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Natural Frequencies Calculators







Vibration of a Simply Supported Beam with Set of Concentrated Masses

In this calculation, a beam with the moment of inertia of the cross section Ix and own mass M is considered. The ends of the beam are simply supported. Concentrated masses m1 ÷ m5 are located at the distances L1 ÷ L6 along the length of the beam. For the calculation, the elastic modulus E of the beam should be specified. As a result of calculations, the natural frequency of the beam f is determined for the first vibration mode. To determine the natural frequencies for a smaller number of concentrated masses, enter zero values in the corresponding fields of the program.

Calculation of natural frequency of beam
Natural frequency of beam

INITIAL DATA

L1 ÷ L6 - Distances between concentrated masses;


m1 ÷ m5 - Concentrated masses;


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


E - Young's modulus;


М - Beam weight excluding concentrated loads.

RESULTS DATA

f - First mode frequency.

Length (L1)

Length (L2)

Length (L3)

Length (L4)

Длина (L5)

Length (L6)

Mass (m1)

Mass (m2)

Mass (m3)

Mass (m4)

Mass (m5)

Moment of inertia (Ix)

Young's modulus (E)

Beam weight (М)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of mass m1, acting alone:

fm1 = (1 / 2π)*3EIx / [m1*L12*(L2+L3+L4+L5+L6)2)]0.5;

Natural frequency of the unloaded beam, acting alone:

fbeam = (1.57 / L2)*(EIx / M)0.5;

Total natural frequency of beam of the first mode:

1 / f = (1 / fbeam2) + ∑ (1 / fmi 2) .

INITIAL DATA

L1 ÷ L6 - Distances between concentrated masses;


m1 ÷ m5 - Concentrated masses.


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


E - Young's modulus;


М - Beam weight excluding concentrated loads.

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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