Nonlinear dynamical behaviors of deploying wings in subsonic air flow

In recent years, the morphing wing, which is able to improve the lift-drag characteristics of the aircraft by changing wingspans and aspect ratios, has become a popular research issue. However, aeroelasticity is still a bottleneck for the development of morphing wings. The development of the aerodynamic calculations of deploying wings in subsonic air flow will be further promoted by this research. In this paper, we consider the aerodynamics of a deploying wing and investigate the nonlinear dynamic behaviors of the deploying wing. Under the flow condition of ideal incompressible fluid, the flow field is a potential field and the potential function must satisfy the Laplace linear equation and the superposition principle. When the thickness, curvature and angle of attack of the airfoil are small, the thin airfoil theory can be used to calculate the effects of the mean camber line to obtain the circulation distribution of the deploying wings in subsonic air flow. The steady aerodynamic lift on the deploying wing is derived by using the Kutta-Joukowski lift theory. Then, the aerodynamic lift is applied on a deploying wing, which is modeled as a cantilevered thin shell deploying in the axial direction. The nonlinear partial differential governing equations of motion for the deploying cantilevered thin shell subjected to the aerodynamic force in subsonic air flow are established based on Hamilton's principle. The time-varying dependent vibration mode-shape functions are chosen using the boundary conditions, and then the Galerkin method is employed to transform the partial differential equation into two time-varying nonlinear ordinary differential equations. Numerical simulations are performed for the nonlinear dynamic responses of the deploying wing subjected to the aerodynamic force, and then the influence of different parameters, including the extending velocity and disturbance velocity, on the stability of the wing are analyzed. The effects of deploying velocities on the nonlinear vibrations of the first-order and second-order modes for the deploying wing are studied.