A First-Order Perturbation Theory of Modal Assurance Criterion Degradation: Analytical Derivation and Application to High-Density Aerospace Structures

Abstract

In structural health monitoring, the Modal Assurance Criterion is widely used to evaluate modal fidelity. While empirical studies show that localized stiffness loss induces off-diagonal mode mixing and violent mode veering in high-density frequency bands, a closed-form analytical connection between classical eigenvalue perturbation theory and MAC degradation has been lacking. This study proposes a First-Order Perturbation Theory of Modal Assurance Criterion Degradation to analytically derive the exact mechanics of structural mode mixing. By formulating the damaged structure as a perturbed generalized eigenvalue problem, we establish that mode mixing is explicitly governed by two parameters: cross-modal strain energy and eigenvalue proximity. The theoretical framework was validated on an exact analytical spring-mass system and a multi-degree-of-freedom computational model representing typical aerospace enclosures. Results mathematically prove that mode mixing grows quadratically with localized structural damage rather than linearly. At low degradation levels (10% stiffness loss), the theoretical off-diagonal approximation matches exact non-linear solutions with a minimal absolute error of just 0.0097. Furthermore, the derivation definitively explains why high-density modal regions are inherently vulnerable to severe mode veering: as modal separation narrows (e.g., Δλ= 0.1757), the mixing coefficient amplifies exponentially, resulting in total mode coupling (Theoretical MAC=1.0000). Convergence analysis establishes that the first-order theoretical projection maintains robust predictive accuracy up to a 30% localized stiffness degradation threshold, beyond which secondary mode interactions exponentially increase residual error and necessitate higher-order perturbation terms. Ultimately, this transparent mathematical framework provides a highly efficient computational tool for predicting dynamic instability in high-performance aerospace structures without exhaustive iterative simulations.