Paper received: 2012-03-28, paper accepted: 2012-09-18 © 2012 Journal of Mechanical Engineering. All rights reserved. Design of Space-Curve Meshing-Wheels with Unequal Tine Radii Jiang Ding -Yangzhi Chen* -Yueling Lv South China University of Technology, China In some recent papers a new gear mechanism named Space-Curve Meshing-Wheel (SCMW) has been proposed by the present authors. However, the research presented was limited in the equal tine radius case. This paper presents a method to solve the SCMW design with unequal tine radii. As a consequence, with a pair of contact curves, the tine radii of the driving and driven wheels can be selected independently according to actual need. A design example is illustrated in detail and testified in both simulation and practical experiment. This method provides a design fundamental for the SCMWs to optimize the tine shapes according to strength conditions, and therefore it can extend their application in industrial. Keywords: gear, Space-Curve Meshing-Wheel, unequal tine radius, contact curve 0 INTRODUCTION A gear mechanism named Space-Curve Meshing-Wheel (SCMW) was invented by Chen et al. [1] to [3]. The SCMW is based on the theory of space curve meshing instead of traditional space surface meshing [4] to [8]. Space curve meshing is a transmission through continuous point contacts between two conjugate curves. With a large transmission ratio, small size and light weight, SCMW is highly convenient to be applied in the transmission between two intersecting axes in the small space inside micro machines. After recent research on space curve meshing equations [1] to [3], contact ratio [9], design criterion [10] and manufacture technology [11], the SCMW has possessed an integrated application like micro reducers [12]. However, the research published was limited in the equal tine radius case. Furthermore, the space curve meshing equations obtained at the meshing point were relevant to the tine radii, and therefore the wheel pair of the SCMW must be in a one-to-one correspondence. But in practical application, the demanding tine radii of the driving and driven wheels are not always the same. The tine radii depend on the demanding strength corresponding to their working conditions. If the tine radii of the driving and driven wheels are irrelevant from each other, they can be designed independently. This paper presents a design method to make the tines of SCMW irrelevant in the unequal tine radius case. To testify the method, an example is designed, and both simulation and practical experiment are carried out with the same data in the example. 1 DESIGN FUNDAMENTAL The essence of the space-curve meshing is transmission through continuous point contacts between two conjugate curves [1]. To provide contact curves smooth and slight objectives like tines, for instance, should be designed. For reasons of simplicity, cylindrical tines have been adopted in this paper as in previous papers [2] to [3]. As shown in Fig. 1, two invariant circles are tangent at the meshing point each moment. The circle centers are along the contact vector direction and at the opposite sides of the meshing point (indicated as -y1 and y1 in Fig. 1). After the curves finish the whole meshing, point after point, the two circles form the cylindrical shapes of the driving and driven tines, respectively. In Fig. 1, M is the meshing point at each moment, while M(1) and M(2) are the corresponding points at the driving and driven tines, respectively. The point set of M(1) is the driving contact curve; the point set of M(2) is the driven contact curve. According to the design depicted above, for the same contact curves, the driving and driven tines can adopt circles with different radii, attaining the cylindrical tines with different radii. Even the tines with variable radii can be designed in this way. As the *Corr. Author's Address: School of Mechanical and Automotive Engineering, South China University of Technology, Wushan RD.,Tianhe District, Guangzhou, China, meyzchen@scut.edu.cn circles are always in opposite directions at the meshing point, the driving and driven tines will be irrelevant from each other. In theory, the radii of the tines are free to choose as long as there is enough room. In practical application, the tine radii are determined according to the strength conditions. 2 DESIGN METHOD 2.1 Space Meshing Coordinates The equations of the contact curves and the center curves are all calculated in the space meshing coordinates, which are established as Fig. 2. The fixed coordinates for the driving and driven wheels are given as o -xyz and op -xpyp zp , respectively. Planes xp opzp and x o z are in the same plane. Denote the distance from point op to axis z as a and the distance to axis x as b. Included angle between axis z and axis zp is (n-6), where 6 is the included angle between the angular velocity vectors of the driving and driven wheels. o1-x1y1 z1 and o2 - x2y2 z2 are relatively static with the driving and driven wheels. They are rotating coordinates with respect to o - xy z and op - xpyp zp , respectively. At the initial moment, oj -xjyj zj coincides with o -x yz , while o2 - x2y2 z2 coincides with op - xp yp zp . At any moment, point oj coincides with point o, and axis zj coincides with axis z; point o2 coincides with point op, and axis z2 coincides with axis zp. After meshing begins, oj - xjyjzj rotates around axis zj, while o2 - x2 y2 z2 rotates around axis z2. \zp(zi) z(z,) cozb) // V a \ /6p(OT) Xl 1 9 /f I \ b —........— o(oi) H yp' yi Fig. 2. Space meshing coordinates The transformation matrix from oj - xj yj zj to o - xy z is as Eq. (1) [13]. Mol = cosp1 sinp1 0 0 - sin p1 cosp1 0 0 0 0 10 0 0 0 1 M21 = The transformation matrix from oj - xj yj zj to o2 - x2 y2 z2 is as Eq. (2) [13]. - cos (p1 cos p2 cos 9 - sin p1 sin p2 cos p1 sin p2 cos 9 - sin (p1 cos (p2 cosp1 sin 9 0 - sin (p1 cos (p2 cos 9 + cos (p1 sin (p2 sin (p1 sin p2 cos 9 + cos p1 cos (p2 sin (p1 sin 9 0 - cos (p2 sin 9 -a cos (p2cosd + b cos p2 sin 9 sin (p2 sin 9 a sin (p2cosd - b sin (p2 sin 9 - cos 9 a sin0 + b cosd 0 1 (2) As shown in Fig. 2, suppose: w1 and w2 are the angular velocities of the driving and driven wheels; and are the rotation angles of the driving and driven wheels after the meshing begins; i12 is the transmission ratio. The kinematic relations are obtained as Eqs. (3) and (4): - i 12 w2 , 91 - i12 92 . (3) (4) 2.2 Kinematical Equation According to [1], the motion at the meshing point should satisfy the kinematical equation as Eq. (5): vl2P - 0, (5) where v12 is the relative velocity at the meshing point "A between the driving and driven tines, and p = A P; is the unit normal vector of the driving contact curve in o - xy z. Suppose that yffW is the unit normal vector of the driving curve in o1- x1 y1 z1. Considering Eq. (1), we get the equation of the unit normal vector as Eq. (6): P = P(jl cos^j + fíf1 sin^j -ft® sin^j + py cos^ p« }(j), (6) The velocities at the meshing point of the driving and driven tines are as Eqs. (7) and (8): yu^i yMm2 cos 6 -(zM -b)ct2 sind-(xM + a)cr2 cos0 yMw2sinO (7) (8) The relative velocity at the meshing point is as Eq. (9): V12 = Vi - V2. Considering Eq. (1), we get: xm = xM cos^i + yM sin^i y m = - xM sin^i + yM cosVi. z = zW M M (9) (10) Substituting Eqs. (7), (8) and (10) into Eq. (9), relative velocity at the meshing point as Eq. (11) is obtained: (-x^ sm^j + y« cos^!)( -w2 cos0) cosp + y« sin^ij ) + ( -b) sin0 + ( cos^ + y« sin^ + a) cos0 -(-x« sin^ + y« cos^) sin0 (11) vi = 0 V2 = V12 = Substituting Eqs. (6) and (11) into Eq. (5), the kinematical equation at the meshing point as Eq.(12) is obtained: (W- ^P?) -®2 cos6) + +am2 cos 6 (-pj(l sinp + pj-11 cosp) + + ( - b)m2 sin6 (Py(i) cosp - P(j) sinp) + + (( sin^j -yM cosp ) sin6(pz(l)) = 0. (12) From Eq. (12) above, the relationship between and t can be gained, and then the equations of the contact curves and the center curves can be derived. As neither r1 nor r2 exists in Eq. (12), we can conclude that the kinematical equation is only related to the properties of the driving contact curve (xM, sm, ZM, , Py1} and ) and the kinematic relations (a, b, 6, wh w2, and between the driving and driven contact curves, but not the tine radii. (i) (i) 2.3 Equations of Contact Curves Denote the matrices of M(1) in o1 - x1 y1z1 and M(2) in respectively. The r x(i) i r x(2)i 02 -x2y2 z2 as yM and (2) yM z(2) Z M equation of the driving contact curve is given as Eq. (13): XM = XM ) (13) Considering Eqs. (2) and (13), the equation of the driven contact curve as Eq. (14) can be obtained: x(2) X). -smpsmp2 )xM M -(-cosp cospcostí + (- sin p cos p2 cos tí + cos p sin p>2) yM - cos tp2 sin OzM - a cos p2 cos tí + b cos p>2 sin tí yM - (cosp sinp2 costí - sinp cosp2 'M + + (sin p sin p>2 cos tí + cos p cos p>2) yM + . + sin p>2 sin tízM + a sin p2 cos tí - b sin p>2 sin tí zM - cos p sin tíxM + sin p sin tíyM - cos tízM + +a sin tí + b cos tí (14) 2.4 Equations of Center Curves Denote y and yW as unit binormal vectors of the driving curve in o-xyz and o1 - x1y1z1, respectively. As shown in Figs. 1 and 3, at every meshing moment, if each meshing point M(t) moves a distance of r1 in the direction of -yW, the corresponding point in the driving center curve is obtained; if each meshing point M(t) moves a distance of r2 in the direction of y(t), the corresponding point in the driven center curve is obtained. The tine radii as Eqs. (15) and (16) can be derived: 1 (t) = -r{y (t) = MMi, r2 (t) = r2y(t) = MM2 Denote M1 and MM1 in o1 - x1 y1z1 as xW MM1 yMM1 z(l) x(2) MM 2 (2) yMM2 z(2) M2 and MM2 in o2 - x2y2 z2, as x(j) M1 yMÎ z(j) x(2) ' (2) y m1 z(2) (15) (16) and and , respectively. The vector addition triangles are shown in Fig. 3. From Fig. 3, Eqs. (17) and (18) can be derived. (17) X(1) " M1 \ X(1) 1 M ' X(1) MM1 /i\ = yM + /mm, Z(1) M1 z (1) * M Z(1) * MM j x(2)l r x(2) " M r x(2) mm2 (2) yM2 = (2) yM + (2) yMM2 Z (2) m2 z (2) Z M Z (2) * MM2 (18) According to Eq. (17), the equation of the driving center curves is as Eq. (19): = $ + -x(l] MM1 /Mi) = /Mi + /MM1 ■ (19) z (1) M1 =® + -z(i MM1 According to Eqs. (2) and (18), the equation of the driven center curves is as Eq.(20): x(2) - cos % cos (p2 cos 0 - sm « + r(i) 'M MM, + ^A-fAi I + +(-sinç>i cos^2 cos0 + cos^i sin^2)( + yMM2 - cosP2sin0)ZM)+ 4M2 ) + +)-acosp2 cos0 + bcosp2 sin0) yM2 = )c( sin^2cos0 - sin^i cos^2 )((+xMM2 )+ +(sin Pisin cos o+cos %cos ^2 ) ( ( + y(M2 )+ + sinP2sm0(zM)+ zMM2 ) + +(asin%2 cos9 -bsin%2 sin9) zM2 =cos %sin 9 (( +xMMi2 )+ + sin% sin e(yMl+ yMM1 )- - cos 8(zM + zMMMl ) + ( a sin9 + b cos 9) (20) OMl=OM+MMl Fig. 3. Vector addition triangles 3 DESIGN EXAMPLE 3.1 Equation of Driving Contact Curve In the reference [1] to [3], a helix curve was given as an example. In comparison, the same curve is adopted as the driving contact curve, as shown in Fig. 4a. The Equation of M(1) in o1 - x1 y1 z1 is as Eq. (21): yM = m sm t zM = nn + nt -n< t <-- (21) where m is the helix radius of the driving curve; n is the pitch parameter of the driving curve, denoting the pitch as p, n = p / 2n; t is the parameter indicating the scope of the helix curve. -n < t < -n /2 means a quarter circle of the helix curve. When t = - n, the driving and driven tines begin to mesh; when t = -n / 2, the two tines begin to separate. The lengths of the driving and driven contact curves are directly controlled by the scope of t as needed. b) Fig. 4. a) Driving and b) driven contact curves The unit normal vector and the unit binormal vector of the driving curve are derived as Eqs. (22) and (23). - cos t fif = - sin t 0 (22) yM = Y«' V« ' y V« n sin t V2 . 2 n + m n cos t V2 , 2 n + m V2 , 2 n + m (23) 3.2 Kinematic Equation Considering -n < t < -n / 2 and 0 < the relationship between and t can be deviced: = t + n. 3.3 Equation of Driven Contact Curve (25) According to Eqs. (4), (14), (21) and (25), the equation of M(2) in o2 -x2y2 z2 is as Eq. (26): XM = [(m - a) cos 0 -(trn + nt - b)sin0] ■ cos t—+L hi (2) yM -[(m - a)cos0 -(nn + nt - b)sin0] ■ sint + n . (26) hi zM = -(m - a)sin0-(nn + nt - b)cos0 Obviously, Eq. (26) is a conical helix curve, as shown in Fig. 4b. 3.4 Equations of Center Curves According to Eqs. (15) and (16), the radii of the driving and driven tines are as Eqs. (27) and (28) in 01 - X1 y1 z . xW MM1 yMM1 z (1) XW MM2 y(MM2 zM = -riy (i) (27) = riY (i) (28) From Eqs. (19), (21), (23) and (27), the equation of the driving center curve is derived as Eq. (29): m r n sin t xM = m cos t —/ Vn + m m . rncos t yM = m sin t + - n (29) (1) o rm zM = nn + nt - V2 , 2 n + m Substituting Eqs. (21) and (22) into Eq. (12), the kinematic equation as Eq. (24) is obtained : m2 sin((p1 -1)[acos0 + (nn + nt - b)sin0] = 0. (24) From Eqs. (4), (20), (21), (23), (25) and (28), the equation of the driven center curve is derived as Eq. (30): m (m - a )cosO-| nn + nt - b + r2 m Jn2 sinO t + n r2 n . t + n • cos —— + . 2 - sin —— z'i2 Vn2 + m2 ^12 12 (m - a )cosO-| nn + nt - b + , 2 - I sinO V Vn2+ m2 ) • (30) . t + n r2 n t + n • sin —--+ . 2 - cos —— z'i2 vn2 + m2 z'i2 zm\ =-(m - a)sinO-| nn + nt -b + yjn2 + m2 Comparing Eqs. (21) and (29), it can be noticed that the driving center curve only has a relationship with the driving contact curve and the driving tine radius (r1); comparing Eqs. (26) and (30), the driven center curve only has a relationship with the driven contact curve and the driven tine radius (r2). That is to say, the driving and driven center curves are irrelevant from each other and either r1 or r2 can be selected according to their own demanding strength. 4 VIRTUAL SIMULATION The equations of the driving and driven contact curves are unique if and only if the following six parameters are given: m, n, a, b, 6 and i12. During the virtual simulation, the same pair of contact curves as in the example are used. The six parameters are selected as below: m = 5 mm, n = 4 mm, a = 24 mm, b = 10 mm, 6 = 120° and i12 = 3. Substituting the parameters into Eqs. (21) and (26), the equations of the driving and driven contact curves as Eqs. (31) and (32) are obtained. The virtual simulation is shown in Fig. 5. xM = 5 cos t y{u= 5sin t zM = 4n + 4t -n< t <- — (31) XM y (2)=- M 19(4^ + 4t -10) cos- 19(4n + 4t -10) t + n t + n sin- ". (32) 42 >= ^ +1 (4n + 4t -10) Fig. 5. Simulation of the contact curves Substituting the parameters into Eqs. (29) and (30), the equations of the driving and driven center curves as Eqs. (33) and (34) are obtained. (i) 4k sin t x\J = 5 cos t -- 1 yM = 5sin t + 4r1 cos t vfTF (33) zM = 4n + 4t - 5r V42 + 52 x(2) = M2 19 _ V3 2 2 4^ + 4/ _ 10 + 5r2 • cos- t + n 4r2 (2) _ yM2 _ " 3 V42 + 52 19 2 2 sin- V42 + 52 t + n 3 4n + 4t _ 10 + 5r V42 + 52 . t + n 4r2 (34) • sin- V42 + 5: - cos - t + n z(2) -M2 i 4n + 4t -10 + 5r2 Once the values of r1 and r2 are determined, there are unique center curves corresponding with the contact curves. Therefore, the shape of the driving and driven tines can be simulated in Pro/E. In the industry, the radii of the tines are given according to the actual need. However, to testify to the irreverence of the driving and driven tines, both r1 and r2 with various values are chosen and matched as pairs. To avoid interference, the radii would be assured to be in a reasonable range, which can be easily guaranteed during the simulation. Suppose the numbers of the driving and driven tines are z1 and z2, respectively. If z1 = 5, according to the definition of the transmission ratio, we derive that z2 = 15. As the numbers of the driving and driven tines are defined, then the shape of the SCMW is fixed, as shown in Fig. 6. Fig. 6. Simulation of the driving and driven wheels The meshing of the SCMW is shown as in Fig. 7. The simulation results show that the tines always mesh at the same pair of contact curves and that the value of neither r 1 nor r2 will affect the meshing. Fig. 7. Simulation of the SCMW 5 KINEMATIC EXPERIMENT Using the data from the simulation above (also seen in Table 1), we manufacture some SCMW samples to do experiment to testify the irrelevance of the driving and driven wheels. The samples are produced through Selective Laser Melting (SLM) technology [11]. All the driving wheels have the same driving contact curves, as shown in Table 1, but different tine radii (rj), as shown in Fig. 8a; all the driven wheels have the same driven contact curves, as shown in Table 1, but different tine radii (r2), as shown in Fig.8 (b). Table 1. Uniform parameters of SCMW samples m n a b e i 12 ¿1 ¿2 5 mm 4 mm 24 mm 10 mm 120° 3 5 15 The test rig made by our research group [1] to [3] is adopted to do the experiment. The experimental schematic diagram is shown in Fig. 9, and the test rig is shown in Fig. 10. a) r1 = 0.4 mm : 0.5 mm b) r2 = 0.6 mm r2 = 0.7 mm r2 = 0.8 mm Fig. 8. SCMW samples; a) samples of driving wheels, b) samples of driven wheels Fig. 9. Experimental schematic diagram Fig. 10. Test rig rt = 0.6 mm r 1 In the experiment, to keep the pair of contact curves constant and match the driving and the driven wheels with different tine radii each time, continuous records of contact tines at different locations are captured with high quality camera. For example, as shown in Fig. 11 are continuous records of the mesh between the driving tines with r = 0.5 mm and the driven tines with r2 = 0.8 mm. After recording the rotation speeds of the driving and the driven wheels for a while, the corresponding average ratios are calculated. As shown in Table 2, the data in the same column indicate the meshing between the same driving wheel and different driven wheels, while the data in the same row indicate the meshing between the driving wheels and the same driven wheel. errors. It can be reduced by improving manufacturing technics and experiment conditions. It is noteworthy that the lengths of the tines do not affect the error in theory. However, as the tines are made through SLM technology, their manufacturing error accumulates as they become longer. During the experiment, the meshing of the driving and driven wheels with unequal radii in acceptable accuracy was accomplished. From similar continuous records like Fig. 11, it has been confirmed that the tines obtained always mesh at the same pair of contact curves, so the tine radii of the driving and driven wheels are irrelevant from each other. It can be concluded that the method presented is reliable. 6 CONCLUSION Table 2. Average ratio of SCMW with the same pair of contact curves i 12 r1 [mm] 0.4 0.5 0.6 0.6 2.98 3.00 3.01 r2 [mm] 0.7 3.02 2.99 2.98 0.8 3.01 3.00 3.02 From Table 2, the average transmission ratio measured is from 2.98 to 3.02, while the theoretical transmission ratio is 3. The maximum relative error is 0.6%. This relative error is the combined result of manufacturing errors, assembly errors and measure In this paper, a new method based on the contact curves is proposed to design the SCMWs with unequal tine radii. It is illustrated with a design example in detail, and testified with both simulation and experiment. In comparison with existing design methods, the method presented has two obvious advantages as below: 1) The radii of the driving and driven tine do not affect the meshing process, and the radii of the driving and driven tines can be designed independently according to their working demands. 2) It is the theoretical foundation of the tine shape optimization based on the strength condition. j Ëiliwi^uË flÉÉi M ppHH JÉ \ Bu? M * ijg Fig. 11. Continuous records of contact tines at different locations This method possesses several prospects to conduct further study. With this method, even the SCMW with variable radii and invariable stress can be designed. To identify the optimized radii, the relationship between the stresses and the radii is in progress. The shape selection rule would be established afterward. 7 ACKNOWLEDGMENT It is our pleasure to thank anonymous reviewers and editors for their valuable comments and suggestions. We thank the National Natural Science Foundation of China (No. 51175180) for the support of the research presented in this paper. 8 NOMENCLATURE ri Radius of driving tine r2 Radius of driven tine Rotation angle of driving wheel p2 Rotation angle of driven wheel wi Angular velocity of driving wheel w2 Angular velocity of driven wheel vi Velocity of driving tine v2 Velocity of driven tine vi2 Relative velocity ii2 Transmission ratio 0 Included angle between angular velocity vectors a Distance from point op to axis z b Distance from point op to axis z Moi Transformation matrix from oi - xi yi zi to o - xy z M2i Transformation matrix from oi - xi yi zi to O2 - X2y2 Z2 t Scope parameter of helix curve P Unit normal vector of driving curve Y Unit binormal vector of driving curve M Meshing point MO) Point at driving contact curve M(2) Point at driven contact curve Mi Point at driving center curve M2 Point at driven center curve MM Contact vector from M to Mi MM 2 Contact vector from M to M2 m Helix radius of driving curve n Pitch parameter of driving curve p Pitch of driving curve z1 Number of driving tines z2 Number of driven tines (superscript) Corresponding coordinate 9 REFERENCES [1] Chen, Y.Z., Xing, G.Q., Peng, X.F. (2007). The space curve mesh equation and its kinematics experiment. 12th IFToMM World Congress, Besançon. [2] Chen, Y.Z., Xiang X.Y., Luo, L. (2009). A corrected equation of space curve meshing. Mechanism and Machine Theory, vol. 44, no. 7, p. 1348-1359, D0I:10.1016/j.mechmachtheory .2008.11.001. [3] Chen, Y.Z., Chen, Z., Fu, X.Y. (2011). Design parameters for spatial helix gearing mechanism. Applied Mechanics and Materials, vol.121-126, p. 3215. [4] Litvin, F.L. (2008). Gear Geometry and Applied Theory. Shanghai Science and Technology Publishers, Shanghai. (in Chinese) [5] Bergseth, E. Bjorklund, S. (2010). Logarithmical crowning for spur gears. Strojniški vestnik - Journal of Mechanical Engineering, vol. 56, no. 4, p. 239-244. [6] Fetvaci, C. (2010). Generation simulation of involute spur gears machined by pinion-type shaper cutters. Strojniški vestnik - Journal of Mechanical Engineering, vol. 56, no. 10, p. 644-652. [7] Staniek, R. (2011). Shaping of face toothing in flat spiroid gears. Strojniški vestnik - Journal ofMechanical Engineering, vol. 57, no. 1, p. 47-54, D0I:10.5545/sv-jme.2010.093. [8] Puccio, F.D., Gabiccini, M., Guiggiani, M. (2006). Generation and curvature analysis of conjugate surfaces via a new approach. Mechanism and Machine Theory, vol. 41, no. 4, p. 382-404, D0I:10.1016/j. mechmachtheory.2005.07.008. [9] Chen, Y.Z., Luo, L., Hu, Q. (2009). The contact ratio of a space-curve meshing-wheel. Journal of Mechanical Design, vol. 131, no. 7, p. 074501, D0I:10.1115/1.3116343. [10] Chen, Y.Z., Hu, Q., Su, L. (2010). Design criterion for the space-curve meshing-wheel transmission mechanism based on the deformation of tines. Journal of Mechanical Design, vol. 132, no. 5, p. 054502, D0I:10.1115/1.4001535. [11] Chen, Y.Z., Sun, L.H, Wang, D., Yang, Y.Q., Ding, J. (2010). Investigation into the process of selective laser melting rapid prototyping manufacturing for space-curve-meshing-wheel. Advanced Material Research, vol. 135, p. 122-127, D0I:10.4028/www.scientific.net/ AMR.135.122. [12] Chen, Y.Z., Chen, Z., Ding, J. (2011). Space Curve Mesh Driving Pair and Polyhedral Space Curve Mesh Transmission, PCT/CN2010/078294. [13] Li, G.Y. (2007). Spatial Geometry Modeling and Its Application in Engineering, Higher Education Press, Beijing. (in Chinese)