Limit Cycle Oscillations Research Articles

Nonlinear State-Space Model of Self-excited forces for Bluff Body Aeroelasticity

This paper introduces a novel state-space model of nonlinear self-excited forces designed to capture amplitude dependency and unsteady effect in bluff body aeroelasticity. In the present form, this model represents a theoretical extension of the classical linear state-space model into the nonlinear regime, particularly relevant when a bluff section undergoes large amplitude oscillations. The proposed model incorporates additional state variables to approximate nonlinear convolution-based indicial functions, thereby providing a valuable tool for estimating nonlinear wind-induced instabilities, including nonlinear flutter limit cycle oscillation (LCO), vortex-induced vibration (VIV), and unsteady galloping. Furthermore, we establish the analytical foundation for identifying nonlinear parameters through the equivalent linearization of nonlinear transfer functions and the development of a numerical solving algorithm for the nonlinear governing equation. The feasibility of this model is rigorously validated through experimental results pertaining to nonlinear flutter, unsteady galloping, and VIV of typical bluff sections. Additionally, this model serves as the cornerstone for a nonlinear analytical framework in the time domain, facilitating the integration of both aerostatic and structural nonlinearities into comprehensive structural analyses.

Supercritical and subcritical aeroelastic behaviors of a three-dimensional wing coupled with a nonlinear energy sink

This paper studies the flutter control of a three-dimensional wing using a nonlinear energy sink (NES), which has a purely cubic stiffness and a linear damper. The structural model is derived by Lagrange's equations and the aerodynamics is modeled using the auto regressive with exogenous input method. Both models are solved in a tightly coupling manner to obtain time domain results. Then, the structural describing function and the aerodynamic continuous-time state space model are used to derive the equivalent linearized aeroelastic system, so that eigenvalue analyses can be used to obtain stable and unstable limit cycle oscillations (LCOs). The LCOs obtained by time domain simulations agree well with the equivalent linearization method, while differences are observed for aperiodic motions which are only predicted in time domain for small damping coefficients. Results show the square of aeroelastic amplitude is inversely proportional to the NES stiffness coefficient for both periodic and aperiodic motions. The NES can induce LCOs below flutter speed for small damping coefficients, indicating a subcritical bifurcation which can change to a supercritical one by increasing the damping coefficient. It is discovered that the upper bound of partial suppression region is quantified by a turning point speed. The regions of complete suppression, partial suppression and no suppression are clarified.

Robust network of globally coupled heterogeneous limit cycle oscillators due to inertia

We investigate the phase transition from macroscopic oscillatory state to stable homogeneous steady state in a heterogeneous network of globally coupled Stuart–Landau limit cycle oscillators in the presence of the inertial effect. The phase transition, known as aging transition, onsets above a critical fraction of inactive constituents in the mixed population of active and inactive units. We show that even a feeble increase in the inertial strength increases the critical fraction of inactive units significantly for the onset of the phase transition to the macroscopic steady state thereby resulting in a more robust network, in general. In contrast, a large coupling strength, in the case of a homogeneous network, facilitates the manifestation of the phase transition even for a small fraction of inactive oscillators leading to a more fragile network. Nevertheless, a large coupling strength, in the case of a heterogeneous network, increases the resilience of the network by facilitating the phase transition at a large fraction of inactive oscillators. Furthermore, a larger standard deviation of the natural frequencies always leads to a more fragile network. We derive the macroscopic evolution equations for the order parameters and the stability curve using the first-order moment expansion around the mean-field. In addition, we also deduce the critical fraction of inactive units and the critical inertial strength analytically that matches with the simulation results. Interestingly, we find that the critical inertial strength is reciprocally related to the square of the mean frequency of the network.

Fluctuation and correlation analyses for spontaneous otoacoustic activity from three terrestrial vertebrate groups

Spontaneous otoacoustic emission (SOAE) is a telltale sign of an active ear that commonly manifest as a set of spectral peaks unique to a given individual. These sounds arise from a variety of species, despite dramatic morphological differences considered important for cochlear function. Furthermore, SOAE peaks exhibit amplitude (AM) and frequency modulations (FM), these fluctuations giving rise to their characteristic widths. There is, however, little consensus on how SOAE activity is generated. Here, we provide a systematic comparative study of SOAE peak fluctuations across three different groups: human, owl, and (anole) lizard. Specifically, we focus on analysis of correlative behavior in AM and FM fluctuations within a given peak (intra-peak, IrP) and across peaks (inter-peak, IPP) for SOAE waveforms measured from individual ears. In general, there were numerous similarities and differences across the three groups. To complement the empirical data, we also considered one model class (coupled limit cycle oscillators grouping into “frequency clusters”) to ascertain how well that model captures fluctuations and associated correlative relations. Initial results indicate the model shows some consistencies with data (e.g., IPP correlations strongest for nearest-neighboring peaks) but also inconsistencies (e.g., model commonly exhibits IPP-FM correlations and to a greater degree than AM).

Data-driven control of oscillator networks with population-level measurement.

Controlling complex networks of nonlinear limit-cycle oscillators is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of oscillator populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without prior model assumptions or pre-existing data remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's current dynamics to iteratively learn an appropriate control online without constructing a global model of the system. We illustrate through a range of numerical simulations that the proposed technique can effectively regulate synchrony in various oscillator networks after a small number of trials using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach, illustrate its robustness to system variations, and compare its performance with existing model-based and data-driven approaches.

An Enhanced Nonlinear Energy Sink for Hybrid Bifurcation Passive Mitigation and Energy Harvesting From Aeroelastic Galloping Phenomena

Abstract Galloping is a self-excited vibration problem that structures immersed in fluid flow can experience. Due to its essential nonlinear phenomena, the structure exhibits limit cycle oscillations (LCOs), which, at high levels, can lead to failure of the systems. This work proposes an investigation of electromagnetic-enhanced nonlinear energy sinks (NES-EH) for the hybrid mitigation of aeroelastic LCOs and energy harvesting. The study focuses on a prismatic bluff body with a linear suspension immersed in the airflow, using classical steady nonlinear modeling for aerodynamic loads. The conventional NES approach is adopted, employing cubic stiffness and linear damping. Additionally, a linear electromagnetic transducer is included in the assembly for the energy harvesting process. By combining the method of multiple scales with the Harmonic Balance Method, analytical solutions are derived to characterize the system's dynamics under the influence of the device. The different response domains and their respective boundaries induced by the NES-EH are characterized based on the bifurcation diagrams. Furthermore, a slow invariant manifold (SIM) characterization is presented for each induced response domain, and its significant features are discussed. Parametric studies are carried out based on bifurcation analyses to assess the effect of NES-EH parameters on the galloping system dynamics, which allows for designing the absorber parameters. The electrical resistance is optimized to maximize the harvested power. The optimal design of NES-EH is then compared with classical energy harvesting solutions for the galloping problem. Additionally, a thorough analysis of the Target Energy Transfer phenomenon is performed.

On the nonlinear dynamics and flutter response of hybrid shape memory alloy composite beams

This study presents a theoretical and numerical investigation of the dynamic response of a hybrid SMA composite beam. Particular emphasis is given to the evaluation of the equivalent damping, the aeroelastic stability, and the possibility to tune the composite’s dynamics by means of pre-strain applied locally to the SMA layers. The model of the hybrid composite beam (HCB) accounts for both the material nonlinearity associated with the phase transformation of the SMA and the geometric nonlinearity (in the von Kármán sense) due to potentially large displacements. The nonlinear governing equations of the HCB are solved by the finite element method and the dynamic behavior of the HCB is assessed for different design parameters, such as thickness, position, and pre-strain level of the SMA layers. Results help understand the role played by the different design parameters in improving the dynamic characteristics of the HCB, with particular attention to effective damping. Furthermore, it is of particular interest to understand the aeroelastic stability of hybrid layered composites due to their widespread applications in aerospace engineering. A simplified model of the nonlinear flutter response of the HCB under supersonic flow conditions is presented and analyzed with particular emphasis on bifurcations and limit cycle oscillations.

Optimal phase-based control of strongly perturbed limit cycle oscillators using phase reduction techniques.

Phase reduction is a well-established technique for analysis and control of weakly perturbed limit cycle oscillators. However, its accuracy is diminished in a strongly perturbed setting where information about the amplitude dynamics must also be considered. In this paper, we consider phase-based control of general limit cycle oscillators in both weakly and strongly perturbed regimes. For use at the strongly perturbed end of the continuum, we propose a strategy for optimal phase control of general limit cycle oscillators that uses an adaptive phase-amplitude reduced order model in conjunction with dynamic programming. This strategy can accommodate large magnitude inputs at the expense of requiring additional dimensions in the reduced order equations, thereby increasing the computational complexity. We apply this strategy to two biologically motivated prototype problems and provide direct comparisons to two related phase-based control algorithms. In situations where other commonly used strategies fail due to the application of large magnitude inputs, the adaptive phase-amplitude reduction provides a viable reduced order model while still yielding a computationally tractable control problem. These results highlight the need for discernment in reduced order model selection for limit cycle oscillators to balance the trade-off between accuracy and dimensionality.

Stability analysis of the singular points and Hopf bifurcations of a tumor growth control model

We carry out qualitative analysis of a fourth‐order tumor growth control model using ordinary differential equations. We show that the system has one positive equilibrium point, and its stability is independent of the feedback gain. Using a Lyapunov function method, we prove that there exist realistic parameter values for which the systems admit limit cycle oscillations due to a supercritical Hopf bifurcation. The time evolution of the state variables is also represented.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.

Data-driven nonlinear parametric model order reduction framework using deep hierarchical variational autoencoder

A data-driven parametric model order reduction (MOR) method using a deep artificial neural network is proposed. The present network, a least-squares hierarchical variational autoencoder (LSH-VAE), is capable of performing nonlinear MOR for the parametric interpolation of a nonlinear dynamic system with a significant number of degrees of freedom. LSH-VAE differs from existing networks in two major respects: a deep hierarchical structure and a hybrid weighted, probabilistic loss function. The enhancements result in significantly improved accuracy and stability compared with conventional nonlinear MOR methods, autoencoders, and variational autoencoders. In LSH-VAE, the parametric MOR framework is based on the spherically linear interpolation of the latent manifold. The present framework is validated and evaluated on three nonlinear and multiphysics dynamic systems. First, the present framework is evaluated on the fluid–structure interaction benchmark problem to assess its efficiency and accuracy. Then, a highly nonlinear aeroelastic phenomenon, limit cycle oscillation, is analyzed. Finally, the framework is applied to three-dimensional fluid flow to demonstrate its capability to efficiently analyze a significantly large number of degrees of freedom. The superior performance of LSH-VAE is emphasized by comparing its results against those of widely used nonlinear MOR methods, a convolutional autoencoder, and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-VAE. The proposed framework exhibits significantly enhanced accuracy compared with that of conventional methods, while the computational efficiency continues to remain significantly higher.

Stochastic Response of an Airfoil and Its Effects on Lco’s Behavior Under Stall Flutter Regime

In this work, we investigate the effect of noise on a classical two degree-of-freedom pitch-plunge aeroelastic system. The inlet velocity of the flow is modelled as a stochastically varying parameter by the Ornstein-Uhlenbeck (OU) stochastic process. The system is a 2D airfoil and the elastic problem is simulated using linear springs. We study the manifestation of Limit Cycle Oscillations that corresponds to the varying fluid velocity under the dynamic stall regime. I aim to delve into the unexplored facets of the classical pitch-plunge aeroelastic system, seeking a comprehensive understanding of how parametric noise influences the occurrence of LCO and expands the boundaries of its known behavior.

An improved intermittent control model of postural sway during quiet standing implemented by a data driven approach

This paper proposes a new intermittent control model during human quiet standing, which is consisted of postulated “regular intervention” and “imminent intervention”. The regular intervention is within the main control loop, and its trigger condition is equivalent to the switching frequency of center of pressure (COP) data calculated by wavelet transform. The imminent intervention will only be triggered after the postural sway angle exceeds a certain threshold. In order to prove the effectiveness of the new model, the simulation results of the new model and the model proposed by Asai et al. (2009) are compared with the experimental data. The setting parameters of both models are retrieved by Bayesian regression from the experimental data. The results show that the new model not only could exhibit two power law scaling regimes of power spectral density (PSD) of COP, but also show that indices of the probability density function distance, root mean square (RMS), Total Sway Path, displacement Range, 50% power frequency of center of mass (COP) between the simulation results and the experimental data are closer compared to the existing model. Moreover, the limit cycle oscillations (LCOs) obtained from the simulation results of the new model have a higher degree of matching with those retrieved from the experimental data.

Active flutter suppression for an aircraft wing structure by utilizing FE analysis for Al7075+Sic alloy

An instable stimulated vibration resulting from the combination of aerodynamic, inertia, and aeroelastic forces is known as "active flutter" in the context of aircraft wing structures. The primary destabilizing element that the airplane structures display and which causes active flutter is limit cycle oscillation. Ultimately, it results in fatiguerelated harm and catastrophic aircraft mishaps. Two modern techniques, namely enhanced aero-elastic analysis and flutter control for wings and panel structures, are established by means of these developing technologies. comparable stiffness values and comparable stresses are assessed, together with theoretical and practical approaches, to enhance the features of aero-elastic analysis of aircraft structures. Additionally, it makes reducing flutter mandatory in order to enhance aero-elastic stability. Using Ansys software, a thorough FE analysis is carried out to improve and analyze the benefits of Al7075+ SiC alloy aircraft wing structure. With the aid of Ansys software, a thorough FE study is carried out to enhance their aeroelastic stability and to offer recommendations for the design of revised wings and panel structures. For the chosen NACA4412 aircraft wing, modal analysis is carried out, and frequencies are assessed. The advantages of AL7075+ SiC alloy material will be examined with equal stress, equivalent strain, and equivalent stiffness.

Comparison of pressure-based flame describing functions measured in an annular combustor under self-sustained oscillations and in an externally modulated linear combustor

One important aspect in the analysis of combustion instabilities in multiple-injector annular systems is that of representing such oscillations with a linear array of injectors. This issue is considered in the present work by comparing flame dynamics in a self-sustained oscillations situation in the annular configuration MICCA-Spray, with experiments carried out in TACC-Spray, a rectangular combustor equipped with five injectors externally modulated by driver units. In the annular system, the coupling mode is azimuthal, and in the linear array, the flames are submitted to a transverse acoustic field. An original method is used to change the limit-cycle pressure oscillations amplitude in the annular system and favor a standing mode with a well-defined nodal line position. It consists in using different arrangements of two types of injectors, leading to various azimuthal staging patterns. The range of pressure oscillation amplitudes is swept in TACC-Spray by changing the external acoustic forcing level. It is thus possible to compare the flame dynamics in forced and self-sustained situations using these two experimental facilities. This is done by examining flame describing functions (FDFs) based on downstream pressure fluctuations measured in the two configurations for different pressure oscillation amplitudes. The trends observed in the two systems are in good agreement, and there is a reasonably good match between the values of the gains and phases determined in the two configurations. The pressure-based FDFs are determined for flames located at different positions with respect to the acoustic mode. It is found that the transverse or azimuthal velocity component has only a weak influence on the pressure-based FDF, the strongest impact being observed in the neighborhood of the velocity antinode. This study demonstrates the ability of a linear-array setup modulated by a transverse mode to reproduce conditions of azimuthal coupling leading to self-sustained limit cycles in an annular combustor.

Measurements of flow-induced vibration of a flexible splitter plate mounted on a cylinder in free stream flow

We experimentally study flow-induced vibration (FIV) of a thin, cantilevered, flexible plate attached on the lee side of a circular cylinder in a free stream airflow. The Reynolds number (ReD) based on the diameter of the cylinder was varied in the range of (4 × 103, 5×104). The plates of different lengths and the same span were tested with varying airflow velocities (U) in a wind tunnel. The reduced velocity (UR) based on the length and the second natural frequency of the plate was in the range of (1, 11). We describe the plate dynamics showing displacement, frequency, phase plane, amplitude spectral density (ASD), oscillation envelope, modal energy contribution, and flow structures. Based on the motion of the plate, three FIV regimes are observed, namely, initial excitation, transition, and lock-in. During lock-in, the plate attains self-sustained limit cycle oscillation. We observe the presence of small-amplitude higher harmonics in the ASD plot. The second mode is dominant around the onset of lock-in, while the first mode is dominant later in the lock-in regime. The dimensionless displacement increases with an increase in mass ratio (Ms) in the lock-in regime. With a decrease in Ms, the slope of the dimensionless frequency with UR increases. To explain measurements, we develop a wake oscillator model (WOM) in which the plate and fluctuating lift force are modeled as an Euler–Bernoulli cantilever beam and van der Pol oscillator, respectively. The coupled WOM qualitatively and quantitatively predict the measured amplitude and frequency response, respectively, in the lock-in regime.

Flutter Suppression Effects of Movable Vertical Stabilizers on Suspension Bridges With Steel Box Girders

As the combination of springs and vertical stabilizers, the movable downward vertical central stabilizer (MDVCS) is proposed to further control the nonlinear flutter of super long‐span suspension bridges in this paper. A series of flutter suppression tests of closed‐box girders and twin‐box steel girders with various MDVCSs are conducted. Based on the coupled flutter theoretical method, the sensitivity analysis of two important parameters including the height and stiffness of MDVCS are carried out to compare their nonlinear flutter control mechanism. The results show that the flutter critical wind speed (Ucr) of the closed‐box girder continued to increase with the decrease of the height of the DVCS and the increase of spring stiffness, whereas the Ucr of the twin‐box girder increased at first and then decreased. The cubic polynomial function and quadratic Holliday function are suitable to modify the correction coefficients of Ucr for the closed‐box girder with various stiffnesses and heights of MDVCS, while the Lorentz peak‐value function and cubic polynomial function are suitable to modify the Ucr of the twin‐box girder. Furthermore, the MDVCS significantly changes the rules of two positive and negative aerodynamic damping ratios for the closed‐box girder and two negative aerodynamic damping ratios for the twin‐box girder. Besides, the peak vertical displacement amplitudes of the box girder are about half of the MDVCS, since both the height and stiffness of MDVCS alter the elliptical radius of vertical phase planes to affect the limit cycle oscillation of soft flutter, especially for the leeward MDVCS for the twin‐box girder.

Multilevel Limit Cycle Oscillator for Synchronizing Grid Connected Inverters in Fault Ride-Through Applications

Multilevel Limit Cycle Oscillator for Synchronizing Grid Connected Inverters in Fault Ride-Through Applications

Friction Saturated Limit Cycle Oscillations—Test Rig Design and Validation of Numerical Prediction Methods

Abstract In this paper, an experimental test rig for friction saturated limit cycle oscillations is proposed to provide a validation basis for corresponding numerical methods. Having in mind the application of turbine blades, an instrumented beam-like structure equipped with an adjustable velocity feedback loop and dry frictional contacts is designed and investigated. After dimensioning the test rig by means of a simplified one-dimensional beam model and time domain simulations, the specific requirements of limit cycle oscillations for the design of the frictional contact, the velocity feedback loop and the excitation system are discussed and possible solutions are presented. Also appropriate measuring principles and evaluation techniques are assessed. After commissioning of the test rig, the influence of the negative damping and the normal contact force on the limit cycle oscillations is measured and the practical stability is investigated. The test rig shows linear dynamics for sticking contact and highly repeatable limit cycles. The measured results are discussed regarding the consistency with theory and compared to the predictions of a three dimensional reduced order model solved in frequency domain by the harmonic balance solver OrAgL. It is demonstrated that the numerical modeling strategy is able to accurately reproduce the measured limit cycle oscillations, which stabilized for different contact normal forces and self-excitation levels.

Enhanced vehicle shimmy performance using inerter-based suppression mechanism

Shimmy may occur in the vehicle normal driving process, which is a harmful motion and should be controlled in practical engineering. In order to improve the vehicle driving and handling stability, four kinds of passive inerter-based suppression mechanisms are proposed and applied in the vehicle suspension to improve its shimmy performance. The configurations S1 and S2 are composed of one inerter and one damper, which are in parallel-connected and series-connected, respectively, furthermore, an auxiliary spring is added to constitute configurations S3 and S4. Combined with the magic nonlinear tire model, the vehicle shimmy model with four inerter-based suppression mechanisms is established using Lagrange theory, its shimmy performance is studied using bifurcation analysis method, the stable area and limit cycle oscillation (LCO) magnitude of the system are obtained and compared with those of the original suspension system. The effect of the structural parameters of the inerter-based suppression mechanisms on the vehicle shimmy performance is analyzed, and the structural parameter optimization of the inerter-based suppression mechanism is investigated. The results show that the only parallel-connected and series-connected configurations S1 and S2 have little improvement effect on the vehicle shimmy performance. As the configurations S3 and S4 are used, the unstable area dominated by left and right steering shimmy reduces, the shimmy occurred vehicle velocity range becomes narrower and the maximum LCO magnitude of the front wheel swing motion becomes smaller, which suppresses the vehicle shimmy effectively and clarifies the benefits of the inerter and the necessity of adding the auxiliary spring. In addition, larger inertance of the inerter and smaller stiffness of the auxiliary spring is beneficial to improve the vehicle shimmy performance. The optimized structural parameters of the configurations S3 and S4 are obtained, which maintain the vehicle shimmy system stable, and further shorten the time for the vehicle shimmy system to reach stability. Therefore, the inerter-based suppression mechanism can effectively restrain the vehicle shimmy motion and give guiding significance for the design of vehicle shimmy suppression mechanism.