^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

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 describing function technique [

The approximations obtained by HB1 method are relatively accurate for low wind speeds. However, the errors become larger and larger as the speed increases. In some nonlinear flutter cases, the HB1 method may cease to be valid. In principle, the high-dimensional HB method and the IHB method can give approximate solutions with any desired accuracy as long as enough harmonics are taken into account. Unfortunately, however, it becomes more and more difficult to implement either one of them when the number of considered harmonics increases. Likewise, using the center manifold theory can provide us with satisfactory approximations for the LCOs only in a small range of bifurcation values. When far away from the bifurcation points, results loose accuracy significantly or even become completely incorrect [

Over the past decades, Liao developed the homotopy analysis method (HAM), which does not require small parameters and thus can be applied to solve nonlinear problems without small or large parameters [

In this study, the HAM is employed to propose an efficient and highly accurate approach for nonlinear aeroelastic motions of an airfoil. A major obstacle is met when deducing the high-order deformation equations, because the Taylor expansion of fractal functions is rather cumbersome. An approach is proposed to deal with this problem. This simple yet efficient method ensures an excellent efficiency of the HAM; hence, highly accurate solutions can be easily obtained for both weakly and strongly nonlinear aeroelastic systems.

The physical model shown in Figure

Sketch of a two-dimensional airfoil.

For cubic restoring forces with subsonic aerodynamics, the coupled equations for the airfoil in nondimensional form can be written as follows:

Due to the existence of the integral terms in (

By introducing a variable vector ^{T}” denotes the transpose of a matrix, with

It is assumed that there is no external forces, that is,

Firstly, introduce a new time scale

Let

The homotopy analysis method is based on such continuous variations,

Based on (

When

Expand

Substituting (

Due to the rule of solution expression and the linear operator

Note that when

A key procedure of implementing the HAM is to deduce the high-order deformation equation, that is, to obtain

We take

The system parameters under consideration are

Numerical solutions of (

Using analytical techniques developed for nonlinear dynamical systems, the linear flutter speed is found at

Comparisons of the 50th-order HAM solutions for LCO amplitudes with HB1 results and numerical ones. Dots denote the HAM solutions obtained with

Comparisons of the 50th-order HAM solutions for LCO frequencies with HB1 results and numerical ones. Dots denote the HAM solutions obtained with

In the proposed method, the zeroth-order HAM approximation is essentially given by the HB1 method. The higher-order approximations only contribute a higher precision. Thus, it is not capable of detecting the second bifurcation at the present state. Even so, validity and high efficiency of the proposed method can be observed when

The HAM approximation is based on the first HB method, because the first HAM approximation is the HB1 solution. Since the HB1 method is incapable of tracking the LCOs when

Figures

The phase planes of LCOs of system (

The time history responses of system (

More precisely, the 120th-order HAM solutions shown in Table ^{−8}, 10^{−12}, and 10^{−16}, respectively. Furthermore, as ^{−16}. Roughly speaking, the 120th-order HAM solution can be considered to be correct to 15 decimal places. Note that it is tough to obtain such a highly accurate solution using some numerical techniques, including the RK method.

Comparisons of the amplitudes and frequencies obtained by the HAM (

HAM | Frequency | Pitch | Plunge | Frequency | Pitch | Plunge |

0.077563656128 | 0.13738145786 | 0.356858 | 0.0658609 | 0.2184836 | 0.6945022 | |

0.077563606476 | 0.13738151172 | 0.35685814 | 0.0657867 | 0.2185646 | 0.6964279 | |

0.07756360647090 | 0.13738151173 | 0.35685815 | 0.0657833 | 0.2185685 | 0.6965209 | |

Numerical | 0.0775635 | 0.1373816 | 0.3568590 | 0.0657829 | 0.2185689 | 0.6965298 |

Residues of (

Very interestingly, it is found that

The HAM series are dependent upon the auxiliary parameter

The

In order to achieve faster convergence of HAM series, currently, researchers introduced some optimal approaches and developed the optimal approaches [

Comparisons of the amplitudes and frequencies given by the HAM Pade approximations (

[ | Frequency | Relative error (%) | Pitch | Relative error (%) |

0.07765100977663 | 0.11 | 0.13745051052294 | 0.05 | |

HAM 10 | 0.07770384628505 | 0.18 | 0.13726535168074 | −0.08 |

0.07756746357388 | 5 | 0.13737312674052 | −6 | |

15 | 0.07759388593952 | 4 | 0.13735291378024 | −2 |

0.07756394500764 | 4 | 0.13738142730826 | −6 | |

HAM 20 | 0.07757112740482 | 1 | 0.13737399877222 | −5 |

HAM | 0.07756360647090 | 0.13738151173287 |

Comparisons of the amplitudes and frequencies given by the HAM Pade approximations (

[ | Frequency | Pitch |

0.067011196 | 0.221689949 | |

HAM 10 | 0.0730367 | 0.156371061 |

0.06595560 | 0.21807163 | |

16 | 0.5207506 | 3.48085267 |

0.0658774 | 0.21854332 | |

HAM 20 | 7.63500 | −28.477895 |

HAM | 0.0657829 | 0.2185689 |

Next, we will discuss why it is necessary and worthwhile to employ the approach for series expansion of fractional function, as shown in Section _{n}_{n}

The CPU time needed in seeking the

The | The CPU time needed (second) | |

The routine procedure | The presented approach | |

10 | 14 | 2.8 |

15 | 58 | 4.6 |

20 | 196 | 6.8 |

50 | / | 46 |

80 | / | 239 |

120 | / | 1393 |

The ratio between the CPU times _{n}_{n}

Based on the HAM, we have proposed an approach for obtaining highly accurate approximations for LCOs of strongly nonlinear aeroelastic systems. An easy-to-use approach is proposed to tackle the difficulty in expanding fractional functions into the Taylor series. With the help of this approach, the HAM approximations can be obtained to a very high order and hence can provide solutions to any desired accuracy. The attained HAM solutions are almost the same as the numerical results. Since it is tough to achieve solutions to such high precision, even via the numerical solutions, thus our approaches can be used to validate other solution methods. Also, numerical examples demonstrate that the presented approaches are valid for both weakly and strongly nonlinear aeroelastic systems. These imply that the presented approaches could be applicable in more nonlinear problems, especially those with fractional functions.

As mentioned above, the first HAM approximation is in essence the HB1 solution. Note also that the HB1 method is incapable of obtaining the LCO solutions, when

In addition, both Figure

As for the proposed approach for expanding fractional functions, the problem about the robustness of the calculation should be paid special attention in practicing. For example, the coefficient matrix of

This work is supported by the National Natural Science Foundation of China (10772202, 10102023, 10972241, 11032005), Doctoral Program Foundation of Ministry of Education of China (20090171110042), and Educational Commission of Guangdong Province of China (34310018).