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Application of Differential Transformation Method for Nonlinear Cutting Tool Vibration

Yıl 2021, Cilt: 8 Sayı: 1, 109 - 122, 29.03.2021

Öz

This article aims to solve the mathematical model developed for machine tool vibrations by employing Differential Transformation Method (DTM). Multi-step differential transformation method is used as it has been shown to have good accuracy in physical applications. Nondimensionalization (scaling) technique is used in order to fully understand the physical effects of each varying parameter. Transformed function of delayed nonlinear velocity terms are explained. MatLab® software is used for DTM solutions. The equation of motion is solved with the DDE23 function in MatLab® software as well as with MatLab®/Simulink® software to compare the results. The solution of the fundamental DTM which is obtained by using the constant transformed function values which differ from other methods after a certain time period. However, the results obtained with the transformed functions for different values at each sampling time (Multi-Step DTM) give very close results to Simulink® and DDE23.

Destekleyen Kurum

The Scientific and Technological Research Council of Turkey (TUBITAK)

Proje Numarası

National Scholarships Programme (2211-A) for PhD Students

Teşekkür

The authors would like to thank The Scientific and Technological Research Council of Turkey (TUBITAK) for their support of the National Scholarships Programme (2211-A) for PhD Students.

Kaynakça

  • Abdel, I. H., & Hassan, H. (2002). Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation, 129(2-3), 183-201. doi:10.1016/S0096-3003(01)00037-6
  • Altintas, Y. (2012). Manufacturing Automation. New York: Cambridge University Press.
  • Arıkoğlu, A., & Özkol, İ. (2005). Solution of boundary value problems for integro-differential equations by using differential transform method. Applied Mathematics and Computation, 168(2), 1145-1158. doi:10.1016/j.amc.2004.10.009
  • Arıkoğlu, A., & Özkol, İ. (2006a). Solution of difference equations by using differential transform method. Applied Mathematics and Computation, 174(2), 1216-1228. https://doi.org/10.1016/j.amc.2005.06.013
  • Arıkoğlu, A., & Özkol, İ. (2006b). Solution of differential-difference equations by using differential transformation method. Applied Mathematics and Computation, 181(1), 153-162. doi:10.1016/j.amc.2006.01.022
  • Arıkoğlu, A., & Özkol, İ. (2008). Solutions of integral and integro-differential equation systems by using differential transform method. Computers and Mathematics with Applications, 56(9), 2411-2417. doi:10.1016/j.camwa.2008.05.017
  • Astakhov, V. P. (2004). The assessment of cutting tool wear. International Journal of Machine Tools and Manufacture, 44(6), 637-647. doi:10.1016/j.ijmachtools.2003.11.006
  • Ayaz, F. (2004). Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 147(2), 547-567. doi:10.1016/S0096-3003(02)00794-4
  • Benhammouda, B., & Leal, H. V. (2016). A new multi‑step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations. SpringerPlus, 5, 1723. doi:10.1186/s40064-016-3386-8
  • Bozdogan, K. B., & Ozturk, D. (2014). Free vibration analysis of the tube-in-tube tall buildings with the differential transform method. Advances in Structural Engineering, 17(9), 1271-1279. doi:10.1260/1369-4332.17.9.1271
  • Bozdogan, K. B., & Aydin, S. (2016). Lateral stability analysis of multistory buildings using the differential transform method. Structural Engineering and Mechanics, 57(5), 861-876. doi:10.12989/sem.2016.57.5.861
  • Chen, S-S., & Chen, C-K. (2009). Application of the differential transformation method to the free vibrations of strongly non-linear oscillators. Nonlinear Analysis: Real World Applications, 10(2), 881-888. doi:10.1016/j.nonrwa.2005.06.010
  • Chowdhury, M. A., Khalil, M. K., Nuruzzaman, D. M., & Rahaman, M. L. (2011). The effect of sliding speed and normal load on friction and wear property of aluminium. International Journal of Mechanical & Mechatronics Engineering, 11(1), 53–57.
  • El-Shahed, M. (2008). Application of differential transformation method to non-linear oscillatory systems. Communications in Nonlinear Science and Numerical Simulation, 13(8), 1714-1720. doi:10.1016/j.cnsns.2007.03.005
  • Gokdogan, A., Merdan, M., & Yıldırım, A. (2012). Differential transformation method for solving a neutral functional-differential equation with proportional delays. Caspian Journal of Mathematical Sciences, 1(1), 31-37.
  • Hatami, M., Ganji, D. D., & Sheikholeslami, M. (2017). Differential Transformation Method For Mechanical Engineering Problems. Elsevier Academic Press. WOS:000409365500010
  • Jang, M-J., Chen, C-L., & Liy, Y-C. (2000). On solving the initial-value problems using the differential transformation method. Applied Mathematics and Computation, 115(2-3), 145-160. doi:10.1016/S0096-3003(99)00137-X
  • Karakoç, F., & Bereketoğlu, H. (2009). Solutions of delay differential equations by using differential transform method. International Journal of Computer Mathematics, 86(5), 914-923. doi:10.1080/00207160701750575
  • Kuo, B-L., & Lo, C-Y. (2009). Application of the differential transformation method to the solution of a damped system with high nonlinearity. Nonlinear Analysis: Theory, Methods & Applications, 70(4), 1732-1737. doi:10.1016/j.na.2008.02.056
  • Kurnaz, A., & Oturanç, G. (2005). The differential transform approximation for the system of ordinary differential equations. International Journal of Computer Mathematics, 82(6), 709-719. doi:10.1080/00207160512331329050
  • Maekawa, K., Kitagawa, T., Shirakashi, T., & Usui, E. (1989). Analytical prediction of flank wear of carbide tools in turning plain carbon steels II: prediction of flank wear. Bull. Japan. Soc. Prec. Eng, 23(2), 126-133. WOS:A1989AN12800007
  • Mirzaee, F. (2011). Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Applied Mathematical Sciences, 5(70), 3465-3472.
  • Momani, S., & Ertürk, V. S. (2008). Solutions of non-linear oscillators by the modified differential transform method. Computers and Mathematics with Applications, 55(4), 833-842. doi:10.1016/j.camwa.2007.05.009
  • Rebenda, J., & Šmarda, Z. (2017). A differential transformation approach for solving functional differential equations with multiple delays. Commun Nonlinear Science and Numerical Simulation, 48, 246-257. doi:10.1016/j.cnsns.2016.12.027
  • Süngü, İ. Ç., & Demir, H. (2012). Diferansiyel Dönüşüm/Sonlu Fark Yöntemi İle Denklem Sistemlerinin Çözümleri. e-Journal of New World Sciences Academy, 7(2), 66-73.
  • Šmarda, Z., Diblík, J., & Khan, Y. (2013). Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays. Advances in Difference Equations, 69. doi:10.1186/1687-1847-2013-69
  • Tabatabaei, K., & Gunerhan, E. (2014). Numerical solution of duffing equation by the differential transform method. Applied Mathematics & Information Sciences Letters, 2(1), 1-6. doi:10.12785/amisl/020101
  • Taylor, F. W. (1907). On the art of cutting metals. Transactions of the ASME, 28, 31-350.
  • Thongmoon, M., & Pusjuso, S. (2010). The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations. Nonlinear Analysis: Hybrid Systems, 4(3), 425-431. doi:10.1016/j.nahs.2009.10.006
  • Villafuerte, L., & Chen-Charpentier, B. M. (2012). A random differential transform method: Theory and applications. Applied Mathematics Letters, 25(10), 1490-1494. doi:10.1016/j.aml.2011.12.033
  • Zanger, F., & Schulze, V. (2013). Investigations on mechanisms of tool wear in machining of Ti-6Al-4V using FEM simulation. Procedia CIRP, 8, 158-163. doi:10.1016/j.procir.2013.06.082
  • Zhou, J. K. (1986) Differential Transformation and its Application for Electrical Circuit. Wuhan: Huazhong University Press.
Yıl 2021, Cilt: 8 Sayı: 1, 109 - 122, 29.03.2021

Öz

Proje Numarası

National Scholarships Programme (2211-A) for PhD Students

Kaynakça

  • Abdel, I. H., & Hassan, H. (2002). Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation, 129(2-3), 183-201. doi:10.1016/S0096-3003(01)00037-6
  • Altintas, Y. (2012). Manufacturing Automation. New York: Cambridge University Press.
  • Arıkoğlu, A., & Özkol, İ. (2005). Solution of boundary value problems for integro-differential equations by using differential transform method. Applied Mathematics and Computation, 168(2), 1145-1158. doi:10.1016/j.amc.2004.10.009
  • Arıkoğlu, A., & Özkol, İ. (2006a). Solution of difference equations by using differential transform method. Applied Mathematics and Computation, 174(2), 1216-1228. https://doi.org/10.1016/j.amc.2005.06.013
  • Arıkoğlu, A., & Özkol, İ. (2006b). Solution of differential-difference equations by using differential transformation method. Applied Mathematics and Computation, 181(1), 153-162. doi:10.1016/j.amc.2006.01.022
  • Arıkoğlu, A., & Özkol, İ. (2008). Solutions of integral and integro-differential equation systems by using differential transform method. Computers and Mathematics with Applications, 56(9), 2411-2417. doi:10.1016/j.camwa.2008.05.017
  • Astakhov, V. P. (2004). The assessment of cutting tool wear. International Journal of Machine Tools and Manufacture, 44(6), 637-647. doi:10.1016/j.ijmachtools.2003.11.006
  • Ayaz, F. (2004). Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 147(2), 547-567. doi:10.1016/S0096-3003(02)00794-4
  • Benhammouda, B., & Leal, H. V. (2016). A new multi‑step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations. SpringerPlus, 5, 1723. doi:10.1186/s40064-016-3386-8
  • Bozdogan, K. B., & Ozturk, D. (2014). Free vibration analysis of the tube-in-tube tall buildings with the differential transform method. Advances in Structural Engineering, 17(9), 1271-1279. doi:10.1260/1369-4332.17.9.1271
  • Bozdogan, K. B., & Aydin, S. (2016). Lateral stability analysis of multistory buildings using the differential transform method. Structural Engineering and Mechanics, 57(5), 861-876. doi:10.12989/sem.2016.57.5.861
  • Chen, S-S., & Chen, C-K. (2009). Application of the differential transformation method to the free vibrations of strongly non-linear oscillators. Nonlinear Analysis: Real World Applications, 10(2), 881-888. doi:10.1016/j.nonrwa.2005.06.010
  • Chowdhury, M. A., Khalil, M. K., Nuruzzaman, D. M., & Rahaman, M. L. (2011). The effect of sliding speed and normal load on friction and wear property of aluminium. International Journal of Mechanical & Mechatronics Engineering, 11(1), 53–57.
  • El-Shahed, M. (2008). Application of differential transformation method to non-linear oscillatory systems. Communications in Nonlinear Science and Numerical Simulation, 13(8), 1714-1720. doi:10.1016/j.cnsns.2007.03.005
  • Gokdogan, A., Merdan, M., & Yıldırım, A. (2012). Differential transformation method for solving a neutral functional-differential equation with proportional delays. Caspian Journal of Mathematical Sciences, 1(1), 31-37.
  • Hatami, M., Ganji, D. D., & Sheikholeslami, M. (2017). Differential Transformation Method For Mechanical Engineering Problems. Elsevier Academic Press. WOS:000409365500010
  • Jang, M-J., Chen, C-L., & Liy, Y-C. (2000). On solving the initial-value problems using the differential transformation method. Applied Mathematics and Computation, 115(2-3), 145-160. doi:10.1016/S0096-3003(99)00137-X
  • Karakoç, F., & Bereketoğlu, H. (2009). Solutions of delay differential equations by using differential transform method. International Journal of Computer Mathematics, 86(5), 914-923. doi:10.1080/00207160701750575
  • Kuo, B-L., & Lo, C-Y. (2009). Application of the differential transformation method to the solution of a damped system with high nonlinearity. Nonlinear Analysis: Theory, Methods & Applications, 70(4), 1732-1737. doi:10.1016/j.na.2008.02.056
  • Kurnaz, A., & Oturanç, G. (2005). The differential transform approximation for the system of ordinary differential equations. International Journal of Computer Mathematics, 82(6), 709-719. doi:10.1080/00207160512331329050
  • Maekawa, K., Kitagawa, T., Shirakashi, T., & Usui, E. (1989). Analytical prediction of flank wear of carbide tools in turning plain carbon steels II: prediction of flank wear. Bull. Japan. Soc. Prec. Eng, 23(2), 126-133. WOS:A1989AN12800007
  • Mirzaee, F. (2011). Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Applied Mathematical Sciences, 5(70), 3465-3472.
  • Momani, S., & Ertürk, V. S. (2008). Solutions of non-linear oscillators by the modified differential transform method. Computers and Mathematics with Applications, 55(4), 833-842. doi:10.1016/j.camwa.2007.05.009
  • Rebenda, J., & Šmarda, Z. (2017). A differential transformation approach for solving functional differential equations with multiple delays. Commun Nonlinear Science and Numerical Simulation, 48, 246-257. doi:10.1016/j.cnsns.2016.12.027
  • Süngü, İ. Ç., & Demir, H. (2012). Diferansiyel Dönüşüm/Sonlu Fark Yöntemi İle Denklem Sistemlerinin Çözümleri. e-Journal of New World Sciences Academy, 7(2), 66-73.
  • Šmarda, Z., Diblík, J., & Khan, Y. (2013). Extension of the differential transformation method to nonlinear differential and integro-differential equations with proportional delays. Advances in Difference Equations, 69. doi:10.1186/1687-1847-2013-69
  • Tabatabaei, K., & Gunerhan, E. (2014). Numerical solution of duffing equation by the differential transform method. Applied Mathematics & Information Sciences Letters, 2(1), 1-6. doi:10.12785/amisl/020101
  • Taylor, F. W. (1907). On the art of cutting metals. Transactions of the ASME, 28, 31-350.
  • Thongmoon, M., & Pusjuso, S. (2010). The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations. Nonlinear Analysis: Hybrid Systems, 4(3), 425-431. doi:10.1016/j.nahs.2009.10.006
  • Villafuerte, L., & Chen-Charpentier, B. M. (2012). A random differential transform method: Theory and applications. Applied Mathematics Letters, 25(10), 1490-1494. doi:10.1016/j.aml.2011.12.033
  • Zanger, F., & Schulze, V. (2013). Investigations on mechanisms of tool wear in machining of Ti-6Al-4V using FEM simulation. Procedia CIRP, 8, 158-163. doi:10.1016/j.procir.2013.06.082
  • Zhou, J. K. (1986) Differential Transformation and its Application for Electrical Circuit. Wuhan: Huazhong University Press.
Toplam 32 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makine Mühendisliği
Yazarlar

İbrahim Demir 0000-0002-2128-3949

M. M. Fatih Karahan 0000-0001-9040-5041

Nizami Aktürk 0000-0003-1398-8145

Proje Numarası National Scholarships Programme (2211-A) for PhD Students
Yayımlanma Tarihi 29 Mart 2021
Gönderilme Tarihi 28 Ekim 2020
Yayımlandığı Sayı Yıl 2021 Cilt: 8 Sayı: 1

Kaynak Göster

APA Demir, İ., Karahan, M. M. F., & Aktürk, N. (2021). Application of Differential Transformation Method for Nonlinear Cutting Tool Vibration. Gazi University Journal of Science Part A: Engineering and Innovation, 8(1), 109-122.