key: cord-0332488-30090jej
authors: Singh, Kundan K.; Singh, Ramesh
title: Process Mechanics Based Uncertainty Modeling for Cutting Force Prediction in High Speed Micromilling of Ti6Al4V
date: 2020-12-31
journal: Procedia Manufacturing
DOI: 10.1016/j.promfg.2020.05.048
sha: bbc1680ae71c68c8a58e450f02f67680e530a3c3
doc_id: 332488
cord_uid: 30090jej

Abstract High speed machining is essential to counter the low flexural rigidity of micro-end mill in micromilling process. High speed machining can excite the higher frequency modes and in addition, run-out is also amplified. These effects in high speed micromachining induces the uncertainty in cutting forces. The estimation of cutting coefficients without inclusion of uncertainty cannot give accurate values and hence, the predicted cutting force and machining stability using these cutting coefficient may not be accurate. In the present work, cutting coefficients have been determined using the Bayesian inference which includes the uncertainty in estimated cutting coefficients. Note that, estimated cutting coefficient independent of chip load and cutting velocity does not include the high speed micromachining mechanics like size effect. The cutting coefficients have been estimated as a function of cutting velocity and chip load in the present work. The segmented cutting coefficients for different ranges of cutting speed varying from 10000 rpm to 110000 rpm has been obtained as a non-liner function of chip load. Bayesian inference modelling for cutting coefficients has been carried out using the Metropolis-Hastings (MH) algorithm Markov Chain Monte Carlo (MCMC) method. The convergence of prior and posterior distribution has been verified using correlation and trace of samples used for sampling. The posterior distribution shows that there is a good fitting, which accurately predicts the mean of the cutting coefficients. Finally, predicted cutting coefficients have been compared with cutting coefficients obtained from the deterministic approach using least square method. The experimentally estimated cutting coefficients are found to be lying within the upper and lower limit of predicted cutting coefficients with Bayesian inference approach. The predicted cutting coefficients using Bayesian inference shows the deviation of 0.89% and 8.4% at 40000 rpm from experimentally obtained cutting coefficients for tangential and radial cutting coefficients, respectively. The experimental cutting force is found to be lying within the upper and lower limit of predicted cutting force with Bayesian inference based cutting coefficients fitting at 105000 rpm.

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec-

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim-

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim-

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim-

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim-

High speed machining is essential to counter the low flexural rigidity of micro-end mill in micromilling process. High speed machining can excite the higher frequency modes and in addition, run-out is also amplified. These effects in high speed micromachining induces the uncertainty in cutting forces. The estimation of cutting coefficients without inclusion of uncertainty cannot give accurate values and hence, the predicted cutting force and machining stability using these cutting coefficient may not be accurate. In the present work, cutting coefficients have been determined using the Bayesian inference which includes the uncertainty in estimated cutting coefficients. Note that, estimated cutting coefficient independent of chip load and cutting velocity does not include the high speed micromachining mechanics like size effect. The cutting coefficients have been estimated as a function of cutting velocity and chip load in the present work. The segmented cutting coefficients for different ranges of cutting speed varying from 10000 rpm to 110000 rpm has been obtained as a non-liner function of chip load. Bayesian inference modelling for cutting coefficients has been carried out using the Metropolis-Hastings (MH) algorithm Markov Chain Monte Carlo (MCMC) method. The convergence of prior and posterior distribution has been verified using correlation and trace of samples used for sampling. The posterior distribution shows that there is a good fitting, which accurately predicts the mean of the cutting coefficients. Finally, predicted cutting coefficients have been compared with cutting coefficients obtained from the deterministic approach using least square method. The experimentally estimated cutting coefficients are found to be lying within the upper and lower limit of predicted cutting coefficients with Bayesian inference approach. The predicted cutting coefficients using Bayesian inference shows the deviation of 0.89% and 8.4% at 40000 rpm from experimentally obtained cutting coefficients for tangential and radial cutting coefficients, respectively. The experimental cutting force is found to be lying within the upper and lower limit of predicted cutting force with Bayesian inference based cutting coefficients fitting at 105000 rpm.

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim-

The micromilling process can be used for producing the 3-dimensional product for almost all the materials, which is not possible with the processes like lithography and etching. The lithography and etching based processing is mostly limited to Si-based materials, hence, micromilling is a versatile process which can be used for creating free-form surfaces and miniature product for any materials. The miniature products produced through micromilling process are having application in industries like bio-medical transplantation, micro-mould and die manufacturing, elec- * Corresponding author. Tel.: +91-44-66303-731 tronics, defence, transportation, drug delivery and energy storage systems etc. [1, 2] . The micrommiling process uses the micro cutting tool with diameter varying from 10 µm to 500 µm [2] . The low diameter gives the low flexural rigidity for micro-end mill and hence can result in cutting tool vibration [3] . Note that, if cutting tool vibration is not controlled then catastrophic tool failure can occur. One way to overcome the low flexural rigidity is by using the high speed machining which reduces the chip load at cutting tool-tip and hence, reduction in cutting forces and cutting tool deflection is achieved [4] . However, high speed machining excites the higher frequency modes and increases the dynamic run-out [5] of micro-end mill which makes the machining process instable [6, 7] . The primary requirement for high speed micromachining is doing the machining in stable zone to avoid the chatter. The elim- ination of chatter through selection of stable machining parameters necessities the accurate modelling of cutting forces in high speed micromilling process. The cutting force modelling is not trivial in high speed micromilling due to presence of critical chip thickness effect, size effect and dynamic run-out [8, 9, 10, 11] . The cutting force modelling in micromilling has been carried out using analytical, mechanistic and numerical methods [12, 13, 14, 15] . These cutting forces modelling have also been included to predict the stability through chatter modelling in micro milling process. The effect of high speed micromilling process mechanics in cutting force and stability model has been included via velocity and chip load dependent cutting force model [14, 16, 17, 18] . The ploughing and run-out effect has also been included for cutting force and stability prediction in micromilling [6, 8] . However, these models for cutting force and stability prediction in micro-milling process are based on a deterministic way of estimating the different cutting force coefficients and constants. The inclusion of uncertainties in experimental or numerical cutting force for estimating the cutting coefficients can give robust prediction of cutting force and stability for high speed micromilling process. The complex mechanism of micro-end mill manufacturing through grinding process induces the variation in diameter, edge radius and cutting tool shape which leads to the uncertainty in cutting force. The uncertainty in cutting force can also be induced due to other affects like workpiece material properties, dynamic interaction between cutting tool and workpiece, cutting tool dynamics behaviour and change in process dynamics. Hence, inclusion of uncertainties is necessary for accurate prediction of cutting force and stability in high speed micromilling process.

Uncertainty modelling can be carried out using Bayesian inference where prior information or beliefs is updated based on the process variables uncertainties when new information becomes available [19] . Milling force modelling with uncertainties based on Bayesian inference was carried out by Karandikar et al. [19] . They found that cutting coefficients based on linear regression were changing with a change of radial immersion but there was no change in coefficients predicted using Bayesian inference. Bayesian inference model also reduces the number of experiments to be carried out for estimation of cutting coefficients for mechanistic model as shown by Gozu and Karpat [20] . Mehta et al. [21] carried out the Bayesian model for milling force prediction and found that, Bayesian model predicts the force closer to experimentally obtained cutting forces. They also showed that consideration of uncertainty due to change in speed and feed in Bayesian inference model will predict the force at different speed and feed more accurately compared to deterministic approach of force model. The development of relation between cutting tool wear and milling process parameters is complex due to rapid wear of cutting tool especially for machining of difficult-to-machine materials. Niaki et al. [22] carried out the mechanistic model for tool wear prediction through Bayesian probabilistic approach where model parameters can be generated accurately with fewer experiments. The root cause diagnosis for milling process variation with Bayesian belief network has been carried out by Dey and Stori [23] . Buckner et al. [24] proposes an intelligent feedback loop for controlling the cutting force in single point turning operation where parameters uncertainties were considered to make the process control more robust. Uncertainty diagnosis can be minimised if signal feature extraction technique is used to monitor the microend mill wear as shown by Jemielniak and Arrazola [25] . The uncertainty level in measured data using digital image processing also affects the estimated cutting force and displacement and hence, the cutting forces and displacement estimation need to be inclusive of the measurement uncertainties [26] . Lian et al. [27] proposed a model based fuzzy control system to regulate the cutting force in turning by including the process dynamics uncertainty to minimise the cutting tool wear and failure. The uncertainty in cutting force measurement devices can also induce the local variation in cutting force and the variation in cutting force can easily be observed by enlarging the force signal as explained in work of Wei et al. [28] . The deviation in predicted cutting force and measured cutting force is attribute to uncertainties in updating the cutting force model parameters as analysed by Saglam et al. [29] . Random uncertainty in cutting constants, systematic uncertainty due to variation in cutting force and run-out measurement were included in the cutting force model by Bhattacharya et al. [30] . They concluded that one advantages for inclusion of uncertainty in force model parameters is that force can be predicted at a depth of cut where machine is incapable of carrying out the machining. Uncertainty inclusion in cutting force modelling can predict the cutting forces at very low immersion and hence will save the material and processing cost [30] . Uncertainty in cutting force due to dynamometer calibration and machining process were studied by Axinte et al. [31] . They developed a model where cutting force uncertainty prediction in turning at different cutting process parameters can be carried out. The measured cutting force in micromilling is highly susceptible to uncertainty attributed to limited frequency bandwidth of dynamometer, variation in cutting tool size and shape, size effect, cutting tool dynamics and process mechanics. So, modelling of cutting forces without inclusion of uncertainties in cutting coefficients may not predict the accurate cutting force for different ranges of process parameters. The accuracy of estimated cutting coefficients for cutting force prediction depends on the sufficient experimental data with different cutting conditions which is time consuming and highly costly. In addition, deterministic cutting coefficient based cutting force model cannot be extended to plowing dominant machining regime. Hence, cutting force prediction without inclusion of uncertainties in model parameters may not be most accurate in high speed micro-milling process. Uncertainty based cutting coefficients model requires less experimental data and 2 can be extended to different range of process parameters. Note that, cutting coefficients cannot be assumed constant in high speed micromilling because of variation in cutting force at different chip load and cutting velocity [14, 3, 17] . In the present work, cutting force prediction has been carried out by using the Bayesian inference approach for cutting coefficients prediction as non-linear function of chip load and cutting speeds.

The mechanistic cutting force model has been considered to predict the cutting coefficients with uncertainties attributed to experimental cutting forces. Uncertainties in cutting forces have been observed by carrying out the high speed micromilling experiments at different spindle speed bins and feed rate [18] . The standard deviation of cutting coefficients at different process conditions have been captured and cutting coefficients have been perturbed with noise of measured standard deviation. Bayesian inference has been proposed to predict the cutting coefficients with uncertainties as a non-linear function of cutting velocity and chip load. The posterior distribution of cutting coefficients with Bayesian inference method has been obtained by updating the prior distribution using Metropolis-Hastings (MH) algorithm Markov chain Monte Carlo (MCMC) method. The size of samples for MH algorithm has been selected by analysing the trace of prior and posterior distribution. The convergence of posterior and prior distribution of cutting coefficients for MH algorithm has been been checked with autocorrelation plot. The accuracy of developed model has been verified by comparing the mean value of predicted cutting coefficients with cutting coefficients obtained using the deterministic approach at different cutting conditions. Finally, predicted cutting coefficients have been used to predicts the cutting forces in high speed micromilling of Ti6Al4V. The methodology is shown in Fig. 1 .

The micrmilling process has been modelled as a lumped mass system where tangential and radial force is acting to cutting flutes of micro-end mill in tangential and radial direction, respectively.The direction of force acting on the flutes of micro-end mill is shown in Fig. 2 . The cutting forces are measured in machining direction X and normal to machining direction Y (Fig. 2) . The tangential and radial forces are obtained by carrying out the coordinate transformation of the measured X and Y direction cutting forces. The equations for coordinate transformation are given as: where F t,j and F r,j are the tangential and radial cutting forces, respectively acting on flute j for total flutes N. F x,j and F y,j are the measured forces in X and Y direction, respectively for flue j. φ j is the immersion angle for the individual flute j and it is given as:

where φ p is the pitch angle between flutes. Here, F t,j and F r,j for flute j are directly proportional to cutting area and is given as:

where h(φ j )) is the instantaneous chip thickness removed by flute j and a is the axial depth of cut. K tc and K rc are the tangential and radial cutting force coefficients, respectively. The cutting forces in high speed micromilling are not uniform for different cutting velocity and hence assuming the cutting coefficient independent of cutting velocity will not predict accurate cutting forces at different cutting speeds. Note that, feed rate used for micromilling 3 process is comparable with edge radius of micro-end mill which shows the presence of size effect in micromachining process. The specific cutting energy increases with the decrease of chip load due to size effect. Consequently, cutting coefficients have been assumed as a function of chip load and cutting velocity to account the effect of size effect and cutting velocity. The cutting coefficients is non-linear function of chip load for different cutting speed bins and is given as: (6) where α k , β k , γ k , and δ k are the function of cutting velocity, which is determined via experiments.h is the average chip load in mm and can be obtained by using the feed rate f t .

These equations are transformed to linear function by taking the natural logarithmic and are given as:

The cutting coefficients have been determined by doing the experiments at different cutting speed bins. The experimental set-up is shown in Fig. 3(a) . All the experiments have been carried out at developed high speed micromachining centre in machine tools lab at IIT Bombay. The three axis micromachining centre is having three axes with stacked x and y linear stages driven by DC brushless servo motor through ball screw. The positioning resolution and accuracy for x and y stages are 0.5 µm and ±1 µm, respectively. The third axis is z axis driven by linear motor which is counterbalanced by two pneumatic cylinders to achieve the positional resolution of 0.5 nm. The spindle is mounted on z axis. The maximum rotational speed for ceramic bearing based spindle is 140000 rpm. The complete micromachine tool is placed on a vibration isolation table. All the experiments have been carried out with uncoated two fluted tungsten carbide (WC) micro-end mill of diameter 500 µm and the workpiece used is Ti6Al4V. No lubricants have been used for the experiments. The experiments have been carried out at five different mean speeds 20000 rpm, 40000 rpm, 60000 rpm, 80000 rpm and 100000 rpm for five cutting speed bins; 10000-30000 rpm, 3000-50000 rpm, 50000-70000 rpm, 70000-90000 rpm and 90000-110000 rpm as shown in Fig. 3(b) . Nine different feed rates varying from 2 µm/flute to 10 µm/flute at an interval of 1 µm/flute has been used for the experiments. The axial depth of cut used is 30 µm which has been kept constant for all the experiments. The workpiece vibration and micro-end mill vibration have been monitored with accelerometer and laser sensor, respectively to carry out a chatter free machining. The cutting forces in X and Y direction have been measured using Kistler MiniDyn (Model: 9256C1) dynamometer. The measured cutting forces are transformed to tangential and radial forces (Eq. (1) and (2)). Note that, mean cutting forces will be nearly zero for slot machining carried out with two fluted micro-end mill in the present work and hence, cutting coefficients have been obtained by root mean square (RMS) of the resultant tangential and radial cutting forces due to all the flutes. Equation (4) has been used to get the experimental value of tangential and radial cutting coefficients through below equations of tangential and radial forces due to all the flutes.

The non-linear relationship between cutting coefficients and chip load is usually determined by least square fitment for Eq.(5) and (6) . This least square approach is known as a deterministic approach where fitted cutting coefficient against chip load is passes through a measured data and if fitted line is not completely perfect then fitted line will show some deviation from the measured data. Hence, this fitted line will always give a single value of cutting coefficient against a chip load and will not consider and deviation from the fitted values. However, there can be deviation in cutting forces due to high speed micromachining process characteristics like size effect, ploughing effect, cutting tool wear onset, small amplitude micro-end mill vibration, misalignment, presence of run-out at high speed and micro-end mill manufacturing defect. These uncertainties in cutting forces will lead to variation in cutting coefficients. The deterministic approach which gives the single estimated value against each chip load cannot considers the uncertainty of cutting coefficients.

The Bayesian inference method is a probabilistic approach where prior belief for an uncertain variable is updated and distribution of assumed parameters is obtained with standard deviation. The Bayesian approach is automated with new set of available prior and can reach to reliable prediction of initial belief closer to actual values. Bayes' theorem states that, the posterior distribution is proportional to multiplication of the prior distribution and the likelihood [32, 20, 19] . The updating of prior beliefs with new set of information to get the posterior distribution is referred as Bayesian inference. The modelling 4 approach is based on getting initial prior distribution of cutting coefficients for assumed probability distribution of cutting constants for α k , β k , γ k , and δ k . The probability distribution of cutting constants for α k , β k , γ k , and δ k are generated by assuming the normal distribution with mean zero and different values of standard deviation. Here standard deviation has been assumed in such a way that it covers all the deviation in estimated cutting coefficients via experiments. After initial prior of cutting coefficients, likelihood is generated by perturbing the experimentally obtained tangential and radial cutting coefficients. This perturbation is done by assuming the white noise error in form of normal distribution with standard deviation of five times of variation in experimentally obtained cutting coefficients. Finally, posterior distribution of cutting coefficients is obtained by multiplying the prior and likelihood. Hence, the posterior is given as:

where P (K tc |α k , β k ) and P (K rc |γ k , δ k ) are the posterior distribution of the tangential and radial cutting coefficients, respectively against the constants α k , β k , γ k , and δ k . P (K tc,p ) and P (K rc,p ) are the joint prior distribution of tangential and radial cutting coefficients, respectively for normal distribution of α k , β k , γ k , and δ k . Similarly, P (α k , β k |K tc ) and P (γ k , δ k )|K rc ) are the likelihood of getting α k , β k , γ k , and δ k for given the tangential (K tc ) and radial cutting coefficients (K rc ). In the present work, posterior distribution has been obtained using Metropolis-Hastings algorithm (MH) of MCMC method. MH algorithm enables in obtaining samples from the assumed distribution given that at least a value proportional to assumed distribution is computed. In MH algorithm, a sample x i is taken from random distribution of probability p(x). Then a new value x i+1 is sampled from proposed distribution q(x). The samples from the proposed distribution are either accepted or rejected depending on the MH criteria given by:

If the probability computed from above Eq. (14) is larger than the value taken from a uniform distribution on the interval [0, 1] then we accept this value otherwise, next samples are considered. Finally, at the end of iteration process; trace of the accepted samples is obtained. These samples will be an approximation of the posterior. The initial portion of the trace is also discarded known as burnin process in order to ensure steady state conditions for posterior.

The tangential and radial cutting coefficient obtained from the measured cutting forces at each cutting speed are fitted against chip load to get the non-linear relationship between cutting coefficient and chip load for different speed bins. The power law fitment is based on the minimization of least square error between predicted and measured cutting coefficients. The least square error minimization is based on the below equation.

where n is the number of runs at each feed and m is the number of selected samples for the analysis. respectively.. The segmented cutting coefficients has been obtained by fitting the tangential and radial cutting coefficients against average chip thickness for different cutting speed bins (Fig. 3(b) ). The fitment of tangential and radial cutting coefficients at different cutting speeds is shown in Fig. 4(a) and (b) , respectively. It can be seen that specific cutting energies are increasing with decrease of chip load and hence it captures the size effect of micromilling process. Therefore, assuming the cutting coefficients independent of chip load and cutting speed is not appropriate for micromilling process due to presence of size effect. It can also be seen that there is deviation in fitted curve value from the measured value which cannot be captured by deterministic approach. Therefore, it is necessary to do the probabilistic based modelling for accurate predicting of cutting coefficients and cutting forces in high speed micromilling process. 

The fitment of cutting coefficient as a non-linear function of chip load (Eq. (8) & (9)) has been carried out using the probabilistic based Bayesian inference approach. The prior for cutting constant α k , β k , γ k , and δ k is assumed as a normal distribution for different standard deviation to capture the experimental uncertainty in cutting coefficients. The experimentally estimated cutting coefficients have been perturbed with white noise of standard deviation five times the standard deviation in estimated cutting coefficients. The perturbed cutting coefficients have taken as likelihood. The posterior has been obtained by multiplying the prior and the likelihood. The convergence has been achieved using MH algorithm of MCMC method where different numbers of sample have been used. The traces for different samples are shown in Fig. (5) . The change in samples beyond 20000 is not showing any variation in mean of prior, likelihood and posterior distribution as shown in Fig. 5(a) for log[α] at 40000 rpm. The variation in value of sample at different numbers of samples were consistent without any discontinuity (Fig. 5(a) ). The traces of prior, likelihood and posterior distribution for β also shows no deviation in mean value beyond 20000 samples ( Fig. 5(b) ) and hence 20000 samples have been selected as an optimum sample. The value of sample shows no discontinuity for samples up-to 20000 ( Fig.5(b) ). The selection of 20000 samples size has also been verified using the autocorrelation for both log[α] and β. The correlation of log[α] and β ( Fig. 6(a) and (b)) shows that, there is no correlation beyond initial samples and hence it revalidates the optimum sample size of 20000. Once the sampling size is selected the posterior distribution plot has been obtained to see the deviation of mean of tangential and radial cutting coefficient from the experimentally estimated values. The posterior plot shown in Fig. (7) shows that experimentally estimated value of tangential cutting coefficient at 40000 rpm lies in the range 8.7 to 8.8 ( Fig. 7(a) ) predicted by Bayesian inference. Similarly, the predicted posterior ( Fig. 7(b) for radial coefficients at 40000 rpm predicts the experimentally estimated radial coefficient. The posterior plot shows the high correlation between experimentally estimated and the predicted tangential and radial cutting coefficients. The comparison of cutting coefficients predicted with Bayesian inference and deterministic approach has been carried out with experimentally obtained cutting coefficients. Table 1 show the value of mean of tangential cutting coefficient obtained from Bayesian and deterministic approach along with the experimentally observed value at 40000 rpm spindle speed. The percentage deviation of predicted mean of tangential cutting coefficient using Bayesian inference and deterministic approach from experimentally obtained cutting coefficient is shown in Fig. 8(a) . The prediction error has been reduced to 7.8% and 0.89% from 15.4% and 5.8% at 9 µm/flute and 10 µm/flute feed rate, respectively. This error can further be reduced if the standard deviation predicted by Bayesian inference is considered to predict the upper and lower limit of cutting coefficients as shown in Fig. 9(a) . The experimentally observed tangential cutting coefficients lies within the upper and lower limit of predicted cutting coefficient with Bayesian inference ( Fig.  9(a) ) at 40000 rpm. The predicted mean radial coefficient 6 Fig. 9 : Comparison of upper and lower limit of predicted cutting coefficients by Bayesian inference with experimentally obtained cutting coefficient (a) Tangential cutting coefficient; (b) radial cutting coefficient at 40000 rpm, 30 µm depth of cut predicted with Bayesian inference predicts all the experimentally obtained value for all the feed rates as shown in Fig. 9(b) . Hence, the deterministic approach predicts the single value of cutting coefficients but Bayesian inference predicts the cutting coefficients with standard deviation which can be assumed as robust approach.

The Bayesian inference for cutting coefficients prediction has further been validated by carrying out the experiments at different cutting speeds. Figure 10 (a) shows the predicted normal distribution for tangential cutting coefficients at 100000 rpm and crossed point shows the experimental value of tangential cutting coefficients at 100000 rpm. Hence, experimentally observed value lies closer to predicted mean value. Similarly, for radial cutting coefficient ( Fig. 10(b) ), experimental value lies within the predicted value of radial cutting coefficients from Bayesian inference. However, this variation in cutting coefficients cannot be captured using deterministic approach. The predicted cutting coefficients using Bayesian inference has been used to predict the cutting forces at 105000 rpm, 30 µm depth of cut and 5 µm/flue feed. There is no cutting in alternate cycle due to high run-out than feed rate [8] . 8 The experimental force lies within the force predicted by Bayesian inference. The X direction force ( Fig. 11(a) ) lies within the upper and lower limit of the predicted cutting forces. Similarly, Y direction force( Fig. 11(b) ) lies within the upper and lower limit of predicted cutting force with Bayesian inference. However, the force predicted using conventional deterministic approach shows the deviation from the experimental approach as it predicted the single value of cutting coefficients which has been used to predict the X and Y direction cutting forces.

The uncertainty in experimental cutting coefficients has been included using Bayesian inference to predict the cutting coefficient as a non-linear function of chip load at different cutting speeds. The cutting coefficients as a nonlinear function of chip load capture the characteristics of high speed micromilling process. The conventional approach of predicting cutting coefficients using deterministic approach gives a single value of cutting coefficients against a chip load and cutting speed hence uncertainty in cutting forces used for cutting coefficients determination is not included in the model. Therefore, the present work uses the Bayesian inference to predict the cutting coeffi- • The deterministic approach of cutting coefficients predicts the single value of cutting coefficients against a cutting speed and feed rate. The comparison of fitted and experimentally obtained cutting coefficients show the deviation attributed to single value of predicted cutting coefficients. The Bayesian inference approach of present work predicts the mean value of cutting coefficients with the standard deviation hence considers the uncertainty of high speed micromilling process. • The posterior distribution predicts the mean value of cutting coefficients which is the experimentally obtained cutting coefficients. • The predicted value of tangential cutting coefficients at 40000 rpm with Bayesian inference shows the error of 7.85% and 0.89% for the feed rate of 9 µm/flute and 10 µm/flute, respectively from experimentally obtained value. However, this error in tangential cutting coefficients was 15.41% and 5.8% for feed rate of 9 µm/flue and 10 µm/flute, respectively with deterministic approach. • The predicted value of radial cutting coefficients at 40000 rpm with Bayesian inference shows the error of 6.6% and 8.4% for feed rate of 9 µm/flue and 10 µm/flute, respectively from experimentally obtained value. However, this error in radial cutting coefficients was 22.3% and 22.5% for for feed rate of 9 9 µm/flue and 10 µm/flute, respectively with deterministic approach.

• The predicted mean tangential and radial cutting coefficients with Bayesian inference lies near the experimentally obtained cutting coefficients at 100000 rpm and 5 µm/flute feed rate, which shows the robustness of Bayesian inference approach. • The cutting forces predicted with Bayesian inference predicts the cutting forces with upper and lower limit which covers the experimentally measured cutting forces. However, predicted force with deterministic approach shows the deviation from the experimental force.

A static model of chip formation in microscale milling

On the modeling and analysis of machining performance in micro-endmilling, Part I: Surface generation

Modeling of dynamic instability via segmented cutting coefficients and chatter onset detection in high-speed micromilling of Ti6Al4V

High-speed machining

Runout effects in milling: Surface finish, surface location error, and stability

Effects of micro-milling conditions on the cutting forces and process stability

Modelling and simulation of micro-milling cutting forces

Modeling of dynamic micro-milling cutting forces

Chip formation and minimum chip thickness in micro-milling

Cutting forces in micro-end-milling processes

Modeling of threedimensional cutting forces in micro-end-milling

Modeling micro-end-milling operations. Part II: tool run-out

Chatter modelling in micro-milling by considering process nonlinearities

Chatter stability model of micromilling with process damping

Robust chatter stability in micro-milling operations

Characterization of lubrication sensitivity on dynamic stability in high-speed micromilling of Ti-6Al-4V via a novel numerical scheme

Effect of lubrication on machining response and dynamic instability in highspeed micromilling of Ti-6Al-4V

Modeling dynamic stability in high-speed micromilling of Ti-6Al-4V via velocity and chip load dependent cutting coefficients. International Journal of Machine Tools and Manufacture

Application of Bayesian Inference to Milling Force Modeling

Uncertainty analysis of force coefficients during micromilling of titanium alloy. The International

Mechanistic force model for machining process-theory and application of Bayesian inference

Parameter inference under uncertainty in end-milling γ-strengthened difficult-to-machine alloy

A Bayesian network approach to root cause diagnosis of process variations

Intelligent sliding mode control of cutting force during single-point turning operations

Application of AE and cutting force signals in tool condition monitoring in micromilling

Cutting force sensor based on digital image correlation for segmented chip formation analysis

A grey prediction fuzzy controller for constant cutting force in turning

Cutting force prediction in ball end milling of sculptured surface with Z-level contouring tool path

Investigation of the effect of rake angle and approaching angle on main cutting force and tool tip temperature

A closed form mechanistic cutting force model for helical peripheral milling of ductile metallic alloys

Evaluation of cutting force uncertainty components in turning

Bayesian Analysis with Python