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Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

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Abstract

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.

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Abbreviations

h :

plunge displacement of airfoil

α :

pitch displacement of airfoil

t :

non-dimensional time

Q :

generalized air speed, bifurcation parameter

Q f :

flutter critical point

Q(k 0):

bifurcation point of Eq. (1) corresponding to k 0

ɛ :

the parameter defined as ɛ = Q − Q f

κ 0 :

linear pitching stiffness

e 2 :

coefficient of nonlinear pitching stiffness

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Authors and Affiliations

  1. Department of Mechanics, Sun Yat-sen University, Guangzhou, 510275, P. R. China

    Chen Yan-mao  (陈衍茂) & Liu Ji-ke  (刘济科)

Authors
  1. Chen Yan-mao  (陈衍茂)
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  2. Liu Ji-ke  (刘济科)
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Corresponding author

Correspondence to Liu Ji-ke  (刘济科).

Additional information

Communicated by WANG Biao

Project supported by the National Natural Science Foundation of China (No. 10772202), the Doctoral Foundation of Ministry of Education of China (No. 20050558032), and the Natural Science Foundation of Guangdong Province (Nos. 07003680 and 05003295)

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Chen, Ym., Liu, Jk. Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems. Appl. Math. Mech.-Engl. Ed. 29, 199–206 (2008). https://doi.org/10.1007/s10483-008-0207-x

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  • DOI: https://doi.org/10.1007/s10483-008-0207-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

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