本研究的目的在於建立分析滾珠螺桿內部元件在操作時的幾何變化相互之間的運動關係，藉以了解滾珠螺桿的各項性能參數(包括滾珠、螺桿及螺帽接觸角、正向力、摩擦力、離心力…等)對於滾珠螺桿機械效率的影響。本研究較以往研究最大的不同點在於以下三點：(1) 成功建立了單圈滾珠與雙圈滾珠單螺帽滾珠螺桿複雜的幾何與運動分析模型，同時分析滾珠螺桿中滾珠、螺桿及螺帽的力與力矩平衡關係，據此求得滾珠螺桿的各項性能參數，包括接觸角、正向力、摩擦力、滾珠離心力…等，藉以分析各項性能參數間的關係及對於滾珠螺桿機械效率的影響；(2)考慮了以往研究並未考慮的重要參數，例如離心力、預負荷及潤滑作用等，建立了更完整的滾珠螺桿分析模型；(3)藉由滾珠螺桿的性能實驗驗證理論分析的正確性，同時得到影響滾珠螺桿機械效率的重要參數，包括螺桿螺旋角及預負荷。
滾珠在滾珠螺桿軌道運動時所受的的慣性力為離心力，在以往的研究中大多忽略離心力的作用而將滾珠與螺桿及滾珠與螺帽接觸面的接觸角、正向力及摩擦係數當成一樣來簡化計算。但是接觸角、正向力及摩擦係數均隨軸向負荷及螺桿轉速的變化而明顯變化，若不考慮此三個性能參數隨軸向負荷及螺桿轉速的變化關係會造成計算模型的誤差增加。因此在本文的分析中在滾珠、螺桿與螺帽的力與力矩關係式中加入離心力，反應出離心力對於各項性能參數的影響。在本文中亦分析出軸向負荷小於及大於預負荷時對於滾珠螺桿各項性能參數的影響，發現當軸向負荷的值遠離(遠大於或遠小於)預負荷時滾珠螺桿能有良好的機械性能。
本文將以單圈單螺帽滾珠螺桿模型的建立以了解滾珠、螺桿及螺帽三者的幾何關係及運動模式，可以分析滾珠螺桿在承受軸向負荷的作用後滾珠、螺桿及螺帽位置及幾何參數的變化，以及滾珠與螺桿及滾珠與螺帽之間的運動關係。同時加入高速離心力的作用以了解離心力對於各項性能參數的影響。再來建立雙圈滾珠單螺帽滾珠螺桿的理論分析模型，引用單圈滾珠螺桿的理論模型同時加入預負荷的效應。由於兩圈滾珠的受力方向及幾何關係都是不同的，在此可分析滾珠螺桿的滾珠與螺桿及滾珠與螺帽接觸面上純滾動點的位置與螺桿轉速及正向力的關係，以及軸向負荷改變時預負荷對於滾珠螺桿的各項性能參數的影響。最後在雙圈滾珠單螺帽滾珠螺桿的理論模型中加入潤滑的效應，並以實驗所得到的滾珠螺桿機械效率結果比較以驗證理論分析的正確性。同時發現軸向負荷接近預負荷時滾珠螺桿的機械效率大幅下降的現象。最後可以得到影響滾珠螺桿性能的參數為預負荷、螺桿螺旋角、軸向負荷及螺桿轉速。當軸向負荷小於並遠離預負荷時，滾珠螺桿的機械效率很高，隨著軸向負荷的增加而接近預負荷時，滾珠螺桿機械效率下降。當軸向負荷大於預負荷時，隨著軸向負荷的增加滾珠螺桿的機械效率增加，當軸向負荷為預負荷值的2.83倍以上時，滾珠螺桿有最佳的機械效率。所以預負荷的值由所使用的軸向負荷的大小決定，軸向負荷必須遠大於或遠小於預負荷方能有較佳的機械效率。螺桿螺旋角越大滾珠螺桿機械效率越差，但是較大的螺旋角可以增加滾珠螺桿的進給速率。滾珠螺桿的機械效率亦會隨著螺桿轉速的增加而減少。為使滾珠螺桿具有最佳的機械效率，必須根據使用的軸向負荷與螺桿轉速來選擇適當的預負荷及螺桿螺旋角。

The purpose of this study is to create an effective method to study kinematics and tribological behavior of a ball screw operating under different operating conditions. The analysis is able to realize how the ball screw parameters including contact angles, normal forces, friction forces and centrifugal force of ball screw affect ball screw’s mechanical efficiency. The main distinctions of the present study from the past studies are given as follows: (1) Single-cycle and double-cycle ball screw’s geometrical and kinematical analyses have been developed successfully. The force and torque balance equations of balls, screw and nut have been developed in the analyses. From the solutions of these equations we can get all mechanical performance parameters (including contact angles, normal forces, friction forces, ball’s centrifugal force… et. al.) of a ball screw. The influences of these parameters on the ball screw’s mechanical efficiency are analyzed in the present study; (2) several important factors are considered. These parameters include the centrifugal force, the ball screw’s preload and the lubrication effect. The consideration of these factors can create more perfect ball screw results; (3) the ball screw’s experimental data can verify the ball screw model developed in the present study, and can investigate several important parameters (including the helix angle and the preload, etc.) how they affect ball screw’s mechanical efficiency.
When balls moving in the raceways of a ball screw at high rotational speeds, the centrifugal force acts in the each ball. In past studies, they did not consider the effect of the centrifugal force. They all assume the contact angle, the normal force and the friction coefficient at the ball-screw and ball-nut contact areas are invariant with the each other. Actually, the contact angle, the normal force and the friction coefficient are varying significantly with the axial force and the screw’s rotational speed. The present model finds the effect of the centrifugal force on the ball screw’s performance parameters to be significant. The present model can also find the influence of the preload on the ball screw’s mechanical performance parameters. When the axial load far away (larger and smaller) from the preload is applied, the ball screw shows the better mechanical efficiency.
In this study, single ball cycle-single nut ball screw without the preload is analyzed first. From the model developed in this section one can investigate the geometrical parameters and the contact angles formed at the two contact areas varying with the operating conditions when an axial load is applied to a ball screw. This model also evaluates the influence of the centrifugal force on the mechanical performance parameters. In the second section of this study, a double cycle-single nut ball screw model is established and then analyzed. The analysis in this section has used the concept of single cycle-single nut ball screw method, and the preload applied in the system has been considered too. Because the force direction and the geometrical behavior exhibited at two cycles of balls are different, the pure-rolling points at ball-screw and ball-nut contact areas are varying with the screw’s rotational speed and the normal force. When the axial load is increased, the present model also can find the influence of the preload on the ball screw’s performance parameters. The lubrication effect is considered in double cycle-single nut ball screw model. The ball screw’s experimental results can be compared with that predicted by the theoretical model, and verify the accuracy of the present model. By the present model the ball screw’s mechanical efficiency is decreased greatly when the applied axial load is close to the preload. The preload, the helix angle, the screw rotational speed and the axial load are the four influential parameters to the ball screw mechanical efficiency. When the applied axial load is lower than the preload and is far away form the preload, the ball screw mechanical efficiency is elevated. The ball screw mechanical efficiency is increased when the applied axial load is increased greater than the preload. While the axial load is 2.83 time of the preload s, the ball screw mechanical efficiency is high. A large helix angle is advantageous to the ball screw speed, but the ball screw mechanical efficiency is decreased. The ball screw mechanical efficiency is also decreased when the screw’s rotational speed is increased. In order to get the high ball screw mechanical efficiency, one must choice the proper preload and the screw helix angle dependent upon on the axial load and the screw rotational speed.

參考文獻

1. Shkapenyuk, M. B., 1990, "Ways of Improving the Performance of Ball-screw Drives," Stanki I Instrument, Vol. 61, No. 4, pp. 9-11.
2. Ro, P. I. and Hubbel, P. I., 1993, "Model Reference Adaptive Control of Dual-mode Micro/macro Dynamics of Ball Screws for Nano-meter Motion," ASME Journal of Dynamic Systems, Measurement and Control, Vol. 115, No. 1, pp. 103-108.
3. Yun, W. S., Kim, S. K., and Cho, D. W., 1999, "Thermal Error Analysis for a CNC Lathe Feed Drive System," International Journal of Machine Tools and Manufacture, Vol. 39, No. 7, pp 1087-1101.
4. Spath, D., Rosum, J., and Haberkern, A., 1995, "Kinematics, Frictional Characteristics and Wear Reduction by PVD Coating on Ball Screw Drives," Annals of the CIRP, Vol. 44, pp. 349-352.
5. Bhattacharyya, A., Chatterjee, A. B., and Manwani, G. L., 1971, "Rigidity Analysis of Re-circulating Ball Screw Nut Assembly with a Semi-circular Thread Form," Annals of the CIRP, Vol. 19, No. 1, pp.87-93.
6. Nakashima, K., and Takafuji, K., 1987, "Stiffness of a Pre-loaded Ball Screw,"日本機械學會論文集, Vol. 53, No. 492, pp. 1898-1905.
7. Belyaev, V. G., and Kogan, A. I., 1988, "Effect of Geometrical Errors on Ball Contact Angle in Ball-screw Transmissions," Machines and Tooling, Vol. XLIV, No. 5, pp. 25-29.
8. Weule, H., Golz, H.U., 1991, "Preload-control in Ball Screw – A New Approach for Machine Tool Building," Annals of the CIRP, Vol. 40, pp. 383-386.
9. Cuttino, J. F., Dow, T. A., and Knight, B. F., 1997, "Analytical and Experimental Identification of Non-linearities in a Single-nut, Preloaded Ball Screw," ASME Journal of Mechanical Design, Vol. 119, No. 1, pp. 15-19.
10. Lin, M. C., Ravani, B., and Velinsky, S. A., 1994, "Kinematics of the Ball Screw Mechanism," ASME Journal of Mechanical Design, Vol. 116, pp. 849-855.
11. Lin, M. C., Ravani, B., and Velinsky, S. A., 1994, "Design of the Ball Screw Mechanism for Optimal Efficiency," ASME Journal of Mechanical Design, Vol. 116, pp. 856-861.
12. Levit, G. A., 1963, "Recirculating Ball Screw and Nut Units," Mechines and Tooling, Vol. XXXIV, No. 4, pp. 3-8.
13. Huang, H. T., and Ravani, B., 1997, "Contact Stress Analysis in Ball Screw Mechanism Using the Tubular Medial Axis Representation of Contacting Surfaces," ASME Journal of Mechanical Design, Vol. 119, pp. 8-14.
14. Harris, T. A., 1971, "An Analytical Method to Predict Skidding in Thrust-Loaded, Angular-Contact Ball Bearings," ASME Journal of Lubrication Technology, Vol. 93, pp. 17-24.
15. Boness, R. J., 1981, "Minimum Load Requirements for the Prevention of Skidding in High Speed Thrust Loaded Ball Bearings," ASME Journal of Lubrication Technology, Vol. 103, pp. 35-39.
16. Hiwin Technology Company, Hiwin Ballscrews Technical Information, Tanchung, Taiwan.
17. Hertz, H., 1881, "The Contact of Elastic Solids," J. Reine Angew. Math., Vol. 92, pp.156-171.
18. Harris T. A., 1984, Rolling Bearing Analysis, John Wiley & Sons, New York.
19. Markho, P. H., 1988, "Highly Accurate Formulas for Rapid Calculation of the Key Geometrical Parameters of Elliptic Hertzian Contacts," ASME Journal of Tribology, Vol. 109, pp. 640-647.
20. Harris, T. A., 1973, "Rolling Element Bearing Dynamics," Wear, Vol. 23, pp. 311-337.
21. Goldstein, S., 1938, Modern Developments in Fluid Dynamics, Oxford Press, London.
22. Barus, C., 1893, "Isothermals, Isopiestics and Isometrics Relative to Viscosity," American Journal of Science, Vol. 45, pp. 87-96
23. 溫詩鑄, 楊沛然, 1992, 彈性流體動力潤滑, 清華大學出版社, China
24. Hou, K., Zhu, D., and Wen, S., 1987, "An Inverse Solution to the Point Contact EHL Problem under Heavy Load," ASME Journal of Tribology, Vol. 109, No. 3, pp. 432-436