Cyclic-symmetry analysis#

When to use#

Cyclic-symmetry analysis exploits N-fold rotational symmetry to recover the full-rotor spectrum from a single base-sector model. Use it when:

  • The structure has identical sectors arranged around an axis (impellers, vaned diffusers, cylinder arrays, reciprocating-engine blocks).

  • You’d otherwise be solving an N-times-larger problem on the full geometry — cyclic reduces the DOF count by a factor of \(N\) while recovering exactly the same spectrum.

  • The full-rotor problem doesn’t fit in RAM / wall-time budget but a sector does.

For non-cyclic geometries use the standard Modal analysis (free vibration) path. Cyclic doesn’t apply if the rotor has any deviation from N-fold symmetry (mistuning, intentional asymmetry, broken-blade analysis).

Boundary conditions and loads#

  • Dirichlet constraints on the sector — pin the hub / shaft attachment as you would in a regular modal solve.

  • Cyclic-pairing constraint — auto-applied by CyclicModel from the rotation axis + sector angle. No manual DOF-pair list.

  • No applied loads. Like Modal analysis (free vibration), cyclic-symmetry modal analysis is unforced free vibration.

The math (one paragraph)#

Cyclic-symmetry modal solves the per-harmonic eigenproblem

\[\mathbf{K}_k\, \boldsymbol{\phi}_k = \omega_k^{2}\, \mathbf{M}_k\, \boldsymbol{\phi}_k, \qquad k = 0, 1, \ldots, \lfloor N/2 \rfloor,\]

where \(\mathbf{K}_k\) and \(\mathbf{M}_k\) are the sector matrices augmented with the harmonic-\(k\) cyclic-pairing constraint \(\mathbf{u}_\text{high} = e^{i\, k\, \alpha}\, \mathbf{R}(\alpha)\, \mathbf{u}_\text{low}\). The Grimes-Lewis-Simon real-2n augmentation (Modal eigenvalue problem and shift-invert Lanczos) keeps each per- harmonic eigenproblem real-symmetric. Sweeping \(k\) from 0 to \(\lfloor N/2 \rfloor\) recovers the full- rotor spectrum.

Running the solve#

cyc = femorph_solver.CyclicModel(
    m, n_sectors=16, axis="z",
)
results = cyc.solve_modal(n_modes=4)

results is a list of CyclicModalResult instances, one per harmonic 0 … N/2. Each carries:

  • res.harmonic_index — the \(k\) value.

  • res.frequency — (n_modes,) frequencies at this harmonic.

  • res.mode_shapes — sector-sized eigenvectors (the full-rotor mode is reconstructed via the cyclic-pairing rule).

For a focused recipe, see Cyclic-symmetry modal on a rotor sector.

Travelling-wave mode pairs#

At every \(k > 0\) the eigensolver returns mode shapes in pairs at the same frequency. Real and imaginary parts of the complex eigenvector are quarter-period travelling-wave snapshots — the standing-wave / travelling- wave decomposition of Crandall & Mark 1963 §3.5. See Cyclic-symmetry travelling-wave pair — bladed rotor for the visualisation.

Engine-order resonance check#

For a rotor running at angular speed \(\Omega\), only mode shapes with harmonic index \(k\) matching the engine order \(m\) (modulo N) couple to the excitation (Wildheim 1979). The Campbell-diagram clearance check walks every \((m, k)\) pair and flags resonances within a design-margin band. See Tutorial 5 — Cyclic-symmetry rotor design check for the full design check.

Common gotchas#

  • Cyclic-face auto-detection failure. The CyclicModel constructor auto-detects the low / high cyclic faces from the rotation axis + sector angle. If the geometry is mis-aligned with the axis (off-axis centroid) or the sector angle doesn’t match the geometry, the auto-detector raises. Pass explicit low_face / high_face node-set arrays as a manual override.

  • Mistuned rotors — cyclic-symmetry assumes exactly identical sectors. Real rotors have manufacturing variation that breaks the symmetry; the cyclic spectrum is an idealisation. Mistuning analysis is on the planned roadmap.

  • Forgetting harmonic 0. k = 0 modes are the axisymmetric / breathing-mode family. The clearance check must include them — engine-order m = N, 2N, 3N, … excite k = 0 modes.

End-to-end tutorial#

Tutorial 5 walks the cyclic-symmetry workflow end-to-end: Tutorial 5 — Cyclic-symmetry rotor design check.

Solver mechanics + performance#

For Grimes-Lewis-Simon augmentation details, per-harmonic factor reuse, and the sector-DOF reduction maths, see Cyclic-symmetry modal analysis — that page is the solver-engineering deep-dive on the same CyclicModel.solve_modal() invocation.