The dynamics of laser droplet generation

Abstract

We propose an experimental setup allowing for the characterization of laser droplet generation in terms of the underlying dynamics, primarily showing that the latter is deterministically chaotic by means of nonlinear time series analysis methods. In particular, we use a laser pulse to melt the end of a properly fed vertically placed metal wire. Due to the interplay of surface tension, gravity force, and light-metal interaction, undulating pendant droplets are formed at the molten end, which eventually completely detach from the wire as a consequence of their increasing mass. We capture the dynamics of this process by employing a high-speed infrared camera, thereby indirectly measuring the temperature of the wire end and the pendant droplets. The time series is subsequently generated as the mean value over the pixel intensity of every infrared snapshot. Finally, we employ methods of nonlinear time series analysis to reconstruct the phase space from the observed variable and test it against determinism and stationarity. After establishing that the observed laser droplet generation is a deterministic and dynamically stationary process, we calculate the spectra of Lyapunov exponents. We obtain a positive largest Lyapunov exponent and a negative divergence, i.e., sum of all the exponents, thus indicating that the observed dynamics is deterministically chaotic with an attractor as solution in the phase space. In addition to characterizing the dynamics of laser droplet generation, we outline industrial applications of the process and point out the significance of our findings for future attempts at mathematical modeling.

Dynamics of stainless steel turning: Analysis by flicker-noise spectroscopy

“…Various characteristics/techniques of nonlinear time series analysis [42], including correlation dimension, Lyapunov exponents, Kolmogorov-Sinai entropy, surrogate data tests, singular value decomposition, and many others, have been used to examine the dynamics of steel turning [22] as well as other manufacturing processes, for instance, laser droplet formation in the welding of electrical contacts [43,44]. The estimation of correlation dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy relying on the Takens' embedding theorem, which is valid only for stationary signals (when applied to real signals, it is assumed that both mean and variance are approximately constant with time at any scale) with a large number of sampling points, is complicated in this case due to the apparent nonstationarity of the signals (as clearly seen in Fig.…”

Determining the chaotic behaviour of copper prices in the long-term using annual price data

“…Previous studies have proven the suitability and reliability of small data sets with 50 observations in regards to their calculation of λ for assessing chaotic behaviour of complex dynamic systems (Becks et al, 2005;Blank, 1991;Chen et al, 2016;Gaspard et al, 1998;Navarro-Urrios et al, 2017;Raffalt et al, 2017;Reynolds et al, 2016;Sivakumar, 2000). Moreover, the same methodology has been successfully applied to study the complexity of various dynamic systems, including electrocardiograms, human gait recording, and laser droplet generation (Krese et al, 2010;Perc, 2005aPerc, , 2005b.…”

Nonlinear analysis of laser droplet generation by means of 0–1 test for chaos

“…Over the past years nonlinear time series analysis has proven to be insightful in many fields of science and engineering [9], from biology [10][11][12] and physiology [13,14] to manufacturing processes [15,16]. Results of applying these methods to the laser droplet generation established that the process can be considered as a low dimensional deterministic dynamical system with inherent chaotic behavior [4].…”

“…Initial researches were engaged to analysis and optimization of a single droplet generation [3], while the droplet sequence generation dynamics has only been addressed recently [4]. Usually, dynamics analysis is performed through dynamical system model analysis, but when mathematical model of the observed process is unknown or deficient due to process complexity, one faces the problem of characterizing the process dynamics by means of analyzing experimental data.…”