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 I_x 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;

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

E - Young's modulus;

М - Beam mass excluding concentrated loads.

RESULTS DATA

f - First mode frequency.

Metric Imperial

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 (I_x)

Young's modulus (E)

Beam mass (М)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of mass m₁, acting alone:

f_m1 = (1 / 2π)*3EI_x / [m₁*L₁²*(L₂+L₃+L₄+L₅+L₆)²)]^0.5;

Natural frequency of the unloaded beam, acting alone:

f_beam = (1.57 / L²)*(EI_xL / M)^0.5;

Total natural frequency of beam of the first mode:

1 / f = (1 / f_beam²) + ∑ (1 / f_mi ²) .

INITIAL DATA

L₁ ÷ L₆ - Distances between concentrated masses;

m₁ ÷ m₅ - Concentrated masses.

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

E - Young's modulus;

М - Beam mass 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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