An Improved Cutting Force Model in Micro-milling Considering the Comprehensive Effect of Tool Runout, Size Effect and Tool Wear

: Tool runout, cutting edge radius-size effect and tool wear have significant impacts on the cutting force of micro-milling. In order to predict the micro-milling force and the machining performance related to the cutting force, it is necessary to establish a cutting force model including tool runout, cutting edge radius and tool wear. In this study, an instantaneous uncut thickness (IUCT) model considering tool runout, a nonlinear shear/ploughing coefficient model including cutting-edge radius and a friction force coefficient model embedded with flank wear width, are constructed respectively. By integrating the IUCT, the nonlinear shear/ploughing coefficient and the friction force coefficient, a comprehensive micromilling force model including the tool runout, size effect and tool wear is derived. Experiment results show that the proposed comprehensive model is efficient to predict the micro milling force.

force at the MUCT point are equivalent, Son et al. [17]expressed the stagnant angle corresponding to the MUCT as a function of the friction angle. Based on the minimum cutting energy principle and the infinite stress principle [18], Malekian et al. [19]concluded that the stagnant angle equals to the friction angle.
Under the high rotation speed, the tiny-size micro-milling tool wears rapidly, resulting in a sharp increase in micro milling force. Bao et al. [20] built an experienced relationship between the tool wear and the amplitude of micro-milling force. Based on the friction stress distribution formular [21], Lu et al. [9] built a flank wear-included force model for Niki micro-milling. Compared to the flank wear, the effect of cutting-edge wear on the cutting force is much less studied. By assuming that both of the edge radius and the flank wear width increase with the deteriorating tool wear condition, Zhou et al. [22]attempted to build a cutting force model considering the edge wear and flank wear.
The works mentioned above have deeply investigated the individual effect of the tool runout, edge radius effect and tool wear on micro milling force. Besides, some other studies have explored the influence of the pairwise combination of the three factors on the cutting force of micro-milling. The combined effect of the tool wear and cutting-edge radius on the force was reported in study [22]. Jing et al. [8]considered the edge radius and runout, and built an analytic micro milling force model. Li et al. [23] adopted the spatial analytic geometry method to analysis the combined effect of the tool runout and wear the micro-milling force. Liu et al. [24] built a micro-milling force model including the tool runout and tool wear, and proposed a cutting force model-based tool wear monitoring method under varying tool runout. However, so far, to our best knowledge, no studies have considered the comprehensive effect of tool runout, cutting-edge radius and tool wear on the micro milling force. To bridge this gap, this study builds a mechanics micro milling force model considering the tool runout, cutting-edge radius and tool wear, and investigate the comprehensive effect of the three factors on the micro milling force.
The cutting force modeling process in this study is as follows. Firstly, the IUCT model including the tool runout is constructed. Then, a nonlinear shear/ploughing cutting coefficient model including the IUCT and the cutting-edge radius is derived, and the friction force coefficient in the flank wear region is represented as a function of the flank wear width. Finally, by integrating the IUCT, the shear/ploughing cutting coefficient and the friction force coefficient, a comprehensive micro-milling force model including tool runout, edge radius effect and the tool wear is constructed. This paper evolves as follows. In section 2, the comprehensive micro milling force model is constructed. Section 3 proposes a genetic optimization-based model parameters calibration approach. In section 4, micro milling experiment is conducted to examine the efficiency of the proposed model. The paper is concluded in section 5.

The Improved Cutting Force Model for Micro-Milling
The proposed model is shown in Fig. 1. The input layer of the model consists of the tool runout, cutting edge radius and the flank wear. The second layer is the middle variables including the IUCT, the shear-ploughing coefficient and the friction coefficient. The output layer is the theoretical micromilling force. The comprehensive model in Fig.1 ( 2 ) Notation d c F is the tangential force, d r F is radial force, dz is the unit axial cutting depth. Notation h is the instantaneous uncut thickness, which depends on the tool runout. NotationVB is the flank wear width, e r is the cutting-edge radius related to the flank wear width,  is the rake angle. Notation , cs p K is the shear-ploughing coefficient in tangential direction, , rs p K is the shear-ploughing coefficient in radial direction. The shear-ploughing coefficient varies with the IUCT. If the partial IUCT ' h is greater than the minimum uncut thickness m h , the shear-ploughing coefficient represents the cutting force resulting from the shear effect (Fig.2a). Otherwise, the shear-ploughing coefficient corresponds to the ploughing force (Fig.2b)  The construction of the IUCT model considering tool runout, the shear/ploughing coefficient model considering cutting edge radius and the friction force coefficient model with flank wear width are elaborated in the fowling three subsections.

Uncut Thickness Model considering Tool Runout
By changing the equivalent tool radius and the angle between the radius, axial tool runout makes the trochoidal trajectory and the IUCT of each tooth change. Because the axial cutting depth is much smaller than the length of tool, the radial tool runout could be regarded as the translation of the cutting part at the bottom of the micro-milling tool, and thus the tool runout could be represented by the length o r and the angle o  of the translation vector. According to studies [13,24] The MUCT value is:

Friction Force Coefficient in Flank Wear Region
The relationship between the flank tool wear and the friction force coefficient has been clearly revealed in the existed studies. According to the friction stress distribution formular, the relationship between the friction force coefficient and the flank wear could be written as: Many studies showed that the cutting-edge radius varies with the flank wear width. However, due to the uncertain build up edge and micro chipping that affect the effective cutting-edge radius, it is difficult to build a deterministic analytic model to represent the relationship between the edge radius and the flank tool wear width. To revel the dependency of the cutting-edge radius on the flank wear width, in the experimental validation part, empirical relationship between the calibrated edge radius and the measured flank wear width is analyzed by statistical correlation analysis method.

Calibration of the Model Parameters
Genetic optimization algorithm is adopted to calibrate the model parameters. Besides the two unknown mechanical parameter sets s  and v  , the tool runout and the efficient edge radius are also calibrated via the genetic optimization algorithm. Practically, the static tool runout could be directly measured by optic measurements. However, the optic measurement-based method is incompetent to measure the dynamic rotation-speed-dependent runout in cutting process. Due to the build up edge and microchipping, it is also difficult to directly measure the effective cutting-edge radius by optical microscope or atomic force microscope. Therefore, in this study, the dynamic tool runout and the effective edge radius are also calibrated via the genetic optimization algorithm. The flank wear band is regular and the flank wear width could be directly measured by the optical microscope. In this study, the flank wear width is measure by Olympus Toolmakers microscope during the recess of tool holder.
Including the mechanical parameter sets s  a n d v  , tool runout parameters, effective cutting-edge radius, there are 12 parameters need to be calibrated. The purpose of the optimization-based calibration is to find out the 12 optimum parameters, such that the gap between the theoretical force and the measured force is smallest. The measured forces in feed and normal directions are utilized to calibrate the parameters of the proposed model. In order to reduce the computation cost of the calibration process, instead of jointly optimizing the 12 parameters, the 12 parameters to be calibrated are optimized by three steps. Firstly, the cutting force signal, generated by the fresh tool with known edge radius and flank wear width, is adopted to calibrate the machinal parameters s  , and the parameter set s  is set as a shared input of the subsequent calibration processes with worn tool. Then, with the sharing mechanical parameters s  , the friction force coefficient is calibrated under the worn tool. Finally, the parameters set v  that reflects the relationship between the flank wear and the friction force coefficient, is calibrated with the calibrated friction force coefficients and the measured flank wear widths. The parameters need to be calibrated are listed in Table 1. The stepwise calibration process is shown in Fig.4.

Parameter calibration results
The cutting force signal acquired during the first cutting pass is adopted to calibrate the four mechanical parameters: the shear stress, friction angle in shear region, normal and tangential cutting stresses in ploughing region. It is assumed that the tool at the first cutting pass is fresh. The tool edge radius at the first cutting pass is the initial cutting-edge radius (2μm), and the flank wear width at the first cutting pass is 0μm. By taking the initial cutting-edge radius and the flank wear width into the proposed cutting force    Table 5. As the tool is re-clamped at the beginning of each cutting pass, the runout parameter cannot keep constant for different cutting passes. This could be noticed from Table 5. The varying process of the flank wear width and the effective cutting-edge radius is shown in Fig.5. It clearly shows that the cutting-edge radius increases as the flank wear width increases. The correlation coefficient of the cuttingedge radius and the flank wear width is calculated as 0.8842, implying that the effective cutting-edge radius highly depends on the flank wear width.

Micro milling force prediction results
The cutting forces at the 10 passes are predicted via the proposed comprehensive model. The residual force is defined as the predicted force minus to the measured force. The estimated error is defined as the ratio of the second norm of the residual force to the second norm of the measured force. The predicted error is listed in Table 6. It shows that the proposed comprehensive model is accurate to predict the micro-milling force. Three conventional cutting force models are also utilized to predict the micro milling force in this study. Different from the proposed comprehensive model, the three conventional models only consider two factors. The first conventional model includes tool runout and cutting-edge radius. The second model considers tool runout and tool wear. The third one considers the cutting-edge radius and tool wear. Ta b le 7 lists the prediction results of the three conventional models. It could be found that the proposed comprehensive model is more accurate than the three traditional models. This could also be concluded from Fig.7. Table 7 shows that, the model without considering the cutting-edge radius has a much higher prediction error than the other three models. This implies the cutting-edge radius has the most significant effect on the micro-milling force.

Conclusion
In order to accurately predict the micro-milling force, this study construct a micro-milling fore model show that the proposed model is efficient to predict the micro-milling force under varying tool runout, cutting edge radius and tool wear condition. Some conclusions are as follows.
1) Including tool runout, cutting edge radius and tool wear, the proposed comprehensive model is more accurate than the conventional models.
2) The cutting-edge radius increases as the flank wear width increases. The effective cutting-edge radius could be utilized to indicate the tool wear condition of micro milling.
3) The cutting stress in ploughing region is much higher than the shear stress and the friction stress in the flank wear region. The micro-milling force is most concentrated in the ploughing region.
4) Among the tool runout, cutting edge radius and tool wear, the cutting-edge radius has the most significant impact on micro-milling force.