Tools · Structural dynamics
Plate natural frequency
First modes of a flat rectangular panel: duct walls, enclosure panels, tank shell sections, deck plates. Useful when a "drumming" panel lines up with a running speed or blade pass.
Edges simply supported on all four sides. Fully clamped edges raise the fundamental by roughly 1.8× for a square plate.
| Mode (m, n) | Hz | CPM | vs excitation |
|---|
The formula
fmn = (π/2) · [ (m/a)² + (n/b)² ] · √( D / (ρh + ma) )
D = E·h³ / (12(1 − ν²))
m and n count the half-waves along each side. This is the exact Kirchhoff thin-plate solution for simply supported edges. Real duct and enclosure panels sit somewhere between simply supported and clamped, so treat this as the lower end of the range. Stiffeners, internal pressure and flow all shift the answer; for a fix, an FEA model tuned to a bump test is the reliable route.
Practical fixes for a resonant panel: stiffen it (moves the frequency up), brace it to a nearby structure, or add damping. Thicker plate helps less than you'd think: bending stiffness grows with h³, but mass grows with h too, so frequency only rises in proportion to thickness. Doubling the plate doubles the frequency and doubles the weight. Stiffeners usually win.
Method & sources
What it calculates
- Kirchhoff thin-plate theory, all four edges simply supported: exact Navier solution fmn = (π/2)[(m/a)² + (n/b)²]·√(D/μ).
- Flexural rigidity D = Eh³/12(1 − ν²); μ = ρh + added mass per unit area (smeared uniformly).
Assumptions & limits
- Thin plates only (side length more than about 20 × thickness). Stiffeners, in-plane loads and fluid loading are not included.
- Clamped edges raise the fundamental (by about 1.8× for a square plate); Leissa tabulates every edge condition.
Verified: 1000 × 800 × 6 mm steel gives 36.89, 80.08 and 104.4 Hz for the first three modes, matching Python.