i.e., the other contact surface, is determined. The generated surface is solved by the equation of meshing at the point of contact. Furthermore, the curvatures of contact surfaces should be analyzed. On one hand, the dimensions of these contacting surfaces are limited in order to avoid surface interference and inappropriate tool cutting. On the other hand, the relative curvature between contacting surfaces influences on the stresses and deformations of the contacting bodies at the point of contact. The curvature properties of cam profiles and pitch curves were derived by graphical methods or analytical geometry l-4. These equations derived for the curvatures of cam profile are restricted to various particular types of cam mechanisms. For specified generating surfaces, the curvatures of generated surfaces can be derived from differential geometry and kinematic relations between mutually contacting surfaces. Litvin 5-71 presented a method to determine the principal curvatures and the principal directions of the generated surfaces, and the relative normal curvatures of mating surfaces in terms of those of generated surfaces for gear mechanisms. Dhande and Chakraborty &lo used the same concept to derive the curvature relations of various planar and spatial cam mechanisms. Chen ll used the geometric and kinematic concepts to obtain the formula of reduced curvature of two conjugate surfaces with conjugate motions of Curvature Analysis 71 two degrees of freedom. Wu and Luo 12 used the concept of differential geometry of surfaces to develop a systemic theory of conjugate surfaces. The limit functions of the first kind and the second kind are defined to analyze the curvature relations and the limit points of conjugate surfaces. This paper intends to analyze the curvature properties of the roller-follower cam mechanisms. We investigate the following three types of rollers: revolution surface, hyperboloidal surface, and globoidal surface. The first type is the general one of the roller surface, and the surfaces of the latter two types are generated by a straight line and a circular arc individually. The general coordinate systems of the roller-follower cam mechanisms are given. Additionally, the surface geometry of the rollers are educed. In general, the curvatures of the cam surface, the relative curvatures of the contacting surfaces, and the condition of undercutting depends on the meshing function and the limits functions 12. Here, we only evaluate these functions that are determined by the geometry of the roller surface, the relative location of the screw axes, and the relative motion between the cam and the follower. Finally, three particular cam mechanisms with hyperboloidal roller are illustrated. Figure 1. Coordinate systems. TRANSFORMATION MATRICES AND LIMIT FUNCTIONS For the roller-follower mechanism shown in Figure 1, the roller surface Cs contacts to the cam surface Ci. The settings of the coordinate systems for four-link cam mechanisms are identical. The coordinate systems Si(q, yi, pi) and Sc(zc, yc,zc) are fixed to the frame while the moving coordinate systems Si(zi, yi, .q), &(Q, ys, 4, and Ss(zs,ys, es) are fixed to the cam, the fol- lower, and the roller, respectively. All z-axes are directed along the rotation axis or the screw axis of the moving bodies. Axes xi and xc lie along the common perpendicular between axes zi and to, and axes 22 and x3 lie along the common perpendicular between axes z2 and 3. The distance between axes zi and axis 0 is a, and the twisted angle that screws axis zi into axis zo along axis zi is cr. Similarly, the distance and the twisted angle between axes 2 and 3 are b and p, respectively. The cam screws around the input axis zi with angular displacement 41 and translational displacement 1, whereas the coordinate system Si coincides with coordinate sys- tem Si in the initial position. Likewise, to the motion of the cam, the follower screws around the 72 H.-S. YAN AND W.-T. CHENG output axis zs with angular displacement q5s and translational displacement ss for the coordinate system Ss being transformed to SO. Point M on Cs expressed in coordinate system Ss is rc3) and the corresponding coordinates in coordinate system Sr is r cl). The rotation of the roller has no influence to the input-output relation since the roller has a surface of revolution about its rotation axis. Therefore, the roller is assumed to be fixed rigidly to the follower. By employing 4 x 4 transformation matrix, the transformations between these coordinates systems are represented by 0 a Tiol = 0 Ca -6% 0 1 0 Sa CCY 0 0) 00 0 1 Tll=L O C41 -S& 0 0 Wl Wl 0 0 0 0 ls1 (2) 0 0 01 cdJ2 -w2 0 0 To21 = ;” 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 &3l, (8) where qs is a translational velocity column vector and TK is the derivative of Tls with respect to time. The components of matrices Tls, Wrs, and l&13 are given in the Appendix. These compo- nents are applied to the kinematics of the roller-follower cam mechanisms. Let the homogeneous coordinates of the point on the roller surface in coordinate system Ss be 4% u, = zp(e,u) yf(e,u) zf(e,u) llT, 6-J) where f3 and u are the curvilinear coordinates of the roller surface. Curvature Analysis 73 Using transformation matrix Trs, the corresponding position vector rp) for the cam surface in coordinate system Sr is given by #) = T13 #I . The relative velocity between the roller and the cam is given by 1 v3 t31) = W;, $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& = p T in, 1 Iv = $1) 1 T lid 09) where We” is a 3 x 1 column matrix corresponding to matrix (Qr3. 74 H.-S. YAN AND W.-T. CHENG Again, the principal curvatures of the cam surface Ci and surface as follows 7: the principal directions on this tan(2a) = 25c ( $1 - $1 E + (62 - S2) “y) - $) = ( .f) + I$) )*+(c2-t2) W(2U) . CAM MECHANISMS FOR ROLLERS WITH REVOLUTION SURFACES (20) (21) (22) The general type of the roller surface is a surface generated by rotating a planar curve about its rotation axis. The coordinates of a surface of revolution (Figure 2) and the surface normal are expressed in coordinate system Ss as follows: )(8,“) = R(U)cY R(U)9 21 llT ) (23) (e, IL) = llN(3)ll-1 ce se -R+) IT , where N(3) = (1 + R2)/2 and R(U) is the radius from the point on the surface to the axis 23. Its derivative is exnressed as Figure 2. Surface of revolution. The principal curvatures of the roller surface are c3) = -R-r $3) -I 61 II II , fir = R” llNt3) ll-3 . (25) The corresponding principal directions are given by inI =-se ce 01 T, bn = llN(3)ll-1 R%f3 RS0 llT. (26) Substituting equations (13) (23) and (24) into equation (14), the meshing function is given by = NC31 -l(AsinB+BcosB+C), II II where A = rv - (u + RR)wz, B = rl: + (u + RR)wy, and C = -Rr,. (27) Curvature Analysis 75 The equation of meshing is given by cp = 0 or U&B + v,SB - u,R = 0. (28) Moreover, the limit function of the second hind is given by Qt = II II Nc3) - (At sin 0 + Bt cos 0 + Ct), (2% where At = iy -(u+RR)&, Bt = iz +(u+RR)&, and Ct = -Ri,. Now, we express the relative sliding velocity and the relative angular velocity in coordinate system Sn. Substituting equation (26) into equation (18), we have wrrz = -sew, + cew, , (30) wny = II II Nt3) -l (R (cew, + sew,) + w,) . (31) Combining equations (13) (19), and (23), and using equation (28) we have ez = -seu, + ceu, (32) u&l = II II Nt3) v,. (33) Substituting equations (25) and (30) to (33) into equation (17), we have c = R-l llN(3)i1-1( seu, - ceu, + RR (cewz + sew,) + RU,) , (34) C = R llN(3)11-2 U, + sew, - cew, (35) 9 = R- l/N(3)11-1( -Bu, -Au, - TU, + (l+ Rt2 + RR)u,2) +$. (36) CAM MECHANISMS FOR ROLLERS WITH HYPERBOLOIDAL SURFACES The hyperboloidal surface is generated by a certain motion of a straight line revolved about its rotation axis. The coordinates of a hyperboloidal surface (Figure 3) and the surface normal are expressed in coordinate system Ss as follows: (37) (38) where llN(3)11 = (c2 + (zctanysecy)2)1/2. c Y 0 z XP 2 Figure 3. Hyperboloidal surface. 76 H.-S. YAN AND W.-T. CHENG The principal curvatures of the roller surface are (3) _ 61 - II II N(3) - , I$ = (ctan+y)2 llN(3)1/-3. (3% The corresponding principal directions are given by in = E-12 -y3 x3 olT bn = NC31 II II - Ee1j2 xsutan2 y ysutan2 y EIT , (40) where E = c2 + (utany)2. Substituting equations (13), (37), and (38) into equation (14), the meshing function is given by Cp = Nc3) -l(AsinB+BcosB+C), I/ /I (41) where A = C(T -CL,) + u (tanr(, + a!,) - csec2yw,) + u2 tanysec2yw, B=c(7,+dWy)-U(tanY(7y-_dWI)-csec2YWy)+U2tanysec2yw, C = -u tan2 77,. The equation of meshing is given by Q = 0 or x3uz + y3uzI - u tan2 yv, = 0. (42) From equation (41), the equation of meshing is a quadratic equation when the curvilinear coordinate u is considered as an unknown parameter. If the curvilinear coordinate u appears in the equation of meshing, u can be solved to be function of t and 0 for the contact line. Furthermore, the limit function of the second kind is given by Qt = NC31 - (AtsinB+BtcosO+Ct), II II (43) where At = c(iy - e!&) + u (tan r(iz + ckj,) - c set” y&) + u2 tan y sec2 ytiy , Bt = c(iz + dcj,) - u (tan$ iY - ti,) - csec2 yLjy) + u2 tan y sec2 y3, C, = -u tan2 riz. Substituting equation (40) into equation (18), the relative angular velocity in coordinate sys- tem Sn are given by wnz = E-li2( -y3wz + x3wy), (44) Wry = II II N(3) - E-“2 (u tan2 y (xgwz + yswy) + Ew,) . (45) Combining equations (13), (19), and (37), and using equation (42), we have vns = E-12(-y3, + x3vy), (46) Q-la, = II II N(3) ,9-/2vz. (47) Substituting equation (39) and equations (44) to (47) into equation (17), we have - sucew, - susew, + CWZ, C = -r-l secuu, + Sf3w, - CBw, k= r(c +lrCu) -TCu(&rz + vyrr/ + &rz) +r(cSu - dC2L)(v,wy - vYwz) - csec2 UV + &. (62) (63) (64) ROLLERS OF AXIS-SYMMETRIC QUADRIC SURFACES The roller surface may be generated by a planar quadratic curve revolved about its rotation axis. The implicit form of the axis-symmetric quadratic equation in R and z is represented by aR2+bRz+ccz2+dR+ez+f =O, (65) where R2 = x2 + y2. Equation (65) can be expressed in explicit form as follows: R(z) = alz + a2 f (a3z2 + adz + ab)12, (66) where al = -b/(2a), a2 = -d/(2a), aa = (b2 - 4ac)/(2a)2, a4 = (2bd - 4ae)/(2a)2, and as = (d2 - 4af)/(2a)2. Taking the first derivative of equation (66), we have R = al f 2a3.z + a4 2da3,z2 + a42 + a5 * Taking the second derivative of equation (66), we have R” = f -(2a32 + a4)2 a3 4(asz2 + a42 + as)3/2 +( asz2 + a4.z + as)12 (67) Substituting equations (66) to (68) into the equations related to the cam mechanism with roller of revolution surface, the cam profile and curvature analysis for generating surface being axis- symmetric quadric surface can be derived. In what follows, we shall transform the parametric forms of the hyperboloidal surface and the globoidal surface into axis-symmetric quadric surfaces. Considering equation (37) for the hyperboloidal surface, the distance from the point on the surface to z-axis is R(u) = (c” + u2 tan2 7) 12. (6% Besides, let the curvilinear coordinate u be u=z-d. Substituting equation (70) into equation (69), we have (70) R(u) = (a2 tan2 y - 2d tan2 ye + c2 + d2 tan2 7) 12 . (71) Comparing equation (71) with equation (66), the sign before the root symbol is positive and the coefficients al, a2, as, a4, and as are al = 0, a2 = 0, as = tan2 y, a4 = -2dtan2y, a5 = c2 + d2 tan2 y. (72) Curvature Analysis 79 Considering equation (51) for the globoidal surface, we let the curvilinear coordinate u be (73) Further, the distance from the point on the surface to z-axis is R(z) = c f (-z2 + 2d.z + r2 - d2) 12, where r 0 and Iz - dl 0. 80 H. S. YAN AND W.-T. CHENG The principal curvatures and the principal directions of the roller surface are given by = _A C (3) = (-J K2 ) in = -se ce O T ) bn)=O 0 llT. For the sake of convenience, we assume that $1 = wi = const and & = $&WI, where 4; = g. Using equations (A2) and (A4), the components of the relative velocity matrix Wis becomes w, = WY = 0, W% = (4; - l)Wl, rz = -aSf&wl, 7y = (-a2 +b(laa - l)wl, Tz = 0. From equation (13), the relative sliding velocity between surfaces Cs and Ci is V, = (-as& - c (4; - 1) Se) wi, vu = (-aC& + b (4; - 1) + c (46 - 1) CO) WI, u* = 0. From equation (41), the meshing function is given by = (-aS(0+2)+b(f& - l)se)wl. Furthermore, the limit function of the second kind becomes apt =(-ad;C (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) Cf& - acC (e+$2)+bc(&se -($a - i)ce) 2. Example 2. Conical Cam with a Translating Conical Follower Figure 6 shows a conical cam with a translating conical follower. The conical cam rotates about the input axis while the follower translates along the output axis. The angle of rotation 41 is the parameter of motion of the cam, and the translational displacement s2 is that of the follower. In the meantime, let si = 0 and $9 = 0. The twisted angle from the input axis to the output axis is Q! and a = 0 for the two axes being intersected. Due to the rotation axis of the roller intersected with and perpendicular to the output axis, the distance b = 0 and the twisted angle p = 12. The distance from the origin of the coordinate system Ss to the apex of the conical roller is d and the angle between the rotation axis and the generating line of conical surface is y. And, the specified displacement relation is s2 = ss($i). Figure 6. Conical cam with a translating roller-follower. Curvature Analysis 81 For a conical roller, the value of the parameter c of equations (37) and (38) is zero. Therefore, the coordinates of the conical roller surface and its unit normal in coordinate system Ss are given (3) 1 r3 = us8 tany -uC0tany d+u llT, I 7-p = SBC -cec+y -SylT, where u 0, d 0, and 0 y 7r/2. The principal curvatures and the principal directions of the roller surface are given by KY = -(utanysecy)-, (3) = 0 It2 in = CO se O T ) jn = $SSy -CBSy C+ylT, For the cam mechanism, & = 1, sr G 0, 42 z 0, Q2 = s&r, and i2 = swT, where sb and si are the first and second derivatives of s2 with respect to 41, respectively. Using equations (A2) and (A4), the components of the relative velocity matrix Wrs becomes w, =o, WY = -w1ca, wz = WlSQ, r, = -w1s2sa, Ty = WI&, Tz = 0. From equation (13), the relative sliding velocity between surfaces C3 and Cl is UX I 1 Scy( -s2 + UC8 tan 7) - Ca(d + u) %f = wr s& + US&B tan y v.z ucde tan y From equation (41), the meshing function is given by = - ( s2SaCy + dCcvCy + uCa set y) SO + yZB wl. Furthermore, the limit function of the second kind becomes cpt = (-de - gee) c7-f:. From equations (48) to (SO), th e coefficients and C, and the limit function of the first kind are given by C = (82% + (d+ usec27)ca)c - 4s) utanwylsecy, =chse, Q=-sg-( s2Scy + (d + u see2 y) Ccy) 2 + ( (SSIY + (d + usec2 7) Co) ce - s;SO) 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 = &(&). Equations (37) to (40) g ive the position vector, the unit normal, the principal curvatures, and the principal directions for the hyperboloidal surface. The components of the relative velocity matrix Wra are given as follows: % = -ws42, wy = 4&l, wz = WlG52r 72 = 0, TV = wla, rz = 0. The relative sliding velocity is given by 4a(d + u) - (cS8 - 60 tan y)C& a + (d + u)S& + (cc8 + uS8 tan y)C& . -(cSB - uC0 tan y)Sf#9 - &(cCe + r&3 tan 7) 1 From equation (41), the meshing function is given by = llN(3)i1-1 (Asine + Bcos8 + C), where A = w1 (c (a + dS&) + u (d tan y& + csec2 $742) + u2 tan 7 set” 74;) , B = w1 (cdq5 - 21 (tan 7 (a + dS42) - c sec2 7959 - U2 tan 7 sec2 7S42) , c = 0. From equation (42), the equation of meshing is given by ysC2 + (234; + y32) (d + 215X” 7) = 0. Furthermore, the limit function of the second kind is given by !Bt = II II Nt3) -1(AtsinB+Btcosf3+Ct), a3 where Curvature Analysis At = W: (cdC& + 1 (dtan-& + csec2 C&F&) + u2 tan7 sec2 74;) , Bt = wf (cd& - u (d tan yC& - csec2 y&) - u2 tan y sec2 yC&) , c, = 0. The function & is also expressed as & = w; l(N(3)11-1 ( (z3$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 +