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Modeling surface roughness in the stone polishing process
International Journal of Machine Tools & Manufacture 45 (2005) 365–372 www.elsevier.com/locate/ijmactool
Modeling surface roughness in the stone polishing process Fengfeng Xia,*, David Zhoub a
Department of Mechanical, Aerospace and Industrial Engineering, Ryerson University Toronto, Ont. M5B 2K3, Canada b The University of Western Ontario London, Ont., Canada Received 31 May 2004; accepted 10 September 2004 Available online 2 November 2004
Abstract In this paper, a new method for modeling and predicting the surface roughness of the workpiece in the stone polishing process is developed. This method is based on the random distribution of the stone grain protrusion heights and the force balance by contact grains. To do so, first, the topography of a polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from a given stone grit number. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for a given workpiece hardness (Brinell number). Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the force applied on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, a method is presented for predicting the surface roughness based on the micro depth of cut and contact grains. A good agreement of the prediction results with the experimental data proves the effectiveness of the proposed method. q 2004 Elsevier Ltd. All rights reserved. Keywords: Polishing; Modelling; Surface roughness
1. Introduction Generally speaking, manufacturing processes can be loosely classified into primary manufacturing processes and secondary manufacturing processes. The primary manufacturing processes are mainly concerned about producing the products to satisfy the required dimension and shape accuracy. The secondary manufacturing processes are mainly concerned about finishing the products with the required surface integrity and surface finish. Polishing process is one of the main secondary manufacturing processes, and its purpose is to reduce the surface roughness to a desired amount. Industrial applications of the polishing process span from large products such as automobile, aerospace, to small parts such as optics [1–3]. There are a number of polishing methods used in industry which may be classified into mechanical and non mechanical. Mechanical polishing uses abrasives to smooth the workpiece surface. Non-mechanical polishing uses other means, such as laser, magnetic fluid [4,5]. In most * Corresponding author. Tel.: C1 416 979 5000; fax: C1 416 979 5265. E-mail address: [email protected] (F. Xi). 0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.09.016
applications, however, polishing is done through mechanical means simply because of the low cost. Mechanical polishing can be further classified into bound abrasives and un-bound abrasives. The bound abrasives are the polishing stones for stone polishing. The un-bound abrasives are the abrasive compounds, such diamond compounds, for buffing. In practice, mechanical polishing is still done manually, especially in the die and mold industry and aerospace industry. Nevertheless, research has been carried out to develop automated polishing systems in an attempt to improving the productivity of the current polishing process. The automated polishing systems archived in the literature [6–10] can be loosely classified based on whether a conventional machine tool structure (e.g. computer-numerical control—CNC) or an articulated robot arm (i.e. industrial robots) is being used [11]. While in the manual polishing process the polisher examines the surface finish by comparing to a template, computer simulation and prediction of the surface roughens would be essential for automated polishing systems. This motivates the work reported here. Due to the random distribution of the grains in abrasive tools, polishing research has been mainly experimental. Recently, by applying the stochastic theory the authors of the present paper developed a method for predicting
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the surface roughness for the grinding process [12]. This model is further developed here to model and simulate the surface roughness for the stone polishing process. To do so, first, the topography of a selected polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from a given stone grit number. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for given workpiece hardness (Brinell number). Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the force applied on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, a method is presented for the prediction of the surface roughness based on the micro depth of cut and the contact grains. A good agreement of the prediction results with the experimental data proves the effectiveness of the proposed method.
deviation for generating the topography of a polishing stone is given as m Z dg avg
(3a)
and s Z ðdg max K dg avg Þ=3
(3b)
where m and s are the mean value and standard deviation, respectively. From Eqs. (3a) and (3b), it can be calculated that mZ21.1, 15.5 and 8.8 mm, respectively, and sZ8.8, 7.5 and 5.5 mm, respectively, for MZ320, 400, and 600. To generate the topology, the polishing stone is meshed by a grain interval. The grain interval is the distance between two adjacent grains and it is related to the structure number S of the polishing stone. The structure number indicates the volumetric concentration of abrasive grains in the stone. For a given number of S, the density of the stone, Vg (%), may be approximated as [10]
2. Generation of polishing stone topography
Vg Z 2ð32 K SÞ
The first step in modeling and predicting the surface roughness for the stone polishing process is to generating the topography for a polishing stone. The surface roughness of the polished workpiece could be considered as an imprint created by the polishing stone rubbed against the workpiece. Previous experimental research [13] has revealed that the distribution of the grain protrusion heights of an abrasive tool is of a Gaussian distribution. In the paper [8], the present authors developed a method to generate the topography of a grinding wheel. Since polishing stones are made with finer grains in a similar way to grinding wheels, this method is applied here. To generate a random distribution for the grain protrusion heights of a polishing stone, the mean value and standard deviation are required. The grid numbers (M) of the commonly used polishing stones are MZ320 for the first stage stoning, MZ 400 for the second stage stoning and MZ600 for the final stage stoning. The grain dimension dg is expressed equal to the aperture opening of the screen according to the following relationship [9]
In general the distribution of the grain intervals is assumed uniform [12,13]. The interval D can be derived from the following relation p 3 d Z Vg D 3 (5) 6 g avg
dgmax ðmmÞ Z 15:2MK1
The left-hand side of Eq. (5) indicates the volume of the grains in the polishing stone in terms of the average grain size, and the right-hand side in terms of the density. The two sides should be equal. Therefore, from Eq. (5), the grain interval D (mm) can be determined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 D Z 137:9 MK1:4 (6) 32 K S To generate a polishing stone topography, the stone is meshed using the interval determined by Eq. (6). The distribution of the grain protrusion heights is described by
(1)
This grain size is considered as the maximum grain size, denoted by dg max, for a given grit number M. Furthermore, according to [9], the average grain dimension denoted by dg avg is related to grit number M (for MO40) as dg avg ðmmÞ Z 68 MK1:4
(2)
From Eqs. (1) and (2), it can be calculated that for MZ320, 400, and 600, dg maxZ47.5, 38 and 25.3 mm, respectively, and dg avgZ21.1, 15.5 and 8.8 mm, respectively. As reported in [14,15], the extensive experiment using 3D stylus instruments and SEM micrographs has shown that dg max and dg avg are very close to the maximum grain protrusion height hmax and the mean grain protrusion height havg, respectively. Therefore, the mean value and standard
(4)
Fig. 1. Topography of a polishing stone.
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a Gaussian distribution with the mean value and standard deviation determined by Eqs. (3a) and (3b). Fig. 1 is a simulated polishing stone topography with grit number MZ 400, showing that the surface of a polishing tool is made of randomly distributed protrusion heights.
3. Micro depth of cut of a single grain Prior to studying the surface roughness of the polished workpiece, the problem of determining the micro depth of cut of a single grain is addressed. Fig. 2 shows the three shapes archived in the literature [20] for modeling the abrasive grains, namely, semi sphere shape, cone shape, and prism shape. The semi sphere shape is adopted in this paper, because this shape is the most commonly used for modeling the grains. In the polishing process, the polishing stone is pressed under certain pressure against the workpiece. Though the pressure is applied to the entire polishing stone, it is only concentrated on the grains that are in contact with the workpiece. As shown in Fig. 3, the contact grains are more protruded than the grains that are not in contact with the workpiece. Due to the highly concentrated pressure on the contact grains, plastic deformation occurs in the contact area, thereby creating the micro depth of cut. Therefore, the polishing cutting mechanism may be modeled based on the plasticity theory, similar to hardness testing. The most common form of hardness tests for metals involves standard indenters being pressed into the surface of the material. Measurements associated with the indentation are then taken as a measure of the hardness of the surface. The Brinell test, the Vickers test and the Rockwell test are the main forms of such tests. In this paper, the Brinell test model is used to determine the micro depth of cut of a single grain. In other words, the grain is modeled as an indenter inserted on the workpiece. It is assumed that the grain is much harder than the workpiece. For example, the common polishing grain material is aluminum oxide with HB over 1000, while steel is with HB around 300. In the Brinell test model, a hardened ball is pressed into the surface of the material by a standard force. In our polishing model, the hardened ball is replaced by the semisphere polishing grain. After the load and the ball have been removed, the diameter of the indentation is measured.
Fig. 2. Common grain shapes.
Fig. 3. Contact model of polishing grains.
The Brinell hardness number (HB) is obtained by dividing the size of the applied force f by the spherical surface area of the indentation A HB Z
f ðkgf Þ Aðmm2 Þ
(7)
where f is the force applied on the grain, and its determination will be discussed later. As shown in Fig. 4, the contact area can be obtained by calculation from the values of the radius rg of the ball and the radius rc of the indentation, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Z 2prg ðrg K rg2 K rc2 Þ (8) Note that for the grain model, rgZdg/2, where as mentioned before, dg is the grain diameter. From Fig. 4 the following relation holds among rg, rc, and h rg2 Z rc2 C ðrg2 K hÞ2
(9)
From Eq. (9), the micro depth of cut can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10a) h Z rg2 K rg2 K rc2 Substituting Eqs. (7) and (8) into Eq. (10a) yields hZ
f 2prg HB
Fig. 4. Micro depth of cut of a polishing grain.
(10b)
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Using Eq. (10b), the micro depth of cut of a single grain can be determined once the contact force on the grain and the hardness of the workpiece are known.
4. Micro depth of cut of multiple grains
Table 2 Micro depth of cut-upper bound, lower bound and search method Grit number M
Upper bound (mm)
Lower bound (mm)
Search method (mm)
320 400 600
47.5 38 25.3
0.02–0.03 0.01–0.02 0.009–0.01
0.4–0.8 0.25–0.6 0.2–0.5
4.1. Average method The average method is a simple way of estimating the micro depth of cut of multiple grains. This method assumes that all the grains are in contact with the workpiece. In this case, Eq. (10b) is used to calculate the average micro depth of cut of the multiple grains. For computing Eq. (10b), HB is known if the workpiece material is given, and the average diameter of the grains is equal to m if the grit number M is given. However, the average force f on each grain is not given and needs to be determined as follows. According to [2], the polishing force exerted on the polishing stone is in the range of 2–10 N. The contact area can be calculated by the contact mechanics [17], and the total number of grains, denoted by G, within the contact area may be determined as GZ
Ac D2
(11)
where Ac is the contact area. Then the force on each grain f is obtained by dividing the stone force among all the grains in the contact area as fZ
F G
(12)
where F is the force exerted on the polishing stone. Table 1 shows the computed average micro depth for MZ 320, 400 and 600. The grain diameter is determined by Eq. (2), the grain interval is determined by Eq. (6), the grain force is determined by Eq. (12) and the micro depth of cut is determined by Eq. (10b). However, as shown in Table 2, the results obtained by the average method in fact are the lower bound. 4.2. Search method Compared to the experimental data [18], the micro depth of cut determined based on the average method is very small. This discrepancy results from the assumption that all the grains in the contact area are assumed to be in contact with the workpiece. In reality, however, as shown in Fig. 3, only more Table 1 Micro depth of cut based on average method Grit number M
Grain diameter (mm)
Grain interval (mm)
Workpiece hardness (HB)
Micro depth of cut (mm)
320 400 600
21.1 15.5 8.8
21.8 15.9 9.0
300 300 300
0.02–0.03 0.01–0.02 0.009–0.01
protruded grains that are in contact with the workpiece will contribute to the micro depth of cut. Hence, in order to give a more realistic prediction of the surface roughness, it becomes clear that the grain protrusion heights must be considered with different heights obtained from its random distribution. For this reason, a search method is proposed. The basic idea of the search method is to find a solution that the force exerted on the stone is balanced by a number of more protrusion grains in the contact area. Then these grains are taken as the contact grains to determine the micro depth of cut, while excluding the rest of the grains. Since the grain with the maximum protrusion height cuts off the surface polished by all the other contact grains, it generates the most dominant scallops and its micro depth of cut should be considered first. Then, if the second highest grain does not cut into the scallops generated by the first highest grain, the first scallops will be the final marks. If the second grain does cut, it will change the profile of the first scallops. The process of modifying the surface profile continues with more grains involved in a descending order, and it terminates when no grain cuts into the scallops anymore. Mathematically, the search method is described as follows. First, the micro depth of cut of the most protruded gain is determined as hZ
F pdg1 HB
(13)
where h1 is the micro depth of cut of the most protruded grain. Note that in Eq. (13), the total stone force is initially applied to the most protruded grain with dg1 being its diameter. The depth of cut determined based on Eq. (13) is usually very big because it is based on a single grain contact. If this value is greater than the diameter of the biggest grain, it is considered unrealistic; the reason being that the depth of cut cannot be greater than the size of grains. Therefore, the maximum size of the grain is taken as the upper bound. The average method, on the other hand, represents the lower bound, which is too small because it is based on all grains contact. The search method provides a realistic prediction that lies in between the two bounds, since it is based on the actual number of contact grains. Table 2 shows the results of the upper and lower bounds as well as the search method. The force range considered is from 2 to 10 N. As shown in Fig. 5, after h1 is obtained, the difference of (rg1Kh1) is compared to rg2. If larger, it means that the first grain will balance the entire stone force. Then search
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Fig. 5. Search method model.
terminates and h1 is taken as the final micro depth of cut. If smaller, the second most protruded grain will contribute to the force balance. Since the force is distributed proportionally to the area, the stone force will be divided between the two grains as follows: Frgi fi Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 2 ffi iZ1 rgi
(14)
where iZ1, 2. Under the force re-distribution determined from Eq. (14), the micro depth of cut of the second most protruded grain will be calculated as f2 h2 Z pdg2 HB
(15)
Similarly, the difference of (rg2Kh2) is compared to rg3. If larger, it means that the first and second gain will balance the entire stone force. Then search terminates and h2 is taken as the final micro depth of cut. If smaller, the third most protruded grain will contribute to the force balance. To this end, the search algorithm can be expressed, first for the force distribution as Frgi fi Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk 2 ffi iZ1 rgi
(16)
where iZ1,.,k, and then for the micro depth of cut as hk Z
fk pdgk HB
(17)
where index k indicates a number of k contact grains. The search terminates when rgkKhkOrg(kC1). For the simulation results of the search method provided in Table 2, the number of the contact grains is in the range of 100–120 for MZ320, 150–180 for MZ400, and 220–280 for MZ600.
of the workpiece. For this reason, the polishing stone is controlled to follow the profile of the workpiece closely while it is pressed and rubbed against the workpiece. The polishing process is only meant for smoothing the workpiece surface, not supposed to change the profile of the workpiece. Hence, a realistic way of predicting the surface roughness should be based on the surface roughness of the previous process. In this study, the ball-end milling process is considered as the previous process and the surface roughness of the milled surface can be approximated as [19] Ra Z
0:0321 s2 r3
(18)
where s is the feed or step over (mm) and r3 (mm) is the radius of the milling tool. The surface roughness of the milling process is in the range of 0.8–5 mm [16]. In the actual polishing process, the workpiece is first polished by a fine stone (MZ320) with a number of strokes till the coarse milling marks are removed and covered by the fine marks left by the stone. This is followed by the second polishing using a finer stone (MZ400), and then again followed by the third polishing using an even finer stone (MZ600). Since the polishing marks are created by the stone, it would be reasonable to assume that the surface roughness is very close to the micro depth of cut of the contact grains. Based on this assumption, the number of strokes needed to reduce the previous surface roughness to the current one may be calculated as
G Ns Z 2Gk
RðjÞ a hðjC1Þ
!
5. Prediction of surface roughness As stated in the Introduction section of the paper, the goal of the polishing process is to improve the surface finish
Fig. 6. Cutting marks left by milling.
(19)
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Fig. 7. (a) Topography of the milled surface. (b) Topography of the polished surface.
where Ns is the number of strokes, G is the number of grains in the contact area determined based on Eq. (11), Gk is the number of the contact grains determined from the search method (Eq. (17)), Ra is the surface roughness value of the previous process, h is the micro depth of cut
of the current process, and index j indicates the jth time of polishing. Under this study, jZ0 corresponds to the milling process, and jZ1, 2 and 3 corresponds to the first, second and third polishing using MZ320, 400 and 600, respectively.
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It should be pointed out that in practice, the polishing direction is perpendicular to the cutting marks left by the previous process to ensure the old coarse marks are removed and the new fine marks are created. As shown in Fig. 6, the first polishing direction is perpendicular to the milling scallops. After that, the second polishing direction will be perpendicular to the first polishing and the third polishing direction will be perpendicular to the second polishing. In Eq. (19), the second bracket indicates the number of cutting layers needed to reduce the surface roughness. R(j) a is the surface roughness of the previous process, and h(jC1) is the micro depth of cut of one stroke. Moreover, since polishing only occurs on the contact grains, the percentage of the contact area actually being polished would be the ratio of the contact grains versus the total grains, i.e. Gk/G. So in practice, a certain polishing pattern is generated to ensure the whole area is polished. In our model, this is taken into consideration by adding a factor of G/Gk (the first bracket in Eq. (19)) to calculate a realistic number of strokes. Since a stroke is defined as moving forward and back, it actually cuts twice. Therefore, Eq. (19) is divided by 2.
6. Implementation Software has been developed based on the method described in the previous sections. For implementation, the user can define the milling parameters to obtain the surface roughness value and the topography of the milled surface, as shown in Fig. 7(a). Then the user can define the required surface roughness by polishing to determine the number of strokes needed for the selected polishing parameters including grit number and force. Alternatively, the user can define the number of strokes to predict the surface roughness of the polished surface. In both cases, the topography of the polished surface will be generated, as shown in Fig. 7(b). As an example, assume that the surface roughness after milling is around 3 mm. From Table 2 the micro depth of cut of the first polishing using the stone of MZ320 is around 0.8 mm, which is considered as Ra after the first polishing. The number of strokes is estimated based on Eq. (19). The number of cutting layers (the second bracket of Eq. (19)) is calculated around 4, for the reduction from 3 to 0.8 mm. In addition, the grain ratio (the first bracket of Eq. (19)) is needed. As mentioned before, the number of contact grains is around 100 for MZ320. The simulation shows that there are around 20 K grains within the contact area. Substituting these values into Eq. (19), the number of strokes for the first polishing would be around 400. The similar approach can be used to determine the number of strokes for the second and third polishing. As mentioned before, the surface roughness of the polished surface is approximated by the micro depth of cut of the selected stone. As shown in Table 3, the predicated
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Table 3 Comparison with the experiment data Grit number M
Predicted (mm)
Experiment range [21] (mm)
320 400 600
0.4–0.8 0.25–0.6 0.2–0.5
0.7–0.8 0.5–0.6 0.25–0.3
surface roughness is in the same magnitude of the experiment data obtained from the SPI finishing guide [21], thereby proving the effectiveness of the proposed method.
7. Conclusions In this paper, a new method for modeling and predicting the surface roughness of the workpiece in the stone polishing process is developed. This method is based on the random distribution of the stone grain protrusion heights and the force balance by the contact grains. The development work includes four parts. First, the topography of a selected polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from the given grit number of a stone. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for given workpiece hardness. Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the applied force on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, the method is presented for the predication of the surface roughness based on the micro depth of cut and the contact grains. Software has been developed based on the proposed method. The predicted results are in the same magnitude of the experimental results, thereby proving the effectiveness of the proposed method.
Acknowledgements The first author wishes to thank the Natural Sciences and Engineering Research Council of Canada for support of this work.
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[3] Y. Gharbia, J. Katupitiya, Loose abrasive blasting as an alternative to slurry polishing of optical fibre end faces, International Journal of Machine Tools and Manufacture 43 (2003) 1413–1418. [4] B.H. Yan, G.W. Chang, T.J. Cheng, R.T. Hsu, Electrolytic magnetic abrasive finishing, International Journal of Machine Tools and Manufacture 43 (2003) 1355–1366. [5] Automated and intelligent systems for high performance finishing processes, Proceedings of the Materials and Manufacturing Ontario Workshop, Ryerson University, Canada, Nov. 27, 2003. [6] L. Guvenc, K. Srinivasan, An overview of robot-assisted die and mold polishing with emphasis on process modeling, Journal of Manufacturing Systems 16 (1) (1997) 49–57. [7] J.P. Huissoon, F. Ismail, A. Jafari, S. Bedi, Automated polishing of die steel surfaces, International Journal of Advanced Manufacturing Technology 19 (2000) 285–290. [8] H. Tam, O. Lui, C. A, C.K. Mok, Robotic polishing of free-form surfaces using scanning paths, Journal of Materials Processing Technology 95 (1999) 191–200. [9] O. Einav, Large work envelope fully-automated aircraft panel polishing cell, Proceedings of the International Robotics and Vision Automation Conference, May 9–11, 1995, Detroit, Michigan. [10] T. Sasaki, T. Miyoshi, K. Saito, Knowledge acquisition and automation of polishing operation for injection mold, Journal of the Japan Society of Precision Engineering 25 (3) (1991). [11] K., Kawata, Y., Sawada, M., Yamashita, A new method of teaching and path generation for automatic die and mold polishing system, Proceedings of Japan/USA Symposium on Flexible Automation, v2 (ASME), 1992.
[12] X. Zhou, F. Xi, Modeling and predicting surface roughness of the grinding process, International Journal of Machine Tools and Manufacture 42 (2002) 969–977. [13] S. Malkin, Grinding technology: theory and applications of machining with abrasives, Ellis Horwood, Chichester, 1989. [14] T.W. Hwang, C.J. Evens, S. Malkin, High speed grinding of silicon nitride with electroplated diamond wheels, part 1: wear and wheel life, ASME Journal of Manufacturing Science and Engineering 122 (2000) 32–41. [15] L. Blunt, S. Ebdon, The application of three-dimensional surface measurement techniques to characterizing grinding wheel topography, International Journal of Machine Tools and Manufacture 36 (11) (1996) 1207–1226. [16] X. Chen, W.B. Rowe, Analysis and simulation of the grinding process, part 1: generation of the grinding wheel surface, International Journal of Machine Tools and Manufacture 36 (8) (1996) 871–882. [17] K.L. Johnson, Contact Mechanics, Cambridge University Press, Great Britain, 1985. [18] L. Zhang, H.Y. Tam, C.-M. Yuan, Y.-P. Chen, Z.-D. Zhou, An investigation of material removal in polishing with fixed abrasives, Proceedings of the IMECHE Part B, Journal of Engineering Manufacture 216 (2002) 103–112. [19] J.A. Schey, Introduction to Manufacturing Processes, McGraw-Hill, New York, 1987. [20] X. Zhou, Modeling and predicting surface roughness for the grinding process, M Sc Thesis, University of Western Ontario, 2001. [21] SPI Finishing Guide at www.moldpolishingc.come/Spi.html
Modeling surface roughness in the stone polishing process Fengfeng Xia,*, David Zhoub a
Department of Mechanical, Aerospace and Industrial Engineering, Ryerson University Toronto, Ont. M5B 2K3, Canada b The University of Western Ontario London, Ont., Canada Received 31 May 2004; accepted 10 September 2004 Available online 2 November 2004
Abstract In this paper, a new method for modeling and predicting the surface roughness of the workpiece in the stone polishing process is developed. This method is based on the random distribution of the stone grain protrusion heights and the force balance by contact grains. To do so, first, the topography of a polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from a given stone grit number. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for a given workpiece hardness (Brinell number). Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the force applied on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, a method is presented for predicting the surface roughness based on the micro depth of cut and contact grains. A good agreement of the prediction results with the experimental data proves the effectiveness of the proposed method. q 2004 Elsevier Ltd. All rights reserved. Keywords: Polishing; Modelling; Surface roughness
1. Introduction Generally speaking, manufacturing processes can be loosely classified into primary manufacturing processes and secondary manufacturing processes. The primary manufacturing processes are mainly concerned about producing the products to satisfy the required dimension and shape accuracy. The secondary manufacturing processes are mainly concerned about finishing the products with the required surface integrity and surface finish. Polishing process is one of the main secondary manufacturing processes, and its purpose is to reduce the surface roughness to a desired amount. Industrial applications of the polishing process span from large products such as automobile, aerospace, to small parts such as optics [1–3]. There are a number of polishing methods used in industry which may be classified into mechanical and non mechanical. Mechanical polishing uses abrasives to smooth the workpiece surface. Non-mechanical polishing uses other means, such as laser, magnetic fluid [4,5]. In most * Corresponding author. Tel.: C1 416 979 5000; fax: C1 416 979 5265. E-mail address: [email protected] (F. Xi). 0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.09.016
applications, however, polishing is done through mechanical means simply because of the low cost. Mechanical polishing can be further classified into bound abrasives and un-bound abrasives. The bound abrasives are the polishing stones for stone polishing. The un-bound abrasives are the abrasive compounds, such diamond compounds, for buffing. In practice, mechanical polishing is still done manually, especially in the die and mold industry and aerospace industry. Nevertheless, research has been carried out to develop automated polishing systems in an attempt to improving the productivity of the current polishing process. The automated polishing systems archived in the literature [6–10] can be loosely classified based on whether a conventional machine tool structure (e.g. computer-numerical control—CNC) or an articulated robot arm (i.e. industrial robots) is being used [11]. While in the manual polishing process the polisher examines the surface finish by comparing to a template, computer simulation and prediction of the surface roughens would be essential for automated polishing systems. This motivates the work reported here. Due to the random distribution of the grains in abrasive tools, polishing research has been mainly experimental. Recently, by applying the stochastic theory the authors of the present paper developed a method for predicting
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the surface roughness for the grinding process [12]. This model is further developed here to model and simulate the surface roughness for the stone polishing process. To do so, first, the topography of a selected polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from a given stone grit number. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for given workpiece hardness (Brinell number). Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the force applied on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, a method is presented for the prediction of the surface roughness based on the micro depth of cut and the contact grains. A good agreement of the prediction results with the experimental data proves the effectiveness of the proposed method.
deviation for generating the topography of a polishing stone is given as m Z dg avg
(3a)
and s Z ðdg max K dg avg Þ=3
(3b)
where m and s are the mean value and standard deviation, respectively. From Eqs. (3a) and (3b), it can be calculated that mZ21.1, 15.5 and 8.8 mm, respectively, and sZ8.8, 7.5 and 5.5 mm, respectively, for MZ320, 400, and 600. To generate the topology, the polishing stone is meshed by a grain interval. The grain interval is the distance between two adjacent grains and it is related to the structure number S of the polishing stone. The structure number indicates the volumetric concentration of abrasive grains in the stone. For a given number of S, the density of the stone, Vg (%), may be approximated as [10]
2. Generation of polishing stone topography
Vg Z 2ð32 K SÞ
The first step in modeling and predicting the surface roughness for the stone polishing process is to generating the topography for a polishing stone. The surface roughness of the polished workpiece could be considered as an imprint created by the polishing stone rubbed against the workpiece. Previous experimental research [13] has revealed that the distribution of the grain protrusion heights of an abrasive tool is of a Gaussian distribution. In the paper [8], the present authors developed a method to generate the topography of a grinding wheel. Since polishing stones are made with finer grains in a similar way to grinding wheels, this method is applied here. To generate a random distribution for the grain protrusion heights of a polishing stone, the mean value and standard deviation are required. The grid numbers (M) of the commonly used polishing stones are MZ320 for the first stage stoning, MZ 400 for the second stage stoning and MZ600 for the final stage stoning. The grain dimension dg is expressed equal to the aperture opening of the screen according to the following relationship [9]
In general the distribution of the grain intervals is assumed uniform [12,13]. The interval D can be derived from the following relation p 3 d Z Vg D 3 (5) 6 g avg
dgmax ðmmÞ Z 15:2MK1
The left-hand side of Eq. (5) indicates the volume of the grains in the polishing stone in terms of the average grain size, and the right-hand side in terms of the density. The two sides should be equal. Therefore, from Eq. (5), the grain interval D (mm) can be determined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 D Z 137:9 MK1:4 (6) 32 K S To generate a polishing stone topography, the stone is meshed using the interval determined by Eq. (6). The distribution of the grain protrusion heights is described by
(1)
This grain size is considered as the maximum grain size, denoted by dg max, for a given grit number M. Furthermore, according to [9], the average grain dimension denoted by dg avg is related to grit number M (for MO40) as dg avg ðmmÞ Z 68 MK1:4
(2)
From Eqs. (1) and (2), it can be calculated that for MZ320, 400, and 600, dg maxZ47.5, 38 and 25.3 mm, respectively, and dg avgZ21.1, 15.5 and 8.8 mm, respectively. As reported in [14,15], the extensive experiment using 3D stylus instruments and SEM micrographs has shown that dg max and dg avg are very close to the maximum grain protrusion height hmax and the mean grain protrusion height havg, respectively. Therefore, the mean value and standard
(4)
Fig. 1. Topography of a polishing stone.
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a Gaussian distribution with the mean value and standard deviation determined by Eqs. (3a) and (3b). Fig. 1 is a simulated polishing stone topography with grit number MZ 400, showing that the surface of a polishing tool is made of randomly distributed protrusion heights.
3. Micro depth of cut of a single grain Prior to studying the surface roughness of the polished workpiece, the problem of determining the micro depth of cut of a single grain is addressed. Fig. 2 shows the three shapes archived in the literature [20] for modeling the abrasive grains, namely, semi sphere shape, cone shape, and prism shape. The semi sphere shape is adopted in this paper, because this shape is the most commonly used for modeling the grains. In the polishing process, the polishing stone is pressed under certain pressure against the workpiece. Though the pressure is applied to the entire polishing stone, it is only concentrated on the grains that are in contact with the workpiece. As shown in Fig. 3, the contact grains are more protruded than the grains that are not in contact with the workpiece. Due to the highly concentrated pressure on the contact grains, plastic deformation occurs in the contact area, thereby creating the micro depth of cut. Therefore, the polishing cutting mechanism may be modeled based on the plasticity theory, similar to hardness testing. The most common form of hardness tests for metals involves standard indenters being pressed into the surface of the material. Measurements associated with the indentation are then taken as a measure of the hardness of the surface. The Brinell test, the Vickers test and the Rockwell test are the main forms of such tests. In this paper, the Brinell test model is used to determine the micro depth of cut of a single grain. In other words, the grain is modeled as an indenter inserted on the workpiece. It is assumed that the grain is much harder than the workpiece. For example, the common polishing grain material is aluminum oxide with HB over 1000, while steel is with HB around 300. In the Brinell test model, a hardened ball is pressed into the surface of the material by a standard force. In our polishing model, the hardened ball is replaced by the semisphere polishing grain. After the load and the ball have been removed, the diameter of the indentation is measured.
Fig. 2. Common grain shapes.
Fig. 3. Contact model of polishing grains.
The Brinell hardness number (HB) is obtained by dividing the size of the applied force f by the spherical surface area of the indentation A HB Z
f ðkgf Þ Aðmm2 Þ
(7)
where f is the force applied on the grain, and its determination will be discussed later. As shown in Fig. 4, the contact area can be obtained by calculation from the values of the radius rg of the ball and the radius rc of the indentation, that is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Z 2prg ðrg K rg2 K rc2 Þ (8) Note that for the grain model, rgZdg/2, where as mentioned before, dg is the grain diameter. From Fig. 4 the following relation holds among rg, rc, and h rg2 Z rc2 C ðrg2 K hÞ2
(9)
From Eq. (9), the micro depth of cut can be derived as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi (10a) h Z rg2 K rg2 K rc2 Substituting Eqs. (7) and (8) into Eq. (10a) yields hZ
f 2prg HB
Fig. 4. Micro depth of cut of a polishing grain.
(10b)
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Using Eq. (10b), the micro depth of cut of a single grain can be determined once the contact force on the grain and the hardness of the workpiece are known.
4. Micro depth of cut of multiple grains
Table 2 Micro depth of cut-upper bound, lower bound and search method Grit number M
Upper bound (mm)
Lower bound (mm)
Search method (mm)
320 400 600
47.5 38 25.3
0.02–0.03 0.01–0.02 0.009–0.01
0.4–0.8 0.25–0.6 0.2–0.5
4.1. Average method The average method is a simple way of estimating the micro depth of cut of multiple grains. This method assumes that all the grains are in contact with the workpiece. In this case, Eq. (10b) is used to calculate the average micro depth of cut of the multiple grains. For computing Eq. (10b), HB is known if the workpiece material is given, and the average diameter of the grains is equal to m if the grit number M is given. However, the average force f on each grain is not given and needs to be determined as follows. According to [2], the polishing force exerted on the polishing stone is in the range of 2–10 N. The contact area can be calculated by the contact mechanics [17], and the total number of grains, denoted by G, within the contact area may be determined as GZ
Ac D2
(11)
where Ac is the contact area. Then the force on each grain f is obtained by dividing the stone force among all the grains in the contact area as fZ
F G
(12)
where F is the force exerted on the polishing stone. Table 1 shows the computed average micro depth for MZ 320, 400 and 600. The grain diameter is determined by Eq. (2), the grain interval is determined by Eq. (6), the grain force is determined by Eq. (12) and the micro depth of cut is determined by Eq. (10b). However, as shown in Table 2, the results obtained by the average method in fact are the lower bound. 4.2. Search method Compared to the experimental data [18], the micro depth of cut determined based on the average method is very small. This discrepancy results from the assumption that all the grains in the contact area are assumed to be in contact with the workpiece. In reality, however, as shown in Fig. 3, only more Table 1 Micro depth of cut based on average method Grit number M
Grain diameter (mm)
Grain interval (mm)
Workpiece hardness (HB)
Micro depth of cut (mm)
320 400 600
21.1 15.5 8.8
21.8 15.9 9.0
300 300 300
0.02–0.03 0.01–0.02 0.009–0.01
protruded grains that are in contact with the workpiece will contribute to the micro depth of cut. Hence, in order to give a more realistic prediction of the surface roughness, it becomes clear that the grain protrusion heights must be considered with different heights obtained from its random distribution. For this reason, a search method is proposed. The basic idea of the search method is to find a solution that the force exerted on the stone is balanced by a number of more protrusion grains in the contact area. Then these grains are taken as the contact grains to determine the micro depth of cut, while excluding the rest of the grains. Since the grain with the maximum protrusion height cuts off the surface polished by all the other contact grains, it generates the most dominant scallops and its micro depth of cut should be considered first. Then, if the second highest grain does not cut into the scallops generated by the first highest grain, the first scallops will be the final marks. If the second grain does cut, it will change the profile of the first scallops. The process of modifying the surface profile continues with more grains involved in a descending order, and it terminates when no grain cuts into the scallops anymore. Mathematically, the search method is described as follows. First, the micro depth of cut of the most protruded gain is determined as hZ
F pdg1 HB
(13)
where h1 is the micro depth of cut of the most protruded grain. Note that in Eq. (13), the total stone force is initially applied to the most protruded grain with dg1 being its diameter. The depth of cut determined based on Eq. (13) is usually very big because it is based on a single grain contact. If this value is greater than the diameter of the biggest grain, it is considered unrealistic; the reason being that the depth of cut cannot be greater than the size of grains. Therefore, the maximum size of the grain is taken as the upper bound. The average method, on the other hand, represents the lower bound, which is too small because it is based on all grains contact. The search method provides a realistic prediction that lies in between the two bounds, since it is based on the actual number of contact grains. Table 2 shows the results of the upper and lower bounds as well as the search method. The force range considered is from 2 to 10 N. As shown in Fig. 5, after h1 is obtained, the difference of (rg1Kh1) is compared to rg2. If larger, it means that the first grain will balance the entire stone force. Then search
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Fig. 5. Search method model.
terminates and h1 is taken as the final micro depth of cut. If smaller, the second most protruded grain will contribute to the force balance. Since the force is distributed proportionally to the area, the stone force will be divided between the two grains as follows: Frgi fi Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2 2 ffi iZ1 rgi
(14)
where iZ1, 2. Under the force re-distribution determined from Eq. (14), the micro depth of cut of the second most protruded grain will be calculated as f2 h2 Z pdg2 HB
(15)
Similarly, the difference of (rg2Kh2) is compared to rg3. If larger, it means that the first and second gain will balance the entire stone force. Then search terminates and h2 is taken as the final micro depth of cut. If smaller, the third most protruded grain will contribute to the force balance. To this end, the search algorithm can be expressed, first for the force distribution as Frgi fi Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pk 2 ffi iZ1 rgi
(16)
where iZ1,.,k, and then for the micro depth of cut as hk Z
fk pdgk HB
(17)
where index k indicates a number of k contact grains. The search terminates when rgkKhkOrg(kC1). For the simulation results of the search method provided in Table 2, the number of the contact grains is in the range of 100–120 for MZ320, 150–180 for MZ400, and 220–280 for MZ600.
of the workpiece. For this reason, the polishing stone is controlled to follow the profile of the workpiece closely while it is pressed and rubbed against the workpiece. The polishing process is only meant for smoothing the workpiece surface, not supposed to change the profile of the workpiece. Hence, a realistic way of predicting the surface roughness should be based on the surface roughness of the previous process. In this study, the ball-end milling process is considered as the previous process and the surface roughness of the milled surface can be approximated as [19] Ra Z
0:0321 s2 r3
(18)
where s is the feed or step over (mm) and r3 (mm) is the radius of the milling tool. The surface roughness of the milling process is in the range of 0.8–5 mm [16]. In the actual polishing process, the workpiece is first polished by a fine stone (MZ320) with a number of strokes till the coarse milling marks are removed and covered by the fine marks left by the stone. This is followed by the second polishing using a finer stone (MZ400), and then again followed by the third polishing using an even finer stone (MZ600). Since the polishing marks are created by the stone, it would be reasonable to assume that the surface roughness is very close to the micro depth of cut of the contact grains. Based on this assumption, the number of strokes needed to reduce the previous surface roughness to the current one may be calculated as
G Ns Z 2Gk
RðjÞ a hðjC1Þ
!
5. Prediction of surface roughness As stated in the Introduction section of the paper, the goal of the polishing process is to improve the surface finish
Fig. 6. Cutting marks left by milling.
(19)
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Fig. 7. (a) Topography of the milled surface. (b) Topography of the polished surface.
where Ns is the number of strokes, G is the number of grains in the contact area determined based on Eq. (11), Gk is the number of the contact grains determined from the search method (Eq. (17)), Ra is the surface roughness value of the previous process, h is the micro depth of cut
of the current process, and index j indicates the jth time of polishing. Under this study, jZ0 corresponds to the milling process, and jZ1, 2 and 3 corresponds to the first, second and third polishing using MZ320, 400 and 600, respectively.
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It should be pointed out that in practice, the polishing direction is perpendicular to the cutting marks left by the previous process to ensure the old coarse marks are removed and the new fine marks are created. As shown in Fig. 6, the first polishing direction is perpendicular to the milling scallops. After that, the second polishing direction will be perpendicular to the first polishing and the third polishing direction will be perpendicular to the second polishing. In Eq. (19), the second bracket indicates the number of cutting layers needed to reduce the surface roughness. R(j) a is the surface roughness of the previous process, and h(jC1) is the micro depth of cut of one stroke. Moreover, since polishing only occurs on the contact grains, the percentage of the contact area actually being polished would be the ratio of the contact grains versus the total grains, i.e. Gk/G. So in practice, a certain polishing pattern is generated to ensure the whole area is polished. In our model, this is taken into consideration by adding a factor of G/Gk (the first bracket in Eq. (19)) to calculate a realistic number of strokes. Since a stroke is defined as moving forward and back, it actually cuts twice. Therefore, Eq. (19) is divided by 2.
6. Implementation Software has been developed based on the method described in the previous sections. For implementation, the user can define the milling parameters to obtain the surface roughness value and the topography of the milled surface, as shown in Fig. 7(a). Then the user can define the required surface roughness by polishing to determine the number of strokes needed for the selected polishing parameters including grit number and force. Alternatively, the user can define the number of strokes to predict the surface roughness of the polished surface. In both cases, the topography of the polished surface will be generated, as shown in Fig. 7(b). As an example, assume that the surface roughness after milling is around 3 mm. From Table 2 the micro depth of cut of the first polishing using the stone of MZ320 is around 0.8 mm, which is considered as Ra after the first polishing. The number of strokes is estimated based on Eq. (19). The number of cutting layers (the second bracket of Eq. (19)) is calculated around 4, for the reduction from 3 to 0.8 mm. In addition, the grain ratio (the first bracket of Eq. (19)) is needed. As mentioned before, the number of contact grains is around 100 for MZ320. The simulation shows that there are around 20 K grains within the contact area. Substituting these values into Eq. (19), the number of strokes for the first polishing would be around 400. The similar approach can be used to determine the number of strokes for the second and third polishing. As mentioned before, the surface roughness of the polished surface is approximated by the micro depth of cut of the selected stone. As shown in Table 3, the predicated
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Table 3 Comparison with the experiment data Grit number M
Predicted (mm)
Experiment range [21] (mm)
320 400 600
0.4–0.8 0.25–0.6 0.2–0.5
0.7–0.8 0.5–0.6 0.25–0.3
surface roughness is in the same magnitude of the experiment data obtained from the SPI finishing guide [21], thereby proving the effectiveness of the proposed method.
7. Conclusions In this paper, a new method for modeling and predicting the surface roughness of the workpiece in the stone polishing process is developed. This method is based on the random distribution of the stone grain protrusion heights and the force balance by the contact grains. The development work includes four parts. First, the topography of a selected polishing stone is generated based on a Gaussian distribution with the mean value and standard deviation determined from the given grit number of a stone. Second, the plasticity theory is applied to determine the micro depth of cut of a single grain for given workpiece hardness. Third, a search method is developed to determine the number of the contact grains and the micro depth of cut, based on the force balance principle between the applied force on the stone and the forces transmitted to the grains that are in contact with the workpiece. Fourth, the method is presented for the predication of the surface roughness based on the micro depth of cut and the contact grains. Software has been developed based on the proposed method. The predicted results are in the same magnitude of the experimental results, thereby proving the effectiveness of the proposed method.
Acknowledgements The first author wishes to thank the Natural Sciences and Engineering Research Council of Canada for support of this work.
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