Cantilever beam — higher bending natural frequencies

Extends the fundamental-mode page Cantilever beam — first bending natural frequency with the second and third bending natural frequencies of a clamped-free prismatic beam.

Cantilever bending frequencies follow

\[f_n = \frac{(\beta_n L)^{2}}{2 \pi L^{2}} \sqrt{\frac{E I}{\rho A}},\]

where \(\beta_n L\) are the roots of \(1 + \cos(\beta L) \cosh(\beta L) = 0\):

n	β_n L
1	1.8751040687
2	4.6940911330
3	7.8547574382
4	10.995540735

For our reference geometry (L = 1 m, w = h = 0.05 m, E = 200 GPa, ρ = 7 850 kg/m³) the first four frequencies are f₁ ≈ 40.77 Hz, f₂ ≈ 255.54 Hz, f₃ ≈ 715.39 Hz, f₄ ≈ 1 402 Hz.

Problem

Parameter	Value
Length L	1.0 m
Cross-section	0.05 m × 0.05 m (square)
Young’s modulus E	200 GPa
Density ρ	7 850 kg/m³
Expected f_2	255.54 Hz
Expected f_3	715.39 Hz

Analytical reference

Euler-Bernoulli separation-of-variables solution for a clamped-free prismatic beam (Rao, Mechanical Vibrations 6th ed., §8.5 Table 8.1; Timoshenko, Vibration Problems in Engineering §5.3).

femorph-solver result

Ran by tests/validation/test_cantilever_higher_modes.py on the same SOLID185 enhanced-strain hex mesh the fundamental- mode problem uses. Mode identification uses the UZ-dominant + expected-antinode filter to pick the second and third x-z-bending modes out of the interleaved bending / torsion / axial families a square cross-section admits.

Refinement	Mesh	f_2 (Hz)	f_3 (Hz)
Coarse	20 × 3 × 3	255.654 (+0.06 %)	711.883 (−0.49 %)
Medium	40 × 3 × 3	253.709 (−0.70 %)	700.449 (−2.09 %)
Refined	80 × 3 × 3	253.092 (−0.94 %)	697.282 (−2.53 %)

The small downward drift with refinement is the Timoshenko shear + rotary-inertia correction the pure Euler-Bernoulli reference omits — cantilever higher modes are more sensitive to the correction than the fundamental. Both values stay comfortably under the published tolerances (6 % for f_2, 12 % for f_3).

Cross-references

Source	Reported f_2 (Hz)	Problem ID / location
Closed form (Euler-Bernoulli)	255.54	Rao 2017 §8.5 Table 8.1
Timoshenko (1974) §5.3	255.54	Cantilever char. eq.
femorph-solver (refined)	253.09	test_cantilever_higher_modes.py
MAPDL Verification Manual	≈ 255	VM-57 family (cantilever-shaft modal)
Abaqus Verification Manual	≈ 255	AVM 1.6.x cantilever-bending family
NAFEMS FV-2	255.54	Cantilever transverse modes

Source

Problem class: femorph_solver.validation.problems.CantileverHigherModes.

Backing regression test: tests/validation/test_cantilever_higher_modes.py.