Innovative nonlinear vibration analysis of a double pendulum two-degree-of-freedom system

Despite the extensive literature on nonlinear pendulum dynamics, the analytical treatment of strongly coupled multi-degree-of-freedom systems remains a significant challenge, as classical linearization methods neglect the nonlinear interactions among coupled coordinates and thus fail to capture essential mode-coupling effects. To address this limitation, the present study develops an Extended Equivalent Linearization Approach (EELA) for a planar two-degree-of-freedom double pendulum, incorporating amplitude-dependent weighting functions that explicitly account for the coupling between the two oscillators. In parallel, a coupled Lindstedt-Poincaré (two-time domains) transformation is introduced, assigning independent strained time variables to each degree of freedom and reformulating the governing nonlinear equations into a unified coordinate framework. The combined analytical strategy yields closed-form amplitude-dependent frequency expressions and an explicit stability criterion showing that the stability boundaries are governed exclusively by the initial amplitude ratio and the mass ratio, independently of the individual link lengths. Validation against direct numerical integration confirms that the analytical solutions remain accurate for moderate-to-large oscillation amplitudes, well beyond the small-angle regime, with maximum absolute errors strictly bounded and free of secular growth throughout the considered time domain.