Dynamical analysis of a forced vibrating planar motion of a spring pendulum

This work studies the nonlinear movement of a two degrees-of-freedom (DOF) spring pendulum that is dampened and affected by a harmonic force externally. It is presumed that the spring’s pivot point travels along an elliptic route. Lagrange’s equations are utilized to generate the regulating system of motion. The multiple-scales approach (MSA) is used to gain the system’s analytic solutions up to the third-order approximation. Therefore, all resonance cases that have emerged are categorized, wherein two of them are scrutinized at once. As a result of the removal of secular terms, the solvability constraints are attained and then the steady-state solutions are investigated. The examined motion’s temporal evolution, the resonance response curves, and the solutions at the steady-state are all depicted graphically. In compliance with the Routh–Hurwitz criteria (RHC), all possible fixed points (FPs) for the steady and unsteady cases are found and displayed. The stability zones are examined and analyzed to estimate the effect of various factors on the system’s behavior. This model has gained prominence recently due to its industrial uses in seismic isolation systems for buildings and structures. However, in seismic engineering, a 2DOF vibrating pendulum system can be used as part of a seismic isolation system designed to protect buildings and infrastructure from earthquake-induced vibrations. The pendulum mechanism helps to absorb and dissipate seismic energy, reducing the amount of force transmitted to the structure. During an earthquake, the ground motion acts as an external harmonic force on the building. The distribution of mass and the structural layout can cause rotational moments that act on the building. The pendulum system can be tuned to counteract these moments, helping to stabilize the structure. The pendulum system allows for both horizontal and vertical displacement, providing two degrees of freedom. This capability is essential for accommodating the complex, multi-directional nature of seismic waves.