Nonlinear 2DOF van der Pol vibrations: El-Dib’s optimized frequency formula via the two-time-domains method

This paper presents a systematic decoupling framework for strongly coupled two-degree-of-freedom (2-DOF) nonlinear vibration systems. Here, the coupling is classified as strong rather than weak because the cross-mode interaction terms in the governing equations (e.g., the x ² y-, xy ² -, and xy-type terms), enter with coefficients of the same order as the leading cubic stiffness nonlinearities, with no small bookkeeping parameter scaling them down as would be assumed in a weakly coupled (perturbative)formulation. The proposed approach assigns each mode its own independently stretched time domain, ξ = Ω ₁ t and ζ = Ω ₂ t, through a two-time-domain Lindstedt-Poincaré-type transformation. Combined with mode-specific weighted averaging, this decouples the governing equations into two analytically tractable subsystems, each carrying its own closed-form, amplitude-dependent frequency. The methodology extends El-Dib’s frequency formula to accommodate two-term coupled configurations, producing closed-form frequency-amplitude relationships that accurately characterize the nonlinear system dynamics. Analytical solutions exhibit close agreement with direct numerical simulations in terms of amplitude, phase, and oscillatory envelope; residual discrepancies are attributed solely to higher-order truncation effects. The analysis demonstrates that both inter-mode coupling and modal damping significantly influence the energy exchange between vibrational modes and the overall stability of the system. The proposed methodology is directly applicable to several nonlinear problems arising in fluid structure interaction, rotating machinery, MEMS resonators, and magnetic fluid interfaces, thereby providing a rigorous analytical platform that bridges nonlinear dynamical theory and quantitative engineering predictions.