Complex Mode Interactions in Lightly Damped Structures: Understanding Two‑Mode Coupling and Eigenvalue Behavior

Complex mode shapes can emerge in lightly damped structures even when the overall damping is low. This occurs when two structural modes are closely spaced in frequency and are coupled through nonproportional damping. As the damping interaction intensifies, the modal frequencies move toward each other and the associated mode shapes become increasingly intricate. At a critical interaction level, the two frequencies coalesce into a single eigenvalue. Beyond this point, the frequencies remain approximately equal while their modal decay rates separate. Under very strong interaction, the mode shapes become approximately real again, but they manifest as new linear combinations of the original uncoupled modes. This framework provides a clear physical interpretation of mode veering, complex mode formation, eigenvalue coalescence, and decay‑rate splitting in lightly damped systems.

Traditional modal analysis often treats lightly damped structures as nearly proportional systems with real mode shapes. This simplification can be misleading when two modes are closely spaced in frequency. In modal coordinates, nonproportional damping introduces an off‑diagonal coupling term ε. Consequently, the key question is not simply whether damping is small, but whether the coupling is large relative to the baseline half‑frequency separation δ = (ω₂ – ω₁)/2.

Reduced Perturbation Framework
For a closely spaced mode pair, define the central eigenvalue λ₀ ≈ –ζω + iω and express the coupled eigenvalues as λ₁,₂ = λ₀ ± s. Under the assumptions of light damping and close‑mode spacing, the local quartic eigenvalue problem reduces to the single relation
 s² = ε²⁄4 – δ².
This equation determines the direction and magnitude of eigenvalue separation in the complex plane, as summarized in Figure 1. When the coupling dominates the frequency spacing (ε/δ > 2), the eigenvalues split onto the real axis, while weaker coupling leads to complex conjugate pairs.

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