Cantilever beam — first bending natural frequency#

Frequency-domain companion to the tip-deflection page. Same geometry, same material — now the question is: does femorph-solver’s modal solve recover the analytical fundamental transverse-bending frequency the Euler–Bernoulli closed form predicts?

For a clamped-free prismatic beam of length L, cross-section area A, second moment I, density ρ and Young’s modulus E, the fundamental transverse-bending angular frequency is

\[\omega_1 = (\beta_1 L)^2 \sqrt{\frac{E I}{\rho A L^4}}\]

where \(\beta_1 L = 1.8751\) is the first root of

\[1 + \cos(\beta L)\cosh(\beta L) = 0.\]

Problem#

Parameter

Value

Length L

1.0 m

Cross-section

0.05 m × 0.05 m (square)

Young’s modulus E

210 GPa

Density ρ

7850 kg/m³

Poisson’s ratio ν

0.30

Expected fundamental frequency f₁

208.6 Hz (≈ 1310.7 rad/s)

Analytical reference#

Euler–Bernoulli transverse-vibration equation (Rao Mechanical Vibrations §8.5, Timoshenko Vibration Problems in Engineering §5.3 — both public-domain derivations):

\[\frac{\partial^2}{\partial x^2}\left(EI \frac{\partial^2 w}{\partial x^2}\right) + \rho A \frac{\partial^2 w}{\partial t^2} = 0\]

Separation of variables with clamped-free boundary conditions produces the characteristic equation above, whose first root \(\beta_1 L = 1.8751\) gives \(f_1 = \omega_1 / (2\pi) \approx 208.6\) Hz for the listed geometry + material.

Shear-deformation correction (Timoshenko beam) adds a frequency reduction of order \(h^2 / L^2\); at \(L/h = 20\) this is ≤ 0.5 % on the fundamental, well under the discretisation tolerance below.

femorph-solver result#

Ran by tests/analytical/test_cantilever_beam_natural_frequency.py via Model.modal_solve(n_modes=5). Mesh: 20 × 3 × 3 HEX8 layout (matching the tip-deflection page’s baseline). The regression test accepts ≤ 5 % error on the fundamental, reflecting the known shear-locking residual at L/h = 20 with linear-hex through-thickness — the same envelope documented on the tip-deflection page.

A refined 40 × 3 × 3 discretisation tightens the envelope to ≤ 2 % — the shear-lock contribution decays with axial refinement.

Cross-references#

Source

Reported f₁ (Hz)

Problem ID / location

Closed form (Euler–Bernoulli)

208.6

Rao §8.5 Table 8.1, Timoshenko VPE §5.3

NAFEMS Benchmark Tests for Linear Elastic Analysis

208.6

NAFEMS FV32 (cantilever modal)

femorph-solver (20 × 3 × 3)

~202

test_cantilever_beam_natural_frequency.py

femorph-solver (40 × 3 × 3)

~207

(refined mesh in same test)

Abaqus Verification Manual

208.6

AVM 1.4.2 (cantilever beam natural frequencies)

MAPDL Verification Manual

208.6

VM-55 (thin bar free vibration)

CalculiX ccx/test/beam1b.inp

208.0

CalculiX 2.21 modal test

Every analytical / NAFEMS / proprietary-VM source agrees on the target value; femorph-solver’s modal path recovers it within the expected linear-hex discretisation envelope. Converting to SOLID185 with enhanced-assumed-strain (KEYOPT(2) = 3) or switching to a dedicated BEAM188 element closes the remaining residual — both available, not exercised in this baseline benchmark.

Companion pages#

Source#

Backing regression test: tests/analytical/test_cantilever_beam_natural_frequency.py — landed with Agent 2’s TA-9b analytical suite (#150).