Determination of Flutter Speed of 2D Nonlinear Wing by Using Describing Function Method and State-Space Formulation


40th Conference and Exposition on Structural Dynamics (IMAC), Florida, United States Of America, 7 - 10 February 2022, pp.101-107 identifier identifier

  • Publication Type: Conference Paper / Full Text
  • Doi Number: 10.1007/978-3-031-04086-3_16
  • City: Florida
  • Country: United States Of America
  • Page Numbers: pp.101-107
  • Keywords: Unsteady aerodynamics, Describing function method, Flutter speed, Nonlinear wings, Frequency domain
  • Middle East Technical University Affiliated: Yes


Flutter is a phenomenon that occurs in wings or platelike structures as a result of aerodynamical forces when a certain flow speed, i.e., flutter speed, is reached. Flutter results in severe vibrations which eventually leads to fatigue failure of the wing. Many solutions are suggested against flutter phenomena. Wings or platelike structures under the effect of flowing air may contain nonlinearities due to connections or materials used. In this paper, effect of different structural nonlinear elements on the flutter speed is studied by using a 2D wing model. Aerodynamic lift and moment acting on the airfoil is obtained by utilizing Theodorsen's unsteady aerodynamics which is only applicable to subsonic flow. In this paper, modified Theodorsen model for a 2D wing is used. To solve the flutter equation, several methods are suggested in the literature. Methods like k method and p-k method assume harmonic vibration in the generalized coordinates resulting in an eigenvalue problem. These methods are applied to linear systems. When nonlinearities are present in the system, numerical time marching solutions to differential equations are used; however, they are very costly in terms of computational time. In this study, state-space approach is utilized to obtain the flutter speed in frequency domain by using describing function method (DFM). The nonlinear system of differential equations is converted into a nonlinear eigenvalue problem utilizing state-space approach from which the flutter speed resulting in unstable solutions is obtained. Nonlinear eigenvalue problem obtained can be solved iteratively without time marching methods. This method of finding flutter speed is computationally much faster than solving nonlinear flutter problems in time domain. Free play nonlinearity is a frequently observed nonlinearity where there exists a gap at both sides of the wing after which it is restricted by stiffnesses. Piecewise linear stiffness is a symmetric nonlinearity similar to free play where finite stiffness exists instead of a zero stiffness in between the gap. Softening cubic stiffness is a nonlinearity where stiffness of the structure decreases as the amplitude of the vibration increases. In this study, free play (gap nonlinearity), piecewise linear stiffness, and cubic stiffness nonlinearities acting on the rotational degree of freedom are considered in the case studies. Results obtained for these nonlinearities are presented and compared with each other.