Theory

This page summarises the methods VANE implements. The Validation matrix shows where each is checked against an analytical or reference solution.

Linearization

OpenFAST linearizes the full aero-hydro-servo-elastic system about a periodic operating point, producing a continuous-time state-space model

\[\dot{x} = A\,x + B\,u, \qquad y = C\,x + D\,u\]

at one or more rotor azimuths. For a spinning rotor the matrices are azimuth-dependent: the system is linear time-periodic, not time-invariant.

Multi-blade coordinate (MBC3) transform

The Coleman, or multi-blade coordinate, transform maps the three per-blade degrees of freedom of a three-bladed rotor into the non-rotating frame as a collective, cosine-cyclic, and sine-cyclic triplet. Averaging the transformed matrices over a full revolution yields a single linear time-invariant model that approximates the periodic system. The transform is applied to the A, B, C, and D matrices; blade triplets are detected automatically from the channel descriptions.

For an isotropic rotor the per-azimuth transformed matrices are identical, so the averaged model is exact; anisotropy is what the azimuth-spread uncertainty quantifies.

Eigenanalysis

Modes are the eigenpairs of the averaged state matrix. For an eigenvalue \(\lambda = -\zeta\omega_n \pm i\,\omega_n\sqrt{1-\zeta^2}\):

  • natural frequency \(\omega_n = |\lambda|\),

  • damping ratio \(\zeta = -\mathrm{Re}(\lambda)/|\lambda|\),

  • damped frequency \(\omega_d = \mathrm{Im}(\lambda)\).

Conjugate pairs are collapsed to a single representative (the one with positive imaginary part). Each mode additionally reports an eigenvalue condition number \(\kappa = 1/|y^{H}x|\) (from the left and right eigenvectors) measuring its numerical sensitivity, and a near-degeneracy flag where a repeated eigenvalue leaves the mode shape defined only up to a subspace rotation.

Mode identification

Each mode is labelled from the physical degrees of freedom that dominate its participation. Blade modes are further classified as collective, regressive, or progressive directly from the MBC structure (each transformed triplet is [collective, cosine, sine]) and the phase-invariant whirl sign \(\mathrm{Im}(q_s\,\overline{q_c})\). A Gaussian-process classifier provides an independent, uncertainty-aware label that is fused with the rule-based one.

Cross-operating-point tracking

Across a sweep, modes are linked into the lines of a Campbell diagram. Rather than a greedy adjacent-point match, VANE extracts each line as the globally optimal continuity path over the whole sweep — a deterministic dynamic program over a modal-assurance-criterion / frequency-continuity affinity. Each track carries a confidence (its weakest link) and an ambiguity flag at crossing or veering hotspots. See Limitations for the precise optimality guarantee.

Uncertainty quantification

Every mode receives one calibrated confidence in [0, 1] combining independent reliability factors: eigenvalue conditioning, a degeneracy penalty, an azimuth-stability factor (the first-order eigenvalue perturbation of the averaged model across the per-azimuth matrices), and the tracking confidence. A low confidence inflates the Kalman initial covariance, propagating uncertainty into the digital-twin export.

State-space export

The averaged or modal model is exported as a validated continuous-time system, optionally discretised by a zero-order hold (discrete eigenvalues \(e^{\lambda\,\Delta t}\)), with Q, R, and P_0 matrices built from the per-mode confidence for a Kalman-filter-ready model.