Onset Of Limit Cycle Oscillations Research Articles

Signal-processing-based characterization of thermoacoustic coupling in a lean premix swirling gas turbine combustor

This study investigates the strength of the thermoacoustic feedback coupling between the acoustic field and the thermal field in a combustor with the coherence function between dynamic pressure (DP) and heat release rate (HR) signals. The time domain signals of DP and HR - which are measured synchronously from a lean premix combustor experiencing combustion instability (CI) - were used. The coherence function between DP and HR signals in the time-frequency domain reflects the degree of linear interdependence and the mutual causality of signals from the two fields. The results of the coherence analysis revealed the following: (i) the coherences are far from unity due to random properties in both signals during stable combustion, thus implying decoupling of the two fields; (ii) they are close to unity around the acoustic eigenfrequencies of the combustor during unstable combustion, thus demonstrating the linear interdependence and mutual causality between the two signals frequency-wise; and (iii) the formation of nearly linear interdependence between the two signals at the first eigenfrequency precedes the onset of limit cycle oscillation. Based on these findings, we propose an equivalent harmonic oscillator model for a combustion process under severe instability and formulate the damping ratio expression accordingly.

Limit-Cycle Free, Digitally-Controlled Boost Converter Based on DDPWM

In this paper, a digitally controlled boost DC-DC power converter featuring a high-resolution Dyadic Digital Pulse Width Modulator (DDPWM) is proposed to achieve Limit-Cycle oscillations (LCOs)-free operation, high DC accuracy, low output voltage ripple at high switching frequency. The effectiveness of the DDPWM in suppressing the onset of LCOs, increasing DC accuracy, and reducing output ripple is verified by both Simulink/Modelsim co-simulations and experimental testing on a 7-10V input, 13.8V output boost converter operated at 1.17MHz switching frequency. Thanks to the proposed modulator, improvement of more than 6X DC accuracy is achieved compared to a plain digital PWM. Moreover, approximately 3X less output ripple in comparison to a digital PWM with thermometric dithering is achieved.

Tracking shock movement on the surface of an oscillating, straked delta wing

Limit cycle oscillations (LCO), a mechanical nonlinearity similar to flutter, have affected high performance aircraft, like the F-16 and F-18, for years. The phenomena causes a lateral motion of the cockpit that makes performance of typical pilot tasks difficult. To better understand the nature of LCO and why high performance aircraft are typically afflicted, a computational model was developed to compare to wind tunnel tests and investigate the flow field around a straked, semispan delta wing and monitor the changes as the wing was pitched in an oscillatory fashion. The oscillations were intended to mimic LCO. By understanding the flow field around an oscillating wing, the fluid force that acts as the source of the motion could be discerned. The computational investigation focused on tracking shock movement along the top surface of the wing to confirm the presence of shock-induced trailing edge separation, one possible driver of LCO. The surface pressure was shown to be sensitive to the starting trim angle, oscillation frequency and oscillation amplitude. A separation bubble aft of the shock was observed as the source of sharp, localized negative surface pressure coefficient. The large change in surface pressure caused by the separation bubble could be an important phenomena contributing to the onset of LCO.

Transition to Limit-Cycle Oscillation in Fluid-Structure Interactions: Mutual Correlations and Causal Dependencies

We investigate the dynamic characteristics corresponding to the structural fluctuations of a cantilever suspended in a turbulent flow. To investigate the intricate dynamics of the flow–structure interaction, first, we explore the ability of network analysis to identify the different dynamic states and probe the viability of using quantifiers of network topology as precursors for the onset of limit-cycle oscillations. By increasing the Reynolds number, we observe that the structural oscillations, measured using a strain gauge, transition from low-amplitude chaotic oscillations to large-amplitude periodic oscillations associated with limit-cycle oscillations. We characterize the dynamic states of the system by constructing the weighted correlation network from the time series of strain and identifying the network properties that have the potential to be used as precursors for the onset of limit-cycle oscillations. Furthermore, we use Pearson correlation to illustrate the evolution of mutual statistical influence between the structural oscillations and the flowfield. We use this information and the Granger causality to identify the causal dependence between the structural oscillations and velocity fluctuations. By identifying the causal variable during each regime, we illustrate the directional dependence through a cause–effect relationship in this flow–structure interaction as it transitions to limit-cycle oscillations.

Global linear stability analysis of a flame anchored to a cylinder

This study investigates the linear stability of a laminar premixed flame, anchored on a square cylinder and confined inside a channel. Many modern linear analysis concepts have been developed and validated around non-reacting bluff-body wake flows, and the objective of this paper is to explore whether those tools can be applied with the same success to the study of reacting flows in similar configurations. It is found that linear instability analysis of steady reacting flow states accurately predicts critical flow parameters for the onset of limit-cycle oscillations, when compared to direct numerical simulation performed with a simple one-step reaction scheme in the low Mach number limit. Furthermore, the linear analysis predicts a strong stabilising effect of flame ignition, consistent with documented experiments and numerical simulations. Instability in ignited wake flows is, however, found to set in at sufficiently high Reynolds number, and a linear wavemaker analysis characterises this instability as being driven by hydrodynamic mechanisms of a similar nature as in non-reacting wake flows. The frequency of nonlinear limit-cycle flame oscillations in this unstable regime is retrieved accurately by linear eigenmode analysis performed on the time-averaged mean flow, under the condition that the full set of the reacting flow equations is linearised. If, on the contrary, unsteadiness in the density and in the reaction rate are excluded from the linear model, then the congruence between linear and nonlinear dynamics is lost.

Convolution/Volterra Reduced-Order Modeling for Nonlinear Aeroelastic Limit Cycle Oscillation Analysis and Control

A methodology for leveraging a combination of linear (through convolution integrals) and nonlinear (through Volterra series) reduced-order modeling (ROM) development was demonstrated on both a two- and three-degree-of-freedom (2-DOF and 3-DOF, respectively) aeroelastic system. The linear/nonlinear ROM approach was demonstrated against previous work in the field for a 2-DOF system and subsequently extended to a 3-DOF (flapped airfoil) system. Excellent agreement was demonstrated at limit cycle oscillation (LCO) onset (flutter) prediction between computational fluid dynamics, linear ROMs, and nonlinear ROMs. Although linear ROMs were sufficient to predict LCO onset, the nonlinear Volterra correction was required to estimate the amplitude of post-LCOs. Corrections that accounted for more coupling between degrees of freedom predicted the amplitudes more accurately. A study was undertaken to determine the minimal training set required to produce reasonable LCO amplitude estimates of the 3-DOF airfoil. A controller was also implemented in an effort to mitigate LCO onset and flutter divergence and to demonstrate the usefulness and power of leveraging the ROM for control design. A full-state feedback controller, tuned via a linear quadratic regulator (LQR), was shown to be effective in controlling the system below LCO onset. However, beyond LCO onset, this specific linear controller structure was observed to be ineffective in controlling the system. Future work will evaluate nonlinear control methods in which the ROM can be deployed as a part of the control system to update the LQR gains in a fully coupled approach.

Sloshing reduced-order model based on neural networks for aeroelastic analyses

A thorough understanding of the effects of sloshing on aircraft dynamic loads is of great relevance for the future design of flexible aircraft to be able to reduce their structural mass and environmental impact. Indeed, the high vertical accelerations caused by the vibrations of the structure can lead to the fragmentation of the fuel free surface. Fluid impacts on the tank ceiling are potentially a new source of damping for the structure that has hardly been considered before when computing the dynamic loads of the wings. This work aims at applying recently developed reduced-order models of sloshing to the case of a research wing to investigate their effects on the wing aeroelastic response under pre-critical and post-critical conditions. The vertical sloshing dynamics is considered using neural networks trained with experimental data from a scaled tank and then integrated into the aeroelastic system following a suitable scaling procedure. The results concern the aeroelastic response of the wing to gust input under pre-critical (flutter) conditions as well as post-critical conditions highlighting the onset of limit cycle oscillations caused by sloshing, the only nonlinear phenomenon modelled in the present simulation framework. Moreover, the load alleviation performances will be assessed for a typical landing input.

Causation of Supersonic Limit Cycle Oscillations in Atmospheric Entry Vehicles

A successful atmospheric entry vehicle (AEV) design demonstrates manageable aerodynamic heating, bearable structural load, smooth deceleration, and intended trajectory during a descent into the atmosphere. Interestingly, the supersonic regime of the AEV manifests limit cycle oscillations (LCO) that restrict the maneuver potential and deployment of the drag chute. In this paper, efforts are made numerically and analytically to link the causation of LCO with external geometric variables of AEV. Computational fluid dynamics is used to calculate damping derivatives. The numerical results are validated with the Orion Crew Exploration Vehicle. The multiple time scales method, which belongs to the class of perturbation methods, is explored to develop an approximate closed-form solution of the nonlinear dynamic behavior of AEV. The analytical solution identifies that higher-order nonlinearities associated with pitch damping and static lift govern the onset of LCO. Finally, a parametric interaction study is carried out to determine the effect of two design variables, apex angle and length, on the vehicle’s dynamic stability. The mean data values from the main geometric effects plot show the condition of finite-amplitude oscillations. The results indicate that variation in these two parameters significantly impacts the magnitude of identified higher-order nonlinearities.

Filament-motor protein system under loading: instability and limit cycle oscillations

ABSTRACT We consider the dynamics of a rigid filament in a motor protein assay under external loading. The motor proteins are modeled as active harmonic linkers with tail ends immobilized on a substrate. Their heads attach to the filament stochastically to extend along with it, resulting in a force on the filament, before detaching. The rate of extension and detachment are load-dependent. Here we formulate and characterize the governing dynamics in the mean field approximation using linear stability analysis, and direct numerical simulations of the motor proteins and filament. Under constant loading, the system shows transition from a stable configuration to instability toward detachment of the filament from motor proteins. Under elastic loading, we find emergence of stable limit cycle oscillations via a supercritical Hopf bifurcation with change in activity and the number of motor proteins. The number of motor proteins required at the onset of limit cycle oscillations increases with the increasing stiffness of the elastic loading. Numerical simulations of the system for large number of motor proteins show good agreement with the mean field predictions.

Delay in budget allocation for vaccination and awareness induces chaos in an infectious disease model

In this paper, we propose a model to assess the impacts of budget allocation for vaccination and awareness programs on the dynamics of infectious diseases. The budget allocation is assumed to follow logistic growth, and its per capita growth rate increases proportional to disease prevalence. An increment in per-capita growth rate of budget allocation due to increase in infected individuals after a threshold value leads to onset of limit cycle oscillations. Our results reveal that the epidemic potential can be reduced or even disease can be eradicated through vaccination of high quality and/or continuous propagation of awareness among the people in endemic zones. We extend the proposed model by incorporating a discrete time delay in the increment of budget allocation due to infected population in the region. We observe that multiple stability switches occur and the system becomes chaotic on gradual increase in the value of time delay.

Approximations to limit cycles for a nonlinear multi-degree-of-freedom system with a cubic nonlinearity through combining the harmonic balance method with perturbation techniques

This paper presents an approach to obtaining higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system with a single cubic nonlinearity based on a first approximation involving first and third harmonics obtained with the harmonic balance method. This first approximation, which is similar to one which has previously been reported in the literature, is an analytical solution, except that the frequency has to be obtained numerically from a polynomial equation of degree 16. An improved solution is then obtained in a perturbation procedure based on the refinement of the harmonic balance solution. The stability of the limit cycles obtained is then investigated using Floquet analysis.The capability of this approach to refine the results obtained by the harmonic balance first approximation is demonstrated, by direct comparison with time domain simulation and frequency components obtained using the Discrete Fourier Transform. The particular case considered was based on an aeroelastic analysis of an all-moving control surface with a nonlinearity in the torsional degree-of-freedom of the root support, and parameters corresponding to air speed, together with linear stiffness and viscous damping of the root support were varied. It is also shown, for the cases considered, how the method can reveal further bifurcational behaviour of the system beyond the initial Hopf bifurcations which first lead to the onset of limit cycle oscillations.

Bursting and mixed mode oscillations during the transition to limit cycle oscillations in a matrix burner.

We investigate the route to self-excited thermoacoustic instability in a laminar flow multiple flame matrix burner. With an increase in the equivalence ratio, the thermoacoustic system that is initially quiet (stable operation) transitions to limit cycle oscillations through two distinct dynamical states, namely, bursting oscillations and mixed mode oscillations. The acoustic pressure oscillations transition from quiescence to large amplitudes during bursting oscillations. Such high amplitude bursting oscillations that occur well ahead of the onset of limit cycle oscillations can potentially cause structural damage. The thermoacoustic system exhibits hysteresis. The transition to limit cycle oscillations is replicated in a phenomenological model containing slow-fast time scales.

Effects of the pylon–store system on the nonlinear aeroelastic responses of a slender wing

The aeroelastic behaviors of wing–pylon–store system have been researched for several decades. However, the effects of pylon–store system on the nonlinear aeroelastic characteristics of slender wings have not been sufficiently understood. In this paper, the nonlinear aeroelastic properties of a slender wing with a pylon–store system are discussed. The nonlinear equations of motion about the wing–pylon–store system are established, where the nonlinearities about the wing deformation and store motion are considered. The aerodynamic loads on the wing are determined by using two aerodynamic models with strip theory. One is the linear unsteady aerodynamic model based on Wagner function, the other is the nonlinear ONERA aerodynamic model. By comparing the nonlinear responses obtained with different aerodynamic models, it is found that the interaction between the aerodynamic nonlinearities induced by dynamic stall and the structural nonlinearities may simplify the bifurcation diagram structure of the system. For some typical cases of the wing–pylon–store system, the aeroelastic responses are analyzed. It is observed that the store-induced kinematic nonlinearities may affect the bending oscillation equilibrium position of the wing. This effect will be reduced by using soft spring in the pylon. Additionally, in the case of pylon with soft spring, the limit cycle oscillation (LCO) onset velocity and the post-critical response peak of torsional motion are insensitive to the store mass center position. But the post-critical response of store pitch is violent if the store mass center is not under the elastic center. In the case of pylon with hard spring, the post-critical response of store pitch becomes very complex. Though the response peak of the store is small, the LCO onset velocity changes obviously with the variation of the store mass center position. The store-related kinematics will play an important role on the nonlinear responses of the system in this situation.

Experimental Nonlinear Control for Flutter Suppression in a Nonlinear Aeroelastic System

Experimental implementation of input–output feedback linearization in controlling the dynamics of a nonlinear pitch–plunge aeroelastic system is presented. The control objective is to linearize the system dynamics and assign the poles of the pitch mode of the resulting linear system. The implementation 1) addresses experimentally the general case where feedback linearization-based control is applied using as the output a degree of freedom other than that where the physical nonlinearity is located, using a single trailing-edge control surface, to stabilize the entire system; 2) includes the unsteady effects of the airfoil’s aerodynamic behavior; 3) includes the embedding of a tuned numerical model of the aeroelastic system into the control scheme in real time; and 4) uses pole placement as the linear control objective, providing the user with flexibility in determining the nature of the controlled response. When implemented experimentally, the controller is capable of not only delaying the onset of limit-cycle oscillation but also successfully eliminating a previously established limit-cycle oscillation. The assignment of higher levels of damping results in notable reductions in limit-cycle oscillation decay times in the closed-loop response, indicating good controllability of the aeroelastic system and effectiveness of the pole-placement objective. The closed-loop response is further improved by incorporating adaptation so that assumed system parameters are updated with time. The use of an optimum adaptation parameter results in reduced response decay times.

Aeroservoelastic Test of the Subsonic Ultra-Green Aircraft Research Truss-Braced Wing Model

The Subsonic Ultra Green Aircraft Research Truss-Braced Wing aeroservoelastic wind-tunnel test was conducted in the NASA Langley Transonic Dynamics Tunnel. The primary goals of the test were to identify the open-loop flutter boundary and then demonstrate flutter suppression. A secondary goal was to demonstrate gust load alleviation. Open-loop flutter and limit cycle oscillation onset boundaries were identified for a range of Mach numbers and various angles of attack. Two sets of control laws were designed for the model, and both sets of control laws were successful in suppressing flutter. Control laws optimized for gust load alleviation were not designed; however, the flutter suppression control laws were assessed using the Transonic Dynamics Tunnnel airstream oscillation system. This paper describes the experimental apparatus, procedures, and results of the truss-braced wing wind-tunnel model test. Acquired system identification data used to generate aeroservoelastic models are also discussed.

Transient growth of acoustical energy associated with mitigating thermoacoustic oscillations

Energy conversion from heat to sound is desirable in some practical applications such as thermoacoustic heat engines or cooling systems. However, it is unwanted in gas turbine or aeroengine combustors. In this work, a Rijke-type thermoacoustic model with a linearly varied mean temperature configuration is developed. An acoustically compact heat source is confined and characterized by a modified form of King’s law. Unlike previous models available in the literature, the mean temperature is assumed to undergo not only a sudden jump across the heat source but also linearly increasing and decreasing in the pre- and after-heating regions respectively. Such mean temperature configuration is consistent with the experimental measurement. Coupling the heat source model with a Galerkin series expansion of the acoustic fluctuations provides a platform to gain insight on (1) the nonlinearity of the thermoacoustic system, (2) onset of limit cycle oscillations, (3) predicting its non-normality behaviors, (4) energy distribution and transfer between neighboring eigenmodes, and (5) evaluating the performance of feedback controllers. Pseudospectra and transient energy growth analyses are then performed. It reveals that the system is non-normal. And it is associated with transient growth of acoustical energy. The non-normality is found to be less intensified in comparison with that in a system with an invariant mean temperature from pre- and after-heating regions. To mitigate these limit cycle oscillations, the heat-to-sound coupling is interrupted by implementing multiple monopole-like actuators driven by a LQG (Linear Quadratic Gaussian) controller. For comparison, a pole-placement controller is also implemented. Approximately 76dB sound pressure level reduction is achieved. However, implementing the LQG controller is shown to be associated with transient growth of acoustical energy, which has potential to trigger thermoacoustic instability. The present work opens up new applicable way to model thermoacoustic systems in the presence of a mean temperature gradient. Furthermore, it reveals new potential risk of applying active controllers to stabilize thermoacoustic systems.

A PDE-DDE Model for Cell Polarization in Fission Yeast

We consider a one-dimensional model of cell polarization in fission yeast consisting of a hybrid partial differential equation--delay differential equation system. The model describes bulk diffusion of the signaling molecule Rho GTPase Cdc42 in the cytoplasm, which is coupled to a pair of delay differential equations at the ends of the cell via boundary conditions. The latter represent the binding of Cdc42 to the cell membrane and rerelease into the cytoplasm via unbinding. The nontrivial nature of the dynamics arises from the fact that both the binding and unbinding rates at each end are taken to depend on the local membrane concentration of Cdc42. In particular, the association rate is regulated by positive feedback and the dissociation rate is regulated by delayed negative feedback. We use linear stability analysis and numerical simulations to investigate the onset of limit cycle oscillations at the end compartments for a cell of fixed length, distinguishing between symmetric solutions in which the mean concentration is identical at both ends and asymmetric solutions where the mean concentration at one end dominates. We find that the critical time delay for the onset of oscillations via a Hopf bifurcation increases as the diffusion coefficient $D$ decreases. We then solve the diffusion equation on a growing domain under the additional assumption that the total amount $C_{\rm tot}$ of the signaling molecule increases as the cell length increases. We show that the system undergoes a transition from asymmetric to symmetric oscillations as the cell grows, consistent with experimental findings of “new-end-take-off” in fission yeast. (The latter refers to the switch from monopolar to bipolar growth as the cell grows.) The critical length where the switch occurs depends on $D$ and the growth rate.

Modeling the Effects of Underwing Missile Canards on F-16 Limit Cycle Oscillations

This research evaluated various methods using ZONA’s Euler Unsteady Aerodynamic Solver against flight tests previously flown at USAF Test Pilot School, using underwing air-to-air missiles with removable canards, which allows the aerodynamics of the store configuration to be changed while maintaining identical mass properties. ZONA’s Euler Unsteady Aerodynamic Solver was employed with various linear flutter and nonlinear time-domain analyses. ZONA’s Euler Unsteady Aerodynamic Solver correctly predicted that removing the canards would delay limit cycle oscillation onset. However, all of the methods generally predicted onset earlier than flight test. Consistent with the results found by previous researchers, the nonlinear simulation still predicted divergent classical flutter rather than limit cycle oscillation until nonlinear structural damping was added and tuned to produce limit cycle oscillation. Without known structural damping characteristics, the nonlinear structural damping model needed to be arbitrarily tuned to match flight-test data. This research demonstrated a different tuning method from that had been used by previous researchers. Nonlinear structural damping plays an important role in limit cycle oscillation, but a better understanding of the actual nonlinear aerodynamic and structural effects is needed before reliable limit cycle oscillation predictions can be made for all configurations and throughout the flight envelope.

Response of a three-degree-of-freedom wing with stiffness and aerodynamic nonlinearities at hypersonic speeds

In the design of high-speed aircraft particular attention must be given to the effect of stiffness and aerodynamic nonlinearities. Additionally the thermal effects cannot be ignored since the harsh thermal environment influences the dynamic behaviour of a structure. In the current paper the aeroelastic response of a three-degree-of-freedom wing with a control surface, and stiffness and aerodynamic nonlinearities has been analysed. The nonlinear unsteady aerodynamic forces applied to the model are calculated using third-order piston theory and piecewise linear, and cubic stiffening nonlinearities are implemented in the control surface. The effect of temperature on the model is determined by a steady-state analysis with a predefined temperature distribution. The onset of limit cycle oscillations and the bifurcation behaviours is compared for the different nonlinearities. The effect of initial conditions is determined to quantify their impact on the limit cycle behaviour. This article extends the nonlinear analysis of the aeroelastic response of a wing at hypersonic speeds with an added control surface degree of freedom, with the effect of nonlinear aerodynamics and stiffness included in the model. The results show that different nonlinearities influence the response of the system uniquely and heat plays an important role in the order of excited harmonics.

Investigation of Missile Control Surface Effects on F-16 Limit-Cycle Oscillation

Tactical aircraft with thin wings and heavy external wing stores are susceptible to aeroelastic limit-cycle oscillations. This investigation applies both frequency-domain and time-domain solutions using aeroelastic computational models to predict limit-cycle oscillation onset, frequency, and amplitude on an F-16. The test configuration uses missile shapes with reconfigurable aerodynamic and mass properties. Statistical analysis of flight-test results identifies significant experimental variables of missile aerodynamic configuration, flight condition, and aircraft fuel state. Furthermore, statistical analysis permits the normalization of flight-test results for direct comparison to aeroelastic models. Results of the comparison show that flight-test data analysis mostly supports the prediction trends, but the magnitude of the aerodynamic effects due to the canards and fins is much less significant in the test data.