本研究主要探討刀具底部與工件之犁切現象及其所產生之阻尼效應對銑削穩定性之影響。目前與銑削加工製程阻尼相關的研究多數探討銑刀側邊與工件之犁切現象，其使銑削穩定加工之臨界切深隨著主軸轉速降低而有拉升趨勢。但業界及實驗室過去的經驗顯示，刀具底部刃口與工件犁切所產生的摩擦亦會增加額外的阻尼提升切削的穩定性，使全轉速範圍之臨界切深整體抬升。本文先建立包含刀側剪切和犁切力量及刀底犁切力量之銑削力模式。其中刀底犁切力分別來自於犁切與摩擦效應。犁切效應與未變形切屑厚度成正比;而摩擦效應與刀具底刃與工件接觸長度相關。接觸長度愈大則摩擦效應愈大，而接觸長度受到刀具直徑、刃數及切削徑深等參數之影響。透過切削係數辨識實驗，針對不同刃數、切削徑深和每刃進給觀察其刀底犁切常數之變化。接著探討加工過程刀具底部與工件犁切所產生之庫倫摩擦力對銑削穩定性之影響，並推導出刀底犁切之等效阻尼比，此阻尼比與刀底犁切的切向力成正比，與刀尖振幅成反比。最後以穩定性實驗驗證不包含刀底犁切時的剪切常數所預測穩定葉瓣圖之準確性和探討刀底犁切阻尼比與刀尖振幅對銑削穩定性之影響。

The purpose of this study is to investigate the effect of ploughing between the bottom of the tool and the workpiece on the milling stability. At present, the researches on the process damping of milling mostly focus on the ploughing effect between the side of tool and workpiece, which increases the critical depth of cut while decreasing the spindle speed. However, past machining experience in the industry has shown that the friction generated by ploughing between tool bottom and workpiece adds additional damping. It increases the stability of cutting, and raises the curve of critical depth over all full spindle speed range. In this paper, the milling force mode including the shearing and ploughing force on tool side and the ploughing force on tool bottom is established. The ploughing and friction effects result in the ploughing force on tool bottom. The ploughing effect is proportional to the undeformed chip thickness. The friction effect is related to the contact length between tool bottom and workpiece. The larger the contact length is, the greater the friction effect is. The contact length is affected by parameters such as tool diameter, number of flutes, and radial depth of cut. Through the experimental identification of cutting coefficient, the variation in the ploughing constant of tool bottom was observed for different number of flutes, radial depth of cut, and feed per tooth. Next, we can discuss the effect of the Coulomb friction generated by ploughing between tool bottom and workpiece on the milling stability during the machining process, and the equivalent damping ratio of ploughing on tool bottom is derived. It is proportional to the tangential ploughing force of tool bottom and inversely proportional to the amplitude of the tool tip. Finally, we performed stability experiment to verify the accuracy of the stability lobe predicted by the shearing cutting constant without the ploughing force on tool bottom, and the effect of the damping ratio of the tool bottom and amplitude of the tool tip on the milling stability.

[1] Tobias, S. A. and Fishwick, W., 1958, A theory of regenerative chatter, The Engineer, London.
[2] Opitz, H., 1969, “Investigation and Calculation ofthe Chatter Behavior of Lathes and Milling Machines,”Annals of the CIRP, 18:335-342.
[3] Altintas, Y. Budak, E., 1995, "Analytical Prediction of Stability Lobes in Milling," Annals of the CIRP, 44(1)(357–362).
[4] Budak, E., Altintas, Y., 1998, “Analytical Prediction of Chatter Stability Conditions for Multidegree of Freedom Systems in Milling. Part I: General Formulation, Part II: Application of the General formulation to Common Milling Systems”, Transactions of the ASME, 120:22-36.
[5] Insperger, T., Stepan, G., 2004, “Updated Semi-Discretization Method for Periodic Delay-Differential Equations with Discrete Delay,” International Journal of Numerical Methods inEngineering, 61/1:117-141.
[6] Merdol, S.D., Altintas, Y., 2004, “Multi Frequency Solution of Chatter Stability for Low Immersion Milling,” Journal of Manufacturing Science and Engineering, 126:459-466.
[7] Zatarain, M., Munoa, J., Peigne, G., Insperger, T., 2006, “Analysis of the Influence of Mill Helix Angle on Chatter Stability,” Annals of the CIRP, 55/1:365-368.
[8] Insperger, T., Munoa, J., Zatarain, M., Peigne, G., 2006, “Unstable Islands in the Stability Chart of Milling Processes Due to the Helix Angle,” CIRP 2nd International Conference on High Performance Cutting.
[9] Tunç, L. T., and Budak, E., 2012, “Effect of cutting conditions and tool geometry on process damping in machining,” International Journal of Machine Tools and Manufacture, 57, pp. 10-19.
[10] Tunc, L.T., and Budak, E., 2013, “Identification and Modeling of Process Damping in Milling,” Journal of Manufacturing Science and Engineering, 135(2), pp. 021001-021001.
[11] Ahmadi, K., and Ismail, F., 2011, “Analytical stability lobes including nonlinear process damping effect on machining chatter,” International Journal of Machine Tools and Manufacture, 51(4), pp. 296-308.
[12] Ahmadi, K., and Ismail, F., 2012, “Stability lobes in milling including process damping and utilizing Multi-Frequency and Semi-Discretization Methods,” International Journal of Machine Tools and Manufacture, 54–55, pp. 46-54.
[13] Jun, M. B., DeVor, R. E., Kapoor S.G., 2006, Investigation of the Dynamics of Micro-end Milling—Part II: Model Validation and Interpretation, ASME J. Manuf. Sci. Eng., 128:901-912.
[14] Rahnama, R., Sajjadi, M., Park, S.S., 2009, Chatter Suppression in Micro End Milling with Process Damping, J. Mat. Proc. Tech. 209:5766-5776.
[15] Jin, X., Altintas, Y., 2013, Chatter Stability Model of Micro-Milling with Process Damping, ASME J. Manuf. Sci. Eng., 135:031011-1-9.
[16] Uhlmann, E., Mahr, F., 2012, A time Domain Simulation Approach for Micro Milling Processes, 3rd CIRP Conf. on Process Machine Interactions, Procedia CIRP 4:22-28.
[17] Afazov, S. M., Ratchev, S. M., Segal, J., Popov, A.A., 2012, Chatter Modeling in Micro-Milling by Considering Process Nonlinearities, Int. J. of Mach. Tools Manuf. 56:28–38.
[18] Budak, E., Tunc, L.T., 2009, a New Method for Identification and Modeling of Process Damping in Machining, ASME J. Manuf. Sci. Eng., 131:051019-1-10
[19] Kurata, Y., Merdol, S.D., Altintas, Y., Suzuki, N., Shamoto, E., 2010, Chatter Stability in Turning and Milling with In Process Identified Process Damping, J. Adv. Mech. Des. Sys., and Manuf. 4:1107–1118.
[20] Budak, E., Tunc, L.T., 2010, Identification and Modeling of Process Damping in Turning and Milling Using a New Approach, CIRP Annals - Manufacturing Technology, 59:403-408
[21] J.J. Wang, E. Uhlmann, D. Oberschmidt, C.F. Sung, I. Perfilov, Critical depth of cut and asymptotic spindle speed for chatter in micro milling with process damping, CIRP Annals - Manufacturing Technology, Volume 65, Issue 1, 2016, Pages 113-116
[22] Menq, C, Griffin, J.H., A Comparison of Transient and Steady State Finite Element Analyses of the Forced Response of a Frictionally Damped Beam. ASME. J. Vib., Acoust., Stress, and Reliab. 1985;107(1):19-25.
[23] X. Tan, R.J. Rogers, Equivalent viscous damping models of coulomb friction in multi-degree-of-freedom vibration systems, Journal of Sound and Vibration, Volume 185, Issue 1, 1995, Pages 33-50
[24] K.Y. Sanliturk, D.J. Ewins, MODELLING TWO-DIMENSIONAL FRICTION CONTACT AND ITS APPLICATION USING HARMONIC BALANCE METHOD, Journal of Sound and Vibration, Volume 193, Issue 2, 6 June 1996, Pages 511-523
[25] F. Xia, Modelling of a two-dimensional Coulomb friction oscillator, Journal of Sound and Vibration, Volume 265, Issue 5, 28 August 2003, Pages 1063-1074
[26] Jin-Wei Liang, Identifying Coulomb and viscous damping from free-vibration acceleration decrements, Journal of Sound and Vibration, Volume 282, Issues 3–5, 22 April 2005, Pages 1208-1220
[27] Ziying Wu, Hongzhao Liu, Lilan Liu, Daning Yuan, Identification of nonlinear viscous damping and Coulomb friction from the free response data, Journal of Sound and Vibration, Volume 304, Issues 1–2, 10 July 2007, Pages 407-414
[28] Nicholas D. Oliveto, Giovanni Scalia, Giuseppe Oliveto, Dynamic identification of structural systems with viscous and friction damping, Journal of Sound and Vibration, Volume 318, Issues 4–5, 23 December 2008, Pages 911-926
[29] Wang, J.-J. J., and Zheng, C. M., 2002, “An analytical force model with shearing and ploughing mechanisms for end milling,” International Journal of Machine Tools and Manufacture, 42(7), pp. 761-771
[30] 張煌權, 包含側邊及底邊犁切力之端銑及面銑力模式，國立成功大學碩士論文，台南市，台灣，2001
[31] Wang, J.-J. J., & Zheng, C., An analytical force model with shearing and ploughing mechanisms for end milling. International Journal of Machine Tools and Manufacture, 42(7), 761-771, 2002.
[32] Huang, C., & Wang, J. J., Mechanistic modeling of process damping in peripheral milling. Journal of Manufacturing Science and Engineering, 129(1), 12-20, 2007.