变齿距/变螺旋铣刀切削稳定性理论与实验研究
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摘要
铣削加工作为一种先进的加工技术，广泛应用于航空、航天、船舶、模具及汽车等领域。该领域要求从金属结构中切除大量金属材料且保证加工过程的高效性和精确性。铣削颤振是发生在加工过程中的一种自激振动现象，该现象加剧刀具磨损，减小加工中心的使用寿命，降低工件表面加工精度，严重制约加工中心的使用效率。作为一种较好的颤振稳定性被动控制方法，变齿距或变螺旋铣刀切削稳定性性研究以优化加工参数和刀具结构实现加工过程无颤振材料最大去除率为目标，因此，开展该问题系统、深入的研究对提高实际加工生产率，对于增强我国制造业的竞争力具有重大的实用价值。
铣削动态切削厚度的再生效应是诱发颤振的主因。目前国内外学者对铣削过程稳定性分析的工作主要集中于此主题。鉴于此，本文将借助于理论分析和试验研究，基于再生颤振理论开展变齿距或变螺旋铣刀切削稳定性研究，旨在发展和完善该领域切削理论，实现稳定性快速预测，主要研究内容如下：
(1)考虑到刀齿瞬态切削力与刀具螺旋引发的切削域之间的关系，提出了考虑刀齿螺旋特性的积分切削力简化模型。基于该模型和改进的半离散法提出了适用于变齿距铣刀切削稳定性的预测模型。理论模型的正确性通过与前人工作的对比和铣削实验得到了验证。此外，基于上述预测模型，深入研究了齿距类型、齿数和刀具螺旋角等因素对铣削稳定性的影响。
(2)基于铣削系统状态项的离散与插值，提出了变齿距铣刀切削稳定性高效预测算法。该方法的核心思想是在离散的旋转周期时间域上，对方程中的时滞状态项进行拉格朗日插值，对位移状态项进行埃尔米特插值，获得方程的离散映射形式进而确定系统的稳定性域。该算法的优势是：1)在保证数值精度的前提下，较之于现有通用算法的半离散法，该算法对于变齿距铣刀切削系统稳定性预报的计算效率可以提高约78%；2)该方法也适用于其他含有多时滞项的动力学系统的稳定性求解。
(3)提出了适用于变螺旋铣刀切削稳定性的半解析预测模型。该模型主要由系统变时滞向多时滞转化方法和为适应时滞随离散点时变而提出的半离散求解算法两部分组成。数值计算结果表明，较前人的算法，该算法具有更好的计算精度和收敛速度。模型的有效性通过实验得到了验证。同时，基于该预测模型，考察了铣刀端部齿距线性变化和交替变化、变螺旋角和齿数等因素组合效应对铣削稳定性的影响。此外，以该预测模型为基础，将切削力方向系数平均思想融入其中，提出了高效变螺旋铣刀切削稳定性预测模型，该模型较原有模型计算效率进一步提高，但由于该模型未考虑切削力方向系数的时变特性，因此在稳定性预测过程中会造成一些误差。
(4)基于Altintas铣削稳定性解析预测理论，在频域中提出了变螺旋铣刀切削稳定性解析预测模型。该理论首次将变螺旋铣刀切削稳定性预测时间提升到以秒为量级。该方法的正确性和高效性已经在与前人工作的对比中得到了验证。鉴于该理论模型预测效率的高效性，其非常适合快速获取变螺旋铣刀切削稳定性叶瓣图和刀具的几何优化。
Milling is a very commonly advanced manufacturing process, and widely utilized in key industries such as aerospace, shipping, die and mold, and automotive. What is more, manufacturing process in these industries needs a high productivity and precision. Milling chatter is a form of self-excited, unstable vibration, which may result in low machining quality, poor accuracy and surface finish, as well as accelerated tool wear, aside from damaging the cutting spindle and machined part. The use of variable pitch or helix milling tools is a known means to influence and prevent chatter vibration during milling. Due to that its aims are to maximize the material removal rates without chatter through exploring the vibration mechanism and optimizing the machining conditions and tool geometries, the systematic and deep investigation about related problem is very important for increasing the productivity in machining and has great practical value to strengthen the competitiveness of manufacturing industry in China.
Since the regeneration effect of the dynamic chip thickness is the most powerful source of self-excitation, the researcher on the stability of milling processes are mainly focused on this problem. For this reason, in this paper, based on the regeneration theory, theoretical analysis and experimental method are used to investigate the milling stability for variable pitch or helix cutters, and aim to develop or perfect the theory in this realm. The main research content are as follows.
(1) Considering the different cases of piecewise continuous cutting regions regarding the helix angle, a straightforward analytical integral force model is develop. Based on this model and an up-dated semi-discretization method, a method is proposed to predict the stability lobes for variable pitch cutter. The proposed approach has been verified with the comparisons with prior works, time domain simulations and cutting tests. In addition, considered that there is a few research about the influence of tool geometries on milling stability, the method is also applied to examine the effect of pitch variation types, number of the teeth and helix angle on the stability trends for variable pitch milling.
(2) Based on the interpolation to the state items in milling dynamic system, an efficient method for the stability prediction of variable pitch cutter milling is presented. With the help of discretizing the Spindle rotation period and approximating the delay items by Lagrange interpolation and the displacement items by Hermite interpolation, a discrete map form can be obtained, thus the stability can be determined via Floquet theory. The advantage of this algorithm is that:1) compared with the well-known semi-discretization method, this method has much higher computational efficiency (by78%) without loss of any numerical precision.2) The method is also suitable to solve other dynamic systems contained multi-delays.
(3) An alternative semi-analytical method for the prediction of the stability lobes of variable helix cutters is proposed. This method contains two key parts, i.e., the technique of transferring the system with distributed delays to that with multiple delays and an updated semi-discretization method which is suitable for the case of system multiple delays changing with time. Meanwhile, based on this method, stability trends as a function of tooth pitch variation and helix angle variation are investigated. Moreover, an efficient method is presented to stability prediction for variable helix cutters after the author apply the time-averaged cutting force coefficients to system equation. Noted that although this method has a better computational efficiency to obtain the stability lobes than that of above method, it may lead to error in some spindle speed domain because of the neglection of the time variation of the cutting force coefficients.
(4) Based on the principles of variable pitch model developed by Altintas, an efficient frequency-domain solution to predict the stability for variable helix cutters is developed. The proposed technique has been verified with the comparison with other methods. The results show that the proposed method has high computational efficiency. Thus, it is suited to calculate optimal geometries of milling tools and beneficial for application.
铣削动态切削厚度的再生效应是诱发颤振的主因。目前国内外学者对铣削过程稳定性分析的工作主要集中于此主题。鉴于此，本文将借助于理论分析和试验研究，基于再生颤振理论开展变齿距或变螺旋铣刀切削稳定性研究，旨在发展和完善该领域切削理论，实现稳定性快速预测，主要研究内容如下：
(1)考虑到刀齿瞬态切削力与刀具螺旋引发的切削域之间的关系，提出了考虑刀齿螺旋特性的积分切削力简化模型。基于该模型和改进的半离散法提出了适用于变齿距铣刀切削稳定性的预测模型。理论模型的正确性通过与前人工作的对比和铣削实验得到了验证。此外，基于上述预测模型，深入研究了齿距类型、齿数和刀具螺旋角等因素对铣削稳定性的影响。
(2)基于铣削系统状态项的离散与插值，提出了变齿距铣刀切削稳定性高效预测算法。该方法的核心思想是在离散的旋转周期时间域上，对方程中的时滞状态项进行拉格朗日插值，对位移状态项进行埃尔米特插值，获得方程的离散映射形式进而确定系统的稳定性域。该算法的优势是：1)在保证数值精度的前提下，较之于现有通用算法的半离散法，该算法对于变齿距铣刀切削系统稳定性预报的计算效率可以提高约78%；2)该方法也适用于其他含有多时滞项的动力学系统的稳定性求解。
(3)提出了适用于变螺旋铣刀切削稳定性的半解析预测模型。该模型主要由系统变时滞向多时滞转化方法和为适应时滞随离散点时变而提出的半离散求解算法两部分组成。数值计算结果表明，较前人的算法，该算法具有更好的计算精度和收敛速度。模型的有效性通过实验得到了验证。同时，基于该预测模型，考察了铣刀端部齿距线性变化和交替变化、变螺旋角和齿数等因素组合效应对铣削稳定性的影响。此外，以该预测模型为基础，将切削力方向系数平均思想融入其中，提出了高效变螺旋铣刀切削稳定性预测模型，该模型较原有模型计算效率进一步提高，但由于该模型未考虑切削力方向系数的时变特性，因此在稳定性预测过程中会造成一些误差。
(4)基于Altintas铣削稳定性解析预测理论，在频域中提出了变螺旋铣刀切削稳定性解析预测模型。该理论首次将变螺旋铣刀切削稳定性预测时间提升到以秒为量级。该方法的正确性和高效性已经在与前人工作的对比中得到了验证。鉴于该理论模型预测效率的高效性，其非常适合快速获取变螺旋铣刀切削稳定性叶瓣图和刀具的几何优化。
Milling is a very commonly advanced manufacturing process, and widely utilized in key industries such as aerospace, shipping, die and mold, and automotive. What is more, manufacturing process in these industries needs a high productivity and precision. Milling chatter is a form of self-excited, unstable vibration, which may result in low machining quality, poor accuracy and surface finish, as well as accelerated tool wear, aside from damaging the cutting spindle and machined part. The use of variable pitch or helix milling tools is a known means to influence and prevent chatter vibration during milling. Due to that its aims are to maximize the material removal rates without chatter through exploring the vibration mechanism and optimizing the machining conditions and tool geometries, the systematic and deep investigation about related problem is very important for increasing the productivity in machining and has great practical value to strengthen the competitiveness of manufacturing industry in China.
Since the regeneration effect of the dynamic chip thickness is the most powerful source of self-excitation, the researcher on the stability of milling processes are mainly focused on this problem. For this reason, in this paper, based on the regeneration theory, theoretical analysis and experimental method are used to investigate the milling stability for variable pitch or helix cutters, and aim to develop or perfect the theory in this realm. The main research content are as follows.
(1) Considering the different cases of piecewise continuous cutting regions regarding the helix angle, a straightforward analytical integral force model is develop. Based on this model and an up-dated semi-discretization method, a method is proposed to predict the stability lobes for variable pitch cutter. The proposed approach has been verified with the comparisons with prior works, time domain simulations and cutting tests. In addition, considered that there is a few research about the influence of tool geometries on milling stability, the method is also applied to examine the effect of pitch variation types, number of the teeth and helix angle on the stability trends for variable pitch milling.
(2) Based on the interpolation to the state items in milling dynamic system, an efficient method for the stability prediction of variable pitch cutter milling is presented. With the help of discretizing the Spindle rotation period and approximating the delay items by Lagrange interpolation and the displacement items by Hermite interpolation, a discrete map form can be obtained, thus the stability can be determined via Floquet theory. The advantage of this algorithm is that:1) compared with the well-known semi-discretization method, this method has much higher computational efficiency (by78%) without loss of any numerical precision.2) The method is also suitable to solve other dynamic systems contained multi-delays.
(3) An alternative semi-analytical method for the prediction of the stability lobes of variable helix cutters is proposed. This method contains two key parts, i.e., the technique of transferring the system with distributed delays to that with multiple delays and an updated semi-discretization method which is suitable for the case of system multiple delays changing with time. Meanwhile, based on this method, stability trends as a function of tooth pitch variation and helix angle variation are investigated. Moreover, an efficient method is presented to stability prediction for variable helix cutters after the author apply the time-averaged cutting force coefficients to system equation. Noted that although this method has a better computational efficiency to obtain the stability lobes than that of above method, it may lead to error in some spindle speed domain because of the neglection of the time variation of the cutting force coefficients.
(4) Based on the principles of variable pitch model developed by Altintas, an efficient frequency-domain solution to predict the stability for variable helix cutters is developed. The proposed technique has been verified with the comparison with other methods. The results show that the proposed method has high computational efficiency. Thus, it is suited to calculate optimal geometries of milling tools and beneficial for application.
引文
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[75] Wang J and Lee K, Suppression of chatter vibration of a CNC machinecentre—an example. Mechanical Systems and Signal Processing,1996,10(5):551-560.
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[77] Semercigil S and Chen L, Preliminary computations for chatter control in endmilling. Journal of Sound and Vibration,2002,249(3):622-633.
[78] Kim NH, Won D, and Ziegert JC, Numerical analysis and parameter study of amechanical damper for use in long slender endmills. International Journal ofMachine Tools and Manufacture,2006,46(5):500-507.
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[83] Budak E, Analytical models for high performance milling. Part II: Processdynamics and stability. International Journal of Machine Tools andManufacture,2006,46(12-13):1489-1499.
[84] Wan M, Zhang W-H, Dang J-W, et al., A unified stability prediction methodfor milling process with multiple delays. International Journal of MachineTools and Manufacture,2010,50(1):29-41.
[85] Sellmeier V and Denkena B, Stable islands in the stability chart of millingprocesses due to unequal tooth pitch. International Journal of Machine Toolsand Manufacture,2011,51(2):152-164.
[86] Song Q, Ai X, and Zhao J, Design for variable pitch end mills with highmilling stability. The International Journal of Advanced ManufacturingTechnology,2011,55(9-12):891-903.
[87] Zhang X, Xiong C, and Ding Y, Improved full-discretization method formilling chatter stability prediction with multiple delays, In Intelligent Roboticsand Applications, Springer,2010,541-552.
[88] Zhang X, Xiong C, Ding Y, et al., Variable-step integration method for millingchatter stability prediction with multiple delays. Science China TechnologicalSciences,2011,54(12):3137-3154.
[89] Turner S, Merdol D, Altintas Y, et al., Modelling of the stability of variablehelix end mills. International Journal of Machine Tools and Manufacture,2007,47(9):1410-1416.
[90] Sims ND, Mann B, and Huyanan S, Analytical prediction of chatter stabilityfor variable pitch and variable helix milling tools. Journal of Sound andVibration,2008,317(3-5):664-686.
[91] Yusoff AR and Sims ND, Optimisation of variable helix tool geometry forregenerative chatter mitigation. International Journal of Machine Tools andManufacture,2011,51(2):133-141.
[92] Dombovari Z and Stepan G, The Effect of Helix Angle Variation on MillingStability. Transactions of the ASME-B-Journ Manufacturing ScienceEngineering,2012,134(5):051015.
[93] Chiou CH, Hong MS, and Ehmann KF, The feasibility of eigenstructureassignment for machining chatter control. International Journal of MachineTools and Manufacture,2003,43(15):1603-1620.
[94] Olgac N and Hosek M, A new perspective and analysis for regenerativemachine tool chatter. International Journal of Machine Tools and Manufacture,1998,38(7):783-798.
[95] Dohner JL, Lauffer JP, Hinnerichs TD, et al., Mitigation of chatter instabilitiesin milling by active structural control. Journal of Sound and Vibration,2004,269(1):197-211.
[96] Huyanan S and Sims N. Active vibration absorbers for chatter mitigationduring milling. In: Proc ninth international conference on vibrations in rotatingmachinery.2008.
[97] Takemura T, Kitamura T, Hoshi T, et al., Active suppression of chatter byprogrammed variation of spindle speed. Annals of the CIRP,1974,23(1):121-122.
[98] Sexton J, Milne R, and Stone B, A stability analysis of single-point machiningwith varying spindle speed. Applied Mathematical Modelling,1977,1(6):310-318.
[99] Sexton J and Stone B, The stability of machining with continuously varyingspindle speed. Annals of the CIRP,1978,27(1):321-326.
[100] Jayaram S, Kapoor S, and DeVor R, Analytical stability analysis of variablespindle speed machining. Journal of Manufacturing Science and Engineering,2000,122(3):391-397.
[101] Insperger T and Stepan G, Stability analysis of turning with periodic spindlespeed modulation via semidiscretization. Journal of Vibration and Control,2004,10(12):1835-1855.
[102] Zhang H, Jackson MJ, and Ni J, Stability analysis on spindle speed variationmethod for machining chatter suppression. International Journal of Machiningand Machinability of Materials,2009,5(1):107-128.
[103] Sastry S, Kapoor SG, and DeVor RE, Floquet theory based approach forstability analysis of the variable speed face-milling process. Journal ofManufacturing Science and Engineering,2002,124(1):10-17.
[104] Zatarain M, Bediaga I, Munoa J, et al., Stability of milling processes withcontinuous spindle speed variation: analysis in the frequency and timedomains, and experimental correlation. CIRP Annals-ManufacturingTechnology,2008,57(1):379-384.
[105] Radulescu R, Kapoor S, and DeVor R, An investigation of variable spindlespeed face milling for tool-work structures with complex dynamics, part1:simulation results. Journal of Manufacturing Science and Engineering,1997,119(3):266-272.
[106] Radulescu R, Kapoor S, and DeVor R, An investigation of variable spindlespeed face milling for tool-work structures with complex dynamics, Part2:physical explanation. Journal of Manufacturing Science and Engineering,1997,119(3):273-280.
[107] Seguy S, Insperger T, Arnaud L, et al., On the stability of high-speed millingwith spindle speed variation. The International Journal of AdvancedManufacturing Technology,2010,48(9-12):883-895.
[108] Ganguli A, Deraemaeker A, Horodinca M, et al., Active damping of chatter inmachine tools-demonstration with a 'hardware-in-the-loop' simulator.Proceedings of the Institution of Mechanical Engineers, Part I: Journal ofSystems and Control Engineering,2005,219(5):359-369.
[109] Ganguli A, Chatter reduction through active vibration damping, UniversiteLibre de Bruxelles,2005.
[110] Wang M, Zan T, Yang Y, et al., Design and implementation of nonlinear TMDfor chatter suppression: An application in turning processes. InternationalJournal of Machine Tools and Manufacture,2010,50(5):474-479.
[111] Shamoto E, Mori T, Nishimura K, et al., Suppression of regenerative chattervibration in simultaneous double-sided milling of flexible plates by speeddifference. CIRP Annals-Manufacturing Technology,2010,59(1):387-390.
[112] Brecher C, Manoharan D, Ladra U, et al., Chatter suppression with an activeworkpiece holder. production engineering,2010,4(2-3):239-245.
[113]宋清华，高速铣削稳定性及加工精度研究：[博士学位论文]，山东；山东大学，2009.
[114] Powell KB. Cutting performance and stability of helical endmills withvariable pitch:[Ph.D. thesis] Florida; University of Florida.2008.
[115] Smith S and Tlusty J, Efficient simulation programs for chatter in milling.CIRP ANN-MANUF TECHN,1993,42:463-466.
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