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In this paper, Recurrence Quantification Analysis ( RQA) is set as a practical nonlinear data tool to establish and compare surface roughness (Ra) through percentage parameters of a dynamical system: Recurrence (% REC), Determinism (% DET) and Laminarity (% LAM). Variations in surface roughness of different machining procedures from a typical metallic casting comparator are obtained from scattering intensity of a laser beam and expressed as changes in the statistics of speckle patterns and profiles optical properties. The application of the analysis ( RQA) by Recurrence Plots ( RPs), allowed to distinguish between machining procedures, highlighting features that other methods are unable to detect.

Surfaces generated by machining processes contain topographic characteristics determined by different working conditions [_{a}) and mean-square roughness (R_{rms}) to detect changes related to profile amplitudes, and for this reason an alternative technique which does not depend on such parameters is also necessary.

Chaos theory may be considered for surface height fluctuations treatment, as it takes phenomena, which in spite of their apparent random nature, they are governed by deterministic laws which produce complex results as they are mutually combined. Among different chaotic signals analysis methodologies, Echmann et al. proposed the so called Recursive Plots (RPs) method [

The development of several non-contactive methods for surface roughness quantification is a well known topic, being optical techniques the most widely used to measure surface features [_{a}) from different machining processes with known surface roughness. Analysis data, obtained from a scattered laser beam, reveal changes on statistical properties of optical speckle patterns as related to corresponding mechanical changes.

A λ = 533 nm, 50 mW output power green DPSS laser was used as the light source. A 40× microscope objective combined with a f = 50 mm biconvex lens were used to expand and collimate the laser beam. An Alfa a 200 CCD camera with a 3872 × 2592 pixels resolution and 6.095 × 7.253 mm pixel size was used for capturing and recording of speckle patterns. A roughness comparator (Microfinish,

collected by means of a biconvex lens (f = 200 mm) and directed to a power photodetector coupled to a data acquisition unit, which was also coupled to a PC in order to record it for latter analysis. For speckle optical patterns obtaining (^{2} area of the comparator was illuminated with the laser beam considering that its corresponding reflected speckle pattern, recorded with a CCD Alfa a 200 camera, contains the variations and irregularities existing on this surface area.

An application software was used for digital data processing of each recorded speckle pattern. Intensity profiles (1 × 1616 pixel vectors) corresponding to ~78.53 mm^{2} illuminated cross-section were obtained; the resulting vector represents the image average intensity profile which also contains information about surface variations and irregularities of the considered surface area.

The point-to-point and surface speckle optical profiles were used to perform a qualitative analysis of the considered surface section of the comparator by means of RPs; for a quantitative analysis the RQA technique were also used. The Visual Recurrence Analysis application software (VRA Ver. 4.9) was used for RPs building [

In this work, the Commandline Recurrence Plots application software was used for complex structures quantification [

Implicit periodic processes are characterized by higher %REC values. Determinism is the percentage of recurrence points forming parallel segments on the main diagonal, which length reaches or exceeds minimum length threshold; %DET allows to tell between dispersed recurrence points and those points organized in diagonal patterns, and contains information about duration of a stable interaction as longer is an interaction, as higher is the %DET value:

where l is diagonal line length parallel to the identity line; P(l) is the frequency distribution of diagonal lines and R_{ij} is the recurrence of a state. Laminarity is the percentage of recurrence points included in linear segments vertical to the diagonal; %LAM measures chaotic transitions and is related to the quantity of laminar phases in the system (intermittence):

The Kolmogorov Entropy is a measure of information loss index (o gain) along an attractor of certain data set. Numerically speaking, K may be estimated as the Rényi Entropy, being the Information Theory of Shannon Entropy a particular case. On the other hand, the product of the Shannon Entropy and the Boltzmann constant is the Thermodynamic Entropy [

where {X = x_{i}} is the random variable and_{1}, x (t = 2Dt) in i_{2}, and x (t = NDt) in i_{N}. Here, application software RRCHAOS [

The Approximate Entropy (ApEn) quantifies the regularity and complexity in real time series, where time series with higher ApEn values may present considerable irregularity fluctuations (more complex), while lower ApEn values would indicate more regular time series (less complex). Given its simplicity, ApEn has been applied in many fields of science and engineering including psychology, geophysics and financial systems. A detailed description of the method may be found on specialized literature [

For calculation of ApEna Matlab code implementation by Danny Kaplan was used here [

In _{a} = 0.2 - 12.7 mm.

For qualitative (RPs) and quantitative (RQA) optical and speckle profiles analysis, a mutual information function analysis was performed to obtain the optimal time-delay value (τ) and consequently, by using the false neighbor method, it was used to obtain the embedding dimension value (m) where maximum dimension value was set on 10 and then, from obtained τ and computed m, RPs were generated with VRA software [

The analysis consisted in generating RPs from the optical and speckle patterns of different machining processes for different surface roughness degrees. In

The information about the dynamics of a time series/data is usually obtained from the density of points and line structures in a RPs [

Process | Surface | Speckle pattern resolution: 1616 × 1080 pixels | R_{a} [mm] | Optical profiles | Speckle profiles | ||||
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t | m | e | t | m | e | ||||

Lapped | 0.2 | 29 | 10 | 3.54 | 5 | 4 | 2.04 | ||

Rectified | 0.4 | 23 | 10 | 3.54 | 4 | 4 | 2.76 | ||

0.8 | 29 | 5 | 2.55 | 5 | 5 | 3.03 | |||

Profiled | 1.6 | 26 | 10 | 3.45 | 3 | 3 | 2.44 | ||

3.2 | 38 | 4 | 2.16 | 5 | 5 | 3.13 | |||

6.4 | 30 | 7 | 2.46 | 3 | 3 | 2.43 | |||

12.7 | 46 | 4 | 2.55 | 2 | 2 | 1.97 |

As it can be observed on Fig. 4, surface roughness R_{a} offers an initial visual inspection for identification of changes on dynamics of the structure, reflected on RPs textures which are defined as small scale patterns and revealed by the color and/or texture type (generated by the machining process) in the RP. The optical profiles (

Although RPs offer an initial visual comparison of surface roughness, it is necessary to evaluate it in a quantitative way. In order to perform Recurrence Quantification Analysis (RQA), the m, τ, and ε parameters (indicated in _{a}; the average %REC for the profiles was 47.69% y 60.23%, respectively, which implies that analyzed surface textures were generated by processes with certain degree of periodicity.

The calculated Determinism (%DET) is shown in _{a} degrees. The average %DET was 99.756% for optical profiles and 91.534% for speckle ones; this is reflected on surface texture nature of machining processes (see

In

_{a} = 0.2 µm) correspond to the zone I (lapping process); at zones II (rectified process) and III (profiled process), increments with maxima values at KML = 1.237 bits/seg (R_{a} = 0.8 µm), KML = 1.043 bits/seg (R_{a} = 1.6 µm) and KML = 1.13 bits/seg (R_{a} = 3.2 µm) were obtained. Such increments may be associated with a high sensitivity to external conditions or perturbations as the alternation of regular and irregular texture surfaces, for example, presented on speckle pattern intensities.

Regarding Approximate Entropy (ApEn) calculation shown in

The RQA, KML and ApEn analysis methods can be suggested as practical comparison tools for the surface roughness (R_{a}) on different reliefs. From the presented results and considerations, one might conclude that such techniques are very sensitive to changes on mechaning conditions and can be thoroughly employed to establish

and compare the machinability of different types of metallic materials. A combined application of optical methods together with those based on nonlinear data analysis is here outlined.

Oscar Sarmiento Martinez,Darwin Mayorga Cruz,Jorge Uruchurtu Chavarín,Estela Sarmiento Bustos, (2016) Recurrence Quantification Analysis of Rough Surfaces Applied to Optical and Speckle Profiles. Journal of Applied Mathematics and Physics,04,720-732. doi: 10.4236/jamp.2016.44083