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Simply supported 20 × 10 × 0.1 in aluminium plate: the first six natural frequencies by Leissa's exact formula, with Warburton's estimate and the in-plane-force shift

Tool	Plate Vibration
Source	Leissa, A. W. (1969). Vibration of Plates. NASA SP-160, National Aeronautics and Space Administration, Washington, DC. Chapter 4, eq. (4.20) (all edges simply supported, ω_mn = π² [(m/a)² + (n/b)²] √(D/ρh)), eq. (4.16) and Table 4.1 (Warburton's formula), Chapter 10 (in-plane forces).
Level	Undergraduate

Problem

A rectangular aluminium plate 20 in by 10 in and 0.1 in thick (E = 10.5 Msi, ν = 0.33, ρ = 0.100 lb/in³) is simply supported on all four edges. Find the flexural rigidity, the mass per unit area, the first six natural frequencies with their mode indices and Leissa's frequency parameters λ = ω a² √(ρh/D), compare with Warburton's formula, and find the fundamental frequency and the critical force when a uniform in-plane compression N_x = 100 lb/in acts along the long side.

Inputs to Enter

Shape rectangular, a = 20 in along x, b = 10 in along y, t = 0.1 in; E = 10.5 Msi, ν = 0.33, ρ = 0.100 lb/in³; all four edges simply supported; 6 modes
In-plane forces N_x = N_y = 0 for the default; the second part enters N_x = −100 lb/in (compression)

Open the Plate Vibration tool (loaded with this example)

Answers

D = E t³ / [12 (1 − ν²)] = 981.9 lb-in; ρh = 0.0100 lb/in² (7.031 kg/m²); √(D/ρh) = 6,157 in²/s
λ_mn = π² [m² + n² (a/b)²]: (1,1) 49.348, (2,1) 78.957, (3,1) 128.30, (1,2) 167.78, (2,2) and (4,1) 197.39 (a degenerate pair)
f_mn = λ √(D/ρh) / (2π a²): 120.90, 193.43, 314.33, 411.05, 483.59, 483.59 Hz (ω_11 = 759.6 rad/s)
Warburton's one-term formula with G = H = √J = m − 1 reproduces every simply supported value exactly (difference 0.00 %)
With N_x = −100 lb/in: ω² = [D (α² + β²)² + N_x α²]/ρh gives f_11 = 110.47 Hz; the fundamental buckles at N_x,cr = −D (α² + β²)²/α² = −605.7 lb/in

Notes

The tool's default inputs are this plate; press Submit, then enter −100 in Force N_x for the second part. The values are checked in test_plate_vibration.php against the closed form and the independent numpy script References/Plate-Vibration/checks/platevib_check.py (f_11 = 120.8966 Hz, f_11 = 110.4669 Hz with the load, N_x,cr = −605.7053 lb/in), together with 553 further checks of the Lévy, Rayleigh-Ritz and circular-plate solutions against SP-160 Tables 2.1, 2.3, 2.5, 2.6, 4.3, 4.6, 4.8, 4.10, 4.13, 4.15 to 4.18, 4.22, 4.25, 4.27, 4.28, 4.33, 4.35 to 4.37, 4.40 to 4.45, 4.48, 4.61 and 4.65.