Modal analysis (free vibration)#

When to use#

Modal analysis recovers the natural frequencies and mode shapes of an unconstrained or constrained linear-elastic structure. Use it when:

You need to know whether the structure’s natural frequencies fall inside any operating-band excitation (the Campbell-diagram clearance check).
You’re sizing modes for a downstream response-spectrum or modal-superposition transient analysis (cumulative effective mass ≥ 90 % per direction is the standard gate).
You want a pre-stress baseline for buckling or a sanity check for BC completeness (the rigid-body-mode diagnostic in Boundary-condition pitfalls).

For static deflection under fixed loads, use Linear static analysis. For frequency-domain forced response, use Harmonic (frequency-response) analysis. For time-domain transient response, use Transient analysis.

Boundary conditions and loads#

Dirichlet constraints via Model.fix(nodes=..., dof=...) — identical to Linear static analysis. Most modal solves are run on a fully-constrained structure; for free-free modal analysis (a structure floating in space — a spacecraft, an unconstrained rod), see the sigma discussion below.
No applied loads. Modal analysis is the unforced free-vibration response — nodal-force / pressure / gravity are ignored.
Pre-stress (geometric stiffness from a prior static load) is on the planned roadmap for buckling / pre-stressed modal; not yet shipped.

The math (one paragraph)#

Modal analysis solves the generalised symmetric-definite eigenvalue problem (Modal eigenvalue problem and shift-invert Lanczos)

\[\mathbf{K}\, \boldsymbol{\phi}_i = \omega_i^{2}\, \mathbf{M}\, \boldsymbol{\phi}_i,\]

returning eigenpairs \((\omega_i^{2}, \boldsymbol{\phi}_i)\) where \(\omega_i\) is the angular natural frequency in rad/s and \(\boldsymbol{\phi}_i\) is the mode shape. femorph-solver dispatches the lowest n_modes via shift-invert Lanczos (scipy.sparse.linalg.eigsh); see the theory chapter for the full algorithm.

Running the solve#

The high-level path:

res = m.solve_modal(n_modes=12)

Returns a ModalResult:

res.frequency — (n_modes,) natural frequencies in Hz, ascending.
res.omega_sq — (n_modes,) squared angular frequencies in (rad/s)².
res.mode_shapes — (n_dof, n_modes) mass- orthonormalised eigenvectors.

Mode shapes are mass-orthonormalised by default (\(\boldsymbol{\phi}_i^{T}\, \mathbf{M}\, \boldsymbol{\phi}_j = \delta_{ij}\)), which means post-processing recipes (modal participation, mode-shape animation) work directly without re-normalising.

Free-free analysis#

A free-free structure has six rigid-body modes at \(\omega = 0\), which makes \(\mathbf{K}\) rank-deficient. Pass a non-zero shift to move the spectrum off the zero-eigenvalue ridge:

res = m.solve_modal(n_modes=10, sigma=-1.0)

The shift-invert path then factors \((\mathbf{K} - \sigma \mathbf{M})\) which is positive-definite for any \(\sigma < \omega_1^{2}\). See Free-free axial rod — natural frequencies for the canonical free-free pattern.

Targeting a frequency band#

For an off-zero shift (modes near a target frequency rather than the lowest), set sigma to the squared angular frequency of interest:

# Modes near 2 kHz:
sigma = (2 * np.pi * 2_000.0) ** 2
res = m.solve_modal(n_modes=10, sigma=sigma)

Post-processing recipes#

Modal participation factors + effective mass — Modal participation factors + effective modal mass. The recipe used by Tutorial 2 to pick n_modes for a 90 %-cumulative-mass response-spectrum analysis.
Mode-shape rendering — Plotting mode shapes.
Mode-shape animation (one mode swept over a period) — Mode-shape animation — cantilever bending modes.

Common gotchas#

Surviving rigid-body modes when the structure should be constrained. Run the Boundary-condition pitfalls rigid-body-mode check on the unloaded model — every near-zero frequency is an unconstrained DOF.
Bi-degenerate transverse pairs on symmetric cross- sections. A square-section cantilever has f_y = f_z for every transverse-bending mode; the eigensolver returns them as two modes at the same frequency. The translation-energy mode-classification recipe in Tutorial 2 distinguishes them.
Stress-recovery vs displacement accuracy. Nodal averaging stress on a recovered mode shape is meaningless — mode shapes carry no absolute scale. Use stress only on a static or transient solve, or on a participation- factor-scaled mode for response-spectrum analysis.

End-to-end tutorial#

Tutorial 2 walks the modal-survey workflow on a cantilever bracket: Tutorial 2 — Modal survey of a cantilever bracket.

Verification examples that exercise this analysis type:

Cantilever natural frequency — Rao 2017 Table 8.1 — first cantilever NF
Cantilever beam — first four bending modes — first three cantilever modes
Clamped-clamped beam — first three bending natural frequencies — clamped-clamped beam modes
Simply-supported beam — first three bending natural frequencies — simply-supported beam modes
Fixed-free axial rod — natural frequencies — fixed-free axial rod
Free-free axial rod — natural frequencies — free-free axial rod
Simply-supported plate — first transverse bending mode — simply-supported plate

Solver mechanics + performance#

For shift-invert Lanczos algorithm details, dense vs sparse path selection, eigen-solver backend tradeoffs, and the tol parameter, see Modal analysis (free vibration) — that page is the solver-engineering deep-dive on the same Model.solve_modal() invocation.