Abstract
In order to increase the calculation speed of the semi-discretization method (SDM) without accuracy loss, this paper reconstructs the SDM for predicting the stability lobes of the dynamic milling process, mainly considering the regenerative effect. The model of the dynamic milling process is expressed as the linear delay-differential equations (DDE). The fast calculation method is established by reconstructing the SDM based on the Shannon standard orthogonal basis (SSOB). First, the delay term of DDE is constructed without information loss based on Shannon interpolation functions, and SSOB is derived. Secondly, the closed form expression for the transition matrix of the system is constructed based on the SSOB, and the stability limit is predicted based on the Floquet theory. The transition matrix-based SDM and SSOB are theoretically compared, and it shows that the SDM is a special case of the method based on SSOB when the SSOB is regarded as the average in the sampling interval. The fast calculation method is established by using the variable sampling numbers during the period of the delay time in which the variable sampling numbers are determined by the condition which is used to construct the SSOB. Finally, this proposed fast method is used to the one and two degrees of freedom milling model, and the results show that the calculation accuracy is not reduced, and the calculation speed based on the proposed method can be improved nearly five times on the one degree of freedom model and 2.6 times on the two degrees of freedom model, compared to the semi-discretization method.
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References
Li Z, Liu Q, Ming X, et al (2014) Cutting force prediction and analytical solution of regenerative chatter stability for helical milling operation. Int J Adv Manuf Technol. 1-10
Zhang XJ, Xiong CH, Ding Y et al (2014) A synthetical stability method for cutting parameter optimization to assure surface location accuracy in flexible part milling. Int J Adv Manuf Technol 75(5-8):1131–1147
Yang Y, Liu Q, Zhang B (2014) Three-dimensional chatter stability prediction of milling based on the linear and exponential cutting force model. Int J Adv Manuf Technol 72(9-12):1175–1185
Song Q, Liu Z, Shi Z (2014) Chatter stability for micromilling processes with flat end mill. Int J Adv Manuf Technol 71(5-8):1159–1174
Iglesias A, Munoa J, Ciurana J (2014) Optimisation of face milling operations with structural chatter using a stability model based process planning methodology. Int J Adv Manuf Technol 70(1-4):559–571
Du H, Zhao C, Wu W (2014) Stability criteria based on argument principle of a general dynamical system in cutting process. Int J Adv Manuf Technol 70(1-4):747–753
Ozoegwu CG (2014) Least squares approximated stability boundaries of milling process. Int J Mach Tools Manuf 79:24–30
Tlusty J, Ismail F (1981) Basic non-linearity in machining chatter. CIRP Ann-Manuf Technol 30(1):299–304
Tlusty J, Ismail F (1983) Special aspects of chatter in milling. J Vib Acoust 105(1):24–32
Smith S, Tlusty J (1993) Efficient simulation programs for chatter in milling. CIRP Ann-Manuf Technol 42(1):463–466
Campomanes ML, Altintas Y (2003) An improved time domain simulation for dynamic milling at small radial immersions. J Manuf Sci Eng 125(3):416–422
Davies MA, Pratt JR, Dutterer B et al (2002) Stability prediction for low radial immersion milling. J Manuf Sci Eng 124(2):217–225
Davies MA, Pratt JR, Dutterer B et al (2002) Stability prediction for low radial immersion milling. J Manuf Sci Eng 124(2):217–225
Altintaş Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann-Manuf Technol 44(1):357–362
Budak E, Altintas Y (1998) Analytical prediction of chatter stability in milling—part I: general formulation. J Dyn Syst Meas Control 120(1):22–30
Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci Eng 126(3):459–466
Insperger T, Stépán G (2002) Semi‐discretization method for delayed systems. Int J Numer Methods Eng 55(5):503–518
Insperger T, Stépán G (2004) Updated semi‐discretization method for periodic delay‐differential equations with discrete delay. Int J Numer Methods Eng 61(1):117–141
Dombovari Z, Altintas Y, Stepan G (2010) The effect of serration on mechanics and stability of milling cutters. Int J Mach Tools Manuf 50(6):511–520
Ahmadi K, Ismail F (2012) Stability lobes in milling including process damping and utilizing multi-frequency and semi-discretization methods. Int J Mach Tools Manuf 54:46–54
Ding Y, Zhu LM, Zhang XJ et al (2010) A full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 50(5):502–509
Ding Y, Zhu LM, Zhang XJ et al (2010) Second-order full-discretization method for milling stability prediction. Int J Mach Tools Manuf 50(10):926–932
Insperger T (2010) Full-discretization and semi-discretization for milling stability prediction: some comments. Int J Mach Tools Manuf 50(7):658–662
Liu Y, Zhang D, Wu B (2012) An efficient full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 63:44–48
Bayly PV, Halley JE, Mann BP et al (2003) Stability of interrupted cutting by temporal finite element analysis. J Manuf Sci Eng 125(2):220–225
Mann BP, Young KA, Schmitz TL et al (2005) Simultaneous stability and surface location error predictions in milling. J Manuf Sci Eng 127(3):446–453
Bayly P V, Mann B P, Schmitz T L, et al. (2002) Effects of radial immersion and cutting direction on chatter instability in end-milling[C]//ASME 2002 International Mechanical Engineering Congress and Exposition. Am Soc Mechanical Eng 351-363
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Dong, X., Zhang, W. & Deng, S. The reconstruction of a semi-discretization method for milling stability prediction based on Shannon standard orthogonal basis. Int J Adv Manuf Technol 85, 1501–1511 (2016). https://doi.org/10.1007/s00170-015-7719-5
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DOI: https://doi.org/10.1007/s00170-015-7719-5