exploring forced spring-pendulum dynamics: from mathematical analysis to chaotic behavior

This article examines the vibrational movement of a system with two degrees-of-freedom (2dof). it is represented by a spring pendulum (sp) connected to an oscillating support in the presence of a moment and external harmonic forces. the system's governing equations are formulated using lagrange's equations (le) and solved analytically using the multiple-scales perturbation approach (mspa) up to the third approximation. a systematic classification of resonance scenarios is provided, and the modulation equations are established following the solvability conditions (sc). a stability analysis of steady-state solutions is conducted based on the routh-hurwitz criteria (rhc). a comparison has been made between the obtained analytical and numerical solutions (ns) derived by the fourth-order runge-kutta (rk4) methodology, demonstrating their remarkable concordance. the time-dependent analytic solutions (as) modified amplitudes and phases, the curves of frequency response (cfr), and the regions of stability are graphed and analyzed to demonstrate the positive effect of different system parameters. the bifurcation diagrams, the corresponding poincaré maps (pm), and the phase portraits are presented to demonstrate various kinds of the system's motion. the largest lyapunov exponents (lle) are calculated and give good agreement with bifurcation diagrams. the proposed sp framework can be extended to practical systems such as offshore platforms, robotic manipulators, and aerospace structures where coupled oscillations play a critical role. the novelty of this research lies in offering a comprehensive analytical–numerical treatment that not only captures the system's complex dynamics but also establishes a foundation for optimizing real-world engineering designs.