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V4.1 — Cantilever beam natural frequencies

The first three natural periods of a uniform cantilever beam — a standard modal-analysis benchmark.

Time
About 5 minutes
Level
All levels
You'll need
None — this is a reference document
You're done when
You can see how V4.1 compares to its reference
On this page
  1. Description
  2. Reference
  3. Model
  4. Comparison
  5. Conclusion

← Verification — Modal

V4.1 — Cantilever beam natural frequencies

Acceptable

Description

The first three natural periods of a uniform cantilever beam — a standard modal-analysis benchmark.

Reference

Blevins, Formulas for Natural Frequency and Mode Shape, Table 8-1: ωn = (βnL)²√(EI/(m̄L⁴)), βnL = 1.875104, 4.694091, 7.854757 (closed-form eigenvalues for a uniform cantilever).

Model

An elastic frame cantilever, fixed at the base, meshed into 60 elements with mass distributed along its length, solved as an eigenvalue (Modal) load case for its lowest three natural periods and mode shapes.

Length, L240 in
Elastic modulus, E29,000 ksi
Section width, b12 in
Section depth, d24 in
Moment of inertia, I13,824 in⁴
Mass per length, m̄0.001 kip-s²/in²
Elements60

Comparison

QuantityReferencevfopro% diffStatus
Mode 1 period (s)0.16260.1626+0.01%Acceptable
Mode 2 period (s)0.02590.026+0.04%Acceptable
Mode 3 period (s)0.00930.0093+0.07%Acceptable

Conclusion

All three periods matched the closed-form values to within about 0.07%. The result is classified Acceptable rather than Exact only because a 60-element mesh is itself a discrete approximation of the continuous beam the closed-form solution assumes — the residual difference is mesh discretization, not solver error, and is far inside the 5% tolerance.

Last updated September 22, 2026 · applies to vfopro 20260922.0.0 or later