A comparison of classical Runge-Kutta and Henon's methods for capturing chaos and chaotic transients in an aeroelastic system with freeplay nonlinearity

A typical two-dimensional airfoil with freeplay nonlinearity in pitch undergoing subsonic flow is studied via numerical integration methods. Due to the existence of the discontinuous nonlinearity, the classical fourth-order Runge-Kutta (RK4) method cannot capture the aeroelastic response accurately. Particularly, it is because the RK4 method is incapable of detecting the discontinuous points of the freeplay that leads to the numerical instability and inaccuracy. To resolve this problem, the RK4 method is used with the aid of the Henon's method (referred to as the RK4Henon method) to precisely predict the freeplay's switching points. The comparison of the classical RK4 and the RK4Henon methods is carried out in the analyses of periodic motions, chaos, and long-lived chaotic transients. Numerical simulations demonstrate the advantages of the RK4Henon method over the classical RK4 method, especially for the analyses of chaos and chaotic transients. Another existing method to deal with the freeplay nonlinearity is to use an appropriate rational polynomial (RP) to approximate this discontinuous nonlinearity. Consequently, the discontinuity is removed. However, it is demonstrated that the RP approximation method is unable to capture the chaotic transients. In addition, an efficient tool for predicting the existence of chaotic transients is proposed by means of the evolution curve of the largest Lyapunov exponent. Finally, the effects of system parameters on the aeroelastic response are investigated.