Curvature analysis of roller-follower cam mechanisms.pdf

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 P 13l 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 v l 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 y3 2 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 54 2 ya d 4h 55 42 150 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 Sj3SaS 1 Z3 I c42 CaCd1S42 Pwiw42 CffC41C 2 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 w 2 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 lS wJ2 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 C SaS42 qi 1S2 42Bl 2 aC CcxS bSaS W 2 s C SCYC A5 1 a SaSP CPYC b CaCp S k16 4 s2CPSaSqi2 bC 3 51 CCYSP SCYCPG J i2Sp iz S 3SaSqs2 182 2 41 142 aSpCcuS bSaCPS s2S S aC 1 a SaCP SPCaC b CdV3 CPSaC42 s S M bS 3 lil C j3 SCYS C s2Cp REFERENCES 1 M L Baxter Curvature acceleration relations for plane cams ASME Z unsactions 483 469 1948 2 M Kloomok and R V Muffley Determination of radius of curvature for radial and swinging follower cam systems ASME Transactions 795 802 1956 3 F H Raven Analytical design of disk cams and three dimensional cams by independent position equations ASME IPransactions Journal of Applied Mechanics 18 24 1959 4 S Yonggang Curvature radius of disk cam pitch curve and profile In Proceedings of the ph World Congress on Theory of Machines and Mechanisms pp 1665 1668 1987 5 F L Litvin Theory of Gearin