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.     

Leaf flutter by torsional galloping: Experiments and model

When wind blows on trees, leaves flutter. The induced motion is known to affect biological functions at the tree scale such as photosynthesis. This paper presents an experimental and theoretical study of the aeroelastic instability leading to leaf flutter. Experiments in a wind tunnel are conducted on ficus leaves (Ficus Benjamina) and artificial leaves. We show that stability and flutter domains are separated by a well-defined limit depending on leaf orientation and wind speed. This limit is also theoretically predicted through a stability analysis of the leaf motion.     

PrandtlPlane Joined Wing: Body freedom flutter,limit cycle oscillation and freeplay studies

Dynamic aeroelastic behavior of a joined-wing PrandtlPlane configuration is investigated herein. The baseline model is obtained from a configuration previously designed by partner universities through several multidisciplinary optimizations and ad hoc analyses, including detailed studies on the layout of control architecture. An equivalent structural model has then been adopted to qualitatively retain similar aeroelastic properties.Flutter and post-flutter regimes, including limit cycle oscillations (LCOs), are studied. A detailed analysis of the energy transfer between fluid and structure is carried out; the areas in which energy is extracted from the fluid are identified to gain insights on the mechanism leading to the aeroelastic instability. Starting from an existing design of control surfaces on the baseline configuration, freeplay is also considered and its effects on the aeroelastic stability properties of the joined-wing system are investigated for the first time.Both cantilever and free flying configurations are analyzed. Fuselage inertial effects are modeled and the aeroelastic properties are studied considering plunging and pitching rigid body modes. For this configuration a positive interaction between elastic and rigid body modes yields a flutter-free design (within the range of considered airspeeds).To understand the sensitivity of the system and gain insight, fuselage mass and moment of inertia are selectively varied. For a fixed pitching moment of inertia, larger fuselage mass favors body freedom flutter. When the moment of inertia is varied, a change of critical properties is observed. For smaller values the pitching mode becomes unstable, and coalescence is observed between pitching and the first elastic mode. Increasing pitching inertia, the above criticality is postponed; meanwhile, the second elastic mode becomes unstable at progressively lower speeds. For larger inertial values “cantilever” flutter properties, having coalescence of first and second elastic modes, are recovered.     

A theoretical and experimental investigation of the effects of a steady angle of attack on the nonlinear flutter of a delta wing plate model

Limit cycle oscillations (LCO) of wings on certain modern high performance aircraft have been observed in flight and in wind tunnel experiments. Whether the physical mechanism that gives rise to this behavior is a fluid or structural nonlinearity or both is still uncertain. It has been shown that an aeroelastic theoretical model with only a structural nonlinearity can predict accurately the limit cycle behavior at low subsonic flow for a plate-like wing at zero angle of attack. Changes in the limit cycle and flutter behavior as the angle of attack is varied have also been observed in flight. It has been suggested that this sensitivity to angle of attack is due to a fluid nonlinearity. In this investigation, we study the flutter and limit cycle behavior of a wing in low subsonic flow at small steady angles of attack. Experimental results are compared to those predicted using an aeroelastic theoretical model with only a structural nonlinearity. Results from both experiment and theory show a change in flutter speed as the steady angle of attack is varied. Also the LCO magnitude increased at a given velocity as the angle of attack was increased for both the experiment and theory. While not proving that the observed sensitivity to angle of attack of LCO in aircraft is due to a structural nonlinearity, the results do show that a change in the aeroelastic behavior at angles of attack can be caused by a structural nonlinearity as well as a fluid nonlinearity. In this paper, only structural nonlinearities are considered, but an extension to include aerodynamic nonlinearities would be very worthwhile.     

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.     

Parametric aeroelastic modeling based on component modal synthesis and stability analysis for horizontally folding wing with hinge joints

To investigate the aeroelastic stability of a folding wing effectively, a parametric aeroelastic analysis approach is proposed. First, the fixed interface component modal synthesis is used to derive the structural dynamic equation for a folding wing, in which the elastic connection is considered. The unsteady aerodynamic model is established by the doublet lattice method (DLM), and the aeroelastic model is achieved from integration of the DLM with the component modal analysis. The generalized aerodynamic influence coefficient matrix is established by modes kept and constraint modes of each component. The aeroelastic stability of a folding wing is investigated based on the Gram matrix in control theory. The effectiveness of the proposed method is verified via comparison with traditional flutter eigenvalue analysis for both extended and folded configurations. The proposed method identifies coupled modes and improves computational efficiency when compared to classical aeroelastic stability analysis methods, such as the p–k method.     

Nonlinear dynamics of an aeroelastic airfoil with free-play in transonic flow

Nonlinear dynamic behaviors of an aeroelastic airfoil with free-play in transonic air flow are studied. The aeroelastic response is obtained by using time-marching approach with computational fluid dynamics (CFD) and reduced order model (ROM) techniques. Several standardized tests of transonic flutter are presented to validate numerical approaches. It is found that in time-marching approach with CFD technique, the time-step size has a significant effect on the calculated aeroelastic response, especially for cases considering both structural and aerodynamic nonlinearities. The nonlinear dynamic behavior for the present model in transonic air flow is greatly different from that in subsonic regime where only simple harmonic oscillations are observed. Major features of the responses in transonic air flow at different flow speeds can be summarized as follows. The aeroelastic responses with the amplitude near the free-play are dominated by single degree of freedom flutter mechanism, and snap-though phenomenon can be observed when the air speed is low. The bifurcation diagram can be captured by using ROM technique, and it is observed that the route to chaos for the present model is via period-doubling, which is essentially caused by the free-play nonlinearity. When the flow speed approaches the linear flutter speed, the aeroelastic system vibrates with large amplitude, which is dominated by the aerodynamic nonlinearity. Effects of boundary layer and airfoil profile on the nonlinear responses of the aeroelastic system are also discussed.     

Numerical identification of mechanisms triggering 2D choke flutter in transonic fan

Blade vibration may trigger a self-induced aeroelastic instability (flutter). In turbomachinery choke flutter appears when a strong shock-wave chokes the blade passage. The aim of this study is to identify mechanisms responsible for the instability. An innovative methodology relying on the splitting of the emitter and receiver role of the blade is presented. It is successfully applied to 2D linearized RANS computations of choke flutter. The emission splitting shows that the vibration of the blades downstream of the shock-wave generates a backward traveling pressure wave triggering the aeroelastic instability. The reception splitting demonstrates the destabilising contribution of the shock-wave / separated boundary layer interaction. The source of flutter is finally a combination of inviscid (regressive waves) and viscous (unsteady separation) mechanisms.     

Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunities

The identification of nonlinear aeroelastic systems based on the Volterra theory of nonlinear systems is presented. Recent applications of the theory to problems in computational and experimental aeroelasticity are reviewed. Computational results include the development of computationally efficient reduced-order models (ROMs) using an Euler/Navier–Stokes flow solver and the analytical derivation of Volterra kernels for a nonlinear aeroelastic system. Experimental results include the identification of aerodynamic impulse responses, the application of higher-order spectra (HOS) to wind-tunnel flutter data, and the identification of nonlinear aeroelastic phenomena from flight flutter test data of the active aeroelastic wing (AAW) aircraft.     

Implicit Calculations of an Aeroelasticity Problem

The purpose of this work is to show that a linearized implicit scheme for the flow resolution can be an efficient and accurate method for solving fluid-structure interaction. The fluid is modeled by the Euler equations in two dimensions and the structure by a one (free piston) or a two (NACA0012 airfoil) degrees of freedom system. The schemes are developed using a finite volume/finite element formulation and, stating the moving boundary problem in the space-time domain, the Riemann solver is generalized in a suitable manner. Assuming a modal decomposition for the structure's response, an analytical solution to the equation of motion is obtained.

The effects of the linearized implicit scheme on the aeroelastic response are demonstrated on the free piston and the NACA 0012 airfoil problems. In the latter case, we focus on the capability of the linearized implicit scheme to accurately predict the stability limit of the coupled response (wing flutter analysis). Although the above analysis is performed using a rigid transformation, a robust moving mesh strategy is presented for more general 2-D and 3-D deformations.     

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.     

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.     

Aeroelastic analysis and active flutter control of nonlinear lattice sandwich beams

Nonlinear aeroelastic characteristics of sandwich beams with pyramidal lattice core are investigated, and the active flutter control of the nonlinear structural system is also studied using the piezoelectric actuator/sensor pair. In the structural modeling, Reddy’s third-order shear deformation theory is applied. Aerodynamic pressure is evaluated by the supersonic piston theory. Hamilton’s principle and the assumed mode method are used to derive the equation of motion. The proportional feedback and the optimal H _∞ control methods are performed to design the controller. In the robust control, the uncertainty caused by omitting the nonlinear terms of the control equation is taken into account, and the mixed sensitivity method is used to solve the problem. The nonlinear aeroelastic property of the sandwich beam is analyzed and is compared with that of the equivalent isotropic beam with the same weight to show the superior aeroelastic characteristics of the lattice sandwich beam. Controlled vibration responses under the two different controllers are calculated and compared. Simulation results show that the robust controller is much more effective than the proportional feedback controller in the flutter suppression of the nonlinear sandwich beam.     

Effects of multiple structural nonlinearities on limit cycle oscillation of missile control fin

Aeroelastic analyses are performed for a 2-D typical section model with multiple nonlinearities. The differences between a system with multiple nonlinearities in its pitch and plunge spring and a system with a single nonlinearity in its pitch are thoroughly investigated. The unsteady supersonic aerodynamic forces are calculated by the doublet point method (DPM). The iterative V-g method is used for a multiple-nonlinear aeroelastic analysis in the frequency domain and the freeplay nonlinearity is linearized using a describing function method. In the time domain, the DPM unsteady aerodynamic forces, which are based on a function of the reduced frequency, are approximated by the minimum state approximation method. Consequently, multiple structural nonlinearities in the 2-D typical wing section model are influenced by the pitch to plunge frequency ratio. This result is important in that it demonstrates that the flutter speed is closely connected with the frequency ratio, considering that both pitch and plunge nonlinearities result in a higher flutter speed boundary than a conventional aeroelastic system with only one pitch nonlinearity. Furthermore, the gap size of the freeplay affects the amplitude of the limit cycle oscillation (LCO) to gap size ratio.     

Aeroelastic stability of a cantilevered plate in yawed subsonic flow

The aeroelastic stability of cantilevered plates with their clamped edge oriented both parallel and normal to subsonic flow is a classical fluid–structure interaction problem. When the clamped edge is parallel to the flow the system loses stability in a coupled bending and torsion motion known as wing flutter. When the clamped edge is normal to the flow the instability is exclusively bending and is referred to as flapping flag flutter. This paper explores the stability of plates during the transition between these classic aeroelastic configurations. The aeroelastic model couples a classical beam structural model to a three-dimensional vortex lattice aerodynamic model. The aeroelastic stability is evaluated in the frequency domain and the flutter boundary is presented as the plate is rotated from the flapping flag to the wing configuration. The transition between the flag-like and wing-like instability is often abrupt and the yaw angle of the flow for the transition is dependent on the relative spacing of the first torsion and second bending natural frequencies. This paper also includes ground vibration and aeroelastic experiments carried out in the Duke University Wind Tunnel that confirm the theoretical predictions.     

Amplification and amplitude limitation of heave/pitch limit-cycle oscillations close to the transonic dip

Recent results from flutter experiments of the supercritical airfoil NLR 7301 at flow conditions close to the transonic dip are presented. The airfoil was mounted with two degrees-of-freedom in an adaptive solid-wall wind tunnel, and boundary-layer transition was tripped. Flutter boundaries exhibiting a transonic dip were determined and limit-cycle oscillations (LCOs) were measured. The local energy exchange between the fluid and the structure during LCOs is examined and leads to the following findings: at supercritical Mach numbers below that of the transonic-dip minimum the presence of a shock-wave and its dynamics destabilizes the aeroelastic system such that the decreasing branch of the transonic dip develops. At higher Mach numbers the shock-wave motion has a stabilizing effect such that the flutter boundary increases to higher flutter-speed indices with increasing Mach number. Amplified oscillations near this branch of the flutter boundary obtain energy from the flow mainly due to the dynamics of a trailing-edge flow separation. A slight nonlinear amplitude dependency of the shock motion and a possibly occurring boundary-layer separation cause the amplitude limitation of the observed LCOs. The impact of the findings on the numerical simulation of these phenomena is discussed.     

Evaluation of linear,inviscid, viscous,and reduced-order modelling aeroelastic solutions of the AGARD 445.6 wing using root locus analysis

Reduced-order modelling (ROM) methods are applied to the Computational Fluid Dynamics (CFD)-based aeroelastic analysis of the AGARD 445.6 wing in order to gain insight regarding well-known discrepancies between the aeroelastic analyses and the experimental results. The results presented include aeroelastic solutions using the inviscid Computational Aeroelasticity Programme–Transonic Small Disturbance (CAP-TSD) code and the FUN3D code (Euler and Navier–Stokes). Full CFD aeroelastic solutions and ROM aeroelastic solutions, computed at several Mach numbers, are presented in the form of root locus plots in order to better reveal the aeroelastic root migrations with increasing dynamic pressure. Important conclusions are drawn from these results including the ability of the linear CAP-TSD code to accurately predict the entire experimental flutter boundary (repeat of analyses performed in the 1980s), that the Euler solutions at supersonic conditions indicate that the third mode is always unstable, and that the FUN3D Navier–Stokes solutions stabilize the unstable third mode seen in the Euler solutions.     

基于气动力降阶模型的跨音速气动弹性稳定性分析

基于离散型输入输出差分模型,运用非定常CFD方法训练信号,然后运用最小二乘方法进行参数辨识,得到降阶的非定常气动力模型,再将该离散差分模型转换为连续时间域内的状态方程。耦合气动状态方程和结构状态方程,得到耦合系统的气动弹性状态方程。求解不同动压下状态矩阵的特征值,根据根轨迹图分析系统的稳定性特性。分析结果与直接耦合CFD/CSD方法结果相吻合,可以计算跨音速非线性气动弹性问题。其计算效率比直接耦合CFD/CSD方法提高1~2个数量级。针对Isogai wing在跨音速出现的S型颤振边界进行了较为细致的分析,阐述了该现象是由于系统诱发颤振的分支随着速度(来流动压)的提高而发生转移所导致的。     

CFD／CSD紧耦合及新型动网格方法在气动弹性模拟中的应用

研发出一套基于紧耦合的CFD／CSD耦合方法和程序。非定常流场求解采用混合网格有限体积方法,时间离散采用基于LU-SGS隐式格式的双时间步长法。通过求解雷诺平均Navier—Stokes方程模拟了三维机翼的跨音速气动弹性现象。得到了其颤振边界,与风洞实验结果吻合较好,验证了方法和程序的有效性和实用意义。同时将Delau...     

Nonlinear aeroservoelastic analysis of a controlled multiple-actuated-wing model with free-play

In this paper, the effects of structural nonlinearity due to free-play in both leading-edge and trailing-edge outboard control surfaces on the linear flutter control system are analyzed for an aeroelastic model of three-dimensional multiple-actuated-wing. The free-play nonlinearities in the control surfaces are modeled theoretically by using the fictitious mass approach. The nonlinear aeroelastic equations of the presented model can be divided into nine sub-linear modal-based aeroelastic equations according to the different combinations of deflections of the leading-edge and trailing-edge outboard control surfaces. The nonlinear aeroelastic responses can be computed based on these sub-linear aeroelastic systems. To demonstrate the effects of nonlinearity on the linear flutter control system, a single-input and single-output controller and a multi-input and multi-output controller are designed based on the unconstrained optimization techniques. The numerical results indicate that the free-play nonlinearity can lead to either limit cycle oscillations or divergent motions when the linear control system is implemented.