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Mathematical Model of Forecasting Laser Marking Eperiment Results Pavels Narica Artis Teilans Lyubomir Lazov Pavels Cacivkins Edmunds Teirumnieks Faculty of Engineering, Rezekne Academy of Technologies,

Mathematical Model of Forecasting Laser Marking Eperiment Results Pavels Narica Artis Teilans Lyubomir Lazov Pavels Cacivkins Edmunds Teirumnieks Faculty of Engineering, Rezekne Academy of Technologies, Latvia, Abstract Method allows for modelling of the anticipatory results of colour laser marking eperiments. The process of calculating epected results takes into consideration the construction specifics of laser system being used and displays the results in compact form of a set of parameter matrices that have their values conditionally formatted as colour maps for easy identification of comple patterns. The complete set of all the related parameter matrices, both technical and derived, as well as the specific relations between them form the mathematical model of forecasting laser marking eperiment results. Because the mathematical model is implemented within spreadsheet processor, it can be instantiated multiple times for any number of eperiments. Keywords: mathematical model, modelling method, laser system construction specifics, visualization of parameters, matrices, eperiments, colour laser marking 1 Introduction The main goal of this research is to simplify the process of forecasting and visualizing the distribution of values of eperimental laser parameters. Laser systems are widely used and provide some spectacular capabilities in many different fields. However lasers are complicated systems and operating a laser system requires a very good understanding of how lasing is actually done and how it can affect surroundings. One case of common use of laser systems is colour laser marking on metal. Pulsed fiber lasers are usually used for producing markings of different colours on metal workpiece. Such laser systems are rather small in size, very precise, relatively inepensive, and provide easy to use computer software for plotting laser marking elements. When plotting different elements in marking software, different technical laser parameters can be assigned to them in order to produce different marked colours (Laasko et al, 2008; Antonczak et al, 2014; Veiko et al, 2016; Qia et al, 2003; Antonczak et al, 2013; Amara et al, 2015; Lehmuskero et al, 2010; Veiko et al, 2014). In many cases while operator is testing the laser system for its marking capabilities he/she marks some matrices of different combinations of laser's technical or other physical parameters. While this is logical approach, in most such cases it has serious issues, because the construction of laser system may behave differently under different set of parameters and thus produce unepected or difficult to interpret marking results. Because metal workpieces used for laser marking tests usually have rectangular flat two-dimensional surface (Figure 1), plotting on them rectangular matrices of rectangular elements is very efficient. This way an operator can assign for all the rows of such matri some distinct technical or other test parameter and then assign all columns some other distinct test parameter while at the same time varying linear or logarithmic values of both assigned test parameters for each row and each column. Figure 1. Eperimentally produced sample marking colours on stainless steel workpiece. When both dynamic test parameters are assigned to rows and columns of the matri, all remaining parameters for all matri elements are set to have some constant value. Such plotted matri is then marked and can be further analyzed. This way one should be able to see how varying values of only two test parameters affect the produced marking colours. Though, as it was mentioned above, in some cases the construction of laser system itself can indirectly affect some additional parameters thus making analysis of marking colours against selected dynamic test parameters of marked matri much more difficult and unintuitive. 800 When output marking samples are analyzed, the corresponding output colours are usually mapped against technical and other physical parameters set dynamically before marking test as well as other constant technical laser parameters and parameters related to the nature of marking process itself. This approach helps one better see which parameters cause marking colours to change. Yet such mappings usually are in form of linear flat tables requiring lots of rows for each eperimentally marked colour and columns for listing all parameters to display all the necessary data. Within such table it may become very hard for one to efficiently identify important parameters that actually affect the formation of marked colour. Finally, it is common practice to produce marking eperiment first and then to analyze resulting colours against parameters afterwards. Such practice can often result in regions within marked test matri that do not have many distinct and quality colours in them. The present method allows for modelling colour laser marking eperiments while considering the construction specifics of a laser system. In particular, it is a novel method that deploys built-in spreadsheet processor's features in order to automatically generate informative numeric data matrices and their visualizations by means of conditional formatting (Figure 2, Figure 3). Figure 2. Distribution of planar energy densities for sample. 2 Materials and methods Eperiments were carried out using PowerLine F-20 Varia series pulsed fiber laser system produced by Rofin-Sinar Laser GmbH. It emits photons of wavelength equal to 1064 nm, has maimum average lasing power of 20 W, pulse repetition rate of khz, adjustable pulse width of ns. The stainless steel sample used for colour laser marking was E 2R. Table 1 provides information used by current implementation of the mathematical model. It uses such common technical parameters as f for pulse repetition rate, v for scanning speed, P for average lasing power, Δ for line step (distance between two lines), and y for width and length of individual rectangular marking element (Bliedtner et al, 2013; Eichler, 1998). Table 1. Formulas used within proposed mathematical model as an eample. Description Formula Base unit f 1 Pulse overlap (linear pulse density) N y [ mm ] v Line overlap (linear line density) Pulse energy Planar pulse density Planar energy denstiy Total energy delivered to element Total marking time of element N 1 1 [ mm ] P E P [ J] f N N y N E N N P y [ mm 2 ] [ J / mm E y [J ] N y t [s] v The method of modelling colour laser marking eperiments takes into consideration the construction specifics of laser system. The method then displays anticipated results in compact form using equally sized matrices that have conditional formatting applied to their values. Present method allows for quick visual relation of parameters and their values as well as identification of patterns associated with the eperiment (Figure 3). The aim of the method is to make analysis of the colour laser marking eperiments easier to understand, define, quantify, visualize, teach, and simulate by referencing it to eisting and usually commonly accepted knowledge and formulas. The method offers a new way of looking at what actually happens to different eperimental parameters before the actual marking of eperiments. Therefore, as one identifies the set of the most important eperimental parameters affecting the colour laser marking process, one can then better optimize the colour laser marking process itself. The method implements one primary mathematical model of forecasting results for colour laser marking eperiments that incorporates three secondary interconnected mathematical models (Figure 4). The primary mathematical model is a set of related matrices (Figure 5) with relations being defined by lasing formulas shown in Table I. The present method allows for addition of new as well as removal of unnecessary eperimental parameter matrices, both technical and derived. The present method allows for very fast and fleible input of technical parameters and their values for the eperiment as well as immediate output of the epected results. 2 ] 801 Figure 3. Overview of proposed mathematical model for eperimentally produced sample (with constant pulse width of 4 ns). Once all three secondary mathematical models are defined, the primary mathematical model is set and can be instantiated for any different eperiment. Thus, before one proceeds with actual marking of test matrices, it is possible to efficiently plan the eperiments and see epected distribution of values of output parameters in advance. 3 Results and discussion The method maps and visualizes parameters of colour laser marking eperiments. The method displays all the necessary eperimental parameters, both technical and derived, that are associated with marking eperiment of a test matri in a compact form of a set of equally sized matrices where each such matri is assigned its own eperimental parameter and stores that parameter's values positionally related to elements of test matri to be marked as well as the values of every other eperimental parameter matri. Thus the number of rows and columns of each such eperimental parameter matri is identical to those of test matri itself. The method of mapping values of necessary eperimental parameters within their own matrices to elements of the test matri lets one immediately start seeing eperimental values in contet of two dimensions corresponding to those of test matri, also in the contet of values belonging to all other eperimental parameter matrices, and, more importantly, in the contet of marking colours to be produced. Thus one might think of the set of all eperimental parameter matrices including the test matri to be marked itself as a three-dimensional stack of matrices where each matri cell that corresponds to specific row and specific column is directly associated with every other such cell up and down the stack. 802 Figure 4. Schematic view of primary mathematical model implemented by the present method: (1) secondary mathematical model of input of eperimental data; (2) secondary mathematical model of output of eperimental data; (3) secondary mathematical model of laser system's construction specifics; (4) instances of laser system's technical parameter matrices; (5) instances of derived parameter matrices. Figure 5. Schematic view of parameter's matri: (6) header associated with parameter's matri containing information about the parameter in question, its commonly accepted physical symbol and base unit; (7) two-dimensional array of distribution of values of the parameter. By applying spreadsheet processor conditional formatting feature to the values of each eperimental parameter matri, one can see the planar distribution of relative magnitudes of these values coded with a colour scale. Thus each eperimental parameter matri becomes a colour map and can be analyzed when spreadsheet processor worksheet is zoomed out. Each such colour map helps efficiently spot contrasting values within all matrices just by quickly looking through them. Interestingly enough, distinct patterns may emerge within some of such eperimental parameter colour maps. Such patterns may even sometimes resemble the pattern seen within the marked test matri itself. Though, more often similar patterns can be observed within the set of eperimental parameter matrices. Thus if any two such patterns look similar or even identical one might assume these are somehow proportional to each other. The method implements the primary mathematical model of forecasting results for colour laser marking eperiments which in turn comprises three secondary 803 interconnected mathematical models. The secondary mathematical model of laser system construction specifics allows user to define any conditions where laser system may behave differently by setting specific relations between the set of technical parameter matrices and the set of derived parameter matrices. The set of all laser system's technical parameter matrices represents the secondary mathematical model for input of eperimental data. The set of all derived parameter matrices forms the secondary mathematical model for output of eperimental data. Thus the method allows for modelling distribution of the values within eperimental laser parameter matrices for any particular colour laser marking eperiment. Both secondary mathematical models for input of eperimental data and for output of eperimental data operate with numeric data. 4 Conclusions The user's overall understanding of how the present method's primary mathematical model of forecasting laser marking eperiment results works can help the user etract even more useful information out of it. Therefore the method can be used for interactive teaching purposes or to assist advanced users. Finally, the method can help its users better interpret and test common lasing formulas as well as produce new ones, and no similar modelling method, which allows that, yet eists. There is always a problem of eperiment repeatability when testing produced marking colours for technical parameters mentioned in research papers of other authors. The proposed mathematical model should be able to help authors to present their eperimental parameters and their values as well as produced marking colours in a very compact form that is both easy to understand and easy to implement. During testing of the mathematical model it was found that different formulas which define relations between the technical parameter set and the derived parameter set are easy to view, edit and copy. The method can also be useful for researchers testing different theories about colour laser marking, since adding new or eliminating unnecessary parameter matrices is easy and different emerging colour patterns help quickly spot parameters that matter in the contet of colour laser marking eperiments. Acknowledgements This research was supported by laser system equipment from Rofin-Sinar Laser GmbH. References E. H. Amara, F. Haïd, and A. Noukaz. Eperimental investigations on fiber laser color marking of steels. Applied Surface Science, 351:1-12, doi: /j.apsusc A. J. Antonczak, D. Kocoń, M. Nowak, P. Kozioł, and K. M. Abramski. Laser-induced colour marking Sensitivity scaling for a stainless steel. Applied Surface Science, 264: , doi: /j.apsusc A. J. Antonczak, B. Stepak, P. E. Kozioł, and K. M. Abramski. The influence of process parameters on the laser-induced coloring of titanium. Applied Physics A, 115(3): , doi: /s J. Bliedtner, H. Muller, and A. Barz. Lasermaterialbearbeitung. Fachbuchverlag Leipzig im Carl Hanser Verlag, doi: / J. Eichler and H. J. Eichler. Laser. Springer-Verlag Berlin, doi: / P. Laakso, H. Pantsar, and V. Mehtälä. Marking decorative features to stainless steel with fiber laser. IMD/ALAC, A. Lehmuskero, V. Kontturi, J. Hiltunen, and M. Kuittinen. Modeling of laser-colored stainless steel surfaces by color piels. Applied Physics B, 98(2-3): , doi: /s J. Qi, K. L. Wang, and Y. M. Zhu. A study on the laser marking process of stainless steel. Journal of Materials Processing Technology, 139(1-3): , doi: /S (03) V. Veiko, G. Odintsova, E. Ageev, Y. Karlagina, A. Loginov, A. Skuratova, and E. Gorbunova. Controlled oide films formation by nanosecond laser pulses for color marking. Optics Epress, 22(20): , doi: /oe V. Veiko, G. Odintsova, E. Gorbunova, E. Ageev, A. Shimko, Y. Karlagina, and Y. Andreeva. Development of complete color palette based on spectrophotometric measurements of steel oidation results for enhancement of color laser marking technology. Materials & Design, 89: , doi: /j.matdes

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