The estimation of cutting force coefficients in milling of thin-walled parts using cutter with different tooth radii

Abstract

The precise estimation of the cutting force coefficients in milling is very important, which has great impact on the precise calculation of the milling forces and the stability lobes of the system. On the basis of the imprecise estimation of cutting force coefficients in milling of thin-walled part using cutter with different tooth radii, the calculation process of actual cutting force coefficients is proposed in this article. First, based on the different amplitudes of milling forces caused by the long and short teeth of cutter, the nominal milling forces and nominal cutting force coefficients are constructed, and the radii error of the long and short teeth of cutter is calculated. Then, actual cutting force coefficients are derived based on radii error of cutter teeth. Finally, an experiment is performed to verify the validity of the calculation process of actual cutting force coefficients proposed in this article, and the results show that the actual cutting force coefficients obtained by consideration of radii error are more effective than theoretical cutting force coefficients obtained without consideration of radii error in the calculation of milling forces.

Introduction

Based on the advantages of lightweight, saving in materials, lower cost and so on, thin-walled structures have been widely used in the fields of aerospace, automotive and so on. But the stiffness of thin wall structures is relatively poor in the machining process, and therefore, chatter and large deformation are easy to happen during machining. The machining of thin wall structures has been considered to be a complicated and difficult-to-cut process. Milling is one of the most common machining operations to machine the thin wall structures. In milling operation, the milling forces have direct influence on the machining stability of the system, deformation and surface finish, and therefore, the precise calculation of milling forces is particularly important.^1–5

At present, the most common method for the modeling of milling forces is based on the ideas of Martellotti, ⁶ which has the advantages of simple mathematical model, high calculation precision, wide application range for various milling operations and so on. The cutting force coefficients in milling are important parameters in the above model, which have important influence on the calculation and prediction of the milling forces. Currently, the methods to estimate the cutting force coefficients in milling are mainly based on the measured average milling forces, that is, the components of resultant milling forces in different directions are measured, and cutter coefficients are derived based on measured forces. For example, cutting force coefficients in milling are expressed as nonlinear functions of uncut chip thickness in literatures.^7–13 Considering the dependence of other milling parameters on cutting force coefficients in the literatures,^14–16 cutting force coefficients are expressed as quadratic functions of milling speed, uncut chip thickness, axial and radial milling depth.

Because of the manufacturing errors of milling cutter and machine tool spindle, the rotation radii of each tooth of cutter are different. And the theoretical cutting force coefficients calculated directly by the measured average milling forces when the tooth radii are different are imprecise. Based on the above reasons, actual cutting force coefficients are derived in this article and are compared with theoretical cutting force coefficients. The results show the actual cutting force coefficients are more effective than theoretical cutting force coefficients.

Theoretical foundation

Identification of theoretical cutting force coefficients

According to the literature, ¹⁷ the theoretical cutting force coefficients neglecting the influence of edge forces are given as

$$\begin{array}{cc}\multicolumn{1}{c}{}& \left[\begin{array}{c}\hfill K_{\mathrm{tc}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{rc}}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& ={\left\{\frac{8\pi }{\mathrm{Nac}}{\left[\begin{array}{cc}\hfill \cos 2\phi \hfill & \hfill -(2\phi -\sin 2\phi )\hfill \\ \multicolumn{1}{c}{(2\phi -\sin 2\phi }& \hfill \cos 2\phi \hfill \end{array}\right]}^{-1}\left[\begin{array}{c}\hfill {\overline{F}}_x\hfill \\ \multicolumn{1}{c}{{\overline{F}}_y}\end{array}\right]\right\}|}_{{\phi }_{\mathrm{st}}}^{{\phi }_{\mathrm{ex}}}\hfill \end{array}$$

(1)

where φ is the angle of immersion; ${\phi }_{\mathrm{st}}$ and ${\phi }_{\mathrm{ex}}$ are the cutter entry and exit angles, respectively; N is the number of cutter teeth; $a$ is the axial milling depth; $c$ is the feed rate (mm/rev-tooth); ${\overline{F}}_x$ and ${\overline{F}}_y$ are the average milling forces per rotation of horizontal (i.e. feed direction) and normal components of the resultant cutting forces, respectively; and $K_{\mathrm{tc}}$ and $K_{\mathrm{rc}}$ are cutting force coefficients contributed by the shearing action in tangential and radial directions, respectively.

Identification of actual cutting force coefficients

The actual milling operation with different tooth radii of cutter is shown in Figure 1. In this figure, $r_l$ and $r_s$ are the radii of the long and short teeth of cutter, respectively; $c$ is theoretical feed rate and $c_l$ and $c_s$ are the actual feed rates of long and short teeth. $r_{\mathrm{deviation}}$ is the difference between long and short teeth. As can be seen from Figure 1, actual feed rates of long and short teeth are different from theoretical feed rate, which lead to different milling forces on each cutter tooth. If average forces per rotation (see equation (1)) is used to calculate the cutting force coefficients, equation (1) will produce the calculation distortions of cutting force coefficients. In the following, the calculation of actual cutting force coefficients will be derived.

Figure 1.

Milling operation with different tooth radii of cutter.

Cutting force coefficients of nominal milling forces

Horizontal, normal components of the resultant cutting force are marked as $F_x$ and $F_y$ , respectively. In horizontal force $F_x$ , the cutting forces of same tooth of cutter are grouped together to construct the nominal milling forces of long and short teeth (i.e. $F_{\mathrm{xl}}$ and $F_{\mathrm{xs}}$ ), and the same principle is applied to normal force $F_y$ to obtain $F_{\mathrm{yl}}$ and $F_{\mathrm{ys}}$ , as shown in Figure 2.

Figure 2.

Nominal milling force of long and short teeth in horizontal and normal directions.

As can be seen from Figure 2, the amplitudes of the milling force $F_{\mathrm{xl}}$ are same, so are the amplitudes of nominal milling forces $F_{\mathrm{xs}}$ , $F_{\mathrm{yl}}$ and $F_{\mathrm{ys}}$ . Substituting the average values of single tooth of nominal milling forces $F_{\mathrm{xl}}$ , $F_{\mathrm{yl}}$ , $F_{\mathrm{xs}}$ and $F_{\mathrm{ys}}$ in equation (1), the tangential and radial cutting force coefficients of long and short teeth (i.e. $K_{\mathrm{ti}}$ and $K_{\mathrm{ri}}$ ) are obtained as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& \left[\begin{array}{c}\hfill K_{\mathrm{ti}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{ri}}}\end{array}\right]\hfill \\ \multicolumn{1}{c}{}& ={\left\{\frac{4p}{\mathrm{ac}}{\left[\begin{array}{cc}\hfill \cos 2\phi \hfill & \hfill -(2\phi -\sin 2\phi )\hfill \\ \multicolumn{1}{c}{(2\phi -\sin 2\phi }& \hfill \cos 2\phi \hfill \end{array}\right]}^{-1}\left[\begin{array}{c}\hfill {\overline{F}}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{{\overline{F}}_{\mathrm{yi}}}\end{array}\right]\right\}|}_{{\phi }_{\mathrm{st}}}^{{\phi }_{\mathrm{ex}}}\hfill \end{array}$$

(2)

where $i=l,s$ represent the long and short teeth of cutter, respectively, and p is the immersion angle of single tooth, as shown in Figure 3.

Figure 3.

Cutter–workpiece contact zone and corresponding parameters.

In Figure 3, $\lambda$ is helical angle, $a$ is the axial milling depth, ${\theta }_b$ is the immersion angle of the bottom end of cutter and $\phi$ and $b$ are instantaneous immersion angle and axial milling depth, respectively. In up-milling operation, the values of ${\theta }_b$ and p are derived as follows

$${\theta }_b={\cos }^{-1}\left(\frac{r-a_e}{r}\right),p={\theta }_b+\frac{a}{\tan (90-\lambda )\times r}$$

(3)

where $a_e$ is the radial milling depth and r is the radius of cutter.

Radii error of long and short teeth of cutter

Equation (2) is simplified as follows

$$\left[\begin{array}{c}\hfill K_{\mathrm{ti}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{ri}}}\end{array}\right]=\left\{\frac{4p_i}{{\mathrm{ac}}_i}\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \multicolumn{1}{c}{C}& \hfill D\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\overline{F}}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{{\overline{F}}_{\mathrm{yi}}}\end{array}\right]\right\}$$

(4)

where

$$\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \multicolumn{1}{c}{C}& \hfill D\hfill \end{array}\right]={{\left[\begin{array}{cc}\hfill \cos 2\phi \hfill & \hfill -(2\phi -\sin 2\phi )\hfill \\ \multicolumn{1}{c}{(2\phi -\sin 2\phi }& \hfill \cos 2\phi \hfill \end{array}\right]}^{-1}|}_{{\phi }_{\mathrm{st}}}^{{\phi }_{\mathrm{ex}}}$$

Expanding the equation (3) yields

$$K_{\mathrm{tl}}=\frac{4p_l}{{\mathrm{ac}}_l}(A{\overline{F}}_{\mathrm{xl}}+B{\overline{F}}_{\mathrm{yl}})$$

(5)

$$K_{\mathrm{rl}}=\frac{4p_l}{{\mathrm{ac}}_l}(C{\overline{F}}_{\mathrm{xl}}+D{\overline{F}}_{\mathrm{yl}})$$

(6)

$$K_{\mathrm{ts}}=\frac{4p_s}{{\mathrm{ac}}_s}(A{\overline{F}}_{\mathrm{xs}}+B{\overline{F}}_{\mathrm{ys}})$$

(7)

$$K_{\mathrm{rs}}=\frac{4p_s}{{\mathrm{ac}}_s}(C{\overline{F}}_{\mathrm{xs}}+D{\overline{F}}_{\mathrm{ys}})$$

(8)

As can be seen in equations (4) and (6), the tangential cutting force coefficients of long and short teeth (i.e. $K_{\mathrm{tl}}$ and $K_{\mathrm{ts}}$ ) are different, so are the radial cutting force coefficients $K_{\mathrm{rl}}$ and $K_{\mathrm{rs}}$ . The difference between $K_{\mathrm{tl}}$ and $K_{\mathrm{ts}}$ (or between $K_{\mathrm{rl}}$ and $K_{\mathrm{rs}}$ ) is attributable to the radii deviation $r_{\mathrm{deviation}}$ (as shown in Figure 1) of long and short teeth of cutter. But in a special process system, the cutting force coefficients have no concern with the radii of each tooth cutter. Under this condition, the equations (9) and (10) are derived as follows

$$\frac{\frac{4p_l}{{\mathrm{ac}}_l}(A{\overline{F}}_{\mathrm{xl}}+B{\overline{F}}_{\mathrm{yl}})}{\frac{4p_s}{{\mathrm{ac}}_s}(A{\overline{F}}_{\mathrm{xs}}+B{\overline{F}}_{\mathrm{ys}})}=1$$

(9)

$$\frac{\frac{4p_l}{{\mathrm{ac}}_l}(C{\overline{F}}_{\mathrm{xl}}+D{\overline{F}}_{\mathrm{yl}})}{\frac{4p_s}{{\mathrm{ac}}_s}(C{\overline{F}}_{\mathrm{xs}}+D{\overline{F}}_{\mathrm{ys}})}=1$$

(10)

Defining $K_1=(4/a)(A{\overline{F}}_{\mathrm{xl}}+B{\overline{F}}_{\mathrm{yl}})/(4/a)(A{\overline{F}}_{\mathrm{xs}}+B{\overline{F}}_{\mathrm{ys}})$ and $K_2=(4/a)(C{\overline{F}}_{\mathrm{xl}}+D{\overline{F}}_{\mathrm{yl}})/(4/a)(C{\overline{F}}_{\mathrm{xs}}+D{\overline{F}}_{\mathrm{ys}})$ . Under the constraint condition $r_l+r_s=D$ , equation (11) is derived as follows

$$\{\begin{array}{c}\hfill K_i\frac{p_l}{p_s}\frac{c_s}{c_l}=1\hfill \\ \multicolumn{1}{c}{r_l+r_s=D}\end{array}$$

(11)

where $i=1,2$ represent the long and short cutter teeth, $c_s/c_l=c_l-(r_l-r_s)/c_l=1-((r_l-r_s)/c+(r_l-r_s))$ is the ratio of the feed rate of the long tooth to the feed rate of the short tooth and $p_l/p_s=({\theta }_b)+(a/\tan (90-\lambda )\times r_l)/({\theta }_b)+(a/\tan (90-\lambda )\times r_s)$ is the ratio of the immersion angle of the long tooth to the immersion angle of the short tooth. Nonlinear equation (11) is solved by Newton’s iterative method, and the radii of the long and short teeth of cutter are obtained, that is, $r_l$ and $r_s$ .The radii error of long and short teeth of cutter can be calculated as follows

$$r_{\mathrm{deviation}}=r_l-r_s$$

Actual cutting force coefficients

Depending on the calculated radii error $r_{\mathrm{deviation}}$ , the actual feed rates are calculated as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& c_l=c+r_{\mathrm{deviation}}\hfill \\ \multicolumn{1}{c}{}& c_s=c+r_{\mathrm{deviation}}\hfill \end{array}$$

(12)

where $c$ is the theoretical feed rate of each tooth of cutter and $c_l$ and $c_s$ are the feed rates of long and short teeth of cutter, respectively. Substituting the equations (3) and (12) in equation (4) yields actual cutting force coefficients in milling, and the tangential and radial cutting force coefficients of long tooth are equivalent to that of short tooth of cutter, that is, K_tl  = K_{t s} and K_rl  = K_rs .

Experiment

Experimental platform

Experiment platform consists of a vertical machining center, thin wall aluminum alloy structure (height: 60 mm; width: 40 mm; thickness: 5 mm, machined by up-milling operation), cylindrical end milling cutter (helical angle: 30°; diameter: 8 mm; the lengths of milling edge and cutter are 35 and 80 mm, respectively) and Kistler milling dynamometer, as shown in Figure 4.

Figure 4.

Experimental platform in up-milling thin wall structure.

Experimental verification

In order to exhibit the advantage of actual cutting force coefficients in milling, in this article, the simulated milling forces which are simulated based on actual cutting force coefficients and theoretical cutting force coefficients are compared with the measured milling forces in two experiments. Milling parameters are as follows: ν_c  = 15.702 m/min, f = 150 mm/min, a_e  = 0.2 mm, a_p  = 7 mm; ν_c  = 20.096 m/min, f = 250 mm/min, a_e  = 0.3 mm and a_p  = 10 mm. Depending on equations (4) and (11), actual milling coefficients under different milling parameters are obtain, that is, K_{t 1} = 781.55 N/mm², K_r1 = 671.92 N/mm², K_{t 2} =  513.61 N/mm² and K_{r 2} = 466.9 N/mm². According to the above actual milling coefficients, instantaneous milling forces are simulated, as shown in Figure 5

Figure 5.

Comparison of simulated forces based on actual and theoretical milling coefficients.

As can be seen from Figure 5, the simulated forces based on actual milling coefficients show good agreement with measured milling forces in experiments 1 and 2. However, the simulated forces based on theoretical milling coefficients without consideration of radii error have a large difference with measured forces. Depending on the above comparison between actual and theoretical milling coefficients, it shows that the actual milling coefficients can more accurately reflect the milling forces in practical machining, and the construction process of actual milling coefficients proposed in this article is effective for the identification of cutting force coefficients in milling using different tooth radii of cutter.

Conclusion

To avoid the distortion of the identification of cutting coefficients in milling caused by different tooth radii of cutter, nominal milling forces and nominal milling coefficients are constructed, and the calculation formulas of radii error and actual cutting force coefficients in milling are derived. Finally, simulated milling forces based on actual and theoretical milling coefficients are compared with measured milling forces, and the results show that the actual milling coefficients are more precise than theoretical milling coefficients, and the calculation process of actual milling coefficients is effective for the identification of milling coefficients using cutter with different tooth radii.