This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.
Issue Section:
Technical Brief
References
1.
Tlusty
, J.
, 2000
, Manufacturing Processes and Equipment
, Prentice Hall
, Upper Saddle River, NJ
.2.
Budak
, E.
, 2006
, “Analytical Models for High Performance Milling. Part II: Process Dynamics and Stability
,” Int. J. Mach. Tools Manuf.
, 46
(12–13
), pp. 1489
–1499
.10.1016/j.ijmachtools.2005.09.0103.
Schmitz
, T. L.
, and Smith
, K. S.
, 2008
, Machining Dynamics: Frequency Response to Improved Productivity
, Springer
, New York
.4.
Altintas
, Y.
, 2000
, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design
, Cambridge University
, Cambridge, UK
.5.
Schmitz
, T. L.
, Davies
, M. A.
, and Kennedy
, M. D.
, 2001
, “Tool Point Frequency Response Prediction for High-Speed Machining by RCSA
,” ASME J. Manuf. Sci. Eng.
, 123
(4
), pp. 700
–707
.10.1115/1.13929946.
Schmitz
, T. L.
, and Duncan
, G. S.
, 2005
, “Three-Component Receptance Coupling Substructure Analysis for Tool Point Dynamics Prediction
,” ASME J. Manuf. Sci. Eng.
, 127
(4
), pp. 781
–790
.10.1115/1.20391027.
Ghanati
, M. F.
, and Madoliat
, R.
, 2012
, “New Continuous Dynamic Coupling for Three Component Modeling of Tool-Holder-Spindle Structure of Machine Tools With Modified Effected Tool Damping
,” ASME J. Manuf. Sci. Eng.
, 134
(2
), p. 021015
.10.1115/1.40060948.
Law
, M.
, Phani
, A. S.
, and Altintas
, Y.
, 2013
, “Position-Dependent Multibody Dynamic Modeling of Machine Tools Based on Improved Reduced Order Models
,” ASME J. Manuf. Sci. Eng.
, 135
(2
), p. 021008
.10.1115/1.40234539.
Karandikar
, J. M.
, Schmitz
, T. L.
, and Abbas
, A. E.
, 2014
, “Application of Bayesian Inference to Milling Force Modeling
,” ASME J. Manuf. Sci. Eng.
, 136
(2
), p. 021017
.10.1115/1.402636510.
Budak
, E.
, and Tekeli
, A.
, 2005
, “Maximizing Chatter Free Material Removal Rate in Milling Through Optimal Selection of Axial and Radial Depth of Cut Pairs
,” CIRP Ann. Manuf. Technol.
, 54
(1
), pp. 353
–356
.10.1016/S0007-8506(07)60121-811.
Kurdi
, M. H.
, Schmitz
, T. L.
, Haftka
, R. T.
, and Mann
, B. P.
, 2009
, “Milling Optimisation of Removal Rate and Accuracy With Uncertainty: Part 1: Parameter Selection
,” Int. J. Mater. Prod. Technol.
, 35
(1–2
), pp. 3
–25
.10.1504/IJMPT.2009.02520912.
Merdol
, S. D.
, and Altintas
, Y.
, 2008
, “Virtual Simulation and Optimization of Milling Applications—Part II: Optimization and Feedrate Scheduling
,” ASME J. Manuf. Sci. Eng.
, 130
(5
), p. 051005
.10.1115/1.292743513.
Altintas
, Y.
, Stépán
, G.
, Merdol
, D.
, and Dombovari
, Z.
, 2008
, “Chatter Stability of Milling in Frequency and Discrete Time Domain
,” CIRP J. Manuf. Sci. Technol.
, 1
(1
), pp. 35
–44
.10.1016/j.cirpj.2008.06.00314.
Quintana
, G.
, and Ciurana
, J.
, 2011
, “Chatter in Machining Processes: A Review
,” Int. J. Mach. Tools Manuf.
, 51
(5
), pp. 363
–376
.10.1016/j.ijmachtools.2011.01.00115.
Altintas
, Y.
, and Budak
, E.
, 1995
, “Analytical Prediction of Stability Lobes in Milling
,” CIRP Ann. Manuf. Technol.
, 44
(1
), pp. 357
–362
.10.1016/S0007-8506(07)62342-716.
Budak
, E.
, and Altintas
, Y.
, 1998
, “Analytical Prediction of Chatter Stability in Milling—Part I: General Formulation
,” ASME J. Dyn. Syst. Meas. Contr.
, 120
(1
), pp. 22
–30
.10.1115/1.280131717.
Tunc
, L. T.
, and Budak
, E.
, 2013
, “Identification and Modeling of Process Damping in Milling
,” ASME J. Manuf. Sci. Eng.
, 135
(2
), p. 021001
.10.1115/1.402370818.
Zheng
, C. M.
, Wang
, J.-J. J.
, and Sung
, C. F.
, 2014
, “Analytical Prediction of the Critical Depth of Cut and Worst Spindle Speeds for Chatter in End Milling
,” ASME J. Manuf. Sci. Eng.
, 136
(1
), p. 011003
.10.1115/1.402545219.
Insperger
, T.
, and Stépán
, G.
, 2002
, “Semi-Discretization Method for Delayed Systems
,” Int. J. Numer. Methods Eng.
, 55
(5
), pp. 503
–518
.10.1002/nme.50520.
Insperger
, T.
, and Stépán
, G.
, 2011
, Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications
, Springer-Verlag
, New York
.21.
Bayly
, P. V.
, Halley
, J. E.
, Mann
, B. P.
, and Davies
, M. A.
, 2003
, “Stability of Interrupted Cutting by Temporal Finite Element Analysis
,” ASME J. Manuf. Sci. Eng.
, 125
(2
), pp. 220
–225
.10.1115/1.155686022.
Butcher
, E. A.
, Bobrenkov
, O. A.
, Bueler
, E.
, and Nindujarla
, P.
, 2009
, “Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels
,” ASME J. Comput. Nonlinear Dyn.
, 4
(3
), p. 031003
.10.1115/1.312408823.
Butcher
, E. A.
, and Bobrenkov
, O. A.
, 2011
, “On the Chebyshev Spectral Continuous Time Approximation for Constant and Periodic Delay Differential Equations
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(3
), pp. 1541
–1554
.10.1016/j.cnsns.2010.05.03724.
Ding
, Y.
, Zhu
, L.
, Zhang
, X.
, and Ding
, H.
, 2010
, “A Full-Discretization Method for Prediction of Milling Stability
,” Int. J. Mach. Tools Manuf.
, 50
(5
), pp. 502
–509
.10.1016/j.ijmachtools.2010.01.00325.
Schmitz
, T.
, and Ziegert
, J.
, 1999
, “Examination of Surface Location Error Due to Phasing of Cutter Vibrations
,” Precis. Eng.
, 23
(1
), pp. 51
–62
.10.1016/S0141-6359(98)00025-726.
Schmitz
, T. L.
, and Mann
, B. P.
, 2006
, “Closed-Form Solutions for Surface Location Error in Milling
,” Int. J. Mach. Tools Manuf.
, 46
(12–13
), pp. 1369
–1377
.10.1016/j.ijmachtools.2005.10.00727.
Mann
, B. P.
, Young
, K. A.
, Schmitz
, T. L.
, and Dilley
, D. N.
, 2005
, “Simultaneous Stability and Surface Location Error Predictions in Milling
,” ASME J. Manuf. Sci. Eng.
, 127
(3
), pp. 446
–453
.10.1115/1.194839428.
Mann
, B. P.
, Edes
, B. T.
, Easley
, S. J.
, Young
, K. A.
, and Ma
, K.
, 2008
, “Chatter Vibration and Surface Location Error Prediction for Helical End Mills
,” Int. J. Mach. Tools Manuf.
, 48
(3–4
), pp. 350
–361
.10.1016/j.ijmachtools.2007.10.00329.
Insperger
, T.
, Gradisek
, J.
, Kalveram
, M.
, Stépán
, G.
, Winert
, K.
, and Govekar
, E.
, 2006
, “Machine Tool Chatter and Surface Location Error in Milling Processes
,” ASME J. Manuf. Sci. Eng.
, 128
(4
), pp. 913
–920
.10.1115/1.228063430.
Bachrathy
, D.
, Insperger
, T.
, and Stepan
, G.
, 2009
, “Surface Properties of the Machined Workpiece for Helical Mills
,” Mach. Sci. Technol.
, 13
(2
), pp. 227
–245
.10.1080/1091034090301216731.
Ding
, Y.
, Zhu
, L.
, Zhang
, X.
, and Ding
, H.
, 2011
, “On a Numerical Method for Simultaneous Prediction of Stability and Surface Location Error in Low Radial Immersion Milling
,” ASME J. Dyn. Syst. Meas. Contr.
, 133
(2
), p. 024503
.10.1115/1.400337432.
Eksioglu
, C.
, Kilic
, Z. M.
, and Altintas
, Y.
, 2012
, “Discrete-Time Prediction of Chatter Stability, Cutting Forces, and Surface Location Errors in Flexible Milling Systems
,” ASME J. Manuf. Sci. Eng.
, 134
(6
), p. 061006
.10.1115/1.400762233.
Ding
, Y.
, Zhu
, L.
, Zhang
, X.
, and Ding
, H.
, 2013
, “Stability Analysis of Milling Via the Differential Quadrature Method
,” ASME J. Manuf. Sci. Eng.
, 135
(4
), p. 044502
.10.1115/1.402453934.
Horn
, R. A.
, and Johnson
, C. R.
, 1991
, Topics in Matrix Analysis
, Cambridge University
, Cambridge, UK
.35.
Farkas
, M.
, 1994
, Periodic Motions
, Springer-Verlag
, New York
.36.
Kurdi
, M. H.
, 2005
, “Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling Under Uncertainty
,” Ph.D. thesis, University of Florida, Gainesville, FL.37.
Rao
, S. S.
, 2009
, Engineering Optimization: Theory and Practice
, Wiley
, Hoboken, NJ
.38.
Yuan
, Y.
, and Sun
, W.
, 1997
, Optimization Theory and Methods
, Science Press
, Beijing, China
.39.
Lagarias
, J. C.
, Reeds
, J. A.
, Wright
, M. H.
, and Wright
, P. E.
, 1998
, “Convergence Properties of the Nelder–Mead Simplex Method in Low Dimensions
,” SIAM J. Optim.
, 9
(1
), pp. 112
–147
.10.1137/S105262349630347040.
Kolda
, T. G.
, Lewis
, R. M.
, and Torczon
, V.
, 2003
, “Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods
,” SIAM Rev.
, 45
(3
), pp. 385
–482
.10.1137/S00361445024288941.
Ma
, L.
, Melkote
, S. N.
, and Castle
, J. B.
, 2013
, “A Model-Based Computationally Efficient Method for On-Line Detection of Chatter in Milling
,” ASME J. Manuf. Sci. Eng.
, 135
(3
), p. 031007
.10.1115/1.402371642.
Hynynen
, K. M.
, Ratava
, J.
, Lindh
, T.
, Rikkonen
, M.
, Ryynänen
, V.
, Lohtander
, M.
, and Varis
, J.
, 2014
, “Chatter Detection in Turning Processes Using Coherence of Acceleration and Audio Signals
,” ASME J. Manuf. Sci. Eng.
, 136
(4
), p. 044503
.10.1115/1.402694843.
Striz
, A.
, Wang
, X.
, and Bert
, C.
, 1995
, “Harmonic Differential Quadrature Method and Applications to Analysis of Structural Components
,” Acta Mech.
, 111
(1
), pp. 85
–94
.10.1007/BF0118772944.
Wang
, Y.
, 2001
, “Differential Quadrature Method and Differential Qudrature Element Method—Theory and Application
,” Ph.D. thesis, Nanjing University of Aeronautics & Astronautics, Nanjing, China.Copyright © 2015 by ASME
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