Distribution Of Natural Frequencies Research Articles

Dynamics of Oscillator Populations Globally Coupled with Distributed Phase Shifts.

We consider a population of globally coupled oscillators in which phase shifts in the coupling are random. We show that in the maximally disordered case, where the pairwise shifts are independent identically distributed random variables, the dynamics of a large population reduces to one without randomness in the shifts but with an effective coupling function, which is a convolution of the original coupling function with the distribution of the phase shifts. This result is valid for noisy oscillators and/or in the presence of a distribution of natural frequencies. We argue also, using the property of global asymptotic stability, that this reduction is valid in a partially disordered case, where random phase shifts are attributed to the forced units only. However, the reduction to an effective coupling in the partially disordered noise-free situation may fail if the coupling function is complex enough to ensure the multistability of locked states.

Fault Diagnosis for Wind Turbine Flange Bolts Based on One-Dimensional Depthwise Separable Convolutions

In this paper, a new bolt fault diagnosis method is developed to solve the fault diagnosis problem of wind turbine flange bolts using one-dimensional depthwise separable convolutions. The main idea is to use a one-dimensional convolutional neural network model to classify and identify the acoustic vibration signals of bolts, which represent different bolt damage states. Through the method of knock test and modal simulation, it is concluded that the damage state of wind turbine flange bolt is related to the natural frequency distribution of acoustic vibration signal. It is found that the bolt damage state affects the modal shape of the structure, and then affects the natural frequency distribution of the bolt vibration signal. Therefore, the damage state can be identified by identifying the natural frequency distribution of the bolt acoustic vibration signal. In the present one-dimensional depth-detachable convolutional neural network model, the one-dimensional vector is first convolved into multiple channels, and then each channel is separately learned by depth-detachable convolution, which can effectively improve the feature quality and the effect of data classification. From the perspective of the realization mechanism of convolution operation, the depthwise separable convolution operation has fewer parameters and faster computing speed, making it easier to build lightweight models and deploy them to mobile devices.

Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions.

We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.

Reconstruction of phase dynamics from macroscopic observations based on linear and nonlinear response theories.

We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.

Exploring the phase diagrams of multidimensional Kuramoto models

The multidimensional Kuramoto model describes the synchronization dynamics of particles moving on the surface of D-dimensional spheres, generalizing the original model where they are characterized by a single phase. Particles are represented by D-dimensional unit vectors and the coupling constant can be extended to a coupling matrix acting on the vectors. The system has a large number of independent parameters, given by the characteristic widths of the distributions of natural frequencies and the D2 entries of the coupling matrix. Moreover, as the coupling matrix breaks the rotational symmetry, the average values of the natural frequencies also play a key role in the dynamics. General phase diagrams, indicating regions in parameter space where the system exhibits different behaviors, are hard to derive analytically. Here we obtain the complete phase diagram for D=2, for arbitrary coupling matrices and Lorentzian distributions of natural frequencies. We show that the system exhibits four different phases: disordered and static synchrony (as in the original Kuramoto model), rotation of the synchronized cluster (similar to the Kuramoto-Sakaguchi model with frustration) and active synchrony, a new phase where the module of the order parameter oscillates as it rotates on the sphere. We also explore the diagrams numerically for higher dimensions, D=3 and D=4, for particular choices of coupling matrices and frequency distributions. We find that the system always exhibits the same four phases, but their location in the space of parameters depends strongly on the dimension D being even or odd, on the coupling matrix and on the shape of the distribution of natural frequencies.

Amplification Analysis and Seismic Vulnerability Index of Air Tawar Barat Village of Padang City Using HVSR Method

Padang City is an area that has a very high level of seismic activity in Indonesia. The city of Padang is also an area severely affected by the earthquake in the Subduction Zone. Considerable earthquakes impact building damage and infrastructure, one of the factors that can see the vulnerability of an area with a seismic vulnerability index. Therefore, a study was conducted to determine and analyze the amplification and seismic vulnerability index in the Air Tawar Barat Village of Padang City and create a map of the natural frequency distribution, amplification, and seismic vulnerability index. Microtremor data retrieval was carried out in February-May 2022 using the Seismograph Sysmatrack MAE and S3S Sensors at Air Tawar Barat Village, Padang City. The method used in this study is the HVSR (Horizontal to Vertical Spectral Ratio) method using geopsy software to remove noise in the data and see the natural frequency value and the value of the amplification factor. Furthermore, from the data, contour maps were made using Surfer 13, where the type of research carried out was in the form of descriptive research. Based on the results of the study found that the natural frequency value (f0) in the study area ranged from 0.59141 Hz – 12.9172 Hz, the amplification (A0) ranged from 2.39825 to 13.7759, and the seismic susceptibility index (kg) was 0.4802 – 320.885. Based on the data processing results, it can be concluded that fresh western water is an area with a high seismic vulnerability index, and the location is very vulnerable when shaken by an earthquake.

Relaxation dynamics of phase oscillators with generic heterogeneous coupling.

The coupled phase oscillator model serves as a paradigm that has been successfully used to shed light on the collective dynamics occurring in large ensembles of interacting units. It was widely known that the system experiences a continuous (second-order) phase transition to synchronization by gradually increasing the homogeneous coupling among the oscillators. As the interest in exploring synchronized dynamics continues to grow, the heterogeneous patterns between phase oscillators have received ample attention during the past years. Here, we consider a variant of the Kuramoto model with quenched disorder in their natural frequencies and coupling. Correlating these two types of heterogeneity via a generic weighted function, we systematically investigate the impacts of the heterogeneous strategies, the correlation function, and the natural frequency distribution on the emergent dynamics. Importantly, we develop an analytical treatment for capturing the essential dynamical properties of the equilibrium states. In particular, we uncover that the critical threshold corresponding to the onset of synchronization is unaffected by the location of the inhomogeneity, which, however, does depend crucially on the value of the correlation function at its center. Furthermore, we reveal that the relaxation dynamics of the incoherent state featuring the responses to external perturbations is significantly shaped by all the considered effects, thereby leading to various decaying mechanisms of the order parameters in the subcritical region. Moreover, we untangle that synchronization is facilitated by the out-coupling strategy in the supercritical region. Our study is a step forward in highlighting the potential importance of the inhomogeneous patterns involved in the complex systems, and could thus provide theoretical insights for profoundly understanding the generic statistical mechanical properties of the steady states toward synchronization.

Сучасний стан підходів до моніторингу технічного стану лопатей вітрових турбін з використанням інформаційних технологій

Nowadays wind energy is one of the most important and promising sources of environmentally clean renewable energy. Wind turbine blades are among the most expensive components. Depending on the size, their manufacturing costs range between 10 % and 20 % of total manufacturing costs. Moreover, the size of blades has increased in recent years, leading to greater efficiency and energy production, but presenting higher failure probability. It is extremely important to avoid critical blade failures, because when damaged blades liberate, they have the potential to damage not only the turbines they were attached to, but also other turbines in their vicinity. In order to increase the reliability and safety of wind turbine operation, as well as to reduce costs due to maintenance and downtime in a non-working state, it is necessary to apply modern methods of monitoring the condition of large-sized and highly loaded parts of wind power plants using information technologies. The main types of defects and their classification are considered. The influence of the rotation speed of the turbine and the presence of a damage in the blade on the oscillation natural frequencies was analyzed. The main types and methods of non-destructive testing (NDT) are presented. The acoustic method is considered in detail, as it is rapidly developing and is promising for the field of green energy. The classification of acoustic methods of NDT is provided based on the studied literature. An analytical review of publications considering NDT methods for diagnosing wind turbine blades, including the ones which use unmanned aerial vehicles (UAVs), was conducted. The advantages and disadvantages of each method are shown. The analysis of NDT approach of wind power plants using machine learning based on Gaussian processes to predict natural frequencies of one blade based on the statistical data of the distribution of natural frequencies of neighboring blades and ambient temperature was carried out. The description of the full cycle of the system's functioning, from data collection to decision-making about the possible presence of a defect in the structure, is provided. This paper has summarized and analyzed the most important advances done in the field of NDT in the last few years. The considered approaches can serve as a basis for building new highly reliable methods for detecting dangerous defects in the blade material at the early stages of their development.

Nonlinear time series analysis of limestone rock failure process

Nonlinear analysis of the rock mass failure process is a powerful tool for the study of instability and prevention of disaster in rock mass engineering. Nonlinear analysis has rapidly developed in recent decades. We propose a new method to determine the fractal characteristics of the failure process using time series analysis based on an iterative cycle division of the box dimension of fractal theory. We use AE technology to carry out time series analysis of fracture sources in the failure process of rock mass. We show that the disordered failure of rock mass exhibits notable fractal characteristics in the time series. The advantage of this method is that only the time and frequency of AE monitoring are used, resulting in a better identification and classification of damage, compared with traditional natural sequence frequency distribution. In addition, MATLAB numerical simulation analysis shows that these time series fractal dimensions can accurately characterize the failure characteristics of rock mass. In brief, we show the potential for future orderly time analyses of the complex and disorderly process of rock mass failure.

Correcting Stellar Flare Frequency Distributions Detected by TESS and Kepler

The habitability of planets is closely connected with stellar activity, mainly the frequency of flares and the distribution of flare energy. Kepler and TESS find many flaring stars via precise time-domain photometric data, and the frequency and energy distribution of stellar flares on different types of stars are studied statistically. However, the completeness and observational bias of detected flare events from different missions (e.g., Kepler and TESS) vary a lot. We use a unified data processing and detection method for flare events based on the light curves from Kepler and TESS. Then we perform injection and recovery tests in the original light curve of each star for each flare event to correct the completeness and energy of flares. Three samples of flaring stars are selected from Kepler and TESS, with rotation periods from 1 to ∼5 days. Adopting a hot-blackbody assumption, our results show that the cumulative flare frequency distributions (FFDs) of the same stars in Kepler and TESS bands tend to be consistent after correction, revealing a more natural flaring frequency and energy distribution. Our results also extend the low-energy limit in cumulative FFD fitting to 1031.5−33 erg on different types of stars. For solar-type stars, the average power-law index of cumulative FFD (α cum) is −0.84, which indicates that low-energy flares contribute less to the total flare energy. With a piecewise correlation between α cum and T eff, α cum first rises from M2 to K1 stars, then slightly decreases for stars hotter than K1.

SOIL VULNERABILITY ASSESSMENT USING THE HORIZONTAL AND VERTICAL SPECTRUM RATIO (HVSR) METHOD OF AMBIENT NOISE WAVES, EAST JAVA ,INDOSEIA

The local site effect determines the risk level of an earthquake. East Java has a high level of seismicity. One of the earthquake mitigation efforts is to estimate the dynamic characteristics of grounf using microtremors. The purpose of this study was to determine the natural frequency distribution, amplification factor, and seismic vulnerability index. Seismic signals are obtained from stationary and temporal seimograph records from 100 measurement locations installed in the East Java and Madura regions. This study uses the horizontal and vertical spectrum ratio (HVSR) curve analysis method popularized by Nakamura (2000). The results of the study showed that the seismic vulnerability index varied from 0.25 to 54.2. Areas that have a high level of earthquake risk include Bangkalan, Gresik, Jombang, Lumajang, Malang, Mojokerto, Pasuruan, Ponorogo, Sampang. Sidoarjo, and Surabaya. Based on the history of destructive earthquakes, this area has experienced earthquake intensity ranging from III - IX MMI.

Entrainment degree of globally coupled Winfree oscillators under external forcing.

We consider globally connected coupled Winfree oscillators under the influence of an external periodic forcing. Such systems exhibit many qualitatively different regimes of collective dynamics. Our aim is to understand this collective dynamics and, in particular, the system's capability of entrainment to the external forcing. To quantify the entrainment of the system, we introduce the entrainment degree, that is, the proportion of oscillators that synchronize to the forcing, as the main focus of this paper. Through a series of numerical simulations, we study the entrainment degree for different inter-oscillator coupling strengths, external forcing strengths, and distributions of natural frequencies of the Winfree oscillators, and we compare the results for the different cases. In the case of identical oscillators, we give a precise description of the parameter regions where oscillators are entrained. Finally, we use a mean-field method, based on the Ott-Antonsen ansatz, to obtain a low-dimensional description of the collective dynamics and to compute an approximation of the entrainment degree. The mean-field results turn out to be strikingly similar to the results obtained through numerical simulations of the full system dynamics.

Robust optimization of MTMD systems for the control of vibrations

Multiple Tuned Mass Dampers (MTMD) is one of the simplest and most reliable solution to control the vibrations of a structure, e.g. allowing to deal with a wide distribution of structural natural frequencies and usually showing inherent stability. Nevertheless, uncertainties in the behavior of structures under random dynamic input may induce implications on the MTMD effectiveness, e.g. detuning or amplifications. Typically, uncertainties in the input are exclusively considered, while uncertainties affect as much also the mechanical parameters.This paper proposes a robust optimum design of the MTMD, considering uncertainties both in the structural parameters and in the earthquake input. At this aim, a random vibration analysis of the response is adopted, and a direct linear perturbation method is applied on the uncertain parameters. Results show that a deterministic optimization is inappropriate for sensible systems as TMDs, while a robust optimization aims to better control the response with low computational efforts. Furthermore, the selection of the objective functions significantly affects the optimal design parameters, as the number of TMDs influences the robust designs.

Synchronization in the Kuramoto model in presence of stochastic resetting.

What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott-Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.

Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.

Partial locking in phase-oscillator populations with heterogenous coupling.

We consider a variant of the mean-field model of coupled phase oscillators with uniform distribution of natural frequencies. By establishing correlations between the quenched disorder of intrinsic frequencies and coupling strength with both in- and out-coupling heterogeneities, we reveal a generic criterion for the onset of partial locking that takes place in a domain with the coexistence of phase-locked oscillators and drifters. The critical points manifesting the instability of the stationary states are obtained analytically. In particular, the bifurcation mechanism of the equilibrium states is uncovered by the use of frequency-dependent version of the Ott-Antonsen reduction consistently with the analysis based on the self-consistent approach. We demonstrate that both the manner of coupling heterogeneity and correlation exponent have influence on the emergent patterns of partial locking. Our research could find applicability in better understanding the phase transitions and related collective phenomena involving synchronization control in networked systems.

Optimizing synchrony with a minimal coupling strength of coupled phase oscillators on complex networks based on desynchronous clustering.

Finding the globally optimal network is an unsolved problem in synchrony optimization. In this paper, an efficient edge-adding optimization method based on global information is proposed. The edge-adding scheme is obtained through the eigenvector corresponding to the maximum eigenvalue of the system's Jacobian matrix. With different frequency distributions, we find that the optimized networks have similar features, concluded as three conditions: (i) The deviations of nodes' frequencies from the mean value are linear with the nodes' degrees, (ii) the oscillators form a bipartite network divided according to the frequencies of the oscillators, and (iii) oscillators are only connected to those with sufficiently large frequency differences. An optimal network can be constructed directly based on these three conditions for a given distribution of natural frequencies. We show that the critical coupling strengths of these constructed networks approach the theoretical lower bound. The constructed networks are or at least close to the globally optimal ones for synchrony.

Uncertain Parameters Estimation using Multi-Dimensional Analysis and Stochastic Model Updating

Stochastic model updating based on perturbation theory has been widely applied to quantify uncertain parameters in structural systems due to its simplicity and straightforward approach. Nevertheless, the significant requirements for establishing a good correlation in the initial prediction of structural responses and small perturbations in uncertain parameters have become influential in stochastic model updating. The initial assumptions of structural parameters are often unavailable to quantify the input properties due to insufficient information about the structural system. These problems contribute to large errors in initial prediction, causing ill-posedness in sensitivity matrices and convergence difficulties caused by the local minima function in the stochastic model updating approach. In these circumstances, this study attempts to propose a novel scheme to overcome the ill-posed and converging problems in the stochastic model updating by quantifying structural parameters of the assembled structure encompassing high uncertainties such as the stiffness term of the contact joint interface by using a combination of the lattice-based exploration approach and the perturbation-based stochastic model updating method. The lattice-based exploration approach is adopted for generating samples of predicted responses from the assumed initial distribution of random parameters in the interest of improving the initial correlation of the predicted responses for producing well-condition sensitivity. Responses from each sample are evaluated in light of their experimental counterparts to estimate the optimum initial distribution of the random parameters. Then, the initial statistical properties of the parameters can be estimated by rerunning the sampling approach using the optimum distribution. As a result, stochastic model updating using the perturbation approach can be applied efficiently with the new initial distribution. The proposed scheme has been demonstrated on an assembled bolted joint structure, focusing on the contact interfaces. It is found that the proposed scheme managed to produce satisfactory predictions on the distribution of natural frequencies with only 12.5 % of total errors are recorded in comparison with the experimental data.

Spatial Assessment of Heterogeneous Tissue Natural Frequency Using Micro-Force Optical Coherence Elastography.

Analysis of corneal tissue natural frequency was recently proposed as a biomarker for corneal biomechanics and has been performed using high-resolution optical coherence tomography (OCT)-based elastography (OCE). However, it remains unknown whether natural frequency analysis can resolve local variations in tissue structure. We measured heterogeneous samples to evaluate the correspondence between natural frequency distributions and regional structural variations. Sub-micrometer sample oscillations were induced point-wise by microliter air pulses (60–85 Pa, 3 ms) and detected correspondingly at each point using a 1,300 nm spectral domain common path OCT system with 0.44 nm phase detection sensitivity. The resulting oscillation frequency features were analyzed via fast Fourier transform and natural frequency was characterized using a single degree of freedom (SDOF) model. Oscillation features at each measurement point showed a complex frequency response with multiple frequency components that corresponded with global structural features; while the variation of frequency magnitude at each location reflected the local sample features. Silicone blocks (255.1 ± 11.0 Hz and 249.0 ± 4.6 Hz) embedded in an agar base (355.6 ± 0.8 Hz and 361.3 ± 5.5 Hz) were clearly distinguishable by natural frequency. In a beef shank sample, central fat and connective tissues had lower natural frequencies (91.7 ± 58.2 Hz) than muscle tissue (left side: 252.6 ± 52.3 Hz; right side: 161.5 ± 35.8 Hz). As a first step, we have shown the possibility of natural frequency OCE methods to characterize global and local features of heterogeneous samples. This method can provide additional information on corneal properties, complementary to current clinical biomechanical assessments, and could become a useful tool for clinical detection of ocular disease and evaluation of medical or surgical treatment outcomes.

Hierarchy of Exact Low-Dimensional Reductions for Populations of Coupled Oscillators.

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an Ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the Ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)CHAOEH1054-150010.1063/1.2930766]. Dynamics on the extended manifolds facilitate higher-dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies.