### Resonant Frequency Flat Plate Electronic Circuit Equations and Calculator

**Vibration Design and Engineering**

Vibration Resonant Frequency of a Flat Plate / Electronic Circuit Card Equation and Calculator.

Equations and calulator will determine resonant frequency of a rectangular plate or PCB that is subjected to a know G peak sinusoidal vibration. With a low input G level, the four edges of the circuit board will act as though they were simply supported. The plate board has a uniformly distributed weight.

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Preview: Resonant Frequency of a Flat Plate / Electronic Circuit Card Calculator

Natural frequency Vibration equation derived with the use of a trigonometric expression (Equation a),

f_{n} = Π / 2 ( D/ρ )^{1/2} ( 1/a^{2}+ 1/b^{2 })

Natural frequency equation shown above is exact using the Rayleigh method, because the deflection curve used for the vibrating plate is exact for the boundary conditions.

D = E h^{3} / [ 12 (1 - µ^{2}) ]

ρ = W / g a b

Where:

a = Plate length (in)

b = Plate Width (in)

h = Plate Thickness (in)

W= Plate Weight (lbs, N)

g = 386 in./s^{2} (gravity)

E = Modulus of Elasticity ( lbs/in^{2} )

µ = Poisson’s ratio (dimensionless)

ρ = Mass per Unit Area
( lb · s^{2 }/ in^{3}
)

D = Plate Bending Stiffness Factor (lb · in)

f_{n} = Natural Frequency (Hz)

Or, the natural frequency can also be determined from the equation derived with the use of a polynomial expression (Equation b):

f_{n} =5.31 / Π · { D / ρ [ 1/ (15 c^{4)} + 1/ (9 c^{2} d^{2} ) + 1/ ( 15 d^{4}) ] } ^{(1/2)}

Where:

c = a / 2 (in)

d = b / 2 (in)

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