Experimental rotating modal analysis

This experimental route is part of QF Solver 0.2.11. Its bounded scope and maturity are distinct from package publication and do not change the immutable 0.2.10 release contract.

This page documents the bounded rotating_modal capability introduced for QF Solver 0.2.11. It describes the numerical route; it does not assign a qualification level. The route remains EXPERIMENTAL.

Scope

The initial implementation accepts an undamped, unprestressed, linear and serial model consisting of a straight, connected, unbranched BEAM2 shaft and one or more centered rigid axisymmetric disks attached to existing beam nodes. Beam sections must be circular and isotropic, and the spin axis must be common, fixed in the global spatial frame, and parallel to the shaft. The solver uses a dense SciPy generalized eigenvalue solve.

The analysis is not a general rotating-machinery solver. It rejects shells, solid rotor meshes, eccentric or non-axisymmetric disks, generic duplicate disk masses, unsupported constraints, contact, damping, centrifugal stress-stiffening, variable speed, distributed beam gyroscopic terms, nonlinear rotor response, and sparse/PETSc/SLEPc/MPI backends.

Model and sign convention

For small perturbations around the undeformed, unprestressed state, the equation is

M q_ddot + Ω G q_dot + K q = 0
q(t) = Re(φ exp(λ t))
(λ² M + λ Ω G + K) φ = 0

Ω is a signed scalar in rad/s and appears once in the quadratic eigenvalue problem. The configured axis is an explicit, normalized global vector using the right-hand rule; the frame convention is global_fixed_right_hand_rule. RPM conversion is available only through an explicit API and is never inferred from an input value.

For a disk with unit axis a, mass m, diametral inertia Jd, and polar inertia Jp, its nodal mass block is

M_disk = diag(m I, Jd (I - a aᵀ) + Jp a aᵀ)
G_disk = diag(0, -Jp [a]×)

G_disk is the unit-speed contribution; the assembler does not multiply it by Ω or project it to skew symmetry after assembly. The rotating disk owns both its translational/rotary mass and its gyroscopic term. A generic concentrated mass cannot also represent that disk at the same node.

Numerical route and results

The route reuses structural K and M assembly, adds disk mass and a separate skew G, and applies the same homogeneous fixed-DOF reduction to all three matrices. It checks symmetry, skew symmetry, positive definiteness of reduced M and K, and finite values before solving. It then solves the dense generalized companion pencil with scipy.linalg.eig(A, B); it does not use a symmetric eigensolver or explicitly invert M.

The solver retains every raw complex root and mode. The convenience view selects positive-imaginary oscillatory roots deterministically, while raw roots remain available for inspection. Modes are mass-normalized with φᴴ M φ = 1, phase-normalized without changing the represented eigenspace, and checked against the original quadratic polynomial. Serialized complex values carry explicit real and imaginary components.

At Ω = 0, the route is checked against the classic modal solve for the same physical K/M; degenerate eigenspaces are compared as subspaces, not by column-wise vector equality. Passing numerical checks does not imply qualification or maturity promotion.

Verification status and limitations

The 0.2.11 WP05 verification records report the exact cases, source identity, environment, tolerances, and evidence availability. The initial checks cover local disk formulation, zero-speed modal recovery, an independent analytical disk oscillator, signed-speed spectrum behavior, complex result serialization, and a bounded dense-backend characterization. Read those records for the measured size limit; no limit is extrapolated beyond tested sizes.

This single-speed route does not implement speed sweeps or modal tracking. The separate experimental campbell route orchestrates multiple rotating_modal solves and reports modal association and ambiguity; it does not change the WP05 physical formulation or imply general rotordynamics support.