Modal Analysis — Natural Frequency and FEA Comparison

1. Problem

A simply supported steel beam: L = 500 mm, rectangular cross-section 50 × 25 mm. Determine the first three natural frequencies analytically, predict the FEA result with different mesh densities, and compare.

2. Given

3. Required

• Section properties (I, ρA)
• First three natural frequencies (analytical)
• FEA prediction at two mesh densities
• Comparison and error assessment

4. Assumptions

• Euler-Bernoulli beam theory (plane sections remain plane, no shear deformation)
• Simply supported (pinned-pinned) boundary conditions
• No damping
• Prismatic uniform cross-section
• No axial preload

5. Governing Equations

n-th natural frequency (simply supported beam):
          (nπ)²      ╱  EI
fₙ  =   ──────── × ╲  ─────
          2πL²          ρA

Mode shapes:  φₙ(x)  =  sin(nπx/L)

I  =  b·h³/12  (bending about strong axis)
ρA  =  ρ × b × h  (mass per unit length)

6. Calculation

Step 1: Section properties.

I  =  50 × 25³ / 12  =  50 × 15,625 / 12  =  65,104 mm⁴

ρA  =  7,850 × 10⁻⁹ × 50 × 25  =  7,850 × 10⁻⁹ × 1250  =  9.813 × 10⁻³ kg/mm

EI  =  200,000 × 65,104  =  1.302 × 10¹⁰ N·mm²

6. Calculation (continued)

Step 2: Natural frequencies. The factor (nπ)²/(2πL²) × √(EI/ρA) simplifies for n = 1, 2, 3.

√(EI / ρA)  =  √(1.302 × 10¹⁰ / 9.813 × 10⁻³)
            =  √(1.327 × 10¹²)
            =  1,152,000 mm/s  =  1152 m/s

f₁  =  π / (2 × 500²) × 1,152,000  =  3.142 × 1152 / 500,000  =  327 Hz

f₂  =  4 × f₁  =  1,308 Hz
f₃  =  9 × f₁  =  2,943 Hz

6. Calculation (continued)

Step 3: FEA models. Two mesh densities: coarse (5 elements) and refined (20 elements), using B31 beam elements.

7. Result

f₁ = 327 Hz, f₂ = 1,308 Hz, f₃ = 2,943 Hz. The refined FEA model (20 elements) matches the analytical solution to within 0.4%. The coarse mesh (5 elements) overestimates by 2-7%.

8. Check

• The ratio f₂/f₁ = 4 and f₃/f₁ = 9 — exactly as expected for a simply supported beam (n² relationship)
• The coarse mesh overestimates frequencies — this is because a coarse mesh is too stiff (fewer elements cannot represent the curvature)
• The refined mesh converges to the analytical solution — confirms both the hand calculation and the FEA model
• Total mass check: ρ × L × b × h = 7850 × 0.5 × 0.05 × 0.025 = 4.9 kg — should match the FE model mass
• Dimensional check: (N·mm²) / (kg/mm) → mm²/s² → √ → mm/s → × (1/mm²) → Hz — consistent

9. Interpretation

The analytical and FEA results agree to within 0.4% for the refined mesh. The coarse mesh overestimates frequencies because a low-element-count beam model is artificially stiff — it cannot represent the smooth curvature of the mode shape. This is a mesh convergence demonstration: the results converge as the mesh is refined.

For practical FEA models, 10-20 elements per mode shape half-wavelength are typically sufficient. For the third mode (3 half-waves over the span), 20 elements gives ~7 elements per half-wave — adequate for <1% error.

10. Limitations

• Euler-Bernoulli theory neglects shear deformation — for deep beams (h/L > 0.1), Timoshenko theory is needed
• No damping — real structures have 1-5% damping which slightly reduces the natural frequency
• No rotational inertia (Euler-Bernoulli assumption)
• Simply supported boundary conditions are idealised — real supports have some rotational stiffness
• No axial preload effect — tensile preload increases natural frequency, compressive preload decreases it

11. Related Resources

• Dynamics and Vibration Formulae (Reference Library)
• How to Perform a Modal Analysis
• How to Check Modal Effective Mass
• How to Choose Structural Damping