Nonlinear dynamical systems with structural damage present significant challenges for analysis and monitoring due to their complex responses, the presence of multiple interacting nonlinearities, and the sensitivity of system behavior to small changes in conditions. These complexities are further compounded when the system exhibits both inherent nonlinear effects, such as those induced by magnetic forces, and localized damage mechanisms like breathing cracks, which can introduce time-varying stiffness and intermittent contact phenomena. This study investigates a mechanical system exhibiting both magnetically induced nonlinear behavior and damage in the form of a breathing crack. By relying solely on system output measurements, the research eliminates the need for input excitation data, offering a practical approach for complex or inaccessible systems. Transmissibility and coherence functions are then employed to exploit nonlinear dynamics, accounting for potential damages and uncertainties. These functions facilitate the identification and characterization of harmonics and other nonlinear features, while also suppressing the influence of unknown input excitation effects. Experimental investigations involve measuring outputs from multiple locations along the structure to explore the distinct features identified by a combination of each pair of output signals used in computing the transmissibility and coherence functions. The results demonstrate the capability of these functions to reveal key signatures of nonlinearity and damage, supporting their use in structural health monitoring and fault diagnosis for complex systems in modern engineering applications.