Beam Vibration

Beam Vibration (structures/fem/beam-vibration)

Use when the task is computing the natural frequencies of a continuous Euler-Bernoulli beam member: exact bending frequencies in hertz for pinned-pinned, cantilever, clamped-clamped and free-free end conditions, obtained from the transcendental characteristic-equation roots rather than a discrete mass model. The beam is treated as a distributed-parameter member with bending stiffness EI (N m^2) and mass per unit length m (kg/m); every end condition reduces to a root of cos x cosh x = -1 or cos x cosh x = 1, and each root feeds the shared frequency law f_n = (beta_n L)^2 sqrt(EI / (m L^4)) / 2pi. This leaf is the continuous-member complement to the discrete 2-DOF modal-analysis leaf, and it supplies the member natural frequencies that a random-vibration or vibration-clearance check consumes.

Domain quick reference

• Transverse free vibration of a uniform Euler-Bernoulli beam obeys EI y'''' + m y_ddot = 0; separation of variables gives spatial solutions built from cos, sin, cosh and sinh of beta x, and the end conditions select the allowed beta_n L.
• Pinned-pinned (simply supported): beta_n L = n pi exactly, so f_n = n^2 pi^2 sqrt(EI / (m L^4)) / 2pi, with the exact spacing f_n = n^2 f_1. No root search is needed.
• Cantilever (fixed-free): cos x cosh x = -1. Bisection roots: 1.87510407 (n=1), 4.69409113 (n=2), 7.85475744 (n=3).
• Clamped-clamped and free-free: cos x cosh x = 1, roots 4.73004074 (n=1), 7.85320462 (n=2). A free-free beam also carries two zero-frequency rigid-body modes (translation, rotation); only the elastic modes are reported, so the first elastic free-free root is 4.73004074, identical to clamped-clamped.
• Shared frequency law for every end condition: f_n = (beta_n L)^2 sqrt(EI / (m L^4)) / 2pi in hertz, where the characteristic roots are non-dimensional.
• Rayleigh quotient for non-uniform members: omega^2 = int_0^L EI phi''^2 dx / int_0^L m phi^2 dx. With the uniform-cantilever parabola phi = (x/L)^2 this gives omega^2 = 20 EI / (m L^4), an upper bound about 1.272x the exact fundamental (ratio sqrt(20) / 3.51602).
• FAR-25 (25.301-25.307) sets the certification context for structure loads and proof; the natural frequencies computed here feed vibration-clearance and excitation checks, not the loads themselves.

Workflow

1. Gather the member data: bending stiffness EI (N m^2), mass per unit length m (kg/m) and length L (m), all SI.
2. Name the end condition: pinned-pinned, cantilever, clamped-clamped or free-free.
3. For a pinned-pinned member call pinned_pinned_frequency(mode_n, ei, mass_per_len, length_m); the closed form is exact and the mode ratios come out as exact squares.
4. For the other end conditions the module first re-derives the characteristic root with characteristic_root(mode_n, bc, tol=1e-12) by bisection: the cantilever root n=1 is bracketed in (1.8, 2.0), cantilever n>=2 in ((n-1) pi, n pi), and the clamped-clamped/free-free roots in (n + 0.5) pi +/- 0.8, brackets verified to straddle the published values. Pass any root into beam_frequency(beta_n_L, ei, mass_per_len, length_m), or call the convenience functions cantilever_frequency, clamped_clamped_frequency and free_free_frequency (elastic modes only) directly.
5. Sweep modes with beam_mode_frequencies(bc, n_modes, ei, mass_per_len, length_m) to get the first n_modes frequencies as an ascending list for one boundary condition.
6. For a non-uniform member or a quick shape-based estimate, call rayleigh_fundamental(ei, mass_per_len, length_m) with the stored cantilever-parabola shape and report the result as an upper bound to the exact fundamental.
7. Confirm the deterministic checks with the contract test scripts/test_beam_vibration.py.

Worked example

Aluminum beam: E = 73.1 GPa, rho = 2780 kg/m3, cross-section 50 x 100 mm, L = 2 m. Then I = b h^3 / 12 = 4.1667e-6 m^4, EI = 3.0458e5 N m^2 and m = rho b h = 13.9 kg/m. Module outputs:

• Characteristic roots by bisection: cantilever 1.87510407, 4.69409113, 7.85475744; clamped-clamped and free-free 4.73004074, 7.85320462 (all match published Blevins values).
• Cantilever: f1 = 20.7089 Hz, f2 = 129.7803 Hz, f3 = 363.3887 Hz (about 20.709, 129.780, 363.389 Hz).
• Pinned-pinned: f1 = 58.1307 Hz, and f2 = 232.5228 Hz = 4 f1 exactly.
• Clamped-clamped: f1 = 131.7758 Hz; free-free first elastic mode is the same 131.7758 Hz (shared root 4.73004074).
• Rayleigh cantilever-parabola estimate: 26.3403 Hz = 1.272x the exact 20.7089 Hz, confirming the upper-bound property.

Pitfalls

• Reading the wrong characteristic equation: cantilever roots come from cos x cosh x = -1 while clamped-clamped and free-free share cos x cosh x = 1; swapping them swaps the frequency sets.
• Counting free-free rigid-body modes as elastic: a free-free beam carries two zero-frequency rigid-body modes, and the module reports elastic modes only - its first elastic root 4.73004074 matches clamped-clamped, so expecting a lower first free-free frequency is wrong.
• Using the Rayleigh estimate as exact: the cantilever-parabola quotient gives 26.3403 Hz against the exact 20.7089 Hz fundamental in the worked example (an upper bound at 1.272x), so it is a screening bound, not the frequency.
• Forgetting pinned-pinned is closed form: pinned-pinned needs no root search, f_n = n^2 f_1 exactly; forcing it through the transcendental machinery invites rounding where none is needed.
• Feeding the shape guess into the wrong law: the shared law f = (beta_n L)^2 sqrt(EI / (m L^4)) / 2pi needs the NON-dimensional characteristic root, not a radian frequency or a beta in 1/m.
• Asking for invalid modes or members: a mode number below 1 (or fractional), non-positive EI, mass per length or length, and unknown boundary-condition names all raise ValueError.

Verification

• Cantilever roots land within 1e-6 of 1.87510407 / 4.69409113 / 7.85475744 and clamped roots within 1e-6 of 4.73004074 / 7.85320462 (contract requires 1e-5).
• Pinned-pinned closed form: f_n / f_1 = n^2 exactly, asserted for several modes.
• Rayleigh bound: rayleigh_fundamental returns 26.3403 Hz, above the exact cantilever fundamental 20.7089 Hz by the expected 1.272x.
• Free-free reports elastic modes only: its mode list equals the clamped-clamped list, and no zero-frequency rigid-body rows appear.
• Determinism: no random numbers anywhere; identical floats run to run.
• beam_mode_frequencies returns n_modes entries in ascending order for every boundary condition.
• Every non-positive ei, mass per unit length or length, every mode number below 1 (and every fractional mode), and every unknown boundary-condition name raises ValueError.
• Run the contract test offline: python3 scripts/test_beam_vibration.py (32 tests, deterministic, under one second).

Related leaves

• structures/fem/modal-analysis: the discrete 2-DOF sibling; this leaf is its continuous-member complement for distributed-parameter beam members.
• structures/loads/random-vibration-analysis: consumes member natural frequencies for random excitation response assessment.
• cross-cutting/numerics/eigenvalue-decomposition: generic eigensolvers for discretized models; a continuous beam member instead uses the closed-form characteristic roots above.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_beam_vibration.py

The test covers the worked aluminum beam frequencies inside the spec bounds, the published characteristic roots to 1e-5 or better, the bisection brackets and tolerance behavior, the exact pinned-pinned n^2 spacing, the shared beam_frequency law and its round trip, the Rayleigh upper bound at 1.272x, the free-free elastic mode identity with clamped-clamped, ascending mode lists for every boundary condition, determinism run to run, and ValueError rejection of non-positive stiffness/mass/length, invalid mode numbers and unknown boundary-condition names.

Compliance

• Standards referenced, not reproduced: FAR-25 is public-domain US government work (17 U.S.C. 105) but standards-map.yaml marks it gated: false and reference-only: true, so only the summary paraphrase above is used, never standard text. The characteristic roots and frequency laws are standard engineering methodology (Blevins-style formulas), summary-only.
• compliance: STANDARDS-REF, gated: false.