Modified Mechanistic Model Based on Gaussian Process Adjusting Technique for Cutting Force Prediction in Micro-End Milling

Liao, Xiaoping; Zhang, Zhenkun; Chen, Kai; Li, Kang; Ma, Junyan; Lu, Juan

doi:https://doi.org/10.1155/2019/7468698

Mathematical Problems in Engineering

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Abstract Introduction Results and Discussion Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 7468698 | https://doi.org/10.1155/2019/7468698

Modified Mechanistic Model Based on Gaussian Process Adjusting Technique for Cutting Force Prediction in Micro-End Milling

Xiaoping Liao,¹Zhenkun Zhang,¹Kai Chen,¹Kang Li,¹Junyan Ma,¹and Juan Lu^2,3

Academic Editor: Xiaoliang Jin

Received12 Nov 2018

Revised10 Jan 2019

Accepted07 Feb 2019

Published28 Feb 2019

Abstract

Micro-end milling is in common use of machining micro- and mesoscale products and is superior to other micro-machining processes in the manufacture of complex structures. Cutting force is the most direct factor reflecting the processing state, the change of which is related to the workpiece surface quality, tool wear and machine vibration, and so on, which indicates that it is important to analyze and predict cutting forces during machining process. In such problems, mechanistic models are frequently used for predicting machining forces and studying the effects of various process variables. However, these mechanistic models are derived based on various engineering assumptions and approximations (such as the slip-line field theory). As a result, the mechanistic models are generally less accurate. To accurately predict cutting forces, the paper proposes two modified mechanistic models, modified mechanistic models I and II. The modified mechanistic models are the integration of mathematical model based on Gaussian process (GP) adjustment model and mechanical model. Two different models have been validated on micro-end-milling experimental measurement. The mean absolute percentage errors of models I and II are 7.76% and 6.73%, respectively, while the original mechanistic model’s is 15.14%. It is obvious that the modified models are in better agreement with experiment. And model II performs better between the two modified mechanistic models.

1. Introduction

Recently, important applications of miniature parts and devices are growing widely in the fields of watches, medical devices, electronics optics, and biotechnology, to mention a few. The micro-end milling is an effective process to produce miniature components with feature sizes of microlevel accuracy. Micro-end milling is similar to macro milling and is usually viewed as a downscaled version of the conventional end-milling with only dimensional differences; however, significant differences between micro-end milling and macro milling in some aspects exist [1, 2]. Although the diameters of micromilling tools are small varying from 25 μm to 1.0 mm [3], most micro-end-milling processes still keep reasonable productivity, which demands feed per tooth to tool radius (fx/r) ratio and spindle speed of micro-end-milling process is selected much larger than that of the conventional end-milling process [4]. Moreover, cutting edge radius has an important influence on chip formation in micro-end milling [5]. Ploughing action is more obvious than cutting action when the uncut chip thickness equals to the cutting edge radius leading to a minimum uncut chip thickness value where continuous chip formation ceases [6, 7] and the surface roughness becomes worse [8]. For the synthesis effects of these factors, poor surface quality, diminished tolerances of the micro-milled components, and shorter tool life take place.

In order to improve productivity and quality at a lower cost in micro-end-milling process, accurate and reliable predictive models of the mechanics and dynamics of micro-end-milling process become necessary. In micro-end-milling process, the machine and tool deflect based on the behavior of cutting forces, which impacts the geometrical accuracy of the processing feature, and the scientific prediction of cutting force can not only optimize cutting parameters and ensure the accuracy of processing but extend tool life [9]. Researchers have used artificial intelligence techniques and statistical methods to predict cutting force. These models are valid within only cutting parameters (cutting speed, feed rate, and depth of cut). Many attempts have been made to develop mechanistic models to predict cutting force from basic principles considering mechanical properties of materials and established principles of metal cutting. Development of mechanistic model is important to analyze the cutting process and get a clear understanding of the mechanism involved [10].

Many researchers have applied the mechanistic models to predict cutting force in micromilling with various assumptions. Srinivasa et al. [11] proposed a new methodology taking into account material strengthening effects and the edge radius to calculate the cutting coefficients. And a mechanistic model was presented based on the cutting coefficients; the model considered the effect of overlapping tooth engagement, meanwhile. Ravi et al. [12] developed a mechanistic model by including the minimum chip thickness size effects and cutting edge radius. Wang et al. [13] investigated the fiber cutting angle based on cutting principle of helical milling and modeled the cutting force. The result indicated that the new model could predict cutting forces with higher accuracy. Han et al. [14] developed a modified mechanistic model which is based on the initial developments in [15]. The motivation for the newly proposed model was to more accurately reflect the actual machining situation at the edge of the tool. The modification was reflected in two aspects: firstly, the partial effective rake angle was used to determine the components of the cutting force vector; secondly, instantaneous cutting force coefficients were considered in the proposed model. Shunmugam et al. [16] presented a mechanistic model to predict thrust and torque in micro-drilling process. The model took into account material removal in indentation zones, major cutting edges, and major cutting edges chisel edge. And the predicting values had a better agreement with the experimental results. Okafor et al. [17] integrated emulsion cooling strategy with mechanistic model to predict cutting forces. The mechanistic model was simulated by MATLAB codes and validated by validation experiments. It was found that the predicted values could capture the qualitative trends fairly well.

Although the overall magnitudes of predictive values obtained by these mechanistic models for cutting force match well the experiment values, the maximum errors and the mean absolute error are still large between predictive values and experiment values, because the cutting forces are closely related to various phenomena found during machining process (e.g., cutting temperature, tool deflection, run-out, wear, and chattering), and these factors will lead to the change of the cutting forces. Moreover, these factors are hardly considered in the mechanical models. In order to solve this problem, many sophisticated modeling approaches which compose the simulation data which have different levels of accuracy are available to improve the accuracy of the predicting models [18, 19]. Kennedy and O’Hagan [20] developed an autoregressive structure to connect the low-accuracy data (Y_low) to high-accuracy data (Y_high) by Y_high(x) = ρY_low(x) + δ(x), where ρ denotes a constant and is used for the scale adjustment and δ(x) is a Gaussian process model (GP) to adjust location tend. Qian et al. [21] expanded [20] research and adopted a linear regression model ρ(x) to replace constant ρ,]. Based on [21], Qian and Wu [22] used GRs to synchronous implement the scale and location adjustments. Xia et al. [23] further developed the approach and used it to integrate misaligned two-resolution data. Inspired by the above literatures, B Shanet al. [18] got a statistical model that could predict the machining forces based on analytical models and finite-element models. The numerical results indicated that the fitted integrated meta-model could approximate the machining forces more precisely.

Gaussian process models can flexibly represent the complex nonlinear relationships without presuppose model and are the common approaches that are used to local adjustment in the method which composes the simulation data with different levels of accuracy to improve the prediction accuracy [24]. Micro-end-milling is a complicated process, and cutting force during micro-end-milling process has nonlinearity characteristics [25]. The mechanistic model of cutting force cannot fully express the relationship between parameters and cutting force. The data gained by physical experiment can truly reflect the actual micro-end-milling processing; however, the cost of the experiment is expensive. Based on the above research, to improve the prediction accuracy of the mechanistic model of cutting force during micro-end-milling, this paper proposes a modified mechanistic model based on Gaussian process. In the modified model the predicted values of cutting force gained by mechanistic model and experiment values are integrated, Gaussian process is used to predict the deviation between experimental and predicted values to implement location adjustment, and the mechanistic model captures the global trend. Moreover, in order to provide more choices for the selection of predictive models, this paper compares the common integrated forms (logarithmic residuals and residuals) of data with different levels of accuracy.

2. Validation of the Mechanistic Model

In order to highlight the primary coverage of this paper, the deduction process of the mechanistic model is put in Appendix A. Figure 1 shows the comparison of cutting forces obtained by experiment and mechanistic model. It is found that the mechanistic model can capture the trend of experimental results and the amplitudes of predicted values compare well with experimental results. However, there are still cases which deviate from experimental results. And the absolute percentage errors of those cases are almost all above 15%, and the maximum absolute error is closed to 37%, which are shown in Figure 2. It can be seen that mechanistic model is little numerically unstable.

3. Analyses

There are several factors that lead to the deviation between mechanistic model and experimental results:

The input parameters of the mechanistic model, such as yield strength, hardness, or Young’s modulus, are deduced from experimental tests. However, there are many factors that affect the precision of the input parameters. For example, the run-out of the tool can lower the precision of the identification of the cutting forces parameters [26].

These mechanistic models do not take into account the effect of temperature rise due to the heat generated from plastic deformation. The temperature directly affects the tool wear and ultimately affects the machining accuracy of the workpiece and cutting forces [18].

Compared with macro machining, the tool diameter is smaller in micro-machining processes. To achieve the recommended cutting speed and keep the reasonable level of productivity, spindle speed is higher which exacerbates tool wear. Tool wear leads to the change of the cutting forces. However, mechanistic models do not take into account the tool wear [4].

When the uncut chip thickness is comparable to the tool edge radius, the change of tool edge radius could result in the decrease of the uncut chip thickness, and it will lead to the decrease of the effective rake angle, which makes the nonlinear increase of the specific cutting energy. The nonlinear increase of the specific cutting energy makes the accuracy of a mechanistic cutting force model considerably poor, especially in micro-machining [27].

The ratio of feed tooth to tool radius is much larger than macro machining operation, due to which the stress variation on the tiny body of the microtool is much higher; it leads to the deflection of the tool tip and the change of cutting force.

As is mentioned above, the cutting forces are closely related to various phenomena found during machining (e.g., cutting temperature, tool deflection, run-out, and tool wear). However, these factors are hardly considered in the mechanical model; as a result, the mechanical model is generally less accurate, which means that it is necessary to modify the mechanical model. Therefore, the paper develops a framework in a way that can combine results from mechanical model and experimental results to create surrogate models that are as accurate as possible, given the resources available.

4. Using Experimental Data to Adjust the Mechanistic Model

The paper integrates mechanistic model with experiments to get a mathematical model and adopts the GP adjustment model based on the data from experimental results and mechanistic model outputs to adjust the mathematical model. Finally, the surrogate model is developed. Figure 3 illustrates the proposed strategy.

Gaussian process regression (GPR) algorithm is very adaptable to deal with complex problems such as high dimension, small sample, nonlinearity, and so on. If the underlying model is linear, quadratic, cubic, or even not polynomial, choosing underlying model among the various possibilities based on model selection principles is done usually. Gaussian process regression can determine model obliquely, but rigorously, by the data which contains rich information [24]. Gaussian process provides a principled, practical, probabilistic approach to learning in kernel machines. This gives advantages with respect to the interpretation of model predictions and provides a well-founded framework for learning and model selection.

A Gaussian process (GP) is an extremely concise and simple way of placing a prior on functions. In the simplest form GPs are limited by the nature of their simplicity: the target data is assumed to be distributed as a multivariate Gaussian, with Gaussian noise on the individual points. This hypothesis makes the matrix operation in the method easy and convenient, and its predictive results also satisfy the Gaussian distribution. However, it is unreasonable to assume that and many practical situations do not satisfy this hypothesis. For example, when the observations are positive and vary between several orders of magnitude, it is difficult to directly assume a Gaussian noise with the same variance. Generally it is standard practice in the statistics literature to take the log of the data. Then modeling proceeds assume that this transformed data has Gaussian noise and will be better modeled by the GP [28]. Hence this paper applies this common data processing method to develop the mathematical model I. At the same time, it should be noted that the cutting forces we obtained have no difference between orders of magnitude. In view of this, the mathematical model II without logarithmic transformation is adopted.

The paper takes two mathematical models applied as follows:

where corresponds to the experimental results, represents the mechanistic model outputs, and are the random errors uncorrelated at different input locations, and are assumed to be a realization from a stationary GR adjustment model. Since the cutting forces are nonnegative, log values of the corresponding responses are used in the model [18]. are used to express and because they have the same distribution function and can be expressed as

where is mean function with and is covariance function with . Here is the squared exponential (SE) covariance function and represents the degree of influence of the distance between two sample points on their relevance and indicates the degree of coverage of the sample on the y-axis [29]. For training points, the outputs are known. Equation (3) is a random Gaussian vector with this covariance matrix:

Then a random Gaussian vector with the same covariance matrix for testing points is generated

And the test outputs according to the prior are

If there are training points and test points then denotes . Matrix of the covariance is evaluated at all pairs of training and test points, and similarly for the other entries , and , is the error variance. Therefore, the corresponding predictor at a new input location can be expressed as

where

The two integration surrogate models for the turning forces (at a new input location ) are given by

where are mathematical models used to modify mechanistic model.

The experimental results selected from the paper [11] are shown in Table 1. The first 20 data sets are chosen to get model parameters and the remaining seven for validation. The details of the tool material, cutting conditions, and experimental setup are given in [11].

5. Results and Discussion

5.1. Predicting Performance Evaluation

To compare the performance of the predicting models quantitatively, three performance metrics, the mean absolute percentage error (MAPE), root mean square error (RMSE), and the mean absolute error (MAE), are used [30, 31]. These indexes embody the deviation between the predicted values and the experimental results. And the smaller they are the better predictive performance it indicates. This paper applies these three criteria to evaluate the predicting models. The performance metric is defined as follows:

where is the number of test sample, is the predicted value, and is the testing data acquired from experiment. The MAPE is a unit-free measure of accuracy for the predicted values and is sensitive to small changes in the data. The RMSE represents the deviation between predicted values and experimental results. The MAE indicates similarity between predicted values and experimental results [30].

5.2. Comparison Results

The data in Table 2 is a comparison of amplitudes of forces predicted with mechanistic model and modified model, from which we can see that the modified models are closer to experiment. Figures 4 and 5 show the comparison of errors of predicted cutting forces obtained from original mechanistic model and modified models. It is found that the absolute percentage errors of modified mechanistic models are smaller than original mechanistic model’s, which means that the results obtained by modified models are much closer to experimental values. The performance metrics shown in Table 3 also indicate that modified mechanistic models perform better. This may be due to the fact that only peripheral cutting edges of micro-end mill are considered in the mechanistic model of micromilling. In practice, end-cutting edge and back-cutting phenomenon introduce additional cutting forces [11], although attempts have been made to develop mechanistic model to predict cutting force from basic principles considering mechanical properties of materials and established principles of metal cutting. Micromilling is characterized by multifactor feature and uncertainty; the cutting forces produced by the machining process contain synthetic feature, such as linear and nonlinear characteristics [25, 32], while a single prediction model is unable to capture multiple data features at the same time leading to unsatisfactory prediction precision. The modified models combine the mechanistic model with Gaussian process adjustment model; on the one hand, the mechanistic model is used to control the global trend; on the other hand, the Gaussian process regression (GPR) algorithm is suitable for nonlinearity capturing and location adjustment to reduce the deviation between the experimental and the predicted values [33]. Therefore, the proposed modified models perform better than the mechanistic model in cutting force prediction. The mechanistic model we combined in the paper ignores the run-out effect though it is less than 0.1um. This is part of the explanation for the deviation of the modified models and the mechanistic model.

In practice, there are some other continuous transformations, which can transform the data of observation space into a space that can be modeled well by using the method. Log transformation is one of them. It is noted that, for inference made on log-transformed data, biases may be introduced when retransforming back to the original scale and care should be taken to include appropriate bias correction factors when presenting and interpreting the results on the nontransformed scale [34, 35]. The original information of data can be well preserved without data transformation at the cost of complex computation. Figure 6 indicates comparison of error of predicted cutting forces obtained using modified mechanistic model I and modified mechanistic model II. It can be seen that modified mechanistic model II is much more stable than I. And the performance metrics shown in Table 3 further confirm that modified mechanistic model II precedes model I.

6. Conclusions

Due to the complexity of the micro-end-milling process, the mechanical model cannot describe the detailed characteristics between parameters and cutting forces and the paper proposes two modified mechanistic models integrating mechanistic model with mathematical model based on Gaussian process instead of using the mechanistic model singly to predict micro-end milling forces. As the basis of predicting micro-end milling forces, the mechanistic model of milling force based on cutting mechanisms takes into account of edge radius, material strengthening effects, uncut chip thickness variation along the cutting edge, effect of shear band spacing on strain rate, and principles of metal cutting. And Gaussian process is used to predict the combination of mathematical model to achieve local adjustment. The modified models have been validated with full factorial design of experiments. The comparative results show that the developed models are of higher prediction accuracy than the original mechanistic model, since the Gaussian process used for local adjustment is able to capture the more characteristics between machining parameters and milling force. Further, two modified models get better prediction accuracy, and the prediction effect is not much different; the prediction accuracy of modified mechanistic model II is slightly better than modified mechanistic model I. The two new modified prediction methods can complement the relationship between parameters and cutting force that cannot be expressed by mechanical models and make the prediction results more in line with the actual processing. These modified models can provide the guide for selection of suitable predictive models and initial processing parameters for a new machining process.

It is the fact that when the training data is more, the data relationship mined by predictive model is more adequate and the generalization ability of the predictive model is stronger. When the amount of data is small, for the prediction model obtained by training limited data, if the machining parameters change, it is possible that the feasibility of the prediction model is not sure. In actual machining process, to ensure the accuracy of prediction model, it is necessary that collecting data on site or new experimental data updates the sample to retrain the prediction model. At present, this paper mainly explores modification of mechanical model and verifies the feasibility of proposed two modified mechanical models on existing data. The adaptability and robustness of machining parameters change for the two modified mechanistic models are further explored in subsequent studies.

Appendix

A.

A.1. Mechanistic Model Formulation

In the analysis, each tooth cutting edge is partitioned into a series of elemental oblique cutting tools which are shown in Figure 7. The cutting forces on these oblique tools are determined from the mechanics of oblique cutting analysis. A Cartesian coordinate system is defined with its origin located at the centre of the cutter in order to denote the cutting forces. The X axis is along the feed direction with respect to the workpiece, Y axis is transverse to the feed direction, and Z axis is along the cutter axis [11].

The oblique elemental forces are shown in Figure 7, where tangential force () and radial force () acted toward to the centre of the tool and axial force () is along the Z-axis, respectively [10]. are expressed [36] as

represents the uncut chip thickness, represents the width of cut, represent the cutting coefficients and represent the edge coefficients, the values of the cutting coefficients and the edge coefficients are given in Table 4, and is the rotation or immersion angle measured clockwise from the positive Y-axis to the reference flute-j [11]. Figure 8 shows the geometry of uncut chip in slot end-milling. In instantaneous uncut chip thickness given by

is the feed per tooth. Then a transformation matrix is used to convert from oblique cutting forces to the elemental forces (machine reference forces) given in (A.4) [37].

The force components after solving the above matrix will be as follows:

where , , and are the elemental forces acting along x-, y-, and z-direction, respectively. More details about the mechanistic model formulation are given in paper [11].

Figure 9 shows the flowchart of predicting machining forces using the above mechanistic models.

A.2. Cutting Force Coefficients

The cutting force coefficients are based on the oblique cutting model in [38]. The expressions are given below:

where is the shear stress in the shear plane, is the normal shear angle, is the normal rake angle, i is the obliquity or inclination angle, is the normal friction angle, and is the chip velocity angle measured on the rake face.

A.3. Edge Force Coefficients

Edge force coefficients can be found from experiments [39] or estimated from the analytical model proposed in [40]. This model is developed for machining with tool having large edge radius at low depth of cut which takes into account the rubbing forces due to the material ploughing action at and beneath the rounded cutting edge. The following equations are used in the present work:

where is the cutting edge radius of the tool, is the shear strength of the material, and is the stagnation or neutral point angle. In the present model the value of stagnation angle is taken as because at low depth of cut and uncut chip thickness, the rake angle becomes negative due to the edge radius of the cutting tool [10]. More details about the mechanistic model formulation are given in paper [11].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare there are no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the support of National Natural Science Foundation of China (No. 51665005), Innovation Project of Guangxi Graduate Education (YCBZ2017015), the Project of Guangxi Colleges and Universities Key Laboratory Breeding Base of Coastal Mechanical Equipment Design, and Manufacturing and Control (GXLH2016ZD-06) and the support from Guangxi Key Laboratory of Manufacturing Systems and Advanced Manufacturing Technology (Grant No. 17-259-05S008). Additionally, the authors give sincere thanks to Y. V. Srinivasa and M. S. Shunmugam for their work.

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