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Abstract

When a soft polishing tool is compressed on a stiff workpiece of curved surface, the contact area is a piece of the curved surface. In the process of aspheric surface polishing, the machining speed is always provided by the rotational speed of the spindle of a computer numerical control lathe. Yet, the polishing tool is usually made rotating to remove cutting scraps and broken abrasives from the contact area. The rotational speed of the polishing tool would change the distribution state of the relative speed in the whole contact area, and it would definitely change the volume of material removal in the surface of the workpiece. This article studies how the rotational speed of the polishing tool changes the distribution state of the relative speed in the whole contact area and material removal in the volume of the surface of the workpiece. The computing results show that the volume of material removal increases with the increase in both the rotational speed of polishing disk and the rotational speed of lathe spindle, but the polishing quality is totally not in this case.

Keywords

Compliant polishing aspheric surfaces relative speed material removal

Introduction

Aspheric optics are always of extremely high shape precision and low surface roughness, which are widely used in the military field, the aeronautics and astronautics field, and some high-tech fields. These workpieces usually have to be polished to reach ultra-smooth surfaces at nano-scale.^1–3 After they have been shaped by advanced computer numerical control (CNC) lathes, aspheric optics still need to be polished to reduce surface quality and improve shape precision.^4,5 Based on the geometric characteristics, the CNC equipments on aspheric surface compliant polishing still have all the same essential functions as ordinary CNC lathes.^6–8 By the existing theories on curved surfaces compliant polishing, the polishing process is controlled by adjusting the product of normal contact force, relative speed, and machining time.^7–9 When a workpiece of aspheric surface is machined with ordinary CNC lathes, the cutting speed, that is, the above relative speed, is the same as the rational speed of the spindle of the machine tool. Theoretically, polishing tools, including polishing wheels and polishing disks, do not have to rotate, and the workpiece could definitely be shaped and polished. But in reality, polishing tools are always made to rotate to keep a good condition for abrasive.^10–12 If the polishing tool is made static, this tool will always have the same area keeping contact with the workpiece, and thus, the contact area would be filled with cutting scraps, broken abrasives, and polishing paste in a few minutes.^9–11 In such case, the workpiece cannot be polished any more.

Based on kinematics, when the polishing tools obtain a rotational speed, the relative speed would be changed because the relative speed is the resultant speed of the rotational speed of workpiece and the rotational speed of polishing tool.^13,14 If the polishing tool rotates faster, it would play the dominant role especially in the center of workpiece where the rotating radius is small. But this is totally not the case on the edge of the workpiece. Moreover, if the workpiece is a large one in size, a large polishing disk would be needed, as large workpieces always obtain more unneeded materials needed to be polished, and hence, large polishing disks could remove material much more efficiently. By the way, large polishing disks could effectively eliminate surface chatter marks caused by the high-frequency vibrations of the tool system, so they could significantly improve the surface’s quality and have been widely used in large aspheric optics polishing. But when a large aspheric optic of large curvature is polished by a large soft polishing disk, how the rotational speed of the disk changes the polishing process has achieved worldwide admiration and interest down to present time. Changed relative speed would result in changed material removal.^10–12 It is generally believed that it is the convolution of the rotational speed of polishing disk and the rotational speed of the spindle of lathe that dominates the curved surface polishing process. But for the large surface of large curvature, the resultant speed would have a component in the normal direction at the cutting point and no longer lies in the tangent plane of working point, so the traditional convolution algorithm could not describe the relationship between the rotational speed of polishing disk and the rotational speed of the spindle of lathe. No research work has been done on how the machining speed has been changed for the large aspheric surface with large curvature yet; therefore, we shall study how the material removal would be changed for the same. In this article, the kinematics model is established based on the deformation analysis of soft polishing disk in the vector space, the relationship between the material removal process and the relative rotational speed is built, and how the removal process is affected by the rotational speed of polishing disk are studied. The computing results show that the volume of material removal increases with the increase in both the rotational speed of polishing disk and the rotational speed of lathe spindle, but the polishing quality is totally not in this case.

Kinematics model

When an aspheric part is polished with a CNC lathe, the machine tool has its accredited coordinate system. For the sake of simplification, define a coordinate system of $xoy$ to take place of that of a CNC lathe, and set the right direction of y-axis as the direction of the rotational speed of spindle. If it is compressed by a normal force $F_{n}$ , a soft polishing disk is kept in touch with the workpiece at a small area. Suppose the point of $q_{0}$ is the cutting point in the workpiece’s surface and define $ϖ_{0}$ as the rotational speed of the spindle of the lathe and $ϖ_{1}$ as the rotational speed of the polishing disk. The kinematics model of the process of aspheric surface compliant polishing could be defined by Figure 1(a).

Figure 1.

Kinematics model of aspheric surface polishing with soft pads: (a) 2D model and (b) 3D model.

Define $Curve 0$ as the intersection curve between the machined aspheric surface and the plane of $xoy$ and $curve 1$ as the intersection curve between the machined surface and the polishing disk. $Curve 1$ is also the boundary line of the contact area between the machined surface and the polishing tool. In the contact area, there is a random point of $q_{1}$ . In the contact area of the polishing disk, $q_{1}$ is a point in the circle of $Curve 2$ . And in the machined aspheric surface, $q_{1}$ is a point in the circle of $curve 3$ . In the circle of $curve 2$ , define $r$ as its rotational radius and $v_{1}$ as its line speed. Similarly, in the circle of $curve 2$ , define $R$ as its rotational radius and $v_{0}$ as its line speed. Then, the kinematics model of the process of aspheric surfaces compliant polishing could be described by Figure 1(b) in detail.

Define $v$ as the real relative speed of $q_{1}$ , which is used to machine the workpiece, then¹⁵

{\begin{matrix} {\vec{v}}_{0} = \vec{R} \times {\vec{ϖ}}_{0} \\ {\vec{v}}_{1} = \vec{r} \times {\vec{ϖ}}_{1} \\ \overset{⇀}{v} = {\vec{v}}_{0} - {\vec{v}}_{1} = \vec{R} \times {\vec{ϖ}}_{0} - \vec{r} \times {\vec{ϖ}}_{1} \end{matrix}

(1)

Computing method

In equation (1), all variables are vectors which should be described by both values and directions. All the values and directions are determined by the location of the cutting point $q_{0}$ and the location of the studied point $q_{1}$ . Suppose $q_{0}$ locates at $q_{0} (x_{0}, y_{0}, z_{0})$ and $q_{1}$ locates at $q_{1} (x_{1}, y_{1}, z_{1})$ . In Figure 1, $z_{0} = 0$ .

Computing method of $\vec{R}$

In Figure 1(b), $curve 3$ is a circle in the surface of the aspheric part which results from the studied point $q_{1}$ rotating around the y-axis. Suppose the center of this circle is the point $O_{R}$ , which is a point on the y-axis. $O_{R}$ , the origin of coordinate $O$ , and the observing point $q_{1}$ make up a right triangle. In the $RT Δ O_{R} {Oq}_{1}$ , then¹⁵

{\begin{matrix} \vec{R} = {Oq}_{1}^{\to} - \vec{{OO}_{R}} \\ {Oq}_{1}^{\to} = x_{1} \cdot \vec{i} + y_{1} \cdot \vec{j} + z_{1} \cdot \vec{k} \\ \vec{{OO}_{R}} = y_{1} \cdot \vec{j} \end{matrix}

(2)

\vec{R} = x_{1} \cdot \vec{i} + z_{1} \cdot \vec{k}

(3)

In the above two equations, $x_{1}$ , $y_{1}$ , and $z_{1}$ are variables of $q_{1} (x_{1}, y_{1}, z_{1})$ , and $\vec{i}$ , $\vec{j}$ , and $\vec{k}$ are the unit vectors in the x-, y-, and z-axes, respectively.

Computing method of $\vec{R}$

In Figure 1(b), $curve 2$ is a circle in the surface of the polishing disk which results from the studied point $q_{1}$ rotating around the u-axis. Suppose the center of this circle is the point $O_{r}$ , which is the projected point $q_{1}$ to u-axis, then

\vec{R} = \vec{q_{1} O_{r}}

(4)

The right direction of u-axis is exactly the right directions of the grad at the cutting point $q_{0}$ of the machined surface. Suppose the direction components of the grad at the cutting point $q_{0}$ in the x-, y-, and z-axes are $p_{u}$ , $q_{u}$ , and $r_{u}$ , respectively. Then, u-axis could be defined by¹⁵

\frac{x - x_{0}}{p_{u}} = \frac{y - y_{0}}{q_{u}} = \frac{z - z_{0}}{r_{u}}

(5)

And the projected vector of u-axis could be defined by¹⁵

\vec{r} = \frac{1}{\sqrt{p_{u}^{2} + q_{u}^{2} + r_{u}^{2}}} | \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ p_{u} & q_{u} & r_{u} \\ x_{1} - x_{0} & y_{1} - y_{0} & z_{1} - z_{0} \end{matrix} |

(6)

where $x_{0}$ , $y_{0}$ , and $z_{0}$ are variables of $q_{0} (x_{0}, y_{0}, z_{0})$ ; $x_{1}$ , $y_{1}$ , and $z_{1}$ are variables of $q_{1} (x_{1}, y_{1}, z_{1})$ ; and $\vec{i}$ , $\vec{j}$ , and $\vec{k}$ are the unit vectors in the x-, y-, and z-axes, respectively.

Suppose the machined aspheric surface is an ellipse, which could be defined by

F (x, y, z) = \frac{x^{2} + z^{2}}{a^{2}} + \frac{{(y + b)}^{2}}{b^{2}} - 1 = 0

(7)

where $a$ and $b$ are the short and the long axes of the ellipse, respectively. Then¹⁵

{\begin{matrix} p_{u} = \frac{\partial F (x, y, z)}{\partial x} \\ q_{u} = \frac{\partial F (x, y, z)}{\partial y} \\ r_{u} = \frac{\partial F (x, y, z)}{\partial z} \end{matrix} | (\begin{matrix} x = x_{0} \\ y = y_{0} \\ z = z_{0} \end{matrix})

(8)

Computing method of ${\vec{ϖ}}_{0}$ and ${\vec{ϖ}}_{1}$

Let $∥ \vec{X} ∥$ denote the value of $\vec{X}$ . $∥ {\vec{ϖ}}_{0} ∥$ and $∥ {\vec{ϖ}}_{1} ∥$ are always monitored online, which could be regarded as well known. But we should still get to know their directions. According to Figure 1(b), ${\vec{ϖ}}_{0}$ and ${\vec{ϖ}}_{1}$ could be defined as

{\vec{ϖ}}_{0} = ‖ {\vec{ϖ}}_{0} ‖ \cdot \vec{j}

(9)

{\vec{ϖ}}_{1} = ‖ {\vec{ϖ}}_{1} ‖ \cdot \frac{\vec{u}}{‖ \vec{u} ‖}

(10)

According to (5)

\frac{\vec{u}}{‖ \vec{u} ‖} = \frac{p_{u} \cdot \vec{i} + q_{u} \cdot \vec{j} + r_{u} \cdot \vec{k}}{\sqrt{p_{u}^{2} + q_{u}^{2} + r_{u}^{2}}}

(11)

Relative speed computing

For the sake of flexibility and efficiency, the polishing tool would be shaped as a disk, and it is usually very soft and thick, so that it would contact with the workpiece with its whole area when it is compressed by an average normal force $F_{n}$ . Suppose the radius of the disk is $r_{0}$ , then the $curve 1$ in Figure 1 is a circle with the radius of $r_{0}$ . Define $λ = ϖ_{1} / ϖ_{0}$ , and using equations (1), (9), and (10), the relative speed could be defined as

\begin{matrix} v & = ‖ \vec{R} \times \vec{j} \times ‖ {\vec{ϖ}}_{0} ‖ - \vec{r} \times ‖ {\vec{ϖ}}_{1} ‖ \times \frac{\vec{u}}{‖ \vec{u} ‖} ‖ \\ = ‖ \vec{R} \times \vec{j} - λ \cdot \vec{r} \times \frac{\vec{u}}{‖ \vec{u} ‖} ‖ \cdot ϖ_{0} \end{matrix}

(12)

When $λ > 0$ , the angle between ${\vec{ϖ}}_{0}$ and ${\vec{ϖ}}_{1}$ is smaller than $90^{\circ}$ , and when $λ < 0$ , the angle between ${\vec{ϖ}}_{0}$ and ${\vec{ϖ}}_{1}$ is greater than $90^{\circ}$ . $λ = 0$ means ${\vec{ϖ}}_{1} = 0$ . In the case of $a = 250 mm$ and $b = 40 mm$ in equation (7) and $r_{0} = 30 mm$ , the curve of $v = ∥ \vec{R} \times {\vec{ϖ}}_{0} ∥$ on the polishing disk could be calculated and shown in Figure 2.

Figure 2.

Curves of $v = ∥ \vec{R} \times {\vec{ϖ}}_{0} ∥$ at different $λ = ϖ_{1} / ϖ_{0}$ : (a) $x_{0} = 125 mm$ and (b) $x_{0} = 220 mm$ .

Influence of ${\vec{ϖ}}_{1}$ on material removal

Based on the most accepted Preston polishing theory and other studies,⁹ the volume of material removal at a single point is just in direct proportion to the product of the contact stress in the normal direction, the machining time, and the relative speed between the polishing tool and the workpiece at the machining point. Define $Vol (x, y, z)$ as the volume of material removal at the point of $(x, y, z)$ , then it could be described by⁹

Vol (x, y, z) = \int_{0}^{Δ t} k \cdot p (x, y, z, t) \cdot v (x, y, z, t) dt

(13)

where $p (x, y, z, t)$ is the contact stress in the normal direction at the point of $(x, y, z)$ , $v (x, y, z, t)$ is the relative speed at the point of $(x, y, z)$ , $k$ is the material removal coefficient determined by cutting conditions, and $Δ t$ is the real machining time.

In the process of compliant polishing, the speed of feed movement is always much more lower than the cutting speed. In this way, any single point in the machined surface would be polished again and again. When the aspheric surface rotates around the y-axis, $q_{1}$ becomes all points in the circle of $curve 3$ in the plane of $y = y_{0}$ . With the polishing tool moving under the feed movement and rotating around its own u-axis, the cutting point $q_{0}$ would be polished by the whole contact area. Define $h (x, y, z)$ as the depth of material removal at the point of $(x, y, z)$ , that is, the volume of material removal per single-unit area, then it could be described by

h (x, y_{0}, z) = \frac{k}{2 π \cdot R} \int \int_{A} p (x_{1}, y_{1}, z_{1}) \cdot v (x_{1}, y_{1}, z_{1}) dA

(14)

where $dA$ is a small strip in the contact area at a certain $R$ . Take $S$ for the whole contact area which is a piece of curved surface surrounded by the $curve 1$ in Figure 1(b). Figure 3(b) shows the relationship between $S$ and $A$ .

Figure 3.

Contact area in the aspheric surface polishing with a lathe: (a) model of contact condition and (b) enlarged contact area.

When the polishing tool is a ballonet wheel, the value of the contact stress is almost the same as everywhere in the whole contact area. Define $p_{0}$ as the contact stress, and by equation (12), then

\begin{matrix} h (x, y_{0}, z) = \frac{k \cdot p_{0}}{2 π \cdot R_{0}} \\ \int \int_{S} ‖ \vec{j} \times \vec{R} - λ \cdot \frac{\vec{u}}{‖ u ‖} \times \vec{r} ‖ \cdot ϖ_{0} (x_{1}, y_{1}, z_{1}) dS \end{matrix}

(15)

The rotational speed of the spindle of a lathe is always too high to be changed abruptly online. As a matter of fact, it is usually set as a constant. So

h (x, y_{0}, z) = \frac{k \cdot p_{0} \cdot ϖ_{0}}{2 π \cdot R_{0}} \int \int_{S} ‖ \vec{j} \times \vec{R} - λ \cdot \frac{\vec{u}}{‖ u ‖} \times \vec{r} ‖ dS

(16)

In a real system, $ϖ_{1} < ϖ_{0}$ , that is, $- 1 < λ < 1$ . Under the conditions of $p (x, y, z, t)$ , $ϖ_{0} (x, y, z, t)$ are set constant, and so does the speed of feed movement. When an aspheric surface of equation (17) is polished with a soft polishing disk of $r_{0} = 12 mm$ , the computed depth of material removal $h (x, y, z)$ could be computed and shown in Figure 4

\frac{x^{2} + z^{2}}{75^{2}} + \frac{{(y + 30)}^{2}}{30^{2}} = 1

(17)

Figure 4.

Computed depth of material removal with different $λ = ϖ_{1} / ϖ_{0}$ .

Analysis and discussion

According to Figure 4, in the usual cases of $∥ {\vec{ϖ}}_{1} ∥ < ∥ {\vec{ϖ}}_{0} ∥$ , the rotational speed of the polishing tool has some influence on the process of material removal:

When $λ = 0$ , $ϖ_{1} = 0$ . The curve of the depth of material removal, which locates at the lowest position, is almost a smooth line while the polishing tool moves from the center of the workpiece to the edge. It means that it removes few materials from the surface of the workpiece but more smooth;

When $λ = - 1$ , $ϖ_{1} = - ϖ_{0}$ . The curve of the depth of material removal, the most fluctuant red curve, changes acutely while the polishing tool moves from the center of the workpiece to the edge. It means that material in the workpiece’s surface would be removed more at some places and less at other places. It would result in higher surface roughness and not be used in a real system;

When $λ \neq 0$ and $λ \neq - 1$ , it depends. The curves of the depth of material removal locate above that of $λ = 0$ . They come together and fluctuate seasonally. Material would be removed quickly but not so smooth as $λ = 0$ . While $∥ λ ∥$ changes from 1 to 0, the curves become more and more smooth, but the value of the depth of material removal has not been changed so much;

For the sake of chip removal, $∥ λ ∥ \to 0$ . It is to say, the rotational speed of the polishing tool should be small, whether it rotates in the same direction or in the reserve.

Polishing experiments

On an ordinary CNC lathe of two-axis interpolation theory, this article has developed a compliant polishing system for aspheric surface precision machining under the hybrid motion or force control policy,^16–19 in which a torque-servo device is used to control the normal force, as shown in Figure 5. The system includes three subsystems.^17,18 The first one is the machining movement controlling subsystem, which is composed of 1 and 2. The machining movement is driven by the rotational motion of the spindle of the CNC lathe. The second one is the feed movement controlling subsystem, made up of 4 and 5. The polishing tool holder could rotate around 9, while 5 provides the feed movement in two orthogonal directions. By moving 5 and turning 4, we could keep the normal force just in the normal direction. The third one is the normal force controlling subsystem, composed of 6, 7, 8, and 9. A stepper motor, the device 6, transmits the required rotational displacement on 7, and the output torque of 7 would then be transmitted to 9 through 8. A torque sensor installed on 7 would detect the output torque. When the output torque changes from its required value, the input current would be adjusted to drive the stepper motor change its output displacement automatically to make the normal force at its expected value.

Figure 5.

Aspheric surface polishing system.

Polishing experiments are performed as shown in Table 1 with the parameters shown in Table 2. The surface roughness is scanned and shown in Table 3, and the photograph of the aspheric workpiece is shown in Figure 6. After being polished 22 times, the surface roughness of the aspheric surface of equation (17) has been changed from $R_{a} = 0.3 μ m$ to $R_{a} = 0.06 μ m$ uniformly.

Table 1.

Polishing procedures.

	Step 1				Step 2
	Metallographic sandpaper				Polishing paste
Size of abrasive	400	1000	2000	3000	W0.5
Polishing times	3	3	3	3	10

Table 2.

Parameters of experiments.

Rotational speed of spindle	600 rad/min
Rotational speed of polishing disk	60 rad/min
Feed speed	0.06 mm/rad
Surface roughness before polished	R_a 0.3 μm
Material of abrasive	Diamond powder
Material of workpiece	Steel 45

Table 3.

Surface roughness measured.

Position	Cutoff of scanning	Length	Surface roughness
Position	Cutoff of scanning	Length	R_a (μm)	R_z (μm)	R_y (μm)
Center	0.25 mm	5 × cutoff	0.062	0.438	0.555
Middle	0.25 mm	5 × cutoff	0.059	0.439	0.556
Edge	0.25 mm	5 × cutoff	0.055	0.376	0.524

Figure 6.

Photograph of an aspheric workpiece after being polished.

Conclusion

In the process of aspheric workpiece polishing, the polishing tool is usually made rotating to remove cutting scraps and broken abrasives from the contact area. Even if the rotational speed of the spindle of a CNC lathe provides the machining movement for aspheric surfaces polishing, the rotational speed of the polishing tool would increase the efficiency in material removing. The higher the speed of the polishing tool, the greater the material removal. But higher rotational speed of the polishing tool would result in nonuniform material removal. In general, the rotational speed of the polishing tool is small, and it does not have great influence on the volume of material removed from the workpiece, but it could have significant influence on surface roughness.

In detail, the rotational speed of the polishing tool has almost the same influence on the volume of material removed whether it rotates in the same direction or in the reserve. But when $λ = - 1$ and $ϖ_{1} = - ϖ_{0}$ , the depth of material removal changes acutely while the polishing tool moves from the center of the workpiece to the edge. It means bad surface roughness, and it could not be used in reality.

Footnotes

Appendix 1 Acknowledgements

The authors extend our thanks to all the graduate students in our team for their great jobs: Jianbo Zhang, Bida Lv, Sihai Yu, Mingqin Zhang, and Xiangen Ying.

Academic Editor: Ramiro Martins

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

All the research works are supported by the Natural Science Foundation of Zhejiang Province of China under granted nos LY12E05007 and LY12E05014, the Industrial Major (Key)-Commissioned Research of Ningbo City of China under granted no. 2012B10057, and K.C. Wong Magna Fund in Ningbo University.

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Study on the influence of the rotational speed of polishing disk on material removal in aspheric surface compliant polishing

Abstract

Keywords

Introduction

Kinematics model

Computing method

Computing method of R →

Computing method of R →

Computing method of ϖ → 0 and ϖ → 1

Relative speed computing

Influence of ϖ → 1 on material removal

Analysis and discussion

Polishing experiments

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of conflicting interests

Funding

References

Computing method of $\vec{R}$

Computing method of $\vec{R}$

Computing method of ${\vec{ϖ}}_{0}$ and ${\vec{ϖ}}_{1}$

Influence of ${\vec{ϖ}}_{1}$ on material removal