Natural Frequencies Calculators

Vibration of a Rectangular Plate with Fixed Edges

In this calculation, a rectangular plate with the dimensions a × b, thickness s, and own mass m is considered. The plate edges are fixed. For the calculation, the elastic modulus E and Poisson's ratio ν of the plate should be specified. As a result of calculations, the natural vibration frequency of the plate f is determined for the first vibration mode.

Natural frequency of rectangular plate with fixed edges

Calculation of Natural frequency of rectangular plate with fixed edges

INITIAL DATA

a - Length of the long side of the plate;

b - Length of the short side of the plate;

s - Plate thickness;

m - Plate weight;

E - Young's modulus;

ν - Poisson's ratio.

RESULTS DATA

f - First mode frequency.

Metric Imperial

Length (a)

Width (b)

Thickness (s)

Plate mass (m)

Young's modulus (E)

Poisson's ratio (ν)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of the first mode:

f = K₁*[[Es³ / 12(1 - ν²)]*[a/(m*b³)]]^0.5;

K₁ - Coefficient, depending on a / b ratio.

INITIAL DATA

a - Length of the long side of the plate;

b - Length of the short side of the plate;

s - Plate thickness;

m - Plate weight;

E - Young's modulus;

ν - Poisson's ratio.

RESULTS DATA

f - First mode frequency.

MATERIALS PROPERTIES

Material	Young’s modulus Pa (psi)	Poisson’s ratio
Steel	1.86÷2.1×10¹¹ (2.7÷3.05×10⁷)	0.25÷0.33
Cast iron	0.78÷1.47×10¹¹ (1.1÷2.1×10⁷)	0.23÷0.27
Copper	1.0÷1.3×10¹¹ (1.45÷1.9×10⁷)	0.34
Tin bronze	0.74÷1.22×10¹¹ (1.1÷1.8×10⁷)	0.32÷0.35
Brass	0.98÷1.08×10¹¹ (1.4÷1.6×10⁷)	0.32÷0.34
Aluminum alloy	0.7×10¹¹ (1.0×10⁷)	0.33
Magnesium alloy	0.4÷0.44×10¹¹ (5.8÷6.4×10⁶)	0.34
Nickel	2.5×10¹¹ (3.6×10⁷)	0.33
Titanium	1.16×10¹¹ (1.7×10⁷)	0.32
Lead	0.15÷0.2×10¹¹ (2.2÷2.9×10⁶)	0.42
Zinc	0.78×10¹¹ (1.1×10⁷)	0.27
Glass	4.9÷5.9×10¹⁰ (7.1÷8.5×10⁶)	0.24÷0.27
Concrete	1.48÷2.25×10¹⁰ (2.1÷3.3×10⁶)	0.16÷0.18
Wood (along the grain)	8.8÷15.7×10¹⁰ (12.8÷22.8×10⁶)	-
Wood (across the grain)	3.9÷9.8×10¹⁰ (5.7÷14.2×10⁶)	-
Nylon	1.03×10¹⁰ (1.5×10⁶)	-

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