*Proceeding Paper* **Proposal of a Computational Algorithm for Calculating Material Ratio of Surface Texture †**

**Hirokazu Machida 1, Ichiro Yoshida 2,\* and Yuki Kondo <sup>2</sup>**


**Abstract:** The material ratio curve (hereafter referred to as MRC) of ISO 13565-2 and ISO 4287 is widely used in industrial fields. The computational algorithm of MRC proposed in ISO has a problem of long calculation time, because of a method of slicing the roughness profile. Therefore, in this study, a sort method was proposed as a computational algorithm for time reduction. However, depending on the form of the surface profile, the algorithm of the proposed sort method has a problem in that calculation errors occur. Therefore, in this paper, we report a new improved algorithm that solves this problem. In this paper, a new and improved algorithm for calculating MRC has been researched and developed. The proposed algorithm in this paper succeeded in reducing the computing time to derive MRC compared with the calculating algorithm of MRC proposed in the ISO standard. This algorithm is expected the efficiency improvement of quality control.

**Keywords:** material ratio curve; Abbott–Firestone curve; roughness; surface texture; computational algorithm

#### **1. Introduction**

Surface textures are geometric features that collectively refer to surface roughness and scratches. This is described in the ISO standard [1]. Since the surface textures are greatly related to the geometrical specifications of the product from an industrial point of view, their quality is required to be controlled quantitatively.

An example of where surface textures play an important role is in bearings. Bearings are used in automobiles, aircraft, and household appliances such as refrigerators and air conditioners because they have the role of preventing energy loss due to frictional resistance by rotating the shaft smoothly.

Because the frictional resistance of a bearing depends on the surface texture, the bearing is closely related to the surface texture.

ISO 4287 [1] contains various surface texture parameters. Typical parameters of surface texture include arithmetic mean roughness *Ra*, maximum height *Rz*, etc. Among them, the material ratio curve (MRC) and MRC parameters can be mentioned as effective methods for evaluating surfaces with excellent lubrication and friction characteristics [2,3]. MRC is a curve that expresses the ratio of the material and void parts of the surface profile with respect to the height direction [4]. MRC and MRC parameters are used in the industrial world as an effective evaluation method for the quality control of automobile parts and bearings.

A method for calculating MRC from the roughness profile has been proposed in ISO 13565-2 [5]. This calculation method has a problem that it takes a long time to calculate as the number of times of slicing the roughness profile increases. Therefore, in a previous

**Citation:** Machida, H.; Yoshida, I.; Kondo, Y. Proposal of a Computational Algorithm for Calculating Material Ratio of Surface Texture. *Eng. Proc.* **2021**, *11*, 22. https://doi.org/10.3390/ ASEC2021-11168

Academic Editor: Filippo Berto

Published: 15 October 2021

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**Copyright:** © 2021 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/).

paper [6], the sort method, which is a new MRC calculation algorithm that can shorten the calculation time compared to the ISO standard method, has been reported. However, it was clarified that this calculation method causes problems in the derivation of MRC depending on the profile shape of the measured data [6]. Therefore, in this paper, we report a new improved algorithm that does not cause problems in deriving the MRC. If the improved algorithm by this research is completed, it can be expected to contribute to the efficiency of quality control in actual industrial sites.

#### **2. The Calculation Method of MRC by ISO 13565-2**

The calculation method of MRC proposed by the ISO standard is as follows: First, a slicing level *ZL* is set parallel to the reference line for the evaluation of the roughness curve. The slicing level is the height of slice (hereafter referred to as the slice height) [5,7]. Next, the ratio of the material and void parts when the surface profile is sliced at a certain height is calculated, and this calculation is performed sequentially over the whole height direction (henceforth referred to as the slicing method). In this research, the algorithm for calculating MRC with the image shown in Figure 1 [7] is coded using MATLAB.

**Figure 1.** Slice method: derivation method of MRC [5,7].

The procedure of the slicing method is as follows.


Figure 2 shows an adaptation of the slice method procedure (1) to the roughness profile. The blue line shows the roughness curve (extract) and the red line shows the slice height in Figure 2. The slicing method has a problem in that it is time consuming when the number of slicing levels increases.

**Figure 2.** Roughness profile (blue line) and slice height (red line).

#### **3. New Proposed Algorithm for Calculating MRC**

*3.1. Sort Method for MRC*

In this research, the sort method is proposed as a new method of calculating MRC which can reduce the calculation time [6,8]. The sort method is a method for sorting the height data of the roughness profile in ascending order. Figure 3 shows the MRC of the data points sorted by the sort method. The procedure of the sort method is as following steps.

**Figure 3.** MRC using the sort method.


The measured data of a roughness profile may have the same value as height data. In the sort method algorithm, it was clarified that if the same height value exists in the measured data, counting is performed multiple times, which causes a problem in the calculating of MRC (hereinafter referred to as the overlap counting). The overlap counting is one of the factors that obstructs the correct calculation of MRC in the sort method.

Figure 4 shows MRC where the overlap counting occurs. Since the height values from 0.22 to 0.24% are the same, the sort method results in the same number of data points. Therefore, the values from 0.23 to 0.24% are not output on MRC in Figure 4. Therefore, this research develops a new algorithm to solve the problem of overlap counting.

**Figure 4.** Enlarged view of MRC with overlap counting.

#### *3.2. Improved Sort Method for MRC*

The time to calculate MRC was successfully reduced by using the sort method [6]. However, some conditions need to be satisfied so as to use the sort method. If the values of the surface profile are randomly arranged as shown in Figure 5a, the sort method can be applied. If the sort method is applied to the case in which values of the same height occur consecutively, as shown by the red dots in Figure 5b, a problem occurs in the derivation of MRC.

**Figure 5.** Random and rectangular waves: (**a**) random wave; (**b**) rectangular wave (Red dots show that values of the same height occur consecutively).

Therefore, in this research, in order to solve these problems, we develop an algorithm that does not cause problems in the derivation of MRC even if data with the same height value exists. The green circles in Figure 6 show the measured data where the same three values appeared consecutively. In this study, when the same values appear consecutively as shown in Figure 6, we have developed an algorithm that does not cause problems in the derivation of MRC even if up to three points appear consecutively.

**Figure 6.** Example of three consecutive points with equal values (green circle) at a certain slice height (red line).

#### **4. Experiment**

*4.1. Experimental Results by the Sort Method*

Figure 7a shows MRC calculated by the slice method and MRC calculated by the sort method before the improvement of the overlap counting. Figure 7b is an enlarged view of the part where the overlap counting occurs in Figure 7a. The blue line in Figure 7b is MRC by the slice method, and the red line is MRC by the sort method. The slice method (blue line) shows that MRC is parallel to the *X*-axis from 0.22 to 0.24%; therefore, measured data with the same height values exist. On the other hand, the sort method (red line) shows that MRC is a diagonal straight line from 0.22 to 0.25%.

**Figure 7.** Comparison of calculated MRCs: (**a**) MRC by sort method before improvement and slice method; (**b**) enlarged view of (**a**).

The above results show MRC by the sort method is different from that by the slicing method when the measurement data of the same height value exist.

#### *4.2. Experimental Results Obtained Using the Improved Sort Method*

In the sort method before improvement, the locations with the same value of height and the locations where the same value occurs consecutively are recorded in a variable by specifying the condition. In addition, the sort method calculates the number of data points between data points. Therefore, the sort method is improved such that the derivation of MRC does not have problems by modifying the algorithm not to recognize the data points where the same value appears consecutively.

Figure 8a shows the deviations between MRC by the sort method before improvement and the theoretical values based on the total number of data points. The red circle shown in Figure 8a is the error when three same values appear consecutively. Figure 8a shows the deviations between MRC by the improved sort method and the theoretical values. The results in Figure 8b show that the error in Figure 8a has disappeared.

**Figure 8.** Deviations from theoretical value: (**a**) before improvement (Red circle shown the error when three same values appear consecutively); (**b**) after improvement.

From the above results, the development of an algorithm that does not cause problems in the derivation of MRC is succeeded when the same value that appears continuously is up to three. However, if four or more of the same values appear consecutively, a problem still occurs in the derivation of MRC; therefore, further improvement of the algorithm is necessary.

#### **5. Conclusions**

The results and new knowledge of this study can be summarized as follows:


**Author Contributions:** Conceptualization, I.Y.; methodology, H.M. and I.Y.; software, H.M. and I.Y.; validation, H.M. and I.Y.; formal analysis, H.M., I.Y. and Y.K.; investigation, I.Y. and H.M.; resources, I.Y.; data curation, H.M.; writing—original draft preparation, H.M. and I.Y.; writing—review and editing, H.M., I.Y. and Y.K.; visualization, H.M.; supervision, I.Y.; project administration, I.Y.; funding acquisition, I.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Precise Measurement Technology Promotion Foundation (PMTP-F) grant number 1-38.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

