Elastic Axis Location Research Articles

The effect of airfoil elasticity on quasistatic aerodynamics and divergence

Shape-morphing airfoils are of interest due to their benefits for aerodynamic efficiency and commonly involve chordwise elasticity of the airfoil. We aim to examine the influence of chordwise elasticity on the aerodynamic properties and stability in the quasistatic limit. We model a shape-morphing airfoil as two, rear and front, Euler–Bernoulli beams connected to a rigid support at an arbitrary location along its chord. This setup is mounted on a torsion spring and is exposed to a uniform flow. We model aerodynamic forces acting on the wing via thin airfoil theory and obtain the solution of the elastic deflection via regular asymptotic expansions. Then, we substitute this result into the moment balance equation to find the rotation angle. This procedure allows us to examine the influence of chord-wise elasticity on the lift, aerodynamic center, twist angle, and the onset of divergence. We examine these results in terms of the elastic axis location and the dimensionless ratio between aerodynamic moment to bending stiffness. The lift curve slope is smaller than 2π up to the hinge location of about 0.53 chord and is greater than 2π elsewhere. The chord-wise elasticity moves the aerodynamic center forwards and increases the twist angle. In addition, chord-wise elasticity is found to increase the onset of divergence relative to the rigid case. Moreover, unlike the rigid case, divergence also exists for hinges in the front quarter chord. Finally, we present several possible cases of an actuated airfoil validated by numerical simulations.

On the Aeroelasticity of the Active Span and Passive Pitching Polymorphing Wing: Effect of Morphing Rate

This paper studies the effect of morphing rate on the aeroelasticity of a polymorphing wing capable of active span extension and passive twist/pitch. A variable domain size finite element model is developed to capture the dynamic effects due to the presence of a variable span in the Euler–Bernoulli beam model, which introduces a structural damping term in the equations of motion. The effect of various morphing rates on the aeroelastic boundaries of the system, namely, flutter velocity and flutter frequency, is examined for a beam undergoing span extension and retraction, from baseline span to 25% span extension and vice versa, respectively. Three points of interest are analyzed: at the start of the span morphing, at the mid-point of morphing, and just before the morphing process ends. The parametric analysis is carried out to determine the effect of varying critical parameters, such as the elastic axis location of the outboard wing section and adjoining spring torsional rigidity on the aeroelastic boundaries of the polymorphing wing.

X-ray Computed Tomography Method for Macroscopic Structural Property Evaluation of Active Twist Composite Blades

This paper describes an evaluation of the structural properties of the next-generation active twist blade using X-ray computed tomography (CT) combined with digital image processing. This non-destructive testing technique avoids the costly demolition of the blade structure. The CT scan covers the whole blade region, including the root, transition, and tip regions, as well as the airfoil blade regions, in which there are spanwise variations in the interior structural layout due to the existence of heavy instrumentation. The three-dimensional digital image data are processed at selected radial stations, and finite element beam cross-section analyses are conducted to evaluate the structural properties of the blade at the macroscopic level. The fidelity of the digital blade model is first assessed by correlating the estimated blade mass with the measured data. A separate mechanical measurement is then carried out to determine the representative elastic properties of the blade and to verify the predicted results. The agreement is found to be good to excellent for the mass, elastic axis, flap bending, and torsional rigidity. The discrepancies are less than 2.0% for the mass and elastic axis locations, and about 8.1% for the blade stiffness properties, as compared with the measured data. Finally, a sensitivity analysis is conducted to clarify the impact of modeling the sensor and actuator cables, nose weight, and manufacturing imperfections on the structural properties of the blade.

Stochastic Modeling and Reliability Analysis of Wing Flutter

In this work, a physics-based first-order reliability method (FORM) algorithm is proposed for the flutter reliability analysis of an aircraft wing in the frequency domain. The limit state function, which is an implicit function of random variables, is defined in terms of the damping ratio of the aeroelastic system in a conditional sense on flow velocity. Two aeroelastic cases, namely, an airfoil section model and a cantilever wing model, are considered for carrying out the studies. These aeroelastic models have well separated mean bending and mean torsional modal frequencies. The geometric, structural, and aerodynamic parameters of airfoil and wing systems are modeled as independent Gaussian random variables. The effects of these on the statistics of frequency and damping ratio, and the cumulative distribution functions (CDFs) of flutter velocity are studied. In the case of the wing, the effects of modeling stiffness parameters as Gaussian random fields on the CDFs of flutter velocity are also studied. Here, spectral stochastic finite element method (SFEM) based on Karhunen–Loeve (K–L) expansion is used to discretize the random fields into random variables. From the study of an airfoil system, it is observed that parameters like torsional stiffness, elastic axis location, free stream density, and mass moment of inertia are more sensitive as compared with other parameters. However, in the case of the wing parameters such as torsional stiffness, free stream density, mass moment of inertia, and mass are observed to be more sensitive. The CDFs of flutter velocity obtained using the proposed algorithm are compared with Monte Carlo simulations (MCS) and found to be accurate. A comparative study of aeroelastic reliability for the wing is also carried out by treating stiffness parameters as random variables and random fields. It is observed that the CDFs of flutter velocity in the tail region are conservative when stiffness parameters are treated as random variables.

Flow physics and dynamics of flow-induced pitch oscillations of an airfoil

We conduct a computational study of flow-induced pitch oscillations of a rigid airfoil at a chord-based Reynolds number of 1000. A sharp-interface immersed boundary method is used to simulate two-dimensional incompressible flow, and this is coupled with the equations for a rigid foil supported at the elastic axis with a linear torsional spring. We explore the effect of spring stiffness, equilibrium angle-of-attack and elastic-axis location on the onset of flutter, and the analysis of the simulation data provides insights into the time scales and mechanisms that drive the onset and dynamics of flutter. The dynamics of this configuration includes complex phenomena such as bifurcations, non-monotonic saturation amplitudes, hysteresis and non-stationary limit-cycle oscillations. We show the utility of ‘maps’ of energy exchange between the flow and the airfoil system, as a way to understand, and even predict, this complex behaviour.

Hydroelastic analysis of two degree of freedom hydrofoil using a reduced-order hydrodynamic model considering unsteady partial sheet cavity flows

In the present study a new, fast, precise algorithm for studying hydroelastic stability of a typical section with two degrees of freedom (2DOF) is proposed based on the finite element method (FEM), while the partial sheet cavitation effects are considered. For this means, the steady cavity boundary is calculated by some conventional iterative procedures, developed based on the potential flow simulation. Thereafter, assuming that the amplitude and the frequency of the body oscillations are altered so that the cavity length in unsteady flow remains unchanged, the simulation of the unsteady hydrodynamic flow is performed by imposing some velocity fluctuations over the rigid cavity boundary. For the next step, according to the modal analysis of this model, the hydrodynamic eigen modes, which are employed to construct governing hydrodynamic system of equations, are calculated. Using this model along with the structural model the governing hydroelastic set of equations are introduced. According to this simulation approach, numerous calculations are preformed over the considered hydrofoil and the effects of various parameters including the cavitation number, initial angle of attack and structural parameters such as the elastic axis location, radius of gyration, static unbalance, structure to fluid density ratio, and the frequency ratio, over the stability range are surveyed and some conclusions are outlined.

A rapid aeroelasticity optimization method based on the stiffness characteristics

A rapid aeroelasticity optimization method based on the stiffness characteristics was proposed in the present study. Large time expense in static aeroelasticity analysis based on traditional time domain aeroelasticity method is solved. Elastic axis location and torsional stiffness are discussed firstly. Both torsional stiffness and the distance between stiffness center and aerodynamic center have a direct impact on divergent velocity. The divergent velocity can be adjusted by changing the correlative structural parameters. The relation between structural parameters and divergent velocity is introduced to aeroelasticity optimization design as a constraint condition. After optimization, the structural and aerodynamic characteristics have a large improvement while satisfying the constraint conditions. The optimization method can be well used in high aspect ratio wing and has great computational efficiency.

Stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method

An adaptive stochastic spectral projection method is deployed for the uncertainty quantification in limit-cycle oscillations of an elastically mounted two-dimensional lifting surface in a supersonic flow field. Variabilities in the structural parameters are propagated in the aeroelastic system which accounts for nonlinear restoring force and moment by means of hardening cubic springs. The physical nonlinearities promote sharp and sudden flutter onset for small change of the reduced velocity. In a stochastic context, this behavior translates to steep solution gradients developing in the parametric space. A remedy is to expand the stochastic response of the airfoil on a piecewise generalized polynomial chaos basis. Accurate approximation andaffordable computational costs are obtained using sensitivity-based adaptivity for various types of supersonic stochastic responses depending on the selected values of the Mach number on the bifurcation map. Sensitivity analysis via Sobol' indices shows how the probability density function of the peak pitch amplitude responds to combined uncertainties: e.g. the elastic axis location, torsional stiffness and flap angle. We believe that this work demonstrates the capability and flexibility of the approach for more reliable predictions of realistic aeroelastic systems subject to a moderate number of uncertainties.

Global Robust Control of an Aeroelastic System Using Output Feedback

A EROELASTIC systems exhibit a variety of phenomena including instability, limit cycle, and even chaotic vibration. Researchers in aerodynamics, structures, materials, and control have made interesting contributions to the analysis and control of aeroelastic systems [1]. Readers may refer to an excellent article by Mukhopadhyay [2] for the analysis and control of aeroelastic systems. A benchmark active control technology (BACT) windtunnel model has been designed at the NASA Langley Research Center and control algorithms for flutter suppression have been developed [3–5]. At Texas A&Ma 2-degrees-of-freedom aeroelastic model has been developed and tests have been performed in a wind tunnel to examine the effect of nonlinear structural stiffness, and control systems have been designed using linear control theory, feedback linearizing technique, and adaptive control strategies [6– 10]. Based on the Euler–Lagrange theory, control of an aeroelastic model has been considered [11]. The state variable and output feedback designs of [8–10] are based on adaptive control techniques. The synthesis of adaptive controllers is not simple because a large number of parameters must be updated in the dynamic feedback loop. It is also well known that unmodeled dynamics of the system can cause parameter divergence and instability in the closed-loop system. Therefore, it is interesting to design nonadaptive control systems for uncertain aeroelastic models which can be synthesized relatively easily. The contribution of this paper lies in the design of a robust control system for the global regulation of an aeroelastic system which describes the plunge and pitch motion of a wing. The model has polynomial type structural nonlinearity and only the pitch angle is measured for feedback. It is assumed that all the system parameters are unknown to the designer, but the bounds on uncertainties are known. For the purpose of design, afirst-order dynamic compensator is introduced and a “chained” structure of the aeroelastic model including the dynamic compensator is obtained by an appropriate coordinate transformation. Then using the Lyapunov stability theory, a control law for robust output regulation of the transformed system including the compensator is derived. In the closed-loop system, the controller accomplishes global robust stabilization of the aeroelastic system and system trajectories converge to the origin. Simulation results for various flow velocities and elastic axis locations are obtainedwhich show that the control system is effective in flutter suppression in spite of the large parameter uncertainties. An attractive feature of this control system lies in its simplicity from the point of view of implementation.

Control of Unsteady Aeroelastic System via State-Dependent Riccati Equation Method

A nonlinear control system for the flutter control of an aeroelastic system with unsteady aerodynamics is designed. The model describes the plunge and pitch motion of a wing. In this model both plunge and pitch structural nonlinearities are included. A single control surface is utilized for the flutter control. For the purpose of design, it is assumed that there exists a specified hard magnitude constraint on the control input. For the synthesis of the controller, only the plunge displacement, pitch angle, and control-surface deflection are measured. The control system design is based on the state-dependent Riccati equation method. A slack variable is introduced to transform the constrained control problem into an unconstrained problem and then a suboptimal nonlinear control law is designed. An observer is constructed to estimate the unavailable state variables of the system for the synthesis of the control system. In the closed-loop system, including the observer and nonlinear controller, the zero state is (locally) asymptotically stable, and the state vector asymptotically converges to the origin. Simulation results for various flow velocities and elastic axis locations are presented, which show that the designed control system is effective in flutter suppression.

Limit Cycle Oscillation and Orbital Stability in Aeroelastic Systems with Torsional Nonlinearity

The paper treats the question of the existence of limit cycleoscillations of prototypical aeroelastic wing sections with structuralnonlinearity using the describing function method. The chosen dynamicmodel describes the nonlinear plunge and pitch motion of a wing. Themodel includes an asymmetric structural nonlinearity in the pitchdegree-of-freedom. The dual-input describing functions of thenonlinearity are derived for the limit cycle analysis. Analyticalexpressions for the average value, and the amplitude and frequency ofoscillation of pitch and plunge responses are obtained. Based on ananalytical approach as well as the Nyquist criterion, stability of thelimit cycles is examined. Numerical results are presented for a set ofvalues of the flow velocities and the locations of the elastic axiswhich show that the predicted limit cycle oscillation amplitude andfrequency as well as the mean value are quite close to the actualvalues. Furthermore, for the chosen model with linear aerodynamics, itis seen that the amplitude of the pitch limit cycle oscillation does notalways increase with the flow velocity for certain elastic axislocations.

Stability and Control of a Structurally Nonlinear Aeroelastic System

The authors examine the stability properties of a class of nonlinear controls derived via feedback linearization techniques for a structurally nonlinear prototypical two-dimensional wing section. In the case in which the wing section has a single trailing-edge control surface, the stability of partial feedback linearization to achieve plunge primary control is studied. It is shown for this case that the zero dynamics associated with the closed-loop system response are locally asymptotically stable for a range of e ow speeds and elastic axis locations. However, there exist locations of the elastic axis and speeds of the subsonic/incompressible e ow for which this simple feedback strategy exhibits a wide range of bifurcation phenomena. Both Hopf and pitchfork bifurcations evolve parametrically in terms of the e ow speed and elastic axis location. In the case in which the wing section has two control surfaces, the global stability of adaptive control techniques derived from full feedback linearization is studied. In comparison with partial or full feedback linearization techniques, the adaptive control strategies presented do not require explicit knowledge of the form of the structural nonlinearity.

Flutter stability of a detuned cascade in subsonic compressible flow

A mathematical model is developed to predict the unsteady aerodynamics of a detuned two-dimensional flat plate cascade in subsonic compressible flow. Aerodynamic detuning is introduced by nonuniform circumferential spacing and chordwise offset. Combined aerodynamic-structural detuning is accomplished by replacing alternate airfoils with splitter blades. A torsion mode stability analysis that considers aerodynamic and combined aerodynamic-structural detuning is developed by combining the unsteady aerodynamic model with a single degreeof-freedom structural model. The effect of these detuning techniques on flutter stability is then demonstrated by applying this model to a baseline unstable 12-bladed rotor and detuned variations of this rotor. This study demonstrates that detuning is a viable passive flutter control technique. Nomenclature A = cascade A airfoil index a — freestream speed of sound B - cascade B airfoil index CL = lift coefficient CM - moment coefficient C, = ratio of chord length of cascade B to cascade A CA = chord length of cascade A CB = chord length of cascade B ea = elastic axis location k - reduced frequency a>c/W n = airfoil index os - chordwise offset of cascade B relative to cascade A

Aeroservoelastic DAP missile fin development

The development of an active aeroservoelastic missile fin using directionally attached piezoelectric (DAP) actuator elements is detailed. Several different types of actuator elements are examined, including piezoelectric polymers, piezoelectric fiber composites and conventionally attached piezoelectric (CAP) and DAP elements. These actuator elements are bonded to the substrate of a torque plate. The root of the torque plate is attached to a fuselage hard point or folding pivot. The tip of the plate is bonded to an aerodynamic shell which undergoes a pitch change as the plate twists. The design procedures used on the plate are discussed. These include an optimization of the actuator element orientation, substrate material type and thickness, as well as a determination of the optimum elastic axis location. A comparison of the various actuator elements shows that DAP elements provide the highest deflections with the highest torsional stiffness. A torque plate was constructed from 0.2032mm thick DAP elements bonded to a 0.127mm thick AISI 1010 steel substrate. The torque plate produced static twist deflections in excess of ±3º. An aerodynamic shell with a modified NACA 0012 profile was added to the torque plate. This fin was tested in a wind tunnel at speeds up to 50ms-1. The static deflection of the fin was predicted to within 6% of the experimental data.

Static aeroelastic characteristics of circulation control wings

Circulation control airfoils can develop lift coefficients far in excess of conventional airfoils through the use of tangential blowing and thus have potential applications for V/STOL aircraft. The static aeroelastic effects of circulation control on swept wings are examined using two analytical models: a simple two-degree-of-freedom model with linearized aerodynamics and an elastic beam model coupled with nonlinear two-dimensional airfoil data. The static divergence instability and a circulation control reversal phenomenon are investigated through the use of lift and control effectiveness ratios. Effects of wing sweep angle, elastic axis location, blowing level, and spanwise blowing distribution are presented. Linear, nonlinear incompressible, and nonlinear compressible aerodynamic representations are compared. Significant differences were observed between the linear and nonlinear aerodynamic results. Tailoring of the spanwise blowing distribution is shown to be an effective means of improving undesirable aeroelastic characteristics. The results indicate that the aeroelastic behavior of circulation control wings can be quite different from that of conventional wings.

Optimization of cascade blade mistuning. II - Global optimum and numerical optimization

The values of the mistuning which yield the most stable eigenvectors are analytically determined, using the simplified equations of motion which were developed in Part I of this work. It is shown that random mistunings, if large enough, may lead to the maximal stability, whereas the alternate mistunings cannot. The problem of obtaining maximum stability for minimal mistuning is formulated, based on numerical optimization techniques. Several local minima are obtained using different starting mistuning vectors. The starting vectors which lead to the global minimum are identified. It is analytically shown that all minima appear in multiplicities which are equal to the number of compressor blades. The effect of mistuning on the flutter speed is studied using both an optimum mistuning vector and an alternate mistuning vector. Effects of mistunings in elastic axis locations are shown to have a negligible effect on the eigenvalues. Finally, it is shown that any general two-dimensional bending-torsion system can be reduced to an equivalent uncoupled torsional system.

The coupled response of turbomachinery blading to aerodynamic excitations

An energy balance technique is developed which predicts the coupled bending torsion mode aerodynamic forced response of an airfoil The effects of the various aerodynamic parameters are then considered utilizing a subsonic compressible flow/flat plate cascade gust analysis The increased coupling between the torsion and translation modes as the natural frequencies approach one another is shown It is also demonstrated that the coupled response amplitudes increase with: 1) decreased structural damping 2) increased solidity stagger angle, and Mach numbers 3) interblade phase angles corresponding to forward traveling waves, and 4) shifting of the elastic axis location aft