What the calculation gives you
A flat plate has natural frequencies at which it vibrates freely, and if anything drives it near the lowest of them it resonates: a cover panel drums, an electronics chassis amplifies the shaker input, a machine guard cracks at its welds. The calculation gives that first frequency from the plate's dimensions, thickness, material and the way its edges are held, so that you can move it away from the excitation, usually upward, by changing thickness, support or stiffeners.
The formula
f = (λ2 / 2π a2) √(D / γ)
a plate length · λ2 dimensionless frequency parameter for the boundary condition and aspect ratio · D flexural rigidity · γ mass per unit area
D = E h3 / [12 (1 − ν2)]
E modulus of elasticity · h thickness · ν Poisson's ratio, 0.3 for steel, 0.33 for aluminium
γ = ρ h
ρ density; add the mass of anything attached to the plate, spread over its area, to γ
Since D goes with h3 and γ with h, the frequency is proportional to the thickness and inversely to the square of the length: doubling the thickness doubles the frequency; halving a span by adding a stiffener quadruples it. The material enters through √(E/ρ), which is nearly the same for steel, aluminium and titanium, so a change of metal does little; a change of thickness or support does a lot.
The frequency parameter
Everything about the edges and the shape is in λ2, tabulated by Leissa and in Blevins' formulas for the common cases. For a square plate (a = b), first mode:
- Simply supported on all four edges: λ2 = 19.74 (exactly 2π2; for a rectangle, π2 [1 + (a/b)2]).
- Clamped on all four edges: λ2 = 35.99.
- Clamped on one edge, free on three (a cantilever plate): λ2 = 3.49.
- Supported at the four corners only: λ2 = 7.12; the worksheet below uses a slightly different published value for its case.
- Free on all edges: λ2 = 13.49 for the first elastic mode (the rigid-body modes are at zero).
Real supports fall between the ideal ones. A bolted edge with bolts at wide spacing is nearer to simply supported than to clamped; a continuous weld or a clamped flange approaches clamped. The ratio between the two, a factor of 1.8 on frequency for a square plate, is the uncertainty you carry when the support is not well defined.
Worked example
The worksheet below is a 36 × 36 in aluminium 6061-T6 plate, 1/8 in thick, supported at its four corners. It defines E, the density, the mass per unit area and Poisson's ratio, writes the frequency as a function of λ, takes the published parameter for the corner-supported first mode, and gives the base frequency. Change the thickness to 1/4 in or the span to 24 in and see the frequency follow the rules above; replace λ with the simply supported or clamped value to bracket a real support.
Common mistakes
- Wrong boundary condition. It is the largest single uncertainty; bracket it with the two nearest ideal cases rather than picking one.
- Attached mass forgotten. A board, a transformer or a coating adds mass without stiffness and lowers the frequency; add it to γ.
- Mixed units. E in psi with density in g/cm3 and thickness in inches gives nonsense unless the units are carried; the worksheet does, by hand you must convert first.
- Stiffener counted as a boundary. A stiffener raises the frequency but is not a rigid support unless it is much stiffer than the plate; its own flexibility must be included, or the sub-panel frequency is overestimated.
- Thin-plate theory stretched. The formula assumes the thickness is small against the span (under about 1/20) and the deflection small against the thickness; thick plates and large amplitudes need shear deformation or non-linear theory.
- Only the first mode checked. A broadband excitation, random vibration from a vehicle for instance, excites the second and third modes too; their parameters are in the same tables.
Questions
- How far from the excitation should the frequency be?
- A factor of 2 above the highest forcing frequency is the usual target for a lightly damped metal plate; a factor of 1.4 (the square root of two) is the point at which the amplification stops being larger than one.
- How accurate is the formula?
- Within a few percent for an ideal support; the deviation of the real support, the attached mass and the plate's own tolerances usually matter more than the theory. A measured tap test on a prototype settles it.
- What about a circular plate?
- The same form, with the radius in place of a and its own parameters: 10.22 clamped at the edge, 4.98 simply supported at the edge, for the first mode with ν = 0.3.
