The gyroscopic velocity term makes the eigenproblem quadratic and generally complex. It is not the classical symmetric K–M modal solve.
01 Model and scope
Substituting an exponential mode into the rotating equation of motion yields a quadratic pencil because the gyroscopic term multiplies velocity. The skew gyroscopic matrix makes the spectrum generally complex; the solve is distinct from a symmetric K–M modal analysis.
Route assumptions and calculation checks
- Ω is signed in rad/s and appears once in the quadratic pencil; the axis is an explicit normalized global vector.
- The same fixed-DOF reduction is applied to K, M, and G; symmetry, skew symmetry, definiteness, and finite values are checked.
- Raw complex spectrum and QEP residual are retained; a convenience view selects positive-imaginary oscillatory roots.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- M, K
- Assembled mass and stiffness.
- Signed prescribed spin speed.
- Gyroscopic matrix, skew-symmetric under the declared disk model.
- Complex eigenvalue and mode vector.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Add gyroscopic inertia to the rotating equations
For the documented undamped linear rotor model, mass multiplies acceleration, stiffness multiplies displacement, and spin couples through a skew gyroscopic matrix multiplying velocity. Signed Ω is measured in rad/s and depends on the declared global spin axis.
2 · Use an exponential eigenmode
Assume q(t)=φe^{λt}. The first and second time derivatives introduce λ and λ², respectively, yielding a quadratic matrix pencil. Unlike the symmetric K–M problem, this pencil is generally complex and its eigenvalues need not be purely imaginary numerically.
3 · Build the centered disk inertia and gyroscopic block
For a centered axisymmetric rigid disk with unit axis a, transverse inertia Jd and polar inertia Jp, translational mass is mI. The rotational inertia distinguishes the axis-parallel and transverse directions; spin angular momentum generates the skew coupling.
4 · Reduce supports and validate the raw spectrum
Apply the same fixed-DOF reduction to M, K, and G. Check finite values, symmetric M/K, skew G, the zero-speed limit, and the scaled QEP residual. The presentation may filter oscillatory roots, but the stored raw spectrum remains the basis for review.
04 Algorithm and implementation
Read the linked versioned source for the actual assembly and solver behavior. The derivation above explains the mathematics and is not a verbatim source excerpt.
05 Verification and limitations
EXPERIMENTAL · serial dense SciPy QEP
At zero speed, compare against the classical modal solve on the same K and M; compare repeated modes as subspaces.
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.