Limit Cycle Oscillations Research Articles

Synchronized shock wave and compliant wall interactions: Experimental characterization and aeroelastic modeling

The static and dynamic interaction of a normal shock wave (upstream Mach number 1.35) with a compliant wall is characterized experimentally by schlieren visualizations and an optical displacement sensor. Depending on the location of the shock wave along the compliant wall, three different regimes of interaction are found: large-amplitude synchronized regime, small-amplitude synchronized regime and unsynchronized regime. The regime of large-amplitude synchronized oscillations is found for shock locations close to the mid-point of the compliant wall along the streamwise direction; at this location, the coupled system locks to the second vibration frequency of the structure. Three regimes of small-amplitude synchronized oscillations are found depending on the shock position. When the shock is located upstream the center of the compliant wall, the shock may oscillate either periodically at the frequency of the first vibration mode or quasi-periodically with highest amplitudes at the three frequencies of the vibration modes. When the shock is located downstream the center of the compliant wall, the shock oscillates periodically at the frequency of the third vibration mode. Finally, close to the trailing edge of the compliant wall, the shock oscillation is not synchronized with the compliant wall which oscillates with a very small amplitude. An empirical model is proposed to investigate the energy exchange between the flow and the compliant wall during the limit cycle oscillations. A negative aerodynamic damping – and, hence, the possibility of a limit cycle – is observed when a sufficiently extended separation is considered in the model for the pressure distribution at the wall.

Validation of a Hybrid Domain Overlapping Coupling Between SAM and CFD Against the TALL-3D Transients

The system thermal-hydraulic (STH) code SAM has been coupled to the computational fluid dynamics (CFD) code Simcenter STARCCM+ utilizing a hybrid domain overlapping coupling method in space and an explicit coupling in time. The coupling aims to extend the STH code’s applicability to scenarios where local momentum and energy transfers are important yet difficult for STH standalone models to capture, such as three-dimensional (3D) mixing and thermal stratification. The coupling method’s numerical stability was previously verified against multiple test cases including two closed-loop configurations, and it was validated against a double T-junction experiment with 3D scalar mixing. In the present work, the method is validated against the TALL-3D STH/CFD coupling benchmark facility. TALL-3D is a liquid-metal facility with a large, pool-type enclosure (test section) that exhibits 3D flow effects to be modeled by a CFD code. The rest of the system exhibits approximately one-dimensional behavior that is well predicted by a STH code. First, a STAR-CCM+ CFD model of the 3D test section is validated against experimental temperature data. Then a SAM-STARCCM+ coupled model is validated against six different TALL-3D steady states and includes SAM standalone results for comparison. Last, the coupled model is validated against two TALL-3D transients: one exhibiting flow reversal in the test section and one exhibiting nonlinear, limit cycle oscillations (LCOs). For the first transient, the SAM-STARCCM+ coupled model properly predicts an increase in the test section’s inlet temperature during flow reversal, and this increase is not predicted by the SAM standalone model. The coupled model also better predicts initial flow recovery and subsequent oscillations as the system approaches a final natural circulation state. For the second transient, no true final steady state is observed due to LCOs. Neither the SAM-STARCCM+ coupled model nor the SAM standalone model can perfectly capture the experiment’s changing oscillation frequency during the transient. Nevertheless, the coupled model does reproduce the general oscillatory feedback observed. This is a significant achievement because the present coupling only uses an explicit coupling in time, as opposed to a tighter, semi-implicit coupling. In comparison, STH/CFD coupling efforts previously performed by other authors required a semi-implicit coupling to obtain similar results to this work. The strengths of the novel coupling scheme validated in the present paper lie in the superior convergence characteristics and the much simpler, nonintrusive implementation.

Prediction of flame transfer function and combustion instability on a partially premixed swirling combustor by the system identification and CFD methods

Combustion instability can lead to high amplitude oscillations of pressure in the combustion chamber, which would seriously affect the normal operation of the combustion system. In order to reduce the cost and time of combustion chamber design and testing, combustion instability of the partially premixed swirl combustor was predicted, by solving the Helmholtz equation coupled the flame transfer unction (FTF). In the experimental study, by changing the equivalence ratio, the pressure in the combustion chamber appeared to oscillate in different forms including combustion noise, intermittent oscillation and limit cycle oscillation, while the frequency of the self-excited oscillations was obtained to be 165 Hz at the equivalence ratio of 0.7. In Computational Fluid Dynamics (CFD) Simulations, by applying Discrete Random Binary Signal (DRBS) excitation at the inlet, the FTF has been calculated by system identification method, where the excitation results of perturbation signals with different magnitudes were studied. And the result was basically consistent with the FTF calculated by the single-frequency excitation method, which means that the system identification method is suitable for calculating the FTF of the partially premixed swirl combustor. The pulse excitation method has been proposed to determine the selection of the filtering order, and the accuracy of the method has been validated through comparing the FTF results obtained by using different filter orders for discrimination. Moreover, coupling the FTF into the Helmholtz equation, the oscillation frequency of the system was predicted as 159.7 Hz, compared with the experimentally measured oscillation frequency of 165 Hz, which verified that the model could be effectively used for the prediction of oscillation frequency in the partially premixed combustion systems.

Auto-tuning method for PI controllers under Symmetric-Send-on-Delta sampling strategy

In this work, an auto-tuning method for PI controllers applied in a loop under a Symmetric-Send-On-Delta sampling strategy is proposed. Auto-tuning algorithms usually implement an identification phase to obtain some information about the system dynamic that is used for tuning the controller. In our proposal, two frequency response points are estimated through an iterative procedure. With this information, a simple tuning rule is applied which guarantees robustness against limit cycle oscillations that can be induced by the SSOD sampler. Both the robustness measure to limit cycles and the identification procedure are based on the describing function technique, whose validity in the networked control systems under study requires a low incidence of losing packets as well as the transmission delays to be known or negligible compared to the system’s dynamic. Comparatives with other classical tuning rules support the suitability of the proposed method. The auto-tuning procedure has been validated through a simulation study reflecting its applicability for the most common dynamics found in industrial processes.

Maximisation of synchronisability under low injection power

Synchronisability of limit cycle oscillators has been measured by the width of the synchronous frequency band, known as the Arnold tongue, concerning external forcing. We clarify a fundamental limit on maximizing this synchronisability within a specified extra low power budget, which underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronisation of weakly forced nonlinear oscillators. In this letter, injection-locked Class-E oscillators are considered as a practical case study, and we systematically analyse their power consumption; our observations demonstrate the independence of power consumption in the oscillator from power consumption in the injection circuit and verify the dependency of power consumption in the oscillator solely on its oscillation frequency. These systematic observations, followed by the mathematical optimisation establish the existence of a fundamental limit on synchronisability, validated through systematic circuit simulations. The results offer insights into the energetics of synchronisation for a specific class of injection-locked oscillators.

Phase autoencoder for limit-cycle oscillators.

We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

Dynamics of minimal networks of limit cycle oscillators

Dynamics of minimal networks of limit cycle oscillators

Limit Cycle Oscillating Detonation Wave Behavior Analysis Within a Rotating Detonation Engine

Limit cycle oscillation (LCO) detonation wave behaviors are presented and analyzed for test times exceeding 20 s in a water-cooled rotating detonation engine (RDE). LCO detonation waves exhibit cyclic acceleration and deceleration, resulting in oscillating wave spacing at unique process conditions. In previous RDE studies, similar behaviors have been studied as microsecond-scale instabilities leading to ascending or descending modal transitions. In the current work, however, LCO waves are considered a persistent wave mode, occupying unique portions of the operational envelope adjacent to those of their equally spaced counterparts. These occurrences of LCO waves are repeatable, enduring behaviors. A method to generate shifted contour surfaces specifically intended to extract and analyze wave spacing variation through time, termed limit cycle oscillation visualization (LCOV) surfaces, is presented. LCOV surfaces transform data into the reference frame of a primary traveling wave and are used to analyze quasi-steady, short-timescale, and transitional LCO modes. Results are leveraged to understand the relationship between fill height, wave strength, local wave acceleration, and subsequent LCO wave spacing for individual wave sets. Quasi-steady LCO waves display wave spacing oscillations between equal spacing values associated with ±1 wave across runs exceeding 18 s.

Nonlinear vibrations, bifurcations and chaos of piezoelectric composite lattice sandwich plate with four simply supported edges

This paper is devoted to investigate the nonlinear dynamics, bifurcation and chaos of the piezoelectric composite lattice sandwich plate with 1:3 internal resonance. Galerkin method is used to discretize the nonlinear partial differential governing equations of motion into the ordinary differential equations. The multi-scale method is used to obtain the modulation equations of the piezoelectric composite lattice sandwich plate in the polar coordinates and Cartesian coordinates in the primary and parametric resonances, respectively. Using Newton-Raphson method, the bifurcation diagrams of the steady-state equilibrium solution are obtained for the modulation equation with the change of the system parameters. The stability of the steady-state equilibrium solution is analyzed for the modulation equation. The existence of the static bifurcations, such as the saddle-node bifurcation, pitchfork bifurcation and Hopf dynamic bifurcations are found. The complex nonlinear jump phenomena caused by the existence of the multiple nonlinear steady-state solution regions are analyzed in detail. Fourth-order Runge-Kutta method is used to continue tracing the nonlinear periodic solution of the modulation equation in Cartesian coordinates. The time-history diagram, phase portrait and Poincaré map are obtained for the piezoelectric composite lattice sandwich plate in the nonlinear resonance. The dynamic process of the nonlinear periodic solution is described for the modulation equation from the limit cycle oscillation to the chaotic attractor in detail.

Designing 2-D Limit-Cycle Oscillators With Prescribed Trajectories and Phase-Response Characteristics

Designing 2-D Limit-Cycle Oscillators With Prescribed Trajectories and Phase-Response Characteristics

Numerical Analysis of Free Play-Induced Aeroelastic Phenomena: A Numerical Approach With Adaptive Step Size Control

This study presents a detailed numerical analysis of nonlinear aeroelastic behavior in a two degree of freedom (DOF) model, focusing on plunge and pitch motions and employing the continuation method (CM) with an adaptive step size control algorithm. The research incorporates free-play nonlinearity at the plunge hinge, a common structural nonlinearity in aeronautics that can induce detrimental limit cycle oscillations (LCOs) during flight. By examining three scenarios—linear response, unhindered plunge motion, and nonlinear stiffness behavior—the study assesses the effects of free play on flutter and LCO phenomena, including discontinuity-induced bifurcations like grazing bifurcation. Additionally, the study explores parameter variation for nonlinear flutter analysis, revealing the dynamics of grazing bifurcation and its impact on LCO behavior. The research also demonstrates the method’s superior accuracy in flutter speed estimation and mode-switching identification, despite higher computational demands. The findings underscore the diminishing influence of nonlinear free-play behavior on LCO amplitude, providing insights with significant implications for aeroelastic design and aircraft safety.

Limit cycle oscillation and dynamical scenarios in piecewise-smooth nonlinear systems with two-sided constraints

Limit cycle oscillation and dynamical scenarios in piecewise-smooth nonlinear systems with two-sided constraints

Cluster formation due to repulsive spanning trees in attractively coupled networks.

Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.

Effects of structural geometric nonlinearities on the transonic aeroelastic characteristics of wing

The structural geometric nonlinearity has a great influence on the transonic aeroelastic characteristics of aircraft wings, which is an important problem for the aerodynamic and structural design of wing. In this paper, a CFD/CSD coupling method based on the high precision in-house solver is first established, and the reliability of the CFD and the CFD/CSD coupling method is then verified. Subsequently, the effects of geometric nonlinearity on the transonic aeroelastic characteristics of wing under different dynamic pressures are studied. The results show that: (1) At low dynamic pressure conditions, for geometric linear wing structure, limit-cycle oscillation occurs at various angles of attack, and the limit-cycle oscillation amplitude decreases with the increases of angle of attack. At low angle of attack, the wing deformation characteristics are dominated by first bending and first torsion, and the first torsion weaken as the angle of attack increases. For the geometric nonlinear wing structure, the limit-cycle oscillation occurs only at an attack angle of 0° As the angle of attack increases, the structural displacement eventually exhibits static deformation characteristics. (2) Under high dynamic pressure conditions: the displacement of geometric linear wing structure presents a divergent trend. For geometric nonlinear wing structure, the limit-cycle oscillation amplitude decreases first and then increases with the increase of the angle of attack. At low angle of attack, the deformation characteristics of the wing are dominated by first bending and first torsion. With the increase of the angle of attack, the first torsion weakens, while the second bending strengthen. With the further increase of the angle of attack, the second bending weaken and become co-dominated by the first bending and first torsion. It can be seen from the above results that the aeroelastic characteristics of both the geometric linear structure and the geometric nonlinear structure wings are significantly different. These findings provide valuable insights for the meticulous design of aircraft wings.

Acoustic Design Parameter Change of a Pressurized Combustor Leading to Limit Cycle Oscillations

When aiming to cut down on the emission of nitric oxides by gas turbine engines, it is advantageous to have them operate at low combustion temperatures. This is achieved by lean premixed combustion. Although lean premixed combustion is a proven and promising technology, it is also very sensitive to thermoacoustic instabilities. These instabilities occur due to a coupling between the unsteady heat release rate of the flame and the acoustic field inside the combustion chamber. In this paper, this coupling is investigated in detail. Two acoustic design parameters of a swirl-stabilized pressurized preheated air (300 °C)/natural gas combustor are varied, and the occurrence of thermoacoustic limit cycle oscillations is explored. The sensitivity of the acoustic field as a function of combustion chamber length (0.9 m to 1.8 m) and reflection coefficient (0.7 and 0.9) at the exit of the combustor is investigated first using a hybrid numerical and analytical approach. ANSYS CFX is used for Unsteady Reynolds Averaged Navier-Stokes (URANS) numerical simulations, and a one-dimensional acoustic network model is used for the analytical investigation. Subsequently, the effects of a change in the reflection coefficient are validated on a pressurized combustor test rig at 125 kW and 1.5 bar. With the change in reflection coefficient, the combustor switched to limit cycle oscillation as predicted, and reached a sound pressure level of 150 dB.

Fluid–structure–surface interaction of a flexibly mounted pitching and plunging flat plate in proximity to the free surface

This experimental study investigates the fluid–structure–surface interactions of a flexibly mounted rigid plate in axial flow, focusing on flow-induced vibration (FIV) response and vortex dynamics of the system within a reduced velocity range of $U^*=0.29\unicode{x2013}8.73$ , corresponding to a Reynolds number range of $Re=518\unicode{x2013}15\,331$ . The plate, with one and two degrees of freedom (DoFs) for pitching and plunging oscillations, is examined at various submerged heights near the free surface. Results show that the plate exhibits divergence instability at low reduced velocities in both 1DoF and 2DoF systems. As the flow velocity surpasses a critical reduced velocity, periodic limit-cycle oscillations (LCOs) occur, increasing in amplitude until a second critical reduced velocity is reached. Beyond this point, LCOs are suppressed, and the plate experiences an increased static divergence angle with further flow velocity increase. The proximity to the free surface significantly influences the FIV response, with decreasing submerged heights leading to reduced LCO amplitudes and a shift of instabilities to higher reduced velocities. Vortex dynamics are analysed using time-resolved volumetric particle tracking velocimetry and hydrogen bubble flow visualisation. The analysis reveals disruptions in the symmetric flow field near the free surface, causing elongation and fragmentation of vortices in the wake of the plate, as well as vortex coupling. Proper orthogonal decomposition (POD) identifies dominant coherent structures, including leading-edge and trailing-edge vortices, captured in the first and second paired modes. On the other hand, higher POD modes capture the interaction of vortices in the wake and near the free surface.

Nonlinear aeroelastic analysis of a twin-tail boom UAV with rudder freeplay nonlinearities

The issue of nonlinear structural freeplays in aircraft has always been a significant concern. This study investigates the aeroelastic characteristics of a twin-tail boom Unmanned Aerial Vehicle (UAV) with simultaneous freeplay nonlinearity in its left and right rudders. A comprehensive Limit Cycle Oscillation (LCO) solution route is proposed for complex aircraft with multiple freeplays, which can consider both accuracy and efficiency. For the first time, this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and provides simulations and explanations of interesting phenomena observed during actual flight. The governing equations are established using the free-interface component mode synthesis method, and the LCOs of the system are mainly solved through the improved time-domain numerical continuation method and frequency-domain numerical continuation method. Furthermore, the study investigates the influence of the left and right rudder freeplay size ratio on the LCO characteristics. The results demonstrate that the twin-tail boom UAV exhibits two stable LCO types: close and differing left and right rudder amplitudes. The proposed method successfully describes the complete LCO behaviors of the system. Overall, this study makes significant contributions to our understanding of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

Experimental and numerical investigations on non-synchronous vibration and frequency lock-in of transonic compressor rotor blades

Non-synchronous vibration (NSV) problems in axial compressors are challenging and frequently result in high vibration stress or structural failure within the frequency lock-in region. A deeper understanding of the lock-in phenomenon in NSV is necessary to improve blade reliability and safety. In this study, an experiment was conducted on a multistage transonic axial compressor to obtain frequency lock-in characteristics. The enforced motion of the flexible blade simulation method was used to study the frequency lock-in mechanism between blade vibration and tip unstable flow. The results showed that the aerodynamic frequency characteristic changed from a broadband spectrum to a distinct frequency peak as the NSV onset. The nonlinear vibration pattern could maintain the limited cycle oscillation for 26 s. The vibration stress of the frequency locked-in case was approximately 19 times that of the frequency unlocked one. The pressure fluctuations generated by the large-scale radial separation vortex structures were stronger than those of the tip leakage flow, and the third-order harmonic frequency of the circumferential instability flow was close to the first-order bending model natural frequency. The periodic shedding and reattachment process of flow separation provided the initial non-synchronous aerodynamic excitation source. The frequency lock-in phenomenon occurred at a maximum vibration amplitude of 2 % and 3 % tip chord length. The tip unstable flow characteristics were consistent and entirely dominated by blade vibration.

Tuning limit cycles with a noise: Survival and collapse.

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equationof motion for slow dynamics after appropriate averaging over the fast motion. The equationfor slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

Data-Driven Bayesian Inference for Stochastic Model Identification of Nonlinear Aeroelastic Systems

The objective of this work is to propose a data-driven Bayesian inference framework to efficiently identify parameters and select models of nonlinear aeroelastic systems. The framework consists of the use of Bayesian theory together with advanced kriging surrogate models to effectively represent the limit cycle oscillation response of nonlinear aeroelastic systems. Three types of sampling methods, namely, Markov chain Monte Carlo, transitional Markov chain Monte Carlo, and the sequential Monte Carlo sampler, are implemented into Bayesian model updating. The framework has been demonstrated using a nonlinear wing flutter test rig. It is modeled by a two-degree-of-freedom aeroelastic system and solved by the harmonic balance methods. The experimental data of the flutter wing is obtained using control-based continuation techniques. The proposed methodology provided up to a 20% improvement in accuracy compared to conventional deterministic methods and significantly increased computational efficiency in the updating and uncertainty quantification processes. Transitional Markov chain Monte Carlo was identified as the optimal choice of sampling method for stochastic model identification. In selecting alternative nonlinear models, multimodal solutions were identified that provided a closer representation of the physical behavior of the complex aeroelastic system than a single solution.