Complexity

Volume 2017, Article ID 8017138, 10 pages

https://doi.org/10.1155/2017/8017138

## Determining the Coupling Source on a Set of Oscillators from Experimental Data

^{1}División de Investigación y Posgrado, Facultad de Ingeniería, Universidad Autónoma de Querétaro, 76010 Santiago de Querétaro, QRO, Mexico^{2}Departamento de Ingeniería Mecánica, Facultad de Ingeniería, Universidad Autónoma de Querétaro, 76010 Santiago de Querétaro, QRO, Mexico^{3}Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

Correspondence should be addressed to Juan Carlos Jauregui-Correa; xm.qau@iugeruaj.cj

Received 9 January 2017; Accepted 5 April 2017; Published 4 May 2017

Academic Editor: Jia Wu

Copyright © 2017 Juan Carlos Jauregui-Correa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Complex systems are a broad concept that comprises many disciplines, including engineering systems. Regardless of their particular behavior, complex systems share similar behaviors, such as synchronization. This paper presents different techniques for determining the source of coupling when a set of oscillators synchronize. It is possible to identify the location and time variations of the coupling by applying a combination of analytical techniques, namely, the source of synchronization. For this purpose, the analysis of experimental data from a complex mechanical system is presented. The experiment consisted in placing a 24-bladed rotor under an airflow. The vibratory motion of the blades was recorded with accelerometers, and the resulting information was analyzed with four techniques: correlation coefficients, Kuramoto parameter, cross-correlation functions, and the recurrence plot. The measurements clearly show the existence of frequencies due to the foreground components and the internal interaction between them due to the background components (coupling).

#### 1. Introduction

Complex systems are common in engineering, examples being communication networks, computer networks, and electronic devices, as well as mechanical systems such as manufacturing plants, aircrafts, and automobiles. Determining the dynamic behavior of these mechanical systems requires the use of advanced analytical techniques. As the number of elements increases, simple systems evolve into ones that are more complex. Concepts such as correlation dimension, Lyapunov exponents, fractal properties, and complex network analysis are some of ideas that are used to determine dynamic properties. The problem is to identify the source of synchronization from field data.

In addition to other behavioral characteristics, complex systems can present synchronization. In most cases, the system deploys two behaviors: one that is easily identified at a foreground level and another that remains at background level [1]. The foreground behavior is seen at frequency responses with high amplitudes and narrow bands, whereas the background behavior presents a wide band response and low amplitudes. The study of synchronization has been an exciting and interesting problem since Huygens first wrote about his experiments in 1665 [2]. Peña Ramirez et al. [3] analyzed Huygens’ experiment, modeling the system as a lumped mass model, and found that synchronization depends on the relationship between the masses, the stiffness of the connecting beam, and the amount of damping in the system.

Keeffe et al. [4] presented a model for the analysis of pulse coupled oscillators. As they described it: in some systems synchronization starts at a certain location and expands forming clusters. They suggested that clusters formed, evolved, and collapsed. For this purpose, they developed a model that provides an idea of the physical phenomena and how clusters evolved into synchronization. They described different systems that behave as pulse coupled oscillators. They presented two types of oscillators, systems with local coupling and systems with global coupling. There is still a question how transient dynamics leads to synchronization [4]. Ulrichs et al. [5] presented the analysis of metronomes as nonlinear periodic oscillators. The metronomes behave asymptotically and they show higher dimensional attractors. Metronomes have the same behavior as Huygens clocks, and their synchronization can be modeled using Kuramoto’s parameter. They found small traveling waves through the system support that lock the phase difference among all metronomes and they model the connections in terms of a Van der Pol oscillator. Aragonès and Guasch [6] presented a theoretical analysis of path propagation in a set of systems with similar structures. Woodhouse [7] and Langley et al. [8, 9] introduced statistical energy analysis. They modeled the system as a set of springs and represented it as a block diagram. The diagram represented subsystems as blocks and power flows as connections among subsystems. For the net analysis, they assigned weight factors relating a coupling loss factor to a total loss factor. Moreover, they dealt with the net analysis including a probabilistic density function and computed the weight factors as a statistical ratio. They defined the length of a particular link as a random variable, and they defined this function as a mean function plus a function of the standard deviation. A similar approach was presented by Llerena-Aguilar et al. [10]. They reported a hybrid method for the analysis of wave propagation and synchronization of a Wireless Acoustic Sensor (microphones) Networks. The problem they solved is the inverse of what we are trying to understand. In fact, they studied the desynchronization. The phenomenon is caused by two factors, namely, the clock phase offset of each sensor and the propagation delay due to the distance between them. These factors are critical in the selection of the processing algorithm. The classical solution was the implementation of clock synchronization protocols. The phenomenon is simulated assuming that the source signals are received by an array of sensors. Then, the signals are combined adding a Gaussian noise and convoluting the signal with an impulse function for each sensor. They discretized the signal using the Short Time Fourier Transform and they aligned the mixture of signals using the cross-correlation and the Log Energy Cross-Correlation functions. Ray et al. [11] investigated the Noise Induced Synchronization in two identical time delay systems. They compared numerical results with experimental data and they analyzed the effect of white, green, and red noises. Messina and Vittal [12] reported a method for analyzing dynamic patterns from measurements. They applied the Hilbert transform and proper orthogonal analysis technique. This technique is based on the relation of the energy and phase embedded into the mode function. Rosenblum and Pikovsky [13] presented a method for detecting weak couplings between two oscillators. If two oscillators are weakly coupled, they assume that the two signals contain enough data from the weak interaction. The relationship in a driven-response from noisy data could be unreliable and the interpretation is difficult. They discussed the effect of noise as follows: “in case of perfect synchrony, we are not able to separate the effect of interaction from the internal dynamics of autonomous systems. In order to obtain the information on the direction of coupling we need to observe deviations from the synchrony, either due to noise or due to onset of quasiperiodic dynamics outside the synchronization region.” They propose a method for determining the direction of the coupling and they quantified the cross-dependencies of the phase dynamics. They illustrate their method with two coupled Van der Pol oscillators.

Friedrich et al. [14] presented an approach for the study of complex systems using stochastic methods. Complex systems can be traced back to rather simple laws because of the existence of self-organized processes. While the analysis of nonlinear dynamics is traditionally conducted using time series, complex systems show stochastic data in time and length scales. It is clear that the synchronization of complex systems is a function of time and space. In Friedrich’s work, the dynamic model is represented as a Langevin equation with white noise. Since the dynamic model can be represented as a deterministic set of nonlinear differential equations, they added a stochastic term with white noise. They considered a Gaussian distribution for time propagation as a Markov process. A model for self-similarity was included, meaning that at a certain time two elements should have the same statistics. In a previous work, Ghasemi et al. [15] presented the steps below for developing the stochastic equation. First, review if the data follow a Markov chain. Then, estimate the Markov time scale, which is the minimum time interval for a Markov process. For this analysis, it is necessary to use a joint probability density function. This can be simplified with the Chapman-Kolmogorov equation, which is the probability of having two similar distributions in a time interval. Then, they derived the stochastic equation and a regeneration of the stochastic process. They applied the method to heart beat signals. In complementary analyses, Siefert et al. [16] and Shirazi et al. [17] classified complex systems as networks with fluctuating stochastic processes. For this problem, they y represented spatial networks of complex dynamics in phase space. They proposed a method for mapping a stochastic process onto a complex network. As for recurrence plots, they determined the Markov time scale, and they based their analysis on the Markov process.

Marwan et al. [18] presented a review of recurrence plots applied to dynamic systems. Recurrence is one of the basic properties of dynamic systems and the recurrence plots can help visualize dynamic system behavior. They described the use of recurrence plots for identifying synchronization in the sense that two systems synchronize if their recurrence in phase space is linearly dependent. In another work, Donner et al. [19, 20] studied the dynamics of complex systems using the recurrence concept and network theory. They demonstrated that exploiting the duality of the recurrence matrix and the adjacent matrix of a complex network improved the determination of quantitative characteristics of recurrence networks. They divided them into three types of recurrence networks, that is, local, intermediate, and global. All their results were obtained from simulations.

Other researchers have applied different techniques to identify synchronization. Huo et al. [21] analyzed the synchronization of a couple of calcium oscillators. They proposed a mathematical model based on a set of nonlinear first-order differential equations. They found synchronization through the analysis of phase diagrams. Kim and Lim [22] worked on burst and spike synchronization of neurons. Their interest was aroused by the analysis of electroencephalogram (EEG) data. Researchers believed that synchronization of neuron activities could help in understanding sensory and cognitive processes. From an engineering point of view, it can be understood as periods of high frequency impulse generation and periods of no impulse generation. Xu et al. [23] and Rosário et al. [24] described motif synchronization. They studied temporal and spatial synchronization from EEGs and applied time-varying graphs, static network, and motif synchronization. Motif synchronization is a sequence of small patterns in a particular order of occurrence. With three first-order differential equations, they simulated the propagation of an external stimulus in time and space.

Childs and Strogatz [25] analyzed a periodically forced Kuramoto model. They defined a rich dynamic system when it has a strong interconnection among randomness, force, and coupling. Randomness is associated with variations in the natural frequencies; this concept is important because it determines the desynchronization of the oscillators. The coupling is related to the oscillator’s phase and the force to the external excitation. Thus, the system synchronizes and desynchronizes depending on the relative magnitude of these three effects. Another important concept described by Childs and Strogatz is that a subset of the oscillators is “phase-locked” to the force drive, whereas the rest of the oscillators are on the tails of the frequency distribution outside of synchronization. Their method is applied to a set of Van der Pol oscillators and they proposed that the oscillators are sinusoidal coupling; in our case it is assumed that the coupling has a very low frequency compared to the natural frequency and the amplitude is low.

The objective of the present paper is to determine, using experimental data, if there is a way to identify the background effects that enable synchronization in complex systems and coupling. It is assumed that a coupling source exists and is the cause that synchronizes all the elements. The difficulty of identifying this coupling source rests on the fact that its dynamic response is of low frequency and low amplitude. First, different techniques that describe synchronization are applied; then, data evolution is analyzed in order to determine if there is a very slow dynamic behavior. The results show that synchronization is easily identified, but the source of synchronization is not; the reason is its low energy content and its larger fundamental frequency. Direct FFT, or other traditional dynamic analysis techniques, is unable to process this type of dynamics. The background modes are, however, responsible for internal interaction, communication, and energy transfer between the natural frequencies of the foreground components. The background vibrations are transmitted through somewhat undefined neighboring linkages at very low frequencies, which may be orders of magnitude lower than the natural frequencies of the foreground element.

#### 2. Experimental Setup

The experimental setup consisted of a stationary rotor with 24 blades. In order to reduce the number of variables, the blades were built with a rectangular cross section, as shown in Figure 1, and they were mounted with the thin side of the blade facing the front. At the tip of each blade, a MEMS accelerometer was mounted; data from 22 accelerometers were recorded. These accelerometers were double-axis Analog Devices ADXL321 type with a sensitivity of 10 g. Their masses were small enough so they had a negligible contribution to the natural frequency of the blades. A laser Doppler vibrometer, similar to that used by Oberholster and Heyns [26], was used to calibrate the individual accelerometers that were mounted on each blade. Figure 2 shows the experimental setup and the laser vibrometer used for calibration. The accelerometers were connected to a data acquisition system, and the data were recorded simultaneously.