Natural Frequencies Calculators

Vibration of a Cantilever Beam

The appearance of vibrational loads is a result of a cyclical movement of mechanical systems. Due to the elastic properties of structures, the dynamic influences result in vibrations of their elements and, as a consequence, to vibrational loads.

An exact analytical calculation of the natural vibration frequencies of products of an arbitrary shape is complex and performed using the FEA. However, it is often possible to represent the structure as a beam with concentrated masses. In this case, the frequency of natural vibrations will be equal to:

f = [K / m₀] ^1/2

K - structure stiffness;
m₀ - reduced mass of the structure.

In this calculation, a cantilever beam of length L with a moment of inertia of the cross-section I_x and own mass m is considered. For the calculation, the elastic modulus E of the beam should be specified. As a result of calculations, the natural vibration frequency of the beam f is determined for the first vibration mode.

Calculation of natural frequency of beam

Natural frequency of beam

INITIAL DATA

L - Beam length;

I_x - Moment of inertia of the beam in the direction of vibration;

E - Young's modulus;

m - Beam mass.

RESULTS DATA

f - First mode frequency.

Metric Imperial

Length (L)

Moment of inertia (I_x)

Young's modulus (E)

Beam mass (m)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of the first mode:

f = (0.56 / L²)*(EI_xL / m)^0.5

INITIAL DATA

L - Beam length;

I_x - Moment of inertia of the beam in the direction of vibration;

E - Young's modulus;

m - Beam weight.

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)	-

OTHER CALCULATORS

AREA MOMENTS OF INERTIA

BEAM CALCULATORS

TORSION OF BARS

CIRCULAR FLAT PLATES

BUCKLING

ELASTIC CONTACT

IMPACT LOADS

NATURAL FREQUENCIES

PRESSURED SHELLS

FLUID DYNAMIC

COMPOSITES

SPRINGS

THREAD CONNECTIONS

SHAFT CONNECTIONS

BEARINGS

DRIVES

FATIGUE

HEAT TRANSFER