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Flutter control and mitigation of limit cycle oscillations in aircraft wings using distributed vibration absorbers

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Abstract

In this work, we demonstrate the application of the conserved-mass metamaterial concept to control the flutter onset in aircraft wings and mitigate their induced vibrations. The numerical study is conducted on a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. The aeroelastic system is attached to an array of passive vibration absorbers. The mass of the vibration absorbers is deducted from the mass of the aeroelastic structure, which makes the total mass of the system conserved. The equal mass constraint is imposed so that improvements in the aeroelastic performance are due to the addition of the vibration absorbers and not the added mass attached to the main aeroelastic system. The linear analysis of the aeroelastic system reveals that a proper selection of the stiffness and position of the vibration absorber leads to 23.4% increase in the flutter speed when deploying one single absorber. Placing the vibration absorber closer to the leading edge is observed to delay the occurrence of flutter. Furthermore, considering an array of distributed vibration absorbers with a proper selection of their stiffness is found to result in an increase of the flutter speed by 84%. We also develop the normal form of the aeroelastic system equipped with the vibration absorber using the method of multiple scales to investigate the limit cycle oscillations (LCOs) beyond the flutter boundary (Hopf bifurcation). The analytical results are verified against their numerical counterparts. The normal form is used to examine the effects of the vibration absorber stiffness nonlinearities on the dynamic behavior of the aeroelastic system. Incorporating quadratic and cubic stiffness coefficients is found to degrade the aeroelastic system by amplifying the amplitude of LCOs beyond flutter. However, soft pitch spring proved to be beneficial to decrease the LCO amplitude. Moreover, the increase in the number of the vibration absorbers while imposing the equal mass constraint is observed to lead to relatively lower LCO amplitudes in the post-flutter regime.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The author M. Ghommem gratefully acknowledges the financial support via the American University of Sharjah Faculty research grant FRG19-M-E26 (fund number EN6001).

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  1. Department of Mechanical Engineering, American University of Sharjah, Sharjah, 26666, UAE

    Ehab Basta, Mehdi Ghommem & Samir Emam

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  1. Ehab Basta
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Appendix

Appendix

The notations presented in Eq. (17) are expressed as:

\(B_{1} = - 2U_{f}^{2} \left( {k_{1} I_{1} + k_{2} I_{2} } \right)\)

\(B_{2} = \overline{{k_{h0} }} \left( {c_{1} I_{3} + c_{2} I_{4} } \right)\)

\(B_{3} = \overline{{k_{\alpha 0} }} \left( {c_{5} I_{5} + c_{6} I_{6} } \right)\)

\(B_{4} = \overline{{k_{ab0} }} \left( {c_{6} I_{7} + c_{7} I_{8} + c_{8} I_{9} } \right)\)

\(I_{1} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{2} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{3} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{4} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{5} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{6} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{7} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{8} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right)\)

\(I_{9} = \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ \end{array} } \right)\)

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Basta, E., Ghommem, M. & Emam, S. Flutter control and mitigation of limit cycle oscillations in aircraft wings using distributed vibration absorbers. Nonlinear Dyn 106, 1975–2003 (2021). https://doi.org/10.1007/s11071-021-06889-z

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