〰️ Modal Analysis
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Mode shape
Natural frequencies — FEM vs exact closed-form solution
Mode
FEM (Hz)
Exact (Hz)
Error
Setup
Boundary conditions
Cantilever (fixed – free)
Simply supported (pin – pin)
Fixed – fixed
Fixed – pinned
Free – free
Material
Steel (E = 210 GPa)
Aluminium (E = 70 GPa)
Titanium (E = 116 GPa)
Carbon/epoxy, UD (E = 135 GPa)
Length (m)
Elements
Cross-section
Solid rectangle
Circular tube
I-beam
Width b (mm)
Height h (mm)
Thickness t (mm)
Tip mass (kg) — cantilever only
Beam