The analysis offers a thorough analytical examination of the auto-parametric pendulum vibration damper, focused on enhancing vibration mitigation in engineering structures via accurate modelling of nonlinear systems and optimization of energy transfer processes. The work aims to enhance the practical application of auto-parametric pendulum vibration dampers in engineering systems. The non-perturbative approach offers a distinctive framework that does not depend on Taylor series expansions. The primary objective of the non-perturbative approach is to transform nonlinear ordinary differential equations of time varying control systems into analogous linear ones. Approximate solutions are obtained without employing series expansions, unlike conventional perturbation methods. The study seeks to advance beyond traditional perturbation methods to evaluate the behaviour of systems exhibiting small-amplitude parametric fluctuations. To generate successive approximations that define the system’s nonlinear parametric behaviour, it is crucial to precisely delineate the relationship between frequency and amplitude. The numerical implementations are employed to validate the derived parametric formulation, which demonstrates significant agreement with the original governing ordinary differential equation. The influence of several factors on the stability of equilibrium states is systematically analysed. The findings indicate that the proposed strategy is very successful, thoroughly validated, and intuitive. Moreover, it could be extended too many applications in dynamical systems and fluid dynamics. The multiple-time scales method is employed to assess the stability structure for a more comprehensive analysis of the system’s behaviour. The morphology of the bifurcation curves alters significantly with variations in the bifurcation parameters. Phase portraits, Poincaré maps, and bifurcation diagrams are employed to do a comprehensive bifurcation analysis. As the excitation amplitude rises, it detects changes from periodic to chaotic motion. The method facilitates the differentiation and classification of the many dynamic reactions exhibited by the system.
Almutlg et al. (Wed,) studied this question.