An Improved Mathematical Model of Cutting Temperature in End Milling Al7050 Based on the Influence of Tool Geometry Parameters and Milling Parameters

Excessively high temperature during milling will shorten the life of the milling tools, reduce the surface quality of the workpiece, and increase production costs. In this paper, a novel cutting temperature prediction model for milling Al7050 is proposed, which considers both tool geometry parameters and milling parameters. The aim is to adjust the milling conditions in advance according to the predicted temperature, thereby prolonging the tool life and improving the machining quality. Through single-factor experiments, it is found that there is a polynomial relationship between tool geometric parameters and cutting temperature. Based on the empirical formula of cutting temperature, an improved mathematical model of cutting temperature is proposed. In this model, eight variables including tool geometry parameters and milling parameters are considered. The coefficients of the mathematical model are determined by the cyclic fitting method. The simulation data and experimental data show that compared with the temperature prediction model considering only milling parameters, this model has better prediction accuracy. This idea can also be used to establish the cutting temperature prediction model of other materials.


Introduction
Widely used in high-speed train, aerospace, aircraft, and other manufacturing industries, milling plays an important role in manufacturing, mainly due to its advantages in high production efficiency and excellent surface processing quality, which leads to lower production costs and higher economic efficiency in mass production. In the milling process, cutting force, cutting temperature, cutter displacements, etc., which are affected by many factors, are directly related to the final machining quality. It is very important to model them and predict the possible situation [1,2]. e highest cutting temperature produced during milling is an important analytical and control parameter, because excessive high temperature not only accelerates tool wear and shortens the tool life, but also impairs the surface quality of the workpiece. Detailed study on the prediction of cutting temperature would facilitate the cutting condition controlled during milling, thereby preventing the tool from working at excessively high temperature. Currently, the prediction methods of cutting temperature can be summarized into the following four types: empirical equation, analytical method, numerical method, and neural network method [3,4].
Kadirgama et al. [5] established a first-order tool temperature mathematical model for determining tool temperature by using the response surface method. Yi-jian et al. [6] developed a nonlinear curve using particle swarm optimization to determine the cutting temperature. Bai et al. [7] detected the change of cutting temperature during high-speed milling of Al7050-T7451 aluminum alloy and established an empirical equation of cutting temperature through experimental analysis. eoretical prediction yielded by this equation match well with the measured temperature. Findes et al. [8] studied the influences of cutting speed, feed rate, and cutting depth on cutting force and cutting temperature when machining AISIH11 steel with composite ceramic tools. It has been found that the cutting speed affects the temperature in the cutting zone more significantly than the cutting depth and feed rate do. Richardson et al. [9] established a temperature prediction model in the dry milling process of aerospace aluminum alloys and found that higher cutting speed and feed rate can reduce the temperature rise of the workpiece. Toh et al. [10] compared the surface temperature of the chip of hardened steel during down milling and up milling by infrared technology. Spânu and Iliescu [11] used a full-factor experimental design method to develop a mathematical model of cutting zone temperature during the milling of GFRP materials via polynomial regression technique. Cutting speed, feed rate, and cutting depth are changed during the experiment and the corresponding temperatures are recorded by a thermal camera. Chien et al. [12] used the AdvantEdge software to simulate ball end milling of Inconel 718 with a coated carbide tool. With the cutting speed, feed rate, and axial cutting depth taken into consideration, they designed a three-factor and three-level experiment using Box-Behnken method and developed a linear polynomial model of the maximum temperature of the tool using multiple linear regression. Abhang and Hameedullah [13] used the tool-work thermocouple technology and conducted an experimental study on the average temperature of the chip-tool contact surface. ey believe that factors such as cutting speed, feed rate, depth of cut, and blunt radius have a certain impact on the chip-tool contact surface temperature during EN-31 steel alloy turning. Based on this conclusion, the first-and secondorder empirical models have been established. Saglam et al. [14] studied the influencing factors of cutting force component and tool tip temperature and found that the rake angle was effective on all cutting force components, while the cutting speed was effective on the tool tip temperature. Santhanakrishnan et al. [15] studied the influences of the blunt radius, rake angle, cutting depth, feed rate, and cutting speed on the temperature rise during the milling of the Al6351 with high-speed steel. Sheng et al. [16] studied the influences of cutting speed and tool wear rate on cutting temperature in milling.
e authors obtained an empirical equations of cutting temperature, involving feed rate, axial cutting depth, cutting speed, and radial cutting depth, via an orthogonal regression test. Kaushik et al. [17] studied the effects of helix angle, rake angle, cutting speed, feed rate, and axial cutting depth on cutting temperature when milling Al7068 with highspeed steel and milling and developed a multivariate secondorder mathematical model. Rana and Kumar [18] developed a mathematical model of the cutting temperature when using a tungsten carbide tool to turn EN19 steel by central composite design method. Using a central combination design method, Zeelanbasha et al. [19] designed the cutting experiment of cutting Al6006-T6 with high-speed steel end milling and established a second-order mathematical model of dependency of cutting temperature on spindle speed, feed rate, axial cutting depth, radial depth of cut, and rake angle. Zheng et al. [20] adopted the method of combining simulation and experimental verification to establish the 2D milling simulation model, studied the influence of milling parameters (feed rate, milling depth, spindle speed, milling width, and milling depth) on milling force and milling temperature during the milling process of 7075 aluminum alloy, then carried out milling parameter optimization. Ping et al. [21] studied the influences of milling parameters and tool geometric parameters on milling force and milling temperature through single-factor experiment.
In the analytical prediction of cutting temperature, Aneiro et al. [22] conducted experiments on hardened steel with multilayer coated carbide tools and studied the influencing factors of tool temperature, tool wear, cutting force, and surface roughness. Under conditions of fixed cutting speed and metal removal rate, Cui et al. [23] studied dependencies of the average instantaneous temperature of the tool on the radial depth of cut and the feed rate when the AISI H13 tool steel was face-milled with an uncoated cemented carbide tool. Zhou [24] proposed an analytical model for predicting workpiece temperature in a torus end mill with 4-axis milling and verified the validity of the model. ere are mainly two methods for numerical studies on turning and milling temperature: the finite element method (FEM) and the finite difference method (FDM). Lin et al. [25] proposed analytical model of workpiece temperature using the finite difference method (FDM), which takes into account the complex tool geometric and the effect of friction on the flank of the tool on the temperature rise. Sreeramulu et al. [26] used FEM software (Deform-3D) to simulate three-dimensional tilt cutting operations and studied the influences of cutting speed, feed rate, and cutting depth on cutting temperature. Lacalle et al. [27] performed finite element simulation of dry milling of forged aluminum alloys. e authors found that the cutting temperature rises as the cutting speed and inclination angle increase.
In neural network prediction, Rai et al. [28] established a new temperature prediction model of shear plane during end milling through a feed-forward back propagation neural network. Korkut et al. [29] predicted the temperature of the cutting-edge contact interface of the tool by regression analysis and artificial neural network. Considering the cutting speed, feed rate, and the cutting depth, Masoudi et al. [30] used artificial neural network (ANN) and adaptive neurofuzzy inference system (ANFIS) to predict cutting temperature. e above researches on cutting temperature prompt the development of the study on cutting temperature, but most of them focus on the relationship between cutting parameters (less than five) and cutting temperature; it is not clear how the tool geometric parameters affect the cutting temperature. e model proposed in this paper considers eight parameters including milling parameters and can better reflect the variation of cutting temperature during milling.

Workpiece Material, Tool Material, and Tool Geometry Parameters.
is study uses aluminum alloy 7050 as the workpiece material. Aluminum alloy 7050 has been widely used in aerospace, automotive industry, and lightweight construction due to its high strength, high toughness, and excellent resistance to stress corrosion.
Ultrafine grained carbide materials, widely used in aviation enterprises, are selected as tool materials. e cemented carbide tool is one of the main cutting tools for cutting aluminum alloys because of its sharp cutting edge, good antiflaking performance, and good wear resistance.
During the milling process, tool geometry parameters play an important role. ey should be taken into consideration when choosing the cutter, such as diameter, blunt radius, helix angle, rake angle, and relief angle. In the singlefactor experiment, the nonvariable tool geometry parameters are set as follows: the milling cutter diameter is 16 mm, the blunt radius is 0.01 mm, the helix angle is 30°, the rake angle is 15°, and the relief angle is 8°.

Finite Element Milling Model.
During the milling process, the cutting part can be treated as turning. e plane perpendicular to the axial direction of the tool is selected as the reference place, as shown in Figure 1(a). In Figure 1(b), keeping the cutting length constant, the undeformed chip thickness varied with the cutting position. So, the milling can be completely converted to orthogonal turning with varying undeformed chip thickness. To make the cutting mechanism easier and more intuitive, the cutting process can be simplified to 2D orthogonal cutting. e projections of the rake face and flank surface are simplified to straight lines and a nose arc. In general, due to the influence of helix angle, milling is an oblique cutting process. e active cutting edge and the cutting speed direction are not perpendicular to each other, which results in changes in the rake angle and the relief angle. In the 2D milling simulation process, the equivalent angle of the tool is calculated by the following formula [31]: where r oe is the turning tool's equivalent rake angle, α oe is the turning tool's equivalent relief angle, ψ A is the flow chip angle, λ s is the shear angle, c n is the rake angle, and α n is the relief angle.

Influence of Tool Geometry on
Cutting Temperature e single-factor simulations of tool geometry parameters were carried out to study the relationship between the changes of tool geometry parameters and the corresponding cutting temperature when machining aluminum alloy 7050. e milling parameters are as follows: spindle speed is 15000 r/min, feed per tooth is 0.15 mm, the cutting depth is 1.5 mm, and the cutting width is 1.5 mm.

Tool Diameter.
e parameter range of the tool diameter is set to 6-24 mm with increments of 2 mm. e relationship between the cutting temperature and the diameter of the milling cutter is shown in Figure 2. e standard deviation of the data in Figure 2 is 37.80, which means that the tool diameter has a great influence on the cutting temperature. Figure 2 shows that the cutting temperature gradually increases as the tool diameter increases. Polynomial fitting (T � −0.2487x d 2 + 13.911x d + 209.99) shows higher accuracy for the relationship between tool diameter and cutting temperature among fitting ways like exponential, linear, logarithm fittings, with the fitting accuracy R 2 of 0.9975 (Table 1). Based on the above analysis, it can be concluded that selecting a smaller tool diameter is advantageous for reducing the cutting temperature. Nevertheless, it should be noted that the smaller tool diameter, the greater the cutting force, so a trade-off must be made between the cutting force and cutting temperature.

Blunt Radius.
e parameter range of the blunt radius is set to 2.5-25 μm with increments of 2.5 μm. e relationship between the blunt radius and the cutting temperature is shown in Figure 3. e standard deviation of the data in Figure 3 is 7.72. e cutting temperature fluctuates with the change of blunt radius. Figure 3 demonstrates that as the blunt radius increases, the cutting temperature tends to increase.
Polynomial fitting (T � −0.0041x r 2 + 1.1545x r + 349.89) shows higher accuracy for the relationship between blunt radius and cutting temperature among fitting ways like exponential, linear, logarithm fittings, with the fitting accuracy R 2 of 0.9975 (Table 1). In the actual cutting process of aluminum alloy, the cutting edge of the cemented carbide tool is generally not rounded, which also conforms to the rule of the blunt radius and the cutting temperature.

Helix Angle.
e parameter range of the helix angle is set to 12-40°with increments of 4°, and the relationship between the helix angle and the cutting temperature is shown in Figure 4. e standard deviation of the data in Figure 4 is 12.61. e cutting temperature fluctuates with the change of helix angle. Figure 4 shows a nonlinear relationship between the cutting temperature and the helix angle; the cutting temperature decreases with the increase of the helix angle of the milling cutter. Polynomial fitting (T � −0.036x l 2 + 0.5372x l + 382.92) shows higher accuracy for the relationship between tool helix angle and cutting temperature among fitting ways like exponential, linear, and logarithm fittings, with the fitting accuracy R 2 of 0.9821 (Table 1). According to Figure 4, it can be seen that when milling aluminum alloy 7050 at high speed, a larger helix angle should be selected, and a larger helix angle is often used in actual production.

Rake Angle.
e single-factor analysis parameter of the tool rake angle is set to 6-26°with an increment of 2°, and the variation of cutting temperature with tool rake angle is shown in Figure 5. e standard deviation of the data in Figure 5 is 11.33. e cutting temperature fluctuates with the change of rake angle. Figure 5 shows that there is a decreasing relationship between them. e reason is that the shear angle decreases with the increase of rake angle, and the plastic deformation of metal and the friction of rake face Mathematical Problems in Engineering 3 decrease, which leads to the decrease of cutting temperature. Polynomial fitting (T � −0.0123x q 2 − 1.3561x q + 389.38) shows higher accuracy for the relationship between tool rake angle and cutting temperature among fitting ways like exponential, linear, and logarithm fittings, with the fitting accuracy R 2 of 0.9545 (Table 1). Ensuring sufficient strength of the milling cutter blade, a large rake angle helps to reduce      the cutting temperature. Generally, the cemented carbide tools selected have large rake angles (12°-20°), due to the low hardness of aluminum alloys.

Relief Angle.
e parameter range of the tool relief angle is set to 2°-16°with increments of 2°. e relationship between the tool relief angle and the cutting temperature is shown in Figure 6. e standard deviation of the data in Figure 6 is 2.17, which means that the data fluctuates slightly. As the relief angle increases, the cutting temperature remains almost unchanged, which is in line with the rule that the highest cutting temperature occurs where the chip meets the rake face of the tool. erefore, the effect of the relief angle on the cutting temperature can be ignored.

Empirical Equation of Cutting Temperature.
According to the metal cutting principle, the mathematic relation between the cutting temperature and the milling parameters is governed by an exponential equation. e general form of the cutting temperature empirical equation is [32] T � Cv i f j z a m p a n e , where v is the cutting speed, f z is the feed per tooth, a p is the cutting depth, a e is the cutting width, i, j, m, and n are the exponential coefficients corresponding to each parameter, and C depends on the workpiece material and cutting condition.
To fit the empirical model, a milling parameter orthogonal experiment of four factors and four levels was carried out. In the experiment, the blunt radius is 0.01 mm, the helix is 30°, the rake angle is 15°, and the relief angle is 8°. Table 2 shows the simulation results under different milling parameters.
By fitting the above simulation results, the empirical model of cutting temperature for cemented carbide tool milling aluminum alloy 7050 is obtained: (3)

Improved Model of Cutting
Temperature. According to the above single-factor test results and the relationship obtained by fitting, we can infer that there is a nonlinear relationship of polynomial between the cutting temperature and tool geometry parameters during milling. Based on the empirical equation of cutting temperature, the parameters of the four tool geometry parameters and the cutting temperature empirical model are combined to obtain the following relationship: where x d is the tool diameter, x r is the blunt radius, x l is the helix angle, x q is the rake angle, k 0 is a constant, and k i (i � 1, 2, 3, . . ., 12) is the coefficient corresponding to each parameter.
To fit the improved model, a tool geometry parameter orthogonal experiment of four factors and four levels was carried out. e milling parameters are as follows: the cutting speed is 750 m/min, the feed per tooth is 0.1 mm, the axial depth is 0.15 mm, and the radial depth is 0.15 mm; the geometry parameters and simulation results of the end milling are shown in Table 3.
Based on the simulation results shown in Table 2, the following relationship is obtained via cycle fitting, the process of which is shown in the flowchart in Figure 7: e tool diameter, blunt radius, helix angle, and the rake angle of equation (6) do not completely conform to the     No. Simulation Improved model Empirical equation

Simulation Data Verification
Five sets of parameters are selected for the simulation test, for verifying the prediction accuracy of the mathematical model; the parameter settings are shown in Table 4; the simulation results and the prediction results of the improved model and empirical equation are shown in Figure 8. e prediction errors of the improved model for the simulation temperature are 1.28%, 1.36%, 1.57%, 1.34%, and 2.44%, and the prediction errors of the empirical equation for the five simulation temperatures are 11.43%, 5.84, %, 3.11%, 8.71%, and 12.02%. e prediction accuracy of the empirical equation fluctuates sharply because the tool geometry parameters are not considered. According to the previous single-factor analysis, the geometry parameters of the tool have a significant influence on the cutting temperature. When the geometry parameters of the tool change, the cutting temperature changes as well, but the empirical equation cannot reflect this phenomenon, resulting in poor prediction accuracy. In contrast, the improved model has a high prediction accuracy because it takes the relationships between all tool geometry parameters and the cutting temperature into consideration.

Experimental Data Verification
e orthogonal machining tests with an infrared (IR) camera system were used to verify the improved model. e material is aluminum alloy 7050, and the tool is WC tool. e infrared camera ( Figure 9) used in the experiments has an uncooled focal plane array (FPA) detector. e IR camera has the spectral responsivity of 7-14 mm, an interlace scanning property of 2 : 1, and a minimum resolvable temperature difference of approximately 0.2°C at 25°C [33]. e cutting parameters are set as follows: spindle speed is 8000 r/min, axial depth is 2 mm, the radial depth of cut is 16 mm, tool diameter is 16 mm, blunt radius is 0.01 mm, and helix angle is 30°. Test with various rake angles and feed rates. e test conditions are shown in Table 5. Figure 10 shows the prediction of the experimental data by the improved model of cutting temperature and the empirical model of cutting temperature. e prediction errors of the two models for the experimental temperature are shown in Table 5. e results show that the improved model proposed in this paper has a higher prediction accuracy compared with the traditional empirical model.

Conclusion
In the process of Al7050 with cemented carbide tools, the influences of tool geometry parameters on the cutting temperature were studied, and the corresponding mathematical relationship was established for each parameter. e relief angle has little effect on the cutting temperature, and there is a polynomial relationship between the geometry parameters and the cutting temperature. An improved mathematical model of cutting temperature, which involves tool geometry parameters and milling parameters, is established based on the empirical equation. We can use this model to estimate the cutting temperature of the tool under different tool geometric parameters and milling parameters. For excessively high cutting temperature, the model can well reflect the optimization direction of variables. According to the prediction temperature, we can also change cutting condition, such as the consumption of cutting fluid, to lower cutting temperature, which can prolong tool life and improve machining quality.
e simulation data and experimental data show that the cutting temperature model considering the tool geometric parameters has better prediction accuracy. e average prediction error is 1.6% for simulation temperature and the average prediction error is 8.6% for actual cutting temperature. is modeling idea can be used with other types of tools and cutting materials.  Data Availability e relevant data of calculation used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.