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 Poincare 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 Poincare section also approves the previous conclusion.

Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity

We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring （freeplay） in the pitch degree of freedom and a linear fiexural spring in the plunge degree of freedom. By attaching a piezoelectric material （PSI-5A4E） to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations （catastrophic for real aircrafts）, harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum value of the load resistance exists, at which the level of the harvested power is maximized.

Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the non- linearity of the airfoil section＇s freeplay. There are two crit- ical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincar6 map is constructed for the limit cycle os- cillations by using piecewise-linear solutions with and with- out contact in the system. Through analysis of the Poincar6 map, a series of equations which can determine the frequen- cies of period-1 limit cycle oscillations at any flight veloc- ity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agree- ment is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. More- over, there exist multi-periods limit cycle oscillations and even complicated ＂chaotic＂ oscillations may occur, which are usually found in smooth nonlinear dynamic systems.

Dynamic analysis of two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flow

The nonlinear aeroelastic response of a two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flows is investigated. The second-order piston theory is used to analyze a double-wedge airfoil. Then, the fold bifurcation and the amplitude jump phenomenon are detected by the averaging method and the multi-variable Floquet theory. The analyticall results are further verified by numerical simulations. Finally, the influence of the freeplay parameters on the aeroelastic response is analyzed in detail.

Oscillation quenching and physical explanation on freeplay-based aeroelastic airfoil in transonic viscous flow

Limit Cycle Oscillation(LCO)quenching of a supercritical airfoil(NLR 7301)considering freeplay is investigated in transonic viscous flow.Computational Fluid Dynamics(CFD)based on Navier-Stokes equations is implemented to calculate transonic aerodynamic forces.A loosely coupled scheme with steady CFD and an efficient graphic method are developed to obtain the aerodynamic preload.LCO quenching phenomenon is observed from the nonlinear dynamic aeroelastic response obtained by using time marching approach.As the airspeed increases,LCO appears then quenches,forming the first LCO branch.Following the quenching region,LCO occurs again and sustains until the divergence of the response,forming the second LCO branch.The quenching of LCOs was addressed physically based on the aerodynamic preload and the linear flutter characteristic.An“island”of stable region is observed in the flutter boundary,i.e.the flutter speed versus the mean Angle of Attack(AoA).The LCO quenches when the aerodynamic preload crosses this stable region with the increasing of airspeed.The LCO quenching of this model in transonic flow is essentially induced by destabilizing effect from aerodynamic preload,since the flutter speed is sensitive to AoA due to aerodynamic nonlinearity.

Nonlinear flutter wind tunnel test and numerical analysis of folding fins with freeplay nonlinearities

The flutter characteristics of folding control fins with freeplay are investigated by numer- ical simulation and flutter wind tunnel tests. Based on the characteristics of the structures, fins with different freeplay angles are designed. For a 0° angle of attack, wind tunnel tests of these fins are conducted, and vibration is observed by accelerometers and a high-speed camera. By the expansion of the connected relationships, the governing equations of fit for the nonlinear aeroelastic analysis are established by the free-interface component mode synthesis method. Based on the results of the wind tunnel tests, the flutter characteristics of fins with different freeplay angles are analyzed. The results show that the vibration divergent speed is increased, and the divergent speed is higher than the flutter speed of the nominal linear system. The vibration divergent speed is increased along with an increase in the freeplay angle. The developed free-interface component mode synthesis method could be used to establish governing equations and to analyze the characteristics of nonlinear aeroe- lastic systems. The results of the numerical simulations and the wind tunnel tests indicate the same trends and critical velocities.

Nonlinear aeroelastic analysis of the folding fin with freeplay under thermal environment

The nonlinear aeroelastic behavior of a folding fin in supersonic flow is investigated in this paper.The finite element model of the fin is established and the deployable hinges are represented by three torsion springs with the freeplay nonlinearity.The aerodynamic grid point is assumed to be at the center of each aerodynamic box for simplicity.The aerodynamic governing equation is given by using the infinite plate spline method and the modified linear piston theory.An improved fixed-interface modal synthesis method,which can reduce the rigid connections at the interface,is developed to save the problem size and computation time.The uniform temperature field is applied to create the thermal environment.For the linear flutter analyses,the flutter speed increases first and then decreases with the rise of the hinge stiffness due to the change of the flutter coupling mechanism.For the nonlinear analyses,a larger freeplay angle results in a higher vibration divergent speed.Two different types of limit cycle oscillations and a multiperiodic motion are observed in the wide range of airspeed under the linear flutter boundary.The linear flutter speed shows a slight descend in the thermal environment,but the effect of the temperature on the vibration divergent speed is different under different hinge stiffnesses when there exists freeplay.

Nonlinear system identification framework of folding fins with freeplay using backbone curves

Due to wear and manufacturing tolerance,the freeplay is unavoidable in the hinges of folding fins,which exerts significant effects on the aerodynamic characteristics.This paper proposes a backbone-curve-based framework for the dynamical identification of folding fins containing the freeplay nonlinearity.With no need to measure the input force signal and the response signals of nonlinear related Degrees of Freedom(DOFs),the proposed method is more direct and elegant than most existing nonlinear identification approaches,and it contains three steps:Firstly,the underlying linear model of the folding fin structure is obtained through the modal test on its linear sub-parts,and then,the harmonic approximation solves the analytical expressions of the backbone curves of measurable DOFs.Secondly,response data measured from the sine-sweep test are used to extract the fitting points of backbone curves for these DOFs.Finally,the curve fitting approach is applied to identify the freeplay parameters.A series of numerical experiments verify the effectiveness of the proposed method.A real-life folding fin structure is also employed to illustrate how the method can be applied.These examples demonstrate that the identification framework can give an accurate dynamic model of the folding fin structure.

Active flutter suppression of nonlinear fin-actuator system via inverse system and immersion and invariance method

Structural nonlinearities such as freeplay will affect the stability and even flight safety of the fin-actuator system.There is a lack of a practical method for designing Active Flutter Suppression (AFS) control laws for nonlinear fin-actuator systems.A design method for the AFS controller of the nonlinear all-movable fin-electromechanical actuator system is established by combining the inverse system and the Immersion and Invariance (I&I) theory.First,the composite control law combining the inverse system principle and internal model control is used to offset the nonlinearity and dynamics of the actuator,so that its driving torque can follow the ideal signal.Then,the ideal torque of the actuator is designed employing the I&I theory.The unfavorable oscillation of the fin is suppressed by making the output torque of the actuator track the ideal signal.The simulation results reveal that the proposed AFS method can increase the flutter speed of the nonlinear finactuator system with freeplay,and a set of controller parameters is also applicable for wider freeplay within a certain range.The power required for the actuator does not exceed the power that can be provided by the commonly used aviation actuator.This method can also resist a certain level of noise and external disturbance.