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Stabilization of turning processes using spindle feedback with state-dependent delay
Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Road, FO. 35, Richardson, TX, 75080, USA |
We develop a stabilization strategy of turning processes by means of delayed spindle control. We show that turning processes which contain intrinsic state-dependent delays can be stabilized by a spindle control with state-dependent delay, and develop analytical description of the stability region in the parameter space. Numerical simulations stability region are also given to illustrate the general results.
References:
[1] |
Y. Altintas and E. Budak,
Analytical prediction of stability lobes in milling, CIRP Annals - Manufacturing Technology 44, 1 (1995), 357-362.
|
[2] |
D. Bachrathy, G. Stépán and J. Turi, State dependent regenerative effect in milling processes,
J. Comput. Nonlinear Dynam. 6, 4 (2011), Article Number: 041002.
doi: 10.1115/1.4003624. |
[3] |
B. Balachandran and M. X. Zhao,
A mechanics based model for study of dynamics of milling operations, Meccanica, 2 (2000), 89-109.
|
[4] |
Z. Balanov, Q. Hu and W. Krawcewicz,
Global Hopf bifurcation of differential equations with threshold-type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670.
doi: 10.1016/j.jde.2014.05.053. |
[5] |
D. E. Gilsinn,
Estimating critical hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter, Nonlinear Dynam., 30 (2002), 103-154.
doi: 10.1023/A:1020455821894. |
[6] |
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Chapter 5: Functional differential equations with state-dependent delays: Theory and applications. In Handbook of Differential
Equations: Ordinary Differential Equations, P. D. A. CaÑada and A. Fonda, Eds., vol. 3.
North-Holland, 2006,435–545.
doi: 10.1016/S1874-5725(06)80009-X. |
[7] |
J. Hale and S. Verduyn Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[8] |
Q. Hu, W. Krawcewicz and J. Turi,
Stabilization in a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1-24.
doi: 10.1137/110823468. |
[9] |
Q. Hu, W. Krawcewicz and J. Turi,
Global stability lobes of a state-dependent model of turning processes, SIAM Journal on Applied Mathematics, 72 (2012), 1383-1405.
doi: 10.1137/110859051. |
[10] |
T. Insperger and G. Stépán,
Stability analysis of turning with periodic spindle speed maching, J. Manuf. Sci. Eng., 122 (2000), 391-397.
|
[11] |
T. Insperger, G. Stépán and J. Turi,
State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283.
|
[12] |
F. Ismail and E. Soliman,
A new method for the identification of stability lobes in machining, Int. J. Mach. Tools Manufacture, 37 (1997), 763-774.
|
[13] |
F. Koenigsberger and J. Tlusty,
Machine Tool Structures, vol. 1. Pergamon Press, 1970. |
[14] |
W. Krawcewicz and J. Wu,
Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts. Johns Wiley & Sons, New York, 1997. |
[15] |
X. Long and B. Balachandran,
Stability of up-milling and down-milling operations with variable spindle speeds, J. Vibration and Control, 16 (2010), 1151-1168.
doi: 10.1177/1077546309341131. |
[16] |
A. Otto and G. Radons,
The influence of tangential and torsional vibrations on the stability lobes in metal cutting, Nonlinear Dyn., 82 (2015), 1989-2000.
|
[17] |
M. Pakdemirli and A. G. Ulsoy,
Perturbation analysis of spindle speed variation in machine tool chatter, J. Vibration and Control, 3 (1996), 261-278.
|
[18] |
J. S. Sexton, R. D. Milne and B. J. Stone,
A stability analysis of single point machining with varying spindle control, Appl. Math. Modeling, 1 (1977), 310-318.
doi: 10.1016/0307-904X(77)90062-2. |
[19] |
S. Smith and J. Tlusty,
Update on high-speed milling dynamics, ASME Journal of Engineering for Industry, 112 (1990), 142-149.
|
[20] |
H. Smith,
Existence and uniqueness of global solutions for a size-structured population model of an insect population with variable instar duration, Rocky Mountain J. Math., 24 (1994), 311-334.
doi: 10.1216/rmjm/1181072468. |
[21] |
G. Stépán,
Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Sci. tech., UK, 1989. |
[22] |
E. Stone and S. Campbell,
Stability and bifurcation analysis of a nonlinear dde model for drilling, J. Nonlinear Science, 14 (2004), 27-57.
doi: 10.1007/s00332-003-0553-1. |
[23] |
F. W. Taylor, On the art of cutting metals, Oscillation and Dynamics in Delay Equations, Contemporary Mathematics, 1907. |
[24] |
S. A. Tobias,
Machine Tool Vibration, Blackie, London, 1965. |
[25] |
S. A. Tobias and W. Fishwick,
Theory of regenerative machine tool chatter, The Engineer, London, 205 (1958), 199-203.
|
[26] |
H.-O. Walther,
The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65.
doi: 10.1016/j.jde.2003.07.001. |
show all references
References:
[1] |
Y. Altintas and E. Budak,
Analytical prediction of stability lobes in milling, CIRP Annals - Manufacturing Technology 44, 1 (1995), 357-362.
|
[2] |
D. Bachrathy, G. Stépán and J. Turi, State dependent regenerative effect in milling processes,
J. Comput. Nonlinear Dynam. 6, 4 (2011), Article Number: 041002.
doi: 10.1115/1.4003624. |
[3] |
B. Balachandran and M. X. Zhao,
A mechanics based model for study of dynamics of milling operations, Meccanica, 2 (2000), 89-109.
|
[4] |
Z. Balanov, Q. Hu and W. Krawcewicz,
Global Hopf bifurcation of differential equations with threshold-type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670.
doi: 10.1016/j.jde.2014.05.053. |
[5] |
D. E. Gilsinn,
Estimating critical hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter, Nonlinear Dynam., 30 (2002), 103-154.
doi: 10.1023/A:1020455821894. |
[6] |
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Chapter 5: Functional differential equations with state-dependent delays: Theory and applications. In Handbook of Differential
Equations: Ordinary Differential Equations, P. D. A. CaÑada and A. Fonda, Eds., vol. 3.
North-Holland, 2006,435–545.
doi: 10.1016/S1874-5725(06)80009-X. |
[7] |
J. Hale and S. Verduyn Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[8] |
Q. Hu, W. Krawcewicz and J. Turi,
Stabilization in a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1-24.
doi: 10.1137/110823468. |
[9] |
Q. Hu, W. Krawcewicz and J. Turi,
Global stability lobes of a state-dependent model of turning processes, SIAM Journal on Applied Mathematics, 72 (2012), 1383-1405.
doi: 10.1137/110859051. |
[10] |
T. Insperger and G. Stépán,
Stability analysis of turning with periodic spindle speed maching, J. Manuf. Sci. Eng., 122 (2000), 391-397.
|
[11] |
T. Insperger, G. Stépán and J. Turi,
State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283.
|
[12] |
F. Ismail and E. Soliman,
A new method for the identification of stability lobes in machining, Int. J. Mach. Tools Manufacture, 37 (1997), 763-774.
|
[13] |
F. Koenigsberger and J. Tlusty,
Machine Tool Structures, vol. 1. Pergamon Press, 1970. |
[14] |
W. Krawcewicz and J. Wu,
Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts. Johns Wiley & Sons, New York, 1997. |
[15] |
X. Long and B. Balachandran,
Stability of up-milling and down-milling operations with variable spindle speeds, J. Vibration and Control, 16 (2010), 1151-1168.
doi: 10.1177/1077546309341131. |
[16] |
A. Otto and G. Radons,
The influence of tangential and torsional vibrations on the stability lobes in metal cutting, Nonlinear Dyn., 82 (2015), 1989-2000.
|
[17] |
M. Pakdemirli and A. G. Ulsoy,
Perturbation analysis of spindle speed variation in machine tool chatter, J. Vibration and Control, 3 (1996), 261-278.
|
[18] |
J. S. Sexton, R. D. Milne and B. J. Stone,
A stability analysis of single point machining with varying spindle control, Appl. Math. Modeling, 1 (1977), 310-318.
doi: 10.1016/0307-904X(77)90062-2. |
[19] |
S. Smith and J. Tlusty,
Update on high-speed milling dynamics, ASME Journal of Engineering for Industry, 112 (1990), 142-149.
|
[20] |
H. Smith,
Existence and uniqueness of global solutions for a size-structured population model of an insect population with variable instar duration, Rocky Mountain J. Math., 24 (1994), 311-334.
doi: 10.1216/rmjm/1181072468. |
[21] |
G. Stépán,
Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Sci. tech., UK, 1989. |
[22] |
E. Stone and S. Campbell,
Stability and bifurcation analysis of a nonlinear dde model for drilling, J. Nonlinear Science, 14 (2004), 27-57.
doi: 10.1007/s00332-003-0553-1. |
[23] |
F. W. Taylor, On the art of cutting metals, Oscillation and Dynamics in Delay Equations, Contemporary Mathematics, 1907. |
[24] |
S. A. Tobias,
Machine Tool Vibration, Blackie, London, 1965. |
[25] |
S. A. Tobias and W. Fishwick,
Theory of regenerative machine tool chatter, The Engineer, London, 205 (1958), 199-203.
|
[26] |
H.-O. Walther,
The solution manifold and $C^1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, 195 (2003), 46-65.
doi: 10.1016/j.jde.2003.07.001. |








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