Aeroelastic Stability Boundaries Research Articles

Effect of variable thickness on the aeroelastic stability boundaries of truncated conical shells

The flutter characteristics of a truncated conical shell with variable thickness exposed to a supersonic fluid flow are studied in this study. The first-order shear deformation theory (FSDT) is employed to model the shell and the linear piston theory is utilized to evaluate the aerodynamic pressure. The boundary conditions and governing equations are formulated based on Hamilton’s principle. An exact solution is presented in the circumferential direction and the differential quadrature method (DQM) is hired to present an approximate solution in the meridional direction. A general two-parameter power-law equation is considered for thickness variation in the meridional direction and the effects of these coefficients on the flutter boundaries are examined. It is observed that utilizing the conical shells with uniform thickness is not an optimum design and to improve the aeroelastic stability of the conical shells it is better to increase the thickness of the shell from the small radius of the shell to the large one. As the novelty of the presented work, it can be claimed that it is the first attempt to examine the flutter characteristics of conical shells of nonuniform thickness and the results can be utilized to provide optimum designs for aerovehicles.

Time-domain Analytical Method for Aeroelastic Analysis of High Aspect Ratio Wing Using Unsteady Indicial Aerodynamics

This paper deals with the development of a novel time-domain analytical method to perform an aeroelastic stability analysis of a high aspect ratio wing. The aerodynamic model in this paper is built upon the frequency domain of Theodorsen aerodynamics. An inverse Fourier transform technique is applied to convert the Theodorsen frequency domain transfer function to the Wagner time-domain response function. Later, lift and moment expressions, containing aerodynamic lag states are obtained using the indicial function approach which performs the convolution of the unit step response Wagner lift function with arbitrary input and its time derivative using the principle of superposition. Additional equations containing the derivative of aerodynamic lag states are coupled with previously obtained lift and moment expressions to fully get the complete unsteady aerodynamic model in the time-domain. This time-domain unsteady aerodynamic model is then coupled with a combined bending torsion beam finite element model, using the state space approach. A complex eigenvalue analysis of this state space system is performed, using MATLAB, to determine both the static and dynamic aeroelastic stability boundary. Finally, in this paper, aeroelastic stability analysis of the high aspect ratio wing is performed using the current method and results are validated with MSC Nastran aeroelastic analysis and those available in the literature.

Flutter Analysis of a 3D Box-Wing Aircraft Configuration

The aim of this paper is to present a flutter analysis of a 3D Box-Wing Aircraft (BWA) configuration. The box wing structure is considered as consisting of two wings (front and rear wings) connected with a winglet. Plunge and pitch motions are considered for each wing and the winglet is modeled by a longitudinal spring. In order to exert the effect of the wing-joint interactions (bending and torsion coupling), two ends of the spring are located on the gravity centers of the wings tip sections. Wagner unsteady model is used to simulate the aerodynamic force and moment on the wing. The governing equations are extracted via Hamilton’s variational principle. To transform the resulting partial integro-differential governing equations into a set of ordinary differential equations, the assumed modes method is utilized. In order to confirm the aerodynamic model, the flutter results of a clean wing are compared and validated with the previously published results. Also, for the validation, the 3D box wing aircraft configuration flutter results are compared with MSC NASTRAN software and good agreement is observed. The effects of design parameters such as the winglet tension stiffness, the wing sweep and dihedral angles, and the aircraft altitude on the flutter velocity and frequency are investigated. The results reveal that physical and geometrical properties of the front and rear wings and also the winglet design have a significant influence on BWA aeroelastic stability boundary.

A Parametric Study on the Aeroelasticity of Flared Hinge Folding Wingtips

This paper presents a parametric study on the aeroelasticity of cantilever wings equipped with Flared Hinge Folding Wingtips (FHFWTs). The finite element method is utilized to develop a computational, low-fidelity aeroelastic model. The wing structure is modelled using Euler–Bernoulli beam elements, and unsteady Theodorsen’s aerodynamic strip Theory is used for aerodynamic load predictions. The PK method is used to estimate the aeroelastic boundaries. The model is validated using three rectangular, cantilever wings whose properties are available in literature. Then, a rectangular, cantilever wing is used to study the effect of folding wingtips on the aeroelastic response and stability boundaries. Two scenarios are considered for the aeroelastic analysis. In the first scenario, the baseline, rectangular wing is split into inboard and outboard segments connected by a flared hinge that allows the outboard segment to fold. In the second scenario, a folding wingtip is added to the baseline wing. For both scenarios, the influence of fold angle, hinge-line angle (flare angle), hinge stiffness, tip mass and geometry are assessed. In addition, the load alleviation capability of FHFWT is evaluated when the wing encounters discrete (1-cosine) gusts. Finally, the hinge is assumed to exhibit cubic nonlinear behavior in torsion, and the effect of nonlinearity on the aeroelastic response is assessed and analyzed for three different cases.

On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Comparative Assessment

This work reviews different analytical formulations for the time-dependent aerodynamic load of a thin aerofoil and clarifies numerical flutter results available in the literature for the typical section of a flexible wing; inviscid, two-dimensional, incompressible, potential flow is considered in all test cases. The latter are investigated using the exact theory for small airflow perturbations, which involves both circulatory and non-circulatory effects of different nature, complemented by the p-k flutter analysis. Starting from unsteady aerodynamics and ending with steady aerodynamics, quasi-unsteady and quasi-steady aerodynamic models are systematically derived by successive simplifications within a unified approach. The influence of the aerodynamic approximations on the aeroelastic stability boundary is then rigorously assessed from both physical and mathematical perspectives. All aerodynamic models are critically discussed and compared in the light of the numerical results as well, within a comprehensive theoretical framework in practice. In all cases, results accuracy depends on the aero-structural arrangement of the flexible wing; however, simplified unsteady and simplified quasi-unsteady aerodynamic approximations are suggested for robust flutter analysis whenever the wing’s elastic axis lies ahead of the aerofoil’s control point.

Theoretical analysis for the effect of static pressure differential on aeroelastic stability of flexible panel

The static pressure differentials have been neglected in almost all of theoretical analysis models for panel flutter in the supersonic flow, which are established on the basis of the classic aeroelastic model by Dowell in 1966. However, in actual engineering, flexible panels are still subjected to the static pressure differentials under some circumstances. By using the numerical methods, Dowell et al. found that the static pressure differentials as small as 10−2 psi may significantly alter the flutter boundary. In this paper, a novel strategy for a theoretical analysis model is proposed to study the effect of static pressure differential on aeroelastic stability of flexible panel in the supersonic airflow. We regard the final total displacement of the panel as the superposition of dynamic displacement and static deformation, which is in accord with the actual situation of physicality. The static deformation is obtained by solving the static aeroelastic equation. Using the curve/surface fitting method, the static deformations are described by two approaches with/without considering the interaction between the static deformation and steady aerodynamic loads. And then the static deformation is introduced into the dynamic aeroelastic equations in the form of the stiffness through the nonlinear induced loading. The dynamic aeroelastic equation is solved by using the Galerkin discrete method, which can truncate the partial differential equations into a set of ordinary equations. Lyapunov indirect method is utilized to evaluate the aeroelastic stability of the flexible panel. According to the known expression of static deformation and Routh-Hurwitz criterion, a theoretical solution for the aeroelastic stability boundary under the action of the arbitrary static pressure differentials is derived. The theoretical results show that (1) The static pressure differential can significantly enhance the aeroelastic stability of the flexible panel, which is consistent with the numerical results. (2) Whether the actual coupling between the static deformation and aerodynamic loads is considered or not, the stability boundaries obtained are quite different. (3) The increase of the stiffness of structure induced by the static deformation can increase the critical flutter dynamic pressures, but it is not the dominant factor to lead to such a remarkable improvement of the stability of the flexible panel under the action of the static pressure differential. The actual coupling between the static deformation and aerodynamic loads plays a significant role in substantially improving the aeroelastic stability of the flexible panel.

Multifidelity Sensitivity Study of Subsonic Wing Flutter for Hybrid Approaches in Aircraft Multidisciplinary Design and Optimisation

A comparative sensitivity study for the flutter instability of aircraft wings in subsonic flow is presented, using analytical models and numerical tools with different multidisciplinary approaches. The analyses build on previous elegant works and encompass parametric variations of aero-structural properties, quantifying their effect on the aeroelastic stability boundary. Differences in the multifidelity results are critically assessed from both theoretical and computational perspectives, in view of possible practical applications within airplane preliminary design and optimisation. A robust hybrid strategy is then recommended, wherein the flutter boundary is obtained using a higher-fidelity approach while the flutter sensitivity is computed adopting a lower-fidelity approach.

Supersonic flutter analysis of annular/circular sandwich panels containing magnetorheological fluid

The present study investigates the effect of magnetorheological fluids on the aeroelastic stability and flutter boundaries of the circular/annular magnetorheological (MR) sandwich plates in a supersonic airflow. The linear piston theory is employed to formulate the external force applied to the structure due to the aerodynamic pressure. The classical plate theory along with Hamilton’s principle is used to derive governing equations of motion of the sandwich plate. The assumed mode method is utilized to solve the equations and identify the flutter boundaries of the structure. In order to demonstrate validity of the developed model, an experiment is conducted on a circular MR sandwich plate to characterize dynamic properties of the structure, under different levels of the magnetic field, in terms of the resonant frequencies. Then, the verified model is employed to investigate the effects of MR fluid, magnetic field, boundary condition and inner/outer radius of the circular/annular MR sandwich plate on the aerodynamic instability of the structure. The results highlight the significance of the MR fluid in extending flutter boundaries of the annular/circular sandwich plates in the supersonic airflow.

Effects of damage parametric changes on the aeroelastic behaviors of a damaged panel

In this study, the aeroelastic responses and stability boundaries of a simply supported supersonic plate with structural damage are investigated to assess the effects of damage parametric changes on the stability regions as well as to explore some potential tools for damage detection. In the modeling, structural damage is a local bending stiffness loss with various levels, extents and positions. The effects of damage level, extent and position are presented via exploiting nonlinear tools such as bifurcation diagrams, stability regions, Poincare maps and Lyapunov exponents. Specially, the proper orthogonal decomposition (POD) method is applied to extract the POD modes to detect the damage parametric variations. It is determined that (1) structural damage has a notable influence on the aeroelastic stability of the panel; (2) the damage level and extent affect in a similar way that a larger damage level/extent tends to reduce the flutter boundary for a flat plate, but conversely increase the flutter boundary for a buckled plate; (3) the damage occurring around the leading of the panel corresponds to the least stable panel compared to the other positions along the chordwise; (4) the stability region as a novel way for damage detection is proved to be sensitive and effective, and the largest Lyapunov exponent as a quantitative measure is powerful to reveal the subtle differences in the chaos induced by damage changes; (5) the higher-order POD modes are more sensitive to the subtle damage than the primary POD modes.

Local Aeroelastic Instability of High-Speed Cylindrical Vehicles

Different levels of structural modeling fidelity are evaluated against experimental results for the local aeroelastic stability boundaries of an internally pressurized circular cylindrical shell. Finite-element analysis is used to inform a modal approach taken to model the structural dynamics. Third-order piston theory is used to model the external and internal surfaces’ unsteady aerodynamic pressures. Results are used to drive model improvements of free-flight aerothermoelastic simulation in order to predict aeroelastic instabilities in a representative supersonic/hypersonic vehicle. The stability of a finite-element cylindrical structure when inclined to the flow is also considered to inform future work in which a cylindrical high-speed vehicle is required to perform maneuvers.

Aeroelastic stability analysis of curved composite panels with embedded Macro Fiber Composite actuators

This work investigates the aeroelastic stability boundary of curved composite panels with embedded Macro Fiber Composite (MFC) actuators in the supersonic airflow. Prescribed voltages are statically applied to the MFC actuators, inducing a pre-stress field which results in an additional stiffness effect on the curved panels, thus changing the aeroelastic stability boundary of curved composite panels. The principle of virtual work is applied to develop the equations of motion for the nonlinear flutter of curved composite panels with embedded MFC actuators. The Von Karman large deflection panel theory and the first order quasi-steady piston theory are adopted in the formulation. The Newton-Raphson method is employed to determine the static aeroelastic deflection under the applied voltages, and an eigenvalue solution is adopted to predict the aeroelastic stability boundary of the curved panels. Numerical results show that the influence of the applied voltages is distinct for the curved composite panels with different curvatures. In addition, the lamination angles of the MFC actuator and the temperature elevations will also significantly affect the aeroelastic stability boundary of curved composite panels in the supersonic airflow.

Viscous damping effect on the aeroelastic stability of subsonic wings: Introduction of the U–K method

This study aims at introduction of a novel method for evaluating the effect of viscous damping on the aeroelastic stability boundaries. The K-method is well-known for being one of the fastest methods in determining the instability conditions (i.e. critical speed and its corresponding frequency). However, formulation of the K-method is developed for aeroelastic systems without viscous damping and solution is valid where the introduced artificial damping is zero. Taking into account the framework of the K-method in general, this study has tried to remove the major shortcoming of the K-method, i.e. investigation of the effect of viscous damping on the aeroelastic stability boundaries. The aeroelastic feasibility of the introduced method is studied via the analysis of typical sections and subsonic high-aspect ratio wings. Parameter studies provided in the present investigation, indicate that viscous damping may alter the stability conditions significantly.

Aeroelastic behavior of composite laminated shells with embedded SMA wires under supersonic flow

This work investigates the aeroelastic stability boundary of flutter in Shape Memory Alloy Hybrid Composite laminates (SMAHC). The SMAHC consists of SMAs wires and continuous carbon fibers embedded into a polymeric matrix resulting in a three constituent composite material. The derivation of the effective mechanical properties of the SMAHC is based on micromechanical model which accounts for temperature and fraction of martensite/austenite transformation phases of the shape memory alloy. Hamilton's principle is used for the formulation of the energy functional and to obtain the equilibrium equations and boundary conditions of the aeroelastic problem. The finite element method is employed to numerically solve the equations. Different geometric configuration, laminate stacking sequence, boundary conditions and curvatures are investigated. The study shows that the stiffening effect induced by the changes in the fraction of martensite/austenite transformation phases of the shape memory alloy increases the rate of occurrence of flutter, stabilizing the plate. Thus, one can control the occurrence of flutter speed by controlling the temperature of the SMA wires and the proper design of the geometric properties of the panel and tailoring of the composite laminate.

구조 불확도를 고려한 강건 공탄성 해석

공력탄성학적 안정성 해석에 있어서 모델링 오차 및 구조 불확도에 의해 결과의 정확도는 떨어질 수 있다. 따라서, 이러한 모델링 오차 및 구조 불확도를 고려한 공탄성 안정성 경계를 예측할 필요가 있다. 이러한 모델링 오차 및 불확도를 고려한 공탄성 안정성 예측을 위해 강건 공탄성 해석이 제안되었다. 본 연구에서는 <TEX>${\mu}$</TEX> 해석기법과 모달접근법과 MSA를 사용한 조종날개의 공탄성 모델로 부터 강건 공탄성 모델링과 해석을 수행하였다. 강건 공탄성 해석 프로그램이 개발되었고, 기존의 공탄성 해석 결과와 비교/검증하였다. An aeroelastic stability can be degraded due to an aeroelastic modeling error and a structural uncertainty. Therefore it is necessary to predict the aeroelastic stability boundary considering an aeroelastic modeling error and a structural uncertainty. Robust aeroelastic analysis was proposed to predict the aeroelastic stability boundary considering these error and uncertainty. In the present study, the robust aeroelastic modeling and analysis were performed by using the <TEX>${\mu}$</TEX> analysis technique and the aeroelastic model of the control fin with modal approach and MSA. The computer program for the robust aeroelastic analysis was developed and verified by comparing its results with those of conventional aeroelastic analysis methods.

Multibody analysis of whirl flutter stability on a tiltrotor wind tunnel model

Tiltrotor aircraft are equipped with large rotors, usually driven by heavy engines in nacelles located at the tip of the wings. This combination of large rotors and high masses on wings which form a relatively elastic support makes whirl flutter a critical phenomenon for the design of the aircraft and its performance. In this article, a multibody-based simulation model of a tiltrotor wind tunnel model is presented, combining a mixed finite element/multibody representation of the support, a detailed kinematic model of the rotor hub and a strip-wise definition of air loads on the blades. Investigations on the effect of parameter changes and non-linearities on the calculation of the aeroelastic stability boundary are executed. The selected operational points are taken from wind tunnel experiments performed by a consortium of partners in the course of the European ADYN project. The regarded wind tunnel model is a half model of a tiltrotor wing, simplified for the use in whirl flutter investigations. The rotor is four-bladed; the rotor hub is of a gimbal type. Structural dynamics properties of the model are known from detailed experimental investigations. Numerical analyses of the dynamic behaviour of the model are performed in the frequency and in the time domain. Results show considerable differences for linear and non-linear models. This paper describes the simulation approach, the model setup and the comparison of the data with selected experimental results.

Development and Validation of a Fluid–Structure Solver for Transonic Panel Flutter

A partitioned fluid–structure coupling code for transonic panel flutter has been developed and validated. The Reynolds-averaged Navier–Stokes equations are solved numerically by means of an implicit finite volume method to account for nonlinear aerodynamics, as there are shock waves and a viscous boundary layer at the panel surface. An implicit finite element formulation of the structural equations, as well as a Galerkin solution of the von Kármán plate equation are employed to solve elastic panel deformations with respect to geometric nonlinearities. A detailed validation process has shown good agreement with results from the literature for high subsonic and low supersonic Mach numbers. Thereby, a recent comparison between theory and experiment is confirmed. The validated solver is then used for further studies focusing on the impact of turbulent boundary layers on aeroelastic stability boundaries and instabilities. An increase in aerodynamic damping due to a viscous boundary layer is identified by an increase of the aeroelastic stability boundary. Furthermore, a significant damping on high flutter frequencies and mode shapes is revealed. The application of either a one-equation or two-equation turbulence model did not cause any major deviations in the results.

Nonlinear and chaotic vibration and stability analysis of an aero-elastic piezoelectric FG plate under parametric and primary excitations

In this study, in the presence of supersonic aerodynamic loading, the nonlinear and chaotic vibrations and stability of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer have been investigated. It is assumed that the plate is simultaneously exposed to the effects of harmonic uniaxial in-plane force and transverse piezoelectric excitations and aerodynamic loading. It is considered that the potential distribution varies linearly through the piezoelectric layer thickness, and the aerodynamic load is modeled by the first order piston theory. The von-Karman nonlinear strain–displacement relations are used to consider the geometrical nonlinearity. Based on the Classical Plate Theory (CPT) and applying the Hamilton׳s principle, the nonlinear coupled partial differential equations of motion are derived. The Galerkin׳s procedure is used to reduce the equations of motion to nonlinear ordinary differential Mathieu equations. The validity of the formulation for analyzing the Limit Cycle Oscillation (LCO), aero-elastic stability boundaries is accomplished by comparing the results with those of the literature, and the convergence study of the FGP plate is performed. By applying the Multiple Scales Method, the case of 1:2 internal resonance and primary parametric resonance are taken into account and the corresponding averaged equations are derived and analyzed numerically. The results are provided to investigate the effects of the forcing/piezoelectric detuning parameter, amplitude of forcing/piezoelectric excitation and dynamic pressure, on the nonlinear dynamics and chaotic behavior of the FGP plate. It is revealed that under the certain conditions, due to the existence of bi-stable region of non-trivial solutions, system shows the hysteretic behavior. Moreover, in absence of airflow, it is observed that variation of control parameters leads to the multi periodic and chaotic motions.

Aeroelastic Stability of Thermal Protection System for Inflatable Aerodynamic Decelerator

A theoretical aeroelastic stability analysis has been performed on the flexible thermal protection system for an inflatable aerodynamic decelerator. Structural models consist of one or more truncated conical shells of the Donnell type, which may be elastically supported along the middle surface. The aerodynamic model is first-order piston theory. The Lagrangian of the system is formulated in terms of the generalized coordinates for all shell displacements, and the Rayleigh–Ritz method is used to derive the equations of motion. The aeroelastic stability boundaries and mode shapes are found by calculating the eigenvalues and eigenvectors of a large coefficient matrix. When the thermal protection system is approximated as a single conical shell, circumferentially asymmetric coalescence flutter between the second and third axial modes is observed. When many circumferential elastic supports are included, the shell flutters symmetrically in zero circumferential waves, with the first, second, and third axial modes being the most critical. In this case, the flutter boundary, flutter mechanism, and critical modes may change significantly with the addition of structural damping. Aeroelastic models that consider the thermal protection system as multiple interacting shells tend to flutter asymmetrically at high dynamic pressures relative to the single shell models, with higher axial modes being more critical. It is also found that tension applied at the shell edges, orthotropicity, and elastic support stiffness are important parameters that can dramatically affect the shell’s flutter behavior.

Integrated analysis on static/dynamic aeroelasticity of curved panels based on a modified local piston theory

A flow field modified local piston theory, which is applied to the integrated analysis on static/dynamic aeroelastic behaviors of curved panels, is proposed in this paper. The local flow field parameters used in the modification are obtained by CFD technique which has the advantage to simulate the steady flow field accurately. This flow field modified local piston theory for aerodynamic loading is applied to the analysis of static aeroelastic deformation and flutter stabilities of curved panels in hypersonic flow. In addition, comparisons are made between results obtained by using the present method and curvature modified method. It shows that when the curvature of the curved panel is relatively small, the static aeroelastic deformations and flutter stability boundaries obtained by these two methods have little difference, while for curved panels with larger curvatures, the static aeroelastic deformation obtained by the present method is larger and the flutter stability boundary is smaller compared with those obtained by the curvature modified method, and the discrepancy increases with the increasing of curvature of panels. Therefore, the existing curvature modified method is non-conservative compared to the proposed flow field modified method based on the consideration of hypersonic flight vehicle safety, and the proposed flow field modified local piston theory for curved panels enlarges the application range of piston theory.

Effect of thrust on the aeroelastic instability of a composite swept wing with two engines in subsonic compressible flow

This paper aims to investigate aeroelastic stability boundary of subsonic wings under the effect of thrust of two engines. The wing structure is modeled as a tapered composite box-beam. Moreover, an indicial function based model is used to calculate the unsteady lift and moment distribution along the wing span in subsonic compressible flow. The two jet engines mounted on the wing are modeled as concentrated masses and the effect of thrust of each engine is applied as a follower force. Using Hamilton's principle along with Galerkin's method, the governing equations of motion are derived, then the obtained equations are solved in frequency domain using the K-method and the aeroelastic instability conditions are determined. The flutter analysis results of four example wings are compared with the experimental and analytical results in the literature and good agreements are achieved which validate the present model. Furthermore, based on several case studies on a reference wing, some attempts are performed to analyze the effect of thrust on the stability margin of the wing and some conclusions are outlined.