An instability endemic to aircraft, resulting in possible catastrophic destruction, is 'Wing Flutter', which sets a limit to attainable speed at any altitude known as the Flutter Boundary. The FAA mandates a 15% Flutter safety margin. Our primary objective is to study control design for Flutter Boundary Expansion based on mathematical models not available hitherto because of the level of mathematics required.
Intellectual Merit:
The precise dependence of the Flutter Boundary on the altitude is naturally of great interest While the flutter speed generally decreases with altitude, it was recently discovered in computation that this need not be the case if the 'angle of attack' (the angle made by the aircraft velocity (vector) with the wing) is taken into account and can in fact start to increase at a high enough altitude, known as the 'Transonic Dip' which is taken advantage for instance in the design of the Air Bus 380. The analytical theory we are developing is able to predict this phenomenon. Such a theory is essential for studying a novel control concept of current interest: Morphing -- changing the shape of the wing dynamically.
At normal altitudes the air-friction (viscosity) is small enough to be neglected, so that the aerodynamic flow model is much simpler. But the effect of viscosity is little understood and one of our objectives is to extend the theory to include viscosity and explore phenomena unique to it.
Broader Impact:
Although flutter phenomena were observed as early as the 1930's (by stunt pilots), many basic questions still remain unanswered. Remarkable progress has been made in numerical computational schemes, but these require approximations for specific parameters, provide little insight into phenomena, and above all are inadequate for Control Design. The PI plans to educate students and transition the research results to aerospace companies for the design of high performance aircrafts.
P I A Balakrishnan U C L A The subject matter of this research is Aeroelastic Flutter –the mathematical theory. Aeroelasticity is concerned with fluid-structure interaction and combines Aerodynamics with Structure Dynamics without being either. The main area of interest is Flutter-an endemic instability that occurs at high enough speed that can potentially cause catastrophic destruction of the wing in fixed wing aircraft where the wind flow is in plane normal to the wing.This was noticed in the 1930’s by the stunt fliers. As a safety issue, the FAA mandates a 15% ‘flutter margin’. . The determination of the Flutter speed is thus an essential activity at every Flight Center. Flight Tests are most expensive, mathematical analysis is the least expensive. Currently the work is largely computational—CFD for the aerodynamics and FEM for the structure. The aeroelastician usually begins with a pde description but discretises it immediately for digital computer calculation,leaving the nature of the approximation vague. The feature of this work is that we stay with the continuum model throughout until the very end- before computation. The continuum approach offers many advantages besides potential higher accuracy-it can answer ‘What If?’ questions which numerical codes cannot. It can provide a precise definition of ‘Flutter Speed’ for example . By providing generality , it can help develop intuition—as in any scientific discipline. It can provide simple closed form solutions . But the use of continuum models comes with a price—higher level of mathematics and most Recently developed mathematics –abstract Functional Analysis/Semigroup Theory -often beyond the scope of current training in Aeroelasticity. ACCOMPLISHMENTS We show for example that a precise statement of the problem requires Hilbert Space theory-that the description of aeroelastic dynamics takes the form of a nonlinear convolution-evolution equation in a Hilbert Space. Using this we are able to show that Flutter is an LCO-Limit Cycle Oscillation consistent with the Hopf Bifurcation Theory—where the Flutter speed is determined by the linearised model—linearised about the steady state solution and so is the period of the LCO. Determination of the amplitude is still an open problem. The analysis shows only odd harmonics. We are also able to provide a digital computer algorithm based only on the continuum model. We take an Input-Output point of view—the structure motion is the input and the pressure jump across the wing structure is the output. This is embodied by the Possio Integral Equation – a singular equation which plays a key role in the theory, and the solution allows us to compute the aerodynamic loading –Force and Moment. The structure model is the uniform beam model of Goland with 2 modes –bending along the beam axis and torsion about the elastic axis. For the aerodynamics we consider first inviscid flow,characterized by the Euler Full Potential Equation . The fluid-structure boundary conditions are the Flow Tangency and the Kutta Joukowsky . For the Typical Section case where the flow along the beam axis is uniform,so that it is 2D,we are able to obtain a series solution bootstrapping on the Linear which shows that the flow can be decomposed ,one in which there may be shocks but develops no lift on the wing,but the other which no shocks and provides the lift. Another important finding is that flutter suppression-increasing the flutter speed –control need only be linear and can be based solely on the linearised model. Publications: Book: Aeroelasticty:The Continuum Theory-Springer-August 2010. 400 p. Students:Graduate: Iylene Patino Oleg Melnikov PostDoc: Amjad Tuffaha
