Nonlinear vibration analysis of a 3DOF double pendulum system near resonance

The primary goal of this work is to analyze the energized motion of three-degrees-of-freedom (3DOF) dynamic system consisting of a coupled double pendulum with the damped mass under the external harmonic forces and moments. Lagrangian equations are employed to derive the differential governing equations of motion (GEOM) based on the system generalized coordinates. The approximate solutions (AS) of these equations are generated through the utilization of the multiple scales technique (MST) at the third-order level of approximation. These solutions are ascertained by contrasting them with numerical solutions (NS) that are derived utilizing the fourth-order Runge-Kutta algorithm (RKA-4). The modulation equations are constructed, and the principal external resonance cases are scrutinized concurrently based on the solvability constraints. The steady-state solutions are studied. Based on Routh-Hurwitz criteria (RHC), the stability and instability zones are examined and assessed in the line with the steady-state solutions. The amplitudes and phases over a specific period of time have been graphed to illustrate the movement at any given instant. Furthermore, in order to evaluate the advantageous effects of different values pertaining to the physical parameters on the system behavior, the graphed representations of the obtained results, resonance reactions and areas of the stability are provided. The significance of the research model stems from its numerous applications such as the gantry cranes, robotics, pump compressors, transportation devices and rotor dynamics. It may be applied to the study of these systems' vibrational motion.