Cantilever natural frequency

The same rectangular steel bar and the same clamped end as the cantilever deflection case — no load, solved for its own natural frequencies instead.

Running the same geometry through a different analysis type is the point. Anything wrong with the material, the restraint or the mesh would show up in both; anything that shows up in only one of them is in the analysis path, which is a much smaller place to look.

Every mesh degenerates the first two modes correctly. f₁ and f₂ agree to 0.25 % at the coarsest mesh and to 0.002 % at the finest — the square cross-section’s two orthogonal bending modes, as theory requires. A solver whose stiffness or mass matrix had an axis wired wrong would not produce this, and it is a check the frequency alone does not give you.

The deflection case’s residual converged to zero as the mesh refined. This one converges to about −2.1 % and stops there. That is not an unconverged run — the last refinement, 4 162 to 31 940 nodes, moved the answer by 0.09 percentage points — it is a real gap between the finite-element answer and pure Euler–Bernoulli theory.

Euler–Bernoulli ignores shear deformation. The standard first-order Timoshenko correction for a cantilever’s fundamental frequency, evaluated for this geometry, gives 1632.09 Hz. The converged answer of 1635.52 Hz sits +0.21 % above that and −2.13 % below the uncorrected prediction — so the correction accounts for essentially all of the residual.