Localized Basis Function Method for Computing Limit Cycle Oscillations

An alternate approach to the standard harmonic balance method (based on Fourier transforms) is proposed. The proposed method begins with an idea similar to the harmonic balance method, i.e. to transform the initial set of differential equations of the dynamics to a set of discrete algebraic equations. However, as distinct from previous harmonic balance techniques, the proposed method uses a set of basis functions which are localized in time and are not necessarily sinusoidal. Also as distinct from previous harmonic balance methods, the algebraic equations obtained after the transformation of the differential equations of the dynamics are solved in the time domain rather than the frequency domain. Numerical examples are provided to demonstrate the performance of the method for autonomous and forced dynamics of a Van der Pol oscillator.     

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.     

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.     

Limit Cycle Oscillation and Orbital Stability in Aeroelastic Systems with Torsional Nonlinearity

The paper treats the question of the existence of limit cycleoscillations of prototypical aeroelastic wing sections with structuralnonlinearity using the describing function method. The chosen dynamicmodel describes the nonlinear plunge and pitch motion of a wing. Themodel includes an asymmetric structural nonlinearity in the pitchdegree-of-freedom. The dual-input describing functions of thenonlinearity are derived for the limit cycle analysis. Analyticalexpressions for the average value, and the amplitude and frequency ofoscillation of pitch and plunge responses are obtained. Based on ananalytical approach as well as the Nyquist criterion, stability of thelimit cycles is examined. Numerical results are presented for a set ofvalues of the flow velocities and the locations of the elastic axiswhich show that the predicted limit cycle oscillation amplitude andfrequency as well as the mean value are quite close to the actualvalues. Furthermore, for the chosen model with linear aerodynamics, itis seen that the amplitude of the pitch limit cycle oscillation does notalways increase with the flow velocity for certain elastic axislocations.     

基于传递函数法的带多个外挂机翼颤振分析

基于带外挂机翼结构和气动特点，使用带有半解析半数值特性的传递函数方法进行处理．首先，通过机翼的运动微分方程、二元机翼非定常气动力Therdorson 模型、结合外挂挂载处的内力平衡与位移状态条件，得到了三维的带多个外挂大展弦比机翼的颤振微分方程．进而，使用传递函数方法，先将颤振计算方程整理成为状态空间方程形式，结合求解复特征值的方法，完成了带多个外挂大展弦比机翼的动气动弹性稳定性分析．对比已有文献的计算结果以及通过有限元方法进行的仿真结果，证实了文章所提计算方法的准确性和高效性，结合传递函数方法的优势，进一步将文章方法拓展到机翼的固有频率和发散速度求解．文章结尾，分析了外挂数量、外挂质量、转动惯量及位置分布等变量对带多个外挂大展弦比机翼的动气动弹性稳定性的影响．     

带外挂三角机翼极限环颤振的实验研究

本文从实验方面来研究室外挂俯仰有预加载间隙对颤振的影响。风洞吹风中测出了单稳定极限环、双稳定极限环颤振的情况。实验结果与谐波平衡法计算值进行了比较。     

Studies of Store-Induced Limit-Cycle Oscillations Using a Model with Full System Nonlinearities

The authors investigate limit-cycle oscillations of a wing/store configuration. Unlike typical aeroelastic studies that are based upon a linearized form of the governing equations, herein full system nonlinearities are retained, and include transonic flow effects, coupled responses from the structure, and store-related kinematics and dynamics. Unsteady aerodynamic loads are modeled with the equations from transonic small disturbance theory. The structural dynamics for the cantilevered wing are modeled by the nonlinear equations of motion for a beam. The effects of general store-placement are modeled by the nonlinear equations of motion related to the position-induced nonlinear kinematics. Chordwise deformations of the wing surface, as well as pylon and store flexibility, are assumed negligible. Nonlinear responses are studied by examining bifurcation and related response characteristics using direct simulation. Particular attention is given to cases for which large-time, time-dependent behavior is dependent on initial conditions, as observed for some configurations in flight test. Comparisons of results in which selective nonlinearities are excluded indicate that the accurate prediction of nonlinear responses such as limit cycle oscillations (LCOs) may depend upon consideration of all nonlinearities related to the full system.     

Robust Limit Cycle Suppression for Control Systems with Parametric Uncertainty and Nonlinearity

A systematic limit cycle suppression method is presented to predict and suppress the sustained limit cycles persisting in a class of uncertain control systems containing inherent nonlinearity. The aim is to determine the feasible controller parameter sets in the parameter plane for the entire uncertain system. A family of constant limit cycle amplitude loci is plotted. Successful application to pitch orientation control system with parametric uncertainties and nonlinearity is demonstrated. System output response is further investigated.     

A Reduced Order Cyclic Method for Computation of Limit Cycles

A reduced order cyclic method was developed to compute limit-cycle oscillations for large, nonlinear, multidisciplinary systems of equations. Method efficacy was demonstrated for two simplified models: a typical-section airfoil with nonlinear structural coupling and a nonlinear panel in high-speed flow. The cyclic method was verified to maintain second-order temporal accuracy, yield converged limit cycles in about 10 Newton iterates, and provide precise estimates of cycle frequency. This method was projected onto a low-order space using a set of variables governing the amplitudes of empirically derived modes, which were computed with the proper orthogonal decomposition. In this reduced order form, the cyclic Jacobian was greatly compressed, allowing accurate limit cycle solutions to be very efficiently computed.The U.S. Governments right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

某飞机部件测力风洞试验天平研制与应用

为满足某飞机部件高速风洞测力试验的需要,研制了四台部件测力天平,用来同时测量飞机机翼不同部件(全翼、外翼、翼尖弹、弹架组合)所受的气动载荷。由于是全模试验,天平的设计载荷不匹配,安装空间极度受限。设计时,全翼天平采用了机身中部安装方案、外翼天平与机翼加工成一体并且采用了新型多柱梁结构;翼尖弹天平和弹架组合天平采用了杆式天平方案。上述措施不仅满足了天平的测量需要,也确保了天平在模型中的安装位置以及测力的有效性。该部件测力试验已经在中国空气动力研究与发展中心FL-26风洞完成。试验结果表明,天平测力结果准确可靠,精准度高,达到了预期的目标。     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Averaging Oscillations with Small Fractional Damping and Delayed Terms

We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.     

快速俯仰三角翼的非定常空气动力实验研究

展弦比为１．５和２．０的两个平板三角翼作快速俯仰振荡运动的非定常空气动力实验研究在南京航空航天大学３×２．５米低速风洞中进行。俯仰运动的振幅为６０°和９０°，无因次俯仰率参数范围为０．０１￣０．０３。以三角翼根弦为参考长度的雷诺数为０．７２×１０＾６。用六分量应变天平测量模型的空气动力和力矩。实验结果表明：法向力和俯仰力矩系数相对于静态值的过冲量以及滞环大小与无因次俯仰率、振幅及展弦比密切相关，而     

扑翼飞行器带翼刀机翼气动特性实验研究

通过进行微型扑翼飞行器低速风洞试验,研究了带弯度机翼下翼面翼刀对扑翼飞行器升阻特性的影响。文中进行了带翼刀机翼和不带翼刀机翼在不同迎角下的风洞吹风试验。试验结果表明,带翼刀机翼升力系数大于不带翼刀机翼升力系数,从而证明了翼刀可以阻止机翼下表面气流展向流动,起到增加机翼升力的作用。当扑翼在小迎角飞行时,带翼刀机翼可以有效地提高扑翼的气动效率,改善扑翼的飞行性能。研究结果可为带翼刀机翼在扑翼飞行器上的应用提供技术支持。     

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.     

A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.     

热环境下壁板非线性颤振分析

基于一阶活塞气动力理论,采用Von Karman大变形应变-位移关系建立了无限展长壁板热环境下颤振方程,采用伽辽金方法对方程进行离散处理.取温度为分叉参数,研究壁板颤振时的分叉及混沌等复杂动力学特性.结果表明:温度载荷降低了系统的颤振临界动压,改变了颤振特性.在整个分岔参数范围内,系统呈现出较为复杂的变化,包括衰减振动、极限环振动、拟周期振动和混沌型振动.当考虑材料热效应时,系统的颤振动压将进一步降低,其响应也表现出更为丰富的非线性动态力学行为.     

颤振的增益调度控制实验研究

本文对机翼外挂系统颤振的半主动抑制实验进行了研究分析,在引入电磁阻尼器对风振控制的同时,根据增益调度控制原理设计并制造了控制系统,在风洞实验中实现了电磁阻尼器对系统的颤振半主动抑制,为颤振抑制的实现又提供了一种实验的方法。     

鸭式旋翼/机翼飞机悬停及小速度前飞气动干扰实验研究

鸭式旋翼/机翼飞机是一种新概念可垂直起降高速飞行器,为了解该飞机在悬停及小速度前飞时的全机气动干扰特性,在南京航空航天大学开口风洞中进行了飞机全机气动力实验,实验采用多台测力天平分别测量主机翼和机身的气动力.结果表明,悬停时受主机翼高速旋转产生的下洗尾流影响,机身产生了较大的法向力和低头力矩;前飞时下洗尾流对机身的法向力和俯仰力矩有比较严重的干扰,对滚转力矩和偏航力矩干扰较小,对侧向力有一定影响.实验结果为飞机的飞行动力学特性研究以及控制律设计提供了参考.＃     

On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems

In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [Somieski, G., Nonlinear Dynamics 26(1), 2001, 3–22]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing computational efficiency by avoiding a so called “frequency iteration”. An erratum to this article is available at .