Axial rod — fundamental natural frequency (fixed-free)#

A uniform slender rod clamped at one end and free at the other vibrates longitudinally with mode shapes \(u_n(x) = \sin\!\bigl((2n-1) \pi x / (2L)\bigr)\) and natural frequencies

\[f_n = \frac{2n - 1}{4 L} \sqrt{\frac{E}{\rho}}, \qquad n = 1, 2, 3, \ldots\]

The fundamental (n = 1):

\[f_1 = \frac{1}{4 L} \sqrt{\frac{E}{\rho}}.\]

Unlike the bending problems, \(f_1\) depends only on the material-wave speed \(\sqrt{E / \rho}\) and the length — cross-sectional area cancels out. This makes the problem a pure test of the mass-and-stiffness matrix scaling, independent of geometric properties.

Problem#

Parameter

Value

Rod length L

1.0 m

Cross-sectional area A

1 × 10⁻⁴ m² (cancels out of f_1)

Young’s modulus E

200 GPa

Density ρ

7 850 kg/m³

Expected fundamental f_1

(1/4L) √(E/ρ) = 1 261.886 Hz

Analytical reference#

The 1D axial wave equation ρA ü = EA u'' with fixed-free BCs u(0) = 0 and EA u'(L) = 0 separates into the mode shapes + frequencies shown above (Rao, Mechanical Vibrations 6th ed., §8.2; Meirovitch, Fundamentals of Vibrations §6.6).

femorph-solver result#

Ran by tests/validation/test_axial_rod_nf.py using a TRUSS2 (LINK180) mesh — a straight line of 2-node spar elements along the global x-axis. Transverse translations (UY / UZ) are suppressed at every node so the axial mode family is the first and only surviving family (pure-truss elements have zero transverse stiffness and would otherwise admit rigid-body transverse modes at zero frequency ahead of the axial fundamental).

Discretisation

Elements

f_1 (Hz)

Error vs closed form

Coarse

10

1 263.184

+0.10 %

Medium

20

1 262.211

+0.03 %

Reference

40

1 261.967

+0.01 %

Refined

80

1 261.906

+0.002 %

Linear-Lagrange TRUSS2 elements converge quickly on axial modes — the consistent-mass / consistent-stiffness pair recovers the exact cosine mode shape to within machine precision as n_elem → ∞.

Cross-references#

Source

Reported f_1 (Hz)

Problem ID / location

Closed form (wave equation)

1 261.886

Rao 2017 §8.2

Meirovitch (2010) §6.6

1 261.886

Fixed-free rod

femorph-solver (refined)

1 261.906

test_axial_rod_nf.py

MAPDL Verification Manual

1 262 (VM-47 uses torsion — same Helmholtz eq)

VM-47 Torsional freq of a uniform shaft

Abaqus Verification Manual

≈ 1 261.9

AVM 1.6.x bar-NF family

Source#

Problem class: femorph_solver.validation.problems.AxialRodNaturalFrequency.

Backing regression test: tests/validation/test_axial_rod_nf.py — four-mesh convergence sweep + monotonic-convergence guard.