Theoretically Investigation On The Roughness Prole of The Machined Surface After Flank Milling

In this study, the roughness profile of a machined surface obtained via a flank milling process is thoroughly investigated through theoretical modeling and experimental demonstrations. First, the roughness profile of a machined surface generated by a single - tooth end milling cutter along a straight path is considered (without helical angle). The trajectory of a point on the cutting edge is constructed according to the cutting kinematics, and the roughness profile of the flank surface is theoretically extracted from the trajectory. The surface topography is constructed by integrating the roughness profile along the axial direction of the cutter. Based on the constructed roughness profile model, the effects of cutting parameters on the roughness profile are discussed, including those of the cutting speed, radial depth of the cut, and feed rate. In addition, the effects of cutter geometries including the cutter tooth number, tooth spacing angle, and helical angle on the roughness profile and surface topography are discussed. Further, roughness profiles are constructed for cutter feeds along different tool paths, such as round and curved paths. Finally, experiments are conducted to verify the method developed in this study. The results show that the roughness profile obtained from testing matches well with the theoretically modeled profile. Moreover, the methodology for constructing the roughness profile is compared to an existing approach, which shows that the method in this study is significantly faster.


Introduction
Surface topography or texture of a workpiece is related to its surface quality. It will affect the tribology as well as wear properties of the workpiece. Smooth surface of a workpiece is always required in many engineering practices, especially in the precision engineering. Thus, it is necessary to investigate the topography or texture of a workpiece after being manufactured.
In recent years, the surface topography of a workpiece obtained via milling process, a widely used material removal technique in the aerospace, mold industry as well as other metal cutting industries, has gained more and more attention. Many investigations were dedicated to develop approaches to construct the machined surface topography after milling process and to analyze the factors that affect the surface topography. As to the methods of constructing the surface topography, the one that based on milling kinematics is commonly used. In this method, a set of planes that vertically pass through the cutting tool axis were constructed at first. Then, the roughness profile determined by the trajectories of the cutter edge points in each plane was computed. Finally, the surface topography was obtained by fitting all the surface profiles. For this method, the key to construct the surface topography is to compute the roughness profile in each plane.
So far, different methods to compute the roughness profile were proposed. For example, Arizmendi et al. [1] proposed a discretization method to construct the roughness profile. In their method, a set of lines were equally distributed along the milling path, and the lines were perpendicular to the cutting path. The lines intersected the trajectories and resulted in a set of intersection points. Then, intersection points that belong to the roughness profile were extracted and used to fit the profile. The schematic of the method is show in the Fig. 1. The roughness profile is consisted of a bunch of small arcs that belong to trochiodal trajectories between two adjacent intersection points. Similar to the method developed by Arizmendi et al. [1], Arizmendi et al. [2], Quinsat et al. [3], Layegh and Lazoglu [4] modeled the machined surface topography after ball end milling process. Arizmendi and Jiménez [5] modeled the surface topography generated after face milling process. Instead of trochoidal arcs, Buj-Corral et al. [6] approximated the surface profile by circular arcs that equally distributed along the milling path. The distance between two adjacent arcs equals to the feed rate. Fig. 1 The schematic to obtain the surface roughness profile [1].
Based on the roughness profile construction methods as reviewed, the effect of cutting parameters such as feed rate [7,6], the effect of tool geometry parameters including helical angle [7,6], the effect of tool error induced by the tool wear [8,9] or tool deflection [7,9], the effect of tool setting error including tool axis offset [10,6,2], tool tilt [10,4] and run out [7,4,9], the effect of tool posture such as tool inclination [9,4], the tool vibration [1,11] on the surface topography were discussed.
In the literatures mentioned above, different methods to construct the roughness profile were developed. Some of the methods are numerical while the rest of them are semi-analytical. Yet none of them tried to understand the roughness profile as well as to analyze the influence of the cutting parameters and the cutter geometries theoretically. Besides, in all the investigation the cutter was assumed to move along a straight path. However, the cutter will move along a curve path when milling a workpiece with complex surface. In addition, the existed methods are time cost to construct the surface topography. For example, it took 5 h to construct the surface topography within an area about 6 mm 2 [4]. Henceforth, this investigation focuses on the problems as mentioned and tries to develop an analytical method to construct the roughness profile and to analyze the influence other factors such as cutting parameters and cutter geometry based on the developed method. The rest of the paper is arranged as follows: Section 2 constructs the roughness profile of a very simple case. Based on the simple case, the effect of cutting parameters on the roughness profile is discussed in Section 3, the effect of the cutter geometry on the roughness profile is discussed in Section 4 while the roughness profile when the cutter moves along curve is discussed in Section 5. Experiments are conducted in the Section 6 to verify the developed roughness profile model. The whole paper is concluded in Section 7.

Roughness profile generation by the cutter with a single straight tooth
After flank milling process, two surfaces are generated, namely a bottom and a side face (as shown in Fig. 2 (a)). Only the roughness profile of the side face will be considered in this paper.
For simplicity, the milling cutter is assumed to be with a single and straight tooth. The cutter radius is assumed to be r. The cutting parameters are assumed to be axial depth of cut ap, the radial depth of cut ae, the feed velocity vf, the spindle speed n, the angular rotational speed ω respectively. Down milling is employed during modelling. During milling process, the tool edge interacts with the workpiece. The surface is generated after the cutting edge sweeps the workpiece materials. Thus, the roughness profile of the machined surface is related to the trajectory of the cutting edge points.

The trajectory of a point on the cutting edge
As mentioned before, the roughness profile is related to the trajectory of the cutting edge point.
Thus, the trajectory equation of a point on the cutting edge is derived. Let P denote a point on the cutting edge as shown in the Fig. 2 (a). Let a plane A vertically pass point P. The trajectory of the point P locates in this plane A. To derive its trajectory, a coordinate system x-O-y is constructed as shown in Fig. 2 (b). Another coordinate system x1-O1-y1 is placed at the cutter center with its x axis align with the feed direction, as shown in the Fig. 2 (b). The displacement vector of point P is obtained according to following relationship in which θ0 means the angle between the point P and the y1 direction at the initial moment as shown in Fig. 2. x0 means the distance between the cutter center and the workpiece at the initial moment.  Fig. 3 shows the trajectory of the single tooth cutter when the initial start angle is 0°. From the figure, it can be seen that the roughness profile is consisted of the trochoidal arcs among the lowest self-intersection points of the trajectory (only when the self-intersection points locate under the un-machined surface). Henceforth, it is necessary to solve the coordinates of the lowest self-intersection points of the trajectory for construction of the roughness profile.

Solve the lowest self-intersection points of the trajectory
Assuming that the coordinate of a self-intersection point is denoted by (x1, y1), the initial time instant that this self-intersection occurs is t1 (t1>0). After a time period Δt (Δt>0), the trajectory will come back to itself (x1, y1) because of self-intersection. Substituting the self-intersection point coordinate (x1, y1), the time instant t1 and the time period Δt into Eq.(2) results in From the lower equation of Eq.(4), it can be obtained that which results in Δφ=2kπ is an inappropriate solution because it will lead to vfΔt=0 and thus Δt=0. This is contradict to the fact that Δt>0. Therefore, only Δφ=2(kπ-φ) is the solution of Eq.(5).
From Eq.(4), the following relationship can be obtained in which vc=ωr. Detail derivation of this equation is given in the appendix. From Eq.(6), Δφ =2(kπ-φ), substitute it into Eq.(7) leads to The trajectory of the point P has self-intersection points as long as Eq.(8) has solutions.
Clearly, φ =kπ is an solution of the Eq.(8), but it means not that tooth trajectory will intersect itself when the cutter rotates an angle of kπ, because when kπ is substituted into the first equation of Eq.(6), it will lead to Δφ =0, which is contradict to the fact that Δφ >0. Thus, there are no self-intersection points when the cutter tooth rotates kπ.  In the real milling process, it is unlikely to use a feeding speed vf that larger than or equal to the cutting speed vc since the machined surface is very rough, as shown in the Fig. 5. Besides, it may break the cutting tool. Therefore, small feed speed (vf<vc) is usually used in real milling process. This investigation will also focus on the case when vf<vc. Under this condition, the machined surface generation is related to the lowest self-intersection point as shown in Fig. 3 and Fig. 6. Henceforth, it is necessary to obtain the coordinates of the lowest self-intersection points and the corresponding the time instant that the lowest self-intersection occurs.  Define a function as: The solutions of Eq. (8) can be thought as the zero points the above function. From Fig. 6, it can be seen that lowest self-intersection points of the trajectory is close to the lowest points of the trajectory. The lowest points occurs when φ=(2k1-1)π where k1 is a positive integer. Thus, the values of φ at which the line intersect with the sinusoidal curves, also corresponding to the lowest self-intersection points of the trajectory occur, can be approximated by the first order Newton method, in which the initial estimation of the φ0 can be chosen as φ0=(2k1-1)π. Namely Then the rotation angle at which the lowest self-intersection point occurs is Thus, the coordinates of the lowest self-intersection points of the trajectory are obtained by taking φ1 into Eq.(2), which are written as

Construct the roughness profile
The cutter rotation angle range φe between two adjacent self-intersection points is 2×(2k1-1)π-φ1≤ φe ≤ φ1. The roughness profile of the cutter edge point P can be obtained by taking φe into its trajectory equation. This is true when all the lowest self-intersection points are located under the un-machined part surface as shown in Fig. 3, namely y1<-ae (ae>r[1+cos(φ1)]/2). However, when the lowest self-intersection points of the trajectory are located above the un-machined workpiece surface as shown in Fig. 7, namely y1>-ae (ae< r[1+cos(φ1)]/2), there will be some area remain uncut. In this condition, the profile is related to the intersection points between the trajectory and the un-machined workpiece surface. These points can be obtained by letting the y coordinate of the trajectory equal toae, namely r-ae+rcos(ωt+θ0) = r-ae+rcosφ=0, which results in φ=arccos(-1+ae/r)+2kπ (k=1,2,3…). The roughness profile is consisted of the trochoidal arcs parts between the two adjacent points intersected by the workpiece surface and the trajectory plus the line that uncut, as shown in Fig. 7 (b).

Construct the surface topography
The surface topography is obtained by integrating the roughness profile along the axial direction of the cutter. The axial depth of cut ap is divided into many small pieces as shown Fig. 8 (a). The surface generated by each small cutter edge that corresponding to the small piece of small axial depth of cut can be approximated by the roughness profile on the any point of the small cutter edge. A bunch of roughness profiles along the axial direction can be obtained as shown in Fig. 8 (b). The whole surface topography is obtained by fitting all roughness profiles. Fig. 8 (c) shows surface topography when r=4 mm, n=1000 rpm, vf=0.05vc.

The effect of cutting parameters on roughness profile
The cutting parameters during milling process include axial depth of cut, radial depth of cut, feed speed and spindle speed. For all these cutting parameters, the axial depth of cut has no effects on the roughness profile due to the straight tooth cutter. The spindle speed will decide the cutting speed according to the relationship vc=nπr/60. According to the lower equation of Eq.(11), the height of the lowest self-intersection points (also the height of the roughness h as shown in Fig. 9) is decided by the cutting speed and the feed speed simultaneously. The equation can be rewritten From this equation, it can be seen that the height of the roughness profile is decided by the ratio between the feed speed and the cutting speed. The height of the roughness profile respect to the speed ratio is shown in Fig. 9. From the figure, it can be seen that the height increases with the ratio. Roughness profiles of different ratio are shown in Fig. 10. The figure shows that the surface profile is consisted of a series of fundamental element. The fundamental element looks like a part of the parabola. The width of the opening of the element is denoted by w while its height is denoted by h. When the ratio increases, the roughness profile becomes higher (h in the figure becomes larger) while the width becomes wider (w in the figure becomes larger). All these mean that the generated surface will become rougher. Thus, small feed speed should be used during precision stage.
As the increasing of the height h, it may exceed the un-machined surface. Under this condition, part of the surface will remain un-machined as shown in Fig. 7. The surface will be very rough. Thus, to avoid this situation, the ratio between feed speed and the cutting speed should be chosen so that yn=h<-ae if the radial depth of cut ae is given, as discussed in the end of Section 2.3. Fig. 9 The height of the roughness profile respect to the ratio between feed speed and cutting speed (r=4 mm, ae=1 mm, n=1000 rpm).

The effect of cutter geometry parameters on roughness profile
The geometry parameters of an end milling cutter include cutter radius (r), tooth number (z), tooth spacing angle (β) and helical angle (η). The roughness profile is related to trajectories of the cutting edge points which are affected by the cutter radius, the tooth number, tooth spacing angle and helical angle. The effect of radius on the roughness profile will be reflected by the cutting speed together with the spindle speed as discussed in the Section 3. Henceforth, the effect of tooth number, tooth spacing angle and helical angle will be discussed in detail.

The trajectories of points on different cutting edge
In practical milling process, the milling cutter usually has two or more teeth. Thus, it is practical to investigate the machined surface topography generated by the milling cutter with more than one tooth. Assuming that all the teeth are averagely distributed around the cutter, which means that the tooth spacing angle (or pitch angle) is β =2π/z (z is tooth number). For simplicity, the helical angle is still assumed to be zero. The tooth is numbered from 1 to z in a clockwise way. Similar to the procedure in the Section 2.1, let a plane A vertically cross the cutter axis at some height. The plane will intersect with the z cutter edges and thus results in z points, which is marked as P1, P2, …, Pz in a clockwise way, as shown Fig. 11 (a). The trajectories for all teeth have the same form but different start position due to the lag angle caused by the tooth spacing angle. In the coordinate system as shown in Fig. 11, the trajectory equations for all the cutter edge points is expressed as

Solve the lowest self-intersection points of the trajectories
The roughness profile is related to the lowest intersection points of all the cutter teeth trajectories as shown in Fig. 13. The lowest points are generated by intersecting the current tooth trajectory with the succeeded one. Its coordinates can be obtained by solving the intersection points of the corresponding trajectories equations. Assuming that coordinate of intersection point is (xi, yi), take it into to the corresponding trajectories (Eq.(13)) results in Solve the lower equation of Eq. (14) results in If ω(t1-t2)/2-β/2=kπ, then t1=(2kπ+ β)/ω+t2, take it into upper equation of the Eq. (14) results which means that t1=t2, however, t1=t2 is impossible. Thus, only when ω(t1+t2 from which one can solve t2 and thus the t1.
The function f(φ) has many zero points. However, only the one corresponding to the lowest intersection points occur is meaningful since only the lowest intersection points as shown in Fig.   13 is related to the roughness profile. Each lowest intersection point is close to the location when the cutter tooth reaches to the lowest position at y direction, at which the rotation angle of the cutter equals to (2k-1)π, which means that the zero points are close to (2k-1) π. The approximation values of the zero points can be obtained according to the Newton formulation. The first iteration is written as Because that φ= (2k-1)π is close to the zero points, it can be regarded as initial value of first iteration, namely, φ0=(2k-1)π. The zero point value of the first iteration is obtained as The coordinates of the lowest intersection points can be obtained by taking the φ1 into the trajectory equation (Eq. (14)).  After the coordinates of the lowest intersection points are obtained. The rotation angle of the cutter between two adjacent lowest intersection points can also be obtained and thus the roughness profile can be constructed. The surface topography can be constructed by the similar procedures as described in the Section 2.3.

The effect of tooth number on roughness profile
From Eq.(20), it can be seen that the angle at which the lowest intersection points between two adjacent trajectories occurs will be affected by the tooth spacing angle, which is affected by the cutter tooth number. Take it into the y coordinate of the trajectory, the height of the lowest intersection points (also the height of the roughness profile h as shown in Fig. 16 According to the above equation, the height of the lowest intersection point respect to the tooth number can be analyzed as long as the ratio between the cutting speed and the feed speed is given. Fig. 15 shows the varying trend when the cutter radius r=4 mm, speed ratio vf/vc =0.05, spindle speed n=1000 rpm. The figure shows that the height becomes lower and lower when the cutter teeth number increases, which is similar to the effect of feed speed. Thus, more cutter teeth number will result in smoother surface when the cutting speed is given. Fig. 16 shows the roughness profile of different tooth number. The profile has the same shape as shown in Fig. 10. The more the teeth number is, the lower the height (h) is and the narrower the opening is (w). Thus, the surface will become smoother.

The effect of tooth spacing angle
Recently, the variable pitch cutter is widely investigated because its effectiveness on chatter suppression [12][13][14][15]. Based on the method proposed in the above mentioned sections, this part tries to construct the roughness profile obtained by the variable pitch cutter. Fig. 17 shows the cutter with constant and variable tooth spacing angle respectively. Fig. 17 (a) is the cutter with constant tooth spacing angle, in which the angle between two adjacent cutter teeth is constant. Fig. 17 (b) is the cutter with variable tooth spacing angle, in which the angle between two adjacent teeth is not equal to each other. For the cutter with variable tooth spacing angle, the tooth spacing angle has the relationship that β1= β3 and β2= β4 for dynamic balance. Fig.   18 shows the trajectories of the cutting edge points and its generated roughness profile for a cutter with variable tooth spacing angle. From the figure, it can be seen that the roughness profile is still related to the lowest intersection points between the trajectories. The height of the lowest intersection points and the distance between two adjacent lowest intersection points are changed due to the variable tooth spacing angle according to Eq.(20).
For simplicity, this part compares the roughness profile of two kinds of four-tooth cutter, one with constant tooth spacing angle while the other one with variable tooth spacing angle, the tooth spacing angle is 77°-103°-77°-103° and 70°-110°-70°-110° respectively according to [12]. The roughness profiles of all these three cutters are listed in the Fig. 19. From the figure, the wave crest of the roughness profile for the constant tooth spacing angle has the same height, unlike the one for the variable tooth spacing angle in which the height of the wave crest has different height.
Besides, the total height of the roughness profile for the variable pitch cutter is larger than the one for the constant pitch cutter. It means that the surface topography generated by the cutter with constant tooth spacing angle is smoother.

The effect of helical angle
In the above investigation, the cutting edge is assumed to be with zero helical angle. Actually, cutter with helical cutting edge is often used in real milling process. This part will consider the effect the helical angle on the roughness profile and the surface topography. x v t r t y y r t  The roughness profile in plane A can be obtained according to the trajectory of P (Eq.(24)) and the procedures presented in Section 2. As to the whole surface topography, the axial depth of cut is divided into many small parts averagely. At each small part of depth of cut, the cutting edge is assumed to be upright. Thus, the profile for this small part can be obtained according to the procedures presented in Section 2. The whole surface topography can be obtained by integrating all the roughness profiles. As such, it can be seen that the helical angle will affect the phase of trajectory of the cutter edge point, which will affect the position of the lowest intersection points but not the height (h) and the distance (w) between two adjacent intersection points as shown in Fig. 22. There is a shift between the two roughness profiles that generated at two different heights.
This shift will be reflected on the surface topography. The peak of the generated surface topography will lie on an oblique line as shown in Fig. 23, which is very different from the surface topography generated with straight cutter tooth as shown in Fig. 8 (c).

The effect of tooth path on roughness profile
In the above mentioned investigation, the cutter moves along a straight path. However, the cutter will move along curve path during milling the workpiece with complex surface. Therefore, it is significant to construct the roughness profile of the workpiece with curve surface after milling.
This part will try to construct the roughness profile that generated by the cutter moves along a curve path and to discuss its effect.

Milling along a round path
Cylindrical workpiece is widely used in the engineering practice. When machining the outer surface of a cylindrical workpiece, the cutter will move along a round path. This part will discuss the roughness profile of the outer surface of a cylindrical workpiece after flank milling process, as shown in Fig. 24 (a). Assume that the radius of the cylindrical workpiece after machining is R. For simplicity, the cutter is assumed to be with a single and straight tooth. Two coordinate systems (x-O-y and x1-O1-y1) as shown in the Fig. 24 (b) are constructed.

The trajectory of a cutting edge point
To obtain the trajectory of a point P on the cutter edge, following vector relationship must be satisfied during milling process according to Fig. 24 (b).

Solve the self-intersection points and construct roughness profile
Suppose that the self-intersection occurs at (x1, y1), the corresponding time instant is t1. After time period Δt, the trajectory will go back to itself. Thus, for the trajectory, it will satisfy The roughness profile is related to the self-intersection points of the trajectory as shown in which means that the t1and Δt has a linear relationship. cutting speed (rv) cannot be too large. There exists a critical value that decides the roughness profile. The critical ratio rvcritical lies between 0.2 and 0.3 for this specific simulation case. At this critical value, there exists just one intersection point between the two sinusoidal curves, namelysin(  Δt/2) and (-1) k r/(R+r)sin(ωΔt/2) (k is even). When the ratio is larger than this critical value, there will be no self-intersection points on the right hand side of the trajectory. At this condition, there will be part of workpiece surface remain to be uncut (as shown in the third row of Fig. 26), which make the surface is very rough. In the real machining, it is impossible to choose a cutting condition in which the ratio is larger than critical ratio rvcritical. Henceforth, this paper focus on the condition of rv< rvcritical. Under the condition of rv< rvcritical, the desirable point that related to the roughness profile is found to be the first intersection point between the two sinusoidal curves (namely, -sin(  Δt/2) and (-1) k r/(R+r)sin(ωΔt/2), k is even)) as shown in the Fig. 27. This intersection can be easily computed numerically. The corresponding t1 can also be obtained according to Eq. (28). The combination of Δt and t1 (Δt, t1) that related to the roughness profile is shown in the last column of Fig. 26. As long as the desirable t1 is obtained, the intersection point of the trajectory can be obtained and thus the arcs between two adjacent self-intersection points. The roughness profile can be finally constructed. Fig. 28 shows the machined roughness profile when r=4 mm, n=1000 rpm, vf /vc =0.05. The influence of the ratio between feed speed and cutting speed rv on the roughness profile when milling along a round path can be analyzed according to the above descripted method. The roughness profile of when rv has different values is shown in Fig. 29. From the figure, it can be seen that the profile becomes rough when the rv increases. It is similar to the case when milling along a straight path. Fig. 29 The influence of ratio between the feed speed and cutting speed on roughness profile.
In Eq.(29), r/(r+R) can be rewritten as 1/(1+1/ r/R), which means that the roughness profile is depend not only upon the speed ratio of rv but also the ratio between the cutter radius and workpiece radius, namely rr=r/R. The effect of radius on the roughness profile can also be numerically investigated according to the previous described method. Fig. 30 shows the roughness profile when rr has different values. From the figure, it can be found that the profile becomes rough when the rv increases, which means to obtain smoother surface when milling round workpiece cutter has smaller radius should be recommended. Fig. 30 The influence of ratio between the cutter radius and workpiece on roughness profile.

Milling along a general curve path
Workpieces with curved surface are also widely used in the engineering practice. The cutter will move along a curve path when milling the workpiece with curved surface (As shown in Fig. 31 (a)). In this part, the roughness profile of the workpiece with curved surface after milling is constructed. For simplicity, the cutter is assumed to be with a single and straight tooth. The moving path of the cutter center is assumed to be (x(u), y(u)).Two coordinate systems (x-O-y and x1-O1-y1) as shown in the Fig. 31 (b) are constructed.

The trajectory of a point on the cutting edge
According to Fig. 31 (b) from which one can solve the relation between t and u. Usually, it is impossible to get the analytical solution of t respect to u, which means it is also impossible to get analytical expressions of x(u) and y(u).

Solve the self-intersection points and construct roughness profile
To construct the roughness, the self-intersection points of the trajectory needs to be solved. Then the self-intersection points that related to the roughness profile needs to be extracted. Assuming that the coordinate of a self-intersection point is (x1, y1), the time instant that the intersection occurs is t1. After a time period of Δt, the point will go back to itself, which means that The relationship between the t1 and Δt can be obtained from Eq.(36). It is really difficult to give an analytical expression between t1 and Δt because of the existence of a transcendental function and the unknown analytical expressions of x(u) and y(u). All one can do is using numerical method.
Instead of trying to solve Eq.(36) analytically or numerically, this paper proposes an approximation method to obtain the self-intersection points that related to the roughness profile.
The method is based on the simple case that discussed in the Section 2.
The curve path is divided into many small parts. Each small curve path has two end points. A line segment can be determined from these two end points. Thus, the curve milling path is now approximated by a bunch of line segments. As such, the self-intersection points of the trajectory obtained when milling along a curve path can be approximated by the ones when milling an approximated cutting path, which is composed by a bunch of small line segments. The self-intersection points of the trajectory when milling along a small line segment can be easily obtained by referring the simple case discussed in Section 2. Once the self-intersection points are obtained, the roughness profile can be constructed. The key for this method is to determine the length of the small line segment. It should large enough to let the trajectory when the cutter moves along the line segment to have self-intersection points, but it should be kept to be small to guarantee the approximation accuracy of the curve path. This paper recommends that the line segment has a length that the cutter moves in two periods of the cutter rotation. Fig. 32 The relationship between the u and t. To construct the roughness profile, the self-intersection points that are related to the roughness profile are solved according to the proposed method. The results are shown in Fig. 34.
The trochoidal arcs between two adjacent self-intersection points as shown in Fig. 34 can also be obtained according to the trajectory Eq.(32), the results are shown in Fig. 35 (a) and (b). The roughness profile is finally constructed as shown in Fig. 35 (c) and (d).

Experimental demonstration
The verification of the modeled roughness profile is conducted according to the following procedures: (i) several cutting experiments with different cutting parameters (as shown in Table 1) are carried out; (ii) the roughness profiles of the machined surface are tested; (iii) roughness profile under same cutting conditions used in the cutting experiment is simulated according to the procedures presented in the previous sections; (iv) compare the tested roughness profiles with the simulated ones.
A new end mill with radius equals to 3 mm is used in cutting experiments. The tool has four teeth and a helical of 45°. The material for the cutter is carbide. The workpiece used in the experiment is Aluminum blocks. After cutting, the roughness profile is tested under a telescope (Nano View). Meanwhile, the profile is simulated according to the procedures presented in the Section 4.1. And the simulated roughness profile and the tested roughness profile are compared.
The comparison results are shown in Fig. 36. For all these tests, the predicted profile matches well with the tested one, which validates the rightness of the profile modeling method presented in this paper.  respectively, which is much longer than the method developed in this paper. The roughness profiles obtained under these three different distances are listed Fig. 37 and compared to the one obtained by the method developed in this paper. It can be seen that the accuracy of the roughness profile obtained for the method developed by Arizemendi et al. [2,1] is related to the vertical distance. To obtaine accurate roughness profile, small distance is needed and thus more time will be consumed to construct the profile. vertical distance equals to 0.05 mm; (c) vertical distance equals to 0.01 mm.

Conclusions
In this paper, the roughness profile generated by flank milling process is theoretically investigated.
For the first time, the roughness profile is modeled and analyzed analytically. This is the main difference between this investigation and the existed ones. Based on the constructed the roughness profile model, the effect of the cutting parameters, the cutter geometry parameters on the roughness profile is discussed. Besides, the roughness profile of the machined surface obtained when the cutting tool feeds along curve paths are also constructed. At last, experiments are conducted to verify the developed method. The main conclusions can be summarized as following:  Two parameters, namely w and h can be defined to descript the roughness profile obtained when the cutter moves along a straight path, in which w means the width of the elemental profile and h means the height of the roughness profile.
 The height of the roughness profile h is related to the ratio between feed speed and the cutting speed. h will increase with the speed ratio. This is true when h is smaller than the radial depth of cut. When the h is larger than the radial depth of cut, the shape of roughness profile will be decided by the ratio as well as the radial depth of cut.
 Both w and h will decrease as the increasing of tooth number for a milling cutter with equal tooth spacing angle.
 The roughness profile of the milling cutter with variable pitch is also simulated. The height of the roughness profile h becomes larger compared to the cutter with equal pitch.
 The helical angle has no effect on the profile parameters w and h. But it will affect the surface topography. The peaks and valley of the surface topography lie on a bunch of slope lines.
 The roughness profiles when the cutter feeds along a round path and a curve path are also simulated. It is impossible to analytically solve the self-intersection points that are used to construct the roughness profile. Thus, a numerical method is proposed. And the roughness profile is successfully constructed.
 The roughness profile measured from the machined surface matches well with the simulated one, which demonstrates the correctness of the roughness profile construction method developed in this paper.
 The method to construct the surface roughness developed in this paper is compared to the one developed by Arizemendi et al. The method in this paper consumes much less time to construct the surface roughness.
In the future investigation, more attention will be focused on the effect of other factors such as the tool vibration, tool wear, runout etc. on the roughness profile and surface topography based on the method proposed in this paper.