# Gloss and Modelling Studies of Stone Polishing Using Linear Polishing Machines with Rotating Heads

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{L}, the transverse head velocity, V

_{T}, and the rotational velocity, ω. These studies were carried out by measuring the surface gloss (as roughness measurements proved ineffective for the highly polished surfaces that were analyzed) and comparing it with computer simulations of the same polishing procedures, to better understand the geometrical and kinematic issues determining the final results.

## 2. Theoretical Background and Adopted Polishing Cycle

_{L}and V

_{T}. Condition 1 states that after a complete zigzag movement (with two linear segments), the tool center should be closer to its initial position than the tool diameter [1]: l < d, with d = 2r = tool diameter. Otherwise, there will be gaps of size l − d where the polishing head does not interact with the surface.

_{L}, the transverse head velocity, V

_{T}, and the rotational head velocity, ω.

_{T}= b/V

_{T}and the belt shift after a zigzag becomes l = 2bV

_{L}/V

_{T}≤ d (limited by condition 1).

_{S}moved by the polishing head in one linear segment (half of a complete zigzag) is given by:

_{S}= ωt

_{T}as the number of rotations executed during the time t

_{T}by the tool head rotating at ω rotations per unit of time, the distance x that the tool head moves in a single rotation is given by (limited by condition 2):

## 3. Computational Details

## 4. Experimental Procedure

_{L}= 600 mm/min, the head pressure P = 2 bar, and the water flow Q = 30 litter/min. These values were chosen to maximize quality and reproducibility of the results, and to emulate industrial data. For example, a minimum coefficient of variation (standard variation/mean value) of 6% in gloss measurements was achieved for a head pressure P = 2 bar. Six slabs of limestone, as equal as possible, were selected for these tests due to the high homogeneity of its surface. This is required to simplify the comparison between experimental and simulation data, as the simulator assumes a perfect, two dimensional, stone surface. After applying the sequence of abrasives, the surface gloss is measured and compared with the abrasion predicted in the same conditions by the polishing simulator (see Figure 7 and Figure 8). To make the measurements more precise, we used a physical grid made of wire to precisely define the sections where gloss was measured for comparison with simulated values. Throughout this work, sections with homogeneous simulated abrasion values were identified as green cells (see Figure 7) while sections with simulated abrasion values too different were marked red and discarded in the subsequent analysis and comparison with experimental results. The gloss values reported represent average values over the various green cells (typically 10, depending of each experimental pattern). After each experiment, the slab surface was reset, using the more abrasive tool head (320 grit), thus slabs were only discarded after significant wear.

_{L}= 600 mm/min as stated above, the transverse speed V

_{T}must be V

_{T}≥ 20 mm/s. To obey condition 2 (x < r), with V

_{L}= 600 mm/min and V

_{T}≥ 20 mm/s, the head rotation speed must be ω ≥ 10 rpm. Table 1 shows the various operational conditions tested in this work, whereas l values are determined by condition 1 and ω values are determined by condition 2. Clearly, setting conditions 1 and 2 leads to an increase of the transverse velocity (to decrease l) and an increase of the rotational velocity (to decrease x). The simulated linear velocity (V) matches the vectorial combination of both cross and conveyor belt velocities from the polishing machine, according to the following equation:

## 5. Results and Discussion

_{L}= 600 mm/min = 1 cm/s) and the relatively large tool head diameter (d = 435 mm), condition 1 is easily obeyed, even for small transverse speeds (see Table 1). As expected, increasing the transverse speed requires larger rotational speeds to fulfil condition 2. Decreasing x, from x = r to x = r/8, to further obey condition 2, requires even larger rotational speeds.

_{T}, for x = r, x = r/2, x = r/4, and x = r/8 conditions, show a significant increase in gloss from 40 to 200 mm/s for all four x conditions. For higher transverse velocities, gloss remains essentially constant, showing that a maximum threshold has been attained and further increasing the energy spent in the polishing process (by increasing the transverse and rotational speeds) does not lead to a refinement of the surface quality.

_{T}= 150 mm/s (w = 330 rpm, from Table 1) provides essentially the same final quality (Figure 10) as working at x = r/2 with V

_{T}= 200 mm/s (w = 110 rpm). On the other hand, working at low V

_{T}speeds, such as 90 mm/s, a good polished surface is never attained, even for x = r/8 and high rotational speeds (ω = 200 rpm). Transverse and rotational speeds must be expertly combined to achieve adequate levels of polishing at low energy consumption.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Roman, A.L.P.D.; Chaves-Jacob, J.; Linares, J.M.; Arrazola, P.J. Analytical-method for polishing-surface prediction considering tool flexibility and grain-material interaction. J. Mater. Process. Technol.
**2021**, 295, 117208. [Google Scholar] [CrossRef] - Sousa, F.J.P.; Weingaertner, W.L.; Alarcon, O.E. Computational simulation of the polishing process of stoneware tiles. In Proceedings of the World Congress of Quality in Ceramic Tiles, QUALICER 2008, Castellón, Spain, 10–13 February 2008; pp. 359–367. Available online: https://www.qualicer.org/recopilatorio/ponencias/pdfs/0833103e.pdf (accessed on 12 July 2022).
- Sousa, F.J.P.; Aurich, J.C.; Weingaertner, W.L.; Alarcon, O.E. Optimization of the kinematics available in the polishing process of ceramic tiles by computational simulations. J. Am. Ceram. Soc.
**2009**, 92, 41–48. [Google Scholar] [CrossRef] - Xi, F.; Zhou, D. Modeling surface roughness in the stone polishing process. Int. J. Mach. Tools Manuf.
**2005**, 45, 365–372. [Google Scholar] [CrossRef] - Wang, G.; Wang, Y.; Xu, Z. Modeling and analysis of the material removal depth for stone polishing. J. Mater. Process. Technol.
**2009**, 209, 2453–2463. [Google Scholar] [CrossRef] - Amaral, P.M.; Rosa, L.G.; Pinto, S.; Pozo, D. New line of diamond tools raise productivity in polishing stone. IDR Ind. Diamond Rev.
**2004**, 3, 33–37. Available online: https://scholar.tecnico.ulisboa.pt/records/3f3e886c-472e-4261-9cf7-b6938a6dff86 (accessed on 12 July 2022). - Yang, H.D.; Li, H.C.; Zhu, C.J.; Fang, H.; Li, J. A process parameters selection approach for trade-off between energy consumption and polishing quality. Int. J. Comput. Integr. Manuf.
**2018**, 31, 380–395. [Google Scholar] [CrossRef] - Soares, J.E.; Aurich, J.C.; Sousa, F.J.P.; Nascimento, R.M.; Paskocimas, C.A. Estimation of the minimum material removal thickness during the polishing process of ceramic tiles by laser triangulation. Ceram. Int.
**2018**, 44, 4646–4652. [Google Scholar] [CrossRef] - Sani, A.S.A.; Sousa, F.J.P.; Hamedon, Z.; Azhari, A. Contact pressure distribution during the polishing process of ceramic tiles: A laboratory investigation. IOP Conf. Ser. Mater. Sci. Eng.
**2016**, 114, 012008. [Google Scholar] [CrossRef] - Nascimento, A.S.B.D.; Sousa, F.J.P. Distribution of contact pressure over the surface of ceramic floor tiles during the polishing process. J. Eur. Ceram. Soc.
**2014**, 34, 3209–3215. [Google Scholar] [CrossRef] - Barbosa, A.R.J. Validação Experimental de Simulador de Polimento em Rocha Ornamental. Master’s Thesis, Instituto Superior Tecnico, Technical University of Lisbon, Lisbon, Portugal, 15 December 2014. Available online: https://fenix.tecnico.ulisboa.pt/cursos/memat/dissertacao/846778572210523 (accessed on 8 July 2022).
- Barbosa, A.R.; Coelho, A.; Fernandes, J.C.; Amaral, P.M.; Rosa, L.G.; Pereira, J.C. A contribution for an optimization of the polishing quality of stone slabs: Simulation and experimental study using a single-head polishing machine. In Proceedings of the International Conference on Stone and Concrete Machining (ICSCM), Bochum, Germany, 2–3 November 2015; Volume 3, pp. 178–187. [Google Scholar] [CrossRef]
- Frankfurt 5 Extra for Marble. Difa Stonetech Co., Ltd. Available online: http://www.difastone.com/e_productshow/?44-Frankfurt-5-Extra-for-Marble-44.html (accessed on 8 July 2022).
- Hanson, A.R. Good Practice Guide for the Measurement of Gloss; Measurement Good Practice Guide, No. 94; National Physical Laboratory: Teddington, UK, 2006; Available online: https://www.npl.co.uk/special-pages/guides/gpg94_gloss.aspx (accessed on 12 July 2022).
- Carvalho, D.L.S. Determinação de Parâmetros do Polimento, em Três Tipos de Rochas Graníticas. Master’s Thesis, Escola de Engenharia de São Carlos, Universidade de São Paulo, Campus USP de São Carlos, São Carlos, Brazil, 17 August 2014. Available online: https://teses.usp.br/teses/disponiveis/18/18132/tde-02122011-154618/fr.php (accessed on 8 July 2022).
- Yavuz, H.; Ozkahraman, T.; Demirdag, S. Polishing experiments on surface quality of building stones tiles. Constr. Build. Mater.
**2011**, 25, 1707–1711. [Google Scholar] [CrossRef] - Ranjan, P.; Balasubramaniam, R.; Jain, V.K. Molecular dynamics simulation of mechanical polishing on stainless steel using diamond nanoparticles. J. Manuf. Sci. Eng.
**2019**, 141, 014504. [Google Scholar] [CrossRef] - Maekawa, K.; Itoh, A. Friction and tool wear in nano-scale machining—a molecular dynamics approach. Wear
**1995**, 188, 115–122. [Google Scholar] [CrossRef] - Qin, K.; Moudgil, B.; Park, C.-W. A chemical mechanical polishing model incorporating both the chemical and mechanical effects. Thin Solid Films
**2004**, 446, 277–286. [Google Scholar] [CrossRef]

**Figure 3.**Sequence of the polishing cycle abrasives used in this work: (

**a**) 320 grit; (

**b**) 400 grit; (

**c**) Frankfurt 5 Extra grit.

**Figure 5.**Simulated tool heads used in this work, containing six abrasive elements of: (

**a**) 320 grit; (

**b**) 400 grit; (

**c**) 5 Extra grit.

**Figure 6.**Tool head with six 320 grit elements rotating at 10 rotations per second, simulated with: (

**a**) a time step of 0.01 s, corresponding to an increment angle of 36° per step; (

**b**) a time step of 0.001 s, corresponding to an increment angle of 3.6° per step.

**Figure 7.**Output image showing the simulated abrasion with homogeneous (green) cells and heterogeneous (red) cells.

**Figure 8.**Experimental setup used to measure gloss data, using a wired grid with cells of 30 × 30 mm

^{2}, where each gloss measurement takes place.

**Figure 9.**Gloss as a function of transverse velocity V

_{T}, for x = r, x = r/2, x = r/4, and x = r/8.

**Figure 10.**Abrasion as a function of transverse velocity V

_{T}, for x = r, x = r/2, x = r/4, and x = r/8.

**Figure 11.**Abrasion images from polishing simulation, for three transverse velocities V

_{T}: (

**a**) 40 mm/s; (

**b**) 90 mm/s; (

**c**) 200 mm/s, with x = r/8.

**Figure 12.**Comparison of abrasion and gloss results, both as functions of transverse velocity V

_{T}, for x = r/8.

**Table 1.**Twenty-four polishing conditions used in experimental and modelling work, obeying conditions 1 and 2.

l (mm) | V_{T}(mm/s) | V (mm/s) | ω (rpm) | ||||
---|---|---|---|---|---|---|---|

x | r | r/2 | r/4 | r/8 | |||

120 | 40 | 41.58 | 10 | 20 | 45 | 90 | |

64 | 75 | 75.80 | 20 | 40 | 80 | 165 | |

53 | 90 | 90.67 | 25 | 50 | 100 | 200 | |

48 | 150 | 150.33 | 40 | 80 | 165 | 330 | |

32 | 200 | 200.25 | 55 | 110 | 220 | 440 | |

24 | 300 | 300.17 | 80 | 165 | 330 | 660 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Coelho, A.; Pereira, J.C.G.; Amaral, P.M.; Rosa, L.G.
Gloss and Modelling Studies of Stone Polishing Using Linear Polishing Machines with Rotating Heads. *Appl. Sci.* **2022**, *12*, 7521.
https://doi.org/10.3390/app12157521

**AMA Style**

Coelho A, Pereira JCG, Amaral PM, Rosa LG.
Gloss and Modelling Studies of Stone Polishing Using Linear Polishing Machines with Rotating Heads. *Applied Sciences*. 2022; 12(15):7521.
https://doi.org/10.3390/app12157521

**Chicago/Turabian Style**

Coelho, Adriano, José Carlos Garcia Pereira, Pedro M. Amaral, and Luís Guerra Rosa.
2022. "Gloss and Modelling Studies of Stone Polishing Using Linear Polishing Machines with Rotating Heads" *Applied Sciences* 12, no. 15: 7521.
https://doi.org/10.3390/app12157521