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December  2018, 23(10): 4329-4360. doi: 10.3934/dcdsb.2018167

## 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

* Corresponding author: Qingwen Hu

Received  March 2017 Revised  February 2018 Published  June 2018

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.

Citation: Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167
##### References:
 [1] Y. Altintas and E. Budak, Analytical prediction of stability lobes in milling, CIRP Annals - Manufacturing Technology 44, 1 (1995), 357-362. Google Scholar [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. Google Scholar [3] B. Balachandran and M. X. Zhao, A mechanics based model for study of dynamics of milling operations, Meccanica, 2 (2000), 89-109. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] T. Insperger and G. Stépán, Stability analysis of turning with periodic spindle speed maching, J. Manuf. Sci. Eng., 122 (2000), 391-397. Google Scholar [11] T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. Google Scholar [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. Google Scholar [13] F. Koenigsberger and J. Tlusty, Machine Tool Structures, vol. 1. Pergamon Press, 1970.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] S. Smith and J. Tlusty, Update on high-speed milling dynamics, ASME Journal of Engineering for Industry, 112 (1990), 142-149. Google Scholar [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. Google Scholar [21] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Sci. tech., UK, 1989. Google Scholar [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. Google Scholar [23] F. W. Taylor, On the art of cutting metals, Oscillation and Dynamics in Delay Equations, Contemporary Mathematics, 1907.Google Scholar [24] S. A. Tobias, Machine Tool Vibration, Blackie, London, 1965.Google Scholar [25] S. A. Tobias and W. Fishwick, Theory of regenerative machine tool chatter, The Engineer, London, 205 (1958), 199-203. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [3] B. Balachandran and M. X. Zhao, A mechanics based model for study of dynamics of milling operations, Meccanica, 2 (2000), 89-109. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] T. Insperger and G. Stépán, Stability analysis of turning with periodic spindle speed maching, J. Manuf. Sci. Eng., 122 (2000), 391-397. Google Scholar [11] T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. Google Scholar [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. Google Scholar [13] F. Koenigsberger and J. Tlusty, Machine Tool Structures, vol. 1. Pergamon Press, 1970.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] S. Smith and J. Tlusty, Update on high-speed milling dynamics, ASME Journal of Engineering for Industry, 112 (1990), 142-149. Google Scholar [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. Google Scholar [21] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Sci. tech., UK, 1989. Google Scholar [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. Google Scholar [23] F. W. Taylor, On the art of cutting metals, Oscillation and Dynamics in Delay Equations, Contemporary Mathematics, 1907.Google Scholar [24] S. A. Tobias, Machine Tool Vibration, Blackie, London, 1965.Google Scholar [25] S. A. Tobias and W. Fishwick, Theory of regenerative machine tool chatter, The Engineer, London, 205 (1958), 199-203. Google Scholar [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. Google Scholar
Turning model
The graphs of $y = f(\beta) = \frac{\sin\beta}{\cos\beta(1-\cos\beta)}$ and $y = g(\beta) = -\frac{2q}{\beta}$
Graphs of $y = \delta(\beta) = \frac{\xi\beta^2(1-\cos\beta)+\beta^3\sin\beta}{2q(1-\cos\beta)+\beta\sin\beta}$ and $y = h_2(\beta) = -\frac{q\beta(1-\cos\beta)-\xi\beta-q\xi\sin\beta}{\xi\beta(1-\cos\beta)+\beta^2\sin\beta}$ with $\xi<2q$
The graphs of $y = F(\beta) = \beta\cot\frac{\beta}{2}$, and those of $y = G(\beta)$, $y = H(\beta)$ which are the upper and lower part of the curve $\frac{(y+q)^2}{q^2}+\frac{\beta^2}{\left(\frac{q^2}{1-\frac{2q}{\xi}}\right)} = 1$, $\beta>0$, respectively
The curves of $(\delta, \, h_1)$, $\delta>0$ where $\beta_1\frac{\sqrt{2\sqrt{5}-2}}{4\sqrt{5}-8} = 8.955929<q$
The shaded region shows the details of the stability region of Figure 5 near the origin $(0, \, 0)$
The curves of $(\delta, \, h_1)$, $\delta>0$ where $\beta_1\frac{\sqrt{2\sqrt{5}-2}}{4\sqrt{5}-8} = 8.955929>q$. The shaded region near the origin $(0, \, 0)$ is the stability region
The curves of $(\delta, \, h_2)$, $\delta>0$, $h_2>0$ where $\xi<2q$. The shaded region near the origin $(0, \, 0)$ is the stability region
The curves of $(\delta, \, h_2)$, $\delta>0, \, h_2>0$ with $\xi>2q$. The connected region near the origin $(0, \, 0)$ without crossing the lobes, or the vertical line $\delta = 0$ or the horizontal line $h_2 = 0$ is the stability region
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