Nonlinear Response Analysis of an Airfoil Using Multistep Differential Transform Method

The analysis of limit cycle oscillations (LCOs) and flutter behavior in hypersonic aeroelastic systems presents significant challenges owing to their strong nonlinearities. This study demonstrates the applicability of the Multistep Differential Transform Method (MSDTM) as an efficient semi-numerical approach for modeling and characterizing LCOs. A two-degree-of-freedom airfoil model is considered, with nonlinear torsional stiffness and aerodynamic loading modeled using third-order piston theory. The analysis spans flow regimes from Mach 5 to Mach 20, where both structural and aerodynamic nonlinearities significantly influence dynamic behavior. The MSDTM is systematically formulated and applied to derive aeroelastic responses without discretization or linearization, preserving the inherent nonlinear characteristics of the system. The results show that MSDTM effectively captures the onset of flutter and the transition from supercritical to subcritical bifurcation with increasing Mach number. The impact of initial perturbations on flutter speed and the sensitivity of LCO behavior to varying structural nonlinearity parameters are thoroughly examined. It is observed that, at higher Mach numbers, reducing the structural nonlinearity factor leads to subcritical bifurcation, indicating increased susceptibility to dynamic instability. The study emphasizes the necessity of including aerodynamic nonlinearities in hypersonic flutter analysis to ensure accurate prediction of system stability and avoid catastrophic failures. MSDTM proves to be a simple, yet robust tool for nonlinear aeroelastic analysis, offering high accuracy without compromising computational efficiency.