Mathematical Problems Arising in Aircraft Modeling

Texas Tech University, Lubbock TX

Investigators

Abstract

0072247 Shubov The 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.