Computational Investigation of Wing Flutter

Numerical aeroelastic simulation of a high-aspect-ratio transport type wing model in transonic region is presented. The aeroelastic responses of the wing are extracted by integrating compressible thin-layer Navier-Stokes equations coupled with the equations of motion of the wing structure, in a time dependent manner. The Yee-Harten implicit TVD scheme and the Wilson's θ method are employed to integrate these equations, respectively. Flutter boundaries were found for Mach number range, 0.7 to 0.85 and the results were compared with experimental flutter boundaries. Futhermore, Limit Cycle Oscillations were found and the characteristics of the flutter and limit cycle oscillations are investigated and discussed.     

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.     

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.     

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.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.     

Nonlinear natural responses of a rotating circular string

This study investigates the nonlinear dynamics of a rotating circular string subjected to a spring force fixed in space. The governing equation for out-of-plane vibration is developed using Hamilton's principle. The nonlinearities of the string deformation and the spring stiffness are considered in the governing equation. Applying Galerkin's method, the governing equation is transformed from a nonlinear partial differential equation into a set of coupled nonlinear ordinary differential equations through orthogonal trigonometric shape functions. Butenin's method is adopted to develop a closed-form analytical solution for single-mode oscillations of the system. Comparison shows that the closed-form solution is in a good agreement with the numerical results over a wide range of the nonlinearities. Multi-mode responses of the string are investigated by means of numerical integration. Based on the results, the nonlinear dynamics of the string are discussed.     

Output Feedback Modular Adaptive Control of a Nonlinear Prototypical Wing Section

The paper presents nonlinear adaptive control systems for the control of limit cycle oscillations of a prototypical wing section with structural nonlinearities using only output feedback. The chosen model describes the plunge and pitch motion of a wing. The model includes plunge and pitch nonlinearities, and has a single control surface for the purpose of control. Using a canonical representation of the aeroelastic system, a modular output feedback adaptive control system consisting of an input-to-state stabilizing controller and a passive identifier (an observer and adaptation law) is derived. In the closed-loop system, asymptotic stabilization of the pitch and plunge motion is accomplished. Simulation results show that the control system is effective in regulating the state vector to the origin in spite of large parameter uncertainties.     

Computational aeroelastic investigation of a transonic limit-cycle-oscillation experiment at a transport aircraft wing model

Aeroelastic measurements of a three-dimensional wing model, the so-called Aerostabil wing, were conducted in the Transonic Windtunnel Göttingen. This clean, backward-swept wing allowed the experimental investigation of limit cycle oscillations in a certain transonic parameter range. In this paper, a detailed insight into the observed physical phenomena, especially the measured limit cycle oscillations, is presented by means of CFD–CSM coupled simulations. These simulations on the basis of a detailed structural finite element model reveal the specific properties of the Aerostabil wing and furthermore allow investigating the unstable behavior of this windtunnel model for transonic flow settings. The aerodynamic characteristics include a two-shock system and large flow separation areas, further increasing the complexity of the aeroelastic problem. A structural single degree-of-freedom system is used for the prediction of the experimental stability range and the limit cycle oscillation investigations. Due to the good agreement of simulation and experiment the limit cycle oscillations can be explained by means of nonlinear aerodynamic effects.     

Effects of damping on the speed of increase and amplitude of limit cycle for an aircraft braking system subjected to mode-coupling instability

A nonlinear model of an aircraft braking system is presented and used to investigate the effects of damping on the stability in Chevillot et al. (Arch Appl Mech 78(12):949–963, 2008). It has been shown that the addition of damping into the equations of motion does not lead systematically to the stabilization of the system. In the case of a mode-coupling instability, there is indeed an optimal ratio between the modal damping coefficients of the two modes in coalescence, that maximize the stable area. But the stable area is not a sufficient criterion. In dynamics, the amplitude of the vibrations and the transient behavior characterized by the speed of increase of the oscillations are best indicators. In this paper, the same nonlinear model of the aircraft braking system is used to compute time-history responses by integration of the full set of the nonlinear dynamic equations. The aim of the study is to evaluate the effects of damping on the nonlinear dynamics of the brake. It is shown that damping may be very efficient to significantly reduce and slow down the increase of the friction-induced vibrations. But, in the same way as for the stability area, there exists a value of the damping ratio that optimizes the effects of damping.     

亚音速气流中复合材料悬臂板的非线性振动响应研究

随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和 Kutta--Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯$\cdot$卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.      

跨音速极限环型颤振的高效数值分析方法

事先建立一个低阶的非线性、非定常气动力模型是开展非线性流场中气动弹性问题研究的一个捷径. 基于CFD方法, 通过计算结构在流场中自激振动的响应来获得系统的训练数据. 采用带输出反馈的循环RBF神经网络, 建立时域非线性气动力降阶模型.耦合结构运动方程和非线性气动力降阶模型, 采用杂交的线性多步方法计算结构在不同速度(动压)下的响应历程, 从而获得模型极限环随速度(动压)变化的特性. 两个典型的跨音速极限环型颤振算例表明, 基于气动力降阶模型方法的计算结果与直接CFD仿真结果吻合很好, 与后者相比其将计算效率提高了1~2个数量级.      

Identification of limit cycles for piecewise nonlinear aeroelastic systems

This paper describes two methods for the analysis of aeroelastic systems with complex piecewise nonlinear structural stiffness. These methods are tested and compared for low speed incompressible and transonic flows. The first technique employed in this paper uses a new application of the analytical solution of linear algebraic systems, the second technique utilises logarithmic and tanh functions to both represent discrete nonlinearities and to act as a switch between different nonlinear areas. The transonic aerodynamic models used are generated using an eigenvalue realisation algorithm (ERA) which produces reduced order models (ROMs) from the pulse responses of time linearised Euler simulations. It is shown that such aerodynamic models are well suited to use with continuation methods. Flutter boundaries and limit cycle oscillations can then be rapidly identified with good accuracy.     

Limit cycle oscillations of rectangular cantilever wings containing cubic nonlinearity in an incompressible flow

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of rectangular cantilever wings with a cubic nonlinearity are investigated. Aeroelastic equations of a rectangular cantilever wing with two degrees of freedom in an incompressible potential flow are presented in the time domain. The harmonic balance method is modified to calculate the LCO frequency and amplitude for rectangular wings. In order to verify the derived formulation, flutter boundaries are obtained via a linear analysis of the derived system of equations for five different cases and compared with experimental data. Satisfactory results are gained through this comparison. The problem of finding the LCO frequency and amplitude is solved via applying the two methods discussed for two different cases with hardening cubic nonlinearities. The results from first-, third- and fifth-order harmonic balance methods are compared with the results of an exact numerical solution. A close agreement is obtained between these harmonic balance methods and the exact numerical solution of the governing aeroelastic equations. Finally, the nonlinear aeroelastic analysis of a rectangular cantilever wing with a softening nonlinearity is studied.     

Nonlinear Analysis of Store-Induced Limit Cycle Oscillations

Flight tests of modern high-performance fighter aircraft reveal the presence of limit cycle oscillation (LCO) responses for aircraft with certain external store configurations. Conventional linear aeroelastic analysis predicts flutter for conditions well beyond the operational envelope, yet these store-induced LCO responses occur at flight conditions within the flight envelope. Several nonlinear sources may be present, including aerodynamic effects such as flow separation and shock-boundary layer interaction and structural effects such as stiffening, damping, and system kinematics. No complete theory has been forwarded to accurately explain the mechanisms responsible. This research examines a two degree-of-freedom aeroelastic system which possesses kinematic nonlinearities and a strong nonlinearity in pitch stiffness. Nonlinear analysis techniques are used to gain insight into the characteristics of the behavior of the system. Numerical simulation is used to verify and validate the analysis. It is found that when system damping is low, the system clearly exhibits nonlinear interaction between aeroelastic modes. It is also shown that although certain applied forcing conditions may appear negligible, these same forces produce large amplitude LCOs under specific realizable circumstances.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.     

CHARACTERIZATION OF THE BEHAVIOUR OF A SIMPLE AEROSERVOELASTIC SYSTEM WITH CONTROL NONLINEARITIES

The characterization of the behaviour of nonlinear aeroelastic systems has become a very important research topic in the Aerospace Industry. However, most work carried to-date has concentrated upon systems containing structural or aerodynamic nonlinearities. The purpose of this paper is to study the stability of a simple aeroservoelastic system with nonlinearities in the control system and power control unit. The work considers both structural and control law nonlinearities and assesses the stability of the system response using bifurcation diagrams. It is shown that simple feedback systems designed to increase the stability of the linearized system also stabilize the nonlinear system, although their effects can be less pronounced. Additionally, a nonlinear control law designed to limit the control surface pitch response was found to increase the flutter speed considerably by forcing the system to undergo limit cycle oscillations instead of fluttering. Finally, friction was found to affect the damping of the system but not its stability, as long as the amplitude of the frictional force is low enough not to cause stoppages in the motion.     

Limit cycle oscillation behavior of transonic control surface buzz considering free-play nonlinearity

The limit cycle oscillation (LCO) behaviors of control surface buzz in transonic flow are studied. Euler equations are employed to obtain the unsteady aerodynamic forces for Type B and Type C buzz analyses, and an all-movable control surface model, a wing/control surface model and a three-dimensional wing with a full-span control surface are adopted in the study. Aerodynamic and structural describing functions are used to deal with aerodynamic and structural nonlinearities, respectively. Then the buzz speed and buzz frequency are obtained by V-g method. The LCO behavior of the transonic control surface buzz system with linear structure exhibits subcritical or supercritical bifurcation at different Mach numbers. For nonlinear structural model with a free-play nonlinearity in the control surface deflection stiffness, the double LCO phenomenon is observed in certain range of flutter speed. The free-play nonlinearity changes the stability of LCOs at small amplitudes and turns the unstable LCO into a stable one. The LCO behavior is dominated by the aerodynamic nonlinearity for the case with large control surface oscillation amplitude but by the structural nonlinearity for the case with small amplitude. Good agreements between LCO behaviors obtained by the present method and available experimental data show that our study may help to explain the experimental observation in wind tunnel tests and to understand the physical mechanism of transonic control surface buzz.     

Amplification and amplitude limitation of heave/pitch limit-cycle oscillations close to the transonic dip

Recent results from flutter experiments of the supercritical airfoil NLR 7301 at flow conditions close to the transonic dip are presented. The airfoil was mounted with two degrees-of-freedom in an adaptive solid-wall wind tunnel, and boundary-layer transition was tripped. Flutter boundaries exhibiting a transonic dip were determined and limit-cycle oscillations (LCOs) were measured. The local energy exchange between the fluid and the structure during LCOs is examined and leads to the following findings: at supercritical Mach numbers below that of the transonic-dip minimum the presence of a shock-wave and its dynamics destabilizes the aeroelastic system such that the decreasing branch of the transonic dip develops. At higher Mach numbers the shock-wave motion has a stabilizing effect such that the flutter boundary increases to higher flutter-speed indices with increasing Mach number. Amplified oscillations near this branch of the flutter boundary obtain energy from the flow mainly due to the dynamics of a trailing-edge flow separation. A slight nonlinear amplitude dependency of the shock motion and a possibly occurring boundary-layer separation cause the amplitude limitation of the observed LCOs. The impact of the findings on the numerical simulation of these phenomena is discussed.     

Nonlinear vibrations of a shell-shaped workpiece during high-speed milling process

This paper is focused on nonlinear dynamics of a shell-shaped workpiece during high speed milling. The shell-shaped workpiece is modeled as a double-curved cantilevered shell subjected to a cutting force with time delay effects. Equations of motion are derived by using the Hamilton principle based on the classical shell theory and von Karman strain-displacement relation. The resulting nonlinear partial differential equations are reduced to a two-degree-of-freedom nonlinear system by applying the Galerkin approach. The averaging method is used to obtain four-dimensional averaged equations for the case of foundational parametric resonance and 1:2 internal resonance. Using a numerical method, the dynamics of the cantilevered shell-shaped workpiece is studied under time-delay effects, parametric excitation, and forcing excitation. It is found that time-delay parameters have great impact on chaotic motion. With increasing amplitude of forcing and parametric excitations, the shell-shaped workpiece exhibits different dynamic behavior.