粗銑製程經常採用高切深的方式以提升加工效率，此種加工方式下切削力的變動較小所引發之強迫振動也較小。而在精銑、薄板加工或高速銑削製程中則多採用低徑向切深的方式，以避免顫振及降低刀具磨耗速度，然而此種低徑深之間歇式切削易產生高頻之諧合銑削力，進而激發劇烈的強迫振動，因此如何選擇適當的切削條件以降低其強迫制動是重要的研究議題。此外就銑削顫振而言，低徑深之間歇式切削屬週期性時變系統，傳統以線性非時變系統預測顫振條件的方式並不適用於預測間歇式切削的顫振行為，因此如何分析間歇式切削的的顫振特性也是一必要之研究議題。而刀腹與工件表面所產生製程阻尼是影響切削振動的重要因素，此種阻尼對戰強迫振動皆有重要之影響，然而由於其機制較複雜，較少有文獻對其作系統性的分析。根據以上論述本論文主要再研究以下關於銑削的三大主題1.製程阻尼2.強迫振動3.間歇式切削之顫振。
　　本論文首先提出包含切削阻尼的銑削動態模式。模式中包含剪切、犁切兩種切削機制，以及切削力大小及方向變動所產生的阻尼效應，配合銑削製程間歇式切削的時變特性建立線性週期性時變的銑削動態模式。以本模式為基礎，本文提出以結構振動信號辨識切削阻尼係數方法，此法可求取各種刀具/工件組合的切削阻尼係數，進而推測切削阻尼的大小。此外本文也提出平均切削阻尼的解析模式，並以此模式為基礎分析切削阻尼與切削條件的關係及其占切削系統整體阻尼的比例，文後輔以切削振動實驗來驗證本模式。在強迫振動分析方面，本文結合結構撓性與銑削力模式建立強迫振動之動態解析模式。再藉由強迫振動函數之頻譜分析，瞭解銑削過程中各加工條件如軸深、徑深、轉速、進給等對工件振動的影響。研究中發現在特定的軸向切深或主軸轉速下，結構共振頻率的附近激振力最小，此時強迫振動也相對減小，本文依此特性提出選取低強迫振動之加工條件的方法。最後對間歇性切削的顫振特性進行分析，文中將銑削系統的週期性時變參數以傅立葉級數展開，去除高頻部份級數以簡化模式為有限維度，結合考慮結構撓度函數建立二維銑削系統的顫振模式。利用此分析模式，本文系統性地分析各切削條件對間歇性銑削之動態諧合力與切削穩定性的影響。

　The cutting conditions with high radial immersion are usually used in rough milling to increase productivity, which leads to smaller variation of milling forces and also smaller cutting vibration. On the other hand, the cutting conditions with low radial immersion are usually used in finish milling, thin plate milling and high speed milling to reduce the tool wear rates, and this type of interrupted machining can easily induces large cutting vibration. Therefore, how to choose the proper cutting conditions to reduce the cutting vibration is an important subject for low immersion milling process. Furthermore, early researchers usually simplify the milling process as a linear time-invariant system to study milling stability, but the method is not suitable to analyze low immersion milling process whose system parameters is highly time-variant. However, the interrupted milling process with low radial immersion as a periodic time-varying system has not been clearly understood. In addition, process damping significantly affects the cutting vibrations and machining stability, but up to now no study has proposed a systematic method to analytically quantify its effect and to investigate effects of cutting conditions on process damping. Hence the research areas of this paper mainly focus on the three subjects below: (1) process damping, (2) forced vibration and (3) chatter vibration caused by the periodical time-varying parameters in milling operation.
　This thesis extends analytical modeling of the milling process to include process damping effects. Two cutting mechanisms (shearing and plowing mechanisms) and two process damping effects (directional and magnitude effects) are included. The analytical nature of this model makes it possible to determine unknown process damping coefficients from measured vibration signal during milling. The effects of cutting conditions on process damping are systematically examined. In addition, an analytical model for the forced vibration in an end milling process is derived and the criteria in selecting cutting conditions to reduce the forced vibration are presented. The analytic expression for the forced vibration due to the periodic milling force is obtained as the product of the Fourier transform of the milling force and the frequency transform of the structure dynamics. Analysis of the vibration model shows that the structure vibration can be reduced by selecting proper cutting conditions. A design equation in terms of cutter geometry, axial depth of cut and structure natural frequency is obtained for the conditions when the forced vibration can be minimized. Furthermore, this thesis investigates the effects of time-varying forces on the regenerative stability of an end milling process. By representing the milling force pulsation in a Fourier series expansion form, the dynamic force components and the average forces due to bi-directional dynamic feed rates are both included in the generalized system dynamics formulation. In the resulting expression for the stability criterion, the spectral features of the milling forces are integrated with the dynamics of the structure, showing the significance or insignificance of the dynamic components of the milling forces in affecting the stability of the milling process. The effects of the cutting parameters on milling stability are systematically discussed.

[1]Tlusty, J. and Ismail, F., 1983, “Special Aspects of Chatter in Milling,” ASME Journal of Engineering for Industry, 105, pp. 24-32.
[2]Jemielniak, K. and Widota, A., 1989, “Numerical Simulation of Non-linear Chatter Vibration in turning,” International Journal of Machine Tools and Manufacture, 29(2), pp. 239-247.
[3]Tarng, Y. S., Young, H. T. and Lee, B. Y., 1992, “An Analytical Model of Chatter Vibration in Metal Cutting,” International Journal of Machine Tools and Manufacture, 34(2), pp. 183-197.
[4]Shaw, M. C. and DeSalvo, G. J., 1970, “A New Approach to Plasticity and Its Application to Blunt Two Dimension Identers,” ASME Journal of Engineering for Industry, 92, pp. 472-479.
[5]Wu, D. W., 1989, “A New Approach of Formulating the Transfer Function for Dynamic Cutting Processes,” ASME Journal of Engineering for Industry, 111, pp. 37-47.
[6]Chiou, Y. S., Chung, E. S. and Liang S. Y., 1995, “Analysis of Tool Wear Effect on Chatter Stability in Turning,” International Journal of Mechanical Science, 37(4), pp. 391-404.
[7]Montgomery, D. and Altintas, Y., 1991, “Mechanism of Cutting Force and Surface Generation in Dynamic Milling,” ASME Journal of Engineering for Industry, 113, pp. 160-168.
[8]Elbestawi, M. A., Ismail, F., Du, R. and Ullagaddi, B. C., 1989, “Modeling Machining Dynamics Including Damping in the Tool-Workpiece Interface,” ASME Journal of Engineering for Industry, 116, pp. 435-439.
[9]Ranganath, S., Narayanan, K. and Sutherland, J. W., 1999, “The Role of Flank Face Interference in Improving the Accuracy of Dynamic Force Predictions in Peripheral Milling,” ASME Journal of Manufacturing Science and Engineering, 121, pp. 593-599.
[10]Tobias, S. A. and Fishwick, W., 1958, “A Theory of Regenerative Chatter,” The Engineer, London.
[11]Tlusty, J. and Polacek, M., 1963, “The Stability of Machine Tools Against Self Excited Vibration in Machining,” International Research in Production Engineering, ASME, pp. 465-474.
[12]Smith, S. and Tlusty, J., 1990, “Update on High Speed Milling Dynamics,” ASME Journal of Engineering for Industry, 112, pp.142-149.
[13]Smith, S. and Tlusty, J., 1991, “An Overview of Modeling and Simulation of the Milling Process,” ASME Journal of Engineering for Industry, 113, pp.169-175.
[14]Altintas, Y., Montegomery, D. and Budak, E., 1992, “Dynamic Peripheral Milling of Flexible Structure,” ASME Journal of Engineering for Industry, 114, pp. 137-145.
[15]Minis, I. and Yanushevsky, T., 1993, “A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling,” ASME Journal of Engineering for Industry, 18, pp.1-8.
[16]Altintas, Y. and Budak, E., 1995, “Analytical Prediction of Stability Lobes in Milling,” Annals of the CIRP, 44, pp. 357-362.
[17]Budak, E and Altintas, Y., 1998, “An Analytical Prediction of Chatter Stability in Milling – Part I: General Formulation,” ASME Journal of Dynamic Systems, Measurement and Control, 120, pp. 22-30.
[18]Budak, E. and Altintas, Y., 1998, “An Analytical Prediction of Chatter Stability in Milling – Part II: Application of the General Formulation to Common Milling Systems,” ASME Journal of Dynamic Systems, Measurement and Control, 120, pp. 31-36.
[19]Jensen, S. A. and Shin, Y. C., 1999, “Stability Analysis in Face Milling Operations, Part 1: Theory of Stability Lobe Prediction,” ASME Journal of Manufacturing Science and Engineering, 121, pp. 600-605.
[20]Abrari, F., Elbestawi, M. A. and Spence, A. D., 1998, “On the Dynamics of Ball End Milling: Modeling of Cutting Forces and Stability Analysis,” International Journal of Machine Tools & Manufacture, 38, pp. 215-237.
[21]Altintas, Y. Shamoto, E., Lee, P. and Budak, E., 1999, “Analytical Prediction of Stability Lobes in Ball End Milling,” ASME Journal of Manufacturing Science and Engineering, 121, pp. 586-592.
[22]Altintas, Y., Engin S. and Budak, E., 1999, “Analytical Stability Prediction and Design of Variable Pitch Cutter,” ASME Journal of Manufacturing Science and Engineering, 121, pp. 173-178.
[23]Budak, E., 2003, “An Analytical Design Method for Milling Cutter with Nonconstant Pitch to Increase Stability, Part I: Theory,” ASME Journal of Manufacturing Science and Engineering, 125, pp. 29-34.
[24]Schmitz, T. L., 2002, “Automatic Trimming of Machining Stability Lobes,” International Journal of Machine Tools & Manufacture, 42, pp. 1479-1486.
[25]Davies, M. A., Pratt J. R., Dutterer, B. and Burns, T. J., 2002, “Stability Prediction for Low Radial Immersion Milling,” ASME Journal of Manufacturing Science and Engineering, 124, pp. 217-225.
[26]Insperger, T., Stépán G., Bayly P. V. and Mann B. P., 2003, “Multiple Chatter Frequencies in Milling Processes,” Journal of Sound and Vibration, 262, pp. 333-345.
[27]Insperger, T., Mann B. P., Stépán G. and Bayly P. V., 2003, “Stability of Up-milling and Down-milling, Part 1: Alternative Analytical Methods” International Journal of Machine Tools & Manufacture, 43, pp. 25-34.
[28]Insperger, T., Mann B. P., Stépán G. and Bayly P. V., 2003, “Stability of Up-milling and Down-milling, Part 2: Experimental Verification” International Journal of Machine Tools & Manufacture, 43, pp. 35-40.
[29]Corpus W. T. and Endres W. J., 2004, “Added Stability Lobes in Machining Processes That Exhibit Periodic Time Variation, Part 1: An Analytical Solution, ASME Journal of Manufacturing Science and Engineering, 126, pp. 467-474.
[30]Budak, E. and Altintas, Y., 1995, “Modeling and Avoidance of static Form Errors in Peripheral Milling of Plates,” International Journal of Machine Tool and Manufacture, 35, pp. 459-476.
[31]Budak, E. and Altintas, Y., 1996, “Cutting Force and Dimensional Surface Error Generation in Peripheral Milling with Variable Pitch Helical End Mills,” International Journal of Machine Tool and Manufacture, 35, pp. 587-584.
[32]Engin, S. and Altintas, Y., 2001, ”Mechanics and dynamics of general milling cutters. Part 1:helical end mills,” International Journal of Machine Tools and Manufacture , 41, pp. 2195-2212.
[33]Wang, J.J., and Liang, S.Y., and Book, W.J., 1994,“Convolution Analysis of Milling Force Pulsation,” ASME Journal of Engineering for Industry, 116, pp. 17-25.
[34]Liang, S. Y. and Zheng Li, 1998,“Analysis of End Milling Surface Error Considering Tool Compliance,”ASME Journal of Manufacturing Science and Engineering, 121, pp. 207-210.
[35]Wang, J. -J. Junz 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, pp. 761-771.
[36]Endres, W. J., Devor, R. E. and Kapoor S.G., 1995, “A Dual-Mechanism Approach to the Prediction of Machining forces, Part 1: Model Development,” ASME Journal of Engineering for Industry, 117, pp. 526-533.
[37]Wang, J. -J. Junz and Zheng, C. M., 2003, “On-line Identification of Shearing and Plowing Constants in End Milling,” ASME Journal of Manufacturing Science and Engineering, 125, pp. 58-65.
[38]Koenigsberger, F. and Tlusty, J., 1967, Machine Tool Structures–Vol. I: Stability against Chatter, Pergamon Press, New York, USA.
[39]Ismail, F. and Vadari, V. R., 1990, “Machining Chatter of End Mills with Unequal Modes,” ASME Journal of Engineering for Industry, 112, pp. 229-234.
[40]Altintas, Y., Engin, S., and Budak, E, 1999, “Analytical Stability Prediction and Design of Variable Pitch Cutters,” ASME Journal of Manufacturing Science and Engineering, 121, pp. 173-178.
[41]Jayaram, S., Kapoor, S. G., and DeVor, R. E., 2000, “Analytical Stability Analysis of Variable Spindle Speed Machining,” ASME Journal of Manufacturing Science and Engineering, 122, pp. 391-397.
[42]Tlusty, J., 1997, “Current Trends in High-Speed Machining,” ASME Journal of Engineering for Industry, 119, pp. 664-666.