Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow

Y P Zhang,J K Liu,Y M Chen,G Meng

doi:10.1155/2011/926271

Abstract

The homotopy analysis method (HAM) is employed to propose a highly accurate technique for solving strongly nonlinear aeroelastic systems of airfoils in subsonic flow. The frequencies and amplitudes of limit cycle oscillations (LCOs) arising in the considered systems are expanded as series of an embedding parameter. A series of algebraic equations are then derived, which determine the coefficients of the series. Importantly, all these equations are linear except the first one. Using some routine procedures to deduce these equations, an obstacle would arise in expanding some fractional functions as series in the embedding parameter. To this end, an approach is proposed for the expansion of fractional function. This provides us with a simple yet efficient iteration scheme to seek very‐high‐order approximations. Numerical examples show that the HAM solutions are obtained very precisely. At the same time, the CPU time needed can be significantly reduced by using the presented approach rather than by the usual procedure in expanding fractional functions.

Highlights

Predicting amplitude and frequency of flutter oscillations of an airfoil via analytical and/or semianalytical techniques has been an active area of research for many years
The incremental harmonic balance (IHB) method is a semianalytical method for nonlinear dynamic systems
Based on the homotopy analysis method (HAM), we have proposed an approach for obtaining highly accurate approximations for limit cycle oscillations (LCOs) of strongly nonlinear aeroelastic systems

Summary

Introduction

Predicting amplitude and frequency of flutter oscillations of an airfoil via analytical and/or semianalytical techniques has been an active area of research for many years. The incremental harmonic balance (IHB) method is a semianalytical method for nonlinear dynamic systems It was used by Shahrzad and Mahzoon [7] and Cai et al [8], respectively, to predict the amplitudes and frequencies of the LCOs of an airfoil in steady impressible flow. It is necessary to develop new easier-to-use methods which can guarantee accuracy for high flow speeds and in more flutter cases, for example, weakly and strongly nonlinear systems. The HAM is employed to propose an efficient and highly accurate approach for nonlinear aeroelastic motions of an airfoil. An approach is proposed to deal with this problem This simple yet efficient method ensures an excellent efficiency of the HAM; highly accurate solutions can be obtained for both weakly and strongly nonlinear aeroelastic systems

Equations of Motions

Homotopy Analysis Method

Expansion of Fractional Functions

Numerical Examples

Conclusions