Effects of actuator nonlinearity on aeroelastic characteristics of a control fin

The influences of actuator nonlinearities on actuator dynamics and the aeroelastic characteristics of a control fin were investigated by using iterative V-g methods in subsonic flows; in addition, the doublet-hybrid method (DHM) was used to calculate unsteady aerodynamic forces. The changes of actuator dynamics induced by nonlinearities, such as backlash or freeplay, and the variations of flutter boundaries due to the changes of actuator dynamics were observed. Results show that the aeroelastic characteristics can be significantly dependent on actuator dynamics. Thus, the actuator nonlinearities may play an important role in the nonlinear aeroelastic characteristics of an aeroelastic system. The present results also indicate that it is necessary to seriously consider the influence of actuator dynamics on the flutter characteristics at the design stage of actuators to prevent aeroelastic instabilities of aircraft or missiles.     

Response analysis of a pitch–plunge airfoil with structural and aerodynamic nonlinearities subjected to randomly fluctuating flows

This study focuses on numerically investigating the response dynamics of a pitch–plunge airfoil with structural nonlinearity under dynamic stall conditions. The aeroelastic responses are investigated for both deterministic and randomly time varying flow conditions. To that end, a pitch–plunge airfoil under dynamic stall condition is considered and the nonlinear aerodynamic loads are computed using a Leishman–Beddoes formulation. It is shown that the presence of structural nonlinearities can give rise to a variety of dynamical responses in the pre-flutter regime. Next, a response analysis under the presence of a randomly fluctuating wind is carried out. It is demonstrated that the route to flutter occurs via a regime of pre-flutter oscillations called intermittency. Finally, the manifestation of these stochastic responses is characterized by invoking stochastic bifurcation concepts. The route to flutter via intermittency is presented in terms of topological changes occurring in the joint-probability density function of the state variables.     

Spectrographic analysis for the modal testing of nonlinear aeroelastic systems

The spectrograph is a signal-processing tool often used for the frequency domain analysis of time-varying signals. When the signal to be analyzed is a function of time, the spectrograph represents the frequency content of the signal as a sequence of power spectra that change with time. In this paper, the usefulness of the technique is demonstrated in its application to the analysis of the time history response of a nonlinear aeroelastic system. The aeroelastic system is modelled analytically as a two-dimensional, rigid airfoil section free to move in both the bending and pitching directions and possessing a rigid flap. The airfoil is mounted by torsional and translational springs attached at the elastic axis, and the flap is used to provide the forcing input to the system. The nonlinear system is obtained by introducing a freeplay type of nonlinearity in the pitch degree-of-freedom restoring moment. The airfoil is immersed in an aerodynamic flow environment, modelled using incompressible thin airfoil theory for unsteady oscillatory motion. The equations of motion are solved using a fourth-order Runge–Kutta numerical integration technique to provide time-history solutions of the response of the airfoil in the pitch and plunge directions. Time-histories are obtained for the nonlinear responses of the linear and nonlinear aeroelastic systems to a sine-sweep input. The time-histories are analyzed using the spectrographic technique, and the frequency content of the response is plotted directly as a function of the input frequency. Results show that the combination of the sine-sweep input with the spectrographic analysis permits a unique insight into the behavior of the nonlinear system with a minimum of testing. It is shown that the frequency of the nonlinear system response is a function of the input frequency and one other characteristic frequency that can be associated with the limit cycle oscillations of the same nonlinear system subject to a transient input.     

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 flexural 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.     

考虑几何非线性的复合材料机翼气动弹性分析

本文研究结构几何非线性与气动力非平面效应对大展弦比复合材料机翼的气动弹性行为的影响．将非线性有限元法与曲面涡格法结合,计算机翼静气动弹性变形;通过曲面偶极子格网法结合静气动弹性平衡位置处的结构切线刚度,建立气动弹性方程并求解得到机翼颤振速度．针对板模型机翼,分析了迎角对机翼几何非线性气动弹性特性的影响．结果表明：本文复合材料板模型机翼的颤振形式不受水平弯曲模态影响,属于经典弯扭颤振;在几何非线性的影响下,机翼扭转频率随结构变形增大而明显减小,颤振速度随迎角增大而减小．     

Camber effects in the dynamic aeroelasticity of compliant airfoils

This paper numerically investigates the effect of chordwise flexibility on the dynamic stability of compliant airfoils. A classical two-dimensional aeroelastic model is expanded with an additional degree of freedom to capture time-varying camber deformations, defined by a parabolic bending profile of the mean aerodynamic chord. Aerodynamic forces are obtained from unsteady thin airfoil theory and the corresponding compliant-airfoil inertia and stiffness from finite-element analysis. V–g and state-space stability methods have been implemented in order to compute flutter speeds. The study looks at physical realizations with an increasing number of degrees of freedom, starting with a camber-alone system. It is shown that single camber leads to flutter, which occurs at a constant reduced frequency and is due to the lock in between the shed wake and the camber motion. The different combinations of camber deformations with pitch and plunge motions are also studied, including parametric analyses of their aeroelastic stability characteristics. A number of situations are identified in which the flutter boundary of the compliant airfoil exhibits a significant dip with respect to the rigid airfoil models. These results can be used as a first estimation of the aeroelastic stability boundaries of membrane-wing micro air vehicles.     

Analytical formulation of 2-D aeroelastic model in weak ground effect

This paper deals with the aeroelastic modeling and analysis of a 2-D oscillating airfoil in ground effect, elastically constrained by linear and torsional springs and immersed in an incompressible potential flow (typical section) at a finite distance from the ground. This work aims to extend Theodorsen theory, valid in an unbounded flow domain, to the case of weak ground effect, i.e., for clearances above half the airfoil chord. The key point is the determination of the aerodynamic loads, first in the frequency domain and then in the time domain, accounting for their dependence on the ground distance. The method of images is exploited in order to comply with the impermeability condition on the ground. The new integral equation in the unknown vortex distribution along the chord and the wake is solved using asymptotic expansions in the perturbation parameter defined as the inverse of the non-dimensional ground clearance of the airfoil. The mathematical model describing the aeroelastic system is transformed from the frequency domain into the time domain and then in a pure differential form using a finite-state aerodynamic approximation (augmented states). The typical section, which the developed theory is applied to, is obtained as a reduced model of a wing box finite element representation, thus allowing comparison with the corresponding aeroelastic analysis carried out by a commercial solver based on a 3-D lifting surface aerodynamic model. Stability (flutter margins) and response of the airfoil both in frequency and time domains are then investigated. In particular, within the developed theory, the solution of the Wagner problem can be directly achieved confirming an asymptotic trend of the aerodynamic coefficients toward the steady-state conditions different from that relative to the unbounded domain case. The dependence of flutter speed and the frequency response functions on ground clearance is highlighted, showing the usefulness of this approach in efficiently and robustly accounting for the presence of the ground when unsteady analysis of elastic lifting surfaces in weak ground effect is required.     

FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a two-dimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a fourth order Runge-Kutta scheme. Results for soft and hard springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit-cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analysed using analytical techniques developed for nonlinear dynamical systems. The Hopf bifurcation point is determined analytically and the amplitude of the limit-cycle oscillation in post-Hopf bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit-cycle oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit-cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom.     

An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system

We perform an analytical and experimental investigation into the dynamics of an aeroelastic system consisting of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The experimental results show that the onset of flutter takes place at a speed smaller than the one predicted by a quasi-steady aerodynamic approximation. On the other hand, the unsteady representation of the aerodynamic loads accurately predicts the experimental value. The linear analysis details the difference in both formulation and provides an explanation for this difference. Nonlinear analysis is then performed to identify the nonlinear coefficients of the pitch spring. The normal form of the Hopf bifurcation is then derived to characterize the type of instability. It is demonstrated that the instability of the considered aeroelastic system is supercritical as observed in the experiments.     

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.     

Assessment of added mass effects on flutter boundaries using the Leishman–Beddoes dynamic stall model

We consider the dynamics of a typical airfoil section both in forced and free oscillations and investigate the importance of the added mass terms, i.e. the second derivatives in time of the pitch angle and plunge displacement. The structural behaviour is modelled by linear springs in pitch and plunge and the aerodynamic loading represented by our interpretation of the state-space version of the Leishman–Beddoes semi-empirical model. The added mass terms are often neglected since this leads to an explicit system of ODEs amenable for solution using standard ODE solvers. We analyse the effect of neglecting the added mass terms in forced oscillations about a set of mean angles of incidence by comparing the solutions obtained with the explicit and implicit systems of ODEs and conclude that their differences amount to a time lag that increases at a constant rate with increases of the reduced frequency. To determine the effect of the added mass terms in free oscillations, we introduce a spring offset angle to obtain static equilibrium positions at various degrees of incidence. We analyse the stability of the explicit and implicit aeroelastic systems about those positions and compare the locations of the respective flutter points calculated as Hopf bifurcation points. For low values of the spring offset angle, added mass effects are significant for low values of the mass ratio, or the ratio of natural frequencies, of the aeroelastic system. For high values of the spring offset angle, corresponding to stall flutter, we observe that their effect is greater for large values of the mass ratio.     

The post-Hopf-bifurcation response of an airfoil in incompressible two-dimensional flow

A bifurcation analysis of a two-dimensional airfoil with a structural nonlinearity in the pitch direction and subject to incompressible flow is presented. The nonlinearity is an analytical third-order rational curve fitted to a structural freeplay. The aeroelastic equations-of-motion are reformulated into a system of eight first-order ordinary differential equations. An eigenvalue analysis of the linearized equations is used to give the linear flutter speed. The nonlinear equations of motion are either integrated numerically using a fourth-order Runge-Kutta method or analyzed using the AUTO software package. Fixed points of the system are found analytically and regions of limit cycle oscillations are detected for velocities well below the divergent flutter boundary. Bifurcation diagrams showing both stable and unstable periodic solutions are calculated, and the types of bifurcations are assessed by evaluating the Floquet multipliers. In cases where the structural preload is small, regions of chaotic motion are obtained, as demonstrated by bifurcation diagrams, power spectral densities, phase-plane plots and Poincaré sections of the airfoil motion; the existence of chaos is also confirmed via calculation of the Lyapunov exponents. The general behaviour of the system is explained by the effectiveness of the freeplay part of the nonlinearity in a complete cycle of oscillation. Results obtained using this reformulated set of equations and the analytical nonlinearity are in good agreement with previously obtained finite difference results for a freeplay nonlinearity.     

Nonlinear Analysis of Store-Induced Limit Cycle Oscillations

Flight tests of modern high-performance fighter aircraft reveal the presence of limit cycle oscillation (LCO) responses for aircraft with certain external store configurations. Conventional linear aeroelastic analysis predicts flutter for conditions well beyond the operational envelope, yet these store-induced LCO responses occur at flight conditions within the flight envelope. Several nonlinear sources may be present, including aerodynamic effects such as flow separation and shock-boundary layer interaction and structural effects such as stiffening, damping, and system kinematics. No complete theory has been forwarded to accurately explain the mechanisms responsible. This research examines a two degree-of-freedom aeroelastic system which possesses kinematic nonlinearities and a strong nonlinearity in pitch stiffness. Nonlinear analysis techniques are used to gain insight into the characteristics of the behavior of the system. Numerical simulation is used to verify and validate the analysis. It is found that when system damping is low, the system clearly exhibits nonlinear interaction between aeroelastic modes. It is also shown that although certain applied forcing conditions may appear negligible, these same forces produce large amplitude LCOs under specific realizable circumstances.     

Limit-cycle oscillations in unsteady flows dominated by intermittent leading-edge vortex shedding

High-frequency limit-cycle oscillations of an airfoil at low Reynolds number are studied numerically. This regime is characterized by large apparent-mass effects and intermittent shedding of leading-edge vortices. Under these conditions, leading-edge vortex shedding has been shown to result in favorable consequences such as high lift and efficiencies in propulsion/power extraction, thus motivating this study. The aerodynamic model used in the aeroelastic framework is a potential-flow-based discrete-vortex method, augmented with intermittent leading-edge vortex shedding based on a leading-edge suction parameter reaching a critical value. This model has been validated extensively in the regime under consideration and is computationally cheap in comparison with Navier–Stokes solvers. The structural model used has degrees of freedom in pitch and plunge, and allows for large amplitudes and cubic stiffening. The aeroelastic framework developed in this paper is employed to undertake parametric studies which evaluate the impact of different types of nonlinearity. Structural configurations with pitch-to-plunge frequency ratios close to unity are considered, where the flutter speeds are lowest (ideal for power generation) and reduced frequencies are highest. The range of reduced frequencies studied is two to three times higher than most airfoil studies, a virtually unexplored regime. Aerodynamic nonlinearity resulting from intermittent leading-edge vortex shedding always causes a supercritical Hopf bifurcation, where limit-cycle oscillations occur at freestream velocities greater than the linear flutter speed. The variations in amplitude and frequency of limit-cycle oscillations as functions of aerodynamic and structural parameters are presented through the parametric studies. The excellent accuracy/cost balance offered by the methodology presented in this paper suggests that it could be successfully employed to investigate optimum setups for power harvesting in the low-Reynolds-number regime.     

LIMIT CYCLE FLUTTER CHARACTERISTICS RELATED TO CONTROL SURFACE OF LARGE AIRCRAFT1)

操纵面间隙会引起结构非线性,容易诱发极限环颤振。根据设计需要,考虑操纵面中心间隙的影响,采用最小状态拟合技术对频域非定常 气动力进行有理函数拟合,采用分段函数描述间隙引起的非线性刚度,研究操纵面在间隙作用下的极限环颤振响应的行为特点。结果表明,由于 中心间隙的影响,系统会在低于线性颤振速度时就产生极限环振荡,同时,振荡幅值随飞行速度或中心间隙的增大而增大。     

Dimensionless modeling and nonlinear analysis of a coupled pitch–plunge–leadlag airfoil-based piezoaeroelastic energy harvesting system

In this paper, an airfoil-based piezoaeroelastic energy harvesting system is proposed with an additional supporting device to harvest the mechanical energy from the leadlag motion. A dimensionless dynamic model is built considering the large-effective-angle-of-attack vibrations causing (1) the nonlinear coupling between the pitch–plunge–leadlag motions, (2) the inertia nonlinearity, and (3) the aerodynamic nonlinearity modeled by the ONERA dynamic stall model. Cubic hardening stiffness is introduced in the pitch degree of freedom for persistent vibrations with acceptable amplitude beyond the flutter boundary. The nonlinear aeroelastic response and the average power output are numerically studied. Limit cycle oscillations are observed and, as the flow velocity exceeds a secondary critical speed, the system experiences complex vibrations. The power output from the leadlag motion is smaller than that from the plunge motion, whereas the gap is narrowed with increasing flow velocity. Case studies are performed toward the effects of several dimensionless system parameters, including the nonlinear torsional stiffness, airfoil mass eccentricity, airfoil radius of gyration, mass of the supporting devices, and load resistances in the external circuits. The optimal parameter values for the power outputs from the plunge and leadlag motions are, respectively, obtained. Beyond the secondary critical speed, it is shown that the variations of the power outputs with those parameters become irregular with fluctuations and multiple local maximums. The bifurcation analysis shows the mutual transitions between the limit cycle oscillations, multi-periodic vibrations, and possible chaos. The influences of these complex vibrations on the power outputs are discussed.     

Modeling and analysis of piezoaeroelastic energy harvesters

This work investigates the influence of structural and aerodynamic nonlinearities on the dynamic behavior of a piezoaeroelastic system. The system is composed of a rigid airfoil supported by nonlinear torsional and flexural springs in the pitch and plunge motions, respectively, with a piezoelectric coupling attached to the plunge degree of freedom. The analysis shows that the effect of the electrical load resistance on the flutter speed is negligible in comparison to the effects of the linear spring coefficients. The effects of aerodynamic nonlinearities and nonlinear plunge and pitch spring coefficients on the system’s stability near the bifurcation are determined from the nonlinear normal form. This is useful to characterize the effects of different parameters on the system’s output and ensure that subcritical or “catastrophic” bifurcation does not take place. Numerical solutions of the coupled equations for two different configurations are then performed to determine the effects of varying the load resistance and the nonlinear spring coefficients on the limit-cycle oscillations (LCO) in the pitch and plunge motions, the voltage output and the harvested power.     

Bifurcation analysis of an aeroelastic system with concentrated nonlinearities

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system??s instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed.     

Output Feedback Modular Adaptive Control of a Nonlinear Prototypical Wing Section

The paper presents nonlinear adaptive control systems for the control of limit cycle oscillations of a prototypical wing section with structural nonlinearities using only output feedback. The chosen model describes the plunge and pitch motion of a wing. The model includes plunge and pitch nonlinearities, and has a single control surface for the purpose of control. Using a canonical representation of the aeroelastic system, a modular output feedback adaptive control system consisting of an input-to-state stabilizing controller and a passive identifier (an observer and adaptation law) is derived. In the closed-loop system, asymptotic stabilization of the pitch and plunge motion is accomplished. Simulation results show that the control system is effective in regulating the state vector to the origin in spite of large parameter uncertainties.     

Limit cycle oscillation behavior of transonic control surface buzz considering free-play nonlinearity

The limit cycle oscillation (LCO) behaviors of control surface buzz in transonic flow are studied. Euler equations are employed to obtain the unsteady aerodynamic forces for Type B and Type C buzz analyses, and an all-movable control surface model, a wing/control surface model and a three-dimensional wing with a full-span control surface are adopted in the study. Aerodynamic and structural describing functions are used to deal with aerodynamic and structural nonlinearities, respectively. Then the buzz speed and buzz frequency are obtained by V-g method. The LCO behavior of the transonic control surface buzz system with linear structure exhibits subcritical or supercritical bifurcation at different Mach numbers. For nonlinear structural model with a free-play nonlinearity in the control surface deflection stiffness, the double LCO phenomenon is observed in certain range of flutter speed. The free-play nonlinearity changes the stability of LCOs at small amplitudes and turns the unstable LCO into a stable one. The LCO behavior is dominated by the aerodynamic nonlinearity for the case with large control surface oscillation amplitude but by the structural nonlinearity for the case with small amplitude. Good agreements between LCO behaviors obtained by the present method and available experimental data show that our study may help to explain the experimental observation in wind tunnel tests and to understand the physical mechanism of transonic control surface buzz.