翼型厚度对气动弹性特性的影响

结合基于$k$-$\omega$的SST两方程湍流模型,求解雷诺平 均Navier-Stokes方程获得定常和非定常气动力,耦合翼型弹性运动方程,在时间 域内模拟了不同厚度对称翼型在不同迎角下的气动弹性动态过程, 并重点研究了较大迎角下的不同厚度翼型流场特征和气动弹性的性质,研究结果表明：在论 文所涉及的参数情况下,对于迎角从零到大迎角范围,翼型颤振临界速度随迎角的变化不是 单调的. 翼型颤振临界速度迅速下降的起始迎角比最大升力系数对应的迎角小很多.     

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

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

Transonic limit cycle oscillation behavior of an aeroelastic airfoil with free-play

The limit cycle oscillation (LCO) behaviors of an aeroelastic airfoil with free-play for different Mach numbers are studied. Euler equations are adopted to obtain the unsteady aerodynamic forces. Aerodynamic and structural describing functions are employed to deal with aerodynamic and structural nonlinearities, respectively. Then the flutter speed and flutter frequency are obtained by V-g method. The LCO solutions for the aeroelastic airfoil obtained by using dynamically linear aerodynamics agree well with those obtained directly by using nonlinear aerodynamics. Subsequently, the dynamically linear aerodynamics is assumed, and results show that the LCOs behave variously in different Mach number ranges. A subcritical bifurcation, consisting of both stable and unstable branches, is firstly observed in subsonic and high subsonic regime. Then in a narrow Mach number range, the unstable LCOs with small amplitudes turn to be stable ones dominated by the single degree of freedom flutter. Meanwhile, these LCOs can persist down to very low flutter speeds. When the Mach number is increased further, the stable branch turns back to be unstable. To address the reason of the stability variation for different Mach numbers at small amplitude LCOs, we find that the Mach number freeze phenomenon provides a physics-based explanation and the phase reversal of the aerodynamic forces will trigger the single degree of freedom flutter in the narrow Mach number range between the low and high Mach numbers of the chimney region. The high Mach number can be predicted by the freeze Mach number, and the low one can be estimated by the Mach number at which the aerodynamic center of the airfoil lies near its elastic axis. Influence of angle of attack and viscous effects on the LCO behavior is also discussed.     

变角度纤维复合材料层合斜板的颤振分析

假设纤维方向角沿层合板的长度方向线性变化，研究了变角度纤维复合材料层合斜板的颤振．通过坐标变换将斜板变换为正方形板，采用层合板表面连续变化的速度环量来模拟空气对其的作用，速度环量分布利用Cauchy积分公式计算．建立了系统的Lagrange方程并采用Ritz法得到了层合板的自振频率和颤振/不稳定性分离临界速度．通过数值算例验证了本文模型和方法的正确性和收敛性，分析了各个铺层内纤维方向角的变化对自振频率和颤振/不稳定性分离临界速度的影响．研究结果表明，通过纤维的变角度铺设，可有效地提高层合板的基频和颤振/不稳定性分离临界速度．经合理设计的变角度复合材料层合板具有抑制颤振的作用．     

Precursors to flutter instability by an intermittency route: A model free approach

The aeroelastic response of a NACA 0012 airfoil in the flow regimes prior to flutter is investigated in a wind tunnel. We observe intermittent bursts of periodic oscillations in the pitch and plunge response, that appear in an irregular manner from a background of relatively lower amplitude aperiodic fluctuations. As the flow speed is increased, the intermittent bursts last longer in time until eventually transitioning to a fully developed periodic response, indicating the onset of flutter. The repeating patterns in the measured response are visualized using recurrence plots. We show that statistics of the recurrence states extracted from these plots can be used to develop model-free precursors that forewarn an impending transition to flutter, well before its onset.     

The post-Hopf-bifurcation response of an airfoil in incompressible two-dimensional flow

A bifurcation analysis of a two-dimensional airfoil with a structural nonlinearity in the pitch direction and subject to incompressible flow is presented. The nonlinearity is an analytical third-order rational curve fitted to a structural freeplay. The aeroelastic equations-of-motion are reformulated into a system of eight first-order ordinary differential equations. An eigenvalue analysis of the linearized equations is used to give the linear flutter speed. The nonlinear equations of motion are either integrated numerically using a fourth-order Runge-Kutta method or analyzed using the AUTO software package. Fixed points of the system are found analytically and regions of limit cycle oscillations are detected for velocities well below the divergent flutter boundary. Bifurcation diagrams showing both stable and unstable periodic solutions are calculated, and the types of bifurcations are assessed by evaluating the Floquet multipliers. In cases where the structural preload is small, regions of chaotic motion are obtained, as demonstrated by bifurcation diagrams, power spectral densities, phase-plane plots and Poincaré sections of the airfoil motion; the existence of chaos is also confirmed via calculation of the Lyapunov exponents. The general behaviour of the system is explained by the effectiveness of the freeplay part of the nonlinearity in a complete cycle of oscillation. Results obtained using this reformulated set of equations and the analytical nonlinearity are in good agreement with previously obtained finite difference results for a freeplay nonlinearity.     

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．     

激波主导流动下壁板的热气动弹性稳定性理论分析

针对激波主导流动下弹性壁板的热气动弹性稳定性分析问题,建立了基于当地活塞流理论的分析模型,并用数值仿真方法来验证其正确性. 首先基于Hamilton原理和Von-Karman大变形理论,建立壁板的热气动弹性运动方程,其中假设壁板受热后温度均匀分布,激波前后区域的气动力模型采用当地一阶活塞流理论;利用Galerkin方法将具有连续参数系统的偏微分颤振方程离散为有限个自由度的常微分方程;基于李雅普诺夫间接法将非线性颤振方程组在平衡位置处进行线化,再用Routh-Hurwits判据来判断线性系统的稳定性,从而来推论出非线性颤振系统的气动弹性稳定性. 在时域中采用龙格--库塔法对非线性颤振方程进行数值积分,得到壁板非线性颤振响应的时间历程,与理论分析结果进行对比. 研究结果表明,壁板受到斜激波冲击时,更容易发生颤振失稳,并且激波强度越大,极限环幅值和频率越大;激波主导流场中的壁板失稳边界不同于传统单纯超声速气流中壁板颤振的失稳边界;只有在斜激波前后不同的动压值都满足颤振稳定性边界的条件下,壁板才可能保持其气动弹性稳定性.      

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

粘弹性夹芯层合旋转圆板的气动弹性动力稳定性分析

基于哈密尔顿原理,考虑作用于圆板上的空气动力载荷以及粘弹性复模量本构模型,本文建立了含有粘弹性夹芯层的旋转圆板的空气动力学模型,并给出了相应的稳定性的控制方程及边界条件。运用Galerkin方法,文中数值求解并给出了夹芯层合旋转圆板的动力学基本特征,以及结构几何参数、材料参数对于其前、后行波振动频率、阻尼和动力稳定性的影响。结果表明：通过优化合理选取粘弹性夹芯层合旋转圆板的几何参数和材料参数,可提高旋转圆板的临界转速和颤振速度,从而增强其稳定性。     

FLUTTER OF AN AIRFOIL WITH A CUBIC RESTORING FORCE

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a two-dimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a fourth order Runge-Kutta scheme. Results for soft and hard springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit-cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analysed using analytical techniques developed for nonlinear dynamical systems. The Hopf bifurcation point is determined analytically and the amplitude of the limit-cycle oscillation in post-Hopf bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit-cycle oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit-cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom.     

An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system

We perform an analytical and experimental investigation into the dynamics of an aeroelastic system consisting of a plunging and pitching rigid airfoil supported by a linear spring in the plunge degree of freedom and a nonlinear spring in the pitch degree of freedom. The experimental results show that the onset of flutter takes place at a speed smaller than the one predicted by a quasi-steady aerodynamic approximation. On the other hand, the unsteady representation of the aerodynamic loads accurately predicts the experimental value. The linear analysis details the difference in both formulation and provides an explanation for this difference. Nonlinear analysis is then performed to identify the nonlinear coefficients of the pitch spring. The normal form of the Hopf bifurcation is then derived to characterize the type of instability. It is demonstrated that the instability of the considered aeroelastic system is supercritical as observed in the experiments.     

Eigenvalue problem for integro-differential equation of supersonic panel,flutter

The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue prob-lem for a non-self-adjoint fourth-order integro-differential equation of Volterra’s type.Exact solutions of the aeroelastic system is obtained. In contrast to various approximate analyses, our critical curve agrees satisfactorily with experimental data, being free from divergence in the low supe’rsonic region. Moreover, we observe some notable physical behaviors: (1) mutual separation of flutter and vacuum frequency spectrums, (2) degeneracy of critical Mach number. The present method may be generalized in solving the supersonic flutter for 3-dimensional airfoil model as well as blade cascade in turbo-generator.     

Post-flutter response of a flexible cantilever plate in low subsonic flows

Numerical simulations on the post-flutter response of a flexible cantilever plate are carried out by establishing a nonlinear aeroelastic model. The present study shows that chaotic movements may exist in the three-dimensional panel flutter problems in case of low subsonic flows. In the analysis, time traces, phase-plane plots, Poincare maps as well as power spectral densities are employed to identify the dynamic behavior of the system. It is observed that the plate undergoes period-1, period-3 and non-periodic motions with the increase of inflow velocity. The post-flutter behavior is dominated by both geometric and aerodynamic nonlinearities. Numerical results show that wingtip vortexes are in fact an important source of aerodynamic nonlinearities, which have not been fully studied before. The study also provides a criterion on how to choose a coupling strategy in the nonlinear aeroelastic simulation of a low-aspect-ratio flexible structure in low subsonic flows when the dominant nonlinear effect is different in the post-flutter response.     

Camber effects in the dynamic aeroelasticity of compliant airfoils

This paper numerically investigates the effect of chordwise flexibility on the dynamic stability of compliant airfoils. A classical two-dimensional aeroelastic model is expanded with an additional degree of freedom to capture time-varying camber deformations, defined by a parabolic bending profile of the mean aerodynamic chord. Aerodynamic forces are obtained from unsteady thin airfoil theory and the corresponding compliant-airfoil inertia and stiffness from finite-element analysis. V–g and state-space stability methods have been implemented in order to compute flutter speeds. The study looks at physical realizations with an increasing number of degrees of freedom, starting with a camber-alone system. It is shown that single camber leads to flutter, which occurs at a constant reduced frequency and is due to the lock in between the shed wake and the camber motion. The different combinations of camber deformations with pitch and plunge motions are also studied, including parametric analyses of their aeroelastic stability characteristics. A number of situations are identified in which the flutter boundary of the compliant airfoil exhibits a significant dip with respect to the rigid airfoil models. These results can be used as a first estimation of the aeroelastic stability boundaries of membrane-wing micro air vehicles.     

Porous panels with fixed ends lose stability by divergence

The aeroelastic stability of one-dimensional porous panels with a Darcy boundary condition on its surface is examined theoretically. Analytical and numerical analyses demonstrate that a porous panel in a uniform, single-sided, incompressible flow becomes aeroelastically unstable via divergence. This primary route of instability is identical to the well-known mechanism for non-porous panels. However, the divergence speed of a porous panel is always greater than the non-porous limit and increases with a dimensionless porosity parameter formed by the aeroelastic system. Various chordwise porosity distributions along the panel are also investigated, where the uniformly-porous panel is shown to be the most stable configuration. The generality and robustness of the primary divergence instability for porous panels is established analytically using a simple but general flutter analysis approach based on the Routh–Hurwitz stability criterion.     

Aeroelastic stability of a cantilevered plate in yawed subsonic flow

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

Interaction between a cantilevered-free flexible plate and ideal flow

We develop a new computational model of the linear fluid–structure interaction of a cantilevered flexible plate with an ideal flow in a channel. The system equation is solved via numerical simulations that capture transients and allow the spatial variation of the flow–structure interaction on the plate to be studied in detail. Alternatively, but neglecting wake effects, we are able to extract directly the system eigenvalues to make global predictions of the system behaviour in the infinite-time limit. We use these complementary approaches to conduct a detailed study of the fluid–structure system. When the channel walls are effectively absent, predictions of the critical velocity show good agreement with those of other published work. We elucidate the single-mode flutter mechanism that dominates the response of short plates and show that the principal region of irreversible energy transfer from fluid to structure occurs over the middle portion of the plate. A different mechanism, modal-coalescence flutter, is shown to cause the destabilisation of long plates with its energy transfer occurring closer to the trailing edge of the plate. This mechanism is shown to allow a continuous change to higher-order modes of instability as the plate length is increased. We then show how the system response is modified by the inclusion of channel walls placed symmetrically above and below the flexible plate, the effect of unsteady vorticity shed at the trailing edge of the plate, and the effect of a rigid surface placed upstream of the flexible plate. Finally, we apply the modelling techniques in a brief study of upper-airway dynamics wherein soft-palate flutter is considered to be the source of snoring noises. In doing so, we show how a time-varying mean flow influences the type of instability observed as flow speed is increased and demonstrate how localised stiffening can be used to control instability of the flexible plate.     

Vibration characteristics and flutter analysis of a composite laminated plate with a store

The effects of an external store on the flutter characteristics of a composite laminated plate in a supersonic flow are investigated. The Dirac function is used to formulate the interaction between the plate and the store. The first-order piston theory is used to describe the aerodynamic load. The governing equation of the composite laminated plate with an external store is established based on the Hamilton principle. The mode shapes are constructed by the admissible functions which are a set of characteristic orthogonal polynomials generated directly by the Gram-Schmidt process, and the boundary constraint is modeled as the artificial springs. The frequency and mode shapes of the plate under different boundaries are determined by the Rayleigh-Ritz method. The validity of the proposed approach is confirmed by comparing the results with those obtained from the finite element method (FEM). The effects of the mounting position, the center of gravity position and the mounting points spacing of the external store on the flutter boundary are discussed for both the simply supported and cantilever plates, respectively, which correspond to the two installation sites of the external store, i.e., the belly and wings of the aircraft.     

壁板颤振的分析模型、数值求解方法和研究进展

研究壁板颤振问题需要计及大挠度变形下结构的几何非线性效应,不仅涉及气动弹性稳定性,而且关心结构的非线性颤振响应.该文回顾了飞行器壁板颤振问题的国内外研究情况,评述了在壁板颤振研究中采用的分析模型、数值求解方法以及在理论分析和试验方面的研究成果,并提出了今后壁板颤振问题的4个研究方向.