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Development of Automatic Chatter Suppression System in Parallel Milling by Real-Time Spindle Speed Control with Observer-Based Chatter Monitoring

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Abstract

To maximize the potential for high material removal rates in simultaneous processes such as parallel milling, developing strategies for successful chatter suppression/avoidance is an important concern for manufacturers. In this study, the effectiveness of the spindle speed difference method (SDM) for chatter suppression is discussed in a parallel end-milling process where a flexible workpiece is machined by two tools rotating in opposite direction. The process model is developed, considering that the dynamic variation due to the regenerative effect occurs on a plane perpendicular to the tool axis direction. Through the process simulations and the experiments, this study provides informative discussion for comprehending the process behavior. Additionally, a real-time active chatter suppression system with adaptive SDM, where the spindle speed difference is sequentially optimized during the process according to the tracked chatter frequency, is developed by integrating a chatter monitoring system based on sensorless cutting force estimation with sliding discrete Fourier transform. The results show that the developed real-time adaptive system of spindle speed suppresses chatter vibrations more effectively than non-adaptive SDM system; hence, the integrated system can contribute self-optimizing machining systems oriented to Industry 4.0.

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Abbreviations

a p :

Axial depth of cut

a t :

Acceleration of table

c f :

Feed per tooth

\(dF_{t} , dF_{r} , dF_{a}\) :

Minute cutting force in tool tangential, radial, and axial directions

\(dF_{\text{x}} , dF_{y} , dF_{z}\) :

Minute cutting force in Cartesian coordinate system

F cut :

Cutting force

F stat :

Non-process-related force including friction and gravity terms

\(F_{x} , F_{y} , F_{z}\) :

Cutting force in Cartesian coordinate system

g(θ):

Unit step function to judge tooth engagement

G(iω):

Frequency response function

G LPF(s):

Low-pass filter

h :

Uncut chip thickness

I ref a :

Motor current reference

J r :

Total inertia of motor, coupling, and ball screw

k c :

Chatter lobe number

K t :

Torque coefficient

\(K_{te} , K_{re} , K_{ae}\) :

Edge force coefficient in tangential, radial, and axial directions

\(K_{tc} , K_{rc} , K_{ac}\) :

Cutting force coefficients in tangential, radial, and axial directions

L :

Number of divisions along axial depth of cut

M t :

Movable mass

n :

Arbitrary integer

R :

Transform coefficient for rotational motion to translational motion

S :

Spindle speed

T z :

Tooth-pass period (= 60/(ZS))

T c :

Chatter vibration period (= 2π/ωc)

Z :

Number of teeth

α 0 :

Time-invariant average directional dynamic milling force coefficient

α r :

Angular acceleration of motor

Δa p :

Thickness of each minute disk element

ΔT z :

Difference of tooth-pass period between two tools

Δɛ :

Phase difference between tool 1 and 2

ɛ :

Total phase shift between present and previous vibration before one tooth pass

θ :

Rotation angle

\(\lambda , \lambda_{R} , \lambda_{I}\) :

Eigenvalue of the characteristic force equation and its real and imaginary part (λ = λR + λI)

ω c :

Chatter frequency

\(_{j} , _{m}\) :

Value at j-th teeth and/or m-th minute disk

\(_{n}\) :

Nominal value

\(_{pq}\) :

Value from q to p direction (\(p, q = x, y, z\))

\(^{{{\text{t}}1}} , ^{{{\text{t}}2}}\) :

Value for tool 1 or tool 2

\(^{\text{w}}\) :

Value for workpiece

\(\left( \wedge \right)\) :

Estimated value

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Acknowledgements

This work was partially supported by the SIP Innovative Design and Production Technology Project commissioned by the New Energy and Industrial Technology Development Organization (NEDO), JSPS Grant-in-Aid for Fellows Grant Numbers JP19J13204, and Keio University Doctoral Student Grant-in-Aid Program. The authors thank Mr. Okuma and the OMRON Corporation for their support and assistance.

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Authors and Affiliations

  1. Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Shuntaro Yamato & Yasuhiro Kakinuma

  2. Nakamura-Tome Precision Industry Co., Ltd., Ro 15 Netsuno, Hakusan, Ishikawa, 920-2195, Japan

    Kenichi Nakanishi

  3. Department of Mechanical Aerospace Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan

    Norikazu Suzuki

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  1. Shuntaro Yamato
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Correspondence to Shuntaro Yamato.

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Appendix

Appendix

Equation (10) can be rearranged as follows:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} + \frac{1}{2}a_{p} K_{tc} \left[ {DM_{\Delta } } \right]\left[ {TF} \right]\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} = \frac{1}{2}a_{p} K_{tc} \left\{ {\left[ {A_{0} } \right] - e^{{ - i\varepsilon^{t1} }} \left[ {DM_{\Delta } } \right]} \right\}\left[ {TF} \right]\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} + \frac{1}{2}a_{p} K_{tc} \left[ {DM_{\Delta } } \right]\left[ {TF} \right]\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} \\ & \quad \to \left\{ {\left[ I \right] + \frac{1}{2}a_{p} K_{tc} \left( {\left[ {DM_{\Delta } } \right] - \left[ {A_{0} } \right]} \right)\left[ {TF} \right]} \right\}\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} = \frac{1}{2}a_{p} K_{tc} \left( {1 - e^{{ - i\varepsilon_{1} }} } \right)\left[ {DM_{\Delta } } \right]\left[ {TF} \right]\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} \\ & \quad \to \left\{ {\left[ I \right] + \frac{1}{2}a_{p} K_{tc} \left( {\left[ {DM_{\Delta } } \right] - \left[ {A_{0} } \right]} \right)\left[ {TF} \right]} \right\}\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} = \lambda \left[ {DM_{\Delta } } \right]\left[ {TF} \right]\left\{ {\begin{array}{*{20}c} {F_{z}^{w} } \\ {F_{y}^{w} } \\ \end{array} } \right\} \\ \end{aligned}$$
(17)

By solving the general eigenvalue problem of Eq. (17) and obtaining the eigenvalues, the critical axial depth of cut, ap(lim) can be derived similarly as in the conventional milling process, as follows:

$$a_{{p\left( {\lim } \right)}} = \frac{{\lambda_{R} }}{{K_{tc} }}\left\{ {1 + \left( {\frac{{\lambda_{I} }}{{\lambda_{R} }}} \right)^{2} } \right\}$$
(18)

The axial depth of cut term is included in the left-hand side of Eq. (17). Therefore, the eigenvalues were searched such that the initially set value of the axial depth of cut was equal to the value calculated using Eq. (18) while updating the initial value.

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Yamato, S., Nakanishi, K., Suzuki, N. et al. Development of Automatic Chatter Suppression System in Parallel Milling by Real-Time Spindle Speed Control with Observer-Based Chatter Monitoring. Int. J. Precis. Eng. Manuf. 22, 227–240 (2021). https://doi.org/10.1007/s12541-021-00469-2

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