Skip to main content
Log in

Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process

Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper is focused on nonlinear dynamics of a shell-shaped workpiece during high speed milling. The shell-shaped workpiece is modeled as a double-curved cantilevered shell subjected to a cutting force with time delay effects. Equations of motion are derived by using the Hamilton principle based on the classical shell theory and von Karman strain-displacement relation. The resulting nonlinear partial differential equations are reduced to a two-degree-of-freedom nonlinear system by applying the Galerkin approach. The averaging method is used to obtain four-dimensional averaged equations for the case of foundational parametric resonance and 1:2 internal resonance. Using a numerical method, the dynamics of the cantilevered shell-shaped workpiece is studied under time-delay effects, parametric excitation, and forcing excitation. It is found that time-delay parameters have great impact on chaotic motion. With increasing amplitude of forcing and parametric excitations, the shell-shaped workpiece exhibits different dynamic behavior.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. Taylor, F.W.: On the art of cutting metal. Trans. ASME 28 (1907)

  2. Tobias, S.A.: Machine-Tool Vibration. Wiley, New York (1965)

    Google Scholar 

  3. Tlusty, J., Ismail, F.: Basic nonlinearity in machining chatter. CIRP Ann. 30, 299–304 (1981)

    Article  Google Scholar 

  4. Kim, H.S., Ehmann, K.F.: A cutting force model for face milling operations. Int. J. Mach. Tools Manuf. 33, 651–673 (1993)

    Article  Google Scholar 

  5. Budak, E.: Analytical models for high performance milling: Part II: Process dynamics and stability. Int. J. Mach. Tools Manuf. 46, 1489–1499 (2006)

    Article  Google Scholar 

  6. Sutherland, J.W., DeVor, R.E.: An improved method for cutting force and surface error prediction in flexible end milling systems. J. Eng. Ind. 108, 269–279 (1986)

    Article  Google Scholar 

  7. Altintas, Y., Lee, P.: A general mechanics and dynamics model for helical end mills. CIRP Ann. 45, 59–64 (1996)

    Article  Google Scholar 

  8. Shirase, K., Altintas, Y.: Cutting force and dimensional surface error generation in peripheral milling with variable pitch helical end mills. Mach. Tools Manuf. 36, 567–584 (1996)

    Article  Google Scholar 

  9. Zhao, M.X., Balachandran, B.: Dynamics and stability of milling process. Int. J. Solids Struct. 38, 2233–2248 (2001)

    Article  MATH  Google Scholar 

  10. Balachandran, B.: Nonlinear dynamics of milling process. Philos. Trans. R. Soc. 359, 793–819 (2001)

    Article  MATH  Google Scholar 

  11. Nayfeh, A.H.: Perturbation Method. Wiley, New York (2008)

    Google Scholar 

  12. Nayfeh, A.H.: Nonlinear Oscillations. Wiley, New York (1974)

    Google Scholar 

  13. Kalmar-Nagy, T., Stepan, G., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gradisek, J., Govekar, E., Grabec, I.: Time series analysis in metal cutting: chatter versus chatter-free cutting. Mech. Syst. Signal Process. 12, 839–854 (1998)

    Article  Google Scholar 

  15. Gradisek, J., Govekar, E., Grabec, I.: Using coarse-grained entropy rate to detect chatter in cutting. J. Sound Vib. 214, 941–952 (1998)

    Article  Google Scholar 

  16. Stepan, G.: Delay-differential equation models for machine tool chatter. In: Moon, F.C. (ed.) Dynamics and Chaos in Manufacturing Processes, pp. 165–192. Wiley, New York (1998)

    Google Scholar 

  17. Insperger, T., Stepan, G.: Stability of the milling process. Period. Polytech. 44, 47–57 (2000)

    Google Scholar 

  18. Nayfeh, M.A., Hamdan, A.M.A., Nayfeh, A.H.: Chaos and instability in a power system-primary resonant case. Nonlinear Dyn. 1, 313–339 (1990)

    Article  Google Scholar 

  19. Nayfeh, M.A., Hamdan, A.M.A., Nayfeh, A.H.: Chaos and instability in a power system-subharmonic-resonant case. Nonlinear Dyn. 2, 53–72 (1991)

    Article  Google Scholar 

  20. Insperger, T., Stepan, G., Bayly, P.V., Mann, B.P.: Multiple chatter frequencies in milling process. J. Sound Vib. 262, 333–345 (2003)

    Article  Google Scholar 

  21. Davies, M.A., Balachandran, B.: Impact dynamics in milling of thin-walled structures. Nonlinear Dyn. 22, 375–392 (2000)

    Article  MATH  Google Scholar 

  22. Neyfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. Appl. Mech. Rev. 42, 175–201 (1989)

    Article  Google Scholar 

  23. Grabec, I.: Chaotic dynamics of the cutting process. Int. J. Mach. Tools Manuf. 28, 19–32 (1988)

    Article  Google Scholar 

  24. Nayfeh, T.A., Asrar, W., Nayfeh, A.H.: Three-mode interactions in harmonically excited systems with quadratic nonlinearities. Nonlinear Dyn. 3, 385–410 (1992)

    Article  Google Scholar 

  25. Nayfeh, A.H., Chin, C.: A parametrically excited system with widely spaced frequencies and cubic nonlinearities. Nonlinear Dyn. 336, 405 (1994)

    Google Scholar 

  26. Stepan, G., Szalai, R.: Nonlinear vibrations of highly interrupted machining. In: Proceedings of Dynamics and Control of Mechanical Processing, Budapest, Hungary, pp. 59–64 (2001)

    Google Scholar 

  27. Bailey, T., Elbestawi, M.A., El-Wardany, T.I.: Generic simulations approach for multi-axis machining, parts 1 and 2. ASME J. Manuf. Sci. Eng. 124, 624–642 (2002)

    Article  Google Scholar 

  28. Gradisek, J., Kalveram, M., Weinert, K.: Mechanistic identification of specific force coefficients for a general end mill. Int. J. Mach. Tools Manuf. 44, 401–414 (2003)

    Article  Google Scholar 

  29. Pratt, J.R., Nayfeh, A.H.: Design and modeling for chatter control. Nonlinear Dyn. 19, 49–69 (1999)

    Article  MATH  Google Scholar 

  30. Faassen, R.P.H., Van de Wouw, N., Oosterling, J.A.J., Nijmeijer, H.: Prediction of regenerative chatter by modeling and analysis of high-speed milling. Int. J. Mach. Tools Manuf. 43, 1437–1446 (2003)

    Article  Google Scholar 

  31. Jayaram, S., Kapoor, S.G., Devor, R.E.: Estimation of the specific cutting pressures for mechanistic cutting force models. Int. J. Mach. Tools Manuf. 41, p265–281 (2001)

    Article  Google Scholar 

  32. Chiang, S.T., Tsai, C.M., Lee, A.C.: Analysis of cutting forces in ball-end milling. J. Master Process Technol. 47, 231–249 (1995)

    Article  Google Scholar 

  33. Zhu, R., Kapoor, S.G., Devor, R.E.: Mechanistic modeling of the ball end milling process for multi-axis machining of free-form surfaces. ASME J. Manuf. Sci. Eng. 123, 369–379 (2001)

    Article  Google Scholar 

  34. Yun, W.S., Cho, D.W.: An improved method for the determination of 3D cutting force coefficients and run-out parameters in end milling. Int. J. Adv. Manuf. Technol. 16, 851–858 (2000)

    Article  MATH  Google Scholar 

  35. Yun, W.S., Cho, D.W.: Accurate 3D cutting force prediction using cutting-condition-independent coefficients in end milling. Int. J. Mach. Tools Manuf. 41, 463–478 (2001)

    Article  Google Scholar 

  36. Ko, J.H., Yun, W.S., Cho, D.W., Ehmann, K.F.: Development of a virtual machining system, part 1: approximation of the size effect for cutting force prediction. Int. J. Mach. Tools Manuf. 42, 1595–1605 (2002)

    Article  Google Scholar 

  37. Engin, S., Altintas, Y.: Mechanics and dynamics of general milling cutter: Part II: inserted cutters. Int. J. Mach. Tools Manuf. 41, 2213–2231 (2001)

    Article  Google Scholar 

  38. Azeem, A., Feng, H.Y., Wang, L.: Simplified and efficient calibration of a mechanistic cutting force model for ball-end milling. Int. J. Mach. Tools Manuf. 44, 291–298 (2004)

    Article  Google Scholar 

  39. Axelrad, E.L.: Theory of Flexible Shells. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  40. Rao, S.S.: Mechanical Vibrations. Prentice Hall, Singapore (2005)

    Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11290152, 11072008 and 10732020, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), and the Natural Science and Engineering Research Council for Discovery Accelerator Supplement of Canada (NSERC).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to W. Zhang.

Appendices

Appendix A

The coefficients presented in (27a)–(27d) are as follows:

(A.1)

Appendix B

The coefficients presented in (32) are as follows:

(B.1)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, W., Zhou, R. & Zu, J.W. Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process. Nonlinear Dyn 72, 767–787 (2013). https://doi.org/10.1007/s11071-013-0752-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-0752-8

Keywords

Navigation