Frequency of a Beam with Fixed Ends | Online Calculator

Natural Frequencies Calculators







Vibration of a Beam with Fixed Ends and Concentrated Mass

In this calculation, a beam of length L with a moment of inertia of the cross-section Ix and own mass m is considered. The ends of the beam are fixed. At the distance L / 2 from the end of the beam, a concentrated mass M is located. 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 excluding length of concentrated mass;


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


E - Young's modulus;


M - Concentrated mass;


m - Beam weight excluding concentrated mass.

RESULTS DATA

f - First mode frequency.

Length (L)

Moment of inertia (Ix)

Young's modulus (Е)

Concentrated mass (M)

Beam weight (m)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of the mass M, acting alone:

fM = (1 / 2π)*[192EIx / (M*L3)]0.5;

Natural frequency of the unloaded beam, acting alone:

fbeam = (3.57 / L2)*(EIx / m)0.5;

Total natural frequency of beam of the first mode:

1 / f 2 = (1 / fbeam 2) + (1 / fM2)

INITIAL DATA

L - Beam length excluding length of concentrated mass;


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


E - Young's modulus;


m - Concentrated mass;


М - Beam weight excluding concentrated mass.

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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