Limit Cycle Oscillations Research Articles

Fundamental investigation into output-based prediction of whirl flutter bifurcations

This paper investigates an approach for predicting whirl flutter bifurcations using pre-flutter output data. The approach leverages the critical slowing down phenomenon, which makes aeroelastic systems recover to equilibrium from disturbances more slowly when closer to the flutter boundary. By quantifying this phenomenon based on output data from pre-flutter transient responses, one can predict the flutter onset and the amplitude of limit-cycle oscillations (bifurcation diagram). The approach is demonstrated for a propeller-nacelle system using output data from transient simulations. Bifurcation diagrams are predicted within a few percentages of reference results using output data at two forward speeds, as low as 25% to 30% below the whirl flutter speed. Extensive sensitivity analyses highlight the accuracy and robustness of predictions for various output data choices and levels of nonlinearity, providing insights for future applications to higher-complexity systems. This work opens a new path to predicting whirl flutter bifurcations during the design of aeroelastic systems prone to this instability.

Iontronic Nanopore Model for Artificial Neurons: The Requisites of Spiking.

Brain-inspired neuromorphic computing is currently being investigated for effective artificial intelligence (AI) systems. The development of artificial neurons and synapses is imperative to creating efficient computational biomimetic networks. Here we propose the minimal configuration of an effective iontronic spiking neuron based on a conical nanofluidic pore ionic diode. The conductance is composed of a Boltzmann open channel probability and a blocking inactivation function, forming the structure of a memristor. The presence of a negative resistance and the combination of activation-deactivation dynamics cause a Hopf bifurcation. Using the characteristic frequencies of small perturbation impedance spectroscopy, we discuss the conditions of spiking, in which the system enters a limit cycle oscillation. We arrive at the conclusion that an excitable neuron-like system can be made with a single active channel instead of the more complex combination of multiple channels that occurs in the Hodgkin-Huxley neuron model.

Experimental investigation of combustion instability in a centrally staged combustor under self-excited oscillation conditions

This paper presents an experimental investigation of combustion instability and flame dynamics in a laboratory-scale lean premixed prevaporized centrally staged combustor under self-excited oscillations. The macrostructure of the flame is captured by CH* chemiluminescence images using a high-speed camera, and the two-dimensional flow field of the combustor center plane is obtained by using particle image velocimetry. The effects of the pilot stage swirl number and main stage equivalent ratio on the thermoacoustic oscillations and flame dynamics are analyzed. The results indicate that there are intermittent oscillations, limit cycle oscillations, and mode switching in the combustor. As the pilot stage swirl number increases, the system transitions from intermittent oscillation to limit cycle oscillation. Additionally, the case with a pilot stage swirl number of 0.5 also occurs mode switching. Based on dynamic mode decomposition, the heat release fluctuation is primarily concentrated in the shear layers under different operating conditions. Furthermore, the intensity of thermoacoustic oscillations in a system is determined by the coupling strength between pressure and heat release fluctuations. From the results of the flow field, it is observed that as the pilot stage swirl number increases, the time-averaged axial strain rates and vorticities increase, but the time-averaged axial velocity is generally lower for the swirl number of 0.7 compared to the other two cases. On the other hand, the time-averaged axial strain rates and vorticities for the case with a swirl number of 0.5 decrease with the increase in the main stage equivalence ratio before and after the mode switching.

Application of heat transfer at the combustor wall to attenuate self-excited thermoacoustic oscillations

Self-sustained thermoacoustic oscillations (also known as combustion instability) arise in combustors when pressure and heat release fluctuations couple to produce high-energy sound waves. The need for diverse strategies to control these oscillations will increase as future design moves towards highly efficient, lean combustors. In this work, the impact of heat transfer at the combustor wall on the generation of thermoacoustic oscillations is examined through computational fluid dynamics. Characteristics of the thermoacoustic oscillations generated across cases with a range of thermal boundary conditions are compared to an adiabatic reference case. The amplitude of limit cycle oscillations in a Rijke tube is seen to decrease as wall temperatures increase. A stable combustor with a negative growth rate is observed when the full outer wall is heated to 900 K, and a sound pressure level reduction of 75 dB is achieved. The most efficient stable configuration is achieved when only the inlet duct wall is heated to 1000 K, requiring a minimum of 430 W of input power. The present work introduces an alternative stable combustor design and demonstrates that optimizing the temperature distribution of the combustor wall can effectively dampen thermoacoustic oscillations.

Non-linear Dynamic Characteristics of High-Speed Wheel-Rail Interface Considering Surface Topography and Liquid Medium

In this work, the force-deflection characteristics, and the nonlinear vibration of the wheel-rail interaction with surface roughness and liquid medium are studied. The statistical micro-contact Greenwood and Williamson model is used to characterize the rough surface. The load sharing theory is used to determine the analytical elastic contact force-deflection and damping force-deflection relationships, which are further approximated by power law functions. The nonlinear contact stiffness and damping characteristics are illustrated for different rough surface topographies and running speeds. The natural frequency is solved by assuming small covariates and defining time variables of different scales. Effects of surface topography and liquid medium on the natural frequency of the wheel-rail interface system are studied. The first-order harmonic responses under harmonic excitation are further determined as well as the jump-up and jump-down responses for different rough surface topographies, excitation loads, liquid mediums and running speeds. It is shown that the resonance region and the peak value of the vibration amplitude increase as the surface roughness, running speed and excitation load increase; whilst the jump-up and jump-down frequencies decrease. In addition, the system can change from a monostable to a bistable structure when surface roughness and excitation force increase and large limit cycle oscillation can occur.

Flutter and Limit-Cycle Oscillations of a Panel Using Unsteady Potential Flow Aerodynamics

The potential aerodynamic theory is implemented to model the flow over a flexible panel. The present study compares the more complete unsteady version of the potential flow theory with its simplification known as the linear piston theory for different Mach numbers and the two- and three-dimensional aerodynamic models. The piston theory is “local” in space and time because it assumes the pressure at a spatial point and time only depends on the panel deformation at the same point and time. The full potential flow model includes the effect of the past history of the panel deformation and the spatial distribution of the panel deformation on the pressure at any instant in time and at any point in space. Aeroelastic analysis is made to trace the flutter onset critical condition based on the limit cycle oscillation amplitudes, and the results are compared with the more traditional implementation using piston theory. Conclusions are made based on the use and application of this more complete aerodynamic theory, particularly for near-transonic and hypersonic flow regimes. Subsonic results are also presented in this study.

The static and dynamic regimes of the bistable asymmetric cross-ply composite laminated plates

For the static and dynamic regimes of the bistable asymmetric cross-ply composite laminated plates, the theoretical model and the finite element model (FEM) are established. The 2D and 3D diagrams for the supercritical pitchfork bifurcation, which are dominated by the length, the width, the aspect ratio, the ratio of side to thickness, the number of layers and the temperature difference, can be used to describe the static regimes. The restoring force curves (RFCs) and the potential energy curves (PECs) are shown. The single-well vibration, the limit-cycle oscillations, the multiple-period snap-through, and the chaotic snap-through are the dynamic regimes that predominate.

Mixed synchronization in multiplex networks of counter-rotating oscillators

In this study, we introduce a novel concept known as ”mixed synchronization” within the context of two indirectly coupled counter-rotating oscillators. Our investigation begins by exploring the dynamics of two co-rotating oscillators when they are connected through a counter-rotating relay oscillator. Remarkably, this coupling mechanism leads to the emergence of what we term ”relay synchronization,” which signifies complete synchronization between the outer oscillators. This intriguing phenomenon lays the foundation for our subsequent exploration. We delve deeper into the dynamics of counter-rotating oscillators and discover that when two such oscillators are linked through relay oscillators, a distinct form of synchronization arises, which we aptly term ”relay mixed synchronization.” Notably, this reveals that even if counter-rotating oscillators are part of a larger network and indirectly coupled through a simple diffusive mechanism, it is possible to induce mixed synchronization. We extend the same innovative concept to the realm of multiplex networks, which consist of a finite number of counter-rotating oscillators in each layers and we explore both intralayer and interlayer mixed synchronization states. We verify the mixed synchronization state by master stability function analysis. We numerically verify the results by considering limit cycle and chaotic oscillators. This extension of our concept to multiplex networks showcases the versatility and applicability of mixed synchronization in diverse scenarios.

Nonlinear interactions between the fundamental and its harmonics of self-excited azimuthal thermoacoustic instabilities

Thermoacoustic instabilities in combustors are inherently nonlinear processes, manifested by multiple peaks in the spectrum. Analyzing these signals is crucial to understand the mechanism of nonlinear limit cycle oscillations. The goal of this study is to investigate the acoustic modal dynamic relation between the fundamental of azimuthal instabilities and its first harmonic. The paper first demonstrates that the nonlinear interaction of the fundamental azimuthal mode produces not only higher order azimuthal modes but also an axial mode in the harmonic. This is confirmed by the experimental data showing a correlation between the axial mode in the harmonic and two counter-rotating modes in the fundamental. Furthermore, the study shows that neglecting this axial mode may result in inaccurate estimation of the maximum pressure in the chamber, potentially leading to safety concern. The combustor is operated under various conditions to examine the correlation between the fundamental and its harmonic in terms of dominant spinning direction, orientation of the acoustic structure, and the maximum pressure magnitude. It is found that the azimuthal components of the fundamental and its harmonic rotate in the same direction under a wide range of operating conditions. Additionally, the maximum pressure magnitudes are positively correlated with each other despite the difference in the order of magnitude between them. Specifically, the harmonic's magnitude shows quadratic dependence on the fundamental's, even when the mode transitions from a standing wave to a spinning wave. However, the anti-node locations of the fundamental and its harmonic exhibit no correlation, and their difference in location varies with operating conditions.

Flame dynamics and combustion instability induced by flow-flame interactions in a centrally-staged combustor

This study mainly investigates the flame and flow characteristics under combustion instability in a centrally-staged swirl spray combustor, using 5 kHz CH* chemiluminescence (CL) and 15 Hz particle image velocimetry (PIV). The inlet air total temperature and total pressure are up to 600 K and 1 MPa, respectively. A dynamic pressure sensor with an acquisition frequency of 20 kHz is located downstream of the combustor casing. The pressure signal shows large-amplitude limit cycle oscillations when the fuel split ratio of the pilot stage decrease. The phase-averaged CH* CL images indicate that the periodic ignition/extinction of the main stage flame triggers combustion instability. A proper orthogonal decomposition (POD) analysis based on the reacting flow field further explains this phenomenon. The first two POD modes containing three coherent vortex pairs are associated with the extinction and ignition of the main stage flame, respectively. The vortex pairs located at the central shear layer (CSL) generated by Kelvin-Helmholtz (K-H) instability affect the primary recirculation zone (PRZ). The change in the size of the PRZ influences the main stage flame dynamics. In a cycle of oscillation, ignition happens when the size of PRZ increases and the coupling between the main and pilot stages enhances. Extinction takes place when the opposite happens. The flow-flame interaction mechanism demonstrated in this study provides strong experimental support for understanding combustion instability in aero-engine combustors.

Correlation of supersonic wind tunnel measurements with a nonlinear aeroelastic theoretical/computational model of a thin plate

Supersonic wind tunnel test data is compared with a nonlinear aeroelastic computational model, considering a freestream flow with Mach nearly 2, coupled cavity, and no-shock impingement. The measurements include Limit Cycle Oscillations (LCO) for the given set of flow and structural parameters. The effect of a static pressure differential, temperature differential, and the type of LCO is also studied: periodic or chaotic. A fully coupled aero-thermal–acoustic–elastic analysis was carried out, where the strong dependency of the dynamic instability upon the in-plane boundary stiffness was discovered. Associated with aerodynamic and acoustic formulations, the nonlinearities from the structure are found to be the strongest factor determining the occurrence and character of LCO. At the same time, the cavity effect was found to be negligible for this experimental configuration. Additionally, a comparison analysis was performed on the aerodynamic model for this flow condition between the Linear Piston Theory and the Full Potential Aerodynamics. The calculation of the critical flutter boundary was also performed in terms of the static pressure differential between the cavity pressure and the panel surface pressure, presenting regions where the panel is in static deformation due to the pressure differential versus LCO response. The comparison with the analytical model showed good agreement with the measured data, achieving a similar panel displacement at three-quarters of the panel length as well as the expected behavior of periodicity or chaos for specific values of temperature and static pressure differential.

Spherical formation control of mobile target by multi-agent systems with collision avoidance: A limit-cycle-based design approach

This paper focuses on the control of spherical formation for multi-agent systems with double-integrator dynamics in three-dimensional space. To achieve the desired formation pattern on a spherical surface, N agents must be distributed evenly around a target. Based on the information gathered about the target and its immediate neighbors, all agents can either remain stationary or rotate around the target at the same velocity on a spherical surface while the target moves in three-dimensional space. A decoupled control method with limit cycle oscillator characteristics is explored to achieve the desired formation geometric pattern. Accordingly, three subproblems are proposed to achieve the spherical formation more effectively. The first subproblem, target-spherical-surface, requires the agents to converge on a specified spherical surface with the target as its center. The second subproblem, target-circling, requires the agents to follow three prescribed orthometric circular orbits on the spherical surface to avoid collisions while moving. The final subproblem is distributed-adjustment, which involves agents to maintaining a prescribed angular distance from their immediate neighbors while remaining relatively stationary or rotating around the target. By addressing the first two sub-problems simultaneously, the agents can be guided to the desired position. The properties of the limit-cycle-based spherical formation controller are analyzed theoretically, and its effectiveness is demonstrated through numerical simulations.

Shock migration on an oscillating straked delta wing using an unsteady Euler solver

Many investigations, including flight tests, wind tunnel tests, and computational methods have attempted to determine the drivers of limit cycle oscillations (LCO) on the F-16 fighter aircraft. One area of interest has focused on shock-induced trailing-edge separation (SITES) as a driver of limit cycle oscillations. The investigation presented herein aims to expand the understanding of unsteady, nonlinear aerodynamics by studying the flow over a wing undergoing pitch oscillations using Euler computational fluid dynamics with a boundary layer coupling scheme capable of estimating viscous flow effects within the boundary layer. A straked NACA 64A204 delta wing model was used for this analysis. Parameters such as trim angle of attack, pitch amplitude, and Mach number were varied to create 71 unique test cases. Results demonstrate that the same single shock systems at low angles of attack and two shock systems at moderate to high angles of attack that were observed by Cunningham in his wind tunnel tests were also present in this investigation. Patterns of significant shock movement and disappearance were observed within the transonic flow region. Hysteresis of the pitching moment was also found to be present. These findings support the theories that shock movement and SITES may play significant roles in driving LCO.

Study of Nonlinear Aerodynamic Self-Excited Force in Flutter Bifurcation and Limit Cycle Oscillation of Long-Span Suspension Bridge

This article establishes a nonlinear flutter system for a long-span suspension bridge, aiming to analyze its supercritical flutter response under the influence of nonlinear aerodynamic self-excited force. By fitting the experimental discrete values of flutter derivatives using the least squares method, a polynomial function of flutter derivatives with respect to reduced wind speed is obtained. Flutter critical value is determined by the linear matrix eigenvalues of a state-space equation. The occurrence of a supercritical Hopf bifurcation in the nonlinear system is determined by the Jacobian matrix eigenvalues of the state-space equation and the system’s vibrational response at the critical state. The vibrational response of the supercritical state is obtained through Runge–Kutta integration, revealing the presence of stable limit cycle oscillation (LCO) and unstable limit cycle oscillation in the system, and through analyzing the relationship between the LCO amplitude and wind speed. Considering cubic nonlinear damping and stiffness, the effects of different factors on the nonlinear flutter system are analyzed.

Synchronization of spin-driven limit cycle oscillators optically levitated in vacuum

We explore, experimentally and theoretically, the emergence of coherent coupled oscillations and synchronization between a pair of non-Hermitian, stochastic, opto-mechanical oscillators, levitated in vacuum. Each oscillator consists of a polystyrene microsphere trapped in a circularly polarized, counter-propagating Gaussian laser beam. Non-conservative, azimuthal forces, deriving from inhomogeneous optical spin, push the micro-particles out of thermodynamic equilibrium. For modest optical powers each particle shows a tendency towards orbital circulation. Initially, their stochastic motion is weakly correlated. As the power is increased, the tendency towards orbital circulation strengthens and the motion of the particles becomes highly correlated. Eventually, centripetal forces overcome optical gradient forces and the oscillators undergo a collective Hopf bifurcation. For laser powers exceeding this threshold, a pair of limit cycles appear, which synchronize due to weak optical and hydrodynamic interactions. In principle, arrays of such Non-Hermitian elements can be arranged, paving the way for opto-mechanical topological materials or, possibly, classical time crystals. In addition, the preparation of synchronized states in levitated optomechanics could lead to new and robust sensors or alternative routes to the entanglement of macroscopic objects.

Toward an Efficient Method for F-16 Limit Cycle Oscillation Prediction

This study presents the development and validation of a computationally efficient prediction framework for the well-known nonlinear F-16 limit cycle oscillation (LCO) phenomenon. The framework relies on a simple physical working model that has been suggested and demonstrated in the past according to which LCO is primarily a flutter instability that is bounded by the existence of nonlinear structural damping (NSD), although potentially affected by nonlinear aerodynamic effects as well. In the framework developed herein, the NSD model is derived and calibrated using a novel method that simplifies the process and allows applicability of the derived NSD models for multiple aircraft download cases. Good LCO prediction capabilities are obtained using the suggested method in terms of LCO levels and trends with flight conditions, as demonstrated using four F-16 test configurations. This framework also allows several practical benefits, which makes it particularly suitable for industrial-level applications.

Nonlinear dynamic characteristics of self-excited thermoacoustic instabilities in premixed swirling flames

The nonlinear behavior of thermoacoustic oscillation for premixed swirl flames fueled with CH4/air mixtures is experimentally studied by changing the equivalence ratio Φ, combustor inlet bulk velocity Uin and swirl number S. The nonlinear dynamic characteristics of the system are analyzed by the methods of phase space reconstruction, recurrence plot, Poincaré section and 0–1 test for chaos. Five kinds of flame modes are observed in the combustion system with the change of Φ, Uin, and S, which are lean/rich blow-off, stable combustion, quasi-periodic oscillation, and limit cycle oscillation. Subcritical Hopf bifurcation of the oscillation is captured either by changing Φ or Uin, and Φ and Uin are found to have similar effects on the flame mode transition under specific operating conditions. It is observed that the appearance of flame in the outer circulation zone (ORZ) and the enhancement of its burning intensity are matched with the occurrence of quasi-periodic and limit cycle oscillations, respectively. It is examined that the lean/rich blow-off limits and oscillation range are extended with a larger S, and the stable combustion regime is wider with a smaller S. This study contributes to understanding the nonlinear thermoacoustic instability of swirl flames under broad flame conditions, which helps to develop effective instability prediction and control methods in stable, efficient, and low-emission gas turbines.

Cooling the optical-spin driven limit cycle oscillations of a levitated gyroscope

Birefringent microspheres, trapped in vacuum and set into rotation by circularly polarised light, demonstrate remarkably stable translational motion. This is in marked contrast to isotropic particles in similar conditions. Here we demonstrate that this stability is obtained because the fast rotation of these birefringent spheres reduces the effect of azimuthal spin forces created by the inhomogeneous optical spin of circularly polarised light. At reduced pressures, the unique profile of these rotationally averaged, effective azimuthal forces results in the formation of nano-scale limit cycles. We demonstrate feedback cooling of these non-equilibrium oscillators, resulting in effective temperatures on the order of a milliKelvin. The principles we elaborate here can inform the design of high-stability rotors carrying enhanced centripetal loads or result in more efficient cooling schemes for autonomous limit cycle oscillations. Ultimately, this latter development could provide experimental access to non-equilibrium quantum effects within the mesoscopic regime.

Delta wing rocking motion suppression by deep reinforcement learning

The wing rock phenomenon is a self-excited roll motion occurring on the aircraft at high angles of attack (AoAs), which negatively impact safety and maneuverability. The limit cycle oscillation is a typical characteristic of this self-excited roll motion. A controller employing a model-free training approach is constructed on the basis of deep reinforcement learning (DRL) to address this severe nonlinear control problem. For an 80° swept delta wing model, the analytical model describing the nonlinear behavior of wing rock is presented and implemented in the simulation environment. The DRL-based controller is trained with the proximal policy optimization algorithm. A reward function is carefully designed for training to achieve excellent convergence stability. Various simulations were conducted at a series of unlearned initial conditions (roll angles, AoAs) to demonstrate the effectiveness and generalization capability of the proposed method and verify and evaluate the performance of the trained DRL controller. Finally, the scenario that undergoes disturbance is added to confirm the effectiveness and robustness of the proposed DRL-based controller. Results show that the DRL-based controller could effectively regulate the oscillation and retain the capability to suppress the un-wanted behavior with favorable generalization capability and robustness. Therefore, delta wing rocking motion suppression based on DRL has powerful intelligent control performance and provides a new idea for predicting the evolution of aerodynamics in rocking motion suppression.

Spatiotemporal motor learning with reward-modulated Hebbian plasticity in modular reservoir computing

Generation of complex patterns at a specific timing is crucial to most forms of learning and behavior, which are acquired through dopamine-modulated plasticity in the striatum. However, the neural mechanisms of such reward-based spatiotemporal processing remain unknown. Inspired by the cortico-striatal circuits, this study developed a new reservoir computing method, a class of recurrent neural networks, based on reward-modulated Hebbian learning (RMHL) for the spatiotemporal motor learning. We utilized a reservoir of basal dynamics (reBASICS), which generated self-sustained limit cycle oscillations with various frequencies, as a reservoir structure. Then, the oscillations were linearly integrated as readout output, in which readout weights were modulated with RMHL. The simulations showed that reBASICS-based RMHL was able to accomplish both motor timing and pattern drawing tasks, for which existing reservoir-based RMHL failed. Further, introducing an eligibility trace mechanism into RMHL allowed the model to learn motor timing even when reward-based modulation was delayed. In conclusion, this model is proposed as a new computational model of temporal processing of the striatum, where the cortical areas generate stable oscillations. From the oscillatory dynamics, spatiotemporal patterns are learned using RMHL in the striatum.