Complex MAC, lineage, and ambiguity

Analysis methodEXPERIMENTAL · ambiguity preserved

A Hermitian mass-weighted MAC and global one-to-one assignment associate modes; near-degenerate modes are compared as subspaces.

01 Model and scope

The mass-weighted complex MAC measures shape correlation while ignoring eigenvector phase and scale. A one-to-one assignment links modes between speeds, but near-degenerate subspaces or weak correlations remain ambiguous rather than being forcibly joined into a smooth curve.

Route assumptions and calculation checks

The speed list is explicit and strictly increasing; the sweep does not mutate model properties.
When evidence cannot distinguish a branch, the result keeps a gap or ambiguity instead of forcing continuity.
A 1× intersection is a frequency coincidence in a linear modal model, not forced-response amplitude, operating risk, or instability.

Useful foundations and prerequisites

Sparse generalized eigenproblem Modal sparse/dense eigensolvers

02 Essential equations and symbols

\operatorname{MACM}(\varphi{}_{i},\varphi{}_{j}) = |\varphi{}_{i}^H M \varphi{}_{j}|^2 / [(\varphi{}_{i}^HM\varphi{}_{i})(\varphi{}_{j}^HM\varphi{}_{j})]

Definitions and conventions

$\varphi{}_{i}^H$: Conjugate transpose of complex mode i.
$M$: Mass weighting used in the MAC.
$MACM$: Normalized mass-weighted modal correlation.
$\Omega{}/(2π)$: Spin frequency used only for the optional 1× reference line.

01Signed speed sweep

02Complex MAC

03One-to-one assignment

04Preserve ambiguity

Complex MAC, lineage, and ambiguity · method mapMass-weighted complex correlation assigns modes across speeds; uncertain matches remain explicitly ambiguous.

4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Solve at an explicit signed speed grid

At each strictly increasing spin speed Ω, solve the quadratic pencil and retain the complex eigenpairs. Each speed is an independent solve from the frozen model; the sweep does not silently mutate material or support properties.

\Omega_0<\Omega_1<\cdots<\Omega_m,\qquad\mathcal E_j=\{(\lambda_i^{(j)},\boldsymbol\phi_i^{(j)})\}

2 · Compare complex shapes with a mass-weighted MAC

The Hermitian inner product accounts for complex phase. The normalized mass-weighted MAC is unchanged by a nonzero complex scale and lies between zero and one for positive-definite weighting. It measures shape correlation, not eigenvalue proximity alone.

\operatorname{MAC}_M(\boldsymbol\phi_i,\boldsymbol\phi_j)=\frac{|\boldsymbol\phi_i^H\mathbf M\boldsymbol\phi_j|^2}{(\boldsymbol\phi_i^H\mathbf M\boldsymbol\phi_i)(\boldsymbol\phi_j^H\mathbf M\boldsymbol\phi_j)}

3 · Assign lineages without hiding ambiguity

A global one-to-one assignment avoids pairing every new mode greedily with the same previous mode. If correlations are weak or repeated modes exchange within a nearly degenerate subspace, keep the match marked ambiguous or leave a gap rather than forcing a smooth branch.

\max_{\pi\;\mathrm{one-to-one}}\sum_i\operatorname{MAC}_M(\boldsymbol\phi_i^{(j)},\boldsymbol\phi_{\pi(i)}^{(j+1)})

4 · Interpret Campbell intersections narrowly

Plot modal frequencies against signed spin and optionally overlay the 1× line |Ω|/(2π). A crossing is only a frequency coincidence in this linear modal model; it is not forced-response amplitude, operating risk, instability, or a critical-speed certification.

f_i(\Omega)=\frac{|\operatorname{Im}\lambda_i(\Omega)|}{2\pi},\qquad f_{1\times}=\frac{|\Omega|}{2\pi}

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

EXPERIMENTAL · ambiguity preserved

The high-frequency pair remains ambiguous at 100 rad/s; the plot intentionally preserves a missing branch point.

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

Recorded GYRO06 mesh case and limits Verification summary Frozen Campbell contract

06 Sources and related sheets

Tracking limits Verification summary Frozen Campbell contract

Continue the reasoning

Quadratic eigenvalue problem Frozen experimental boundary

Back to sheet contents ↑