104 ˙Izzet Ulug˘ and Ender Cigeroglu expensive, especially for systems with nonlinearities like free-play. As shown by Conner et al. [3] and Trickey et al. [5], solving nonlinear equations in the time domain often requires significant computational resources due to the complexity of the simulations. In contrast, frequency-domain methods, such as the DFM, are more efficient for analyzing systems with periodic responses, providing accurate predictions of flutter behavior. This method linearizes the nonlinear system around limit cycles, allowing for the calculation of stability boundaries and LCOs without extensive simulations. Studies by Dowell and Tang [1] and Yurkovich [6] have demonstrated the advantages of frequency-domain approaches in handling nonlinear aeroelastic problems, offering accurate predictions of flutter dynamics. In a recent study, Frey et al. [7] compared timedomain and frequency-domain methods on turbine blades with nonlinear unsteady flow conditions in case, and demonstrated that frequency-domain solvers not only reduce the computational cost but also effectively model aeroelastic instabilities, especially under transonic flow conditions where strong shock interactions impact system dynamics. The flutter characteristics of an airfoil with free-play nonlinearity at the control surface are explored using a three-degreeof-freedom (3-DOF) model. Free-play nonlinearity is represented by a piecewise linear stiffness element and the behavior of the system is studied in the frequency domain. This approach effectively predicts the flutter boundary and the onset of LCOs, providing a clearer understanding of how free-play affects system stability and post-flutter dynamics. Mathematical Modelling The 3-DOF airfoil model used in this study is given in Fig. 1. The airfoil exhibits three degrees of freedom: vertical motion (plunge), rotational motion around a horizontal axis (pitch), and control surface motion (flap), denoted by h, α, and β, respectively. The moments of inertia about the elastic axis for the airfoil and control surface are represented by Iα and Iβ, respectively. Nonlinearity is introduced through torsional free-play at the control surface hinge, as previously studied and tested by Conner et al. [2]. The numerical parameters used in this paper were also adopted from Conner et al.’s study. Simple models, like the one shown in Fig. 1, are commonly used in similar analyses for their practicality. As shown in Fig. 1, the model represents a two-dimensional airfoil placed in a steady horizontal airflow. Key parameters include the freestream velocity U, the semi-chord length b, stiffnesses in pitch, plunge, and control surface hinge directions (kα, kh, kβ), as well as the mass of the wing mw, and the total mass of the systemmT. Additionally, the damping coefficients for the pitch, plunge, and flap motions are denoted as cα, ch and cβ, respectively, which describe the energy dissipation in each mode of motion. The structural behavior is governed by linear bending and torsional springs along the airfoil’s elastic axis and control surface hinge line, which simulate the restoring forces from the structure. Additionally, the distances between key points in the system —such as from elastic axis to the mid-chord a the mid-chord to the hinge line c, the elastic axis to the center of mass of the airfoil xα, and the hinge line to the center of mass of the control surface xβ—are also considered. Finally, the total unsteady aerodynamic forces in the vertical direction, as well as the aerodynamic moments about the elastic axis and flap hinge, are represented byL, Mα and Mβ, respectively. Fig. 1 Schematic of a 3-DOF aeroelastic wing section with pitch, plunge, and control surface spring restraints.
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