The landing gear system plays a critical role in maintaining aircraft stability during taxiing, takeoff, and landing. Traditional approaches to estimating landing impact forces often suffer from inaccuracies, stemming from either overly simplistic assumptions or the lack of detailed structural data during early design stages. To address this, we develop a four-degree-of-freedom nonlinear dynamic model of an aircraft’s landing system, incorporating both main and nose landing gears modeled as spring-damper assemblies with nonlinear tire stiffness. Two common landing scenarios, three-point and two-point landings, are simulated using numerical methods based on the fourth-order Runge–Kutta algorithm. The system’s behavior was analyzed under varying stiffnesses and initial landing velocities using phase space diagrams, Poincaré maps, and fast Fourier transforms to identify dominant frequencies and potential chaotic responses. The findings underscore the significant impact of tire stiffness on vibration amplitude and the smoothness of landing. High stiffness (3 × 108 N/m) accelerates system stabilization but causes excessive force peaks, while low stiffness (3 × 104 N/m) leads to persistent oscillations and limit cycles. This model enables efficient prediction of landing gear responses and offers insight into optimal parameter selection for safe and comfortable aircraft landings.