##
**Equivalent linearization method for the flutter system of an airfoil with multiple nonlinearities.**
*(English)*
Zbl 1426.74125

Summary: The equivalent linearization method was extended to analyze the flutter system of an airfoil with multiple nonlinearities. By replacing the cubic plunging and pitching stiffnesses by equivalent quantities, linearized equations for the nonlinear system were deduced. According to the linearized equations, approximate solutions for limit cycle oscillations (LCOs) were obtained in good agreement with numerical results. The influences of the linear and cubic stiffnesses on LCOs were analyzed in detail. Reducing linear pitching stiffness leads to decreasing of the critical flutter speed. For linear plunging stiffness, the opposite is true. Also, it reveals that the bifurcation could be supercritical or subcritical, which is related to the ratio between the coefficient of cubic pitching stiffness and that of plunging one.

### MSC:

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |

70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |

### Keywords:

cubic stiffness; limit cycle oscillation; subcritical bifurcation; supercritical bifurcation
PDF
BibTeX
XML
Cite

\textit{F. X. Chen} et al., Commun. Nonlinear Sci. Numer. Simul. 17, No. 12, 4529--4535 (2012; Zbl 1426.74125)

Full Text:
DOI

### References:

[1] | Lee, B.H.K.; Price, S.J.; Wong, Y.S., Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos, Prog aerosp sci, 35, 205-334, (1999) |

[2] | Gao, C.; Luo, S.J.; Liu, F., Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions, Acta mech sinica-prc, 19, 427-436, (2003) |

[3] | Ding, Q.; Cooper, J.E.; Leung, A.Y.T., Application of an improved cell mapping method to bilinear stiffness aeroelastic systems, J fluid struct, 20, 35-49, (2005) |

[4] | Lan, S.L.; Sun, M., Aerodynamic force and flow structures of two airfoils in flapping motions, Acta mech sinica-prc, 17, 310-331, (2001) |

[5] | Yang, Y.R., KBM method of analyzing limit cycle flutter of a wing with an external store and comparison with a wind-tunnel test, J sound vib, 187, 171-181, (1995) |

[6] | Lee, B.H.K.; Liu, L., Bifurcation analysis of airfoil in subsonic flow with coupled cubic restoring forces, J aircraft, 43, 652-659, (2006) |

[7] | Awrejcewicz, J., Bifurcation and chaos in coupled oscillators, (1991), World Scientific Singapore · Zbl 0824.58034 |

[8] | Lee, B.H.K.; Gong, L.; Wong, Y.S., Analysis and computation of nonlinear dynamic response of a two-degree-of-freedom system and its application in aeroelasticity, J fluid struct, 11, 225-246, (1997) |

[9] | Awrejcewicz, J., Numerical investigations of the constant and periodic motions of the human vocal cords including stability and bifurcation phenomena, Dynam stabil syst, 5, 11-28, (1990) · Zbl 0694.34032 |

[10] | Liu, L.P.; Dowell, E.H., The secondary bifurcation of an aeroelastic airfoil motion: effect of high harmonics, Nonlinear dynam, 37, 31-49, (2004) · Zbl 1078.74016 |

[11] | Awrejcewicz, J.; Krysko, V.A., Introduction to asymptotic methods, (2006), Chapman & Hall/CRC London · Zbl 1110.34001 |

[12] | Andrianov, I.V.; Awrejcewicz, J.; Barantsev, R.G., Asymptotic approaches in mechanics: new parameters and procedures, Appl mech rev, 56, 87-110, (2003) |

[13] | Chung, K.W.; Chan, C.L.; Lee, B.H.K., Bifurcation analysis of a two-degree-of-freedom aeroelastic system with freeplay structural nonlinearity by a perturbation-incremental method, J sound vib, 299, 520-539, (2007) |

[14] | Chen, Y.M.; Liu, J.K., Homotopy analysis method for limit cycle flutter of airfoils, Appl math comput, 203, 854-863, (2008) · Zbl 1262.74015 |

[15] | Mickens, R.E., A combined equivalent linearization and averaging perturbation method for non-linear oscillator equations, J sound vib, 264, 1195-1200, (2003) · Zbl 1236.34053 |

[16] | Liu, J.K.; Zhao, L.C., Bifurcation analysis of airfoils in incompressible flow, J sound vib, 154, 117-124, (1992) · Zbl 0922.76215 |

[17] | Shahrzad, P.; Mahzoon, M., Limit cycle flutter of airfoils in steady and unsteady flows, J sound vib, 256, 213-225, (2002) |

[18] | Lim, C.W.; Wu, B.S., A new analytical approach to the Duffing-harmonic oscillator, Phys lett A, 311, 365-373, (2003) · Zbl 1055.70009 |

[19] | Chen, Y.M.; Liu, J.K., On the limit cycles of aeroelastic systems with quadratic nonlinearities, Struct eng mech, 30, 67-76, (2008) |

[20] | Cai, M.; Liu, J.K.; Li, J., Incremental harmonic balance method for airfoil flutter with multiple strong nonlinearities, Appl math mech-engl, 27, 953-958, (2006) · Zbl 1137.70359 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.