Effects of surface splicing characteristics of the hardened steel mold on the machined surface topography

Abstract

The car cover mold has characteristics of large structure size, complicated surface shape, dimensional precision, and high surface precision. The milling process of the splicing transition zone having different hardness can easily produce load mutation, which has an impact on the tool in the form of vibration. This vibration affects the surface topography of the splicing transition zone. In this article, we propose the surface motion model for ball-end milling cutter with impact vibration and the surface topography prediction model for splicing transition zone which is established based on the Non-Uniform Rational B-Splines (NURBS) surface control point reconstruction. Then, the effects of surface splicing characteristics (different hardness, over-seam scale, etc.) and processing parameters on the surface topography and on the residual height have been analyzed. Finally, based on the hardened steel surface splicing milling experiment, the accuracy of the surface topography model in which vibration impact has been incorporated is verified. At the same time, the corresponding roughness value and fractal dimension are calculated on the basis of two-dimensional contour graph under different splicing characteristics. The results show that the non-linear characteristics of the surface topography of the hardened steel surface are significant, and the best processing parameters can be obtained under different splicing characteristics.

Introduction

The car cover mold has the characteristics of large structure size, complicated surface shape, dimensional precision, and high surface precision. In order to meet the processing precision of large stamping molds and to improve the processing efficiency, large cover parts mold finish machining mostly adopts the method that overall milling after insert block mold splicing. Due to the large proportion of non-uniform high hardness surfaces in the processing area, the hardness difference between the inserts is between HRC5 and HRC10, and the hardness difference between the inserts and the mold body reaches to HRC15. The splicing mold is shown in Figure 1. In the process of milling, the milling of different hardness splicing transition zone with different hardness can easily lead to the load mutation, which has obvious vibration impact on the tool. The size and position of the tool cut interface change constantly in the surface machining, which will directly affect the three-dimensional (3D) surface topography modeling. Therefore, in the modeling of complex surface splicing mold surface topography, it is necessary to consider not only the influence of the tool forced vibration, wear, impact deformation, and other factors on the surface topography but also the influence of the mold surface features on the mold surface topography.^1,2

Figure 1.

The mold of side enclosure of a car with splicing seam and inserts.

The research work on the modeling techniques for the surface topography of ball-end milling can be divided into two categories. The first technique uses the envelope constraint formed on the tool cutting edge trajectory to model the maximum height of the residual material on the workpiece surface to realize the simulation of the machined surface profile. ³ The second technique uses Z-map method.^4,5 None of these techniques include time variable. They cannot be used to model the dynamic response in the milling process and are not suitable for dynamic error profile modeling application.

At present, some tentative studies have been made on the 3D analysis and machining error prediction of machined surface produced at home and abroad. Zhang et al. ⁶ and Dang et al. ⁷ have studied the influence of feed mode, angle of inclination and feed rate on the surface topography, and roughness of the workpiece. Peng et al. ⁸ have used the cutting edge kinematics and cutting mechanism of machining process to study the forming process of workpiece surface topography. The 3D trajectory of the tool and the 3D machined surface topography model are developed. The model incorporates the main factors such as cutting parameters, tool tip geometry, and cutting vibration. The surface of the cutting workpiece is tested by non-contact surface profilometer. Liang and Yao ⁹ have presented a double buffer discrete mechanism to buffer the residual height at the equidistant between the workpiece surface and the buffer response at the tool end and they also find the same for other time intervals using the 3D trochoid trajectory of the ball-end milling edge. The nominal machining surface modeling and dynamic response trajectory simulation of five-axis milling are performed.

To study the influence of the surface curvature processing characteristics of high-speed milling of machined surface topography of hardened steel, Chen et al.^10,11 have used the geometric modeling method to analyze the contact area of workpiece and the influence of different aspects of tool on machining characteristics. It is found that the tool posture with a composite inclination angle with a negative inclination angle, and a positive lead angle are helpful to improve the cutting efficiency and maintain higher cutting speed. The surface performance of the multi-spindle ball-end milling is studied to improve the high performance of the multi-spindle ball-end mill. The cutting speed and machining characteristics of the meshing cutting edge under different dip angles are analyzed. Gray et al. ¹² have applied this method to evaluate the influence of tool position on the machined surface profile of five-axis milling. It is found that the equivalent method of tool path envelope surface is only applicable to the milling conditions in which the tool real distance is much larger than the feed per tooth and it is not possible to describe the details of the shape of the adjacent cutting edge in the process of turning.

To study the influence of tool characteristics on the surface topography of hardened steel at high-speed milling, Senatore et al. ¹³ have used this method for the tool selection and tool path planning in surface machining. Zhang et al.^14,15 have established an online simulation model for surface topography of curved surface by considering the cutting edge locus equation which is affected by the tool wear and the side slip angle. Compared with the surface topography of B-spline surface, the prediction of the surface topography model is in good agreement with the experimental results. Xu et al. ¹⁶ have established a new simulation model by combining the sweep surface model and the N-buffer method, which can be used to study the effect of cutting parameters and tool runout on the surface topography of the milling. Wan et al. ¹⁷ have studied the spectra of the machined surface and pointed out that in the case of stable milling, the only significant spectral property is related to the marks belonging to feed rate. However, in the case of chatter vibration, an inclined pattern of dense spectral property is formed, which is associated with the chatter frequency.

In the article, as for the instantaneous cut out the impact force and instantaneous cut into the impact force of the free form surface cover splicing, the main objective of the study is to provide the machining model for weak rigid milling system and rigid complex splicing surface. We develop the 3D surface topography simulation model for the ball-end milling cutter. The model is capable to analyze the influence of surface splicing characteristics (different hardness, over-seam size, etc.) and processing parameters on machined surface topography and residual height. The accuracy of surface topography prediction model in which impact vibration has been incorporated is verified by performing the hardened steel surface splicing milling experiments, and the non-linear characteristics of machined surface topography are analyzed in the same experiment.

The cutting motion model for the cutting edge considering the impact vibration scheme

The geometric model of the cutting edge of the ball-end milling cutter is shown in Figure 2. The tool coordinate system $O_t-X_t{Y}_t{Z}_t$ is defined for the ball-end milling cutter. In the tool coordinate system $O_t-X_t{Y}_t{Z}_t$ , any point (e.g. point $B$ ) on the cutting edge line satisfies the following conditions

$$\{\begin{array}{c}{x}_0=R_0\sin \theta \cos \left(\tan {\beta }_0\cos \theta \right)\hfill \\ y_0=-R_0\sin \theta \sin \left(\tan {\beta }_0\cos \theta \right)\hfill \\ z_0=-R_0\cos \theta \hfill \end{array}$$

(1)

Figure 2.

Integral ball-end milling tool.

In order to facilitate the calculation, any point (e.g. point $B$ ) on the cutting edge line can be expressed by the following matrix form in the tool coordinate system

$${[x_0,y_0,z_0,1]}^T=\left[\begin{array}{c}{R}_0\sin \theta \cos \left(\tan {\beta }_0\cos \theta \right)\hfill \\ \multicolumn{1}{c}{-R_0\sin \theta \sin \left(\tan {\beta }_0\cos \theta \right)}\\ \multicolumn{1}{c}{-R_0\cos \theta }\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(2)

where $R_0$ is the radius of ball-end milling cutter, $\theta$ is the angle between cutting edge point and center of the line with the tool axis, $\phi$ is the spiral lag angle, and ${\beta }_0$ is the spiral angle of the cylindrical part of ball-end milling cutter. A cutting motion model in which impact vibration has been incorporated is established. The tool produces impact vibration in the process of milling and splicing the mold, which is due to the existence of splicing seam. The impact vibration seriously affects the quality of milling. The vibration of the center point of the tool is assumed to be the vibration of the tool; the vibration in $X$ , $Y$ , and $Z$ directions is defined on the tool coordinate system $O_t-X_t{Y}_t{Z}_t$ ; and the cutting motion model is developed under the condition of impact vibration.

As shown in Figure 3(a), the coordinates of any point (e.g. point $B$ ) on the cutting edge are converted to the workpiece coordinate system according to the movement of the tool; we define the tool coordinate system as $O_t-X_t{Y}_t{Z}_t$ , the spindle coordinate system as $O_a-X_a{Y}_a{Z}_a$ , and the workpiece coordinate system as $O_w-X_w{Y}_w{Z}_w$ , respectively. The tool coordinate system $O_t-X_t{Y}_t{Z}_t$ is fixed on the tool, which does not only rotate but also move along with the spindle. The spindle coordinate system $O_a-X_a{Y}_a{Z}_a$ is fixed on the main spindle of the machine tool and can only translate without rotation. The workpiece coordinate system $O_w-X_w{Y}_w{Z}_w$ is fixed on the workpiece. When the tool moves to point $A$ , the position information is represented as the side view of the workpiece, as shown in Figure 3(b). Then continuously compare the sizes of $z_i$ and $z_j$ .

Figure 3.

Cutting motion coordinate system: (a) Geometries in milling of the model; (b) Projections directed along the z-axis.

When the cutting edges are more than one, there is an angle transition between each cutting edge and the first cutting edge. The transition matrix is then expressed as follows

$$M_{\mathrm{jt}}=\left[\begin{array}{cccc}\cos {\varphi }_i\hfill & \sin {\varphi }_i\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin {\varphi }_i}& \cos {\varphi }_i\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(3)

where ${\varphi }_i=(2\mathbf{\pi }/{\mathbf{T}}_{\mathbf{z}})(\mathbf{i}-1),\mathbf{i}=1,2,3,\dots ,{\mathbf{T}}_{\mathbf{z}}$ . $T_z$ is the number of teeth of the tool.

In the cutting process, the tool is represented by its own rotary motion. The rotation of the tool coordinate system is converted to the spindle coordinate system. The rotation matrix is expressed as follows

$$M_{\mathrm{ta}2}=\left[\begin{array}{cccc}\cos \omega t\hfill & \sin \omega t\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \omega t}& \cos \omega t\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(4)

When the tool produces impact vibration, the impact vibration displacement matrix is expressed as follows

$$M_{\mathrm{ta}1}=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & V_x\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & V_y\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & V_z\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(5)

$$\mathrm{Where}\{\begin{array}{c}\multicolumn{1}{c}{V_x\left(t\right)=(V_{x0}\left(t\right)\cos {\omega }_{0}t+\frac{{\stackrel{·}{V}}_{x0}}{{\omega }_0}\sin {\omega }_{0}t-\frac{F_x}{k_x}\frac{\omega {\omega }_0}{{\omega }_0^2-{\omega }^2}\sin {\omega }_{0}t)r\left(t\right)+\frac{F_x}{k_x}\frac{{\omega }_0^2}{{\omega }_0^2-{\omega }^2}\sin \left(\omega t\right)}\\ \multicolumn{1}{c}{V_y\left(t\right)=(V_{y0}\left(t\right)\cos {\omega }_{0}t+\frac{{\stackrel{·}{V}}_{y0}}{{\omega }_0}\sin {\omega }_{0}t-\frac{F_y}{k_y}\frac{\omega {\omega }_0}{{\omega }_0^2-{\omega }^2}\sin {\omega }_{0}t)r\left(t\right)+\frac{F_y}{k_y}\frac{{\omega }_0^2}{{\omega }_0^2-{\omega }^2}\sin \left(\omega t\right)}\\ \multicolumn{1}{c}{V_z\left(t\right)=(V_{z0}\left(t\right)\cos {\omega }_{0}t+\frac{{\stackrel{·}{V}}_{z0}}{{\omega }_0}\sin {\omega }_{0}t-\frac{F_x}{k_x}\frac{\omega {\omega }_0}{{\omega }_0^2-{\omega }^2}\sin {\omega }_{0}t)r\left(t\right)+\frac{F_x}{k_x}\frac{{\omega }_0^2}{{\omega }_0^2-{\omega }^2}\sin \left(\omega t\right)}\end{array}$$

As the mass of the workpiece is far greater than that of the tool, only the milling cutter is considered to be a low rigidity system.^18–20 Furthermore, assuming that the system is undamped, $V_x(t)$ , $V_y(t)$ , and $V_z(t)$ are the vibration displacements in the $X$ , $Y$ , and $Z$ directions of the ball-end milling at time t in the tool coordinate system. Because the size of the tool vibration in the milling convex surface splicing tool through the area is mainly analyzed and the time is short in the milling splicing seam area, the vibration damping attenuation is not considered. ${\omega }_0$ is the natural frequency of the tool, $\omega$ is the rotation frequency of the tool, $V_{x0}(t)$ is the initial displacement through the splicing seam, and ${\stackrel{·}{V}}_{x0}(t)$ is the initial speed through the splicing seam. The function $r(t)$ is a unit step function for determining whether the ball-end milling is in the region of the splicing seam. $F_x$ , $F_y$ , and $F_z$ are the transient milling force in the $X$ , $Y$ , and $Z$ directions of the ball-end milling; and $k_x$ , $k_y$ , and $k_z$ are the rigidity in the $X$ , $Y$ , and $Z$ directions of the ball-end milling.

When the tool is given a feed motion, the amount of translation in the three directions is only related to the feed direction. The translation matrix is expressed as follows ²¹

$$M_{\mathrm{aw}}=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & S_x\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & S_y\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & S_z\left(t\right)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & V_{f}t\cos \alpha \hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & V_{f}t\sin \alpha \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(6)

At this moment, the translation transformation matrix is only suitable for the convex arc surface machining with a constant curvature.

In the cutting process, the cutting method is unidirectional, and there is a change of line feed in the direction of row spacing. The line feed transformation matrix is expressed as follows

$$L={\left[\begin{array}{cccc}0\hfill & \mathrm{Np}\hfill & 0\hfill & 0\hfill \end{array}\right]}^T$$

(7)

where $N$ is the Nth intermittent feed along the Y-axis, and $p$ is the row spacing between two adjacent intermittent feeds.

In order to facilitate the calculations without affecting the results, it is assumed that every cycle of the knife starts at time 0. When the tool reaches the same $X$ and $Z$ coordinate values as point $A$ , the time is t. The time needed for each circulation feeding is $T$ , so

$$T=\frac{\psi \pi R_1}{180V_f}$$

(8)

where $R_1=R+R_0$ . $R_0$ is the radius of ball-end milling cutter, $R$ is the radius of circular arc after machining, and $V_f$ is the feed rate. It can be obtained by the geometric relationship expressed as

$$\frac{t}{T}=\frac{\delta }{\psi },t=\frac{\pi R_1\delta }{180V_f}$$

(9)

In summary, the coordinate value of any point of the cutting edge of the three-axis ball end in the workpiece coordinate system is expressed as follows

$${[x,y,z,1]}^T=M_{\mathrm{aw}}{M}_{\mathrm{ta}1}{M}_{\mathrm{ta}2}{M}_{\mathrm{jt}}{[x_0,y_0,z_0,1]}^T+L$$

(10)

$${[x,y,z]}^T=\left[\begin{array}{c}{R}_0\sin \theta \left[\cos (\omega t+{\varphi }_i+\tan {\beta }_0\cos \theta )\right]+V_{f}t\cos \alpha +V_x(t)\hfill \\ \multicolumn{1}{c}{\mathrm{Np}-R_0\sin \theta \left[\sin (\omega t+{\varphi }_i+\tan {\beta }_0\cos \theta )\right]+V_y(t)}\\ \multicolumn{1}{c}{-R_0\cos \theta +V_{f}t\sin \alpha +V_z(t)}\end{array}\right]$$

(11)

where $\alpha$ is the angle between the tool and the $O_w{X}_w$ axis in the workpiece coordinate system when it goes to point $A$ , $\delta$ is the angle between the tool and the $O{O}_w$ when it goes to point $A$ , and $\psi$ is the angle between the two ends of the workpiece and the connection of point $O$ .

Simulation of surface topography considering forced vibration

A simulation algorithm of NURBS surface topography is conducted based on the surface control point reconstruction. First, the 3D model of splicing mold is developed. Then, the mold surface is divided into $(m-1)\times (n-1)$ matrix meshes. Given the $X$ coordinates and $Y$ coordinates of each grid point, the vibration displacement is solved. The cutting edge and the vibration displacement are discretized by equal time interval as required. In the process of cutting motion, compare the size of the $Z_i$ coordinate value of any point (e.g. point $B$ ) of cutting edge at any time t and the $Z_j$ coordinate value of grid point $P$ on the grid surface. In addition, update the coordinate value of grid point $Z$ in time. Finally, the coordinate values of the reserved grid points are obtained.

In this article, the surface topography is reconstructed based on NURBS surface control points according to the coordinate values of the residual grid points. Using this method, the required surface can be accurately fitted out with less points; the simulation efficiency is greatly improved.^22–24 Assuming that NURBS curved surface meshed into a $k\times l$ grid is represented as follows

$$p(u,v)=\frac{\sum _{i=0}^m\sum _{j=0}^n{\omega }_{i,j}{d}_{i,j}{N}_{i,k}\left(u\right)N_{j,l}\left(v\right)}{\sum _{i=0}^m\sum _{j=0}^n{\omega }_{i,j}{N}_{i,k}\left(u\right)N_{j,l}\left(v\right)}$$

(12)

where $d_{i,j}(i=0,1,2,\dots ,m;j=0,1,2,\dots ,n)$ is the control point of the surface, and ${\omega }_{i,j}$ is the weight factor associated with control point $d_{i,j}$ . When ${\omega }_{i,j}=1$ , $N_{j,k}(u)$ is the normal B-spline basis of $u$ toward $k$ times and $N_{j,l}(v)$ is the normal B-spline basis of $v$ toward $l$ times. They are piecewise polynomials of $k$ and $l$ that are determined by node vectors $U$ and $V$ $(U=(u_0,u_1,\dots ,u_{m+k+1}),V=(v_0,v_1,\dots ,v_{n+l+1}))$ . It can be obtained using the recursive formula of Boolean Cox

$$\{\begin{array}{c}{N}_{i,0}\left(u\right)=\{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\begin{array}{c}{u}_i⩽u⩽u_{i+1}\hfill \\ \multicolumn{1}{c}{\mathrm{else}}\end{array}\hfill \\ \multicolumn{1}{c}{N_{i,k}=\frac{u-u_i}{u_{i+k}-u_i}{N}_{i,k-1}\left(u\right)+\frac{u_{i+k+1}-u}{u_{i+k+1}-u_{i+1}}{N}_{i+1,k-1}\left(u\right)}\end{array}$$

(13)

It is shown in equation (12) that the shape of NURBS surface is related to the control points, the weight factors, and the node vectors. In this article, the control points are obtained from the previous discrete points, then the control points of the final control surface topography are inverted, and the surface topography of the machined surface is fitted out. The point $p$ on the theoretical surface corresponds to the sampling point $q$ , and the distance from $p$ to $q$ is set as s. If $q$ is between $p$ and $d_{i,j}$ , then the surface is pulled toward the control point, and the s is positive at this time and negative for all other times. The specific process is shown in Figure 4. The schematic diagram of surface topography fitting is shown in Figure 5.

Figure 4.

Flowchart of reconstruction of the surface topography based on NURBS surface control points.

Figure 5.

Surface topography control point reconstruction diagram.

The normal distance between the surface and the surface topography is defined as the normal error of fitting. The relative error rate of the maximum of the normal error of each point on the surface is less than 6.0% and the relative error rate at the minimum of the normal error is less than 4.5%. The coordinates of the control points are shown in Table 1.

Table 1.

The coordinates of NURBS surface control points.

Control point	$X(\mathrm{\mu }\mathrm{m})$	$Y(\mathrm{\mu }\mathrm{m})$	$Z(\mathrm{\mu }\mathrm{m})$	Control point	$X(\mathrm{\mu }\mathrm{m})$	$Y(\mathrm{\mu }\mathrm{m})$	$Z(\mathrm{\mu }\mathrm{m})$	Control point	$X(\mathrm{\mu }\mathrm{m})$	$Y(\mathrm{\mu }\mathrm{m})$	$Z(\mathrm{\mu }\mathrm{m})$
1	0	0	4.3241	9	160.3241	33.1203	1.8579	17	316.3562	66.3215	2.7896
2	84.5563	0	3.1161	10	245.1256	33.1203	1.8570	18	400	66.3215	3.1244
3	160.3241	0	2.8366	11	316.3562	33.1203	2.7892	19	0	100	4.3240
4	245.1256	0	2.8342	12	400	33.1203	3.6235	20	84.5563	100	3.1542
5	316.3562	0	3.1512	13	0	66.3215	3.7114	21	160.3241	100	2.8356
6	400	0	4.3241	14	84.5563	66.3215	2.7891	22	245.1256	100	2.8341
7	0	33.1203	3.6238	15	160.3241	66.3215	1.8586	23	316.3562	100	3.1212
8	84.5563	33.1203	2.7880	16	245.3562	66.3215	1.8573	24	400	100	4.3245

Instead of measuring acceleration signals traditionally required in existing researches,^20,25,26 formulas for the determination of the forced vibration in X, Y, and Z directions are theoretically solved based on the stable solution and the transient solution. The surface topography is reconstructed based on NURBS surface control points according to the coordinate values of the residual grid points with considering the forced vibration in X, Y, and Z directions. The proposed method can also be used to simulate the surface topography with considering the impact vibration.

Simulation of surface topography considering impact vibration

For the splicing mold, the influence of impact vibrations on the surface topography under different feed rate is analyzed under the condition of constant speed. The Fast Fourier Transformation (FFT) transform of the milling vibration signal measured at the speed of 4000 r/min and the feed speed of 800 mm/min is performed before and after 0.5 mm seam. The spectrum analysis is shown in Figure 6.

Figure 6.

Milling vibration spectrum before and after the splicing seam: (a) milling vibration spectrum before splicing seam and (b) milling vibration spectrum after splicing seam.

The main frequency of milling vibration spectrum before splicing seam is $127.4\mathrm{Hz}$ , which is the rotation frequency of spindle. After splicing, due to the existence of impact vibration, milling vibration spectrum of the main frequency concentrated at $414\mathrm{Hz}$ , which is considered as the main frequency of the impact vibration, and the energy accumulates here. ²⁷

Then, the single-factor experimental method was used to simulate the surface topography of the workpiece after four groups. The four groups of simulation are selected on the principle of the pre- and post-stitched points. The point before the cutting is the point where the tool has not been sewn and the edge of the seam is selected after the cutting of the tool. After the sewing point is the point where the tool passed through the seam. Due to the smaller regional choice for simulation, the minimal impact point of minimum deviation in the simulation results before and after the splicing seam can be neglected. The simulation points are shown in Figure 7 before and after the seam. The snapshots of the surface topography of simulations are shown in Figures 8–11. The simulation parameters are shown in Table 2.

Figure 7.

Position of the simulation points before and after the seam.

Figure 8.

Feed per tooth, f_z = 0.05 mm/tooth—the simulation of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 9.

Feed per tooth, f_z = 0.10 mm/tooth—the simulation of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 10.

Feed per tooth, f_z = 0.15 mm/tooth—the simulation of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 11.

Feed per tooth, f_z = 0.20 mm/tooth—the simulation of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Table 2.

Specific simulation parameters.

Parameters and equipment	Numerical and model
Spindle speed	4000 r/min
Cutting feed rate	400, 800, 1200, and 1600 mm/min
Cutting depth (mm)	0.3
Milling spacing (mm)	0.3
Milling cutter material	Carbide
Tooth number of milling cutter	2
Helix angle of milling cutter	30°
Milling cutter diameter (mm)	10
Splicing workpiece material	Cr12MoV
Stitching width (mm)	0.5 mm
Hardness	From hardness HRC50 to hardness HRC59

It can be seen from Figure 8, with the feed per tooth of $0.05\mathrm{mm}$ , that the residual surface shows severe fluctuations in the simulation of the surface topography. The unit of surface topography after splicing seam is longer than that before splicing seam, also with the larger residual height. In Figure 9, it can be seen from the simulation results, when the feed per tooth is $0.10\mathrm{mm}$ , that the phenomenon of single tooth cutting and the forced vibration is not so serious before the seam. The surface after the seam is formed by single tooth cutting obviously, due to the influence of the impact vibration. In Figures 10 and 11, it can be seen that the single tooth cutting phenomenon occurs before and after the seam when the feed rate is $0.15$ and $0.20\mathrm{mm}$ , and the vibration after the seam is always larger than the vibration before the seam.

In general, when the amount of feed per tooth is moderate, the residual height of the surface topography increases with the increase in the feed per tooth. Especially, when the feed per tooth is larger than 0.15 mm, the reason for the phenomenon of single tooth cutting is the forced vibration, which has negative impacts on the quality of both surface before and after the seam. Only the forced vibration should be considered in milling of splicing mold, when the feed per tooth is less than 0.15 mm.

Milling experiment of surface topography with splicing characteristics

In this article, VDL-1000E three-axis vertical milling machine is used, the diameter is 10-mm two-edged integral carbide ball-end milling, and the workpiece consists of hardened steel of different hardnesses. The measuring instrument includes data acquisition system, dynamometer, eddy current sensor, three-direction acceleration sensor, and white light interferometer. The experiment is carried out using the above instruments. The experimental site is shown in Figure 12.

Figure 12.

Milling experiments.

First, the mold model is imported into the numerical control (NC) machining module of UG. Then, set up the corresponding process parameters and generate NC code. The code is input to VDL-1000E three-axis vertical machining center. Rough milling, semi-finished, and finished machining are carried out by the way of forward milling and unidirectional milling, and then the machined surface is obtained. The specific experimental parameters and experimental conditions are also shown in Table 2.

Then, the single-factor experimental method was used to compare the surface topography of the machined workpiece with the white light interferometer based on the influence of variation of surface curvature on the appearance of the surface. The position points of the actual surface topography are obtained according to the position points of simulation of surface topography after the seam. The two-dimensional (2D) contour and the measured surface topography of the position point are obtained and then compare it with the 2D contour that has been obtained from the simulation.

The measured surface topography is shown in Figures 13–16, and the simulated and measured 2D contours are shown in Figures 17–20. By comparing the surface topography of the preceding simulation with the measured surface topography, it can be found that the change rule and the trend of the surface topography are basically consistent. The measured surface topography also appears to be the phenomenon of single tooth cutting. In order to further illustrate the change rule of the surface topography of the simulation and the measured surface, the 2D contours are generated. It can be seen from the contours that the change rule is basically consistent, but the measured 2D profile of non-linear fluctuation is relatively strong. It is due to the fact there are other factors that are not considered. Overall, the simulation model can be used to predict the actual machined surface topography.

Figure 13.

Feed per tooth, f_z = 0.05 mm/tooth—the measurement of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 14.

Feed per tooth, f_z = 0.10 mm/tooth—the measurement of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 15.

Feed per tooth, f_z = 0.15 mm/tooth—the measurement of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 16.

Feed per tooth, f_z = 0.20 mm/tooth—the measurement of the surface topography before and after the splicing seam: (a) surface topography before the seam and (b) surface topography after the seam.

Figure 17.

Feed per tooth, f_z = 0.05 mm/tooth—the simulation and measured 2D contours before and after the splicing seam: (a) 2D contours before the seam and (b) 2D contours after the seam.

Figure 18.

Feed per tooth, f_z = 0.10 mm/tooth—the simulation and measured 2D contours before and after the splicing seam: (a) 2D contours before the seam and (b) 2D contours after the seam.

Figure 19.

Feed per tooth, f_z = 0.15 mm/tooth—the simulation and measured 2D contours before and after the splicing seam: (a) 2D contours before the seam and (b) 2D contours after the seam.

Figure 20.

Feed per tooth, f_z = 0.20 mm/tooth—the simulation and measured 2D contours before and after the splicing seam: (a) 2D contours before the seam and (b) 2D contours after the seam.

In Figure 13, it can be seen that the surface topography shows obvious fluctuations when the feed per tooth is $0.05\mathrm{mm}$ , but the volatility is not severe and the surface quality is relatively good. In Figure 14, it can be seen that when the feed per tooth is $0.10\mathrm{mm}$ , the single tooth cutting occurs after the seam, the phenomenon of cutting edge is partially cut before the seam, the residual surface of single edge cutting is large, and the quality is poor. In Figures 15 and 16, it can be seen that when the feed per tooth is $0.15$ and $0.20\mathrm{mm}$ , the single tooth cutting occurs before and after the seam, but the vibration after the seam is larger than the vibration before the seam, and the residual height is higher after the seam. In general, with the increase in feed per tooth, the residual height of the surface topography increases gradually when the feed per tooth is between $0.05$ and $0.20\mathrm{mm}$ . In addition, the phenomenon of one tooth cutting, other part cutting, and single tooth cutting will have an adverse effect on the surface quality. Comparing Figures 13–16 with Figures 8–11, the relative morphological error of the simulated surface topography and the measured surface topography is less than 14%, and the residual height is also very small. It indicates that the simulation model is accurate and the proposed method can be used to predict the surface topography of milling of splicing mold with considering forced and impact vibrations efficiently.

In order to further illustrate the change rule of the surface topography of the simulation and the measured surface, a 2D contour line is made. The position of the 2D contour line is a vertical plane along the feed direction at a distance of 200 µm. The 2D contour formed by the intersection of the vertical plane and the surface is the required contour line, as shown in Figures 17–20.

In Figure 17, it can be seen that the 2D contour of the residual surface fluctuates more frequently for the feed per tooth of $0.05\mathrm{mm}$ , which is due to relatively large frequency of vibration when the feed of each tooth is small and the non-linear of the 2D contour is stronger. In Figure 18, it can be seen that when the feed per tooth is $0.10\mathrm{mm}$ , the single tooth cutting occurs after the seam, the cutting edge is partially cut before the seam, and the 2D contour line is more obvious. From Figures 19 and 20, it can be seen that when the feed per tooth is $0.15$ and $0.20\mathrm{mm}$ , the single tooth cutting occurs before and after the seam, but the vibration after the seam is larger than the vibration before the seam. The up and down lines of the contour after the seam are relatively large, which is due to the impact. In general, with the increase in feed per tooth, the residual height of the surface topography increases gradually when the feed per tooth is between $0.05$ and $0.20\mathrm{mm}$ . The phenomenon of one tooth cutting, other part cutting, and single tooth cutting will have an adverse effect on the surface quality. In addition, the variation tendency of the simulation and measured 2D contour is basically consistent. It indicates that the simulation model is accurate and it is efficient to predict surface topography to evaluate the residual height which is the critical parameter of surface quality for milling of splicing mold.

Fractal analysis of surface topography with splicing characteristics

As for the study of the splicing characteristics in this article, we still need to analyze the influence of impact vibration on surface topography for different parameters. It reflects that the quality is the size of the surface roughness. The surface roughness represents the arithmetic mean of the peaks and troughs in a certain region and it is a 2D representation. In this article, we use the white light interferometer for 3D representation of surface topography. In addition, the fractal dimension is introduced to analyze the non-linear characteristics of 2D contours under impact vibration and to analyze the fine structure of surface topography.

The fractal dimension reflects the validity of the space occupied by the complex shape and it is a measure of irregularity of the complex shape.^28,29 The surface topography has the fractal features of statistical self-similar and self-affine. It is inadequate in evaluating the surface topography only by surface roughness frequency. ³⁰ Therefore, the fractal dimension of the contour curve can be used to characterize the machined surface. The surface contour curve is covered with a square box with side length equal to 1 and the box is divided into a grid set containing 2n small square boxes. The side length of the small square box is 2^–n and the grid covers the contour curve. Then the number $M(n)$ of small boxes that intersect with the contour is counted. The fractal dimension of the curve is given by

$$D=\underset{n\to \infty }{lim}\frac{\log M\left(n\right)}{n\log 2}$$

(14)

The surface roughness and the fractal dimension of the 2D contour are analyzed under the same rotational speed and at different tooth feed rates. The variation trend chart is made. It can be seen from the chart that the fractal dimension shows a decreasing trend before and after the seam with the increase in feed per tooth (0.05–0.20 mm) under the same rotational speed of $4000$ and $5000r/\min$ . Moreover, the fractal dimension after the seam is smaller than the fractal dimension before the seam. This is because the splicing joint and hardness characteristics make the tool produce impact vibration, the 2D contour fluctuates more after the seam than before the seam, and the surface quality is poor. It is also consistent with the meaning of fractal dimension, which further illustrates the influence of the impact vibration on the surface topography. The variation trend of fractal dimension is shown in Figure 21.

Figure 21.

The fractal dimension of the surface two-dimensional contours before and after the splicing seam: (a) speed 4000 r/min and (b) speed 5000 r/min

In Figure 22, we can see that at the spindle speed of $4000$ and $5000r/\min$ before and after the seam, the 3D arithmetic mean deviation of the surface with feed per tooth (0.05–0.20 mm) is increased and the 3D arithmetic mean deviation is higher after the seam than before the seam. It indicates that the quality of the process is affected by the splicing seam and hardness characteristics. As can be seen from Figure 23, the peak residual height of the surface before and after the seam at the speed of $4000$ and $5000r/\min$ also increases with the increase in the feed per tooth, and the residual height of the surface after the seam is significantly higher than that before the seam, which is related to the seam and hardness characteristics, and at the speed of $5000r/\min$ , the machine itself has more obvious vibration. According to the properties of fractal dimension, the surface topography before the splicing seam with relatively large fractal dimension means the surface microtopography is multifarious and disorderly. Due to the impact vibration, the fractal dimension of the surface topography after the seam is relatively low, and the residual height after the seam is larger than that before the splicing seam. Therefore, the impact vibration and the forced vibration of the machine together make the surface quality after the splicing seam worse.

Figure 22.

The variation of the three-dimensional arithmetic mean deviation of the surface before and after the splicing seam: (a) speed 4000 r/min and (b) speed 5000 r/min.

Figure 23.

Peak height of residual surface before and after the splicing seam: (a) speed 4000 r/min and (b) speed 5000 r/min.

Conclusion

As is well known, milling is a kind of intermittent machining process, in which the tooth is sometimes in cut while sometimes out of cut. Because of the periodical change of the cutting force, it is easy to lead to forced vibration. There are limited works that were carried out to obtain the forced vibration from experimental measurement. In addition, it is very limited for studies on the impact vibration caused by the seam. The coupling of the impact vibration and the forced vibration has great effects on the surface roughness. Therefore, a simulation model of the 3D surface topography is developed for ball-end milling of splicing transition mold with considering the coupling of the impact vibration caused by the seam and the forced vibration caused by multi-edge interrupted cutting.

1. Instead of measuring acceleration signals traditionally required in existing researches,^20,25,26 formulas for the determination of the impact vibration and the forced vibration in X, Y, and Z directions are theoretically solved based on the stable solution and the transient solution during milling splice mold. The motion trajectory of cutting edge is established with considering the vibration.

2. In this article, the surface topography is reconstructed based on NURBS surface control points according to the coordinate values of the residual grid points. Using this method, the required surface can be accurately fitted out with less points, and the simulation efficiency is greatly improved compared with methods proposed in references.^14–16,24 Based on the NURBS surface control point reconstruction, the surface topography prediction model of the splice transition zone is established with considering the influence of the coupling of the impact vibration and the forced vibration on the surface topography.

3. After the seam, at the speed of 4000 r/min, with the increase in feed per tooth, surface topography residual height increased gradually, the 3D arithmetic mean deviation of the seam is higher than that before the seam, and the phenomenon of the cutting of the other tooth and the phenomenon of the single tooth cutting occurs, which adversely affect the surface quality.

4. At the speed of 4000 and 5000 r/min, the fractal dimension of the 2D contours before and after the seam decreases with the increase in the feed per tooth (0.05–0.20 mm). It means that the fractal dimension after the seam is smaller than the fractal dimension before the seam, which indicates that the impact vibration of the tool is caused by the splicing seam and hardness characteristics. The 2D contour line after the seam fluctuations relative to the 2D contour line before the seam fluctuations is relatively large, and regularity is enhanced.