Stable Limit Cycle Oscillations Research Articles

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.

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

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

Enhanced vehicle shimmy performance using inerter-based suppression mechanism

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

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.

Shimmy Suppression of an Aircraft Nose Landing Gear Using Torsional Nonlinear Energy Sink

Abstract Shimmy motion may occur for an aircraft nose landing gear (NLG) during its takeoff, landing, and taxiing procedure, which is an undesirable motion and should be suppressed. Here, a novel torsional nonlinear energy sink (NES) based on the cam-roller mechanism is proposed and applied in the NLG to improve its shimmy performance. The torsional NES is connected with the upper and lower struts of the NLG and its mechanical characteristic is studied. A seven-dimensional dynamic model of the NLG coupled with NES is built, which considers the NLG torsional and lateral motions, and the nonlinear tire model. The numerical continuation method is applied to analyze its shimmy performance, the stable area and limit cycle oscillation (LCO) amplitude are acquired, further compared with those of the original NLG without using the NES. The results show that as the NES is used in the NLG, the torsional shimmy dominated unstable area reduces significantly and the lateral shimmy dominated unstable area decreases slightly, which results in the increase of the stable area of the NLG; the maximum LCO amplitudes for the NLG torsional and lateral shimmy become smaller, the forward speed ranges as the torsional and lateral shimmy occur become narrower, which indicates that the stable forward speed ranges of the NLG become wider; the NLG shimmy performance improves as the NES has larger torsional inertia and damping, and smaller torsional linear stiffness. Thus, the NES can suppress the NLG shimmy motion and enhance its shimmy performance effectively.

Experimental and Numerical Nonlinear Stability Analysis of Wings Incorporating Flared Folding Wingtips

Recent studies have considered the use of wings incorporating flared folding wingtips (FFWTs) to enable higher aspect ratios (reducing overall induced drag) while also reducing gust loading and meeting airport operational requirements. This paper presents the first experimental research into the nonlinear dynamic behavior of a wing incorporating an FFWT. Wind-tunnel tests were conducted at a range of velocities below and beyond the linear flutter boundary. The experimental findings are compared with results obtained from continuation and bifurcation analyses on a representative low-fidelity numerical model. The results show that beyond the linear flutter boundary, stable limit cycle oscillations form, which is dependent on the flare angle, are bounded by either geometric or aerodynamic nonlinearities. Also presented is the effect of a wingtip trim tab on the stability boundary of a wing incorporating FFWTs. It is found that the tab angle can significantly alter the stability boundary of the system, indicating that the choice of camber is an important parameter when considering the stability boundary of FFWTs and that a moveable control surface on an FFWT could be used “in flight” to extend the stability boundary of an aircraft.

Numerical study on the nonlinear characteristics of shock induced two-dimensional panel flutter in inviscid flow

For an aeroelastic system of a two-dimensional elastic panel subjected to an impinging inviscid oblique shockwave, the nonlinear flutter characteristics are affected by many factors such as shock impingement location, cavity pressure and initial perturbation. The effects of the above factors on the variation of system bifurcation type and dynamic behaviors are investigated numerically. A low-fidelity computational method coupled with local piston theory and van Karman plate model, and a high-fidelity computational method coupled with Euler equations and finite element model are used for fluid-structure interaction simulations. Two sets of new findings are unveiled. First, either the variation of shock impingement location or cavity pressure can induce the aeroelastic system to transition between a subcritical bifurcation and a supercritical bifurcation. For some cases, the system bifurcation characteristics exhibit strong sensitivity to these two factors. Second, it is found that in addition to the limit cycle oscillation in the form of a combination of the second and third structural modes, multiple stable limit cycle oscillations due to the coupling of higher-order modes can be triggered by suitable initial perturbations. These limit cycle oscillations have high frequencies and amplitudes, which suggest a higher risk of structural fatigue failure.

An Integrated Test-Based Nonlinear Aeroelastic Analysis Method to Freeplay-Induced Limit-Cycle Oscillations

The nonlinear limit-cycle oscillations induced by aileron rotational freeplay are investigated by numerical simulation and flutter wind tunnel tests. An integrated nonlinear aeroelastic analysis method and procedure based on the nonlinear system identification technology is proposed. Based on the characteristics of the structure, a low-speed nonlinear flutter wing-aileron model that considers both the linear flutter test and the nonlinear limit-cycle oscillation test with aileron rotational freeplay is designed. The wind tunnel tests are conducted with the model, and the stable limit-cycle oscillation is observed by accelerometers under the linear flutter boundary with a 0° angle of attack. Higher-order harmonic vibration is also observed with a low proportion compared to the fundamental oscillation, and the limit-cycle oscillation grows gradually with the increase of airspeed until flutter occurs. The comparison of numerical and experimental investigation shows high consistency.

Insight into the intrinsic time-varying aerodynamic properties of a truss girder undergoing a flutter with subcritical Hopf bifurcation

In this study, the investigation on the intrinsic time-varying nonlinear aerodynamic properties and actual energy feedback mechanism of limit cycle oscillation (LCO) and subcritical Hopf bifurcation, which is of great importance of the flutter design of bridges, but once rarely be conducted, was comprehensively carried out. The response characteristics of the nonlinear flutter were experimentally investigated. The dynamic mechanism of the subcritical Hopf bifurcation was firstly introduced in terms of the equivalent modal damping ratio as a function of amplitude. Further, a modified nonlinear self-excited force model was proposed to investigate the intrinsic time-varying characteristics of the aerodynamic properties and the real energy feedback mechanism of the subcritical Hopf bifurcation. Based on the model, the characteristics of nonlinear self-excited forces and nonlinear aerodynamic damping coefficients, as well as their contribution to the energy exchange behaviors, were studied. Subsequently, the energy feedback mechanisms of LCO and subcritical Hopf bifurcation were qualitatively discussed in detail considering the hysteresis loop of nonlinear self-excited forces (NSEF), which highlighted the important role of the linear self-excited-moment (including damping and stiffness) in the generation of a subcritical Hopf bifurcation, and the higher-order force components on the generation of stable LCO. Finally, the driving mechanisms of LCO and subcritical Hopf bifurcation were also quantitatively explained via an energy budget analysis, wherein the total work applied by structural and aerodynamic forces was presented as a function of amplitude and time.

Deconstructing the role of myosin contractility in force fluctuations within focal adhesions

Force fluctuations exhibited in focal adhesions that connect a cell to its extracellular environment point to the complex role of the underlying machinery that controls cell migration. To elucidate the explicit role of myosin motors in the temporal traction force oscillations, we vary the contractility of these motors in a dynamical model based on the molecular clutch hypothesis. As the contractility is lowered, effected both by changing the motor velocity and the rate of attachment/detachment, we show analytically in an experimentally relevant parameter space, that the system goes from decaying oscillations to stable limit cycle oscillations through a supercritical Hopf bifurcation. As a function of the motor activity and the number of clutches, the system exhibits a rich array of dynamical states. We corroborate our analytical results with stochastic simulations of the motor-clutch system. We obtain limit cycle oscillations in the parameter regime as predicted by our model. The frequency range of oscillations in the average clutch and motor deformation compares well with experimental results.

A modal self-excitation method of tensegrity robots

A tensegrity robot is a type of soft-rigid system, whose high compliance is prone to induce structural vibrations which traditionally should be avoided. However, a stable limit cycle oscillation at a natural frequency can be exploited to improve motion efficiency of tensegrity robots. By constructing a modal self-excitation system, we can produce limit cycle oscillation for tensegrity robots. Therefore, in this paper, we propose a method to realize modal self-excitation of under-actuated tensegrity robots. A task-space second-order structural model of a tensegrity robot is developed and a mapping between forces of cable-space and that of task-space is derived. Meanwhile, we propose a cable sensitivity vector from modal shape to guide the selection of active cables in the tensegrity robot. A modal self-excitation system is designed by combining a bandpass filter with a describing function into the control system. In this way, a limit cycle oscillation at a frequency can be formed close to the desired natural frequency. The stability of this limit cycle oscillation is analyzed by perturbation method. Modal switching excitation is also achieved by switching between different modal excitation loops online. The effectiveness of the proposed modal self-excitation method for tensegrity robots is verified by several numerical simulations and tests.

EXTENDING A VAN DER POL BASED REDUCED-ORDER MODEL FOR FLUID-STRUCTURE INTERACTION APPLIED TO NON-SYNCHRONOUS VIBRATIONS IN TURBOMACHINERY

Abstract This paper expands upon a multi-degree-of-freedom, Van der Pol oscillator used to model buffet and Nonsynchronous Vibrations (NSV) in turbines. Two degrees-of-freedom are used, a fluid tracking variable incorporating a Van der Pol oscillator and a classic spring, mass, damper mounted cylinder variable; thus, this model is one of fluid-structure interaction. This model has been previously shown to exhibit the two main aspects of NSV. The first is the lock-in or entrainment phenomenon of the fluid shedding frequency jumping onto the natural frequency of the oscillator, while the second is a stable limit cycle oscillation (LCO) once the transient solution disappears. Improvements are made to the previous model to better understand this aeroelastic phenomenon. First, an error minimizing technique through a system identification method is used to tune the coefficients in the Reduced Order Model (ROM) to improve the accuracy in comparison to experimental data. Secondly, a cubic stiffness term is added to the fluid equation; this term is often seen in the Duffing Oscillator equation, which allows this ROM to capture the experimental behavior more accurately, seen in previous literature. The finalized model captures the experimental cylinder data found in literature much better than the previous model. These improvements also open the door for future models, such as that of a pitching airfoil or a turbomachinery blade, to create a preliminary design tool for studying NSV in turbomachinery.

Experimental investigation on post-flutter characteristics of a typical steel-truss suspension bridge deck

A testing device that was specially developed for large torsional amplitude vibrations was adopted to investigate the nonlinear post-flutter behaviors of a typical steel-truss girder bridge deck. Firstly, a series of free decay vibration tests of a narrow rigid rod model and a section model of the bridge were conducted in still-air to obtain mechanical properties and still-air induced aerodynamic properties. Then, post-flutter experiments of the bridge section model under initial attack angles of −3°, −0°, +3°, +5°, +7°and with different mechanical damping ratios were carried out. Finally, nonlinear and coupling behaviors of the bridge section model during post-flutter were described and analyzed in detail from several different aspects such as aeroelastic center, vibration mode, and wind-induced aerodynamic damping. The results showed that the truss girder section exhibited significant vertical-torsional coupled soft flutter behavior at various attack angles and there was a stable limit cycle oscillation (LCO) and an unstable LCO under a given wind speed near the critical speed. Moreover, the added aerodynamic damping, the coupling vertical deformation, the aerodynamic torsional center, the wind-induced aerodynamic damping, and the soft flutter mode in the post-critical state all exhibited the characteristic of amplitude dependence. Meanwhile, it was found that the coupling vertical deformation was strongly related to the torsional amplitude and wind attack angle. Finally, the effects of mechanical damping on the post-flutter performance were discussed and the results showed that some nonlinear dampers whose damping increased with the increasing amplitude can be adopted to improve the critical flutter speed without deteriorating or even enhancing the post-flutter performance.

Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters

Abstract In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings.

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.

Environment coupled piezoelectric galloping wind energy harvesting

As a stable limit cycle oscillation occurred after subcritical Hopf bifurcation, galloping was preferred in scattered wind energy harvesting. For practical applications across seasons and regions, environmental adaptability of galloping piezoelectric energy harvester on atmospheric temperature and wind speed is investigated. A multi-field coupled geometric and aerodynamic nonlinear distributed-parameter model is established. The method of multiple scales is used to derive the analytical solutions of the nonlinear coupled continuous model. Wind-tunnel tests of the piezoelectric galloping energy harvesters are performed for model validation. The empirical coefficients of nonlinear piezoelectric parameters varied by the temperature are identified. The optimal load resistance tailored for the atmospheric temperature and airflow speed is determined. With the environment adaptable load resistance, the onset speed of energy harvesting is minimized to be lower than 1 m/s for the galloping to occur. High temperature benefits galloping energy harvesting begin at low wind speed. Excellent environmental adaptability of the piezoelectric galloping energy harvester is realized in atmospheric temperature of −40 °C to 50 °C and breeze to gale wind speed of 1–24 m/s.

Experimental investigations of equivalence ratio effect on nonlinear dynamics features in premixed swirl-stabilized combustor

In this work, the dynamics of the pressure oscillations and flame behaviors in a premixed swirl-stabilized combustor were experimentally investigated. As equivalence ratio Φ varied between 0.55 and 1.35, five combustion states were distinguished based on the characteristics of the acoustic pressures and flame dynamics. The permutation entropy (PE) and proper orthogonal decomposition (POD) were used to analyze the fluctuating pressure signal complexities and spatial CH⁎ chemiluminescence images, respectively. The results showed that the magnitude of the PE was at a differentiable level. In addition, the POD modes exhibited distinguishable features when the system was in different combustion states. Further analyses of the state transitions indicated that the system underwent a supercritical Hopf bifurcation and a pair of fold bifurcations when Φ was in the lean condition. The state transition from quasi-periodic oscillation to limit-cycle oscillation was attributed to the movement of the flame location, which enhanced the coupling of the acoustics and combustion by reducing the phase difference between the pressure oscillations and heat release fluctuations. In addition, a hysteresis phenomenon was observed for the trajectory of the fluctuating pressure root-mean-square (RMS) value in the forward and backward processes of the equivalence ratio under the lean condition. A low-order model was used to capture the bifurcations; the theoretical results agreed well with the experimental data. As for the rich condition, a supercritical Hopf bifurcation was captured when the system switched between the dual-frequency oscillation mode and the rich stable state. Furthermore, the influence of combustion chamber length Lc on the combustion stability in relation to Φ was investigated. Parameter Lc was divided into steady and unsteady regions depending on whether the combustion system could reach limit-cycle oscillation. The PE was demonstrated to be effective in distinguishing the stable state and limit-cycle oscillation even for combustors of different lengths. Understanding the nonlinear dynamics by analyzing experimental data could be critical to developing models for the prediction and suppression of thermoacoustic instabilities.

A numerical study on flapping dynamics of a flexible two-layered plate in a uniform flow

Over the past few decades, energy generation from piezoelectric patches mounted on a flexible flat plate exhibiting flapping motion has gained attention. Piezoelectric patches are generally multilayered consisting of piezoelectric, substrate, and electrode layers placed on top of each other. Although the flapping dynamics of single-layered structures have been extensively studied, understanding the flapping dynamics of multilayered structures is minimal. We propose a quasi-monolithic formulation with exact interface tracking to simulate the fluid–multilayered structure interactions. The proposed formulation is validated by considering a simple two-layered plate-like structure with identical material properties against a single-layered plate. We then use this formulation to perform parametric simulations by providing different material properties to each layer of the plate to understand the effect of differences in the material properties on the flapping dynamics. The simulations are performed by selecting different values of Young’s modulus and density for each of the layers such that the average structure-to-fluid mass ratio m*avg=0.1 and the average non-dimensional bending stiffness KBavg=0.0005 remain constant for a Reynolds number Re = 1000. First, the effects of difference in elasticity between the two layers on the flapping amplitude, frequency, forces, and vortex shedding patterns are investigated. Following this, the effect of differences in elastic properties on the onset of flapping is investigated for a case with Re = 1000, m*avg=0.1, and KBavg=0.0008, for which a single-layered plate does not undergo self-sustained flapping. Two distinct response regimes are observed depending on the difference in elastic properties between the two layers: (I) fixed-point stable and (II) periodic limit cycle oscillations. Finally, we look into the effects of structural density differences on the flapping dynamics of a two-layered plate.

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.

Nonlinear aeroservoelastic analysis of a supersonic aircraft with control fin free-play by component mode synthesis technique

To investigate the nonlinear aeroservoelastic behaviors of a three-dimensional supersonic aircraft with control fin free-play, a modeling method based on the component-mode synthesis technique is utilized. The most distinctive feature of this method is that the structural nonlinearity can be expressed explicitly in the reduced-order aeroservoelastic model. The unsteady aerodynamic model with discrete gust loads is formulated using the supersonic lifting surface theory and the minimum state approximation. The results validate the feasibility of the reduced-order aeroelastic model for gust response analysis. A hybrid adaptive feedback control algorithm is proposed for the gust load alleviation by integrating the positive position feedback (PPF) and the filtered-x least-mean-square (FxLMS) algorithm. The comparative study demonstrates that the designed PPF-FxLMS algorithm has a better control performance for the alleviation of the pitch attitude angle induced by the gust loads, and it remains effective in the post-flutter regime. Moreover, the effects of free-play nonlinearity on the dynamic behaviors of the aeroservoelastic system are also studied, and numerical results show that the presence of free-play can lead to a larger peak value of gust response. The nonlinear vibration of the control fin induces the aircraft pitch attitude to produce stable limit cycle oscillations with high frequency, and variation in the free-play magnitude significantly influences the gust responses of the nonlinear aeroservoelastic system.