Asymptotic analysis of aircraft wing model in subsonic air flow
Asymptotic analysis of aircraft wing model in subsonic air flow
复制标题
亚音速气流中飞机机翼模型的渐近分析
DOI：
10.1093/imamat/66.4.319
复制
发表时间：
2001
影响因子：
1.2
通讯作者：
M. Shubov
中科院分区：
文献类型：
--
作者：
M. Shubov
本文是致力于亚音速气流中飞机机翼的渐近和谱分析的一系列工作中的第一篇。该模型由加州大学洛杉矶分校飞行系统研究中心开发，并在Balakrishnan的作品中提出。该模型是由两个耦合的积分微分方程和两个参数的边界条件模拟的自应变致动器的行动的系统。未知函数（弯曲和扭转角）取决于时间和一个空间变量。上述方程的微分部分形成耦合线性双曲系统;积分部分是卷积型的。运动方程组等价于系统状态空间中的一个单算子演化-卷积型方程，该方程具有所谓的能量度量。这个方程的解的拉普拉斯变换可以用所谓的广义预解算子来表示。广义预解算子是谱参数的算子值函数。该广义预解算子是复平面上的有限亚纯函数，其分支沿沿着负真实的半轴切割。广义预解式的极点正好是气动弹性模态，在这些极点处的留数是广义特征空间上的投影。在本文中，我们感兴趣的主要对象是系统的微分部分的动力发生器。它是状态空间中的非自伴算子，具有纯离散谱。在本文中，我们表明，谱由两个分支，我们得到了精确的谱渐近。基于这些结果，在下一篇论文中，我们将推导出气动弹性模态的渐近性和模态形状的近似。
This paper is the first in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works of Balakrishnan. 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 the self-straining actuators. The unknown functions (the bending and the torsion angle) depend on time and one spatial variable. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution type equation in the state space of the system equipped with the so-called energy metric. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator. The generalized resolvent operator is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite meromorphic function defined on the complex plane having the branch cut along the negative real semi-axis. The poles of the generalized resolvent are precisely the aeroelastic modes, and the residues at these poles are the projectors on the generalized eigenspaces. In this paper, our main object of interest is the dynamics generator of the differential parts of the system. It is a non-selfadjoint operator in the state space with a pure discrete spectrum. In the present paper, we show that the spectrum consists of two branches, and we derive their precise spectral asymptotics. Based on these results, in the next paper we will derive the asymptotics of the aeroelastic modes and approximations for the mode shapes.