# Surface Roughness Modeling Using Q-Sequence

## Abstract

**:**

## 1. Introduction

_{a}, peak-to-valley height R

_{z}, fractal dimension, and so on [8]. Sometimes, non-conventional parameters, e.g., entropy [8], are also used for quantifying the surface roughness. Some authors have worked on the uncertainty modeling of surface roughness using probability-distribution-neutral representation, e.g., possibility distributions and fuzzy numbers [8]. Nowadays, we have Web-based systems, e.g., the system developed by the National Institute of Standards and Technology (NIST, Gaithersburg, MD, USA) [9], by which one can determine the standard roughness parameters online [1]. In addition, to share the evaluated surface roughness, efforts have been made to create web-based data, e.g., XML data incorporating the information of R

_{a}[10]. Even though we characterized a surface by using R

_{a}, R

_{z}, entropy, fractal dimension, and possibility distribution, an individual/system who/that receives this information will not be able to recreate the surface heights. This means that we lose the surface height dynamics if we use the conventional quantification process described above. Thus, we need another means to capture the dynamics of surface roughness so that the individual/system) who/that) receives it can recreate it whenever necessary. It has been stressed that the manufacturing phenomena (e.g., the cutting force and even the surface roughness) are highly non-linear and can be modeled, in principle, by stationary and non-stationary Gaussian processes [11,12,13]. A surface roughness profile, in particular, consists of certain stochastic features [11], and to recreate the heights of a surface roughness profile, i.e., to capture the dynamics underlying a surface roughness profile, one needs to model these features using stochastic formulations [11,14]. This issue of dynamic representation of surface roughness has become even more important due to the advent of Internet of Things (IoT) [15], as schematically illustrated in Figure 1.

## 2. Q-Sequence

^{+}for n ≤ 2000 (i.e., relatively small values of n). Pinn [25] studied the chaotic behavior that Q(n) exhibits for both small and large values of n. Some of the characteristics reported in [25] are as follows: (1) On short scales, Q(n) looks chaotic having groups of generations of sequences. (2) The k-th generation has 2k members which have “parents” mostly in the generation k − 1, and a few from the generation k − 2. In other words, in a short scale, a segment of Q(n) consists of integers that are close to each other. (3) Q(n) has bursts with increasing amplitude and length. (4) The trend in Q(n) disappears if the integer part of n/2 is subtracted. This is true if n is relatively small.

## 3. Modifying the Q-Sequence

^{2}| t = 0,1, …, ∃j ∈ {1, 2, …}} created from a time series y(t) ∈ ℜ, t = 0, 1, etc. It can represent the hidden order underlying the parameter considered, which is difficult to grasp from the time series plot alone. For this reason, when one studies a non-linear behavior or dynamical system, e.g., S(n), return-maps are prepared along with the time series to attain more insights into it [26]. This role of the return-map can be understood clearly when one compares the time series plots and the return-maps shown in the remainder of this article.

## 4. Modeling Surface Roughness

_{max}= max(z(t) | t = 0, 1, …), z

_{min}= min(z(t) | t = 0, 1, …), r

_{max}= max(R(t,3) | t = 0, 1, …), and r

_{min}= min(R(t,3) | t = 0, 1, …). If needed, one can add noises using a stochastic process, e.g., a noise that follows normal distribution [28], to V(t) to make it even more meaningful. The idea of adding such a stochastic noise is kept out of the scope of this article, however.

_{max}= 2.260563913, z

_{min}= −2.020452268, r

_{max}= 81, r

_{min}= −72, and α = −0.1. As seen from time series plots in Figure 10, both segments show a similar kind of variability. The similarity between the variability is more evident in the plot of the return-maps. In particular, both return-maps overlap and have almost the identical returns from one point to another.

_{a}, R

_{z}, and Entropy are calculated as described in [8]. The results are summarized in Table 1. As seen from Table 1, R

_{z}is the same for both model and real surface roughness profiles, whereas R

_{a}of the model surface roughness is greater than that of the real surface roughness profile, and Entropy of the model surface roughness is less than that of the real surface roughness profile. This difference in Entropy refers to the fact that the real surface roughness heights exhibit more complexity than those of the model.

## 5. Discussion and Concluding Remarks

- Step 1
- Generate Q(n) (Equation (1)).
- Step 2
- Generate S(n) (Equation (2)).
- Step 3
- Select a segment of S(n), S(n1,n2) (Equation (3)).
- Step 4
- Expand S(n1,n2) by a linear interpolation operation and generate R(t,1) (Equation (4)).
- Step 5
- Expand R(t,1) by successive linear interpolation and generate R(t,i), i = 1, 2, … (Equation (5)).
- Step 6
- Select one of the R(t,i), ∃i ∈{1, 2, …}.
- Step 7
- Scale and shift R(t,i) selected in Step 6 and generate V(t) (Equation (6)).
- Step 8
- Compare V(t) with the given surface profile z(t) or with one of its (z(t)’s) linear interpolated profiles, z(t,j), j = 1, 2, etc.
- Step 9
- If the comparison is satisfactory in terms of qualitative and quantitative measures, then accept V(t) as a model of the roughness profile. Create semantic web to use V(t) in the framework of IoT. Otherwise, go back to Step 3 and continue.

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**The time series and return-map of R(t,1) corresponding to S(1000,1200): (

**a**) time series; (

**b**) return-map.

**Figure 6.**The time series and return-map of R(t,2) corresponding to S(1000,1200): (

**a**) time series; (

**b**) return-map.

**Figure 7.**The time series plot and return-map of R(t,3) corresponding to S(1000,1200): (

**a**) time series; (

**b**) return-map.

**Figure 9.**The time series plot and return-map of the linear interpolated roughness profile: (

**a**) time series; (

**b**) return-map.

**Figure 10.**Comparison between the model and real surface roughness profiles: (

**a**) time series; (

**b**) return-map.

**Figure 11.**Comparison between real surface roughness and its model in terms of the degree of possibility.

Parameters | Model (V(t)) | Real (z(t,2)) |
---|---|---|

R_{a} (μm) | 0.471 | 0.419 |

R_{z} (μm) | 4.281 | 4.281 |

Entropy (Bits) | 3.209 | 4.063 |

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Ullah, A.M.M.S.
Surface Roughness Modeling Using Q-Sequence. *Math. Comput. Appl.* **2017**, *22*, 33.
https://doi.org/10.3390/mca22020033

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Ullah AMMS.
Surface Roughness Modeling Using Q-Sequence. *Mathematical and Computational Applications*. 2017; 22(2):33.
https://doi.org/10.3390/mca22020033

**Chicago/Turabian Style**

Ullah, A.M.M. Sharif.
2017. "Surface Roughness Modeling Using Q-Sequence" *Mathematical and Computational Applications* 22, no. 2: 33.
https://doi.org/10.3390/mca22020033