Novel methodology of a moving cart on a horizontal plane coupled with an inclined damped nonlinear spring

This work introduces a comprehensive framework in analysing nonlinear dynamics of a moving cart with an inclined smooth surface on a horizontal plane featuring a damped nonlinear spring. It integrates nonlinear elastic behaviour and damping effects to depict mechanical engineering systems. The study aims to enhance the comprehension of intricate dynamic reactions affected by gravity, nonlinear stiffness, and energy dissipation, which are relevant to vibration isolation and mechanical transport. The issue involves a two-degrees-of-freedom of dynamical model with parametric excitation. The methodology employs He’s frequency formula that implemented via a non-perturbative approach. This method aims to convert ordinary differential equations that are nonlinear into linear ones. To confirm the significant accuracy between nonlinear and linear/nonlinear ODEs, Mathematica Software was adopted. The distinctive methodology is defined by its simplicity, attractiveness, potential, and efficiency, rendering it beneficial for highly nonlinear ordinary differential equations. It can be employed in different classes’ fields of dynamical systems as well as fluid mechanics. The stability analysis illustrates several non-dimensional physical concerns. The system’s erratic behaviour is analysed to understand its dynamics and transition from a periodic to a chaotic state. This organization utilizes bifurcation diagrams and the largest Lyapunov exponent which communally provide a comprehensive depiction of the system’s behaviour across several stages. Moreover, the system’s complexity landscape is visualized through 2D chaos maps based on Kaplan–Yorke dimension, pinpointing the exact parameter shifts where simple order collapses into high-dimensional fractal chaos.