Closed-form solution for vibration analysis of structures with localized features via Metaheuristic-enhanced Homotopy Mapping

This paper presents a novel closed-form analytical methodology for predicting the global modal parameters, specifically natural frequencies and corresponding mode shapes, of one- and two-dimensional (1D/2D) structures containing localized features. These features are defined as discrete irregularities, such as cracks, bolted joints, or abrupt stiffness changes, which introduce concentrated changes to the system’s properties. The core of the approach is an integrated framework combining Homotopy Mapping with Metaheuristic Optimization. Homotopy Mapping constructs a continuous transformation from the known linear modal properties of the intact structure to those of the structure with localized features. Metaheuristic Optimization is then employed to determine a set of auxiliary convergence-control parameters for each mode order in this transformation. Based on the transformation along with pre-computed auxiliary parameters, an explicit closed-form expression is derived, linking the physical parameters (e.g., location and stiffness) of the localized features to the global modal behavior of the structure. A key advantage is that the derived explicit closed-form expression naturally avoids the iterative numerical re-modeling inherent in traditional methods, leading to substantially improved computational efficiency. The accuracy of the proposed method is validated through experimental tests. When benchmarked against semi-analytical, data-driven, and numerical approaches, the method exhibits natural frequency prediction errors below 1.5 % for both fundamental and higher-order modes, across the full range of variations in localized features. Moreover, the method offers physical interpretability by explicitly revealing how feature parameters influence global modal behavior. This work provides a computationally efficient and robust analytical tool for the forward analysis of structures with localized features, effectively balancing theoretical innovation with engineering applicability.