Sage Journals: Discover world-class research

Abstract

A novel milling force model for cutting aviation aluminum alloy 7075 using carbide end mill is established in this article. A two-dimensional end-milling model is set up to investigate the influence of tool geometric parameters on milling force with the single-factor analysis. The relationship between milling forces and tool geometric parameters is obtained by nonlinear regression fitting method. Based on the existing empirical model of milling force, quadratic polynomial factor is taken into consideration to explore the influence of tool geometric parameters on milling force. Thus, a novel milling force model is built up which includes tool geometric parameters and milling parameters. The coefficients of the novel model are identified by the direct method and the loop method. The precisions of the coefficients obtained by the two methods are compared between prediction values and experiment values. After comparison, the model whose coefficients are obtained by loop method has higher prediction ability. End-milling experiments were carried out to verify the prediction accuracy of the novel milling force model. The result shows that the novel model of milling force has high accuracy in prediction. The method of building the milling force model proposed in this article can be applied to other types of milling cutter.

Keywords

Milling force tool geometric parameters milling parameters end mill simulation experiment

Introduction

Milling process has been extensively applied for die components, and aeronautical or biomedical parts made of metal alloys, such as Ti-, Ni-, and Al-based alloys, due to their excellent mechanical properties.^1,2 During high-speed milling, the milling force has an important influence on the tool life, surface quality, vibration stability, cutting tool, and workpiece machining deformation. This is why a precise prediction of milling force is so essential to select machining conditions.³

The existing milling force models were divided into following categories: empirical model, analytical model, artificial intelligence model, and finite element method.^4–7 In the empirical model, the related factors influencing milling force are expressed in the form of coefficients and exponents.⁸ Ren J et al.⁹ combined the grey correlation with Taguchi’s method to analyze the relationship between tool geometric parameters and cutting forces, surface roughness, and acceleration. That is to say, the accuracy of the empirical model depends on a lot of experimental data. The analytical model is based on the shear theory proposed by Merchant.^10,11 He deemed that cutting process was the shear deformation of the metal layer of workpiece generated along the shear plane under the effect of tool rake face. In the analytical model, cutting area, friction angle, and shear angle are calculated to simulate the mechanical mechanism of milling process. Chen D et al.¹² predicted the milling force of variable helical angle milling cutters using analytical method and verified with experiments. However, due to the influence of high strain rate, elastic–plastic deformation, and high rate of temperature change, the related angles and the mechanical mechanism of cutting face could not be obtained accurately. Artificial intelligence model used artificial neural network technology to effectively deal with the related parameters which affected the cutting force. Wang and Huang¹³ used the improved back propagation (BP) neural network to establish the neural network model of the milling force and optimized the process parameters. Zheng¹⁴ applied the particle swarm optimization of artificial neural network theory to the high-speed milling force model. The finite element model discretizes the continuum into a set of finitely sized elements to solve the continuum mechanics problem. Wang L et al.¹⁵ used the professional finite element software, providing access to varying speed cutting force changes.

In the models above, milling force is a function established respectively with the parameters of milling three elements, shear angle, cutting force coefficient,and so on. On some occasions, these models could predict the milling force accurately. However, few models have taken into consideration the influence of the cutter geometrical structure on the milling force, and few scholars have done research on the relationship between the cutter geometrical structure and the milling force. Actually, in all the models mentioned above, it is important to note that the coefficients of models have to be determined in advance for each specific cutter and then to be used under different cutting conditions. In other words, cutting force coefficients have to be renewed whenever the geometry of cutter is changed.¹⁶ This article is based on a two-dimensional (2D) milling simulation model of solid carbide end-mill milling aviation aluminum alloy 7075. The influence of tool geometric parameters on milling force is studied; the corresponding mathematical model and a novel milling force model which contain tool geometric parameters and milling parameters are built through the finite element simulation. And a milling experiment has been set up to verify the predictive accuracy of the novel model by comparing the predicted values and the experimental values.

Finite element analysis on milling force

Tool material and geometric angle

Aviation aluminum alloy 7075 has several advantaged properties such as low yield strength, less hardness, good thermal conductivity, and low cutting force. So, there are a lot of cutting tool materials suitable for processing aviation aluminum alloy 7075, such as high-speed steel, cemented carbide cutting tools, and coated tools. However, high-speed steel is unsuitable in high-speed milling process because of its bad wear resistance; coated tool would reduce the surface quality of workpiece because of the diffusion from coating to workpiece. Therefore, ultrafine-grain-cemented carbide material, which is widely used in the actual aviation enterprise, is selected as the tool material in this article.

Geometrical structure parameters of tool edge have extraordinary important impact on milling process. The selection of milling tool includes not only cutter’s material and type but also some other important geometrical parameters: diameter, blunt round radius, helix angle, rake angle, and relief angle. For the convenience to carry out single-factor analysis, geometrical structure parameters of end mill in this article are 16 mm in milling cutter diameter, 0.01 mm in blunt round radius, 30° in helix angle, 15° in rake angle, and 8° in relief angle.

Finite element milling model

No matter how complex the tool structure is, the cutting part can be regarded as the evolution of the turning tool. In milling, the main section reference is produced in the process of cutting out perpendicularly to the axial direction of the tool, as shown in Figure 1(a). After that, the cutting length is kept unchanged, while the feed thickness changes with the change in cutting position, as shown in Figure 1(b). Milling can be roughly transformed into orthogonal turning of cutting thickness variation. In order to express the cutting mechanism more simply and intuitively, we have simplified the cutting process to 2D orthogonal cutting. In 2D orthogonal cutting, the cutter is simplified to a straight line and a nose arc for the projection of the rake face and the flank surface.¹⁷ Generally speaking, milling is a kind of oblique cutting process, because the helix angle of milling cutter is usually not 0 degree. The main cutting edge and cutting speed direction are not perpendicular, which lead to a change in front and rear corners in the process of actual cutting. In the process of 2D milling simulation, the equivalent tool angle is obtained by tool angle transformation, and the equivalent equation is shown below¹⁸

{\begin{matrix} \sin γ_{0 e} = \sin λ_{s} \sin ψ_{A} + \cos λ_{s} \cos ψ_{A} \sin γ_{n} \\ \cos α_{0 e} = \sin λ_{s} \sin ψ_{A} + \cos λ_{s} \cos ψ_{A} \cos α_{n} \end{matrix}

(1)

where $r_{0 e}$ is the equivalent rake angle and $α_{0 e}$ is the equivalent relief angle. According to the laws of Stabler test, chip flow angle $ψ_{A}$ is approximately equal to inclination angle, namely, the helix angle of end mill.

Based on cutting simulation software, the models of workpiece and tool are established, as shown in Figure 1. In order to obtain the optimal mesh, the adaptive re-meshing technique is adopted.

Figure 1.

Milling model: (a) Section model; and (b) 2D simplified model.

Influence of tool geometric parameters on milling force

In order to study the law of change of milling force with the change in tool geometric parameters when cemented carbide tool is used in machining aviation aluminum alloy 7075, the single-factor simulation analysis of the geometric parameters of the cutting tool was carried out. The geometric parameters of the cutter were given as mentioned above. Different tool geometric parameters were chosen to carry out single-factor analysis, and then the law of effects on milling force could be concluded. The milling parameters are as follows: the spindle speed was 15,000 r/min, the feed per tooth was 0.15 mm, the axial cutting depth was 1.5 mm, and the radial cutting depth was 1.5 mm.

Diameter

The scale of milling cutter diameter is set at 6–24 mm and the increase in each scale is 2 mm. The tangential force F_x and the radial force F_y represent the cutting force F_t and the feed force F_f in the orthogonal cutting simulation, respectively. In this article, in order to facilitate comparison with experimental data, the cutting forces are replaced by F_x, while the feed forces are replaced by F_y. Figure 2 shows the relationship between the milling force and the diameter of end mill.

Figure 2.

Relationship between milling force and diameter.

In Figure 2, it could be concluded that with the increase in the milling cutter diameter, the tangential force F_x decreases obviously and so does the radial force F_y, but with a small range. And the milling forces fluctuate slightly between 10 and 16 mm. The regression method was applied to fit the relationship between the milling cutter diameter and the tangential force F_x. It could be observed that quadratic polynomial fits well and the result could be expressed as $F_{x} = 0.232 x_{d}^{2} - 10.48 x_{d} + 261.5$ with the corresponding R² value up to 0.98. Through the above analysis, a conclusion that the bigger diameter is conducive to reduce milling force has been made. However, the bigger diameter means the production cost is higher because the tool material is made of cemented carbide which is relatively expensive. Therefore, without decreasing the machining accuracy and the machining demands, the smaller diameter is preferred for lower cost.

Blunt radius

The scale of the blunt radius of cutting edge is set between 2.5 and 25 μm and the increase in each scale is 2.5 μm. Figure 3 shows the curves between the blunt radius and the milling force which are obtained by simulation.

Figure 3.

Relationship between milling force and blunt radius.

In Figure 3, it could be seen that the milling force and the blunt radius linearly increased. Due to the increase in the radius, the cutting edge of the end mill is no longer sharp. If the blunt radius is greater than the instantaneous uncut chip thickness, the practical rake angle is negative. When the cutter tooth presses and slides along the surface of the workpiece, it causes work-harden and is not conductive to the next cutting process. However, appropriate blunt radius is necessary, for excessively sharp cutting edge can easily lead to edge collapse. As a result of fitting the two curves in Figure 3, the regression equations could be written as $F_{x} = 1.365 x_{r} + 138.7$ and $F_{y} = 2.751 x_{r} + 6.077$ , and the linear regression R² values are 0.965 and 0.988, respectively. The growth is relatively large.

Helix angle

The scale of the helix angle is from 12° to 40° and the increase in each scale is 4°. Figure 4 shows the curves between the helix angle and the milling force which were obtained by simulation.

Figure 4.

Relationship between milling force and helix angle.

Figure 4 shows that the milling force and the helix angle are in monotonous relations. The milling force decreases linearly with the increase in helix angle. The results are as follows: $F_{x} = 1.075 x_{l} + 185$ and $F_{y} = 0.029 x_{l}^{2} - 2.368 x_{l} + 77.88$ , and the regression R² values are 0.987 and 0.982, respectively. From the regression analysis and Figure 4, it is wiser to adopt bigger helix angle when the aviation aluminum alloy 7075 is under high-speed milling. It is in accordance with practical choice and the appropriate scale of angle is 25°–40°. Because the helix angle can prolong the length of cutting edge, reduce the unit cutting force on the cutting edge, stabilize the cutting process, and increase the tool life. Small helix angle is used in rough machining because of the benefit of chip removal, whereas big helix angle is chosen in finish machining to improve quality.

Rake angle

The scale of the rake angle is set from 6° to 26° and the increase of each scale is 2°. Figure 5 shows the curves between the rake angle and milling force which were obtained by simulation.

Figure 5.

Relationship between milling force and rake angle.

Figure 5 shows that the rake angle and the milling force do not completely linearly decrease. The tendency of F_y changes when the rake angle reaches 20°–22°, but the increasing amplitude is not large. The reason is that when the rake angle increases, the shear angle increases, the plastic deformation of the metal decreases, the deformation coefficient decreases, and the friction along the rake face decreases, so that the milling force also decreases. However, too large rake angle leads to small wedge angle, the intensity of the tool becomes weaker, it easily causes tipping, and finally, it may affect the life of milling cutter. By linearly fitting the simulation results, it holds that $F_{x} = - 1.404 x_{q} + 175.6$ , with the regression value R² equal to 0.982, and $F_{y} = 0.063 x_{q}^{2} - 2.6 x_{q} + 60.51$ , with the regression value R² equal to 0.85. So, if it is guaranteed that the cutting edge of milling cutter has sufficient intensity, the larger rake angle is beneficial for reducing milling force. Since the hardness of aviation aluminum alloy is small, wider scale of rake angle is chosen for hard alloy milling cutter, for example, 12°–25°.

Relief angle

The scale of the relief angle is set at 2°–16° and the increase in each scale is 2°. Figure 6 shows the curves between the relief angle and the milling force which were obtained by simulation.

Figure 6.

Relationship between milling force and relief angle.

Figure 6 shows that with the increase in the relief angle, the milling force changes unstably, but within a small range. So, the effect of relief angle on milling force can be neglected. In practical cutting process, the relief angle of end mill is relative to the anti-vibration problem and the friction force between the rear face and the machined surface. Increasing the relief angle can reduce the contact area between the rear face and the surface of the workpiece. As a result, the friction reduces and the blade becomes sharper, and it does good to improve the tool life. However, an oversize relief angle will lower the pressing of the flank to the workpiece and it is not conductive to the suppression of vibration or flutter in machining process which will lead to poor processing quality.

A novel milling force model of end mill

Empirical model of milling force

According to the research conclusion of metal cutting theory, elastic–plastic deformation and friction condition would be affected by different cutting parameters and tool angles during milling process. They will change the chip-forming process, and thus change the milling force. What mentioned above could reach a conclusion that there is a complex index empirical equation between the cutting forces and the cutting parameters when specific machine tool and cutting tool geometric parameters are decided. The equation can be expressed as follows

F = k v^{a} a_{p}^{b} f_{z}^{c} a_{e}^{d} D

(2)

where F is the cutting force, k is the coefficient determined by the cutting material and grinding condition, v is the milling speed, a_p is the milling depth, f_z is the feed, a_e is the milling width, and D is the diameter. It could be seen from equation (2) that the relation between the diameter and the milling force was linear, which is not quite accurate.

In order to accurately obtain the coefficients of the empirical model, orthogonal cutting simulation experiments were performed. Five cutting parameters were considered and four levels were set up. The structural parameters of milling cutter were as follows: blunt radius was 0.01 mm, helix angle was 30°, rake angle was 15°, and relief angle was 8°. The milling parameters and simulation results are shown in Table 1.

Table 1.

Milling parameters and simulation results.

No.	D(mm)	v(m·s⁻¹)	f_z(mm·z⁻¹)	a_p(mm)	a_e(mm)	F_x(N)	F_y(N)
1	8	250	0.05	1.5	1.5	54.22	30.96
2	8	500	0.1	3	3	222.03	126.59
3	8	750	0.15	4.5	4.5	495.79	278.64
4	8	1000	0.2	6	6	811.46	196.91
5	12	250	0.1	4.5	6	456.8	178.27
6	12	500	0.05	6	4.5	192.72	66.16
7	12	750	0.2	1.5	3	297.98	133.68
8	12	1000	0.15	3	1.5	140.98	25.87
9	16	250	0.15	6	3	305.22	69.52
10	16	500	0.2	4.5	1.5	173.64	46.21
11	16	750	0.05	3	6	234.68	78.81
12	16	1000	0.1	1.5	4.5	249.64	67.09
13	20	250	0.2	3	4.5	445.74	188.2
14	20	500	0.15	1.5	6	424.44	385.25
15	20	750	0.1	6	1.5	100.75	24.16
16	20	1000	0.05	4.5	3	118.01	42.38

The coefficients of the empirical milling force model could be obtained as follows after nonlinear fitting of the above simulation results:

{\begin{matrix} F_{x} = 0.9055 v^{0.377} f_{z}^{0.8836} a_{p}^{0.7068} a_{e}^{1.2874} D \\ F_{y} = 1.2235 v^{0.2487} f_{z}^{0.9046} a_{p}^{0.0594} a_{e}^{1.5657} D \end{matrix}

(3)

Novel milling force model of end mill

It could be learned from the above analysis that the diameter of end mill, blunt radius, helix angle, and rake angle had significant impact on milling force. Based on the fitting situation between the tool geometric parameters and the milling forces, it could be inferred that their relation might be either linear or quadratic polynomial. Because quadratic polynomial contains linear relation, the quadratic polynomial regression model about the four tool geometric parameters and empirical milling force model were gathered together in order to obtain a milling force model. The new model contains tool geometric parameters and milling parameters, which means that once the new model is established, it is needless to carry out new experiments and calculations to obtain the coefficients of the model but apply directly to this model when tool geometric parameters and milling parameters change. In the empirical model of milling force, the relation between the milling cutter diameter and the milling force was defined as linear which contradicted with the above analysis. Therefore, the diameter was classified as a parameter of quadratic polynomial to establish the new milling force model, and it is expressed as follows

\begin{matrix} F = (k_{1} + k_{2} x_{d} + k_{3} x_{r} + k_{4} x_{l} + k_{5} x_{q} + k_{6} x_{d}^{2} + k_{7} x_{r}^{2} \\ + k_{8} x_{l}^{2} + k_{9} x_{q}^{2}) v^{a} f_{z}^{b} a_{p}^{c} a_{e}^{d} \end{matrix}

(4)

where k₁, k₂, …, k₉ are the impact factors of tool geometric parameter.

In order to accurately obtain the coefficients of the new model, four structure parameters of end mill were set into four levels to carry out the orthogonal experiment simulation analysis. The cutting parameters were as follows: each tooth feed was 0.1 mm, axial cutting depth was 1.5 mm, and radial cutting depth was 1.5 mm. The geometric parameters of end mill and simulation results are shown in Table 2.

Table 2.

Geometric parameters and simulation results.

No.	x_d (mm)	x_r (µm)	x_l (°)	x_q (°)	F_x (N)	F_y (N)
1	8	5	20	5	231.87	85.03
2	8	10	25	10	212.5	62.34
3	8	15	30	15	200.56	54.54
4	8	20	35	20	198.28	70.09
5	12	5	25	15	166.75	25.83
6	12	10	20	20	168.42	37.25
7	12	15	35	5	178.39	58.75
8	12	20	30	10	187.39	64.69
9	16	5	30	20	143.69	16.77
10	16	10	35	15	152.32	29.37
11	16	15	20	10	186.28	67.74
12	16	20	25	5	194.66	86.83
13	20	5	35	10	142.61	18.85
14	20	10	30	5	165.73	48.56
15	20	15	25	20	154.82	46.28
16	20	20	20	15	173.93	67.21

With the method of nonlinear quadratic regression fitting on the results of Table 2, the new milling force model can expressed as

{\begin{matrix} F_{x} = (179.4705 - 10.6497 x_{d} - 205.7822 x_{r} \\ - 0.9437 x_{l} - 0.9422 x_{q} + 0.2678 x_{d}^{2} \\ + 20436.5885 x_{r}^{2} + 0.0082 x_{l}^{2} + 0.0135 x_{q}^{2}) \cdot \\ v^{0.377} f_{z}^{0.8836} a_{p}^{0.7068} a_{e}^{1.2874} \\ F_{y} = (100.8819 - 4.8414 x_{d} - 90.636 x_{r} \\ - 2.1692 x_{l} - 1.9695 x_{q} + 0.1337 x_{d}^{2} \\ + 32445.68 x_{r}^{2} + 0.0319 x_{l}^{2} + 0.0544 x_{q}^{2}) \cdot \\ v^{0.2487} f_{z}^{0.9046} a_{p}^{0.0594} a_{e}^{1.5657} \end{matrix}

(5)

The values of R² in the X and Y directions were 0.9847 and 0.9512, respectively, which indicated that the new model is good fitting precision. However, the former empirical equation (3) was changed because of the increased geometrical factors of quadratic polynomial and the different relationship of diameter. It means that the fitting precision of equation (5) does not conform to the data of Table 1. In order to obtain high fitting precision of the new milling force model to the data of Tables 1 and 2, the direct method and loop method were adopted to obtain the corresponding coefficients.

Direct method

The data in Tables 1 and 2 were combined to obtain 32 groups of data and the fitting was carried out towards the coefficients of the new milling force model with nonlinear least-squares method. The fitting results are shown as follows

{\begin{matrix} F_{x} = (3430.8695 - 42.5732 x_{d} - 46213.7383 x_{r} \\ - 122.6707 x_{l} - 64.0865 x_{q} + 1.0597 x_{d}^{2} \\ + 2221432.226 x_{r}^{2} + 2.037 x_{l}^{2} + 2.0956 x_{q}^{2}) \cdot \\ v^{- 0.0481} f_{z}^{0.6744} a_{p}^{0.1127} a_{e}^{0.8545} \\ F_{y} = (2268.9799 - 188.1324 x_{d} + 19244.2462 x_{r} \\ - 15.602 x_{l} - 37.1805 x_{q} + 6.0205 x_{d}^{2} \\ + 138916.5977 x_{r}^{2} + 0.0478 x_{l}^{2} + 0.8315 x_{q}^{2}) \cdot \\ v^{- 0.2711} f_{z}^{0.3938} a_{p}^{- 0.6121} a_{e}^{1.5695} \end{matrix}

(6)

The fitting precisions of F_x and F_y were 0.9197 and 0.7892, respectively, and the fitting precisions were high.

Loop method

First, the empirical equation of milling forces could be inferred from Table 1. Second, the impact factors of tool geometric parameters were fitted according to Table 2 without changing the index coefficient values of milling parameters. Third, new indexes were fitted according to Table 1 without changing the impact factors of tool geometric parameters. After hundreds of times of circulation, the coefficient values would be stable. The final equation of milling force was obtained as follows

{\begin{matrix} F_{x} = (1445.0346 - 55.1554 x_{d} - 260.1047 x_{r} \\ - 9.088 x_{l} - 10.3518 x_{q} + 1.4733 x_{d}^{2} \\ + 197739.9578 x_{r}^{2} + 0.0565 x_{l}^{2} + 0.1331 x_{q}^{2}) \cdot \\ v^{- 0.0388} f_{z}^{0 . . 6939} a_{p}^{0.1722} a_{e}^{0.961} \\ F_{y} = (2268.9799 - 188.1324 x_{d} + 19244.2462 x_{r} \\ - 15.602 x_{l} - 37.1805 x_{q} + 6.0205 x_{d}^{2} \\ + 138916.5977 x_{r}^{2} + 0.0478 x_{l}^{2} + 0.8315 x_{q}^{2}) \cdot \\ v^{- 0.2711} f_{z}^{0.3938} a_{p}^{- 0.6121} a_{e}^{1.5695} \end{matrix}

(7)

After circle fitting, the fitting precisions of F_x and F_y were 0.9771 and 0.9196, respectively. It was obvious that the fitting effect is better to use direct method.

Experimental verification

Simulation experimental verification of the novel model

The milling force models which were established by direct method and loop method, respectively, were validated by the following three groups of new parameters as shown in Table 3. By comparing the experimental results and model prediction values, it could be seen from Figures 7 and 8 that the models had good prediction within parameters and the precision of loop method is higher.

Table 3.

Data of three groups of forecast models.

No.	x_d(mm)	x_r(µm)	x_l(°)	x_q(°)	v(m·s⁻¹)	f_z(mm·z⁻¹)	a_p(mm)	a_e(mm)
1	8	0.015	35	10	500	0.2	3	2
2	16	0.005	32	12	800	0.05	3	5
3	18	0.008	25	16	900	0.08	6	2

Figure 7.

Comparison tangential milling forces.

Figure 8.

Comparison radial milling forces.

Processing experimental validation of the novel model

In order to verify the prediction accuracy of the milling force model, nine different geometric parameter carbide cutters which were used for aerospace aluminum alloy 7075 end-milling experiment were grinded. Considering four structural parameters of the milling cutter, three levels were set up as shown in Table 4.

Table 4.

Three levels of the milling cutter geometric parameters.

Parameters	1	2	3
Diameter (mm)	12	16	20
Helix angle (°)	30	38	45
Rake angle (°)	10	14	18
Relief angle (°)	10	12	14

The nine different carbide cutters were named from d1 to d9, and their main geometric parameters are shown in Table 5. Their carbide cutters’ structure model is shown in Figure 9.

Table 5.

Main geometric structure parameters of a milling cutter.

No.	Diameter, D_o(mm)	Helix angle (°)	Rakeangle, a (°)	Reliefangle, b (°)	Corediameter,D_i (mm)	Teeth	Transitionradius,R_f (mm)	Bladewidth, w (mm)
d1	12	30	10	10	6	3	1	0.6
d2	12	38	14	12	6	3	1	0.6
d3	12	45	18	14	6	3	1	0.6
d4	16	30	14	14	8	3	1	0.8
d5	16	38	18	10	8	3	1	0.8
d6	16	45	10	12	8	3	1	0.8
d7	20	30	18	12	10	3	1	1
d8	20	38	10	14	10	3	1	1
d9	20	45	14	10	10	3	1	1

Figure 9.

Carbide cutter geometry model.

Under the dry cutting condition of MAZAK UHS FJV-200, aerospace aluminum alloy 7075 was machined with nine different geometry cutters. In the milling test platform shown in Figure 10, data acquisition software DynoWare is used to select milling peak force during the process of stable milling. Milling parameters were as follows: milling speed = 200 m/min, axial cutting depth = 6 mm, radial cutting width = 3 mm, and feed per tooth = 0.05 mm/s. The test results are shown in Table 6.

Figure 10.

Schematic diagram of milling test.

Table 6.

Milling force values under different rotation speed.

No.	v (m·min⁻¹)	n (r·min⁻¹)	F_x (N)	F_y (N)	F_z (N)
d1	200	3979	160.04	275.79	82.40
d2	200	5968	175.17	228.98	95.32
d3	200	7958	151.81	209.95	86.24
d4	200	9947	188.39	251.09	64.98
d5	200	11937	186.68	212.29	83.25
d6	200	13926	166.85	212.04	93.05
d7	200	15915	142.44	215.28	42.88
d8	200	17905	146.05	233.78	79.25
d9	200	19894	155.70	192.18	89.51

Since the milling force in 2D milling simulation only contains the component force in the X and Y directions, the results did not consider the milling force in the Z direction. The milling force expression is shown as follows

F = \sqrt{F_{x}^{2} + F_{y}^{2}}

(8)

In the same milling condition and the same milling parameters, the results which contain the milling force prediction values and the experimental values are shown in Figure 11.

Figure 11.

Comparison of milling forces between experimental values and predicted values.

Conclusion

The following conclusions are drawn:

A 2D simulation model of milling force is established and the equivalent transformation equation of rake and relief angles is given.

At aviation aluminum alloy 7075 milling with a cemented carbide tool, the influence law of geometric parameters of solid carbide end mill on milling force is researched and the corresponding mathematical relationship is established. Besides relief angle, the influence of other geometrical parameters on milling force is obvious.

According to the relationship between tool geometric parameters of end milling and milling force, a new milling force model is established on the basis of empirical model. The diameter is classified as a parameter of quadratic polynomial. The model contains the tool geometric parameters and the milling parameters, which means that it is needless to carry out new experiments and calculations to obtain the coefficients of the model. When tool geometric parameters and milling parameters change, it is just applied directly to this model.

By simulation experimental verification, the accuracy of milling force model established by the loop method is higher than that established by the direct method.

Through the milling test, the proposed new model has better prediction accuracy.

The new modeling method can be extended to milling force modeling of other types of cutter.

Footnotes

Handling Editor: Jining Sun

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (Nos.51605337,51505338), the Zhejiang Provincial Natural Science Foundation of China (Nos.LQ17E050003, LQ16E050004).

ORCID iDs

Xianglei Zhang

Yan Ren

References

Sun

Guo

YB.

A new multi-view approach to characterize 3D chip morphology and properties in end milling titanium Ti–6Al–4V. Int J Mach Tool Manu 2008; 48: 1486–1494.

Wang

Wei

Positive excess volume of liquid Fe–Cu alloys resulting from liquid structure change. Phys Lett A 2010; 374: 4787–4792.

Janez

Martin

Klaus

Mechanistic identification of specific force coefficients for a general end mill. Int J Mach Tool Manu 2004; 44: 401–414.

Kline

DeVor

Lindberg

JR.

The prediction of cutting forces in end milling with application to cornering cuts. Int J Mach Tool D R 1982; 22: 7–22.

Yang

Park

The prediction of cutting force in ball-end milling. Int J Mach Tool Manu 1991; 31: 45–54.

Tuysuz

Altintas

Feng

HY.

Prediction of cutting forces in three and five-axis ball-end milling with tool indentation effect. Int J Mach Tool Manu 2013; 66: 66–81.

Man

Ren

Usui

et al . Validation of finite element cutting force prediction for end milling. PROC CIRP 2012; 1: 663–668.

Wang

Huang

ZG.

Aviation aluminum alloy 7050—T7451 milling force model of experimental research. Chin J Mech Eng 2003; 14: 1684–1686.

Ren

Zhou

Wei

Optimization of cutter geometric parameters in end milling of titanium alloy using the grey-Taguchi method. Adv Mech Eng 2015; 7: 721093–721093.

10.

Merchant

ME.

Mechanics of the metal cutting process. J Appl Phys 1945; 16: 318–324.

11.

Merchant

ME.

Basic mechanics of the metal cutting process. J Appl Mech 1944; 67: 168–175.

12.

Chen

Zhang

Xie

et al . A unified analytical cutting force model for variable helix end mills. Int J Adv Manuf Techno 2017; 92: 1–19.

13.

Wang

Huang

HH.

Milling force of plastic mold parts in high speed hard milling based on BP neural net work. J Cent South Univ T 2010; 41: 2218–2223.

14.

Zheng

JX.

Application of particle-swarm-optimization-trained artificial neural network in high speed milling force modeling. Comput Integr Manuf 2008; 14: 1710–1716.

15.

Wang

Huang

Xian

Milling force of aerospace aluminum alloy thin-wall parts in high-speed machining. J Cent S Univ 2017; 48: 1756–1761.

16.

Wan

Pan

Zhang

et al . A unified instantaneous cutting force model for flat end mills with variable geometries. J Mater Process Tech 2014; 214: 641–650.

17.

Zhang

JF.

Parameter optimization and database implementation based on physical model of cutting process. Kunming, China: Kunming University of Science and Technology, 2017.

18.

Zhang

XH.

Parameter optimization and database implementation based on physical model of cutting process. Shanghai, China: Shanghai Jiao Tong University, 2009.

No.	Diameter, D_o(mm)	Helix angle (°)	Rakeangle, a (°)	Reliefangle, b (°)	Corediameter,D_i (mm)	Teeth	Transitionradius,R_f (mm)	Bladewidth, w (mm)
d1	12	30	10	10	6	3	1	0.6
d2	12	38	14	12	6	3	1	0.6
d3	12	45	18	14	6	3	1	0.6
d4	16	30	14	14	8	3	1	0.8
d5	16	38	18	10	8	3	1	0.8
d6	16	45	10	12	8	3	1	0.8
d7	20	30	18	12	10	3	1	1
d8	20	38	10	14	10	3	1	1
d9	20	45	14	10	10	3	1	1

No.	Diameter, D_o(mm)	Helix angle (°)	Rakeangle, a (°)	Reliefangle, b (°)	Corediameter,D_i (mm)	Teeth	Transitionradius,R_f (mm)	Bladewidth, w (mm)
d1	12	30	10	10	6	3	1	0.6
d2	12	38	14	12	6	3	1	0.6
d3	12	45	18	14	6	3	1	0.6
d4	16	30	14	14	8	3	1	0.8
d5	16	38	18	10	8	3	1	0.8
d6	16	45	10	12	8	3	1	0.8
d7	20	30	18	12	10	3	1	1
d8	20	38	10	14	10	3	1	1
d9	20	45	14	10	10	3	1	1

A novel milling force model based on the influence of tool geometric parameters in end milling

Abstract

Keywords

Introduction

Finite element analysis on milling force

Tool material and geometric angle

Finite element milling model

Influence of tool geometric parameters on milling force

Diameter

Blunt radius

Helix angle

Rake angle

Relief angle

A novel milling force model of end mill

Empirical model of milling force

Novel milling force model of end mill

Direct method

Loop method

Experimental verification

Simulation experimental verification of the novel model

Processing experimental validation of the novel model

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

No.	Diameter, D_o(mm)	Helix angle (°)	Rakeangle, a (°)	Reliefangle, b (°)	Corediameter,D_i (mm)	Teeth	Transitionradius,R_f (mm)	Bladewidth, w (mm)
d1	12	30	10	10	6	3	1	0.6
d2	12	38	14	12	6	3	1	0.6
d3	12	45	18	14	6	3	1	0.6
d4	16	30	14	14	8	3	1	0.8
d5	16	38	18	10	8	3	1	0.8
d6	16	45	10	12	8	3	1	0.8
d7	20	30	18	12	10	3	1	1
d8	20	38	10	14	10	3	1	1
d9	20	45	14	10	10	3	1	1