滾齒凸輪製造組裝預壓條件之分析與設計

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(2) !"#$%&'() Analysis and Design of Pre-load Condition for Manufacturing and Assembling on Roller Gear Cam NSC-88-2212-E-009-007 87 8 1

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(8) DE56F. . Abstract Positive-drive spatial cam mechanism, the roller gear cam mechanism, with. followers and their mating cam surfaces is. mechanical. and. influences of two main mechanical errors,. investigated for preload condition analysis in this paper. In the traditional model, cutters generate the roller gear cam surfaces with the same geometrical shapes as the mating roller followers. In this proposed model, cylindrical roller followers are replaced by crowned cylindrical roller followers with hyperboloidal shapes as the spatial cam surface mating part. The meshing condition between two crowned roller . the manufacturing errors and assembly errors, on preload conditions are studied and discussed in three-dimensional space. This study can be a guide for adjusting the preload condition when a spatial cam mechanism is assembled with mechanical errors.. errors. is. modeled. Since all industrial products have some mechanical errors, line contacts on cam. PQ!YWZ[$.

(9) . modeled and analyzed by means of dual surface contact analysis (DSCA). The. Keyword\Roller Gear Cam Mechanism, Preload, Dual Surface Contact Analysis surfaces often become point contacts (Rothbart H. A., 1956). When the contact point moves to the edge of the cam or the.

(10) follower, it becomes edge contact, and can. generated. cause undesirable conditions such as stress concentrations and excessive wear on the. geometrical shapes as the mating roller followers. In this proposed model,. contacting edges of the cam and its mating. cylindrical roller followers are replaced by. follower. Because adjustment of the relative position of the turret and the. hyperboloidal roller followers of the cam mechanism.. globoidal cam can introduce assembly. The mathematical model of the cam. errors, edge contact often occurs as a result of adjusting the driving or driven shafts of a. mechanism was first derived from the desired displacement function by using. cam mechanism for proper preload.. theory of gearing (Litvin, 1994) and. In. by. cutters. with. the. same. order to reduce the occurrence of edge. differential geometry theory.. contact, the outer rolling surfaces of cylindrical roller followers are crowned. surface contact analysis (DSCA) is used to investigate the preload condition of. slightly, on the order of R=500 mm. positive-drive spatial cam mechanisms.. (MISUMI, 1993) along the axial direction in actual industrial applications. These crowned surfaces are helpful in minimizing. The theory of surface contact analysis was originally proposed for gearing, and is called tooth contact analysis (TCA) (Litvin,. the disadvantages caused by misalignments in cam mechanisms (Rothbart H. A., 1956). Many types of roller followers can be. 1994) in that field. Different from TCA and SCA, DSCA considers dual contact conditions on upper and lower cam surfaces. selected for cam mechanisms. Generalized mathematical expressions of. simultaneously. The manufacturing errors on cam profiles will result when grinders. surface geometry for globoidal cams with cylindrical, conical, and hyperbolic meshing roller followers (Yan and Chen, 1996) have been proposed. In this paper, an other type of roller follower, a crowned cylindrical roller follower with a hyperboloidal shape is chosen as the mating part. Two roller followers on the turret of the roller gear cam contact, respectively, the upper and the lower surfaces of the globoidal cam. In the traditional model, the cam surfaces are !"#$ The symbol Σ cu(l) , used later, represents. with inaccurate radii are used. As with manufacturing errors, assembly errors can also cause different preload conditions in cam mechanisms with positive-drive follower motions. Such mechanical errors are defined and analyzed using DSCA in this study. The influences of mechanical errors and follower crowned radii on preload condition are analyzed and demonstrated using three numerical examples.. the geometrical surface of the upper/lower cam surface in the X2-Y2-Z2 coordinate system, and was derived by applying the. The dual. theory of conjugate analysis and differential geometry (Wang et al., 1992) as Σ cu(l) =.

(11) The new calculated follower positions δ u ( φ) and δ l ( φ) must satisfy the.  Lu(l) cosSu(l) cosφu(l) +r sinSu(l) cosθu(l) cosφu(l) -    X cosφu(l) +r sinθu(l) sinφu(l)     - Lu(l) cosSu(l) sinφu(l) - r sinSu(l) cosθu(l) sinφu(l) + .   X sinφu(l) +r sinθu(l) cosφu(l)     Lu(l) sinSu(l) - r cosθu(l) cosSu(l)        . . ' φu(l), ζ, uu(l), ρu(l), δu(l), X ).. Since each two contact surfaces between two followers and their mating cam surfaces must be in continuous tangency, that is, their position and normal vectors must coincide at any instant in the same coordinate system, the two surfaces ( Σ cu(l) , Σ hu(l) ) and their unit normal vectors ( n cu(l) , n hu(l) ) were transformed into the same fixed coordinate system Xf-Yf-Zf with manufacturing and assembly errors as The. surface contact condition between the upper roller and its mating cam surface is decided by following equations: Σ. cu f. (Lu, φu, ζ) = Σ. hu f. (uu, ρu, δu),. n cu (Lu, φu, ζ) = n hu (uu, ρu, δu), f f n cu f. =. n hu f. =. 1.. (13) The equations to determine the surface contact condition between the lower roller and its mating cam surface are as follows: Σ clf (Ll, φl, ζ) = Σ hlf (ul, ρl, δl), n clf (Ll, φl, ζ) = n hlf (ul, ρl, δl), n clf = n hlf. = 1.(14). (15). Eqs. (13)~(15) yield eleven independent scalar equations with twelve unknowns (Lu(l),. . ( Σ cu(l) , Σ hu(l) ) and ( n cu(l) , n hu(l) ). f f f f. constraint V = δ u ( φ) − δ l ( φ) .. Variable. X ' (φ ) denotes the new relative distance between the two rotation axes of the globoidal cam and the turret in DSCA. Eleven unknowns (Lu(l), φu(l), uu(l), ρu(l), δu(l),. X ' ) can only be determined by the above eleven scalar equations when the cam rotation angle ζ is given. Thus, the contact conditions between two followers and their mating cam surfaces can be sequentially ' calculated after the values of X (φ ) have been determined by DSCA at different cam rotation position. The preload condition index ∆X( φ) can be derived by ∆X( φ) = X ' ( φ) − X .. (16). The mechanical errors will cause different contact conditions when the cam rotates to different position. The cam and the turret will attempt to push away from each other when the preload exists in the assembly. Because all the components of the mechanism are assumed to be rigid, the new assembly should increase the relative distance between two rotation axes to avoid preload, and the preload index becomes positive. On the contrary, the relative distance between two rotation axes should be reallocated closer in the mathematical model if backlash appears in real assemblage. This will obtain a negative.

(12) preload index. Example 1 : Influences of Different Manufacturing Errors on Preload Condition Example 2 : Influences of Different Assembly Errors on Preload Condition In industrial applications, the radius of the crowned roller surface Rc is set at 500 mm. In this example, the effect of crowned radius Rc on preload condition is studied. It contains the three main combined mechanical errors in Example 1, the manufacturing error ∆r =-0.05 mm, the translational error ∆ z =0.1 mm and the o rotational error ∆γ y =0.1 .. shown in Fig. 4.. The data for this example. are the same as those in example 2, except that radius Rc is set to three different values (100 mm, 500 mm, 1000 mm). In Fig. 4, the preload index increases when the crowned radius Rc is raised. Another result indicates that the variation in preload condition near the dwell period becoming larger as the crowned radius Rc is increased.. It is concluded that preload. condition is less sensitive to the same mechanical errors when a smaller Rc is chosen.. The results are.

(13) In this paper, two crowned cylindrical roller followers with hyperboloidal shapes were selected for mating with upper and lower surfaces generated by a cylindrical cutter in the oscillating-type of roller gear cam mechanism. The line contact between each pair of contact surfaces became point contact when crowned cylindrical roller followers were substituted for the cylindrical ones. These hyperboloidal roller followers are superior to conventional cylindrical roller followers because they provide the 1. Angeles, J. and Lopez-Cajun, C. S., Optimization of Cam Mechanisms, Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands, pp. 10 (1991).. capacity to eliminate the edge contact caused by assembly errors when adjusting the preload in the cam mechanism. Because the shape of the crowned surface on the hyperboloidal roller follower is one segment of a circle, the cutting mechanism and manufacturing processes for crowning the cylindrical roller followers do not need to be changed. These kind of crowned roller followers have been widely used in industry. Industrial demands have been considered in this geometrical model of spatial cam mechanism. 2. Ardayfio, D. D. and Trower, P. S., "Kinematic Analysis of Three Dimensional Cams, " ASME Design Engineering Division Conference and Exhibit on Mechanical Vibration and Noise, Cincinnati, Ohio, pp. 1-11.

(14) (1985).. Chia-Yi, Taiwan, ROC, pp. 171-179. 3. Chakraborty, J. and Dhande, S. G., Kinematics and Geometry of Planar. (1995). (in Chinese) 12. Tsai, Y. C. and Cheng, M. S., "An Study. and Spatial Cam Mechanisms, New. on Thrust Force on Roller Gear Cam,". York: Wiley (1977). 4. Chen, F. Y., Mechanics and Design of. Proceedings of the 12th National Conference of the CSME, Chia-Yi,. Cam. Mechanisms,. York:. Taiwan, ROC, pp. 181-188 (1995). (in. Cam. Chinese). 13. Wang, W. H., Tseng, C. H. and Tsay, C.. Mechanisms, Published by Mechanical. B., "Analytical Design of a Spatial Cam. Engineering Publications Ltd. for the. Profile,". Institution of Mechanical Engineers, London & Birmingham, Alabama. National Conference of the CSME, Kaohsiung, Taiwan, ROC, pp. 297-304. (1978).. (1992). (in Chinese).. Pergamon Press, (1982). 5. Jones, J. R., Cams. New and. Proceedings. of. the. 9th. 6. Litvin, F. L., Gear Geometry and Applied Theory, NJ: Prentice Hall (1994).. 14. Wang, W. H., Tseng, C. H. and Tsay, C. B., "Optimum Design of a Spatial Cam Mechanism," Proc. of the Int. Conf.. 7. MISUMI, MISUMI Production Catalog 1993-1995, 4-43, 2-Chome, Toyo, Koto-Ku, Tokyo, 135 Japan, MISUMI. Modeling, Simulation and Optimization, Gold Coast, Australia, Paper #: 242-068 (1996).. Corporation, pp. 509-514 (1993). 8. Oizumi, T. and Emura,. T.,. "Globoidal-cam Type Gearing," Proc. 92 Int. Power Transm. Gearing Conf. ASME Design Engineering Division, New York, Vol. 43(2), pp. 535-541 (1992). 9. Rothbart, H. A., Cams Design, Dynamics and Accuracy, New York: Wiley (1956). 10. Tsai, D. M. and Huang, N. J., "Geometrical Design of Roller Gear Cam Reducers," Power Transm. and Gearing Conf. ASME, DE-Vol. 88, pp. 153-160 (1996). 11. Tsai, Y. C. and Chang, C. S., "An Analysis on the Contact Path of Roller Gear Cam," Proceedings of the 12th National Conference of the CSME,. 15. Wang, W. H., Tseng, C. H. and Tsay, C. B., "Surface Contact Analysis for a Spatial Cam Mechanism," ASME Journal of Mechanical Design, Vol. 119 (March 1997). 16. Yan, H. S. and Chen, H. H., “Geometry Design and Machining of Roller Gear Cams with Cylindrical Rollers,” Mech. Mach. Theory, Vol. 29, No. 6, pp. 803-812 (1994). Yan, H. S. and Chen, H. H., “Geometrical Design of Globoidal Cams with Generalized Meshing Turret-rollers,” ASME Journal of Mechanical Design, Vol. 118 , pp. 243-249 (1996)..

(15) . Z f(h). Y1(u). Z2 ,Zf. Y1(l). Yf(h). RT -Xf(h). V. O. 1. Y2 Y f. d r. Su(φ) Σr S l(φ). Σcu. X1(u). φ. X2. φ. So. Xf. Of ,O2. Σcl X1(l). X. . Fig.1. The coordinate systems of the cam mechanism. (h) Zf(h). Zf ∆γ z (h) Yf(h) (h). Xf(h) Turret. ∆γy Yf y ∆ ∆z ∆x Cam. Xf ∆γ x. Fig.2. Definition of possible assembly errors. .

(16) ∆ X(mm). 0.30. ∆γx = 0.1(Deg.) ∆γ = -0.1(Deg.) x. 0.25. ∆γy = 0.1(Deg.) ∆γy = -0.1(Deg.). Preloading Index. 0.20. ∆γ z = 0.1(Deg.). 0.15. ∆γz = -0.1(Deg.). 0.10 0.05 0.00 -0.05 -0.10. φ (Deg.). 0. 30 60. 90 120 150 180 210 240 270 300 330 360 Cam Rotation Angle. Fig.3. Influences of rotational errors on preload condition. . .

(17) ∆ X(mm). 0.50. Rc = 1000(mm) Rc = 500(mm). 0.45. Rc = 100(mm). 0.40. Preloading Index. 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00. φ (Deg.). 0. 30. 60 90 120 150 180 210 240 270 300 330 360 Cam Rotation Angle. Fig.4. Influences of crowned radius Rc on preload condition.. .

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