Mathematical Problems Arising in Aircraft Modeling

本提案的主要目标是为三种越来越完整和复杂的飞机机翼在周围气流中的谱、渐近和稳定性分析。前两个模型（分别为一维和二维）已经在加州大学洛杉矶分校的飞行系统研究中心（FSRC）与加利福尼亚州爱德华兹市的NASA Dryden飞行研究中心合作开发。三维模型的设计正在进行中。在大量现代气动弹性文献中存在的机翼模型中，上述模型在物理上是最完整的。1999年11月，在加利福尼亚州爱德华兹空军基地进行了一系列的四次飞行实验，实验结果与模型的理论预测非常吻合，至少在低能气动弹性模式下是如此。目前，该合作得到了NSF基金DMS-9972748（数学科学跨学科基金）的支持。该基金将资助首席研究员对该中心进行为期一年的访问（1999年秋季至2000年春季），以深入研究飞机机翼建模的工程和物理原理，并继续与该中心的研究人员一起开展联合项目。近年来，研究者的研究主要集中在两个方向上：(a) Hilbert空间中非自伴随算子的谱和渐近分析，这些算子是双曲方程和系统的动力学生成器，包含阻尼项和耗散边界条件；(b)将分析结果应用于由这些方程和系统控制的分布参数系统的控制。在这项研究中开发的一系列结果和方法，现已在上述振动飞机机翼的一维模型上达到高潮。已经取得了实质性的进展：PI能够在文献中首次获得高频气动弹性模态和模态振型的显式渐近公式。该项目的目标包括：(a)获得一维模型解的时空域表示；(b)得到最新二维模型解的谱渐近性和表示；(c)将渐近和频谱结果应用于颤振抑制问题；(d)参与机翼三维模型的设计，并将上述分析推广到该模型。本项目可视为上述中心研究人员开展的宽翼建模项目的理论部分。整个项目的最终目标是为从事飞机机翼和尾翼颤振抑制工作的飞机工业工程师提供具体的实用建议。颤振是飞机以特定速度飞行时发生的一种动态不稳定性，这种速度被称为颤振速度。颤振造成的损害给飞机工业造成了巨大的损失。本项目的目标是对飞机机翼模型进行严格的数学分析，并将分析结果应用于颤振控制问题。工程界已经认识到，这种分析的结果可以提供从实验或数值模拟中无法获得的新见解。除了上述项目的技术目标外，首席研究员还计划为数学和工程专业的学生开发一个新的关于飞机工程数学方法的研究生课程。

0072247ShubovThe primary goal of this proposal is to develop the spectral, asymptotic, and stability analysis for three increasingly more complete and complicated models of an aircraft wing in a surrounding airflow. The first two of these models (1-dimensional and 2-dimensional respectively) have been developed in the Flight Systems Research Center (FSRC) at UCLA in collaboration with NASA Dryden Flight Research Center, Edwards, CA. The designing of the 3-dimensional model is in progress. Among the wing models existing in the extensive modern literature on aeroelasticity, the aforementioned ones are most physically complete. In November 1999, the 1-dimensional model was tested in a series of four flight experiments at Edwards Airforce Base, CA. The experimental results are in excellent agreement with the theoretical predictions of the model at least for low - energy aeroelastic modes. Currently, the collaboration is supported by NSF Grant DMS-9972748 (Interdisciplinary Grants in the Mathematical Sciences). This grant provides the support for a one year visit (Fall 1999 - Spring 2000) of the principal investigator to the Center in order to study in depth the engineering and physical principles of aircraft wing modeling and to continue work on the joint project with the researchers of the Center. During recent years, the investigator's research has been focused on two main directions: (a) spectral and asymptotic analysis of non-self-adjoint operators in a Hilbert space, operators which are the dynamics generators of hyperbolic equations and systems containing damping terms and subject to dissipative boundary conditions; (b) applications of the results of this analysis to the control of distributed parameter systems governed by those equations and systems. The series of results and methods, developed in this research, has now culminated in the work on the aforementioned 1-dimensional model of a vibrating aircraft wing. Substantial progress has been made: the PI was able to obtain first in the literature explicit asymptotic formulas for the high-frequency aeroelastic modes and mode shapes. The objectives of this project include: (a) obtaining space-time domain representations for the solutions of the 1-dimensional model; (b) obtaining spectral asymptotics and representations for the solutions of most recent 2-dimensional model; (c) applying asymptotic and spectral results to the flutter suppression problem; (d) participating in the designing of a 3-dimensional model of a wing and extending the above analysis to this model.The present project can be considered as a theoretical part of the broad wing modeling project conducted by the researchers at the aforementioned Centers. The ultimate goal of the entire project is to give specific practical recommendations to aircraft industry engineers working on flutter suppression in aircraft wings and tails. Flutter is a dynamic instability occurring in an aircraft in flight at a specific speed which is called a flutter speed. Damage inflicted by flutter results in significant cost to the aircraft industry. The objective of this project is to carry out a rigorous mathematical analysis of the aircraft wing model and to apply the results of this analysis to the problem of flutter control. It has been recognized in the engineering community that the results of such an analysis can provide new insights which are not available from experiments or from numerical simulations. In addition to the above technical objectives of the project, the principal investigator is planning to develop a new graduate program on mathematical methods in aircraft engineering for both mathematics and engineering students.