1. Introduction
The operational performance of parts (fatigue performance, wear resistance, corrosion resistance) is controlled by surface integrity [
1,
2]. The formation mechanism of machined surface topography is an important part of surface integrity research. Ball end milling cutter is widely used in the complex surface finishing of important parts in aerospace, die and automobile industry because of its strong adaptability. Due to the high surface quality requirements of these products, extensive research work has been carried out around this milling process [
3,
4]. The cutting edges geometry of the ball-end milling cutter is complex, and the contact point between the cutting edges and the workpiece is constantly changing during milling. In addition, the dynamic characteristics of the milling process are complex [
5]. It is difficult to study the surface quality of the machine by traditional experimental methods. Therefore, it is necessary to simulate the surface topography.
Mizugaki et al. [
6,
7] established a theoretical model for predicting the surface topography during milling. The effects of the shape of cutting edge, the tilting tool orientation, the tool eccentricity and the interference area of tool flank face on the surface topography were studied. However, the simulation results of surface topography considering the shape of cutting edge, the tilting tool orientation and the tool eccentricity were not verified by experiments. Bouzakis et al. [
8] developed a simulation algorithm considering milling kinematics to determine the shape of the undeformed chip, cutting force, tool deformation and surface topography. The effects of different milling methods, the tilting tool orientation, feed rate and radial cutting width on surface morphology were investigated. Based on the assumption that the envelope of the cutting edge of the ball-end milling cutter is approximated, a hemisphere when the spindle speed is much larger than the feed speed, Jung et al. [
9,
10] defined three types of ridges to represent the scallop-height of surface, and proposed a ridge model to predict the surface morphology and roughness. The surface morphology characteristics of one-way feeding and reciprocating feeding are studied. It was found that one-way feeding can obtain smaller surface roughness than reciprocating feeding. Based on a Z-map model, Liu et al. [
11] developed a comprehensive simulation system for predicting surface topography and roughness characteristics during finishing milling. The influence of tool eccentricity and wear on surface morphology and roughness was studied by modifying tool center movement and the shape of cutting edge. It was found that tool wear and tool eccentricity have little effect on surface morphology and roughness, but the prediction model of surface quality considering tool wear and tool eccentricity is closer to the experimental value. Buj-Corral et al. [
12] established a mathematical model for predicting the surface topography and roughness by constructing surface topography into a functional relationship between cutting parameters and tool geometry parameters. Xu et al. [
13] proposed an improved surface topography simulation algorithm based on a Z-map model. The effects of cutting feed rate, cutting width, the tilting tool orientation and tool runout on surface quality were studied. By solving the cycloidal equation of cutting edge and the feed plane equation, the intersection points of all feed planes and the cycloidal trajectory of cutting edge are determined. Then, the surface morphology of five-axis milling process is obtained by an intersection points matrix [
14]. The effects of the tilting tool orientation, tool lead angle and tool eccentricity on surface morphology were studied. However, the proposed model had lower accuracy. Li et al. [
15] established a five-axis milling surface topography analysis model considering positioning error, spindle error, machine tool geometry error and tool deformation. The above surface morphology model does not take into account the tool vibration with time-varying characteristics, and can not correlate it with the dynamic response of the milling process.
Cutting vibration and tool wear have a significant influence on the surface quality in cutting process, which has attracted the wide attention of researchers. Arizmendi et al. [
16] and Jiang et al. [
17] used eddy current displacement sensors to obtain tool vibration displacement during milling. Then, the equation of cutting edges trajectory was modified based on the vibration displacement data, and the prediction model of surface topography considering tool vibration was established. Costes et al. [
18] used a laser displacement sensor to obtain tool vibration displacement in the cutting process, and established the milling morphology prediction model considering the tilting tool orientation and the tool vibration. Whether an eddy current displacement sensor or a laser displacement sensor, the accuracy of data acquisition is affected by spindle speed. Therefore, this method is not suitable for high-speed and ultra-high-speed machining. Based on the tool-tip trajectory equation, Peng et al. [
19] established a micro-ball-end milling morphology prediction model considering the tilting tool orientation, vibration, eccentricity and deformation. However, the vibration of the cutter in the axial direction was only considered in the surface topography model, while the vibration in the radial direction was ignored. Since the tool stiffness in the axial direction is much greater than that in the radial direction, the vibration amplitude in the radial direction is larger. If the radial vibration is neglected, the simulation precision of the surface morphology will be reduced. Chen et al. [
20] established a surface morphology prediction model for micro-milling considering kinematics, tool eccentricity and dynamic regeneration. Lu et al. [
21] established a surface morphology prediction model for micro-milling considering scale effect, multiple regenerative effect and tool deformation. Yang et al. [
22] investigated the influences of milling vibrations on the machined surface topography. Zhang et al. [
23,
24] proposed a simulation model of surface topography in multi-axis ball-end milling considering tool wear. The effects of feed rate and the tilting tool orientation on surface topography were studied, but the effects of different tool wear on surface topography were not clearly revealed, which need to be further explored. Omar et al. [
25] established a mathematical model that can predict cutting force and surface morphology at the same time. The effects of tool eccentricity, tool deformation, system dynamics, flank wear and the tilting tool orientation on surface quality were studied. The simulation result was in good agreement with the experimental result, which verified the accuracy of the simulation mathematical model.
Milling is a kind of interrupted cutting. Cutting impact vibration and cutting vibration caused by a nonlinear change of chip thickness exist in the process of cutting tool material removal. Meanwhile, tool wear will occur gradually on a flank face with the cutting progress. The morphological characteristics of machined surfaces are controlled by the dynamic characteristics of the cutting system and the tool flank wear of time variation. The influence of dynamic characteristics of cutting system and tool wear on surface morphology has not been taken into account in the morphology prediction model established in the existing literature. In the present study, the machined surface topography considering the dynamic characteristics of cutting system and tool flank wear is established. The research results have important guiding significance for the reasonable selection of processing parameters in actual production.
2. The Kinematic Trajectory Equation of Cutting Edges
The establishment of the trajectory equation of any cutting point on the cutting edge in the milling process is the basis and key to the simulation modeling of the surface topography of the ball-end milling cutter. Therefore, the rectangular coordinate system as shown in
Figure 1 is introduced.
The workpiece coordinate system OWXWYWZW ({W} for short) is a reference coordinate system fixed on the workpiece. The tool rotation coordinate system OTXTYTZT ({T} for short) rotates with the tool. The tool coordinate system OGXGYGZG ({G} for short) translates and vibrates with the tool. The spindle coordinate system OSXSYSZS ({S} for short) moves along the feed direction with the spindle. is the normal contact force. is the tangential contact force. and are the entry and exit angles of the cutter to and from the cut, respectively. is the axial position angle.
Taking ball-end milling cutter as the research object, the homogeneous coordinates of any point P on the cutting edge in {
T} can be expressed as follows:
where
is axial position angle,
is helix angle of tool, and
is tool radius.
Meanwhile, tool wear will occur gradually on the flank face and rake face with the cutting progress. Since tool flank wear has a significant influence on the surface topography, the effect of tool flank wear on the surface topography is researched. The geometric relationship between the flank wear and the discrete points of the cutting edge is shown in
Figure 2.
The homogeneous coordinates of any point P on the cutting edge with wear in {
T} can be expressed as follows:
where
and
are the clearance angle and the rake of the cutting tool.
The coordinate transformation matrix from {
T} to {
G} at any point P on the cutting edge can be expressed as follows:
where
,
is the initial entrance angle of the tool,
is the teeth number, and
is the angular velocity of the tool.
The stiffness of workpiece is much higher than that of the cutter-spindle cutting system for the milling process of automobile panel die. Therefore, the vibration of machining process mainly occurs in the tool-spindle cutting system. Meanwhile, because the stiffness of the ball-nose cutter in the axial direction is far greater than that in the radial direction, the vibration of the cutter in the axial direction is neglected:
where
and
are the vibration caused by cutting impact.
and
are the vibration caused by stable cutting. Detailed solutions of the two kinds of cutting vibration are given in Parts 3 and 4.
The coordinate transformation matrix from {
G} to {
S} can be expressed as follows:
The most basic of the tool path mode is unidirectional or bi-directional in the milling process. The tool path mode of one-way is selected in this paper. The transformation matrix from the spindle coordinate system to the workpiece coordinate system is shown in Equation (9):
where
is the number of feeds in the cross-feed direction.
is the feed rate of the tool.
is the cut width.
is the tilt angle of the workpiece.
is the coordinate value of the origin of the machine tool spindle coordinate system in the workpiece coordinate system during the first feed of the tool.
The machined surface topography is obtained by cross-calculation between the space motion envelope of the tool and Boolean of the workpiece. The space motion envelope of the tool can be expressed by Equation (10):
where
represents the product of the transformation matrix of tool motion space at each moment.
and
are window functions, which are used to judge the tool wear area. The expression is shown in Equation (11):
where
and
are the minimum and maximum axial position angles of tool wear, respectively.
The precise sweep surface mathematical model can be obtained by substituting the sweep surface model of the cutting edge in the tool coordinate system into Equation (10), as shown in
Figure 3.
3. Milling Dynamics Equation
The cutter has more flexibility than the workpiece in the process of milling automobile mould, so the cutter is considered a flexible body and the workpiece a rigid body. A two-degree-of-freedom spring-damper-mass system model is established, as shown in
Figure 4:
where
,
,
,
,
,
are vibration acceleration, vibration velocity and vibration displacement of the cutting system.
,
,
,
,
, and
are the mass, structural damping, and stiffness matrices of the cutting system in the
x and
y direction.
The cutting vibration in the
x and
y directions is obtained by solving the milling process dynamics with the fourth-order Runge–Kutta method. Taking the differential equation in Equation (12) as an example, the solving process of vibration in
x direction is as follows:
We set
equals
, Equation (13) can be descended to one-order differential equation:
The function Ode45 in MATLAB (R2014a, MathWorks, Natick, MA, USA) is used to solve differential equations. The function Ode45 belongs to the explicit Runge–Kutta algorithm and has fourth-order accuracy. Choosing a small time interval
can obtain a more accurate solution, but it will increase the operation time. Therefore, the optimum selection principle of
is as follows:
where
is the maximum natural frequency of the milling system.
7. Results
The cutting impact vibration with the cutting parameters (Test 1) in the the
x and
y directions are shown in
Figure 9.
The simulation results of maximum cutting vibration values with/without impact vibration are compared with the experimental results for cutting parameters (Test 1–5) as shown in
Figure 10. The maximum cutting vibration values in the
x direction were changed into absolute values.
As seen in
Figure 10, the simulation results of cutting vibration considering impact vibration (Simu.2) are more precise. The simulation error of cutting vibration in the
x direction can be reduced by 33.13% in maximum (Test 5). The simulation error of cutting vibration in the
y direction can be reduced by 25.02% at the maximum (Test 2).
The simulation results of total cutting vibration with cutting parameters (Test 1) in the
x and
y directions are shown in
Figure 11a,c. Cutting vibration obtained from cutting experiments is shown in
Figure 11b,d.
It can be found from
Figure 11 that the vibration signal obtained by simulation and experiment has a good consistency in both trend and value. Therefore, the vibration simulation data can be used to simulate the dynamic characteristics of the machining process, which lays a foundation for studying the influence of milling vibration on the surface morphology.
The simulation and experimental results of surface morphology and roughness for cutting parameters (Test 1) are shown in
Figure 12.
It can be seen from
Figure 12 that the dynamic characteristics of the cutting system affect the texture structure of the machined surface. Due to the influence of cutting system vibration, the cutting contour of adjacent surface topologies in the feed direction is different, which is caused by the slight up–down displacement of the two edges of the ball-end milling cutter, as shown in
Figure 12a. A is the surface profile formed when the vibration of the cutting system is maximum, and B is the surface profile formed when the vibration of the cutting system is minimum. It can be seen from
Figure 12 b–d that, when the vibration of cutting system is considered, the surface topography obtained by simulation is closer to that obtained by actual machining process. Meanwhile, it is found that the impact vibration has little effect on the texture structure of the surface morphology but affects the scallop-height of the machined surface. The reasons for the difference in surface morphology between the simulated and experimental are as follows: (1) the cut-in phase angle is random in the experiment process; and (2) the surface topography model does not take into account the tool deformation. When the dynamic characteristics of cutting system are not considered, the simulation error of surface roughness is 19.05%. When only the vibration caused by a nonlinear change of chip thickness is considered, the surface roughness error is 12.09%. When the vibration of cutting system with cutting impact vibration is considered, the error of surface roughness is 9.52%.
The simulation and experimental results of surface roughness at different tool wear for cutting parameters (Test 1) are shown in
Figure 13.
It can be seen from
Figure 13 that the surface roughness increases with the increase of tool wear. When the tool wear amount is less than 0.164 mm, the surface roughness increases more slowly. When tool wear increases from 0.164 mm to 0.21 mm, the increase rate of surface roughness is close to 50%. When the tool wear is less than 0.164 mm, the maximum error between the surface roughness simulation and the experimental measurement is 12.3%, which is within the acceptable range. When the tool wear is 0.21 mm, the maximum error between the surface roughness simulation and the experimental measurement is 21.9%. The reason is that the cutting edge near the tool tip induces micro-chipping.
The simulation and experimental results of surface morphology at different tool wear for cutting parameters (Test 1) are shown in
Figure 14.
It can be seen from
Figure 14 that, with the increase of tool wear, the texture structure of the workpiece surface changes and the maximum scallop height on the surface increases accordingly. Meanwhile, the texture asperities grow and the grooves deepen with tool wear in the experiments. The worn tool causes scratches on the machined surface, which leads to deterioration of the machined surface. It is found that the scratch width of simulation is larger than that of the experiment. The reason is that the tool wear is assumed to be the same in the wear area in the simulation process, which is inconsistent with the actual situation.