Stable Limit Cycle Oscillations Research Articles

Energy harvesting for actuators and sensors using free-floating flaps

A novel configuration of an energy harvester for local actuation and sensing devices using limit cycle oscillations has been modeled, designed and tested. A wing section has been designed with two trailing-edge free-floating flaps. A free-floating flap is a flap that can freely rotate around a hinge axis and is driven by trailing edge tabs. In the rotational axis of each flap a generator is mounted that converts the vibrational energy into electricity. It has been demonstrated numerically how a simple electronic system can be used to keep such a system at stable limit cycle oscillations by varying the resistance in the electric circuit. Additionally, it was shown that the stability of the system is coupled to the charge level of the battery, with increasing charge level leading to a less stable system. The system has been manufactured and tested in the Open Jet Wind Tunnel Facility of the Technical University Delft. The numerical results could be validated successfully and voltage generation could be demonstrated at cost of a decrease in lift of 2%.

On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise biological parameters

In this paper we have analyzed the stability and Hopf-bifurcation behaviors of a multi-delayed two-species competitive system affected by toxic substances with imprecise biological parameters. We have exercised a method to handle these imprecise biological parameters by using parametric form of interval numbers. We have studied the feasibility of various equilibrium points and their stability. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation due to variation in the delay parameters which lead to Hopf-bifurcation. Numerical simulations with a hypothetical set of data have been done to support the analytical findings.

Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis

We have studied the combined effect of toxicant and fluctuation of the biological parameters on the dynamical behaviors of a delayed two-species competitive system with imprecise biological parameters. Due to the global increase of harmful phytoplankton blooms, the study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research now days. The ordinary mathematical formulation of models for two competing phytoplankton species, when one or both the species liberate toxic substances, is unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic model never predicts the sudden localized behavior of certain species. These obstacles of mathematical modeling can be overcomed if we include interval variability of biological parameters in our modeling approach. In this investigation, we construct imprecise models of allelopathic interactions between two competing phytoplankton species as a parametric differential equation model. We incorporate the effect of toxicant on the species in two different cases known as toxic inhibition and toxic stimulatory system. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Analytical findings are supported through exhaustive numerical simulations.

Airfoil-based piezoelectric energy harvesting by exploiting the pseudoelastic hysteresis of shape memory alloy springs

The modeling and analysis of an electromechanically coupled typical aeroelastic section with shape memory alloy springs for wind energy harvesting is addressed in this paper. An airfoil with two-degrees-of-freedom, namely pitch and plunge, is considered and piezoelectric coupling is added to the plunge degree-of-freedom. A load resistance is assumed in the electrical domain of the problem in order to estimate the electrical power output. Shape memory alloy coil springs are modeled in the pitch degree-of-freedom of the typical section. A nickel–titanium alloy that exhibits pseudoelasticity at room temperature is assumed. The constitutive model for the shape memory alloy is based on classical phenomenological models. The unsteady aerodynamic loads are obtained by Jones’ approximation to Wagner’s indicial function. The resulting nonlinear electroaeroelastic model is cast into a state-space representation and solved with a Runge–Kutta method. The effects of preload values of the shape memory springs and resistive power generation on the aeroelastic behavior of the wind energy harvester are investigated at the flutter boundary and in a post-flutter regime. The nonlinear kinetics of the austenite-to-martensite phase transformation changes the typical linear flutter behavior to stable limit-cycle oscillations over a range of airflow speeds. Such nonlinear aeroelastic behavior introduced by the hysteretic behavior of the SMA springs provides an important source of persistent electrical power.

Post-Critical Behavior of Suspension Bridges Under Nonlinear Aerodynamic Loading

Abstract The limit cycle oscillations (LCOs) exhibited by long-span suspension bridges in post-flutter condition are investigated. A parametric dynamic model of prestressed long-span suspension bridges is coupled with a nonlinear quasi-steady aerodynamic formulation to obtain the governing aeroelastic partial differential equations adopted herewith. By employing the Faedo–Galerkin method, the aeroelastic nonlinear equations are reduced to their state-space ordinary differential form. Convergence analysis for the reduction process is first carried out and time-domain simulations are performed to investigate LCOs while continuation tools are employed to path follow the post-critical LCOs. A supercritical Hopf bifurcation behavior, confirmed by a stable LCO, is found past the critical flutter condition. The analysis shows that the LCO amplitude increases with the wind speed up to a secondary critical speed where it terminates with a fold bifurcation. The stability of the LCOs within the range bracketed by the Hopf and fold bifurcations is evaluated by performing parametric analyses regarding the main design parameters that can be affected by uncertainties, primarily the structural damping and the initial wind angle of attack.

Three-dimensional autonomous pacemaker in the photosensitive Belousov-Zhabotinsky medium

In experiments with the photosensitive Belousov-Zhabotinsky reaction (PBZR) we found a stable three-dimensional organizing center that periodically emits trigger waves of chemical concentration. The experiments are performed in a parameter regime with negative line tension using an open gel reactor to maintain stationary non-equilibrium conditions. The observed periodic wave source is formed by a scroll ring stabilized due to its interaction with a no-flux boundary. Sufficiently far from the boundary, the scroll ring expands and undergoes the negative line tension instability before it finally develops into scroll wave turbulence. Our experimental results are reproduced by numerical integration of the modified Oregonator model for the PBZR. Stationary and breathing self-organized pacemakers have been found in these numerical simulations. In the latter case, both the radius of the scroll ring and the distance of its filament plane to the no-flux boundary after some transient undergo undamped stable limit cycle oscillations. So far, contrary to their stationary counterpart, the numerically predicted breathing autonomous pacemaker has not been observed in the chemical experiment.

Experimental Small-Amplitude Self-Sustained Pitch–Heave Oscillations at Transitional Reynolds Numbers

An experimental investigation of a free-mounted rigid NACA0012 airfoil in pitch–heave is conducted at transitional Reynolds numbers. Previously, small-amplitude self-sustained oscillations in pure pitch were observed for a specific range of Reynolds numbers: . They were attributed to laminar boundary-layer separation, which leads to negative aerodynamic damping at a 0 deg angle of attack. In this phase of the research, the investigation is extended to 2 deg of freedom by enabling the heave motion. Depending on initial conditions, two competing stable limit-cycle oscillations, small and large, are observed. It is determined that the origin of the large-amplitude limit-cycle oscillations is coalescence flutter for which the exponentially growing amplitude is limited by flow separation at large angles of attack. On the other hand, the small-amplitude oscillations are the same laminar separation flutter type as the pitch-only case. For the most part, the heave motion plays no fundamental role in the dynamics, as the amplitude of the pitch oscillations is not drastically affected in comparison with the single-degree-of-freedom pitch-only case nor is the range of Reynolds numbers at which the limit-cycle oscillations are observed. Nonetheless, the heave motion contributes to the net transfer of energy from the flow to the structure, as the mean kinetic energy present in the oscillations is increased significantly compared to the pitch-only case.

Aeroelastic Systems with Softening Nonlinearity

Aeroelastic systems with softening nonlinearity are not fully understood. The literature is not definitive, contains partial explanations, and even conflicting results. This problem is addressed for the case of a simple pitch-flap wing, with softening cubic nonlinearity in the pitch stiffness. Complex dynamical behavior is revealed. The study is carried out in three parts, the first of which is the identification and stability analysis of limit cycles in the frequency domain using describing functions combined with the Sherman–Morrison formula. Numerical integration of the nondimensionalized governing equations in the time domain is then carried out and, in addition to confirming the frequency domain results, new behavior is revealed, including asymmetric limit cycles and chaos. Finally, bifurcation analysis is undertaken using numerical continuation methods to reveal Hopf, symmetry breaking (pitchfork), fold and period doubling (flip) bifurcations. The effects of initial conditions and the proximity of limit cycle oscillation and chaos to static divergence are considered. Sections of the basin of attraction are presented for different wing configurations to show that the boundaries separating the different regions of dynamic behavior may be simple or nonsimple. It is found that some regions may appear where predicted stable limit cycle oscillation is free from the destabilizing effect of softening nonlinearity.

Nonlinear aeroelasticity of high-aspect-ratio wings excited by time-dependent thrust

Effects of engine placement on flutter characteristics of a very flexible high-aspect-ratio wing are investigated using the code NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Aircraft). Gravity for this class of wings plays an important role in flutter characteristics. In the absence of aerodynamic and gravitational forces and without an engine, the kinetic energy of the first two modes are calculated. Maximum and minimum flutter speed locations coincide with the area of minimum and maximum kinetic energy of the second bending and torsion modes. Time-dependent dynamic behavior of a turboshaft engine (JetCat SP5) is simulated with a transient engine model and the nonlinear aeroelastic response of the wing to the engine’s time-dependent thrust and dynamic excitation is presented. Below the flutter speed, at the wing tip and behind the elastic axis, the impulse engine excitation leads to a stable limit cycle oscillation; and for the ramp kind of excitation, beyond the flutter speed, at 75 % span, behind the elastic axis, it produces chaotic oscillation in the wing. Both the excitations above the flutter speed are stabilized, inboard of the wing.

Low speed flutter and limit cycle oscillations of a two-degree-of-freedom flat plate in a wind tunnel

This paper explores the dynamical response of a two-degree-of-freedom flat plate undergoing classical coupled-mode flutter in a wind tunnel. Tests are performed at low Reynolds number (Re~2.5×104), using an aeroelastic set-up that enables high amplitude pitch–plunge motion. Starting from rest and increasing the flow velocity, an unstable behaviour is first observed at the merging of frequencies: after a transient growth period the system enters a low amplitude limit-cycle oscillation regime with slowly varying amplitude. For higher velocity the system transitions to higher-amplitude and stable limit cycle oscillations (LCO) with amplitude increasing with the flow velocity. Decreasing the velocity from this upper LCO branch the system remains in stable self-sustained oscillations down to 85% of the critical velocity. Starting from rest, the system can also move toward a stable LCO regime if a significant perturbation is imposed. Those results show that both the flutter boundary and post-critical behaviour are affected by nonlinear mechanisms. They also suggest that nonlinear aerodynamic effects play a significant role.

Aeroelastic characteristics of linear and nonlinear piezo-aeroelastic energy harvester

In the present study, a piezo-aeroelastic energy harvester using nonlinear aeroelastic behaviors is proposed, and their characteristics and performance are investigated. The energy harvester is modeled by a two-dimensional typical section airfoil. The nondimensional parameters of the harvester are introduced, and the nondimensional piezo-aeroelastic equations are formulated. For the piezo-aeroelastic analysis, the root-locus method and time-integration method are used, and the present method is verified with experimental and analytical results. The iterative method is introduced to calculate the frequency response functions of a nonlinear piezo-aeroelastic energy harvester. The aeroelastic characteristics of a linear piezo-aeroelastic energy harvester and the effects of parameters are investigated. The results show that the linear piezo-aeroelastic energy harvester can be used to generate electricity only at the vicinity of flutter speed. It is assumed that the nonlinear piezo-aeroelastic energy harvester has free play and cubic hardening in pitch. For free play, nonlinear aeroelastic results show that stable limit cycle oscillations are observed in the wide range of air speed below flutter speed when the frequency ratio is 1.3, and unstable limit cycle oscillations are observed at air speeds over flutter speed when the frequency ratio is 0.3. For cubic hardening, unstable limit cycle oscillations are observed at air speeds below flutter speed when the frequency ratio is 0.3, and stable limit cycle oscillations are observed at air speeds over flutter speed when the frequency ratio is 1.3. Finally, the authors discussed how to use these aeroelastic responses for piezo-aeroelastic energy harvesting.

Probabilistic parameter estimation of a fluttering aeroelastic system in the transitional Reynolds number regime

We present a Bayesian parameter estimation results of a self-sustaining aeroelastic oscillator. The system consists of an elastically mounted rigid wing on a rig fixed in a wind tunnel. For certain flow conditions in the transitional Reynolds number regime, i.e. 10,000<Re<1,000,000, the wing absorbs energy from the flow and settles in a stable limit cycle oscillation (LCO). The LCO originates from laminar boundary layer separation which leads to negative aerodynamic damping at small angles of attack. We propose an empirical model of the aeroelastic system in the form of a generalized Duffing–van der Pol oscillator. A statistical technique is developed to estimate the linear and nonlinear parameters of the aeroelastic oscillator. In addition, the model noise term, in part accounting for the amplitude modulation of the LCO, is also estimated. In particular, we apply a Bayesian inference technique for parameter estimation that involves a state estimation problem using the measurement data. We exploit the extended Kalman filter for the state estimation. To generate samples from the posterior joint probability density functions of the aeroelastic parameters, we use the Markov Chain Monte Carlo (MCMC) simulation. Finally, the confidence interval of the static aerodynamic moment coefficient is presented.

Experimental study on the flight dynamics of a bioinspired ornithopter: free flight testing and wind tunnel testing

This study experimentally shows the flight dynamics of a bioinspired ornithopter using two different types of approach: (1) free flight testing, and (2) wind tunnel testing. An ornithopter is flown in straight and level flight with a fixed wingbeat frequency and tail elevation angle. A three-dimensional visual tracking system is applied to follow the retro-reflective markers on the ornithopter and record the flight trajectories. The unique oscillatory behavior of the body in the longitudinal plane is observed in the free flight testing and the detailed wing and tail deformations are also obtained. Based on the trim flight data, a specially devised tether device is designed and employed to emulate the free flight conditions in the wind tunnel. The tether device provides minimal mechanical interference and longitudinal flight dynamic characteristics similar to those of free flight. On introducing a pitching moment disturbance to the body, the oscillation recovered to the original trajectory turns out to be a stable limit-cycle oscillation (LCO). During the wind tunnel testing, the magnitude of LCO is effectively suppressed by active tail motion.

Comments on local power oscillation phenomenon at BWRs

Under the framework of BWR stability analysis, local neutron-flux oscillation events have attracted the attention of a number of researchers. In 1996, an unusual instability event occurred at Forsmark-1 in which an irregular oscillation pattern with highly localized, relatively large-amplitude oscillations were measured. Some authors assumed that this behaviour was caused by the superposition of stable spatial mode limit cycle oscillations, where the BWR core as a neutron kinetics/-thermal-hydraulic coupled system is unstable. Subsequent time-series analysis of the local power range monitor (LPRM) signals resulted in a space-dependent decay ratio, an inexplicable result. Furthermore, noise analysis-based localization techniques pointed towards the existence of two strong “perturbation sources” in one of the two halves of the core, one of them coinciding with the radial position of an unseated bundle. In the scope of theoretical work, the possibility of a space-dependent decay ratio was discussed but not comprehensively understood. Motivated by these findings, the effect of local neutron-flux oscillations on the stability behaviour of BWR is discussed, and one possible interpretation is proposed which is able to explain the space-dependent decay ratio as well as the long term oscillation pattern. The RAM–ROM method is applied to a Forsmark measurement case, where an irregular oscillation pattern was found and to an operational point (KKB-B8) of NPP Brunsbüttel, where a local neutron-flux oscillation is superimposed on an unstable global power oscillation. The effect of the local neutron flux oscillating sources on the space- and time-dependent neutron field is described by a rigorous application of the mode expansion approach. The consequences to signal analysis are then discussed. It will be pointed out in the paper that when a BWR system is stable with regards to power oscillations but is driven by local neutron-flux oscillating sources, the decay ratio does not indicate the real BWR stability behaviour.

Static/Dynamic Edge Movability Effect on Non-Linear Aerothermoelastic Behavior of Geometrically Imperfect Curved Skin Panel: Flutter and Post-Flutter Analysis

This paper addresses the problem of the aerothermoelastic modeling behavior and analyses of skin curved panels with static and dynamic edge movability effect in high supersonic flow. Flutter and post-flutter behavior will be analyzed toward determining under which conditions such panels will exhibit a benign instability, that is a stable limit cycle oscillation, or a catastrophic instability, that is an unstable LCO. The aerothermoelastic governing equations are developed from the geometrically non-linear theory of infinitely long two dimensional curved panels. Von Kármán non-linear strain-displacement relation in conjunction with the Kirchhoff plate-hypothesis is adopted. A geometrically imperfect curved panel forced by a supersonic/hypersonic unsteady flow is numerically investigated using Galerkin approach. These equations are based on the third-order piston theory aerodynamic for modeling the flow-induced forces. Furthermore, the effects of thermal degradation and Kelvin’s model of structural damping independent of time and temperature are also considered in this model. Computational analysis and discussion of the finding along with pertinent conclusions are presented.

Flow-induced control of chemical turbulence

We report spatiotemporal chaos in the Oregonator model of the Belousov-Zhabotinsky reaction. Spatiotemporal chaos spontaneously develops in a regime, where the underlying local dynamics show stable limit cycle oscillations (diffusion-induced turbulence). We show that spatiotemporal chaos can be suppressed by a unidirectional flow in the system. With increasing flow velocity, we observe a transition scenario from spatiotemporal chaos via a regime of travelling waves to a stationary steady state. At large flow velocities, we recover the known regime of flow distributed oscillations.

A drive-free vibratory gyroscope

Computational and analytical works have shown that certain coupling schemes can lead to significant enhancements in sensitivity, accuracy, and lower costs for a wide range of sensor devices whose output and performance depends directly on the ability of individual units to generate stable limit cycle oscillations. Vibratory gyroscopes are very good candidates for this new paradigm as their accuracy and sensitivity are directly dependent on the ability of a driving signal to produce and maintain oscillations with stable amplitude, phase, and frequency. To achieve higher accuracy, we show proof of concept of a novel scheme: a drive-free coupled gyroscope system in which the coupling alone can lead to self-regulated limit cycle oscillations in the drive- and sense-axes with stable constant amplitude and phase-locking.

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

SUZAKUOBSERVATION OF GRS 1915+105: EVOLUTION OF ACCRETION DISK STRUCTURE DURING LIMIT-CYCLE OSCILLATION

We present results from the Suzaku observation of the microquasar GRS 1915+105 performed during the 2005 October multiwavelength campaign. The data include both stable state (class \chi) and limit-cycle oscillation (class \theta). Correct interstellar absorption as well as effects of dust scattering are fully taken into account in the spectral analysis. The energy spectra in the 2-120 keV band in both states are all dominated by strong Comptonization of disk photons by an optically thick (\tau ~7-10) and low temperature (T_e ~2-3 keV) hybrid plasmas containing non-thermal electrons produced with 10-60% of the total power input. Absorption lines of highly ionized Fe ions detected during the oscillation indicate that a strong disk wind is developed. The ionization stage of the wind correlates with the X-ray flux, supporting the photoionization origin. The iron-K emission line shows a strong variability during the oscillation; the reflection is strongest during the dip but disappears during the flare. We interpret this as evidence for "self-shielding" that the Comptonizing corona becomes geometrically thick in the flare phase, preventing photons from irradiating the outer disk. The low-temperature and high luminosity disk emission suggests that the disk structure is similar to that in the very high state of canonical black hole binaries. The spectral variability during the oscillation is explained by the change of the disk geometry and of the physical parameters of Comptonizing corona, particularly the fractional power supplied to the acceleration of non-thermal particles.

Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch

The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincaré mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincaré section also approves the previous conclusion.