Tightly coupled aeroelastic model implementation dedicated to fast aeroelastic tailoring optimisation of high aspect ratio composite wing

This paper presents the development of a code, called GEBTAero, dedicated to very flexible aircraft (VFA) aeroelasticity and especially the evaluation of aeroelastic tailoring effect on critical speeds. GEBTAero is an open source code consisting in a tightly coupling between a geometrically exact beam theory -and a finite state induced flow unsteady aerodynamic model, including an homogenisation tool. This model has been implemented in Fortran using GEBT code and optimised open source libraries with particular focus on computation speed. Besides a non linear transient dynamic simulation capacity, a particular focus is put on the fast critical speed computation strategy using a non-iterative modal approach about the geometrically non linear deformed shape of the wing with the computation of only a few aeroelastic modes. Computation speed and accuracy of this implementation is assessed using widely used aeroelastic test cases and compared successfully to other aeroelastic codes. Configurations using aeroelastic tailoring, which are the core target of this solver, are then evaluated numerically on a representative high aspect ratio anisotropic composite wing and a simple 2-ply composite laminates with both variable ply orientations. It illustrates the strong correlation between the structural bending/twisting coupling of an unbalanced composite laminates and its critical aeroelastic speed. It also shows the high sensitivity of ply orientation on the aeroelastic behaviour.     

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.     

Reduced order nonlinear aeroelasticity of swept composite wings using compressible indicial unsteady aerodynamics

Nonlinear dynamic aeroelasticity of composite wings in compressible flows is investigated. To provide a reasonable model for the problem, the composite wing is modeled as a thin walled beam (TWB) with circumferentially asymmetric stiffness layup configuration. The structural model considers nonlinear strain displacement relations and a number of non-classical effects, such as transverse shear and warping inhibition. Geometrically nonlinear terms of up to third order are retained in the formulation. Unsteady aerodynamic loads are calculated according to a compressible model, described by indicial function approximations in the time domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to derive the final explicit form of the coupled fluid-structure equations of motion. The final nonlinear governing aeroelastic system of equations is solved using the eigenvectors of the linear structural equations of motion to approximate the spatial variation of the corresponding degrees of freedom in the Ritz solution method. Direct time integrations of the nonlinear equations of motion representing the full aeroelastic system are conducted using the well-known Runge–Kutta method. A comprehensive insight is provided over the effect of parameters such as the lamination fiber angle and the sweep angle on the stability margins and the limit cycle oscillation behavior of the system. Integration of the interpolation method employed for the evaluation of compressible indicial functions at any Mach number in the subsonic compressible range to the derivation process of the third order nonlinear aeroelastic system of equations based on TWB theory is done for the first time. Results show that flutter speeds obtained by the incompressible unsteady aerodynamics are not conservative and as the backward sweep angle of the wing is increased, post-flutter aeroelastic response of the wing becomes more well-behaved.     

Numerical analysis on flutter of Busemann-type supersonic biplane airfoil

The Busemann-type supersonic biplane can effectively reduce the wave drag through shock interference effect between airfoils. However, considering the elastic property of the wing structure, the vibration of the wings can cause the shock oscillation between the biplane, which may result in relative aeroelastic problems of the wing. In this research, fluid–structure interaction characteristics of the Busemann-type supersonic biplane at its design condition have been studied. A theoretical two-dimensional structure model has been established to consider the main elastic characteristics of the wing structure. Coupled with unsteady Navier–Stokes equations, the fluid–structure dynamic system of the supersonic biplane is studied through the two-way computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling method. The biplane system has been simulated at its design Mach number with different nondimensional velocities. Different initial disturbance has been applied to excite the system and the effects of the position of the mass center on the system’s aeroelastic stability is also discussed. The results reveal that the stability of the airfoil in supersonic biplane system is decreased compared with that of the airfoil isolated in supersonic flow and such stability reduction effect should be given due attention in practical design.     

Aeroelastic tailoring of a plate wing with functionally graded materials

A functionally graded material (FGM) provides a spatial blend of material properties throughout a structure. This paper studies the efficacy of FGM for the aeroelastic tailoring of a metallic cantilever plate-like wing, wherein a genetic algorithm provides Pareto trade-off curves between static and dynamic aeroelastic metrics. A key comparison is between the effectiveness of material grading, geometric grading (i.e. plate thickness variations), and using both simultaneously. The introduction of material grading does, in some cases, improve the aeroelastic performance. This improvement, and the physical mechanism upon which it is based, depends on numerous factors: the two sets of metallic material parameters used for grading; the sweep of the plate; the aspect ratio of the plate; and whether the material is graded continuously or discretely.     

Coherent structures and their influence on the dynamics of aeroelastic panels

A panel forced by a supersonic unsteady flow is numerically investigated using a finite difference method, a Galerkin approach, and proper orthogonal decomposition (POD). The aeroelastic model investigated is based on piston theory for modeling the flow-induced forces, and von Karman plate theory for modeling the panel. Structural non-linearity is considered, and it is due to the non-linear coupling between bending and stretching. Several novel facets of behavior are explored, and key aspects of using a Galerkin method for modeling the dynamics of the panel exhibiting limit cycle oscillations and chaos are investigated. It is shown that multiple limit cycles may co-exist, and they are both symmetric and asymmetric. Furthermore, the level of spatial coherence in the dynamics is estimated by means of POD. Reduced order models for the dynamics are constructed. The sensitivity to initial conditions of the non-linear aeroelastic system in the chaotic regime limits the capability of the reduced order models to identically model the time histories of the system. However, various global characteristics of the dynamics, such as the main attractor governing the dynamics, are accurately predicted by the reduced order models. For the case of limit cycle oscillations and stable buckling, the reduced order models are shown to be accurate and robust to parameter variations.     

An efficient method for nonlinear aeroelasticy of slender wings

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.     

Nonlinear aeroelastic analysis of an airfoil-store system with a freeplay by precise integration method

The aeroelastic system of an airfoil-store configuration with a pitch freeplay is investigated using the precise integration method (PIM). According to the piecewise feature, the system is divided into three linear sub-systems. The sub-systems are separated by switching points related to the freeplay nonlinearity. The PIM is then employed to solve the sub-systems one by one. During the solution procedures, one challenge arises when determining the vibration state passing the switching points. A predictor-corrector algorithm is proposed based on the PIM to tackle this computational obstacle. Compared with exact solutions, the PIM can provide solutions to the precision in the order of magnitude of 10⁻¹². Given the same step length, the PIM results are much more accurate than those of the Runge–Kutta (RK) method. Moreover, the RK method might falsely track limit cycle oscillations (LCOs), bifurcation charts or chaotic attractors; even the step length is chosen much smaller than that for the PIM. Bifurcations and LCOs are obtained and analyzed by the PIM in detail. Interestingly, it is found that multiple LCOs and chaotic attractors can exist simultaneously. With this magnitude of precision and efficiency, the PIM could become a solution technique with excellent potential for piecewise nonlinear aeroelastic systems.     

True damping and frequency prediction for aeroelastic systems: The PP method

This paper presents a numerical scheme for stability analysis of the aeroelastic systems in the Laplace domain. The proposed technique, which is called the PP method, is proposed for when the aerodynamic model is represented in the Laplace domain and includes complicated transcendental expressions in terms of the Laplace variable. This method utilizes a matrix iterative procedure to find the eigenvalues of the system and generalizes the other methods such as the P and PK methods for prediction of the flutter conditions. The major advantage of this technique over the other approximate methods is true prediction of subcritical damping and frequency values of the aeroelastic modes. To examine the present technique for stability analysis, some typical examples are used which illustrate its applicability and advantages.     

Aeroelastic analysis of cantilever plates using Peters’ aerodynamic model,and the influence of choosing beam or plate theories as the structural model

In this paper, the aeroelastic analyses of a rectangular cantilever plate of varying aspect ratio is presented. The classical plate theory has been selected as the structural model. The main point that distinguishes this study from previously reported research is employing Peters’ theory to model aerodynamic effect which is not straightforward. The Peters’ aerodynamic model was originally developed to provide lift and moment, which is only applicable to the structural model based on the beam theories. In this study, using the basic concept of the Peters’ aerodynamic model in addition to utilizing the Fourier series, the pressure distribution is derived, which makes Peters’ model applicable to structural models based on plate theory. This combination provides a much simpler state–space aeroelastic model for plates in comparison to the prevalent panel methods, which could lead to a significant reduction in computational time. In addition, the aeroelastic response of the plate with respect to changes in the structural model from the beam theory to the plate theory is evaluated. By using data from an experiment carried out at Duke University, the theoretical results are evaluated. Furthermore, the differences in structural models obtained from the plate and beam theories can be divided into two distinct parts, which are responsible for differences in bending and torsional behaviors of the structure, separately. This approach enables us to measure the effects of differences of each behavior separately, which could provide with a new insight into the problem. It has been determined that the flutter speeds obtained from the beam and plate aeroelastic models are little affected by the difference in bending behavior, but rather is mainly caused by the difference in torsional frequencies.     

考虑几何非线性的复合材料机翼气动弹性分析

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

基于流形切空间插值的折叠翼参数化气动弹性建模

变体飞行器的气动弹性力学建模是当前先进飞行器设计的研究热点和难点. 然而传统的气动弹性动力学建模方法对于具有结构参变特性的变体飞行器气动弹性力学研究存在建模效率低、计算复杂等问题. 本研究提出了一种基于流形切空间插值的可折叠式变体机翼参数化气动弹性建模方法. 首先, 该方法建立若干个典型折叠角下的折叠翼结构有限元模型, 通过流形切空间插值方法建立折叠翼参数化结构动力学模型. 其次, 采用偶极子网格法得到参数化非定常气动力模型, 进而建立气动和结构相互耦合的折叠翼参数化气动弹性模型. 为了验证该参数化建模方法在折叠翼气动弹性分析中的准确性, 本文以一小展弦比折叠翼为研究对象, 从折叠翼自由振动时的参变模态特性、颤振边界预测两方面进行了算例验证, 并与直接计算方法进行了对比, 进一步验证了参数化气动弹性模型的有效性. 研究结果表明, 该参数化气动弹性模型对上述两类问题的计算精度与直接计算方法一致, 并且有着计算效率更高的优势.      

基于POD和代理模型的静气动弹性分析方法

静气动弹性问题考虑弹性结构与定常气动力间的相互耦合作用,对飞行器的性能和安全具有显著的影响.在现代飞行器设计阶段,计算流体力学(CFD)/计算结构力学(CSD)直接耦合方法是精确考察静气动弹性影响的重要手段.然而,基于CFD技术的气动力仿真手段在耦合过程中计算量大且耗时长,难以满足设计阶段的需求.因此,为了兼顾计算精度与效率,文章采用本征正交分解(POD)和Kriging代理模型相结合的模型降阶方法,替代CFD求解过程并耦合有限元分析(FEA)方法,建立了高效、准确的静气动弹性分析框架.相较于传统的以模态法为主的静气动弹性分析方法,该方法能够解决更为复杂的静气动弹性问题以及提供静气动弹性变形过程中的气动分布载荷.针对典型三维跨声速HIRENASD机翼模型开展的马赫数、迎角变化的算例验证表明:由建立的静气动弹性分析方法与CFD/CSD直接耦合方法计算得到机翼翼梢处的静变形量间的相对误差在5%以内;同时该方法预测静平衡位置处的气动分布载荷的误差在5%以内,静气动弹性分析的计算效率至少提升了6倍.     

A discrete 3D model for bridge aerodynamics and aeroelasticity: nonlinearities and linearizations

A framework for the numerical analysis of bridges under wind excitation is outlined. It is based on structural finite element scheme and cross-sectional wind load models. Two aspects are investigated: (1) how considering the mean steady configuration in the aerodynamic stability calculation; and (2) the effects of load nonlinearities on structural response. A quasi-steady load model is adopted, which is able to deal with the considered problems by using experimental data easily available in the practice. By means of numerical examples, it is pointed out (1) that both the modifications in structural tangential stiffness and in the aerodynamic coefficients due to the mean steady deformation may affect the aeroelastic stability threshold and (2) that load linearization may produce an underestimation of the structural response.     

Aircraft T-tail flutter predictions using computational fluid dynamics

The paper presents the application of computational aeroelasticity (CA) methods to the analysis of a T-tail stability in transonic regime. For this flow condition unsteady aerodynamics show a significant dependency from the aircraft equilibrium flight configuration, which rules both the position of shock waves in the flow field and the load distribution on the horizontal tail plane. Both these elements have an influence on the aerodynamic forces, and so on the aeroelastic stability of the system. The numerical procedure proposed allows to investigate flutter stability for a free-flying aircraft, iterating until convergence the following sequence of sub-problems: search for the trimmed condition for the deformable aircraft; linearize the system about the stated equilibrium point; predict the aeroelastic stability boundaries using the inferred linear model. An innovative approach based on sliding meshes allows to represent the changes of the computational fluid domain due to the motion of control surfaces used to trim the aircraft. To highlight the importance of keeping the linear model always aligned to the trim condition, and at the same time the capabilities of the computational fluid dynamics approach, the method is applied to a real aircraft with a T-tail configuration: the P180.     

Aeroelastic instability of composite wings by the consideration of different structural constitutive assumptions

The aeroelastic instability of composite wings modeled as Circumferentially Asymmetric Stiffness (CAS) thin-walled composite beams with closed cross-section is carried out. The objective has been to investigate the effects of different assumptions of constitutive equations on the aeroelastic instability behavior. Non-classical effects such as restrained warping and transverse shear are included in the beam model. The unsteady incompressible airloads are presented using Wagner׳s function. A comparison of the results based on different constitutive equations for a number of configurations including three types of stacking sequence for a box cross-section and two types of stacking sequence for a biconvex cross-section, is performed. The effects of the values of twist as well as twist-bending stiffness coefficients have been studied carefully on the results. As an outcome of this investigation it is revealed that the different choices of structural constitutive equations which result in different values of stiffness quantities; namely, twist and twist-bending stiffness, significantly affect the predicted results. For example, a difference of up to 45% in the aeroelastic critical speed has been observed between different sets of constitutive equations in some cases.     

Aeroelastic response of an airfoil to gusts: Prediction and control strategies from computed energy maps

A method to predict the aeroelastic pitch response of an airfoil to gusts is presented. The prediction is based on energy maps generated by high-fidelity fluid dynamic simulations of the airfoil with prescribed pitch oscillations. The energy maps quantify the exchange of energy between the pitching airfoil and the flow, and serve as manifolds over which the dynamical states of aeroelastic airfoil system grow, decay and attain stationary states. This method allows us to study the full nonlinear response of the system to large gusts, and predict the growth and saturation of aeroelastic pitch instabilities. We also show that the manifold topology in these maps can be used to make informed modifications to the system parameters in order to control the response to gusts.     

Wind-structure interaction simulations of ovalling vibrations in silo groups

During a storm in October 2002, wind-induced ovalling vibrations were observed on several empty silos of a closely spaced group of silos in the port of Antwerp (Belgium). In this paper, three-dimensional numerical simulations are used to investigate this complex case of wind-structure interaction. The computed amplitude of the ovalling vibrations of the silos is similar to that in the observations, indicating that the adopted modelling approach can be suitable for the analysis of new silo groups.Both one-way and two-way simulations are presented, for a single silo and for the silo group. In the one-way simulations, the wind pressure is applied on the structure, disregarding the structural displacements in the wind flow simulation. By contrast, the two-way simulations also take into account the effect of the structural motion on the wind flow. For a single silo, the one-way and two-way simulations yield similar results. Conversely, for a silo in the group, the ovalling vibrations are significantly larger in the two-way simulations than in the one-way simulations. Consequently, aeroelastic effects and/or interactions between the wake-induced excitation and the vibration are present in the silo group for the investigated case.Furthermore, it is shown that the aerodynamic loading and vibration amplitudes are considerably larger for silos in the group than for a single isolated silo.     

细分曲面边界元法的黏附吸声材料结构拓扑优化分析

等几何分析采用样条基函数构造几何模型和实施变量近似,实现了计算机辅助设计和辅助工程的无缝连接,并已广泛应用于弹性力学、电磁场和位势问题等领域.然而直接采用等几何方法难以构造复杂模型,限制了该方法在大规模实际工程问题上的应用.细分曲面法可用于克服这一问题,该方法对传统模型的离散网格进行细分和拟合操作,构造出极限光滑曲面,连续性更高,对复杂结构的适用性更强.该方法主要有以下优点:(1)适用于任意拓扑结构;(2)数值计算稳定;(3)实施简单;(4)局部细化与连续性控制.由于该方法在复杂结构模型构造方面具有较强的灵活性和便利性,已被广泛应用于航空航天、汽车、动画、游戏制作等建模领域.将细分曲面法与边界元法相结合进行结构声学分析,几何场与物理场均采用箱样条基函数进行插值近似.以黏附吸声材料结构的声散射问题为例,建立吸声材料分布拓扑优化数学模型,并采用移动渐进线算法进行设计变量更新,最终获得最优材料分布.      

AEROELASTIC OPTIMIZATION DESIGN OF COMPOSITE WING WITH CONSIDERATION OF STRUCTURAL INSTABILITY

从工程数学求解和有限元分析角度对复合材料结构的稳定性分析方法进行研究,基于这两个方面分别建立了同时考虑壁板稳定性约束和气动弹性约 束的气动弹性优化技术,并以大展弦比复合材料机翼为对象,进行气动弹性综合优化设计。研究表明,机翼气动弹性优化中若不考虑稳定性约束条件,虽然可以获得较小结构重量,但往往不满足稳定性要求;相比从有限元角度考虑结构失稳特征的气动弹性综合优化设计方法,通过工程数学方法对机翼结构进行分区失稳分析优化可以更加精准地控制变量,在满足各项性能指标,特别是稳定性约束的同时,进一步减轻了结构重量,提高了结构失稳因子。