Mathematical Theory of Aeroelasticity

目的和知识价值本研究的目的是系统地发展气动弹性数学理论，该理论以抽象泛函分析为基础，特别是非线性气动弹性，以颤振的典型问题为中心：在亚音速范围内足够高的速度下飞机特有的不稳定性，M 1。特别是，这将意味着发展有关气动弹性现象的数学模型，并对反馈控制器的设计进行分析研究--包括自应变作动器，颤振边界扩展的“智能控制器”。（包括无人机）是理论的主要领域，有一个新的新兴应用领域：替代能源，如风力涡轮机-其中叶片的气动弹性稳定是一个越来越重要的问题。较长的叶片显然更有效，但减轻重量意味着更高的灵活性，具体任务：第一年分析研究非零攻角和非零弯度对颤振速度作为马赫数函数的影响，特别是在跨音速范围内，这是一个目前尚未解决的问题。这意味着特别要研究非线性气动弹性方程的线性化形式，从而推广Possio积分方程。反过来，我们将需要研究相应的卷积演化方程的根轨迹问题，包括自应变作动器模型的性能--仍然使用Goland悬臂梁模型的机翼。具体任务：第二年将理论扩展到结构是均匀薄板而不是梁的地方，这样我们就可以考虑机翼前缘和/或后缘的控制。然后我们可以研究自应变致动器--压电条弦向和展向，消除副翼。对于大展弦比机翼，我们仍然可以使用典型截面空气动力学--这样我们开发的非线性空气动力学理论就可以在不需要额外努力的情况下使用。具体任务：第三年将气动弹性理论扩展到有限机翼，最终消除大展弦比约束。

Purpose and Intellectual MeritThe objective of this research is the systematic development of a mathematical theory forAeroelasticity drawing on Abstract Functional Analysis -- in particular nonlinear Aeroelasticity-- centered on the canonical problem of flutter: the instability endemic to aircraft at high enoughair speed in the subsonic range, M 1. In particular this would mean the development ofmathematical models for aeroelastic phenomena of interest on their own and also exploreanalytically the design of feedback controllers -- including self-straining actuators, "intelligentcontrollers" for Flutter Boundary Expansion.Broader ImpactWe should point out that while aircraft wings (including UAV's) are the primary domain of thetheory, there is a new emerging area of application: Alternate Sources of Energy such as WindTurbines -- where the aeroelastic stabilization of the blades is an increasingly important issue.Longer blades are apparently more efficient but reducing weight implies higher flexibility, just aswith UAV's.Specific Tasks: First YearAnalytical study of the effect of nonzero angle of attack and nonzero camber on flutter speed as afunction of Mach number, especially in the transonic range -- a currently unsettled problem. Thiswould mean in particular studying the linearized version of the nonlinear aeroelastic equation,leading to generalization of the Possio Integral Equation. In turn we will need to study the rootlocus problem for the corresponding convolution-evolution equation, including the performanceof self-straining actuator models -- still using the Goland cantilever beam model for the wing.Specific Tasks: Second YearExtend the theory to where the structure is a uniform thin plate rather than a beam so that we canconsider controls on the leading and/or trailing edge of the wing. We could then investigate self-strainingactuators -- piezo-strips chordwise as well as spanwise, eliminating ailerons. For ahigh-aspect-ratio wing we could still work with typical-section aerodynamics -- so that thenonlinear aerodynamic theory we have developed can be used without additional effort.Specific Tasks: Third YearExtend the aeroelastic theory to finite wings finally eliminating the high-aspect-ratio constraint.