<p>This study presents a nonlinear dynamic analysis and closed-form modelling framework for vibration isolation systems incorporating negative stiffness elements under bidirectional excitation. A two-degree-of-freedom model is formulated to capture the coupled dynamics of the primary structure and isolation mechanism, incorporating nonlinear restoring forces within a consistent analytical framework. The governing equations are developed using classical principles and subsequently simplified through Taylor series expansion and statistical linearization to obtain tractable closed-form solutions. Unlike conventional optimization-based approaches, the proposed methodology enables direct determination of optimal system parameters, including frequency and damping ratios, without reliance on iterative procedures. The dynamic response of the system is evaluated using harmonic balance and transfer function analysis, providing insight into resonance characteristics, stability behaviour, and energy dissipation mechanisms under coupled excitations. Comparative analyses demonstrate that the incorporation of negative stiffness significantly enhances vibration mitigation performance. The proposed closed-form solutions show good agreement with hybrid optimization methods while offering improved computational efficiency and deeper physical insight into system behaviour. The results highlight the effectiveness of negative stiffness mechanisms in modifying system dynamics and reducing response amplitudes. The developed framework contributes to the fields of nonlinear dynamics and vibration control by providing a generalized analytical tool for the design and analysis of coupled mechanical systems with nonlinear stiffness characteristics.</p>