Dynamical analysis of a four-degree-of-freedom vibratory structure: Bifurcation, stability, and resonance exploration

This study introduces a novel approach to analyzing a four-degree-of-freedom (DoF) nonlinear system by leveraging advanced numerical and analytical techniques to comprehensively examine its dynamic behavior. The system’s nonlinear differential equations (DEs) are obtained through the application of Lagrange’s equations (LE). The solutions are obtained using the fourth-order Runge–Kutta method (4-RKM). The investigation involves analyzing the relationships between the angular solutions and their corresponding first-order derivatives, commonly referred to as phase plane analysis. The study aims to examine bifurcation diagrams and Lyapunov exponent spectra to reveal the various modes of motion within the system and visualize Poincaré maps. These tools are used to analyze a unique system configuration. Lastly, the conditions for solvability and the characteristic exponents are identified by examining resonance scenarios. The examination of resonance scenarios through characteristic exponents and solvability conditions, coupled with the application of Routh-Hurwitz criteria (RHC) for stability evaluation, provides an innovative framework for understanding frequency response and nonlinear stability across stable and unstable ranges. By exploring both theoretical and practical aspects of vibrational dynamics in applications like aviation, robotics, and underwater exploration, this work offers a significant advancement in analyzing complex systems, with wide-ranging implications for various engineering fields, including aerospace, structural mechanics, and energy harvesting.