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Predicting the dynamics of thin-walled parts with curved surfaces in milling based on FEM and Taylor series

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Abstract

The effect of material removal on workpiece dynamics has a great influence on suppressing resonance and regenerative vibrations in milling of thin-walled structures. Although some works have been done to obtain the varying dynamics of the workpiece, many researchers take plate structures as research objects. In this paper, a method based on FEM and Taylor series is proposed to predict the time-varying dynamics of thin-walled blade structures in milling. The input data in the method is the mass and stiffness matrices of the initial workpiece and the change of the matrices caused by material removal. The initial matrices can be obtained through the FEM model established by the mechanical analysis of the eight-node shell element, and the change of the matrices can be obtained by a third-order Taylor series. In comparison with three-dimensional cube elements, the shell elements can reduce the number of degrees of freedom of the FEM model by 74%, which leads to about 9 times faster computation of the characteristic equation. Meanwhile, the proposed method eliminates the necessities of re-building and re-meshing the FEM model at each cutting position, then the computational efficiency of workpiece dynamics is improved. Finally, the result of experiments shows that the maximum prediction error of the proposed method is about 4%.

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References

  1. Tobias SA (1965) Machine-tool vibration. Blackie and Sons Ltd, New York

    Google Scholar 

  2. Smith S, Tlusty J (1993) Efficient simulation programs for chatter in milling. CIRP Ann Manuf Technol 42(1):463–466

    Article  Google Scholar 

  3. Insperger T, Stépán G (2002) Semi-discretization method for delayed systems. Int J Numer Methods Eng 55(5):503–518

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding Y, Zhu LM, Zhang XJ, Ding H (2010) A full-discretization method for prediction of milling stability. Int J Mach Tools Manuf 50(5):502–509

    Article  Google Scholar 

  5. Bayly PV, Halley JE, Mann BP, Davies MA (2003) Stability of interrupted cutting by temporal finite element analysis. Trans ASME J Manuf Sci Eng 125(2):220–225

    Article  Google Scholar 

  6. Altintaş Y, Budak E (1995) Analytical prediction of stability lobes in milling. CIRP Ann Manuf Technol 44(1):357–362

    Article  Google Scholar 

  7. Merdol SD, Altintas Y (2004) Multi frequency solution of chatter stability for low immersion milling. Trans ASME J Manuf Sci Eng 126(3):459–466

    Article  Google Scholar 

  8. Yan BL, Zhu LD (2019) Research on milling stability of thin-walled parts based on improved multi-frequency solution. Int J Adv Manuf Technol 1–11

  9. Comak A, Budak E (2017) Modeling dynamics and stability of variable pitch and helix milling tools for development of a design method to maximize chatter stability. Precis Eng 47:459–468

    Article  Google Scholar 

  10. Bravo U, Altuzarra O, Lacalle De López LN, Sánchez JA, Campa FJ (2005) Stability limits of milling considering the flexibility of the workpiece and the machine. Int J Mach Tools Manuf 45(15):1669–1680

    Article  Google Scholar 

  11. Seguy S, Campa FJ, Lacalle De López LN, Arnaud L, Dessein G, Aramendi G (2008) Toolpath dependent stability lobes for the milling of thin-walled parts. Int J Mach Machinabil Mater 4(4):377–392

    Google Scholar 

  12. Thévenot V, Arnaud L, Dessein G, Cazenave-Larroche G (2006) Influence of material removal on the dynamic behavior of thin-walled structures in peripheral milling. Mach Sci Technol 10(3):275–287

    Article  Google Scholar 

  13. Song QH, Ai X, Tang WX (2011) Prediction of simultaneous dynamic stability limit of time–variable parameters system in thin-walled workpiece high-speed milling processes. Int J Adv Manuf Technol 55(9-12):883–889

    Article  Google Scholar 

  14. Crichigno Filho JM, Negri D, Melotti S (2018) Stable milling of cantilever plates using shell finite elements. Int J Adv Manuf Technol 99(9-12):2677–2693

    Article  Google Scholar 

  15. Meshreki M, Attia H, Kövecses J (2011) Development of a new model for the varying dynamics of flexible pocket-structures during machining. J Manuf Sci Eng 133(4):041002

    Article  Google Scholar 

  16. Song QH, Liu ZQ, Wan Y, Ju GG, Shi JH (2015) Application of sherman-morrison-woodbury formulas in instantaneous dynamic of peripheral milling for thin-walled component. Int J Mech Sci 96:79–90

    Article  Google Scholar 

  17. Song QH, Shi JH, Liu ZQ, Wan Y (2017) A time-space discretization method in milling stability prediction of thin-walled component. Int J Adv Manuf Technol 89(9-12):2675–2689

    Article  Google Scholar 

  18. Ahmadi K (2017) Finite strip modeling of the varying dynamics of thin-walled pocket structures during machining. Int J Adv Manuf Technol 89(9-12):2691–2699

    Article  Google Scholar 

  19. Tuysuz O, Altintas Y (2017) Frequency domain updating of thin-walled workpiece dynamics using reduced order substructuring method in machining. J Manuf Sci Eng 139(7):071013

    Article  Google Scholar 

  20. Yang Y, Zhang WH, Ma YC, Wan M (2016) Chatter prediction for the peripheral milling of thin-walled workpieces with curved surfaces. Int J Mach Tools Manuf 109:36–48

    Article  Google Scholar 

  21. Tuysuz O, Altintas Y (2018) Time-domain modeling of varying dynamic characteristics in thin-wall machining using perturbation and reduced-order substructuring methods. J Manuf Sci Eng 140(1):011015

    Article  Google Scholar 

  22. Dang XB, Wan M, Yang Y, Zhang WH (2019) Efficient prediction of varying dynamic characteristics in thin-wall milling using freedom and mode reduction methods. Int J Mech Sci 150:202–216

    Article  Google Scholar 

  23. Luo M, Zhang DH, Wu BH, Tang M (2011) Modeling and analysis effects of material removal on machining dynamics in milling of thin-walled workpiece. Adv Mater Res 223:671–678. Trans Tech Publ

    Article  Google Scholar 

  24. Tian WJ, Ren JX, Zhou JH, Wang DZ (2018) Dynamic modal prediction and experimental study of thin-walled workpiece removal based on perturbation method. Int J Adv Manuf Technol 94(5–8):2099–2113

    Article  Google Scholar 

  25. Zienkiewicz OC, Taylor RL (2000) The finite element method: solid mechanics, vol 2. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  26. Budak E, Tunç LT, Alan S, Özgüven HN (2012) Prediction of workpiece dynamics and its effects on chatter stability in milling. CIRP Ann Manuf Technol 61(1):339–342

    Article  Google Scholar 

  27. Logan DL (2011) A first course in the finite element method. Cengage Learning, New York

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 51775444).

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Authors and Affiliations

  1. The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, Box 552, Xi’an, Shaanxi, 710072, People’s Republic of China

    Dazhen Wang, Junxue Ren, Weijun Tian, Kaining Shi & Baogang Zhang

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  1. Dazhen Wang
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  2. Junxue Ren
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  3. Weijun Tian
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  4. Kaining Shi
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  5. Baogang Zhang
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Correspondence to Junxue Ren.

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Appendix A: The partial derivatives of the matrices K n and M n

Appendix A: The partial derivatives of the matrices K n and M n

According to Eq. 26, the first-order partial derivatives of the matrices Kn and Mn with respect to thickness ti can be described as:

$$ \begin{array}{@{}rcl@{}} \frac{{\partial {{\mathbf{K}}_{n}}}}{{\partial {t_{i}}}}=\int\!\!\!\int\!\!\!\int{ {\left[ \begin{array}{l} \frac{{\partial {{\mathbf{B}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{DB}}\left| {\mathbf{J}} \right| + {{\mathbf{B}}^{\mathbf{T}}}{\mathbf{D}}\frac{{\partial {\mathbf{B}}}}{{\partial {t_{i}}}}\left| {\mathbf{J}} \right| \\ + {{\mathbf{B}}^{\mathbf{T}}}{\mathbf{DB}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} \end{array} \right]}}{ds dt dn} \end{array} $$
(A.1)
$$ \frac{{\partial {\mathbf{M}_{n}}}}{{\partial {t_{i}}}}=\rho\int\!\!\!\int\!\!\!\int{ {\left[ \begin{array}{l} \frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{N}}\left| {\mathbf{J}} \right| + {{\mathbf{N}}^{\mathbf{T}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i}}}}\left| {\mathbf{J}} \right| \\ + {{\mathbf{N}}^{\mathbf{T}}}{\mathbf{N}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} \end{array} \right]}}{ds dt dn} $$
(A.2)

The second-order partial derivatives of the matrices Kn and Mn with respect to ti can be described as:

$$ \frac{{{\partial^{2}}{\mathbf{K}}}}{{\partial {t_{i}^{2}}}} = \int\!\!\!\int\!\!\!\int{\left[ \begin{array}{l} 2\frac{{\partial {{\mathbf{B}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{D}}\frac{{\partial {\mathbf{B}}}}{{\partial {t_{i}}}}\left| {\mathbf{J}} \right| + 2\frac{{\partial {{\mathbf{B}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{DB}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}}\\ + 2{{\mathbf{B}}^{\mathbf{T}}}{\mathbf{D}}\frac{{\partial {\mathbf{B}}}}{{\partial {t_{i}}}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} + {{\mathbf{B}}^{\mathbf{T}}}{\mathbf{DB}}\frac{{{\partial^{2}}\left| {\mathbf{J}} \right|}}{{\partial {t_{i}^{2}}}} \end{array} \right]}{ds dt dn} $$
(A.3)
$$ \frac{{{\partial^{2}}{\mathbf{M}}}}{{\partial {t_{i}^{2}}}} = \rho\int\!\!\!\int\!\!\!\int{\left[ \begin{array}{l} 2\frac{{\partial {\mathrm{N}^{\mathrm{T}}}}}{{\partial {t_{i}}}}\frac{{\partial \mathrm{N}}}{{\partial {t_{i}}}}\left| {\mathbf{J}} \right| + 2\frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{N}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}}\\ + 2{{\mathbf{N}}^{\mathbf{T}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i}}}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} + {{\mathbf{N}}^{\mathbf{T}}}{\mathbf{N}}\frac{{{\partial^{2}}\left| {\mathbf{J}} \right|}}{{\partial {t_{i}^{2}}}} \end{array} \right]}{ds dt dn} $$
(A.4)

According to Eqs. A.1 and A.2, we can obtain:

$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}{\mathbf{K}_{n}}}}{{\partial {t_{i}}\partial {t_{i + 1}}}}=\int\!\!\!\int\!\!\!\int{ {\left[ \begin{array}{l} \frac{{\partial {{\mathbf{B}^{\mathbf{T}}}}}}{{\partial {t_{i}}}}\mathbf{D}\frac{{\partial \mathbf{B}}}{{\partial {t_{i + 1}}}}\left| \mathbf{J} \right| \\ + \frac{{\partial {{\mathbf{B}^{\mathbf{T}}}}}}{{\partial {t_{i}}}}\mathbf{D}\mathbf{B}\frac{{\partial \left| \mathbf{J} \right|}}{{\partial {t_{i + 1}}}} \\ + \frac{{\partial {{\mathbf{B}^{\mathbf{T}}}}}}{{\partial {t_{i + 1}}}}\mathbf{D}\frac{{\partial \mathbf{B}}}{{\partial {t_{i}}}}\left| \mathbf{J} \right|\\ + {{\mathbf{B}^{\mathbf{T}}}}\mathbf{D}\frac{{\partial \mathbf{B}}}{{\partial {t_{i}}}}\frac{{\partial \left| \mathbf{J} \right|}}{{\partial {t_{i + 1}}}}\\ + \frac{{\partial {{\mathbf{B}^{\mathbf{T}}}}}}{{\partial {t_{i + 1}}}}\mathbf{D}\mathbf{B}\frac{{\partial \left| \mathbf{J} \right|}}{{\partial {t_{i}}}} \\ + {{\mathbf{B}^{\mathbf{T}}}}\mathbf{D}\frac{{\partial \mathbf{B}}}{{\partial {t_{i + 1}}}}\frac{{\partial \left| \mathbf{J} \right|}}{{\partial {t_{i}}}}\\ + {{\mathbf{B}^{\mathbf{T}}}}\mathbf{D}\mathbf{B}\frac{{{\partial^{2}}\left| \mathbf{J} \right|}}{{\partial {t_{i}}\partial {t_{i + 1}}}} \end{array} \right]}}{ds dt dn},\\ (i = 1,2, {\ldots} ,7) \end{array} $$
(A.5)
$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}{{\mathbf{M}}_{n}}}}{{\partial {t_{i}}\partial {t_{i + 1}}}}=\rho\int\!\!\!\int\!\!\!\int{{\left[ \begin{array}{l} \frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i + 1}}}}\left| {\mathbf{J}} \right| \\ + \frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i}}}}{\mathbf{N}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i + 1}}}} \\ + \frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i + 1}}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i}}}}\left| {\mathbf{J}} \right| \\ + {{\mathbf{N}}^{\mathbf{T}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i}}}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i + 1}}}}\\ [6pt] + \frac{{\partial {{\mathbf{N}}^{\mathbf{T}}}}}{{\partial {t_{i + 1}}}}{\mathbf{N}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} \\ + {{\mathbf{N}}^{\mathbf{T}}}\frac{{\partial {\mathbf{N}}}}{{\partial {t_{i + 1}}}}\frac{{\partial \left| {\mathbf{J}} \right|}}{{\partial {t_{i}}}} \\ + {{\mathbf{N}}^{\mathbf{T}}}{\mathbf{N}}\frac{{{\partial^{2}}\left| {\mathbf{J}} \right|}}{{\partial {t_{i}}\partial {t_{i + 1}}}} \end{array} \right]}}{ds dt dn},\\ (i = 1,2, {\ldots} ,7) \end{array} $$
(A.6)

The first-order partial derivatives of the inverse matrix of the Jacobian matrix with respect to ti can be expressed as:

$$ \frac{\partial{{\mathbf{J}}^{- 1}} }{{\partial {t_{i}}}} = - {{\mathbf{J}}^{- 1}}\left( \frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}\right){{\mathbf{J}}^{- 1}} $$
(A.7)

The first and second orders partial derivatives of the determinant |J| with respect to ti can be expressed as:

$$ \frac{{{\partial}\left| {\mathbf{J}} \right|}}{{\partial t_{i}}} = \left| {\mathbf{J}} \right|Tr\left( {{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}\right) $$
(A.8)
$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}\left| {\mathbf{J}} \right|}}{{\partial {t_{i}^{2}}}} &=& \left| {\mathbf{J}} \right|{\left[ {Tr\left( {{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}\right)} \right]^{2}} \\&&+ \left| {\mathbf{J}} \right|Tr\left[ { - {{\mathbf{J}}^{- 1}}\left( \frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}\right){{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}} \right] \end{array} $$
(A.9)

The second-order partial derivatives of the determinant |J| with respect to ti and ti+ 1 can be expressed as:

$$ \begin{array}{@{}rcl@{}} \frac{{{\partial^{2}}\left| {\mathbf{J}} \right|}}{{\partial {t_{i}}{t_{i + 1}}}} &=& \left| {\mathbf{J}} \right|Tr\left( {{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i + 1}}}}\right)Tr\left( {{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}\right) \\ &&+ \left| {\mathbf{J}} \right|Tr\left[ { - {{\mathbf{J}}^{- 1}}\left( \frac{{\partial {\mathbf{J}}}}{{\partial {t_{i + 1}}}}\right){{\mathbf{J}}^{- 1}}\frac{{\partial {\mathbf{J}}}}{{\partial {t_{i}}}}} \right]\\ &&(i = 1,2, {\ldots} ,7),\\ \end{array} $$
(A.10)

In the above equations i = 1, 2,…, 8, when the value of i is not marked. Similarly, the third-order partial derivatives of the mass and stiffness matrices with respect to ti can be obtained.

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Wang, D., Ren, J., Tian, W. et al. Predicting the dynamics of thin-walled parts with curved surfaces in milling based on FEM and Taylor series. Int J Adv Manuf Technol 103, 927–942 (2019). https://doi.org/10.1007/s00170-019-03585-6

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