A 2D Flutter Equation Transformation Using Chebyshev Expansion Method

The classical two-dimensional airfoil flutter equations can be established by many ways. For the simplicity of solving it, a sinusoidal structure motion hypothesis and some kind of aerodynamic theory must be proposed in advance. However, when a wing flutter occurs, its structural movement is likely to be more complex. Furthermore, the well-known harmonic balance method may not sufficiently accurate due to those higher order terms are ignored, which might lead to larger errors. In this paper, we propose a parametric method such that the original equations are parameterized, namely Chebyshev expansion method. A simple example is used to illustrate our strategy. Introduction As is known to all, it’s easy to induce the continuous or divergent vibration form when the elastic structure of the aircraft in uniform flow is impacted coupling by the air force, elastic force and inertial force. This phenomenon is called “flutter”, and it is one of the most important questions in the pneumatic elastic mechanics [1]. In recent years, with the development of computer hardware and software technologies, the coupled flutter computation of the research based on the Computational Fluid Dynamic (CFD) and the Computational Structure Dynamics (CSD) began to prevail. The flutter calculation and research based on the two-dimensional airfoil can be divided into categories, the qualitative and the quantitative, which is similar to the three-dimensional airfoil flutter problem on the mechanism [2]. The former makes research on the stability of the system, and the latter focuses on the flutter amplitude, frequency, phase and so on. For a 2D wing prediction of flutter, the structural motion was scribed in a sine function in the pass, and the aerodynamics model established by the aid of Theodorsen unsteady aerodynamic force theory. Calculation results can be used in the primitive engineering practice [3]. Based on the sine motion hypothesis and the common harmonic balance method, the accuracy of quantitative calculation may not be very high [4], namely large errors may occur [5] because of the standard harmonic balance method ignoring the higher frequencies. Other works of modeling an aeroelasticity system or calculating air forces can be found in references [6,7] behind. This paper proposes a parametric approach to transform the original flutter equations: using Chebyshev expansion method. The method doesn’t only limit the form of structure movement, but also it can be applied to those nonlinear aeroelastic problems. Finally, one can obtain sufficiently precise result when solving the new equations expressed by Chebyshev series. Properties of the First Chebyshev Polynomial Since Chebyshev polynomial is put forward, it is widely used in academic fields such as in system analysis, parameter identification, optimal control, model reduction and so on[8]. In the field of aeronautics and astronautics, for example, the model identification problem, we can effectively improve the accuracy and efficiency by using the good properties of Chebyshev orthogonal basis functions. The first Chebyshev polynomial n T which are defined in [0,1] satisfy orthogonal relationship according to weight function 2 / 1 )] 1 ( [ ) (    t t t w . International Conference on Modeling, Analysis, Simulation Technologies and Applications (MASTA 2019) Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Advances in Intelligent Systems Research, volume 168