Amplitude Of Limit Cycle Oscillations Research Articles

Amplification of weak magnetic field effects on oscillating reactions

We explore the possibility that chemical feedback and autocatalysis in oscillating chemical reactions could amplify weak magnetic field effects on the rate constant of one of the constituent reactions, assumed to proceed via a radical pair mechanism. Using the Brusselator model oscillator, we find that the amplitude of limit cycle oscillations in the concentrations of reaction intermediates can be extraordinarily sensitive to minute changes in the rate constant of the initiation step. The relevance of such amplification to biological effects of 50/60 Hz electromagnetic fields is discussed.

Influence of Bistable Plunge Stiffness on Nonlinear Airfoil Flutter

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry's influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation (LCO) amplitudes and also delays the occurrence of the snap-through.

Dynamics of Coupled Thermoacoustic Oscillators Under Asymmetric Forcing

Quenching of limit cycle oscillations (LCO), either through mutual coupling or external forcing, has attracted wide attention in several fields of science and engineering. However, the simultaneous utilization of these coupling schemes in quenching of LCO has rarely been studied despite its practical applicability. We study the dynamics of two thermoacoustic oscillators simultaneously subjected to mutual coupling and asymmetric external forcing through experiments and theoretical modeling. We investigate the forced response of both identical and non-identical thermoacoustic oscillators for two different amplitudes of LCO. Under mutual coupling alone, identical thermoacoustic oscillators display the occurrence of partial amplitude death and amplitude death, whereas under forcing alone, asynchronous quenching of LCO is observed at non-resonant conditions. When the oscillators are simultaneously subjected to mutual coupling and asymmetric forcing, we observe a larger parametric region of oscillation quenching than when the two mechanisms are utilized individually. This enhancement in the region of oscillation quenching is due to the complementary effect of amplitude death and asynchronous quenching. However, a forced response of coupled non-identical oscillators shows that the effect of forcing is insignificant on synchronization and quenching of oscillations in the oscillator which is not directly forced. Finally, we qualitatively capture the experimental results using a reduced-order theoretical model of two coupled Rijke tubes which are coupled through dissipative and time-delay coupling and asymmetrically forced. We believe that these findings offer fresh insights into the combined effects of mutual and forced synchronization in a system of coupled nonlinear oscillators.

Validation of a new fluid—structure interaction framework for non-linear instabilities of 3D aerodynamic configurations

A new framework developed at CIRA to deal with Fluid–Structure Interaction phenomena in a partitioned approach has been developed. A multi-block structured flow solver for unsteady Navier–Stokes equations, UZEN, was updated and tightly coupled with the open-source FEM code CalculiX in a modular approach.The solvers are interfaced through the open source library preCICE (Bungartz, 2016), that takes care of exchanging data, synchronization and controls the executing processes by means of specific adapters.The framework was designed with the capability to simulate large variety of configurations with complex motions, including rotorcrafts; it has growth potential to include multi-body dynamics and possibly other tools for multi-disciplinary analyses.The present work is focused on numerical validation, by demonstrating the solution of external aerodynamics problems in Euler and viscous flows. In details, instabilities in subsonic and supersonic Euler flows of 2D and 3D panels are verified, comparing the results in terms of static divergence and Limit Cycle Oscillation amplitude with literature works. Furthermore, the AGARD 445.6 wing flutter behavior is investigated in transonic turbulent flow.

Acoustic Energy Balance During the Onset, Growth, and Saturation of Thermoacoustic Instabilities

Abstract Thermoacoustic instabilities can occur in a wide range of combustors and are prejudicial since they can lead to increased mechanical fatigue or even catastrophic failure. A well-established formalism to predict the onset, growth and saturation of such instabilities is based on acoustic network models. This approach has been successfully employed to predict the frequency and amplitude of limit cycle oscillations in a variety of combustors. However, it does not provide any physical insight in terms of the acoustic energy balance of the system. On the other hand, Rayleigh's criterion may be used to quantify the losses, sources and transfers of acoustic energy within and at the boundaries of a combustor. However, this approach is cumbersome for most applications because it requires computing volume and surface integrals and averaging over an oscillation cycle. In this work, a new methodology for studying the acoustic energy balance of a combustor during the onset, growth and saturation of thermoacoustic instabilities is proposed. The two cornerstones of this new framework are the acoustic absorption coefficient Δ and the cycle-to-cycle acoustic energy ratio λ, both of which do not require computing integrals. Used along with a suitable acoustic network model, where the flame frequency response is described using the weakly nonlinear Flame Describing Function (FDF) formalism, these two dimensionless numbers are shown to characterize: 1) the variation of acoustic energy stored within the combustor between two consecutive cycles, 2) the acoustic energy transfers occurring at the combustor's boundaries, and 3) the sources and sinks of acoustic energy located within the combustor. The acoustic energy balance of the well-documented Palies burner is then analyzed during the onset, growth and saturation of thermoacoustic instabilities using this new methodology. It is demonstrated that this new approach allows a deeper understanding of the physical mechanisms at play. For instance, it is possible to determine when the flame acts as an acoustic energy source or sink, where acoustic damping is generated, and if acoustic energy is transmitted through the boundaries of the burner.

Bifurcation Characteristics of Airfoil-NESs coupled System

Aeroelastic instability has become a hot research topic. It has increasingly high-lighted its importance in the aircraft and fluid machinery industry and brings challenges to the safe flight of various aircraft and the safe operation of fluid machinery. Numerical simulations were performed to study the flow-induced oscillation of a two-dimensional airfoil with two non-linear energy sinks (NES). The main system has two degrees of freedom-pitch and heave. These two NESs are considered as subsystems, with the first NES at the leading edge and the second NES at the trailing edge. The bifurcation behaviour and the mechanism of the system leading to chaos of the coupled system under different parameters and initial conditions is studied by nonlinear analysis, and the relationship between large amplitude limit cycle oscillation and the chaotic occurrence point of the main system is revealed, this corresponding relationship has important theoretical significance for mastering and adjusting system parameters to make NES work within its ideal oscillation suppression parameters.

Experimental and numerical investigations on the nonlinear aeroelastic behavior of high aspect-ratio wings for different chord-wise store positions under stall and follower aerodynamic load models

This study performs an experimental and numerical investigation on the nonlinear aeroelastic response of cantilever high-aspect-ratio beam-like wings with a ballast at their free tips, emulating the effects of a store. As an extent, the effects of different chord-wise ballast positions are experimentally examined for two highly flexible rectangular wings. Furthermore, the numerical model proposed brings forward a nonlinear finite element beam model accounting for aerodynamic nonlinearities, via stall and follower forces models, along with geometrical nonlinearities due to large displacements and rotations. A great variety of analyses were performed: First, the flutter boundaries of the wings were analyzed; second, the limit cycle oscillation amplitudes and frequencies in the oscillating wings were evaluated; third, the coupling behavior and the nonlinear responses obtained were discussed under several attributes. The geometrical nonlinearities were taken into account by a total Lagrangian formulation based on a straightforward and consistent interpolation field in order to describe the exact kinematics of a Timoshenko’s beam. Nonlinear aerodynamic loads were computed via an unsteady strip theory in the time-domain with the Jones approximation for the Wagner’s function along with a follower aerodynamic loads assumption. Additionally, a non-usual stall model based on an experimental quasi-static stall curve for flat plates was used to interpolate the lift-curve slope. The experimental and numerical results indicated a minimum flutter speed for ballast positions about of −5 mm toward the leading edge. Next, different nonlinear post-flutter LCO behaviors were obtained for the different ballast positions tested. To conclude, the good correlation between model and experiments indicated that the nonlinear modeling approach proposed herein was capable to predict the aeroelastic behavior of the tested high aspect-ratio wings.

Analytical study of time-delayed feedback control of rectangular prisms undergoing subcritical galloping

In this study, time-delayed feedback control is investigated for an elastically mounted rectangular prism undergoing subcritical galloping in the transverse direction, when subjected to wind excitation. The mathematical model of the galloping system under consideration is established by using the quasi-steady aerodynamic theory. The control performance in terms of the galloping onset speed of the time-delayed displacement, velocity and acceleration feedback is investigated via linear stability analysis, respectively. Subsequently, the method of multiple scales is implemented for nonlinear analysis in order to derive the analytical expression of the vibration amplitude of the galloping system and determine the criticality curve that is the boundary of the subcritical and supercritical bifurcation regions. The results show that the hybrid objective of increasing the galloping onset speed, changing the Hopf bifurcation behavior from subcritical to supercritical and reducing the amplitude of limit-cycle oscillations can be achieved by means of delayed acceleration feedback. This study provides an analytical tool and procedure for time-delayed feedback control design of such a kind of flow–structure interaction system.

Nonlinear analysis of functionally graded skewed and tapered wing-like plates including porosities: A bifurcation study

In the present study, nonlinear panel flutter and bifurcation behavior of functionally graded ceramic/metal wing-like tapered and skewed plates are investigated. Porosities are distributed over the cross-section of the functionally graded structure. The flutter speed, limit cycle oscillations, and bifurcation diagrams of the functionally graded plate with two types of geometrical non-uniformities being skewness and taperness are explored. Nonlinear structural model is utilized based on the virtual work principle by including the von-Karman nonlinear kinematic strain assumption. The first order shear deformation theory is employed to consider the transverse shear effect in the structural model. First-order linear piston theory is used to model the aerodynamic loading while the generalized differential quadrature method is employed to solve the governing equations of motion. Time integration of the final ordinary equations of motion is carried out using the Newmark average acceleration method. Different volume fractions are investigated to enhance the flutter instability margins and post-flutter behavior of functionally graded plates. Results demonstrate that the volume fraction and porosity coefficients have significant effects on dynamic behavior and limit cycle oscillation amplitudes.

Three dimensional aeroelastic analyses considering free-play nonlinearity using computational fluid dynamics/computational structural dynamics coupling

Most of the previous studies on free-play aeroelasticity were based on the linear aerodynamic theory, which cannot consider the aerodynamic nonlinear effects. This paper proposes a framework of computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling approach that can deal with the three dimensional aeroelastic problems with both the free-play nonlinearity and the aerodynamic nonlinearity. The fictitious mass method is used to construct the reduced structural equations of motion and the switching point is detected using the bisection method. The adaptive time step obtained by the bisection method is returned to the CFD solver so that both the structural and the fluid equations are integrated using the same time step. An all-movable wing with free-play at the root is considered for numerical studies. Results demonstrate the CFD/CSD coupling method can predict the stable limit cycle oscillation (LCO) effectively. The initial condition study shows that the LCO behavior is subcritical and the hysteresis response can be predicted in time domain effectively by the presented method. The viscous effect is shown to increase the LCO boundary and shift the LCO amplitude to a larger velocity in transonic regime. The LCO boundary is determined from subsonic to transonic Mach numbers. The transonic dip in LCO boundary is found by the CFD/CSD coupling method, but the equivalent linearization with doublet lattice method fails to predict this phenomenon. From the results of this study, the LCO boundary is shown to be 43.5% below the flutter boundary at most.

Aeroelastic inverse: Estimation of aerodynamic loads during large amplitude limit cycle oscillations

This paper presents an algorithm to compute the aerodynamic forces and moments of an aeroelastic wing undergoing large amplitude heave and pitch limit cycle oscillations. The technique is based on inverting the equations of motion to solve for the lift and moment experienced by the wing. Bayesian inferencing is used to estimate the structural parameters of the system and generate credible intervals on the lift and moment calculations. The inversion technique is applied to study the affect of mass coupling on limit cycle oscillation amplitude. Examining the force, power, and energy of the system, the reasons for amplitude growth with wind speed can be determined. The results demonstrate that the influence of mass coupling on the pitch–heave difference is the driving factor in amplitude variation. The pitch–heave phase difference not only controls how much aerodynamic energy is transferred into the system but also how the aerodynamic energy is distributed between the degrees of freedom.

Nonlinear supersonic post-flutter motion of panels with adjacent bays and thermal effects

Aerospace vehicle structures in the supersonic regime flight have their outer skin subjected to unsteady aerodynamic and thermal loading, which may lead to aeroelastic instability. Typical aerospace structures are built as multiple adjacent panels, the so-called multi-bay configuration, but early efforts to predict the supersonic flutter were based on a single panel arrangement. This work evaluates the aeroelastic behavior of supersonic multi-bay fluttering panels under thermal effects, aiming to improve the understanding of adjacent panels interaction in the nonlinear regime. The aeroelastic model is established by using the first-order quasi-steady piston theory in conjunction with isotropic panel model using the von Kármán’s assumptions to account for geometrical nonlinearities. The Newmark time-integration method is used to evaluate the resulting equations of motion. The Hopf bifurcation behavior that determines the flutter onset, thermo-buckling loading, phase portrait plots, and bifurcation diagrams for two adjacent panels are presented. The numerical results show the detrimental aspect of thermal loading in the aeroelastic behavior of fluttering panels, and the new findings corroborate with some recent studies that highlight the difference in the nonlinear flutter behavior between a single panel and multi-bay panels. Moreover, the existence of limit cycle oscillations amplitude jumps from a certain level of flow dynamic pressure is also observed. The multi-bay panels configuration also shows the anticipation of the buckled to the limit cycle oscillation solutions, when compared with a single panel analysis. Results indicate that simplified single bay panel assumptions can underestimate the post-flutter oscillations amplitudes of the adjacent bay. Such dynamic behavior may lead to a negative impact on aircraft structural design and fatigue life estimation.

Experimental Nonlinear Flutter Analysis of a Cantilever Wing/Store

This paper studies experimentally the nonlinear aeroelastic and flutter behavior of a cantilever plate wing with an external store. The wing model that is constructed from plexiglass sheet is designed and tested in a closed-circuit subsonic wind tunnel. To deal with the structural nonlinearities of the model, various analysis tools such as time history plots, phase-plane projections and Fast Fourier Transform (FFT) have been used for detecting the critical and post-critical behaviors of the structure. The results show that flutter takes place by the coupling between the torsional and bending modes. A good correlation between the present experiments and previous numerical results is obtained. The nonlinear aeroelastic response and flutter boundary are investigated for different sweep angles. The flutter velocity and amplitudes of limit cycle oscillations (LCOs) increase rapidly with increasing sweep angle. The nonlinear response of the wing with an external store is also investigated, with the effect of store location on the nonlinear flutter boundary evaluated.

Vibration suppression of nonlinear rotating metamaterial beams

In this paper, we investigate the nonlinear vibration of a metamaterial structure that consists of a rotating cantilever beam attached to a periodic array of spring–mass–damper subsystems deployed for vibration suppression. The full nonlinear model of the system is developed. The nonlinear response due to a primary resonance excitation is investigated, and the capability of the metastructure to suppress vibration is examined. The mass of the resonators (absorbers) comes at the expense of the host structure’s mass itself, which makes the total mass of the system conservative. Free and forced vibration analyses are performed. We first use the method of multiple scales to analyze the nonlinear behavior of rotating beams. The perturbation solutions are validated against their numerical counterparts. Results show the presence of a critical rotational speed at which the beam undergoes bifurcation and starts to flutter. The addition of the absorbers is observed to slightly reduce this critical speed. Nevertheless, the amplitude of limit-cycle oscillations beyond bifurcation is found to decrease when equipping the rotating beam with local absorbers. The results demonstrate the capability of the metamaterial structure as an efficient damping treatment. Furthermore, we show that careful placement of the absorbers along the cantilever beam (close to the tip) enables further vibration mitigation.

Interplay of unsteady aerodynamics and flight dynamics of transport aircraft in icing conditions

Airframe icing causes significant degradation of aerodynamic characteristics and influences the flight safety. Wind tunnel study of longitudinal steady and unsteady aerodynamic characteristics of a transport aircraft in icing conditions is carried out in order to develop mathematical model of aerodynamics in the extended flight envelope. The wind tunnel results are validated through flight tests conducted for the real aircraft. Large, glaze-horn ice shapes, corresponding to holding flight phase, are considered. Influence of an ice protection system as well as its failure is examined. Effect of icing on the unsteady aerodynamics characteristics is studied not only through wind tunnel tests but also via analysis of subsequent influence on the flight dynamics of the aircraft. The conducted study shows that the ice shapes of the holding phase lead to reduced stall angle of attack (AoA), maximum lift, and longitudinal damping. Flight dynamics analysis demonstrates that dangerous aircraft behaviour in the form of high AoA departure and limit cycle oscillations (LCO) can be observed at smaller elevator deflections for the iced aircraft. The account for icing influence on the unsteady aerodynamics in the flight dynamics simulations revealed degradation of the dynamic response and deterioration of phase portraits of the system. Even for small AoA and elevator deflection, the aircraft might be trapped into the basin of attraction of high-AoA LCO. In addition, incorporating icing effects in unsteady aerodynamics manifest larger amplitude of LCO.

Limit cycle characterization of an aeroelastic wing in a bluff body wake

This paper presents an experimental investigation aimed at characterizing the kinematics of a pitching-heaving aeroelastic wing placed downstream of a rectangular bluff body. The influence of the bluff body wake on the wing is twofold: a viscous wake which produces a velocity deficit downstream and an oscillating induced velocity field due to periodic vortex shedding. The latter effect is the focus of this paper, specifically, the interaction between the wake frequency and the wing limit cycle oscillation (LCO) frequency. Wind tunnel experiments showed that the presence of the upstream bluff body causes modulation of the LCO amplitude. The modulation resembles a beat phenomenon, however the modulation frequency is related to the third harmonic of fLCO rather than the fundamental frequency. The modulation behavior also differs from that of a beat in that the spectral content contains sideband frequencies, characteristic of a multiplication between a carrier wave and a modulation wave rather than a simple sinusoidal superposition. Additionally, the streamwise spacing between the bluff body and the wing significantly influences the wing kinematics, with a closer spacing between the two bodies increasing the intensity of the amplitude modulation. For shedding frequencies sufficiently close to the LCO third harmonic, reducing this streamwise distance was shown to induce an alternation between two distinct modes of amplitude modulation, each with its own intensity and frequency.

Enhancing flutter stability in nanocomposite thin panels by harnessing CNT/polymer dissipation

Numerical and analytical investigations into the aeroelastic response of thin nanocomposite panels are discussed. The mechanical data of the nanocomposite plates with different weight fractions of randomly oriented carbon nanotubes (CNT) are obtained via an extensive experimental campaign based on dynamic mechanical analysis (DMA). Panel flutter induced by high supersonic flows is studied through both linear and nonlinear dynamic analyses aimed to investigate the effect of the CNT weight fraction onto the flutter and post-flutter condition. To this end, multiple scales perturbation analysis is carried out to characterize the flutter boundaries and the type of Hopf bifurcation at flutter. It is shown that a few percent CNT weight fraction, while entailing negligible weight penalty, can considerably increase the flutter speed as well as reduce the limit cycle oscillations amplitude.

A theoretical computational model of a plate in hypersonic flow

A theoretical model of an elastic panel in hypersonic flow is derived to be used for design and analysis. The nonlinear von Kármán plate equations are coupled with 1st order Piston Theory and linearized at the nonlinear steady-state deformation due to static pressure differential and thermal loads. Eigenvalue analysis is applied to determine the system’s stability, natural frequencies and mode shapes. Numerically time marching the equations provides transient response prediction which can be used to estimate limit cycle oscillation amplitude, frequency and time to onset. The model’s predictive capability is assessed by comparison to an experiment conducted at a free stream flow of Mach 6. Good agreement is shown between the theoretical and experimental natural frequencies and mode shapes of the fluid–structure system. Stability analysis is performed using linear and nonlinear methods to plot stability, flutter and buckling zones on a free stream static pressure vs temperature differential plane.

Active dynamic vibration absorber for flutter suppression

A novel flutter suppression technique is proposed using an active dynamic vibration absorber (ADVA). The ADVA is introduced by adding an active element to a classical mass-spring-damper system. This active element drives the mass by a controlling force using feedback signal from the response of the aeroelastic system. An aeroelastic mathematical model of a 2 degrees of freedom (DOFs) airfoil equipped with the proposed ADVA is established based on unsteady aerodynamics theory. The proposed ADVA is experimentally realized by a cantilever beam with a bonded Macro-fiber Composite (MFC) and a lumped mass. MFC nonlinearities are compensated by applying Prandtl-Ishlinskii approach to design a proportional integral (PI)-based hysteresis compensator controller. A test rig is designed, fabricated and the parameters of the physical airfoil model and the connected ADVA are identified. The feasibility of the ADVA for flutter suppression is assessed via a wind tunnel testing. The experimental results show that applying the closed-loop ADVA increases the airfoil flutter speed by 42.9%. Whereas the open-loop ADVA results in 26.6% improvement in the flutter speed. Moreover, if the airfoil experiences a high amplitude limit cycle oscillation (LCO), the ADVA efficiently suppresses up to 93.3% of its amplitude when turned to the closed-loop control.

Vortex-induced vibration of bridge decks: Describing function-based model

A describing function (DF)-based model is introduced for the simulation of vortex-induced vibration (VIV) of bridge decks. Similar to the linear frequency response function, the DF is the complex ratio of the first-order component of the nonlinear output to the harmonic input. The DF can be either identified using the forced vibration technique or based on the VIV nonlinear response time history. An iterative procedure is accordingly developed to predict the VIV response with the DF-based model, and an equivalent-damping-ratio-based simplified method is further proposed to efficiently obtain the limit cycle oscillation (LCO) amplitude of VIV. It is demonstrated that the conventional van der Pol-type model is equivalent to a special case of the DF-based model for VIV. Three case studies involving various cross-sections are utilized to validate the simulation accuracy and efficiency of the proposed DF-based model for typical features at VIV lock-in such as LCO and hysteresis phenomena. The vertical VIVs can be well captured by the DF-based model, while its capability of simulating the torsional VIVs requires further improvement. Furthermore, the predictive capability of DF-based model for vertical VIVs of bridge decks within a wide range of mass-damping conditions is highlighted.