Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow
Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow
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亚音速气流中飞机机翼模型气动弹性模态的渐近行为
DOI：
10.1098/rspa.2003.1217
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Shubov
中科院分区：
文献类型：
--
作者：
By A. V. Balakrishnan;M. Shubov
本文研究了亚音速气流中飞机机翼模型的渐近分析和谱分析。这个模型是在加州大学洛杉矶分校的飞行系统研究中心开发的。该模型由两个耦合的积分-微分方程组和一个模拟自应变作动器作用的双参数边界条件组成。这些方程的微分部分形成一个耦合的线性双曲系统；积分部分是卷积型的。运动方程系统在能量空间上等价于一个单算子演化-卷积方程。该方程解的拉普拉斯变换可以用所谓的广义解析算子来表示，它是谱参数的算子值函数。该广义解析算子是复平面上沿负实半轴有分支割的有限亚纯函数。它的极点恰好是气动弹性模态，这是本文的主要研究对象。第二作者在一系列著作中系统地研究了系统微分部分的动力学产生。该发生器是能量空间中具有纯离散谱的非自伴随算子。在上述系列文章中，证明了谱由两个分支组成，并推导了谱关于特征值数的精确渐近。得到了模态振型的渐近近似。基于渐近结果，证明了动力发生器的广义特征向量集在能量空间上形成Riesz基。在本文中，我们考虑控制模型的整个积分-微分系统。也就是说，我们研究了原始系统的积分卷积型部分的性质。特别地，我们证明了气动弹性模态的集合是渐近地接近于与微分部分相对应的动力学发生器的离散谱的。本文的结果对于从拉普拉斯变换中重建原初边值问题的解以及对颤振现象的分析具有重要意义。
The present paper is devoted to the asymptotic and spectral analysis of a model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of the University of California at Los Angeles. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modelling the action of self-straining actuators. The differential parts of these equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function on the complex plane having a branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes, which are the main object of interest in the present paper. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-self-adjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it was shown that the spectrum consists of two branches, and the precise spectral asymptotics with respect to the eigenvalue number was derived. The asymptotic approximations for the mode shapes have also been obtained. Based on the asymptotic results, it has been proved that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space. In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of the aeroelastic modes is asymptotically close to the discrete spectrum of the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.