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Closed-Form Time-Domain Solutions of Arbitrary-DOF Forced Vibrations and of Surface Location Error for General-Helix Cylindrical Milling Tools

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Abstract

Background

Periodic cutting forces are inevitably generated during milling processes and the associated forced vibrations leave behind an imprint called surface location error (SLE) on machined parts causing tolerance concerns. Numerical prediction of SLE is extensively demonstrated in literature.

Purpose

However, closed-form models are unarguably preferred over numerical and surrogate models when available due to higher effectiveness and accuracy for design analysis. The few existing closed-form models of forced vibrations and SLE are limited to 1 and 2-DOF systems excited by 1 or 2-dimensional cutting forces (feed and feed-normal directions) for the conventional fixed-helix angle shape even though variable helix angle is a common technical possibility.

Results

This work proposes a closed-form solution that is based on the more realistic 3-dimensional cutting force for the forced vibrations and SLE of arbitrary-DOF milling excited by arbitrary variable helix angle tools. The validity of the proposed model is checked with comparisons to numerical cases of flexible cutting tools (lumped-mass model) and thin-walled workpieces (continuum/finite element model) drawn from literature.

Conclusion

To demonstrate the potential application of the model in SLE suppression without compromising productivity, an optimization problem is formulated and solved for selecting helix angles, and almost complete suppression of SLE is recorded for the illustrative case.

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Data availability

Data sharing not applicable to this article as no datasets were generated during the current study. The predicted numerical results from the proposed closed-form solution were checked using duly cited published predicted/experimental data.

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Funding

The described research was done while Chigbogu Ozoegwu visited the ITM at the University of Stuttgart from 2022 to 2023. This stay was funded by the Alexander von Humboldt Foundation. This support is highly appreciated.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Nigeria, Nsukka, Nigeria

    Chigbogu Ozoegwu

  2. Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany

    Peter Eberhard

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  1. Chigbogu Ozoegwu
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Correspondence to Chigbogu Ozoegwu.

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Appendix

Appendix

A: Coefficients in Equation 11

$$\begin{aligned} \varvec{C}^{(p)}_r=&\varvec{F}^{(p)}_\text {f,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p) +g_{\text {s},r}^{(p)}\sin (r\phi _p)) \end{aligned}$$
(18)
$$\begin{aligned} \varvec{S}^{(p)}_r=&\varvec{F}^{(p)}_\text {f,0}(-g_{\text {c},r}^{(p)}\sin (r\phi _p)+g_{\text {s},r}^{(p)}\cos (r\phi _p)) \end{aligned}$$
(19)
$$\begin{aligned} \varvec{C}^{(p)}_{r+1}=&0.5(\varvec{C}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p) \nonumber \\ {}&+g_{\text {s},r}^{(p)}\sin (r\phi _p))+ \varvec{S}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p)-g_{\text {s},r}^{(p)}\cos (r\phi _p))) \end{aligned}$$
(20)
$$\begin{aligned} \varvec{S}^{(p)}_{r+1}=&0.5(-\varvec{C}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p) \nonumber \\ {}&-g_{\text {s},r}^{(p)}\cos (r\phi _p))+\varvec{S}^{(p)}_{1,0} (g_{\text {c},r}^{(p)}\cos (r\phi _p)+g_{\text {s},r}^{(p)}\sin (r\phi _p))) \end{aligned}$$
(21)
$$\begin{aligned} \varvec{C}^{(p)}_{r-1}=&0.5(\varvec{C}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p) \nonumber \\ {}&+g_{\text {s},r}^{(p)}\sin (r\phi _p))-\varvec{S}^{(p)}_{1,0} (g_{\text {c},r}^{(p)}\sin (r\phi _p)-g_{\text {s},r}^{(p)}\cos (r\phi _p))) \end{aligned}$$
(22)
$$\begin{aligned} \varvec{S}^{(p)}_{r-1}=&0.5(-\varvec{C}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p) \nonumber \\ {}&-g_{\text {s},r}^{(p)} \cos (r\phi _p))-\varvec{S}^{(p)}_{1,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p)+g_{\text {s},r}^{(p)}\sin (r\phi _p))) \end{aligned}$$
(23)
$$\begin{aligned} \varvec{C}^{(p)}_{r+2}=&0.5(\varvec{C}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p) \nonumber \\ {}&+g_{\text {s},r}^{(p)}\sin (r\phi _p)) +\varvec{S}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p)-g_{\text {s},r}^{(p)}\cos (r\phi _p))) \end{aligned}$$
(24)
$$\begin{aligned} \varvec{S}^{(p)}_{r+2}=&0.5(-\varvec{C}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p) \nonumber \\ {}&-g_{\text {s},r}^{(p)}\cos (r\phi _p)) +\varvec{S}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p)+g_{\text {s},r}^{(p)}\sin (r\phi _p))) \end{aligned}$$
(25)
$$\begin{aligned} \varvec{C}^{(p)}_{r-2}=&0.5(\varvec{C}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p) \nonumber \\ {}&+g_{\text {s},r}^{(p)}\sin (r\phi _p)) -\varvec{S}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p)-g_{\text {s},r}^{(p)}\cos (r\phi _p))) \end{aligned}$$
(26)
$$\begin{aligned} \varvec{S}^{(p)}_{r-2}=&0.5(-\varvec{C}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\sin (r\phi _p) \nonumber \\ {}&-g_{\text {s},r}^{(p)}\cos (r\phi _p)) -\varvec{S}^{(p)}_{2,0}(g_{\text {c},r}^{(p)}\cos (r\phi _p)+g_{\text {s},r}^{(p)}\sin (r\phi _p))) \end{aligned}$$
(27)

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Ozoegwu, C., Eberhard, P. Closed-Form Time-Domain Solutions of Arbitrary-DOF Forced Vibrations and of Surface Location Error for General-Helix Cylindrical Milling Tools. J. Vib. Eng. Technol. 12, 3563–3574 (2024). https://doi.org/10.1007/s42417-023-01064-7

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