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A real-time intelligent method to identify mechanistic cutting force coefficients in 3-axis ball-end milling process using stochastic gradient decent: the mechanistic network

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Abstract

The mechanistic force model is one of the most common methods used to predict cutting forces in milling processes. In this model, the cutting forces are considered a function of cutting geometry and the material properties of the tool and workpiece, generally known as cutting coefficients. These coefficients are commonly identified by performing special calibration tests and are applied to predict cutting forces for other conditions. Although the mechanistic model is a powerful tool for predicting cutting forces, its accuracy decreases as the cutter-workpiece engagement geometry differs from the calibration tests. Thus, it is necessary to update the cutting coefficients to preserve the accuracy of the model. This paper proposes a real-time intelligent method, named “the mechanistic network,” for identifying and updating the cutting coefficients. To this end, an analogy between the mechanistic force model and artificial neural networks is identified, in which the weight coefficients of the artificial neural networks have been replaced with the cutting coefficients. To identify and update the cutting coefficients, an algorithm is proposed using stochastic gradient descent, which updates the coefficients in each iteration. In addition, some other important parameters in milling processes, such as the phase shift between the measured and predicted forces and run-out parameters, are calculated using stochastic gradient descent. The good performance of the proposed network is shown through case studies by utilizing reliable data existing in the literature and also by performing ball-end milling experimental tests. The results show that the proposed network can predict the cutting forces with an error of less than 10% and update the cutting coefficients with a calculation speed of 125k iterations per second. The robustness of the network against noise that may arise in real machining conditions is also shown. The proposed mechanistic network is a reliable and efficient tool that can be applied to real-time applications such as cyber-physical manufacturing systems and condition monitoring of machining processes.

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Abbreviations

Subscript: r, t, a :

Representation of radial, tangential, and axial directions

Subscript: i :

Representation of element number

\({z}_{i}\) :

Value of z coordinate with respect to ball-end center (mm)

\(d{S}_{i}\) :

Length of the cutting-edge element (mm)

\({h}_{i}\) :

Undeformed chip thickness (mm)

\(d{b}_{i}\) :

Chip width (mm)

\({K}_{re}\), \({K}_{te}\), \({K}_{ae}\) :

Edge coefficients (N/mm)

\({K}_{rc}\), \({K}_{tc}\), \({K}_{ac}\) :

Shear coefficients (N/mm2)

\({dF}_{ri}\), \({dF}_{ti}\), \({dF}_{ai}\) :

Differential cutting forces (N)

\({C}_{n}^{re}\), \({C}_{n}^{te}\), \({C}_{n}^{ae}\) :

Edge constants (N/mm(n+1))

\({C}_{n}^{rc}\), \({C}_{n}^{tc}\), \({C}_{n}^{ac}\) :

Shear constants (N/mm(n+2))

\(\overrightarrow{{F}_{p}}\) :

Predicted force vector (N)

\(\overrightarrow{{F}_{s}}\) :

Measured force vector (N)

\({F}_{px}\), \({F}_{py}\), \({F}_{pz}\) :

Predicted force components (N)

\({F}_{sx}\), \({F}_{sy}\), \({F}_{sz}\) :

Measured force components (N)

\({\widehat{n}}_{r}\),\({\widehat{n}}_{t}\), \({\widehat{n}}_{a}\) :

Unit vectors

\({\phi }_{i}\) :

Element angular position (rad)

\(\phi\) :

Total rotation of the tool (rad)

\({\Omega }_{s}\) :

Spindle speed (rpm)

\({\beta }_{0}\) :

Helix angle (rad)

\({R}_{0}\) :

Radius of the ball-end tool (mm)

\({N}_{f}\) :

Total number of flutes

\({n}_{f}\) :

Flute number

\(E\) :

Objective error function (N2)

\(e\) :

Run-out value (mm)

\({\overrightarrow{e}}_{ij}\) :

The relative position of the tool center with respect to the ideal position (mm)

\({\phi }_{p}\) :

Run-out angle (rad)

\({\phi }_{p0}\) :

Run-out angle at \(t=0 (s)\) (rad)

\({h}_{ei}\) :

Effect of run-out on the undeformed chip thickness (mm)

\({h}_{0i}\) :

Undeformed chip thickness without the effect of run-out (mm)

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Authors and Affiliations

  1. Advanced Manufacturing Laboratory (AML), Department of Mechanical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, 11365-11155, Iran

    Mahmoodreza Forootan & Javad Akbari

  2. Department of Mechanical Engineering, Shahreza Campus, University of Isfahan, Isfahan, Iran

    Mohammad Ghorbani

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  1. Mahmoodreza Forootan
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Contributions

Mahmoodreza Forootan: conceptualization, methodology, software, validation, writing—original draft, and writing—review and editing. Javad Akbari: supervision and writing—review and editing. Mohammad Ghorbani: supervision, validation, and writing—review and editing.

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Correspondence to Javad Akbari.

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Forootan, M., Akbari, J. & Ghorbani, M. A real-time intelligent method to identify mechanistic cutting force coefficients in 3-axis ball-end milling process using stochastic gradient decent: the mechanistic network. Int J Adv Manuf Technol 129, 2949–2968 (2023). https://doi.org/10.1007/s00170-023-12460-4

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