Linear Unsteady Aerodynamics Research Articles

Characterization of typical aeroelastic sections under combined structural concentrated nonlinearities

Structural nonlinearities are usually present in aeroelastic systems. The analysis of this system commonly comprises a study involving only one type of nonlinearity, influencing a particular motion of the airfoil. However, practical applications of aeroelastic systems can be affected by different types of structural nonlinearities. It becomes essential to study the stability of the aeroelastic system under these conditions to assess more real operational flight procedures. In this context, this article presents an investigation of a typical aeroelastic section response with trailing edge control surface subjected to combinations of concentrated structural nonlinearities. Different nonlinear scenarios involving cubic hardening stiffness in pitching and free play, free play with preload, and slip dry friction in the trailing edge control surface motion are analyzed. The mathematical model is based on linear unsteady aerodynamics coupled to a three-dof typical aeroelastic section. Hopf bifurcations diagrams are obtained from direct time integration of the equation of motion. The post-flutter limit cycle oscillations are investigated, revealing supercritical and subcritical bifurcations. A complete parametric study of the nonlinear parameters is carried out, thereby allowing a sensitivity analysis of each nonlinear scenario. The results show that aeroelastic tailoring considering the mild post-flutter behavior can be achieved through an appropriate choice of combined nonlinear effects. Moreover, combined nonlinearities can mitigate the undesired subcritical aeroelastic responses caused by free play.

Aeroelastic and Trajectory Control of High Altitude Long Endurance Aircraft

We investigate the aeroelastic and trajectory control of an high altitude long endurance aircraft model in the presence of gust and turbulence disturbances. The model is derived from geometrically nonlinear beam theory using intrinsic degrees of freedom and linear unsteady aerodynamics, which results in a coupled structural dynamics, aerodynamics, and flight dynamics description. The control design employs a two-loop PI/linear active disturbance rejection control (LADRC) and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control scheme in both the longitudinal and lateral channels, based on a reduced-order linearized model. In each channel, the outer loop (position control) employs a PI/LADRC technique to track the desired flight routes and generate attitude command to the inner loop, while the inner loop (attitude control) uses H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control to track the attitude command generated from the outer loop and computes the control inputs to the corresponding control surfaces. A particle swarm optimization algorithm is employed for parameter optimization of the weighting matrices in the H∞ control design. The simulation tests conducted on the full-order nonlinear model show that the aeroelastic and trajectory control system achieves good performance with respect to robustness, trajectory tracking, and disturbance rejection.

Vortex Sheet Strength in the Sears, Küssner, Theodorsen, and Wagner Aerodynamics Problems

The seminal aerodynamics literature provides analytic predictions of the loads due to sinusoidal gusts (Sears and von Kármán), sharp-edged transverse gusts (Küssner), sinusoidal motions (Theodorsen), and step-change airfoil motions (Wagner). Although these workers determined the overall loads (circulation, lift, and moment), they did not explicitly derive the vortex sheet strength (which is used to compute these loads). Simply put, vortex sheet strength is the velocity difference above/below the airfoil, so it is related to the pressure distribution and therefore the loads. Knowledge of the vortex sheet strength is important because, to extend this seminal theory to include nonlinearities such as thickness–load coupling, viscous–load coupling, or boundary-layer separation, one needs to inform these calculations with the vortex sheet strength. The main contribution of this paper is a method to theoretically predict the vortex sheet strength in the seminal unsteady aerodynamics problems of Sears, Küssner, Theodorsen, and Wagner. These theoretical calculations are enabled by developing a numerical method for calculating the required Fourier coefficients. In addition, a unified presentation of linear unsteady aerodynamics theory is contributed, and examples are provided to illustrate the vortex sheet strength in each of the four seminal problems.

Effects of combined hardening and free-play nonlinearities on the response of a typical aeroelastic section

This paper presents an investigation about the dynamic response of a three-degree of freedom airfoil with hardening nonlinearity in the pitching stiffness and free-play nonlinearity in the control surface stiffness using bifurcation and HOS analysis. An experimental apparatus was conceived to test an airfoil aeroelastic responses when nonlinearities of varying intensities are present. A numerical model is also used to simulate at the same conditions of the experimental tests. It is based on the classical theory for the linear unsteady aerodynamics with corrections for arbitrary motions coupled to a three-degree of freedom typical aeroelastic section, where the hardening effect is modeled by means of rational polynomial function, while the free-play is represented by hyperbolic functions combination. Aeroelastic responses are analyzed from numerical and experimental results. Hopf bifurcations are identified and diagrams of amplitudes versus airspeeds are used to investigate the conditions in which the system is supercritical or subcritical. Higher-order spectra analysis is also used to check on frequency couplings, thereby allowing to identify quadratic- and cubic-like nonlinear behavior. The study of the phenomena associated with the hardening, free-play and their intensity variation effects may be useful in the mitigation of undesired responses of aeroelastic systems.

Aeroelastic coupling of geometrically nonlinear structures and linear unsteady aerodynamics: Two formulations

Two different time domain formulations of integrating commonly used frequency-domain unsteady aerodynamic models based on a modal approach with full order finite element models for structures with geometric nonlinearities are presented. Both approaches are tailored to flight vehicle configurations where geometric stiffness effects are important but where deformations are moderate, flow is attached, and linear unsteady aerodynamic modeling is adequate, such as low aspect ratio wings or joined-wing and strut-braced wings at small to moderate angles of attack. Results obtained using the two approaches are compared using both planar and non-planar wing configurations. Sub-critical and post-flutter speeds are considered. It is demonstrated that the two methods lead to the same steady solution for the sub-critical case after the transients subside. It is also shown that the two methods predict the amplitude and frequency of limit cycle oscillation (when present) with the same accuracy.

Unsteady-Aerodynamic Shape Sensitivities for Airplane Aeroservoelastic Configuration Optimization

Conversion of a general three-dimensional linear unsteady aerodynamic software system to a configuration shape sensitivity code for planar-lifting-surface configurations at subsonic, sonic, supersonic, and hypersonic flows is described. This was achieved by the accomplishment of the following: A global shape design variable sensitivity scheme was formulated based on the complex variable differentiation (CVD) technique. The CVD technique was incorporated into the software system for shape sensitivity analysis. The new capability was validated with the finite difference method and parametric studies on an AGARD swept wing, AGARD wing‐tail configuration, a 60-deg swept wing, and an F-5 wing with a control surface. This selection of test cases covers subsonic to hypersonic planar cases in which singularities in the integrals involved are strong, with aerodynamic interference between lifting surfaces, and including supersonic cases with discontinuous pressure distributions. The most challenging aspects of shape sensitivity analysis for linear unsteady aerodynamics have, thus, been addressed.

Flutter Analysis of Aircraft with External Stores Using Modal Coupling

A new modal coupling technique for efficient flutter and aeroservoelastic analyses of aircraft with multiple external-store configurations is presented. The aircraft is represented by a set of free-free normal modes obtained with large fictitious masses loading the interface coordinates to yield high-accuracy subsequent coupling results. Each store is represented by its own vibration modes obtained with a subset of statically determined interface coordinates clamped to the ground, and the remaining coordinates (if any) are loaded with large fictitious masses. The technique is formulated in a way that facilitates its application using standard options of common commercially available software packages. A finite element code is used to construct the aircraft and stores modal databases. A linear unsteady aerodynamics code is used to read the modal information, calculate the aerodynamic databases, construct and solve the aircraft-store coupling equations, and perform frequency-domain and state-space flutter analyses with or without control system effects

Optimization of a Wing Structure for Gust Response and Aileron Effectiveness

Reliability-based weight optimization of a generic, e ghter-like wing structure is conducted for gust response and aileron effectiveness constraints. The formulation accounts for parametric uncertainties in these aeroelastic response quantities. Reliability indices measure the probability of satisfying each constraint, and a preliminary design procedure is developed in which constraints are enforced on these indices. This framework integrates ASTROS for structural and loads analysis, object-oriented MATLAB ® tools for reliability analysis, and DOT for optimization and most probable point estimation. The reliability analysis algorithm takes advantage of adaptive nonlinear approximations to compensate for nonlinearity of the failure surfaces. The wing structure is modeled with e nite elements, each of which is assumed to have random thickness of known standard deviation. Young’ s modulus of the wing skin material is also assumed to be random. Mean thickness values are taken as design variables. Linear unsteady aerodynamics is used to estimate frequency response functions caused by continuous gust loads. Reliability index constraints are enforced for gust-induced bending moment and shear at the wing’ s root, and also for aileron effectiveness. Redistribution of structural mass by the optimizer produces designs with improved aeroelastic performance reliability and relatively small weight penalties.

Computational Aeroelasticity: Success, Progress, Challenge

This article asserts a much broader dee nition of the term, one in whichCAEencompassesalllevelsofaeroelasticanalysis.Aeroelastic tools based on both linear unsteady aerodynamics and nonlinear CFD methods have been developed and successfully applied. We refer to both of these methodologies as components of CAE. Likewisestructuralmodelingassimpleasbeamtheorytostate-of-the-art e nite element modeling (FEM) have been incorporated into aeroelastic tools, and these techniques should also be included under the CAE heading. It is not the intention of this article to provide an exhaustive history of the development and application of CAE over the past 70 years. Rather, the subject will be examined in the context of two primary themes: 1) aeroelastic problems requiring theoretical

Aeroelastic Response of Nonlinear Wing Sections Using a Functional Series Technique

This paper addresses the problem of the determination of the subcritical aeroelastic response and flutter instability of nonlinear two-dimensional lifting surfaces in an incompressible flow-field via indicial functions and Volterra series approach. The related aeroelastic governing equations are based upon the inclusion of structural and damping nonlinearities in plunging and pitching, of the linear unsteady aerodynamics and consideration of an arbitrary time-dependent external pressure pulse. Unsteady aeroelastic nonlinear kernels are determined, and based on these, frequency and time histories of the subcritical aeroelastic response are obtained, and in this context the influence of the considered nonlinearities is emphasized. Conclusions and results displaying the implications of the considered effects are supplied.

Analysis of Random Gust Response with Nonlinear Unsteady Aerodynamics

Experimental nonlinear unsteady aerodynamics is employed to determine the maximal aircraft response to random gust through matched filter theory. Two airplane configurations, the F-18 High Alpha Research Vehicle (HARV) and F-16XL, have been tested in forced oscillation to provide the necessary data. The atmospheric turbulence is assumed to be characterized by the von Karman gust power spectral density function. The nonlinear aerodynamic models are set up through Fourier functional analysis of the test data. For comparative purposes, linear aerodynamic models are also calculated by using the unsteady quasi-vortex-lattice method. It is shown that nonlinear unsteady aerodynamics produce the instantaneous maximum lift response to gust significantly higher than that by the linear unsteady aerodynamics

Unsteady Aeroelastic Optimization in the Transonic Regime

A methodology for including transonic e utter requirements in the preliminary automated structural design environment is developed and tested. The problem of minimizing structural weight while satisfying behavioral constraints is stated in nonlinear mathematical programming form and is solved using a gradient-based optimizing technique. The structure is modeled by using e nite elements, and the associated design variables consist of the structural properties: thicknesses of skins, spars, and ribs; cross-sectional areas of posts and spar and rib caps; and concentrated masses. The method requires that the transonic unsteady aerodynamic forces be represented in the frequency or Laplace domain. In this work, the indicial response method is used to transform time-domain aerodynamic forces found by solving the transonic small disturbance (TSD) equations into the Laplace domain. The indicial responses are calculated about static aeroelastic equilibriums found using the TSD equations for the steady aerodynamics. Once in the Laplace domain, the unsteady aerodynamic forces are used to determine system dynamic stability by the p-method and in semianalytic equations for the e utter constraint sensitivities. With constraint values and the required gradients, a Taylor series approximation is used to develop an approximate nonlinear mathematical programming problem for weight minimization. This approximate optimization problem is iteratively solved by the method of modie ed feasible directions until convergence of the exact problem is obtained. Examples of the redesign methodology are given for the simultaneous consideration of constraints on transonic e utter, stresses, and displacements. Results found using nonlinear aerodynamics show that designs can differ considerably from those obtained using linear unsteady aerodynamics when in the transonic e ight regime.

Semianalytical technique for sensitivity analysis of unsteady aerodynamic computations

A semianalytical approach is developed for the sensitivity analysis of linear unsteady aerodynamic loads. The semianalytical approach is easier to implement than the analytical approach. It is also computationally less expensive than the finite difference approach when used with panel methods, which require a large number of panels. The semianalytical approach is applied to an isolated airfoil in a 2-D flow and rotating propfan blades in 3-D flow. Sensitivity coefficients with respect to non-shape-dependent variables are shown for some cases. It is expected that the semianalytical approach will be useful in aeroelastic design procedures particularly when mistuning is present, and that it is potentially useful for shape sensitivity analysis of linear unsteady aerodynamics.